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MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | 6_Charge_Separation_Part_II_Diode_Under_Illumination.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: So we have an interesting class today. We're going to be taking this IV curve that we've so laboriously set up and understood-- sorry about that. And now we will subject it to illumination. So that's the essence of our lecture today, the diode under illumination. And as part of today's lecture, we have some wonderful little kits over there in the corner where we'll actually be testing IV curves of solar cells. So I hope some of you brought the computers today, and if not, we have some extras up here as well we can use. So again, just to situate ourselves. We're here in fundamentals. We're approaching the end of our fundamental section, but we still have a few really important lectures to get through. After we get through the fundamentals, we'll be in a good position to understand the different technologies and finally the cross-cutting themes. And our goal is to, at least for the fundamentals, to understand solar cell conversion efficiency, which is the ratio of output to input energy. And for most solar cells, this breaks down to the following progression, from the solar spectrum to charge collection. And we're going to be focusing on charge separation, incorporating elements of either side but mostly focused on charge separation today. reminding everybody, of course, that the total system efficiency is the product of each individual efficiency. And if any one of these is low, the efficiency for the entire system is low. And since folks are tired of looking at this chart by now, every single lecture I intend to introduce something new that follows a similar pattern. Does anybody recognize what this is about? So you have over here some H2O going into an oxygen-evolving complex, and light is coming in, essentially exciting up an electron, which is being stored in some form of chemical energy. What is that? AUDIENCE: Photosynthesis. PROFESSOR: Photosynthesis, right? And just like a solar cell, the photosynthesis conversion efficiency of the entire system is dictated by the efficiency of each individual part. Roughly it can be broken down to this little pie chart up here. The total system efficiency in blue is somewhere, depending on the plant, somewhere around 1%, maybe as high as 7% or 8%, depending on very specialized plants that are experts at converting sunlight into usable chemical energy. And that, in part, is largely due to optical losses. If you can see the absorption spectrum of chlorophyll, of the different types of chlorophyll here, you'll see large portions of the solar spectrum that go underutilized. So again, another system that's similar to a solar cell, that the total system efficiency is the product of each individual component going on here. All right. So now what we're going to do is just quickly revisit the diode in the dark and construct the energy band diagram for pn-junction in the dark. Each of you should have on your desk these sheets. Oh, we don't have them on the desks. We need to pass those out. We need to pass those out. So we should have sheets that describe essentially the equivalent circuit diagram, the IV characteristics, and the energy band diagram for our pn-junction in the dark. We laboriously filled this out last class. We're just going to refresh ourselves to make sure we're all on the same page and redo it this class right at the beginning because it's that important. Thank you, thank you, thank you for those who came to our office hours and for those who came to the recitations, and we really tried to get this across. For those who are still struggling, let's make sure that you get this sometime between now and, say, the next two weeks because this will feature prominently on the exam, and it's pretty important for understanding how a solar cell works. So if you would not mind working directly with your partner, the person who's sitting directly next to you. Let's walk through the diode in the dark and construct the energy band diagrams for the diode in the dark. I'll walk you through it as soon as you've done. Maybe I'll give you three minutes to complete that. And then we'll progress to the diode under illumination. Should be a lot of fun. I see convergence among several of you, so let's move forward. Just to review quickly, the way I typically think about it, if we set up in the model circuit right here, we have our pn-junction. We have our space-charge region, also known as the quasi-neutral region, also known as the depletion zone. So we have this region right here that represents the space-charge region. So this is in the dark. Now we have the energy band diagram shown right here, where this dashed blue line represents the chemical potential, also called the Fermi energy, throughout the entire device right here in cross-section. And just to be very, very clear, we've so far described the solar cell as like coming in through the top. And now we've rotated this structure by 90 degrees to represent the pn-junction. That's been a little confusing for some folks. So just to be totally clear, in a device like this one, if it were subject to illumination you would have light coming in from the side, right, either from the p side or from the n side. So to transfer this into what we've seen so far with the solar cell devices facing up toward the sun, you'd have to rotate this by 90 degrees, right? Just to make sure we're all clear with orientations. [? Because ?] we have the Fermi energy right here. The drift and diffusion currents for electrons-- electron diffusion, electron drift-- there is an abundance of electrons over here in the n-type side, and so they want to diffuse over to the p-type. That's why the diffusion current is pointing left. Once they do to a certain degree, they set up a field, the electrons and holes, the mobile charges set up a field, and that creates a drift current that counteracts the diffusion. And once these two are in equilibrium, there's no current flowing through our device. That's why current is equal to 0 right here. And there's also no potential difference because the Fermi energy, the chemical potential, is the same on either side of that device, and so the voltage output of that solar cell would also be 0. When we forward bias our device, now we're forcing a separation, or we're forcing a separation of the chemical potential on either side of the device. If you connected this to an external circuit, the electrons would want to flow from this side to that side. But since we're forcing this condition here with a battery, we are reducing the barrier height here. Electrons can now diffuse over from the n-type into the p-type side, and they do. And the diffusion current increases. And that's why we have current now a positive value. We've defined the electrons traveling to the left as being a positive current. We have now electrons traveling from the n-type to the p-type material. When we reverse bias our device, notice the separation of the quasi-Fermi energies. Again, we have here one sign of voltage because the right side is higher than the left. And now the right side is lower than the left, so our voltage sign flips from right to left over here from positive to negative values. So notice the voltage. And now the current as well. The drift current will outweigh the diffusion current in this particular case because now the barrier for electrons to diffuse from the n to the p-type is very large they'll have difficulty going from that side to that side. Whereas the drift current is larger because of the larger electric field. And as a result, the drift current will dominate. And so now instead of electrons flowing from n-type to p-type, where we had defined as positive current, electrons are flowing from p-type to n-type in that, which we defined as negative current. And that's why our current has changed signs. Over here, notice that we're in positive current territory, and over here, notice we're in negative current territory. Also, you'll notice the width of the space-charge region changing as we forward and reverse bias our device. As the barrier height decreases, we have a decrease of the built-in electric field. We have a decrease of the amount of charge on either side of the junction. That's why the depletion width decreases. And the opposite happens here under reverse bias. So getting to the point where you can set up a pn-junction and understand how drift and diffusion currents come into being in the first place and then being able to bias your diode under different conditions is a really important fundamental skill for understanding how a solar cell works. Question. AUDIENCE: On the forward and reverse bias, does the Fermi energy actually continuous, or does it actually [INAUDIBLE]? PROFESSOR: So the Fermi energy, which we defined here as the chemical potential, notice we're avoiding talking about what's happening here in the middle until a couple of lectures from now. That gets into a gray zone where we talk about quasi-Fermi energies. We'll get to that in a minute or maybe a couple of lectures. But yes, the Fermi energy in the extreme sides right here near the contacts, so your contact in your device over here and your contact in your device over here, those Fermi energies are different on an absolute energy scale. So there is an energy difference when you're driving the electrons from one side to the other. Notice in this case right here, you would think that the electrons would want to travel through an external circuit to come back to this side because their energy's higher in this side and lower over here. But we're not illuminating the solar cell yet. We're biasing it using a battery. And this is why we have current flow coming from this side into that side. We're essentially forcing the electrons from the n-type into the p-type material. We're pushing them up that hill with a battery. And that's why we have this diffusion current dominating in the dark, in the dark. So we have a current flowing from right to left. In the illuminated case, we'll have all of our carriers traveling from left to right. And in the dark, this is the only case in which we have carriers traveling from right to left. And it's happening because we're using that battery in the dark to change the chemical potential on either side, which, in effect, reduces this barrier height and allows carriers to diffuse from the n-type into p-type. So you can think about it as forcing carriers up the junction. And this is a very useful technique because, in effect, what it's doing in a real device, when we you have a two-dimensional device within homogeneities, the current will travel through the weakest point of that pn-junction. Wherever the barrier height is lowest, current will begin crowding through that spot. And so it's a way of probing or testing the quality of your junction characteristic in the dark. OK. So this is the basics of pn-junction in the dark. Let's flip our page over and now let's try to imagine what will happen under illuminated conditions, and let's start out in a very simple case. We'll assume that the principle of superposition applies here, that the photo-excited carriers-- in other words, when light shines into our device-- I'm looking at this one right now or this one right here-- and light's coming in and generating electron-hole pairs, essentially exciting electrons across the band gap like we described in lecture 3, so we have carriers now being excited across the band gap, what will happen to those electrons now that they're in the conduction band? Where will they want to go? AUDIENCE: To the right. PROFESSOR: To the right, right? OK. So what you'll do is set up what's called an illumination current. Notice now at the bottom of the second row here, you have electron diffusion, electron drift, and IL current. IL stands for illumination current. So you have a third arrow here that will have to be implemented in some way. And that will-- by the principle of superposition you should think what happens to your IV curve as well. So let's make an estimate of what we think should happen, and then I'll confer some notes, and then we'll measure what actually does happen under illumination. Why don't we give it a shot? What does forward and reverse bias mean when there's no battery? This is a very interesting question. So once you start illuminating your solar cell device and you start injecting carriers into it, what will happen is, very naturally, this band alignment that you see right here, this band diagram, will begin to shift toward the forward bias condition. Rationale? You'll be generating carriers. They'll be swept over into this region here. One way to think about is that you'll be increasing the number of electrons over here and the number of holes over here, which will naturally cause the energy of those electrons to increase on this side, the chemical potential on this side, to increase. So one way to think about it is, as you're illuminating your solar cell more and more and more, you're forcing a forward bias condition on your device. To get the solar cell to go into reverse bias, you really do need to bias your device. You physically need to bias it. Yeah. So when I mentioned illumination current here, we're really talking about the electron illumination current, right, what direction of travel the electrons are taking inside of our system. So yes, the electrons would be traveling from left to right. Very astute observation. Current is generally described as a flow of positive charge. And in the absence of a definer or, say, electrons or holes, you rightly could assume that it would be the flow of positive charges. We're assuming electrons are flowing here in the illumination. current. Good. All right. So This is very positive. I see everybody has settled on the notion that illumination will shift our IV curve down. Because of the way we've defined current, that if current flows from left to right that lands us in negative current territory. So that makes sense. If we shine light on our system, we have electrons flowing from left to right here. That'll put us into negative territory. So that shifts the entire thing down. And we would add illumination current, an arrow pointing to the right right here, which would mean that we would have current flowing through our device, but there's still no difference in the chemical potential on either side. There's still a difference in Fermi energy in the p-type and the n-type, which means our voltage is equal to 0, which means we're intersecting the y-axis right here, and our little x should be marked right down here. So it's really just a superposition. Great. OK. Now what happens if we forward bias our device either because we're adding a resistor in series to our solar cell? So instead of a battery there, you would replace that with a resistor. Or if we're applying a bias voltage as well, we could also do that under illumination. So we'd still have the illumination current, right? And we, through superposition, shift this entire curve down, we'd be operating somewhere in this quadrant right there, right? So this we call IV quadrant, typically I, II, III, IV, IV quadrant. This is where power is coming out of the solar cell device. Because if we imagine instead of having a battery there, we have a resistor, the electrons will travel from the n-type material through that external load to get work the p-type material where the chemical potential is lower, right, because they'll desire to minimize their free energy. And as a result, they'll deposit that power across that resistor, across that external load, in order to get back to this other side because that's the only path that they can travel easily, right? And so this entire curve shifts down. You have your red x somewhere in the quadrant over here. And power is flowing out of the solar cell across that external load. So in the next slide, pretty much everything is right, except that, mea culpa, I forgot to replace the battery up there with a little resistor. So you'll want to correct that in your notes. Instead of having the little batteries up there, you can replace those with resistors or a resistor in series with a battery, if you prefer. Since depending on the illumination condition, the intensity we may have natural for a bias condition, we may need to apply a bias voltage. OK. So we have our IV characteristic like this. We have our red x under forward bias conditions in the IV quadrant, denoting that power would be flowing out of this solar cell device under these conditions. And now the bias is inverted. We have reverse bias conditions. And notice that the current still has the same sign. So the current is negative here, negative here, and negative here. So the net current is always flowing in the same direction in all cases because we have this illumination current, because we have this generation of carriers inside of the material. That's pretty cool. That wasn't the case when we had the device in the dark. Here, in the forward bias conditions, we're actually forcing carriers from the n-type material into the p-type material. But under illumination, now we have all of our carriers traveling from the p-type into the n-type. What's varying is the potential that the carriers have and, of course, the total amount of current. As you forward bias more and more and more, this downhill slope here decreases, right? So there's less of a driving force for the carriers to be going from the p-type to the n-type, and that's why the current approaches 0. Eventually at some point, if you keep forward biasing here, the current will be 0. There will be no net current flow because there will be no driving force for the carriers to go from the p-type into the n-type. There will be no more built-in field. Kind of cool. OK. So now we're beginning to wrap our heads around what's happening to the electrons and holes during solar cell operation. Let me put a little bit of mathematics to this. These are your IV curves. The blue is in the dark, and the red is under illumination. And we're focused on this IV quadrant right here because this is the quadrant in which power is coming out of our solar cell device, which usable power is coming out that we can power an external load. Why? Well, first off, the voltage is such that we can power an external load. We have charge separation. The electrons are accumulated over here. And they have higher potential than they do on the other side, which means that there's an incentive for them to go through the external circuit, deposit the power across that external load to get back to this other side where their potential is lower, right? So the voltage is favorable. And the current is also favorable because now we have light coming in. And this generation current is driving the carriers-- well, or is creating carriers here in the p-type that can then be driven toward the n-type because of the built-in field. So the conditions are just right under illuminated forward bias conditions to drive power through our external load. Under all other conditions of operations of the solar cell, we're putting power into the device, not getting power out of it. This IV quadrant over here, this forward bias illuminated case is the only case in which power is coming, usable power is coming out of our solar cell that we can use. So that's why we focus on this IV quadrant right here. The illuminated IV curve is, to the first order, is just your dark IV curve with a superposition, which we call the illumination current, I sub L. And that's what shifts our entire curve down by this I sub L right here, so this I sub L, that one right there or that right there. Kind of cool. All right. What do we think will happen if our light intensity goes down by a factor of 2? So now if the amount of sunlight falling on our solar cell drops by 1/2, what will happen? What do we predict will happen based on this right here? AUDIENCE: [INAUDIBLE]. PROFESSOR: The curve will shift, the red curve will shift up by about 1/2, right, because the illumination current is now cut in half. What will happen to the voltage intersect right here? What is the relation between voltage and current? It's the logarithmic relation, right? So it won't necessarily be cut in half, right? But it'll be cut by whatever this would be here, a log of 2. So OK. So we're beginning to develop an intuitive understanding of where electrons are flowing inside of our solar cells in the dark and under illumination. In the dark is important because we can test our devices in the dark, and we can still learn a lot about our solar cell device characteristics in the dark. As well, we can, under forward bias conditions in the dark, we can force carriers from one side of the junction to the other the wrong way and probe for weaknesses in the pn-junction regions. That's helpful. And in the illumination condition, obviously we're testing the total amount of power that's coming out of our device. So again, this is very useful because It's off of this red curve here that we defined the efficiency or the performance of the solar cell. So what I'll ask folks to do now is to-- we'll begin passing around these little tools. And I'll ask David Berney Needleman to come up to the front. David is our lab guru. He's the one who helped build these, really was the driving force behind getting them built. These are IV testers that will allow you to measure the current voltage characteristics of solar cells right here in class. And he's going to-- well, we'll pass them out while maybe he comes to the front here and explains how they work. So now that we have the basic IV curves rolling in, what I'd like you to do is modify the height or the intensity of the light. And the easiest way to modify the intensity of light is to move the light position up and down. So modify the intensity of the light and see how this IV curve changes. Note the y-axis scale, which might change as well. It might rescale depending on the condition. But note the y-axis scale and see how the intercept of the y-axis is changing with illumination intensity. Give that a shot. All right, folks. Why don't we circle back real quick. This has been a good experiment. I am very much in favor of multitasking and browsing. So if you want to keep your experiment running over the course of the remainder of the lecture, I will certainly have nothing opposed to testing a few different illumination conditions. And if there's anything really, really important I'm going to emphasize, this is a really important wake-up, folks. The I sub L, just to really recap here, we have this ideal diode equation, the illumination current coming in from our light source. And in our little set-up, how many batteries did you see there? AUDIENCE: One. PROFESSOR: Look closer. How many batteries do you see total in our set-up? Look especially at that light source. Two of them, right? There's a 9-volt and a 1.5-volt. All right. So one of them is powering the light source. And we have, as well, bias to the solar cell device, right? So we have a bit of a combination of the last two slides, right? In this case, in the dark, we were biasing our solar cell using the battery. And in the illumination conditions, the light itself was causing the solar cell to become forward biased. But we can add a battery to sweep the bias condition of the device, right? So even though we have a natural biasing of the solar cell by the light, we can force the solar cell under different operating conditions with the battery. And so that's effectively what's happening right here is you have a combination of both simple scenarios that we just looked at, and we're building on those components to really understand the larger system. Why might that be important, or why might that be realistic? Well, the light itself is biasing that particular solar cell device. But that solar cell might be connected in series with a bunch of other solar cells in a module, right? And those other solar cells might be biasing that one solar cell. So that's why we have to think about the solar cell device both from the perspective of what bias condition is it under, what illumination is it under, and of course, what's happening around it. Is it just powering an external load? Is there a battery connected in series to it? Are there other solar cells connected in series with it? OK. Yes, Ashley? AUDIENCE: So I don't understand still why having a load would bias the device. PROFESSOR: So let's imagine under illumination conditions right here, what I'm going to do very, very quickly is replace this battery manually in my slides, if PowerPoint auto save will allow me, with a resistor. So if you'll excuse my quick introduction here of the resistor. Now, sorry about the little artistic license. OK. So now I have my resistor here. And my illumination is coming into the device from-- say one of the sides is generating electron-hole pairs. I have a multiplicity of electrons that have now gone down the hill, right? It's more energetically favorable for these electrons to be on the other side. And what has that done? It's raised the chemical potential of this side. It's difficult for them to get back the other way. It's not impossible, but it could be more easy for them to flow through an external circuit to get back to the other side. And as they flow through that external circuit, they're depositing their energy across that external circuit. What energy? Well, it's the potential difference from this side to that side. So that's why the solar cell can be thought of, as in forward bias conditions, under illumination with an external load attached to it. Of course, the load has to be well-matched to the output of the solar cell device. AUDIENCE: And is the biasing because there is a voltage drop across the resistor? PROFESSOR: Yes. The biasing is because you have a shift in the chemical potential of this side up relative to-- you have a shift of the n-type side higher than the p-type side. That's a bias. Anytime you have a difference in the chemical potential in one terminal versus the other terminal of your solar cell, you have a bias. Whether that's generated by light, whether it's generated by a battery, right, whether it's the energy input to create that bias is coming from the sun or if it's coming from an external battery, that's a matter of detail. AUDIENCE: So is the sun forward biasing the cell? PROFESSOR: Yes. One can think about this as-- AUDIENCE: I thought you were talking about the difference between light and the LE-- OK. PROFESSOR: Oh. So the LED, in this case, could forward bias your device, too. I mean, it's just photons. Photons are forward biasing the device. AUDIENCE: So then can some ever reverse bias? PROFESSOR: No. That would be very difficult. What you could do, though, is have a bunch of solar cells connected in series with this one, right, that are producing forward power. You could shade this device, and then power could be flowing backward through it, right? In other words, it could be in the dark right here, in a dark condition. And you could be in a reverse bias condition just because of the way the other solar cells around it are behaving, if you have a shaded solar cell, for example, in a module. So imagine a seagull lands and kind of covers up one of the cells. That will be under reverse bias, and that could present problems if the solar cell can't withstand the reverse bias. No, this is a very ideal condition. What happens in the real world is that at some reverse bias condition, you'll just have biased it so much that electrons will be able to tunnel through from the p-type into the n-type right here. The electrons in the valence band will be able to tunnel through into the conduction band. And what happens to this IV curve is it goes zoom, begins dropping. So if the solar cell reverse bias-- let's see, the reverse bias current, or the current at reverse bias voltage, is not low enough, in other words, if the pn-junction is not strong enough, you could have a catastrophic failure of your module by just shading one of your cells. Thankfully, this is one of the testings that are done with the solar simulator to prevent that failure mode. AUDIENCE: OK. PROFESSOR: All right. I'll entertain deeper dives with questions. I'll try to keep the lecture focused on the broader general topics. But if somebody is interested in learning more, I'm happy to kind of dive into there. All right. So readings are strongly encouraged. I have interacted with several of you. Joe has interacted with probably 3n, n being the number of people I've interacted with so far, and really tried to impart the wisdom of pn-junctions. So please, please, please come to us if still things are going over your head. You should be able to explain to your roommates exactly what is going on in a pn-junction. Define parameters that determine solar cell efficiency. So now we have a qualitative sense about where current is flowing, where electrons are moving around, what defines the power output, how the power output is changing with illumination condition. We're getting an intuitive sense of all this. Let's start putting some discrete variables to all of that. And there are a bunch of two-letter or three-letter acronyms with some subscripts here that we'll get to know and we'll become very familiar with over the next few lectures. So how is solar cell conversion efficiency determined from that illuminated IV curve? That's our first question. And what I'm going to do is start with our source IV curve right here. This is just the IV quadrant. OK? So notice the current starts at 0 and goes to some negative value. So we're looking in the IV quadrant. Voltage is going from 0 to a positive value. So again, IV quadrant. We have our ideal diode equation here. And oh, notice one thing. I just changed I to J. What just happened? Well, I and J look very similar, but they're, in fact, two different variables. Most often, PV researchers will report a current density, in other words, a current per unit area instead of the actual current coming out of a solar cell device. So what you've been measuring here off of the DAC has been current, total current output from that device. And it might be a really tiny number, and it might be difficult to compare against other-sized devices. And so what solar cell researchers often do is they say, OK, let's normalize the current by the area to get a current density. And we'll call current density J, and we'll call current I, right? So for calculating power, we'll have to use I. We'll have to multiply I times V. But if we're just looking at one solar cell versus another, we can use J as a very convenient way of comparing one solar cell versus another. Cool. OK. So the illuminated IV curve looks something like this, right? It's in the IV quadrant. It goes out a certain amount, then it the curves up. We just determined that here in class. And that's essentially the ideal diode equation with a superposition term, this J sub L right there. So let's parse through this. We have the y-intercept over here. At the y-intercept, there is maximum current flowing through the circuit but no power because voltage is equal to 0. So remember, the Fermi energy is the same on either side. The chemical potentials are the same on either side of the device. So there's no energy gain of the electron traveling through the external circuit, but there's a maximum current. And no power flowing through that external circuit because there's no potential to be dropped across the external resistor. The opposite happens over here at this point called Voc, which we'll call open-circuit voltage. The oc stands for open circuit, V, voltage. Open-circuit voltage is just as you would think it is. When your solar cell is in an open circuit, when you-- say you took a pair of scissors-- please don't this-- and cut the leads so that your solar cell wasn't outputting the current through an external load, there would be a bias voltage built up across the p and the n side of the solar cell. And it would be the maximum voltage that could be supported by that solar cell device under illumination conditions. That's the open-circuit voltage, open circuit because there's no current, again, traveling through the external circuit. That's why current is 0, open circuit. And voltage because this is-- well, it's an interesting point because it's the maximum voltage here represented in the IV quadrant. And somewhere in between the point of open-circuit voltage and short-circuit current-- short-circuit current, again, because you're short circuiting your device-- current is flowing through, but there's no resistor. There's no external load. There's no power being deposited on external load. Somewhere between these two extreme conditions where there's no power flowing through the external circuit, you have a maximum power point where there is a power being deposited across your external circuit and a lot of it, right? That's the maximum power point. This is the point at which the solar cell is producing the maximum amount of power output. And to represent that slightly differently, what I've done-- so if I were to take current times voltage right here using IV quadrant data, my power would be a negative number. Why? Because voltage is positive, but current is a negative number, right? So I'd multiply a positive and negative number together, you get a negative number, and that just sounds weird. Who talks about power output from solar cells being negative? It almost sounds like power's going into the device. So this is another convention that you're going to have to get used to is looking at the IV curve in the I quadrant. So all we've done is taken the y-axis and multiplied by a negative 1. So we flipped it up, right? So bear with me here. It's a bit tricky to keep all this in your RAM. But here's our short-circuit current point now. Here's our open-circuit voltage point. Our IV curve now is pointed down. Before it was going up because we were in the IV quadrant. Now, we flipped, essentially just done a-- we've done a flip vertical, if you will, on our IV curve, and we have our current increasing here going to higher bias voltages. Now we can take the product of the voltage and the current to determine the power, and we obtain a curve that looks very much like this blue curve right here that you can see. And the maximum power point is truth in advertising. It's at the maximum power. It's where this blue curve reaches a maximum. That is the maximum power point of the solar cell device. That is where the solar cell is outputting the maximum amount of power. And so at this maximum power point, there is a voltage and a current associated with it that you can read right off the IV curve. And this we call Jmp, or current density at the maximum power point, and Vmp, which is the voltage at the maximum power point. So, so far, we've learned essentially four variables here. We have our Jsc, our Voc, and our Jmp, and our Vmp at that data point right there. Questions, since I know you have them. AUDIENCE: To ensure that the device is working in the maximum power point, does an external voltage have to be applied to it? PROFESSOR: So to ensure that the solar cell device is operating right here, a couple of things need to happen. You need to have the right illumination conditions, and you need to have the right load. So the two need to be matched to each other. Absolutely. And that's where some of the power electronics come into play. Yeah? AUDIENCE: So in the last problem set, where [INAUDIBLE], we assumed that output voltage would be [INAUDIBLE] volts. PROFESSOR: Yeah. AUDIENCE: Is that, in general, a safe assumption for [INAUDIBLE] solar cell? PROFESSOR: Yeah, yeah. So it's a very interesting question regarding the homework question. Let me repeat it so that the microphone can hear it. The homework question in the last homework, there was one question that inquired, assume that the voltage at the maximum power point is the band gap voltage equivalent minus 0.5 volts. And the rationale for that assumption is as follows. The open-circuit voltage, this point, is generally between 0.35 and 0.4 volts minus the band gap, or lower than the band gap. So you have the band gap energy minus 0.4 volts. And I can show you a very nice little paper that describes why that is. It essentially has to do, in part, with losses inside of the solar cell at thermodynamic limits of conversion inside of the solar cell device. Then what we've done is we've done another additional discounting from the Voc to the maximum power point, which we've assumed is around 0.1, maybe 0.2 volts. Notice the shape of the IV curve right here. The maximum power point is interesting because the voltage at the maximum power point is almost the Voc, in a good device. And the current at the maximum power point is almost Jsc, but not quite. All right? So the discounting from the Voc to the maximum power point voltage is not that much, as is the discounting from the short-circuit current to the maximum power point in a good device. In a bad device, this maximum power point here could be dragged all way down here. You could have an IV curve that looked something more like this instead, almost like a resistor, at which point the maximum power output would be a lot less, a lot less than what's shown here in blue curve. Cool. All right. So let's continue moving on. The efficiency of the solar cell. Eta, this Greek letter eta, is our power out versus power in. Our power in is the illumination intensity given in units of watts per meter squared. So we calculated this in our very first homework assignment and realized that the AM 1.5 spectrum is around 1,000 watts per meter squared. So that's our input power right here. Our output power is the voltage at the maximum power point times-- whoopsy-- times the current at the maximum power point, not the current density, the current at the maximum power point. So take this current density and multiply by area, and that's effectively-- the units work out better that way. So it would be either V times I at the maximum power point or V times J times the area, the area of the device, the area of the solar cell, at the maximum power point. And that's the total power out. Actually, yeah, yeah. So as long as the units are in units of watts per meter squared-- yeah, down here-- if this is not total watts but, watts per meter squared, you could still use current density. Those units would still work out. So be very careful whether you use total power in or normalized by unit area power, right? Just keep track of your units. Don't do like the professor. OK. So we have efficiency here as power out versus power in, the power out being the maximum power point power and the power in being the illumination from the sun. Now we're really talking. This is starting to get interesting because it's beginning to click. Pieces from lecture number 2 come together with what we're seeing now. So this is solar cell output power at the maximum power point and sunlight coming in. OK. So what I'm going to do next is I'm going to take this maximum power point and I'm going to draw a box that starts at the origin here, and the kitty-corner corner of my box is going to end at the maximum power point. So it'll have some rectilinear shape that will comprise the maximum power point and 0, 0, the origin, as two of its corners. And that box looks like this blue one right here. The area of that box is Jmp times Vmp. OK? And notice I have another box around here. I have this clear box that starts at the Voc point and the Jsc point. And now I have two rectilinear shapes, this blue one and the clear one right here, the bigger one. The bigger one has an area of Jsc times Voc. And I'm going to define a parameter called fill factor, which will be the ratio of these two areas, the ratio of those two boxes, the Vmp times Jmp divided by the Voc and Jsc. If this is 1, which is virtually impossible to do, but if this were 1, it would mean that these two boxes were the same size. And the current and voltage at the maximum power points would be the current and voltage under short-circuit and open-circuit conditions respectively. In real life, this blue box is smaller than the square box right over here. And so the Jmp Vmp product is less than the Jsc Voc product. And by consequence as well, the Jmp is less than the Jsc, Vmp is less than Voc. So the ratio of the two boxes is defined as the fill factor. The fill factor indicates the quality of your diode. If your fill factor is very poor, that means that that sun right over there at its maximum power point is being dragged toward the origin. That means that the area of this blue box is growing smaller relative to the area of this clear box. The fill factor is going down. That means you're filling less of this maximum square box function defined by the Voc Jsc. OK. So we have defined efficiency as power out divided by power in, power out being the current voltage product of the maximum power point divided by the solar insulation, fill factor being defined as the ratio of Vmp Imp product divided by Voc Ioc product. Notice that here I've written this in terms of total current, here in terms of current density. The areas essentially just cancel out because you have an area in the numerator and denominator. They cancel. These ratios should be identical. Thus we obtain an expression for the efficiency in terms of fill factor, Voc, and Ioc. Simply by using this fill factor definition right here, what I've done is I've multiplied this side of the equation-- let's just focus right here-- where we have fill factor equals Vmp times Imp divided by Voc times Ioc. I moved the denominator up to the side over here, multiplied it by fill factor, and that's my Vmp Imp. Now, I go back to that top equation and say my Vmp Imp is now going to be substituted by fill factor times Voc times Ioc, and that's how I get to this equation right here. Why? Why do I go through the effort of this little numerical manipulation? I do it because these parameters right here are fairly easy to measure using the solar simulator that you just put together. So I can measure the point at which my voltage is at open-circuit condition. I can measure the current at short-circuit condition. And simply by taking the ratio of those boxes, I can determine what my fill factor is as well. And these break down roughly into the current is going to be a function roughly-- again, I'm really painting broad brush strokes here-- the current is going to be roughly a function of illumination condition and bulk material quality. The Voc will be roughly a function of the interface and the diode characteristics. And the fill factor is going to be a function of the interface diode characteristics but also of the resistances within the device. And so from an engineering point of view, when we break the solar cell output down into these three parameters so that we can better understand what's going wrong with our solar cell. If we have everything lumped in terms of Vmp Imp, it becomes a little bit more obscure to figure out what exactly is going wrong with our solar cell device. Ashley? AUDIENCE: You said the fill factor also an easily measurable parameter? PROFESSOR: So the fill factor you would measure essentially by doing the little analysis we just did right here. Yeah, exactly. So you'd have to do a voltage current sweep. Mm-hmm. Coolness. OK. So we have an expression for efficiency in terms of fill factor, Voc, Ioc and our incoming power. So power out, this right here again is power out, divided by power in. Why does efficiency matter? Why do we care so much about efficiency? Well, the conversion efficiency determines the area of solar cells needed to produce a certain peak power, or to think of it differently, the area of solar panels that is necessary to produce a certain energy per unit time. And many costs scale with area. You have glass, encapsulants, the absorbent materials within the solar cell devices themselves, the metals that are used to make contacts, the labor that's used to install the panels. If you have a larger panel area, you need more labor to install it. The aluminum and racking and framing materials that go into holding the panels up in the field either on a roof or out in the field. So efficiency affects pretty much everything but the inverter and possibly some of the soft costs of the project. That includes the architect and the people who you pay to handle the money, financing, and the lawyers perhaps. So pretty much all of the real material and labor costs are scaling with area. And so efficiency determines that to a large degree, and hence it's a highly-leveraged way to reduce the costs of solar energy. If you do a sensitivity analysis, which you will do in the second and third parts of the class, and look at the costs of solar and how it scales with efficiency, you'll see that efficiency is one of the determining factors for cost in a solar cell device. And that's why we focus on it a lot. To put it into perspective, if the efficiency up there is determined by the output power versus the input power, if we had 100% conversion efficiency, which is impossible to achieve, thermodynamically impossible to achieve, we would produce a certain amount of energy per unit time, or a certain amount of peak power, with this panel right there. Say that's the size of our field installation. If we had a 33% efficiency cell, which is closer to the space-grade solar cells, we'd need three times that area, so three times the encapsulants, three times the glass, three times the labor to install it. And if we had a 20% efficiency, say, high end but still commercial solar module, not something you'd need to get from NASA, but something that you could buy from a supplier, you'd need five times that area. Whereas if you had a 10% efficiency module, which is more approaching the area of some relatively inexpensive solar cells, you would need 10 times that area. So if you're doing a cost analysis, this is why efficiency matters. It might still be cheaper to use this instead of to use this over here. That might very well be more expensive when you do the math and figure out how much it costs to deposit those materials with a very low throughput deposition process and very high cost. It might still be, but it might not. The material costs might end up whopping you. And so a simple equation that calculates all these parameters in, the material costs, the module efficiency, essentially the material [? and labor ?] costs, are being calculated in dollars per meter squared, just saying, how many dollars go into producing a meter squared of this material? And the efficiency is over here. And this is just a very simple back-of-the-envelope calculation type of way of estimating the cost of a solar system. So if you say, OK, I'm willing to pay more for a high-efficiency cell because I'm using less area, you can use this type of calculation to get to the answer quickly. It's not a levelized cost of electricity analysis. It's not using discounted capital flows and so forth, which we'll get to later on in class. This is a really back-of-the-envelope envelope engineering approach to estimating costs of a solar system. So I think this is a great place to stop. And if anybody has a pitch concerning their project ideas, class project ideas, I'd like to invite them to the front now. The class project, mind you, is really the capstone of this class, 2.626, 2.627. So if you have an idea, a fun idea, for a class project, I'd invite you to give a pitch up here at the front of class, or you're welcome to send it on an email to the class listserv. |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | 7_Toward_a_1D_Device_Model_Part_I_Device_Fundamentals.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: What we're going to be doing over the next few days, today and on Tuesday, is really diving into the device fundamentals, and then on Tuesday, the materials fundamentals, of how a solar cell device really works. And what we've done so far is skimmed along at a very high level using only the necessary physics and nothing more to describe how solar cell works. Because we want to give you an intuition about solar cell device operation. PROFESSOR: The alternative way of teaching PV is that we go heavy on the device physics upfront, you're completely overloaded, your RAM is completely full, and by the time we actually get to IV curves, you're completely lost. So hopefully you have some intuitive sense now about how a PV device works. Now we're going to be doing some deep dives into some advanced concepts so that we really have a sophisticated understanding of how a solar cell works. So let's dive into the lecture material for the 1D device model, we want to be able to create a one dimensional model that describes how a solar cell operates. So we want to capture all of the necessary physics from the materials and the device to describe the IV characteristics of a solar cell. So we'll do at the very beginning is start a little bit of nomenclature so we understand what we're talking about in terms of energy conversion efficiency and quantum efficiency. And then we'll start describing the different parameters that effect energy conversion efficiency and efficiency here. So one of the first key concepts that's very, very important to understand is that energy conversion efficiency is not the same thing as quantum efficiency. You'll here these two terms used quite frequently the PV community. Now, I would say we're really starting to get sophisticated about their use. In the very beginning of the field-- well let's put it this way, in the third wave of PV, which began at around late 1990s early 2000s, which was really when, say, some of the novel materials-- quantum dot-based solar cells, some modern organic material based solar cells-- really started to take off, there wasn't that sophisticated of understanding of some of the nomenclature in PV. And so terms were used in a confusing way. When you read some of the papers back from the early 2000s, you might pick up on some of this. And So that's why I have this slide right here describing very clearly what is quantum efficiency and what is energy conversion efficiency. So energy conversion efficiency is pretty easy to understand. We've been talking about that all class. We've been talking about how photon comes in with a certain amount of energy. And you extract electrons for your solar cell device that of a certain amount of energy. And that ratio would be the energy conversion efficiencies, essentially the energy coming out divided by the energy going in. So if, for example, one photon comes in with three EV and generates an electron hole pair, which is then extracted from the device with a voltage of say 0.6 volts, we would have an energy conversion efficiency less than 1, somewhere around 15% or so. Whereas if we looked at quantum efficiency and define quantum efficiency as the number of electrons out per incident photon, that means if we have one photon coming in and one electron coming out of our device, we have a quantum efficiency of 1, or 100%. So quantum efficiency can be thought of as collection efficiency. It means how many electron hole pairs were generated inside of the device, and how many were collected coming out of the device. There's a little bit more to it than that dealing with reflectivity off the surface. I'll get to that slide or two. But it can be thought of roughly as collection efficiency, as whereas energy conversion efficiency is really what you think of in terms of thermodynamic efficiency of a device. Why is this important? Well, there are papers out there-- this is an old one, I think, in nature of materials, if I'm not mistaken, that started using a bunch of terms, right. So under 5 volts bias and illumination from a 975 nanometer laser-- so nothing like a polychromatic solar spectrum, this is a monochromatic light source-- our detectors show an internal quantum efficiency of 3%. The photovoltaic response under this monochromatic light results in a maximum open circuit voltage of so and so, a short circuit current 350 nanoamps-- very, very tiny-- and a short circuit current internal quantum efficiency of 0.006%. So what they're talking about here is a quantum efficiency of 0.006%. As we've already done just by the simple example the 3 V phonon coming in and exciting electron hole pair, quantum efficiency of 1 but energy conversion efficiency below 20%, we can already guess that the energy conversion efficiency of a device like this, even for monochromatic light, is going to be very low. So the reason I'm highlighting this abstract right here is because it's a great example of the plethora of different terms that you can find in reading a paper. And if you focus on just a few of them, like oh, 3%-- wow, that's awesome, 3% device efficiency. No, it's not energy conversion efficiency. That's the quantum efficiency under a very specific bias condition, reverse bias is like a photo detector, not a solar cell, not under forward bias conditions. OK, so what does this really mean, this 0.006% short circuit internal quantum efficiency? Well, we defined our energy conversion efficiency or solar conversion efficiency as power out divided by power in. That was done in the previous lectures. The power out to would the power and the current and the voltage product at the maximum power point divided by the solar flux coming in. And that would be equivalent to the fill factor times the short circuit current times the open circuit voltage. And typical values for energy conversion efficiency are in the 12% to 20% range, I would say. Maybe less than 10% for emerging technologies. But these are typical values. And the solar flux, the illumination intensity might vary as well. But typically we talk about one sun, or AM 1.5 illumination conditions, right. So that's our spectrum at AM 1.5 conditions, what we did on homework number one. Now the quantum efficiency has two flavors. One we'll call external quantum efficiency and the other internal quantum efficiency. And basically the difference between them is the reflectance off the front surface of the device. The internal quantum efficiency essentially factors out the reflectance. The external quantum efficiency is really taking reflectance into account as well. So external quantum efficiency is defined as electrons out per photons toward the device. We say photons in, but this is really how many photons are impingent upon the device itself. Some of those photons will be reflected off the front surface. Other photons will go into the device and generate electron hole pairs. Or some will go straight through the device and be absorbed in the back surface or so forth. So the EQE, external quantum efficiency, typical peak values range between 60% and 90% depending on the reflectivity for moderate efficiency devices. So the reason peak values ranging 60 to 90 is because there's going to be some band of wavelengths at which the solar cell really responds well. It's really efficient at converting those photons into electron hole pairs. And were the sun-- instead of a beautiful polychromatic blackbody emission source-- if the sun where a monochromatic light source tuned to that particular wavelength, the solar cell would be wicked efficient. But it's not. So this is why EQE is an interesting parameter is because it tells you the response of the solar cell to different spectral conditions. And that's why we sometimes call it spectral response of a solar cell device. By the way, those who need notes, if you would be so kind as to raise your hand, Joe will pass them around as well so we can have enough for everybody to write down. Any questions thus far on EQE, external quantum efficiency, the difference between QE and the difference between ECE, or shall we say, the energy conversion efficiency? Okay. AUDIENCE: Quantum efficiencies, they also include [INAUDIBLE] photons, right? Or do they just include photons which have a potential to make an electron [INAUDIBLE]? PROFESSOR: In principle, QE, or sorry, EQE, should have EQE as a function of lambda, really. And the photon in is going to be, obviously, at a certain wavelength usually. I've rarely heard EQE given in a polychromatic sense where they take an entire solar spectrum and measure the quantum efficiency average. Usually quantum efficiency is measured wavelength by wavelength. And so you might change your monochrometer settings 20 or 30 times over, or even more, over a quantum efficiency measurement to really map out the entire solar spectrum from short wavelengths in the ultraviolet all the way to longer wavelength and the infrared. And you're absolutely right. Some of those wavelengths, the photo response of the solar cell will be very close to 0 because it just doesn't respond well at that wavelength. Maybe the wavelength is too short and is being absorbed by some surface layer. Or maybe the wavelength is too long it's just going straight through [INAUDIBLE] photon. Whereas energy conversion efficiency is really taking the entire solar spectrum and matching or testing the energy, the power coming out of the solar cell versus the power going in. So a typical quantum efficiency curve might look something like this. Let's walk through the axes first and then talk about the curve. So the axes, we have wavelength on the x-axis, on the abscissa. Then on the ordinate, we have quantum efficiency. Wavelength is varying from, let's say, the ultraviolet to the infrared and beyond. Well, beyond the red into the near infrared and probably ending somewhere on the mid-infrared. And the quantum efficiency extends from 0 to 1, in other words from 0% to 100%. And we have this box that's represented by this maroon line right here entitled Ideal Quantum Efficiency. And this would be if at the band gap energy, all of a sudden our quantum efficiency turned on to 100% and cut across. And this was the assumption that we made in our homework, that the solar cell device would respond in this ideal manner. In reality, at longer wavelengths here, what's going on? Why would there be a decrease of the quantum efficiency at longer wavelengths? What effect is happening there? AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah, the absorption coefficient is dropping. And this light is starting to go straight through the device. There's a growing fraction of photons that are not being absorbed by the solar cell device. In the very short wavelengths right here, as I said before, oftentimes there's a dead layer right near the surface of the device that's impeding good electron hole pair charge separation and eventually a collection. And right here in the middle, there's a slow but steady decrease of the quantum efficiency typically. These are photons that are being absorbed in the absorber layer. If you think about your solar cell in cross section, here's your solar cell in cross section, these are photos being absorbed in the absorber layer generating electron hole pairs which are then being separated by the junction. And as we go to longer and longer wavelengths, if you remember your optical absorption coefficient begins dropping, that means that the penetration depth of the light, the average penetration that the light is increasing. And so the average distance that the electron hole pairs are being generated from the junction is increasing. So in other words, as we go from short wavelengths to longer wavelengths, the distance from which the electron hole pairs are being generated from the junction is increasing. That means that the electron hole pairs generated at these wavelengths have further to travel to reach the junction than those electron hole pairs generated at these wavelengths closer to the front surface. And since there's a finite probability that the electron hole pair will recombine as it travels through the material, that's why you get this very slow but steady decrease of quantum efficiency over the mid-wavelength range. And on Tuesday we'll talk more about that. Internal quantum efficiency. So for those who say, well you know, there's a certain delta right here between 1 and my maximum response, here my maximum quantum efficiency. And I suspect that's really due to the fact that a certain percentage of my photons are being reflected off the front surface. I know how to calculate reflectance. We did that in lecture number two or three, I believe. So we know how to calculate reflectance off the front surface. We can also measure reflectance using a spectrophotometer. And what we can do is normalize the effect of reflectance out. By dividing our external quantum efficiency by 1 minus R, what in effect we're doing is we're normalizing the effective reflectance out of our measurement. So now we're only considering those photons they get into the device. If you want, you can think about the extreme cases there. If reflectance is 0, that means your EQE is equal to your IQE, your internal quantum efficiency is equal to the external quantum efficiency. But if reflectance is, say, 50%, if 50% of your photons are being reflected off the front surface of your device, now your IQE is probably going to be double your external quantum efficiency. And so now this is an interesting parameter because, for example, in our lab scale devices when we're testing them, oftentimes we don't optimize for optical properties. Sometimes we do, but many times where we're focused on the material, we're focused on the junction, and we don't bother to put the perfect entry reflective coating on it. It would take too long to fabricate it. And so IQE gives us a good metric of how the solar cell is responding if we were to go through all the effort to make the light capture ideal for our device. Okay? And that's why we sometimes use a report IQE in studies. And typical peak values are between 80% and 98% for moderate efficiency devices. So it would essentially-- the curve would look very much like the EQE curve. It would just be boosted up by 1 over 1 minus R. Okay? So any questions so far? Make sure we're on the same page here. Yeah? AUDIENCE: [INAUDIBLE]? PROFESSOR: Absolutely. So IQE is a function of wavelength. R, reflectivity, is also going to be a strong function of wavelength because the real component of the refractive index is changing as a function of wavelength. Yeah, absolutely. OK, so this gives you a sense. Sorry for the poor choice of the reflectivity colors right here. This baby blue, which is almost impossible to see, but I've just traced it out with my finger, that denotes the reflectivity as a function of wavelength from some data that it took as a graduate student. So you have some typical quantum efficiency curves right here. And then the quantum efficiency curves that are shown are for a pretty poor quality device. And in this particular case, I measured quantum efficiency under short circuit conditions, but with different illumination intensities. So going from 0 sun illumination all the way up to 1 sun illumination. But the device was still in short circuit conditions. So what I was doing, in effect, was flooding the device with more and more and more carriers, and I got this boost here in the middle wavelength range. I still was limited by my surface. I was still limited by my band gap and non-absorption effects over here. But in the middle, I was able to get this boost in performance simply because I was filling in trap states inside of material. This is an advanced concept that we're going to get to over the next few lectures. But it gives you a sense of what a QE curve might look like for device, what the reflectivity might look like, and these curves right here were IQE, or internal quantum efficiency. OK, so take efficiency with a grain of salt. When you hear somebody reporting an efficiency value, ask-- think about what efficiency is being measured. Is it energy conversion efficiency?4 is it quantum efficiency? Internal, external? What wavelengths? What is the nature of the light being used, right? Is a 1 sun illumination source that has been well calibrated to within 2% spectral fidelity, spatial uniformity, temporal stability? Is it a really bona fide good light source? Or is this some monochromatic light source that somebody set up because they knew, aha, my solar cell really responds well to, say, 500 nanometer lights, right? So I'm going to test my-- I'm going to report my values right at that wavelength. You have to take it with a grain of salt. Yeah, exactly. And what is the intensity of the light that's being used as well. That's very important as we just saw right over here because the bulk material is going to respond differently. There are certain trap states for electrons inside of the material. And as you add more and more light, you're flooding those traps. They become filled, and the electrons can move through the material more easily. That's why you get this boost of response as you increase the illumination intensity in this particular experiment. So these are all questions that you should ask yourselves when you see quantum efficiency-- internal quantum, external quantum efficiency values reported and/or energy conversion efficiencies. What is really happening? What are they measuring? This right here, a paper out of Paul Alivisatos' lab from 2002 was, in my humble opinion, an excellent example of honest efficiency reporting. First off, they picked their highest values and put their best foot forward, and said look we have a quantum efficiency over 54% , and a monochromatic power conversion efficiency of 6.9% under very low intensity light at 515 nanometers, right. So first off, they put their best foot forward and said, under optimal conditions monochromatic light that's what we get. But also under air mass 1.5 global solar conditions, under the standard solar spectrum, we obtained an energy conversion efficiency, or a power conversion efficiency of 1.7%. So both values are put there out on the table, and they say look, here's a characterization of our device. So I really like this paper. It was back in 2002 right at the beginning of the new wave of new materials in PV, so to speak, this third wave of PV in the United States. And it was a beautiful example of a really well-reported, carefully described efficiency. Any further thoughts or comments before we jump on to the next topics of the day? AUDIENCE: How do you have a QE curve for 0 suns, I noticed on your plot? Or is that kind of an extrapolated? PROFESSOR: Yeah, so this is the bias condition-- sorry the bias lighting condition, not the bias voltage as in battery, but bias lighting as in I shine a light source, a polychromatic light source at my material. I believe in that case it was a halogen lamp with a series of filters to simulate the solar spectrum. And obviously you can't measure a quantum efficiency without having some light involved. And so the light that was being generated by the monochrometer was much lower in intensity than these illumination intensities here. And so it was more of a perturbation on top of a steady-state background. And we ran the measurement exactly those conditions. We had a steady-state background light, and then we chopped the monochrometer light coming in , and used a lock in amplifier to track that particular frequency. So we were able to detect the small perturbation due to the monochromatic light coming in on top of the steady background. Yeah? AUDIENCE: There's no international centers to report efficiency or convention? PROFESSOR: There are. There are conventions, international conventions, to report energy conversion efficiencies. And I believe there might also be for quantum efficiencies as well. Let me look into that. Why don't we take note of that and bring back the precise ASDM standards next time. And I believe also there are other entities like IEC that have their own systems for it. There about four laboratories worldwide that are authorized to give standard efficiency measurements. So they have very carefully calibrated quantum efficiency measurement devices, very carefully calibrated solar simulators. And if you really think you have a record efficiency cell, you should get it checked at one of those laboratories. You shouldn't just report what you have coming up your lab-based system. You may even want to go around to a few different labs at MIT first because one might have a better calibrated solar simulator than another. What we're going to do now is dive into some of the common factors that cause solar cell IV curves to deviate from normal ideal diode model. We're going to talk about shunt and series resistance, recombination currents-- maybe we'll get to current crowding, some other effects. Basically what we're doing now in number two, is we're beginning to say, OK, now that we understand an ideal diode, now that we have an ideal picture of how our solar cell works, let's bring it down to real life and talk about all the things that can go wrong. Because when you try to make a solar cell in the lab, these are the things that are going to start nipping at your heels, so to speak. So we'll start out with an equivalent circuit diagram of a solar cell device. And this might be new to some folks, but it's all stuff you've seen so far. What we have over here is the current generation source of the solar cell operating under forward bias conditions, under illumination, generating a voltage and a current. We have a diode here that prevents the back flow of current, and forces the current to go out through the external circuit loop. And it forces it to drive the external load, which would be attached to that little circle over there, and that other little circle over there. So you can imagine external circuit attaching itself right there. So the diode-- if the diode quality is high, if it's good, it will prevent the backflow of current. If the diode quality is poor, you will have so-called leakage current, current leaking into the diode back-- if you want to think about under forward bias conditions, here's your solar cell forward bias. You have light, your electrons coming in, separated by the electric field. Instead of powering the external load and coming back into the device through the back, leakage current would be the electrons going back up the barrier, going back into the base of the device, the so-called diffusion current. So this is that diode right here representing the diffusion current, or in that particular case, in the ideal, the equivalent circuit diagram, this is representing the resistance to diffusion, if you will. So we have this voltage and current being generated by the solar cell, and that would be described by the ideal diode equation. My apologies, there should be actually a x of this minus 1 right there. We should take note of that and correct it. I just happened to notice that. OK, so the ideal diode equation is right here. And this is the current density versus voltage. So we have in linear log scale or axes, we have a straight line. That would mean on a linear scale, we would have an exponential curve. So an exponential function, exponential curve up here, and a straight line in linear log scale. I'm trying to sensitize you to the two ways of looking at this, both a linear scale and a log scale of current in the y-axis because both will tell us something. From the linear scale up here in the upper right, it will be very easy for us to see the fill factor. We can just glance at that curve and see, oh, it looks really box like. It must have a large fill factor. From this curve down here we can begin to see deviations from ideality. If anything causes this curve to bend up or down, either down here or up here, we're going to see it really quick on the log scale. And that's why looking at the IV curve in both ways is helpful. It's a method of diagnosing the problem quickly. So what we're going to do now is to take our equivalent circuit and begin adding a few things to it that represent realistic effects that we might see in a real device. So the first thing that we might do is say, well listen, a real device has a finite series resistance. And that can come from a variety of sources. We'll talk about those in a few minutes. But we might add a series resistance component right up there. So let's look back. I'm going to flip back and forth between the two slides just so folks can really see what the difference is. This was the ideal case. That's the with series resistance, again ideal with series up there in the equivalent circuit. Great. So what have we done? Well, we've added a series resistance component which is essentially dropping the voltage output of our solar cell. And notice what else happened. Now we have a nonlinear equation. Our current density is a function of our current density. So this is becoming more difficult to solve analytically. And what is happening to our IV characteristics? Well, let's look-- we'll go back once again to our ideal diode equation right here. We have a very sharp box-like IV curve under the ideal conditions. Now when we introduce the series resistance component, we begin smoothing out that IV curve. The fill factor has dropped. Take a look. Ideal conditions, sharp, very large fill factor. Now we've added the series resistance component, the fill factor has begun dropping. If we look at it in log scale, it's very easy to detect that series resistance. We can see it right here. This used to be our curve, right? So again, this was the ideal case, straight line. Again, this is log of current verses linear voltage. So the x-axis of the two plots here are the same. I'm only changing the y-axis from linear scale up there to log scale down here. And notice what happens. If I add the series resistance component, now I begin dropping off. So this is a telltale sign when you take an IV curve and you see that, then you have some series resistance effect inside of your external circuit or your solar cell, or some combination of the two. Yes, Ashley? ASHLEY: So from this curve it looks the series resistance really only starts to have an effect once you have a higher voltage. PROFESSOR: Absolutely. And that, you can begin to see directly from the equation right here as well. Take the limit of low voltage and the limit of high voltage. Or another way to put it would be the high current, really. And the high current to voltage ratio only begins to take on up at higher voltages because of the exponential function of that curve. So that's a very, very important and astute observation that the series resistance begins to kill you at high forward bias voltage conditions. You hope that you design your solar cell in such a way that the series resistance only really begins to kick in above the maximum power point voltage. That's the goal. AUDIENCE: This is not voltage out, this is voltage applied to the [INAUDIBLE]? PROFESSOR: This is the voltage that is being measured across the solar cell. AUDIENCE: The output voltage. PROFESSOR: The output voltage, yeah. Yes, you can think of this as the biasing condition of your solar cell device. The two are almost equivalent. The only difference is you have that series resistance in series there. So you have to be careful whether you're talking about the voltage here at the solar cell itself across a junction, or the voltage coming out of your device that is effected by that series resistance as well. So think of it is doing that IV sweep that we did in class the other day. So you're taking the IV sweep of your device, and you're seeing the impact of the series resistance in the output voltage you're measuring of your solar cell device, just like we did in class. So again, series resistance will effect you at some point. It is almost inevitable. But you have to design your solar cell such that it effects you only after the maximum power point. Likewise, we're going to introduce another resistance term inside of our equivalent circuit diagram, what we call a shunt resistance. Let me back up one step here. This was just a series resistance, now I've added a shunt resistance term here as well. And whereas we wanted our series resistance to be small, right? The smaller our series resistance, the less this effect would be, the more ideal this curve would be. We want our shunt resistance to be large, because we want to prevent the current from flowing inside of the device back to the base. Instead, we want the current to flow through the external circuit. So we want the shunt resistance to be large. In other words, we don't want our solar cell device to full of shunts. That's pretty obvious. OK, so what does shunt resistance do to you? Well, if we compare the IV curves once again between the solar cell just with the series resistance, we had a very low forward current here under low forward bias conditions. If we have a very poor shunt-- if we have a plethora of shunt pathways in our device, we're going to get current flowing through our device even at low bias conditions. That's because the pn-junction is weak. And even though we have a very large field in there, there are some regions where the field is much smaller because of the shunt pathway, and current is going to be flowing through there. Diffusion current will be flowing through that shunt pathway. So think of this as being a very strong diode, right. In the absence of shunt resistance, you have a very strong diode. You have in your p-type region and your n, and that large barrier is preventing the diffusion of electrons back into the device. It's preventing this from rising. It's keeping it low. And that can only happen when the pn-junction barrier is uniform throughout your entire device. If you have one region if your device that is poor that has a shunt pathway, maybe you have a piece of metal that fired through the junction and is making good ohmic contact to both sides. If you have a shunt pathway, now the current can flow even under small bias conditions. The diffusion current can flow into your device. And that, of course, drops on the linear log scale right up here, you hardly detect it. Notice, if we go back and forth between with and without shunt resistance, you can hardly detect it right here until you start getting to some really leaky diodes, in which case you begin to impact your fill factor as well. So shunt resistance is important for a solar cell device that it not be too, too high. Because at some point it does begin impacting your fill factor. Sorry, the shunt resistance not be too, too low. Yes, your shunt resistance can't be too, too low because at some point it will begin impacting your fill factor. But more importantly, shunt resistance is usually indicative of some localized failure in your pn-junction. It's usually not homogeneously distributed throughout the entire pn-junction. Maybe your first solar cell device you ever make might be effected by that. But as you get better and better, you'll start making higher quality junctions. And usually the shunt resistance is indicative of one little spot. And what happens at that one little spot? Well, current is flowing in from the entire emitter region into that one little shunt and locally heating up your device. And so that's why people worry about shunts in industry, is because if you have current crowding into that one little spot, it's heating up, it's becoming a hotspot, and it can even melt the encapsulated materials in your module. Perhaps even create a fire in extreme conditions. And so when you test the solar cell devices, typically the IV tester will measure the shunt resistance in a solar cell by looking at what's happening under very small bias conditions right here, and comparing it against the healthy cell that it would expect. And if it finds cells with high shunt leakage current, in other words, low shunt resistance, it will advise the tester and sorter to throw out those cells, and so they'll be put into the scrap pile. And so the shunt resistance manifests itself right here in the ideal diode equation in this term right there. And so it's essentially an add-on term after your exponential, looking something like this. Essentially, at low bias conditions, you can extrapolate this point right here, determine the slope of that point, and find the shunt resistance. And likewise, you'll notice that the shunt resistance really isn't impacting you up here. The shunt resistance really isn't impacting you at higher forward bias conditions. That's mostly series resistance in the illumination current that's overwhelming any shunt pathway. Ashley, yeah? ASHLEY: Do you know for, I don't know, a high-quality manufacturer, what percentage of produced cells get tossed because [INAUDIBLE]. PROFESSOR: Very low. The manufacturing yields of a well-oiled manufacturing line- I say oiled in a figurative sense. We're not typically-- yeah, some of the parts, I suppose, use oil. In a well-functioning manufacturing line, the yields are upwards of 95% for cell fabrication. So it's a very small percentage. You typically see this when you're developing a new process. And that's why I'm informing all of you about this is because we're all developing new processes here, right? We're all working on new solar cell devices. And it's very important to be aware of what's going wrong in your device, not just that you're getting a low efficiency because this is the first step to troubleshooting. Yep? AUDIENCE: [INAUDIBLE] but why is it bad that we have a high current density? [INAUDIBLE] current density [INAUDIBLE]? PROFESSOR: Why is it bad that we have a high current density? Why don't we go back to simple case right here where we have a series instance, for instance, right? And so at, say, 0.5 volts forward bias, we now have with the series resistance, a higher forward currend-- we'll call a dark forward current because we're measuring this in the dark. That's why we have 0 here. And it's forward current, meaning it's like a diffusion current going from the emitter back into the base. And so we have a higher dark forward current here in the dark with the series resistance, then if we didn't have the series resistance, right? So let me do that again. We're up here at, say, 1 times 10 to minus 2 milliamps per square centimeter, whereas before we were down at around a quarter. So why is that bad? That's bad because when we illuminate our solar cell device, we are going to be shifting that curve down, and that's going to reduce the total current output at the maximum power point. So anything that goes up here when it's shifted or transposed by a finite fixed amount, it will be closer to the 0 point of current than it used to be. That's why it's bad. So anything that increases that blue curve and shifts it up in the dark is going to be bad for our solar cell. In an ideal case, we want our dark forward current to be as small as possible. We want our J0 to be as tiny as possible for a good solar cell device. OK. Good let me show you dynamically what happens in a solar cell. Oops, sorry about that. Here we go. So this is a PV CD-ROM again in a beautiful example of, in this particular case, looks like series resistance. So now we're at 1 ohm series resistance for the entire device. The ideal solar cell is shown in red. The IV curve for the ideal solar cell is shown in red. It would be a short circuit current of 35 milliamps per square centimeter, an open circuit voltage of 624 millivolts, and a fill factor of 83%, ideal solar cell. The real solar cell now with 1 ohm series resistance, the fill factor has dropped by 5% absolute down to 78%. And now I'm going to increase the fill factor. What would you expect to happen to that curve? Flattens, right? The fill factor drops. So let me start increasing the series resistance by manually dragging this forward. So now we're at around 2, 3, 4, 5, 7, 8, 9, 10. So now we're at 10 ohms series resistance. That means we have a large resistance inside of our solar cell device at some point. And our fill factor has now dropped precipitously. And if you remember that efficiency is proportional to fill, factor that means we have a problem with our device. We're really dropping the maximum power point. We're really dropping the operating point of our solar cell. So if you were to do an extreme condition, you might have so much series resistance that your IV curve looks like a straight line, you have an ohmic resistor now. The resistance component is swamping out any diode-like behavior of your solar cell. And that's why you have a line. Likewise, you're dropping the total current output of your device, right, because you have a resistor in series now that's preventing the current flow. OK, so that's, again, an extreme case of what can happen in a shunt resistance. Let me show you that as well. So we have another beautiful example shunt resistance here. Now we start out with a very high shunt resistance, because in our equivalent circuit diagram, if we go back up here, we have a large barrier that prevents the current from flowing back internally inside of the device. The current is forced to go outside through the external circuit. But as we drop that red shunt resistance component right there, as we decrease the magnitude of the shunt resistance, we will allow the current to flow inside of the device. Now we have internal current loops, and that means that we no longer are forcing the current go outside through the external circuit. And somewhere around between 1,000 and 100 ohms for our shunt resistance, we really begin to see that drop in fill factor. And as expected, it's really begin to impact us at the lower bias voltages down here like we saw. And at some point, we drop the shunt resistance so much that now the voltage across our solar cell is even suffering because there's not enough separation of charge sustained to make that voltage large because the current is being shunted back inside the device. And again, we have what appears to be very linear IV curve. So you can see when you measure your solar cell and you test it, and you get a linear IV curve instead of that nice exponential, you have to do a little bit of troubleshooting to figure out what exactly is going on. OK. Good. So key concepts so far. We can add the effects of parallel resistance. What we typically refer to a shunt resistance is also called parallel resistance because it appears in parallel with the diode inside of the device, right. So we can have a parallel or shunt resistance and series resistance in our devices. And as an advanced concept, I really wanted to expose you once, and we can talk more about this in office hours, to the notion that in addition to just having one saturation current, you could envision difference saturation currents coming from different regions of your solar cell device. Let me walk you through that. Under low bias conditions where we have a large built in field, and we have very low bias conditions. Before bias, obviously the field is going to be decreasing, the barrier height's going to be decreasing as well. So under low forward bias conditions, current is going to have a hard time getting all the way up into the base and recombining inside of the base, but some of the current could recombine in the space charge region of our device. And that's why under low forward bias conditions, we might have what is called a J02 current, or recombination current in the space-charge region. And then there are high forward bias conditions when a lot of the carriers can diffuse into the bulk, and recombine inside of the bulk, we might be driven by different recombination mechanisms, bulk recombination mechaniss. So that's sometimes where you see what is called a two-diode model for a solar cell. And the two-diode model for a solar cell would look very much like this, except you'd have two diodes one right next to the other. One that would effect you at low bias conditions, another one at large bias conditions. You might see this in your research. It's really an advanced concept more targeted towards the graduate students here. Another thing you might see pop up is this ideality factor showing up right down here. The ideality factor is a slope factor. What that means is-- here if we go back to this, if you change your ideality factor, you're changing the slope in log scale. It'd still be a straight line, but if your ideality factor increases, you would get something like this. If it decreases, the slope would increase, right? Because you have the equation dependent on 1 over this parameter. So the ideality factor is important as well because it indicates the type of recombination mechanism that's driving the current inside of that region of the device. If anybody ever comes across this in their research, come talk to me. I'd be happy point to the right directions. Or you could look up the thesis of Keith McIntosh from University of New South Wales. Excellent, excellent thesis entitled, "Humps, Lumps and Bumps: Three Dimensional Effects of the Current Voltage Curve of Silicon Solar Cells." Some of which might apply as well to other material systems as well. And you can tell where the lumps, humps and bumps comes from. It's really these lumps, humps and bumps in the IV curve he's talking about. Excellent read, very good thesis. Highly recommended for those who want to look into their IV curves a little bit more. It's kind of like looking to the tea leaves, right? Because there are a number of things that can be effecting the IV curves. We'll get to some of those on Tuesday as well. These are IV curves, again in log scale on the ordinate, linear scale on the abscissa. You notice there's little commas here because I took this data in Germany so the decimal place is written as a comma. The IV curves vary in shape and in style. Let's look at this one right here, this black one. It starts up like this, and then it goes flat for a while, and then it becomes series resistance dominated way up at higher bias voltages. Whereas this blue curve right here, this one again, you have the effect of shunt resistance at lower buys voltages. Then you have one diode, two diodes, and eventually the series resistance component coming in as well. Or perhaps it's a large shunt as well. You don't really know until you start fitting it, and modeling it, and maybe measuring the temperature dependence and so forth. So this gives you a sense of the real life diode IV curves that you might get over the course of your research, and it gives you some tools to use to begin parsing through and figuring out what's going wrong. Let's talk about that. Let's talk about solving problems. So fill factor. What can impact the fill factor? Well, we talked about the need for a high fill factor to produce a high power cell. Very easy to understand. The larger the fill factor, the larger the maximum power point. If you have a low fill factor, your maximum power point is going to drop. Let me show you back and forth. Good device, bad device, large fill factor, low fill factor, high maximum power point, lower maximum power point. OK, we talked about this. OK, so causes of shunt resistance, physical causes. I alluded to this earlier, but let's imagine you have that weak spot in your pn-junction. This is meant to represent pn-junction in two dimensions now. So, so far we've only taken a cross section like this. Now you have a real device in 2D. And what this is meant to represent to some local weakness in the pn-junction. And the current is now flowing through that local weakness. So that's a representation in a 2D energy band diagram E versus x versus y of a shunt in realistic conditions. So shunt series resistance, on the other hand, we talked about the effect of high series resistance, and we're going to calculate the series resistance for a solar cell. In part, because this is asked for you on your homework assignment. At least the graduate students have a problem pertaining to this. So let's talk about the different components of series resistance. We have a bulk current and then a lateral current inside of a typical solar cell device. So sunlight comes in, shines, generates electron hole pairs. These electrons will diffuse to the junction region, and then they'll drift across the junction, wind up in this emitter front surface region. And eventually through lateral diffusion reach the context and be pulled out of the device. So we have to consider these two different components, and pretty much the current all the way from the back contact to the front contact. Any one of those things could contribute to series resistance in an additive sense. So to put that pictorially, we have bulk resistance, emitter sheet resistance, the contact resistance, and in line losses out of plane as we travel along those front contact metallization. So bulk resistance. This is another way to state how to choose an absorber thickness. So we've talked about choosing absorber thicknesses, in other words, we've talked about choosing the thickness of our solar cell device based on how much light we want to capture, based on the optical absorption coefficients. Well guess what? This is a co-optimization problem. We also have to worry about the finite resistance of the bulk material, and we have to spec the thickness accordingly so that we don't wind up with too thick of a bulk material and the series resistance component will be too large. So how to choose the appropriate absorber thickness. Let's talk about that for a minute. So you can measure a parameter in the bulk called resistivity. Resistivity will be given as a function of q, which is the charge, u, which is the carrier mobility, which means how easy is it for that carrier to move around the lattice, and n being the carry density. So let's think that through. Resistivity-- if our carrier density goes up, if we have more carriers to transport the charge, the resistive should go down. That makes sense. Check. If the mobility goes up, meaning it's easier for the carriers to move around the lattice, then the resistivity should go down. Check. OK, so that makes good, intuitive sense. Now the base resistance, in other words, the resistance that we should have as current travels across the base, across the absorber of our material, which we also call the base of our solar cell. So as the total resistance, as current goes across the base, should be given as Rho, which is a fundamental material parameter combined with some geometric parameters that define the size of our solar cell. So l is going to be the length of the conductive of path which typically would be given as the thickness of the device, and A is the area of current flow. In other words, the dimension or size of our solar cell device. So to minimize base resistance, we want to have a very thin solar cell device. But wait a second. If we make it very thin, then what happens? AUDIENCE: [INAUDIBLE]. PROFESSOR: You get less absorption. So you have to make it a certain thickness to get a good enough absorption. Then we can compensate. If we still need to drop the base resistance, we can increase the area of the base. And so far that makes good sense. There's nothing else tugging for the area to be the other direction, right? So we can make this solar cell infinitely large, thick as we want to be to absorb all the light. Great, OK. Hold on to that thought because we're going to have another constraining parameter that pushes us in the opposite way in a few slides. All right, so the emitter sheet resistance. In other words, how to design front contact metallization. Emitter sheet resistance refers to the resistance of lateral current that the carriers will experience as they move laterally to reach the front contacts. And again, we go back to this fundamental material parameter where we have resistivity as a function of mobility, and total dopant concentration. Sorry, for some reason, I used the big N here and a little n on the previous slide. But it's the same thing. It's carrier density. So again, our resistivity is a function of carrier mobility and carrier concentration, how easy it is for the carriers to move, and how many of them are there. For a thin layer, a sheet resistance can be described as the integral of the resistivity by the thickness of that layer. Or to put it more simply, if you have a uniform layer, you simply multiply the denominator here by thickness. So this can be thought in the following way. If we have a thicker emitter, we will be dropping the emitter sheet resistance. We will be dropping the resistance component that carriers experience as they travel laterally through this. And that makes sense because, in effect, as we increase the thickness of this emitter region, we're increasing the total cross sectional area for the charges to travel through. So folks with me so far. For the emitter sheet resistance to be low, we need the thickness to be good enough, the thickness to be thick enough, the carrier density to be high enough, and the mobility to be large enough. And oftentimes, the resistivity of a material, you might spend months trying to optimize the resistivity of your emitter region. For those doing thin film materials, you might be working on zinc oxide, aluminum doped, or fluorine doped zinc oxide, for instance. And after months of tuning the temperature and the oxygen partial pressure during the deposition process, you're there and you have what you have. You have a certain carrier density and a certain mobility. And you can't do much more than vary the geometric parameter varying the thickness. So what next? If my thickness is too big on my emitter, if my thickness is too big, I'm going to be absorbing all of my short wavelengths photons in the front surface of my device. And maybe the current collection probability from that layer up here, that has a very high carrier density, is not going to be as large-- in other words, the current collection efficiency won't be as large as if I generated the carriers here in the bulk because of an effect called Auger recombination, which we'll get to a few lectures. So this front surface region here oftentimes is considered a dead layer to a solar cell. Its electrical properties are very, very poor. So if I make this layer too thick, sure I get rid of my emitter sheet resistance term. But at the same time I'm killing my quantum efficiency in the short wavelengths. Because I'm not able to absorb those short wavelength photons and convert them efficiently into electron hole pairs. So this is a story about co-optimization of different device parameters. So let's keep this thought for a minute. And let's now try to calculate the total power loss due to the emitter sheet resistance, because now we have this weird unit of ohms per square. In reality, it's ohms. But when we think about emitter sheet resistance, we're thinking about a square of the material. We have units of ohms per square, square being a unit-less parameter. So it has, let's see, resistivity had units of ohm centimeter, we divided our resistivity by our thickness of that emitter region, so our units became ohms. And ohms per square here is the units typically given for emitter sheet resistance. And now we have to convert this into some meaningful parameter for actual solar cell device. So if we want the calculate the fraction of power lost due to this emitter sheet resistance, it's fairly straightforward. We would have to think about the current. Since power is I squared R, we have to think about the current as a function of the resistance, right? As a function of the position here, we're integrating over y, y being the lateral dimension. And we're integrating-- imagine the separation between contacts is S. So here's one contact metal finger. Here's the next contact metal finger. It's extracting charge from the material. And we have a certain distance, S, right here between them. So the maximum distance that the current would travel in principle would be S divided by 2, since any current that reaches the emitter over here will be collected by this contact grid, and any electrons reaching the emitter over here will collected by that contact grid finger. OK, so now we have our current. We can translate that into here where we have our b, b being the vertical dimension, y being the horizontal dimension here. And we would get this equation coming out right in the other side, which is a function of the separation of our contact metallization fingers to the cubed, to the third power. So the power loss coming from the solar cell device, as we separate our contact metallization by small amount, this power loss is going to go up very quickly. That's a really deep point to recognize here because what this means, in effect, is that we can lose a lot of power very quickly in our solar cell device if we place our contact fingers too far apart, due to this emitter sheet resistance effect, due to the row sub s, which is our emitter sheet resistance right here. OK, so then if we divide the power loss by the maximum power point, we essentially get a normalized power loss. And so if we want to, say, limit this ratio to a certain fraction, say 4% of total power is lost due to emitter sheet resistance, now we can define the geometric parameter, the distance between the contact metal that we need to do an actual solar cell. And if you run through the calculations with a typical set of parameters for a silicon-based solar cell, you'll find out that your metallization fingers should be separated on the order of 4 millimeters. And if you remember the real solar cell devices that you saw embedded in that module, the separation of the fingers is just about 4 millimeters. So this is where that finger separation comes from. In one case, you'd want to have your fingers spaced really, really close together to minimize the emitter sheet resistance. But on the other hand, if you place your fingers too close together, your shading the front surface of your solar cell with metal and light's not going to get in to generate electron hole pairs. So that's the optimization function that's run right there. Wow, so far I'm counting three or four of these different parameters tugging in different directions, right? So you can begin to sense how constrained the solar cell device is in terms of optimization of all these parameters, and how easy it is to make a very low efficiency device as a result, how difficult it is to make a high efficiency device. So this is interesting. So we have now the spacing, the appropriate spacing between fingers. It's about 4 millimeters. Which means if we're going to be making a solar cell device, say a thin film solar cell device on a sheet of glass with some transparent conducting materials [INAUDIBLE], and then we have our source of device, and we have their front surface here, and we have, let's say, something else on top, we have to limit how wide we make that device. Otherwise, we could be limited by our sheet resistance in, say, the zinc oxide layer of our device. And this is interesting now because previously for the bulk resistance-- let's think back over to our bulk resistance. We wanted our area to be large because we wanted to minimize the bulk resistance. Now we want the area to be small so that we minimize our emitter sheet resistance. So you can see how the current traveling in two orthogonal directions inside of our device is really causing us problems because we're having to optimize the same parameter in different directions for each. It's tricky. And I'm providing you the tools here to be able to sit down and calculate out what is the optimal thickness, what is the optimal thickness of your emitter, of your base, and what is the optimal lateral dimensions of your solar cell device to simultaneously minimize your bulk resistance and your emitter resistance instead of a device. And you can do it all now with equations that I just gave you. So it's pretty cool. At least to first order. AUDIENCE: How did you define B, again, in that? PROFESSOR: Yeah, absolutely. So B was a vertical dimension. B is just given as the distance here. Yeah. AUDIENCE: And that's going across like if you look down on your model? PROFESSOR:Yeah, you're looking down on your solar cell. So the pn-junction would be planar to this view right here. AUDIENCE: [INAUDIBLE]? PROFESSOR: Yep. It's a bird's eye view on the solar cell. Any other questions? OK, so contact resistance. This is the next thing that can kill a solar cell device. Contact resistance-- we want to minimize the contact resistance, but the first step to minimizing is to measure, to know what your contact resistance really is. And one simple way to measure contact resistance is to take a cross section of your solar cell device. We have deposited, say, four metal fingers on it, and they're appropriately spaced by, I don't know, how many millimeters or so. So you have a certain contact resistance right here between the metal and your semiconductor material that's denoted as R sub C. And then there's a certain emitter resistance that we just talked about, R sub em right here. And if you were to measure the current passing through here to here, or another way to say it is, the resistance for current to pass from this point to that point, you might get, let's say, this data point right here, a total resistance value of some value, and a distance between the two probes of, say, 1 times d sub f we're d sub f is the unit distance between two adjacent contact metallization fingers. Now if we measure the resistance between this point and that point, we might wind up with that data point right here. And if we measure the resistance between this point and that point-- we already did that-- to this point and that point, we might wind up with that data point right here, and so forth. And what's happening? Well, as a current goes into the device, it passes once through the contact resistance, a certain n times R sub em, n being the number of metallization, number pops, the number of times we travel distance df. And then it goes out the other side, again traveling through the contact resistance. So the slope of this line here should be equal to the emitter resistance. And the offset, the intercept at x equals 0 here, the intercept should be 2 times the contact resistance. Because in principle, that would be the equivalent of current going in and going back out through this contact resistance right here, through the same one, not traveling at all through the emitter. Obviously since that's impossible to measure, we have to determine that through extrapolation back to the y-axis. So this so-called TLM method, the transmission line method, is a method that is used to determine the contact resistance of a solar cell device. You could either perform it linearly in this fashion right here, or using a circular, essentially an array of concentric circles. And the latter method is typically preferred, especially in thin film devices. So this is how you measure the contact resistance. And then there are line losses. So what can-- let me back up one step-- what can effect the contact resistance? Typically, what effects contact resistance is the atomic interface between your metal and your semiconductor, whether it's an organic or an inorganic semiconductor. Bless you. So the interface here is what determines the contact resistance, typically. And of course, the dopant density within your semiconductor. So components of the series resistance in the line losses. So line losses are essentially the losses in the contact metallization lines out of plane. What can cause a resistance in the contact metallization line? Well, intuitively we know that, OK if the line is too small, if the cross section that line is too tiny, then there's going to be a high resistance to current travel along that line. And indeed that's the case. Whoa, we have the same exact equation as we had for our base resistance. Now it's applied to the line, because in principle it's the same thing. We have current traveling through in one direction along a constrained cross sectional area, a certain distance, a certain length. So we have now a resistivity given by the metal obviously, because now we're talking about the metal itself, not the semiconductor anymore. Currents traveling along the contact metal so that the row becomes the resistivity of the metal, the A becomes a cross sectional area of the contact finger, and the l is the length of the conductive path. In other words, how long does the current have to travel along that contact metal before it reaches a highway, a bus bar, one those really thick strips of metal on our social device, in which case it's home free. So the R, the row rather, the metal resistivity, resistivities of several metals and non-metal materials are given here. That's typically why you see silver contact metallization used for solar cells. It's because it has a very low resistivity, which means that the resistance line losses will be small. Now, what is the problem with silver? Does anybody know what percentage of the world's silver right now manufactured each year is currently being used for contact metallization on solar cells? Take a number. Guess. AUDIENCE: 30. PROFESSOR: A little lower, but in that order of magnitude. So he said 30. Somewhere upwards of 10%, right, would be your guess? So around 10%. And of course PV is growing at a cumulative annual growth rate in the several 10s of percents per year. So silver utilization is expected to continue to increase. So either we mine more silver, or we venture to another contact metal material. And that's why you see folks looking into copper. Nickel, which is similar in electronic structure. It's one element to the left of copper on the periodic table, and another metal alternatives. Yes? AUDIENCE: Yeah, where does nickel fall on the resistivity chart? PROFESSOR: I have to give you the exact number, but if I were to venture a guess, what you're noticing over here are several of the metals that fall in the noble metal category. These are far to the right on the 3D series. So if you picture the periodic table again, far in the 3D series you have that d-shell orbital completely filled. And those outermost electrons are then screened by all the other electrons in the system and very loosely bound. And so you can remove the electrons fairly easily. It's one of the reasons why these elements in that class have a higher conductivity, a lower resistivity. Yeah, question? AUDIENCE: Copper doesn't seem that much worse that silver. [INAUDIBLE]. PROFESSOR: So, yes, absolutely. Copper doesn't seem so much worse than silver. So the contact metallization pastes to date have mostly been using silver. Copper can oxidize fairly easily. So that's one downside of copper. Another problem with copper is that it's a very fast diffuser. It's very tiny. And so the elastic strain energy turn as it tries to move through a lattice is very small. And as a result, it goes deep. So when you fire your contact metallization when you create the contact between your metal and your semiconductor underneath, you run the risk of having atoms diffuse into your device and hurting the efficiency, hurting the ability of the solar cell to collect carriers. We'll are more about that on Tuesday. AUDIENCE: So it becomes an impurity. PROFESSOR: Yeah, exactly. it becomes an impurity. In this case, unintentional. So just remember where the contact is right here in the emitter. OK, if it diffuses into the emitter, it's OK. The electrons at this point are the majority carriers. But if the metal diffuses all the way into the bulk, the electrons inside of the bulk in here are minority carriers. And they can be impacted by the presence of that copper there. All right, so I wanted to get through at least one more point to really set us up well on Tuesday. We want to be able to calculate the Fermi energy of a solar cell as a function of dopan concentration, illumination condition and temperature. So, so far we've really talked about the Fermi energy, the chemical potential inside of a solar cell in hand wavy terms. We've talked about general trends, about how as we add more electrons to our system, say by doping, we shift the Fermi energy up because the average ensemble energy of the electrons is increasing. But now we're going to be calculating it and determining how to calculate it. So the question is calculating Fermi energy. So let's introduce a new concept here. So far we've talked about and energy band diagram, and drawn our valence band, and drawn our conduction band, and of course the band gap in between them. We've drawn the valence band and conduction bands as it they were just continuum density of states, as if they were just these blocks where electrons could pile in. In reality we have what's called a density of states in the valence band and a density of states in the conduction band. If you want to think about electrons as cars, the density of states could be considered the number of parking spaces there are per floor in a parking garage. As you increase the energy in the case of a parking garage, you increase your potential energy as you move your car higher and higher. In here, you would be increasing the energy of the electrons as you go higher and higher. The increase in the energy of holes as they go lower and lower. So we have a certain density of states that is allowed inside of our semiconductor. And there's a very small density of states right at the band edge. And that density of states typically increases as you go deeper and deeper into the bands. So now at absolute zero, and the person who probably knows more about absolute zero than anybody else here in this room is Kristy Simmons in the back, who for her Ph.D. Would work on very low temperature experiments down into the, what, 10s of millikelvin range. 10s of millikelvin range-- it's very, very cold. And then if you cool things down a lot, all the electrons that were excited will drop down, say, into the valence band right here. And what this is represented is as a filled valence band right there. And as you begin heating your system up, some of the electrons will have the thermal energy to be excited across the band gap. So thermally excited electrons here represented in your conduction band. [WHISPERING] So we have a certain number of thermally excited electrons here up into the conduction band. Oh, and these carriers we'll call intrinsic carriers because they're not added by dopans. We're not adding anything to the semiconductor to create dopans to generate new carriers. There are intrinsic carries because they came from the valence band, went up to the conduction band. So if we plot intrinsic carrier concentration as a function of temperature, we see a curve that looks like this. And it follows this expression right here, which is called in an Arrhenius equation in a generic form. In this case it would be intrinsic carrier concentration as a function of temperature. We'd have to substitute this big N here for intrinsic carrier concentration, some exponential prefactor, and then an activation energy, the activation energy being the energy needed to create that free carrier. The band gap energy in this case. The activation energy divided by KbT, the KbT being Boltzmann's constant times temperature in Kelvin. Now, if we plot the same functions that we just saw in the previous slide a little differently, we plot 1 over T-- see 1 over T right -- and there-- and we plot the log of this value. So again, log of this value, we can read the activation energy off of the slope of this graph. So that's why you typically see Arrhenius plots plotted in this way-- 1 over T verses log of parameter-- because you can read the activation energy straight off of the graph. It becomes very easy to see the carrier concentration. So again, at absolute zero we have, here the valence band completely filled, the conduction band completely empty in our semiconductor. And the probability of occupancy of a state inside of our semiconductor is given by this box function right here. Now let's look at what happens when we heat this up. When we heat the sample up, some of the electrons that were in the higher energy states are going to be excited. So they used to be here in lower energy states. They are going to be excited to higher energy states. In the area of this little shape right here is equal to the area of that little shape right here because of conservation of number of electrons in our system. If we multiply this by the density of states, this is the density of states in the valence band minimized here at the band edges, and increasing as we go deeper into the bands for the valence band and the conduction band. So we take our probability distribution function, multiply it by the density of states, and we get the occupied density of states in our semiconductor. So now we see that there's some number of missing electrons that are now in the conduction band. They used to be in the valence band at very low temperature. But now as we increase the temperature, they've had the thermal energy to excite across given this probability distribution function. And these carriers are now free to conduct charge inside of our system. So what do we do with this framework, with this understanding? Well, we can look at this function here and ask ourselves, what is the name of that function. It's called the Fermi-Dirac distribution, and it's given by this equation right here. We'll notice in the Fermi Dirac distribution, we'll notice that this equation approximates to an exponential equation as we move further and further away from the Fermi energy, the Fermi energy being defined as this energy right here. If we move further way down here, or further way up here, this starts looking an awful lot like an exponential. Meaning this right here and this right here start looking like exponential functions. OK, so let's do a quick little Gedanken experiment is a quick little thought experiment, about what happens when we heat up a semiconductor. And how would this impact, how these free carriers right here impact the noise, say, in a camera, in a CCD-based camera. A charge couple display camera, which functions not so different from a solar cell device, except that it's biased a little differently than an actual solar cell device. But it could be like a pn-junction, p-i-n, to be more precise. So we have our little camera right here, and the camera's typically placed inside of a liquid nitrogen dewar if we want very high performance. These are not your typical handheld cameras that you would use for shots in the street. These are cameras that we use in the laboratory for detecting very faint signals inside of our semiconductor systems. And we typically cool our cameras down. Why? Because we want to minimize the density of these intrinsic carriers. Because any photon that comes into our system, we want to be able to count it and to detect it. And not have to detect a very small signal on top of a large background noise of thermally excited carriers. In a solar cell device, the intrinsic carrier concentration is typically much, much less than the dopant concentration. So the intrinsic carrier concentration matters less for our semiconductors than it does for a photo detector like this. And that's why, thankfully, we don't have to cool our solar cells with liquid nitrogen. So in terms of have intrinsic carries, let's dwell there just a little bit longer. Transistors are made of what semiconductor material to have less electronic noise to experience? OK, so what semiconductor material, germanium or silicon, would experience less electronic noise at room temperature? And let me give you a hint. Silicon has a band gap of 1.1 eV. Germanium has a band gap of 0.67 eV. Which of the two is going to have greater electronic noise at room temperature? AUDIENCE: Germanium. PROFESSOR: Germanium because it has smaller band gap, it's going to be easier for the carriers-- we go all the way back here-- it's going to be easier for carriers to hop across because the band gap is smaller. Another way to look at it is on the Arrhenius plots, you'll have a higher carrier concentration because your slope is smaller, your activation energy is smaller. And as a result, you'll have a higher-- if the band gap shrinks-- you'll have a higher carrier density here in the conduction band. Let me show you that in pictorial form. So this is the silicon band gap. This is the density of thermally excited carriers for silicon. Now we shrink the band gap. Boom, we shrunk it. Same probability distribution function because we're at the same temperature. But now we have a smaller band gap so we have a higher density of free carriers. And true story, germanium was one of the first crystals to be purified during the early days of semiconductors, and silicon was following germanium. And silicon was ultimately chosen as the material of choice for transistors because the intrinsic carrier concentration of germanium was just too high for most applications. And so you can see the intrinsic carrier concentration at room temperature of silicon is around 10 to the 10 whereas for germanium is around 10 to the 13, three orders of magnitude higher. So, yes, question? AUDIENCE: [INAUDIBLE]? PROFESSOR: Yeah, it would be most costly. Absolutely. And so you do see solar cells like Concentrics is one company that's commercializing gallium arsenide-based solar cells. But they have to concentrate the sunlight into very small area devices because the devices cost so much to make per unit area. So they take cheap plastic and concentrate the sunlight down. Yeah, and here's a gallium arsenide. The intrinsic carrier concentration is even less than that of silicon because the band gap is 1.4 eV instead of 1.1. OK, so I think this is a good place to stop. I'm still not through all the material. Oh, we wanted to do one demo. JOE: Yeah. PROFESSOR: Do you want to do it first thing next lecture? JOE: That's fine. We've got five more minutes. Ah, we do have five more minutes before 10 after. OK, why don't we do a quick demo then. So what we're going to do as Joe sets this up, is I'll verbally describe what's going to happen here. We are going to take this-- actually let me go back to be Fermi-Dirac distribution function. This function right here, as we increase temperature, what happens to this? This will look more and more, shall we say, flat. It won't actually go flat, but it'll become more and more diagonal opposed to box-like in function. What that means is the density of carriers, the density of free carriers will increase with temperature. And because of the Arrhenius dependence here, we have an exponential increase of the number of free carriers with temperature. Because the function right around this region right here, which impacts the electron concentration, the conduction band, this small portion of the Fermi-Dirac equation can be approximated by an exponential. If we have an increasing temperature, we should have a drastically increasing carrier density. And let's see if that's indeed the case. What we have is a piece of intrinsic silicon just with a battery pack attached so we have a field across that piece of intrinsic silicon. Whoops. Here we go. And we are going to apply, again, that field across the intrinsic piece of silicon. A certain current will flow through it. And then Joe has a heat gun, so he's going to heat up the silicon and see what happens as the Fermi-Dirac equation here is effected by temperature, we should have an increasing density of carriers in the conduction band. So let's see if that's indeed the case. Can somebody read off what the current is currently measuring there on our current meter? AUDIENCE: It's about 9.8 microamps right now. PROFESSOR: 9.8 microamps. So this battery pack of 3 volts right here is it is forcing a current of 9.8 microamps across that little slab of silicon that is sitted right here. And so Joe is going to apply the heat gun, and if none of the contacts pop off, let's see what will happen. So we expect the resistivity to go what? AUDIENCE: [INAUDIBLE]. PROFESSOR: Down. And the conductivity to go up. All right, so let's see here. So what's happening there folks? It's going up? So that's the -- JOE: So it went up to around 50 microamps, and can see it's dropping rather dramatically now [INAUDIBLE]. PROFESSOR: So the current that the material's able to pass through is increasing as a result of the increasing temperature, because you're exciting more carriers into the conduction band, leaving more holes in the valence band, and these are free charge carriers are able to conduct charge through the material. Cool, OK. So with that, I will leave you, and I will return Tuesday. And Joe give a fantastic lecture on device physics as well since I will be at a kickoff meeting for a big NSF-related center in Phoenix, Arizona. But it will be great lecture. You won't want to miss it, and look forward to seeing you on Tuesday. |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | 9_Charge_Extraction.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. TONIO BUONASSISI: Why don't we go ahead and get started, folks. Just some small talk to get us started. I promise to tell you the stories about contamination and unintentional contamination. Before we dive in, what a life, right? Steve Jobs. That was really something. Moments like that, I think the best you can possibly do is to celebrate the person. And well, in honor of his inventiveness and the way he really turned Apple around, I just wanted to focus a minute on that. Speaking of contamination and unintentional contamination, the effect on processing, the history books are full of this folklore, if you start talking to people who grow crystals and who manufacture solar cells. The growth of the crystals that are used to actually make the wafers that ultimately wind up being solar cells is a little bit of a-- how would you say-- a little bit of an art. It is being codified rather well now. And there's some strong science behind it. But by and large, up to very recently, it was more of an art than a science. The best crystal growers are people who are very observant it and who are able to look out and see correlations where they weren't immediately visible to others. So in one particular factory, they started noticing that they had yield losses during the winter time. And in particular they had extreme yield losses whenever there was a big snowstorm. And it was somebody, one of the shift leads, I believe, who traced it back to the salt that was marching in on people's boots as they were coming into the factory. And that sodium was contaminating the silicon melt and resulting in the poor quality of the crystals that were being grown. Flash forward several years at Fraunhofer Institute for Solar Energy Systems in Freiburg, Germany where the observant cell manufacture started noticing, well, gee, our efficiencies are always lower on Fridays. Why is that? And eventually, they traced the problem down to the fact that, on Fridays, the technicians would go out to lunch at a Chinese restaurant. And they would come back, and their breath and their hands would be MSG. And that was, again, the sodium was downing performance of the devices. As so it's really interesting. This kind of sounds like something you might see in a TV show where that scientist in the lab coat waltzes into a room and says, it must be the sodium. Right? And then they test it. And oh, my gosh, it's the sodium. Now, problem solving the real world doesn't quite happen that way. It involves a very methodical approach to testing a variety of hypotheses, first brainstorming in a structured manner, identifying the most likely candidates, and going about solving the problem, performing a series of design experiments to really get to the root cause. So it's an interesting story. It's an interesting aside. Solving contamination problems, though, is very difficult. And the best thing you can possibly do if you're growing crystals is to keep your system more clean than you think you need it to be. That's the best advice I can give. I was just talking with some folks in Caltech the other day. They were running into contamination issues. And that advice goes a long way. OK. Well, let's go ahead and get started in the topic today. Back from Phoenix and eyes are red from irritation of having been awake for too many hours. So that could only mean one thing. Contacts, right? So we're going to be talking about contacts today, charge extraction, as well. And to situate us on the roadmap, we have our fundamentals right here. We're about to jump forward into the technology, which is really exciting. We're about out of the woods here, folks. And finally, into the cross-cutting themes. And so, again, every photovoltaic device must obey this general equation right here where the output energy over the input energy equals a conversion efficiency. And for most solar cells, this is represented by this right here. And we have now tackled every single one of those, at least in a very fundamental way. And now, finally, we are focused on the output, the charge collection, in other words, the contacting of the device. Again, the total cell efficiency is going to be the product of each of these individual cell efficiencies here. And contacts are a very, very easy way to kill your device in a variety of ways. And we'll get to some of these points right here. As a matter of fact, this s back, this surface recombination velocity on the backside of the device, is a contact-related phenomenon. And that's exactly where the water is spilling out of this bucket right here. Learning objectives. The idea is to start out with describing the purpose of contacts, and their most common types, then to describe the impact of good and poor contacts on I-V characteristics, in other words, to describe the device impact of contacts. So we're convinced that this is really something we should be spending a lot of time thinking about. Then we're going to sketch the I-V characteristics of Schottky and Ohmic contacts. We'll learn what those mean. I hope you know what ohmic means. Describe what fundamental material parameters determine I-V characteristics of a contact/semiconductor junction, sketch common band alignments, and sketch common solar cell device architectures, once we're done. And then we can look back and gaze over the fundamentals and say, Ha, ha, how far we've come. And now that you look at a solar cell device, hopefully, you'll look at it in a much more profound way than you did in the first day of class. The more you know, the more you see. So contacts, why do we need them? We need to extract the carriers from the device. We need to prevent the back-diffusion of carriers into the device. These contacts, in general, have been studied very extensively in the semiconductor industry. Why is that? Why do we need contacts in the semiconductor industry to make integrated circuits? Obvious answer. We need wires. We need to extract the charges from our transistors, right? So we're injecting charge, pulling out charge. So contacts have been pretty extensively studied in the semiconductor industry. And we're going to leverage that a lot. We're not going to reinvent the wheel where we don't need to. Contacts are semiconductor-specific. Fundamentals apply broadly, but the specifics pertain to-- or the devil's in the details, in other words. So there are very specific effects that occur, depending on the precise semiconductor metal combination. And lastly, the contacts are heavily influenced by the interface between the semiconductor and the metal. We're going to be talking about that as well. So typical materials used for contacts include metals, transparent conducting oxides. And also, we can find heavily doped organic materials as well. Metals, we understand. OK, they're optically opaque and electrically conductive. That means they conduct electricity very nicely. And so our series resistance along the metal wire should be low. But they're optically opaque, meaning they're resulting in a shading loss. So if we cover entire front side with metal, we're not going to have a very efficient solar cell, because our absorption of light is going to be very poor. On the other hand, transparent conducting oxides are optically transparent and electrically conductive, not quite as conductive as metals, but pretty close. So let me think about that for a minute. A material that is optically transparent-- that means it must have a very large band gap to let the light through, does interact with it, but is electrically conductive. In other words, it has a high concentration of free carriers that can move around the material and conduct charge. How is that possible? I hope you're asking that question, because this perplexed me for a long time as well. And we'll answer that in a couple of slides. First, the properties of TCOs. What makes a transparent conducting oxide a transparent conducting oxide? I figure I'll spend a couple slides on this, since metals are pretty self-evident. But TCOs, this might be the first time you're encountering them. They're present in a variety of devices. If you have an iPhone, or if you've ever seen certain types of military aircraft, TCOs are involved. So the material is very transparent. You can see the transparency over quite a broad wavelength range here, cutting off only in the several eV range here. Transparency begins to drop, because you're able to excite carries across the band gap. So it's a very large band gap material, and yet, the conductivity is very good. So here, we can see that the conductivity of ITO, that's indium tin oxide, that's a particular type of transparent conducting oxide, is almost as good as silver. Not quite, but it's approaching it. And so again, the conductivity, which is 1 over the resistivity is related to the carrier concentration and the mobility, as we've seen before. And therefore, there must be a lot of free carriers. And they must be fairly mobile, to move around the material. So how does that work? Band gap greater than 3.1 eV, transmittance very high, as a result, but still a large number of carriers. Well, if we consider our band gap right here-- and we'll consider floating Fermi energy right now, we'll explain why that's pegged there in a minute-- we assume that the band gap is very large, so that light can go through, and it's optically transparent. But now, by doping the material with a specific type of dopant that forms mid-gap states, we're going to create a new energy level here that is partially filled. We can then excite, with relatively little energy, carriers into the excited states within that orange band, and they can transport charge throughout the material. On the other hand, if we come in with a high energy photon or a moderate energy photo, let's say, in the visible, it doesn't have quite enough energy to excite from that orange band into the conduction band, nor from the valence band into the orange band, the intermediate band there. And that's the general principle of how transparent conducting oxides function. You can have indium doped tin oxide, for instance, ITO, as a classic example. And this is the basic premise. Any questions so far, because this is kind of important, before we launch into-- yeah. AUDIENCE: So how big are those, that E1, E2? TONIO BUONASSISI: Yeah. So this E3 right here is around 0.4 eV, for a typical TCO. And E1 and E2, the sum of both of them could be on the order of three to four eV. Maybe higher. Well actually, no. Sorry. Just one of these transitions right here could be on the order of three eV. So the entire band gap could be quite large. If you think of metal oxides-- most metal oxides have very large band gaps. And most TCOs fall into that category. They could be tin oxide or zinc oxide, but heavily doped with an element, like indium, or fluorine, or aluminum, to provide that intermediate band. AUDIENCE: What are the doping concentrations required [INAUDIBLE]? TONIO BUONASSISI: Yes. So this is a deep dive, and I'm happy to entertain that question. For those who it kind of goes over your head, don't worry about it. We'll return, so plant a flag post in your mind where we're at. Remember the hydrogenic donor model for a dopant atom. You introduce it into the lattice. And that loosely bound electron is there, kind of like a hydrogenic donor. It has a certain radius. And if you begin doping at a very high concentration, those electrons will begin interfering with each other. They'll begin interacting. And so instead of forming one isolated defect level, they'll start splitting and forming a band. And the density at which you have to dope depends on that donor radius. So you're typically talking about concentrations in the 10 to the 20 per cubic centimeter, or one atomic percent or higher. So that's the dopant density that's required to force this transition. It's pretty cool. I mean, there are people who do their PhDs on transparent conducting oxides. Maybe some of you are. You could probably come up and give a lecture about it. So it's a fascinating subject. And because ITO, or indium tin oxide, is just so amazing in terms of its conductivity-- it's almost up there buttressing again silver-- it's very difficult to replace with something else. Zinc oxide and its variants get pretty close, like aluminum dope zinc oxide, sometimes called AZO in the PV community. They get pretty close, but they're not quite as good in terms of optical transparency and conductivity as ITO. Now ITO is a problem. It contains indium. And the world supply of indium is limited. It's not the most abundant element on the Earth's crust. If you look at a periodic table, it's pretty far down there. And as you know, the elemental abundance, it has a relative decay from-- a power law decay-- from light elements to heavy elements with some fluctuations in between. But in general, the star dust that we have here is less abundant in the heavier elements. And indium is one of those elements. And so people are searching for alternatives to ITO. And it's an active area of research right now. OK. So for now, let's go back to the flag post that we planted just a few minutes ago. For everyone who didn't quite follow the detailed explanation, the important thing to keep in mind is that TCOs are conductive and transparent. And they are so, because of the unique band structure. So we can create contacts to semiconductors and extract charge using either a metal or a TCO. And right now, we're going to dive into the impact of good and bad contacts on device characteristics. So during our last-- I believe it was two lectures ago, we talked about the equivalent circuit diagram of a solar cell, this being the simplest case, now corrected with our minus 1 term right here. And we can see our I-V curve in linear, linear scale and log, linear scale there, log current linear voltage. And because this is an exponential, you would expect a straight line, and you do see that indeed. Now the I-V curve, in the upper part, you can really see has a very nice fill factor as a very sharp kink. Pops straight up. And so the fill factor of that I-V curve, when illuminated, will be very high. And hence, the solar cell efficiency will be high. Not so when you introduce a series resistance. So when you have that series resistance component, now, at higher bias voltages, you begin to be series resistance limited, which causes the fill factor to decrease. And you can see that by going back and forth between these two slides. High fill factor, no series resistance. Lower fill factor, higher series resistance. Especially, if you focus right here at this point right there, you can really see the decrease of fill factor as a result of adding a series resistance component. So contacts, when improperly performed, can add a series resistance component, a rather significant one, and drop the performance of your device. Likewise, shunt resistance. We saw a simple effect right here of shunt resistance. If the shunt resistance goes down further and the saturation current goes up even higher, it begins to affect the fill factor as well. And so shunting can also be impacted by contacts. Particularly when you fire your contacts, in other words when you heat your sample up, or you heat your stack up to create good contact between the metal and the semiconductor underneath, or the TCO and the semiconductor underneath, a couple of things can happen. If you don't get the chemistry just right at the interface-- maybe didn't clean your sample properly, maybe you didn't heat it up high enough, so that the atoms really started to interdiffuse and interact-- you can under-fire your contact. And that results in poor contact with your semiconductor and large series resistance. So large series resistance will, essentially, impede charge flow and result in a fill factor loss. On the other hand, if you over-fire your contact-- let's say you heat it up, but fire it a little too long-- then the contact material can drive too far into your semiconductor, and you wind up shunting your device. What does a shunt mean? A shunt mean that the metal goes straight through the PN junction and contacts the base material on the other side. So remember, the PN junction region of the solar cell is only about a micron away from the surface. It's really close. So if you have a fast diffusing metal species and you heat it up too much, you get the metal going straight through, in this case, the n+ layer into the P, you'll shunt your device. So this is the fine balance that you have to walk when you're putting contact metallization on the device. You can't over-fire it, because we wind up with shunts. You can't under-fire it either, because you might wind up with a very large contact resistance. So it's a very tricky thing to get right. We just finished installing a contact metallization printer in our lab. And Joe, how many human hours total have been spent trying to optimize the firing process? JOE: Probably about 30 or 40 now? TONIO BUONASSISI: 30 or 40 human hours? So it's still a work in progress, but it takes a while to really nail that. Yes, Ashley? ASHLEY: So would you want your TCO to melt at a lower temperature that your semiconductor? Is that another consideration? TONIO BUONASSISI: So let's back up a step. For the metal case, it's pretty straightforward. This is a homogeneous element. Let's call it a unary material, meaning one element comprising that metal, typically. And it's very simple for us to see, OK, if we keep it up and it reacts with our material, it drives in. For a TCO, it's a little less straightforward, because you typically have a multinary compound, meaning several elements comprising your TCO. The formation of this intermediate band right here that forms, really, the conductivity of the TCO, is predicated upon the principle that you have these dopant atoms homogeneously spaced throughout your material. If you heat it up too high and they begin to cluster, in other words, precipitate out of phase, you can wind up destroying your TCO, or increasing the resistivity, maybe even decreasing the optical transparency. And so it really is material-specific when we start talking about TCOs. I'd rather not generalize about them. About metals is more straightforward. ASHLEY: OK. I guess then, for metals, you would have to choose a metal that melts before your semiconductor melt-- or [INAUDIBLE]. TONIO BUONASSISI: Typically, you wouldn't have your mental melting, per se. What you're looking for is a chemical reaction here at this interface. You're looking for the metal, most often, to form a binary compound between your metal and the semiconducting material underneath. Let's say you have nickel and silicon. You'd be looking for it to form a nickel silicide at that interface. ASHLEY: OK. TONIO BUONASSISI: And that's where you go to your phase diagrams and figure out what temperature those, say, intermetallics should form. And then you're targeting that particular temperature during your ramp. ASHLEY: OK. TONIO BUONASSISI: And of course, it's not only phase diagrams which are under equilibrium conditions, kinetics are involved as well. So it becomes a rather tricky process. Yeah, question in the back. AUDIENCE: This picture is drawn where it says doping is concentrated underneath [INAUDIBLE]. TONIO BUONASSISI: Yep. AUDIENCE: How do we manage to concentrate the dopant in just that certain section [INAUDIBLE]? TONIO BUONASSISI: Very good question. So we'll explain toward the end of lecture why dopant is concentrated underneath there. For those who are a little bit more advanced, it relates to a tunneling junction effect or a field emission effect. But we'll explain what that is for everyone else. The way you typically get an enhanced dopant concentration right underneath the metal, first, let's appreciate the length scales involved here. The width of that contact metal is probably on the order of somewhere between, let's say, 80 and 120 microns, maybe a little more. And so, at those length scales, we don't need photo lithography to really nail a particular location. We're not talking about tens of nanometers. It's really something more macroscopic. So we can do it in a few ways. On the high end, we could use an ion implantation tool to pattern our material with a mask. On the low end, we might diffuse in the emitter very deeply, to create a highly doped region everywhere, and then create a mask using not exactly a photo lithography process. It could be a much simpler contacting process, and then etch away some of the emitter in other regions to make it more lightly doped or, even in silicon's case, create a poor silicon layer and etch that off, to distinguish between the heavily doped and the lightly doped regions. There are a few different ways to skin that cat. And that's one of the beauties of solar cell processing is that it really pertains to the specific material system involved. But the length scales are such that you're really not typically limited by photolithography. You can probably get around that and use other techniques for masking it off. Mm-hm. Great. I see folks are awake today, despite the P-set due. Sketch the I-V characteristics of Schottky and Ohmic contacts. OK, so what we're going to do here is talk about how Schottky and Ohmic contacts come into being. And first off, define them. So let's define them. An Ohmic contact is one in which I sweep my voltage, and I obtain a linear current response. So this is pretty straightforward. It's Ohm's law, follows Ohm's law, so V equals IR. This is a linear relationship then between voltage and current, the slope of which is dictated by the resistance. A Schottky contact, on the other hand, follows this beautiful exponential curve that we've come to know and love from our ideal diode equation. As a matter of fact, it follows that same expression rather nicely. There's an exponential relation between voltage and current. So whenever we have an exponential relation between voltage and current, now that we know how a PN junciton works, and we recall that it's really that diffusion current that's driving the forward current right here under forward bias conditions. We've learned that, when the barrier's too large, the electrons just can't make it over. But as a barrier begins dropping, an exponentially increasing number of electrons can jump over that barrier and move from the region of high concentration to the region of low concentration, or from the n-type material to the p-type material. So when we see an exponential current voltage response, we should immediately think about some barrier involved in our system. And as we begin biasing this device-- in this case, a semiconductor metal junction-- as we begin biasing the semiconductor versus the metal, we begin seeing an exponential current response. We should imagine that there must be some barrier in between the metal and the semiconductor. And we'll explain how that barrier comes into being. So again, just to recap, making sure we hit all the points. Ohmic, a linear I-V curve. And Ohmic contacts are typically used when charge separation is not the goal for your metallization. Let's say you've already achieved your charge separation through the PN junction, and now, you just want to contact your semiconductor to extract the charge, you'd use an Ohmic contact. But you would use a Schottky contact when you want to enforce some charge separation. Typically, not quite as good as, say, a PN junction for separating charge, not quite as high voltage. But it's a useful tool, for example, in research. So we're going to describe what fundamental material parameters determine the I-V characteristics of both the Ohmic and Schottky junctions, more generally, a contact semiconductor junction. And this dives into Schottky band theory. This is a very idealized version of the real world. It's an idealized version, because it gets us started and points us in the right direction. Then we'll add layers of complexity, until we build up to really understand what's going on inside of these devices. So let's start here. This is a very nice, elegant view of contact theory. We have our semiconductor over here. Let's walk through this. So this is our energy band diagram. We have E on the vertical axis that represents the energy of the electron, versus X, some real space parameter. We have our semiconductor material here. We have our valence band, our conduction band, and our Fermi energy. This looks to be an n-type semiconducting material. And we have our vacuum level. The vacuum level is essentially the energy at which we remove an electron from the semiconductor. If the semiconductor were placed, let's say, in a vacuum, and we were to excite an electron out of the material through some, let's say, a photoemission effect, we would have to overcome the energy between the Fermi energy and the vacuum level. This xi right here is essentially the electron affinity of the semiconductor. That's the delta in energy, or q times xi is the delta in energy between the vacuum level and the conduction band. So we have all of the variables mapped out for the semiconductor. Most of them should be familiar for us already. We've added the vacuum level for completeness. The metal, we also have a Fermi energy, we also a chemical potential of the metal. And we have a work function of the metal, right here. So this work function is defined as the energy necessary to remove an electron from the metal. Say again, by if you shined a light on the material, it has a high enough energy, you can begin exciting electrons off of the metal with a photoemission effect. And so we have our vacuum levels in this particular diagram lined up. And our chemical potentials are a little bit different between the metal and the semiconductor. So what do you think will happen when we put the two together? If we put the two together, the chemical potential throughout the entire system has to be the same, right? Because now, they're in good contact with one another. But what about that vacuum level? What's going to happen? Will there be discontinuity, first of all, in the vacuum level? Can there be a discontinuity in the vacuum level? If I move one little delta x from the semiconductor into the metal, do I expect there to be a discontinuity in the vacuum level? No, probably not. So there has to be some smooth change in the vacuum energy, relative to the Fermi energy. And if the vacuum level changes, that means that the conduction band and valence band energies are also going to change, relative to the Fermi energy, because the valence band and conduction band energies track with the vacuum level-- the vacuum level being defined as the amount of energy necessary to remove the electron from the system. So what you get when you put these two materials together is something like this right here. This dashed blue line representing the Fermi energy, or the chemical potential throughout the entire system, is the same. In this particular case, we didn't bias our device. We're not applying a battery or some bias voltage between one side and the other. So we have the same chemical potential throughout. If we have the same chemical potential throughout, notice that we had to push this up, or push the Fermi energy up, to reach-- and of course, the work function in the metal's not changing, and so the vacuum energy also goes up. But since there can't be a discontinuity in the vacuum energy, you see a little bit of a rise of the vacuum energy in the semiconductor. And of course, the valence band and conduction bands follow. And that results in this little rise right here, right next to the interface between the semiconductor and the metal. Any questions so far? AUDIENCE: Is there a discontinuity in the first derivative of the vacuum level? Is there a kink in it? TONIO BUONASSISI: Ha. A very interesting question. So typically, we draw it as having a discontinuity in the first derivative. We would draw a very sharp interface right there. But in a practical sense, I'll reserve judgment on that, until I think a little bit more deeply about it. It's difficult for me to believe that there would be, if you zoom in very, very closely, very finely. The bending, let's think about it from another way, from what we already know. This is essentially the second integral of the charge distribution across that interface. And if the charge distribution is sharp enough, sure, I suppose you could get some difference. But I think-- I'll have to research judgement, but my gut is telling me that there would not be at a very fine microscopic level. In general, though, we draw it looking much like this. OK? Any other questions about this, about how we constructed it, using the Anderson method? No? OK. Why don't we continue? What we've done right here is fill in those diagrams, the same ones that you were looking at, with many more variables. The purpose of this is to get to the point where we can describe the built-in voltage. Remember, we mentioned that, if we have some exponential current voltage response, there should be some barrier embedded inside of our system. And this right here is that barrier. So these series of variables are teaching us how to derive it, how to discover how large that barrier is. And this other component right here, which is the width of the space charge region, which is the width of this, so-called, curved region, if you will, is also going to be of importance. So first off, let's think about it this way. If we are looking at this system from the electron's perspective, we have an n-type semiconductor right here coming up against the metal. If our work function of the metal is larger than the work function-- or the so-called work function of the semiconductor, which is really the electron affinity plus the delta in energy between the conduction band of the Fermi energy. But if the work function of the metal is larger than that, we can see that the electrons should very easily plop down into the metal. And there shouldn't be-- let's see, in that particular case, work function of the mental should result in an Ohmic contact for the minority carriers, in that case. But in terms of the barrier height, this barrier height right here would be determined by that equation there. The contact potential, the built-in bias voltage would be here, and the barrier height there. And the width of the space charge region, once again, is dependent on the dopant density inside of the semiconductor. Here, we've defined it as Nd, because we're assuming that this is an n-type semiconductor with donor atoms. But of course, if this were a p-type semiconductor, we'd just replace this with Na to determine the width. And if the dopant density is small, this means that the width of the space charge region is going to be large. And that makes sense, because there has to be an equal amount of charge on either side of the junction. And if there is less charge per unit volume here, you need a larger volume to compensate the charge on the other side. So this is, again, a similar expression to the one that we saw when we were studying the PN junctions. Now, the important thing to recognize here is that, if we have the bands bending this way, for electrons in our system, there may be a barrier to hop out. For holes in the system, though, this could be a favorable exit strategy, if the bands are bent that way. If the bends are bent the other way, electrons can rather easily roll down. But holes will have a barrier to go out of the material. So the same type of metal that might make an Ohmic contact for an n-type material might make a Schottky contact for a p-type material. Yeah, that's about it. So we have to be very careful in terms of our matching or pairing between the specific type of metal and the specific semiconductor involved. Let me put that into perspective here with something we've already seen. Again, the Ohmic contact is one in which we have a linear IV characteristic. And the electron barrier height has to be less than or equal to 0, to have the electrons very easily pop down into the contact. Whereas, for a Schottky contact, there should be some barrier height. It should be finite. And we should have a difficulty of moving the electron from the semiconductor into the metal, in that particular case. So the natural question is, what metals should we use? Or what is the range of metals that are available? If the barrier height is a function of the work function of the metal, what is the range of work functions that are available to us? On this green arrow right here, which is Q times the work function of the metal, we have a range of different metals, from aluminum to platinum, and titanium, zinc, tungsten, molybdenum, copper, nickel, gold, in between. And of course, there are other metals, too, in the periodic table that you can add to this chart. But this gives you a sense of where the Fermi energy of the metal would lie, relative to the vacuum level, and how that would match up against your arbitrary semiconductor. This one here happens to be a silicon carbide material. But if you happen to be working on a different semiconductor and place it right here, you can see where your Fermi energy would lie, relative to the work function of the metal, if you line up the vacuum levels. And this gives you an idea already of how the bands will bend, once you adjust the position of the metal to match up the Fermi energy inside of your semiconductor material. Yeah. AUDIENCE: So on the left is an intrinsic semiconductor? Is it? TONIO BUONASSISI: Yeah, it's pretty close to intrinsic. It's a little n-type, it looks like. But it's pretty close to intrinsic. This, in this case, just a variant of silicon carbide. There are many polymorphs of silicon carbide, meaning same composition, same silicon carbon ratio, but different crystal structures, depending on how it was grown. Yep? AUDIENCE: Do you know where silver would fall? TONIO BUONASSISI: Silver? AUDIENCE: It's not listed [INAUDIBLE]. TONIO BUONASSISI: No, it's listed. AUDIENCE: [INAUDIBLE] a lot for [INAUDIBLE]. TONIO BUONASSISI: Yeah. I'm going to have to guess that it's going to fall around here, just from a similarity of other elemental species in that list. Do you happen to remember the work function of silver, Joe? Been working with it a lot lately. JOE: Yeah. I don't know. TONIO BUONASSISI: OK. Quick, interesting aside about silver, since it was brought up. It is used very often in contact metallization in solar cells. As a matter of fact, a bit too much now. So much so that it's driving up the price of silver on the market. We talked about this, I think, in a few classes ago, right? 10%, approximately, of all silver worldwide is currently being used in contact metallization in solar cells. That's a pretty high number. OK, so we have a variety of metals to choose from, almost like a menu. And then the semiconductor here, and we can see, OK, if we make contact-- and let's say we pick nickel, in this case-- if the Fermi energies line up, that means we shift the metal down. That means that our vacuum level is going to fall down here in the semiconductor. And for electrons, at least, will be very easily contacting this silicon carbide material. So from a simple energy band diagram point of view, we can begin to see which contacts will create Ohmic contacts, and which contacts will create Schottky contacts. That's in the ideal world. In reality-- well, let me make sure that we grasp this before we move on. Any questions concerning this so far? Yeah? AUDIENCE: How would we be able to tell if it's Ohmic or Schottky in this sense? TONIO BUONASSISI: Sure. Sure. AUDIENCE: [INAUDIBLE]. TONIO BUONASSISI: Absolutely. So from this diagram right here, let's walk through it. So if I, say, have a semicondting-- let's first line up the vacuum levels, like we've done right here. That's a good first step on, what we call, the Anderson method of identifying with the band diagram or band structure will look like. We start with the vacuum levels lined up. I'm going to write to this vacuum. And I'm going to dray my semiconductor on one side. So let's say I have conduction band and bands bent like so. And I have my Fermi energy like so, Ef. So this is my semiconductor. And then I have an imaginary barrier, and I have my metal on the other side. So now, let's say we have a metal that has a work function somewhere up here. So this is the work function of the metal. We'll call it the Fermi energy inside of the metal. What happens when I put these two materials together? Well, step one is to remove that imaginary barrier in between them. And now, you begin to look at this and say, well, if they're in contact, in good electrical contact, and there's no resistance throughout the material, the chemical potential should be identical all throughout the material, from left to right. And so now, I merge into phase two, which is to say, I've removed this barrier here in the middle, and now my Fermi energy is going to be equilibrated throughout. So my Ef is going to be the same throughout. And so, far away from the junction region here in the middle, I'm going to draw my conduction band here and my vacuum level here. And now, far away from the junction on the other side in the metal, if my Fermi energy is here and my vacuum energy is going to be somewhere down around here. So semiconductor and metal. Now, what happens closer to the interface right here-- this is the Fermi energy. This is the conduction band. This is the valence band. This is the vacuum level in the semiconductor, the vacuum level in the middle. And like we just described, the vacuum level has to be continuous. There has to be a continuity throughout. And so what we see here at the interface, since the density of charge is very high in the metal, but lower in the semiconductor, there's going to be band bending at the interface. And there's going to be a lot more bending on the bands in the semiconductor than there will be in the metal, as a result of the difference in charge densities. And so what you'll see is something that looks like this. The vacuum level will drop as it reaches that semiconductor metal interface, in order to match either side. And the conduction band and valence bands will track the vacuum level on either side of that junction. So this is the junction right here, in between. And now I look at this situation right here, and I say, OK, if I have an electron approaching this junction right here, there is no barrier to going from the semiconductor into the metal. As a matter of fact, there's almost an energy gain to be had from going from the semiconductor into the metal. But now, from the hole's perspective, if I'm a hole right here in the valence band and approaching this junction, there's actually a repulsion away from that interface, if I'm a hole. And so this particular junction here allows electrons to pass, but repels holes. This is looking an awful lot like a diode. So it's this construction right here that allows you to tell-- asterisk, tell, asterisk, according to the Schottky band diagram-- whether or not this should behave in an Ohmic fashion or a Schottky fashion for the particular karyotype that you're probing. And I say asterisks, because, in real life, contacts are never that simple. This is an example right here of the calculated-- make sure I get all my variables right. Yeah. So work function in the middle. The calculated barrier height for different metal species here on a certain type, a certain polymorphous silicon carbide. And these are the major barrier heights for different metals, titanium, nickel and gold, at different faces of the silicon carbide. So if you have a binary semiconductor, meaning a semiconductor comprised of two elements, if you terminate at a surface and you cut the plane just right, you could wind up exposing a plane of silicon atoms or of carbon atoms, if you have silicon carbide, let's say, depending on where you cut. So if you cut right here, you might expose a plane of silicon atoms. You cut one atomic layer above, you might expose a row of carbon atoms. Cut a row above that, it's silicon again. And so depending on what face you expose and depending on the orientation of that face, your barrier height, relative to the metal, could be different. So, hm. You glance at that, and you begin thinking to yourself, OK, this is macroscopic right here. This tells you, kind of from a continuum point of view, what the interface should behave like, what the context should be. But what I'm seeing right here from the data is that the atomic configuration at the interface really matters. So there must be something going on in an atomic level, as well, that's determining the barrier at that interface. In fact, there is. The dipole at the interface between the metal and that last layer of semiconductor, in other words, the distribution of charge between those two sets of atoms that are meeting at an interface determines, in part, the barrier height. We'll get to that in a few slides, as well. So this is meant to shine light on a much broader picture, which is to say that there can be substantial deviations from Schottky theory at the interface. And some of the effects are due to orientation-dependent surface states. It could be due to the specific elemental nature of the terminated surface, as we just discussed, whether you're terminating at the silicon plane or the carbon plane. Orientation-dependence means, gee, if I have an anisotropic crystal, meaning a crystal that looks a little bit different, if I rotate it around in different orientations, if I make cuts in different directions exposing different planes, they're going to be a different density of atoms in that plane. There's going to be a different charge distribution in that plane. And hence, I would expect there to be some different interaction between the semiconductor and the metal at that interface. And finally, interface dipoles, this relates back up to the first one in part. This just goes to say that, if I have an ionic material or an element there at that interface that is grabbing charge, I could result in a small region of negative charge and positive charge forming a dipole right there at the interface. And if I have a buildup of charge, I have a field. If I have a field, I have a potential. And if I have a potential, I'm going to be disrupting this precise interface right here. And we can, as well, vary the density of interface states-- I got you, Ashley, I'll be there in a second-- so we can, as well, vary the density of defect states here at this interface that can trap charge. And again, with the accumulation of charge, comes a field. With a field comes a deviation of the potential. So what we've done for ourselves is, effectively, using the Schottky model, we've set up for ourselves the energy band diagram of a metal semiconductor contact in its simplest case. And what we're going to do over the next few slides is explore some of the more complexity behind what goes into a semiconductor metal contact. Ashley? ASHLEY: So going back that with the green line, so these metals, given some [INAUDIBLE] up there on the left, would all result in Schottky contacts? Is that right? Because all of their work functions are above the Fermi level? TONIO BUONASSISI: Yeah. ASHLEY: Assuming [INAUDIBLE]. TONIO BUONASSISI: So yeah, it depends on what is the minority carrier in this particular junction that's driving it. Because this is intrinsic, it's a little difficult to say, actually. But if it was clear there was n- or p-type, and the minority carrier density at that interface was determining the current flow across it, then we'd be able to say with certainty whether it was one type or another. ASHLEY: OK. TONIO BUONASSISI: But, yeah. ASHLEY: So if it were-- TONIO BUONASSISI: So in this case right over here-- let's make it specific. ASHLEY: Yes. TONIO BUONASSISI: In this case right over here, let's make this an n-type material. So my Ef is now higher. Ew. OK. That actually had the opposite effect. Let's exacerbate this a bit. I'm actually going to invert that and make this a p-type material. The reason is I want a big delta in my Fermi energies right here, if my vacuum level is constant, so to exacerbate that band bending at the interface. And so now, I'm going to get a pretty drastic band bending. It'll be even more extreme than what I drew here. But let's say it's fine. So what we would have to do is we would have to move this down further, right? Actually, no. Sorry. This wouldn't change, because this is still the same. What we've done here is essentially shifted everything up further. So to be more precise, we'd have to move this up, move this up. And we'd have to move this up, as well. Again, p-type material. Excellent. And now, again, just going through the logic of the process here, we have to make the vacuum levels match. And there's going to be an equal amount of charge on either side of that junction. Now, when there's a charge, there's a field. When there's a field, there's a potential. And the potential will be manifested in the bending of those bands approaching the interface. The bands will bend a lot more in the semiconductor than they will in the metal, because the free charge density here in the semiconductor is a lot less than it is in the metal. And going back to this equation right here, the width of the region of bent bands is going to depend inversely on the dopant density or the amount of charge in that region. So this is to say that we'll have a much more extreme bending here in the semiconductor than we will in the metal. Virtually imperceptible there in the metal. And we drop this down, as well. And we drop this down, as well. So some really interesting things have happened right here to the position of the Fermi energy in the band gap, right? Over here, we have a p-type material. And over here, at the surface layer, it almost looks like this material is going n-type. ASHLEY: Right. TONIO BUONASSISI: Let me pause right here, because this is just utterly fascinating, and tell you the story about indium nitride. It's a common material that many people-- I see a few knowing nods here in the audience. So indium nitride was a semiconducting material that people really didn't have a good handle on for a variety of reasons. And there was a renewed interest in it, because people finally started to understand why the band gap was the way it was. And many groups started studying it. And they also reporting this n-type indium nitride behavior, very highly n-type material, highly doped n-type material. It turned out that there was this band bending here at the surface. And they were measuring a high electron concentration at the surface, because of the bending of bands. It was in reality the intrinsic material deep in the bulk was p-type. But they just couldn't see it because their probes were touching the semiconductor at the surface. It took a certain scientist by the name of Becca Jones in Berkeley, Lawrence Berkeley Lab, I believe, at the time. She immersed the sample into a liquid solution of an acid-- I believe it was hydrofluoric acid-- and managed to relieve the pinning of the bands at the surface. And so they returned to normal. And she was able to probe using a semiconductor liquid junction, the true conductivity type of the bulk of the material. Becca Jones then went on to produce what is now the-- oh, she was on the team that produced one of the highest efficiency cells on the efficiency versus time plots. So she's doing well for herself there in the PV world. Yeah. AUDIENCE: [INAUDIBLE]? TONIO BUONASSISI: Sorry. Sorry. So let me get back to that. What would happen if you were an electron right here and you were approaching this junction? This is E. This is the energy of electrons. So the electron will seek to minimize, it will seek to encounter a lower energy state. So if you're approaching this junction as an electron, what would happen? AUDIENCE: [INAUDIBLE]. TONIO BUONASSISI: Zoom. Yep. So it's go down. And there wouldn't be any barrier for it to do so. AUDIENCE: Right. TONIO BUONASSISI: It would just go, [MAKES WHOOSH SOUND] straight down. Now, let's inverse the situation and imagine, what if you're a hole right here? What now? If you think of electrons as bowling balls that want to roll down hills, holes would be the opposite, which would be like balloons that want to rise, right? AUDIENCE: Right. TONIO BUONASSISI: And so this is actually a barrier for a hole. AUDIENCE: Right. TONIO BUONASSISI: You need a certain amount of energy to get the hole across. And so the hole naturally will be repelled by this junction. The hole will want to go this way. The electron will want to go that way. AUDIENCE: OK. So this is Schottky for electrons, but Ohmic-- sorry, Schottky for holes and Ohmic for [INAUDIBLE]. TONIO BUONASSISI: Let's imagine we have a device that is driven by the minority carrier flux here at this interface, right? AUDIENCE: Right. TONIO BUONASSISI: And so what is the minority carrier in this particular case? AUDIENCE: So the left is p-type, and so the minority carrier would be electrons? TONIO BUONASSISI: Yep. Exactly. So what would you expect this contact to behave like? AUDIENCE: So it'd be Schottky for-- no, Ohmic for electrons. TONIO BUONASSISI: So this one right here, the electrons would be able to flow over quite easily. So in the illuminated case where the current would be driven by the minority carrier flux, you would expect to see a behavior without a barrier. AUDIENCE: OK. So Ohmic for-- TONIO BUONASSISI: Yep. AUDIENCE: OK. TONIO BUONASSISI: Now, it can become a little bit more complicated, but let's leave that right there. AUDIENCE: Right. OK. AUDIENCE: If you have a Schottkys barrier like that, it's just extracting electrons or holes, then is the relevant quantity on the left a quasi-Fermi energy for electrons or holes? Or is it it's always the overall Fermi energy? TONIO BUONASSISI: Yeah. So you raise an interesting point, which is the question of quasi-Fermi energy. So if you illuminate your device or you forward-bias it, what happens? Now, you start injecting carriers. And as we discussed during class the time, the total carrier density is going to be equal to, let's call it, the intrinsic carrier concentration plus the dopant density, if this n would be the density of donors, plus my delta n, which is the injected carrier concentration. So in this particular case-- let's parse this through-- this is going to be small, so we can ignore that. Well, actually, no. For the electrons in this case, this will actually be quite big. This is going to be smaller, rather negligible. And let's think this through carefully here. So under equilibrium conditions, under equilibrium conditions-- let's describe this as an n0, briefly. So under equilibrium conditions, my n0 is going to be rather small because, if I have a high dopant density of p-type material, that means that my electron and my n0 is going to be equal to ni squared divided by p0. And if p0 is in the order of, say, 10 to the 16, ni would be nearer to 10 to the 10. We'd have something in the order of 10 to the 4 for n0. That would be a rather small number. Now, if I inject light on it, I could have, say, 10 to the 14 electrons being created. So this drowns out everything else. So now, my electron concentration is going to be driven by the free carrier concentration, the photo-excited carrier concentration. Now, on the p, on the other hand, this is going to be equal to p0 plus delta p. So let me drag this over here. The delta p is still on the same order as the delta n, let's say, 10 to the 13, somewhere in that range. But my p0 is equal to the dopant density, acceptor density, which could be on the order of 10 to the 16. So this will be somewhere in the range of, say, 10 to the 16, driven by the acceptor concentration. And so now, I have a situation in which I have 10 to the 13 electrons and maybe 10 to the 16 holes. And if I multiply the two together, I'm not getting the intrinsic carrier concentration squared, because I'm not under dark conditions. I'm not under equilibrium conditions. I'm in a steady state condition. I'm shining light. I'm exciting carriers. They're recombining, but at steady state. I'm shining light, generating the carrier again. And we reach this quasi-equilibrium. And so what we have, at the end of the day, is a separation of the Fermi energy between electrons and holes. The Fermi energy for holes is going to be relatively constant. But the electrons for the minority carrier, you'll have a rise of that quasi-Fermi energy, because now you're filling in more of your electrons up here. And so the Fermi energy, as defined as the energy state that has a 50% occupancy probability, is now at a different energy for electrons or for holes. Awesome. It's really one of those things that takes a while to really wrap your mind around. And so now, we might have a situation which my electron quasi-Fermi energy is somewhere up here. Let's make it a little bit-- I don't want to get into what's called a two-dimension electron gas. I'll draw it somewhere down here. So now, I have this distance here is defined by the minority current diffusion length, typically, from the junction. AUDIENCE: And as the electrons drop from that quasi-Fermi energy to the Fermi energy of the metal, what happens? TONIO BUONASSISI: So the important thing to think about is current flow can be reduced to electrons within the conduction band, discrete particles that are seeking to minimize their free energy. The voltage or the potential across is dictated by the ensemble. And that's where the Fermi energies come into play. So in that particular case, it might even cause the Fermi energy over here to rise a bit. And then you have a potential across. If you had a wire connecting this to an external load, you'd actually be able to power it. The electrons would be photo-excited in here, would be driven over into the metal, because of the built-in field. And then they'd be able to power an external circuit and come back into the back. So you're beginning to see something that looks a lot like a PN junction. And in this particular case, the dark current for electrons would be dictated by their flow back into the semiconductor. So yes, in this particular case here-- let me correct myself-- in this particular case here, for the electron, you would have a barrier for the diffusion current, which would be driven from the metal into the base right here. And so the diffusion current of the electrons would be facing this barrier here. And this would be Schottky-like in nature. If you illuminate it, your illumination current would-- so let me back up one step right here. So in the dark, a completely dark sample right here, that I-V characteristic is driven by the diffusion current of electrons to the metal and to the semiconductor. And that is, in this case, with the barrier. And as you begin forward-biasing it, you would have more and more carriers coming across. And that would drive the diffusion current forward. The illumination current would be driving the opposite way. And so, if you illuminated this entire system, your blue curve would shift down. Let's say, if you're photo-exciting carriers within the base right here, they can very easily reach the metal. But if you're putting contacts and probing the total current flowing through your system, in the dark, you would be measuring the diffusion current from the metal into the semiconductor. There's some positive electrons over here. And so the electrons will be driven from the metal into the semiconductor. And they would be facing this barrier. And hence, you would see a Schottky-like behavior. AUDIENCE: OK. So in the dark, it would be Schottky-like for electrons. And in the [INAUDIBLE]-- TONIO BUONASSISI: So you think of it in terms of total current. Yes, you're absolutely right. AUDIENCE: Right. TONIO BUONASSISI: And the holes would be equal and opposite. Then, when you illuminate it, current would start flowing the other direction. So in the dark, the current is flowing, say, in the positive direction. And let's define the positive direction as electrons flowing from here to here, which means that positive flow is going from left to right. So that makes sense. And when we illuminate it, now, instead of the electrons flowing from the right into the left, we have the illumination current driving electrons from the left and to the right. So the direction of current is reversed, when we illuminate this. And that's the same thing as saying, I'm going to shift this down under illumination and wind up with something that behaves very similar to a PN junction. AUDIENCE: Wait. I thought we said that, under illumination conditions, this situation is Ohmic for electrons. Or is that not true? TONIO BUONASSISI: Sorry. The electrons have no barrier to travel over here for the illumination current. AUDIENCE: Right. Oh, so this-- TONIO BUONASSISI: So the illumination current's going to be constant throughout. AUDIENCE: So it's-- [INAUDIBLE]. TONIO BUONASSISI: But for the diffusion current-- the net current behavior, the net I-V characteristic of this will be Schottky-like. AUDIENCE: OK. TONIO BUONASSISI: But the diffusion current is the one that's driving the current in that case. And the illumination current is being driven by the minority carrier flow right here at the edge of the metal semiconductor contact. AUDIENCE: OK. TONIO BUONASSISI: Joe, why don't we flag this? I sense this is going to be an item for discussion during the recitation session. JOE: Sorry. [INAUDIBLE]? TONIO BUONASSISI: Just the metal semiconductor interface, I think, we're going to have to go through it from scratch again. This is very, very similar to a PN junction. AUDIENCE: Yep. TONIO BUONASSISI: So I think where folks are getting tripped up is thinking about-- and even I sometimes get tripped up-- is thinking about what happens to the illuminated current and what happens to the drift current and the diffusion current. AUDIENCE: Right. TONIO BUONASSISI: So it's all very similar. It's all stuff we've seen before. We just have to think through it step by step and make sure we don't get tripped up along the way. AUDIENCE: Yep. TONIO BUONASSISI: OK. So let's talk about some of the non-idealities because this is where the fun stuff lies. When we have a metal on a semiconductor, a lot goes into it. For example, if we have interface states or surface states, we can form band bending there at that interface. Let me demonstrate how. This is an article that everyone should have read already. It was part of the assigned reading today. And if my past experience is a guide, there is a fixed percentage of you who haven't seen it yet, so I'm going to pass around the article that you should have read. So this is something-- feel free to skim through it, gain an appreciation for, really, the spectacular nature of that article. And then pass it around your colleagues. Make sure everybody gets to see them. On the left-hand side, we have an example of a semiconductor, let's say, surface. The surface is represented by this really large band gap material right there, that box, that rectilinear box, very thin and skinny. It's thin because the surface layer is very thin. It's maybe a few nanometers, typically, in a semiconductor. And it's very large and band gapped because we formed a semiconductor and oxide, let's say, a silicon oxide, a silica layer at that surface. So if we have pure silicon and we expose it to air, the silicon will react with air and form a very thin SiO2 layer or silicon oxide layer. And that's what this represents. This is a very large band gap material, and it's very thin. And so if there are no interface states, if we have an ideal material, we have our semiconductor over here and our surface layer there. And at that interface, boom. You just have a discontinuity in the conduction bands. And of course, the vacuum levels are matching up. But in the case where you have surface states at that interface, what happens? Well now, those surface states can be filled by electrons in the neighboring region. And as a result, you'll have a natural band bending at that interface. So as you begin filling in those states, another way to think about it is that you're pulling the Fermi energy toward mid-gap, because now, as the electrons are depleted from that near-surface region, as they fall into those surface states, the Fermi energy is going to shift towards mid-gap. And if you indeed look at the semiconductor, right here at the surface, you'll see the Fermi energy is almost at the mid-gap. And so this is a very curious phenomenon that occurs in many semiconductors. There's a natural band bend toward the surface, depending on the surface chemistry. If you have a semi-conducting oxide forming at the surface, or if you happen to be exposing it to a sulfur based gas right after you grow it, and you form some sulfide at the surface, the surface chemistry is very important for dictating the band bending at the surface. Why is that important? That's important because, when you go ahead and deposit a metal on top of it, now, you not only have to contend with what the semiconductor is doing, but what that semiconductor surface layer is doing. Now, the situation is really becoming complex. So this is meant to represent right here what happens when you have a metal-- in this case, a semi-conducting oxide semiconductor interface, also known as a MOS, metal oxide semiconductor-- what happens? Well, in an ideal case, you would have the bands bending a certain way. And then, with the metal oxide layer in between, you can have a definite shift. So what this is meant to represent right here is a metal with a work function that is larger than the electron affinity, and a work function that is smaller than the electron affinity. And so, given what we've seen before, given the Schottky method of producing the band diagrams here, we would expect that, in this case, we would see bands bending, say, down. And in that case-- let's see. No. In this case, we'd see bands bending down in the semiconductor. And in this case, the bands would bend up in the semiconductor as we reach that interface. But because the Fermi energy is pinned by these surface states here, we have very little motion of those bands. And so we wind up in a case where we pin the Fermi energy at the surface and, pretty much, no matter what metal we deposit on top of our sample, we can wind up with a certain type of behavior of the semiconductor, a certain current voltage response. And this can drive someone absolutely insane when you're in the laboratory trying to get a repeatable contact and everything is changing all the time. And you walk through the math, and you know that you should be getting, let's say, an Ohmic contact. You walk through the math, and you derive the energy band diagram of the interface. And you go to the lab, and over, and over, and over again, you obtain a Schottky contact. And it just drives you insane. The trick is to always use repeatable surface preparation. Always prepare your surfaces using the same chemistry, the same way, every single time. And if you do that, most likely, in many cases, you can get rid of this native surface oxide on your semiconducting sample. But if you take your sample out of the growth chamber and move it around in air and then put it into the metallization chamber where you evaporate your contacts, you're exposing your sample to the air. You'll form some surface layer on it, and your contacts will be different. There was a colleague of mine, who's now a professor at a university. And during his PhD, I think he spent about a year on a problem like this, trying to make good contacts. And unfortunately, contacts, it's such a basic thing that it's one of the first skills to go when the field moves on. So for example, people really knew how to make good contacts at MIT 30 years ago, when the IC industry was the hot topic of the day. Then, as that phased out, the knowledge of how to make good contact onto silicon disappeared. There was maybe one or two professors who really had that knowledge embedded in their minds. And so, when a new PhD student came along at Harvard University trying to make Ohmic contact into silicon, there wasn't anybody he could go to and talk to. He didn't know where to start or who to talk to, because this is kind of base knowledge. It's really fundamental, instrumental knowledge that is going to be infinitely useful. But it quickly gets lost when a field moves on, when the next material system becomes hot, or the next topical area is the key research area of the day. If you know how to make good contacts onto a semiconductor, I guarantee you, you will find someplace in industry or someplace in academia where you'll be well-sought out, and that skill will be put to good use. Let's talk about this space-charge region width very briefly, because we're running out of time. We've talked about this barrier as being this insurmountable barrier, let's say, for an electron to travel over, in this case, to get to the metal. But what happens if we increase the dopant density within the semiconductor? If we increase the dopant density in the semiconductor, then the amount of charge per unit volume at that interface or per unit thickness at the interface increases. So to compensate some buildup of charge in the metal, we would need a thinner region of the semiconductor, because it's more heavily doped, because there are more dopants per cubic centimeter. And hence, there are more free carriers per cubic centimeter as well. And as a result, the width of this space-charge region will decrease. And if the width of the space-charge region decreases, what happens? This barrier height still stays the same. The barrier height doesn't change, but the width changes. And what do we know from quantum mechanics, if our barrier is very, very thin or very narrow? What happens? Electrons will? AUDIENCE: Tunnel. TONIO BUONASSISI: Tunnel. And that's exactly what happens right here. So if we have a very narrow space-charge region, we'll have a tunneling junction, field emission effect, it's also known as. And irrespective of the barrier height, you will get electrons moving straight across. Now, if we have a wide barrier-- represented there at the top-- there is an energy barrier for the electrons to overcome before they reach the metal on the other side. And so there, we have a thermionic emission, which means that you need to have an excited electron, thermally excited-- that's where the name therm comes from-- so thermally excited electron, before it manages to hop across. And because there's and exponentially decaying number of electrons with higher and higher energies, we'll get a very small number of electrons that can actually make it. So the thermionic emission case will, in effect, manifest itself as a higher series resistance than the field emission case. You can have the same metal contacting a semiconductor. But if the dopant density is changing, you will make a better Ohmic contact. And of course, when you're actually trying to measure the contact resistance here or measure the barrier height, you really have to know this interface really, really well. You can measure experimentally. But to determine from theory, you get a good start by using the Schottky models we described right here. But to really nail it, you need to perform density functional theory and determine the charge distribution across that interface. It's really the best way to go about it. So in terms of experiment, thankfully, there are big tables that we can look up. If we have a particular semiconducting material that's well-studied, we can look up the contact materials that would make the best contacts, and what techniques we can use to deposit them, what temperature we should anele at to form that alloy. These are very practical look up tables that exist in reference books, trusted websites-- not any website, but the trusted ones-- review articles. But let's say you're working on a semiconducting material that's not on this list. Let's say you're working on cuprous oxide, for instance, and you have to start from scratch. The best way to go about it would simply be to measure the contact resistance using the techniques that we described two lectures ago where we described the TLM method or the-- sorry, not much sleep-- the Transfer Line Method. So the important thing here is to know that you do have a few tools at your disposal to measure barrier heights experimentally. And at the end of the day, you can obtain precise numbers. Just going quickly into heterojunctions. This is similar to the Schottky junctions we've been talking about so far, Schottky, Ohmic contacts between metals and semiconductors. This, very, very briefly, is what happens when a semiconductor meets a semiconductor. Not always can you form a homo-type junction, meaning not always can you have n- and p-type dopants in a semiconductor like silicon. Sometimes, it just won't dope both n- and p-type. And so you have to make contact with another semiconducting material. And for that, we call it a heterojunction. And there are three general types of heterojunctions. One in which you have a wide band gap material on the left, a narrow band gap material on the right. The two are in contact with each other. And the vacuum levels line up into such a way as to wind up with this band alignment. Now, let's say I excite an electron hole pair in this wide band gap material. The hole will be able to move across. The electron will be able to move across. So both the electron and hole have moved across the junction, and we have not separated charge. So type one, very bad from the photovoltaic point of view. Type two, this is more similar to the PN junction, what we've seen so far. And in this particular case, if you excite an electron-hole pair, the electron will move across because of the field, but the hole will be repelled. And so the type two junction is actually preferred, from a photovoltaic point of view. Type three junction will still separate charge. The electron will move from the left-hand side to the right-hand side. And the hole will be repelled. But now that the electron has moved to the right-hand side, it actually has a lower energy than on the right-hand side. So there is no driving force. There's no voltage built up for the electron to go across, outside of an external circuit and power and external load. And so, from a photovoltaic point of view, the type three, unless under very high illumination conditions, is generally considered not desirable, as well. So the type two junction is the preferred junction type for photovoltaic devices. And yeah, that's pretty much it. So you can go through the exercise shown right here on your own and determine, for this particular system right here, what will happen at that junction. How will those bands align? You have the cheat sheet shown right after. But an open question for you is, using what you know from today where we first lined up the vacuum levels, drew out the band diagrams independently for material one and material two, just like we've done there, now we eliminate that barrier in between them. We put them together, and we match the Fermi energies. In other words, this Fermi energy, Ef 1 has to equal Ef 2. So there has to be this shift going on like that. What happens then? How will the bands align? Follow the vacuum level. That's my best advice I can give you. Let the conduction band and valence bands follow the bending of the vacuum level. And then, from there, you should be able to determine the final structure that'll look something like that. But that's a take-home assignment, just to see if the concepts really jelled today. So what did we learn how to do? Or at least, what do we have a greater appreciation for? We have a greater appreciation for all of these different components, how to make different types of contacts on a material, what they're good for, and ultimately, how this band alignment that we've learned in our PN junction expropriations can be applied as well to metal contacts, and ultimately, how this can also be applied to semiconductor, semiconductor contacts where we might have two different types of semiconductors coming together and forming a junction. How do we predict what the energy band diagram is going to look like, once we have our final junction right here? And what do we expect that to mean, in terms of charge transport across that layer? So those are the building blocks, the basic building blocks that you will need to be able to predict how two materials will behave when they encounter each other and so that you can make solar cell devices. And I think, with that, we can pretty much wrap up our fundamentals portion of the class. Woo-hoo! We get to talk about technologies next. |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | Tutorial_Doping.txt | [MUSIC PLAYING] PROFESSOR: Hello, everyone. Today we'll talk about doping, which is the process of intentionally adding impurities to a semiconductor in order to change its electrical properties. Doping is a critical process in the tech world. It's used in manufacturing almost all semiconductor technologies today. Without doping, the solar industry would not exist, but even though doping is common today, the effects of impurities confused semiconductor physicists in the 1950s, who had trouble reproducing results. Eventually, they realized that contamination levels, as low as 1 in a billion, were vastly changing the electrical properties of their samples. Today, we'll show you how this works with a very simple experiment. We'll be measuring the electrical conductivity of two silicon slabs using an ohmmeter. One is doped with impurities, phosphorus in our case, and the other is ultra-pure, or what we call intrinsic. Let's go over our experiment. We'll start with a slab of silicon, which we attach metal contacts to. We'll use an ohmmeter, that we connect to our sample with metal wires to measure the conductivity. The conductivity describes how well electricity can flow through the material. The measured resistance from our ohmmeter is related to the inverse of the conductivity. The resistance also varies according to the physical size and shape of our sample, which adds a length over area term to our equation, like so. Rearranging this equation, gives is what we're looking for, the conductivity. Let's measure our samples, and estimate the conductivity. Here are two samples, notice that the doped sample looks identical to the intrinsic one, or undoped sample. Because we've only added trace impurities, the optical properties are nearly identical between the two samples. Let's hook up the ohmmeter to the intrinsic sample. We can see that the resistance is 130,000 ohms, which roughly corresponds to a conductivity of 0.0002 inverse ohm centimeters. Let's compare this to the doped sample. We read a resistance of 34 ohms, which corresponds to roughly 0.6 inverse ohm centimeters. So we can see that the dope sample is around 3,000 times more conductive. But why would adding small amount of our doping, about one phosphorus atom for every million silicon atoms, make our sample 3,000 times more conductive? On the periodic table, we see that silicon is in the fourth column, which means it has four valence electrons. Phosphorus, which is just to the right in column five, has five valence electrons, one extra compared to silicon. I'd also like to point out boron in column three, with one fewer valence electron than silicon. Later, I'll explain what happens when you add boron as a dopant. We'll start with a 2D representation of a single silicon atom, with the nucleus in the center, and its four valence electrons in a silicon crystal, each silicon atom bonds to four other silicon atoms around it. These rigid covalent bonds, shown here, keep all of the electrons effectively immobile, and are therefore, unable to aid in the full electricity. Our intrinsic silicon, or undoped example, has this material structure, which is why it has a very low conductivity. Let's quantify this relationship between conductivity and mobile electrons. Conductivity is defined as n times mu times e. n is a number of free or mobile electrons. Again, in this drawing of intrinsic silicon, all electrons are covalently bonded so there are no mobile electrons, and n is 0. The symbol mu represents the mobility, a material parameter which you can look up in a textbook, or online, and it basically describes how well the charge can move around in the material. e is simply the amount of charge that each mobile particle possesses, which in all of our cases, is simply the charge of an electron. So let's ask, what happens when we add dopants like phosphorus and boron to the silicon lattice? Now, let's dope our material by replacing one of the silicon atoms with a phosphorus atom. First, we'll remove a silicon atom, and for contrast, we'll dim the background silicon lattice so we can emphasize the dopant atom. Notice that the inserted phosphorus atom has five valence electrons, four of which form four covalent bonds with their neighboring silicon atoms and are immobile. The fifth electron is not bonded, and as a result, is free to move around the lattice. When the negatively charged electron leaves, the phosphorus dopant is now positively charged. So we see that each phosphorus atom that is added will contribute a single mobile electron. So basically, in our case, the number of mobile electrons is roughly equal to the number of phosphorus atoms in our system. Now, let's remove our phosphorus atom and put in an element with three valence electrons, such as boron. We see here that boron lacks the necessary valence electrons to form covalent bonds to its four neighboring silicon atoms. This missing electron is actually referred to as a hole, and is represented by an H+ symbol. This hole acts as a mobile positive charge because it can swap places with neighboring covalently bonded electrons and move around the crystal. When the positively charged hole leaves its nucleus, the boron atom becomes negatively charged. So we've demonstrated that introducing atoms that have one more, or one less, valence electron than silicon, can add mobile charges and make the material more conductive. In our examples, the conductivity of silicon is proportional with the density of either phosphorus or boron atoms. While phosphorus and boron both affect the conductivity in a very similar manner, they introduce mobile and static charges of the opposite sign. Phosphorus introduces mobile negative charges and immobile positive charges, while boron creates mobile positive charges and immobile negative charges. This subtle difference between phosphorus and boron dopants will be crucial in our final video when we discuss solar cell operation. Today we learned that we can use doping to control the conductivity of semiconductors by changing the number of mobile charges in the material. When we look at the range of conductivities that silicon possess, it is truly amazing. Through doping, we have a very powerful way of varying the conductivity of semiconductors. This is something that is not possible in other classes of materials, like metals. Next time, we'll be discussing how light can be used to generate mobile charges in silicon, so watch our next video. I'm Joe Sullivan, and thanks for watching. [MUSIC PLAYING] |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | 16_Solar_Cell_Characterization.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: So I'm going to describe the basic classification of solar cell characterization methods. And then I'll describe some of the characterization tools that are used to measure Jsc losses and other tools are used to measure VOC and fill factor losses. So we're getting a sense of the lay of the land. Let me ask you how would you characterize-- how would you create a taxonomy of solar cell characterization techniques? Those of you might have a little bit more experience might actually have used a few characterization techniques in your laboratories. What taxonomy would you use? How would you slice the pie of all these characterization techniques? Go. What are ways to divide the characterization universe into distinct groupings? How about we start-- go ahead. Yeah? AUDIENCE: Performance and performance losses. PROFESSOR: Yeah so we can describe based on the economic variables like efficiency and then within efficiency Jsc, Voc, fill factor, mechanical yield, reliability. These are all parameters that can lump into a cost model and ultimately impact the cost-- dollars per watt, let's say of a PV panel. Excellent, so we can describe it based on performance. If we dive into performance a little more, what are some of the properties that affect performance? We have electrical properties, optical properties, mechanical, thermal. So we can break up characterization techniques based on the properties that they probe. And we can also characterize-- we can create a taxonomy of characterization techniques based on how fast they are, whether it's something that you have to sit around and twiddle your thumbs for a half or two hours to get done or something that gets done in 10 milliseconds of which there are several characterization techniques. What would the 10 millisecond variety enable you to do? In a manufacturing environment? Yeah? AUDIENCE: Sort them. PROFESSOR: Sort them. Test them. Measure them-- every single cell coming through. So you might have a bar code like one of those fancy two-dimensional bar codes laser marked on each single wafer going through your line so you can trace it back in your MES system all the way back to the crystal that was grown or from the thin film growth chamber where it was deposited. So, yes, you have inline and shall we say, the inline characterization techniques and what are called offline characterization techniques, which tend to be lower throughput. You can think of the inline characterization techniques as being in the line of the manufacturing environment and the offline as being those techniques that are sitting in your laboratory waiting to be used in the R&D lab, which might be next door to the manufacturing line. Device performance metric affected-- that addresses the efficiency point up here. And then by property tested-- electrical, structural, optical, mechanical. So if you talk to somebody who works in a PV company, she will likely give you a break down based on number three right here. OK, these are the techniques that we have in our inspection system and those are the techniques that I have over there in the R&D lab. If you talk to somebody who is giving a fundamental course in materials science, they are likely to pick part number one way up there and give you the breakdown based on the properties that are probed. I'm going to opt for number two today in lecture not because it's any better or worse than all the others, but just because that's the metric that we care about right now as we're trying to probe dollars per watt that relates to our quiz and our homework, but also relates to the ultimate economic driver of solar. So we will keep this in mind that we might need to ferret out references for the different techniques in different textbooks or different papers based on whether they're probing electrical structural optical mechanical properties of our solar cells. But we're going to be focused on efficiency-- short circuit, current, open circuit, voltage, fill factor. So let's go ahead and dive into that. We are going to first start with techniques to measure Jsc or short circuit current losses. And some of these slides are going to be repeats. And the reason they're repeats is because the first time you saw it OK, you sort of kind of got it. You went to the lab, made your devices, tested them. And now all of a sudden, you have a much stronger background with which to understand. We're going to start by discussing the optical components just very briefly and then spectra response and minority carrier diffusion length, revisiting some of these concepts that we have already seen, but now with the benefit of having all of our background knowledge. The spectrophotometer measures specular and diffuse reflectance and transmission. All right, let's break that down. Specular reflectance-- specu lar-- Latin. So specular reflectance means light comes in and out pretty much at the same angle relative to the surface normal. So if you come in from here, it's going to bounce out light there right at the same angle relative to the surface normal. Diffuse reflectance means that if you shine light in a certain angle, it's going to reflect back not necessarily at that angle. You could have a distribution. A Lambertian scatterer might qualify. And reflectance in transmission-- we talked about different optical losses of a solar cell material. Reflectance means that light comes back off the surface. Transmission means that light did not get absorbed. So it went through the material and didn't get absorbed. That is an optical loss as well. So the spectrophotometer is useful for measuring these different loss mechanisms. And it can tease apart the specular from the diffuse reflectance, giving you some indication, some idea, of how the surface is behaving and what you might do to improve it. So the spectrophotometer is useful in that regard. In terms of increasing absorption, we talked about various methods to increase absorption. Those who attended Eli Yablonovitch's talk heard about many more. The goal here is essentially to increase the optical path length by texturing your surface, for instance. The physical thickness can remain very low. And, again, just to refresh ourselves if we decrease the thickness of our devices but manage to have very good like trapping, what happens to our excess carrier density? It goes up, right? Because now we have more carriers being generated in a smaller volume so the carrier density increases. And as the carry density goes up, that means the separation of the quasi-Fermi energies increases, which means the maximum voltage extractable from or devices increases as well. So there's a strong push right now in the field to go thinner and thinner devices. That also has the added benefit economically of using less material. So we want to decrease the thickness by improving our optical trapping-- our light trapping. Another benefit is it allows carriers to be absorbed closer to the junction, which increases the probability of collection. And we talked about this during the very beginning of class how you might texture your surface. But now we actually saw it in producing our cells. So this is an example-- an SCM image-- of textured silicon. In that particular case, this was an alkali etch on a single crystalline sample probably of 1 0 0 orientation so that these edges of the pyramids are 1 1 1 planes. You could also achieve a similar, although not identically, looking result if you have performed an acidic etch, which would be isotropic in nature. So the light comes in for the textured surface. Some of it goes into the device. Notice that Snell's law is in effect. So the light bent or was refracted. And some of the light is reflected off the surface. Now, because of the texturization, you get that second-chance absorption. So the light has two bounces before leaving, which means that the probability of the light getting absorbed is 1 minus r quantity squared. And so you get an enhanced absorption. And for those who looked at the samples before and after texturization, you could visibly see that they looked darker-- the reflectivity had gone down. I see some smiles over here. There are probably some initial etching processes that didn't quite work out, but eventually the process was controlled and worked out fine. So we have as well other mechanisms to trap the light besides just texturing our front surface. These we didn't get to do an actual cell fab. But we could envision putting a reflective or defuse scatter on the back and reflective layer so that the light that goes all the way through the material once gets bounced back in perhaps at an angle to increase the trapping in the back. And that I think is represented on the previous slide. It is indeed. So we have a textured back surface. And we also have this layer-- probably in this case, it would most likely be a dielectric material with the refractive index that is significantly different than the absorber itself. So if your absorber material is an organic, you might have a refractive index somewhere between 1.5 and 2 maybe on the high end. And if you have an inorganic material, you could have refractive indices as high as 3, 3.5. And the material you put in the back would probably be as close as you could possibly get to air to get as a large reflection off the back as possible. In the case of silicon devices, oftentimes you have a dielectric material with a refractive index somewhere between 1.5 and 2. And that serves also to passivate the surface to prevent carriers from recombining in the back side. So a collection probability-- we talked about this earlier now that we've performed quantum efficiency and we've tested our own devices, I'm going to walk through it again. It's an important concept to really grasp. I want to make sure that everybody got it. We have here a p-n junction. Here's our P-type material. Here's our N-type material. Shown at the blue little dots-- those are our holes-- free holes-- mobile holes able to move around the material. And on the n-type side, we have here the red dots. Those are also free to move around the materials. Those are electron-- negative charged carriers. Omitted from this diagram-- omitted from this diagram is the fixed charge associated with each connectivity type. We have, for example, fixed negative charge from the ionized acceptors and the p-type and fixed positive charge from the ionized donors. And the n-type-- we've omitted it for clarity. We note that an electric field builds up here at the junction between. And that sweeps carriers into one side or another so that if light comes in and generates a pair of charges-- due to charge neutrality, it generates a pair-- a positive and a negative-- the charge carrier type will be swept across the junction and the other will remain inside of the material. And so to be more precise about the exact motion that occurs with the carriers, we have at first a diffusion process until the electric field starts becoming large and then finally drift across that junction. And that's what generates the current inside of a solar cell device. So collection probability means that light-- a light-generated-- a photo-generated minority carrier can readily recombine. But if the carrier reaches the edge of the space charge region, the minority carrier can be swept across and collected. So the probability of collection means for every electron hole pair that I generate at a certain depth, what is the probability that the minority carrier will make it across a junction. And this is what we're probing in spectral response. Yes? AUDIENCE: How long is the exciton diffusion in silicon? PROFESSOR: Great question. The exciton in silicon has a binding energy less than kt, which means at room temperature, the exciton readily dissociates. So the exciton diffusion length can only be measured at lower temperatures. At room temperature, you essentially just discuss minority carriers. And the minority carrier diffusion length-- let's revert it in terms of lifetime and then do the translation to diffusion length. In terms of lifetimes, one typically has a range between a microsecond to a millisecond-- millisecond being on the high end. It can go as high as five milliseconds even. And so if you revert that into a diffusion length, one microsecond lifetime would correspond to a 50-micron diffusion length. And then you can do the rest of the conversion from there just taking the square root sign into account. So it can be rather long on the order of the thickness of the solar cell devices. Now those length skills, which we were talking about hundreds of microns-- maybe even a thousand microns. Compare that now to 10-nanometer diffusion length say for an exciton in a polymer blend material. Then you have to reconsider your device architecture. And how you're going to actually collect them. So this is a very generic picture, which could easily be substituted. Instead of n-type, p-type, it could be easily substituted by say polymer 1 and polymer 2 that just have the right band alignment to separate charge. Then it gets a little more complicated as well. Here we're assuming that the carriers that are swept across the junction can move easily away from the junction. In a polymer, not always so. You might have charge accumulation right at the edge of the space charge region, which creates its own field, which counteracts the built-in field here and also inhibits current flow. So things get a lot more complicated as you venture away from the simple case of say a well-behaved what it was called in this case a homo junction, meaning a p-n junction created from the same material on both sides-- silicon just n-type an p-type. If you have a hetero junction comprised of two different materials, you may have additional effects occurring. But it's always helpful to think about the motion of carriers from the perspective of drift and diffusion. And if you're ever in doubt, you can begin thinking about the process in terms of delta T's-- small increments of time where you imagine light coming in generating an electron hole pair, imagining what happens as that carrier moves to the junction then moves across the junction. And then what happens? Does it immediately get collected? Does it stay there? What happens next? Is there a potential-- an attractive potential between the carriers on either side of that junction. So asking those sorts of questions can help you walk through and troubleshoot what exactly is going wrong with your device. Collection probability-- so we take this diagram that we just on the previous slide-- this one right here. And we say, if light were to come in and generate an electron hole pair right here, then I would expect very high probability of collection, meaning the minority carrier would be swept across. The majority carrier would stay on this side. And presuming that they can get from here to the contacts-- big assumption-- but assuming that they can-- then the probability of collection of those carries will be very, very high. But if my light comes in deep within the device and generates an electron hole pair, the minority carrier will have to cruise through-- you can think about it. It may be a nice, tasty chicken in the Everglades National Park trying to make its way over from that side all the way over to the junction and across. The minority carrier is at risk of recombining. And many of them don't make it. So we have the following graph-- the collection probability versus distance into the device where this yellow region represents the space charge region-- where the probability of collection is very high. And as you move away from the space charge region, the probability of collection decreases. And you can see it decreases exponentially as a function of distance from the space charge region. The reason you have almost-- not exactly symmetric-- these are two different decay constants, but they're both exponential functions on either side-- is because over here on what ostensibly would be the-- what did we call it-- the p-type side-- the electrons would be the minority carriers. And over the n-type side, the holes are the minority carriers. And each one is trying to get to the other side. So the collection probability is actually the collection probability of minority carriers-- the precise definition of which is changing from one side of the junction to the other. On one side of the junction, the electrons are the minority carriers. And on the other side, the holes are the minority carriers. So by probing at different wavelengths of light, we essentially change the optical absorption coefficient of the material, which means that the light will get absorbed at different depths. So we might have a very short wavelength light that probes up here. And then, say light is coming in from this side-- so short wavelengths light gets absorbed near the front surface. And then longer, longer, longer, and longer, and longer wavelengths until we're deep within the material. And we can flush out exactly what the current collection probability would be as a function of depth. Then we essentially take that data knowing the generation rate and the collection probability. We can tease out the minority carrier diffusion length using the spectrophotometer. So let me show you examples of good and bad cells. This would be a bad solar cell and a good solar cell device right over here. The good solar cell has a high internal quantum efficiency out to longer wavelengths. And as we know from the very second lecture of our class, the longer wavelengths get absorbed deeper into the device further from the junction. And so you can see this tail off occurring. Actually, the tail off is somewhere in this region right here for this device is occurring within the bulk. At some point, the device just isn't thick enough to absorb all the light. So there is an inevitable drop toward longer wavelengths. But the bad cell doesn't an even poorer job of collecting these longer wavelengths. And that's in part because the minority carrier diffusion length of that particular cell was much lower or save the minority carrier diffusion length of the material which comprise the cell is much lower than the good cell. So if you come in with a longer wavelength for light-- let's say maybe 933 nanometers in silicon that would give an absorption depth of around 100 microns-- that's pretty far from the junction-- now you would have a much lower probability in the bad cell of collecting than the good cell. And that's precisely because the diffusion length is lower. So integrated over all of the wavelengths, one obtains the short-circuit current. And the method of measurement was the spectrophotometer. Now, I understand most of you opted to choose Sun's Voc for your one characterization tool to apply to your cells. Did anybody choose a spectrophotometer? Show of hands. Anybody? AUDIENCE: It was broken. PROFESSOR: It was broken, yeah, so the filter wheel was down. That was it. So let's see, the filter wheel-- the filter wheel. Let's go fixing this. Here it is. So the filter wheel was broken. So essentially, the polychromatic light source was shining through a filter wheel which was selecting one wavelength of light. The monochrometer, of course, was helping as well and eventually on to the solar cell sample, which was shown in front of the light. So that's the way in principle spectral response works. And one variant of spectral response-- that's for the full device. You typically have an illuminated area of a few millimeters squared or maybe even a few centimeters squared. But let's say that wasn't good enough for you. You knew you had an inhomogeneous material and you want to probe the inhomogeneity across your device. You had a large device maybe about that big-- a few 10s of centimeters squared. And you want to probe the distribution of minority carrier diffusion lengths across your device. How would you do that? Well, in one incarnation, you would use a much more finely-focused beam. Instead of having something on the order of a millimeter, you might shrink the beam spot size down to the range of single microns. And then use an xy stepper motor to scan across your device. And that's precisely what this does right here. This white piece is the xy stage. And this black head right here is essentially comprised of several lasers that will shine on to the sample. Now, you need a few different wavelengths of light to really flesh out this curve to determine the minority carrier diffusion length from this curve. And so the laser light is typically chosen at very auspicious wavelengths to maximize utility in this particular type of evaluation. And so we have usually four-- a minimum three. You can always fit a line through two data points-- so a minimum three laser wavelengths and usually about four or more to process the quantum efficiency as a function of position. And so the xy stage moves the sample around and the laser head-- or moves the laser head around while the sample stays fixed and you map out the current response at each point on the device, obtaining a map that looks much like this right here where this is on the y scale or the z scale, sorry, you have from 0 to 120-micron diffusion length on that particular solar cell device. And points 3, 1, and 2 represent regions in which the minority carrier diffusion with was calculated using the method that we have discussed in the previous lectures-- the Paul Basore method. So we have a map of minority carrier diffusion length across our device. Some regions are underperforming. Other regions are performing rather well. And we can see that in general, these lower-cost materials tend to be fairly inhomogeneous in terms of their performance response. And if you really want to get fancy and say, my goodness, if I spend all this time mapping my device, I'm not going to get through many devices during my Ph.D. I'm going tour have to make do with very poor statistics. How do I speed this up? Some very clever people have figured out that if you fire your laser diode simultaneously but with different frequencies-- each of them with its own frequency-- you can deconvolute the current response of your device using a Fourier transform to pick out the current response at each portion of the frequency space-- in other words, decouple the different wavelengths of light. So it's a little clever method involving Fourier transforms and lock-in method to speed up the measurement a little bit-- small aisde-- small footnote there. Important-- minority carrier diffusion length. That's of utmost importance. So we can relate the diffusion length directly back to our IV curve via the saturation current-- via the J nought or I nought, depending on whether you're looking at density or absolute amount. Let's shift gears a bit. I want to talk about the Voc measurements and fill factor loss measurements. So how do you determine losses in Voc and fill factor? Before I jump into this, does anybody have any questions about Jsc loss mechanisms? So sad that the spectrophotometer wasn't up and running. Sniff. It should be fixed relatively soon, right Joe? It's fixed now? All right. So we have I think David Berney Needleman is back on board, right? AUDIENCE: Rupac is. PROFESSOR: Rupac as well, yeah. So if we have folks who still want to measure their devices using spectrophotometer measurement, we can probably make arrangements for say what, 10 cells to be measured? Something in that range? AUDIENCE: There were only three people who want to do it originally. PROFESSOR: OK, let's fit them in-- the three people who wanted to do them. And we'll have our full data set. Good. So let's describe fill factor and VOC losses. And this is venturing into operating conditions. Under short circuit conditions, how much power is running through the solar cell device? Power defined as current voltage product. AUDIENCE: Zero. PROFESSOR: Zero, why is that? AUDIENCE: No voltage. PROFESSOR: No voltage, right? So you're not testing your solar cell under ideal operating conditions. It's an artificial operating condition simply to probe a bulk property characteristics-- to minimize the effect of the junction. Now, if you really want to see how does is the junction behaving-- how that charge separation going-- you might want to venture into forward bias conditions and eventually into open circuit voltage conditions to really test what the junction quality is instead of just probing bulk properties. So let's talk about some of the measurement techniques to really get to the heart of how are solar cells performing. It's helpful to do the Jsc measurements. It helps you predict what might be some of the loss mechanisms later on. But for this really gets at the meat of it. IV curve measurements-- check. We studied that. We did that in the laboratory. And I think we have a pretty good sense of what's going on there now. Series resistance losses-- we talked earlier in the class about context and sheet resistance losses. Now we'll go back to it again just to revisit so we have the materials necessary to write up the report here on this project. Shunt resistance-- specifically in shunt resistance we'll talk about lock-in thermography and electroluminescence. And finally, we'll have a small slide on Suns-Voc and talk about that as well since I know the majority of you selected that as your measurement tool. All right, so open circuit voltage-- reading straight off the slide here. "If the collected light-generated carriers are not extracted in the solar cell, but instead remain, then a charge separation exists across, meaning there's a buildup of charge. And the charge separation reduces the electric field in the depletion region, which reduces the barrier to diffusion current and causes the diffusion current to flow. In words, if you have light coming into your device, but you're not able to extract the charges from either side to make sure that the chemical potential is equal on both sides, the chemical potential will change on either side. You'll have a buildup of one charge type on one side of the junction and build up of the opposite charge type on the other side of the junction. And so that will shift the Fermi energy on the other side of the junction and result in a bias of your device and eventually counteract the built-in electric field to the point where the diffusion current is now enabled to flow. All this sounding familiar to folks. It might be a little rusty. But it's this all there. So the idea now is to begin probing that junction condition. We talked about the ideal diode equation. And now we're going to piece through it once again. This is your current density here. And I believe if you really want to be precise about this, you should use straight current and not current density-- the reason being if you look at current resistance product, that gives you voltage. But current density resistance product gives you essentially a voltage density, which is not a unit that we typically use. So just keep that in mind as a small, little asterisk. This equation is often used in PV. That little unit conversion issue is generally ignored, but you might want to flag it for your own benefit. So we have the current output of our device as a function of voltage. And this is a rather complicated expression going well beyond the ideal diode equation-- the ideal diode equation, which would consist of the J sub L-- the illumination current. This is the light coming in creating the carriers that are then swept across the junction . The J sub L is going to be the integral current under a spectral response curve at each wavelength weighted for the wave length of the incoming light or the intensity at that particular bandwidth of the incoming light. Figure it this way. You're measuring the spectral response at each particular wavelength. what the collection probability is. Then if you multiply by the total number of photons in that wavelength range, you're determining the total number of carriers generated within that frequency range of light. And if you add up all the little bins, you get all of the carriers generated by all of the light. And you can adjust depending what spectral conditions you have-- say 8.15 spectrum. So the J sub L here-- this one, is our short circuit current effectively and what is derived from a spectral response measurement. The rest of these terms here are of interest. So the J-- let's break down into pieces here. This term over here is essentially to take into account shunt resistance losses. If we have shunts in our device, we're going to be reducing the total power output. These two components-- what are usually called diffusion current and recombination current-- this one, you have to take me for my word at it right now-- this is the recombination within the space charge region. And this over here is a recombination within the bulk of the solar cell. Here's a good way to think about it. If our solar cell is not forward biased-- if it's very, very slightly forward biased-- just a little bit-- there's going to be a large barrier for majority carriers one side the junction to overcome-- to drive a drift current-- sorry, diffusion current-- a very large barrier to overcome. And hence, the recombination is most likely to occur within the space charge region. The carriers aren't likely to overcome that barrier, get into the bulk, and recombine there. Whereas if we are under large forward bias conditions, now the carriers can very easily go over that barrier and recombine in the bulk. So this term right here dominates under low forward-bias conditions. And this term over here dominates under larger forward bias conditions. And that's why you see when you plot your IV curve on log linear scale, you see essentially two flat points on your IV curve. Let me forward that to the next slide right here. So this is log of current versus linear voltage. And you see one flat portion right here and another flat portion right here before the series resistance begins to dominate at too high voltage. And your shunt resistance dominates at too low voltage. This flat portion right there is being driven by recombination within the space charge region. Why? Well, the forward bias voltage isn't enough to really lower that barrier enough to allow the carriers deep within the bulk to recombine. So the carriers are recombining in the space charge region. Whereas here, they are recombining within the bulk. So this is a more complicated expression for the IV characteristics of a solar cell device. Not always are these precise mechanisms at work. If you have a new material system that you're working with, there might be different re combination mechanisms at work. There might be charged accumulation effects at work. But they can all be encapsulated in some form of current voltage expression. Think of the IV characteristics of the solar cell similar to say the constitutive relations of a complex viscoelastic material where you have say a nice, linear elastic component, a dashpot component. So you have a very complex expression describing the stress-strain relationship in the material. A similar thing goes on here with the solar cell where you're describing the current voltage relationship if you understand all of the mechanisms driving carrier recombination in your material and charge separation. You can come up with an expression that describes very precisely the IV characteristics. But if you change material systems, you may not necessarily be able to transition the same models over. It's helpful to start with them. But sooner or later you might decide, well, gee, I have to make some modifications. All right, OK, series resistance-- we already studied this before, but just a quick refresher. We have the series resistance of the bulk. We have the series resistance of the emitter. And then we have the series resistance of the contacts as we contact the device and then the series resistance within the contact itself. So at least four different components of series resistance in our device that are all lumped together when we express our IV curve in this manner. We all lump them together in that R sub S. Question? AUDIENCE: Without going through the math, is there a simple explanation where the 2 comes from? PROFESSOR: Yeah so simple explanation, if you would like, is mostly where the Fermi energy is sitting relative to the conduction or valence band. And the space charge region-- the quasi neutral region is closer to mid gap as opposed to closer to bandage. Hand wavy-- if you really want to get into it, I spent about three months studying this as a graduate student. The 1 and the 2-- the ideality factor, which is to show more precisely. It's the number that comes before the kT for each of these. That's called the ideality factor. It typically expresses N sub 1 or N sub 2. In this case I just substituted straight out for a 1 and a 2. Those numbers aren't always 1 and 2. As a matter of fact, especially the G nought 1-- the bulk recombination current, depending on where the defect levels lie, it can be somewhere between say 1.1 and 1.4 typically. And beautiful thesis work done by a fellow in Australia hold on-- Keith McIntosh And the title of the thesis is called "Humps, Bumps, and Lumps." It's all about non idealities within an IV curve. So if you're really curious about that, "Humps, Bumps, and Lumps," by McIntosh I believe was the name. So we talked about series resistance already. Hold that-- put that in your RAM. That's our R sub S right over here. That's our R sub S. And our shunt-- you want to hold this other component in your RAM. So that's the picture to have of our series. This is our shunt. This is a p-n junction. This is your n plus region. And this is your p. And this is looking at the electron energy in 3D. So we typically take a slice out like that just like this. And we draw the electron energy as a function of position in a nice little wavy form that we typically do to describe the p-n junction. What's being described right here is adding a leader dimension-- a spatial dimension in what's shown as the abscissa in this plot and showing what is a weakness in the p-n junction. It could be a localized defect. This could be a shard of metal that happened to lie in the wrong place in your device and fire through, essentially when you did the firing step to adhere the metal to create the contact. It could have gone straight through the p-n junction and contacted the other side. It could be a charged dislocation. It could be structural defect, but a local reduction in the barrier height. Now where is the electron going to go if you have a diffusion current? It's going to be crowding through that little, localized, reduction in the energy barrier. And you're going to have a higher current flowing through the specific region. So how to measure that? Well, if I make sure my full IV curve like this, it might manifest itself as some reduced shunt resistance. So you'll have a larger current flowing through it at lower bias voltages. But how do I know what's going on? Without some spatial-- a spatially-resolved measurement tool, I am in the dark-- no pun intended. So I have to figure out how to measure the current flow as a function of position. Now somebody thought about this a while. And they said, ha, well, let me get this straight. If I apply a known bias voltage to my device, I'm going to get some current flowing through it. And that current-- what I read out of the whole device microscopically has some spatial distribution to it. Let me think about this. OK, so current times voltage is power. And power generates heat. So if I have a current crowding at some portion of my device, I'm going to be generating a large amount of heat as the carriers flow through that specific region. So if I have some method of measuring the heat-- the heat distribution across my device is very, very sensitive-- I might be able to divide that heat measurement-- if calibrated properly to determine total power-- I might be able to divide that heat measurement by voltage so I know the applied bias voltage to my device. I take my power, divide it by the voltage, and I get the current as a function of xy position. So I can map out my dark IV curve at any arbitrary bias voltage condition. Let's say I want to do the measurement at 0.4 volts forward bias. I apply 0.4 volts across my device. And I just measure the heat distribution. And in the calibrated measurement, you can see how the dark current is distributed-- how the diffusion current is distributed. And that's what lock-in thermography is all about. This is a somewhat typical image. It's still very noisy because it was a very fast image. But you have here a solar cell device. You can barely make out the contact grid right here. And then the fingers are moving down separated by about this much. On this particular image, you can see the fingers or the dark shadow of the fingers right here. And you see bright spots 1, 2, 3, 4, 5, 6, 7 across the device, indicating regions where more current is flowing. Aha! So I'm beginning to figure out that, well, this dark IV curve that I have right here-- this large amount of current flowing through my device at lower bias voltages. That's not all uniformly distributed. But it's rather concentrated in specific areas where I have weaknesses in my p-n junction under low forward bias voltages. And then at larger forward bias voltages once this barrier is significantly reduced, I may have more carriers going through in regions of smaller minority carrier diffusion length. Why? Well, because if the carrier goes in and recombines quickly because the diffusion length is very short, that means there's one less carrier in there, which means that now there's a greater driving force for diffusion. And another care is going to come in and recombine too and another carrier and recombine too. And so now you have a larger diffusion current going into the regions of lower minority carrier diffusion length. Let's look at a few lock-in thermography images and see how this plays out. So under low-bias conditions-- so I'm still-- still have a very large barrier there in the p-n junction region. I'm typically going to see isolated hot spots-- not always, but typically isolated hot spots. And these most often are defects within the p-n junction. Somebody must have scratched the junction or maybe dropped their tweezer on it or maybe a piece of metal during the contact metalization fired through at that particular region. And so we have what are typically these point shunts. Sometimes you see shunts around the edge as well if you didn't isolate the edge well. And now as we forward bias, forward bias, forward bias, those barriers become less important. And now what's driving the current-- the dark current in the device is a recombination of the bulk. So the regions of higher recombination will be regions of higher current flow. They'll drive more recombination current. So this is a higher bias voltage-- 560 millivolts or 0.56 volts as opposed to 0.36 volts. We can still see the shadow of the big three right here. But they're much diminished. Now instead what we see are these wispier features right here. Now I think gee, if that really is-- if those really are regions of lower minority carrier diffusion length-- I just learned about a methods to measure minority carrier diffusion length, didn't I? What was that? Laser beam induced current method? Spatially-resolved laser beam in this current? So I have at least one method of measuring minority carrier diffusion length. Let's put one and one together and see what we get. So we'll take this image right here and put it aside, rotate it, and put it right beside the minority carrier diffusion length. And voila, you see how those wispier features are corresponding to regions of lower minority carrier diffusion length-- once again, the explanation. If you have large forward bias condition, the barrier is lower. Carriers can easily diffuse into the bulk. The carriers defusing into the bulk will go a certain distance before recombining. If the diffusion length is short, they'll recombine quickly, which means now there's no more carrier-- which means the diffusion current will push another carrier into that spot, which means they'll recombine. More current is flowing through if the diffusion length is shorter. The alternative is if the diffusion length is long and one carrier makes it over and then takes its time getting all the way across before recombination occurs. So what you're seeing here is essentially an effect-- the current the dark, forward current, meaning you're measuring the IV curve in the dark. You're obtaining an IV curve. Here it is. You're obtaining an IV curve for your device. You're measuring in the dark under larger forward bias conditions somewhere around here where bulk recombination is dominating. And you're able to visualize the current loss mechanisms in your device. So you say, well, gee if I want in the dark-- if I want this to be as small as possible so that when I transpose this in the light and I shift everything into negative-- into fourth quadrant territory because I illuminate it now-- if I want there to be a small, dark forward current as possible so as I maximize my fill factor when I illuminate my device, I want all of these current loss mechanisms to go away. Let me repeat that one more time so people get it. So what I'm doing right now is I'm measuring this device in the dark. And this is my IV curve right here. And I have an example of bad device-- good device. The bad device has-- so bad and good. The reason this is bad and good is because when I transpose this curve under illumination-- this is now under illumination-- my good and bad right here-- you'll see that the bad has a lower Voc. So the intersection between these illuminated curves-- this is now illuminated. And this is dark. Right over here-- these are dark curves. These here are illuminated curves. The intersection between the illuminated curve and the x-axis denotes the open circuit voltage. And you see that the larger this current is in the dark the earlier you're going to intersect with your x-axis-- the lower the voltage output of your device will be-- the lower the maximum power point will be. So want in the dark when you measure the IV curve, you want that current to be as small as possible in the dark. And, obviously, when you illuminate it, you want this jump to be as large as possible. And you want this to almost look like a box to have a large fill factor. So if we say, OK, we want this to be small because we want a large fill factor, somehow we have to know where the current loss mechanisms are occurring. And that's where we have lock-in thermography available to us we can visualize it. You say, well, to do lock-in thermography well, I'm going to need an-- indium antimonide-- sorry, yeah that would be it. It would be an indium antimonide camera sensitive to the 3 to 5 micron wavelength range that might cost $70,000 to get a good one with high frame rates. The lock-in system-- I don't know if I have that money. I can go over and borrow Professor Buonassisi's system and use it there. Or if my shunts are big enough, I can just use straight-out thermionic sheets or thermochromic sheets, rather, sorry. So you can take these liquid crystals and slap them because your device and measure the heat distribution straight up just by making sure you have good thermal contact between your cell and your thermochromic sheets, which cost about $100. Now the reason we use the lock-in thermogrpahy and not these thermochromic sheets is this curve right here. This is lock-in thermography sensitivity versus integration time. And we're getting into the 10 microKelvin range, which is rattlesnake territory. That's how the rattlesnakes are able to sense heat and reach out to you is because they have those little organs that look like little, black-- they're integrating spheres to be honest-- small, little spot to open up. Rattlesnake-- that's why the sidewinder missile. Anybody? F 16s? Hornets? No. If you're into aviation, there's a special type of heat seeking missile developed I think in the 1950s that uses a device not dissimilar from the rattlesnakes heat sensing organs. Regardless, I digress, we're looking at about a 10 microKelvin sensitivity in this lock-in thermography technique, which is very sensitive-- can measure under low forward bias conditions and hence useful. So that pretty much sums up lock-in thermography. It is a difficult technique to wrap your brain around. So for those of you who got 75% of it, congratulations. That was really, really well done. For those who got about 25% of my explanations today, don't worry. You're not alone. It just takes a lot of thinking it through exactly where is the current flowing as a function of bias condition? What does that do to my IV curve? And then what does it do in a two-dimensional regime when I'm measuring heat output? It's is a very complicated thing. Thankfully there is an entire book lock-in thermography written by Otwin Breitenstein. I gave to you today a paper-- in one of the papers that I distributed today-- the one that has the PSSA written on it-- this one. That paper is by Otwin Breitenstein from the Max Planck Institute of Microstructure Physics in Halle, Germany about two a half and a half hours south of Berlin. And this describes in more detail that which I attempted to get across in class today. He with Martin Langenkamp have also written a book on lock-in thermography. So if you're interested in more information, that's where to go. In your handout set today, we also have a classic paper on series resistance effects of solar cell measurements. This one is from '63-- yes, 1963. Classic paper worth the read if you're in that regime making devices that are series resistance limited. It's more optional reading for folks. And let's venture forward. Folks who did these scanning Elbit techniques for a while really found value in measuring as a function of spatially-resolved manner down to a micron in spatial resolution the electrical output of a solar cell device. And then they thought to themselves, well, goodness, if we measure with one micron spot size, it's going to take us a week to get a measurement across a full area solar device, if not longer. And if we measure say with the 200 nanometer spot size using some sort of X-ray technique and we accumulate for say two seconds a point, it can take 10,000 years to map out a full-size solar cell device, literally, do the math. And they said is there an easier way for us to acquire information that does not rely on scanning-- on point by point, step by step measurements. And they thought about it long and hard they said, well, gee, these new, handheld, digital cameras coming out on the market are pretty cool. What technology do they rely on? Oh, it's a CCD-- Charge Couple Display that visualizes or images over a large area the output of a device. And that's exactly how the lock-in thermography techniques were developed here with this imaging camera-- in that case, an indium antimonice CCD. And they said, well, what other wavelengths emit-- in what other wavelength does a solar cell emit? We know that it emits heat. And heat is typically in the 3 to 5 micron wavelengths regime, maybe even as high as 10 microns depending on what temperature. And we know that as well we can have band to band emission inside of a solar cell that would emit around the band gap energy. So let's set up a camera to image the band-to-band recombination intensity or the band-to-band illumination intensity. And that's what this paper is about. This is wavelength. This is intensity. This is the emission spectrum of a solar cell. So if you were to measure what colors-- what color light does a solar cell emit after you pump current into it, this isn't is more or less the spectrum. So we have the thermal radiation shown over here. This is longer wavelength light. That's the heat that it gives off. That's what indicates current flow inside of the device. There is the band to band luminescence right here. And we'll neglect the other two for now. Those are more advanced concepts. But the band to band luminescence is essentially just the recombination of a carrier from the conduction band to the valence band-- recombination-- very straightforward. And the recombination intensity is inversely proportional to the minority carrier lifetime. Why is that? This was that quote remember who attended Eli Yablonovich's talk. He was saying, it's completely counter intuitive. Do you remember specifically what he was referring to in this? Let me show you an image here. AUDIENCE: That you want your device to emit more light for higher device performance. PROFESSOR: Exactly, yeah, so what is going on here? We have our tau-- one of our tau effective-- or one of over lifetime what we're measuring. This is our effective lifetime that we're measuring. And we have one over tau. Let's call it non-radiative. So we have non-radiative recombination mechanisms. This comprises Shockley-Read-Hall recombination-- Auger recombination-- a bunch of different recombination mechanisms. And then we have one of our tau radiative, which is essentially this recombination mechanism shown right here, which is just the band-to-band recombination. So if you have defects around, that will lower the lifetime-- the non-radioactive lifetime. If you have tons and tons of defects, the carriers will most likely recombine through the defects and not through the band-to-band They will only recombine with band-to-band if there is nothing-- no other recombination path left available to them. It's a question of probability. So if this is really, really short-- non-radiative recombination lifetime is really, really short because a defect density is very, very large. So the lifetime due to non-radiative recombination mechanisms is very short, then that's tau effective is also going to be very short. And very few carriers are going to recombine radiatively. Conversely, if we have a very clean material-- virtually defect-free-- this non-radiative lifetime is then going to be out the roof. And now we're going to be limited instead by the radiative lifetime. And so tau effective will be more similar to radiative. Now an interesting thing happens when you have a radiative recombination event. As the name would suggest, it emits light. And that light can be detected by a CCD camera on your body. So what you have-- there you go. What you have right here is a solar cell device that is being biased so current is flowing into the device. And now the map what you're seeing is indicative of radiative recombination, why? Because nearby the cell, it's essentially a very similar mechanism or similar set up to what we saw right here where we had bias voltage across the device and imaging not with an infrared camera, but now with a camera matched to the band gap energy. This could be an indium gallium arsenide camera or say silicon camera, depending on the precise band gap. And you're imaging the radiative recombination as a function of xy position across your wafer. So these darker regions here on the bottom and on the right-- sorry left-- the other right. They indicate regions of lower minority carrier diffusion length. Why? Because there's less radiative recombination because this term in those regions is very, very small so that there's not much radiative recombination going on. Whereas these other regions here that appear brighter have a lot more radiative recombination going on. Any questions about that? Yeah? AUDIENCE: What fraction of these radiative recombinations are reabsorbed by the material? Is it possible to figure that out to a high degree so you can get a quantitative sense of how much is radiating? PROFESSOR: Sure. The question pertains to photon recycling and how many photons that are radiatively emitted get reabsorbed by the material. That would depend on the optical path length inside of the material of carriers being generated in the material. And you assume that carrier generation is fairly isotropic in nature-- that there is not any preferred direction unless you have a dyed molecule that's designed in a certain way to emit light in a certain orientation. So you can assume that the carrier mission in most materials as isotropic in nature. And then it's just a question of ray tracing to figure out what the optical path length is. Compare that to the optical absorption coefficient in the material, and you get your answer-- what fraction of light gets reabsorbed. All right, I can sense where we're reaching threshold here in terms of new information gathered. I'm just going to briefly go over the Suns-Voc at the very end . And the Suns-Voc technique is useful as we saw during class because as this paper right here by Ron Sinton and Andres Cuevas-- this one-- I know you definitely have this one. This paper essentially describes the functioning of the technique. On page number two on the upper, left-hand side, so figure 2, we essentially have the figure that was determined on the screen during the measurements. So for those of you close enough to the monitor to actually watch the measurement being taken in the Suns-Voc, you saw the light intensity decaying exponentially. So the light intensity is a flash bulb, poof, and decayed exponentially. The open circuit voltage measured by the device-- so you had this solar cell device sitting on the platen with a little probe coming in and a 50 megaohm resistor connected in series-- very little current flowing through that external circuit. But it's measuring the voltage. So it's essentially maintaining the solar cell in open circuit voltage conditions while the flash is decaying. And it's measuring the open circuit voltage across the device at each illumination intensity as the pulse decays. So you can see the time scale here is on the order of a few 10s of milliseconds-- so a very fast measurement. The electronics response time has to be very fast. It tells you what the RC time constant of the circuit is. And you can notice that the voltage-- the open circuit voltage decays linearly when the illumination intensity decays exponentially. Why is that ? AUDIENCE: Natural log. PROFESSOR: This is a natural log-- so what Joe showed to you yesterday or on Thursday in class is how the voltage output of a solar cell varies by illumination intensity. So there's a logarithmic dependence there. So that's why you take the log of the exponentially decaying light intensity and you obtain a linear relation that's the open circuit voltage decline as a function of time. So we can derive an IV characteristic by translating that open circuit voltage into an implied bias voltage. And knowing the short circuit current density up front essentially by measuring it by measuring the solar cell device in a solar simulator, we can obtain the short circuit current density. We pin both curves up here at the measured short circuit current density. And then we plot both curves out-- the IV curve and the Suns-Voc curve. Now the Suns-Voc curve is similar to the IV curve with only one difference. There's no current or very, very little current flowing through the external circuit, which means there isn't any real power flowing through the external circuit, which means if we go back to our expression right here-- if we have no current flowing through, that J is equal to zero. And now that R sub S term disappears. So we've come up with a clever way to drop all of our R sub S out of that expression. And the only thing that matters essentially is to some degree shunt. But this is essentially the highest IV curve that one could possibly obtain in the absence of series resistance. And that's what you see right here. That's this delta in between your illuminated IV curve-- what is measured using the solar simulator-- and your Suns-Voc curve, which is measured in the absence of series resistance. Now this works well with most materials. If you have a very high capacitance in your device, you'll want to look very carefully at this decay-- at the light intensity decay. If your capacitance and your device is very large, it might actually flatten out the voltage response. So you want to make sure that the time-- the RC time constant of your device is matched to the flash bulb decay time. That's just one word of warning for those working on high voltage or organic materials. Any questions on the Sun-Voc measurement? This gives you what's called an implied IV curve. And that is-- we could say in the best case scenario what you would get from the solar cell in the absence of any series resistance losses. AUDIENCE: The last question-- Voc-- that's not a solar simulator in any way is it? PROFESSOR: No. AUDIENCE: Is that important? PROFESSOR: Yeah, it is important. It would be different-- in essence, it would be a different current response if you didn't have the same exact solar spectrum. Now, that's a very important point. If you're venturing outside of, say, silicon-based devices, and you're more sensitive to the infrared or the UV, you really want to make sure that you measure the intensity as a function of wavelength output of that bulb. Otherwise, you might be higher or lower during the Suns-Voc measurement. For a silicon-based device, the majority falls within the specified regime. Down here-- actually right here on this diagram-- that a little spot right there-- that is a small calibration solar cell of known short-circuit current and open circuit voltage. That's used to calibrate for other silicon devices if you're moving away from silicon, that becomes an issue. Correctable-- you can make sure the spectral response-- or the spectral radiance of that light source and compare it to the solar simulator and do the normalization accordingly. Here's what I recommend. With the last 15 minutes of class, I wanted to catch each of the teams and talk about your class projects just to make sure that things were rolling along. I have feelers out to some of the mentors who haven't responded yet. And so I just wanted to touch base with each of you for about three minutes just make sure everything is rolling along. |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | Tutorial_Solar_Cell_Operation.txt | [MUSIC PLAYING] PROFESSOR: Hello everyone, today we're going to learn how a Solar Cell is able to turn light generated mobile charges into electricity. Today's lesson will use everything we've learned in the past videos to understand this effect. So, make sure you understand the material from the previous videos before watching. First, let's go over the structure of a Solar Cell. Here's a cell that I made. And we can see that a metal ribbon is connected to the top metal contacts, which form a grid. The spaces between the grid lines allow light to enter the cell. If we flip over the cell we see the entire back surface is coated with metal, which allows easy extraction of charge from the back surface. Additionally, we have another metal ribbon that's connected to the backside. Now, let's hook up our Solar Cell to an ammeter to measure the current. So, here we have an ammeter connected to our Solar Cell and our light source which will simulate the sun. And we can see that if we turn on our light source we start to read a current flowing out of our Solar Cell. In this case about 0.12 amps or 120 milliamps. Now, if we turn off the light the output of the cell drops to zero and we no longer read any current. We know from our last demo the light generates mobile charges and silicon. But how do these mobile charges become a electric current coming out of our Solar Cell? The secret has to do with doping. The top layer is doped with phosphorus, shown in blue. Well the bottom layer is doped in boron, shown in red. The different dopants interact in a way, which we'll describe shortly, to create an electric field in our device where the Boron-doped and Phosphorous-doped regions meet. It is this electric field that acts as a one way valve in our Solar Cell for electrons and holes. An electric field is created when positive and negative charges are separated. As you know, opposite charges attract and like charges repel. We'll exploit this property by creating a sheet of positive charges on the left and negative charges on the right, thus producing an electric field, which we denote with the Greek letter z. If you were to insert a negatively charged particle, such as an electron, into this field it would move toward the positive charges. Alternatively, if you put a positively charged particle it would move toward the negative charges. We're able to create an electric field inside our Solar Cell by using different dopants on either side of the device. Here we have our silicon lattice, which is un-doped. We'll start by replacing some of the silicon atoms with phosphorus atoms on one side. On the opposite side we'll put in boron atoms. To focus on our dopant atoms and the mobile charges they introduce we'll fade out the silicon lattice. Recall that phosphorus atoms introduce static positive charges in mobile negative charges. While boron atoms introduce static negative charges in mobile positive charges. All the mobile charges are free to move around at random. A process known as diffusion. Here we see a single electron moving around on its random walk. During this random motion if an electron and hole encounter each other they neutralize and effectively vanish. As this process of holes and electrons randomly defusing and neutralizing as the interface continues, the total number of mobile charges in the device decreases. This leaves a region at the interface of immobile static charges where the net charge is negative on one side and positive on the other. These opposing sheets of charge create an electric field of the interface, which at this point is very weak. As charges continue to diffuse, they're still able to move across this weak electric field and neutralize. As this happens, the sheets of net positive and net negative static charges widen and the electric field grows in strength. Now that the electric field is stronger, as other mobile charges continue to move and diffuse around the lattice, they're now repelled by the field and electrons to the left and holes stay to the right. It is this electric field that separates light generated mobile charges and pushes them to the extreme ends of the device. The image we see now is our Solar Cell in the dark. However, recall that our silicon atoms are still present. And if light strikes our silicon atom, a mobile hole and electron is generated. As these mobile charges move around randomly there's a chance that they will randomly encounter the electric field. The mobile electron will get repelled by the electric field. However, the mobile hole will get swept to the other side by the electric field. Now, let's zoom out. We can see that after the electric field has pushed our light excited electron and hole to the left and right respectively, we now have an extra negative charge on the left and extra positive charge on the right. If we connect a wire to short the two opposite sides together the excess electrons are attracted to the excess holes on the opposite side. This attraction is what drives electricity through our wire. As light continually shines on the Solar Cell charges are constantly being pushed out of the device and driving the electric current. Now hopefully you understand the basics of how these amazing, but rather simple, devices work. We hope that this knowledge will provide the basic foundation while tackling more difficult and abstract concepts while you learn the material in this course. I'm Joe Sullivan, thanks for watching. |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | 8_Toward_a_1D_Device_Model_Part_II_Material_Fundamentals.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: All right, so let's get started. So today-- our last lecture we talked about different device parameters, mainly our series resistance and shunt resistance, and how that affects our efficiencies. Today we're going to talk a lot about different material properties and how they affect certain device characteristics, and mainly just affect our output efficiency. So we've been talking a lot about the fundamentals. I'm sure you guys are loving this right now. So we're going to complete this in probably the next three lectures and then move on to a lot of the kind of cross cutting themes in PV-- some advanced concepts, different device architectures, and that kind of thing. And so that is coming in the future, but we're still doing fundamentals today. And so I know you're probably all aware of this equation. And again, this is kind the progress we've made, so we're almost all the way through explaining basic device physics and basic semiconductor physics so you can understand simple devices. And it's always important to remember that your device is like a leaky bucket, and you're limited by the largest hole in that bucket. So the weakest aspect of your solar cell is really what's going to limit your device performance, and especially if any of you here are trying to make devices, it's really important to think about all of these things when building it. And this is very difficult to do. I can certainly tell you that. So I kind of like this slide. What this is saying what's the thing we care about most in our solar cells? Well, as scientists, other than dollars per watt, we want to maximize our efficiency for a certain price. And our efficiency, there's several parameters that go into it. Again, we talked about our GSC, our short circuit current, our open circuit voltage, and our fill factor. And that gives our output energy, or output power, and we divide that by the input power, which is the solar insulation. Now we can split up that again into open circuit voltage, short circuit current, and fill factor. We talked a little bit last lecture about fill factor and how that's influenced by different resistive losses in our solar cell. Today we're going to mainly be focused on short circuit current and things like internal quantum efficiency, which are highly affected by our diffusion length. And the diffusion length is often limited by certain defects in our materials, and we're going to get into why that is. And to a certain extent, we'll talk about open circuit voltage, because your GSC really has a large effect on your open circuit voltage. So what we're going to learn today is what is minority carrier diffusion length. It was in the homework. Hopefully you guys have some idea coming to lecture, but today we're going to talk about it a little more in depth, and why it's important, and how it's affected. What are the parameters of determinants? So mainly diffusivity and lifetime. We're going to describe how it's actually measured in a solar cell, which is actually a really cool measurement, and we actually have the capabilities in our lab. And possibly some of you, when you're making cells, will be able to do that measurement. We're also going to look at some of the things that limit lifetime, some of the basic recombination mechanisms. Also look at how your excess carrier concentration changes as a function of lifetime and generationally. And then also talk about the last material parameter, which is mobility, which discusses of how well these excited charges can move around in your material. So without further ado, here are minority curve, diffusion length. The definition is really simple. If you generate-- let's say, a photon comes in and hits a silicon atom and generates an electron pair over here. How far or how much volume can it explore? And that volume it can explore is described by some characteristic radius, and that radius is known as the diffusion length. And it's really important to solar cells, because when you think about these carriers that you're generating. If they can only explore a very short area, they're not going to make it. This is a very good solar cell, so the diffusion length is really long, and all these carriers that are just generated there will be able to make our junction. So again, just so we're familiar, this is our base. In the top, we have our emitter. So the junction would be at this line right here on that plane. If we have a really bad solar cell-- so let's say a lot of defects present, a lot of areas for these excited carriers to recombine-- they won't make it to the junction. They'll have a very short diffusion length, and as a result, your short circuit current and your VOC will suffer dramatically. So it's really important to get good crystal quality and good material quality, but up to a certain point. There's kind of diminishing returns as you go to higher and higher qualities, so we'll talk about that in a second. So if we assume-- what this is showing is how our short circuit current scales with our diffusion length. So we have something called the generation rate, and this is often proportional to the photon flux on your material. So the number of photons hitting your solar cell. And this generation rate is something-- the number of carriers produced per second in some given volume. So it's a volumetric term. And if we assume that everything within one diffusion length of our junction gets collected, that'll all be counted as short circuit current. So basically your JSC has this kind of linear dependence on your diffusion length, but that's only true up to a certain point. So for example, if we have a diffusion length that is much longer than the device thickness. It's really not going to be-- you get, again, diminishing returns as you go to longer and longer diffusion lengths. So this is a calculation I did using a 1D simulation program called PC1D, which if you want to play around with, it's free. It's a lot of fun to use, actually, and you can put in things like lifetimes-- and very basically, the lifetime, which we'll get to in a second-- how that changes the diffusion length. And you can see that there actually is a linear relation until the diffusion length is about on the order of 300 microns, which is the device thickness. So you can see this kind of trailing often and become sub linear in its response. Yeah? AUDIENCE: Can you clarify why when the minority carrier flux at the edge of the space charge region matters, because I'm thinking about there's the back contact and the front contact, and there's a junction right near the front project. They're not [INAUDIBLE] connecting. PROFESSOR: So let's think about it one step back. Why do we care about minority carriers? AUDIENCE: Because those are the ones that are actually generating the current. PROFESSOR: Right, so if, let's say, you generate an electron hole pair and n-type material, the hole wants to move to the p-type side. And the electric field will actually repel and keep the electron on the n-type side. So it's your minority carriers the matter in terms of the separation, and we'll talk about that again if that's still fuzzy in people's heads. And so what matters in terms of-- you're talking about deriving the ideal diode equation? AUDIENCE: Well, no. [INAUDIBLE]. So it seems like the important thing is that some carrier gets to the metallization? PROFESSOR: Well, yeah, but in order to be separated, which is the first part that we care about, it matters that it's reached the junction. And so it's that concentration at the junction that determines the flux across the junction. Does that make sense? AUDIENCE: Yeah. PROFESSOR: And so there's also this other loose dependence on VOC on your diffusion length. And so if you recall from a few lectures ago, this is your equation for VOC. It's dependent on your short circuit current, temperature, and your saturation current, which you can often think also think of as your reverse bias current. And your saturation current is dependent on your diffusion length. JSC, if you recall, was linearly proportional to the diffusion length, so the VOC actually scales with the natural log of the square of the diffusion length. And if you pull out that exponent, it just squares with the natural log of your diffusion length. And again, very, very simple analogy-- we're assuming, again, that fill factor is not really affected by your diffusion length and that your efficiency is proportional to the product of your JSC and your VOC, so your short circuit current and your open circuit voltage. And you can see as go to longer diffusion lengths, there's this area of diminishing return. And again, there's two different regimes here. One is when your diffusion length is, again, much less than your device thickness. You can see that there's a huge increase in efficiency, but as your diffusion length gets well, well above your device thickness, it becomes less and less important. So how do you deal with this? So if you have, let's say-- suppose you want to make a really cheap solar cell, and you have a very dirty material with very short lifetime shown here. There's several ideas of what you can do. You can have a very, very thin device. Now, that's a problem if you can't absorb it very well. So you can have some optical tricks. You can do surface texturing. There's all sorts of other ideas for basically having the carriers coming in at angles or having a good reflector on the back. So that's ways you can work with very low diffusion length materials and still get a device efficiency that's not too bad. Now we're going to talk quickly about how our minority carrier diffusion length is measured, and again, this is something we're able to do in our lab. And one of the important things to say is defined is collection probability. So for example, if this is our junction right here-- so our space charge region-- if you generate a carrier, let's say very, very close to the junction, it's going to have a high or near unity probability of making it to the junction and being collected. If it's generated very, very, very far away, it's going to have a lower probability. And that's what this graph is trying to show-- is that right near the junction you have near unity collection probability, and as you go way, it comes down. And the different colored lines are showing different ways of increasing that diffusion length. Service passivation is important. We'll talk about what that is soon. And also diffusion lengths-- if you're limited by diffusion length, you can see this very, very sharp drop off. AUDIENCE: Just a follow up. So is collection meaning collected by the front contact to be used in your external circuit? PROFESSOR: Mm-hm. Right here it's technically defined as reaching the junction and being separated, which, in essence, will hopefully be the same thing. AUDIENCE: So is surface passivation [INAUDIBLE] sort of like diffusion? PROFESSOR: It can affect the diffusion length, and we'll-- or the affect of diffusion length. We'll to that in a second. It'll be a bit later. This is kind of just showing that there's a lot of different material parameters-- not just your diffusion length-- that can really affect your device performance and your collection probability. So if you recall a few slides back, we said that our short circuit current is directly proportional to our diffusion length. And the reason is that, if you're generating carriers within the diffusion length, you generally think of that as the region of which you're actually collecting those carriers. And if you go back to the next slide, that kind of makes sense. So this is highlighted. So this is our junction here. We have our n-type material on top, our p-type below. And within a diffusion length of your minority carrier-- so on your p-type type side you care about the diffusion length of your electrons. On the n-type side, you care about the diffusion length of your holes, and that's the region where you're really collecting carriers. Of course, there's a tail off, but the first order approximation-- this is actually a very good one. And so now putting those all together. If we know how our carriers are generated as a function of x, and we what the collection probability is as a function of x based on diffusion length, surface passivation, and other parameters, we can multiply them and then integrate to actually get what our illuminated current will be. Yeah? AUDIENCE: [INAUDIBLE] have like three regimes because the diffusivities are different at each [INAUDIBLE]. PROFESSOR: The emitter? AUDIENCE: [INAUDIBLE] the junction afterwards? PROFESSOR: Mm-hm. For most models you pretty much assume that anything absorbed-- first of all, the space charge region is very, very narrow. It's almost assumed just to be negligibly thin. And the emitter is generally very, very short, and the diffusion lengths are so poor in the emitter for reasons we'll get to soon that it's almost a dead layer. Your response, the very short wavelengths, where you're absorbing most of your light and emitter generally don't add to your short circuit current, and we'll talk about that soon. So excellent question. Yeah? AUDIENCE: Is that first region right there-- PROFESSOR: Wait, sorry. Say that again. I couldn't-- AUDIENCE: People just approximate the first region there and not care about [INAUDIBLE] afterwards? PROFESSOR: So for certain wavelengths of light, let's say, that are-- this curve, this is kind of representing Beer Lambert's law. If it attenuates less drastically, that characteristic absorption length, if it's a lot longer than this length here, then yes. You can assume that this emitter region is negligibly thin, and that's a little bit what the homework goes over as well in problem three. Excellent question. And so what we care about is the spectral response of our short circuit current. So what this is is this a quantum efficiency tool. This is something we have in our lab. It's a really fun tool to play around with. How it works is you basically have-- we have a light source that's white light, and it goes through a series of filters. There's a monochromator, which basically-- it diffracts the light, so it spatially separates the light, kind of like a rainbow. And then you can focus that onto your solar cell and measure the current output under short circuit conditions. And that will tell you-- because you know the carrier generation profile for different wavelengths of light, because you know what alpha is for silicon, you can actually pretty well calculate what your diffusion length is, and we'll talk about on the next slide. So this is kind of what your quantum efficiency will look like. This is, I think-- I actually don't know if this internal or external, but they're just related by a factor of 1 over r. So here you're blue response-- so everything that's absorbed right in the near surface region in your emitter generally doesn't get collected, and this is due to really bad diffusion lengths in the emitter region. As you go to longer wavelengths, the absorption length is much, much deeper. You're collecting a lot of it, and then what do you think is happening here? Generally these longer wavelengths-- your alpha is so low that you're not actually absorbing much of this light, which is part of the reason that you're not collecting it, or that you're absorbing it so far away from the junction that it's not being able to diffuse there. And so that's kind of how you can look at these quantum efficiency curves, and it's this region here that's really limited by diffusion length. And again, you have a homework problem discussing how that's done. Some of the cool other tools-- the lab can kind of do this, but it requires a little more-- I don't know-- hand work on the operator. But if you can take different EQE or I IQE curves at different points on your solar cell and you get those spectrums, you'll get a whole bunch of different spectrums. You'll scan with your light beam. You can actually get diffusion lengths as a function of position and get a spatial map of your different diffusion lengths. And it's really helpful if you're trying to fine spatial inhomogeneities in your cell. So I believe that this is some multi-crystalline cell, and see all sorts of grain boundaries. And those are areas of very short diffusion length, and we'll talk about why that's the case, actually, on the next couple slides. Question? Yeah? AUDIENCE: Could you just clarify the difference between diffusion length, absorption length, and band depth energy? Because I thought that the middle weight ones were more absorbed because they were bent? Their higher weight [INAUDIBLE] for bigger-- to cross bigger band depth energies. I didn't think it was because of the absorption [INAUDIBLE]. PROFESSOR: So alpha is what tells you is your absorption coefficient, and it's often in units of one over centimeters, so inverse centimeters. And so if you plot what that looks like, it's an exponential function with x. And the point to which it is attenuated by a factor of one over e, that point is 1 over alpha. And that's often called the absorption depth or-- what am I thinking of? Absorption length of that wavelength. And alpha will vary as a function of wavelength. If you recall that, if you look at lambda, for silicon, it's something that continues to go down at longer wavelengths. So the short wavelengths are absorbed very strongly, and so most of the light is absorbed very, very close to the surface, where the longer wavelengths-- most of the light is actually absorbed rather far from the surface. Does that make sense? And so there's a difference between your absorption length and your diffusion length, and that ratio is what's really important. If you're absorbing really far away from the junction, but you have a long diffusion length, there's a greater chance of it making there. And it's that ratio that's really, really important. Does that answer your question? AUDIENCE: Yeah. PROFESSOR: OK. So what limits this minority carrier diffusion length? We're going to get to the equation in a second for the minority carrier diffusion length, but basically when you excite an electron hole pair, you have this mobile electron, and it's in this excited state. You've given it this energy from a photon, and now it can move around, and it can only exist for a certain amount of time before it finds another whole and recombines. And that event, again, is called recombination. And a lot of this is actually determined by the size of grains in your material. If you've seen-- I think on the cell that [INAUDIBLE] brought in earlier-- sorry I don't have a good example-- it didn't just look like one kind of flat plane. You could see different grain orientations, and the edges of those grains are called grain boundaries, and those can act as recombination centers and actually reduce your-- it's called your lifetime, which we're going to get to on this slide. So this slide has a lot of stuff going on. What it's saying is that your diffusion length is characteristic of the diffusivity-- the square root of the product of your diffusivity in your lifetime. The way I like to think of diffusivity is that it goes up with temperature, and it's also affected by your mobility. The mobility is saying that, if you apply, let's say, an external field, and electric field, how well can those electrons move around? So a really high mobility means that those electrons can move really easily, and you'll accelerate them really quickly, where very low mobility means they're going to continue to hit into things, and bump around, and not move around too well. And your diffusion is this kind of thermal process. If you think of, let's say, gases in a room, and you have a hot gas, that's going to diffuse a lot faster. So it's the product of these two things. It's how well it can move around times it's thermal energy that it has to move around, the energy it has for moving. And so that's kind of what the diffusivity means to me. The lifetime, again, is what I mentioned earlier-- is that when you create this excited electron that's now free to move, this mobile electron, it can move around and explore a certain area, that area. Volume is defined by the diffusion length. And it exists in that excited state for some amount of time, tau, and that tau is not-- not every carrier that is generated necessarily has that lifetime, but it's a characteristic lifetime that it could have. And then they're pointed out here what they are. So that's what affects our diffusion length. So in the next bunch of slides, we're going to talk about mainly how we can affect tau. So tau is mainly affected by recombination centers-- so defects, and semiconductors, and a few other things that we're going to talk about. And then this is almost limited depending on what kind of materials you're using, the mobility, and so we'll talk about that, as well. So what affects lifetime? We're going to go over, again, basic recombination mechanisms in semiconductors. There's a lot of them. A lot of them have some rather complex equations behind them. We're not going to delve too deeply into how to derive them. You're welcome to do that. It was actually kind of fun to do on my own and refresh myself, so it was really useful-- and also be able to calculate our excess carrier concentration, which we're going to do in the next couple slides. So n-- let's say for n-type material, the number of mobile electrons you have is defined as n. n0 is very frequently your doping density-- these come up one at a time. Sorry. Wrong direction-- are generally the doping concentration. So if you're putting in phosphorus atoms into your silicon, it would be the concentration of phosphorus atoms. Your delta n is how many extra electrons are you adding, mobile electrons, due to the photo excitation of light. And so that's what this is saying-- is that you have some-- your delta n is equal to your generation rate. So your generation rate is generally in units of carriers per volume per second-- so how many carries you're generating in a certain volume. And because delta n is actually a density, you need to say, OK, how long do those carriers last once they're excited? And so it's the g tau product is what gives you your delta n. Now, when working with silicon, it's really important to understand what the different ratios are of n, n0, delta n, the doping concentrations. So getting these relative numbers in your head is an important step in moving forward. So let's say we subject a piece of silicon to AM1.5 spectra. So your G-- sorry. This is a little off. There we go. Sorry about that. When I added some equations-- so your generation rate is on the order of 10 to the 16th. A care lifetime for silicon-- this is not a great lifetime, but an OK one-- is about 10 microseconds. And so as a result of that, you'll get about 10 to the 11th excess carriers per centimeter cubed. And if we compare that-- so for every excess electron we make, remember we also leave behind a hole. So we have delta n is generally equal to delta p. And that's about 10 to the 11th. If we look at our intrinsic carrier concentration-- so if we had no dopants, how many carriers would we have just for thermal excitation? And that's about 10 to the 10th. And so you can see that delta n is actually larger than your intrinsic carrier concentration for silicon under normal illumination conditions. Now, let's take an example. Suppose we add phosphorus at the order about 10 to the 16th, and so that's generally about what a base doping concentration should be-- in that realm. It might be a little high, but you can see that your delta p is much greater than p0. So p0 would be how many holes do you have, which is a ratio of your intrinsic carrier squared over your doping density. And that's about 10 to the fourth, so it's way, way less than what was there without excitation. So the number of holes without any light shining on it is p0. You generate a bunch of holes, delta p, by shining light on it, and you can see that these numbers are drastically different. And of course, your doping density is actually much, much larger than your delta n. Your majority carriers don't change very much, but your minority carriers change very, very drastically. That's really what this is trying to say here under excitation. What is lifetime? So that bubble shouldn't be up yet. So we measure tau by creating some excess carrier population and then watch them decay. And they decay at some rate, recombination rate, and under steady state conditions-- so under constant illumination, we're not looking at transients-- your recombination rate is actually equal to your generation rate. So if you compare the two equations on the previous slide, they're true under steady state conditions. [INAUDIBLE] going to pop up. And so your lifetimes add up like parallel resistors. So we have tau bulk, which is kind of the effective lifetime of these photo excited carriers. 1 over tau bulk is equal to 1 over tau radiative, so this is radiative recombination. And this has to do with-- basically if you read the Shockley-Queisser efficiency limit paper, this is the lifetime that they assume was limiting. And for silicon, this is absurd. This is never ever the limiting factor. And a lot of direct band gap materials-- for those of you who don't know what that is, don't worry about it. That's often the limiting factor, and it has to do with the absorption is always equal to the emissivity in a material. AUDIENCE: [INAUDIBLE]? PROFESSOR: For a direct band gap material, radiative recombination can be an issue, and I'll talk about that in a second. And there's also another combination called Auger recombination. It's not "Oger" like I thought when I first came here. It's "O-jay," kind of like-- I don't know-- OJ Simpson, I guess. [LAUGHTER] And it's dominant only under very high injection conditions or very high doping density. So in your emitter layer where there's really, really high doping densities, you're going to have a lot of Auger recombination. And the last one is Shockley-Read-Hall So these are three guys. They came up with this kind of a model for how recombination happens in defective materials-- so materials with levels, electronic levels, in the mid-gap. And again, these add like parallel resistors, so you're always-- so if you remember back to the leaky bucket, these your leaky buckets for diffusion length. You're always limited by your shortest diffusion length-- or, sorry, your shortest lifetime. And this is often your limiting lifetime-- is your Shockley--Read--Hall recombination. AUDIENCE: Is it for silicon and for [INAUDIBLE]? PROFESSOR: For silicon-- AUDIENCE: [INAUDIBLE]. The Shockley-Read-Hall is the [INAUDIBLE]. PROFESSOR: So radiative recombination. So you can probably guess from the name that radiative recombination involves a photon. The ability to absorb photons also means you have the ability to emit them, and so silicon, or many semiconductors, will emit photons when you get a recombination event across the band. And when that happens, you emit a photon under equilibrium. So equilibrium means no outside excitation. It doesn't mean steady state, so this is, let's say, in the dark. You get your recombination rate is equal to your generation rate, because it's in thermal equilibrium with the area around it. So it's absorbing protons and emitting them at the same rate. And this is equal to B. So some material parameter times your hole in a electron concentration. And again, under equilibrium conditions, that's equal to your intrinsic carrier concentration. np product is equal to ni squared. Now, when you shine light on it, your n, which is to n0 plus delta n-- so this is you excited carriers, your excess carriers. So your n now increases and is greater than n sub i, and your net recombination rate is determined by this equation right here. So B np minus B ni squared-- so the difference between your equilibrium and your now excited carrier concentrations. AUDIENCE: I'm not sure if I missed this. Is B just a proportionality? PROFESSOR: It's a material parameter, yeah. It depends on-- for silicon, I forget. It was on the next slide. It's 10 to the minus 15th. I don't know for other materials, but I presume that that would change-- probably be a lot higher for other materials. And it turns out your radiative recombination lifetimes, when you plug these numbers in and you make some assumptions about what's really small compared to each other, you get these equations here. And again, this is tau is equal to delta n over R. And so you get that. And now if you look at for silicon, you get B is about 2 times 10 to the minus 15th. Your delta n I put-- n I determined just was 10 to the 16th-- some doping concentration. And your radiative lifetime is incredibly, incredibly long-- about 50 milliseconds. And if you remember from before when I was calculating a generation rate for silicon, we used about 10 microseconds. So this is really, really long. And so radiative recombination is very, very slow, and it's rarely ever the limiting lifetime in silicon solar cells. However, as you mentioned earlier, it's actually a big problem in direct band gap materials. And if you think of there's some materials we actually want a very short radiative recombination time. So for example, if you're trying to make an LED, you inject carriers using a voltage that recombination emits photons, and then you get light. And that's basically how an LED works. So now we'll talk a little about Shockley-Read-Hall recombinations. So this is something that our lab works, I think, very, very well in. We do a lot of defects in semiconductors, specifically iron. And so iron is one of these really, really awful contaminants in solar cells. Just a little bit of iron, I think-- [INAUDIBLE], correct me if I'm wrong. I don't remember what year production this was. Maybe it was 2009, but two grams of iron could contaminate the entire year supply of silicon detrimentally. So that's a lot. AUDIENCE: I think you actually calculate this in your [INAUDIBLE]. PROFESSOR: Yes, you do for a single panel, and it shocked me. So basically what we're trying to say here is that you have iron atoms and it can sit in different areas of your lattice, but you have these defects that exist, and they introduce different energy levels within the band gap. So the outer electrons of iron can either sit at these sites-- so these blue sites where they're donors, or they can create acceptor states kind of like boron does, but they're much higher up into the gap. And these act as recombination centers, and we'll talk about why that is in a second. So these trap levels can interact with mobile carriers in a whole bunch of different ways. They can capture an electron. That electron can then sit there. If there's enough heat energy, it might actually get promoted and jump out of that, and then which case it wouldn't have actually decreased your excess carrier population. It can also capture holes, and it can also emit holes. And there's a bunch of things that go into these equations here. It depends on a lot of the energy of this trap state. It depends on the carrier excitation. We'll talk about that in another few slides. And it also depends on these-- what are effective-- I forget the exact word, but the effective hole in electron lifetimes. And those are limited by your trap density. So these can be thought of-- suppose they're each-- let's say each iron atom is introducing one trap level. It would be the number of trap levels, the density of trap levels within your system, times some thermal energy, and then a capture cross section, which is saying, OK, that trap state exists in one location. How much area can it see in terms of what effective area is it capturing electrons? And oftentimes under the right conditions-- so with very, very deep traps, so traps mid-gap under low injections, your Shockley-Read-Hall recombination is actually one of those two lifetimes, and it's a very simple equation. And under very, very high injection conditions, it's actually summing them up, and if you look in the previous slide, if you go to delta n goes to infinity, you can see that this becomes true. Joel? AUDIENCE: The energy, the thermal energy type [INAUDIBLE]? PROFESSOR: That's a good question. That would be my guess. Sorry, there's another question back there? AUDIENCE: Yeah, so is iron worse or is gold worse? PROFESSOR: I can't hear you. AUDIENCE: Is iron a worse dopant, or is gold your worst dopant in terms of [INAUDIBLE]? PROFESSOR: Oh, in terms of capture cross section? AUDIENCE: Yeah. PROFESSOR: I know they're both bad, really bad. [INAUDIBLE], do you know that off the top of your head? AUDIENCE: Sure, so gold has a larger lifetime impact at lower concentrations than iron, but it's perhaps one of worst, and that's why you're not allowed to wear gold jewelry at the cleaners. They ask you to take off your wedding bands and other jewelry before entering the [INAUDIBLE] cleaners. PROFESSOR: So no bling in the cleaner. There you go. And so for the material scientists and physicists in the room, if that does not apply to you, don't worry. This is just to explain what is going on. Often when you want recombination to happen-- so this is a momentum, or k, versus energy-- it requires not only the emission of a photon, but also a phonon to change its momentum. When you introduce a trap level, or these localized impurities, because it's localized in real space, it's delocalized in k space, so you have this kind of flat level in k, and you have these very, very efficient pathways for recombination. If that doesn't resonate with you, don't worry about it right now. So here we see that really the impurities can have a very, very large effect. If you think of the doping densities that we put in, we've been using about 10 to the 15th, 10 to the 16th for our doping density. This is on the order of like a million less, and it can have a huge impact on lifetime. So one of the worst, again, as we said, was iron, and these interstitial irons are especially bad. And at 10 to the-- I don't know-- looks like 10 to the 11th. Very, very low concentration. That's one in 10 to the 12th. So that's one in a trillion atoms are iron-- cane detrimentally impact your solar cell. So that's really, really bad. So keeping fabs clean-- so again, no jewelry and other things can really-- that can actually have a very large effect on your device performance. And if you plot that versus your dislocation density, you can see that if you have-- it's especially bad for very, very high lifetime silicon. Just a few dislocations can actually really, really hurt it, but the effect is mitigated if you're already starting with very low lifetime silicon. And again, this is something our lab works on quite extensively. It's not just the number of iron atoms. So if you take a piece of silicon, and you want to know its lifetime or how iron impacts its lifetime, it's not just the total number of iron atoms in it. It's also how they're distributed. So if you have, let's say, clusters of iron atoms, that would count as one defect, or effectively less than the number of atoms in it. And so clustering these things can actually be really, really a good way of cleaning up your solar cell material, and this is an effect called gettering. And if you can getter these impurity atoms into one location, they'll have less of a detrimental impact on your material. So this is kind of a tricky one to explain, but this is-- so Shockley-Read-Hall recombination can also show up in something called service recombination. So if you look at your silicon lattice, each silicon atom has four valence electrons, and it bonds to four silicon atoms around it, and it has satisfied covalent bonds. So this silicon atom has all its satisfied, all its satisfied, until you get to the surface. And at the surface, you have what are called dangling bonds. And these dealing bonds can actually introduce traps states, and so you can see that actually introduces a whole ton of levels within the band gap that can provide Shockley-Read-Hall recall recombination pathways for your carriers. And so surfaces are incredibly important, and the way we tend to think of it-- and this is a concept that might be difficult to grasp at first, but it scales with two things. There's two things going on. You can think of that this is the width of your cell, and you have some service recombination velocity, which is some kind of characteristic of how well they can recombine at the surface. And at let's say infinite surface recombination velocities, it means that any carrier that comes and hits that surface will most surely recombine. So this drops to zero. So then you're limited by, OK, how well can they actually diffuse to your surface? So it's again some kind of ratio of your self thickness squared over the diffusivity, your carrier diffusivity, and that that gives you an idea of what your limiting factors are. So under very low surface recombination velocities, you're limited by this term here, the first one, and then the very high ones, you're limited by this term. And do there's two effects going on there. And it's summarized as well here. At very, very low surface recombination velocities, your tau surface almost goes to infinity-- very high. And you can passivate these bonds using hydrogen. So for example, if you use hydrofluoric acid, what it does is it etches away the silicon oxide layer that sits there, and you have these hydrogen atoms that now sit and satisfy these bonds. And if they're perfectly satisfied, you'll have a mobile carrier. It'll actually elastically scatter off, not lose any energy, and not recombine through these trap states. AUDIENCE: I have a question. I thought [INAUDIBLE] is a very good passivation barrier for silicon? PROFESSOR: What is? AUDIENCE: I thought silicon dioxide is a really good passivation barrier for silicon. PROFESSOR: I've seen some diagrams of what silicon oxide looks like on silicon. It passivates many of the bonds. You're absolutely right. HF is actually probably the best. The problem is that it etches glass and other things that are in your source material, and it's incredibly dangerous. It can kill you rather dramatically. So it's only used in laboratory settings. If you're trying to actually take lifetime measurements and negate the effect of surface recombination, silicon oxide can be a good one. If you actually look at the structure, there's a few dangling bonds in there, but it can passivate most of them-- just not all of them. Another good passivation technique is actually the silicon nitride ARC coating. That passivates the surface very well. So yeah, good question. There's other ways to mitigate surface recombination, as well, and we'll talk about that, I think, in either next lecture or the one after that. So yeah, this slide is just telling us that, if we vary our thickness of our silicon, we can actually measure our surface recombination velocity, and we can fit it so we can figure out our tau surface, which is a really important material parameter. Generally, I think good surface recombination velocities are anywhere from like 10 to maybe in the 100ths for centimeters per second, and really bad ones are much, much higher. And the last type of recombination mechanism we're going to talk about today is Auger recombination, and this looks like-- when I first saw this, I'm like, why on earth would this ever happen? And the fact is it does until you get to very, very high carrier concentration. So you can see that it involves three particles. Let's say an n-type silicon. It needs two electrons in the hole. What happens is that you get the simultaneous relaxation and excitation. So you get this relaxation of this excited electron into a hole, and then you get this excitation of this other electron into a higher energy state, and then it thermalizes down and releases a phonon, releases heat. Yeah, Jessica? AUDIENCE: [INAUDIBLE] terms of p-type here? PROFESSOR: So this is going to be our n-type material. And again, because this type of recombination event requires two particles, it requires two electrons, so it's n squared in one hole. So the recombination rate goes up with pn squared at high enough concentrations. And so your tau goes out with the 1 over n squared. And this is particularly bad, again, only at very-- because it goes up with the square of the carrier concentration, it's really bad at very high carrier concentrations. So if we look at this plot, we can see that our minority care lifetime drops well below a microsecond around 10 to the 18th, so you generally want to stay out of that range of doping concentrations in your base, because you'll have very, very bad lifetimes. AUDIENCE: [INAUDIBLE] are both sides-- is that the top blue part and the bottom blue part n-type? PROFESSOR: I didn't label this. You're right. This is the valence band. That's the conduction band. And so this is kind of driving home, again, that leaky bucket idea-- that is, you're really limited by your worst lifetime. And so if you remember, we're thinking about parallel circuits here. Your bulk, or your effective lifetime of each carrier, is determined by these three here-- your radiative. Sorry, that should be tau band, which is the same as your radiative recombination-- Auger and Shockley-Read-Hall. And so at different excess carrier densities-- so we're varying delta n by shining various intensities of light. And we can kind of activate different limiting lifetimes. So at very low, you're limited by Shockley-Read-Hall. And Shockley-Read-Hall can actually go down with higher illumination conditions. And your Auger, remember it gets really, really bad around 10 to the 18th-- becomes your limiting. So your tau bulk is never doing better than your worst lifetime. AUDIENCE: So on that graph is tau and emitter also the same thing as-- PROFESSOR: Tau Auger? AUDIENCE: [INAUDIBLE] tau [INAUDIBLE]. PROFESSOR: This came out of Daniel McDonald's thesis. Tau emitter-- it's complicated because it has other influences in it. Auger, I always think of recombination in the emitter as Auger, but you also have other carriers coming in due to injection currents. And so your excess carrier population is also a function of that, and so the equation for it gets a little more complicated, but it would make sense that, as you increase illumination, you're increasing your current into the emitter. And you would get more carriers, and it would decrease. And it looks almost exactly like Auger in his regime over here. I'm not sure of the other things that go into it, but if you look up his thesis, I think it gives a pretty good description. So you're probably bored of hearing me say this, but again, we're always limited by our weakest. So in defect mitigated recombination materials-- so where your shortest lifetime is due to some kind of Shockley-Read-Hall recombination. Your lifetime for Shockley-Read-Hall is always going to be much, much shorter than your radiative lifetimes, which is a characteristic we can exploit for measuring the lifetimes of our materials. So because very, very few carriers will actually radiatively recombine and emit a photon, that, if more of them are radiatively recombining, then we know that it's a very high lifetime material. If there's a lot of defects and we have very, very short lifetimes, very few of them will radiatively recombine. And so this emission of photons with energy at the band gap can give us an idea of the lifetime within our material, and that's how we measure. It's a technique called photoluminescence, and what you do is you shine light on it, generally with a laser. We put a diffuser in front of the lasers so the laser beam spreads its photons over a large area. We excite all these carriers. So this laser has very, very short wavelengths, and I think in our-- well, very short. It's 900 nanometers. Silicon's band gap corresponds to about 1,108 nanometers. And so 900 is easily absorbed, not just right near the emitter, but also somewhat well deep below the junction as well. And then as you begin to see recombination, when radiative recombination happens, we emit a photon. And that can happen in certain areas better than others, and a lot of it depends on, like I said, defect density and other lifetime eliminating defects. And it's a good way to spatially locate where problems are in your solar cell. Any other questions? Ben? AUDIENCE: How good is spatial resolution on [INAUDIBLE]? PROFESSOR: It depends on your camera. So we use, I think, a germanium camera to detect those photons, because Silicon won't have a very good response. And it depends on the CCD array within your camera. It can be really good. I do a similar technique to measure shunts, and we have microscope objectively. We just have to zoom in really, really far, and then we just scan over an area. And last little thing is that, if you recall-- hold on. Let me-- this slide. Traps cannot only trap an electron, but you can also emit an electron, assuming it has enough thermal energy, and you can see that on this plot. And so this is an Arrhenius plot. So again, high temperatures are in this direction. Low temperatures are over here. And you can see that your lifetime actually increases at higher temperatures because electrons that see that trap fall into it, then can easily come back out of it because they have enough thermal energy to escape. And that's really what this is depicting. So one of the things I think for researchers in the room who are studying these types of materials, varying temperature is often a really, really good way of looking at electronic structure materials. And it can be very, very powerful, and this is one example of a tool to look at these types of traps. Oh, good. We have plenty of time. We actually might end early. So now we're going to talk about mobility. We've given a lot of-- sorry? AUDIENCE: I was just wondering, is it possible to somehow introduce defects that are at the energy levels [INAUDIBLE]? PROFESSOR: That's a good question. So phosphorus actually has, if you draw it on an e versus x diagram-- so if we have our conduction band here, our valence band here, we said iron puts states in the middle of the gap. Phosphorus and boron actually put states very, very, very close to the valence band and conduction band. If you go to low enough temperature-- so let's say a below 100 Kelvin-- you can actually freeze out those donor electrons onto the phosphorus atoms. And below certain concentrations of phosphorus atoms, for example-- so below like 10 to the 18th-- at 0 Kelvin, you cannot conduct electricity. It actually becomes a total insulator. That's an excellent question. But at room temperature, when kt is on the order-- so kt is your thermal energy, and at room temperature, if you put that for electrons, it's 0.026 electron volts or 26 millielectron volts. This is a good number to have in mind, by the way. When that number is on the order of this binding energy for phosphorus, they're almost always fully ionized and free, but that's a very good question. Anyone else? So if we remember our definition mobility is related to or diffusivity, and again, our mobility is saying how well these excited charges can move around. And it's related to how much thermal energy these charges have, so that's why we have this kbt factor. And what's plotted on the right is the Shockley-Queisser efficiency limit, which are the stars. And then how if you-- let's say you reduce your mobility by, let's say, a factor of 10 or 100. What's the impact on the overall efficiency? And you can see that, if you detrimentally impact your mobility, you can really have a large effect on your diffusion length, and it can really hurt your device performance. So it's a really important material parameter to think about. So there's lots of ways that these mobile electrons can do what's called scattering. So if I'm a mobile electron, I'm moving down through the silicon lattice. And let's say I see a defect, and this defect, because it has these extra electrons, it creates this kind of area of charge. It can see that it can scatter off of it and lose its energy, and so that's called a scattering event. Not really lose its energy. Sorry. It'll change direction. It kind of impacts the movement of that carrier. And there's all sorts of other defects scattering mechanisms. You can also scatter with an oscillating atom or a phonon. There's another type of scattering mechanism, and it's heavily dependent on what you put into your material, and we'll talk about that in a second. And for a lot of materials that are, let's say, porous or amorphous in some way, or even a lot of, let's say, organic semiconductors, having a good percolation network is really important to transport these charges. And often it's a very limiting factor in, let's say, like organic photovoltaics. And so this is a relatively simple scattering mechanism. What time-- oh, we have plenty time. What's going on here is that we can see that, as we add carriers-- so this is n is 10 to 14th. Very, very low concentration of dopants-- as we increase the number carriers, are scattering off of those ionized impurities. So every time you add a phosphorus atom, lets say, you introduce a static positive charge and a mobile negative charge when that electron leaves. And so you now have all of these scattering centers of positive charge. And so as you increase the number of dopants-- this for silicon-- you decrease the mobility of your material, and it's also greatly a function of temperature. I think that's mostly due to either phonon scattering-- is that right? Is there any other mechanism I'm missing, [INAUDIBLE], if you're still there? AUDIENCE: Sorry, I was on mute. Yes, I think you're good so far. We'll keep it simple, and use the simplest case first. I think that makes [INAUDIBLE]. PROFESSOR: But importantly is that higher temperatures, you generally get a much lower, lower mobility. And again, hitting home for-- this is not true necessarily for silicon, but for a lot of these heterojunction devices-- so for example, organics have very, very low diffusion lengths, and a lot of it's limited by mobility. And so what you do is you make these Interdigitated-- what I would call p and n. I forget the organic analogy, but p and n layers that interdigitated so that they only have to diffuse not the width of the device, but the length of those fingers. So you effectively need a much shorter diffusion length. And so this is talking about some of those other different ideas, and-- AUDIENCE: I'm sorry, that was a heterojunction? [INAUDIBLE]. PROFESSOR: A heterojunction is two different materials. AUDIENCE: OK, what was the thing you were just describing with the-- PROFESSOR: That's an interdigitated pn structure. Yeah, so what we're going to be talking about is the product of n and mu. And if you recall that your conductivity-- so hold on. Let's go back. What we have here is that we have a highly doped semiconductor. So this is about 10 to the 16th, and then we have our intrinsic silicon. So this has no dopants in it whatsoever. And now, you remember from last lecture when we applied a voltage across the terminals, a current started to flow, and when we heated it up, what had happened? Who remembers? AUDIENCE: Current increased. PROFESSOR: Current increased, and that's due to more thermally excited carriers. And so your intrinsic carrier concentration goes up. And so what was-- for room temperature, what's the intrinsic carrier concentration? It's about 10 to the 10th-- in that range. And so in increase-- so a small increase in temperature can greatly increase the intrinsic carrier concentration-- maybe something like 10 to the 12th. Now in a doped semiconductor, is that going to affect it as much? How about you guys think about that for a little bit. Mute and talk to your neighbor. And so I'm going to heat both of these up, one with a high dopant concentration, and one with a low. And which one you think will have the highest relative change in connectivity and in which direction? So I'll give you three minutes. So we're now going to subject. You've seen this demo before. We're now going to subject our intrinsic carrier to my hair dryer. And right now we're getting-- let's see. It's about 10 microamps. And if we heat this guy up-- did I mix these two up? Ah, there we go. So you can see we get a rather large increase in current. That was up to 100 microamps, so a factor of 10. So quite a large increase. So let's see a show of hands. So right now we're getting-- maybe we need to put this on milliamps. So we're getting about 58 milliamps of current through the semiconductor. Who thinks that the current is going to increase when we add more thermal carriers? This is where the doped one. We just saw the intrinsic. Do you think it'll go up, the connectivity? So this is the doped semiconductor. Now who think it's going to stay the same? Who thinks it's going to go down? All right, so this is split. Wow. So right now we're getting about 57 milliamps, and let's heat this guy up and see what happens. And so you can see, it's actually going down. It's now 52, 50, 49. So what's important is that we're measuring conductivity. It's not only how many carriers we have, but also how well they can move around. And it's, again, that product of number of carriers times the mobility. And again, each of those carriers carries an electric charge. So you put the electric charge of an electron in front of it. This is what I was supposed to have up in the background while that was happening. AUDIENCE: The intrinsic still had to go change, right? [INAUDIBLE]. PROFESSOR: Yeah, so again, remember there was huge changes. One was measuring milliamps. One was measuring microamps. So if we look at room temperature, we have about 10 to the 10th intrinsic carriers. So this is for intrinsic silicon with no dopants whatsoever. As we increase the heat, I don't think we're going to 500 degrees, but let's say we get the 400. We're only going up by a factor of 100, which is substantial, but if we had 10 to the 16th carriers originally from our dopants, these added number of intrinsic carriers aren't going to have really much of an effect at all in terms of the connectivity. So what's really affecting the dope case is that our mobility actually goes down with temperature due to the scattering events with temperature. And for intrinsic silicon, you get a little bit better mobility, so it's a little bit higher, but they both have the same general trend of lower mobility, but only by, let's say, this is about a factor of 10, where it was about a factor of 100 for the intrinsic carrier. So again, our carriers in the intrinsic case go up by a factor of 100. Our mobility goes up by a factor of 10, so the conductivity then has to increase by a factor of 10. And for our doped semiconductor, really the heat is just hurting our conductivity because of the decrease in mobility, and the thermal carriers don't really add-- they're washed out by the sea of dopant atoms that are really adding all the carriers. Yeah? AUDIENCE: So if you were to obviously heat up [INAUDIBLE] more-- [INAUDIBLE] the doped more, it would probably eventually get to the intrinsic case where there are constant increases, but would you ever want your solar cell that hot? PROFESSOR: Would you ever you your solar cell that hot? That's a good question. AUDIENCE: Yeah, I don't know how [INAUDIBLE] would it melt [INAUDIBLE]. PROFESSOR: So let's go-- there's an equation for that, and we'll talk about that later, too. So if you look at our VOC, if we have a large saturation current, that means that we're going to have a very low VOC. We have this reverse current that's going in the opposite direction of our illumination current that's opposing that illumination current. And so if it's large, then it'll hurt our VOC. And you can see that it scales with d, which scales with kt, and so we get this increase in temperature is increasing this J0. And so for most types of cells, heat is very, very bad, especially for crystalline silicon. For amorphous silicon, it's different. Well, I'm not going to get into that today, but for crystalline silicon, heat is generally very bad for the performance. And when you do testingg-- so for example, when NREL does testing, they'll rate all your cells at AM1.5G, some calibrated solar simulator that's illuminating your sample. And they're kept at constant temperature, so the temperature is always reported, and it's generally kept at the 25 C. So the temperature is a very important characteristic. But to answer your question, if, again, we looked at an Arrhenius plot-- so this is 1/kt. So this is high temperatures over here, low temperatures over here. And this is carrier concentration. At very, very high temperatures, when your thermal carrier-- so if you extend this out to higher temperatures, you can see that this will actually surpassed the number of dopant atoms, and you'll actually increase. So this is high temperature over here. Your carrier concentration will actually increase at much, much higher temperatures, and this is what's the extrinsic region. So your donor concentration is pretty much only determined by your dopant density, so this is ND. And then at low enough temperatures-- so this is very low temperatures-- you actually start freezing out you're donor electrons into these donor states. So that actually sums it up. If you guys have other questions, feel free to ask them, but we actually can end a little bit early. I do have your homework, so if you want those, come up here. And I think [INAUDIBLE] posted the projects online so you can finish homework number three. That's it. |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | 4_Charge_Excitation.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from 100's of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Why don't we go ahead and get started. What we're going to be talking about today, is what happens after that photon gets absorbed. So we spent a great deal of time in our last lecture talking about light absorption, the interaction between light and the semiconductor. Now we're going to talk about what happens once that light is absorbed. So we're in the fundamental section. We're right here. Later on, we'll get into the technologies and the cross cutting themes. As we go through the fundamentals, I'm going to attempt to relate those fundamentals to real solar cell technology. So you're not left kind of floating, wondering why it is that we're studying this stuff, but you're really seeing the connection to solar cell devices. So first if we remind everyone of the broader picture that the conversion efficiency, the ultimate performance of the device, is defined as the output energy versus the input energy. And for most solar cells, this breaks down into the inputs and the outputs. The input being the solar spectrum and the output being the collection of charge. And there are multiple processes that occur here in the middle. Light absorption, we talked about last time. We're going to be talking about charge excitation in charge transport today, mostly charge excitation. So we're right here. We're going to be marching steadily down toward the right over the next few lectures. And let me remind you, everybody, that the efficiency of the device is the product of each individual efficiency of each different step. So if anything is going wrong here, it will be limiting device performance. And the image you should definitely have in your mind is of that bucket, with each plank representing a different component of the device, perhaps a different physical process or perhaps a different physical component of the device itself. And whatever is the poorest, is going to be limiting overall performance. Your efficiency is going to be flowing out from that low plank and you will have a low efficiency device. So the art of making a high efficiency solar cell is really understanding everything that goes into the physical processes, but also the devices. And that's why we break it down like this. We have the physical processes, and finally we get into some of the devices in the architectures of manufacturing methods. So learning objectives today. Oops. That was a little bit of a typo. It's not the solar resource. We're talking about charge excitation today. We will be talking phenomena logically how this thing called a band gap forms. It's a very important physical concept. We'll lead off the lecture with it, and then very quickly go into applications that you see why it's important, returning back to the fundamentals, how it actually forms, going back and forth until we have a pretty solid understanding. From the background surveys, I understand that about 50% of you understand what a band gap is. So 50% of you may be a little bored, or a little entertained, at my hand wave explanations during the first part of class. I encourage you to think about the band gap from the perspective of the solar cell device, because this is probably not something you've done before. You've probably understood band gap from the perspective of a semiconductor device as packaged in the dark, perhaps in a little gadget like this. But not something that's exposed to light with the addition of a generation current. Or at least not in detail. So I welcome you to think about that as we go through the explanations in the beginning of class. Then we'll describe how optical absorption in semiconductors represents the transitions of charge in an energy band diagram. In other words, if we imagine a given space of the semiconductor divisive, say a surrounding of a given atom, and we imagine that different orbitals, different electron orbitals will have different energies associated with them. We are going to be talking about how light gets absorbed in transitions electrons between those different energy orbitals. That's important because then we'll be able to calculate the fraction of photons lost, not absorbed, by a given semiconductor material. We'll be able to calculate the fraction of incident solar energy that is lost as well, due to a phenomenon called thermalization. And finally, we're going to be able to plot efficiency versus band gap, and denote specific materials on it. In other words, we are going to be doing our first higher level efficiency calculations for a solar cell device by the end of today's lecture. And the idea is to expose a little bit on the technology side as well, so that you have an appreciation for what's up and coming in the field, what are some of the ways to enhance the performance of social devices. So band gap. Very, very basic description. If you've never been exposed to band gaps before, the reason it's important is because the band gap is going to define what color, what portion of the solar spectrum that material absorbs light most efficiently. How a band gap forms is related to the atomic structure. Think of bonds as essentially why stuff is tough. If the mechanical engineers in the room, if you remember linear elasticity, those bonds are what are essentially forming those springs between the atoms that are keeping them, the material, from flying apart. There is also a very interesting property here, that if you have a bound electron, it's usually not moving very far that atom. It's usually in a very localized electronic state close to that atom. So if you were to apply a resistance meter, an Ohm meter, to your device to measure the resistance across it, and you had just bound electrons, you would measure a very, very low current passing through that material. Because there would be very few free charge carriers to move in that applied field, when you apply the two probes and there's a little battery in here that applies a field across the material, you wouldn't be able to measure much current flow, because there wouldn't be too many free electrons to carry that current. So the bound electrons essentially enhance the strength of the material. But it doesn't help us from a semiconductor point of view. What we need are excited electrons. And excited electrons are why materials conduct. So let's imagine we have a bond. For example, in this material right here, these gray lines are really two electrons that are covalently bonded. Essentially two electrons-- one electron from each little black circle representing a carbon atom. And each electron associated with that carbon atom is shared with the other, in a covalent bond. And for the chemists in the room, more precisely, you have sp3 hybridized orbitals in this diamond cubic crystal structure. For everyone else, these are shared electrons in a covalent bonding configuration. Now if light comes in with enough energy, it can excite an electron from that covalently bonded state into an excited state, where it can then roam freely throughout the lattice. And that's the nature of charge excitations. So the big question that we have to ask ourselves is, what color of light will this material absorb most effectively? Will those electrons in the covalent bonds absorb light and be excited into a state where they can roam freely around the material, and ultimately conduct electricity? OK. So the answer to that question is not simple. Understanding how a band gap forms can be an entire semester of quantum physics. But what we're going to do is do it in a three step approach. Simple, very, very simple explanations that hopefully everyone will get. The next most simple explanation, which hopefully 80% of you will get, and then perhaps a more detailed explanation that only a few of you will get. But the idea is to really progress in levels of explanation. So let me go back one step. Yeah. The band gap energy can be most simply understood as a finite amount of energy needed to excite a highly localized electron into a de-localized excited state in the semiconductor where it can move around the crystal. OK. So this is the description of the band gap, how a band gap forms that you'll see in many chemistry textbooks. And I like it, because it's something that you can really grasp and understand. We have to start with the simple premise that electrons are a type of particle called a fermion. What is a fermion? Let's refresh our basic physics here. What means a fermion? Spin one half. And it also means-- can two fermions occupy the same state? No, they can't. Bosons can. But fermions cannot. And so what happens when you put two fermions together in a system? And say we have discretized quantum states. We begin filling up from the bottom up, right? Typically. So the lowest energy states get occupied first. You have, for example, spin up, spin down in your s orbital. And then they begin populating the next electron orbital. Once that's filled, you go to the next one, and the next one. So if we have one atom in isolation, the easiest example is that hydrogen atom, for those who took quantum mechanics, who actually calculated the energy of the bound electron there in the ground state of the hydrogen atom, around 13.6 eV. So imagine now you take a silicon atom, or carbon atom, or something else, a little bit bigger, more electrons in its structure. So you have the core, the 1s orbital filled. And then you start progressing outward. 2s2p and so forth. And so you begin filling up the orbitals. Let's see atomic separation. So imagine that we have our atoms infinitely separated. So we have just one atom in the middle of an infinite lonely space. One atom. And we would have discrete energy levels corresponding to the different orbitals, the different orbitals around the nucleus. Now imagine the atom is no longer lonely. It now has a partner. And you bring those two atoms closer and closer and closer together. That's this d getting closer and closer together. What happens? Well, electrons are fermions. They don't like occupying the same electronic state. So first, the first shell that's going to quote, unquote see each other, the first electron shell that will interact is what? Is it the outermost, or the innermost? Outermost. Right? Because they're the closest together. So you're moving the two atoms closer together, that outermost shell will begin to quote, unquote see the other atom. And those electrons will say, hey, you're occupying my state. And what will happen is they'll begin splitting, in energy level, and that's what you see right here. What this is representing is not just two atoms coming together, but a multitude of atoms coming together. And we have to really imagine, for example, and in most matter, we have something in the order of 10 to the 22, 10 to the 23 atoms per cubic centimeter. So atoms per cubic centimeter. Fun math, just total aside. If anybody talks about teletransporttion, think about the number of atoms in your body. And assign an xyz-coordinate to them, and then calculate the amount of data in terabytes. And then calculate our Ethernet speeds, and try to figure out how long it would take to transmit that data to the other side. Anyway. So this is to give you an impression of the density of atoms within the cubic centimeter. So you have a lot of atoms coming together, even though the electron wave function can be highly localized, they still interfere with each other. There's still enough atoms coming together that you have this interference. And it's not just two electron states, but many of them that are splitting, that are forming a band of states because no two electrons can occupy the same energy level as these atoms are coming together. Now, as they become closer and closer and closer, it's not only that outer shell that begins interacting. It's the next most outer shell. This one, here, that begins interacting. And finally another shell, and another shell. And so at some equilibrium distance, the equilibrium distance here defined as the attractive potential between atoms. There's an energy gain by forming that bond. And the repulsive force of the nuclei, their large concentration of positive charge in the nucleus, that if you tried pushing them too close together, they'll repel. This balance of forces, the attractive potential due to that bond formation, the repulsive force to the nucleic, results in an equilibrium bond distance, d, shown here as this dotted line. And at that equilibrium bond distance, some of the orbitals will be forming bands and other orbitals will still be discrete, namely the core orbitals will still be discrete. The outermost orbitals here will be forming bands. Now an interesting thing happens when it just so occurs, and say, for example, in the system right here, this shell is filled. This shell is filled. This shell is completely filled. And let's imagine that this orbital exists, but it's unpopulated. You ran out of electrons. And you started filling in the orbitals of this atom over here. You had enough for this orbital, that orbital, that orbital, but then you ran out of electrons. You didn't have anything more. And this band of states was empty. So the band of allowed states exists up there, but it's empty. And this band right here exists, and it's filled. These core states exist, and they're filled as well. So the action is really happening right here. If light comes in, if visible light comes in, it's probably going to excite an electron from this filled band right here to somewhere in this unoccupied band up there. And this gap, right here, is an interesting thing. There is no stable electron orbital. If you were somehow to take an electron with that given energy, this energy here, from the vacuum and stick it inside of the system here, it would quickly decay into a stable orbital. So this is a prohibited band of states. There is no stable orbital for that electron to exist at that energy. And hence we call it a band gap. The reason band gaps are interesting, is because different materials have different, or can have different, size band gaps. Some don't have a band gap at all. Most of those that don't have a band gap, because you have a band of states like this one that's partially full. Let's imagine we have three electrons in this band right here. One, two, three. And then we have these other four empty states about it. In fact, you have a partially filled band. And let's imagine, instead of just having seven states within that band, we had a multitude of states within that band. Almost too many to count. And now, with even the slightest of energy, we could excite an electron from this state into that state plus little delta. That would be a metal. Now if we had a really big band gap, we wouldn't be able to absorb much light. Because you'd need a large amount of energy to excite across that band gap. And that would be an example of, say, glass. Where light would come in. Most of our visible spectrum makes it through the glass, and that's why we see through the other side. And diamonds as well. OK. So let's review. An atom in isolation has discrete electron energy levels. And on this diagram right here, it's represented with this d, the atomic separation being way out here. We have discrete energy levels. And that we can remember from our basic physics, especially the hydrogen atom. And as atoms move closer together, as in a crystal, where you have a regularly repeating arrangements of atoms, the electron wave functions begin overlapping, and since electrons are fermions, meaning that the two electrons can't occupy the same state, you have a splitting of those energy levels forming bands. And you can see the splitting as you move from right to left, as you move from further to closer, on these diagrams right here. First starting with the outermost states, the most valence of electron orbitals, and then slowly moving into the core states as well. And the gap between bands denoting an energy range in which no stable orbital exists is called a band gap. Sometimes you'll see it all written as one word. Sometimes there will be a space between the band and the gap. Any questions so far? OK. So now this explanation should be relatively straightforward. What I want you to log into your ram, because we're going into some more detailed explanations, and I want to set a marker, a flag, so we can come back here afterward. And even if you're lost with the subsequent explanations, you still get this one. Right? So remember these three points. If you make a note of them in your notes, and we're going to move on to a few more explanations here. OK, so another way of looking at this which really builds on the folks who have had a more advanced physics backgrounds, is that the wave function of an electron, inside of a crystal-- if you recall the wave function of an electron in free space, we have what is called the plane wave equation for the electron. It's this regular repeating, nice stable function. Now if we introduce that electron into a crystal, it's almost like an infinitely repeating system of atoms. And so you could envision that you could describe the wave function of that electron as a combination of a plane wave, but perturbed by that locally repeating potential from the atomic nuclei. And let me be more specific about regular repeating potential. What I mean is that you have a series of nuclei here, and if you imagine the electron potential around these nuclei, there is a propensity for the election to be bound by the atom. And so as it's moving along, instead of just a free plane wave, now it's almost like driving along a bumpy road, bump, bump, bump. And so the wave function of the electron can be described as this combination or product of the wave function of a plane wave, envelope function describing the electron localization. So you have this localized function here and the delocalized function describing the plane wave behavior of the electron as it moves through space inside of the crystal. Now what we have as an easy way to describe this mathematically, is we make an approximation. Instead of describing this potential in detail, the easiest way to do this is to describe this square well potential. We have a so-called Kronig-Penney idealization of the repeating Coulombic potential. It's easier to solve numerically. And then essentially what we do is we solve Schrodinger's equation for two possible solutions. One is for the electron wave function centered on the atoms themselves. So that would be the bound state. And the other is we solve the equation for electron function centered between the atoms. So essentially in the unbound state, right here. Which do you think will have the higher energy level? In between the states, right? Because if the electron wave function is centered over here, it's further away from the positive charge of the nuclei. You need more energy to put it into that position. It's an unbound state, but it's also at a higher excited energy level. Whereas if it's bound around the atoms themselves, you would have a lower energy state. In this delta of energy levels between the bound and the excited states represents the band gap of the semiconductor in this simple approximation right here. Now you can take that one step further and say, well, gee, if I could solve that in one dimension in a very simple case, why don't I put a three dimensional structure in place, such as a real crystal, and then solve the Schrodinger equation? And sure you can do that. You can definitely do that. It becomes much more complicated solution numerically, but the general principle stays roughly the same. You can add several more tricks, and bells, and whistles. And of course, I would probably emphasize that the solution could either be done numerically or you could use the symmetry relations of the crystal to develop a very simple expression of the crystal, minimizing the redundant directions inside of the crystal, collapsing it down into the bare essential to describe that crystal structure. And we would have some uniform energy space that looks, for example, represented by this yellowish surface, where you'd have an isopotential inside of your crystal structure. So these are different ways of looking at the band gap inside of a semiconductor crystal. If you're really interested, and for the advance reading, I would definitely suggest this book called, Fundamentals of Semiconductors, by Professor Peter Yu, now retired at UC Berkeley. Very nice explanation. Uses an elegant application of group theory to derive the band structure of a semiconductor in detail. So let's hop back to this one, real quick-- actually, yeah? AUDIENCE: I'm just curious if you could tell us the name of the equation that was at the top of-- PROFESSOR: Sure. Absolutely. So what we're describing here is essentially the e to the ikx, or e to the ikr, essentially describing that periodic potential. And then in essentially an envelope function. So, yeah, that would be the wave function that would be introduced into the Schrodinger equation. So let's go back to here really quickly, because I don't want to get us lost in the weeds. I want us to focus on the main concepts at hand. Let's, from an engineering point of view-- given this formation of bands inside of real materials, since the atoms are coming together and forming crystals, we can envision a few different scenarios. Let's focus on the this band here, and that band here, and the space in between them, the band gap. And let's assume that we have this band here either completely filled or partially filled, and talk about what happens in three different extreme scenarios. In one case, when we have the band partially filled, we have what is known as a metal. And the reason this case is interesting is because a very small amount of energy is all that it would take to excite an electron and make it to move around the crystal. As a matter of fact, a metal has what's called electron c, that serves to conduct electricity at room temperature and even at very, very, very low temperatures. Semiconductor, on the other hand. Now semiconductor has a finite band gap, and we'll get to this in a minute. Let's start with the insulator, because that's a more easy to understand case. If the band gap is very, very, very big, then a large amount of energy is required to excite an electron into an unbound state, so it can move freely across the crystal. And as a result, you're not going to have a very large population of electrons up here. The photo excited carrier population, meaning the population of charge carriers, electrons, that are excited into here from light, is very small. And the thermally excited carrier concentration, just some background thermal energy, kt, Boltzmann's constant times temperature, that energy is also going to be insufficient to drive electrons across this band gap. And as a result, you will have a very, very small population of carriers up here. And this will be an insulator, because you will have very few charges to transport current. If you apply a potential across an insulator, you will not have very much current flow. And that's the principle of insulating materials around wires. Actually this is in the semiconductor. This is a polymer. But if we had, for example, glass or diamond, that would serve as a nice insulator as well. Now a semiconductor material is somewhere between a metal and an insulator here. And that the band gap is now shrinking. The band gap is small enough to interact with lights, typically. For example, this little piece of silicon, which I'll show in detail in a minute, towards the end of class. It doesn't look clear, like glass does. It looks opaque. And that's because it's absorbing photons in the visible spectrum. But it's letting some of the infrared photons go through it. Some of the very low energy photons can go through the material, because those photons have insufficient energy to excite carriers across that band gap. So we have a semiconductor material, defined here as having a band gap. That band gap defines the energy of the light that is most efficiently absorbed. And any photon with energy in excess of the band gap can also be absorbed by that semiconductor. Any questions so far? AUDIENCE: Are there any natural materials where the spacing of the bands is such that three bands are active, or is it always just two bands which are active? PROFESSOR: Interesting. So the question is, are there more bands above here that you could excite into? Absolutely. So as you can envision, back here for instance, you could continue drawing more and more and more bands. They're unoccupied, but they exist. And as the atoms come closer and closer together, those begin interacting as well. And so in a few lectures we'll look at a band diagram of a semiconductor, and we'll see all the occupied bands, and all the unoccupied bands. And you can excite into any one of them if you have the right excitation condition. Let's put it that way. Any other questions? Yeah. AUDIENCE: I'm having trouble seeing the chemist's description along with the physicist's description along with this slide all together. How does, once it's in the crystal structure and it's in its lattices and everything, how does that look in this picture? Where is that in this picture. PROFESSOR: Sure. So this one here, let me describe the axes, because that might make things a little easier to understand. In the vertical axis here, we have energy. And that's the same axis as we have right here, energy. In this axis right here, we had inter atomic separation. So this was meant to demonstrate what happens when we bring atoms closer and close together. In this description right here, this is an unlabeled axis that could be x real space, for instance. Why do we plot an x in the x-axis? Why do we plot real space? Well sometimes, when we talk about solar cell devices, we're talking about bringing charge from deep within the device to the front surface where the contact is. And so we want to talk about the flow of charge in real space. So that's oftentimes why you see in these band diagrams for solar cells, you'll see e versus x, versus real space. Now let me show you what these two levels here represent. This is the top of the filled band and the bottom of the empty band. And those correspond over here. I'm going to use the mouse now, because I'm going to be stretching. Let's imagine that this level on down was filled. And that this level up here is empty. So if I may, these are all filled. All filled. And this one here is empty. Gonna take a little bit. Here we go. All right. So now this band is going to be filled, because we're really looking at the equilibrium inter atomic separation in a crystal. We're looking right here this d. That's the spacing right now of those atoms inside of that piece of semiconductor material, inside of that silicon. So what we care about is this right here. And this would be the top of the filled band, and this would be the bottom of the empty band, corresponding to the top of the filled band right here and the bottom of the empty band right there. So that's how it all fits together. This diagram here can get kind of confusing when you're looking in the abscissa, essentially the x-axis. Because you're thinking in terms of real space. Well, it is real space, but you're bringing atoms together. So it's really meant to be a phenomenological tool to describe how band gaps form. This diagram over here, we're now starting to get into the engineering diagrams that can help explain how solar cells work. And from here on out, we'll be abandoning that chemists diagram and simply using this one because we assume that we're not applying a significant strain to our semiconductor material to get significant deviations here. We'd have to apply strains of several percent, and probably result in fracture of our semiconductor before we ended up changing that inter-atomic distance. We could also heat it up. It's another way of introducing strain, thermal strain, but not mechanical. OK. So did that help answer the question? OK. So the second point here, we want to first off describe phenomenological band gap and understand how that works. Secondly, we want to describe optical transitions in the semiconductor and understand how charge carriers, in other words, how electrons on an energy band diagram are moving around. Why do we call electrons charge carriers once again? Remind me, from this diagram here. Exactly. It's brilliant. I just want to make sure that those two things are synonymous in people's minds, that we're not getting caught up on the language. Thank you. So again, charge carriers, electrons, being free electrons, unbound electrons, not any electron, not the ones down here, but the ones that are excited. OK, so let's go back to lecture number two, I think it was, where we talk about the duality of light. It's a wave and a particle. Well it can be thought of, can be described mathematically as waves and particles. We've, in last lecture, when we talked about lights and the interaction with lights with materials, we referred to light mostly as a wave. This was very easy for us to describe interference. It made it convenient to do the homework problems, especially for the graduate students, the last problem. That was a lot of fun But now we will be talking about life in terms of discrete quanta of energy, because one photon is going to be absorbed by an electron, an excited electron, into another state. So now we think of the photon as having a certain energy, defined by the wavelength of light, and that energy is going to be given to the electron, which will then be excited inside of our semiconductor crystal. So what happens? Well we have our sun. Here we have our semiconductor crystal. Again, I'm representing it in terms of e versus x. I'm saying that these states here are filled. And these states here are mostly empty. And these are mostly filled. Mostly empty. And this is my band gap right here. I'm plotting e in the ordinate and x real space in the abscissa. And again, x because, ultimately, we'll be looking at the cross section of the solar cell in x, in describing how charge gets extracted from the device. So that's why we're going to be using x on this axis. Now we have sunlight here, outside of our device. And it's going to be shining light onto the semiconductor crystal. So what happens? Well, we know intuitively that charge should be excited from a bound states to an unbound state, where it can move around the material. But not all light serves this function. If we have a photon that has an energy above the band gap energy, so e photon, meaning the energy of the incident photon is greater than e gap, the energy band gap, then we'll have an excitation of charge. We'll have an electron taken from somewhere here, brought up to a higher excited states, and it might come to settle down at the bottom of that band of states that is allowed. A settling, if you will. But if our photon energy is less than our band gap, we won't have this process occur. Instead we'll have the light going straight through it. And this is brilliant, because you can begin to understand the behavior of many solids in this way. Let's take a glass, for instance. How many you have noticed that you never get sunburned when the windows are up? But you get sunburned when you forget to put the windows up, when you're driving around and your arm's sticking out, you get that nice truckers tan. That's because the ultraviolet photons have enough energy in glass to excite electrons across the band gap. So the ultraviolet photons gets absorbed by the glass, but the visible photons instead are coming straight through, much like this. Now glass has a very large band gap. Now silicon, much smaller band gap. And in that case, in the case of silicon, even the visible gets blocked. So silicon would make a really crummy window. But it makes a really great infrared window. So if you're detecting infrared light, and want to block out all the visible, you might consider sticking up a filter made of silicon. And that's regularly done in laboratories. So you can begin to understand how light interacts with matter, with solids, using this energy band diagram. And ultimately how these optical absorption spectra come into being. So we studied these optical absorption spectra in our last class. We just took them as a given. We didn't question where they came from. Now we're going to be focusing on this so-called turn on energy, or turn on wave length. Remember wavelength and energy are interchangeable. I've written down below here the energy corresponding to a given wavelength, around 2000 nanometers. We have an energy of around 0.62 eV. At around 200 nanometers, we have an energy of 6.2 eV. Since one is the reciprocal of the other, related by hc, it's only natural that if we change one by an order of magnitude, we're changing the other by an order of magnitude in the opposite direction. So we're wrapping our minds around this interchangeability of wavelength and energy. That's important, because we'll be referring to them very naturally as a photon either having a certain wavelength or a certain energy. You notice that there are certain turn on energies. So energy is increasing in that direction there. So as we go from infrared to ultraviolet, the energy of the incoming light is increasing. And if the energy is too low, the semiconductor crystal just won't absorb. The absorptance is 0, or very, very small. It doesn't even appear on the log plot here. And as we begin increasing the energy, suddenly there's like a turn on. The semiconductor begins to absorb that light, and you begin to have this process occur right here. You begin to go from that red light that went straight through the material, to this blue light example here where we have absorption event. Yeah? AUDIENCE: This is probably a minor question, but where does the energy go-- that it's released when your electron settles back down? PROFESSOR: We're going to get to that. We're taking it step by step. That's a really good question. You're one step ahead of me. We're going to be describing the actual shape and character up here, because this is really equally important for the functioning of a solar cell device, over the next few lectures. But that gets really complicated. So for now, we're going to just focus on this turn on energy. And we're going to make some assumptions from here on out that the semiconductor doesn't absorb, and all of a sudden it absorbs, and the absorption goes high. It begins absorbing all of our light, for purposes of our homework assignments. So again, just to put this all into one big picture, so far we have the case out here where very little light is absorbed by our semiconductor. We have the case of glass, light coming straight through it, visible light rather. Because the incident photon energy is less than the band gap, there is insufficient energy to excite into the conduction band. Now as we transition from a photon energy less than the band gap to photon energy larger than the band gap, we can begin exciting electrons or charges across the band gap, and that's why we get absorption inside of our semiconductor materials. Any questions so far? OK. Good. Everybody's still with me. We're going to now calculate the fraction of photons lost, not absorbed by a semiconductor material with a given band gap, thickness, and reflectivity. And this gets to Ashley's question right here. So again, we ran through a thickness estimate in class the last time. And we assumed for a semiconductor, I think it was around 800 nanometers. For gallium, arsenide, and silicon. Or maybe it was a little less, somewhere around 550. What was the thickness necessary to absorb say 90% of the light at a given wavelength? And we, in the back of our minds, have the solar spectrum here as a reference point to begin visualizing the number of photons that occur within each of these delta wavelengths down here. Essentially what portions of the spectrum really matter? Obviously you're not going to optimize your solar cell to absorb light way out here, because there's really not too much of the solar spectrum way out there. You're probably going to optimize it somewhere around the peak of the solar spectrum, somewhere near there. So we walked through that in the last class. I want to have that in your ram as we move forward. We want to calculate the fraction of incident solar energy lost to this thing called thermalization that Ashley was mentioning before. We have, in this simplified diagram right here, an electron popping up to a higher energy level and then settling down to the bottom of this conduction band, we call it, this band of states that's empty right here, or largely empty. And this settling down to the bottom of what is this mostly empty band of states is a process called thermalization. And it occurs, well, because there are a series of states the electron can very easily move between them by emitting phonons, or lattice vibrations, or heat. And very quickly that electron will settle down into what is called or referred to as the conduction band minimum, the minimum energy level within the conduction band. And this little arrow shown in red right here has a component in the y-axis, in e space here, which is a finite energy. We can quantify that. That thermalization loss right there is the difference between this energy level and that energy level of the electron. And so that's the thermalization loss. That is another type of loss of a semiconductor crystal. So we've discussed two so far. We've discussed non-absorption of light. There are certain long wavelengths, low energy light, that go straight to our material and doesn't get absorbed, because the photon energy is less than the band gap. And the second major loss mechanism is when we have too much energy of the incoming photon, we have an excitation of a bound electron into a very highly unbound state, and then a quick loss of that excess energy due to a process called thermalization. Any questions so far? AUDIENCE: You said that the thermalization can result in heat being released or phonons-- PROFESSOR: Mh-hmm. Yeah. So not the prime energy that we extract from the solar cell of electricity. In principle, if you could develop some clever way of extracting heat, perhaps by slapping on a thermoelectric device in the back of your solar cell, then you could extract that energy as well. So there's a potential to recover that heat. It's not total loss, but from a mechanical engineering point of view, if you think about the thermodynamic efficiency of a Carnot engine, where your t high is this, and your t low is your ambient temperature, that delta really doesn't look that good. Yeah. It's really tiny. So you have to think of a more clever mechanism, perhaps through a thermoelectric device, or some other mechanism to extract that extra thermalization loss. I wanted to-- so what we're doing is we're taking the simplest picture, and then adding layers of complexity. And you'll hear me stumbling over my language, as I sanitize it in my head, to remove all of the complex techno lingo, and reduce it to its essence. And so right now I'm going to add one more layer of complexity, and ultimately build up to the point where I can feel conversant again. We have incoming light, not exciting the electron all the way up to there from the top of the valence band, but typically somewhere within the valence band. Not really at the valence band maximum or the maximum energy level within the valence band, these being the filled states. So this is a more realistic representation of charge excitation. We have an electron being excited up. And this little h plus that's right here, who did the reading and know what that h plus represents? A hole. So what is a hole? It's just what it sounds like. It's the absence of an electron. There used to be electron here. Now it's up there. It left behind a hole. An easy way to think about a hole is if you're in a big traffic jam, bumper to bumper, something bad happened up ahead, and backed up the traffic. Now imagine just removing one car from that highway. You've created a hole. So the car that used to be behind that hole moves forward. And the car that used to be behind that one moves forward. And the car that used to be behind that one moves forward. So three cars moved forward. And the hole moved backward three places. Right. So you can either describe the dynamics of the system as n number of cars moving forward or one hole moving backwards n places. It's much simpler from an accountability point of view to be talking about one quasi particle, a hole, rather than talking about n cars moving forward, n electrons moving. So when we talk about this hole right here, thermalization is going to drive this electron to the lowest energy level within the conduction band. And it's also going to drive the electrons above the hole, essentially to fall down into the hole. In other words, the hole is going to move up to the top of the valence band. Yes. Question. AUDIENCE: Why isn't the hole originally created at the top of the band? PROFESSOR: Well, it really depends on what's called the matrix element of the absorption process. In other words, what electrons have the largest capture cross section for that incident photon. And it is a probability distribution function. So you will get some electrons further down from the top of the valence band absorbing light, as well as some of the electrons right there at the tippy top of the valence band absorbing light. So I'm representing, say for example, a typical case, where you have an electron that's not right at the top of the valence band, and certainly not deep in the core level, but nearish enough to the top of the valence band absorbing that light being excited across. So in reality, this is meant to be an arbitrary scale here, but more representative of a whole process. And to think about this in a more realistic sense, what you would do is you think, OK, I've this incoming photon, and there's a certain probability distribution function that represents the probability of absorptance by different electrons in my system. Which one is going to absorb the light, let's roll the dice. Do a Monte Carlo. OK. That one absorbed it this time. That one got excited up. OK. Another photon with identical energy coming in. Which electron absorbs it? That one. So that's the way I would say it really works. This is a simple representation on the lecture slide. So we have the hole going up, and the electron going down. Both particles are relaxing. Right. So the electron is moving to its lowest energy state and the hole is also moving to its lowest energy state. So if you want to think about the electron as a bowling ball, and the hole as a balloon, you're welcome to. Whatever mechanism helps you think through this process, use that as a crutch right now as we move forward. Always remember that in the system, these are electrons essentially moving down, and filling up that hole, and the hole is moving in the opposite direction, much like the vacancy in a traffic jam. So again, thermalization losses can be described by both electrons and holes in our system, by both the rattling around of an electron in the conduction band and the settling of electrons here in the valence band as well, or the rising of that hole toward the valence band maximum and the settling of that electron toward the conduction band minimum. So a natural question is, how fast is this process? Can it be reversed? If we're losing all this energy due to thermalization, can somehow we halt it and stop it from happening? There are people trying. It is a valiant fight. This plot right here represents what's called the density of states. We're going to get to that in another lecture here, but I'm going to expose you to the rough concept here. Y-axis is energy. So again, we have our valence band maximum and our conduction band minimum up there. X-axis here is representing time. So we're going from the excitation event, which occurs right at t equals zero. Zero plus represents what this is on the positive side, moving time forward from the excitation event, so instantaneously after the execution event. We had to the equilibrium population of holes and of electrons inside of our semiconductor system. There weren't absolutely zero electrons in the conduction band, because of heat. There was enough thermal energy in our system to excite some of those electrons across. And that's why we had that small population of electrons there and holes down here, the holes they left behind. Now we had a photon coming in, high energy photon, that excited these electrons down here up there into the valence band. Up into here, in the valence band. And over time, this excited population will decay down to the conduction band minimum and valence band maximum, the electrons and the holes respectively. And over time, when I say over time, I'm referring to something in the range of one picosecond, 10 to the minus 12 seconds. That's like that, but faster. So we have a quick decay of these photoexcited carriers, and so extraction of that energy is going to be nearly impossible, unless we do something very clever with our crystal to prevent that decay from happening. Somehow we suppress the phonons from being emitted at that frequency, at least, or at that energy. So there are a lot of people, clever people, working on this problem and trying to prevent the decay. The next challenge, of course, is extracting those so-called hot carriers, the excited carriers, from your device and keeping that energy when they're going into the metal. That's a whole another can of worms. So we have the time scales of thermalization, and this is why, at least from the perspective of your homework, we're going to treat thermalization losses as inevitable. As Harvard and MIT students, I would urge you to never consider anything is inevitable. If we understand the physics well enough, we can probably engineer a solution. But for now, let's consider this a reality, a loss mechanism. So if we start putting things together, if we put non absorption, which we talked about last lecture, we actually ran some calculations, back of the envelope, and we calculated the thickness necessary to absorb 90% of the photons at a given wavelength. We also know how to calculate reflectance off of a front surface. And so we get this third point right here. We actually got it last class. And the fourth point right here, we intuitively understand now that if the photon comes in with more energy than the band gap, that excess energy is going to be lost due to thermalization. So now if we have the solar spectrum, and we know the band gap of a semiconductor, we should be able to do a very cursory plot of the efficiency versus band gap, versus energy. So again, we have thermalization losses. The band gap is too small. And non absorption losses, if the band gap is too large. And let's just do an absurd thought experiment. If we say, OK, I'm really, really scared of thermalization losses, so I'm going to make the biggest band gap material I possibly can, I'm going to have a very low efficiency in my cell, because I'm not absorbing any light. The solar cell will be transparent. If I'm scared of non-absorption losses, I say, OK, I'm going to make my band gap really, really, really tiny. I'm going to be losing a heck of a lot of energy due to thermalization, due to this loss mechanism right up here. And so there has to be some happy optimum, somewhere between the two, where we have the maximum potential efficiency of a solar cell device, given our solar spectrum. We're going to walk through that right now. So approximating non-absorption losses, very first step. What we're going to do, for our non-absorption losses, is run a very quick approximation that any photon with photon energy corresponding to the band gap energy is going to be absorbed. So if our photon has a larger energy than the band bap, it will be absorbed. If it has an energy less than the band gap, it will not be absorbed. And that's what this plot right here is representing. We have wavelength right here, longer wavelength, lower energy. Smaller wavelength, larger energy. And at some point, we have the turn on of absorption of our device, because we have band gap at that energy. And the y-axis here, I've plotted EQE, which is External Quantum Efficiency. I've coined the term over here on the left-hand side. It's the efficiency at which free charge carriers are generated by an incident photon on the device. So one way to think about it is if I have a EQE of 100%, that means that for each photon that I throw at my solar cell device, I'm generating and collecting one free carrier from that device. One electron hole pair, if you will, from that device. So that's plotted right here. That makes intuitive sense. This is an approximation though. And I wanted to take it one step further, again, planting the flag right here. Because we're going to come back to this, we're going to use this. But I wanted to go one level deeper into the trees, for everyone else who wants some more advanced concepts. This is the reality of how quantum efficiency looks. Yes, we have a turn on, depending on the material that's involved. We have silicon, gallium arsenide, copper indium gallium diselenide, amorphous silicon, dye sensitized solar cells, organic solar cells, a variety of different materials right here. And their different turn on wavelengths right here. So we have wavelengths, different energies. Some turn on at lower energies, others at higher energies. So this represents, more or less, the band gap turn on energy, more or less. It also has to do with the thickness, how it absorbs light. At the shorter wavelengths though, instead of just having a QE of one, going all the way out to x-ray territory, we have a turn off at some point. Why is that? Why in a real device would we-- Yeah. We have glass absorbing, on the front side, that's one real big reason. Do we have another idea? Yep. Glass absorption is one real big one. There are other dead layers inside of a solar cell device in the near surface region that also absorb the light in some architecture. So it's not a perfect QE spectrum that looks like that. But why don't we care? For the purposes of just engineering approximation, why aren't we bothered by these photons that we're losing down there? AUDIENCE: Because the percentage of the solar spectrum that falls in that region is small. PROFESSOR: Absolutely. So what we're doing is an engineering approximation right here to get to a very first cursory efficiency calculation, neglecting things at the extremes. Two short wavelengths, because there's just not a whole lot of solar flux down there. And too long wavelengths, because again, not a whole lot of solar flux right there. So we're going to be using this approximation in our homeworks and that will get us somewhat close. If we really want it to be true, instead of using a box function like this, an absorption box function, we would use an absorption spectrum that looks something more like that. We've used a two absorption spectrum. And then convolute that with the thickness of the device to calculate what fraction of photons at each energy is absorbed inside of my device. And that would give you a plot that looks something more like this. Instead of this box plot up here, it might give you plot that looks something more like this. So you'd have instead of just a sharp turn on, if you had a one micron thick silicon wafer, and we calculated last class that we need somewhere around 10 microns or 100 microns to absorb light well, if we reduce the thickness of the wafer to 1/10 or 1/100 of what's necessary to absorb light well, we don't see that sharp turn on at the band gap. We see rather a gradual turn on of our silicon as we move to shorter and shorter wavelengths that are absorbed more and more efficiently by the device. So again, just wrapping our head around the basic concept that we're going to use for the purposes of our calculation, but, again, some of the more fine structure, some of the more advanced concepts that come into play when we're doing more detailed calculations. Approximating thermalization losses now. We want to calculate the amount of energy lost of incident sunlight lost due to thermalization, due to heat. And we're going to say here that if the photon energy is greater than the band gap, than the photon energy is approximately the band gap plus thermalization. Another way to put it would be the thermalization is approximately the photon energy minus the band gap energy. I kind of wanted to hide all this down here. Forget this exists right now. Let's focus on that top part. So that's the easy way of thinking about it. We're going to be thinking about this in terms of just any photon coming into our device with a larger energy than the band gap is going to generate one electron hole pair and lose some energy due to thermalization. The reality is that for very high energy photons, say for example photons with three times the band gap energy or more, you could have electron-electron interactions. So if you excite one electron to very highly excited state, it can bounce around and in the process excite another electron across the band gap. That's the case when x-rays are incident on a piece of silicon. For instance, if you have a 10 kilo-electron volt, so a 10,000 eV x-ray incident on a piece of silicon, you will generate approximately 3,000 electron hole pairs with that one x-ray. Because that one x-ray is going to take an electron and excite it to very highly excited state, and that's going to excite further electrons across the band gap as that excited carrier decays, as it loses it's energy and settles at the conduction band minimum. So we have what's called multiple exciton generation. Sometimes we've heard it as multiple free carrier generation. In semiconductors, it's a hot topic, because folks would like to take the ultraviolet portion of the spectrum, the high energy, short wavelength, portion of the solar spectrum, and use it to excite multiple carriers. One photon exciting multiple carriers across the band gap. And so there's some work, or was some work, it was a hot topic for a while. It's kind of decayed a bit. It comes in waves. It cycles, kind of like a plane wave. There is an interest, a general interest in the field, of how do you capture these higher energy photons and convert them in some usable way, instead of just having thermalization loss, instead of just having heat. And then for really high energy photons, gosh, you can really get into some neat physics. I've given you a link and cross section here. This is the electron cross section, or actually really what it is is an anatomic cross section, because of the electrons in the material versus photon energy going through 10 eV out to 10 to eV 11. A very high photon energy. We're transiting from the visible spectrum over here deep into the infrared, and then going into the x-ray, and finally to the gamma ray regime. There you can have a number of interesting phenomena. You can even have an electron hole, or electron-positron pair generation with gigaelectron-volt incident radiation or above. We're not gonna even talk about this regime, because the total amount of solar flux in that regime is really tiny compared to visible. But it's a lot of interesting, fascinating physics. So I put it on here anyway just to keep our approximation in focus. So again, I'm kind of mixing the forest in the trees here to give you a sense of the complexity. But also to give you the tools necessary to do simple calculations. We're going to combine this simple approximation right here of the non-absorption losses. We're going to say that any photon with energy less than the band gap doesn't get absorbed by our crystal. It's gone to us. And we're going to make the approximation that any photon with energy above the band gap will have that excess energy loss due to thermalization. And we put those two things together. And one very convenient way of representing this is shown in a paper in 1980. Let me walk you through it. So this curve right here, this outermost curve, represents the solar spectrum, energy versus number of photons. So we have essentially a photon density. This is number of photons per centimeter squared, per second. So photon flux, if you will. And over here we have the energy of those photons. And so what this is meant to represent is the cumulative solar spectrum, going from a high energy here all the way down to the lowest energy. So we're essentially adding the different components, cumulation plot. And if we have a photon energy in excess of the band gap energy, as would be the case, say, for example three eV photon, and let's say the band gap is 1.35 eV, represented by this straight line right here, this energy is going to be lost. And so by drawing this cumulation plot, we're plotting the area here, the energy area that is lost to us due to thermalization. We could also say that if the photons have an energy less than the band gap, that is going to be lost to us as well. And what we're left with is a little box that represents the total usable photons, the usable photons that we can extract device. This particular plot is a little bit complicated, because it contains two curves. One is this one right here. The other one is this one right here. This second curve represents another realistic loss mechanism that we didn't talk about yet. And that is why the extractable work, represented as w, is going to be less than the band gap energy of our solar cell device. So if the band gap is 1.35 eV, the total usable work would be 0.9 eV, we'll get to that in a lecture or two. And that's why the white box here is going to be smaller than this horizontal line box over there. But it gives you a very first approximation, a very easy way to describe the different losses that can occur in a solar cell, and quickly visualize them here, thermalization. Over here, non-absorption. And plot them out very nicely as the total area and representative area fractions. And that's why you can see here that the total usable solar energy is going to be somewhere around 30%. This band gap right here, around 1.35 eV is actually pretty near optimal for the solar spectrum. You can imagine if we increase the band gap energy, we would be doing something like this. So we would draw another box right around here. This would be the new band gap, instead of 1.35, that would be out here. If we increase the band gap further, we might have something that looks like that. And the area of this little rectangle would be much smaller than the area of this rectangle. Likewise, if we said, OK, we're going to shrink our band gap energy to avoid non-absorption loss, we're going to have an excess of thermalization loss. We'll have a very narrow rectangle, right, like this, and all of this would be loss. The solar spectrum, obviously, does not change. This line right here this is fixed, it's constant. You folks kind of see it? Beginning to grasp it? This is a convenient way of representing the total usable portion of the solar spectrum from a given semiconductor material. You can begin seeing the trade offs between different material systems. You can say, what happens if I change my band gap? How would that change the fraction of photons, or fraction of photon energy not absorbed? Or fraction of photon energy lost due to thermalization. And you can also see what happens if you-- what would be an easy way to extract more potential for the sun? Instead of just using one semiconductor material, you? More than one. More than one. Right. So now you're drawing different boxes that represent the different materials. And if you stack them right, you're able to capture more of this total area, more of the total solar spectrum. So this is a really cool plot. A lot of good work back in the 1970s, 1980s. If you recall from lecture number one, this was the first wave of real hardcore solar science, after the initial Bell Labs invention. When we had the OPEC oil crises, and there was a large rush of funding into solar research. So you had a lot of good work coming out from those days. This next plot that I'm going to show you represents the area of this box relative to the solar spectrum, the percentage of light that is usable, the balance between non-absorption thermalization losses as a function of band gap energy. So if the band gap is too small, again, we're losing a lot of energy due to thermalization. If the band gap's too large, a lot of energy we're losing to non-absorption. And somewhere in the middle is our happy medium. These little symbols here represent different materials. Cadmium sulfide, cadmium telluride, gallium arsenide, indium phosphide, silicon, germanium, and so forth, and calculated efficiencies. So I'm going to go into one advanced concept, the multi-junction devices. And we just talked about this box here representing the usable energy from the solar spectrum. Now if we go to an absurd case of 36 band gaps. If we imagine 36 materials with graded band gaps, starting from large band gap at the top to small band gap at the back, you can envision almost approximating the entire curve. And you can estimate what the upper efficiency limit would be with many band gaps inside of your material. So for one band gap, somewhere around 37, we'll have an entire lecture dedicated to calculating that number. For two band gaps somewhere around 50. For three band gaps, 56. For 36 band gaps around 72. So you can begin seeing how the efficiency of solar energy conversion changes as you change or add materials, you move materials to your system. So how do you practically do that in a real device? Well one method is to put one material on top of another, much like it's demonstrated right there. You have E versus x. So I have my E versus x, and I have multiple materials. Another way to do it would be to use optics to split our light into different colors. So if we have polychromatic or multi-colored light coming into our system, we somehow have a set of optics put up that reflect one color, or one band of the solar spectrum, while letting the other portion of the solar spectrum through it. And we have different solar cells with different band gaps, each matched to the incident light. Each matched to the particular color of light. Notice this paper, 1982, this concept was out. So again, during that first wave of real photovoltaics innovation in the late 1970s and early 1980s. And this was coming right out of Lincoln Laboratory. This is about 35 minutes north of here. And this group at the time, the group number continues to exist and is in operation at Lincoln Labs. So most of the people have moved on obviously, but several of the ideas continue to this day. There was a more recent incarnation in 2009 of a spectral splitter, a very similar concept as you can see, optics to concentrate the light, a dichroic mirror, meaning it reflects certain wavelengths, let another portion of the spectrum past, and solar cell devices down below absorbing efficiently in that region of the spectrum and avoiding thermalization losses, and avoiding non-absorption losses. This was a $50 million DARPA project, in fact, involving University of Delaware and several other companies. And they published the results in 2009. It was one of their wrap up papers that described the results of the project. So concepts. To this day, this spectral splitter it's not in commercial production. What is in commercial production are the multi-junction devices, where you stack one on top of another. And we'll actually be having the benefit of using some of them in class. Boeing spectral lab was kind enough to donate a set of them for our class purposes. Yes, question. AUDIENCE: This is also basically what plants do by having multiple different pigment molecules in leaves, right? That would absorb the different wavelengths? PROFESSOR: Well, it's not exactly separating light by optics in that case. So the question was-- AUDIENCE: No, I mean multi-junction. PROFESSOR: Oh, the multi-junction idea. Yes, if you consider that there is a reaction process occurring in series. The important distinguishing feature of the multi-junction device is that the same current is flowing through the entire device. So each sub cell has to be current matched to the others, because the cumulative current output is a harmonic mean of each one. So in other words, you're limited by the worst resistor in series, if you will. So it's very tricky to engineer properly these multi-junction devices. You have to be thinking about the current output for each sub cell and match the currents. That means you match the geometry, the thickness, but also the resistivity of each layer and so forth. So similar to a plant, but not quite. There are some special characteristics of the inorganic system. So we have a small demonstration. I'd like to welcome Joe up to the front as well. We have a small demonstration of a few of the concepts that we've covered today in class. We're going to be exciting silicon with light, and we're going to be monitoring the current output using this little resistance measurement device right here. So actually I think it's set to a current readout. Yes, current readout. So it's in microamps right now, probably going to milliamps. Is that right? OK, so milliamp current readout. Let me explain to you what we have right here. We have a bare piece of silicon, and two electric leads on either side. So a bare piece of silicon, no device. Just a piece of silicon. And we have electrical leads coming out of either side. We have a very low resistance contact on either end. For those who are curious, we used a mixture of indium and gallium, soldering iron, scratch the surface, penetrated the native surface oxide, and put the electrical leads on either side. And we have this connected in series here to our current readout device. So now what we're going to do is we're going to illuminate this. And again, bare silicon leads on either side, and we're going to be passing the current through the current measurement system. Before we turn it on, we're going to ask people, how many people expect there to be current driving through the system? We have light incident on the material. We're exciting charges. Based on today's lecture, there should be free charges moving around that material now. How many expect a current to flow? A few. But a lot of people shaking their heads. Why do you think a current won't flow? There's no electric field. There's no potential. Why don't we give it a shot, and why don't we see. If we turn this on right now, what do we see? We see zero. Right. Can some of the folks right here in the front see? Says zero. All right, so what this is telling us again, you're welcome to come up after class and take a closer look. What this is telling us here is that yes, what we talked about in class is important today. Yes, it's all very important and it's the foundation of calculating ultimate solar cell efficiency, how the material absorbs the light, how charge is excited. But we need something else too. Once the charge is excited, somehow we have to give it an incentive to leave the material. We have to have a field. In this case, we're going to be using an applied field, a couple of batteries. So total voltage around three volts, applied across that small material. But in a solar cell device, we're going to have a built in electric field that we'll engineer into the device. And we'll talk about that over next lecture. So just for purposes of sanity, we have our batteries now connected in series, and is now applying a potential across those two leads. And now if we turn on the light, what do we expect? We expect to see some current going through. And voila, we have a current running through it. So now, what is the nature of that current? If we move the light closer, we see the number go up. What is the nature of that current? That current is photo excited. That current is exciting electrons from the valence band, from bound states, into the conduction band, it's unbound states, where they can move freely across the material. And now, because we have that potential applied across the material, there's an incentive for them to drift in one direction. The net flow of electrons is in one direction. We have a drift current in our material, in our semiconductor, and hence we have the flow of charge that is readable by this current meter right here. So without the light, we have no current. With the light, we have current flowing through the material. But current and light-- actually light is not the only thing necessary to create that output current. We also need the potential. So next class we'll be talking about the potential and how that's created inside of a solar cell device. I've included a few slides extra for people to see through. I'm going to explain once again the experiment. We just had light coming in. It was in the visible. We were able to excite carriers, but no current was observe because we didn't have a potential. When we applied the battery, or the potential across, now when we shone light of the sample, we had a current flowing through. And we even have a current, a very small one, flowing through when we have the battery applied without the light on. And you need microamp detection I think to really see it. Yeah. Just from the battery. These are a very small population of excited carriers, either thermally or donated to the electron conduction band. Any questions before we close for the day? Yes. AUDIENCE: Yeah, so I guess-- I'm the devices person in here, so also the notion of the direct band gap, versus an indirect band gap, are we going to discuss that? Or is that something that's not relevant? PROFESSOR: We will definitely discuss direct and indirect band gaps. So this goes back to describing the notion of a direct and an indirect band gap is fundamental for describing the reason why these curves have the shape they do. For example, this one right here, if you plotted as alpha-- if you plot it in a certain way, you'll be able to see a very characteristic shape, indicative of direct transition into a direct band gap. The reason we aren't launching into several of those terms right now is because several of your colleagues, more than half, don't have a semiconductor physics background. And the beauty of this approach is that you will be able to understand most things about a solar cell without really diving deep into the semiconductor physics until it's necessary. And hopefully we keep everybody with us. So for those who have a strong semiconductor physics background, I bid you, I urge you to have patience, because we will get to the interesting stuff. But we're working on that fundamental background right now. Thanks. |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | 12_Thin_Films_Material_Choices_Manufacturing_Part_I.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: So folks, we're going to get started into thin films for a moment, but I saw two of you, at least, in the class at Eli Yablonovitch talk on, gosh, what was it, Tuesday? Tuesday is was. How many attended the talk-- show of hands? Three? OK, three, awesome. I must have missed one of you-- very interesting talk. This was a talk about solar cells given from the perspective of somebody who does light management. And so I wanted to share with you a book that is essentially from where he takes his efficiency calculations, which are based largely on thermal dynamics and less on the continuity equations-- Peter Wurfel's book Physics of Solar Cells, a brilliant, brilliant book. I'm going to pass it around. On page 33, very easy number to remember-- 2 times 3, 3. On page 33, he starts delving into the derivation that Eli Yablonovitch presented during his talk, so folks can follow along from a thermodynamics point of view and maybe read up a little more and understand that perspective. But he very, very briefly touched upon essentially the same physics but from the perspective of what we've been talking about a class in terms of carrier densities and current flows. He had it on the bottom of a slide, perhaps halfway through the talk, on four different bullet points. Does anybody remember what those were? Why did he achieve such a high efficiency conversion efficiency with the gallium arsenide cell? Anybody remember that one? He had a thin device, so by thinning the device down, if he's able to concentrate the carriers, in other words, if he's able to collect all of the charge carriers inside of that very thin layer, he'll have a higher charge carrier density. And the charge carrier density is what influences the separation of the quasi Fermi energies, which is what influences the voltage output of the device. So he was able to obtain a higher voltage output because he had a thinner solar cell. He was able to concentrate the carriers in that thinner region by light trapping, by light management. And so as a result of having a higher carrier concentration, he had a higher separation of the quasi Fermi levels and hence a higher voltage output of his device. So in reality, it was very simple from the perspective of what we've been learning in class here, how he was able to obtain the very high efficiencies of gallium arsenide. The physics is well known; it's not new physics. It's actually quite old physics, and that that approach has been used within the crystalline silicon solar cell community for some time as well. The back surface reflectors off of the devices are highly optimized and the texture, as well, to scatter the light. So I would invited you to take a look, and this is another example of how technologies can flow from one photovoltaic system into another. So you can learn a lot from material systems that you aren't working on necessarily yourself. That's another take-home message from the talk, at least what I walked away with. Any other impressions that folks would like to share before we dive into the lecture? Yeah. AUDIENCE: I just have question about carrier collection. How is it possible to extract any energy from carriers which are generated in front of the junction? Because even if they diffuse another junction, they have nothing to fall down? PROFESSOR: OK, so you have to think about it always from the perspective of the minority carrier. So if you generate an electron-hole pair, your minority carrier is now a hole, in the n plus region. And that hole diffuses across the junction. The electron stays. AUDIENCE: I see, OK. PROFESSOR: So did anybody else pick up on the point at the very beginning of his presentation? He said a P-N junction isn't necessary to separate charge. OK, that's fine. We've talked about heterojunctions. We all agree there are other ways to separate charge. And he said an electric field is not necessary to separate charge, but then he immediately went into discussing how the chemical potential was slightly lower in the contact than it was in the semiconductor, which would result in a charge imbalance, which would result in a field. And I think Gene Fitzgerald from material science and engineering department-- Professor Fitzgerald called him out on it and said, isn't there a field there at the metal contact. He said, quiet, wise guy, we'll get back to you later. But essentially his point was a very small electric field is necessary. So his point was a matter of degrees, that you don't need a massive electric field. A very slight field is all that's necessary to start driving a current through your system. I just wanted to make sure we didn't leave that talk thoroughly confused with our head on backwards. We're going to talk about thin film materials today. Why thin film solar cells? Well, we've been talking about crystalline silicon solar cells that have a lower optical absorption coefficient, so you need a larger amount of material, or a larger optical path length, to absorb a significant fraction of the light. Already, in lecture number two, we saw how other material systems that have higher optical absorption coefficients are able to absorb this equivalent amount of light in a thinner amount of material and less material. So to put this in perspective, what we're talking about on one hand with the crystalline silicon devices is we might have a device that's maybe three or four times the thickness of your hair in crystalline silicon, and for the other materials, so these thin film materials, you might be talking about a material absorber that has maybe 100th the thickness, so something under a micron or a 50th of the width of your hair. So that's the perspective of scale that we want to have in mind. When we're talking about thin films, we're talking about thin materials, really on the order of one micron or so. And even brittle materials, at one micron thickness, if deposited on compliant substrates, can be flexible. Another thing to keep in mind in thin film technology is that the scale of the thin films industry is about 1/10 that of silicon industry right now. So the crystalline silicon industry is going full force, gangbusters right now, and the thin film is a growing fraction, but it's on the order of 10% of the total world market. And many, many, many startup companies, which are young, dynamic, fun-- and that's why today I'm not wearing a tie; I'm in startup mode. I'm a lot more relaxed. We're going to be talking about thin film technologies and diving into some fun work. So we'll talk about these specific technologies of thin film materials, and before we get into those, I'm going to address some general topics about deposition and, of course, general parameters that affect all thin film material systems. We have to appreciate the sheer diversity of technologies that are out there on the market. We have a variety of different solar cell materials that are available, some of which are thin films, other ones, wafer-based crystalline silicon. And all of these technologies have to consider cost resource availability and, eventually, environmental impact as well. So these or some of the things I'd like you to keep on the forefront of your mind as we talk about these different technologies. Think about the broader picture, and ultimately, this cost, or the amount of money per unit energy produced, is really paramount in determining marketability and determining the scales to which they'll penetrate the market. This is the one slide that you have printed out. You have one per pair of students, or you should. So if you don't have access to that particular slide, feel free to share it with the person next to you. This is representing as a function of time, going back to the 1970s, the record solar cell conversion efficiency. The chart is maintained by a certain Larry Kazmerski at NREL. Actually, he used to be the head of NREL's solar program. He stepped down a few years ago after a very successful run-- many years. And he's a bit of a father of the US PV industry. He's been around for a long time and has been tracking the growth of the PV industry and, of course, the improvement of performance over time. Many of these devices-- many of these record efficiency devices-- are very small area, and many of them were actually grown for the intent, the explicit purpose, of getting onto this chart. And so when you're trying to make a record efficiency device, you do things a little differently. Let me give you an example; you'll optimize your anti-reflection coding for air not for glass, right? So if you want to minimize the reflectance off the front surface, you'll be optimizing it for air, which has a refractive index of one, as opposed to glass, which has a refractive index of 1.5. So there are some tricks that one does and engages in, some are a little bit under the table, too. It has been done in the past that people would do an HF dip of their silicon-based solar cells right before measuring the efficiency. The hydrofluoric acid would result in surface pacification, but, of course, it would result in a very low surface adhesion of the metal, and so the metal flake off afterwards. It wouldn't pass the tensile, but you would nevertheless achieve instantaneously higher efficiency. Those practices have largely been weeded out. These were the early days, when it was a wild west of solar cells. In more recent times, there are some very strict standards, and there are only a few laboratories around the world where you can take these standard measurements. The one at NREL is extremely well staffed in terms of the quality of the people. They're notoriously under resourced, but that's another issue. But in terms of the quality of the people there, very, very good, very thorough, very pedantic and careful about taking their measurements. And if you ever have a question about how to perform a solar cell efficiency measurement, they're a very good resource. Their website would be an excellent place to go. So these data points here versus time represent the record cell efficiencies. They may be on very, very small pieces of material. They may be on a centimeter squared, perhaps even smaller, so they're not necessarily representative of what is in commercial production today. Let me give you one example; the record crystalline silicon cells, which are in blue here, has been around 25% for about a decade-- actually, a little more than a decade-- and the record efficiency crystalline silicon device has, in essence, not been so planted for many, many years. There are a number reasons for that. It's very much approaching its theoretical efficiency limit. People haven't necessarily tried specifically to get a record efficiency crystalline silicon device. They're more intent on making lower cost silicon devices than record efficiency ones, and the average module efficiencies are somewhere down at around here-- actually, somewhere in the 13% to 15% range for module, average module efficiency. You have some modules that are in the 18%, 19% 20% range, but most of them are significantly lower. And the record cell efficiencies, as you can see, is 25%. So there's a significant delta between what is commercially available and what the record cell efficiency is. There are several reasons for this. To make a record efficiency cell, you have to throw everything at Liebig's law of the minimum. You have to make sure that every plank is really, really high. That costs a lot of money typically, and so doing that cheaply is a big challenge. Some companies, like First Solar, for instance, has some of the lowest cost models in the market. We'll describe how they're made in a few slides. First Solar forwent the anti-reflective coating on their glass for many years, because it just didn't make cost sense. It didn't help optimize this function right here. Although, you'd get more energy out, the dollars that it took to add that component just didn't make sense for them. So you have to think about a few different perspectives. You have to think both in terms of cost, and in terms of performance. The performance, what it does or what it tells you is that this material system has potential. It has been demonstrated we can get the high performance. It's a proof of concept. The trick now is to get there at low cost, and that's pretty much what you should walk away from this chart having seen. Another thing to keep in mind is that it takes a long time to improve the performance of a new material. If you're starting out somewhere down around here, it's going to take you a while to reach higher efficiencies. Granted we can learn a lot from the previous material systems. We could learn a lot by reading those old NREL project reports that are available online of all the people who were working towards these record efficiencies-- what they did differently, how they advanced, and how they improved cell performance-- and leverage that information as you try to develop your material. But the fact of the matter is, it'll still take a bit of time to develop new technologies, and you can see that by some of the newer materials that are coming along down here, for example, the organic-based solar cells. So thin films, general issues-- so we talked about the advantages here, that we're squeezing the cost of the absorber layer out of the module, which is excellent from a cost point of view. But obviously, there are trade-offs involved. If it was all a walk in the park, we would be 100% thin films and have abandoned silicon by now. There are both advantages in this advantages with thin films. Instead of disadvantages, perhaps a happier way of looking at this is challenges and opportunities for getting PhDs and other advanced degrees. So let's go up into advantages. The advantages of thin films, quite simply, is that you're using a very thin amount of material, so thin, in fact, that it's virtually insignificant in terms of the total cost structure of your module. One perspective is that if you're depositing a thin film using a fairly low-cost technique, like a c spaced sublimation type process, you may be able to deposit the material for as much cost as it takes the cardboard that separates the modulus from each other in the stack that's being loaded onto the 18-wheeler out of the factory, to put things in perspective. It's very cheap to deposit these thin layers. Now let's hop down to the disadvantages real quick. If you're depositing a very thin layer, and it's not high efficiency, then you need more glass, more encapsulants, more framing materials, more labor, and everything for the same amount of power out. If your efficiency is low, your costs will be higher. Even if you have a dirt cheap absorber layer, you might as well get the absorber for free. If your efficiency is too low, all the other commodity materials are going to outweigh that cost advantage because the commodity materials scale with area. If you have low efficiency, you need a larger area module to make the same amount of power. So, at some point, if you look at the cost of the material versus efficiency, you start entering negative territory. You actually have to be paid. If you're producing like an 8% or a 7% module, typically, you would have to pay your customer for them to accept your module. So you really have to achieve a minimum efficiency target to be cost competitive, and as a rule of thumb, that's typically 10% to 12% for today's cost of glass encapsulance framing materials and labor and installation and so forth. So back up to the advantages-- there's a potential here for a very low thermal budget. If we're able to print, say, a micron-thick layer onto a substrate, remember, we go back to that the high speed printer analogy, there's a potential for a low thermal budget, which means thermal budget is the amount of heat that you're introducing during the processing. As a result of a very low thermal budget, you have a potential cost decrease. Instead of heating things up to 1,400 degrees C over several hours, having all that massive amounts of electricity that go into producing the crystalline silicon wafers, here, potentially, we could be printing stuff on flexible substrates. So that's the thermal budget argument. In terms of conformal deposition and flexible substrates, there's a potential here for roll-to-roll deposition. Picture a newspaper plant, where you have one roll of paper on one side being pulled on to another spool in the other, with some deposition process happening in between. If you can deposit on a flexible substrate, this is the vision. And if you're not depositing onto a flexible substrate but onto hard substrate like this one right here-- this is glass, a thin film material deposited on glass right here, a very small one. Oops, some tape on the front. Let me get rid of that for you. Here we go. It's in a nice little protective coating here, so you can have a look at it without worrying about getting your fingerprints all over it. And the company name is fully removed-- check. This is an example of a thin film material deposited on glass without any anti-reflective coating, just the absorber material, so you can get a sense. It looks great. It's about a micron thick. It's about 170 times thinner than those wafers that you saw on Tuesday. So that's an example of a thin film material. It will be making its rounds. There's a large amount of technology transfer with a thin film display, the flat panel display industry, with deposition on glass like that one right there. And there's a potential it'll be very nice for building integrated PV applications. If you're able to get rid of the glass and deposit on a conformal substrate, you could envision roof shingles or other flexible substrates that would allow you conformal coverage on undulating roof tops and so forth. Radiation hardness-- this is just a small aside, but there are some materials that have better radiation hardness than silicon. What does radiation hardness mean? It means that if I send something to outer space, where we don't benefit from the radiation shield of our own atmosphere in the Van Allen belts on earth, and we have proton bombardment and other forms of radiation striking are module and creating damage within the absorber layer, some compounds are naturally better at resisting degradation of performance than others, and that's what radiation hardness means. So there are some thin film materials that are exceptional for space applications. The challenges and-- oh, go ahead, Ashley. ASHLEY: Is gallium arsenide one of them? PROFESSOR: We're going to show you in a few slides. We'll compare them all as a function of radiation exposure time. The disadvantages, or shall we say challenges and opportunities for PhD and master's students, lower efficiencies in crystalline silicon potentially larger module costs. If you're able to improve the performance of these thin film materials, wow, you have now equivalent performance of crystalline silicon but at much lower cost. Good for you-- you have a marketable product. Potential for capital intensive production equipment-- not all of the production equipment is as low cost and as low thermal budget as simply printing on a piece of paper. As a matter of fact, that's one of the more avant garde and R&D type of deposition processes. Most deposition processes and the vast majority of companies used are actually quite capital intensive, and the cost of the equipment can add up. Sometimes, not always, but sometimes scarce elements are used. We're going to have a debate about that on next class, on Tuesday. Put an asterisk next to that. I'll get back to those as soon as this slide is over. And spatial uniformity is a challenge during deposition. Imagine trying to deposit a film one-micron thick over glass that is one meter in size. You're talking about a six order of magnitude aspect ratio here. So we have to somehow deposit a film a micron thick in layers that are even thinner, that might be only a few tens or hundreds of nanometers on top of that and below that absorber layer to separate charge, for instance, and that's really challenging to do on a very large scale, and that is an engineering challenge or a process engineering challenge that had many startup companies flailing for a long time. Think of spatial homogeneity in the following manner; if you have one region of your solar cell that's producing a lot of power, and the region next to it is not, and they're connected in parallel through the contacts, power will flow from the good region into the bad region. So you have internal current loops inside of your module. That is essentially decreasing the power output of your module itself. So that's why homogeneity is important. This is just to represent the vision of a roll-to-roll process in the upper right-hand side there. Kind of a visionary cartoon that is being enacted by one company, in particular, Uni-Solar, based out of Michigan. They do have a roll-to-roll process and PCBD-- we'll describe what that is in a second-- deposition of this material, so-called amorphous silicon. And here are some building integrated solutions, just showing you what you can accomplish or what the vision would be. If had have this really flexible substrate that you could literally take it as a roll from Home Depot, bring up to your rooftop, splay it out on your roof, much like you'd lay down a piece of tarp or plastic, and take a staple gun or a nail gun and drill it into location, that would be an example of a much reduced installation cost. So you have the potential here of reducing the installation cost of solar as a result of the form factor of your module. And this here is another example of a building integrated photovoltaic solution within films. The fact that it looks really nice, is really sleek, you'd never guess that those are solar panels there, and that's, of course, from an aesthetic point of view, a huge benefit. Common growth methods-- how do we make that sample of copper indium gallium diselenide, that thin film material that happens to be making its way around the classroom right now, how do we actually make it? Well, not only the material I just described, there are other materials as well. We'll talk about the general classes of growth method. So this is the material science processing class condensed into a few slides. Bear with me; this very high level, but it aims to highlight the techniques that are most commonly used in PV today. We're going to start with what are called vacuum-based thin-film deposition technologies. And the reason I'm separating vacuum from non-vacuum is because if you have a system that is comprised of these large stainless steel chambers that you typically see when you go walking in the physics building, if you have a vacuum chambers, those are typically quite costly, at least the large scale ones that are in commercial production. As the name would suggest, you need to have pumps to suck out the air inside of the chamber, and that's how you create the vacuum. The vacuum is necessary because typically you're transporting atoms from some sort of source, either gas or a solid target, onto the substrate. So you're transferring individual atoms or clusters of atoms from some source onto the substrate that will ultimately hold your thin film device. And that process requires a limited number of interactions of those atoms or clusters of atoms, in other words, a large mean-free path, as these make their way to your substrate. And that's why the vacuum is typically required in these deposition systems. There are a variety of ways to accomplish this goal. One class of techniques is called Chemical Vapor Deposition, often referred to as CVD. This typically involves flowing in some form of gas into your chamber and then allowing that gas to react on the surface of your sample or above the surface of your sample and ultimately depositing on the surface. The chemistries involved in CVD processes can be quite complex, and the reaction process itself can be very difficult to master. So you might have some friends who are involved in spectroscopy shining lasers at their system and looking at the absorption lines and trying to figure out how these molecules are evolving between when they're inserted into the chamber and when they actually wind up as your film, because understanding the reaction, the chemical reactions, that take place is essential, is key, to really controlling the CVD process. The other class of technologies involved is called PVD, or Physical Vapor Deposition, and this tends to be a bit more straightforward. We tend to have atoms of a specific type. They may be ionized, or they may be charge neutral, and they're making their way to your substrate. And the chemistry tends to be much more simple, but the apparatus around it to give the incentive for the atoms to leave the target and deposit on your substrate, that tends to be more complex. And so some of these tools, especially molecular-beam epitaxy can be very expensive, very slow, but very high quality, but very expensive as a result. And so a very simple way to think about the vacuum-based deposition technologies is a compromise-- this is an oversimplification indeed, but it's an easy way to get started about thinking of the parameter space of all these techniques. It's a compromise between speed and quality. Some of the techniques that are fastest also tend to be the lowest quality materials, and the other ones that tend to be the slowest tend to produce the highest quality materials. How do you optimize somewhere in between, somewhere in that parameter space, to get reasonably high material, just enough that you can produce a high efficiency device-- remember that saturation of device performance versus diffusion length. At some point, it just doesn't make sense to keep optimizing your material. You've got it good enough. You're good to go. So that's one of the things to consider when you're choosing your deposition system. So let's go into a few examples of these vacuum-based deposition systems. Within the PVD techniques, within the Physical Vapor Deposition techniques, one of the most commonly used in manufacturing, at least in some startups-- you have examples like MiaSole-- is sputtering. And this sputtering process is essentially very, very straightforward. You have a plasma. The plasma consists of atoms that are charged. These are accelerated toward your target, which is comprised of the elements that you want to deposit onto your substrate. Your substrate is sitting up top. And this target material is sputtered off and eventually makes its way up and sticks to and eventually grows the film on that orange platen up here at the top. That is your substrate. The substrate is facing down. Why is the substrate looking down? Why wouldn't you invert this and put the target on top and in the substrate in the bottom? What could happened then in terms of purity of the deposition process? Let's go to Kristy. AUDIENCE: Things could fall onto it. PROFESSOR: So stuff, gunk, could fall onto your substrate. You're trying to grow a thin film a micron thick, and you're trying to avoid any imperfection, and now gravity is working against you in that case. Because, if you were to invert this, your target would be on top. You could have stuff raining down onto your substrate. There are a few people who sputter down. It's very tricky. You have to be able to control your process very well and avoid flakes from coming off. There are folks who sputter sideways, saves some ground space in their factory. They might load things vertically, put them in. And many people, at least in R&D, sputter up. So again, you're creating this plasma. The charged species are accelerated toward the target. They sputter off atoms, which are then deposited on to your substrate, which is there at the top. And the film that was just being passed around is an example of a sputtered film. The spatial uniformity of sputtering over large area depositions can be in the order of a few percent. So the ability to control this process in terms of spatial uniformity is fairly good. You could also employ radio frequency modulations to the bias voltage. That's called RF sputtering for Radio Frequency. Industrial applications usually involve large rotating targets. So for those of you-- how many people actually work with some sputtering materials or have done it in the past? One, two, three, four, five, six, OK. So you know that, at least in the laboratory, if you have a fixed target, you wind up with that race track, right? So if you have a fixed target in the lab, and you're trying to deposit your films, if you wear it down several hours, eventually the metal that you're trying to deposit, or the ceramic that you're trying to deposit, will usually wind up having a bit of shape to it. Instead of being flat on the surface, you'll have what's called a race track; it'll dipped down near the edges, and that can result in a change of the deposition rate of the species that you're trying to deposit. And from a homogeneity point of view, that might be disastrous in the company, and so there are methods to move your target to avoid that sort of effect from happening. And when we talk about large targets, we're really talking about large targets, right? These aren't your lab scale two-inch or three-inch, these are much, much bigger in commercial production. So in terms of comparing sputtering against other growth technologies, there are technologies that are more conformal. Because this is more of a line-of-sight deposition technique, the atoms are moving toward your substrates. But if you have some shape to your substrate, maybe you have a ledge or a ridge, in that case, you won't necessarily coat that uniformly. You might have less being deposited on that edge rather than the flat sections. And so conformality of coverage, or conformal surface coverage, can be an issue with sputtering. Let's talk about the next technique that is commonly used in inorganic thin-film deposition. Excuse me. This is called metalorganic chemical vapor deposition. So again we notice the CVD appearing at the end. We know it's a Chemical Vapor Deposition process. MO in this case, standing for Metalorganic. The reason metalorganic is because we typically have a metal, like this representing the indium right here, and then little organic compounds on the outside. Those are methyl groups. The little gray and the two white dots, those represent three methyl groups around the indium, so trimethylindium. And what we do is we flow these molecules into our reaction chamber and control the temperature gradients inside in such a way to have those molecules deposit on the surface, leaving the indium behind, or the metal behind, and the reaction products flow away out the back, and that is represented chemically here on the surface. This is zooming in right at the surface of our sample so right where the gas interacts with the thin film material that you're depositing. This is representing the incoming metalorganic molecule reaching the surface. This represents, right here, the separation where we have the indium shown in black right here, and then the methyl groups are moving off, and essentially, those will be sucked out of the chamber, leaving behind, in this particular case, you have a layer of indium forming, probably another layer of material underneath. Say, for example, your other species comprising the thin film may be phosphorus, so it would be indium phosphide growth. This metalorganic chemical vapor deposition is very nice from the point of view that you tend to form homogeneous films-- very good surface coverage. The disadvantages would be that many of the inputs and outputs are toxix-- not always, but many of them are. They have to be volatile and reactive so that you can crack the metal on your surface and create the thin film. If it wasn't reactive, you would just have it flowing through and leaving, not having a reactant with your substrate. But because of the reactivity involved, oftentimes these are not very friendly for human beings or for other organisms. It was not uncommon in the early days of MOCVD reactor development where they'd have this little stack going up to the roof, and then when they'd do maintenance on the roof, they'd find all these dead birds lying around. That obviously has improved since people have put up the appropriate filtration on the output of their growth system, so-called scrubbers, to prevent toxic gases from being released into the atmosphere. But you do have some old stories. So the proper design of metalorganic precursors is essential. You can easily see how if you change the molecule that you're bringing in, all of a sudden now, your reaction temperatures are changing. The rate of deposition is changing, and you have to optimize your growth process all over again. So part of the trick of doing good MOCVD is knowing your chemistry, being able to design or synthesize these metalorganic precursors. And the deposition process is very sensitive to temperature, pressure, the precise surface orientation, and preparation, what carrier gases, as well, are mixed in with the metalorganic precursor that you're putting in, and the byproducts obviously need to be managed. So that's MOCVD in a nutshell. Yes? AUDIENCE: And pure quality is much better with MOCVD than it is for sputtering, right? PROFESSOR: It depends on a lot of factors. So the reason the purity of MOCVD is generally better than sputtering is because the mass flow controllers necessary to control the gas flow specific for particular types of gases. Now, in sputtering, because of the versatility of the sputtering chamber, you could take this target out and put-- maybe Ashley comes along into your sputtering chamber, and she puts in another target of another metal. And now you're depositing two different metals in the same sputtering chamber. You're going to get cross contamination. There are things you can do to minimize cross contamination. You can have a chimney around your target to prevent flakes from coming down. You could sandblast the sidewall coating and so forth to prevent stuff, gunk, from building up around the side, but you're still going to get a lot of cross contamination here. And furthermore, the purity of your film is dictated by the purity of your re-target. And if you go online and look at [INAUDIBLE] or CERAC or some of the big metal selling firms, which are essentially from where the target manufacturers are purchasing their precursors and they compact them and make their targets, the target purity, or the metal purity, is only on the order of maybe 2/9 to 6/9 pure, typically within that range. So from an MOCVD point of view, you could do a distillation process and increase the purity of your precursor gas and avoid that. So I think two big reasons why MOCVD can produce higher purity films in the sputtered system, one is the quality of the target, and the other, I think, bigger parameter, at least in our growth system, is cross contamination. And whenever you deposit, say, an EML-- they have a sputtering system there from AJA, or over at Harvard CNS, there's another AJA sputtering system there-- you're going to get cross contamination. Just look to the log book and see what people have tried to deposit. It gets kind of scary. PECVD, Plasma Enhanced Chemical Vapor Deposition-- so similar to the previous variety right here, but instead of saying, OK, we're going to put the burden of the design, the scientific design, onto this interface right here and on to the chemist, who has to design this molecule that reaches a surface and breaks up in just the right way in an orderly fashion, leaving behind the metal and letting the other gases go away, what we're going to do here instead is to shift the burden of separation onto the plasma. So the centers around the physicists. We can flow in gases. We can break them up inside of a plasma, atomize them or, at least, create radicalized versions of them and then allow them to it on to the substrate-- very simple in theory. In practice, what happens inside that plasma, depending on the temperature, depending on the frequency and other factors, you'll get different types-- and the pressure, especially the pressure-- you'll get different types of molecules forming in the plasma. They may be charged, and they'll be accelerated toward your substrate and eventually grow and form a thin film. But depending on what species you have up there that is being deposited on your surface, you'll get different types of thin films growing-- different quality material. And so, again, this shifts the burden back to the spectroscopist to measure what is exactly the composition of that plasma. What is the active molecule that's being accelerated and deposited on the surface? And usually it's some probability distribution function of varied species. The plasma is created by this radio frequency. Let's put it this way; usually you have a plasma frequency of around 13.56 megahertz. Does anybody know why this 13.56 keeps on coming up over and over again? Yeah? AUDIENCE: [INAUDIBLE] energy to the ionized hydrogen, right? PROFESSOR: Well, if we're thinking about eV, that would certainly be the energy necessary to remove the electron from the hydrogen atom, but this is another reason. Yeah? AUDIENCE: It's a special bend that's dedicated for these crazy noise-emitting medical and industrial purposes. PROFESSOR: Exactly, so this is falling within the radio frequency regime, which would affect communications. And if everybody was allowed to run rough shod around, creating these very high intensity emission sources of radio frequency waves, we would very likely have interruptions to our police communications or maybe even our radios or cell phones. And so, at some point, they had to say, look, we have to assign definite bands within the radio frequency space and allocate them to specific purposes. In one band, they allocated to all the scientists and medical personnel and said, you have to operate your equipment in these specific bands, and we'll give you a few of them, because we know that one frequency doesn't work for all the things you're trying to do. But for medical equipment, for scientific equipment, and I believe even some home electronics, like microwaves, there are specific bands dedicated to them. And that's why we have this 13.56 number popping up over and over again. The reality is that if you change the frequency, you'll change the nature of your plasma. You may change the deposition rates and the quality of your film as well. And so there are people who get special permits and have these radio frequency shielded rooms, where they do experiments outside-- or excursions outside-- of the 13.56 megahertz range. So this is PCBD-- excellent conformal surface coverage again. Because you're biasing your substrate, you're able to conform. The electric field is usually always perpendicular to the surface, and so the angle of entry of those atoms or molecules, the ionized species, entering the surface is going to be normal to that surface. And you can get good coverage around rough textured surfaces. The deposition is very sensitive to temperature, pressure, power, carrier gases. Power of the-- here, as well, shown. And the byproducts, as well, need to be managed because sometimes you're sucking out-- in this particular case, you could be pulling out silane, as shown right there, and we talked about all of the risks involved a silane in our last class. So, as you could guess, each of those different deposition techniques is used or is favored for specific material systems. And we shouldn't forget, as we talk about all these fancy vacuum equipment that look nice and cool as you walk through the labs, and you see these big stainless steel chambers, we shouldn't forget about the simpler, lower cost, lower thermal budget, lower capital equipment cost techniques-- the solution-based deposition methods. And these involve printing. They involve a electrodeposition, spin casting, colloidal synthesis, layer-by-layer deposition-- developed here at MIT-- and other technologies as well. I want to point out two technologies, in general, the first of which is still under some development printing. Obviously, we have inkjet printers. That's pretty straightforward. But printing fractional solar cells is something being commercialized by only a handful of companies, Nanosolar being one of them. And I would say there aren't any authoritative textbooks that will describe for you their technology, because it's largely under wraps. They're a startup company, and it's not publicly available. Electrodeposition, on the other hand, is fairly well known. You're, again, applying a voltage difference between two electrodes, one of which will be your substrate, and depositing a species contained within your electrolytic solution onto that substrate, growing your layers. Because you're growing it at room temperature, these films, I would say, tend to have rough surfaces. That could be a downside of electrodeposition. They might have some pinholes as well. But you do get fairly large grained materials. It can be a very gentle growth process, and, of course, the advantage is lower temperature. So you have a variety of different growth techniques. Let's talk about the general issues involved with thin films in general, and then we'll dive into the specific materials. So taking the same tact as we've taken the full class, going from fundamentals toward the technologies. Yes? AUDIENCE: I'm just curious; do you know of any companies that actually use electrodeposition? PROFESSOR: I know of some companies. Let me think which I can talk publicly about. So IBM, they presented at the Electrochemical Society meeting last Monday here in Boston. They're an example of a company that is developing electrochemical deposition processes for material systems, including copper zinc tin sulfide and copper indium gallium diselenide. We'll talk about the latter in a few slides, but that's one example of a company. So general issues in thin films-- thin film compounds are typically, not always, but typically, binary, ternary, quaternary, or multinary semiconductors. Meaning you don't have just one element comprising the semiconductor species. You might have several, and they form a crystal structure with repeating structure but alternating atoms typically. And so, if you have multiple atoms in one compound, a couple of issues could arise and need to be controlled to grow good films. The first involves phase stability. What is shown right here in mulitnary parameter space, this is the chemical potential zinc, copper, and tin in a so-called zinc copper tin sulfide material system. This red fin right here is showing you the parameter space within which this compound is stable. If your stoichiometry takes an excursion from that red fin, you could wind up in a bi-phase regime. Meaning you have CZTS and something else, a copper tin sulphide, a zinc sulfide, or some other species that happens to be nearby in phase space. One way to think about this is it's just you have a homogeneous material. If you exceed a solubility limit in one direction or another, you'll have precipitation of a secondary phase. So you have to make sure that in a gross perspective, on a percents basis, you're in the right regime of stoichiometry. Stoichiometry being the ratio of different elements in your system. So it's like cooking; you need the right set of ingredients to make the right material. Now, that has to do with-- large excursions from stoichiometry can result in phase decomposition. Small excursions from stoichiometry, a much more subtle effect can occur. Let's imagine for a moment that we have two species comprising are binary material. One species has three valence electrons. The other species has five valence electrons. Now, because of a small error in stoichiometry, maybe something in the order of a few tens or hundreds of parts per million in stoichiometry, we didn't get the ratio just right. We were off by a little bit. Now we have one of our compounds in excess and the other one in deficiency. If we have a different number of electrons surrounding the atoms, we could wind up with an excess free carrier density. In other words, you could self-dope your material if you're unlucky, in other words, if the material system has a propensity for this. And you can change the free carrier concentration, and because the free carrier concentration is changing, you might even change your mobility. So there are some effects that can occur as a result of small excursions from stoichiometry. As a result of the self-doping, you're shifting the Fermi energy inside your semiconductor. And as a result of shifting the Fermi energy, it might lead to a cascade series of events. There could be other defects that form as a result of the Fermi energy change. You could have other so-called antisite defects. Atoms could switch positions inside of your lattice, and as a result of that, have very low minority carrier lifetime in certain materials. So nailing the stoichiometry both from a very large sense, to avoid phase decomposition, instead of having a dalmatian film, you have phase pure film, and from a local perspective, once you get on to this phase space where you can grow your film well, you want to make sure that your stoichiometry is controlled to avoid self-doping and to prevent certain types of intrinsic point defects from forming that might lower minority carrier lifetime or change carrier concentration, change other properties of your film. For those who are working on these sorts of materials, I'm happy to talk ad nauseam about these topics, maybe after class, since this is a more detailed subject. Another topic of interest in thin films is grain size. At some point, grains don't matter anymore. The grain size, typically if you exceed the thickness of your film by about a factor five. In other words, the grain diameter's about five times wider than the thickness of your film, grain size is not as much of an issue. But if you do have very small grains, they can impact performance, because carriers will interact with those grain boundaries. And depending how recombination active they are or where the grain boundary is pinning the Fermi energy, the density of state at that, at the grain boundary, will dictate the effect on device performance. So these are some very rough plots in crystalline silicon for thin film devices and for some thicker ones as well. So performance is a function the grain size. And I show crystalline silicon because the data is really well developed for it, but you see similar types of plots for organic materials, for some inorganic thin film materials, like CIGS and so forth. And this convolutes a few different parameters. You have to take into account that the recombination activity of the grain boundary is also a factor. The next topic, general topic of interest, another tool that we'll want to have an our material science toolkit as we start designing these materials, we have to think about the interfaces between the different materials. Especially in thin films, interfaces are so important because we don't much bulk anymore. So the device could really be affected or device performance really reduced if we don't pay proper attention to our interfaces. What are these plots over here? These plots are used to grow some very high efficiency materials, for example, by MOCVD or molecular-beam epitaxy. And what is represented on the horizontal axis is lattice constant. Lattice constant refers to the equilibrium spacing of atoms inside of your material. So this regular repeating unit cell that defines a crystal has a certain lattice constant, a certain distance-- physical distance-- shown here in angstroms. The energy of the gap is shown on the vertical axis. And if we want to select two or three of these materials to stack on top of one another to absorb well at different portions of the solar spectrum, we'll be choosing, for example, one band gap at around 1.9 eV, another band gap of 1 eV, or maybe if we want three materials, we'll go even higher at the top end and lower at the low end. So we'll stack different materials on top of each other to absorb preferentially in different regions of the solar spectrum and hence exceed the Shockley-Queisser efficiency limit, because now we're absorbing well in two or three different colors as opposed to just one. And the energy gap here is important because you want maybe one material at 1 eV, one material at about 1.9 eV. But you also want to make sure that they can grow on top of each other, that you're not going to get a mismatch of that interface, that the lattice constants aren't so different that you wind up with these dangling bonds at the interface, where you have an atom coming down and nothing on the other side for it to bond to. And so you need to make sure that the materials that you grow are matched in lattice constant but varying in band gap, if you're trying to grow a multi-junction device, if you're trying to grow a very high efficiency solar cell device. And so the growth or matching of materials one on top of another is important, especially for the multi-junctions, also for some of the single junction materials if you really want to minimize the interface recombination. So let's look at this growth system up here, the one that is typically used in high efficiency solar cell materials. We have germanium right here, gallium arsenide, and indium gallium phosphide, which is essentially a mixture between gallium phosphide up here and indium phosphide down here. You can alloy the two together and get an indium gallium phosphide mixture and stack these three materials on top of one another-- germanium, gallium, arsenide, and indium gallium phosphide. They have three different band gaps. The absorb in three different regions of the solar spectrum. But they have a very similar lattice constant, and so the interfaces will be very well maintained. That's an example of using a chart like this to design your solar cell materials. Next topic is material abundances. If we're trying to engineer all of these other parameters that we've been talking about-- the lattice constant, the band gap, the grain size that also is a function of how the material grows, the ability to self-dope. We have all of these material issues that we have first and foremost in our minds. We go to the periodic table. We find some compounds that work. We're really happy about it. But then, all of a sudden, life comes along and slaps us the face and says, well, we don't have enough of this material to really scale to get all the way to the terawatt cell [INAUDIBLE]. Oh, I wish I had known about this before when I first got started. So we're presenting to you upfront the state-of-the-art of what is known about material abundances. And these last two studies right here, APS Energy Critical Elements and the DOE Critical Material Strategy, both of them represent a synthesis of the information, essentially the equivalent to the IPCC reports in climate change, but the best synthesis that we have right now about the abundances of different elements out there. There are as well a variety of different papers that have been published in the subject over the last couple of decades or even earlier. So what we have to keep in mind is that our stardust out there is not in infinite supply. Every element we have on the planet that we know of came from fusion reactions in stars, and there was a probability distribution function of the appearance of those elements as a function of z on the planet as a result biased toward the lighter elements. And some of the heavier elements are in lesser supply, that we know of, on the Earth's crust. Not to say that the deposits don't exist. Not to say that these studies right here are the authoritative end-all and be-all. We might discover next year or next month for tomorrow huge deposit of a particular element at a specific spot, let's say, under the Arctic. But from what we know right now, that's the stardust that we have to work with. These are our abundances. So if you'd like to design around it, I'd advise looking into those reports as well. And finally, radiation hardness, getting back to Ashley's question, gee, what are the most radiation hard species? This is the efficiency of solar cell performance normalized at the very start of a test, and this is the equivalent radiation damage. You could also think about this as the amount of momentum or energy depending transferred to the atoms inside of your semiconductor that would result in lattice damage that would result in a decrease of minority carrier lifetime or mobility, which ultimately would impact cell performance and efficiency. We can see that different material systems have different degrees of radiation hardness. Some maintain their high efficiency until very high radiation dose, and others degrade much quicker. And look at this. This is a dose in orbit per year, right around there. And you can already begin to see that some of our most common compounds are not doing too well out there-- not doing too well in outer space. So we have the radiation hardness to take into account if we're putting these solar panels out there into outer space. This is one older study I would definitely encourage you to look. There may be some newer studies. As the solar cell efficiency improves, they become more sensitive because you begin decreasing your efficiency with smaller variations in the minority carrier diffusion length. So those charts make look a different as time goes by. AUDIENCE: Is the effect of the radiation cumulative? So for example, gallium arsenide or any of these would just continue to degrade as they're out in space? PROFESSOR: So is effective radiation dose cumulative? I am not the expert on this particular topic. But from what I know about radiation exposure of detectors at synchrotrons, which is a little similar, not quite the same, the mechanisms involved with this essentially involve atomic displacements within the lattice. You have atoms physically being displaced from their equilibrium positions as they interact with this incoming radiation. And the probability that it occurs is a function of time, will increase per unit volume, and hence it can be thought of as accumulative exposure effect. The first order impact would be on minority carrier diffusion length, impacting both lifetime and mobility. And to the effect that you have a relationship between exposure time-lattice displacement, lattice displacement-minority carrier diffusion length, minority carrier diffusion length and efficiency, you might be able to model this effect. That would be my uninformed answer. Again, you might want to look this up yourself. Question? AUDIENCE: Do you know why the cadmium telluride improves? PROFESSOR: Oh, why does it improve with efficiency? I don't know specifically about why that is for this particular case, but do know that some materials are what are called defect tolerant. Some are more naturally able to withstand antisite defects or a certain concentration of damage, internal surfaces, voids, grain boundaries. Cad-tel, cadmium telluride, is fairly defect tolerant. It's one of nature's gifts to humanity in that regard. The degree to which a material can be defect tolerant depends partly on the carrier density. If you have very high carrier density, you tend to screen defects. Another reason why they could be defect tolerant-- all of these compounds are somewhere between an ionic and a covalent semiconductor. In a covalent semiconductor, those materials tend to be very defect intolerant because there's the conduction band and valence band as a function of position, tend to be very flat. The material tends to be fairly homogeneous throughout the electron densities, fairly well distributed in a covalently bonded material. In an ionic material, you tend to have charge localization. So energy as a function of position might look like this on an atomic scale as you go from one atom to the next atom on your lattice, to the next one, to the next one, to the next one, and that reflects the localization of charge in your material. Those materials can be more defect intolerant, because conduction can happen more from a hopping mechanism than from a band conduction mechanism, and this is really a gradient between, and most materials, they tend to be partially ionic partially covalent going down the list here. And there's a bit of a shift between cadmium sulfide in zinc sulfide in terms of the ionicity covalence. So cad-tel would be with the chalcogen two lower than sulfur on the periodic table. I would imagine it would be at this transition as well, but I'd have to look that up. Gives a place to get started and read more about it. And then reliability and degradation-- this is important. This is a crystalline silicon module being loaded into environmental testing chamber. Inside of that chamber, the module is going to be put through hell and back. The temperature is going to be raised. The humidity is going to be pumped in. Sometimes ultraviolet light is even put in there in some of the more modern advanced ones. And then the temperature can drop down to temperatures as low as minus 40 degrees C, depending on what the environmental chamber is designed to do, exactly how it's designed to stress or test your module. And the idea is to promote an accelerated degradation of the module on purpose to test what its failure modes will be, and we'll see these we go take a tour of Fraunhofer CSE in a couple weeks. This is a crystalline silicon module being loaded in. If you were to put a thin film material-- and crystalline silicon are materials very, very thick, and we said at the native oxide was very tenacious. It was only a few tens or maybe hundreds of angstroms thick, and the junction depth was about on the order of a micron. If you have water attacking the surface of you silicon wafer, water vapor, really not too much of an issue, and silicon's is fairly inert anyway. But now if you take a fairly reactive material, a thin film material that might react with air or might react with water, and it's so thin that the these rusting modes, or reaction modes, the weathering modes, can really impact a large fraction of the thickness of your device. Now you've become a lot more sensitive to accelerated degradation. Now you've become a lot more sensitive to the elements, and this includes both oxygen and water. So if the ambient is able to penetrate through the encapsulate and get to the active absorber material, you may have accelerated degradation of module performance as a result. And so there also some unique failure modes within thin film modules. If you have two different species comprising your compound, one of them might be prone to move in electric field. For example, copper is notorious for zipping along in electric field, in electromigrating. And so that's a failure mode that doesn't exist in large thick crystalline silicon modules but could existent in thin films. And so because of all of this, and because of the growing realization that the way we test crystalline silicon modules and drive them failure is not the same that we might be able to achieve failure in a thin film module. There are newer testing protocols, such as this IEC 61853, that have been introduced in an attempt to do test appropriately thin film modules for their respective failure modes. And this is, I would say, still work in progress. So much so, that we have a group project focused in part on this. It's still a work in process to try to figure out how do we appropriately test these thin film modules toward the point where they can fail. Any questions so far on these topics? Because these are general issues that will affect all thin film materials, I wanted to make sure that we were comfortable with these general topics before we dove in any detail into the technologies themselves? Yes? AUDIENCE: A question about lattice matching-- is it important to lattice match the semiconductor to the contact as well? Or is that not as important PROFESSOR: So the question was is it important to lattice match the semiconductor to the contact? So let me emphasize that in many semiconductor contact combinations you would have a highly doped semiconductor right before the contact, a very localized region of highly doped semiconductor that would create a tunneling injunction. In that case, the density of states at the interface doesn't matter because you have a tunneling junction. You're be able to tunnel straight from the semiconductor into the contact. The lattice matching would matter, though, if you didn't have a tunneling junction. If you had a regular Schottky ohmic contact, then you might have to worry about the density of interface states, which would be regulated by the number of dangling bonds, and then you might want every single atom pairing up with a neighbor on the other side. So lattice matching would be important for the contacts there. All right, fun stuff-- wow, we've had a good dose of material science of the day. Thin film cost structure-- I just wanted to highlight one quick thing. This material right here, that's not the absorber material. The absorber material is a really tiny fraction of the material. This comprises the other materials within the module as well. So the encapsulates, the glass, and so fort, just keep that in mind as a kind of asterisks. So it's typical thin film cost structure. It evolves with time. This is a few years old, this slide, but it gives you a sense, a feeling. In terms of global production, this is a year-old data now from 2010. During this past year-- so this was 2009 data shown in 2010. The 2010 market was very harsh for the thin film producers, many of which tend to be in the United States and in Europe. In 2010, prices dropped quite a bit, and that favored the low-cost Chinese solar cell manufacturers, many of which were invested in crystalline silicon technology. So by no detriment to the technology itself, market dynamics worldwide, due to other factors, tended to favor crystalline silicon in the past year. And the market share of thin films contracted a bit so it's now about 90% crystalline silicon, 10% then films worldwide in 2010. But the break down between the different thin film technologies, we had the so-called cadmium telluride, CIGS, so that's Copper Indium Gallium Diselenide, and amorphous silicon. And the dynamic between 2009-2010 was that the amorphous silicon shrank a bit. CIGS grew, and cad-tel continued growing, but more marginally because it was already big to begin with. So you could think of this red portion growing at the expense of the green, if you want to translate this into 2010 numbers. So what is CIGS, cad-tel, amorphous silicon-- what are those materials? Well, we'll get into that. I think the best thing to do is to leave this for next class. I'll briefly go over cad-tel just because it is so important. It is the biggest-- the single biggest US solar cell manufacturer is producing cadmium telluride solar cells, or cad-tel for short. And to make just a description of what you're solar panel would look like in cross section, this is your glass on the backside here. This ITO Indium Tin Oxide. It is a what? A transparent conducting oxide, very good. So the ITO is a transparent connecting oxide. Your light, your sunlight, is coming in through here. So this is electrically conductive layer, but it's transparent, so it's allowing the sunlight through. Tin oxide, we'll get to that in a second. Cadmium sulfide and cadmium telluride-- so the cadmium telluride layer is a layer that's absorbing most of the sunlight and producing electron hole pairs. The cadmium sulfide is providing the header junction separating charge at the header junction between the cad-tel and the cad-sulfide. This tin oxide is generally an intrinsic layer. It's assisting here with the ITO on the front contact, and then you have your back contact that extracts charge from the back. There are a couple more tricks to creating a good cad-tel device that involve chlorine treatment and passivation of defects. That's where some of the activation comes in. This is another view of the cad-tel device in cross section, another example. This transparent conducting oxide, in this case, is fluorine doped tin oxide, another TCO material. But very similar structure here, the cad-tel being the p-type material, and cad-sulfide the n-type material. The thicknesses of these different layers you can see. The cad-tel is only a few microns thick, and the cad-sulfide this is even thinner. It's a very thin layer just serving to separate the charge. The band diagram of a cad-tel solar cell is shown right here. We have the cad-tel here and the cad-sulphide right here. So you can see the junction. Notice, because of the thickness of this layer, the band bending at these interfaces extends quite an appreciable percentage of the total thickness of your device. Whereas in crystalline silicon, we have the band bending right near the junction region, so right within a few hundreds of nanometers, maybe a micron away from the junction, and the device is 100 microns thick. So we had 100 microns approximately of this flat band condition, at least in the dark. Here we have bending extending an appreciable percentage of the total thickness of our device, just by virtue of the fact that we have such a thin film. And the characteristics, the deposition technology of cad-tel, as I said, it's nature's gift to humankind. If you put cadmium and tellurium in together in a pot and start heating it up, what evaporates is cad-tel. It congruently evaporates. So you could use a process called close space sublimation, where you essentially sublime your cad-tel, and you deposit it onto your substrate. If you try to do this with most other compounds in the periodic table, you'll get either one element or the other element evaporating first. They'll create an overpressure. They'll evaporate off, and you won't get your compound depositing, but you'll get one element depositing on your substrate preferentially. cad-tel, again, nature's gift to humankind. The two come off together and form a cad-tel layer and so that congruent evaporation makes it very nice from a deposition point of view-- very low cost, high throughput deposition process for solar benefits from. Environmental concerns-- cadmium has raised quite a bunch of concerns amongst folks in environmental groups because it's a known carcinogen. It is responsible in Japan for, I believe it was called the itai-itai ban, which means "the ouch-ouch disease." It was a disease that was acquired by folks exposed to cadmium during manufacturing. And as a result, cad-tel solar panels are not allowed to be installed in Japan. So First Solar cannot sell its cad-tel products in Japan. There are very tightly regulated emissions laws in the EU and the United States, especially in the EU, where cradle-to-grave recycling of cad-tel solar panels are necessary. So you'll see a lot of cad-tel solar panels in large field installations or in commercial buildings where it's very easy to collect them all after their 20- or 25-year life span and bring them back to the factory, as opposed to having them distributed amongst hundreds of thousands of smaller systems deposited on rooftops. The arguments in favor of having cadmium inside of solar panels is the following. It's better to tie up cadmium inside of a relatively inert compound, cad-tel, then to have it go off and cause problems on its own. If you heat it up, the cad-tel would evaporate congruently. Typically cadmium is so-called "negligible" amounts are released during fires, and they put it in between quotes because these are studies, very good studies, and I trust the work coming out of the [INAUDIBLE] group very much. His critics would argue that, well, the studies were paid for, in part, by First Solar, so how do you trust studies like that? I would counter and say, this is a pretty good group. Out of all the people to life cycle analysis, he's within the top tier. So it's some question as to that. People do question were the temperatures used in these studies representative of what you would actually get in the hot zone of a fire and so forth. There's a public fear and perception issue. It's a big deal. And the folks would argue that much less cadmium is released per kilowatt hour than, say, in a battery, where we would use a nickel cadmium battery, for instance. And safe production methods now-- fully automated, and recycling is guaranteed by law. So have arguments in favor and against. I'm going to stop right here, and I'm going to pull aside the teams during the last 15 minutes of class. I emailed to you. If those of you had checked your email before last night at 5 o'clock, should have received an email saying you're on a particular project team. Find your partners, cluster together. Joe and I are going to come around to spend a few minutes with each of you just to make sure that our first steps are clear and that we have a forward path and we gain some momentum. So self-assemble and don't leave the room before the chance to come talk to you. |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | 1_Introduction_2627_Fundamentals_of_Photovoltaics.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Ladies and gentlemen, thanks for coming today. I'd like to formally start the course, The Fundamentals of Photovoltaics. That's 2.626/2.627. Why don't we dive quickly into the syllabus, and then, a few slides of motivation, why we're here, why we're studying photovoltaics. Hopefully, get you excited for the course. The syllabus that you have before you should outline the course objectives and the course learning objectives. At the end, during the background assessment survey, we'll take the last 10 minutes of class for you to provide your feedback to us, the teaching staff, to make sure that we're addressing your needs and your interests. So take a quick moment to read over that while I describe the overall flow of the course. The course roadmap, this little diagram right here, is essentially a three step component. We first instill the fundamentals of how light is absorbed into a material, how charge is excited, how then charge is separated and a voltage created, and finally, how a charge is collected. And that is the essence of a photovoltaic device. In 30 years time, photovoltaic devices probably will still be using that combination of physical processes. So understanding these fundamentals will arm you-- will give you the information needed to be able to assess any photovoltaic technology that might be presented to you. Then, in the second component of the course, we'll discuss the technologies, the specific technologies that are out there in the market today, and those that are up and coming that have the potential to replace them. And as a third part of the course, we'll be discussing cross cutting themes. These include the policy, economics, and social aspects of photovoltaics that, of course, are of general interest and are particularly interesting for scientists and engineers, who spend most of their time thinking about the fundamentals, to take a step back and look at the broader picture. A note on the fundamentals. I recognize that many of you come from diverse backgrounds, some from nontechnical backgrounds, many from mechanical engineering who never really have looked into semiconductors or semiconductor devices. Not to worry, as you'll see on page number-- page number 2, meeting times, class recitation, and office hours. We provide a number of opportunities for you to get more closely engaged with us, the teaching staff, and to work through some of the fundamentals as you might experience difficulties in the learning process. Let's take a quick look at the course schedule just to situate ourselves. So the course schedule follows that three step process very closely. The first component of the course, the first third, roughly, is focused on the fundamentals. So we'll learn about light absorption, charge excitation, charge separation, and charge collection. And the recitation times will be used to discuss those fundamentals because, for many of you, this is the first time you're working with this material. The second third of the course, on PV technologies-- when we discuss the industry that's out there today, how it's evolving, how the different technologies are evolving, this is when we get to experience some of the industry pain upfront, up close and personal. We'll be making solar cells. And as part of your take home quiz number two-- as you'll notice, take home quiz number two is distributed right in the beginning of October-- middle of October. And then it's due in the middle of November. So it's almost a month. And the reason it's a month long take home quiz is because, during the recitation times, we will be making solar cells with you. And it will be a little bit of a challenge. It's not only to make the most efficient solar cell, but the most cost effective solar cell. And so we'll be making technology choices as we go along, processing our solar cells, deciding whether we do process A or process B. We'll be doing the calculations that we learned how to do during the fundamentals section to predict what the efficiency gains should be. And it will have costs associated with each of the different process steps as well. So it will be a little bit of a game, a little competition within the group, as well, to see who can make the most cost effective solar cell in terms of dollars per unit power output. And finally, in the last third of the course, this is really when the projects kick off in earnest. We have several really interesting projects lined up as well as we're open to hearing your own project ideas. This is when you form teams of three, four, perhaps five, but hopefully three or four. And you will be addressing some of the most important questions of the day, obviously, in a very bound, well-defined way. And some of the projects that we have lined up include looking at actual photovoltaic installer data coming from houses with temporal resolution on the order of five minutes. So you can obtain a huge database of maybe 10,000, 15,000 homes distributed geographically, and determine to what degree is the ensemble of photovoltaic systems predictable. Obviously, if a cloud goes over one home, power output drops pretty dramatically. But if you begin averaging over several homes, how predictable is the solar power output of that ensemble? And that's going to be very important as photovoltaics scales up and assumes a greater percentage of the total grid. Another interesting project we have lined up is with the World Bank. This is with folks in Washington DC who are looking into a project called Lighting Africa. And they're installing PV on small little lights and distributing those to folks in sub-Saharan Africa. And their big question to the MIT audience is, with some of the newer up and coming technologies out there, how will this impact their technology? How will this impact their lighting? And so the deliverable at the end will be a technology perspectus-- one page. A lot of thought has to go into it. That will be delivered to companies that will be selling their products in Africa to guide them and to inform them about some of the up and coming technologies and how their markets will be impacted. Like those two projects, we have several others. And we're open to your ideas as well. So if you're really jazzed about one particular topic, there will be opportunities to let us know, specifically on homework number 2, when there will be a specific question there, are you interested in a particular topic of your own. We'll assemble-- begin creating teams early on so that there's some bonding going on, especially during the cell fabrication part during the second third of the course when we make the actual solar cells. But then, the third part of the course will be really focusing on the class projects themselves. So that's the lay of the land. And I want to give you some motivation as to why we're here and why this is really a special time in the field of photovoltaics. This is not your parents' solar energy anymore. Things have changed quite a bit. And hopefully, over the course of these slides, I'll be able to convey that message loud and clear. We'll go ahead and get started. So first question is why photovoltaics, or why solar. Photovoltaics is one particular embodiment of solar energy where we convert sunlight into electricity. And in most photovoltaic panels-- I'll definitely let you guys come up and have a look at it afterwards. In most photovoltaic panels, you have two leads coming out, basically, the equivalent of a positive and a negative. And you have a bunch of cells here that are converting the sunlight into electricity. It's different than, let's say, solar thermal, which is converting sunlight into heat, or solar to fuels, which is converting sunlight into chemical energy. And the reason we're studying photovoltaics as a starting point is because PV, photovoltaics-- PV for short-- is the most widespread technology, widespread solar conversion technology out there today. So the big question is why solar in general. Why are we at all interested in this? Can anybody tell me what this is a picture of? It's obviously not from the United States. Does anybody recognize the language here written on the side of the boat? It's very small. [SPEAKING PORTUGUESE] AUDIENCE: Portuguese. PROFESSOR: It's Portuguese. It's from Brazil. It's form the northeast of Brazil. It's a small island called Morro de Sao Paulo. It's located about an hour south of Salvador in Bahia. These are folks arriving at the island with gas cylinders. There is no underwater cable linking the island with the mainland. So they're arriving by boat with gas cylinders. They're tossing them into the salt water. They're pushing them onto the beach, rolling them on the beach, until they get to the little sandy roads-- of course, getting grains of sand embedded inside of the nozzle and so forth. This illustrates to me the great risks that we go through to supply ourselves with energy. It's just one, what might be considered by our safety standards here, extreme example of associated risk with supplying of energy and effort, of course. But if you look at our energy supply to the United States, it's no less heroic. It just has different dimensions. And so the energy that we use today is often produced in some faraway land, not always, but often, transported, sometimes over thousands of miles, and brought home at significant risk and peril. And the question is, why do we go to such extremes. And second question is, is there a better way. So to answer the first question, here, why we go to such extremes, if you look at the world at night, and then look at our human development map, which I use Facebook-- what better indicator of human development is there than Facebook? This map right here shows you the number of linkages between people on Facebook. And of course, the density of the bright lights there is representing the number of users. And you can see that the two maps, the electricity consumption and the technology adoption map very closely, one on to another. And it's almost down to the specific region once in a specific country. This is especially noticeable in some of the developing world where you see these pockets of high concentration of people, essentially capital cities. You have Lagos, Nairobi, and so forth-- Jakarta. And you have this huge concentration of people that, of course, are using electricity. And more and more people flock to those cities, especially in developing countries, because the standard of living tends to be higher. There is a certain indicator, called Human Development Index, that was put together by the World Bank, which pulls together a number of factors, including expectation of life, infant mortality, and so forth-- education levels. So in some hand wavy way, comes up with a metric that indicates quality of life, roughly. And on the x-axis here, we have annual per capita electricity use-- not energy, but electricity specifically. And we see some form of correlation between the two. So one could naturally conclude from this that energy is fueling development, and energy is also fueling per capita income, as a result. This little bubble chart hear, courtesy of UC Berkeley, is showing you the size of the bubble here, indicating the size of the population, and of course, the position on this graph indicating the per capita energy use and per capita income. The reality is that many of the up incoming energy consumers aren't quite there yet in terms of their energy use. There will be a drastically increasing demand as several regions of the world turn on as they begin plugging in and demanding more electricity. So somehow we have to satisfy that growing demand. So to put things in perspective as well, here we have the world somewhat at night. World population in millions. And so we have somewhere around 10 billion approaching by 2050. And you can see that the majority of the growth, what's driving world population, is Asia and Africa. Those are the two lines. My apologies for the small text. But that's the yellow line right here. And the black line right here. They're the two largest bars in that Pareto chart. And the projected human energy use is only going to go up as a result. So again, we look at the world at night. Now instead of looking at the bright areas, we're going to focus on the dark areas instead. The regions of the world where we do have high population densities-- some of the regions, not the deserts, obviously. Some of the regions we do have high population densities, like sub-Saharan Africa, but don't have a whole lot of electricity use right now. Then we'll take another map, which is the solar resource. Again here, the red is indicating a lot of solar resource. And the blue is indicating not so much. But still, it's pretty amazing that the entire world is falling within about a factor of two, maybe a factor of three. So even if you compare Scandinavia against-- let's say, Scandinavia against Kenya, you're still looking at about a variation of a factor of three, right? So the solar resource is pretty well matched with the regions of the world that don't have electricity right now, where the demand will be coming online. And to put that into another nice chart, I don't think this is very common yet. You've seen the HDI versus per capita income. But this is HDI versus insulation, showing that those regions of the world that are ranked lower on the HDI scale are precisely those regions that have higher insulation, that have greater access to that solar resource. Now the big question is, is that solar resource big enough to supply necessary energy needs. And this is a quick intro to next lecture, where we discuss the solar resource in detail. But the short answer is absolutely, yes, by orders of magnitude. The volumes of these cubes represent the volume of either energy resource or energy need. Energy need here, on the far right-- that little blue cube represents the human energy use. Some are very small compared to the solar resource on the Earth's surface. This obviously is including the ocean as well. If we're to be realistic, instead of calling this planet Earth, we should probably call it ocean or water since oceans do comprise about 2/3. But even if we discount this for usable land area, we're still an order of magnitude greater than total human energy use. So the resource base is there. It's available. It's up to us to figure out how to use it, up to us scientists and engineers. So the potential for solar energy is represented on this chart. I'm not a huge fan of this chart, and I'll explain why in a minute. But there is something very valuable to be taken away from here. These black dots, one, two, three, four, five, six, represent around 18 terawatt equivalent, which is total human energy use in a few years time. And you can see the total land area there is not astronomical. The reason I don't like this chart so much is because we're not going to cover up vast swaths of Nevada, for instance, with solar panels for the benefit of the rest of the country. We're going to distribute those solar panels over larger areas. But this is just meant to emphasize the point that the land area usage does work out in our favor. So the way we distribute solar panels typically is either on residential installations, like this one, or in large field installations. This one, the Sarnia Solar Farm in Ontario is currently the largest solar farm in the world. We call it a solar farm because it's just a massive land area comprised of solar panels. This is the covering half of Nevada scenario, right? This here on the left hand side, on the other hand, is a residential neighborhood in California indicating the more distributed variety. And both have their distinct strengths and weaknesses. So solar isn't about those small, little, rinky dinky, 20 or 30 watt panels that are sitting on a remote thatched hut. Solar is really growing up to be a grid tied, grid integrated, renewable energy source that is now probably skirting a $100 billion industry worldwide. So it's growing up, and certainly professionalizing quite a bit. Historical perspective. It's time to take a look back and trace through some of the technical history of how solar cells came into being. And that really will inform why it is we're at where we are today, why the industry has some of the biases it has today, and what are some of the intangible barriers that could be needed to be overcome if we are to develop new technologies. So aside from just general knowledge and general edification, this has an important technical aspect as well. So historical perspective. We credit the discovery of the photovoltaic to this gentleman here, Edmond Becquerel, shown here in his more mature years. When he wrote this article, right here-- I'll probably butcher it, but it's "sur les effets electriques produit sous l'influence des rayons solaires." Basically, the electrical effects produced by the influence of solar rays on a contraption that looked very similar to this. He noticed a current flowing, essentially a photovoltaic, a photon induced, a light induced effect current. And he was very smart to decouple the effect of heat from the light. So his experiment involved selective filters that prevented massive amounts of heat from getting through. And he essentially produced what is a spectral response. Varying the filter color, he was able to trace out the response of this apparatus to the solar light as a function of wavelength. This was a clever experiment. He wrote it up. It's more of an electrochemical device rather than the solid state photovoltaic device, like the one we know now. But nevertheless, it earned him the credit of being the discoverer of the photovoltaic effect. Does anybody happen to know how old he was in 1839, when he discovered this or when he published this work? It's a rather nice article. Very eloquent, very detailed. He was 19. He was born in 1820. Anyway, small aside. The field evolved from 1839, when that first article came out. Folks began refining and-- well, first of all, discovering new elements during that period in the 1800s, refining them and then testing their properties. And this was before we really understood what semiconductors were. They were a little bit of a black box, a big mystery. Their physical, electrical properties were all over the map. We'll explain why over the course of the next 10 lectures. And they began refining these materials and putting them in various contraptions testing them with light. And lo and behold, they would get the photovoltaic effect again, maybe photoelectric effect first, and then, the photovoltaic effect, finally, when they set up the experiment properly. And selenium was a popular material at the beginning. So was cuprous oxide, Cu2O. That was a very common material. And I love pointing this out. This is a little contraption, a vice. To hold the contact onto the device. And as Joe can tell you, contacting a solar cell is not the easiest thing in the world. So it's a pretty funny picture, especially in light of our current difficulties in 2011 on resolving some contact issues, especially with new materials. But that gives you a little bit of a historical perspective. And the references are there. In 1954, the first embodiment of what we consider the modern solar cell came into being. This was driven by the purification, crystallisation, and growth of silicon, which is the second most abundant element on the Earth's crust. It was noted to be superior to germanium for electronic devices because of its larger band gap, less leakage current. We'll get to that in a few lectures. It had superior properties. And it was engineered into, I would say, the first what we call a, homojunction p-n junction based solar cell device in Bell Labs by those three gentleman there, on the upper left. And in 1954, the paper came out in the Journal of Applied Physics. And that really spurred a lot of interest in the field. Why? Because 6% efficiency was about a factor 15 higher than anything that had come before it. And now, people could see the potential of this technology to drive things. At the time, within a few years, within a decade or so, folks were more interested in sending satellites into space than they were, perhaps, powering terrestrial objects. But we'll get to that in a second. But some of the first examples here, in Bell Labs in New Jersey-- they had a small little radio communicating with this little device, over here. And the solar cell was powering the gadget. And it's interesting to note here, the New York Times article from that time, "with this modern version of Apollo's Chariot, the Bell scientists have harnessed enough of the sun's rays to power the transmission of voices over telephone wires." And they speculate that at some point-- obviously this was written in the 1950s, keep that social context in mind. "But eventually leading to the realization of one of mankind's most cherished dreams, the harnessing of the almost limitless energy of the sun for the uses of civilization." They saw the opportunity there. It was not lost to them. But of course, a lot of development had to come under the bridge. A lot of water had to go under the bridge before they were able to make solar really cost effective from 1954 at almost 60 years later. The way that basic solar cell device worked-- I'm going to introduce you to the full picture now. And I will begin dissecting it piece by piece, over the next lectures, so that we really understand each component of how the solar cell works. And we'll put it all back together again. We'll actually make it, literally. So the sunlight comes into this device. This is a cross section of a solar cell device. And today's modern solar cells are about four times the thickness of your hair. So if you can imagine 200 microns in thickness, that's the thickness here, the cross section of this solar cell device. Light comes inside, excites bound charge, and makes it mobile, so it can move around the material. There's a built in electric field, which serves to separate that charge and create the voltage. And so one of the charges goes here. The other charge goes to the back. So you have a voltage or a potential difference across these two terminals, across the front terminal and across the back one. And then, if they're connected by an external circuit to an external load, current will flow through that external load to complete the circuit. And that's essentially how the solar cell device works. So three basic steps, there's charge generation. So light is exciting charge within the material. The second important step, up there in the upper right, is charge separation. Somehow, you have to induce a voltage inside of your material. And the third very important step is somehow you have to collect the charge coming out of it. That's why those folks in the earlier days had that big vice over here. They were trying to really make sure that the metal was in good contact with the material so they could extract the charge. And so that's essentially it. The advantages of a solar cell devices is that there are no moving parts and no pollution created at the site of use. There is, obviously, the manufacturing of the module itself. And we'll get into detail about that, and begin quantifying the amount of energy, the cost to manufacture it. Bottom line is that the CO2 production per unit energy output from the solar panel is on the order of 10 times less than coal, 5 times less natural gas, so significantly less than fossil fuel. It is not a zero energy system. The reason why the majority-- where the majority of that CO2 comes from is actually the energy used to produce the solar panel. So as we transition to solar panels made from other solar panels, as the solar industry ramps up, obviously the carbon intensity of producing the solar panels will go down, as well. Likewise, it matters where you produce the panels. There's some active research going on at MIT to decide where in the world it's optimal to produce the solar panels and where it's optimal to actually install them. The disadvantages, which really embody why we haven't seen a mass of adoption of solar to date and why there are technical and nontechnical challenges for you here to resolve is because there's no power output at night. In other words, when the sun's not shining, it's not producing electricity. And there's lower output when weather's unfavorable. And thirdly, today there's a high cost. We'll get to that in a few slides as well. So it's not economically competitive in most markets. In some, there are. In 1.5 out of the 50 states here in US, solar is cost competitive, today. But in the remainder, it's not. So this is the really fun part. This is why when you pick up your phone, and text your parents, and say I'm in a PV course. And they write back, ah, PV, I've heard about that for decades. That's an old hat. That's not going anywhere. You can write back and say it's very different today than it was then. And here's the reason why. In the 1970s, when PV really started to take off for civilian purposes-- obviously, they had put satellites up into space. They had proven that it worked. It was robust. In microwave relay stations up in remote locations, that they didn't want to service, they also would place PV panels. But in terms of civilian purposes on houses and so forth, really late 1970s, early 1980s were where things were beginning to take off. And driven by the oil crisis. The OPEC oil crisis of 1970s. This is a New York Times article describing the state of the art of solar. This is taking a look some 20 years later at solar and saying how far have we come. And one of the interesting things of note in this article, right here, is that it cost upward of $10 a watt for the solar panels, in that day in 1979. Meaning it would take, roughly, $12,000 to run an ordinary household toaster. So that was the impression that folks had of solar in the 1970s. And for good reason. This is the cost of electricity produced of solar versus time. In reality, the x-axis, if you look closely, it's cumulative PV electricity production. That means for each new panel we make and for each new unit of energy that that panel is producing, the cost of electricity is coming down. That's because we learned how to make panels better. We learned how to make cheaper panels faster, with less cost. So the cost of the electricity reproduced, over here, is showing going down with time. And this is a little bit of apples to oranges comparison, that's why they're two different colors for the two different dots. The black dots represent the average retail electricity prices. Not costs, but price. This is going to be a repeating theme throughout the entire course. I'm going to emphasize it now. Can somebody tell me what the difference between cost and price is? AUDIENCE: Price is going to be more than cost because the company wants to make a profit on the product. PROFESSOR: Yeah. So let's see I make a gizmo-- this is a great example. I make a gizmo that costs a certain amount, x, let's say. And now, I sell it for 3x. And I make 2x profit. So the price would be 3x. The cost would be x. And so the cost of solar is shown here in the white dots. And the price of retail electricity price is in black. Why is this comparison made right here? Why would somebody do that sort of apples to oranges comparison? What point are they trying to make? AUDIENCE: Because we need to bring down how much we need to put into PV to be at to compete with the price that electricity is at, as opposed to cost. PROFESSOR: Exactly. This is a substitution play, right? You're looking at PV substituting what is, in that case, the base load and peaking price of electricity-- probably more driven by the peaking price of electricity. And so what they're doing here is they're saying, OK, how much does it cost to manufacture this panel, and how does that compare against the grid if I were to plug into the wall over there and extract electricity from the grid. How much would that cost me? How much would I have to pay for that electricity? And that's really the comparison that they're trying to drive right here. AUDIENCE: Does that adjust for inflation? PROFESSOR: Yes. The details are in this paper, right here. Again, you can access all this information online. But it is adjusted. These are, I believe, in 2002 prices. I can't remember the exact-- yeah. AUDIENCE: What are some of the assumptions used to compute the cost of PV and electricity? PROFESSOR: Great question. So the higher density of data points, over here, is in part because they get closer together. It becomes harder to drive the cost down. And of course, we were looking at it in a log scale. But also, the quality of the data is much better in recent years because we had access to-- greater number of companies were able to average values coming from multiple sources. Some of the earlier data, especially 1957-- those were some of the first solar cells produced. If they had access to good primary data, those numbers would be highly accurate because it would be one company making it. And that's it. Very little error bar. But if they were making guesstimates based on material cost of the day, then there would be some error bar associated with that data point. These curves are very difficult to produce when you're in academia. But I can say that when we were in industry, we did this for our company just for hahas one day. And it fell on a very similar slope-- with a similar slope and a similar value. So somehow they were getting the numbers right. AUDIENCE: In terms of insulation, what numbers are you using to assume-- like you said, do you use values for Nevada or do you take an average of the summation of the entire US-- US average for the retail electricity prices. PROFESSOR: And so the retail electricity prices in the United States vary quite a bit. You have some coal rich states, like Wyoming, that get $0.05 per kilowatt hour. You have states like Massachusetts at the end of the energy pipeline. If you look at the natural gas pipeline, for example, we're at the very end. We get some of our natural gas even shipped in by boat. $0.18 per kilowatt hour is residential prices. And in California, which has a tiered structure, if you're one of the highest consumers of electricity, you're going to be paying somewhere around $0.30 per kilowatt hour compared to some of the lower use folks down around $0.12. And so, it varies quite a bit. Typically, when you're looking at these sorts of charts, if the chart is produced, say, by the USDOE or some solar promoter, let's say, they will typically be choosing a rosy scenario of the American Southwest because that is-- well, not only do the numbers look better but, more importantly, that's where a lot of the solar is being installed, today, but not all of the solar. Because it is a substitution economics situation, two parameters are really of interest that drive the cost competitiveness of the solar installation. One is the retail price of electricity. How much are you paying out of the wall? What are you substituting it with? And the second is how much sunlight you get locally. So our break even point in the state of Massachusetts is not too far off from Arizona because they have a lot cheaper price of electricity even though we have a lot less sun. So I wanted to emphasize a couple more points. So when Gregory Nemet put together this chart, it was within the context of a really interesting paper in which he attempted to decouple the effects of scale from innovation. Let me emphasize that. So if you are making a widget-- let's imagine a razor, like Gillette does here in South Boston, or if you're making some other high tech product-- razors by the way are very high tech. How many times have you cut yourself by a defective razor? I certainly haven't, and I've probably used tens of thousands in my lifetime of individual blades. And that's because they're examined using laser technology. They're really manufactured in a high tech way. And they get better and better every time they produce one razor blade. And so they follow a learning curve, just like photovoltaics does. With cumulative production, the cost of producing one widget goes down. And likewise, microwave ovens and other high tech products. And so the big question is, how much of this learning curve cost reduction is driven by innovation and how much is driven by scaling-- just learning how to do incremental improvements, tweaking the manufacturing line to make it a little bit more efficient. So Gregorian Nemet, the author of this paper right here, in which this figure appears, looked into that question and came up with some answers. Some of those learnings were incorporated onto this beautiful chat here produced by 1366, a spinoff from MIT focused on commercializing really cool next-gen PV product. They took that learning curve from Gregory Nemet's paper, plotted in a slightly different scale, and showed several of the technology innovations that drove down that learning curve for crystalline silicon. And so those, in the fine text there, represent specific technologies. And we'll be getting to know some of them over the course of the PV course. And so, we're approaching this very interesting point. If you haven't noticed from this chart right here, this ended in 2003. And boy, these two are getting very close to one another. We're entering a very interesting point where the cost of producing PV electricity is rapidly intersecting with the US retail electricity prices. And that is represented in a very broad brush strokes by this DOE chart that was produced in approximately 2006 with the Solar America Initiative, where you have the system price range for PV systems. Again, broad range now, instead of finite data points. Residential and commercial rates and utility generation. For those who have already dealt with electricity markets, the residential commercial rates-- this is the price or the retail price. And the utility generation, this is more the wholesale price over here for utility scale. So again, just showing you the range of substitutions that could be going on. And we're entering the regime now where, finally, solar is starting to be cost competitive. And when you start having this sort of interaction, you can imagine two Gaussian curves, one curve representing the price of solar and the other the price of electricity. And as they begin overlapping, as the price of our electricity goes up-- it went up 15% over the 2000s here in Massachusetts, the retail price of electricity in residential. And as the price of solar comes down, those two Gaussian curves begin moving against each other. And at the edge of a Gaussian, you can model that using an exponential. And so you have two exponential curves overlapping. You have, effectively, an exponentially growing market penetration. In other words, the solar adoption on the grid is following a hockey stick curve. And that's why you hear a lot of interest in solar these days. We had a solar system installed in our house in 2007. And now, our neighbors put them up last year. The folks across are putting them up, actually, just last week. And there's another family down the road. So our little neighborhood is representing this little hockey stick, right here, as is Cambridge as a whole and some of the places in the United States where it does make economic sense. You're beginning to see that take off. And that's why it's such an exciting time right now. This is a much busier chart. There's a lot going on. But to sensitize you, this is the PV residential. In other words, it's either the cost or the price to install PV on a residential home. In other words, it's a smaller system. So there's a larger overhead per system. The architect needs to spend more time per unit energy produced to design your system because it's a smaller one. A lot of people go out there per panel to install it. Whereas PV utility, those large fields filled with PV panels, it's cheaper per unit panel to install. One architect can sit down and design the whole thing-- maybe a team of architects. But the overhead costs are lower. And you can bargain with the module manufacturers to get a better rate on your modules. So get a better price. And as a result, the PV utility costs and prices tend to be lower than PV residential. And the blue and the red, here, just represent the wholesale and retail electricity costs-- what they're substituting here in the United States. So a bit more detailed chart, again, showing the grid penetration down here at the bottom. Also, in terms of percent. So back here, a few years ago, the 0.2% of total worldwide electricity was generated by photovoltaics. And projections are that by 2020, we'll be at around 1%, by 2030 around 5% using these just two overlapping Gaussian curves. And it's interesting to note that this is global. On a local level, Germany has already well surpassed 2% in Bavaria. I think it might be up 3% or 4% now in photovoltaics, in the southeastern region of Germany. There's a small island in the Hawaiian chain that has, I believe in peak days, around 40% of its electricity produced from solar. So there are regions that already, you have a very large percentage of the total electricity being produced by solar because of that distribution of prices. And lastly, this is a really exciting chart. This is the convergence between PV and conventional energy-- essentially, what this chart over here was attempting to capture in its percentages. This is explicitly laid out, now. And I took data going back to the 1970s, and plotted the average terawatts installed of new PV installations versus total primary energy-- new primary energy installations. So for those energy wonks here in the audience, what is the primary energy burn rate in the world, right now, in terms of terawatts average? Around 15, right? And so the average new energy installed each year is represented here. It's somewhere between 100 gigawatts and a terawatt, typically. And this is the new PV installs. You can see that we're within about two orders of magnitude, now, of total new energy installs. This is primary energy. For electricity, it looks even rosier. And so we're rapidly approaching the point where substitution will begin. We're going to start replacing, not only we're going to take a larger share of total new installed energy, but we may even start putting some existing power plants out of business. And we'll get into that in the economic section in the third part of our course. Interesting to note, these three distinct phases of growth of the industry over time. Phase one was right at the beginning, when we had the OPEC oil crises, when people were really interested in solar, but it was really a boutique thing. And solar cute, great PR, but not really impactful. In this regime, right here, where most of you were born, solar went kind of through a down cycle. So while the price of oil was really high, right back here, it crashed in the early 1980s. And symbolically, Ronald Reagan ended up taking down the solar panels from the White House some time in '86, '87. And big oil companies were the ones who kept the solar industry going, interestingly enough. It was Mobil-Tyco. It was BP Solar. The largest companies that were producing solar panels in the world were ones that were small divisions of larger oil companies, which viewed themselves as energy companies. And then finally, this phase three, this really rapid growth here. Again, a cumulative annual growth rate somewhere between 40% and 50% average. That took off when generous government subsidies, whether it be for the manufacturing or the installation. In the case of the United States, it's mostly been on the installation side. In the case of China's, it's mostly been on the manufacturing side. Japan and Germany had a bit of both, but more heavily toward the installation. And we saw a massive growth of the PV industry because, now, the government's realized, well, wait, the cost is coming down, and we will need new electricity coming on board. And our oil supply is a little unreliable. So let's invest in this new technology and see where it takes us. And I think the Germans, now, are paying somewhere on the order of a euro, maybe a little over euro, per month on their electricity bills as a result of having financed a lot of this growth right here in the PV industry, which allowed the costs to come down for the entire world. So it was a successful program. And as a result, many pure play companies saw the financial opportunities. The case Q Cells, which is highlighted down here, is not unusual in those days. In the late 1990s, a group of executives at McKinsey got together and said, wow, the numbers look really promising in the solar business. Why don't we form our own company and only do solar instead of being part of a much larger one where they have their interests dispersed among many different product lines and technologies? Let's focus exclusively on solar, burn our bridges behind us, and just go for it. And they went for it. And for a while, for a few months Q Cells was the largest solar cell producer in the world. It was, I would say, a poster child of this new generation of PV companies coming in this third phase here. And as we'll learn over the course of this semester's course, many of the leading solar producers today are now located in China. So this is, basically, the history of PV development. And the important thing to note is this closing gap, right here. So when folks are saying solar, it's the same old, same old. It's been gimmicky. It's been around for a long time, but it's not going anywhere. You can point to some of this data and, say, no. Actually, it's on the cusp. It really is beginning to take off. And these are some of the data you can point to if you care to do so. Let me spend a few minutes talking about the broader picture beyond just solar photovoltaics into some of the other solar technologies. We won't be addressing too many of these over the course of the lecture because we have to focus and we have to become very good at something, otherwise we spread ourselves to thin. But I did want to give you a sense of what's out there so that you can situate solar photovoltaics within a broader context. And so this is a solar energy technology framework that encompasses all conversion technologies from sunlight into energy. And so first off, I start with a rationale for framework. Why invest the time to come up a the framework? I'll explain why. There are several hundreds of technologies out there that can convert sunlight into energy. And to make sense of the technology space and to provide some meaningful technology assessments, there have to be some performance driven, technology neutral performance metrics that you can use to evaluate one technology against another. And that's why coming up with some sort of framework is very helpful. So the three criteria that I chose together with Vladimir Bulovic when we put the together, to design a technology framework was an exhaustive categorisation. In other words, our framework had to encompass more than 90% of all technologies out there. The 30 years challenge. Again, in 30 years, the PV technologies should be able to fit into this framework still. And it should be a useful analysis tool. It should be able to give insight into the complex space that's out there, and allow folks, like yourselves, to make sense of it, whether you're trying to develop cost models or if you're trying to develop technology prospectus. This should allow you to gain a foothold in it. So we have solar energy conversion technology. And we chose an output oriented rationale for dividing the solar energy conversion space. So the output would be either electricity, heat, or heat which is then used to power, say, a turbine which generates electricity, or fuels. And those are the four primary outputs of solar energy, today. Yes, there are technologies out there, for example, that convert sunlight and store it in some way and convert light on the other end. But we're not including those in here because, again, the 90% rule. We're focusing on the major ones. And then, we do a further subdivision between the non-tracking and tracking. Tracking means if the sun is moving through the sky over the course of the day, your apparatus is following the sun so as to maximize the cross section between the incoming rays in your device. The reason we chose tracking non-tracking is because tracking requires motors, which will add cost and reliability questions to your system considerations. And that's why we chose this further division right here. So on to the assessment. Let's look at the technologies that are out there and try to bin them. Solar to electricity. There are a few embodiments. There's the photovoltaic device, these ones. There's the thermoelectric device as well, which convert solar energy into heat, really, and then heat into electricity. So maybe it should have been in the other category. But it is a device that converts solar energy into electricity. So we've seen a solar cell device. We've learned the three steps, charge generation, charge separation, charge collection. And we look at the existing technologies that are out there, today, and say, all right, let's start to bin them. We have non-tracking systems that can be non-concentrating, like these panels right here. Essentially, they're just flat panels that are receiving the sun's rays. Or, you can have cheap, mirror-like materials that bounce the sunlight off of them into the solar panels and concentrate sunlight. So let's imagine we put a set of mirrors on either side of this panel, right here. And when the sunlight bounced into the mirror, it would reflect back into the panel. That would be a concentrating, but non-tracking, system. And these are common on barriers along the highway in Germany. They're sound barriers. They're preventing the people who live on the other side of that barrier from hearing the noise of the cars going by on the Audubon. They're not meant to be crash barriers. Those are separate, closer to the actual road. But these are examples of concentrating and non-tracking photovoltaics. There are ground mounted and roof mounted systems. So again, another way to split the pie. In the concentrating non-tracking system, there aren't only these types, there are a variety of other species of concentrating non-tracking devices as well. There are so-called sliver cells-- which the light comes in, bounces around a little bit, and then eventually gets absorbed by the device. And that even happens, to some degree, in these modules, too. Because the light comes in-- make sure I don't reflect this into your face. There we go. Point it up. The light can come in sometimes and reflect off of this white back skin. If the light is coming in at an oblique enough angle, total internal reflection by the glass. It'll get a second bounce and go into the device. We'll talk about how that works in a couple of lectures. So internal reflections. And this is particularly timely. Does anybody know-- does the word Solyndra ring a bell for anybody? Yeah. What about Solyndra? AUDIENCE: It went bust. PROFESSOR: It went bust. So it's one of the three photovoltaics start up companies in the United States that went bust over the past few months over this past summer. And that's a really interesting market dynamic, which we'll get to in the third part of this course. And we'll discuss that head on because it's an interesting, and very important dynamic in the evolution of the solar industry. We have some technologies under development at MIT. Marc Baldo's lab and Vladimir Bulovic and others are working on devices that absorb sunlight, reemit the light at a different wavelength, trap it inside of some high index medium, like glass, and then, ultimately, concentrate it on to the cells that are on the corners. So you can imagine a window that absorbs some of the incoming light, bounces light off, and eventually concentrates the light in the corners where you have your solar cell devices. The advantages, or the potential advantage, here is that you can have a very high efficiency, expensive device, but a very small area of it. Instead of covering this entire area right here, you've now reduced the total area. And then, if this is a very small percentage of the total system cost, you can just switch it right out when a new and better technology comes along, almost like you switch out your computer. So if a better solar cell device comes along, you can take this one out and put the next one in. It's almost like an upgradable system because the majority of the embedded cost is in the concentrator and not the solar cell device itself. Again, just really drinking out of the fire hose this morning. We're drilling you with data, but it's meant to begin to sensitize you to some of the terms and some of the ways of thinking here in the field. Tracking. So when we're talking about tracking, there's a rise in the number of tracking systems in the United States. It is shown with high efficiency modules that it can be more cost competitive if you have a large field installation to do one axis tracking. One axis tracking and two axis tracking. Why would you want one or two axis tracking? What are you tracking? One axis tracking. What would make sense to track with a one axis? If you had one axis to choose, would you rotate east west? Would you rotate north south? Would you rotate northwest to southeast? Where would you go? AUDIENCE: East to west. PROFESSOR: East to west. Why is that? [CLASS MURMURS] PROFESSOR: Because you're tracking the sun over the course of the day. And you're tracking, pretty much, every day of the year. So 365 tracks per year. The two axis tracking, what is this other axis? Presumably, it's orthogonal to the east west. In other words, north south. Why would you want to track north south? Seasons, right? Yeah so from winter to summer, you're tracking. So you would always want your solar panels facing south, I guess, right? AUDIENCE: In the northern hemisphere. PROFESSOR: In the northern hemisphere. Exactly. So if you're in Australia or in Brazil, your solar panels are facing north. So let's accustomize ourselves with that. And the two axis tracking, of course, would allow for that adjustment. The reason one axis tracking is taking off as the most common field installation tracking system is because the seasonal adjustment, if it really needs to be done-- it's not a huge energy benefit, but if it really needs to be done, you can probably just crank by hand instead of using a machine or a motor that can break down. And the adjustments still need to be made very often. Non-concentrating and concentrating PV. Tracking. So these are one axis trackers, right here, tracking over the course of the day, but not concentrating. In other words, they're flat panels like this, but just mounted a one axis tracker that follows the sun over the course of the day. The system over here is a two axis tracker that includes little lenses that are focusing the sunlight onto tiny little cells. And again, very similar idea that the solar cell itself is high efficiency, but it is a low percentage of the total system cost. Non-concentrating and tracking. Again, several examples of that. You have fancy systems, two axis trackers, again, most common. Can anybody guess what this little gizmo is, right here? We're going to get to that in next lecture but-- AUDIENCE: A solar sensor that finds the position of the sun? PROFESSOR: Exactly. Somehow, you have to have a measuring device if you have a tracker. It has to tell you where the sun is. So this little gizmo, right here, is just making sure that the panels are facing the right way. Awesome. So concentrating and tracking. Here's a closer look at some of the Frenel lenses that are used to concentrate the light down. On some cheap microscope-- or sorry, cheap magnifying glasses they also use Frenel lenses. And so this is an example of a low cost apparatus here to concentrate the sunlight onto your high efficiency cell. Solar to heat electricity. We're not going to talk too much about this during the course. But just to sensitize you-- that there are technologies out there and some pretty exciting once. There are heat engines. In other words, sunlight heats a fluid, which moves a turbine or a piston, either directly or by heat exchanger. Heat exchangers. Thermoelectrics. Long wavelengths photovoltaics. These are devices that convert the heat portion of the solar spectrum into usable energy. And there are hybrids that are possible with these. So if you heat up a fluid, say, a salt or a glycol solution, then you can store the energy in terms of heat. And if the stored energy begins to decay with time because of poor insulation, you can augment that heat with natural gas or with some other fossil fuel. So you get these hybrid, renewable solar and natural gas power plants that are possible with the solar to heat electricity. And there are some really fancy designs out there. And I'm happy to dive into these in more detail. The most common one are sunlight coming into some sort of reflector, and then concentrating the sunlight into a thin tube that contains your high heat capacity material, liquid usually-- so a glycol based liquid or even a salt, sometimes. It has to have a high heat capacity. In other words, it has to be able to absorb a lot of heat and retain it. But it also has to have, ideally, a minimum amount of corrosion so that the longevity of the parts is sustained. And you can see, here, these tubes that are running along here and going down these fields of collectors all the way to the other side. And somewhere off in the distance is the heat exchanger. So that's solar thermal for you. We have parabolic dishes concentrating sunlight into Stirling engines. That's kind of neat. And so your T high is basically that of generated by the sun. And you T low is the ambient. So typical mechanical engineering there. And you also have solar power towers. There's some work being done at MIT in this as well with Alex Slocum and others that are using fields of mirrors to concentrate the sunlight into a tiny little spot, right here, in a big tower. Say, for example, that spot right there, it's dark. It's not in operation. But if it were, the sun would be concentrated onto that little spot. It'd be really, really bright, indicative of it's very high temperature, on the order of a couple thousand Kelvin. And then the sunlight would either be absorbed up here, with some molten salt, or reflected down underground to a heat reservoir. And that would be your T high running your engine. So your Carnot engine. And then the T low would be the ambient temperature. Solar to heat. This is really important in developing countries. Not to be overlooked, the very simple, low tech conversion of sunlight into heat. You can heat water. This is very, very common on rooftops all throughout the sunbelt of the planet. You'll see these on the roof, painted in black. They contain potable water, typically used for either, say, for example, showers or kitchen use. And the fancier versions that are really marvels of engineering. These materials all have to be coefficient of thermal expansion matched. As it heats up, the glass tubing has to match the expansion of the metal around it. So it is quite a feat of engineering that they make these so well. There are a few companies in Germany that really pioneered this effort right here. Of course, you have tracking versions, like solar ovens. Not too common. You typically find more still in developing countries. Unfortunately, you find a lot of wood burning, which isn't good for the cook, which, unfortunately, most often is female. And so this illustrates some of these societal questions that solar involves. It's not just the technology. This involves gender equality. This involves societal development. This is a much broader topic than just the fundamentals of the physics of how the solar cell device works or how sunlight is converted into energy. And that's why we have the three segments of the course. Lastly, solar to fuels. The way I've traditionally broken it down-- it's a little bit wishy washy-- is into enthalpy and entropy in the sense that, in enthalpy, you're storing the sunlight in bonds-- in chemical bonds. The bonds are forming-- more complex, higher energy molecules are being created. So you're taking water and splitting into the gases. Or you're taking CO2 and water and converting it into hydrocarbons. And those can be used to store the fuel and, ultimately, release it in the form of burning the fuel. So it's a closed loop cycle. And what I refer to as entropy, which I get some flack from the folks in chemistry for, is the separation of phases, in other words, desalination. If you separate your salts from your water, then you're increasing the energy of your system. You're doing a physical separation. And it is a form of energy storage. So this right here is the example of the renewable fuel cycle where you have sunlight coming into your starting compounds. Using some catalyst, typically, you're creating the intermediate compound, which is a solar fuel. Then you burn your solar fuel. Then you have your final compounds. In the ideal world, 5 equals 1. The final compounds are identical to the beginning compounds. And you have a closed loop cycle, a renewable cycle. And so a lot of work is going on here at MIT. This is a recent paper we published together with Dan Nocera. His group is looking to special types of catalysts. Our group makes solar cells. So we work together to make these nifty little devices that convert sunlight into storable fuels. What you see here are little bundles coming up from the water in which the solar cell device is embedded. The water is near pH neutral. Then it's converting that sunlight into gas, into hydrogen and oxygen, which can then be stored. On one side of the device, you could be creating oxygen. On the other side, hydrogen, for instance, if you have a physical separator, you'd be able to store that electricity. This is an example-- a very simple example-- of desalination driven by solar. There are much fancier examples, as well. But that gives you an idea. You have contaminated or salty water. And you're evaporating the water. It dribbles down into this little collector over here, and finally out into your collecting pot, leaving the salty, brine behind. And then in the broader perspective, we have many other issues beside just the conversion technology itself. We have how do we use the electricity and how do we store it. Is the solar power generation centralized and all the users distributed, similar to how we produce power today? Do we have one big solar field that's producing electricity for all of Cambridge, or do we have the individual solar panels in each of our houses that are producing the power locally, and they're all interconnected? In case a cloud goes over one region of Cambridge, there's still coverage. That's a really big question. And the economics are what's driving this right now. These large field installations give you a sense. This is a road right here. These little green specs are trees. These are huge field Installations of solar. The economics are driving it right now. But there are opportunities with commercial buildings. This is the Moscone Center in downtown San Francisco. It's like the Heinz Convention Center equivalent there. This is an example of a house in Rochester, New York. That housing development in Rancho Cordova in California. So you have examples of residential installations as well. Are we just going to let economics drive this? Is there going to be some policy involved? Is there a smarter way to do it, not only from a cost perspective, but from a societal perspective or an energy grid robustness point of view? What are the right choices here? There's a lot of open questions right now in the field. And what about energy storage? Are we going to store it in terms of batteries and fuel? Centralized storage? Are we just going dump it into the grid and be free riders? Let the grid handle it, somehow. Hope that the grid a stable enough that when a lot of solar is being produced and when no solar is being produced, it'll just be able to accommodate. I guess the resistance in the turbines of the fossil fuel plants will either increase or decrease depending on how much energy we're pumping into the grid. And so at the end of the day, we have this very complex space of conversion technologies. The solar electricity, solar to heat, and so forth. And the system itself, whether we have centralized generation of electricity distributed generation, and whether the storage is centralized or distributed, whether you have storage inside of our house on the inverter, let's say, or in the basement, or rather the storage is some centralized storage facility in the center of Cambridge that serves as a buffer. And we have all of this space to play in. We're going to be focusing on solar to electricity. So we'll be focusing on these two columns right here. And specifically, the technologies during the first two thirds, and then, the broader, system level impacts in the third of the course. So that puts it all in perspective, I'm not going to get too much into this. I'm just going to say one quick word about CO2, energy, and climate change. You hear a lot of talk about, at least from the political sector, that scientists are, shall we say, in a lot of debate whether climate change exists or not. That is patently false. The majority of scientists, upwards of 96%, believe that there is strong evidence to support the fact that human energy consumption, especially the high CO2 intensity of our energy consumption, is driving some form of climate change. What the magnitude is and what the impact is-- obviously, that is still under discussion. But the reality that our emission of energy-- our emission of CO2 as a result of energy use, our fossil fuel energy use, is driving some form of climate change that there is widespread consensus among the established scientists in the field. Now if you want to do some back of the envelope calculations just to convince yourself that we, tiny, puny, little human beings are having an impact on our world, do this for me. Take the total energy consumption rate. This is the energy burn rate. So it's the average power-- average rate of electricity use. Look at just the fossil fuel based energy sources. Or if you prefer, take the average CO2 intensity of our energy mix, which somewhere around 600 or maybe 800 grams of CO2 per kilowatt hour. And then look at that amount of CO2 emitted. You can calculate how much CO2 is emitted per unit time from our energy mix knowing the carbon intensity of our energy mix. Then do a quick back of the envelope calculation. Assume that our atmosphere is 30 kilometers thick. It's a generous assumption. The density of the atmosphere dwindles pretty quickly above 10 kilometers. But assume it's 30 kilometers thick. And then dissolve all of that carbon that we're creating from this energy mix into that thin shell surrounding our earth. Our earth is on the order of 6,370 kilometers in radius. And it's only 30 kilometers thick, the atmosphere. That's why those beautiful photos from the space missions, when you see that thin blue shell on the planet, right-- that's the atmosphere. It's really, really thin. Just do a quick carbon density analysis. And you'll see that we're adding hi tens of parts per million of CO2 to the atmosphere. And then you look at the total CO2 in the atmosphere, which is in the order of 400 parts per million, and you'll see that we're adding an appreciable amount, just given the carbon intensity of our energy mix and the total volume of atmosphere into which we're dumping that carbon. And so the question of whether or not we are adding carbon to the atmosphere, I think, is indisputable, based on some quick back of the envelope calculations and, of course, the more in-depth models. The only place where you can have some wiggle room to argue is whether or not CO2 actually influences the climate. And for that, there are a number of studies discussing that point. I would refer you, specifically, to these here, published in Science in 2005, that discuss historical correlations over the last 600,000 years, correlating CO2 and mean global temperatures based on oxygen isotope ratios containing gas bubbles, for example, in ice cores. So I would say if you're arguing whether or not we're having an influence on our atmosphere, I would say that is a difficult position to take. The only room that I would give you some room to maneuver would be if you said, well, you know, CO2 really isn't that bad in the atmosphere, despite what our infrared absorption data seems to indicate, that it really does absorb infrared light and reemit it. So that's what I have to say about the climate, which is a huge motivator for a lot of people taking the course. And you're welcome to talk about that in more detail, but I'd really love to keep this focus on the technology, by and large. And for that, I'd like to hand out these background assessment quizzes for each of you. Please take a few moments to fill these out-- just pass them back-- so we can learn more about your interests. And what I'll also do is pass around this little solar module, right here, so you can get a sense of what a small little solar cell looks like up close and personal. Once you're done, feel free to come up and take a look at the solar module, right here, as well. And thanks. |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | 5_Charge_Separation_Part_I_Diode.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: We'll be talking about the diode and charge separation inside of a solar cell. So at the end of last class, we ended with this little demonstration right here, where after much discussion about band gaps and light absorption, we agreed that that little piece of silicon there should be absorbing the light and free charges should be generated inside the silicon. But without some fixed fields, some built in field inside of the silicon material, it wasn't possible to measure any current output once we shown light on it. And that made sense. There were free charges being generated, but due to Brownian motion, there was no net charge flow. They were just moving around with no net movement. And as a result, there was no current. Now if we attach the batteries in series, the batteries apply to field across this little piece of silicon resulting in current flow-- more precisely, drift current across that piece of silicon. And today we're going to be talking about how the solar cell device has a built in electric field. So obviously we don't want an external power source. We don't want to have a set of batteries attached. That defeats the purpose of an autonomous energy generation device. We'll be talking about how the built in field inside of a solar cell comes into being. As we've followed in the last few classes, we will be discussing physics when it is necessary, taking a very engineering approach to this so as to keep everybody from a business background, from a mechanical engineering background, and from a material science background on the same page. We will have, in the lecture, I think n plus 2. Very physics-y lecture, and we'll be getting to the nitty-gritty of semiconductor physics and how it relates to solar cell devices. So for those of you who are already experts in this material, bear with me. And again, try to relate to the solar cell device. You may not have had that connection in previous lectures in previous classes. So we'll jump right on in. To remind everybody, you're here in the fundamentals. We'll be getting to the technologies and cross cutting themes after we really have a good, solid understanding of how a solar cell works. The conversion efficiency is the output energy versus the input. We have our inputs in the solar spectrum, the outputs in charge collection, and we've been steadily making progress down toward the outputs here. We discussed the solar spectrum, then light absorption, the charge excitation, and finally now we're on to charge drift and diffusion. So we have the total solar cell efficiency as a product of all the individual processes. And any one of these processes can kill the efficiency of the device. That's why it's important to think about your device like so. And just not to make this introduction so boring, I want to really emphasize this point. So I'm making it over and over again. But I wanted to relate this to other engineering devices as well. Namely, today it's a Toyota Prius. These are all the components inside of a Prius that have to work well for the car to function. If one of these components, let's say this one, the inverter, is broken or not functioning well, you're not going to have the car in an autonomous mode. So in a similar manner, in a solar cell we have to have all the different pieces working well together. So the essence of charge separation, we're going to begin our exploration of charge separation using the diode analogy. And just to situate everybody, I brought in a number of small discrete components, small diodes. Those are the ones that are orange-ish that have two leads coming out. There are a bunch of transistors in here as well. Those have three little leads coming out. You'll be able to distinguish them. But just to situate ourselves, these are diodes. The essence of a diode is that you have a dissimilar material on either side of the device. That's why I've worn these dissimilar colors today to denote that on different sides. And of course, you have this mixed region in the middle. We have here an n and a p on either side. And if current is attempting to flow in one direction, it will be barred. But if it attempts to flow in the other direction, it will go through rather easily. That's the essence of what a diode is. How is it made? Well, we're manufacturing materials of the same base element. Let's say, silicon. But we're doing something special to the material to add particular types of charges on either side. We call doping. And we'll get to that in a few slides. Why do we care about diodes? Well, this is the essence of charge separation. And this is what drives the voltage inside of a solar cell device. That's why we care. It's pretty important for at least understanding the traditional semiconductor-based solar cells like this one here. So for those history folks, I figured I would add a quick at description. You typically see the diode represented like this in an equivalent circuit diagram, a little triangle and a line orthogonal to the direction of the current path. And does anybody know where that comes from? No? All right. So back in the day of that, we used to have vacuum tube components for a lot of our-- I would say the discrete components within our circuits. In the case of a diode, you could envision a very simple one where you have a filament that by thermionic emission heats electrons off of the filament, and then they're collected by this other collector up top. If you remember our etymology in Greek, ana is above, cata is below. Catatombs, right? Catatonic, below. So the anode up above is collecting these electrons, and then the electron flow is moving like this, which means that our current flow, defined as the flow of positive charge, is moving the opposite direction. And that's why we have a similar representation right here from our vacuum technology. You could envision also a point where the field is concentrated at that tip and electrons are spreading off. So learning objectives. Today is going to be rather intense, rather dense. And so we will be trying to hold as much as possible in our RAM, so that by the end of class we can really truly understand the solar cell as an entirety. If you get lost along the way, come back to these learning objectives. They're like flag posts along the way. So you can find yourself again. What we're going to do first is describe how the conductivity of a semiconductor can be modified by the intentional introduction of dopants. What this means is we're going to learn how to create the n and the p over here. So if we look at a semiconductor such as silicon-- this is a little piece of silicon right here. Here's a silicon-based solar cell. It's a fairly easy semiconductor to understand. It's what's called a unary semiconductor. It means it's comprised of one element. Silicon right here in the periodic table. This is just an excerpt from the rightmost side of the periodic table. And silicon has 4 valence electrons. So it forms what are called sp3 hybridized orbitals when it's in a crystal structure. It has four bonds with its nearest neighbors, all of equivalent type. And those are covalent bonds. So you can envision that if you were to introduce an atom and substitute out one silicon atom in that lattice for something else that has 5 valence electrons, such as a phosphorus, it's almost of equivalent size. So those material scientists in the room from the Hume-Rothery rules, you should be able to estimate that the miscibility is rather large. In other words, that you could mix in a high concentration of phosphorus into your silicon given the similar size and similar atomic structure. Electronic structure, rather. So you substitute here a group 5 element. Silicon is a group 4 element, as is carbon, germanium, and tin. So we substitute in a group 5 element. Let's say a phosphorus atom in for one of our silicon atoms right here. And this phosphorus atom will have 4 plus 1 valence electrons. So those 4 valence electrons will bond to the silicon atoms, the nearest neighbors. And that one extra valence electron will be left over. It will have nobody to bond to. And we'll describe mathematically what happens to that electron in a minute. But in principle, just intuitively you should be able to understand that that electron should be able to be removed rather easily and move around the lattice with relative ease. Likewise, instead of moving one to the right, if we move one to the left into our group 3 column here, we can dope our material with a boron atom, let's say. Substituting one silicon atom for a boron atom. Boron has 3 valence electrons. Now, those 3 will form bonds with the 3 silicon atoms. And then you'll have a missing bond, a missing electron if you will. That will be a hole. So with phosphorus, or group 5 elements in silicon, we can dope the material with electrons. And with group 3 elements in silicon, we can dope the material with holes. And this is interesting because now we can arbitrarily change the density of charge carriers in our bands. As long as these holes and electrons can become dissociated from the dopant atoms and move freely around the lattice. So that's a big question, how easy is it to remove that electron around the phosphorus atom? If you want to think about this electron just from a quantum mechanical sense, you could almost envision that this electron is attracted to the phosphorus atom because the phosphorus has one extra proton than the silicon, right? So it has a positive charge here in the nucleus. And here's a negative charge. In net, they balance out. But if the electron goes too far away, it will feel that attractive potential back and be drawn back towards the phosphorus atom. So the big question is, what is the binding energy of that electron to the phosphorus atom? And a simple way to think about it is through a hydrogenic model. If you consider a hydrogen atom to have an electron surrounding it, then that binding energy is well-defined. But in the case of a phosphorus atom within a silicon lattice, it's a little trickier. Because now we have to worry about the electron screening coming from the other elements in our lattice. And we also have to worry about a property of the electron in a crystal, which changes the mobility of that electron through the crystal. So what we do is we treat this as a hydrogen atom, 13.6 eV binding energy. But then we do a couple of things. We account for electron screening and we account for what we call effective mass. We'll get back to this in a few lectures and describe exactly what this is here. We need a little bit more semiconductor physics to understand it in its entirety. But think about an electron moving in a crystal as moving differently than moving in free space in a vacuum. That's what this term here is about. This other term, the epsilon, is rather straightforward. It's the electron screening. It's the fact that you have so many other electrons in your system here that screening the charges, screening the electron from the extra positive charge in the nucleus here. And so it becomes easier for that electron to move away. This is about a factor of 0.1, the effective mass. And that's about 1/100. And so overall, we're reducing the binding energy by about a factor of 1,000 down to about 10 m eV in practice. So if we were to run through the calculation for silicon, the binding energy of that electron around the phosphorus atom in the silicon lattice would be around 10 m eV on that order. And that is very, very small compared to the thermal energy in your system, just kt, Boltzmann's constant times the temperature in Kelvin, we have a thermal energy somewhere on the order of 26 m eV. And that's enough to dissociate that electron and allow it to move freely throughout the lattice. Question? AUDIENCE: Is that electron screening related to the [INAUDIBLE] constant at all? PROFESSOR: Yes, absolutely. AUDIENCE: I just had a [INAUDIBLE] question about the holes. So I understand that-- I can imagine an electron moving around in the lattice. But I imagine that a hole moving would be forming a bond-- breaking and forming bonds over and over. Does that actually happen, or are the electrons more free to move? PROFESSOR: Yeah. This is an excellent question. The question, if I may paraphrase, relates to the hole and electron mobilities. How easy is it for them to move around the lattice? And typically, the hole mobility is about a third of the electron mobility in silicon. So absolutely, it is a little bit more difficult for those holes to move around. Holes, you may recall, are quasi-particles. OK, so we have an understanding of doping, a rough understanding at this point. We understand that if we dope a certain type of material, let's say our n-type material, with phosphorus, we have an excess of electrons. And if we dope another type of material with an excess of, say, boron, we would then introduce a large number of holes and the material might have a majority conductivity through our holes. Hence, we call it p-type. n and p comes from negative and positive. Negative being the electron charge and positive being the hole charge. So that's where the n- and p-type come from. Great. So now, pictorially we're going to draw a pn-junction or a junction with our fixed and mobile charges. OK, so let's start out with a simple review of Gauss' law. Spatially variant fixed charge creates an electric field. What do I mean by that? I mean that if you have a high concentration of fixed charge over here and a lower concentration of fixed charge over here, an electric field will develop. And that's the expression right there that relates-- that describes this in mathematical terms where we're using psi here as our electric field. We're using psi instead of e. You probably remember your physics textbooks using e as the electric field because we're going to reserve that variable for the electron energy. That will come in several slides forward. So the charge density will be given by that rho and the material permittivity by our epsilon. And an example is a capacitor here. All I've done is expanded the dimensionality of the derivative here. I'm looking at the capacitor. I have fixed charge on either side. And I have electric field in between. Now, this right here, obviously in the case of a capacity, you're under high vacuum. Just as a very quick aside, this is-- I can't resist. This is really an interesting historical note. This space right here in the diode as we recall, where you have the thermionic emission of the electrons going off to the other plate, this space right instead of here is called the space charge region. Keep that name in mind. We're going to come back to it in several slides. So a bit of history there. So we have the capacitor as our prime example. We have fixed charge on either side and electric field in between. And note that the charge will move parallel to the electric field. If there's no electric field-- in other words, if we're just exciting charge inside of a bare piece of silicon with no electric field applied-- imagine our batteries here are taken out of the circuit. We'll have simple Brownian motion of those photo-excited electrons until they decay back down into their ground state. However, if we apply an electric field, now we have the Brownian motion. But superimposed on top of it, a certain drift of the electrons in response to that electric field. And we can describe this mathematically as well. The drift currents for holes and electrons being described by the charge q. Note that the charge q is going to be different for the electrons and the holes-- the same value, but negative or positive. The mobility of holes and mobility of electrons. The density of holes or electrons. And then finally, the electric field right here. And the reason in this equation you typically see the q's being the same value here, although no minus sign is put in front, is because under the same electric field, you'll have electrons drifting in this direction and holes drifting in the opposite direction. So it's important to keep these signs straight in your mind. Yeah, question? AUDIENCE: [INAUDIBLE]. PROFESSOR: Absolutely. So p and n denote the concentrations of free electrons or free hole-- actually, free holes for free electrons, respectively. So you can almost think of p as being synonymous with the density of boron atoms. If every hole is ionized, meaning dissociated from the dopant atom and free to move around the crystal, then the density of holes and the density of boron atoms is going to be almost identical. Likewise, for phosphorus atoms and/or the donor atoms and the electrons, you'll have almost an equal number. So that's right here. The n is related to the number of electrons that are free to move around the crystal. The p, the density of holes that are free to move around in the crystal. And these are given in units of number of particles charge carriers per unit volume. So per centimeter cubed, let's say. Question. AUDIENCE: [INAUDIBLE] you're creating more holes. So is that not on the same order of magnitude as the number of holes introduced by [INAUDIBLE]? PROFESSOR: Yeah, absolutely. So the density of holes and electrons here in the dark is dictated by the dopant density. Where are we? Right here, for instance, the dopant density. If we add light, that's a complicating factor. The intrinsic population of electrons and holes and the dopant concentration of electrons and holes, we are adding a photo-generated population as well. Let's leave that for next lecture and try to simplify things. So we just discussed the diode in the dark. Perhaps we turn the light off here just for effect. So we're trying to take this piece by piece, so that we can really construct everything and not add too much into the pot at one time. I wanted to discuss the notion of electron drift. And I also wanted to review the concept of diffusion, which everyone should be familiar with. This is the reason why we're breathing right now, and why all the air molecules aren't crowded into that corner of the room. It's because of Fick's law, the process of diffusion. If we have a concentration of a particular species in one spatial location, the natural tendency through random motion is for that concentration to distribute itself equally throughout. And this is described very nicely by Fick's law. If we want to describe a current to it, we would then describe the current of holes and the current of electrons by, again, the charge. The diffusivity, this is a quantity typically given in units of centimeters squared per second. And the gradient. In other words, the concentration gradient. In this case, in one dimension. And that, again, should be review for most folks from physics. And so we can see here two different methods of currents, two different methods of current flow inside of a semiconductor device. We can have diffusion or we can have drift. Drift occurs when there's an electric field present and diffusion occurs when-- well, in this case, diffusion can occur in the absence of an electric field simply when there is a concentration gradient present. So we can envision, just quickly, that if we have an electric field confined to a certain region of our solar cell device, the electric field will be dominant in one portion. And if the rest of our device we don't have a strong electric field, but the electrons nevertheless can sense the electric field far away because of the concentration gradient, diffusion can drive those electrons-- can drive those electron toward the electric field. So in our solar cell device, the ratio of drift and diffusion current will change as a function of distance from the pn-junction, from the built-in field. OK, so I want everybody to upload into their RAM the checkerboard example that we went through. Because this will be important for understanding the next few slides. So let's imagine these n-and p-type materials are in contact, but there's this imaginary barrier right between them. Here are p-type materials, here are n-type materials. And let's parse through this figure. Let's dwell, just a minute. So we have this imaginary boundary between p-type material over here and n-type material. The boundary is right here in the middle. The blue dots here are representing mobile holes, holes that are free to move around the material. And while there's this imaginary barrier here, these holes are just moving in Brownian motion. There's no net current. There's no net charge flow. These minus signs that are left behind are the boron atoms. The boron atoms have one less proton in their nucleus than the silicon atoms do. And so the hole plus the negative charge that's within the boron atom core if you will, that combination is neutral. So we haven't perturbed the net charge of the entire system. But locally, if we draw a little circle around these two for instance, the boron atom that's fixed, that's not moving. The boron atom is bound to the neighboring silicon atoms. It's not moving around. And the hole, which is free to move around the material, now we have net charge neutrality. But if that hole moves too far away, then you could have a field building up. On the other side, very similar. Here we have phosphorus atoms embedded within our silicon lattice. The phosphorus atoms have one extra proton in their nucleus than the silicon atoms do. Hence, we denote that as a positive charge right here. And there's an extra electron associated with the phosphorus atoms. Now as those electrons move around, when it's in isolation, again, there's no net charge flow. There's no current. And we have a situation of net charge neutrality. Question is, what happens if we remove the barrier in between the two? Let me ask you just a basic question. Would the phosphorus atoms diffuse over to this side? AUDIENCE: No. PROFESSOR: Probably not, because the phosphorus atoms are bound to 4 silicon neighbors. Those bonds are really tough and it's probably not going to break. Unless you heat it up to a pretty high temperature. But the electrons are free to move around. They're in the conduction band. They're able to move. And because the binding energy to the phosphorus atoms is on that order of 10 m EV on that order, the thermal energy here at room temperature is enough to dissociate them, and to allow those electrons to move away from the phosphorus atom thanks to the screening potential of the surrounding lattice and thanks to the fact that the electrons inside of a crystal are more easily-- can more easily move than in vacuum. So when we remove that boundary in between, what begins to happen? So we've removed the boundary. We have these two materials in direct contact. The nuclei are not moving. The fixed charge associated with the boron atoms in this side and the phosphorus atoms on that side are not moving. However, the instant you remove this boundary in between, there is a high concentration of electrons over here and a very low concentration of electrons over there, which means that you will have a diffusion process. So the concentration gradient drives the electrons over to this side. Likewise, holes are being driven from here over to here. Now, notice what happens. Holes carry a positive charge. The fixed charge from the phosphorus is also positive. So you have a buildup of net positive charge on this side of the junction. Electrons are negative charge carriers. The boron atoms have a net negative charge here. And so you have a net negative charge building up on this side. And now you have a field beginning to develop, an electric field. And at some point, this built-in electric field will counteract the diffusion process. The diffusion current is driving electrons in this direction, but the field will be driving them back the other way. And it's that equilibrium that establishes the pn-junction. It's that equilibrium that defines the width of this region here, this transition region, which comes in a variety of names. The transition region is also called the depletion region. And it's also called the space charge region. Interesting how these words and terms come back into use. So that's the essence of the formation of this junction. We have this diffusion process that drives the carriers from this side into this side. And then as we have the electrons moving over here, and summing to the boron nuclei that are left behind once the holes, likewise, have moved over to the other side. We have a buildup of net negative charge on this side, a buildup of net positive charge on that side, and the establishment of a built-in electric field. That's cool. It takes a while to really get it. And so the chessboard example was brilliant. I thank Joe for that. And hopefully, this as well reinforces several of those concepts. And it might take some time studying it on your own or setting up special office hours with Joe or coming to my office hours on Mondays. But whatever it takes, make sure that you understand this concept well-- the fundamental-- gaining an intuition about the concept. We'll be getting into the math soon. And if you don't have a good, solid understanding of the intuition of where charges are moving around, it becomes less easy, let's say, to really get the math. So we have a buildup of net charge on either side of the junction. We have a buildup of net negative charge on one side of the junction and a buildup of net positive charge on the other side of the junction. And this line right here is meant to represent that dividing line where the two materials initially came together. So the dashed line is meant to represent the real charge distribution and the boxed colored rectangles, the rectilinear boxes, are meant to represent an approximation that we'll use for the rest of today's class. We're making successive approximations here because we want the mathematics to be manageable. You can, of course, discretize this entire material in one dimension, for instance, and do a finite element solution. We'll show you the equations that you can use to do that later on in today's class. So we have the net charge distribution on either side of the junction shown here. We also have, as a result of that net charge, an electric field. And this goes back to Gauss' law, or the derivative form there of. And so we have here a built-in electric field as you can see. And the field reaches a maximum right here in the middle. So that kind of makes sense. If you're too far away from that junction, you're not going to-- if you're a charge carrier, you're not going to see the field. But if you're right there in the middle of that junction, that field is going to be very strong. You'll be swept out of there pretty quickly. That's why the field reaches a maximum right here in the middle. So that intuitively makes a lot of sense. Now, if we take one further integral of the electric field, here the electric field, arc psi. If we integrate our arc psi, we will get the potential. And this potential right here essentially follows this wave-like curve, where you have a lower potential on this side and a higher potential on that side. And if we take this potential and translate it into something that we can understand, which is electron energy, by multiplying the potential by q. And q in the case of an electron is a negative number. That's why we're flipping this. So we're flipping this upside down because we're multiplying this value here by a negative number. Now, what we can see is that there's an energy gain for the electron by going from the p-side to the n-side. This is the electron energy. Obviously, if you give them an opportunity, all the electrons would want to come down to this side. But why wouldn't all of them come down over here? AUDIENCE: Because of diffusion. PROFESSOR: Because of diffusion. So there's that balance of the two effects. So there is a net energy gain for the electrons, but diffusion is what's driving some of the electrons back across the other side of the junction is that equilibrium, which is establishing the final energy band diagram of the pn-junction, which is really looking something very much like that. So this right here describes for you how you go from atoms and charge carriers up at the top to charge distribution, electric field, potential, and finally electron energy. Yes. AUDIENCE: Are these four diagrams specifically for electrons? Would the hole ones be [INAUDIBLE] the reverse of that? PROFESSOR: So if you wanted to look in terms of hole energy, this equation here would be equally valid, but your charge would be a positive value. And so you'd have a curve that looked very similar to this one right here, where there'd be a net energy gain for the holes to go on to the other side. And that's what establishes the separation of charge. AUDIENCE: So the bottom one is the only one that would change [INAUDIBLE]? PROFESSOR: That's correct. Yep. Everything else identical. So this is important, this electron energy. And this jump right here confuses a lot of folks. Because in books, they don't often describe very clearly what is potential and what is energy, almost using those two terms interchangeably. But it's that cue, the fact that the electron has a negative charge that sets things right. OK, this is pretty important. This is the foundation, the fundamental, of how a pn-junction comes into being. And the summary of our understanding so far is that when the light-- here we go. When light creates an electron-hole pair, a pn-junction can separate the positive and negative charges because of the built-in electric field. Let me repeat that one more time. When light creates an electron-hole pair, a pn-junction potentially can separate the positive and negative charges because of that built-in electric field. For very small light intensities, very small light intensities, such that you can think of that photo-generated carrier as a perturbation to the system, not fundamentally altering the energy levels yet-- it does eventually happen. But if you have, say, one photon coming into your material exciting electron over here, that electron can move down, can be swept out of the device, because of that built-in electric field. Again, assuming that we have a very small impact on the electrostatic [INAUDIBLE] system that it's just a small perturbation, maybe one single electron in our system is not going to affect you much because we have 10 to the 23 or so atoms per cubic centimeter. We have a very small perturbation to our system, that charge carrier will be swept out very quickly. The built-in electric field is established at a pn-junction because of the balance of electron and hole drift and diffusion currents. Let me be more precise about this one point right here. Because the built-in electric field is established at a pn-junction because of the balance of electron drift and diffusion. We talked about that. But hole drift and diffusion as well. So if you go back over to here, notice how the electron drift and diffusion are opposed and equal and opposite to the hole diffusion and drift. So let's think about it from this perspective right back here. This is clear, our electron diffusion is going in that direction because we have a high concentration of electrons that tend to move toward the low-concentration regime. So our electron diffusion current is pointing toward the left. Our hole diffusion current is pointing to the right. And the hole drift current, because of this built-in field, is pointing opposite the hole diffusion. So the hole drift current will be pointing to the left. Whereas, the electron drift current would be pointing to the right. So we have, in total, four currents here. The way to think about it would be, for example, let's take the electrons first. Electron drift or electron diffusion and drift counteract. And then you can take the holes. Hole diffusion and drift counteract. They're all balanced. And this is resulting in zero net current flow when you have no illumination and no external field applied to that device. That's pretty nifty. So the built-in electric field is established at the pn-junction because of the balance of drift and diffusion current for both electrons and holes. So we have a small in-class exercise. I'd like you to take out these. And I'd like you to work in pairs. So that if anybody reaches a small stumbling block, you can help each other work through the problems. So first off, let's focus on this upper left-hand portion right here as shown right there. This is a replica of the sheet that you should have in front of you. So what we're doing is we're dividing-- we're considering the pn-junction under no bias. That means that we're not applying a battery to our pn-junction. So we have a pn-junction and we don't attach a battery in series. We just have the pn-junction under zero bias conditions. So the way we'd represent that in that equivalent or model circuit diagram right here is notice how we have these little lines extending from the p- and the n-regions and going above. Those represent the external circuit outside of the actual device. The device would be this one right here where we have p and our n. We have our positive and negative charges here representing the charges that have built up in that space charge region that are creating the built-in electric field. And under no bias conditions-- I'll give you the answer. It's pretty straightforward. We would just draw a straight line across right there bridging that external circuit because there is no external bias. There is no battery applied. And so from an energy band diagram, e versus x, for e being the electron energy in this case. And our p-type material being on this side and our n-type material being on that side, let's draw the energy band diagram, just like we've done or we've alluded to so far. Sketch out the energy band diagram for this pn-junction. And I'll give you a hint. It's going to look something very similar to that right there, except that you'll have to take into account both the conduction and valence bands, because they're both affected in a similar manner to that. So why don't you go ahead and give it your best? Draw the energy band diagram in terms of the electron energy and position. Notice your p-type on one side, n-type on the other. You can assume that the point of division is like right here in the middle. And for those who are a little quick, you can go on and draw the relative magnitudes of electron drift and diffusion currents. So why don't we give that a quick, little shot? Maybe 30 seconds to think about it and another 30 seconds to draw and chat over it with your colleagues. So we want to draw the band diagram, just to make sure everybody's in the same page. So we're going to be starting out with our valence band and our conduction band over here. And as we move toward our space charge region, as we move toward our depletion region, our transition region here in the middle, we're going to see some effect. We're going to see some bending of the bands, if you will. So your question is to figure out, do the bands do something like this? Or do they do something like that? Do the bands go up or do they bend down? And that should be a relatively easy copy and paste from two slides prior. The correct answer is, indeed, we have the band bending down like this. So we would have the bands higher on one side, lower on the other. The electron energy is higher on the p-type side than the n-type side. So if we add one electron to our system, for example a photo-generated carrier from one photon coming into our device, that electron will have the natural tendency to go onto this side. It will be swept down. Note that the net current flow inside of the device, in the absence of an external excitation, like light or a battery pack over here, an external bias voltage source, the net current flow is zero. Because electron diffusion is pointed in that direction and electron drift pointed in that direction. All right, good. So now that we've done one example, I wanted to-- oh, one last thing. So hole diffusion. So this is electron diffusion pointing to the left and electron drift pointing to the right. Hole diffusion is pointing to the right and hole drift pointing to the left as one might expect. Let's try these two right now, forward bias and reverse bias. And your trick is to figure out, if you have a battery for instance, if the battery looks like this. This would be positive terminal and negative terminal. If you have a battery, which way would you align the battery to induce a forward bias or reverse bias? It's perhaps a little bit beyond my expectation of what you would get. Let me add that for you. Let me just give that to you right here. The forward bias would look like that. The reverse bias would look like that. OK. And now the big question is, what do the band diagrams look like? How would these look? If we forward bias, if we forward bias our device and we inject electrons into this side right here, what would happen to our bands? What would you expect would happen? We'll prove this out in the next few slides, but I want to see how people's intuition is doing this morning. So as you bias a device, you're shifting one level relative to the other. And so the basic question that you have to answer is, should it shift up or should it shift down under Forward and reverse bias conditions? Spend a minute. Talk to your neighbor. Discuss it. All right, folks. Why don't we tie it in? We should be able to get band diagrams that look something like that. I saw several of you have already begun reaching this consensus here. Rationale? An easy way to think about this is, if our battery is aligned with our pn-junction in that way, and we're injecting electrons into one side, we then have a large electron diffusion current. We have the bands shifting in this direction. Now the barrier, the energy barrier for the electrons to overcome, to go from the region of high concentration to the region of low concentration, is smaller. And you'll get a larger diffusion current as a result. More electrons will be moving over that n-type silicon into the p-type silicon. And the drift current will be smaller. Notice here, the balance of the two, if you add them together, you have a net electron flow in that direction, which means you have a net current flow as defined by the flow of positive charges in the other direction. A lot of definitions to keep straight. Whereas, under reverse bias here, you're shifting the bands like so. Because you have the large electric field, the drift current will be increased, but only slightly. Because there's a limited number of carriers here that you can pull from the p-type material. And the diffusion current is going to be drastically reduced because now the energy barrier is so large that the electrons have a very hard time getting from the n-type material into the p-type material. And then that net current flow is in the opposite direction. Does this make intuitive sense to folks? Ashley. AUDIENCE: So when you touch the battery [INAUDIBLE], is the minus side injecting electrons [INAUDIBLE]? PROFESSOR: OK. So in between these two, the easiest way to go from battery to electron energy band diagram is to follow the same sequence that we did over here. And if you're thinking battery, and you're thinking positive and negative, you're shifting the potential of one side relative to the other. Then, take that potential shift and flip it for the electron. AUDIENCE: I follow why the diagrams are the way they are. But is it conceptually accurate to think of the negative side as putting more electrons into the n-side? PROFESSOR: One could. AUDIENCE: Like injecting carriers? PROFESSOR: The way I prefer to think about it is that you're changing the energy levels, the potential one relative to the other. And as a consequence, you have more electrons flowing from that n-side to the p-side. And that's what's pulling the current through the circuit. It's almost a tail wagging dog, dog wagging tail. But I would take a slight preference toward understanding it from the perspective of if you change the potential one relative to the other, and allow for a greater diffusion current, that motion of electrons, the electron moving from here to here, will pull current through the entire circuit. And that's what you'll be measuring with your external-- essentially, all of the electrons in the circuit are moving through. AUDIENCE: [INAUDIBLE] so we've opposed the field that was set up [INAUDIBLE]. PROFESSOR: Exactly. We are opposing the field that was set up by the balance of the drift and diffusion currents initially. Yeah. AUDIENCE: Under the first bias, how come the many electrons in the anti-material aren't [INAUDIBLE] through the entire circuit and injected into the [INAUDIBLE]. PROFESSOR: Certainly. So the reverse bias is such that the battery is aligned to prevent that. perhaps one simple way of thinking about it. Yeah, I think that would be the way I would describe it most accurately. You could-- and you are dragging some of the electrons out through the external circuit. But what is driving the current is mostly the drift current. It's pulling carriers from-- essentially, through your external circle. Because carriers are moving from the p-type to the n-type and creating that charge imbalance resulting in the drift current. Why don't we continue moving on just a little bit because we have some nice demos we'd like to get to. I do know this is really, really important. And I would welcome you, actually urge you, encourage you to come to some of the office hours and our recitation as well. I'd like to continue moving forward for the benefit of several of those who may have seen similar material in the past. But let me, before we move too far ahead, on our I-V characteristics, we still don't know how to map an I-V characteristic of that device. We don't know how the current voltage, I current V voltage, we don't know how the current voltage response is going to look like for that particular device. But we do know that under no bias conditions we're at bias voltage equals 0. So V is at 0. And we know that we have no net current flow. And so our current is going to be at 0 as well. And so we know that we're going to have one point on our I-V curve that's going to be situated at 0, 0. Likewise, in this case over here, we have electron diffusion driving the circuit. And in the other, we have electron drift, which is in the opposite direction. And so you'd expect that the current flow in I under forward bias conditions-- forward bias is positive-- would be somewhere up here. And likewise, under reverse bias, that our current would be opposite the sign. So whereas we had positive current over here, we would expect negative current over there. So our I-V characteristic is going to look something like a combination of one point right here and some function up in this quadrant and some function down in this quadrant, just by glancing at the net current flows and the type of bias that we've applied to our diode. We'll come back to this at the end of class and confirm it using those small apparatus over there. So current flow in a pn-junction. We're going to describe the nature of the drift, diffusion, and illumination currents in a diode. Show the direction and magnitude in the dark. Eventually, we'll do this under illumination as well, but let's just focus in the dark. In fact, we've already done this. I've already-- I've already tricked you into doing learning objective number 3 on your own, before even describing the math. But I will show you the mathematics as well so that you have the complete picture. We have the diffusion current. This is the essence of diffusion current. We talked about Fick's law as well for holes and for electrons. We have our drift current. We spoke about the nature of the drift current as well for holes and electrons. And we know that by combining the two, we can describe the net current flow across a pn-junction. So by combining the two, by combining drift and diffusion-- drift and diffusion for electrons and holes-- we can describe the current flow across that pn-junction. So when the electric field is large, the drift current term is going to dominate. If you have a large field, that's going to be the dominant current source if you will. And when the electric field is small, diffusion is going to dominate. So if we have a device like this right here. And if I told you that the built-in electric field was located pushed up toward the top 1/100 of the device. So it was located very close to the surface. You might surmise that the drift current is going to be having a large influence. If you're a single electron, you'll feel the influence of the electric field near the front surface of the device. And deep within the bulk, the diffusion current is going to be driving the carriers toward that junction. There's a relationship between the mobility and the diffusivity of carriers. It's known as the Einstein relation. And that's given to you over there on the right-hand side for a band conductor. Note that if you have-- this is an advanced concept. But if you have a dispersive hopping mechanism, for instance, you may need to become a little bit more sophisticated in how you relate the diffusivity of a carrier to your mobility. But let's assume that is valid for nice, well-behaved semiconductors. It's just a warning for those who work on organic materials. I'd be happy to talk to you more about that in a bit. So again, we have Gauss' law. The question is, what is that electric field? What is arc psi? What is our built-in electric field? Well, we know that the charge density is going to be comprised of the free hole density, the free electron density, the fixed density of our donor atoms. You could think of these as the phosphorus, the ionized phosphorus atoms, and our fixed density of ionized acceptor atoms. We can also think of this as our boron atoms and silicon. And so that is our fixed-- actually, not our fixed. These are fixed charges. These are mobile charges. But under equilibrium, they've distributed themselves in a certain way. That is our charge density, our rho. And that will figure into Gauss' law and allow us to calculate arc psi. And so in summa, we have here arc psi as a function of dx. We have an expression that relates the electric field to the density of dopant atoms inside of our material. Note that we made one critical assumption going from here to here, where we went from the ionized donor and acceptor concentration to the total dopant concentration. We're assuming that everything is ionized. Again, because the ionization energy, because the binding energy of those free carriers to their dopant atoms is so small that under kt, under thermal energy at room temperature it's enough to dissociate the charge. OK, so we've gotten to here. We can describe quantitatively using this approximation, using the box function approximation. We can describe the electric field quantitatively. And now we have to get to the potential and find the electron energy as well. So we do a little bit of accounting. If we want to account for all of the electrons flowing through our system, we would have to consider as well, in any voxel of our material, in any unit volume of our material, we have electrons going in, electrons coming out. And within it, we could have electrons being generated, say by light, or electrons recombining. Meaning they're losing their energy and falling back into a ground state where they're bound. And that's this U-term right here. And so the continuity equations are essentially a set of accounting equations that just describe the difference, the net change, going in and out. If something changes on either surface, it's because your either added carriers or took them away. And the way you add is by shining light on your sample and generating more carriers. G, for instance. That's one way of generating carriers by adding light. Or recombination. And one way of recombining is, for example, through a defect in our material, where the electron is moving along. It finds a defect and it falls back into a ground state. So the continuity equations are nothing more, nothing less than housekeeping, bookkeeping. But it's very important as you'll see in the next slide right here where we combine five equations that describe current transport in a pn-junction. We have drift and diffusion up here. Those are the equations we just saw. We have the equation describing the electric field. And we have our accounting equations down here. And if we look at these equations in unison, we'll notice that, oh, well, we have our hole current appearing twice here. We have our electric field appearing in two cases. Sorry. Back. Back. There. We have our electric field appearing in those three equations right here. And we have a system of non-linear equations as a result. It is not possible to solve these analytically in their entirety. We'll have to make a series of assumptions to really get to the full solution. But it's possible to solve them numerically using computer simulations. This is important because we want to be able to define the current coming out of our solar cell device. Without solving these equations, we really don't understand what the current is out of a pn-junction, out of a solar cell under different bias conditions. That's why these equations right here are important. What I want to do is not to just dwell on the current. I want to emphasize that to calculate the voltage across the semiconductor-- sorry, a semiconductor device, a pn-junction-based solar cell. To calculate the voltage across a solar cell, we have to understand a little bit more. So far, we've been assuming that each electron is a unique individual. That an electron over here is a small perturbation to a larger system in that it will flow down based on that electric field. For a voltage, we're really talking about an ensemble of electrons. All of the electrons in the system. And so we have to define some chemical potential for our system. We have to define some-- one could think of it as an average energy for our system. So let me define chemical potential for you in a semiconductor. If we describe the semiconductor-- here's our valence band and our conduction band. This is our band gap right here in between the two bands as described by e sub g, our conduction band, valence band. And these little red dots here are representing our covalently bonded electrons in the ground state at 0 Kelvin. If we heat this material up just a little bit, there will be some fraction of carriers moving across into the conduction band just through simple probability. And we'll get to the precise equation that describes the probability distribution function in a few lectures. But for now, let's just assume that a certain number through thermal processes are able to be excited across the band gap. So these are thermally-excited electrons. And we call these-- this is the intrinsic carrier concentration. So these carriers up here are free to move around the crystal. They can conduct electricity. They can conduct charge. We could measure a current flowing through if we applied a bias voltage across material like this. Those are our charge carriers. These holes as well. And so this is the intrinsic carrier concentration. To give you an order of magnitude of the intrinsic carrier concentration in silicon, it's around 10 to the 10 carriers per cubic centimeter. That's 1 over 10 to the 12 per atom. So each atom of silicon is generating roughly 1 over 10 to the 12 free carriers due to this process. Very small concentration of free carriers in the material. Around 5/10 into the 22 atoms per cubic centimeter and only 10 to the 10 free carriers per cubic centimeter due to this thermal process at room temperature. So it's a very small effect in silicon. As you shrink your band gap, the energy necessary to excite across a band gap becomes less since you have more thermal carriers. So the chemical potential is describing the average energy necessary to remove an infinitesimally small quantity of electrons to the system. And again, infinitesimally small quantity meaning it's a mere perturbation. You're not changing the volume. You're not changing the electrostatics of the system. You're just adding one electron to the system, or taking it away, and determining how much energy was required. And so we describe that chemical potential, we call it by a different name. We call it Fermi level, but it is the exact same thing. It's the energy necessary to remove that electron from the material. So you can envision that it's going to be the Fermi level on either side of the pn-junction that will determine the voltage output of our solar cell device. And that's why we want to be able to calculate this. So let's look, first off what happens to our Fermi level when we dope our material, when we intentionally add, in this case, boron atoms to make it p-type or phosphorus atoms to make the material n-type. What happens? Well, we end up shifting our Fermi level toward the valence band or shifting the Fermi level toward the conduction band by the addition of holes or electrons. An easy way to think about it without getting into complicated semiconductor math is the following. If I'm adding more electrons to my system, by doping at n-type, essentially I'm shifting the energy level up. It's an easy way to think about it. If I'm removing the electrons from the system by adding holes down here, then I'm shifting the Fermi level down. Easy way to think about it. We'll get to the math. Actually, if you're really curious, I think I included one slide of the derivation of the precise chemical potential so you can walk through that as well. So voltage across a pn-junction. Under zero bias conditions, this is in the dark and without a battery pack attached to it. So this is an unperturbed pn-junction just in the dark without any external bias. We have the Fermi level constant throughout. Both the p-type and the n-type flat. Notice the bands are bending, so the distance of the Fermi level from the valence band here is smaller than the distance from the Fermi level to the valence band in the n-type material. That's because in the n-type material we have more electrons, and we're pushing the Fermi level higher. The vacuum level follows these bands. It just happens to be way up there. And so the amount of energy necessary to remove the electron from the system, to move it to the vacuum level, changes from the p-type to the n-type. But for the purposes of our diagram right here, we're drawing the Fermi level constant throughout. Now, we have that transition region, the depletion region, the space charge region-- all anonymous, same thing. We have that transition region here in the middle. So if we begin quantifying the different parameters here, we have a Fermi level distance from the valence band. So that's this distance right here. We have the distance between the Fermi level and the conduction band in the n-type material. And then we have our built-in potential. And when we multiply our built-in potential by q, we get an energy. Essentially, an energy gain across that pn-junction. So that quantity here is equal to the band gap minus these two parameters. Minus this, minus that. And so you can see that the built-in potential across the junction is benefited by higher doping. The higher we dope our material, the more we shift our Fermi level toward either band. And the greater the separation we get, the smaller this quantity is, the smaller that quantity is, and the more our built-in potential approximates the actual band gap of the material. This built-in potential will relate to-- it wont' be identical to, but it will be associated with our maximum voltage that we can get out of the device. And so we'll want to engineer our material in such a way so as to maximize that quantity. That's important. The relation between our built-in potential and the dope intensity is shown here. For now, you'll have to take my word for it. I'm sparing you a lot of semiconductor physics. It's written on the next slide right here if you care to look in some detail. It's also described in, I believe, Chapter 2 in Martin Green's textbook. But for now, let's assume that the built-in junction potential is a function of the dopant concentrations just by our intuition. If we're adding more electrons to one side, we're going to be shifting the Fermi energy up. OK, so the voltage across a pn-junction. Now, let's bias our device. We have under zero bias conditions this expression. Sorry, there we go. Under zero bias conditions, we have that expression there. And under biased conditions, now we have an applied biased voltage, our V sub a. We're applying a bias voltage. We've effectively shifted-- notice here, we've shifted our bands when relative to the other. So we've shifted this side up by a certain V sub a. And now if you'll notice the quantities here, we have a separation of the Fermi energy on one side to the other. There is a bias now across this device. There is a driving force for electrons to go and complete an external circuit, to travel through the external circuit. Because the electrons that are over here have a higher energy net than the electrons over there-- the ensemble, on average. The transition region, likewise, will get smaller. Sorry for that animation. But if we keep track here, the transition region becomes smaller under forward bias. Because we're depleting-- we're removing the amount of charge that was over here. We're squeezing it back. We're reducing that barrier height. And so over here, if we go back to this diagram, you can now draw in what's written in these red circles. You can draw in the actual depletion width, the width of the space charge region on your diagrams, just like that. I'll give you just a second to complete that diagram there. Under reverse bias, likewise, the width of the depletion region will increase. And the depletion region is increasing, the built-in charge is increasing, the amount of band bending is increasing, and the amount of drift current also increasing. So it all fits together. It's beginning to really come together nicely in one nice picture in our minds. So yes, one question. AUDIENCE: So I understand that for our solar cell, we wouldn't want to actually use a battery to drive current. PROFESSOR: Let's get to illuminated current next class. For now, we'll just focus on the battery. Yeah. AUDIENCE: So should we not quite yet understand why forward bias and reverse bias applies to [INAUDIBLE]? PROFESSOR: Let's leave that for next class. For now, let's assume that the illumination current-- if you really want to satisfy your curiosity, your illumination current is going to be one additional arrow to this. It's going to be in addition to everything else that's going on. But the majority of the field is going to be created by what is doped into the material. So think of the illumination for now as a small perturbation to the system. That's the easiest way to think about it. To justify to yourself why we need to understand first the solar cell in the dark, and then because of that small perturbation, we can treat it as a linear superposition of effects. And we'll add the illumination next class. But bear with me in the dark first. Because if we really don't understand this, we're not going to understand fully how the solar cell operates in the light either. So next, we'll draw the I-V response. We'll want to really get to this last point right here where we can draw the current voltage response. And we want to recognize that minority carrier flux is what's regulating the current. So to do this well, to do this properly, we have to shift our focus from here where we were talking about ensembles and individual particles. Here we're going to be discussing in terms of carrier densities on either side of the junction. Densities of electrons and holes. And unfortunately, I'm going to have to move rather quickly through these slides. This is the essence of why a solar cell behaves like a diode. And it's really something that is best done by studying on your own. I'm happy to walk you through the most salient points, the most important approximations that we have along the way, but this is best understood by going home, looking up the readings on stellar, and walking through the derivation yourself. It's not something that very easily I can convey a series of equations in the class. So before we go into detail into current flows, I wanted to touch on this with the space charge region. We can describe the width of the space charge region now by the built-in bias across the junction. The applied bias right here and some fundamental material properties as well. And this little epsilon right here is essentially the dielectric constant and the vacuum permittivity, a constant. And so you want to take that into account when you're running your actual calculations. It's very easy to be off on the width of the space charge region by several orders of magnitude if you don't do the proper accounting for those fixed variables. And so if you walk through the equations to describe the width of the space charge region, you'll find that in a typical solar cell device, it's on the order of a micron. And this is related to one of your homework problems, so stash that away somewhere in your brain. It's on the order of a micron. The width of that space charge region typically could be on that order. So it's going to be some multiple. And the reason that's interesting is because the entire solar cell device is about 100 to 200 microns. That's for a crystalline silicon device. That means that in a crystalline silicon device, the diffusion current is what's driving most of the current flow inside of the solar cell. If our solar cell is much, much, much thinner, say in the order of a micron, then our drift current would be dominating. This is a more advanced topic and we'll return to that when we describe the differences between thin film operation and crystalline silicon solar cell operation. Device capacitance. For those who are running experimental measurements and want to determine the RC time constant necessary to have the device settle into a measurable state, that's there. And the capacitance is described, again, by the various properties as well as the built-in potential which is related to the dopant density. OK, so under zero bias, we have a concentration of holes on one side of our junction that's high. In the p-type material, the concentration of holes is high. And it's approximately equal to the acceptor concentration. On the n-type side of our junction, the whole population is drastically reduced. And the flip side, our electron concentration is very high and it drops into the p-type side. So we have predominately holes on this side. Predominately electrons in that side. But a small concentration of holes on the n-type side and a small concentration of electrons in the p-type side. And we call this the minority carrier. And we call that the majority carrier. The majority carrier is in the majority. The minority carrier in the minority. And an interesting thing happens when we bias our device. When we bias our solar cell device, populations of both carriers increase. But because this is on a log scale, we're increasing this-- shall we say the minority carrier concentration in an absolute sense by a lot. And it's because of this drastic uptick in the minority carrier concentration right at the edge of the space charge region that we have current flow across that junction. And that's described by a series of equations here. Let's see, the way to walk through the derivation, there's a series of approximations to make. Think of it in terms of electron and hole fluxes on either side of the junction. You can make a series of assumptions as to what currents matter and which don't. You can, of course, make the assumptions as to the charge distribution as well, fixed using that box potential. And you can consider the cases in which the minority carrier concentration here is dominating to be the regions of interest for a current generation inside of our device. And so if we walk through, again the diffusion equation, the diffusion of carriers at the edge of the space charge region. And from the previous slides here, add our approximations in, we will ultimately derive an expression that has an exponential relation between current and our voltage. And if we continue through the series of calculations, including the continuity equation for accountability, we wind up with an expression that looks like this right here, where the total current flowing through the device will be equal to the electron current at the edge of our space charge region coming from the p-type side. The hole concentration coming from the n-type side. The addition of the two together. And that's effectively, this equation right here, where we have an exponential relation between current and voltage. So again, this we really need to go home and study. And if you'd like to do this pictorially, I provide you with that link right there so you can see it visually. If you're more of a math type, Chapters 3 and 4-- actually, Chapter 4 in Martin Green is probably the best place to go. And the important thing to keep in mind is that in the pn-junction, the current flow across that junction is determined by the minority carrier current flow at the edge of the space charge region. And that's why we wind up with the exponential relation between voltage and current. I decided the best way to emphasize this voltage-current relation is actually to measure it, to run it. To do it. And we have around 10 minutes left before the end of class. Do you think that is time for our demo? Cutting it kind of short. AUDIENCE: Probably wait for Tuesday. PROFESSOR: Probably wait for Tuesday on the demo. AUDIENCE: [INAUDIBLE]. PROFESSOR: So we're going to run the demos on Tuesday, I suppose. Because we're running a little bit short on time. I know David is really disappointed because he's been working hard to get these in perfect ship-shape condition. But we'll have another couple of days to work out some of the bugs in the software, so we don't need to restart it every time we take a new I-V measurement. But by and large, what we'll do is we'll measure the current voltage relation for a real solar cell device. And we'll see that it definitely does follow, at least under forward bias conditions, this exponential relation. So as v goes forward, as v-- this v over here is our applied bias condition. As our v increases, we have that exponential current output from our device. As v goes towards a negative number, we have pretty much a flat lining of our current. It goes into a slightly negative condition and a flat lining of the current. So if we look at this again right here, our I versus V, now we can plot the same equation. Or essentially, the same curve on all three of these. We'll want to plot the same curve-- this one over here. And for now, let's assume that this big lump over here is a constant. And so we just need to plot an exponential function, something similar to this right here. And under zero bias conditions, we'll be right there at 0, 0. Under forward bias conditions will be further up. And the reverse bias conditions will be down there. And let me give you a second just to draw this down. And please do. We'll be validating it to ourselves on Tuesday. My sincere apologies for not having enough time to really go into the demos as well. But suffice to say the following. Under forward bias conditions, when we reduce the barrier height between the n- and the p-type side, we're now allowing that diffusion current from the n to the p to take over. And as we reduce the barrier more and more and more, we have this exponentially increasing density of electrons flowing through our system. And that's why we have this exponentially increasing curve here. As we reverse bias-- our device, we're increasing the barrier height for the electrons to pass over. So our diffusion current practically vanishes. And our drift current stays, more or less, constant all the way throughout because there's only a finite carrier density, a finite minority carrier concentration inside of the p-type silicon. The thermally excited carriers for instance, that are finding their way to the junction and being drifted across. And that concentration is virtually finite. At some point, you'll reverse bias this so much that the carriers will begin tunneling across from the valence band into the conduction band over here and you'll have a catastrophic failure of your device, but that's more of an advanced point. Yeah. AUDIENCE: So does the x indicate the coordinate of I and V at steady state? PROFESSOR: So the entire line represents the I-V characteristic under all biased conditions. The x represents the current and voltage for this particular operating point on the solar cell. On the I-V characteristic. So if you're under forward bias condition. In other words, your v is positive, you will have a positive current flow through your device. Positive current flow, electrons flowing that way. And under reverse bias, notice the sign of the current is flipped because now the electrons used to be going that way. The electrons are predominately moving the other direction. The net current flow is in the opposite direction. And the voltage is also changing sign. AUDIENCE: You can say, OK, I'm going to put myself in this voltage? PROFESSOR: Exactly. And that's important because we want to be able to describe what our current voltage characteristic is of a solar cell because the product of the two is the power. For efficiency, we want to know what the power out of our solar cell is. And it's the product of current and voltage that will give us the power of our solar cell device. Time for a few questions, actually. About five minutes of Q&A since we had to forgo the demo. Promise me you'll do one thing. Promise me you'll go home. And sometime between now and Tuesday, you're going to read through these chapters and do the derivations that will work through to this ideal diode equation right here. Because this is pretty important. And you have to convince yourself that that's indeed the case. I can wave my hands and phenomenologically describe, OK, we have an energy barrier. I kind of get it. I should have an exponential probability of passing over an energy barrier. At least from quantum mechanics that kind of makes sense. So I kind of get how if I forward bias my device, I'll have an exponentially increasing diffusion current. I get that if I go into reverse bias conditions, there's just a finite density of carriers, minority carriers, in my p-type material. If I'm not, say, shining a light on the material, I'm not changing that concentration by much. And I'm going to flat line. Yes, they'll be drifting across, but they'll be limited more by their ability to reach the junction. You can hand wave all you want, but this is really-- whoopsie-- the equations that describe the ideal diode equation. Those are really where it's at. With that equation, you can really understand how material properties will impact solar cell performance, let's say. Yes, question? AUDIENCE: I have a question about the band diagrams. For all three diagrams, the energy levels [INAUDIBLE] the same. We're just changing where the n-type side is. Both of them [INAUDIBLE]? PROFESSOR: Exactly. Yep, absolutely. The point was here, what we've done for simplicity is shifting one side relative to the other. Yeah. In energy-- see, energy is a funny thing because you can redefine your zero. Yeah. So depending on where you're defining your zero point at, you can move things relative to each other. But let's assume that you're changing both sides relative to some universal ground potential that's off on an external component of your circuit. AUDIENCE: [INAUDIBLE]. PROFESSOR: That's a great question. So the question was, when you dope your material, what are the experimental methods that you can use to really determine whether or not the atoms have occupied a substitutional position? One of the methods is called Rutherford backscattering or channeling, IN channeling, where you introduce ions down your lattice. And if there is an atom here, for example, an interstitial site, it will scatter some fraction of those introduced ions back at the detector. And so channeling is one characterization method for determining the interstitial to substitutional ratio inside of a semiconducting material. Another way that we have very strong evidence that boron is occupying a substitutional site and not an interstitial site is we can calculate the binding energy of this hole to the boron atom here on the substitutional site. In principle, if the interstitial was also-- if it was a charge to defect, we could calculate the binding energy of the charge to that defect. And the density functional theory could tell you which is the more likely given your experimental observation that you get about one hole for every one boron anatomy inside of your sample. Boron has been studied to death in silicon. It's pretty well-known that it's substitutional. But if you're working with a new semiconducting material, you never know. You have to run the experiments. Yeah, question. AUDIENCE: Under bias [INAUDIBLE] diagrams, are the p- and n-sides far away from the [INAUDIBLE]. Are those bands actually-- do they actually have a slope to them because of the applied field? PROFESSOR: Yeah. So the question was far away from the space charge region here, do these bands have a slope? My argument would be that if we go back to the way we derive-- there we go. The way we derive this band to begin with, it's predicated upon an electric field, which is predicated upon a charge distribution. So the biggest question is, what does this real charge distribution look like? Most of the time, it's relatively concentrated to the near space charge region. If you get too far away, OK, yes, it decays exponentially. But it falls below the background intrinsic carrier concentration and effectively doesn't matter. AUDIENCE: Under applied bias, [INAUDIBLE]? PROFESSOR: Yes. AUDIENCE: Usually they're just drawing it flat, but is it such a small effect they're just drawing it flat [INAUDIBLE]? PROFESSOR: Yeah. So under applied bias, what happens to these charge distributions? Well, if you reverse bias it, you're essentially increasing the amount of charge on either side of the junction. If you forward bias, you're reducing it. So under forward bias, probably you wouldn't see the effect you're describing. Under reverse bias, the width of the space charge region, let's say, increases from, let's call it 1 micron to 3 microns. And the thickness of the entire device is 200 microns. So still, I think it would be a relatively small effect several 10's of microns away from the junction. It really depends on the length scale, the geometry of your device relative to the length scale of your space charge region. Good? Well, to give you time to reach your next class, thank you. We'll come back on Tuesday, and we'll have a great demo set up for you. |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | 10_Wafer_SiliconBased_Solar_Cells_Part_I.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: But today we're going to be talking about crystalline silicon solar cells. Now, for those of you who do not work in crystalline silicon PV, the reason this topic is important is because crystalline silicon comprises about 90% of all solar cells manufactured today. It's the dominant technology, and the technologies that you're working on are going to displace, or are aiming to displace crystalline silicon, so it's good to know your enemy. For those who are working on crystalline silicon, this is meant to be a background of all of the different aspects-- the entire supply chain of crystalline silicon-- so that you gain insight into the areas that you're not currently focused on. You're getting a perspective of the bigger picture. Crystalline silicon PV has been around since 1954. The original-- well, in its current incarnation. That was when Bell Laboratories announced the development of the modern crystalline silicon PV cell, and that was 6% efficiency in 1954, published in general applied physics, and the cell architecture, it's obviously evolved over the years but it's not entirely dissimilar from what we have today as a cell architecture for our modern PV cells. So, over the course of-- it's almost been 60 years of development of crystalline silicon photovoltaic technology. That means both the cell itself, the materials that go into it, and also the manufacturing, or the methods to produce said materials and device, over the course of those, almost, 60 years much innovation has happened both in terms of manufacturing and technology. So today, we'll be going over kind of a status quo snapshot of where crystalline silicon stands and we brought in a number of show and tell items so that you can see as we talk. So just for the show and tell, we're going to be moving from the feedstock materials over here finally into wafers and cells on that side. All right. So, these lecture notes are going to be valid for both 10 and 11. We're going to split this up over two classes to really dive into some of the details. The first question is why silicon? Why did silicon evolve as what is currently the dominant technology, which is currently 90 percent of the PV market, and I think it boils down to a couple of reasons. One is scalability. If you look at the elemental abundance, on the vertical axis it's abundance, atoms of the element per 10 to the 16 atoms of silicon. The reason that everything is normalized to silicon is because there is, well, quite a lot of it in the earth's crust. As you can see, it's the second most abundant element on the Earth's crust. It just so happens that, out of all the stardust that is here on the planet, we have a high percentage of silicon like the moon and like many other planets in our solar system-- at least the hard ones. You can see oxygen is probably the, well, oxygen is the only element with higher natural abundance in the earth's crust, the upper crust, than silicon and we go down as we go to higher and higher atomic number. The probability of formation due to subsequent fusion reactions in stars decreases and, hence, it follows this almost a power law distribution as you can see there. So it's scalable. It is present in the Earth in high enough capacity to reach terawatt scales. It's nontoxic and, as Don Sadoway likes to say, if you want batteries dirt-cheap, you have to make them out of dirt. A similar expression is used in the crystalline silicon community. I believe the quote in 1366 is, "It's not only good for the planet, it is the planet." A variety of riffs off of this particular chart right here, but from a technological point of view, why did silicon evolve to the point where it is today? It forms a very tenacious surface oxide. So, if you were to expose a piece of pure silicon to air, the surface oxide that forms is very, very strong and very resistant, and very dense. So, unlike some materials that corrode when exposed to atmosphere, silicon oxidizes maybe the first few 10s of angstroms, 100 of angstroms, and then it peters out so it's diffusion-limited oxide growth mechanism that eventually stabilizes at a very thin but very dense and very protective oxide layer. So the risk of having a silicon wafer degrade inside of a solar module is very low. Furthermore, that oxide layer from an electrical point of view it's very passivating. So as we studied on, as we solved in the exam, those interface states or those surface states, the surface of semiconductor, those can be reduced or minimized by the presence of certain passivating layers, and it just so happens that by the benevolence of nature, the silicon oxide, which is shown in these red triangles right here, has a very low surface recombination velocity, passivates a surface very well, and results in high-performing devices. In this particular case, they're plotting emitter saturation current density in femtoamps per centimeter squared-- this is very, very low-- versus sheet resistance. This is essentially the dopant concentration in the emitter, so they're looking at how the passivation quality changes as a function of dopant density and silicon oxide works pretty well, and it's an effective diffusion barrier. And, probably most significantly, those are maybe one looking forward rationale one technological or scientific rationale and as far as the field is concerned, as far as engineering community is concerned, silicon has a lot of momentum. It's the most common semiconductor material, silicon and germanium were both purified, more or less, around the same decades but, because silicon has a wider band gap, you have a lower thermal carrier concentration, lower intrinsic carrier concentration, folks were able to make transistors and devices with lower noise out of silicon as opposed to germanium and silicon technology really took off in terms of the PV industry benefited a lot by that cross-pollination. Many technologies came in from the integrated circuits industry to assist or give a boost to the PV industry. This number is a little outdated, it's now about $100 billion. Hard to keep up with things growing at 68% a year. Technology acceptance results in lower interest rates. So if you have a technology that is well-accepted by the market then you go to a bank and say, hey, I want to install some of those things and the bank says what are those things you say oh, hundreds of thousands of them have been installed already. It's OK. It's a proven technology. The bank says OK, I'll lower your interest rates. That means you pay less money on interest. Your capital is more cheap. It works better in your favor, and the opposite is true with an entirely new technology that's unproven. So that's really summing up why silicon. Momentum, forward motion if you will, some inherent intrinsic technological advantages, some of which are listed here, and I'll get to that in a second, scalability. To get back to the technological advantages, I think it's important to recognize what they are so that when you're thinking of a new material, you can cross check and say, gee, do I have these or do I not have these. If I don't have them, it's not the end of the world. You might have other advantages that overcome the ones that silicon doesn't have. Let's add some more into this list. Just stream of consciousness. Silicon has a very high refractive index near the band gap edge. So, near the band gap edge, it's absorbing light less efficiently. Right? It has a larger attenuation length of the light, a smaller optical absorption coefficient right as you approach the band gap. So silicon absorbs poorly in the infrared because it's an indirect band gap semiconductor, but it also has a very large optical, sorry, a very large real component of the refractive index. Does anybody remember what that refers to? Real component of refractive index. Lesson number two. What does that dictate? AUDIENCE: Reflection. PROFESSOR: Reflection, exactly. So, if I were to tailor and index of refraction grading on the front side of my device, so I allow the light to be absorbed efficiently, on the backside I can put a very large index of refraction mismatch so that the light bounces back. In other words, the light trapping silicon is benefited by the fact that you have this awesome reflection capability. The refractive index is around 3.6, the real component of the refractive index, in the infrared at around 1070 nanometers. Which means that if you design your cell right, you can get an extension of the optical path length by a factor of 50 over the thickness. So if your thickness of the device is d, the optical path length can be increased up to about [? 51d. ?] That's as a result of this great reflectance. Many other materials that are being explored as PV materials have refractive indices around two, which would mean that your optical path length extension is around 16. So that's one thing to keep in mind. even though it doesn't absorb light quite as well, it traps light fairly well. Another advantage of silicon is that it forms sp3 hybridized orbitals, for chemists, it forms-- it's tetrahedrally coordinated, in other words bond to four other neighbors, and most 3D transition metals don't do that. They don't bond in that configuration. Some do but many don't and, as a result, the solid's solubility in other words, the ability to incorporate impurities into a growing silicon crystal is low. It rejects the impurities from the solid into the melt, and you're able to purify the material very efficiently. That's not always the case with most materials. Sometimes they incorporate impurities very readily, up to a few atomic percents. The typical impurity concentration of silicon is in the order of parts per million, parts per billion, parts per trillion. Still can be enough, as you learned during your homework assignments, still could be enough to affect device performance but is very low. It would be a lot worse if silicon were able to absorb more impurities and so forth. So, there are a number of reasons why the silicon PV technology has gained the foothold that it has so to bump it out of its leadership position, one really has to be clever and the parameter of merit is performance per unit cost. Kilowatt hours per dollar, if you will. So, we're going to talk about the current manufacturing methods and materials because this will give you an insight into the dollars per kilowatt hour, the kilowatt hours per dollar. Essentially, the cost per unit energy produced. You can begin to seize opportunities within the crystal silicon world to improve the manufacturing process or you can begin to say OK, you know what, this is way too complicated. Let me take a completely different route. I'm going to develop a new technology instead that will overcome these manufacturing difficulties. So let's explore them in detail. First, the market. This is the evolution of market share from 1980 to mid 2000s. After mid 2000s, the market just continues growing at 68% a year and you really lose resolution to this portion down here so it's to 2006 so that we can actually see what's going on in the earlier days. In the earlier days, 1980, let's pick 1985, the market was split about a third-third-third between thin films, amorphous silicon namely, monocrystalline silicon, and a material called multicrystalline silicon. Now let's go piece by piece. What is monocrystalline silicon, multicrystalline silicon and thin films? Well thin films are materials that are usually between a few hundred nanometers up to about three, maybe five microns thick. To give you size perspective, your hair is about 50 microns in diameter, so we're talking about 1/50 the width of your hair. That's the active absorber layer and of course the plastics and encapsulates and everything else that go around them make it a bit thicker, but the absorber layer is very thin and so you're not spending much on your absorber layer. It absorbs light very efficiently, has a very large absorption coefficient, and is able to absorb photons efficiently. Crystalline silicon, on the other hand, does not absorb light as well as many thin film materials so we need about an order of magnitude to two orders of magnitude thicker substrates, and the crystalline silicon substrates today in commercial manufacturing are typically between 160 to 190 microns, with an average around 170, 180. So about four times the thickness of your hair. Monocrystalline silicon and multi. Let's talk about the difference there. So, monocrystalline silicon, folks are probably familiar seeing pictures, at least something like this. Right? So this right here is an example of a Cherkofsky silicon wafer. Appropriate for integrated circuit work. I'll pass this around so folks can get a sense. So this is an example of a monocrystalline silicon wafer for the integrated circuits industry. Let's analyze it in a little bit more detail. So, the front surface is polished, nicely polished. Polished to, I think, somewhere in the order of a few nanometers mean surface roughness. Using a chemical mechanical polishing mechanism. The thickness is around 700-- or 675 microns. Somewhere in that range. So very, very thick wafer. The objective is not to break. Right? If you're making integrated circuit, this entire wafer that I'm holding right here could be worth a few 10s or 100s thousands of dollars by the end of the processing sequence, so if one of these breaks, that's an awful lot of revenue that the company's losing. So the substrate is thick because they don't want it to break. Silicon is brittle at room temperature. If you were to manufacture solar cell out of this, you could but it would be very expensive. The chemical mechanical polishing that they use to flatten the surface out costs a lot of money, it's very time intensive, and the thickness of the silicon is above and beyond what is necessary to absorb light well. If anything, increasing the thickness is just increasing your emitter saturation current, since you have a higher recombination current being driven by bulk recombination. You have more recombination centers because you have a greater thickness, and it's driving a larger diffusion current from the emitter into the base. So making it this thick really doesn't make sense. So I'll pass this around so folks can kind of get a sense. Make sure this gets the entire round. I'll be recycling those platens. Please hold, if you're going to take it out, which you're welcome to do, please hold it like a photograph. What I don't want to have happen is folks put their fingerprints all over it. The wafers that are used in the PV industry are cut from the same ingot like that one, except that the ingots, essentially, if you were to pack circular wafers into a module, it would look something like this. Here's your module and, mind you, you're spending a lot of money on the glass and the encapsulates and the aluminum framing and so forth, and now your solar cells look like that. There's probably more of them that you can put in here, but what do you notice about this? What is the packing density, or packing fraction. It's very low, right? You're losing all of this material in between. All that space is just going to be blank space. Some of the earliest PV modules actually use circular wafers, but the more modern ones, what they do is a very complicated cost analysis where they say, OK, if I were to chop off the edges of my wafer and completely remove them, I'd be losing a lot of silicon but I'd be increasing the packing fraction. So in the limit that my module materials, the glass, the encapsulant, the framing materials are infinitely expensive and my silicon costs nothing, I want to do this. In the limit that my module materials are free and installation is free but the silicon is super expensive, I want to keep full round wafers, and the reality is that we're somewhere in between. And so, instead of making one or the other extreme, typically what you'll see is something like this chopped off, like that, where you have a pseudo-square. The wafer itself has flat edges on the sides but it also has kind of pseudo-rounded corners here, and Joe did we bring any of those in? The psuedo-squares, the monocrystalline psuedo-squares. These ones. OK. All right. No worries. I'll show them to you next class. So, the idea is to make-- cut it out of the same ingot as that one right there, but make it thinner, on the order of 170 microns thick, and to chop off part of the edge, and how much you chop off depends on the dynamic pricing of silicon versus module materials and installation and whether or not you can sell the module, if there's a certain threshold of performance that it needs to reach because obviously, if you have a bunch of dead space in here, you're losing that to-- you're not producing power out of that. So if somebody wants a module that's yay efficient, you might want to increase the packing density. So that's monocrystalline silicon. Multicrystalline silicon. Let's put it this way for now. We'll describe how multicrystalline silicon is made, but for now I'm going to say that multicrystalline silicon is a crystalline silicon variety that is comprised of many small grains. So if you look at a multicrystalline silicon wafer, something like, let's say, oh this is a perfect example right in here. If you look at a multicrystalline silicon wafer, you can see that it looks nice and-- here maybe, that's probably an OK view of it. You can see individual grains. Right? If you look closely at it. And those are grains of crystalline material that are joined by grain boundaries. So the grain orientation in one region might be pointing in this direction, the grain orientation in the neighboring region like that, and they come together at a grain boundary and, when we have polycrystalline materials like this, it's generally indicative of some faster growth that didn't allow for a nice homogeneous single crystal material to evolve, and that's indeed what happens during the multicrystalline silicon ingot growth. It's occurring under a slightly modified growth condition then, say, that beautiful single crystalline piece over there, and we'll explain how they're made in a second. So those are the technologies in general, the base absorber materials, and then there's ribbon silicon which is a really, really small fraction of the total production in decreasing, but at one time, ribbon silicon was viewed as the up and coming technology. Still today, there are about 20 startup companies around the United States working on some aspect of this and probably about a dozen more around the world. Yeah. AUDIENCE: I had a question about the multi. PROFESSOR: Yeah. AUDIENCE: So for the multi and micro and poly, is that different grain sizes? PROFESSOR: Sort of. So, multicrystalline silicon is a polycrystalline silicon material. The definition of multicrystalline silicon is that the average grain size is about a centimeter squared, or larger, and that's where multicrystalline came about. Polycrystalline silicon, in the silicon community, has a very specific meaning. It means, usually a plasma-enhanced chemical vapor deposited layer, so PCVD-deposited layer of silicon, that has on the order of one to five micron diameter grains. So very, very small grain material. About 1/50 the width of your hair. Maybe 1/10 the width of your hair and, to distinguish it from that really small grain material that will perform very poorly, one calls this multicrystalline silicon. AUDIENCE: And is there microcrystalline silicon? PROFESSOR: There is also microcrystalline silicon and microcrystalline silicon is actually at the phase transition between amorphous and polycrystalline silicon. So as you're going from an amorphous material increasing the temperature, let's say, of growth or increasing other parameters during the deposition process, as you begin to evolve from an amorphous material into a crystalline material, you transition through this microcrystalline regime which is a bit of a hybrid. It has some regions that are amorphous and other regions that are crystalline. In your assigned readings, this book was assigned, and I believe in the syllabus it says read chapter X. Unfortunately, there is no chapter X. I guess you could interpret it as 10, but the essence was that there are two versions of the book. One is version three, which was published about seven years ago, and the newest version just came out last year. The newest addition is addition three. So the chapters have rearranged slightly, but what I'll do is I'll highlight crystalline silicon solar cells and modules in here so that you can get a sense of what is in the chapter and you're welcome to go back and have a look. So I'll go ahead and highlight this chapter right here and pass it around. Feel free to glance through the book as well. It's a great read. It dives into great detail into each of the different technologies. OK. So, let's talk about feedstock refining. We're going to start the silicon value chain from the raw materials and work our way all the way to the final module at the end. So we'll start with the feedstocks themselves. Down here is a rough cost breakdown. Kind of think of it as wafer, cell, module being like a third-third-third of the total module cost and then balance the system components beyond that. So we'll start from our feedstocks and the raw materials in the ground, we'll wind up with systems on the roof, and we'll walk through each of the different steps of current manufacturing process. So raw materials. Shown here is quartz and coal, for a very good reason. The way feedstock refining occurs at the very first stage is to take oxidized silicon, silicon dioxide, quartz and to reduce it to silicon, say, silicon zero. Unoxidized silicon, which is also called silicon metal. It's called a metal because it is very low resistivity. It's very low resistivity because there's a very high impurity content still. The purity of this material coming out here is around 99, 99.9% here. So, it sounds like a high purity but, if we're talking about parts per million of impurities, we have some further refining steps to do after this. So let's walk through this. We start with the raw materials in the upper left. It says raw material inputs. Carbon and SiO2. The SiO2 forms, usually, quartz. That can be some of high purity pegmatite, it could be, for example, a hydrothermal quartz, higher purity varieties of quartz. You could even use, maybe, a metamorphic quartzite material. Let me explain. So, some of the highest purity materials are coming from these veins of magma that float up and then phase separated during millennia. Some of the lowest purity quartz is coming from sand, essentially crushed rock that made its way into, say, a beach-like environment and then rock was deposited on top of that, pressure was increased, and this whole mixture of mica, feldspar, and of quartz got pushed together and formed a solid block. That would be your metamorphic quartz materials, and so you'd have a much higher impurity content in the metamorphic quartz than you would in, say, a high purity pegmatite or hydrothermal quartz. Regardless, depending on the feedstock source of the quartz, and there are people who study this. Believe it or not, there are entire departments dedicated to mining quartz and figuring out where the different veins of the highest purity quartz are, where you get them from. That's the SiO2 input and the C input over here on the left hand side, Carbon. So, typically what is used in the PV industry is either a fast-growing wood source like eucalyptus or southern pine, right? Northern pine tends to be slower growing, but eucalyptus and southern pine both tend to be fairly fast-growing. You can tell by the spacing in the rings, if you chop the tree down and do a cross section, or coal. So carbon, essentially. And the two react inside of this furnace right here and this furnace, just to give you a sense of scale, here's a human being. This is the furnace. So it's about five stories tall, 12 meters in diameter. It's a big, big, big creature. This furnace right here is what is producing the reduced silicon and what's happening is these feedstock chunks are being thrown in at the top and there's an arc going between the electrodes, usually some carbon-bearing material, and a base contact, and so that arc creates a very high temperature. Something in the order of up to 2000 degrees Celsius, near the arc, and the temperature decreases as you go further and further away, so up near the top here it might be even below the melting temperature of silicon, somewhere around 1,200 degrees. So this is an extremely inhomogeneous, messy system. This metallurgical grade silicon refining furnace right here, this arc furnace, also called a carbothermic reduction furnace, a very busy place. Lots going on. Extremely inhomogeneous if you were to take a cross section also in terms of temperature and in terms of the chemical states of the different constituents species, but the general reaction that happens is the carbon would much rather bond to the oxygen than silicon, and so the carbon steals the oxygen from the silicon reduces the silicon to silicon metal and CO2 is released. We'll get to that in a second. Flag that. Put an asterisk next to it. We'll come back to that in a second. Other byproducts of this reaction, so this is the liquid silicon metal coming out here at the bottom. It's essentially liquid molten silicon reduced, so silicon zero, not a silicon oxide, reduced silicon metal, and then finally it's poured into these buckets, also called ladles and solidified, crushed up to size, and then distributed at the end. Other byproducts coming out of this reaction include-- this is liquid silicon up hear. It's very high temperature and there are gases and a lot of oxygen because of the reduction process, and so silica, or SiO gas, can be produced and silica gas can begin aggravating and forming very small particles, almost like shards, of silicon oxide material, and these can be on the order of one to five microns and very rough and jaggedy around the edges. Now, who here has studied public health and knows anything about PM1 or PM1.5 denominations. Do they ring a bell? What are those Ashley? AUDIENCE: It's the size of particles that can get stuck in your lungs. PROFESSOR: Exactly! Right? So PM1 or PM1.5 would refer to the micron diameter, 1 or 1.5 micron diameter particle that would get stuck in the [INAUDIBLE] and result, eventually, in edema or, probably, more of water filling up in the lungs as a result of the body trying to expunge these, and because they're jaggedy and pointy, they get stuck in there and they don't come out and eventually the people can even affixate as a result. So, before in the past, when we had these big smokestacks sitting on the top of these metallurgical grade silicon refineries that would just spew the silica dust into the air, the folks downstream would be affected and this actually did happen, to some degree, in, for, example, Kristiansand in Norway and, as a result, the refineries began putting in filters over here to prevent the silica dust from getting thrown and spewed out into the atmosphere and the filters are a very interesting contraption. A lot of work went into designing them just right to allow the air to go out but the particulate matter to stay behind and once every delta t, maybe in the order of an hour so, the airflow direction inverse and all the dust comes crashing down to the bottom and then gets collected inside of here. It's kind of like pushing air through the different direction through a sock, and all the dust comes out to the bottom, you collect it, and it's sold to the--? AUDIENCE: The footwear industry for absorbing-- PROFESSOR: It might be. I don't know, but I know that the majority of it goes to the cement industry and so, depending on the market rates of silicon, here at the bottom metallurgical grade silicon, versus what the cement industry is willing to pay, you might tune your process to optimize for one industry or another. So, this is to say that early on in refining processes, you're serving multiple industries with one plant and volatility of pricing is affected, in part, by what those other industries are doing. What the demand there is. It's something to be aware of. Let's go back to the CO2 real quick that's being emitted. So that is one of the byproducts of the reaction. In terms of total CO2 content from the production of solar cells, the CO2 produced during the reduction process is a small percentage, I think something under 5% or 10% is the number I pulled out of my head, it's a small percentage of the total CO2 emitted during solar cell manufacturing because the electricity that goes into producing the rest of the solar cells coming from fossil fuel based sources comprises the majority of CO2 emissions during fabrication of these devices. The electricity used to run these electrodes, for instance, the electricity used to melt this silicon byproduct here, or to gasify it in the subsequent reactions, that is the majority of the CO2 coming out of the process. Any questions so far about this? They're fun plants to see. We don't have too many of them in the US. Majority of these carbothermic reduction furnaces are either in China, Norway. Norway has a lot of cheap hydropower so the hydroplant is usually only a few 10s of kilometers away from the refinery and if you go to, say, [INAUDIBLE] in Norway, where they have a number of these plants, you'll see not only silicon being refined there but also magnesium, other elements, aluminum being smelted in the same peninsula-- the same industrial park. AUDIENCE: When general mining of silicon happens or silica, the Chinese have-- PROFESSOR: The reduction process, this carbothermic reduction process here, the majority of it happens at the same places like Norway or China-- places that have cheap electricity. There's also a feedstock refinery. I don't know if it extends all the way back to the metallurgical grade silicon refining, but there's a feedstock refining facility going up in the Middle East right now in Qatar, as a result of the cheap natural gas. So, wherever you have cheap access to energy, you can set one of these plants up and get off and running and your CO2 intensity will be dictated by the fuel source that you're using. Hydro, in that case, it might be low unless you take methane into account that might be emitted in the reservoir, if you have decaying biomass underneath the water, but if you would exclude that and if you look at the CO2 intensity of the fossil fuels that are being burned, it might be better to do it in, say, Norway, from an environmental point of view, than to, say, manufacture this stuff in China. Yeah. AUDIENCE: How many kilowatt hours are we talking [INAUDIBLE]? PROFESSOR: Okay, so what is the energy intensity of this process right here, in other words. Well, why don't I put a flag on that. Why don't we put a flag on that and come back with specific numbers for this process right here. I don't want to say something and regret it later. AUDIENCE: Well, we know the energy intensity of the solar panel itself. PROFESSOR: Yeah. AUDIENCE: But the energy-- PROFESSOR: But specifically what fraction comes from the MGSi refining, I'd rather not pull something out of my head. Any other questions? OK. So somewhere in the order of two million metric tons of metallurgical grade silicon are produced annually. Probably somewhere in the order of 10% of that is destined for the PV industry. The remainder gets split among a variety of different industries. So what I'm talking about here when I say metallurgical silicon, I'm referring to this right here. This stuff coming out. It has about 99% or 99.9% purity and it gets used in a variety of industries. So those industries are: the PV industry, and we'll explain how the rest of the refining happens, the integrated circuits industry, that's the wafer that just went around that's made its way back up here, and silicones those are-- so, a pet peeve of mine is hearing the word silicon and silicone used interchangeably. Silicon is this element-- is an element on the periodic table and it's the element that comprises this wafer right here. Silicone, on the other hand, is an organelle, I guess you could say, it's not exactly organelle metallic, silicon isn't a metal, but it would be a molecule that is comprised of carbon atoms and silicon-- silicon being in the middle and the carbon being on the sides-- and that is used as caulking or sealing agent in your showers, for instance, or in plumbing, round windows. It tends to be very flexible, compliant but yet impermeable, preventing the inflow of gases. So silicones, they're metal alloys including steel and aluminum. Why would you silicon there? What does it have to do with steel or aluminum? Let me ask this. Has anyone ever played with pure aluminum? Highly refined, ultra high purity aluminum. Say five nines or six nines. Yes! What happens to ultra-pure aluminum? AUDIENCE: It's really flexible. PROFESSOR: It's really flexible, you can dent it with your fingernail, and it wouldn't make very great boxes. Right? So we need it to be stronger and scratch-resistant and so we have these additives into the aluminum, silicon being one of them, that increases the strength of the aluminum, essentially preventing plasticity or preventing a dislocation flow into the material. So that's more or less how silicon-- metallurgical grade silicon, also called MGSi as shown up here at the very top-- that's how MGSi gs is distributed worldwide and that's the current production. Now let me ask another question. Steel and aluminum, where are those used the most? What industry uses steel, aluminum the most? AUDIENCE: Construction. PROFESSOR: Constructive industry, automotive industry. How fast are those growing annually? Let's estimate it from GDP. Annual-- worldwide GDP. What's the worldwide GDP growth look like. US is around 1%. China 8%. Let's pick a number somewhere in between. Four, right? All right. So, let's say 4%, 5% worldwide. Silicone's probably on that order. How about the PV industry. How fast is it going right now? Somewhere in the order of, it's a volatile year right now, this one year, but in the past, historically, it's been around 40% to 60% a year. So, where do you think the price pressure for metallurgical grade silicon is going to come from? What industry? It's going to come from PV. It's a small fraction of the pie right now but it's growing fast. Something to keep in mind. So that's why, if you look at pricing of metallurgical grade silicon, yes. Superimposed upon pricing is a function of time. You have the global macroeconomic situation. Right? So that's kind of the dampening function on top of it all, but there's just this general trend toward rising prices as you put increasing price pressure on metallurgical grade silicon. So additional refining capacity will be needed if the current growth keeps up in this industry. So let me talk about going from metallurgical grade silicon about two nines to three nines pure. What I mean two nines means 99%, three nines would be 99.9% pure, to silicon that we can use for solar cells, which typically has to be about six nines pure. And so this is called the Siemens process which is purification through gaseous distillation, and that's the method that is currently used to make most of our silicon. So the way this process works is we start with metallurgical grade silicon at the top, represented by a little sack of metallurgical grade silicon chunks. We produce silane gas out of that metallurgical grade silicon. We essentially- silane gas is SiH4. So it would essentially be this right here. So you'd have a silicon atom here, tetrahedrally coordinated with-- tetrahedrally meaning four bonds with hydrogen atoms on the side-- and this is silane gas-- well, silane-- which, at room temperature, is a gas and that's what happens in this step right here. We're forming-- we're gasifying the silicon. This process is the distillation process. To extract the pure silane gas, it's the distillation process that's used in large towers similar to fractional distillation where we might heat up the material and then, depending on its mass, it settles down to a certain height in that tower and we're able to extract it. The silane gas here has been sold to the photovoltaics industry. LCD. Liquid crystal display. Right? Thin film industries as well, they use silane. If you're depositing the polycrystalline and silicon for your LCDs or if you're making amorphous silicon solar cells, they use silane as well. So this little truck here might go to three different companies, depending on who's willing to pay more. Most of the silane is used for polysilicon. The gas has to be converted back into a solid, and that's where this particular process here, the Siemens process is used. Again, you have a current passing through some seed material and the gas is being cracked onto that seed. You form these rods. The rods are then cracked into chunks and then the chunks are loaded into ingot crucibles. Yes. AUDIENCE: So the silane gas is shipped as a gas in the trucks. PROFESSOR: Sure. AUDIENCE: Or on rails? PROFESSOR: Well it's pyrophoric, as you can guess from just glancing at this chemical structure right here. It's highly reactive. Pyrophoric means that it can combust at room temperature. It can catch on fire, meaning there are more stable compounds than this that can form when you react this gas with air and, during the early days of silane development, folks really didn't know much about it and there's some early research-- some of the earliest research done here at MIT, in fact. They would fill up an evacuated chamber with silane gas and spark and nothing would happen. Spark a second time, nothing would happen. Spark a third time, boom. OK. That's critical limit. Such and such amount. You know, they'd keep increasing the amount and finally it would go boom. Tell you what, lets repeat the experiment since we're good scientists. They'd repeat it and, at low concentrations, click. Boom. That's strange. That was much lower this time. Let's repeat the experiment one more time. Click. Click. Click. Click. Click. Click. Click. Click. Click. Boom. All right. I don't really understand this gas, but I'm going to say it's really dangerous so I'm going to have little warning bells that will detect the silane gas if it's leaking and tell people to get the heck out of the building if it starts being leaked. It's also toxic for humans, by the way. Very small dilute concentrations can kill you and so three buildings on campus, only three to my knowledge, are set up with the proper safety equipment to use silane gas in the laboratory. Building 13, which is the material science building, and then-- MTL and related. So we have this gas right here. Extremely powerful. There are variants thereof. You can replace some of the hydrogens with chlorine, like this and now you have trichlorosilane. It's all one word. So tricholorsilane, I've just replaced three of my silanes-- my hydrogens with chlorine and now I have a different molecule, still silicon bearing, still very reactive, but now reactive at different temperatures and I can modify my process by substituting out some of the hydrogens for chlorines. So we have the silane gas or trichlorosilane or the variants thereof, loaded into some transportation vehicle that is very safe, leak-proof and preventing accidents on the road, to deliver it to where it is going to be consumed, which are these so-called polysilicon, or Siemens reactor as shown here. Excuse me. What happens, or how the process actually flows, let me go back one step. We're going to start from up at the very top of the process and move all the way down, showing you what the manufacturing equipment looks like at each step. So, this is the distillation process used to create the silane and when you see one of these factories just think of a refinery. In fact, the people who don't like this particular process who aren't a fan of the silane refining process and opt for other ways of purifying their silicon, liken this to an oil refinery. The imagery is very stark there. The polysilicon production, this is the Siemens reactor, it's much smaller in comparison to the metallurgical grade silicon furnace. Much smaller than the carbothermic reduction furnace. Here, we have a small human or human next to the small contraption. Here are a series of them lined, almost like little pods and, out of this material, actually inside of the furnace, you have these rods that are passing current and heating up and the silicon is cracking onto the rods. So we wind up with six nines, usually called 6N solar grade silicon as a result of this process. We could also go up to, even, nine nines using the Siemens process. It could be very, very pure depending on how fast you grow, what the purity of your silane gas is. AUDIENCE: Yeah. What is cracking. What does that mean? PROFESSOR: Sure. So what it means is this gas molecule comes in, sees a solid surface, the central atom right here, the silicon atom, gets deposited onto the surface, becomes an adatom, which means it's a surface atom, it's scuttling around and the remaining elements within this molecule are then free to move away as a gas. AUDIENCE: So you've broken those bonds. PROFESSOR: Yes. Effectively, you've added the core constituent of this molecule onto the surface. It's joined the collective if you will and, in this matter, the diameter of those rods grows with time. So what I'm going to do is pass around an example of a chunk coming from this Siemens rod. Be very gentle with it please. On the outside you can see a corrugated, rough, cauliflower-like structure. That's because you're optimizing for deposition speed, not for beauty of the surface. You don't really care how flat it is, unless you're trying to grow a very specific type of material called flotsam, which we get to the second, but in general, if you're trying to crack it up and break it into a smaller piece and into a chunk like this and throw it into a big ingot furnace, it doesn't really matter what the surface looks like. On the inside, it's pretty dense silicon and, if look very carefully, right in the middle there you can see the rod. The initial seeding rod. It's a slightly different color. So I'll pass these around and please be gentle. AUDIENCE: Is the seeding rod just silicon? PROFESSOR: It's actually doped silicon, so it's lower resistivity so you can pass more current through it. This here is chunks, or smaller chunks of the polysilicon so, essentially, just crushed polysilicon and if you're trying to load a crucible with big chunks like this you'll leave a lot of empty space unless you crush some of this up and make finer grains out of it and fill in the gaps. So I'll pass these around right here so you can have a look at them. Those are examples of the Siemens grade polysilicon. This is a bigger rod. Here is the seed coming right through the middle. Here's the surface where you can see it's kind of rough and corrugated and one of the biggest issues with this feedstock refining process is that there are very large plants and long lead times. This is a plant construction going on right now, you can see. Typical lead times are between 18 and 24 months. That's a long time between when the board says yes, we will create new silicon refining capacity and product starts to roll off the production line and into customers' hands. It's a long time and what this results in are drastic oversupply and undersupply conditions in the market. So the silicon feedstock price goes very high during periods of undersupply and very low in periods of oversupply and we're in an oversupply condition right now. Five years ago, let me quantify this. Five years ago if you went to the spot market-- maybe four years ago-- if you went to the spot market, you could pay $100 to $500 per kilogram of silicon. That material that was just right there I bet one you would put it into your bag and run away out the door right now and be able to go to Mexico. Now the polysilicon prices are much, much lower on the spot market. Somewhere in the order of $30 to $50 per kilogram. About an order of magnitude lower. AUDIENCE: Isn't lower cost silicon better for the PV industry, though? PROFESSOR: Is it better for the PV industry? As a customer most definitely, it is good for you. As an installer, it is most definitely good for you. As a polysilicon producer who wants to be a sustained industry presence, it's not good for you. So this wide oscillation between fat cat and scrawny is not very good for any industry. It's unpredictable and it causes some players to drop out. AUDIENCE: OK. PROFESSOR: And the investments are very large as well. As you go from the early stage portions of the value chain toward the module, the investments generally decrease and so this is an outlook coming from last year-- the numbers are still a little bit outdated-- polysilicon production is buttressing up against 200,000 metric tons per year at this point in about 3/4 to the PV industry. The cost of manufacturing is between $20 and $25 per kilogram and 2010 prices were around $50 to $70. Now they're on $30 to $50 in 2011 and the 2008 prices were around $500 per kilogram in the spot market and it really boils down to the inability to adapt to demand. If you have a very large contraption that produces the feedstock materials and it takes a long time to build the factories, you're just not going to be able to adjust fast enough. Here's supply and demand, demand being the red and supply being the blue. You can see how the oversupply-- the undersupply condition of the mid 2000s really led to our current condition. So, alternatives to solar grade silicon feedstock refining. What are some people thinking in terms of other processes that they can use? These are two processes right here and, mind you, when we were in this situation with this price for the silicon, everybody and anybody was coming up with new ideas of how to manufacture the silicon. Now that we're barely selling at cost and in an oversupply condition, many of these ideas are having a struggle-- a hard time in the market. They're struggling right now. So fluidized bed reactor and upgraded metallurgical grade silicon. Let's talk about each of those in turn. So what the fluidized bed reactor folks realized was, gee, if we're depositing on a rod, our surface area to volume ratio is really large-- sorry, is really small. Our surface area to volume ratio is going to be very small. So think of it this way. If we have a sphere, a sphere would be the quintessential example where we'd have a very large surface area to volume ratio. If we had a plate, we would have, as well, a very large surface area to volume ratio and in the case of the condition prior, where you have this rod, you really can't deposit that quickly and so what these folks decided was, what we're going to do is introduce small silicon granules into this vessel, into this evacuated chamber, and-- here's the evacuated chamber right here-- and we're going to flow silane gas into the system right here and the smaller particles are going to go higher up because of this flow of gas coming in the bottom and those will grow and eventually settle down down here where we can extract the bottom. So we'll wind up with these beautiful little silicon granules. These ones shown right here, which I'll pass around as well, those are coming from a fluidized bed reactor, and they're nice beautiful, spherical granules that are grown a lot faster, I mean, a lot more silicon is deposited per unit time than through the Siemens process as shown there in the back. As a result, the energy intensity is lower, the cost is lower, there's a very tricky process to nail to get just right, because you have to get the gas flows right, you have to design the chamber well, redo some purity contents. It's a tricky process, and so this is being produced right now, I believe, by only a few companies. REC has a capability of doing it. MEMC, as well, has the capability of doing this process. By and large, most silicon is coming from the Siemens process. Yup. AUDIENCE: Sorry, both of those companies have the normal refining process? PROFESSOR: They have the normal refining process. AUDIENCE: The [INAUDIBLE] PROFESSOR: Yup, and that's why they developed this new one. They had these smaller, internal projects that we're able to develop. So, yeah. I was just mentioning the energy intensity. This is the kilowatt hours per kilogram, going back to your question about energy intensity. This is trichlorosilane based Siemens process, silane based Siemens process. They're more ore less comparable in terms of energy intensity. And the silance based fluidized bed reactor process. According to internal REC numbers, which are little rosy, but never the less, the trend is correct here. It is lower somewhere in the order of an order of magnitude energy intensity, and cost is lower as well. So let's move away from the silicon refining by distillation process entirely. Let's leave gaseous distillation aside and say, what if we were to take this metallurgical-grade silicon and, through liquid purification routes, result in high purity silicon. How would we do that? Well, if we turn to other industries, the ones that smelter aluminum or refine manganese and so forth, we would see a multitude of different options that we could borrow. Slag refining, bleaching, leaching solidification. Let me walk through them one by one. Leaching-- that's fairly straightforward. So if we put in some acid, for instance, that dissolves the metals but doesn't dissolve the silicon we could leach the metals out of the material, and so that's the essence of leaching. You might crush up your material, in other ways other ways expose the metals, or impurities, to the acids inside of your system. Slag refining says, gee, what if we were to introduce some material that could absorb the metals into it? The solubility of the metals would be higher inside of the slag agent than inside of the liquid. Maybe we throw in calcium oxide or yttrium oxide or some, usually it's a metal oxide that has a very high melting temperature that remains a solid or, at least a glassy solid, and we pour it on top of our silicon and it's able to absorb, say, the phosphorus or the boron that's inside of our silicon so that we reduce impurity content and then we can add the phosphorous and boron later intentionally, but to the concentrations we want not to exuberantly high concentrations that might be found in nature. Solidification-- during this solidification process you're taking your molten silicon and you're solidifying it directionally from the bottom up and, because the solubility of impurities tends to be larger in the liquid than it is in the solid, it's like dragging a comb through the entire material dragging out the impurities. Concentrating them in the liquid and leaving a more pure silicon behind. Obviously, at the very, very end you have this highly concentrated region of impurities which then you have to slice off and remove, so the solicitation process doesn't come without a yield penalty. You still throw away some of your material. So you can't repeat the solidification over and over and over again, I guess you could, but you'd be losing material every step. So some combination of these processes here, and others. Other trickery. Low temperature eutectic formation with other elements, for example. Some combination of this is used to refine the silicon without creating a gas out of it. So wafer fabrication. We're now going from feedstock, we're leaving feedstocking behind, and we're going to be talking about how do you go from the feedstock materials that are being passed around the room right now into a wafer that you can then manufacture a solar cell device out of? One of these for instance. So let's talk about wafer fabrication right here. So again, just to situate ourselves, we've gone from raw materials to silicon feedstock and now we're going to feedstocks to wafers. Any questions right now before we dive into that? Yeah. AUDIENCE: Question about supply. So silicon is very abundant but the high purity silica deposits-- are they really abundant too? PROFESSOR: Great question. So the question was are the high purity silica deposits as abundant as, say, silicon. Certainly. If you bend over and rub your fingers against the ground you're probably going to come up with, probably, millions of trillions of silicon atoms in your fingernails. Those are not very purity. So the highest security quartz deposits are more rare and they are sought after, and so they're are known. Their locations are known. There's one specific one in Norway, one specific one in North Carolina, and so forth around the world and there-- in a sense, they go to places. People have adjusted their metallurgical-grade silicon refineries and their subsequent down process for that particular ore. Once you run out of it, it's not that the world ends, we just have to adjust for the next feedstock source. So, in principle, there are people looking at a variety of silicon inputs. Anything from the dirtier, compressed, metamorphic quartz that I mentioned. Some people looking at rice husks, which are silica rich as well. Other people looking at seashells which, mostly calcium carbonate, but other things as well. I mean, there was a wide range. When the price of silicon was $500 per kilogram, you got a multitude of ideas. When the price comes back down, people tend to be more conservative. AUDIENCE: Is silicon considered a renewable resource? PROFESSOR: Is silicon considered a renewable resource. It is not a renewable resource in the sense that, once you mine it from the ground, you've mined it from the ground and you used in some way. The reason it's considered not an issue is because there's so much of it. Not all of it, though, is in the easy to access form. Right? Some of the silicon might be bound up within heavily contaminated sources and that's where the refining ingenuity comes into play. As long as prices remain low, there's not too much interest, say, for example, that mine in Peru that has titanium oxide needles throughout their silicon because why would you want heavily titanium contaminated silicon? But as the price of silicon, it probably will, rise again then people might take another look at that mine and say gee, how can we phase separate the rutile and anatase from the quartz early on in the process by crushing and etching or something so we can access this feedstock material. We'll see. It really depends on how the market evolves, where people go looking for their silicon, but there's a lot of it in the earth's crust. AUDIENCE: You're not concerned about silicon? PROFESSOR: No. Nope. What is a bigger bottleneck are are the refining steps in between. First it was the reactors and soon it's probably going to be the metallurgical-grade silicon reactors as well. All right. Wafers. How do we get to these from the raw feedstock materials that are being passed around the room right now? So single crystalline silicon ingot growth. Let's walk through that first. How do we get these beautiful ingots? They're about half of all silicon market right now. The biggest growth method, by far, is called Czochralski growth and, named after the Polish physicist there Jan Czochralski. What you do is you have a bath of molten silicon. A crucible, if you will. This tends to be a circular crucible, rounded at the bottom, usually made of quartz with heaters on the outside to heat up the molten silicon. To heat up the silicon chunks in here. Once everything is molten, looking like a big bathtub of silicon, you introduce a small crystalline silicon seed into that molten silicon and then you begin pulling while rotating that seed. So the seed is a single crystal material and what ends up happening is, as you introduce the seed into the material and begin pulling, you start pulling out this crystal. Single crystalline crystal. It's a thing of beauty and this seed is actually very, very narrow in diameter. It might be about that big around so pretty narrow in diameter and it's being able to support this ingot of a few, usually a few, tens to hundreds of kilograms of mass underneath it and that's because silicon is very strong even though it's brittle. So if you weren't to apply, say, for example, a shear force on your silicon but just to apply an axial load, you could support a very, large weight underneath it. So the [INAUDIBLE] of silicon is grown from the bottom and eventually you wind up with this nice ingot, as shown right there. The art that goes into growing this properly is amazing. I'll highlight it with one small little example just to illustrate the bigger picture that a lot of effort goes into making these defect-free, quote unquote, defect-free crystals. They're called defect-free because they contain no grain boundaries and no dislocations. They have impurities, they have intrinsic point defects, meaning vacancies or interstitial atoms, but they don't have grain boundaries or dislocations and so they're called defect-free silicon. You introduce that seed down into the liquid melt. Thermal stress happens. Right? Because you have the shock between the solid silicon seed encountering the liquid for the first time. So this locations [? form ?]. And you have to pull the seed out in such a way, you slowly rotate and make this shoulder. The shoulder has to be as quick as possible because you don't want to waste material. Everything inside this shoulder right here gets thrown away. So that little piece of material right there gets tossed out. So you want to make the shoulders as narrow and as quick as possible so you can utilize the majority of your ingot but, at the same time, you have to make it thick enough so that the dislocations can move all the way and propagate all the way to the outside and end and terminate in the shoulder before propagating into the crystal. So that's just one example of the technology that goes in the growing these. Another might be, gee, we're PV industry, we want to make the stuff fast whereas, in the IC industry you can invest up to a few of dollars per gram of silicon and still make a profit because you're selling a computer at 1,000 bucks. In the PV industry, we can invest, at most, a few tens of cents per gram of silicon. So we have to make this stuff fast. We can't dilly dally. You might want to crank up the growth speed, then you run into issues with defect concentrations, intrinsic point defect concentrations, during the growth. I'm illustrating this just to highlight the complexity of the growth process of making these ingots, and the latter example was one that the PV industry is facing today. It's actually a hot research topic. Yeah. And then Ashley. AUDIENCE: The rotation speed does that just affect time, or does it affect other things? PROFESSOR: So it affects a multitude of things. One of the things that it affects is the flow of, the convective flow, of the melt. So the liquid flow inside of the melt is, in part, determining how much oxygen gets transported from this crucible here into the growing crystal. If you manage to suppress that convective flow in the melt, you will also suppress oxygen transport since the fusion is going to be a lot slower than turbulent transport or [INAUDIBLE] transport or convective transport. Yeah. Question? AUDIENCE: I just have two questions so one is how fast do you rotate it and the other is what that does control-- the diameter because I've heard of 12 inch wafers versus like 18 inch wafers. PROFESSOR: Sure. So one of the things that controls diameter is the balance of heat extraction. So if you cool something down, especially molten silicon, it will freeze, it will grow. If you heat it up, it will shrink. So that's one of the components that controls the diameter. The pull speed and how you grow that shoulder, essentially how you heat up the material and how fast you pull at those initial stages, also dictates the diameter and you can see in the ingots themselves, they're not perfect. They have a little bit of corregation and that's the fluctuations of the temperature of the melt, fluctuations of the heater output, fluctuations of pull speed, maybe what's pulling this entire contraption is kind of a stepper motor that has a certain granularity to it. Results in corregated edges. It's not perfect and so there will be some adjustment made to the form factor of the edge to get this nice round wafer at the end of the day. AUDIENCE: Does the seed rod [INAUDIBLE] all the way to the ingot or just near the top? PROFESSOR: All right. So the entire ingot becomes pattern or templated by the seed rod. So this entire ingot has the same crystalline orientation as a seed. AUDIENCE: And is the seed doped differently than the silicon? PROFESSOR: It might be but I'm not aware that that affects the overall process. It could be that it's one of the critical pieces of the magic sauce that makes it work but I'm not aware. Rotation speed. It's not rotating like this it's a slow rotation so I would-- let's see. How many radians per second-- AUDIENCE: Can you see it? PROFESSOR: You can visually see it if you looked at it long enough. Yeah. Yeah. So one modification, one variant, of the single crystalline growth method is called float-zone growth. You take a rod of poly, much like that right over there that's inside of here, and you pass an RF coil, radio frequency coil, next to the rod and what that does is, essentially, heats up the silicon, if it's doped highly enough. It will melt the silicon locally. Folks have probably heard of fancy high-end stoves that we can only probably hope to afford in 10 or 15 years, but these stoves that are inductive heaters. Right? They're not resistive heating elements, they're inductive heating elements and the way that works is you have a radio frequency source that then is absorbed by, in the case of the RF heater, I believe it's a specific type of iron that the inductive heating ovens need. And so this RF coil here is emitting energy, which is absorbed by the silicon and melting it, and you start with the polycrystalline rod coming from the Siemens process and in that case, this rough, corrugated material right here won't do. Right? This is too rough for that RF coil to pass over and be a consistent distance away. In the case of float-zone growth, you actually have to modify your polysilicon production process. You have to modify the Siemens process so that you get a nice smooth rod, which you can then pass the RF coil next to and melt and you again start with a seed at the bottom, your RF coil starts down here and then the RF coil moves through the material, almost like a comb from the bottom to the top, converting the polysilicon into nice single crystalline material and, in the process, it concentrates impurities in this liquid region. Since the liquids have a higher solubility in the liquid than they do in the solid, the impurities are then aggregated inside of the liquid region and, again, like a comb, they just get swept out of the material. Not all of them, but a large percentage of them, and so you can make multiple passes with this RF coil to further concentrate the impurities and the extremities and remove them from the material. So that's a float-zone method. Very expensive material, very high purity. One of the reasons it has high purity is because you don't have this quartz crucible nearby, you don't have this molten silicon that's absorbing or dissolving the quartz and transporting the oxygen into your crystal. You have much lower carbon and oxygen concentrations to [INAUDIBLE] float-zone material. So if anybody is doing experiments with silicon, for whatever reason, using it as a substrate material, you want to think carefully about what type of silicon you source and from where you source it. You can find some very poor quality silicon out there in the market, especially if you going into the aftersale market, and we know this from some-- AUDIENCE: [INAUDIBLE]. PROFESSOR: --very painful experiences. And so there are some better sources from which to get your wafers and we're happy to talk about that offline. So, again, single crystalline silicon. We're going to venture into the world of multicrystalline silicon ever so briefly here. First, we'll start about cast material and, just to emphasize here, we have regions of crystalline material that have grain boundaries separating the adjacent grains and the reason we go into multicrystalline silicon is really oftentimes, it is a lower cost method of producing a silicon wafer although you have the grain boundaries. So, again, single crystalline, Czochralski and float-zone, you wind up with round wafers, typically single crystalline variety, and multicrystalline silicon wafers tend to be more square-like and more visibly multi-grained, if you will. So let's talk about those for a minute. How do you make a multicrystalline silicon wafer? Again, you would start with the solar-grade silicon that could either be coming from the Siemens process, it could be coming from the fluidized bed reactor, it could be coming from an upgraded metallurgical-grade silicon, the liquid purification route but, somehow, some way, you get chunks of silicon, or granules of silicon, that have a high enough purity for you to make solar cells out of, and high enough purity is typically in the order of one part per million impurity content. So you put your solar-grade silicon into a crucible and then you melt the silicon inside of it. Silicon melts at 1,414 degrees Celsius. It's a very high temperature. So 1,414 degrees Celsius is the melting temperature of silicon. And then it's cooled. Not just randomly, but from the bottom up and the reason it's cooled from the bottom up is because, and here I guess you'll actually have to come up and see this after class, it's rather difficult to see from here, but this is a cross section of a small ingot. This is the outside of the ingot where it was contacting the wall, these little pieces of white stuff that are flaking off, this is the fused quartz silica that forms the crucible wall, and the silicon nitride coating that form the anti-stick coating that prevented the silicon from sticking to the crucible, and so it's kind of rough and corrugated but, if we were to rotate this around and look at the inside, this here is a cross section of the actual ingot from the inside and, if you look carefully, you'll see grains growing from the bottom to the top. You probably can't see them from here, you'll have to come up after class and take a look, but the grains are growing from the bottom to the top and that is called directional solidification, or the result of directional solidification. Directional solidification is when you solidify from the bottom to the top and, typically, your grain boundaries are going to be running perpendicular to the solid-liquid interface, so your grain boundaries will be running up like this as you grow your material from the bottom to the top. If you were to do uncontrolled solidification and all walls would freeze the same time, you'd have grains growing in from the sides, you'd have grains growing in through the bottom and then, when you slice your wafer out horizontally, the grain boundaries wouldn't be running perpendicular to the surface. They might be running parallel to the surface, in which case they could wreck havoc on your minority care diffusion length. Imagine you being an electron having to travel across that grain boundary that's between you and the P-N junction. Whereas, if the grain boundaries are running perpendicular to the surfaces, now they're only affecting very small areas of the entire solar cell wafer. So when I pick up a wafer like this, this wafer was chopped from the ingot this way or, to put it into perspective here, this wafer was sliced out like that from this. So the grain boundaries were running perpendicular to the surfaces and that way they don't impede as much with electron transport. So the multicrystalline silicon ingot is formed. The ingot is then chopped into these blocks, usually between 16 and 24, that means four bricks to an edge or five bricks to an edge. Some folks are exploring six by six, so 36 bricks, and then the bricks are rotated on their side and then sliced into wafers and individual wafers come out. So you can see the wafers I've sliced from the bricks as I showed you right here. Is this diagram clear to folks? In general since-- any confusions? Any questions? No. AUDIENCE: How do they cut the wafers? PROFESSOR: How do they cut wafers! So this is a process called wire sawing sign and this is one of the most beautiful technologies because it was invented in the PV industry and transported back, adopted by the IC industry. So it's one of the few examples of technology that went the other way. Let me get to that point. AUDIENCE: How was it done before? PROFESSOR: It was done by ID saws, for instance, inner diameter saws, that would slice off wafers like a wafer off of a salami. AUDIENCE: So not a wire but like a disk? PROFESSOR: Like a disk saw. Yeah. Exactly. Like the inner diameter meaning your saw is like a rotating blade and you're just using, you know-- Yeah. OK. So directional solidification of multicrystalline silicon. This is a cross section of a furnace that is solidifying an ingot right here. Here's your ingot. This is a liquid silicon and, essentially, it's solidifying from the bottom to the top and, hopefully, we'll have a tour of one of the world's largest ingot solidification furnace manufacturing companies in the world. So, they don't manufacture the silicon, they manufacture the furnace that manufactures the silicon. If that makes sense. AUDIENCE: Do they also make the crucible? PROFESSOR: No. That would be Vesuvius. Yeah. It would be other companies that make the crucibles. And these are some of the furnaces right here. The keyboard and monitor. For size comparison, stairs. So they're about two stories tall. You can go up here to the top and look down into them. It's pretty cool. Using a little infrared lens to block out the heat so you don't get blinded and the furnace itself-- all the action happens inside of here. The top can lift-- typically, they're the bottom loaded. You'll see this little seal right here. So this bottom part typically comes down because you want to trap the heat inside of it so you're not losing all that and the bottom is removed, the forklift comes in, picks up this ingot and crucible which could be a few of kilograms in mass-- up to about 600, maybe even a ton-- and removes it and places in the proper location. It's a pretty dirty environment. The operator will typically take a garden hose and hose it down inside afterward. It's really an antithesis of an IC fab at this stage right here. These are graphite insulation materials on the sides of the crucible and this yellowish dust that you see everywhere is silica, again. That nice fine grained dust that's bad for you lungs. The directional solidification process can be, to some degrees, used interchangeably with the so-called Bridgeman process. It's also a name for a specific type of directional solidification. This is your ingot, this is the ingot chopped into bricks, and then the bricks-- here's an ingot coming out of a furnace. Those are the bricks over here. This is a really tiny one. It's like lab scale. The big ones are about over a meter along the long edge. And then, to saw them into wafers, we use what's called wire sawing. These are several kilometers of wires-- of wire. One continuous wire, several kilometers long, typically of a steel-based composite. Running in these bricks right here, in the presence of a glycol-based slurry, typically, and silicon carbide or diamond grit, and the grit is being pressured by the wire against the silicon. The grit is very small in size-- micron size-- and it's, essentially, chipping out small pieces of silicon as this wire is progressing through and, over a period of around 6 to 8 hours, you saw through the entire brick and you use, maybe, four or eight of them at a time. So if that wire were to snap about halfway through the process, all those bricks are gone. So it's very important that the wire be very robust and able to support the sawing process and, as I said, it's several kilometers long and moving at a speed of a few meters per second. So this is zinging along through your material in the presence of very small grit and slurry, and so the consumables that are used in the wire sawing process are enormous, and you lose about half of your silicon due to sawdust in this process right here. So this is a prime candidate for replacement of the manufacturing process, even though it's so commonly used today. What I'm going to do is give a quick pause right here until our next class, where we'll pick up and talk about ribbon growth, which seeks to get around all the complexities of multicrystalline silicon ingot growth while still keeping the cost advantage. So with that, thank you. |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | 11_Wafer_SiliconBased_Solar_Cells_Part_II.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: All right. Why don't we go ahead and get this started here? We have a cornucopia of different silicon materials out in front here in display, and we'll walk through some of them shortly. What I wanted to do right at the beginning of class was to give a little bit of an update on quiz number two. Some of you have probably seen this already and are aware that on Thursday we're expecting a short little decision tree as to how to process your solar cell to obtain the lowest dollars per watt peak. So this little exercise-- it will last for about a month-- is coincident with our technology section of the class. So remember, we went through the fundamentals. Now we're on the technologies. And then finally, in the cross-cutting themes. So coincident with the technologies portion is designing your own solar cell and optimizing the dollars per watt. So this will entail actually fabricating a solar cell, which is kind of fun. And Joe will be your guide throughout this process, so you'll be able to actually take a piece of bare silicon and finish up with a device, a rudimentary device, but something to take a picture yourself, post on Facebook, that sort of thing. You design your own solar cell. So the idea isn't only to optimize for the performance of the solar cell, but we decided to throw in a little curve ball and design for dollars per watt peak. Now this is a little bit of a contrived exercise since we've arbitrarily chosen what dollars are associated with each different process step, but it's not too unlike what you would face in actual industry if you had real data coming off a production line and knew exactly what it cost for each process step. So instead of having 30 plus components in a more detailed cost model, we've decided to simplify it to this little diagram right here. So this is a flow chart for the fabrication process of your solar cell. You'll start with a wafer. It has a certain cost associated with it. You'll have some decisions to make concerning light management, whether you want to texture your front surface or whether you want to leave it bare and reflective like this right here. So whether you want a reflective front service or you want to texture it, there's a certain costs associated with it. So you can probably go to some online resource, like PVCDROM, and use their simulator or the one you've already constructed for homework number two, and calculate what the predicted efficiency boost should be if you texture your front surface. Keep in mind on this very simple solar cell here, we have no anti-reflection coating. So the texturization is pretty much all you've got for light management. Next, on the emitter, the choice is whether to make a deep emitter or a shallow emitter. The text goes into that in some detail. But your decision is basically if you make a shallow emitter, you have less Auger recombination in that front region. And so your blue response to the device will be better. But you run the risk when you do your contact metalization of firing through that very shallow emitter and shunting your device. Whereas, if you decide to go for a deep emitter, it stays longer inside of the furnace because of the phosphorus will diffuse deeper inside of the device. You blue response will be poorer, but you'll have less risk of shunting. So it's up to you to use all of the tools that you've assembled so far to make a value-based judgment whether or not it makes sense to go with this or that as your selection choice. And finally, narrow and wide fingers, this you can probably guess already pertains to series resistance and shading losses. So these are all representative of trade-offs, trade-offs in terms of the technology and trade-offs in terms of cost. And you have all the tools necessary to calculate or estimate what these outputs should be based on what you've learned so far. And so by Thursday, what we've asked you to do is to make an estimate of what technology pathway your company is going to pursue. Remember, you want to optimize the dollars per watt. You want to minimize that quantity, which means you want to reduce the number of dollars you invest in your solar cell. But you also want to increase the watt peak that you get out of it. And so at the end of the day, it'll be a performance/cost trade-off in each of these different process steps right here. And sometimes it won't be entirely obvious which one to choose because so many factors will converge. And so it'll be up to you to make an engineering decision, a professional judgment, as to which path you should pursue. Since it is kind of-- you know, there's a little element of competition in here, so we decided the dollars per watt peak shouldn't be completely neglected at the end and we all get certificates of merit and all feel good about ourselves. We decided it should be worth some part of the grade, but not such a large portion of the grade that everybody's freaking out and saying, oh my gosh, I don't have the right tools to make this decision. I feel like I'm not being graded fairly. So the portion of dollars per watt is really only going to be affecting 10% of the final grade of quiz number two. And so it will be based on a ranking system where the highest one will be 100% and so forth. But just 10% of your grade. So it's enough to, I would say, create maybe a sting of the pride if you don't happen to hit the highest performance metric, but not enough to sting the actual final grade of your class, which will be one lumped quiz, quiz one, home works, final, and so forth. Right? Any questions about quiz two so far? Yes, Jessica? AUDIENCE: I completely understand, but there's even a note in number three under the deep emitter. And you guys say, any numbers you should give as far as [INAUDIBLE] or are they responsible for [INAUDIBLE]. It seems like its lacking some numbers. And I understand optimization, but I'm having trouble putting just how much better. And it say it'll be much more effective if you do etching. Well, how much is much more effective? PROFESSOR: Oh, the etching for the-- AUDIENCE: For the etching, we gave a rough [INAUDIBLE]. So you can look that up. AUDIENCE: I did look that up. And for the other ones, is there AUDIENCE: So that one, you can get a pretty good estimate for. AUDIENCE: OK. For the other ones, is there going to be a [INAUDIBLE] AUDIENCE: In terms of shunting your device, it's really hard to predict the shock resistance. But if you do shunt your device, you essentially ruin it. So I would just take that into account. You're not going to get exact answers. But you can do your best to estimate [INAUDIBLE] resistance from the [INAUDIBLE] spacing and your emitter thinness. PROFESSOR: Yeah. Believe it or not, you might feel like you don't have the tools right now to get quantitative answers, but you do. You have a number of the tools here to get, say, 90% the way there. And in engineering, 90% of the way there is well beyond what you'll actually face in the field. So that's pretty good. If you have specific questions about what would be a good resource to look up about this, what would be a good resource to look up about that, send an email. And what I'll do, if I receive something in that nature, I'll respond to the class so that everybody has benefit to that information and no one person is particularly advantaged. So it's worth a try. If it's something that was just covered yesterday in lecture, I might be a little bit more reticent. But if it is something to the effect of, gee, how would the lifetime improve with these different gettering scenarios, sure, absolutely. We can give you a little hand there. But everything else, you should definitely have that information available so far. This is meant to be a fun exercise, but also one that illustrates the trade-offs involved with designing solar cells. And trade-offs very similar to this are evaluated on a daily basis in industry, or perhaps not quite as often as they should be in industry. But at some point, they were. And the designer of the manufacturing line made those judgment calls. OK. So again, the pre-analysis, what is due on Thursday is 20% of the grade. The dollars per watt peak metric, at the end of the day, is only 10% of the grade. It's meant to really serve as a stimulus, a little bit of competition, but not meant to really harm you if you happen to not achieve a good value there. And this is meant to be an educational mission, so the solar cell efficiency analysis at the end is really heavily weighted. We'll be walking through some of the characterization tools in the laboratory so you can determine what exactly went wrong with your devices and quantify them. And that'll be a real chance for you to get a tutorial of how solar cells are not only made-- you'll be there when they're actually fabricated-- but also how they're analyzed and how they're assessed. So it's up to you to really grab this opportunity. Maybe if you're working in your own devices and want to bring some of them along, you're welcome to do that as well. We won't take up the time when everyone else is in the room, but we might stay longer afterward and help you walk through the analysis as well. And what we've done, just to resituate ourselves, we talked about the silicon feedstock, right? So we chatted about how you go from quartz in the ground and the carbon-baring feedstock material to the purified, highly purified, silicon feedstock material. This right here is probably on the order of somewhere between 8, 9, or 10 nines pure, very, very pure material, this Siemens-grade polysilicon right here in my hand. And you've taken a look at this during last class, so you have a sense of what it is up close and personal. Silicon, in fact, has been so well refined that, for a period of time, NIST, the National Institutes of Standards in Technologies, they were thinking about redefining the unit of mass in terms of a silicon boule, essentially a silicon sphere, that would be polished down to about 4 nanometers mean surface roughness with a very low defect density, isotopically pure silicon to serve as a new standard for mass because it could just be purified so well and because their standard reference units were beginning to shift relative to all the others around the world, the one in Paris relative to the ones that were stored in Washington and Delhi and others around the world. The values of the mass were shifting as a function of time when they would perform these round-robin. So either the mass in Paris was changing or everybody else was changing. This obviously was unacceptable for an institute that was focused on standards. And so they decided to reformulate the standard for mass. I'm not quite sure where that project currently stands. So if anybody has further information about the NIST unit of mass, I'd be happy to hear it. But that gives you an idea of how well-purified silicon can be and how well-controlled it can be as well. During the integrated circuit fabrication, which uses this ultra purity silicon to produce very nice single crystal wafers like this one right here, the investment per gram of silicon can be on the order a few tens or even low hundreds of dollars per gram of silicon and still turn a profit. But in the solar cell, on the other hand, you can invest, at most, a few tens of cents per gram of silicon. This is because the solar cell has to compete against bulk power. That's its competition coming out of the wall right over there. So the solar cell has to be able to be produced much more cheaply. And as a result, typically thinner wafers are used and less expensive starting materials and faster growth methods, resulting in more defect-rich materials. So one group decided, gee, the embedded cost in the wafer is just so large, it's just so large that we have to make it thinner. And we have to avoid using these ingots, like this one right here, from which these wafers are sawn. So your wafers are sawn out of the ingot like this, like shown. During the process, about 50% of the silicon is lost to sawdust. And they said, well, let's develop a better way. Let's extrude the wafers directly out of liquid molten silicon and make ribbons of silicon instead. That way, we don't have the sawdust, and we don't have to have this expensive ingot solidification step. So ribbon growth has been explored since the 1970s at least. And the advantages is that you have no kerf loss, in other words, no sawdust, due to wire sawing and, hence, more efficient silicon utilization. Immediately out of the gate, if your wafer yields are comparable, you get about a factor of 2 gain because this wafer right here is about 170 microns thick. And the sawdust is around 170 microns as well. So that's about a factor of 2 if you're able to produce a ribbon of silicon directly out of the melt. The disadvantage is that traditionally there's been lower material quality and, hence, lower performance because of the thermal stresses during growth of a very, very thin foil or thin fin. The thermal stresses can be larger, resulting in plasticity, resulting in dislocations and other defects that can reduce minority carrier lifetime. And traditionally, there has been as well a higher capex. And a third disadvantage, traditionally, in ribbon has been that the form factor or the shape of the wafer has just been different than the ingot material. Why is that important? Well, if you're trying to displace the dominant design, the wafer, you would do well to make your wafer the same size and shape as the dominant design. Why is that? Well, if you want to make a cell out of it or solar cell device, you'd want to make sure that you can take advantage of the same manufacturing equipment. And that's just a plug-in-and-play, drop-in replacement. If you require customization of the downstream components on the cell in the module level, you'll wind up having to invest more money in those processes, which might counteract the advantage that you get out of using less silicon. Yes, Ashley? AUDIENCE: What's capex? Is it-- PROFESSOR: Oh, capex. Capex stands for capital expenditure, capital equipment expenditure. And that relates to the cost of the equipment that is typically-- well, in the business world, typically one undergoes what's called an accelerated depreciation where you amortize the cost of the equipment over five years but then assume that it runs over a longer period, maybe 7, 10 years or so giving you profit back. So in layman's terms, what this means is capex is the equipment cost, in other words. And then you just take the cost of the equipment and parse it out. For each wafer you produce, you allocate a portion of equipment cost to that. So let's take a little walk through history and go back to some of the earliest methods of ribbon growth. So one of the earliest forms of ribbon growth was the so-called edge supported ribbon, also known as string ribbon. And there were developments of this general technology in different places. Ely Sachs, former professor here at MIT, now founder and CTO of 1366 Technologies just up the road in Lexington, developed the string ribbon material here at MIT in the early 1980s, late 1970s. And the general idea was to use two filaments like so that would be passed through a crucible. And then the silicon would flow in between those two filaments much like soapy water flows between the little circle when you blow bubbles. So a meniscus would form here and then eventually solidify into a solid piece of silicon, and you'd have edge-supported ribbon, otherwise known as string ribbon because you're using the strings to define the edge of the ribbon. So I have a wafer here, an example of a wafer here, a string ribbon sample. Oh, here it is. It's hiding from me. So this is an example of one of those materials. Here we go. And like usual, it's good to handle these wafers with some care almost like a photograph. So here's an example of a string ribbon wafer, one particular wafer that was laser cut out of a growing ribbon. As you can see, this larger ribbon right here-- these can grow up to be a few meters long. They're rather long. You can pick them up if you have gloves on your hands. And they're quite flexible at that length. You could actually even bend them with a radius of curvature of about a couple of meters. So the reason you wear gloves, obviously, is to prevent your fingers some soiling the wafer. We talked about sodium contamination and other forms of contamination. Silicon is nontoxic, so it won't affect you. It's really you affecting the wafer, much like putting fingerprints all over a nice, clean photograph. So there were similar technologies developed by Ted [INAUDIBLE] at NREL out in Colorado. But the general idea is shown right here. Now some of the earliest edge-supported ribbon samples were developed back in 1970s. It really took a while before they were commercialized in full. And that was done through Evergreen Solar, which was founded in 1994 by Jack Hanoka, Rich Chlebowski, and-- oh, goodness-- Mark Farber. So the three of them a co-founded Evergreen Solar. And they developed the string ribbon growth process shown right over here. Eventually two ribbons face to face, and now four ribbons side by side. So this was called the Gemini because there were two ribbons face to face. And then eventually, the quad process were four ribbons edge to edge. And you can see the conventional ingot multi-crystalline silicon. Here, the different steps forming the ingot, eventually slicing, and so forth and the string ribbon process here being much simplified in correspondence. So not only was the process simpler, but you'd use about half as much silicon. And here's Rick Wallace, the inventor and developer of the Gemini process, up there showing one of these longer meter-length ribbons with some flexibility. So the company had a joint venture with REC and Q-Cells, Norwegian and German companies respectively, to form a factory in Germany. REC would supply the silicon feedstock, Evergreen the growth technology here, and Q-Cell some of the cell fabrication expertise. And very recently, Evergreen Solar encountered some financial difficulties-- we'll get into that during the third section of the course when we talk about cross-cutting themes-- and is in the process of filing for bankruptcy. So this process-- so Sovello is continuing as its own company, but the Evergreen plant here in Massachusetts in Marlborough, about an hour west of here, has effectively shut down. So that was the trajectory of this particular technology through commercialization and ultimately not making it. If you would like, my personal opinion about why Evergreen never quite took off, yes, there are some technical factors, but as well it failed to grow fast enough to keep up with the rest of the industry and scale with the rest of the industry. And part of that can be traced back to the mid 2000s when silicon was scarce, the inability to source the feedstock material. Yeah? AUDIENCE: Excuse me. Can you back one slide? PROFESSOR: Sure. It takes a while. It's a big file. OK. AUDIENCE: How do you seal the space between the filaments and the bottom of the crucible? PROFESSOR: Right there, right? AUDIENCE: Yeah. PROFESSOR: Since this is your graphite crucible right here and these are your filaments popping up through the graphite, the beauty is you don't have to seal that. The surface tension of silicon is greater than that of water. So if you've ever filled up water to the top of a glass and seen that meniscus that forms, the silicon meniscus would be even higher than that. AUDIENCE: Oh, that's cool. PROFESSOR: Yeah. It's pretty nifty. AUDIENCE: So I'm imagining just like molten metal. You don't want that spilling out the bottom. PROFESSOR: No. AUDIENCE: That's really cool. OK. Cool. PROFESSOR: Yeah. AUDIENCE: Are the ribbons a single crystal? Or are there grain boundaries in them? PROFESSOR: Yeah. So let me show you the actual ribbon right here, and you can inspect it first hand. These do indeed have grain boundaries. So what I'll do is I'll place the ribbon inside of here for ease of carrying around. If you'd like to take it out, feel free. They're more where this came from. So in case there was a little accident along the way, don't feel too bad. The growth of an ingot is about one to two days, but you get thousands of wafers out. The growth of a wafer itself-- if the growth rate was around, say, let's pick a number somewhere between 2 and 5 centimeters per minute, then it would take-- let's see, with this, you have a 15 centimeter wafer-- it would take somewhere on the order of four minutes to grow wafer. And you'd have a faster growth of single wafers from the ribbon process, of course, lower throughput. The silicon utilization of the wafer growth process was a lot higher than that of the ingot growth. Some smart people realized along the way that you could grow these ribbons vertically, but you encountered the following problem. During the growth of-- here you go. During the growth of a vertical ribbons, if this was the ribbon growing vertically-- it should be straight. Apologies. There we go. Let's make sure we're good engineers here. And so this is meant to represent a growing ribbon. This is the liquid, and this is the solid silicon right here. The growth velocity would be in this direction right here. So you're growing the ribbon out of the melt. This is your melt. This is the ribbon that's growing up. You're looking at the cross section right here, so looking at the ribbon edge on. So you're pulling it in this direction, so the growth velocity is here. And the direction of latent heat of fusion extraction-- so you have liquid silicon solidifying here. During the solidification process, there's heat released. And that heat has to be conducted up the solid and then radiated outward from the fin, from this thin ribbon. So the direction of heat extraction is also parallel to the direction of growth. What that means is the growth velocity will be limited by the speed at which you can extract heat up the ribbon and then radiated outward. So there are many ideas tossed around about potentially growing in media that are able to extract heat [INAUDIBLE] transport. You can use your imagination. But ultimately, growth continues in air, and you're limited to, at most, around 5 centimeters per minute growth velocity because of the extraction of latent heat. If you try to grow faster than that, you'll eventually just pull the solid off of the liquid. It'll dissociate much like pulling an ice cube off of a top of a glass of water. Surface tension won't be able to hold the two together. So you have here a conundrum. How do you grow faster? If you want to increase the throughput and instead of spending minutes to grow wafer, you'd like to grow a wafer per second, how do you do that? Well, one group of folks thought about this a bit and said, well, what if we do this? If we take our growth velocity and in some way, shape, or form now our growth velocity is going to be perpendicular to the direction of heat extraction, what would that geometry look like? And they came up with something that looked a bit like this right here, a horizontal growth mechanism. So you see the [INAUDIBLE] interface is now at an angle. It's almost vertical at this point, a slight angle. And the pull velocity is almost perpendicular to it. So now, you're able, in theory at least, to grow much, much faster. This was a schematic of the ribbon growth on silicon process. There's also another company called AstroPower that developed silicon film. It was later purchased by General Electric. So these technologies were developed with the intent of pulling very, very fast. And indeed, you can literally extrude the silicon at around 49 meters per second. But the problem about this is that you wind up with very small grains and very poor crystalline quality when you try to grow at the speeds. And so it winds up being a metallurgical problem of how do you ensure the proper grain size when you're growing using these technologies? So there is some work in that regard, but never really took off in commercial production. Yeah? AUDIENCE: So does pulling at a lower speed with the horizontal ribbon increase your quality by increasing your grain size? Or is it not really-- PROFESSOR: If you're able to control the nucleation and growth process at the very beginning, theoretically, that could be possible. AUDIENCE: OK. PROFESSOR: Yeah, question? AUDIENCE: You had mentioned form factor for these wafers before. PROFESSOR: Yeah? AUDIENCE: So is there like a standard form factor for solar cell manufacturing? PROFESSOR: Yep. So the standard form factor today is akin to this one right here. It's about a 15.6 by 15.6 centimeter squared lateral dimension form factor for the wafer. And I can pass this one around as well. This right here is what's called a "pseudo-square." You can see the edges are kind of rounded off. And that's because it came from a CZ wafer like this one. It was just chopped out of it. Let me see if these two are coincidence. It would be a-- oh, yeah. Look at that. So you can see where the solar cell actually came from. So that's the standard diameter of a, say, linear dimension, usually rectilinear shape, a square. And the multi-crystalline silicon ingot material are typically of this size as well. And you can already see that these wafers that I have up here are a bit small. These were the previous generation size. I believe these are 12.5 by 12.5 centimeter squared. Most laboratory devices that you and your colleagues will manufacture are on the order of 1 by 1 centimeter or smaller because-- well, because of a variety of factors. One is the transparent conducting oxide as we saw in our homework problem. We're limited in how big we can make the device by the sheet resistance of that transparent conducting oxide. Another problem that we typically run into is just that we're not able to deposit uniformly over a large area. We don't have a deposition equipment for it in our labs. We're there trying to optimize a new material. We don't necessarily worry about making module-sized devices out of it. Yeah, question? AUDIENCE: Is there a reason why the form factor is different than that used for device manufacturing? PROFESSOR: Sure. AUDIENCE: Like [INAUDIBLE] uses circular wafers. PROFESSOR: Yeah. So if we were to imagine a bunch of circular wafers inside of this module over here, you can imagine the circular wafers side by side. That was how they were done at once upon a time. Obviously you didn't have 8-inch. It was much smaller. Or a 6- or 8-inch. This would be a 6-inch wafer. But the wafers were a little smaller, but you still have circular wafers and a lot of dead space in between. So as you can see, because of the rounded edges, the packing density is very low. The equivalent would be, say, oranges at a market where they're all stacked on top of another and you have all this dead space in between. And so the idea was to optimize between the cost of the silicon and the cost of the encapsulant materials by shaving away a little bit of the silicon and losing that-- and perhaps recycling it, to be honest-- and the encapsulant materials, where you have this dead space in between the wafers, a small amount of it, where you have glass and encapsulant but no active device underneath. Another interesting development, as you can see just from the device point of view-- so this would be an Evergreen string ribbon wafer right here, as you can see. And this right here, a larger area device. Does anybody notice a difference besides the shape? In particular, I lead you to the busbars. How many of those thick, vertical lines appear down the wafer? AUDIENCE: [INAUDIBLE] PROFESSOR: This has two, and this has three, right? AUDIENCE: Yeah. PROFESSOR: So the busbars-- the optimization of these busbars-- this one has three-- that's really to minimize series resistance. Because now that I have a larger wafer, you have so much current flowing through it, being generated, that the series resistance through those very thin metal wires would end up resulting in large power losses, essentially heat instead of electricity. And so they added the third busbar, even though it increased the shading, to reduce the series resistance losses. So you can see these optimization problems are used quite frequently in solar. Let me go back one step. There was an interesting question about could we grow single crystals using the vertical ribbon growth. This is a technology. And I don't know if there are actually any of these, many of these samples left in the world. They're quite rare. So I do ask if you want to come up here, take some care with it. This is a dendritic web sample. So this technology went out of commercial manufacturing, I believe, in 2005. Must have been. Or 2004. It was developed by Westinghouse, which is used to be one of the powerhouses in solar located in Pittsburgh, Pennsylvania. They had a very active solar activity. It was a kind of a crucible out of which many solar experts then went into diaspora around the United States and set up their own activities elsewhere. And one of the technologies that they developed was a single crystalline ribbon technology like this right here. And if you look very closely, it really is a single crystal. The growth methods to make this, though, was extremely intricate. It involved, among other things, control of the temperature, of the liquid silicon to within 1/100 of a degree Celsius at melting temperature, which is an extreme feat of engineering. The uptime of these pieces of equipment, meaning the growth time, was around 50%. And the other 50% of the time, the operators were trying to make it work. So it grew very, very thin material. It wasn't able to scale to the form factors that we see nowadays. The throughput was quite low. The cost was high. And so it didn't quite make it, but from an engineering point of view, it was a marvel in terms of what they were able to accomplish. So history of crystalline silicon development is riddled with these technologies that didn't quite make it with these materials that were extremely inventive, extremely ingenuitive. But at the end of the day, the dollars per watt peak just couldn't continue to justify their existence. And there were a number of factors that could contribute to making that happen. So in terms of wafer fabrication in general-- this includes both the wafers out of ingot materials but also ribbons-- where do I personally see this field going? These are some notes. So in terms of cost, the cost per watt peak can be reduced by using cheaper starting materials. That means instead of using this expensive Siemens poly, perhaps an upgraded metallurgical silicon process. Growing or sawing thinner wafers. Growing, for example, on a ribbon technique. Sawing, maybe making the saws themselves thinner but more robust so that they don't snap as they're pulling through the material at about 5 meters per second in that slurry with the silicon carbide or diamond grit. Very challenging engineering as well. This second bullet point right there can be encapsulated in a larger team called improved silicon materials utilization. In other words, the grams of silicon that you use to produce a watt peak of a solar cell. So improving that number right there. Increasing furnace throughput-- that means increasing ingot size, growth, speed, and so forth. There are many people right now trying to grow these ingot right here up to a ton, one metric ton, so 1,000 kilograms. That would mean for the full-sized wafers, you would have something on the order of 6 by 6 bricks. It's pretty large, a pretty large ingot. Maybe even 7 by 7. And improving the material quality so that you can improve efficiency, efficiency being a huge leverage over the entire cost structure. Because if your solar cell is able to produce more power, that means that you use less encapsulant, and less material, and so forth per unit power produced, and even less labor to install it and less racking and framing materials downstream. The scaling issues, so polysilicon production is currently-- well, this is higher now. It's about 100,000 metric tons per year. And about half of that-- well, about a quarter of that, now, maybe a third is for the semiconductor industry, about 3/4 for the PV industry. The slurry and the silicon carbide grit needed for wire sawing is, at some point, going to become an issue. These are huge volumes of waste that need to be transported through the factories. And of course, the silicon loss due to wire sawing and ingot casting, resulting in only 50% of the silicon here in this ingot being used in the actual wafers to make solar cells. The technology enablers-- using lower-- let's put it this-- lower cost feedstocks. You can't compromise on quality ultimately, so this is a little bit of a false choice right here. Using lower cost feedstocks produced by the upgraded metallurgical route, for example. Producing and handling thinner wafer and growing faster, larger, higher quality ingots. And there's a lot of innovation to be had in this space right here. I believe the numbers in the last quarter, start-up companies raised on the order of $250 million from venture capital. And that wasn't including a new $50 million deal that was just announced of a company attempting to produce upgraded metallurgic grade silicon through liquid routes, purification. This was just announced this past week, if you go to Greentech Media. So there's still a lot of active innovation in this area despite the current market conditions. And those of you who are looking for jobs right now, if you're clever, you'll find them here in this space. Any questions so far about wafers? Yes? AUDIENCE: Does laser cutting cause as much dust? PROFESSOR: Does laser cutting cause as much dust? So let's walk through that. If we're thinking about the ribbon growing from, say-- from this ribbon right here, I'm going to extract this wafer. So I need to make an incision horizontally right around this point right here. If you look at the total height, the wafer's around 15 centimeters long. And the laser cut itself is something on the order of maybe, oh-- I'm going to guess-- a few tens of microns, maybe 100 microns in that order. And so that the amount of kerf loss in that regard would be 100 microns over 15 centimeters, so a relatively insignificant fraction. If you're trying to chop up this using a laser, yes, then you would have significant losses. But since you're growing that ribbon straight out of the melt, the laser cuts themselves are a very small fraction of the total silicon. Yep? AUDIENCE: Can the sawdust be collected and remelted then? PROFESSOR: Wonderful question. Can the sawdust be collected and remelted again? There was a lot of work done to try to figure that out. At that point, the sawdust is mixed with this glycol-based slurry, and with the silicon carbide grit, and with fragments of iron coming from the stainless steel wire, and nickel and chromium and other impurities inside of the wire. And so a lot of the work was focused on separation of those different constituents, shall we say. And when the silicon prices were very high, maybe in 2007, 2008, when the spot prices were $500 a kilogram, there was a large incentive to use every single drop of silicon you had including separation. But in recent years, the incentive to do that has really dropped. And the one company I knew that had a very active slurry recycling program let it go. So there may be companies out there that are looking into it, but I'm not aware of their activities. OK. Let's hop forward into cells and devices. So now we've talked about the market shares of different technologies, feedstock refining, wafer fabrication, how we make these wonderful different pieces of silicon. Now we're going to talk about going from a wafer into a solar cell device. So just to situate ourselves, raw material, silicon feedstock, the module in the system over here. In the middle, we have the wafer to the cell. And this is the portion of discussion forthwith. Cell processing. Let's have a look at this. Again, it's a very different world now in a cell fab line then it was in the crystallisation environment. So in wafer fab, which means wafer fabrication and the section of the company dedicated to producing wafers and ingots, it was a little bit more dirty. You had forklifts moving these big crucibles around with chunks of silicon in it, operators coming by with garden hoses and washing down furnaces after they're finished. Here in the cell fab line, it looks almost more like a clean room. Almost, I say, because these folks aren't in full bunny suits. They're usually just with jackets with booties. Sometimes you see them with hair nets as well to protect from hair and other particulate matter from getting inside of the tools. But by and large, the wafers are brought in. And either in a series of inline processes-- this is a wafer, wafer, wafer, wafer. So there are four wafers across moving through what looks like an etch tank to do the texturization on the wafers. Whereas in wafer fab, it was pretty dirty. In cell fab, it looks pretty clean. You have a combination of these inline processes like this one shown here. We have wafers on conveyor belts moving through lines. And batch processes, where little robots [? pick in ?] places, line wafers up inside of crucibles or boats, and insert them into furnaces for batch processing. So this is the crystalline silicon cell fabrication. In on one side go bare wafers like this, and out the other side come fully processed solar cell devices. So the very first step after wafer sawing is the saw damage etch. After the sawing process, you have subsurface damage, something on the order of 5 to 10 microns deep beneath the wafer's surface. And keep in mind these are only about 170 microns thick. So you have subsurface damage that needs to be removed. And you can take advantage of the subsurface damage by etching it in such a way that you etch along the damage and form texturization. So it's a bit of a two-in-one here. You clean the wafer, you create your texturization, and you remove your saw damage so that when you lift your wafer, the wafer doesn't break because there's some hairline fracture caused by the silicon carbide grit. After you have your wafer-- so you start with your p-type wafer. And this represents the cross section of the wafer from the backside of the eventual cell to the front side of the eventual cell, about 170 microns thick. Wide would be something on the order of 15.6 centimeters in a real device. We're just looking at a small section of it here. So as we walk through the different steps of cell fab, we'll see them evolve over here. The first step after the saw damage etch is to do what's called an emitter diffusion, to create your p-n junction. Straight out of the box, the p-n junction is created after the saw damage etch. And typically, what we do is deposit a lower resistance or more highly doped-- that's why we have the 2 pluses here, that means very highly doped-- emitter right underneath where the eventual contact metalization will go. That's to reduce the contact resistance. That's to create the tunneling junction between the semiconductor and the metal. OK. So we have the high resistance emitter over here. This is representative of a shallow emitter. You remember in your quiz two you have this decision whether to take a shallow or deep. This architecture, which is used in industry, actually combines the best of both worlds. It has a shallow emitter over most of the solar cell device to improve the blue response, minimize Auger recombination. But it also has a deeper emitter right underneath the contact metalization to prevent shunting and to reduce contact resistance. AUDIENCE: I assume we have to choose one or the other. PROFESSOR: You have to choose one or the other unfortunately. To create this-- it's really to create the combination, what's called a selective emitter. It's an emitter because it's the charge separation portion of the device. But it's also selective in the sense that you selectively place these low resistance portions across in a geometric fashion underneath your eventual contact metalization. You have at the end of this diffusion process what's called a phosphorus silicate glass etch, PSG, Phosphorus Silicate Glass etch. After defusing in the phosphorus in the gaseous form, what you'll do-- or actually, you'll watch it being done, since it's happening inside of a furnace. This phosphorus-based gas will deposit a thin glassy layer on the surface of the sample, which then needs to be etched off or removed before you can do further processing. So that's what the phosphorus silicate glass etch is about. Then there's a nitride or a silicon nitride anti-reflection coating that's placed on the front surface. And as we calculated in lecture number two, this silicon nitride coating is only how thick? About? AUDIENCE: 70 nanometers. PROFESSOR: 70 nanometers, right? It's really, really thin. But yet, that's enough to create that quarter wave interference effect that leads to a very blue looking solar cell device. So the reason they looked blue is because of that anti-reflection coating. We are going to omit the ARC coating in our design for quiz number two. It requires silane gas, which we don't have access to down here in the laboratory. We'd have to go to either [? NTL ?] or Harvard CNS to get that deposited. So because we want this to be a hands-on experience, we don't to take the wafers out of your hands, do some magic off to the side, and bring them back and say, oh, here you go. Because your level of ownership in the process just plummets in order of magnitude in the process. We want you to be able to see it every step of the way. So we omit the anti-reflection coating in our quiz number two. But in commercial production, that's done. And people pay a lot of attention to that step. And finally, the metalization is deposited on the sample and fired. Now, the metalization, how is it deposited? We'll see in a few slides what the screen printing process looks like. And then, we'll actually do it ourselves. You'll press the button on the tool and deposit your metal on yourself. But the metalization is typically deposited onto the devices on the front side and on the back. The front side, you have to line up-- in commercial production-- line up with the low resistance portion of the emitter so that you are able to extract the full benefit from the selective emitter design and not shunt your device elsewhere. And on the back contact, this is typically aluminum. The aluminum, some of the aluminum will indiffuse into the silicon and create a p-plus region in the back side here, which is a minority carrier blockade layer. It pushes the electrons away from the back junction and toward the emitter. And so it prevents back surface recombination. So you see every single little step of the solar cell fabrication. A lot of smart people spend a lot of time thinking about, gee, how do I optimize two or three things at once? Question up there? AUDIENCE: So both the front side and backside metalization is [INAUDIBLE] PROFESSOR: No the front side metalization in this case-- thank you for that clarification-- the front side metalization in this case would be silver or silver-based paste. And in most commercial production, this silver-based paste includes metal oxides. It could be glassy frit. It could be lead oxide. It could be some combination of elements. That is able to etch through the silicon nitride anti-reflective coating. This is only 70 nanometers thick, but silicon nitride is a very strong material. It's a ceramic material. So you have to be able to etch through the silicon nitride and make electrical contact with the silicon underneath. And some of the earliest screen printed metalization cells that got in the range of 15% or 16% efficiency only made electrical contact about 10% of the silicon. But it was enough to have these percolation paths for current to flow up into the metalization. It's a miracle that it works at all. But it's a very effective, cheap manufacturing process that, nevertheless, is still being used in commercial production today, even among some of the highest efficiency cell architectures. And so for each of these different processing steps, somebody had to sit there and think deeply about optimization of different functions. The [? aluminium ?] on the backside, somebody had to think about, gee, how do I prevent the wafer from bowing, bowing too much, due to coefficient of thermal expansion mismatch between the aluminium and the silicon? Somebody had to think about, how do I create the right eutectic with the silicon-- the aluminum silicon eutectic is around 577 degrees Celsius-- so that you create a good ohmic contact on the backside? How do I diffuse in a certain amount of the aluminum to create this back surface field to prevent back surface recombination? How do I get the right back surface reflectance of the light coming off of here so that I have multiple optical bounces through my device and so forth? So a lot of optimization goes into making a solar cell device to get the Liebig's law of the minimum, to get each plank Liebig's law as high as you possibly can so you can achieve a high device performance. So hopefully, this walk through now, you can have an appreciation for the difficulty that some of your colleagues face at solar cell fabrication plants. Finally, as last steps-- I mean, this is a real miniature cross section in the lateral dimension right here. We only have two contact metalization fingers. If you look at this solar cell device right here, we have several dozen, right? If I were to make a vertical cross section through it, you'd see several dozen contact fingers. But this is just meant to be a caricature. So on the edge here, we have edge isolation. And what this is doing is preventing shunting pathways from going around to the back. So it's preventing the emitter from being able to make electrical contact to the backside of the device. And this is typically done by inserting a trench, a laser-based trench, just-- gosh, it must be on the order of half a millimeter from the edge. I'm going to pass this around, this solar cell device right here. And if you look very, very carefully, it's literally a few hundred microns from the edge at most. You may be able to see the edge isolation, the trench that is formed by the laser. But it's very difficult to see. So I'll pass this finished device around as well. And feel free to pick it up and look at the backside and the front side. On the back, you'll see some silver paths in the middle of all that aluminum. And if anybody has ever tried to solder to aluminum, you know exactly why those silver pads are there. It's so you can solder to them and make contact to the back of one device and contact it to the front of the next. And you'll notice that they're aligned, so the back pads are aligned with the front. So I'll pass this around right here. Yes, Ashley? AUDIENCE: I assume that in order to go in terms of where you want to put the edge isolation, do you want it as far out as possible so you're not losing that edge part. But you also need to make sure you're actually making a full [INAUDIBLE]. There's some optimization-- PROFESSOR: Exactly. AUDIENCE: [INAUDIBLE] PROFESSOR: Exactly. If your laser edge isolation machine isn't well-calibrated, you're losing area, active area, of your solar cell, hence your current output is going to be lower. Because you know your solar cell has a certain current density, a certain, say, milliamps per square centimeter. But then if your area, if you're square centimeter is smaller, because you're cutting too far away from the edge, you're throwing away good material. This isn't the trench all the way through. It's just electrical isolation. So essentially, this material over here still exists. It's still hanging on to the device, but it's electrically isolated. This trench here is only about a couple of microns deep. And you're losing area. This area over here is not contributing to the photocurrent of your device. Any electron making it up into the emitter over here will just stay there and recombine eventually. It won't be able to be pulled out of the device. A funny, but true story-- there was a company once that I worked with to solve a problem. And they were getting lower efficiencies in their new process. And they couldn't figure out for the life of them why they were getting lower efficiencies. They checked everything, everything, everything, everything. And it turned out that they were cutting their wafers to a slightly larger size than they were before. Actually, it was a slightly smaller size, because it was a lower efficiency. And their tester had embedded in it a fixed number for the area. It wasn't measuring the area of each wafer independently. It just had a fixed number for the area of the cell. And so it was dividing the total current output by a bigger area than what was actually there. And so it was "measuring" a lower efficiency than what actually the cell was outputting. So again, these geometric parameters can come up and bite you if you're so fixated on the electrical performance parameters. Testing and sorting. So after you create your device, you have this beautiful solar cell. And just from simple electrical engineering and maybe as an extreme example if you're stringing Christmas lights together, you know that if you have one bad apple, it can drag down the performance of the entire string, right, if you're connecting these in series. And so it makes sense to test each of the cells individually and make sure that you sort them together with their like cousins. So if you have high performance cells, you bin them all together. And you make models of the high performance cells. These will be high performance modules. The bad apples you put together with the bad apples and so forth. And that way you can extract the maximum value out of the product you've created. You take your good cells, you put them into a higher efficiency module. It looks exactly the same, but its producing more power, so you can sell that module at a higher price than you would, say, a lower power output module. OK. So that's what the test and sort is all about. Turnkey solar cell fabrication lines, very common since the mid 2000s. There are companies-- Centrotherm, gosh, [INAUDIBLE], Roth & Rau, others that were producing either turnkey equipment or even turnkey lines for the entire fabrication line. Even a local company, Spire, just up the road here in Massachusetts. These typically consisted of wafer inspection systems on the input side. You don't want to invest any money in a wafer that's ultimately going to break, so you want to be able to inspect your wafers coming in to make sure that they're high enough quality to be worthy of your cell investment. Next, you have wet processing to do the texturization. That's shown right there. Saw damage texturization. And these are typically inline tools with little ceramic rollers, some pretty nasty acids being use. Silicon is like a rock. And if you want to etch the rock, you need to have some pretty strong solutions, some very high or very low pH, very basic or very acidic respectively. And most of the time, in multi-crystalline silicon, we use an acidic solution. It textures the wafer independent of grain orientation. For the single crystal materials, we use a basic solution that is isotropic or anisotropic in nature. It creates nice little pyramids. So here, you see the wafers being drawn over an etch bath. And very small quantities of liquid are used per wafer in this arrangement. You just coat the wafer's surface, and that's about it. If you were to do it in a batch mode, you need a big bath like a bathtub. And you dunk your wafers inside of it. So that would be the bathtub. This would be the shower equivalent. So more water efficient. In this case, acid efficient. And then the cells come out on the other side and go into the emitter diffusion process. And these are a series of furnaces. We'll see one such furnace over the course of quiz two when we make our solar cells. So this is the phosphorus diffusion furnace right here. The wafers are typically loaded into boats and then inserted into furnace where phosphorus containing gas, POCL3, also called "pah-cul," is flown into the chamber. The chlorine components and the oxygen dissociate from the phosphorus, which is then driven into the wafer. The oxygen reacts with the silicon, creates that phosphorus silicate glass on the surface. And the phosphorus is driven into the solar cell creating the p-n junction, creating your device. And here's an example of Czochralski wafers being loaded into the phosphorus diffusion furnace and then out again, just showing the degree of automation of some of these furnaces. This showing a stack not dissimilar from the one in the laboratory downstairs in building 35 where we'll be doing our phosphorus diffusion. So the next-- after we have our p-n junction-- the next step would be to create the anti-reflection coating. And this is done by a process called Plasma Enhanced Chemical Vapor Deposition, or PECVD for short. And in the PECVD process, you flow in silane gas and ammonia. Silane is silicon with a bunch of hydrogens, four of them. And ammonia is nitrogen with a bunch of hydrogens. And the nitrogen and the silicon react on the wafer's surface and create the silicon nitride coating. The hydrogens, 90% of it, evaporates off. But about 10% of it hangs around. Between 1% and 10% go into the wafer or stay at the interface there and eventually are driven into the wafer passivating bulk defects. So again, a multitude of different things going on at the same time. Eventually, the visible effect is that you've created your anti-reflection coating. The wafers go in looking shiny and come out looking blue. But what's happening underneath the surface is that some of the hydrogen is going into the wafer. Hydrogen, the first element on the periodic table, very tiny. And in the PECVD process, where you have a plasma, you have hydrogen ions, which basically means you have a proton without its electron. And that proton is very fast, moving through the lattice. There's lots of space for it to move through the silicon lattice. And it's also very reactive because it doesn't have that electron. So whenever it finds a defect or a dangling bond, it'll usually lodge itself there, and attach itself, and passivate that defect. And that's what hydrogen passivation is all about during the silicon nitride anti-reflective coating deposition. These are examples of inline processes for doing an anti-reflection coating. I believe there are a few different variants of this inline process, one of which is a sputtering mechanism to deposit this anti-reflective coating. Of course, then during sputtering process, you have to worry, as well, about hydrogen. Do you have the benefit of hydrogen passivation? Perhaps not as much, so additional engineering is needed. But the inline process could be potentially faster and higher throughput than the batch process using the PECVD. So again, manufacturing trade-offs. Next, we have the printing line and screen printing. So this looks very similar to screen printing for a t-shirt. Here is a t-shirt being loaded into a screen printer. And here's a solar cell being loaded into screen printer. This is a close up of the screen, of what the screen actually looks like. Here, the screen, which is comprised of this mesh of metal-- here the screen is bare. And so the metal that's deposited on top can go through those holes in the screen and onto the wafer underneath it. And here, there's a coating, a polymer coating of the screen, which prevents the metal from going through the screen at those places. So it shades the solar cell underneath and prevents metals from being deposited there. And you have fingers and busbars. And those are eventually the thin little fingers that you see right here going sideways and the vertical busbars that you see going vertically right here. Question? AUDIENCE: Yeah. So when you do these [INAUDIBLE] emitter [INAUDIBLE] where you have some areas of high resistance emitters and others of low resistance-- PROFESSOR: Yeah. AUDIENCE: Do you use a screen to shield it. Or do you [INAUDIBLE]. PROFESSOR: Great question. So some of the earliest designs for the selective emitter--if we go back all the way up to here. Yeah. To the selective emitter portion. So the earliest designs used photoresist process. But I would say nowadays, there are a few technology options that are much faster, one of which involves a creation of porous silicon on certain regions of the wafer that you want to etch back and create the shallow emitter leaving the deeper region intact. And that, you could use a form of shading. You could use a wax even for it. There are a variety of technology options for achieving that goal. But in a sense, many of the selective emitter designs involve a deep diffusion first and then a partial etch back. For example, creation of porous silicon and etching that material away. Ashley, you had a question? AUDIENCE: Oh, yeah. So what does "turnkey" refer to? PROFESSOR: Turnkey. Excellent question. So turnkey manufacturing line-- what it refers to is that I'm the vendor of the equipment. In one case, I say, here, Ashley. Here's a piece of equipment. It's going to cost you $1 million. And good luck getting it set up and running. I'm out of here. I'll see you. AUDIENCE: Right. PROFESSOR: A smarter company might come along and say, I'm going to guarantee an output from my piece of equipment. I'm going to guarantee that you'll be able to make 16.7% solar cells, 16.7% efficiency. I will send my engineers to your factory, and they will help you get the equipment set up and running. And once it's running up to spec, then they'll come back home, and you'll be on your own. And you'll be able to optimize it further. And so you walk in knowing that you have this guarantee of a performance. Then you can go to your financing agency. You can go to Joe and say, hey, Joe, give me money for my new factory. I have a guarantee that I'm going to hit 16.7% and have a pathway to get to 17.2. My CTO right here thinks-- she's a really small person, and she has a pathway to get to another 0.5% out of it. And so you can go to your financier and get money more easily than, say, in the first scenario where you're given a piece of equipment and then the person high tails it out of there. AUDIENCE: Right. PROFESSOR: So turnkey refers to the idea that you turn the line on. You essentially turn the key, and you're getting high performance out. In reality, it takes a month or two to ramp up to that point. AUDIENCE: Right. PROFESSOR: To get high yields and to get high performance. But you have the support of the company there on the ground helping you achieve that. And the turnkey lines were actually one of the real reasons why technology flowed around the planet so quickly. Because up until about the mid 2000s, high efficiency cell was limited to a few laboratories and a few companies in the know. But once turnkey equipment manufacturers got into to the mix, they started creating these turnkey lines and selling the equipment around the world and the expertise of how to make high efficiency devices, both the architecture and the processing know-how. And this is how, within in the last 5 to 10 years, you've seen such an explosion of companies around the globe in all sorts of places that traditionally haven't been experts in solar cell manufacturing suddenly knowing how to manufacture solar cells. It flows. The know-how flows through the equipment vendors. So finally, testing and sorting. This is the last stage of the solar cell manufacturing process. Here, we see a little pick-and-place. That means a little robot that picks up wafers and deposits them. The simplest incarnation is just suction cup. The more fancy ones involve Bernoulli lifters, essentially pressure differentials pulling wafers up. So you have wafers being loaded onto a conveyor belt, coming off of one conveyor belt onto another one. And they're moving forward. And what you see right here in very low resolution are two probes coming down. This, evidently, is a two busbar cell, not a three busbar cell like this one. The probes come down and make contact with the busbars. And the probes have multiple contact points, so the series resistance along the busbars is not affecting your measurement. Cell efficiency measurement is always tricky because depending where you put your probes, your measurements are going to change because of the series resistance. So these probes right here are long, and they contain multiple contact points. And they're essentially touching the busbars. And light flashes onto the device simulating the sun, so simulating AM 1.5 conditions. And an IV curve is measured, is swept. I can't really tell from the photograph or from the movie right here whether the IV curve is being swept at illumination, meaning you're sweeping your voltage when the cell is illuminated, or whether the illumination intensity itself is used to vary the forward bias condition of the cell. They could be doing it in one of two ways. But most likely, what they're doing is they're flashing the lights, measuring the IV characteristic of the device, and then sorting the cell based on that performance. It goes into a computer. Efficiency is calculated, just like you did on your homework. And just like that, it's calculated. And then, as the cell moves down the line, the robot knows, oh, that's the cell that got 16.6. We put it over here. Oh, that next cell got 16.8. We put it over there. Some additional companies sort their cells based on color because they want to have the aesthetic appearance of homogeneity within the module. They want every cell to be of uniform aesthetic value inside of a module so that you have a nice, uniform color. AUDIENCE: Is that considered [INAUDIBLE]. PROFESSOR: Whether or not this module right here is considered uniform or different would depend on you, Jessica. You're the customer, and you decide whether this is good enough for you or whether it's not. AUDIENCE: It's not. PROFESSOR: It's not? All right. Well, then we have to work harder. So the customer requirements really drive the industry. So some customers are more discerning. Obviously, if this is going to large field installation, we have big barbed wire around it. Who cares as long as the module's producing high performance? But if it's sitting on the facade of the train station in downtown Freiburg, Germany, where every single person riding the train, entering the station, sees the modules lining the side of Deutsche Bahn's headquarters, you want to make sure that those look nice. So there are differences depending on where they go and where they're installed. High efficiency cell architectures. So there are a plethora of different architectures out there. There are some that, for example, put all their contacts on the backside, so there's no shading. And these are interdigitated positive, negative, positive, negative, positive, negative contacts here. So this is called an interdigitated back contact structure. It's used by the company called Sun Power. And so there's no metalization loss on the front side. All your contacts are on the back. Because lateral carrier diffusion is involved, meaning the carriers have to diffuse laterally, they don't have to diffuse only one dimensionally, you probably can't use PC1D to model this cell. You'll have to use a two-dimensional device simulation like Sentaurus. If anybody has any two-dimensional device simulation questions, Ashley right here in the front is our resident expert, so you're welcome to ask her. AUDIENCE: [INAUDIBLE] PROFESSOR: Yeah. And then there are also other device architectures which we'll get to during our thin films discussions. A couple of ancillary topics, barriers to scale. This is the size of a 1 gigawatt peak plant manufacturing facility for wafers, cells, and modules. This is a palm tree right here. These are roads. So you get a sense of scale. This is located in Singapore. It's a company called REC that has this factory. These are 18-wheelers right here that are taking the materials out and selling them to customers. So you get a sense of the scale of these facilities. They're rather big. And if you say, OK, this is a gigawatt fab, but we need to be producing on the scale of terawatts, which are three orders of magnitude larger in area than this, how big is that factory going to be? It's about half of the state of Rhode Island. Granted, it'll be distributed throughout many different regions, but it's a big, big factory. So one of the interesting questions is, can we produce the silicon in a faster way that involves less area? Because area generally relates to capital equipment costs, not always, but quite typically. If you have a larger area because you need more equipment in there, for more equipment, it's a higher cost. So can the production costs be reduced by a higher throughput growth mechanisms? So instead of using thin film or crystalline technologies that are currently being used today-- apologies for that. Instead, if we used, let's say, a float glass-like process. So these would be extruded pieces of silicon on some bed of--I don't know-- liquid tin would be for float glass, an equivalent for silicon. You could reduce the area by about two orders of magnitude. And if you envision instead these high speed printers that print out your reports for your exam or class notes, they're spitting out 55 pages per minute on 8 and 1/2 by 11 inch squared sheets. If instead those were 15% solar cells being printed, you could envision an area the size of five football fields instead. So this starts opening the mind that, wow, our way of manufacturing these solar cells, this discrete process where it's very segregated-- wafer, cell, and module. Wafer manufacturing almost like a commodity. Ingot of aluminum. The cell like a device. The module-- as we'll see in a second-- like an automobile, an assembly process. If instead we managed to blend these processes together and reduce the barriers, the discrete barriers between these different processes and reinvent the manufacturing process thereof, we stand to make this a lot cheaper, and a lot faster, and a lot smaller to produce. We might even have our own solar cell manufacturing equipment mounted on our desk. When we need to print a solar cell device or power something, we can produce it right there. So that's kind of the vision of the future where this might be going and why bright minds like yourselves are needed. We talked about silver. We know there's a limit for how much silver can be used in the front contact metalization. We're using about 10% of it right now. And if you're looking for environmental impact of crystalline silicon technologies, I've included many different sites right here that talk about the environmental impact of solar cell manufacturing since we have mentioned acids. We have mentioned gases like silane. We've mentioned CO2 production when we produce the wafers. We'll talk about this later on in the class, but in essence, we're looking at around 1/10 or 1/20 the CO2 intensity of coal. So it's not a zero-emission source to produce that module, but it certainly is a lot less than, say, our fossil fuel sources. This declining US market share has really captured the attention of politicians lately, the fact that the US used to comprise 75% of the PV production market. This is to produce and manufacture the modules. And today, it's on the order 5%. This is a risen concern within many in the DOE and today's DOE and government. Meanwhile, the market is growing substantially. And so an open question is, what is the future of US market share? If all goes well, we should have a small Greentech Media article published on this topic probably within about a week or so, so keep your eyes open. And let me briefly jump into module manufacturing. Do we have a question? Oh, we're all set. OK. I'm going to hop into module manufacturing. It'll be the last five minutes. Just to show you how you go from the cell to the module, it's an assembly process, very, very straightforward. We have coming in here sheets of glass, encapsulate materials. And we'll be able to see this up close and personal and feel the materials when we go visit Fraunhofer CSE in the first week of November. We have a field trip going up there. That'll be a lot of fun. And the encapsulants are a lot of fun. They're polymers. They're really tough. You can take the Tedlar back skin, this white stuff here in the back of the device, that white skin right there. That's called Tedlar. As the name would suggest, it comes from DuPont. It's a polymer. Really, really tough. If try to take some in your hand and try to tear it, it's nearly impossible, even for the strongest people here. So it's impermeable, very strong material. The ethyl vinyl acetate, or EVA, is a polymer that infuses the glass in the front side with the cell. And with the Tedlar in the back, it kind of forms this sticky, mushy material when you heat it up above 150 degrees C. And it binds everything together in what's called a laminate. So let's walk through that real quick. To get to the point of a module, we need to take our good apples with our good apples or our bad apples with our bad apples, essentially the like-binned cells, and start stringing them together. That means contacting the front side with the backside of adjacent cells. So the front of one cell is connected to the back of the next. The front of that one is connected to the back to the next, and so forth. And they're connected in series in a big, long string. And that's done at this tabbing, stringing, and layup table. Typically, this is done by an automated solder system. I just put the cells together, and it wires them for you. But usually, there's a manual inspection process afterward because sometimes the soldering isn't perfect. A human being is typically there fidgeting through, making sure that everything is primo. Then we have the lamination process, which takes those strings. They're very fragile at this point. They're just solar cells connected with some solder-coated wire, so they're very fragile at that point. And these are then laid up on the top of sheets of the encapsulant materials and the glass and eventually laminated together to form that nice package. So at the lamination stage, coming out of the lamination, we'd have the glass on one side, the Tedlar in the the back, and the cells in between encapsulated by the ethyl vinyl acetate, the EVA. And we wouldn't have the frame yet around it. And so the put that frame, we would need essentially a large machine that takes those pieces of extruded aluminum and pushes them together around the edges of the laminate, fixing them on there. And this is the examples of the tabbing and stringing right there. And let's see, OK. So the framing materials right here are typically done by machines in places with high labor costs. And they're done by human beings pushing them together at regions of low labor cost. And finally, the junction box is deposited at the end. And the junction box, what it does is it collects the power outputs from each of the cells and very conveniently gives you two leads. So there could be some power electronics inside of here that allows the current to flow around this module if this module's under performing, if it's broken, or if it's shaded. There would be a bypass diode inside of the junction box that allows the power to flow around the module and not get sunk into it. And it also works to collect the power outputs from all the cells and produces two leads, which can then be conveniently plugged into either adjacent modules, which would be strung in series, or into an inverter, which would then take the DC power here and convert it into AC power for your consumption. And that is how a solar cell is made. So I welcome you to spend a few minutes at the very end to come up and take a close look at some of these materials. Ask some further questions. And on Thursday, we'll start diving into thin film technologies and talk about how those are made as well. |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | 3_Light_Absorption_and_Optical_Losses.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. TONIO BUONASSISI: Today we're going to talk-- or it's the first technical discussion of the actual solar cell device itself. We talked last class about the sun and about the nature of the solar resource. Today we're going to be talking about the interaction of light with matter. In particular, we're focusing on light absorption. This lecture could alternatively be called "Light Not Getting Absorbed" or "Optical Losses." Both are important, and both are related, as we'll see. So this is part of the fundamentals of the course. Just to situate ourselves, we're here right now in the fundamentals, the first third of course. Then we'll talk about the technologies and the cross-cutting themes. And what we're going to talk about is extremely important because it allows us to understand the technologies. Once we begin discussing them and we discuss cost trade offs of implementing this particular technique for the way for it to absorb more light, we can appreciate how much we can quantify the impact of that technology development, and we could also later on ascribe a cost to it, to determine the total cost benefit analysis. So conversion efficiency is really what dictates the performance of the device, the solar cell device. It's how the solar cell device converts sunlight, the input energy, to some usable output energy, which is in the form of electricity, typically, from a solar panel. so the electricity coming out of these leads, for instance, right here. And that conversion efficiency, that simple equation, for most solar cells, can break down into the following. You have inputs. Sorry for the small font here. This reads solar spectrum. That's your input. Your output, which is the charge collection, it's a collective charge coming out of your device, and a bunch of steps in between. So from the solar spectrum, we have to absorb that light, then we have to excite charge within the material. Then that charge has to move around inside the material to get to the metallic context in the front side. Charge separation has to occur for there to be a voltage. And finally, the charge collection process. And so the total efficiency of this device is the product of each of these individual processes. And so if you're making a solar cell device, and I know about a third of you are based on your background surveys, this diagram right here will ring true to you. It's Liebig's Law of the Minimum. What this is representing is a barrel that has water being dripped into it. And the water will flow out of whatever piece of wood is the shortest. And in the case of a solar cell device, you can ascribe a certain name to each of these pieces of wood. We'll learn what each of those are with time. But one of the big ones is optical losses. And the optical losses tend to be rather severe on some of our lab scale cells. So one of the easiest ways of boosting efficiency is simply to take care of your optical losses and to minimize the amount of light reflected or not absorbed into maximizing amount of life that's actually absorbed. And so to do that, there are a number of standard techniques and some cutting edge research areas. And I'll attempt to give you a broad overview and survey of both, assuming, of course, you've done your background reading. So the learning objectives, the first is to be able to calculate the reflectance in non-absorption optical losses of a solar cell. So this is essentially all the light that's not absorbed. We want to be able to calculate that. The second is to describe the physical underpinnings and the implementations of four to five-- there are five here. I added one at the end. Four to five advanced methods of reducing optical losses. So there are technologies, techniques that we've used that we've developed over time that we can use to minimize the optical losses, to minimize the amount of light reflected or not absorbed inside of a solar cell device. So to think of this pictorially, we can come up with the following diagram, where we have some incident energy, in this case incident light. Here's our medium. Here's the amount of light that gets absorbed. Here's the amount of light that gets transmitted right through that does not get absorbed within the material upon passing through it. And there's a certain amount of light that just gets reflected off the front of your solar cell device. We want to, obviously, maximize this part right here. So to begin, we give a quick review of light, the nature of light. This is going back to the particle wave duality of light. It will be useful alternatively to think about light as a particle, quant of light, or to think about light as a wave, depending on what light management technique we're going to be describing. And in particular, I'd like to just highlight these equations over here. The notion that one can define the energy of a photon coming in, and that photon has a certain wavelength, a certain frequency, a certain wave length associated with it-- frequency and wavelength-- related, of course, by the speed of light, Planck's constant, and so forth. So just to situate ourselves with broad numbers, so when we dive in and talk about spatial dimensions in relation to the wavelength of the light, we're in a situation where we can actually have a horse sense, a common sense, about it. The visible photon wavelengths are usually in the hundreds of nanometers. And the solar spectrum peaks somewhere around 550, just good numbers to have in mind. So this was that solar spectrum, the integrated solar radiance versus wavelength. And the second point that is equally valid, we can describe the wavelengths of the incoming light, wavelengths of the incoming light lambda, or we can describe the energies of the incoming light, this E sub ph, the energy of the photons. So just to situate ourselves again, the visible photon energies are typically in a range of 0.6 to 6 eV, electron volts, again, with the peak of the solar spectrum at 550 nanometers, somewhere around 2.3 eV. Good. So a simple thing to keep in mind, for those high energy particle physicists in the room, that when we're talking about visible light, we're interacting with a very specific type of electron inside of our system. It's the valence electrons. These are the electrons that are typically most loosely bound inside of a system or I would say in the outer shells of the atoms within the material. You're typically not interacting with core shell electrons with visible light. For that, you need x-rays. So this is just something to keep in mind. When we start looking at the wavelength dependence of absorption inside of a material, you can have, for example, in the visible range, a decreasing depth of penetration of the light with increasing energy, whereas with x-rays, it's the exact opposite. It's because you're dealing with different types of electrons and the material. So just to situate ourselves, I know we have a fair number physicists and chemists in the room. That's a message geared toward them. Let's describe how light interacts with matter. And first off, come up with a few variables. Define a few units that will make it easier for us to understand how light is interacting with matter. And so here what I've done for you is placed the equation that describes the complex index of refraction of a material. What this means, effectively, you can think about this refractive index of the material as being comprised of two different components. For now, it's going to be fairly cerebral, but I'm going to reduce it to practicing in a couple of slides. The real component of the refractive index-- and the refractive index is material-specific property. So if I have, for example, silicon or if I have silicon nitride or if I have a particular type of glass, it'll have a particular refractive index. It's comprised of a real component which indicates the phase velocity inside of the material and an imaginary component, which can be thought of as an extinction coefficient. And it is related to the attenuation of the light intensity as it travels through that material. The measurements for those who have already taken measurements before on a spectroscopic ellipsometer, this is how you measure that parameter up there. We don't have to dive too deeply into that for the purposes of the class. It's just for background. Why these values are important-- these values here describe the interaction of light inside of a medium, inside of a material. And we use that information to calculate engineering relevant parameters such as reflectance of light off of a surface. So if we want to calculate what is the reflectance of light off of the silicon right here, I can calculate it by knowing these properties right here, by knowing the real and imaginary components of the refractive index of silicon, in this case. And the reason that's important is because we want to minimize reflection off of surfaces. So I've come up with the first equation right here which is describing the reflectance from air to a solid, in this case, from air where the refractive index is 1 to a solid, namely, say for example, silicon right here or glass, which has a finite refractive index typically greater than 1. And so I have an equation here that describes the reflectance. Let me dive a little deeper into it and try to understand what exactly that equation is telling me. So from the folks who have studied mechanics, many of you are mechanical engineers in the room, you may recall studying a problem wherein you have two springs that are connected. They have different spring constants, different stiffnesses, shall we say. And you excite a wave over here. It travels down. And when it reaches the interface between the two, part of the wave is reflected back and part continues through. The speed of the wave is changing as it goes from one spring to the other, because the stiffness is changing of the springs. And the amount reflected can be described by this equation right here, which looks awfully like the equation right above it, which is describing the amount of light reflected off of an interface. And in reality, those ends have a very similar meaning, the n and the z. The n, in the case of light, which is the real components of the refractive index. Mind you, this parameter right here, this indicates phase velocity in material. It could also be thought of very loosely as the ability of an electromagnetic wave coming into material to slosh those electrons around. Not exactly a stiffness coefficient, but it bears some rough resemblance. So this is a method for you to gain a foothold in this new area of understanding the refractive index of a material based on something you've already seen before. So I would advise taking this analogy as far as it will go until it breaks down. Push it as far as it goes until it breaks down. And you'll see at some point it actually does, but it's a useful place to start. So I'm going to ask you a couple of questions. This might be rather new for a lot of folks. But the purpose of asking these questions is to get you thinking. And eventually we'll get to a point of heightened understanding as a result. Tinted windows. So if you have a tinted window, what is typically happening at that tinted window? Why can't you see inside? What would you imagine is going on? So let me go back to this reflectance equation right here, this one. How would you modify a reflectance off of a window, let's say? And let's drop the k's for now. Let's leave those aside and just focus on this parameter right here, n minus 1 quantity squared n plus 1 quantity squared. What would increase the reflectance off of that window, if I have a larger n or a smaller n? If I have a bigger n, I would get bigger reflectance. Is that right? r goes up? Well, you'd have to plot it out, I guess. So if I change the refractive index of the material that I am working with, I can change the reflectivity off of that interface, off of that surface. So if I add a coating, for instance, to a window that increases the reflectivity, then the amount of light that is able to escape from the inside to my eyes decreases. Now, with normal incident light, there is a beautiful symmetry involved. That is, the amount reflected off of one side is equal to the amount of light reflected off the other side. So just the same way that I'm losing the ability to see inside, the folks inside are also losing the ability to see out. But they can still see out. Why is that? Why is it that with the same reflectivity they're able to see outside and I'm not able to see in through that tinted window, through that car, for example, that's driving by with the tinted glass? Why can't I see insight but they can see out, what's going on? AUDIENCE: [INAUDIBLE]. TONIO BUONASSISI: Yeah, I hear somebody. AUDIENCE: The light on the outside is much stronger in terms of an absolute amount of light being reflected. TONIO BUONASSISI: Yep, exactly. So yes, the reflectivity as a percentage is the same for both parties. But the amount of light, the magnitude of the light from the outside, is much, much greater than it is on the inside. Can anybody give me just a gut sense. If I'm outside on a sunny day, how much brighter is it outside versus inside right here? Factor of? AUDIENCE: 100, maybe? TONIO BUONASSISI: Maybe a factor of 10, somewhere in that range. And so when you walk outside on a sunny day, you'll notice your eyes adjusting a little bit. It'll take a minute. And when we walk back inside, it will take a minute here for your eyes to adjust as well. That's because of the difference in intensities. So if you imagine being outside of that car and having 10 times the amount of light being reflected, that small amount of light that is actually transmitting through the window from the car to the outside world will be washed out by the amount of reflected light. Whereas if you're inside the car, there's a lot of light coming through that window, even though a lot of it gets reflected, there's still a sizable amount coming through. And the amount of light that gets reflected off that window of the internal light is small in relation to the outside light that is being transmitted through that window. So it's important to think about these processes, both in terms of their reflectance as a percentage but also the magnitudes of the light involved. What if that glass pane was flipped around? Would it change anything? If I took that glass and just flipped it, would it change it? What about the symmetry argument, that the amount of light is reflected, the r reflectance, is the same from both sides? AUDIENCE: Is it not the case that there's a coating on the outside? So if the change in refracted index is an abrupt change from the outside looking at this coating. Because on the inside, you're going through some median glass, which is more index matched than the outside. TONIO BUONASSISI: Yeah, so I would advise you to actually walk through that calculation. And what you'll find is it winds up being the same. And it's because you have to take all reflectances off of all these interfaces into account. There are, in fact, three interfaces-- the air, the glass; the glass, the anti-reflection coding; and the anti-reflection coding, the outside. This, of course, without getting into quarter wave effects, which we'll get into a minute, there's some higher order effects that deal with phase change, which we haven't discussed right now. We're just assuming that all of these layers are well above the wavelength of the light in terms of thickness and that these equations, these linear equations, are valid. Very good. So this is just to get us situated with this new concept of reflectance-- and again, very powerful equation. Keep in mind that this is a very specific form of the reflectance from an air into a solid. If you're going from a solid into a solid, you'll add your n1 and your n2, depending on what material going into and what material are coming from. So we're happy to walk through that perhaps during recitation. OK, so what we're going to do now is we've talked about reflectance off of surfaces. What I'd like to do is talk about a light absorption inside of a material. So let's imagine that through the techniques that we're going to be discussing later on in lecture, we manage to minimize the amount of reflectance off the front surface. And now the light that's incident on the material is actually going to go inside and get absorbed by the material inside. We need to be able to understand how light gets absorbed inside of matter. And for that, we apply a very simple formulation inside of this class, which is called Beer-Lambert's Law, which is a very simple yet very powerful formulation that describes not only the interaction of light with the solar cell material but also light through the atmosphere, light the water, many other forms of optical absorption. And for that, I'd like to call Joe up for a quick demo that he put together that will allow us to actually plot out Beer-Lambert's Law. And I'd like to start with what I would think of as a simple hypothesis. What we're going to be doing, and Joe will explain this a minute, what we're going to be doing is taking many sheets of material. This is just some polyethylene material, a little bit discolored. And we're going to shine a laser down on to this photodiode. The photodiode current will be measured by this little current meter right here. And we'll be inserting these panes of plastic in the middle. And as we increase the thickness of the plastic, applying good pressure in between to minimize the reflectance, the air gap, for instance-- as we increase the thickness of the polyethylene, we will plot the total transmitted light as measured by that photodiode. And so I'm going to come up with a hypothesis of what's going to happen. I'm going to say that if we double the thickness of the polyethylene that we're going to halve the amount of light going through. And if we triple, we're going to reduce it by a third. And if we quadruple, we're going to reduce it by a fourth. And let's see if the hypothesis is correct. It's not. But we're going to test it. And it's a logical thing you might assume. And then we'll walk through a derivation that will correct our missed logic. So go ahead, Joe. Take it away. JOE: Sure, so if you guys want to play along, that's fine too. I know there's lines in the side of your notes. You can make little graph paper, and it comes out looking really nice. So basically what we have is a laser pointed and a photodiode. And the current out of this photodiode is directly proportional to the light hitting that photodiode. And it has a quant efficiency, which we're going to learn what that is in a few lectures, of about 60%. So of the photons hitting it, you'll get a certain number of electrons out, and that ratio's 60%. And so first of all, we're going to see what it's like, what the power of our-- yeah. So right now we're getting about 1.32 milliamps. So Tonio's going to plot that. Then as we keep increasing and put one layer of polyethylene, that drops to 0.75. TONIO BUONASSISI: So before we go onto the next one, what do people predict the next dot is going to drop the total intensity to? Is it going to be kind of a linear line like that? You'd expect it, right, because you're doubling it. So you'd expect the intensity to drop by another factor of 2. Why not? Where am I getting a mistake here? Somebody says exponential. There's kind of this sense that it should be exponential. What don't we add some more filter in front, and we'll see what exactly this comes out to be. JOE: This is with two. TONIO BUONASSISI: Two. JOE: Now we get 0.43. TONIO BUONASSISI: 0.43. OK. All right. Why don't we do one more just to see what sort of trend we're getting. Still 0.26. JOE: 0.26. TONIO BUONASSISI: 0.26. Ah, wow. OK, so it didn't go in a straight line. It's actually starting to curve down. Cool. OK. JOE: And we keep going, 0.16. TONIO BUONASSISI: 0.16 JOE: 0.10 TONIO BUONASSISI: 0.10. OK. Look at that. What sort of curve is it? Exponential. It looks like one at least. And we can test whether or not the hypothesis is correct by an exponential fit, which happens to match pretty well. So-- JOE: Now one other quick thing you notice is that if you look at the fit, the first point's a little bit higher than that fit. Anyone have an idea of why that might be the case? What are we ignoring in this experiment? AUDIENCE: The reflection is [INAUDIBLE]. JOE: The reflections, yeah. So in the first one, you reflect light, and certain amount gets transmitted through that front surface than absorbs. And so right now we're ignoring this is 1 minus r component. But it's so small that it really doesn't matter for this experiment. These things don't reflect a lot of light. TONIO BUONASSISI: Cool. Well, why don't we give a quick rondo. [APPLAUSE] Well done. Can I grab one of those? JOE: Absolutely. TONIO BUONASSISI: This is going to be important for the immersion scattering demo. JOE: Oh, sure. TONIO BUONASSISI: Yep. Cool. OK, so we notice that we have some exponential character to be decay of the intensity of the transmitted light through a medium. And the amount that's absorbed is following another trend, which is just 1 minus that. So it's the amount of light that's absorbed is following a curve looks something like that. OK, so let's look through the formalism of Beer-Lambert Law and try to understand why it is that we come up with that exponential function right here. So if we assume that light is coming in a medium and light is decaying in some function to that medium and a certain amount of light is transmitted, we know, of course, from our little experiment that it follows some exponential function. But how do we justify that to ourselves? Well, first off, we're going to ignore reflections off the front surface. We just talked about them. We can calculate them. Let's leave that aside for now as a parallel calculation. We're just concerning ourselves with what's happening inside of the medium. So if we assume that the change of intensity within that medium in each little delta thickness is going to be affected by some sort of scattering intensity within the medium-- and this sigma here can refer to a variety of processes. That can refer to absorption events that result in the generation of free charge. They can refer to absorption events that just heat the material up and generate phonons, so lattice vibrations. There are a number of processes embedded in the sigma, and that's why this formalism is so powerful, because it doesn't care really what the physical nature of that sigma is. It just matters that there is an absorption per unit distance thickness traveled inside of the material that is constant throughout the entire material. So the sigma here is independent of thickness throughout. And then as you integrate through, you wind up with that beautiful exponential function at the end, the sigma l times n. We collapse the n and the sigma here into an alpha. That alpha is an absorption coefficient. The l is the total length or the total thickness of this medium right here. So if we increase the total thickness, we're going to decrease the total amount of light coming through via that exponential function. The alpha, on the other hand, is not a geometric parameter. It's an intrinsic material parameter. To put that in terms of mechanical engineering, for many of the mechanical engineers in the room, you recall from solid mechanics, 2001, that you have geometric parameters that determine, say for example, structural response and intrinsic material parameters like Young's modulus that determine the structural response of a system. And likewise in here, in the optical, shall we say, response, we have a fundamental intrinsic material parameter, r alpha, the absorption coefficient, and the geometric parameter, rl, which is the thickness. And the beauty of this formalism right here is that we can measure, experimentally just like we did right there, our alphas for materials. And so from an engineering point of view, we don't really-- to first order, it doesn't really matter what sort of scattering or absorption process is happening inside of a material for us to calculate the amount of transmitted light. We just need to know the alpha. We need to know the optical absorption coefficient. This alpha will vary as a function of wavelength inside of a material because, obviously, the physical absorption mechanisms are varying as a function of wavelength. The resonances with different electronic states within the material, that light, depends on the energy of the light, depends on the frequency. So there's a wavelength dependence. Yeah, and that general equation is the same one that drives the reduction of light intensity as it travels through the atmosphere. So if we increase the atmospheric path length, we'll be reducing the amount of light that actually reaches the surface of the earth. That's at air mass two or air mass three, there's less solar flux coming down than at air mass one or air mass zero. The alpha, obviously, is going to be very different for our atmosphere than it was for these little polyethylene sheets. Because the nature of the scattering and absorption processes are very different for the atmosphere than it is for here, the density of the material and so forth. Any questions? Yes? AUDIENCE: What was n? TONIO BUONASSISI: So the n, there's a certain scattering intensity, and then there's a certain number density, for example, of the material. So this alpha here is, I would say, density neutral. What we've done is we have the alpha encapsulating the physical parameters of the material and the absorption processes all in one variable, very nicely and succinctly. And the only geometric parameter that is of essence is really our l. AUDIENCE: It's called an absorption coefficient, but is it more of an extension coefficient, really? Because it's kind of confusing that it includes scattering. TONIO BUONASSISI: The extension coefficient, absorption coefficient, yes, in solar research, when we talk about an absorption coefficient inside of a material. Oftentimes we're operating in a wavelength regime of light wherein free charge is excited. But we can also keep increasing that the wavelength of light, say, out to 10 microns, very long wavelength light, very low energy light. And that can excite free carriers within the material-- carriers that are already excited, essentially excited them further, without generating any new free carriers inside of our material. So we won't necessarily be generating more current by shedding light on it but will be absorbing light, nevertheless, in our material. So it's important to keep, let's say, the underlying physical processes that are occurring distinct. Later on we'll get to that. For now, it's important just to, I would say, recognize that we have an exponential decay of the intensity of the light as it goes through the medium. And then over the next few classes, we're going to get to exactly what physical processes are going on. But I'm glad people are asking those questions. OK, so again, alpha is a function of the wavelength of light and the property of the medium. And let me just flash up some curves of alpha versus wavelength so people have some exposure to those numbers. Again, we're talking about an energy range quite broad here, from about 6.2 eV to 0.62 eV. The visible wavelengths range would be somewhere in this regime right here, so a very limited band. And the infrared out here, ultraviolet over here, and we can see for a variety of different types of materials what the absorption coefficient is. So here we have germanium. The red would be crystalline silicon, gallium arsenide, indium phosphide, and amorphous silicon. So let's do a little quick calculation just to get us a little limber. We're starting to get into the semester, so the energy level starts going down. What we're going to do is we're going to pick a value, say 550 nanometers. Why did I pick 550 again? It's near the peak of the solar spectrum, right? It matters. And we're going to look at two different materials. We're going to look at silicon, and we're going to look at gallium arsenide. And we're going to calculate the thickness necessary to absorb 90% of the incoming light at 550 nanometers. What I want you to do is turn to your neighbor, and once again with your neighbor, calculate what thickness of material, what thickness of gallium arsenide, the yellow curve, and what thickness of silicon, the red curve, is necessary to absorb 90% of the incoming light at 550 nanometers. Why don't you go for it? I'll give you, say, a couple minutes. To make sure people are setting this up right, i divided by i0 to absorb 90% of the light, that would be 0.1, 1 minus 0.9. OK, so as you're finalizing your calculations, I just wanted to make sure set this up right. Again, if we're absorbing 90% of the light, it means only 10% of the light is going out the other side. That means their i is going to be 1/10 of i0 or i divided by i0 is 0.1. And then we would take the log of both sides, typically, and solve for our l based on the alphas that we have here. Again, units of alpha would be in inverse centimeters. And so the l's that you obtained, let's go for gallium arsenide first. Did anybody manage to walk all the way through that calculation? AUDIENCE: 20 micrometers. TONIO BUONASSISI: 20 micrometers. For our gallium arsenide or for our silicon? AUDIENCE: Gallium arsenide. TONIO BUONASSISI: Gallium arsenide. Did anybody get any other numbers for gallium arsenide. AUDIENCE: 0.4. TONIO BUONASSISI: 0.4 microns. Yeah. That's sounding more in the ballpark. Anybody else? AUDIENCE: 23. TONIO BUONASSISI: 23 as well. So I'm getting-- I would have guessed that the number would rather small for gallium arsenide, so something in the range of, say, a micron, in that order. Why don't we give folks enough time to walk through-- I know I rushed you on the calculations here. We have material to get through. And I wanted to see you perform under pressure. But how about the silicon? Is it larger or smaller? Let's just for order of magnitude first and the general trend and then try to pick up the precise number. For silicon, crystalline silicon that is, with an optical absorption coefficient and order of magnitude less than gallium arsenide, is the thickness needed to absorb the same amount of light going to be greater or smaller? AUDIENCE: Greater. TONIO BUONASSISI: Greater. By an-- AUDIENCE: Order of magnitude. TONIO BUONASSISI: Order of magnitude, brilliant. OK. So whatever number you got for your gallium, arsenide you could translate it fairly easily. All right, so that was at 550. And there's a lot of solar radiation right around 550, so the numbers that I have on the top my head work somewhere out to be on the order of a micron, a little less for gallium arsenide, somewhere in the order of 10 microns or so for silicon out here. But now if we go out to 800, there's still a lot of solar flux out there. If you recall the solar spectrum, the folks who have been doing their homework, there's still a lot of flux out around 800. As a matter of fact, it continues going all the way out to here, although decaying intensity a la black body. And at 800 nanometers wavelength light, the optical absorption coefficient is dropped by about an order of magnitude relative to the peak of the solar spectrum. And that's why most of these solar cells that you see of crystalline silicon are on the order of 100 microns, typically a little thicker for some technological reasons, which we'll get to, make it difficult to handle very, very thin stuff. But if you just assume one pass through the material, you'd need about that thickness to absorb a lot of the light. And I'll pass around some of these materials right here just so you can get a sense of how thick they are. Here we go. Actually, here's what I'm going to do. I'm going to take out the big pieces and leave the small ones in here that are already broken. And you can actually pick them up if you like. Just be aware that these little pieces of silicon are-- silicon's brittle material. It's like glass. So if you have a little shard of silicon, it can poke you just like a charge of glass can. So treat it with the same amount of respect that you would a very, very thin piece of glass. But you can see here that if you look at the thickness of these materials inside of that little bin, these are small shards of silicon solar cell wafers. Their thicknesses in the order of 100 microns, those are particularly thin. You have other solar cells that are 170 microns is typical thickness for silicon. And for gallium arsenide, you can deposit thin films that are on the order of a micron thick or less. You can go down to a few hundreds nanometers and still make-- actually the record efficiency of gallium arsenide solar cell is a few hundred nanometers thick. And our calculations right here assumed one pass through the material. That's all we gave the light. We only gave one chance to go through the material and get absorbed. What could you envision would increase the total amount of light absorbed? What could you do to your solar cell device to increase the total amount of light absorbed inside of it? AUDIENCE: Put anti-relfective coating on it. TONIO BUONASSISI: You could put anti-reflective coating on it. Let's do something much more simple. AUDIENCE: Put reflective coating on the back. TONIO BUONASSISI: Reflecting coating on the back, absolutely. Yeah. So if the light goes through the solar cell and doesn't get absorbed, that 10% of the light that didn't make it, that's going to get reflected back. It's going to get another chance to go through. So if you absorb 90% of the light on the first pass, you'll absorb 99% of the light on two bounces, right? Or in one bounce, rather, and two trips, two optical path links through the material. And so the term optical path length is a very important term here, because the optical path length does not have to be the thickness of the material. Ideally, the optical path length through the material is much, much thicker than the actual material itself. And over the next few slides, we're going to learn how we engineer that. So methods to improve optical absorption- generally, these are called light trapping. Not all of these entail trapping the light. Actually, most of them do. We also call them light management as a more general term that includes reflection and absorption inside of the material. So the very simplest thing we can do on the front surface-- so what we're going to do is take this step by step, as light goes into the solar cell from the front side, we're going to take step by step, what can we do to improve the amount of light that is absorbed? The first thing that we can do is texturize our front surface. If we don't have texture on our front surface, if it's absolutely flat, what we call specular surface-- specular coming from the root mirror. In Latin languages, for example, Italian specchio is mirror. So a flat silicon substrate, a specular surface, would reflect some finite amount of light. And we can calculate that now because we know that it relates to the real component of the refractive index of the material. Now if we texturize our surface-- this is representing kind of a pyramid type texturization. If the light comes in and some fraction doesn't go into the material-- there's some component of that ray that's going into the material over here, but we're ignoring it in this drawing. We're just focusing on the lights, the rays that get reflected. That beam that gets reflected off, instead of just going back out toward the sun, it's now going toward the material again. So it has a second chance of getting absorbed. So you just went-- for example, let's say if you have a 10% reflectivity on the surface, you went from a 10% reflectivity over here to a 1% reflectivity over here. Because now you have the total amount of light that gets reflected is 1 minus 0.9 squared as opposed to 1 minus 0.9 to the 1. In this case right here, the amount of light that gets reflected, assuming its 10% reflective, would be 1 minus 0.9, so 10% of light. And over here, the amount of light that gets reflected would be 1 minus 0.9 quantity squared, so 1% instead of 10%. So texturization increases the probability that light will enter the device. And what it also does-- this is a secondary benefit-- is it increases the path length, the effective path length, of the incoming light. And the way to understand that particular phenomena is called Snell's Law. Well, even in the absence of Snell's Law-- no, let's go there. Let's go there. So we have a texturized front surface. What's happening? Well, as the material goes from one medium to another, the refractive index changes. We discussed this right at the beginning of lecture. So the way in which the electromagnetic wave oscillates the electrons instead of the system is changing from one medium to another, let's say from air into the solar cell device from air into our silicon, for example, right here. Now, we can ascribe the refractive indices to air and to our silicon like so. And the light path will obey what is called Snell's Law, which is the product of the refractive index and sine of that angle, the angle relative to the surface normal. So a simple way to think about this is when the light goes from a low index of refraction medium to a high index of refraction medium, light bends toward or away from the normal? So if I'm going from air into silicon, light would bend toward the normal, right? So here my theta 1 is going to be greater than theta 2. My light has bent toward the normal, if this is my silicon and this white stuff over here is my air. So light came in. It encountered the surface. The theta 1 was defined as the angle of the light relative to the surface normal. That was my theta 1. My theta 2 is going to be given as the ratio of the refractive indices. And because the refractive index of silicon is going to be greater than that of air, light would bend toward the normal. And so what I have on a macroscopic view over here, if this is my surface texture, light was coming in, it's now bent. And so the effective optical path length is now larger than the thickness of my device. It's kind of cool. So there are two benefits to texturizing your front surface. One is you have an additional pass, additional bounce, an additional encounter with the material. So that reflected light gets another chance to go in. And the second benefit is that you're able to increase the optical path length by the delta in refractive indices and the fact that the path of the light will be Snell's Law. Now another really interesting aside of Snell's Law is that if light is trying to go from a high index medium to a low index medium, and if it's coming in at a very oblique angle like this, if you run through Snell's Law, you don't get an angle coming out. It actually falls along the surface or actually bounces back in most often, depending on the angle. And you have what is called total internal reflection, which is this case right over here. That little bounce, that friendly bounce, of the light that went in bounced off the back side and then was reflected back in. That's a total internal reflection event. And that happens in solar modules. Right here, when light comes in, bounces off of the white back skin right here, and then gets scattered off at an angle, it can have a total internal reflection off of the front surface glass and have a second chance of getting back into the solar cells inside. So that's one of the reasons why you see this white spacing, the white colored material, in between the cells, is that the light gets reflected off of there. It doesn't make it very aesthetically pleasing. You might want it to look all black. And if you do want it to look all black, what would you do instead? Instead of changing the back skin, what other component might you change? AUDIENCE: The front. TONIO BUONASSISI: The front, right? You might change the nature of the anti-reflection coating on the glass. We'll get anti-reflection coatings in a minute. So even if the panel looks black, there are some really aesthetically pleasing solar panels out there that look completely black. They may still have white back skin, but the glass is just very good at absorbing that light and preventing it from escaping. OK, so to engineer front and back surface reflectances, you really have to carefully select your refractive indices and your materials if you put on either side. And it's very important-- extremely important. To make a long story short, the record efficiency solar cell that was announced this past year in gallium arsenide was achieved because of good light management. And we'll explain how that came about perhaps towards lectures, maybe lectures eight or nine. So I'm going to play a little game with you, which is to look at a swimming pool. This is a pool filled with water, which is refractive index 1.3. Air is 1. And so that's the normal view, what we have. Light bends toward the normal, right? And so you're able to look down inside the pool that stuff that is not in your line of sight, not in your direct line of sight. That's because when you look down, the ray of light is traveling like this and it bends toward the normal and likewise symmetric. So you're seeing material down there. What change of property would give you these two images over here. Let me give you a hint. In one of those two images, the refractive index of the medium inside the pool is not 1.3. It's 0.9. It's 0.9. And in another one of these two, the refractive index of the medium is actually going to be negative. We'll call it a negative refractive index material, a negative index material. So which of these two do you think is which? Why don't you turn to your neighbor quickly and chat about it without peeking at your lecture notes. So let me walk through, as you begin honing in on your answers here. Think about what would happen to the reflectivity of that front surface of the water and what would happen to the angle that the light travels, or the angle of refraction of bending, shall you say, as the light goes from one medium to another. So if we go to a refractive index material of minus 1.3, will we change the reflectivity at all? It depends, but the answers here are shown, for this particular system. It would require sitting down and walking through the equations, but in essence right here, with the pool filled with the negative refractive index material, you're really affecting the angle at which light is coming out of the pool. Here you can see the corner of the pool, which you shouldn't even be able to see. It's just that the light traveled this way and then came back because it was a negative refractive index material. Light actually did something like this, zoop, zoop. AUDIENCE: What's in the pool? TONIO BUONASSISI: Oh, that's just a corner. So what is in that pool? That is a computer generated graphic. This is not a real pool. There exists negative refractive index materials but not in that volume yet. These are relatively small things and very much a study in fundamental science. So in this case right here, we have less of an acute bending of our angle of light. So we don't get to see quite as many features right here toward the edge. And the reflectivity has changed as a result of having drastically modified our reflection condition. AUDIENCE: Why does the reflectivity seem to have gone up and the index has gone down? TONIO BUONASSISI: In that particular case? I think what they were getting at-- this is coming off of an SPI website. I think what they were getting at is mostly just a change in the reflectivity. So they were trying to emphasize that you were modifying the reflection off the surface in addition to the angle at which the light was exiting the material. I'm going to come back to Snell's Law in a minute. But for the time being, I want to move on to the next concept here, which is Lambertian reflector. You'll hear this topic or this word thrown around quite a lot in the solar cell community. And it's used rather liberally to mean a lot of things. Although in optics, it has a very specific meaning. So I'm going to show you that very specific meaning and then describe for you what it has very loosely come to mean in the solar industry. So a diffuse Lambertian reflector will follow a reflectance that follows a cosine theta dependence. So if you have light coming into a sample, the surface normal, and the outgoing light ray form an angle theta. And if the two are perfectly aligned, you get a lot of reflectance off of that angle. If the two are perpendicular to one another, you get zero reflectance in that angle. And so the reflectance parallel to the surface here is zero. In everywhere in between, the magnitude of reflectance is varying as consine theta. That's the, I would say, pedantic definition of an Lambertian reflector. Often in the solar industry you'll hear people, probably because of a lack of optics background, just call any randomly reflecting surface a Lambertian scatter. It's a very loosely used term. And it is wrong by the book, but nevertheless, it's one of these things that live on in our industry. So the difference between a specular reflector, the one that we've just been analyzing right now, and a Lambertian reflector, is that typically the way these are made is that you do have a random texture on your surface. And that's probably where the origin of this misunderstanding comes about. We don't get a random reflectance of the light coming off, but the surface itself can be rather texturized. So, for example, if you suspect that this little material right here might behave like a Lambertian scatter, you might put it inside of a tool and rotate the angle and measure the amount of reflected light as a function of the angle to determine whether or not it follows this cosine theta dependence. And the reason it's important is because the back skins of our solar modules can quite often be Lambertian scatters. And we have a certain amount of light that comes off at some angle here that will get trapped by a total internal reflection inside of a modules. So if, instead of having macroscopic pyramids right here, you had very, very small pyramids-- still not sub-wavelength, but smaller features, for example, the texturization on the back skin right here. An it managed to scatter the light at a particular angle that got caught by total internal reflection. Macroscopically, we might be able to describe the scattering behavior of that surface as Lambertian scatter. But it's those waves, those rays that are bouncing off at those large angles that are causing the total internal reflection event. And so the notion of a Lambertian scatter is important on the backsides of solar cell devices. We would obviously wants to even change the scattering profile. We wouldn't want necessarily specular reflectance. We might want to maximize the amount of light reflected off at particular angles. And there is, of course, research being done to figure out how to make light do that. I'll show you one example at the very end of lecture, a paper that was just published in Science last week, as an example. And so these scattering centers off the backs of the rear sides of cells would operate more or less in the following manner. You'd have incoming light. Let's ignore front surface texturing for now. Let's just focus on the backside. And if you have some random, as we call it, a random reflector, a randomly texturized reflector on the back that reflects off in, say, a Lambertian fashion, you'll have some fraction of that light scattered off at an angle that is large enough relative to the surface normal that it is trapped by total internal reflection. And you don't only have to texture your back skin. You can also texture the bus bars. The bus bars are these little metal wires right here that are collecting the charge from each of the solar cells. And they're connecting essentially the front side of one cell to the backside of the next. If you want to think about it as the cathode to the anode, cathode to the anode, cathode to the anode, stringing all these cells together in series. And this metal right here is just really shiny, and it's reflecting light right back out into space. What if we instead were to texture that metal so that when laser light shined on it a certain amount would be reflected off at an angle and then caught by total internal reflection. And that's exactly what you're seeing right here. The light bounced here on a textured bus bar, bounced off of the glass more or less around here halfway, and then got a second chance to enter the cell over here. Obviously some of it is reflecting off so we can see it. But a lot of it's going in. And that little innovation right there, which was developed in the building right next door by Professor Ely Sachs, can gain module performances somewhere on the order of a few percent relative. So that might not sound like a whole lot, but if you're a $100 billion industry, 1% is a lot of money. So it does add up. So that goes back to the total internal reflection. So there is a limit to all of this texturization. There's a limit to how much we can trap light simply by modifying or corrugating the surfaces to enhance the optical path length with these types of bounces using Snell's Law and of course the general reflectivity equations. And a gentleman by the name of Eli Yablonovitch, who's now a professor in Berkeley calculated these parameters I think back in 1982 and came up with an upper limit to the optical path length. He, after a long series of calculations, derived an expression for the maximum increase of the optical path length due to surface texturing, which was 4n squared. And the Yablonovitch limit to this day is a pretty good litmus test for the ability of a material to trap light. So if you have silicon, for instance, with a refractive index of, let's say, in the infrared some around 3.6, your Yablonovitch limit is around 50, which means that you can increase the optical path length by a factor 50, relative to the thickness of your material. If you have an organic material, which has a refractive index typically of around maximum 2, then that would be squared, 16, somewhere in that range. You can increase probably in the order of 20 the optical path length inside of the material through texturization. So this is a useful parameter for those who are doing research in photovoltaics, the graduate students especially. And so I think the graduate students will have a question at some point on the Yablonovitch limit. And so that's a useful parameter to keep in your mind. Let me touch upon a few other forms of trapping light. We've so far just assumed that light behaves like a continuous wave, doesn't interfere with anything, doesn't interfere with itself. Now we're going to discuss some anti-reflection effects which derives from the notion that light is a wave and can constructively and destructively interfere. What we have right here is a layer of another material with a refractive index, say n1, which is in between our n0 which is at air and our n2 is the absorber material, let's say the silicon. So we have a grading of refractive indices going from air, our anti-reflection coating, to silicon. And right over here we have a certain thickness, d1. And over here we have a certain thickness, d2. So what is happening in these two images? Let me show you with another, a little bit more clear figure, coming from our beloved Wikipedia, and then go back to that other image right there. So what's happening is we have an incoming wave that for some reason is ignoring Snell's Law. It's beyond me. But anyway, the wave is going in a straight line. It should be bending toward the surface norm, obviously. But we have reflections off of this interface and this interface right here. And because the thickness of this layer is in the order of lambda over 4, that means that the wave that's going in will be phase shifted relative to the wave that's reflected off the front surface, first by lambda over 4 then 2 times lambda over 4, in other words, lambda over 2, which means that the two waves are out of phase by lambda over 2, which means that they will destructively interfere. The peak will be at a trough. The trough will be at a peak. So the two waves will are going to be destructively interfering when they come out. If you add these two together, due to the wave nature of light, you get suppressed reflectance. And that's a really interesting property. You can begin varying the thickness of this layer, and of course changing the nature of the reflected light. You can constructively interfere if you like and enhance the amount of reflected light as a result of this interference effect. Obviously, in most solar cells, we want to suppress reflection. And so we go to great lengths to make sure that this thickness as well as the refractive index of the material is optimized for a particular system. And so without going into the hairy math, to calculate this right here, it's definitely possible. It's definitely something that should be done. And I believe the graduate students have it assigned. It's the very last problem in the homework. But for a very simple kind of conceptual understanding that is wavelength independent, if we want to minimize the reflectance at a particular wavelength, let's call it at a lambda 0, which is the photon wavelength at the peak of the solar spectrum, we have to design the thickness of our anti-reflection coating to satisfy that equation right there, essentially lambda over 4. That's the phase shift we want upon one pass of the anti-reflection coating so that two passes, when it goes through and then back, it's phase shifted relative to the surface reflected light by lambda over 2, divided by n, n being the refractive index of the material. Obviously the frequency of light is staying the same as it goes from one material to another. But the wavelength would be changing. So that's why the n parameter appears right here in this equation. The t is the thickness of the optimal anti-reflection coating thickness. So just to give us a sense, kind of an estimate, and to give us some confidence in these engineering methods, what I'd like you to do is to calculate the thickness of an ideal anti-reflective coating. This anti-reflective coating right here on these cells-- I apologize, they also have the metal on the front, so it's a little difficult to distinguish between the two. But in my right hand, this one, I have a piece of bare silicon. And you can see it's rather reflective. In my left hand over here, I have a piece of silicon with an anti-reflective coating as well some contact metalization on the front. So that's why you see those grid lines. But it looks very blue. It looks very blue because the cell is absorbing very well at the peak of the solar spectrum which is in the yellow. So calculate for me what is the optimal thickness of an anti-reflection coating of silicon nitride? And we'll give it a refractive index of, say, 2.1. Let me see if those numbers make sense, so refractive index of silicon nitride somewhere around 550. Let's call it 2, just make our lives simple. And the peak of the solar spectrum we'll again call 550. So why don't we run the numbers quickly. What should that thickness be? AUDIENCE: Tonio, I'm sorry, could you repeat the constant again? TONIO BUONASSISI: Sure. So the n, the refractive index, is going to be around 2 for silicon nitride. So we're going from air, which is around refractive index one, to silicon nitride, the silicon. And the peak of the solar spectrum, our lambda 0, which is the photon wavelength at the peak of the solar spectrum in vacuum or in air, is 550 nanometers. So what thicknesses are folks coming up with? Order of magnitude. AUDIENCE: 70 nanometers. TONIO BUONASSISI: 70 nanometers. That's almost spot on to the actual thickness, to somewhere on the order between 70 to 80 nanometer typically. You're telling me that something that is 1/1,000, the thickness of my hair, is deposited on the surface of this wafer and is absorbing all this light? That's pretty cool. And it's not absorbing the light. The anti-reflective coating is not absorbing the light, which is really important. We want the solar cell underneath it to be absorbing the light. The anti-reflection coding is enabling the light to be absorbed because it's suppressing the reflectance. The reflected modes at that particular wavelength are suppressed because of the destructive interference. That's cool. I really get a kick out of anti-reflective coatings. So they're 70 nanometers thick. And you gain quite a lot in terms of cell performance. I'll show you some slides to drive that point home in a bit. This is really really briefly-- I'll post these slides online so you can have access to them. If you use the matrix transfer method, as described beautifully in [? Gonchen's ?] textbook, you can calculate the amount of light reflected across a broad spectral range for a given thickness of anti-reflection coding. So what we did right now was to calculate a suppression of the light at a particular wavelength. But you can also calculate with the tools that are available to you the reflectance of your particular device over a broader wavelength. Range and that's pretty cool because now you can begin, say, multiplying this function right here against your solar spectrum and begin to calculate the total amount of light entering your sample and the total energy entering your sample. Equations, brilliant. The important thing to note here is that it really, really matters. This is silicon under glass right here, for example, typical solar cell material in blue. It's better than the bare silicon. Why is it better than the bare silicon, silicon under glass? Glass has a refractive index of 1.5 or so. AUDIENCE: The index matching. You go from pairs 1 to 2.3 and then to [INAUDIBLE]. The difference is small between the classes. TONIO BUONASSISI: Exactly. So if we recall that equation that described the amount of reflectance, there was that-- what was it-- n1 minus n2 quantity squared, right? So the bigger the delta between the ends, the bigger the difference in refractive indices between material one and material two, the more the reflectance is going to be off that interface. And so you can begin reducing reflectance off of a stack of light going both ways by grading the refractive index of the material. And that, of course, changes the reflectance in both directions. And so you get a reduction in the total amount of reflected light when you put the silicon under glass because glass has a refractive index somewhere between air and silicon. And then you get a further reduction of the reflectance when you have an anti-reflection coating with a refractive index somewhere around-- for this particular system, silicon again has a higher refractive index. This used in anti-reflective coating of a refractive index of 2.3 of some thickness, probably somewhere around-- let's see, it'd be greater or smaller, probably around 65, 75 nanometer somewhere that range. So what this is saying is that you can minimize the reflection of light off of the front surface of your sample by using an intelligent combination of the very first equation that we're exposed to in the class today, which was the reflected light as a function of refractive index, so essentially refractive index matching and secondly, by engineering by engineering an anti-reflective coating, which oftentimes in the lingo of solar cell science we call it an ARC, an anti-reflective coating. And those two things combined give us very low reflection off of the front surface. Probably 5% of our R&D cells that we make at MIT use these sorts of technologies, which are pretty standard in the industry. And you can see what the hit is, right? Let's see, if I'm just using a bare material, if I'm getting 30% reflection, I'm getting a 30% drop in the current output of my device. That's pretty significant. So these are simple ways to improve performance of devices. If you want to become fancy and actually do what's called a ray tracing to calculate the path of light through a medium, there is software available that will take all of what we've discussed today and calculate it for you so you don't have to walk through the expressions that we just walked through. It is easy. In other words, you plug something in. You get some ray traces. You can calculate reflectance and so forth, transmittance. But it's as smart as what you put into it. It's really important to understand the fundamentals behind any simulation software because you will get out of it what you put into it. You will not be able to pick up on obvious things that you might of-- for example, double clicked on this little material here and find the real component of the refractive index completely wrong. And you might not notice it. You might not pick up on it if you don't have some good intuition which is grounded in the fundamentals. And so it's important that you understand what we've presented today. It's important you understand the reading and, of course, do the p-set as well to really drive those fundamentals home. So to kind of put a big umbrella over the entire lecture, light management ensures that the absorbtance is high. The absorbtance would be, essentially, the amount of light getting absorbed inside of the material, normalized by the amount of light going in, so 1 minus r. So we want to ensure that light enters the absorber. We want to minimize reflection. We want to ensure good light trapping inside of the absorber as well, the absorber being the material, our photovoltaic material, the ones absorbing the sunlight and ultimately going to be generating the charge. So we call it the absorber. So we want to ensure good light trapping inside it. We want to ensure the maximum amount of light gets trapped inside. We want to maximize the optical path length within it. And we want to minimize reflectance off the front surface. There are fancier ways of light management as well that don't involve light trapping necessarily but light manipulation or even semiconductor manipulation. You can, for instance, change the wave length of the incoming light. One very simple example of this is when you shine, say for example, red light on a phosphor and then it glows green in the dark. That's a wavelength change-- maybe not red. You'd probably have to shine blue to have it glow green. That's an example of a spectral down converter where it's taking a higher energy light and converting it into lower energy light. Likewise, there are folks out there trying to do spectral up converters where they take two lower energy photons then somehow convert that into a higher energy photon. And so since our absorption coefficient is dependent on wavelength, if we're able to shift the wavelength of the light around by engineering materials near the surface, we can enhance absorption as well. That is a form-- a valid form-- of light management. It has additional benefits as well. If we can eliminate the longer wavelength stuff out here, which is heat, performance of most solar cell suffers when they get hot. And we'll learn why that is about five or 10 lectures from now. And so if we manage to do spectral up converting or reflect that long wavelength light away from our device, we can improve performance there as well. That's another form of light management that doesn't necessarily involve light trapping. So again, I wanted to really emphasize that light management is necessary devices. This is no light trapping, the blue curve, and with light trapping, light trapping being essentially just an engineered coating on the backside, on the backside of your device, of lights coming in through here. I've engineered a coating on the back to reflect the light back so that it gets a second bounce through the material. I've engineered the front surface, texturized it so that we have not only the benefit of two bounces, double the chance of light going in, but also the Snell's Law working in our favor and increasing the optical pathway. And so all told, the one reason why this boost is so big right here is because I'm increasing the optical path length, the effective optical path length, relative to the thickness of my material. And as a result, I'm getting a much larger current output. I'm generating many more free carriers instead of my material. I'm absorbing much more light inside of my material, just a very simple calculation versus cell thickness. And obviously the thicker and thicker and thicker you go in your device, the less important this becomes. Because the less important light trapping-- I mean, you have the entire thickness. I can absorb the majority of the light in one pass. But if you have a thinner device, it really begins to matter. Once the thickness of your device starts approaching the optical absorption, or 1 over the optical absorption coefficient, which is the extension length, then it really begins to matter in the absorption length. Light trapping can still matter for thick devices, though. Because if you manage to make the light essentially refract or bend, if you will, so that it travels near the surface, the distance that those excited carriers have to travel to be collected is shorter. And so you can get an additional benefit from thicker devices by engineering light trapping as well. OK, any questions about this? This is kind of important. This is why we spent all this time in lecture today talking about light management is because of this plot right here. That's why. I just wanted to show you a cross section of very high efficiency device. This is one of the highest efficiency silicon-based devices are out there. And we have these so-called backside mirror, which is really just a layer of dielectric material, typically, that reflects the light off of that interface using the equation that we saw at the very beginning of class, the r equal to [INAUDIBLE] n minus 1 quantity squared divided by open parentheses n plus 1 quantity squared. So that's benefiting here from the change of refractive index going through your silicon to that dielectric material in the back. They definitely take good advantage of it. Where you have your metal, you're going to be absorbing the light. Or you have a higher probability of absorbing the light than you would if you had a dielectric semiconductor interface. So the device design can get pretty complicated for these super high efficiency devices. And they're worried quite a bit about trapping, other things as well. AUDIENCE: Coefficient, is that one there? TONIO BUONASSISI: This one right here? In the lab, 24.2%, in commercial production, 22% and change. 22.4%, I think. Just to throw some last things out there since we're five minutes to closure. Snell's Law assumes that there's no phase shift of the light as it transfers from one medium to another. If you introduce a phase shift-- this is just a paper published in Science last week by our friends over at Harvard, Federico Capasso. If you introduce a phase shift of the light as it goes through one medium or another, now you can start doing some fun things. If you introduce a constant phase shift gradient throughout the surface of a material, let's say right here, then you can cause each node, each point within your material, to lag by an increasing amount, so that your wave front now bends. You can think of these as kind of a Huygen wavefront forming as a result of these small nodes here. And if you can tailor the phase independently at each one of these points, you can cause an increasing delay as you go across. And that will cause the light, essentially, if you trace through the points of maximum intensity, say the pink, you'll see that the light is bent. And that's pretty cool. Because now we can, in principle, if this is hot off the press-- and then of course there's a whole flurry of researchers out there trying to figure out how to use this to our advantage, but with anomalous refraction, in principle, now you can tailor the angle at which light bends inside of the material. Perhaps you can even exceed the Yablonovitch limit inside of the material as a result of this. And so it's really exciting. There's stuff coming up every day. This is the point. There's stuff coming out every day on light trapping and light management. Mostly it's for photonic devices. But they can be transferred over into solar cells as well. So it's going to keep your eyes open. And another example of the photon up/down converters, there's recent reports in SPIE, a lot of interest in the optics community. There was a TR35 award given to a person who studying this topic. So it is, as well, a very exciting and up and coming field. Again, the opportunities there of manipulating light are large, are vast. So the laws, if you will, that constrain us, that we've discussed today in class, don't let that constrain your thinking. That's my final message. Thanks. |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | Student_Project_Presentations_Part_2.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. STUDENT 1: We would like to present to you PV Next Gen, Earth-Abundant Materials. So the reason we're doing this project is because there are current limits to the high-quality crystalline silicon cells that we're producing. There's too high of an energy cost and too high of a carbon footprint. So we decided to identify likely sources of failure for earth-abundant PV. We focus on tin sulfide, lead sulfide, and cuprous oxide. So we took the approach of subjecting these PV cells into a climate chamber to sort of simulate aging, and accelerated aging, and the effects of long-term operation out in the field. So for a little bit of a background, this is a graph that we'd like to provide you with. If you notice, the graph is centered around zero, which is crystalline silicon. And on the x-axis, it's kind of inverted. So the negative's on the right. But this graph sort of illustrates that where we want to be, in our research, is in that quadrant within the circle. Because that's where we have the lowest materials extraction costs. And we have the highest annual electricity potential. So those are the kind of technologies that we want to be looking at. And as you can see in the arrowhead, cuprous oxide and lead sulfide are in that quadrant. JOEL: So we've established, we've showed you how these kinds of materials can be less expensive and, potentially, produce more electricity than silicon. However, they haven't had the same kinds of decades of experience in testing as silicon has. So we're trying to figure out how they could possibly fail under actual years of operation in the field. First, we're going to talk about tin sulfide. Certainly, tin and sulfur are both elements with which we're very familiar. Tin has a wide manufacturing base. Sulfur is part of all of our biology, pretty easy to find. The band gap of this material is 1.3 electronvolts, a little bit higher than silicon. But that, actually, brings it closer to the optimum of the Shockley-Quiesser efficiency limits for a single-junction solar cell. The maximum possible efficiency is 32%. However, the current record efficiency, for tin sulfide cells, is only 1.3%. So that gives you an idea of the state of development of this field. Often, it seems that devices are limited by their charge extraction. There is an interface between the absorber layer and buffer layers that block electron and hole transport. I'm going to talk about that a little more on the next slide. But this interface has been, unfortunately, not very well characterized to date and is often a site of high recombination rates that can limit the current extracted. However, tin sulfide is a good material for photovoltaics. Because it's crystal structure is amenable to doping. You can add substituent atoms and tune the electronic properties that way. Also, it has a high absorption coefficient. So it can be made into thin film devices. And the device that we're going to show you today was made using atomic layer deposition. Here's that schematic that I was mentioning. So as in this diagram, the light is coming in from the bottom. Don't be alarmed. The bottom substrate, that things are subsequently deposited onto, is a glass pane that is coated with fluorine-doped tin oxide. This is like indium tin oxide. It's another transparent conducting material. And then, this layer here is the tin sulfide absorber. And you can see that on either side of it, there is a hole-blocking layer and an electron-blocking layer, which prevents charge from flowing in the opposite direction that we want it to. And then, both the absorber and the front contact have these metal stack electrodes patterned onto them, onto the same side of gold, copper, and molybdenum. So this is a manufacturing idea that could make tabbing easier and eliminate shading losses. Because all of your electronics contact to one side on the back here. However, as we'll discuss later, it comes with some challenges of its own. JOLENE: The other material that we investigated, for this project, was lead sulfide. Lead sulfide is also an earth-abundant material. Lead is even more abundant than tin. The bulk band gap of lead sulfide is only 0.41 eV, which is a little bit too low to make it a good observer material for photovoltaics. But if you make nanocrystals, you can tune the band gap. So there's confinement effects that make it so that the band gap can actually become larger. So we looked at lead sulfide quantum dot solar cells. And in particular, we looked at lead sulfide quantum dots and a PCBM, which is a fullerene derivative heterojunction, where the lead sulfide is a p-type material. The PCBM is the n-type material. And we just made a bilayer heterojunction. With these materials, the major potential failure mode, that we identified, was oxidation of both the lead sulfide quantum dots and the fullerenes. And then, there's also possibilities of hydrolysis of this PCBM esther group upon reaction with the EVA encapsulant material. That was something that we thought might be a problem. The current record efficiency for these lead sulfide quantum dot solar cells is only 2.1%. Also something, more work needs to be done in improving that efficiency. But a major advantage of this material system is that both the lead sulfide quantum dots and the fullerenes can be solution-processed. So they can be spin cast onto your substrate, which makes the processing cost very low, especially compared to growing a single crystalline wafer. The device architecture that we used is, again, glass coated with a transparent conducting oxide electrode. And then, there's just a bilayer heterojunction with the two observing materials. And then, finally, it's a metal contact on the opposite side. For our case, I've shown the band structure of the devices that we used. And, in particular, I wanted to point out that the top electrode, in our case, was actually aluminum because of the band alignment. The work function of aluminum makes it so that the top band aligns well with the PCBM. And you can get charge extraction the way that you want. And, again, our devices, the quantum dots were tuned so that the band gap was about 1 eV. And these are the parameters for the devices that we used. STUDENT 4: All right, so what we did is our initial plan was to collect IV curves of before and after encapsulation of our solar cells, and after a period of time spent in the environmental chamber. The encapsulation that we chose, from a variety of materials, was EVA. Because it's an industry standard. For environmental chamber, we kept it at 85 degrees Celsius and at 85% relative humidity to accelerate the degradation process. Then, we took photos of our cells over a period of seven days. Here's a schematic of an encapsulated cell. It's a layer of glass followed by a layer of EVA, then the solar cell, another layer of EVA, and then a TPE backsheet-- again industry standard. For our tin sulfide, what we did is we actually omitted the glass layer in our encapsulation. Because the cell came with glass, was made with glass. So to make the contact, so we did is we applied silver paste to attach the tabbing. And we baked them at 45 degrees C to evaporate the solvent. This yielded very delicate contacts. So we had to encapsulate immediately in order to hold the contacts in place. Unfortunately, as Joel was mentioning about how the two electrodes are on the back, during the encapsulation process, we shorted our cell. So we were unable to take an IV curve. For the lead sulfide, initially-- as Jolene said-- we had aluminum contacts. Unfortunately, when exposed to air, this forms a layer of aluminum oxide, which is nearly impossible to solder to. So we switched the aluminum with silver, even though it's less than ideal for a charge extraction. And with these contacts, we were only able to achieve a contact with a large amount of solder. And again, we could not encapsulate. Because we risk breaking the cell with these contacts. What we decided to do, then, was encapsulate unsoldered cells with aluminum contacts, and do a qualitative assessment of the color change, which would indicate a reaction with the encapsulant or with the water oxygen. So basically, unfortunately, despite our initial plan, we were unable to collect IV curves for either type of cell after encapsulation. DANNY: So we did a series of cross-sectional [INAUDIBLE] images on tin sulfide device before and after several days in the environmental chamber. So on the left, you have an image of the tin sulfide layer before exposure to the chamber. And we can see the columnar structure of the tin sulfide layer. However, after several days, on the right is an image of the tin sulfide layer having some changes, having undergone some changes to its structure. This may be due to either thermal degradation or a result of [? cleavage ?] of the cell. We also did a series of photos over the course of several days on the devices. And we noticed that there was some color changes in the device. The tin sulfide layer has undergone, the color of tin sulfide layer darkened over time, especially near the corners of the cell, like around this edge here. For the photo on the extreme right, you may notice some yellowish tinge. And that is due to the lighting the photo was taken in and not an actual change in the cell. So as for lead sulfide, we did a similar series of photos on the device. And you notice a similar color change has occurred across the device. You also notice that at the four corners, there are contacts there. And these contacts have been tarnished significantly. So we think that the color changes, on both devices, may be due to, one, the leakage of the encapsulant, two, the reaction with the encapsulant, and lastly, due to thermal degradation. KATHERINE: So if we're going to achieve terawatt-scale electricity output with solar energy, we need new materials that are cheap and efficient. However, the materials we investigated still have a lot of work to be done before they can achieve that. In particular, they have very low efficiencies. And that's something that absolutely needs to be overcome. However, we focused more on some of the other issues, such as the contacts. Right now, they're still in the research phase. So, for example, using aluminum as a contact is fine. However, when moving into commercialization, aluminum contacts will have to be replaced with something else. Or something else will have to be deposited on top of the aluminum. We saw degradation of the encapsulated cells, which indicates that the current encapsulant will not work with these new materials. In particular, we think that we need an encapsulant that will, does not have an acidic degradation product. Because that was a serious issue with the PbS quantum dot solar cells, with the fullerene part can react with acid and degrade. So we think, we did talk about wanting class Evb? JOEL: PVB. KATHERINE: PVB. But we did not have access to that material. And we think that the method that we came up with of testing materials in the environmental chamber will be useful in the future. And ideally, you could test different encapsulants and see how well the IV curves stay after the environmental testing. So we'd like to thank several people for helping with our project, in particular Andy at Fraunhofer, who's actually here today. We had to go to Fraunhofer several times to-- we had several iterations of getting the cells. And then, once we put them in the chamber, we had to take pictures every day. So he was there and was able to help us with that. Also we'd like to thank Darcy of the Bawendi Group for making the PbS quantum dot cells for us. And also the Gordon Group for making the tin sulfide cells. And then, finally Professor Buonassisi and Joe for teaching us all the fundamentals of the devices. So before we go to questions, we'd like to pass around the cells we had. While people are asking questions, we ask that you don't take them out of the bag or the boxes just because some them contain lead, which it can be harmful. But we can pass this around while we open for questions. PROFESSOR: Thank you, group, [APPLAUSE] JOEL: David. AUDIENCE: So why, does FTO have a different work function from ITO? Or for what was the reason that you used FTO in the consultant itself? JOEL: I'd like to direct that question to Danny. DANNY: FTO is typically much cheaper compared to ITO. So for research purposes, we prefer FTO. JOEL: Is it, light leaves out the indium, right? DANNY: Right. So indium's a rare matel. And it's very expensive and the price is going up. So we try not to use indium in our TCO. AUDIENCE: Is it the same conductivity? DANNY: No. For ITO, the conductivity is much better but, yeah, much better than FTO. JOEL: Still working on the transparent oxide side of earth-abundancy. Different project, though, I guess. AUDIENCE: Can you guys go back to the environmental chamber pictures for, I think, tin sulfide? Yeah. So, yeah, can you just walk me through this again? The yellow is not what my eye should be drawn to? JOEL: Yeah, that's a really unfortunate side effect of the way that the camera was that day. But it's, I think, it would show up better if we turn down the room lights. But there is a darkening from Day 2 to Day 3, and then, also, to Day 6. But that's kind of overwhelmed by the yellow glare. Yeah, maybe now you can see it a little bit better. And we also wanted to point out that although this happens across the whole device, there's particularly more change around the edges. So that to us says that there's definitely something happening here where the atmosphere is leaking in through the encapsulant and reacting with the device. Because that will happen from the outside in, as opposed to a reaction with the encapsulant itself or a thermal process, which would affect the whole device uniformly. AUDIENCE: And this was EVA? JOEL: Yes. AUDIENCE: So, I mean, EVA's a pretty standard encapsulant material. JOEL: Yep. AUDIENCE: Do you think that there is something like the EVA process that you used? Or do you think there's some reaction between the tin sulfide and the EVA that caused these EVA films to not be good encapsulants? JOLENE: So this one, the process that we used to encapsulate these were not at Fraunhofer. These were encapsulated in Joel's lab. And so I think that, and the devices, both devices that we worked with were kind of hard. They aren't meant to be encapsulated. So I think that part of the problem is just the geometry of the device and the presses that we used and everything. We probably just didn't get a very good seal around the outside. And so in this case in particular, I think that it's more likely that it's just not a very good seal. So we're getting some leakage rather than the tin sulfide is reacting with the EVA. For the quantum dots, it's a little bit less clear if there's a reaction with the EVA or not. Because there's a little bit, there's both degradation from, or color changes from the outside that affect particularly the outside. And there's, also, color changes that affect the whole device. And so, then, it's either the quantum dots are somehow changing or there may be some kind of reaction with the encapsulant. But when we took the SEM images, the EDX aspect of it was not working. So it was difficult to do any kind of elemental analysis there. AUDIENCE: And just one other question, in this sort of accelerated aging process, how long would you want to see stability for a, say, a 30-year or 20-year? If you were going to warranty this cell for me, how long would you might say? KATHERINE: I think, usually, they do at least six weeks. AUDIENCE: Six weeks? JOEL: Yeah. JOLENE: Yeah. KATHERINE: And so this was, we didn't have six weeks. But we still [INAUDIBLE] that said. So. JOLENE: More work should be done. AUDIENCE: So how long's your warranty on the cell? JOLENE: Like a day. [LAUGHTER] JOEL: There were other encapsulants available at Fraunhofer. Notably, there's this, a thermoplastic and an ionomer encapsulant. But these are both new and have entirely proprietary chemical composition. So we didn't think it would be very informative to study for our project. Because if we saw something happen, we wouldn't really be able to hypothesize where that came from. But there are other materials out there besides the standard EVA and the polyvinyl butyrate, the PVB that Katherine mentioned at the end. Just not quite as far along. Yeah. AUDIENCE: So how would you compare the degradation between ITO and FTO like-- JOEL: The degradation of ITO and FTO? AUDIENCE: How would you compare this? Which one would you pick [INAUDIBLE]? I mean, in terms of cost, OK. [INAUDIBLE]. In terms of technique, it would be-- AUDIENCE: So can you repeat the question? JOEL: The question is if there's a difference in the degradation between ITO, the indium tin oxide, and FTO, the fluorine doped tin oxide that we used in our tin sulfide cells. Yeah. Do you know anything about that, Danny? DANNY: Yeah, so they normally degrade when you expose them to extended temperatures above 650 degrees Celsius, which is unlikely to happen in a normal day situation. So there's no difference in this [INAUDIBLE]. Yeah, we didn't expose them to 650 Celsius in our accelerated testing either. AUDIENCE: I have a question about the aluminum contacts. First of all, how did you deposit them? JOLENE: The aluminum contacts were thermally evaporated onto the, yeah. AUDIENCE: So would it be possible just to evaporate the [INAUDIBLE]. JOLENE: Yeah, absolutely, absolutely. We were working with cells that had previously been made. That this was not our group, we were getting materials from, generous donations from another group. And so there were cells that had already been made, already had aluminum contacts on them. And so then, it would've been much more involved to get other people in other groups to do stuff for us. So we just used the materials that we had. AUDIENCE: And I'm guessing they don't see these problems because they don't solder their contacts, just scrub them? JOLENE: Yeah. All the tests for the lead sulfide cells are done in a glovebox, an inner atmosphere. And they have pins that just come down. AUDIENCE: Right, so it just [INAUDIBLE]. JOLENE: Yeah. Mm-hm? PROFESSOR: Ashley. AUDIENCE: I'm just curious, more of a logistical question, did you guys have a camera in the environmental chamber? Or do you have to take them out each day? KATHERINE: And so we would have to turn off the chamber, cool it down, take the cell out, take a picture, put the cell back in, then turn it back on. AUDIENCE: OK. JOEL: So in a sense, we actually put extra stress on them. Because there was thermal cycling [INAUDIBLE]. AUDIENCE: So maybe a two-day warranty [INAUDIBLE]. KATHERINE: Mm-hm. AUDIENCE: Did you try leaving the cells [INAUDIBLE] in the glovebox after you [INAUDIBLE] go down and, then, try to [INAUDIBLE]? AUDIENCE: Can he just repeat the question again, too? Sorry. KATHERINE: Yeah, so he was asking about the contacting, the PVS devices. And we did have, we had two things we did. One was we took them in the glovebox and tried to scratch off the aluminum oxide layer, and then put silver paste on, and then make the contacts to that. And that still didn't work. And then, we also got new devices with silver contacts on them. And that also didn't work. Because it either would stick to the device, or it would stick to the contact. But it wouldn't stick to both at once. And they would just peel off. JOLENE: The contacts on those devices are less than 50 nanometers thick. So it, in general, soldering to them was going to be challenging. STUDENT 1: We tried hard. [LAUGHTER] JOLENE: Yep. AUDIENCE: Is there any future plans to continue with this work, with either Bawendi or Gordon Group? Or are they, was just kind of included? JOLENE: Well, I think both of these materials are so much still in the trying to boost efficiency phase that it is less important to be able to say, to talk about the long-term stability than to say that they're even viable materials to begin with. I think that they are interested in the results. But, yeah. JOEL: I mean, this is Danny's thesis, so. [LAUGHTER] DANNY: All right, so [INAUDIBLE] from Gordon's Group, we have much more problems much greater than whether they can last a week or two. [LAUGHTER] PROFESSOR: One more question, guys. Yes. AUDIENCE: Do you IV curves for Day 1 and Day 8? KATHERINE: Yeah, so we weren't able to get IV curves. Because the tin sulfide self-shorted, and we weren't able to make contacts to the lead sulfide, so. AUDIENCE: Make enough as to make up for what would be the drop in-- [INAUDIBLE]. KATHERINE: So Danny has taken tin sulfide IV curves before the encapsulation, right? DANNY: Right. So I [INAUDIBLE] encapsulation and they do what you see here, pretty decent IV curve. But before we did the encapsulation, the cell shorted and [INAUDIBLE] died. So we couldn't-- there's no point in doing IV curve after the test. KATHERINE: So we took them. But it's just a straight line. AUDIENCE: [INAUDIBLE] JOEL: All right, Thanks, guys. Let's thank our speakers again. [APPLAUSE] JACKSON: All right, this is Ron, Rachel, Ben, Kirsten. And I'm Jackson. I'm going to introduce the Lighting Africa project. So the main problem behind our project is the following. There's about 580 million people living off-the-grid in Africa, and about 10 million small businesses operating off-grid in Africa. And as you can imagine, after dark, they have a major lighting problem. So kids need lighting for education, for reading, for studying, et cetera. And then, families need it to interact with one another. And small businesses, a lot of them would like to continue operating after dark. And currently, what they do is they use kerosene. But there's many problems with kerosene lamps. So the first of which is that it's a major health issue. So burning kerosene is about, and like, for one night, is about smoking two packs of cigarettes in terms of health issues. In addition, it's very expensive for these families. They can barely afford to eat. And so buying a weekly supply of kerosene is extremely hard for them. And furthermore, actually the widespread use of kerosene, although low per capita, is causing a large increase in CO2 in the atmosphere. So this brings up a large opportunity for us. And the World Bank Lighting Africa project has proposed the following solution. And that's solar portable lanterns, or SPLs. And these are small devices, maybe this big, that have a compact fluorescent or an LED on them. And they have solar panels on top. And you set the whole device out during the day. And you can bring it in during the night and use it to light your homes. And these are, right now, of a fairly small market share, compared to the kerosene, but are growing very substantially with a 40% to 50% projected annual growth. And as you can see from the graph, I'm sure you guys are aware Africa has a huge solar resource as a great opportunity. So the goals for our specific project are the following. How will next generation solar technologies help bring solar PV lights to rural sub-Saharan Africa? So our specific project is the following. How we looked at the next generation of solar, so future technologies that are emerging in the next six months to 10 years, and how they're going to specifically affect our project. So to do this, we looked at a few different time frames. We looked at the six to 12 month time frame, the one to five year time frame, and the five to 10 year time frame, and made technological prospectus of each one, looked at a wide gamut of different technologies, and selected a few that we thought would most positively affect our project. And we focus on the one to five year time frame. Because we thought that this would be the most important to look at. So to do this, we looked at a few, a couple different consumers. So the first consumer is a low cost consumer. And this is somebody who can barely afford to eat every day and, really, buys kerosene, maybe, on a week-to-week basis. It's extremely expensive for them. So we need to minimize the costs of these devices as much as possible so that they have a low buyback time with compared to kerosene. And these devices are often integrated. So there's a solar panel on the device. And you put the whole device out during the day. And you bring the whole device in at night. The second consumer we looked at is a higher performance device. And so this is a consumer who, maybe, has a little bit more money although still very poor relative to our standards. And then, often these devices are separated. So you'll have a separate solar panel that goes out during the day. And then, you'll bring either the solar panel in or leave it outside. And maybe there's a cable that connects it. And sometimes, these are even on the roof. And these have a little bit some added features that I'll talk about in a little bit. So to look at this, we looked at a few different metrics. So the first, obviously, is cost. And that's the most important metric. So we looked at dollars per watt of all these different devices, projected dollars per watt. And like I said before, minimizing cost is the most important thing. Because these people need an attractive device that will pay itself back fairly quickly in terms of kerosene, and will be affordable. The second thing we looked at is a performance, so efficiency, basically. So we looked at the parameter watts per centimeter squared. And so we need to minimize the size so that it can fit on a device if it's integrated, or minimize the size of a solar panel if you're going to lug it in and out every day. And as performance goes up, you can get a few added features. So one of the, actually, big problems that, maybe, is surprising is there's about 50 million off-grid mobile phones at the current time. So these people live off-grid. They need a place to charge their mobile phones. And often, they have to bring in to the city center, or they use hand cranks or something to charge their mobile phones. And it's expensive and difficult for them to charge. So adding a charging feature, to these devices, can be extremely helpful and save a lot of money. In addition is you get higher performance devices. You can start adding radios, fans, and TV to improve their quality of life. So now, I'm going to pass it off to Kirsten. And she's going to talk about our six to 12 month prospectus. KIRSTEN: Great. Thanks, Jackson. So it's important to note that the Lighting Africa project doesn't actually manufacture photovoltaic devices. That is they actually cooperate with small companies which are risk averse, which means at least in the next 12 months, they're interested in the most established manufacturing infrastructure. For that reason, in the six to 12th month range, we've just looked at silicon. So we've compared mono- and polycrystalline silicon with amorphous silicon. Of course, as you all know, monocrystalline silicon has a higher efficiency but, also, higher price per watt peak. It's important to note that in the Lighting Africa project because the lifetime of these devices is not limited by the photovoltaic component, but rather by the LED, the CFL, the battery, it opens up an interesting opportunity in that the photovoltaic component doesn't have to last as long as an installation you might put on a roof in Cambridge. It could only, maybe, be three to five years in lifetime. And for that reason, we could decrease the cost of the encapsulation materials, the commodity materials, the glass, and the backing that would normally be required. So amorphous silicon might be a better opportunity in this application. So let's compare crystalline amorphous for the two consumers that Jackson mentioned in the beginning. For Consumer 1, who's seeking the lowest cost option, again, we want an integrated module in the solar portable lantern. So for that reason, amorphous silicon doesn't have a high enough efficiency to integrate the device into the SPL. It would require 400 centimeters squared. Because the devices are so small, we're limiting that to the crystalline silicon where, again, you're not going require as many encapsulation materials as you would in a roof insulation in Cambridge. But, still. And then, for Consumer 2, who's seeking higher performance modules, you could either use amorphous silicon, remove the encapsulation materials, and have a very large area solar cell, which you could, again, bring in and out during the day. Or it opens up another opportunity in that you could use crystalline silicon, encapsulate it with standard glass aluminum frame. And this photovoltaic module could actually exceed the lifetime of the device. That is a person could buy a solar portable lantern, to which would only last three to five years, and replace the SPL later, replace the LED in the battery but still use the same photovoltaic module. So for both consumers, low cost and high performance seeking consumers, crystalline silicon presents a better solution. With that, I'll pass it off to Ben, who will talk about the one to five year prospectus. So. BEN: So for the one to five year time frame, we looked at two different technologies, which we've, well, we considered crystalline silicon to offer kind of a baseline for cost. And then, we looked at technologies where we could achieve a cost reduction. We focused on micromorph silicon thin films and organic solar cells. These are both kind of thin film type technologies. You might wonder why we didn't include cad-tel or CIGS. The reason is because since they contain cadmium, they have to be recycled at the end of their life. And we don't think that was very realistic in this context. So first, I'll talk about micromorph silicon. And this is a tandem thin film cell with amorphous silicon and microcrystalline silicon. It's on the verge of commercialization within the next few years. Cost estimates are about $0.70 a watt. And efficiency estimates are about 10%. So first, we considered the low cost option. How inexpensive can we make a basic solar portable lantern? So here, you see the historical cost in 2010. And then, under various scenarios you have the cost estimates for 2015. The crystalline silicon was assumed to be the same as it is today, basically a dollar a watt. Micromorph silicon was assume to be $0.70. And then, on the far right, we have kind of a really optimistic scenario where we've reduced cost even further by reducing encapsulation materials just for this application. And so we're down to $0.55 a watt. What you can see is that the costs are going to go down regardless of what PV platform is used. And micromorph silicon provides about a 5% additional cost decrease compared to crystalline silicon. And that doesn't sound like a lot to us perhaps. But it's not insignificant in this context. Next, we looked at these high-performance options. Because since the cost of the device is dominated by the non-PV materials for these basic devices, as you add more PV material, you leverage the low cost of the PV. So we split us up into three different high-performance products-- the basic, the medium or multifunctional, and the ultra high. And to kind of give you an idea of what these are capable of, the basic might be able to provide about five hours of light a day and a cellphone charge. The medium one could charge 20 cellphones. So this could act as a community sort of cellphone charging center. And then, the high-performance one could even power things like sewing machines, TVs, things like this, so lifestyle changes or entrepreneurial opportunities. So now, Rachel will talk about organic PV. RACHEL: OK, so the second low cost alternative to crystalline silicon, that we considered in the one to five year range, was organic photovoltaics. And organics are unique because they're made up entirely of carbon-based plastic material, which means that they're manufacturing costs can actually be substantially lower than that of silicon. And in fact, they're forecast only cost $0.50 a watt by 2015. And unfortunately, their efficiencies are less promising. Because they're only about 1% to 3%. But Heliatek, which is a Dresden based company, has recently set a new record for almost 10% with their tandem organic cell. So the first thing we're going to look at is how organic PV could play a role in the inexpensive SPL for the first customer. And this is a plot of the module area required for various power generation capacities. And it compares 20% efficient silicon with 3% and 5% efficient organic PV. And basically, what this shows is that to generate the same amount of power as a silicon cell, you need a substantially larger area of organics. Because it's so much less efficient than the silicon. So this affects the first customer. Because in order just to generate the 2.5 watts necessary for the SPL, you actually need over 800 square centimeters of 3% efficient organic material, which is not reasonably something that you could fit onto a lantern that you have to carry around with you. So because the first customer requires a solar panel that could be entirely integrated onto a handheld device, its organic PV doesn't really represent a great option, at least in the one to five year term, because it's efficiencies are so low. OK, so the second thing we're going to look at is how organic PV might be useful to the high-performance modules that are entirely separated from the devices that they charge. This is the second customer. So this is a plot of the manufacturing cost of the entire device and versus the power generation capacity. And it compares a dollar per watt silicon with $0.50 and $0.35 per watt organic PV. And basically, what this shows-- this is pretty much the same plot that Ben just showed except it compares organics-- is that as we look at a more high-performance device, the cost savings, that you get from switching from silicon to organics, are actually quite substantial. And in fact, for the same cost as a given silicon device, you can actually get almost twice the power generation capacity by switching to organics. So for this reason, and also because a module area is not really a constraint in this context, because we're just talking about building a module that you're going to leave in your yard, and not carry around with you on a lantern, organic PV is actually an excellent option as a low cost alternative to crystalline silicon in the one to five year range despite its very low efficiency. OK, so the conclusions for the one to five year term, basically, are that the manufacturing cost of the SPL can be moderately decreased by switching from crystalline silicon to micromorph. And there are greater cost reductions available in the higher performance modules. Because as the lower costs of micromorph and organic PV can be more fully leveraged as we consider a device with greater power generating capacity. So with that, I will give it to Ron for the 10-year. RON : OK, so moving on to the five to 10 year prospectus, this had a unique challenge in that, oftentimes, research cell modules will have a huge barrier to commercialization. And these are things that the probability of them actually realizing commercialization, in the five to 10 year period, decreases significantly. So what we did is we tried to look at modules that we're not like thermal photovoltaics or interband gap transition cells. Because we were afraid that the probability of those actually reaching commercialization were very low. So one option that stood out to us was actually an extension of the one to five year period, was the mature organics as the benefits that both Ben and that Rachel mentioned. Specifically, that as the areas of the cell get larger, they could have a very good cost leverage. Because they could power other household items such as TV, fans, and radios. Granted. also they have extremely attractive dollar per watt peak performances because of their extremely low cost and moderate performances. Also, we expect that commercialization will be set to begin and that, actually, the efficiencies of commercial scale modules will reach 5% to 8%, which are actually pretty good performance modules. Another thing that stood out to us was actually cadmium-free CIGS technology. And as Ben mentioned, cadmium has a huge environmental impact and would require a recycling process, which is not practical given this context. So we looked at a cadmium-free CIGS technology that would actually have the benefits of thin films in their low cost and moderate performance, but not have the actual environmental concerns associated with them. So currently, in lab, there have been a few options to replace the cadmium sulfide buffer layer. There's been zinc oxide, and also zinc indium selenide compounds that can, actually, be grown epitaxially in the manufacturing process, making it really easy for roll-to-roll processing. So currently, in the research cell modules, they are reaching about 10% to 12% efficiencies, which is really promising given they're actually, it hasn't been studied for that long. So we expect that in the five to 10 year period, we could actually realize this high-efficiency cadmium-free CIG cells, especially since manufacturing processes have already been developed for the CIGS cells. So for the long-term conclusions, we think that the cadmium-free CIGS modules be great for Consumer 1 because of their moderate performance and their attractive dollar per watt peak metrics, making them, actually, integratable on a small area, and providing the 2.5 watts for the SPL. For Consumer 2, we actually thought a large, mature organics module in their backyard would be great because of their really low cost, and also because of their ability to possibly charge other household devices as their performance increases. And these are, basically, the conclusions of all my other colleagues in front of me. The six to 12 month mono- and multicrystalline are the best options. In the one to five year period, micromorph and organics will play a larger role. And in the five to 10 year, organics and CF CIGS stand out as very attractive options. So in conclusion, next generation photovoltaic technologies definitely have the capability, especially as alternatives to current crystalline technology, to provide a green and positive difference in the Lighting Africa project. [APPLAUSE] AUDIENCE: Have you worked with anybody, like D-Lab or any other nonprofits that have done this kind of work before? Because, sometimes, it's about the relationships that you can build with the companies, the kind of discounted products that they can give you. And that's, sometimes, kind of takes over. JACKSON: Yeah, so, I guess, we skipped this slide. Or we didn't get to it. But we actually worked with the World Bank, the Lighting Africa is with the World Bank. We also worked with people who were working on it at Humboldt State University, and talked a lot to them. And right now, so one thing that's important is that the Lighting Africa project is, they provide a lot of education and information to companies. But the companies are the one that actually build the devices. Does that answer your question? AUDIENCE: So it's up to the companies to decide which one [INAUDIBLE]. JACKSON: Exactly, yeah. And so the Lighting Africa provides information to them. And our report will be provided to the companies as well. But in the end, it's up to them to decide what's most profitable. KIRSTEN: And also, there's several companies that the Lighting Africa project works with. And each company provides a slightly different solution. So there's a whole spectrum of the number of watts you have, the number of kilowatt hours. So there are several different solutions based on the different companies. AUDIENCE: So when [INAUDIBLE] dollar per watt, what about thermal degradation for all [INAUDIBLE] that could be a major problem, right? I mean, like [INAUDIBLE] itself, I mean, that's why they're not, I think, [INAUDIBLE]. RON : So what's, actually, really positive about this project is that we're not really looking for high lifetime devices. So it's, actually, limited by the batteries and by the other components of the SPL. So that's only, they only have about two to three year lifetime. So we think that even though, obviously, Africa's hot, there the thermal degradation will actually not be the limiting factor in this case, especially if they're encapsulated with relative-- AUDIENCE: So that could increase the cost, right? RON : No, actually if you reduce, since the cost is dominated by non-PV materials, we could actually reduce the encapsulant costs significantly and still have a relatively good lifetime. AUDIENCE: So could you comment on that, actually, a little bit? What kind of lifetimes would you expect out of the current production organic cells? RON : [INAUDIBLE] JACKSON: Yeah, Rachel. [LAUGHTER] RACHEL: OK, great. So, oftentimes, you'll find that current commercialized organic cells only last a couple of days actually. Although it has been proven that certain cells can last up to three years, which would be sufficient for the lifetimes that we're looking at. And this is also in the one to five year range. We're not talking about doing this tomorrow. So, hopefully, that sort of metric would be improved within that time frame. AUDIENCE: And you guys have a sense is it an encapsulation challenge or a fundamental materials challenge? RACHEL: I think it's a fundamental materials challenge. It's the actual-- JACKSON: Yeah, and this actually gives a great place for organics because of our time frame. Usually in the US or, the modules last for 20 or 30 years. But here, the lifetime of the device is really two or three years. So it's a great opportunity for organics. KIRSTEN: Yeah, and then the opportunity for decreasing module costs. It just doesn't go all the way across the board. It's more important for silicon than organics. Because, again, organics aren't as [INAUDIBLE]. RON : And also I know that hydrolysis of the carbon-based compounds is one of the big problems. So they're really not good for, like, relatively humid environments. And in this case, we're going to be in the sub-Saharan African desert. So it's going, that won't be an issue. AUDIENCE: Did you, do you have any idea about what market, what the size of the market would be for different possible costs that you outlined on some of those line graphs? Say your total lantern cost is $40 or something. How many people can buy it if that's what the price is? KIRSTEN: So we've, the top 10% of the population in sub-Saharan Africa earns 40% of the GDP. So there would be very few people in the second customer bracket. Whereas, most people, I guess, earn less than a dollar a day and would be our target for the first archetypal, as the first archetypal customer. AUDIENCE: So if most of the people earn less than a dollar a day, are they going to save up couple months of income somehow in order to buy a $30 lantern. That's what I'm asking. JACKSON: Yeah, so currently, the buy back time on these are about eight months. So it's very significant. And that's why there's not widespread proliferation. But right now, the Lighting Africa project is working on education. And in areas that they've started these educational programs, the proliferation of these products have actually increased dramatically. And people have started saving up for them, and buying them. And actually, there's one of the main problems with the project that there's a wide scale, there's, like, Chinese companies that come in. And they'll sell very, very cheap devices that are extremely shoddy. They last maybe a few weeks or something. And they're really cheap. But they don't really do too much good. So increasing education about those, and the Lighting Africa project gives certification of devices that will last a few years. And so increasing education and decreasing buy back time, which is one of the main things we'll increase. AUDIENCE: Sounds like you do consumer reports. JACKSON: Yeah. RON : And also, one of the, I mean, this is actually more towards customer too. But one of the positive things about this project is that communities could pool resources together and actually have their household items be charged together in one person's house, in the middle of the community. Could have that. AUDIENCE: Yeah, I have a question sort of like all these lights I'm assuming use batteries, like the cadmium batteries or [? lithium ?] batteries, and so on, so forth. And those are hazardous materials. And I'm sure they're either dealing with them or they aren't. So given that fact, doesn't that make your assumption that you ruled out cad-tel or CIGS cells to be incorrect in that you already have hazardous materials in the batteries. JACKSON: Right. So right now, I think, for the most, they are [INAUDIBLE] in these batteries. Most of the batteries are fairly well sealed and such. But I think, ethically, we couldn't suggest more hazardous materials. And so you're saying that just because there's already a problem, why don't we just add another one. I don't think that's necessarily a good option. But you're right about the batteries. AUDIENCE: So for your tier one customer, your lower income customer, you're pretty much only providing them lighting. Is that correct? Or do you think-- JACKSON: Right. AUDIENCE: --that there's options? I mean, you have a lot of graphs that had area of device in cost, and for the different technologies. Do you have any idea, if I wanted to charge a cellphone, where we'd have to lie on those graphs? RACHEL: Yeah. It's a good question. JACKSON: [INAUDIBLE] appendix. KIRSTEN: Should we skip to the appendix? RON: Yeah, of course. BEN: Yeah, so charging a cellphone-- AUDIENCE: I mean, so if you want lighting and cellphone, how much does that add to the cost? BEN: Yeah. AUDIENCE: Is there a larger market for that? BEN: So there could be a very large market for that. Because it doesn't, actually, take that much energy. It takes about four watt-hours to charge a cellphone. So for a very basic device that maybe produces nine watt-hours a day, you could get five hours of light, for instance, depending on the efficiency of your LED, and, then, also charge a cellphone. So it's actually, it's not a great threshold to overcome to be able to charge other things, especially cellphones. JACKSON: And the non-PV aspect of the cellphone charge is also very cheap. AUDIENCE: Could you please comment again on why you say that, or do you see higher potential in organics for the one to five year period than in cadmium-free CIGS cells? So why did you get to the conclusion to recommend organics for the one to five and CIGS for the longer term? RON: We wanted to be conservative. And cadmium-free CIG cells have only been recently not discovered, but researched in the past year or two. So we wanted to be kind of conservative with our estimates and take into account that there might be big hurdles to commercialization. So we put in the five to 10 year period. Thanks. AUDIENCE: So one of the interesting things about electricity generation in Africa, it seems, is the lack of a legacy system. And so you really do have this question of like, what is the most cost effective way of delivering power today rather than what's the most cost effective way of delivering power when you have electricity grid? But what, do you guys have a sense of sort of the cost per person at which electrification by grid becomes a compelling thing to do? JACKSON: Yeah, that sounds something we looked into. [LAUGHTER] JACKSON: Definitely, I think the cost would be pretty high, at least compared to these devices. I mean, of course, you have a lot of, it's very, very spread out, and a lot of small villages, and what not, which these are targeted towards. One of the other ideas, one of the other concepts that was looked at was a larger scale version of this where you have a large PVA ray or a few panels in the city or the city center. And then, all the villagers could come in and charge your batteries that way. And they all have little [INAUDIBLE], which was another model that is being looked at. AUDIENCE: So it sounds like what you're saying is that if you push more towards concentrated populations, then the hereto kind of becomes more favorable where you can have larger installations and central locations that communities pool together and share. JACKSON: Yeah, that's where we're going. BEN: Yeah, there's two different demographics. So you have your off-grid suburban. And then, you have off-grid very rural. So often, in the relatively high income bracket, the tier two, these are people living in kind of off-grid suburban areas. There just outskirts of cities and things like that. JACKSON: And another thing, actually, is that we talk about off-grid a lot. But the actual grid in Africa is so poor that these products are actually very applicable to them. Some people have blackouts daily or weekly, such that these devices become necessary as well. PROFESSOR: All right, let's thank our speakers again. [APPLAUSE] STUDENT 1: So hi, everyone. We are Team 6. And we are going to talk about the potential of simulation softwares to predict the photovoltaic performances. And our team members are [INAUDIBLE], Noami, myself, [INAUDIBLE], and Omar, here. OK, so this is the figure that you guys are very much familiar with. So we all do know that silicon solar cells show the benefit of high efficiency, whereas the efficiency of the thin film, or emerging photovoltaics, are relatively low. But they are definitely under a spotlight. Because they do have the advantage over the silicon such as higher absorption efficient, and lower manufacturing costs, and the flexibility. So we are going to simulate based on the materials used in these types of solar cells. So our model, in general, because they do have a lot of benefits, such as it is very much useful. But it is impossible or impractical to create certain experimental conditions. And also, [? doing ?] assimilation is much more efficient in terms of money and time taken. And also, we can do the fundamental and simple understanding at a very small scale such as atomic scale. But there's one very critical limitation. That is the accuracy problem. So this means in many cases, the accuracy matters. Like it's not really accurate as the [INAUDIBLE] happens. So the system to be modeled will be explained in detail by following speaker. However, if I speak, if I say very briefly, it can be divided into three. First, all organic such as P3HT and PCBM system. And the second one is all inorganic, such as lead sulfide quantum dot and titanium [? fullerene. ?] And the last one is some combination of those, so called hybrid cells. So our aims here, our first, we are going to introduce the two simulation packages and, then, followed by the comparison of the performance by, sorry, performance from the simulation with the values from the literature. And in the detailed study section, we basically change everything in the simulation package, such as layer thickness, temperature, band energy levels including the electron affinity and band gaps, and lastly, the carrier mobility. And then, the presentation will be concluded with sensitivity studies here. STUDENT 2: So the question raised as to which simulation packages we should use. So you all are familiar with PC1D, which is industry standard for simulating crystalline silicon. However, given that we're simulating organic materials, inorganic materials at very low transport, we opted to go with more generic simulators called AMPS, from Penn State University, and SCAPS. And I'll discuss why. So with these organic materials, and these thin film materials, traps and recombination are a huge issue. And the question arises as to how to model these traps. So at the highest level of abstraction, that most top-level view, you can just look at the bulk recombination time and simulate with that, which is the approach that's used by PC1D. The second level of abstraction is to break up your bulk recombination times into the different components. And in particular, Shockley-Read-Hawk recombination is very important for these materials. So as you remember from class, you basically have traps between your conduction band and valence band. You have [? curves ?] that are in these two bands. And when they recombine, that effectively reduces your current. So both AMPS and SCAPS actually allow for simulating these physical traps. So you can simulate it with-- and these traps have different spectrums with respect to energy. So for example, you can have traps that decay exponentially from the band edges. Or you can just have discrete level traps that are certain energy level. And you can have traps that have a Gaussian distribution. Both AMPS and SCAPS allow for that. At the same time, SCAPS allows for spatial non-uniformities for traps. So for example, let's say if I have a layer, and at the top layer, there's a lot of traps. But by the time, by the bottom of the layer, there aren't that many traps, I can model that in SCAPS. However in AMPS, I only have, I can only model a uniform distribution of traps spatially. So I can't have any Gaussian distributions or anything like that. The second key component for solar cells are your contacts. For PC1D, you can only feed it shunt resistances and series resistances, which is nice in that it's convenient. But if you don't know them or if you can't measure them because you don't have the cells, then that's a hindering factor. So therefore, it's nice to be able to model the physics of your contacts by taking the barrier height of your metal and semiconductor, and by looking at the recombination velocity. So both SCAPS and AMPS allows for that. And at the same time, SCAPS also allows for modeling with your shunt resistance and series resistance. So that's nice in that if you have those measurements, you can just do that instead of mucking around with the physics. So other considerations include the total number of flares that you can simulate. AMPS is actually, has an advantage in that you can simulate up to 30 layers. So if you have graded materials, it's a good approximation to be able to just simulate 30 layers. They all pretty much do same basic set of simulations, including simulating under dark conditions, under illuminated conditions, and band calculations. SCAPS, and at PC1D also allow for additional measurements, like capacitance measurements as a function of frequency, or as a function of voltage, which is very nice for physics. Other considerations include documentation and ease of use. AMPS probably had the best documentation. You had an 80-page user manual. SCAPS was this composed of random PowerPoint slides that were thrown on the website. So it's a little bit harder to figure out. And finally, in terms of ease of use, I'd say PC1D is the easiest just because you don't really delve into the physics. You just type in values. And it simulates. Whereas, with AMPS and SCAPS, if you get your parameters wrong, it won't simulate, or it'll freeze, or things like that. So here's this brief run through of the user, of the interface for SCAPS. So this is the main page where you set your measurement parameters like your lighting conditions, your voltages. And then, from there, you can define the structure that you're simulating. So in this case, this is for a p-i-n amorphous silicon structure. And then, you can define the properties for each of those layers. So this, for example, are the properties for the p-type layer. So these are basic material properties like your electron mobility, your band gaps, so on, so forth. And on the left, you can define your traps. As for AMPS, it's got a little bit more of a minimalistic layout, which is more pleasing to the eye. So this is like your basic page. And then, from there, if you can define each of your layers. So with that, I'll pass it on to Naomi. NAOMI: So for six organic materials and six inorganic materials, we found all their physical properties from literature. So we use either well-known properties, or the most common properties for these materials for the most common crystalline structure, or the crystallinity that we would use for solar cells. . And for some materials, it was very hard to find certain properties, such as the density of states, for example. And we had to approximate these numbers. And then, we used these properties, these basic properties to compare the performance from literature versus the values that we obtained from AMPS and SCAPS. So for inorganic solar cells, we can see that there's a good correspondence between literature values and both simulation tools. And this is likely due to the fact that inorganic materials have more similar properties to silicon than organic materials. And the simulation tools were based on silicon when they were created. So we looked at the open-circuit voltage, short-circuit current, the fill factor, and the power conversion efficiency. But then, when we had an organic component to the device, we see that there are more significant discrepancies between literature values and the simulations. So for example, in this case, we had usually an overestimation of the performance with SCAPS and an underestimation with AMPS. And we also have to note that we're expecting lower performance from our simulations than the literature. Because usually in literature, the values are reported for bulk heterojunctions. So a mixture of the two materials while in the simulation, we use only flat bilayer heterojunctions. So here we can see that we have been very little fill factors for SCAPS and very high efficiencies. So that's the main discrepancy. And for organic solar cells, we also obtained a very high efficiencies from SCAP as well as open-circuit voltage, and some very significant underestimations from AMPS, of the short-circuit current, and the efficiency. But due to the [INAUDIBLE] use of SCAP, AMPS was taking longer to run. And it was more sensitive to small changes in property. Sometimes we would have convergence failures. So we decided to look mostly at SCAPS. And we varied some parameter properties to look at how these softwares would respond to changes in the physical properties [INAUDIBLE]. STUDENT 4: So this is a where the detailed study comes. And the first thing we have looked at is the temperature [INAUDIBLE]. So only a representative figure is shown here. So as you learned in the class, if we increase the temperature from 250 to 350, the efficiency and open-circuit voltage is the equation. And this is kind of expected, because decreasing. So we can explain the reduction of open-circuit voltage by, yeah, so if we increase the temperature, the electrons gets easier to excite to the conduction band. So that effective band gap is actually decreasing. And also we didn't show the short-circuit current data here. But there's a slight increase of short-circuit current. And this could be, also, explained by the same thing, like it's easier to get excited to the conduction band. And the second thing we have looked at is layer thickness. And also, here, there only a representative figure is showing here. So if we increase the thickness from 0 to 1,000 for the [INAUDIBLE], the efficiency and the short-circuit current is initially going up. And after certain point, the efficiency, [INAUDIBLE], is slowly going down. And this is also expected. Because for the observing material, when we first initially increased the layer thickness, it gets more, initially more light. But after certain point, absorbing more light effect is already saturated. So increased layer thickness just means the more distance to the electrodes. However, whereas for the acceptor material, there's no significant of effect of the layer thickness on the performances. AUDIENCE: It's like will you help me with this graph, which ones are-- STUDENT 1: Question? AUDIENCE: Are the dotted lines the current? STUDENT 1: Yeah, so the solid line is efficiency. And the dashed line is current. AUDIENCE: And one is for SCAPS and one is for AMPS? STUDENT 4: No, All the simulations done in SCAPS actually. STUDENT 5: So I'm going to talk about how you change your particle size, you will change your band gap, and also change your result of the simulation. So as you know, changing your particle size of the [INAUDIBLE] will change the band gap of the material. And this is very important for organic solar cell. From the table above, you can see when we change the lead sulfide, lead sulfide particle size from bulk material to 2.4 nanometer, you can change your band gap from 0.4 increase to 2.1. So we will use these numbers into the simulation tools to simulate the efficiency of two system. One is lead sulfide zinc oxide. The other one is lead sulfide titanium. So you can see the white dot on the figures. That's mean efficiency. The highest efficiency occur at the band gap is around 1.5. So that's mean the particle size is smaller than the bulk material. So which just mean if you use smaller particle size with the higher band gap, you can get a higher efficiency because two reasons. The first is your [INAUDIBLE] increased because your band gap increased. And the second is that your [? fill ?] factor also increased in both systems. So how is this simulation [INAUDIBLE] of this simulation compared to the literature? So we choose the lead sulfide titanium system. So we used the structure in the right, above figures. Now, there's titanium lead sulfide. And then, the [INAUDIBLE] is gold. And then, the world record efficiency, right now, is around 5.1%. So the particle size of this literature used is five nanometer. So find the nanometer in our simulation. The result is around 4% to 5%, which is, it's very close to each other. But our simulation result can help the future direction of this research regime. Because you can see the simulation result. If you change the particle size to the lower, to the smaller particle, you can increase the efficiency to 8%. So that's how simulation can helps the research. So I would just pass to the sensitivity part. STUDENT 6: We also to determine the critical parameters to which [INAUDIBLE] that the performance [INAUDIBLE] we calculated sensitivity. Sensitivities are calculated from changes in efficiency, [INAUDIBLE] by varying parameters, simulation parameters, or material properties. They were expressed in terms of change, a 1% change of efficiency in the parameter of relative to basis value. So as we can say, we found four major parameters. And the [INAUDIBLE] two things out, there were no band gaps and electron affinity, which are well-known material properties. And others are [INAUDIBLE] thickness and temperature, which can be controlled the [? web ?] by it's [INAUDIBLE]. So from this sensitivity [INAUDIBLE], we strength, I mean, confidence of our simulations. NAOMI: So to conclude, after doing all these simulations, we found that the these simulation softwares were useful tools to simulate efficiency trends as opposed to absolute values of the performance parameters, which remain challenging to simulate, especially for organic compounds. However, the sensitivity analysis was good to validate that the most well-known material properties seem to have the highest influence on the efficiency. And we can usually also control temperature and layer thickness, which we're also very, very influent for the efficiency. So this project was useful to collect a lot of material properties. And we could potentially create a database of properties that could be useful for solar research. And this could be included in such, in a project such as the Materials Project, which incorporates lots of databases about materials for different applications. And then, if you could connect these material databases to the simulation tools directly, it would be good for a user who simply wants to input a pair of materials. And then, we could get the output as performance parameters directly. And finally, we want to thank Professor Buonassisi, as well as the research groups from Gent University and from Pennsylvania State University for the use of their simulations softwares. [APPLAUSE] AUDIENCE: You said you had a problem modeling the bulk heterojunction organics? I mean, because it's bulk heterojunction instead of a bilayer. Did you try just modeling a single layer with the properties that are a mixture of materials? NAOMI: So. STUDENT 1: Yeah, so actually, we tried to do bulk heterojunction modeling as well. But it is limited by the simulation capabilities. So we can define the layer. Yeah, so you can define several layers in here. But we-- STUDENT 4: We can show the software to them. STUDENT 1: Uh-huh. So if we want to do the bulk heterojunction, we could do, just put one layer where the effected material properties. However, the hard thing was, the shifts at the band gap, actually, the parameter. However, you could not define the band gap. Because even if we did just one layer, it is a mixture of two different layers. So it's hard to define band gaps. So it will mess up everything. So we couldn't. AUDIENCE: Your last slide made me think of some recent publicity from a group, Harvard, I think. Some guy trying to use the distributed computing idea similar Folding@home or [? Study at Home, ?] wants people to download his screensaver, and screen thousands of different organic materials, I think, for photovoltaics. NAOMI: Yeah. AUDIENCE: Do You know what software they used for that? NAOMI: Don't know. AUDIENCE: Custom software. AUDIENCE: Custom software? OK. So can you guys go back to the, you had an array of bar charts, at some point, when you were comparing the different simulation tools. |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | 17_Modules_Systems_and_Reliability.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: So modules, systems, and reliability. What we're going to do is talk about how we go from the cells, or from the films, to full modules and, finally, to systems. So our first learning objective is to describe, more or less, the DNA or anatomy of a PV module. This will be a bit of a prep for the visit to Fraunhofer CSE, where we'll get to see the labs and see all the materials that go into a PV module. So just establishing some definitions up front so we're all speaking the same language-- you start out with either a film. Right? If you're depositing a thin film material, which is usually divided into discrete devices using laser processing, or you can start out with a discrete wafer which has been processed into a cell. The module is the combination of many of these cells, shown very generically right here without the interconnects. Because you can tie up the cells together in parallel or in series, depending on what sort of voltage and current outputs you desire from your module. And then finally the modules are situated together in an array. That's showing a combination of six modules forming that array right there. And as you can see, modules are typically this size right here. They're usually about this size or even a little larger. They're rather heavy, so you can get a sense. They're not extremely light. The majority of that mass is coming from the glass in the front, some from the aluminum as well. And they're comprised of many materials. If you were to take this apart-- which we'll actually see the individual components at Fraunhofer CSE-- you'll notice that there are many materials involved in making one of these modules. From the back skin materials, the junction box, the aluminum framing on the side, the glass on the front, the encapsulant materials that fuse everything together-- that bind everything-- and the cells, the solar cell devices themselves. So our first step is to walk through the anatomy of a module, at least in theory, so that when we actually see it at Fraunhofer CSE, we'll have a better sense of what we're looking at that. Yeah. Ashley? AUDIENCE: Here, what determines that specific size? We've already talked about the pseudo-square and how you have to balance those two different cost parameters. But why that size specifically? PROFESSOR: Why that size specifically? We're going to get to that in a few slides. It comes, as you probably guessed, from some historical reasons. AUDIENCE: OK. PROFESSOR: So some basic principles about solar modules-- there's quite a diversity of modules when you look at them. These are examples just within one company, just to highlight the range of module types that you might see. Right here, you can see in this particular module, there's very little anti-reflective coating on the glass. So you're able to look through the front glass and see some white spaces in between the cells. We know, from the lecture on crystalline silicon PV, that when you see these little gaps in between the cells, those are pseudo-squares. Right? Those are those round ingots of Czochralski silicon that have been sliced into wafers. And then the edges have been shaven off, and you have that stop sign type pattern of a solar cell. Right? So that way you know, OK, these are single crystalline cells lined up in a module together. And there's very little anti-reflective coating on the glass, which means that you're losing a fair amount due to reflectance. The module on the right-hand side right here has that anti-reflection coating on the glass. It could be an index of refraction gradient to absorb more of the light. It could also be surface texturing to scatter the light and trap it by total internal reflection once it gets inside. There's a few different ways that one can do that. Usually with a film coating is the most common. And here on the left-hand side, we have a similar sub-cell component but much larger module comprised of a larger number of cells with a higher power output. And we can see quite a wide variance of different types of modules. That will be an important message in a few. The water rinse cycle really depends on where you're at. For the modules on my roof, we just leave them up. Here in New England, it's pretty much OK. If we did have trees that deposited leaves nearby, we'd have to clean it off in the fall, but we don't. If you're mounting modules in the Middle East, where that fine-grained sand exists and, if you're within 10 kilometers from the coast where you have the salt and the seawater in the air, you can get this really hard-to-remove sand caking the modules. And you would have to clean them much more frequently than this. As a matter of fact, I believe the numbers are-- in Abu Dhabi, the capital of one of the Emirates in the UAE, if you were to just leave your modules out near the ocean, near the desert, you would have a 40% drop in module output over a month. So that's what happens in certain environments. In certain others, like New England, we're more buffered in that regard. We have frequent rain, and we don't have the dust coming from any nearby desert. Typically, there are no moving parts, and typically there's a 20- to 30-year manufacturer warranty. Some of the newer materials that have been less tested might give, say, a 10-year manufacturer's warranty and have to offset the risk in years 10 to 20 by lowering the cost up front or lowering the price up front. So we have some basics about solar modules just to situate ourselves. Let's dive into the module DNA, since this is where the rubber hits the road. I'm going to show this to you in theory, and then we're going to see reams of these module innard materials when we visit Fraunhofer. So we start with the solar cells themselves. These are these little blue objects right there. Typically, these cells are already strung together at that point. They have contact metallization on the front side, which we deposit, say, for example, by screen printing, which you've done in the lab downstairs. And then you string them together using a machine called a tabber-stringer to connect one cell to the next, essentially the front of one cell to the back of the next. And they're all lined up like that. And you deposit layers of EVA, which is ethyl vinyl acetate. This is typically the encapsulant material used. It's shown in red right here in this drawing. It's the encapsulant material used for crystalline silicon PV. On the back side, there is a sheet of material called Tedlar. Despite it being drawn in yellow right there, it's the white skin material here. It's this white material right around here. And we have the glass on the front side. And typically, it's low-iron glass so it can transmit the ultraviolet light, or a large portion of it, to give better blue response to your solar cells. So you're capturing a larger portion of the solar spectrum. OK. So we have the different components right there. One little piece of trivia about EVA-- for the chemists in the room, what do you think this would decompose into? Let me just give you a little hand here. If you had some-- let's see. All right. So these represent methyl groups, hydrogens, carbon, carbon, carbon, double bonded to an oxygen. Anybody recognize? AUDIENCE: It looks like acetone? PROFESSOR: Yeah. So if you're looking at the decomposition of this little group right here, you could easily envision it decomposing into acetic acid from the name ethyl vinyl acetate. So if you have a thin film material that could react with the encapsulant, you could decompose your encapsulant and cause a degradation of not only the encapsulant itself, which would block some of the light going through, but of the solar cell absorber material, too. So that's why, in many thin film materials, there are other encapsulants used, like PVB, polyvinyl butyral. And some folks are even talking about getting away from glass and EVA altogether and just putting down, say, an organic resin-type material. Maybe a very hard polymeric material that could be used in a floor in a clean room, let's say. Yeah. AUDIENCE: Is the low iron content of the glass also important to prevent all impurities getting into the silicon or are we past that point? PROFESSOR: So the question was is the low iron content in the glass also useful for preventing impurities from getting in. I would say, if you're producing-- the diffusivity and solubility of impurities in a solid typically follow Boltzmann statistics, meaning they increase exponentially with temperature. It's an entropy-driven effect. And as a result of being at room temperature, the transmission, or the diffusivity and, hence, the solubility of iron inside of the bulk silicon is very low. And, even over a 20-year span, the total diffusion length of the iron from the glass into the silicon would be rather small. Now, point taken, if you were to, say, create a quartz tube to do your phosphorus diffusion at 800 degrees C or 850 degrees C and you had iron inside of that quartz tube, then it could very easily diffuse into your wafer. So it is important to be considerate of impurities and the effects thereof but at low temperatures-- close to room temperature-- there's very little risk of it moving around. Good. So Tedlar, this back skin material right here, forms an almost impenetrable back layer. You might want to put little asterisks around-- sorry, little quotes around impenetrable. Nothing is absolutely impenetrable but it is a very tough, tough material. If you were to take a piece of Tedlar and try to break it, try to rip it or tear it, even the strongest in the class would be challenged in that regard. And we'll have a chance to do that at Fraunhofer. The aluminum frame provides rigidity. Right? So the mechanical engineers in the room would understand that the bending mode of this glass, or the twist, is prevented by these rigid extruded aluminum components right here in the back. And you can see that a lot of the mass is concentrated away from the centroid, which results in a larger stiffness, prevention of a larger bending moment-- second moment, if you will. OK. So there's a fair amount of mechanical engineering that goes into the design because you want to minimize commodity material costs. You want to minimize the amount of materials that go into the module, but you still want it to last for 20 years. You still want it to be resistant to snow loads, to wind loads. You don't want it to break, and you don't want to make good on your 20-year warranty. That costs you money as a manufacturer. So the circuit design in most modules follows something very similar to this. Now, what you see when you glance at this, what you should be looking at is-- here are the contacts. Those are the leads coming out of the module. They're essentially these leads coming out in the back of the junction box. If you want to turn it around again, here we go. So here are the module leads coming out. And the question is, how do I string together the cells inside of my module? How do I string together these cells right in here to provide the greatest value to the customer? Do I connect them all in series and make just one long, snaky electronic path? Do I connect them all in parallel? But that would require a lot of wiring. Do I do something in between? And if you were to come up close to here and inspect it, what you would see is that you have strings in parallel. So you have one going from here, down and around. That's one, two, and three. Right? Three of these strings, and they're connected-- those three strings are connected in parallel. Each string is comprised of several solar cells. Now, for historical reasons, the crystalline silicon solar cell modules have typically had strings of around 36 cells connected in series. What this does is it yields a maximum power point voltage somewhere in the range of 17 to 18 volts. If you remember the VOCs of your solar cells and that maximum power point, the voltage of your solar cells are some around 0.5. And so you take 36 cells, divide by 2-- or multiply by 0.5, if you will. And you're landing somewhere in this ballpark. And what that does is it enables you to hit, even under lower light conditions, the overpotential needed to charge a rechargeable battery. So if there's a 12-volt battery and the voltage output of the battery is between, say, 12 and kind of dropping down to 10 before it starts using its utility-- when you try to start your car and it doesn't really turn over and you can start your engine, it's probably because the voltage in your battery has dropped to about 10 volts. The overpotential needed to charge it, 13-plus, that's what you're hitting with the string. And that's, for historical reasons, why we had this number of cells connected together. As grid-tied systems become more common, though, this constraint is being reduced. Over 90% of all of our solar modules today are connected to the grid, including the ones in my home. So when I'm not there and it's producing excess electricity, it's pumping it into the grid. And maybe we're using some of my electrons here. I don't know. So as grid-tied systems become more common, you care less about meeting the overpotential requirements for a battery. You care more about matching the output of your solar module with whatever inverter technology you're using to convert the DC power into AC power. Number two, why do they spec the modules that size? You notice that I could lift it, just barely. I'm a university professor. I sit in my chair and think most of the time. A person, somebody strong out there in the field mounting modules, say if Jessica in the back were a module installer, she could lift this no problem probably with her pinky finger. And that's why they're bite-sized, right? They're a single-person, liftable, installable modules. You can't make them too big, or else you start needing cranes which, of course, has its own benefits for large field installations. Wouldn't really make sense with our current mode of installation on residential. AUDIENCE: [INAUDIBLE] PROFESSOR: Yeah. AUDIENCE: If you had a battery-- if you were charging a battery with the module, you also wouldn't want to have too many volts, right? Maybe that's a really basic electronics question. PROFESSOR: Yeah, so the question is, do you benefit by having larger overpotential when recharging a battery? Well, I'm not a battery expert, so I'll definitely caution on this. I would imagine that there would be some law of diminishing returns. You would be limited in terms of the recharge rate, not by the overpotential but by some other component like the electrolyte diffusivity, let's say, inside of your system-- as an example. But I would defer that to the battery experts. My impression, though, is that you get a law of diminishing returns. Packing fraction in the modules. So we talked about the different cell sizes if you're processing discrete cells. The higher packing fraction enables lower glass encapsulant and cost per watt peak. The lower packing fraction, you get to play some games with optical concentration. For here, for instance, if you have space between your cells, you can have a diffuse Lambertian scatterer in the back which will reflect the light at a different angle, scattering it. And that light could be trapped by total internal reflection inside of the glass if the angle reaching the glass is, say, 15 degrees or greater. So you can play some games there. That's why the Tedlar back skin being white is typically used in most solar modules, so that we reflect and then scatter and then trap the light that comes in in between the cells. So to put this into another fancier, three-dimensional framework-- we have our interconnected cells. We have our encapsulant above and below. The encapsulant material is an interesting thing. It's solid at room temperature but it begins to flow at around 150, 175 degrees C which means, if you've ever put a plastic bottle in the oven and you've noticed how it begins to melt and sag and droop, that's essentially the type of flow that you would experience inside of this encapsulant material. And it flows around the cells. And it fuses them together with the glass on top, and they stick together once they cool down to room temperature. That forms what's called a laminate. Because now you have the back skin-- the frame isn't shown right here-- you have the back skin material. You have the encapsulant, the cells, the encapsulant on the other side, and your glass on the front side all together forming a nice stack of materials. And they're stuck together, or bound, by the encapsulant. That's called a laminate. It doesn't resist bending very well, or twist very well, because it doesn't yet have the frame on it. But it certainly can be transported around the factory and moved from one station to another without risk of, say, the cells falling out. They're pretty well fused there. In terms of moving to new materials, there's a lot of work right now to squeeze the commodity costs out of solar module manufacturing. If you look at the percentage of cost of commodity materials in the solar module, it is quite high. And so people are focusing on new types of transparent front surface materials that let the light in, glass replacements. Right? You could envision-- we have many examples of really hard, resistant, polymeric materials in our daily lives. People are looking into those. The encapsulant materials-- ethyl vinyl acetate is fairly expensive, and there aren't that many companies that manufacture it. And so people are looking to options to replace it. Tedlar even more so, that back skin material. Although the challenge there is a lot harder, a lot tougher. No pun intended. It's a really strong material. And a lot of our modern chemistry or chemical engineering goes into making and designing it. And then finally finding some replacement for the frame. People are talking about frameless modules. That means taking off this aluminum piece right here on the side and just leaving the laminate and somehow providing the mechanical rigidity to the laminate and avoiding the cost of the extruded aluminum components. So a lot of work going in in module design. This is a picture of folks mounting the aluminum framing onto the module. These are essentially large steel components that are pushing the aluminum into the module and making sure that there's a snug fit between the aluminum and the module itself. As you can see by the size and shapes of these cells, that gives you an indication of what company that is. Module technology-- there's some work as well going into replacing-- let's see, here we go-- replacing the tabbing-stringing step. Right? So this is a machine the tabs and strings the cells to one another. We'll see one at Fraunhofer. It's really impressive when you see it go, but it's also quite expensive. And so if you could get rid of the tabbing stringing-- imagine just lining up the little cells like cookies on a cookie sheet. And they all are interconnected by contact with whatever back skin material is on the back that is printed, let's say, with the circuitry already. You would wind up in a situation like this, where you have your cells in direct contact with some printed circuitry on your back. Boom, boom, boom, boom, boom-- and you're off and running. In this particular case, you see these little white dots here. The grid pattern on that cell is a radial spiderweb pattern that emanates from that central point, collects current from nearby-- essentially, an area about that big is collected, let's say by that little point right there. And then there's a hole drilled through the solar cell where the metal is wrapping through to the back. And that's where it's making contact with the back side right there. And there are other regions of the solar cell that are making contact with the bulk, the base, if you will. So you have both of your n- and your p-type contacts on the back side of your solar cell. And that's what's shown in this drawing. So there are attempts, as well, to put all the contacts, or connect the contacts, onto the back to enable this monolithic manufacturing. Yes. AUDIENCE: How does that not short? PROFESSOR: OK. So how is that not a short? Let me show you how that would not be a short. What I'm going to do is attempt to draw two different cell designs that could lead to not shorting. One cell design that could lead to not shorting would be-- let's say this would be our front surface. Our base is-- I'm going to draw it flat here in the back just for simplicity, but it could be textured as well. I'm going to call this a lightly doped semiconductor material. And then I could have heavily doped-- let's say this would be-- I'm going to exaggerate here just for effect. This could be my heavily n-type doped material. This could be my heavily p-type doped material. So I'm drawing p plus, n plus, another n plus, another p plus. So I have charge separation occurring at these regions right here and then contacts on the other side. This is my electron contact. This is my hole contact. This is my electron contact. This is my hole contact. And now what's happening is photoexcited carriers are being generated in here. But the field that separates them is all in the back. And so now the carriers are being separated, electrons being collected by the darker-shaded, rather, the shaded areas, and the holes being collected by these unshaded contacts here in the back. And, if I'm clever about my contact positioning, here I have kind of an interdigitated electron and hole collectors. Right? I'm able to collect both without having any sort of shunting. Another design might be-- let's see. So this is called an Interdigitated Back Contact structure, or IBC. AUDIENCE: What is it, [INAUDIBLE]? PROFESSOR: Actually, less relevant in this particular case. In most interdigitated back contact cells, it's actually n-type silicon, lightly doped n-type silicon, just because point defects tend to have smaller capture cross sections than n-type silicon and p-type silicon. So minority carrier lifetime tends to be larger. Another cell design, again, looking at cross section, you might have-- this being your solar cell-- you could add texture, but I'm omitting it just for simplicity. Now, let's see, we drill a hole using a laser right here. So we have a very thin hole. Of course, in three dimensions, there's mechanical rigidity by the bulk of the material in the other dimension. But we have this hole drilled through-- and we can make about 18,000 of these holes in about six seconds with modern laser technology, so it's really not that hard to make this happen. Although a lot of engineering effort is put into it, but it is possible. And then you could have your contact metallization, say, for electrons right here. You could have the p-n junction like this. So this, here, could be your n plus material. Again, n plus material, this, by effect, would also be. And you can have your p materials over here. And you could make contact to the back right here and contact to the back right here. And as long as there's no metal short between the electron and hole-- say, for example, you generate an electron hole pair here, the electron goes up to here. It goes through the metal to the back and gets pulled out. And the hole goes to this contact over here, and that's how you separate the charges. So this would be an example of an emitter wrap through or a metal wrap through device technology. Also shown-- MWT, Metal Wrap Through. So we have at least two different device geometries and many more device architectures that could fit in with this future vision. The problem right now is that the printed back skin materials cost about five times more than the non-printed ones just because they're very low scale. People haven't investigated it much. So any time you introduce a new technology into the market-- you'll hear it over and over again in these innovation and entrepreneurship classes here at MIT-- you have to be significantly better than the competition. If you're only a little bit better, you face an enormous barrier to entry because you have the rest of the industry pedaling along on their bicycles in a huge peloton, benefiting from each other's drafting, and you're on your own bike facing your own headwinds. Right? Trying to build everything up-- your own factory, your own expertise. It's challenging. But I'm not trying to put a damper on innovation. I'm just trying to qualify the areas of greatest opportunity and lowest resistance to change. Go ahead, prove me wrong. Make a difference. Without that sort of headstrong mentality of, gosh, I know I have something here and I'm just going to prove the world that I have it, we wouldn't have, for example, the Pilkington plate glass process working-- or float glass process, sorry. We'd still probably be pouring and grinding and polishing our panes of glass and they'd be very expensive. That was a little side story, but it took them about 10 years to develop the float glass process inside of a company owned by a family, as opposed to a publicly traded company. So they could continue losing money over a long period of time until they perfected it. And once they got it right, the whole world copied it, but it took 10 years of good investment for them to get it right. All right. Let's move on. We're going to talk about module spec sheets here. When you came into class today, you should have picked up one of these. This is an example of a First Solar module spec sheet. What it does is it hits you front and center. The company name-- First Solar. FS Series 3-- that's the name of the module, and PV module. So it gives you the mechanical description, the size, the weight. And then, on the back, it gives you all of the technical parameters that you've wanted. So on the backside of the-- we'll go to the First Solar spec sheet. OK. Well, on your spec sheet right here, you can look at some of the parameters. There's power at the maximum power point, voltage and current at the maximum power point, the VOC, the ISC, and so forth. You have a bunch of different parameters including the temperature sensitivity. Right? How much does the power degrade with increasing temperature? So all of these things we've studied in class, and now when you pick up the spec sheet you're a pro. You look at it, and you're like, oh yeah. I know what that does. I know what that means. I know how to model the energy output of the system. And that's essentially what the engineers do when they pick up the spec sheet. They read it. And then they input those parameters into their computer models, and they can predict how these modules will perform out in the field at a given location. OK. And so you have a variety of different spec sheets. These are the modules that are on my roof. They're Sharp 187-watt modules. You have, for example, right here an example of an Evergreen Solar module. Let's spend a minute here looking a little more deeply. These modules, these Evergreen Solar modules, are the same size, but there are three different power outputs. And I think we went through this already. We have good cells, medium cells and bad cells. And they're binned together in the different modules so that they can extract the maximum price from the consumer. So that's why you have the three different ratings of power for three different types of modules with, essentially, the same form factor, the same size and weight. They're just different-quality cells. Yeah. So that's that. Temperature coefficients, so forth. OK. One thing to note right here is, on the back sides of your modules, you typically have a little diagram that shows you how to interconnect one module to the next. And those are the cables that are used to do the interconnection. There are some companies trying to get fancy right now and saying, well, instead of interconnecting and then affixing the modules separately, why don't we just make these modules that kind of click together? And part of the clicking process is the interconnection electrically, and the other part, obviously, is the mechanical rigidity. So there's certainly room for optimization on module design. Yeah. Jessica. AUDIENCE: When they're sold-- when modules are sold, they're advertised for their voltage at maximum power point, right? PROFESSOR: Mm-hmm. AUDIENCE: What's a software that we should use-- I'm actually using this for another class. What's the software that we should use that maximum power point voltage into-- translate that into what it's actually going to be? [INAUDIBLE] PROFESSOR: OK. So the question became how do I go from a module spec sheet, like this, with this data right here, and translate that into number of kilowatt hours produced per unit time-- day, year-- at a given site location. The best program that I know of that encompasses all of this would be a program called PVWatts, produced by NREL. PVWatts, all together, and you can look it up, you can use it online, and you can input your specific module spec parameters into the program. Now, that said, we have a lot of the tools to do the estimates ourselves. If you ruffle back to homework-- I think it was homework number two, we calculated the output of a solar system based on the module spec parameters. And so now that we can pick up a module spec sheet and read it in a bit more detail, we can perform those estimates a little bit more accurately. Yep. Omar. AUDIENCE: Your first solar panel is a lot smaller than that panel right there? PROFESSOR: Yep. AUDIENCE: So is there a reason why First Solar goes with smaller cells? PROFESSOR: Yeah. So why is this First Solar module, the FS Series 3 module, a little smaller than some of the crystalline silicon modules? For thin film deposition, especially for this closed-space sublimation process that First Solar uses, you're limited in terms of the deposition area by the uniformity of deposition over a large region. So if you're sputtering the material, it's typically related to the size of your target and the distance to your substrate. That will limit the maximum size that you can deposit uniformly. And you probably want to deposit within plus or minus 2%, let's say. Maybe 5%. I'm not extremely privy to the precise tolerances in the PV industry in terms of thin film thickness variation, but that would be my estimate. And in terms of these thermal evaporation processes, again, it's how the machine is designed. So I imagine you could probably go bigger. Is it cost effective? Question mark. Somebody at First Solar most obviously did the cost analysis and said that this is the optimum between a variety of different factors, between manufacturability and ease of installation. Yeah. AUDIENCE: Is there any reason why they don't put efficiency on this sheet? PROFESSOR: Is there a reason why they don't put efficiency on the sheet? Well, you could probably very easily calculate the efficiency, right? There are two types of module efficiencies. There is what's called total area efficiency, where you take the total area of your module all the way up to the aluminum frame and calculate that as your area. So you take your power-- your maximum power rated at AM 1.5 sun conditions and divide it by the area. There's also a variety of other less straightforward ways of calculating efficiency, some of which take the active area of the cells only. Others exclude the frame. And so the module efficiency is not extremely straightforward parameter. It has to be specified based on what areas is assumed. That would be my estimate for why they didn't put it right on here. And secondly, their module efficiency, if you do the math, that one probably turns out to be somewhere between 11% and 11.5%. Now they're hitting 12% or so. That's not a very high number compared to, say, a crystalline silicon module. And so it wouldn't be probably something they advertise in their spec sheet. Describe how PV module power output is affected by the cell mismatch losses. Warning-- I'm going to lose some of you on this. Those of you without a very strong electrical engineering or physics background, I'm going to pick you up at point number three, so plant a flag right here. We'll come back to it. But for those who care to follow along, this is some really fun stuff that can get into why we want to match our cells properly in terms of their outputs. So the total current output-- this is the ideal diode equation without all the fancy two-diode model, recombination of the space charge region, recombination of the bulk, without the series resistance and shunt resistance. Just a very simple explanation of this ideal diode model. What we're doing is our M is the number of cells in parallel and our N is the number of cells in series. So we go back to our equivalent circuit diagrams where we have two voltage current sources that are connected in series. If you connect those two in series, now the voltages will add. So the effective voltage across the entire system, if you draw a black box around your two individual voltage producers, the effective voltage across the entire system will be voltage 1 plus voltage 2, in series. The currents-- now, if you think about the current, you can think of current-- the first order-- as a river of electrons trying to flow through your circuit. If there's any bottleneck somewhere, that's going to limit the current that can flow through the entire circuit. So the combined current is going to be limited by the worst performer. In mathematical terms, it would be the harmonic mean. 1 over effective current would be 1 over current 1 plus 1 over current 2. So what we see here is, if we have a number of cells connected in parallel-- now, instead of having them connected in series like this, you connect them in parallel-- now the currents are adding. And the voltage will be limited by the worst performer. Because the potential across both is going to be limited by whatever the lowest potential is. So when we connect in parallel, we add currents. And when we connect in series, we add voltages. That's a very simple way to think about it, and that's why we have the M appearing here up top. If you have a number of cells connected in parallel, you will essentially be multiplying the [? illuminated ?] current of one cell by whatever M it is. And if you have a number of cells connected in series, that will impact the voltage. OK. So that's an expression that is useful for N and M number of identical cells connected in either series or parallel, respectively. And in practice, if we have bad performers in the bunch, if we have one cell that's performing worse than another, either because it is intrinsically worse-- it is a defective cell that somehow got in there or became defective during use-- or it's a temporary effect. Maybe a seagull just landed on top of that cell and is shading it. Right? So whatever the reason is-- temporary or permanent-- that cell number 2, now, is a bad apple. And so if we're looking at parallel mismatches, we take what we know about cells in parallel. OK. The voltage is limited by the worst performer. And we look at the individual IV curves of the good and the bad cell. This is the good cell right here, this curve there. Right? And so this, now, is positive power. Essentially, power coming out of the solar cell is plotted in the first quadrant. We have a lot of power coming out and a large voltage for our good cell. Here's our bad cell right there. So slightly lower current outputs given by the logarithmic effect. And the voltage is lower, significantly lower, than, say, the good cell. And so the combined output right here, the current at short circuit current conditions, is going to be the sum of the good and the bad. So that's there. The voltage is going to be the harmonic mean, so limited by the worst performer. That's how low-voltage cells can really mess up parallel strings. So for example, this set of two rows of cells, this one right here, which is all connected in series. It's connected in parallel to this one and this one. And so if one of these three subsets of cells has a lower voltage than the others, the combined voltage output of the entire module will be lower. Similarly, now we've put them in series, if we have a bad cell that's producing less current than the good cell-- and notice that they're saying open circuit voltage now just for the thought experiment here in the classroom. Typically, if a cell is performing worse than current, it will also have a lower voltage. But just for the thought experiment right here, we have bad cell, good cell. Bad cell has about half the current of the good cell. So the voltage output in series is going to be the addition of the two. Boom, boom. Then you're up to here. But the current output will be limited by the worst performer. And again, the combined output of this set of cells connected in series is going to be right around here. And so based on your cell performance-- if you know that your voltage has a certain variance in your manufacturing line and your current has a certain variance in your manufacturing line, that might entice you to string together your cells in one way or another, depending on how you want to mitigate your losses. Shaded cells-- so up to now, everybody kind of gets based on Kirchhoff's laws and so forth from equivalent circuit diagrams, so we're rolling. Now we're going to get the shaded cells. And here's where it becomes a little bit more difficult to understand absent the reading in that book over there. Did everybody get a chance to see the blue book? Did it make its rounds over here? No, it didn't. Why don't we pass the blue book around? Omar over there didn't get a chance to see it yet. We're going to consider that we have 10 identical solar cells connected in series, and this one over here is shaded. What happens now? So this is described, I believe, in figure 5.7 in the book, one with the green little tab. We've assumed that the combined IV characteristic of the n good cells is this one right here and the one bad cell is this right here. So it's a little bit of a strange IV characteristic. And what happens next? If you combine the two in series and you use the same protocol that we've done so far, the voltage will add. So whatever voltage gain right here has been added over here. But the current is going to be limited by the worst performer, which happens to be this solar cell right here. And so the combined output is going to be lower-- significantly lower-- than those of the good cells. And that's what happens when one cell in a string is shaded if there aren't any fancy electronics to allow the current to bypass it. Furthermore, this amount of power will be dissipated in the bad cell when the string is short circuited. It can change a bit when you start moving toward operating conditions, but it's a sizable amount of power being dumped into that bad cell. And in practice, that cell is reverse biased. Let me show you using our wonderful PV CD-ROM exactly why that is. So here we have two solar cells connected in series. And they're both producing an equal amount of current because they have an equal number of these yellow arrows incident on them. And they're equal in every other aspect. So it's still a thought experiment right now. The current that is cycling through this circuit, in the external circuit right here-- the magnitude of that current is being indicated by the color. And, in this particular case, it's very bright. It's bright green. It indicates a lot of current is flowing through it. So, up to now, everything makes sense. This is in short circuit conditions, so there's no voltage. There's no potential across either cell. It's in short circuit conditions, and current is flowing through this circuit right here. Now I'm going to mismatch the two cells. I'm going to exchange one of the cells for a bad apple. OK. First off, I know, just from what we've been talking about so far that the current is limited by the worst performer, that the current flowing through the external circuit is now going to be lower. And the first order-- it's more or less approximate to what the current is in this bad cell. So the bad cell right here, which is indicated by the two arrows now instead of the four, this shaded cell right here is producing less current. And so I know, based on what we've talked about so far, that the combined current output of the entire circuit is going to be approximately this one. So hence you see a darker color represented, the same color that's flowing through the cell. So what happens to the large amount of current that's being generated inside of this one? That cell still has the capacity to produce a large amount of current. Where is it going? Well, some of that current is going through this cell right here, forward biasing it, which would result in a negative bias, or reverse bias, of the bad cell. Because the potential across this combined system still has to be zero. You're in short circuit conditions, so the potential here has to be equal to the potential there. So what I've effectively done is I've reverse biased my bad cell. And we'll get to why that's a problem later on. Those who are electrical engineers can already begin conjuring up ideas of reverse biased breakdown. We'll explain why reverse bias is bad in a few. Now, if you had a series of these cells connected-- a bunch of these cells connected in series-- and they were all producing copious amounts of juice, and you had one bad apple in your string, you would be dumping a considerable amount of power across that. Or you could be dumping a considerable amount of power across that one cell. And that's where this graph right here comes from, the power dissipated within the bad cell. And that's a problem because if there's any weakness in the p-n junction of that bad cell and you're reverse biasing it, you're going to be flooding a lot of current through that one little spot, which means that that spot's going to heat up. And it's going to form a hot spot, which has the potential to get very hot. And we know that the encapsulant materials flow at around 150 to 175 degrees C, and some other material failures can occur above that. So hot spots are bad. If you reverse bias your devices, you can enter what's called a breakdown regime that, essentially, is driven, oftentimes, in a real solar cell by isolated points and your weaknesses in the p-n junction where current is going to flow. And that leads to the hot spot failure I just mentioned. Here's thermographic imaging-- not using the fancy camera that we have downstairs, using a much simpler camera out in the field, lower cost. These are high enough temperature variances to be able to detect using pretty low-tech technology. It's not a small amount of current going through a solar cell. These are large, massive amounts of current going through defective cells in the field. And so you can use these cameras to visualize the underperforming cells inside of modules. Now, that's pretty nifty. All right. So we planted that flag, and let's come back to it. We're going to describe how microinverters and microelectronics can improve module performance output. Even if you didn't follow all the detailed explanation until now, you can appreciate the fact that a bad apple or bad solar cell, connected in series or in parallel with the rest of them, can cause the combined output of the module to be lower. And so we're going to talk about how microelectronics and eventually microinverters can help resolve that, either on a cell level or on a larger module-to-module level. The simple principle of a bypass diode of a cell is this. If the reverse bias current becomes too large, instead of forcing the current to flow through the bad cell, you can force it to flow around a bypass diode. And therefore, it short circuits-- it leaves the cell out of the circuit. Right? The current is flowing around it, and so you result in less of a degradation of your module. To put it from an equivalent IV curve perspective, instead of adding this IV curve here with the rest of them, you're simply moving the current around it so you would have the n good cells, and that's it. End of story. The 10th cell would just be taken out of your circuit. It would be pushed aside. OK. So on a cell level, that make sense. Let's skip over a couple of these, I think. On a module level, as well, it can make a lot of sense. And here, it really does because the shading losses on a module level tend to be more severe. You can have, for example, a tree in the wrong place or a telephone pole somewhere, shading one particular module in an array, an array of modules. On my roof, for example, there are 12 Sharp 187s. So if you have a series of these modules up on the roof, you can cause the current to circumvent one of the modules that's underperforming, again, using the bypass diodes. And those are fairly standard in most of the junction boxes. So inside of here, there is some very simple electronics to prevent the modules from becoming power sinks and perhaps even drawing power from the grid. OK. So microinverters-- some people say, well, instead of just managing power on a DC basis and either dealing with the mismatch loss or cutting bad apples out of the circuit altogether, let's do something even better. Let's try to-- to put this in childhood education terms-- let's let every module perform up to their greatest potential, and then everybody is happy. Right? So we're going to convert the power directly on the back of the module into AC, alternating current. And then we're going to use electronics from there to essentially add the currents together once we have defined the voltage at 120 volts at alternating current. And the advantage of that is that if one module is underperforming, instead of contributing zero, it's now contributing some fraction. If the mismatch was small enough not to trip any of the bypass diodes but still degrade the power of the overall system, now you're able to recover-- you're able to allow that good module to perform up to its true potential instead of being dragged down by the others. And so the microinverter companies that are proposing, and actually marketing and selling, modules that apply the electronics directly in the back of the solar panels have the advantage of being able to eke out several additional percent of total power from the array, from the system. The disadvantage of distributing these microinverter components over every single module is? AUDIENCE: You have to make a lot of them, and they cost money. PROFESSOR: You have to make a lot of them. They cost money. They're costly, and-- mean time between failures? Increases. Right? Potentially. Potentially. So if you have an inverter-- how many people know how long an inverter lasts before it needs to be replaced, typically? 5 to 10 years. Now it's up to, say, 7 to 12, but yes. Somewhere in that range. Less than the 20- to 30-year life span of your array. So now if you have these distributed little inverters across every solar panel and they fizz out after a certain amount of time, somebody has to go up there and replace them. First, identify where they went down and then replace them. So microinverter longevity is an important part of reducing risk and marketing inverters. So the microinverter community is helping to push, or improve, the quality of inverters over the entire solar spectrum, which is kind of cool. Yeah. AUDIENCE: What's the [INAUDIBLE]? PROFESSOR: So, my understanding from Rajeev Ram, who is actually down in ARPA-E right now-- he's a professor at MIT who's been assisting the DOE over the past couple of years-- most failure in inverters relates to the power electronics itself and the high speed switchers that are used. That's my understanding. I could look into that a little bit more and get you more details about precise inverter failure modes, but that is my rudimentary understanding from discussions with him. Sorry, my screen is still cracked. It's going to be fixed tonight. Advantages of integrated power electronics is maximum power point tracking. This advantage is-- OK. Yep. OK. And so it's still a relatively hot topic. There was an ARPA-E call for improved power electronics across solar and other industries. And there is some discussion in the PV community as to how much to adopt this new technology, especially in residential systems. So trends in installations in general. In the past, installations were heavily labor-intensive. There is now a large amount of pre-assembly done before the modules hit the roof, but there's still a sizable number of components that go into every single solar module. And that takes time, to mount everything out there in the field. So what are some other-- what is the dream of any solar installer? Is to get this roll of material just unrolls, unfurls, onto the roof and using a staple gun-- bang, bang, bang, bang, bang-- or a nail gun. Everything is locked down. It's lightweight. It doesn't underperform when it gets too warm. These are all the specs that would constitute, say, an ideal solar module. We don't get that lucky. We have to make compromises in reality. Say, for example, on my roof there was a soft spot on the roof that had to be worked around. The structural engineer had to be called back and evaluate and figure out whether or not it needed reinforcements and so forth. So we're still far away in terms of where we need to be for solar installations to really be truly low cost. And a lot of it has to do with the balance of system design. So I mentioned a few companies are attempting to integrate the connections into this facile click-lock mechanism. There have been patents filed, I think, since the 1990s on this. Probably even earlier. More recently, some companies have marketed a product in this regard. And of course, that increases the facility of installation. There's even a few that are selling, I think, it was Akeena at Lowe's and, for a time, BP Solar was selling at Home Depot. They're examples of companies beginning to market their product at hardware stores. Question. AUDIENCE: Could you define exactly what you mean by balance of systems? PROFESSOR: Yes. So balance of systems is defined as everything beyond the module. So that would include the wiring, the framing, the racking, the inverter. Yes. That is balance of systems. Excellent question. Excellent question. So we're going to talk about some of the tests that the PV module must pass to ensure reliable multi-decade service life in the field and some of the shortcomings of the tests. I'm going to motivate this by saying there's reason to be concerned, in the early days of the industry, about performance and reliability. This was a study done by Arne Jacobson. The group that's working with the World Bank will be as well working with Arne Jacobson on the class project. He's a professor at Humboldt University in California. During his PhD with Dan Kammen in Berkeley, he did an audit of rated versus actual power outputs of solar modules in the field of Kenya in Africa and found a rather large discrepancy between what was promised and what was actually delivered. This is a great example of field work being done-- good old-fashioned detective work-- that leads to changes in the industry. That got a number of the under-performers to improve their product. So I mentioned there's a question of the customer. Am I getting the real value that I've been promised? Especially in the absence of independent monitoring. If you're not actively measuring the power output and recording it, there's always that question lingering. So module testing and reliability is meant to ensure product quality and output quality. Secondly, it's safety. I mentioned that these thermographic cameras-- this is an example in an array, of a hot spot. It's not too hot right now. It's 36 degrees or so, but everything else is down around 15. If you imagine this being on a sunny day-- a really sunny day-- this hot spot potentially could be getting up much higher. So that's obviously of concern. And there's as well hot spots or heat that can be generated from the junction box on the back. This contains a fair amount of electronics, again. And if there is some failure of the electronics inside of here, you can have the combined power output of the entire module arcing through a given component back here causing sparks to fly. And this is where this Photon International article from 2009 comes in where they say it's extremely important-- the upshot of the entire article, I gave it to you in its entirety. But the upshot is let's not play with fire, meaning let's make sure that the safety standards of solar modules are really up to spec so that the risk of something catastrophically bad happening is negligible. Because there's nothing worse for a nascent industry than to have some, dare I say it, Fukushima equivalent occur on some very high-profile event that causes public opinion to sway against it. So making sure that modules are safe is of utmost importance to preserving the positive image of solar. And that's what this article here is meant to convey, and you're welcome to read it in detail. It cites one particular example where that didn't happen and it did lead to it a little PR event-- well-justified, in fact. Shown here are a series of tests that one can conduct with solar modules. I showed a list of different IEC tests, standardized protocols. These are a series of tests, typically between 12 and 20, that represent a battery of tests for the solar module. And if they pass it, they are IEC certified. And you'll see on the back of certain spec sheets, like this one from First Solar right here, the different testing agencies that have certified the module, these little logos represented. So it says it's UL certified, CE mark, certified according to IEC spec 61646. That's the one right up there at the top and so forth. So certification is very important. What happens during testing? Well, we'll see it because when I designed the Fraunhofer CSE labs, we designed it, Roland Schindler and I, with the IEC tests in mind. So each of the pieces of equipment out there are to test one component of the IEC test. Included in the test is environmental cycling, where you take a module, stick it inside of what looks like an oven. And it heats it up, cools it down, heats it up, cools it down or maintains a high temperature with high humidity. Different types of environmental chambers do different things, but they're meant to simulate an accelerated aging process. Again, since most diffusion processes are driven by Boltzmann statistics, there's an exponential increase with increasing temperature. And hence, we can simulate accelerated degradation at higher temperatures. So it's not uncommon that you hear about 85/85 tests, which mean 85 degrees Celsius at 85% humidity. So that is a massive jungle. We don't have, thankfully, places on earth like that. The maximum temperatures are at around 45, 50, 55 degrees C typically and, in those cases, very dry. But in here, 85 degrees Celsius-- it's still below the boiling point of water. Very high humidity and the humidity, if there is any place for the humidity go inside the module, if you have any failure of your encapsulant material or it can get inside, it will. And it will begin degrading your module. You'll see the power output decrease. And after three months subjected to that sort of torture, if the module is poorly designed, oftentimes it will fail. Now, some of the shortcomings of these tests-- that humidity cycle isn't necessarily, with the module, experiencing an applied bias voltage. So if there's any electromigration affect, you might not detect it with that. So there are people designing new forms of tests that will probe other failure modes, other degradation mechanisms, that aren't necessarily captured in these IEC tests. Do you have a question? No. No question. OK. Yeah. Over here. AUDIENCE: What is the failure mode of degradation by water? PROFESSOR: By water? Yes. So water, humidity-- typically what happens is, in the laminate itself, there will be some part at the edge that water can get in. Not always, but in the poorly designed ones, yes, and so the water can get in and begin diffusing into the module. So if you have a thin film material that reacts with water, you might degrade the actual absorber layer itself. If you have a silicon device, the water could react with the metal and cause it to corrode-- cause it to oxidize, in other words. And of course, in reacting with the ethyl vinyl acetate, it can cause the encapsulant to delaminate from the cells and tear apart. And so you see this discoloration of the module. Those are some of the failure modes that can occur when water gets in through the side. If it managed to permeate the encapsulant materials themselves, those sorts of effects can occur directly as opposed to coming in through the sides. Another question. AUDIENCE: So they presumably test some small fraction of the modules. PROFESSOR: Yes. AUDIENCE: That they [INAUDIBLE]. PROFESSOR: Yep. So they presumably test some small fraction of the modules. And then, if those modules pass the test, which take several months to conduct-- and that's why there's a large barrier for module innovation-- but if they pass those tests, then they assume that every module-- yes. So there are a series of, I would say, definitions of what constitutes a major alteration to the process. So if you majorly alter your process, then you have to get your module recertified. But if you stay within those bounds, then you can continue selling under the old certification. It's not too dissimilar to the way our food is tested by the FDA. It's not every single hamburger. It's a spot test. In this particular case, these are examples of the newest technologies to come off the line, and you get a couple of chances to pass. If you fail the first one, it's not the end of the world. It just means that you have to pass the next time or else you have a stain on your record. And many of the companies now have their own facilities to do these tests in-house. They still send them out to the certified testing bodies that are independent to get the final seal of approval. But they know fairly well before they send them out, are we going to pass or not? In the early days it was-- you always had to hold on to the seat of your pants because you never quite knew what was going to happen when the test results came back. But now it's much better, much more predictable. And my favorite test out of all the ones in there-- there are a number, if you imagine a dozen to 20 tests, from illuminating it and measuring the power output. One of my favorite tests involves a hail gun. So in a similar manner to how the FAA tests windshields of planes and plane engines by shooting frozen chickens or turkeys at them, in this particular case, we're shooting frozen balls of water at the modules and seeing if they crack. That's one of the ones. You can envision, as well, an immersion test underwater combined with bending to see if there's any failure in the module. So they are tested quite extensively using these protocols. And the objective is to ensure quality and safety to the consumer. AUDIENCE: What's the top right test? PROFESSOR: This top right? Oh, top right over here? This is a mechanical loading test, what it appears to be, where you just exert what is typically-- that's what it appears to be in this particular case. I could be wrong. But my understanding is that this module is constrained on the edges right now and that there's a load being pushed down in the center. You could also envision a sandbag test where you load physical sandbags. There are a variety of different types and variants. Yep. OK. So that comes from the First Solar spec sheet. Describe the differences between various types of PV systems. Since we're running short on time, this is a bit of a repeat from what we talked about the very first day in class-- the non-tracking, the tracking. So I'm going to fly through that into grid-tied and stand-alone. So grid-tied and stand-alone systems is very straightforward. If you have a system that is tied to the grid, somehow it has to be able to interface with the grid. Let me go to that in a second. Here. So if you're having a grid-tied system, that means that your house or solar array, whatever it is that's producing the electricity, has to be tied to the grid and interface with it. Since the electricity is alternating current, there has to be a device to convert the direct current power into alternating current. And that's called the inverter. We've already touched upon that a few times over today's class. And these objects range in size. For the ones in the house, you could find inverters that are about yea big that can handle a few kilowatts. The ones that are handling these large solar field installations, or segments thereof-- let me find a large solar field installation. Here we go. Here are some pictures of large solar field installations. This is a road. These little green dots are trees. That's a big field installation right there. Those inverters are warehouse-- can be much larger sized than the smaller ones that you'd find in your house. And so, listing the major balance of system components, the PV array would be connected to the grid by an inverter. The excess power would be sent to the grid, causing the meters to spin backward, which is a very gratifying observation from a homeowner's point of view or, at the very least, a rate payer, utility payer's point of view. The act of sending the power back to the grid is a bit of free riding because I'm not paying for the grid. I'm a consumer, but I'm getting credit for the electricity that I put back in the grid. But I'm not helping to maintain the grid, necessarily. I'm paying some transmission and distribution costs on my utility bill, but that's essentially the same cost that everyone else is paying. Utility companies have begun getting wise about that and have begun leveraging surcharges based on the input of solar electricity into the grid, which I now pay. So I'm no longer a free rider. I was for a little bit. So, mounted onto the roof or the ground. If you mount the solar panels, you need racking and framing materials. That's also balance of systems. OK. So. Check. Check. Check. Check. You typically have, as well, a circuit breaker, or just a very simple lever that disconnects the solar panel from the grid. Why is that important? The utility companies know that, on my house in Cambridge, there's a solar array and that that array is connected to the grid. Let's imagine a tree falls on the power lines during the next storm, and the utility crew has to go out there to repair it during the day. They know that, although the rest of the grid is down, there's that one little power center on my house that's injecting power into the grid. And so they'll come along to my house, disconnect the solar panels from the grid. They'll go to my neighbors, disconnect theirs as well. Go to the person across the street who also has solar panels, disconnect them. Then go to work on the utility line. So that's why the external circuit breaker is very important. Back to inverters. Modern efficiencies of the inverter component range between 94% and 97% typical. Record inverter efficiencies up to 99% have been demonstrated. And they all have what's called maximum power point tracking, which means that they will be able to sense what the combined output is of the array on your roof and adjust the resistance in the circuit to match the maximum power point. And that way you're producing the maximum amount of power that you could be, potentially, under any given illumination condition. So a lot of fancy electronics goes into the inverter. I really have to give them a lot of respect in that regard. Typically five to seven, maybe even 10-year manufacturer warranty. And the mounting methods-- we talked about mounting and racking. This type was fairly common in the early days of solar where you'd have these field installations, especially built on hills in California. I would say most common now is really just mounting directly on a roof-- oftentimes, flat. For large field installations, one-axis trackers are very common nowadays. Tracking as a function of time of day and then manually adjusting the entire array for season, if need be. So the grid-tied systems causing meters to spin backward. Again, very gratifying. And this is an example of my utility bill, where you have a balance of $163 negative, so they're giving me credit over the course of a year. In the winter time-- this was an example of a month in December when a lot of snow happened and the panels were covered for quite a while, or at least partially shaded by the snow for a while. And we drew power from the grid on net that month. But on all the other months we, on net, produced power. And negative numbers aren't shown in the scale here. But if they were shown, you'd see probably greater negative value in the summertime when others were producing a lot of power in the summer-- more sun. And at the end of the year, they draw a line at the bottom. And they say, did you owe us money? No? All right, great. If we owed you money-- tough luck. If we're zero, then everybody's happy. So we typically produce more electricity than we consume in our house. And that's an example of what happens on a practical level. Off-grid grid PV systems, just ever so briefly. So let's imagine that you're running some illicit business, and you want to be off-grid. Or let's imagine that-- instead, a more realistic example, let's say that you're afraid of the next power outage, of being out of power for the next 7 to 10 days after the storm passes through your house. And you want to isolate yourself from the grid, if need be. You would want to have a charge controller that would be able to distribute the power to the DC loads, or to batteries to store them, and then the inverter, going through the AC loads or to the grid thereafter. So it is possible to architect a system that is semi-autonomous. But it does require more components and more cost. And I just wanted to emphasize one thing as we close up. Life is really good as an installer right now. If we flashed back to four or five years ago, life would have been really good as a solar panel producer, especially if you were involved in the feedstock industry. But now the prices of modules have been severely squeezed by low-cost competition from overseas. And the pricing of modules is around $1.03 per watt peak. This is some of lowest price modules that are out there right now. And the price of installing the system on your roof here in the US is around $5.20. Who's pocketing the $4.20? Who gets to keep that money? Is it the installer or the module producer? AUDIENCE: The installer. PROFESSOR: It's got to be the installer. Right? And who's suffering right now in the industry? No matter where you are-- US, China, Germany. Who's suffering right now? The module producers. The people manufacturing this stuff right here. All right. Which brings me to my current topic of the day. Everybody picked up this one right here, this Greentech Media article? This is really current. November 8, that was two days ago. Eric Wesoff. He's higher up there in Greentech Media. And he's writing about the China-US trade war that's evolving pitting-- within the US-- pitting the module manufacturers against the installers. So the installers, these guys who are pocketing a lot of money right now, are saying no, no, no, no. Let the Chinese modules come in. It's great. It keeps our costs low. The folks who are manufacturing the modules, on the other hand, are saying, well, wait a second. They got unfair treatment by their government. And the Chinese module manufacturers, in turn, are saying, well, it's not only us who are getting subsidies from the government. Look at yourselves. You got subsidies, too. And US manufacturers say, well, you got more subsidies than we did. So it's evolving into a little bit of a complex situation. There are some people trying to cut through all of that and say, with very simplistic messages, free trade. Or certain other people saying, made in USA is important. Or other people saying, this isn't fair for one reason or another. But it's important to really understand the complexity of what's going on in the context of a dynamically changing market. The market is changing quite a lot. In the past, the module prices-- and I say past, recent past, as in a year ago-- module prices were upwards of $2.50, $3.00 per watt peak. It's only in the oversupply condition created within the last year and a half where the module prices have dropped precipitously down to $1.00. And that happens when you're in an oversupply condition, whereas these installation prices really haven't budged too much. They're enjoying the fact that it's in an oversupply condition and there's still a market for the modules. So it's complex. We'll explore some of the complex issues in our next few classes, and we'll dive into that in detail. I'll leave, for you to study on your own, life cycle analysis. It's very important. Life cycle analysis-- in a nutshell, it really depends on what you consider in your analysis, where you draw the box. And at the end of the day, module recycling is going to be important as well for determining the ultimate impact, both in terms of CO2 impact, energy payback, and environmental impact of PV in the long run. So I'll leave you with those thoughts. And we'll see us on Tuesday. |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | 13_Thin_Films_Material_Choices_Manufacturing_Part_II.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Last class, we introduced thin-film technologies. We described some of their generic advantages and disadvantages. I showed that particular book that's going around in the back there, The Handbook of Photovoltaics Science and Engineering. That describes several of the technologies that we're discussing today and last class. So last class we covered general thin films and cad-tel-- cadmium telluride. Today we're going to be talking about amorphous silicon, copper indium gallium diselenide, and the last material system, which is actually more of a stack or a composite of several. Then we'll enter a debate, which will be kind of fun. So amorphous silicon-- we talked about crystalline silicon. Crystalline silicon is the crystalline form of the element known as silicon, comprising a diamond cubic crystal structure. Amorphous silicon, on the other hand, is a very broad term catching a range of materials that lack long-range order. Let me be a little bit more specific. Typically, there is some semblance of short-range order in these materials. One can describe the composition of amorphous silicon using, say, for example, a radial distribution function-- the average distance from one atom to the next. And you have these distribution functions describing the material. It gets a little bit more tricky when you try to define it on a more detailed level, with average number of nearest neighbors, the local configuration, the number of stretched bonds locally. And the reality is that with any amorphous material, you can deposit a range of materials. It isn't just one specific amorphous silicon material that you get every time. Depending on your deposition conditions, depending on the pressure, and the temperature, and the power, during the PCVD-- the plasma-enhanced chemical vapor deposition-- you obtain a range of amorphous silicon materials. And within that range, you can find that the mobility of charge carriers can vary by several factors. The mobility, I believe, in amorphous silicon can range easily between 0.001 and 0.1 centimeter squared per volt second. So that's two orders of magnitude easily. And I have heard of amorphous silicon materials that have gone as low as 10 to the minus 4 centimeter squared per volt second, and as high as approaching 1. So we have a range of material qualities as a result, and also a range of performance. Likewise, there's a range of stress in the film, depending on how you deposit your amorphous silicon. So delamination is of concern as well, depending on how you deposit it. And the challenges from a material level is getting consistent quality material, high-quality material every single deposition, and overcoming certain degradation modes that are present in amorphous silicon, not present in other material systems-- certainly not present in crystalline silicon. For example, the Staebler-Wronski defect that's still up for some debate as to what precisely is the atomic nature of that defect. But it is known that when one exposes amorphous silicon to sunlight, that the performance will degrade with a relatively quick time constant over the period of minutes and hours, and finally stabilize. So when you report the cell performance of an amorphous silicon cell, it's always very important to emphasize that this is light-degraded or light-stabilized efficiency, as opposed to the efficiency you get when you take it straight out of the deposition chamber, and it hasn't been exposed to high amounts of sunlight yet. In general, amorphous silicon has low hole mobility. Again, the precise root cause on an atomic scale for the low hole mobility is still up for some debate. There's a professor here at MIT, Jeff Grossman, who has performed a combination of molecular dynamics and density functional theory to obtain amorphous silicon samples in DFT. And through those samples, he and his colleagues were able to determine that the low hole mobility might be caused by stretched bonds, certain types of stretched bonds, within the amorphous silicon. There are other hypotheses out there as well in the literature. Finally, just getting a uniform film deposition at high speed-- that's really key-- a high speed, uniform deposition of this layer has proved challenging in commercial production so far. So all of this has combined so that amorphous silicon has received a lot of attention, but hasn't yet, in a massive way, been economically successful. This is a cross-section SCM of an amorphous silicon device, showing the TCO-- the transparent conducting oxide-- on the top. The TCO was deposited onto the glass, which appears at the very top underneath the scale bar. The amorphous silicon layer right here in the middle is shown. It is in a PIN structure. So we've learned about P-N junctions. The I layer between is an intrinsic layer, meaning lightly doped. And we'll see the band diagram in the next slide. The silver contact in the back here extracting the charge from the backside. So this is an example of the amorphous silicon technology. Again, another cross-section just for your reference, showing the stack there. OK, the PIN device architecture-- right here we can see the P plus layer, or the P layer, the N layer, and then the I here in the middle-- the lightly doped, intrinsic layer. Note the band structure or the band diagram, how it's arranged. And interestingly, the layer thicknesses here are on the order of 300 nanometers for the entire stack. It's rather thin, typically under a micron in thickness. The reason it's very thin is because amorphous silicon absorbs light much, much better than crystalline silicon. For those who have a solid-state physics background and want to know the root cause of that, I'll explain briefly. For those who don't have the background and can follow, don't worry. Plant a flag. We'll come right back to it. So in crystalline silicon, we have crystal symmetry, and hence we have momentum conservation rules when we excite charge carriers, hence the indirect band gap nature of crystalline silicon. When we amorphize our material, we no longer have that crystal symmetry. We no longer have momentum conservation. We can excite directly from your valence band into your conduction band. So that, in a very quick, hand-wavy way is why amorphous silicon tends to behave like a direct band gap semiconductor, absorbing light extremely well with a very high optical absorption coefficient. Hence, instead of 100 microns, we can get away with 300 nanometers of thickness. All right, back to our flag. We see here the PIN structure. Everybody should be able to follow along and understand that if light excites a charge carrier here in the middle, the electrons will be swept the right, the holes to the left. And we'll have charge separation occurring. Sometimes, we can have a stack of materials like shown here, with varying band gaps by, for example, blending germanium into the amorphous silicon. This can modify the band gap of the material. And we can wind up with a stack of layers, or we could simply use two layers of the same material. We wouldn't necessarily be able to absorb more efficiently across the solar spectrum, but we would be able to increase the voltage output of the device. So there's a variety of different amorphous silicon shall we say designs out there in the market. Here's one example of the triple stack. We have amorphous silicon here at the top, with a large band gap. The band gap is somewhere in the range of 1.7 eV. Yeah, question. AUDIENCE: I just have a question of how that-- so does the [INAUDIBLE] pair have to go to the interface between the [INAUDIBLE] to separate? PROFESSOR: Yeah, yeah, so in this particular case, I believe this is a dual terminal device, which would be rather tricky to perform an operation. What you have to imagine-- this is right here with very small perturbation. If you imagine it being biased in sunlight, then you can easily see how a larger voltage would result. Right here, you can imagine a two-terminal device where you extract charge from here and here, or perhaps a four-terminal device-- which is less common because it requires more metal contacts. But that would contact the device here, here, again here, and here. So you'd have the two devices contacted. If you have a two-terminal device, one can typically think about it as adding the voltages together. Then the current would be limited by the worst performer of those two. And that's why you have this thinner than that one over here, because we know that the light tends to be absorbed preferentially near the front surface. We're absorbing an equivalent number of photons in this thickness and in that thickness. So the currents tend to be matched. So that's why I would guess, given the geometry of the fact that we have silver in the back and TCO in the front, and no explicit contacts here in the middle, that this is a two-terminal device as opposed to a four. This stack right here is representative of what is commercialized by Uni Solar, a company in Michigan, where we have an amorphous silicon layer on the top, wide band gap semiconductor-- somewhere in the range of 1.7 to 1.9 eV, depending on the deposition conditions. Again, I mentioned that amorphous silicon is really a range of materials. There is no one amorphous silicon. Depending on the density of atoms inside of the amorphous infrastructure and their local configuration, you can vary the properties quite a bit. So the band gap can vary, but typically it's in the range of 1.7 to 1.9 eV. Blending in germanium-- germanium being a bigger atom-- tends to reduce the band gap of the material. And you have successively higher concentrations moving toward the back, so the band gap shrinks from front to back. And you can wind up with a stack like this. And this is an example here of a three-cell stack, where you have the combined quantum efficiency and the individual quantum efficiencies of each of the sub-cells, showing you how you can really fill up the entire solar spectrum using a combination of these materials. Unfortunately, despite all these efforts, the material is still affected by what's called the Staebler-Wronski effect in honor of the people who initially determined it. The Staebler-Wronski effect typically manifests itself as a degradation of material performance as a function of time. It's also known that as the layer thickness increases, the predominance of this effect also increases, and its impact also increases. So it manifests itself as a reduction of the fill factor, a reduction of performance, and we see the trade-offs during device design of amorphous silicon. So I would say, the essence of amorphous silicon is that it's a very promising material, has a wide degree of tunability. But the low hole mobility limits its performance to somewhere around 10% for just a single amorphous silicon layer, and in the low teens for the stacks of multiple materials, one on top of another. And this right here is showing an interesting effect. When you begin heating up your substrate, when you're depositing amorphous silicon, you have amorphous material. If you heat up your substrate to too high of a temperature during the deposition process-- and this is also impacted by the concentration of silane gas to hydrogen ratio during the deposition process-- one can obtain a phase transition from amorphous material into what's called microcrystalline material. That's shown at the very top. a-Si stands for amorphous silicon, mu-Si for microcrystalline silicon. Microcrystalline silicon is right at the phase transition between an amorphous silicon and a crystalline variant polycrystalline variant of silicon. And in some cases, it can actually contain a mixture of both amorphous and crystalline regions. And so it's very tricky to nail to the position conditions just right. Somebody along the way figured out-- wait a second-- we can actually use this microcrystalline silicon to our advantage, and stack a layer of microcrystalline silicon on the bottom, put a layer of amorphous silicon on the top. And that was called a micromorph technology as a result. And there are companies that have commercialized micromorph. So the micromorph technology commercialized by Oerlikon and previously by a company in the United States called Applied Materials. And we'll get to that in a couple slides as well. This right here is a brief slide just addressing the issue of how growth conditions can impact material quality. There are a number of growth parameters, shall we say, that impact whether you obtain an amorphous material with high mobility, with low mobility, a microcrystalline material, nanocrystalline material, and so forth. And if you're interested in more details, these NREL reports from the late 1990s and early 2000s, mid 2000s, are really a treasure trove of information, if you really want to get up to speed quickly on how to grow high-quality amorphous silicon layers. OK, so these are record amorphous silicon cell efficiencies. This, I believe, was from the Martin Green tables that come out each half year in Progress in Photovoltaics. So he and his colleagues produce an article every half of a year in Progress in Photovoltaics, describing the record efficiencies of lab size cells-- maybe a centimeter squared-- all the way up to full-size modules. So we have here amorphous silicon and the blended amorphous silicon germanium, or SiGe for short. People sometimes call silicon germanium SiGe. So we have a variety of different types shown here, and the areas of the devices shown here. I would caution over-interpretation of the very small area devices. If you're at 0.25 centimeters, you're at a very small linear dimension along the side of the cell, and you can wind up being impacted by edge effects. Sometimes this is the active area of a much larger device, where they've shrunken down the aperture. But depending on the texturization of the surface, you still get light trapping and, essentially, a larger active area than what is actually being shaded off. So I would caution you against over-interpreting these very small area measurements. At 1 centimeter squared larger, you're typically in a regime where the results are more believable. The Jse's shown here, Voc's shown here. Note in certain cases, the Voc you would expect from amorphous silicon, if it's a band gap of 1.7 eV, you'd expect a Voc somewhere in the range of maybe 1.4 volts. And sometimes you get much lower than that. Likewise, the short circuit current is rather low, say, compared to a crystalline silicon device because of that higher band gap-- the inability to capture some of those lower energy photons. And so the record single amorphous silicon device performance somewhere buttressing up against 10%-- not quite. And then if you combine different layers into a stack, for example here, you can move into the low teens for your performance. Flag that-- these numbers here. We're going to get back to this, and use that to understand why some of the technologies didn't make it to commercialization. This represents the largest scale attempt to commercialize amorphous silicon technology. This was a turnkey line. Let's start from over here. This is a human being seated at a computer-- two humans. This pod right here-- this machine-- represents one of these little components right here that fits into a much larger assembly line. This is the PCVD reactor, actually. I believe there were seven of them attached to each of these little pods right here. So this the plasma-enhanced chemical vapor deposition reactors were depositing microcrystalline silicon and amorphous silicon onto sheets of glass that were as large as you could possibly transport. Does anybody just happen to know the number off the top of their head? What is the largest sheet of glass that you can physically transport, say in commercial form, such that you could buy it in bulk without custom design of the Pilkington float glass process? AUDIENCE: It's as big as the cross-sectional area of that. PROFESSOR: It's about, yeah, yeah, definitely. That's a good place to start. It's actually dictated by what can fit on some of the trucks that transport the glass back and forth. You ever seen the glass transport trucks with the sides where they put the panes of glass in? And they have the chassis of the truck like so, and the glass along the edge. It's around 3 by 3.3 meters. And so these deposition systems could incorporate huge sheets of glass. And they would leave it up to the customer whether they wanted to quarter them and make reasonable-sized modules, or whether they want the full-size module and have to use cranes to install them, which would reduce the labor content of installation, but would shift the burden onto the automation. So you see the panes of glass coming along, pre-cleaning, and eventually insertion to these devices. Why so many of those devices? Why so many PCVD reactors like little piglets around a pod here? Each of these reactors would have a cycle time somewhere on the order of a few 10s of minutes-- so my estimate would be somewhere between 20 and 30 minutes, maybe up to 40. Because the microcrystalline silicon layer just took so long to deposit. But without that microcrystalline layer, it was difficult to reach 10% efficiency. With just the amorphous silicon layer, the efficiencies were in the range of 6% to 7%. And if you start doing calculations for a 6% or 7% module, you realize very quickly that the cost of the glass, and the encapsulants, and everything else adds up, because your device is so low in efficiency that you need more encapsulant, more gas, more labor, more installation to produce a panel to produce the same amount of power. And as a result, the system cost was very high. This technology couldn't compete once the Chinese production started really ramping up in the market, and decreasing the cost of traditional crystalline silicon modules. And the SunFab Project, which was Applied Materials name for this amorphous silicon line-- or micromorph line, in reality-- was shut down relatively recently, about a year ago, when it I realized that this was no longer commercially viable in the face of some of the low-cost competition from crystalline silicon. All that could change very quickly, if somehow one of you were to develop a way to, say, double the record efficiency number. So if someone were to figure out why exactly hole mobility is so low, impairing transport inside of these materials, in principle one could then envision rolling it out into the SunFab line. There's already a turnkey production equipment that's been built by Applied Materials, ready to go. And you're off and running. So that's an opportunity out there. This lists some of the commercialization attempts. And the bottom line at the end of the day was that 6% modules were too inefficient to be profitable, unless you reach scales of, say, gigawatts. And believe it or not, there was one company out there called OptiSolar, the business plan was literally to scale up to a gigawatt faster than everybody else. And even though it was lower performance, the sheer size and scale of the manufacturing facility would reduce the cost of the amorphous silicon relative to the crystalline silicon competition. Unfortunately, OptiSolar didn't make it, either. So the companies that are left today producing amorphous silicon include Oerlikon. This is a Swiss company that produces turnkey equipment, and Uni Solar, which is selling the actual modules of amorphous silicon. They have a nice roll to roll deposition process on these stainless steel foils. Very nice factory-- believe it was toured by several members of our current administration. And they're still going. AUDIENCE: Sorry-- PROFESSOR: Yeah. AUDIENCE: Is the roll that that person's holding-- what is he holding? PROFESSOR: I usually think, based on the color, that this already has amorphous silicon deposited onto it. But it could be at some intermediate phase. I don't know for that specific roll, but in principle, yes, at the end of the manufacturing line, they'll have a full stack-- the triple amorphous silicon and then the two layers of SiGe beneath it. AUDIENCE: What's-- the base is flexible, so what's the thing that the film is deposited on, do you know? PROFESSOR: So what they incorporate their flexible material into typically, one of their products is roof shingle. That looks very similar to a standard, asphalt-based roof shingle. And it is very nice building integrated technology. I happened to see one back when I was a graduate student, and I was inspired by the building integrated potential there. So there are attempts to integrate amorphous silicon into regular crystalline silicon technology, as well. Turns out amorphous silicon passivates the surfaces very nicely. And you can create what's called a HIT cell structure, which is a heterojunction with a thin intrinsic layer, using a silicon substrate. Typically what's used is an n-type base. And then the amorphous silicon layers are deposited on either side. And that device is currently being commercialized by a company called Sanyo in Japan. The patents for the base HIT cell structure expire very soon, if they haven't expired already. And so there is a fair amount of interest-- several companies out there right now attempting to reproduce this process. But it is fairly tricky. And again, a lot of it boils down to managing the interfaces-- making sure the interface between the crystal silicon and amorphous silicon are of high quality. So we have our HIT cell structure here. Let me hop over to the last thin-film materials so we have adequate time for our debates. The copper indium gallium diselenide, or SIGS for short-- this is a quaternary phase, sometimes comprised of even five elements. But let's show this is the chalcopyrite structure right here, with copper shown in black, indium or gallium shown in gray-- typically there's some blend between the two-- and the chalcogen-- the selenium or sulfur-- shown in the white right here, typically selenium. So we have here an example of the crystal structure. As I understand the situation, this crystal structure came into being because copper sulfide demonstrated potential as a photovoltaic material early on, say in the late 1970s, early 1980s. There were attempts to grow high-performance copper sulfide cells. They failed because copper electromigrated. In the presence of an electric field, copper, a very small ion far to the right on the 3-d series of elements-- so if you look at the transition metals, copper is on the far right, meaning you're adding more and more and more protons to the nucleus as you go from left to right, and you're pulling the outer shell in. So if you look at atomic radius going from left to right in the periodic table, you'll see that the atoms tend to shrink. So copper, being on the far right-hand side of the 3-d elemental series, has a smaller atomic radius compared to, say, iron, or manganese, or titanium. And as a result, it could move fairly easily throughout the lattice if it is present in a charged state-- if you have, say, copper plus-- and a large electric field distributed across a thin film solar cell device. So copper sulfide devices ended up degrading quickly over time. And it was realized that some heavier elements needed to be included to stabilize the crystal structure. So in the lab, the reason why SIGS-- this particular material right here-- is so interesting is because in the laboratory, efficiencies on the order of 20% have been achieved. For large area-- you'll have to correct this number right here-- you can increase that to 15% plus-- 15.7% in fact, over large area devices. That's a very recent result coming out of MiaSole, a company in California. There are dozens of startup companies focused on SIGS development. And it is a very challenging problem to get the stoichiometry right, and to passivate the defects inside the structure, including surfaces. Where many companies fail or fall short is getting the stoichiometry just right over a large area. It's as simple as a manufacturing challenge. The idealized structure of these thin-film devices-- you have your SIGS layer right here in the middle. In this case, it's shown as CIS or cis, but you can add gallium as well. Typically, gallium is blended in. Moly-- molybdenum-- back contact on the glass-- the glass in this case is just the support for the structure. Molybdenum here would be non-transparent. It'd be an opaque layer. The cadmium sulfide on the front side here forms a very nice interface with the SIGS layer underneath, preventing any interface damage or any segregation, say, of gallium to the surface here. And then the zinc oxide layer on the top, which serves as the buffer window layer, depending, is then deposited on the cadmium sulfide layer. And eventually charge is brought away. This right here is another three-dimensional rendition of the same. What is unique about each of those dozens of startup companies is the process in which they deposit the SIGS layer, typically. So you have companies like MiaSole that are invested in sputtering of SIGS. You have companies like Nanosolar that are invested in ink-based printing of SIGS. So they print an ink down, and then they sinter the ink to form the layer. And like those two, there are many other companies out there that are depositing SIGS in some way, shape, or form. The folks at NREL seem to be more in favor of thermal evaporation based process for SIGS. Many of the more successful industrial renditions of SIGS right now-- at least the ones that appear most promising-- are using other deposition methods. So you'll see a variety out there, of deposition methods for SIGS technology. And one of the things to keep track of, as you watch and monitor all these startup companies, is to bin them by their process, and to begin to discern trends. As we start to see some market leaders evolve in the sector, begin to discern which technologies are the most appropriate, perhaps for large-scale deposition of this material system. Interfaces are critical. Here are some examples of attempts at mapping out-- and quite successful attempts at mapping out the, shall we say, the band diagram of a SIGS-based solar cell. One thing to keep in mind is that, just like amorphous silicon represents a broad class of materials, SIGS also represents a broad class of stoichiometries, meaning the ratio of indium to gallium, the ratio of-- if you include it at all-- sulfur to selenium is critical in determining the properties of both the bulk and the surfaces. Typically at the surfaces, you will get termination by one atom, one atomic species or another-- one elemental species or another. And that will determine, to a large degree, the electronic quality of that surface. So for a long time in literature, you had debating world views of what the band diagram of a SIGS device really looked like. And it turned out a lot of people were right. They just had varying starting materials. So they were comparing apples to oranges. The thing that keeps people going with SIGS is that in the laboratory, efficiencies, I believe, now up to 21% have been demonstrated. So that is really encouraging. People-- venture capitalists-- will look at this and say, wow, with those high efficiencies, you could really make solar cheap. The amount of, again, glass, encapsulants, labor, and so forth that you have to have for watt peak is lower, because the cell can perform at a higher performance. But the downside is, there's a gap between-- a substantial gap between-- what is commercially manufacturable today and that record efficiency device. Sometimes the practical matters of commercialization are the ones that impede a technology from getting to market. And I have a modicum of faith that SIGS will make it. But if it fails to reach its true market potential, if it fails to realize its dream of becoming a significant fraction of all solar panels produced, it will most likely be because of these process engineering issues associated with depositing over a large area, coupled to the scientific gap in managing interfaces properly. So that's the story of SIGS in a nutshell. There are some minor issues, shall we say, associated with cadmium. There are attempts to go to an all zinc oxide or maybe a graded zinc oxysulfide buffering window layer here in the front to get rid of the cadmium. We'll discuss that shortly. And I would say that's probably the biggest near-term challenge. With the removal of cadmium, you can access more markets. And secondly, SIGS does contain indium, which is not in an infinite supply in Earth's crust. It's debatable how much of an issue that really is. And we'll get to that in our discussions as well, in our debates. SIGS commercialization-- several startup companies and I would say a lot of promising research and development going on right now. This particular technology right here-- let's keep this company nameless for now. This company represents some of the difficulties in ramping up large area SIGS. They deposit it sometimes over large areas, but would achieve inhomogeneous results. And so the ultimate form factor for the devices was to chop them up into areas that are not too dissimilar from an actual crystalline silicon device. So to chop up their layers into individual, discrete cells, and then to connect them-- solder them together, much like a crystalline silicon device would be put together inside a module. So the dream of having this inline, thin-film process roll to roll was reaching an abrupt wall, shall we say, in the development process because of the lack of large-scale uniformity. So you need to chop them up into smaller units, which would add to the cost, which would decrease the module performance as well, and now you have gaps in between the cells. So it just goes to show the difficulty of bringing a new technology to market. And hopefully, they're doing better now. Lots of promising news for SIGS, like this particular article-- "The Rise of SIGS, Finally?" question mark. There has always been a lot of promise in the realm of SIGS because of the high performance and high efficiencies reached. The big question is, can you accomplish this at large scale for large area modules? That's been the big challenge so far-- and of course, materials availability. So let me pause right here. What I'm going to do-- we are going to shift gears. And for the next 20 minutes-- unfortunately, not half an hour, but for the next 20 minutes-- we are going to enter debate mode. What we're going to do is divide the group into-- this group right here-- into your project teams. And you'll cluster together and work together on this next exercise. So for the very beginning, we're going to have the cadmium telluride debate. That will be debate number one. We'll be discussing the pros and cons of cad-tel. Keep in mind that First Solar, a gigawatt company, meaning one of the largest companies in the world, and certainly the largest company in the United States, is going full force producing cadmium telluride modules. The debate question that you'll be faced with is, is it a good investment for the United States? Say, for example, you're in charge of US government research and development funds. Should you be putting your money into cad-tel? Or should you be putting your money into something else? And we'll have two sides of the debate-- one that will defend cad-tel and say, no, it's a very promising technology. We should be going all in. And the other side that will be adopting the counter argument saying, well, cad-tel has these concerns. Perhaps we should be looking elsewhere to invest. And that's the debate number one-- cad-tel, no cad-tel. For the folks working on earth-abundant materials, the second debate will be focused on that question. Recently, there's been a lot of interest in developing earth-abundant alternatives to, say, SIGS, because of the indium content, and cad-tel because of the tellurium content. And there's been a huge amount of momentum behind it. But there are few naysayers out there who say, well, wait a second. We've always come up against shortages of one kind or another, since the days of Malthus's warning about the limits of population growth because of limited food supply. But then we developed fertilizers. And that overcame that limit. Likewise, in these resource abundance issues, I'm sure we're going to find some new way to extract the metals. We'll find a way around it. We'll discover new deposits and so forth. You hear similar arguments in the peak oil debate, right? And so what I'd like to do is to set up two teams there as well to debate this particular question. [APPLAUSE] And we furthermore recognize that several of these issues that might appear black and white, in reality have many shades of gray in PV. This cad-tel debate is one of them. We'll hear the second debate the next time. And anything from should we invest in silicon technology or should we develop thin films will have these shades of gray as well. So it's important to recognize that simple fact as you gain maturity, and shades of gray in your own ability to argument one side or another. So thank you, and we'll see us on Thursday. |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | 15_Advanced_Concepts.txt | The following content is provided under a creative commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Welcome everyone, today. Today we're going to be talking about advanced concepts. These are kind of like what I would consider the next generation of solar cells if these ideas pan out. Some of them are very near and dear to my heart because it's what my research is mainly focused on. Also another quick realization I had last week. Probably Tonio mentioned this to me at some point, but do you guys know where 626 comes from, why the course is called that? So 6.26 times 10 to the minus 34th is Planck's constant. So haha, funny. Thought that was kind of cool. It's a little joke put in there. So anyhow without further ado, one of the cool things-- cells are done. Yay. This is great news. I was very, very happy. So it didn't go quite as well as expected. I think most of you are finding out that these are incredibly [INAUDIBLE] limited. I'm going to talk about why that might be in a second. Hopefully that will help you guide you through your analysis, which we'll be doing for the final part of the quiz 2.3, which hopefully I'll be able to post by tomorrow afternoon-ish. I'm still working on that right now. But just to give you guys kind of the processing that you didn't get to see, this is what the contact firing looks like. It's a very, very, very fast process. It's really remarkable to me that this combines so many photlithographic steps that if you wanted to make this in the lab using photlithography, and condenses these six or seven steps into one process. So it's pretty remarkable. So DuPont, the people who made the pace-- so we used PV16A if you guys are curious for our front contacts for silver paste. And we actually get-- the really important part is this peaking temperature, and you can see we actually got pretty close to the ramp times. Where it really starts to deviate is here, and we hold it around 400, a little bit longer, which is several reasons for that, and I'll go into that in a second. But what's important is that all these are fired in an oxygen atmosphere. So normally in these belt furnaces, and this is again what's used in industry-- this is actually the same exact tool that I use at Harvard-- it actually looks like this with the giant CRT monitor on top. The only way to get the data off is with a three and a half floppy drive. I can't tell you how difficult it was to find a working three and a half floppy drive. Most of them are demagnetized at this point. I had to buy new ones. They're still being made, by the way. If you want them, go to Staples. So anyhow, it's done in an oxygen atmosphere, so they generally force airflow into these giant belt furnaces, and these are really long. Like that's probably about a foot wide, maybe 18 inches wide. So this goes on for tens of feet, or several meters, depending on where you're from. And it's done an oxygen atmosphere, an air atmosphere, because it needs to burn off a lot of the binders and some of the organics that might still be there. And so those organics burn off, and what you're left with are the little metal particles, the frit, and some metal oxide glass. It's usually lead oxide, although DuPont will never tell you, but from papers I've read this is generally what's used. So when that happens-- this is that burn-off period. When it spikes, the frit will actually burn through your silicon nitride layer. You guys didn't have a silicon nitride layer, but if it was there, it would help eat through it. And so that way, you've removed the dielectrics so you can actually make metal contact with your silicon surface underneath. That firing will-- during that peak also simultaneously, these middle particles will melt and form this triple eutectic point with the silicon underneath it. And so the triple comes from the silver, the silicon, and the lead. The lead actually dissociates from the glass. This lowers the melting temperature of that mixture, and you can make good ohmic contact with the surface. So it's actually a pretty remarkable, incredible process. It's still almost kind of magic to me on how the whole thing works, and a lot of it's just kind of guess and check and still very proprietary. So the science is still a little lacking there. You don't find a lot of good articles explaining the science of it. I have a few if you guys are interested. NREL's put out a lot of really interesting stuff, and some of the ways they actually figure out the actual profile underneath the contacts versus you measure the temperature on the side of the contacts, so actually touching the silicon or the silicon nitride surface, is you actually measure hydrogen diffusion underneath it. And so they'll actually etch off the contacts and using secondary ion mass spectroscopy measure the hydrogen concentration off the contact and underneath the contact, and that'll actually tell you what temperature it saw based on that diffusion profile. So there is some good science going on, but there's not a lot of great papers on it. So anyhow, that just kind of gives you what I was working on over the weekend, trying to get these cells fired. So I aggregated a lot of the results and tried to make sense of what was going on. I think I showed some of you different firing temperatures and the fill factors that we got and there was kind of just noisy data, so I couldn't get any real trends. The best trend I could find was that if you take the median of all the short-circuit currents of the cells and you look at the ones with four millimeter and two millimeter spacing, the four millimeter spacing has a slightly higher short-circuit current, presumably due to shading losses. If you look at the relative areas that each of these cover up, you would expect a 4% increase. This is about 3%, so I'll take that as fact. It's certainly within the noise and the error of the number of samples that we have, but I thought it was interesting nonetheless. Additionally, I plotted the series resistance that I calculated from your dark IV curves and plotted your maximum power. I also removed all the cells that were broken. I'm sure if I actually normalized them-- some of the cells that were broken actually performed a lot better, and I'll get into maybe why that was the case in a second. But really the take-home message is that our really best performing cells are all clumped over here. And so you can see that our best cells have a very low series resistance. That's not always the case. There's some outliers, like this guy here, and I don't know how that happened. So some really interesting things going on, and I'm still trying to sift through it. So I talked about this I think yesterday in the analysis section, but there's several sources of series resistance. Any of you guys familiar with what they are? AUDIENCE: [INAUDIBLE] resistance and emitter sheet resistance. PROFESSOR: Emitter sheet resistance, and then-- AUDIENCE: [INAUDIBLE]. PROFESSOR: So there's line resistance along the fingers and then there's contact resistance, so I'm going to draw this out really quickly. So if you have-- this is your emitter. You have a contact here and a contact here. If you generate an electron here, it'll diffuse, hit the junction. Then it has to flow through the emitter to the contact. There's an associated resistance with that. That's the series resistance that we taught in class that has to do with the emitter resistance. Additionally to hop over that metal semiconductor junction, there's actually a resistance there. That's called a contact resistance. So this is emitter and this is our contact, and then it has to travel from the line to the busbar here, and that's our line resistance. And I think what's really limiting us, because when I looked at both the shallow and deep emitters, the series resistance didn't really have any noticeable trends. So I really think that this is what's limiting us completely, is our contact resistance, which is kind of too bad. So one of the reasons I think that that's the case additionally is that the cells that performed the best-- I know Joel's did particularly well. What was your fill factor? Do you remember? AUDIENCE: Well, on the one side it was 0.5, and on the other it was 0.64. PROFESSOR: OK, so one side was better. So again, the inhomogeneity of doing the measurement is also another indication that our series resistance across the contacts is really poor. So some of the other contacts if you look on the-- where's my good piece of chalk? Right. If here's our busbar, some people saw delamination. Who saw their contacts delaminated? Yeah, so a couple people. And so what that means is that these contacts actually peeled up, and that's usually showing that the-- what's funny, though, is that these areas in the middle didn't delaminate. It was only the edges here, and so that's showing that some parts are being properly fired and some parts are being underfired because when I fired at higher temperatures, I was getting shunting. And that was due to shunting through the center area. So I think that the RTA that I was using-- this wasn't the case on some of the earlier cells I had processed, but is rather inhomogeneous in terms of the temperature profile it's delivering to our cells. So I think that could be part of our problem, as well. AUDIENCE: How many cells did he fire up at the same time? Just one, or-- PROFESSOR: Yeah, it's one at a time, and there's some variability in the peak firing temperature. So I've learned a lot about when we're going to buy an RTA for our lab, what to look for. The temporal resolution of the data you get out of it and how it records it is 0.6 seconds. And when this thing's ramping at 100 C per second, your variability is quite large. So the temperature I read is not just from the data point, but I do a linear interpolation to actually figure out what the peak is on both sides of that peak, and it ends up being about anywhere from 20 to 40 degrees Celsius higher using that technique. And so the variability varies. It's about 20 degrees plus or minus 10 degrees Celsius in terms of what I want the peak to be and what it actually is. So the RTA that we're using now is not incredibly reputable. Also, better RTAs will have a different heater on top and bottom, and so you can actually heat one side more than the other. And generally belt furnaces, so commercial ones, will have that same kind of control. I know at 1366, they've actually done optimization to do heating the top and bottom differently because they use different pace, they have different heat requirements, thermal requirements. And so nailing that temperature and that profile actually takes a lot of iteration and a lot of time. And it's really just learning the tell as opposed to, the again, good science behind it. So it can be rather difficult and time-consuming to do. So that's what I have to say about the cells. I think overall, though, I think the efficiencies are on the order of 6% to I think 9% for some of the cells, so that's not too bad. I'm still rather pleased with that. Last time I think Tonio stopped right before he was going to talk about performance of modules in the field and kind of what's the difference between your cell efficiency and your module efficiency. Modules are put out in the sun. They heat up. They can get shaded by snow, rain, clouds, trees, and also the modulus are made up of many different cells that are connected in series and parallel, and how does that affect their overall performance of the module? And so that's what we're going to talk about really quickly. So first off, why does temperature matter? So all solar cells when they're measured either at NREL or in our lab, are measured under standard operating conditions-- or standard testing conditions, sorry. Certainly not standard operating conditions. Standard testing conditions are done at 25 C. Generally the cells are actively cooled to maintain that temperature of room temperature, and most semiconductor simulations you perform are generally done at 300 Kelvin. I know all the calculations I did for fitting your [INAUDIBLE] diode curve were done at 300 K. Typical operation can actually be pretty hot, so 50 to 65 degrees Celsius. I actually don't know if that is in Fahrenheit, but it's hot. It's hotter than Phoenix, and I'm sure they get even hotter in Phoenix. So the effective temperature-- so how does it affect Voc? If you recall, this is our diad equation. We have our illumination current, which we'll get to how that's affected by temperature in a second. But for now, just ignore that. Assume it stays constant. So one of things that's really affecting your Voc-- so remember, your Voc is going to make your current go to 0, and the main thing that's being altered is your saturation current, so your intrinsic carrier concentration, your diffusivity. So who here thinks your Voc is going to go up with temperature? Anyone? And who thinks it'll go down? Who has no idea? OK, so who raised their hand for it'll go down? Sorry. Ben, do you have any ideas why that might be the case? AUDIENCE: No. [LAUGHTER] PROFESSOR: So what's going to happen to your I0 with increased temperature? AUDIENCE: The intrinsic carrier concentration, ni, is going to increase, and that should increase I0? PROFESSOR: That should increase I0, so therefore it's going to take a smaller voltage to counteract I sub L to make I equal to 0. And so why does N sub I increase with temperature? AUDIENCE: Greater thermal energy present increases the film probability that electrons can be excited from the [INAUDIBLE]? PROFESSOR: Right, so it increases the number of thermally-promoted carriers. And so that effect can actually be pretty stark, and now we go to a demo. Yay. All right, so what we have here is our favorite solar cell. This came, again, from those small little toy cars that we pulled off, and again, these solar cells, the cars will assume were nothing, and these are about $10 apiece, so pretty cheap. Probably cheaper than your solar cells that you guys made. And, well, certain I know they are cheap. The wafers themselves, by the way, were $16 apiece, so saying they were a dollar was a little inaccurate. So anyhow, we have this hooked up to a multimeter and we're going to measure the voltage off of it. I'm going to illuminate it with our desk lamp, and we can actually get a pretty decent voltage. I can't actually see this. So it's about 0.57 volts, and now we're going to subject it to temperature. This is obviously a gross overestimation of what's actually going to happen, but can someone read off-- Joel, can you read the temperature for me? AUDIENCE: The temperature, or the voltage? PROFESSOR: Sorry, the voltage. AUDIENCE: Yeah. We're at 0.561 and decreasing down to 0.55, 0.54, 0.53-- PROFESSOR: So it's going down. So congratulations the people who said it would go down, correct. Wow, that's actually quite hot. Anyhow, so that's what happens in the field. So when these cells heat up, you actually lose on your Voc. And if you really want to, you can go through all the math. You know what the equation is for your Voc. It's shown here, and you can go through this if you like. It's all on PVCDROM if you want to go through the derivation. You then can take the derivative with respect to temperature. And if you plug in values for crystalline silicon, it comes to about 2.2 millivolts per degree Celsius, which sounds like a small number, but it's at around 0.1 volts if you go up to 65 C, so that's actually quite a substantial amount. And again, when you're going from 0.6 to 0.59, these small margins can make or break a lot of installations, so that's an important one. OK, now this one's a little harder. What do you think will happen with your illumination current, or your ISC? Who thinks it's going to go up? Anyone? Who thinks it's going to go down? Who thinks it's going to stay the same? Who has absolutely no idea and didn't raise their hand? OK, that's totally fine. This confused me, as well, by the way. So when we think about it, what is our short-circuit current proportional to generally? Anyone? AUDIENCE: [INAUDIBLE]. PROFESSOR: Right, so generally your illumination intensity. And what photons generally are we collecting? AUDIENCE: Super bandgap. PROFESSOR: Super bandgap. So what could be happening is our bandgap could be changing. I sub L, as Ben pointed out, should increase the flux of photons above the bandgap, and EG actually decreases the temperature. So your bandgap actually usually increases-- the true bandgap of silicon defined-- most properties of semiconductors are actually defined at 0 Kelvin, and semiconductors are technically insulators because they don't conduct electricity at 0 Kelvin. That just means semiconductor is a class of materials that have a small enough bandgap that they can thermally promote carriers that they can actually conduct some electricity at room temperature. But the true bandgap of silicon's around 1.17. And as you increase temperature, you get a reduction of the bandgap. And so what should happen is your ISC should increase, but only very, very slightly. So if you go over the derivation for the actual decrease in the overall efficiency, there's a few things. One, remember your maximum power is your Voc times your ISC times your fill factor. So if you want the derivative of your max power with respect to temperature, you need to do the partial derivatives and sum them up for all of those different components. So that's what this calculation's doing. And again, this is what it is for fill factor. This is all in Martin Green's paper down here and also on PVCDROM if you guys are interested in that. Martin Green's tabulated a lot of this, making these general expressions and also fitting it to experimental data to get that information. And then it ends up being not a negligible percent, but about half a percent per degree Celsius for silicon. So if you go up by 40 degrees, you can see that temperature's a non-negligible effect in terms of performance. AUDIENCE: [INAUDIBLE] silicon's kind of below the ideal bandgap for a single-junction solar cell? PROFESSOR: Yes. AUDIENCE: If you had a semi-conductor with one that was above that, would an increase in temperature increase the efficiency because it's getting simply closer to that bandgap? PROFESSOR: Oh, that's a good point. Generally temperature's a lot worse because you're really going to hurt your Voc. You would increase your J0 tremendously with temperature because you're creating more thermal carriers, so your diffusion current that's counteracting your illumination current is going to increase dramatically, so temperature is always your enemy. There was a great picture on PVCDROM of a solar install in Antarctica and its perform well above spec, which is kind of cool. So yeah, temperature's generally an enemy. And so a lot of people's ideas have often been like to do active cooling and use that for waste heat, and there's a lot of reasons why that doesn't necessarily economically make sense. But when you do, let's say concentrated solar, for example, and I think Tonio will talk about this in the next lecture, but concentrated solar is you're putting in not just one sun but about 1,000, maybe 10,000 suns on one small device. And the idea there is you can have a really small device, have it be really expensive but incredibly efficient, on the order of 40% efficient because you have these stacks of difference semiconductors that absorb different regions of the light preferentially. And those heat up tremendously when you're being subjective. It's like an ant under a magnifying glass or something. It can burn it. So they actually active cooling to cool those cells. And they also have to track the sun, as well, in order to concentrate it. And the last thing which you guys did for your exam number one was the effect of light intensity, where you looked at light intensity throughout the day, assumed some sinusoid for the incident light, and you measured how your efficiency changes. And you remember, your Voc decreases with light intensity with your JSC, and that's due to the decrease in the photon flux. And your efficiency goes down according to those two equations there, and I think the derating factor if you assume a sinusoid that hits at one sun and then declines, you should derate your-- if you use instead of using peak sun, you want to derate that by about 20%. So it's actually pretty substantial. I was kind of surprised at that finding. So when V cells go out into the field, they are not just one individual cell. So oftentimes you string these in series and then in parallel. So right now we're just going to look at series and in parallel solely, and what you see here are your three different cells all mounted in series. So the top of this cell is connected to the bottom of this cell, the top of this cell is connected to the bottom of this cell. Our bad cell, marked here with the I guess the denim pattern. I don't know what that was in PowerPoint. But what's important here is that because the current flowing to this one is also flowing to this one is flowing to this one, the voltages add and the currents are all matched. So the current out of this stack of cells is limited by your weakest cell. It can't exceed your short-circuit current of your weakest cell. So you can see that your operating point for these three cells is not ideal for this one. It's definitely way off the peak operating point for this cell and way off the peak operating point for that cell. So it can be a pretty detrimental effect, and this is why often in industry when you do a series of testing on your cells, each cell gets tested either using suns Voc or a very, very fast IV measurement. You can [? bin ?] cells based on their performance and then match them so that if you all your cells are perfectly matched, they're all operating at the peak power point and you're getting the most out of all those cells. And so that's why reducing the variability within your process can really, really increase your module performance, and that's something a lot of companies work really, really hard at. Now, the other reason why series is important to study is the effect of shading. So when you decrease-- so let's say a leaf falls on your solar panel or it's partially shaded because you put it near trees. If you look at the BigBelly Solar ones right near campus, they're always in shade. It makes no sense to me. I think they might get one hour of sun a day in certain times of the Anyhow, And so if you partially shade one of the cells, you can see that the IV curve drops down, and now this cell is running in reverse bias. And when you're in reverse bias, you're running a lot of current through your shunts. And for those who were here this morning, we know that shunts are really, really bad because that's where all your current's running through. And if your current's running through one localized spot, that's where it's heating up the most, and it can heat up to temperatures that can actually melt the encapsulant or completely destroy the cell, so this can actually be rather destructive. And generally once your cells are made and encapsulated in the module, it's really hard to remove them and replace that module. Pretty much the whole module's kaput at that point, so it's pretty destructive. So what people generally do for-- yeah? AUDIENCE: So in maintenance, like yearly maintenance, you should clean the panel because of the dirt, or is there any other thing that you should do [INAUDIBLE]? PROFESSOR: There's important things. So you don't want-- it helps if they're angled so snow and other things will fall off of it. You don't want to put it in areas next to a chimney, for example, where certain parts of the day it will be shaded, or things like that. So I've seen some studies on cleaning, and it all depends on if it economically makes sense. If you want to pay someone to clean your panels, that's expensive. Is it worth it for them to do that if you're only saving just a little bit? Obviously, the shading's so bad that it could detrimentally ruin and destroy your module. It's really-- you want to do that? Ways to prevent the destruction is actually a lot of cells will have bypass diodes. So if a string goes down, the current will just [? flow ?] through that diode and won't destroy the cell, and so that's one way of kind of fixing that problem. That's a great question. So now we're stacking ourselves in parallel. Normally, again, in a module, you have them in series and then these stacks of cells in series are actually put in parallel, but there's just an easy illustration. Again, cells in parallel, you're matching in voltage and the currents are what add in. And so if one cell let's say has a low output voltage, it's going to shift the operating power point for the other two cells off of their maximum power point and you're going to get a reduced output. And so this is actually very analogous to this idea that when you make your cells larger, generally their efficiency and performance go down. You have a higher chance of inhomogeneities and other things that might detrimentally hurt your cell. And this is kind of analogous to saying when you hook a whole bunch of cells in parallel, you're limited by your worst cell. And in this case, you're limited by your worst region of that cell, and so that's the same kind of analogy. And I think it's kind of cool, and so you can see this for a lot of different cells. So these are actually getting pretty big. So when you're getting to-- I don't know if they make cells this big. But it's 100 by 100. That's a meter squared. Wow, so maybe Cad-tel can get that big. Actually, they do. Look, here's First Solar right here. AUDIENCE: [INAUDIBLE]. PROFESSOR: I'm sorry? AUDIENCE: [INAUDIBLE]. PROFESSOR: Oh, OK. [INAUDIBLE] modules. So some of the cells you can see. So here's cells-- amorphous, submodule. Anyhow, so the idea is that you're, again, limited by your most defective region in the cell. And again, illustrated by this idea here is that if you take a cell, you split it up into a grid, and you model it as a bunch of many cells operating in parallel, you're again limited by that bad region. And so this looks a little outdated. So this is 2006. So this is cell performance versus module performance, and you can see that it's also quite significant. One of the other things that's different about cells and modules is that you have glass in front. So when you optimize-- for example, in class we optimized our silicon nitride coating for an air-silicon interface. Modules, you optimize at silicon nitride coating for a silicon to glass interface. And the glass, which has an index refraction of about 1.5, you're automatically going to reflect 4% of the light, so that light's lost. But it does help with your reflection losses because you now have a graded index of refraction. So it does reduce some of those reflection losses from that interface. Yeah? AUDIENCE: Is this just silicon? PROFESSOR: I'm guessing yes because this is from Richard Swanson. This is Tonio's slide. My guess would be yes, but don't quote me on that. AUDIENCE: So for the modules, how is voltage and current controlled? I'm assuming that they control it at a specific point. PROFESSOR: Yes, so that's a great, great question, actually. So your modules are hooked up to an inverter. The output of your solar cells are direct current, and so you have to convert that to alternating current. And the inverters are actually rather intelligent. They'll actually constantly scan at a very high frequency where it should be operating and taking the derivative and finding where that optimal point is. And so one of the other ideas is that if you have a bunch of modules, usually those are also hooked up in parallel. And the idea is to have instead of just one inverter, a microinverter-- so small inverters for each one. So that way, each of those are operating at the maximum power point. And I think problem is that inverters are expensive and the larger they are, generally the more efficient they are and cheaper can be. So I don't know how far that idea's been realized, but it's a good question. Does that answer it? So we have-- good, plenty of time. So now we're moving onto advanced concepts, which, again, area that I'm working in. Kind of exciting. These are ideas that are very, very high efficiency, potentially very low-cost, and that's kind of the allure to them. And they're great science and research projects, so they're really exciting to be in. So one of things that kind of motivates this work-- I think you guys have seen this equation. I know I've used it before. But you know that the defining metric for the performance of a cell in terms of its economic cost is dollars per watt, so dollars per watt peak. This is all defined at your peak sun elimination. This makes it location-blind, right? The really important thing when integrating it onto the grid is how much are you paying in terms of cents per kilowatt hour. But to make it location-blind, you put it into dollars per watt, and dollars per watt is a function of many things. It's how much does it cost to make the module itself? So how many dollars per given area? You divide it by the insulation. For dollars per watt peak, you'd set that to aim 1.5 or 1,000 kilowatts per meter squared. Multiply that times your efficiency, and then by your yield, which is a manufacturing parameter. So for some cells where if you can reduce this number quite dramatically, so for example, suppose you decide to go to very, very thin wafers for silicon. You want to move to 20 micron wafers, let's say, which is something that a lot of companies now are working on. What happens is that those become very, very fragile. And when you handle them and using all these pick and place operations to move them to different processes within the cell fab, oftentimes they'll break. And so that yield parameter can come very important and [INAUDIBLE] interplayed with this number here, and that's why it's there. So anyhow, what this graph is is this is the dollars per meter squared, so the cost of producing that cell. And then you have this efficiency here, and each of these lines represents a certain dollars per watt. And so I think the DOE has wanted us to get to $1.00 per watt, and you can do that by either producing incredibly efficient cells, around 50%, which seems a little unreasonable. And they can cost $500 per meter squared, and that would be actually a cheap panel. Or the other idea is to go to very, very low cost and low efficiency, and you can try to hover around in here. And so this nomenclature I'm about to use is a little outdated, but I think still some people use it, although rather loosely. What's known as first-generation-- and again, these terms I've read papers now they claim that they're third-generation where they used to be classified as second-generation. It doesn't really matter, but the ideas behind them are still relevant. So the first-generation with the single bandgap, it's crystalline silicon still [INAUDIBLE] the market leader, and they're on the order of 15% to 20% efficient, but they're still relatively high-cost. And there's actually a lot of work now that says that this can move in this direction, moving to thinner wafers, increased laser processing. Just tighter manufacturing can really bring that cost down. So your second-generation are your thin film. So this is either CIGS, amorphous silicon, organics. A lot of these very, very cheap to deposit, cheap to manufacture, but they generally suffer in efficiency, so they hover around in this area. And so far, this idea hasn't really fulfilled itself except for maybe cad-tel, which is a thin film material. And then what we're about to talk about is our third generation of cells. So these are potentially low-cost and potentially incredibly efficient. So we're going to talk about only a few types today. I encourage you to look up. There's a whole different types of solar cells you can look up, ideas, research being done. But I'm going talk-- this is what I work on, our intermediate-band solar cells, specifically impurity-band photovoltaics. And I'm sure you're all familiar with this picture, but if you want to create free mobile charges within your semiconductor, you have to shine light on it that has energies greater than the bandgap. And so you lose these low-energy photons. They're not collected, and what's a way that we can collect these? Well, one idea this guy had was if you can create a material that has a density of states that looks like this, you can create a stepping stone by putting a half-filled band within your bandgap that allows you to promote to that band and then from that band into the conduction band. So the idea is that your voltage output can still be maintained for the host material, but you can collect that extra current. So your ISC will increase substantially, and so that's the promise. And if you do some theoretical calculations similar what Shockley and Queisser did, assuming the radiative lifetime is your limiting factor-- and again, that's certainly not the case in some of the materials that are worked on for this idea-- and the idea is you can actually get up to around 63% efficient. And so what this graph is here, here's your single gap limit. This is a calculation similar to what Shockley and Queisser did. I think this is using blackbody radiation, which is why the curves are so smooth. And it can actually outperform a tandem cell, which is again a multi-junction cell that are two cells stacked. And it turns out the best is something that has a gap level that's about 0.7 EV from the conduction band and has a larger band gap, so that distance between your valence band and your conduction band of around 1.9 electron volts, and that's kind of the ideal material for an intermediate-band solar cell. So theoretically, this idea is a great idea. Problem is, how do you make a material that has this band structure? And that's really the challenge where people are working right now, and it's actually rather difficult. And so there's three approaches. One is the impurity band, which I personally like because it's a much cheaper method. Other one is this band anti-crossing idea where you can actually split the conduction band, and it's generally done with these mixed metal oxides and these highly mismatched alloys, and they're really cool. And this is currently the most successful material there for intermediate-band solar cells. And then there's quantum dot arrays and quantum structures that are also a possible fabrication method. OK, so idea behind an impurity photovoltaic is you start with let's say a material like silicon or some other high-bandgap gap semiconductor, you put in impurities that have these deep-level states. Iron would be an example. So iron in very low concentrations can be incredibly detrimental. But as you increase that concentration above some critical limit often called the [INAUDIBLE] transition limit, you can actually form a band within the bandgap. And so the idea what's going on here is that each of atoms this has some electron wave function and some radius that it sees. And then as you increase that concentration, these wave functions overlap and you can have conduction through those mid-level states. So that's one idea, and I like it because you use what I could consider very dirty materials. The concentrations you can put in here around are one atomic percent, so your silicon's only 99.9% pure at this point. Other idea is to use either quantum dots or these quantum wells. So I think the idea here is that these are localized and then this becomes delocalized. So as you bring these quantum well structures or quantum dots closer together, you can actually get tunneling through these states, and it basically essentially forms a band within your bandgap. Similar idea is to do this locally-- is that you have one photon promote to one of these confined states in your quantum structure, and then another photon to keep promoting up into your conduction band to create carriers. AUDIENCE: [INAUDIBLE] would there be quantum [INAUDIBLE] embedded in the silicon, or these are solar cells made out of [INAUDIBLE]? PROFESSOR: It's usually stacked between two other semiconductor materials. So you have semiconductor, quantum dot, semiconductor. If you look up this paper among-- if you look up anything-- so Antonio Luque-- sorry, yeah. Luque and Marti are two people in Spain who've kind of promoted a lot of the theory around this. And look up their quantum dot papers, and they'll really go into the device structure if you're curious. I'd recommend doing that, Joel. AUDIENCE: So the x-axis on these figures are-- PROFESSOR: Real space. AUDIENCE: So there will be some portions of the device where this is possible in-- it kind of depends upon how many quantum wells you actually put into your structure. PROFESSOR: Right, and that's one of limiting things, is the EQE of these devices because they can't grow so many are pretty limited. And again, these also provide pathways for traps and other things, which is again a huge problem for these types of devices and ideas. And people are working on that quite extensively from the theory side and from the experimental side, and trying to merge the two. OK. Sorry, Ben? AUDIENCE: The quantum well [INAUDIBLE] on the right, where [INAUDIBLE] functions don't overlap, do those act as recombination centers? PROFESSOR: It seems to me that they would act as traps. Again, I'm not an expert on these types of materials, but that would be my inclination. I guess the idea is that you have a very, very short well and then you have a field across it so there's not something else to hop to. And it just pulls it right out, so that would be one device structure. Again, these are generally just like a few layers of quantum dots. Oh, I didn't put up the [INAUDIBLE]. So this is the band-anticrossing model. So this is they're using these quaternary alloys. So these are, again, very, very hard to grow, and they've done these awesome measurements where they can show your valence band, conduction band, and your impurity bands, and they're using a technique called photomoduled reflectance, so basically the reflectance under oscillating fields. And from that, you can get these kind of resonance points between the different transitions and you can figure out where your energy state lie. It's a really cool technique. But what they did is they've done some very, very careful engineering and grown these very, very carefully. And they can actually demonstrate the sub-bandgap response where if they have, let's say a photon that's able to drive this transition but it's too low in energy to drive that transition, they will get no current out. And then as soon as they add the higher one, the current will increase. And then if they turn off that low-energy photon source, the current decreases but doesn't stop because the other high-energy photon can do both promotions. So this is actually one of the only successful devices of impurity-band solar cells, and it was quite an awesome feat. So I think one of the few ones we'll talk about for another advanced concept is hot carrier cells. This is something I think Martin Green pioneered, and I believe it still working on. I haven't heard much about them recently, so I don't know what the progress is there. So the idea is that one of the biggest losses, as I'm sure you guys know from I think homework 1 or 2, is thermalization in your solar cells. When you promote an electron well up into the conduction band, it then gives off heat or phonons, which are just lattice vibrations, which is another way of thinking about heat. In that process, the problem is that's incredibly fast. If you look at the time scales here, so you come and you promote this electron way up into your conduction band. This is what it looks like prior at that promotion, and then it slowly decays, and that decay happens over about a picosecond. So if you want to move your carrier in a picosecond, if you know what your field is, you can see how long that length is, and that length scale is also incredibly short. And so one of the ideas is, OK, how either we can decrease our path length, or we can somehow slow down thermalization, and that's actually one idea. And to slow down thermalization is something that they call carrier cooling. And basically making certain types of device structures or material structures, you can inhibit certain phonon modes. So when these electrons get promoted, they want to give off heat. They basically give off lattice vibrations, so they want to shake the atoms around them and distribute that motion. If you prevent that from happening somehow, through certain types of structures, you can allow that process to go slower, and that's one of the ideas. Extracting the carriers-- also kind of difficult. You need to have contacts that are selective that can take the hot carriers at all the different energies that they're promoted at, and that's also a really difficult idea. And some of the ideas that they've been working on are these resonant tunneling contacts. So again, this is kind of hairy stuff. I think these slides are rather old and I don't know what's come of this research. So it's a really exciting idea and I encourage you guys to look more into it. OK, so now we're going to move a little bit away from advanced concepts and talk about kind of bulk thin film materials, and I think we're actually going to end quite early, which is fine. So the most common commercially-available type of material are these wafer-based materials, so monocrystalline, which you know, which is what our cells are made of. This CZ, or Czochralski growth, probably pronouncing that wrong. Silicon, multichrystalline silicon, which I think Tonio was showing you what that looks like. You can see the different grains. They're actually quite pretty. Ribbon silicon, which was pioneered by Evergreen, and I think now that technology's gone. Evergreen went out of business, so I don't think that technology's still around. The [INAUDIBLE] still might be making modules. Anyhow, so thin films. Cad-tel is still currently-- or First Solar is still one of the cheapest module makers. Their process was so cheap that their efficiency really suffered because they didn't even have an ARC coating for the first 5 or 10 years of development. And they said we didn't need it, and they were right. They were still outperforming silicon people. That's becoming less so the case and one of the big problems with Cad-tel, as you guys know, is the [INAUDIBLE] and also the toxicity, which some people are really concerned about. Yeah? AUDIENCE: Just a note on that. So if we're debating in this class whether the should be allowed to be imported into Japan. And my understanding, from the reading, was that Japan actually does not allow cadmium telluride imports. Is that correct? AUDIENCE: I guess so, then yes. I don't actually know. I do know that Cad-tel is a very stable compound and I know people who work with it here and they actually work on recycling it, so how to dissolve it and separate the two elements. And they're not worried about the hazards of the actual raw material because it's a pretty stable compound, but a lot of people do worry about it and it's certainly a valid concern. But one of the thin film replacements, well amorphic silicon-- you think you guys know that is, but that's deposited silicon. It has no real like lattice structure to speak of. It's kind of a disordered mess, which means it has a lot of dangling bonds. It has very, very low lifetimes as a result, and also in [INAUDIBLE] low mobilities. You also deposit it and you deposit with hydrogen to passivate all of those dangling bonds, and that's usually done with either plasma-enhanced chemical vapor deposition, and it's done on either metal or glass. And so it has the potential to be very cheap. And then one of things I think was aimed to replace Cad-tel was CIGS-- so copper, indium, gallium, diselenide. I'll get to some of the problems with CIGS in a second, but there are few startups around it. I think Nanosolar is one, HelioVolt, and I think Solyndra was one, although I don't really want to say their name out loud. So problem with CIGS is if you're looking for Earth-abundant films, again if we want to scale-- if we want solar energy to scale to terawatt levels, if we want it to provide all of human energy needs-- then we're going to need to use elements and materials that are Earth-abundant, cheap to find, cheap to produce, and CIGS isn't going to get us there. It's got indium and gallium. Indium, which is highly used and the price has skyrocketed now because I think it's used in your displays and televisions. There's indium in here. AUDIENCE: [INAUDIBLE]. PROFESSOR: OK, so that's where it comes from then. Thank you. So what people have been working or trying to look at replacements-- so this is CZTS. So it's copper, zinc, tin, it's sometimes sulfide. This is selenide, but they are also working on replacing the selenium with sulfur, something a little more Earth-abundant. And this was done at IBM and it was pretty remarkable, actually, because I think within a very, very short period of time, they were able to get efficiencies around 9.6%, which for a fledgling material is incredible, and I'll get to why that was so incredible in a second. But again, when thinking about Earth-abundant materials, I really recommend you guys read this paper. This is a paper by Cyrus Wadia. It's "Environmental Science and Technology" and it looks all of these different semiconductor materials and it looks at in yellow, this is what you could get if you look at annual production of that semiconductor material now in terms of what is being produced to produce the raw materials that make it. And then the known economic reserves, what can we mine and get economically today, how much could it produce? And your worldwide consumption is on this line, and you can see Cad-tel is just barely eking it out. And the ones of note for this discussion are Cad-tel, CIGS, which is slightly better, and then CZTS, which is rather plentiful even with reserves that are currently being mined now, and so that's kind of the take-home message. The other thing also really interesting is the actual raw material cost. Cad-tel, turns out, it's not that great when you compare it to CIGS and then CZTS. It's a whole lot cheaper. So one of the other things about CZTS that's so cool is that there's basically this huge growth parameter space for making these quaternary alloys, and IBM on their first shot just kind of nailed it. So they didn't really look at this whole parameter space. It was just kind of their first shot was right there, and so it might already be optimized, and a lot of people are kind of concerned about that. But I think it's certainly a very interesting field and really cool to work in, so I think hopefully there's hope here. So actually we're ending way early, but that concludes the lecture today. If you guys want to stick around for more questions, that's totally fine, but that's all I had prepared for today. So thank you. |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | 19_Cost_Price_Markets_Support_Mechanisms_Part_II.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: The equivalent cost of manufacturing in China, as best as we can determine, is in the order of $1, $1.30, $1.50-- somewhere in that range with the error bars. So keep those numbers in your head as we dive into price. Because now we're going to be talking about what the market is offering to pay and what that spread is between the cost and the price. So you have, in your slide deck, a list of different websites. I guess you have web [? e-sits. ?] You have websites, email list serves, blogs, and Twitter feeds-- different sources of information for collecting data from PV. And I should add one more to this list, which wasn't on there, but should be there. PVinsights.com. This is where you can find it pretty up-to-date spot prices for wafers, cells, modules, and so forth. Does anybody know what a spot price is? What does it mean, a spot price? Spot price means I'm desperate to buy. I need to buy it right now. I pick up a phone and call somebody and say, I'm willing to pay you. So there's no long term contract involved. It's usually a one-time deal. That's a spot price. A long term contract, on the other hand, says, no, no, really. I have a factory. It's 100 megawatts. And I need wafer supply for five years. Let's lock ourselves into a price. Maybe we visit it every 2 and 1/2 years, but that's a long term contract. So these are spot prices and they provide a variety of information. The information for today is free. If you want historical data, you have to sign up, become a member, and pay. But there are other websites like this-- various consulting groups-- that acquire and gather information, PVinsights. So you have a variety of different sites to grab information that could be useful for your class projects. We're going to talk again about the dynamics of price. And that is driven, in large part, by-- well, we agreed to call it different things. Support mechanisms, tax breaks, incentives-- but in reality, they're support for PV. Recognizing that with $1.30, $1.10 cost, and then you add on top of that the profit margin, even if you assume a very meager margin of 15%, you're-- and then the balances systems, and then the installation costs on top of that-- you're not reaching grid parity in the majority of markets. Not with that sort of cost structure today. With innovation and moving forward into the future and scale-- we'll get there, I'm fairly confident-- but today, we don't have the cost structure necessary to match, say, a subsidized coal-fired power plant. So there are a variety of different support mechanisms, a variety of different subsidies in different countries, and different states within the United States. And you can think about these as the carrot, the stick, and the hybrid. The carrot meaning the margin enhancement, the stick being the penalty if you produce too much carbon, for instance, and some variety of mixtures between the two. So in terms of margin enhancement-- the carrot-- what mechanisms exist? Let me break it down very simply into-- we'll look in two different categories. We'll look at what the United States has mostly done, which are tax relief and grants and soft loans. So let's describe what that means. When I bought the panels on top of the house in 2007, we paid out of the box somewhere in the order of $7 to $8 per watt peak. And after tax rebate coming from the federal government, and some additional support from the state of Massachusetts, the final price tag wound up being from say $18,000 plus down to about $12,000 to $14,000. And then there's revenue coming in from offsets and so forth. So this is a one-time deal. I could have installed those panels in my basement and still gotten the tax break, right? Because it's per watt peak, not per kilowatt hour produced. The rebates based on carbon emissions is based on the amount of energy it produces. That gives you an incentive to maximize the efficiency of the installation. But just a one-time tax rebate doesn't. However, what the one-time tax rebate allows you to do is decrease the upfront sticker price. So if I'm trying to sell you system on your house, US installers are convinced that it's a lot easier to sell you the system if the price tag is lower. If I can say, well this is the real cost, but wait, wait, there's more. We'll give you this tax rebate, this tax rebate, draw a line. The final amount you pay is this lower amount right here. The other mechanism of margin enhancement, if you will, is what's called a feed-in tariff. Now, a feed-in tariff works as follows. A feed-in tariff says, OK, if you're paying-- let's say, in the state of Massachusetts, we don't have one. We don't have a feed-in tariff here in Massachusetts. But imagine if we did. You, as a residential customer, are paying $0.18 per kilowatt hour for your electricity. But if you have solar panels on your roof, the utility, if you will-- the state of Massachusetts-- is willing to pay you $0.30 per kilowatt hour for that PV electricity. Recognizing the additional value that that PV is adding to the state. Reducing the need for additional transmission lines, reducing the amount of investment in new coal-fired power plants, reducing the health detriment to the local communities around the coal plants, and so forth. So a feed-in tariff is meant to give an incentive-- a market pull incentive, if you will-- to install PV on your house. Or in a field. And this is the mechanism that has been in use in Germany. And because it's a market-driven mechanism, it rewards the most efficient systems that are out there. If you install that system in your basement, you're not going to be producing kilowatt hours, and hence you're not going to benefit from the feed-in tariff. Now it's a tricky business to decide where exactly to fit that feed-in tariff. Right? If you go too low, people aren't going to move. They're going to say, eh, no, not enough. Not enough to make me want to install solar. If the feed-in tariff is too high, you're going to get this massive onrush of people coming to install solar. And now you're going to have to finance it, right? And the money has to come from somewhere. In Germany, the money comes from the rate payers, not from the state. Which means that if you install solar panels on your house, all of us have to help pay for the electricity that you sell back to the grid. So our rate goes up from $0.18 per kilowatt hour to, say, 18.2 cents per kilowatt hour. In the beginning, we don't notice it at all. But then if Joe starts putting solar panels up, as well, and then, let's say, 50% of us put solar panels up, now obviously we're paying a lot more. And it gives more of an incentive for more people to put the solar panels up on the roof. And of course, the price goes up. So the feed-in tariff is a very-- it is a market-driven incentive. And hence, it is very skillful at rewarding the most efficient installations. But from a government point of view, it requires very structured, rigorous, and deaf-to-manipulation of the feed-in tariff rate-- the decline of the rate versus time-- to ensure that A, the installers aren't reaping an enormous profit, and B, that the system doesn't become unsustainable over time. That the burden on the rate payers is not so great that they're shouldered in for 20 years paying these excessively high rates. So it rewards first adopters, it allows the market to predict, versus rate of growth, what the reward rate will be. And it allows you to glide into grid parity. So we're at a situation today where we're not at grid parity. In 10, 20 years, we're likely to be there. And so the declining rate of the feed-in tariff allows you to glide back in. In the United States, the tax rebate, unfortunately-- because of the way our political system works-- it tends to get renewed in a very frequent rate. Every two years, it seems, it's going up for debate and discussion. Should we continue it? Should we not continue it? It becomes a big political struggle just to get it passed. And as a result, everybody uses up their energy trying to pass this thing and renew it, as opposed to saying, gee, what's the best way to decrease this over time so that we can kind of glide into grid parity? So we have some issues in the US, more related to how our political system doesn't work. But there are examples of this throughout the world, in terms of what are states and countries doing to enhance the margins to create market-pull incentives to allow solar to be installed on the grid? And the panels that are installed could come from anywhere. They could be produced in Guam and they could qualify for the feed-in tariff. So it doesn't discriminate against particular regions of the world. Another thing to add here is that this is just the support from the state, from the public sector. From the private sector, what is possible? Well what is possible is what's called a power purchase agreement. And this was alluded to during our tour. What is a power purchase agreement? A power purchase agreement is-- instead of you buying a system and putting it up on your roof and having to pay all that money up front, what you do is you enter an agreement with the installer. They will put the panels up on your roof for free because they're getting the financing, say, from Morgan Stanley, from the Bank of Joe. So the Bank of Joe is financing the panels on your roof. So the panels went up on your roof. You didn't pay a penny. But you inked an agreement with me, the installer, because I just borrowed money from Joe at a certain interest rate. And you inked an agreement with me that you'll pay a certain amount for your electricity over the next 10, 15, 20 years, however long it is-- usually 12. And that rate may be a little bit higher than what it is today, but it's certainly going to be lower if current price inflation continues for the price of electricity. It will certainly be lower than what the price of electricity will be in 12 years. And so you'll make money. I'll make money because there's a spread between the rate at which I'm borrowing the money from the bank and what you're paying me for those panels-- for renting the panels on the roof, if you will. For buying the electricity from those panels. So everybody's making money. And the bank's making money, obviously, because they're charging an interest rate on the loan. And so with those financing schemes, where it's called a power purchase agreement, bank loans the money to the installer, the installer loans the panels on the roof of the customer, and the customer pays a fixed price for the electricity. That allows everybody to make money from day one, as long as there's capital in play. It requires capital to be in play, meaning it requires the Bank of Joe to be willing to lend money to me, the installer. If the Bank of Joe doesn't want to lend, then that isn't an option. And so you see many of these very large deals being forged with the investment banks between large installers-- say, SunPower, SunEdison, and so forth-- in New York City. It's becoming an increasingly popular form of financing solar panels. You may stand to make more money as an individual by buying the panels up front because then you reap the entire benefit of your investment. You're not sharing the investment benefit with the installer. You're not sharing the investment benefit with the bank. But that requires, again, access to capital. And not everybody has a spare $14,000, $15,000 lying around to put solar on the roof. Question. AUDIENCE: What's the incentive of the, I guess, whoever's installing the panels on to buy the electricity from you. Why would you buy it from you, [INAUDIBLE]? PROFESSOR: Yeah. So over the past 10 years-- say, from 2000 to 2009-- in the state of Massachusetts, the price of electricity increased by 15%. So if you compare what did it take per kilowatt hour at the beginning and the end-- normalize for inflation, 15% inflation-- in the price of electricity. And there are a variety of reasons for that. We're at the end of a natural gas pipeline, so even if the price of natural gas goes down, it takes a lot to get it to us. Sometimes shipments go in by boat. Other times, up the actual pipeline itself. And then, other forms of fossil fuel-- oil, especially-- has experienced a rise in prices of the last few years. And so for a variety of reasons, including those and including a difficulty in transmission, and including limited new power plants coming online, the price has gone up. And as a result, if you project forward, you could say, OK, let's hedge our bets here. We can estimate that the price is going to go up another 15% between now and the next decade. So what I'm going to do is to say, this is the price today, this is the price tomorrow-- in 10 years, right? I'm going to sell you electricity here. And so it's almost like the deal-- I don't know if anybody signs up for the natural gas lock-in price during winter with NStar. You probably see the envelope in the mail, or maybe your landlord does. But NStar-- the utility company around this area-- will allow you to lock in a price for natural gas per therm-- per unit of natural gas-- over the winter, that is slightly above the market rate in the fall. With the understanding that prices tend to spike during winter, and you're able to hedge, you're able to reduce risk. And so really what it is is a risk mitigation strategy. And it's good enough for most people. I know two people on our street alone have entered power purchase agreements as means of financing their solar installations. So let's discriminate once again between the private sector that's trying to sell you the panels-- I say, Omar, you have to buy my panels. Let me sweeten the deal here. You don't have to pay a penny upfront. We'll introduce a power purchase agreement. Versus what the state is doing, right? Whether that's the national government or the state level is doing to try to get the installers and other industries growing within their organization. And as well, several of the EU states meeting their Kyoto Protocol obligations to reduce carbon emissions by a certain amount by 2020. So we're going to do a deep dive into the German case, just because it is so interesting and so exemplary, in terms of increasing the amount of PV on the grid. And one thing to note, just upfront, this is the insulation map of Europe. Insulation being the solar radiance, the total solar resource available, shown in this barely distinguishable little legend down here. Blue being low, red being high. And Germany is right here, as they would say, from herzen Europas, from the heart of Europe right there. Right in the middle. And this is the insulation comparison, again, between Germany the United States. Same scale over here. A lot less solar resource in Germany, even than in the Northeastern part of the United States. Yet there was about half of all solar panels installed here last year. Why? Well first off, it's high electricity prices. Secondly, there is a feed-in tariff that gives an incentive for solar to be installed on a grid. So what I'm going to do is go over several slides coming from the ministry in Germany, describing the growth of solar and the growth of other renewables on the grid, in response to this feed-in tariff. So the renewable energy resources, if you will, as a share of the total energy supply in Germany-- the goal by 2020 is this white bar right here. And if we look at the share of renewable sources in total gross electricity consumption, you can see that it's getting there, right? 17% versus a minimum of 35%. So working toward those targets pretty well. And climbing from 2000 to 2010-- more than doubling, almost tripling. This is the electricity, heat supply, and fuel supply breakdown. If we just look at the electricity component right here, since that's where PV falls and contributes, you can see it's growing. What you have to keep in mind is this 17% of electricity consumption coming from renewables-- this 17% is this amount here, the yellow bar. And we're looking at something in the range of 103 terawatt hours over the course of a year in 2010. And out of those 103 terawatt hours, breaking it out into PV, biomass, hydro, and wind, you can see that PV has accounted for a relatively small fraction of that total. The largest, by far, has been wind. Wind has reached lower prices of electricity faster than solar has. Differences between the technology, so we say. And as a result, the grid penetration of wind has preceded that of solar. But solar is growing quite a bit. And this is averaged over the entire country. And as I mentioned before, there are certain regions within Germany, as you might guess-- down here, for instance-- that have experienced larger grid penetration of solar than others. Just because they have a larger solar resource available to them. So this is the little fraction here-- growing-- of solar electricity. These here are the different legislations that are being passed, regulating the feed-in tariff. Now, the feed-in tariff is scheduled to reduce gradually for each year. Let me show you how that works. We'll go back to this German energy blog by two of our energy law experts in Germany. And this describes for you the German feed-in tariffs as of 2010. The Renewable Energy Sources act-- essentially, one of those legislations that have passed-- and it just shows you what you can expect over a variety of different sectors. Hydro, landfill, gas. And you can see, it's broken into very specific details. Different sizes of installations, different types of plants and so forth. This is bio, geothermal, onshore wind, offshore wind, solar radiation, roof-mounted facilities, electricity used within the building facility, freestanding facilities, and digression. Digression means, how much does it go down per year? This is based on their best estimate for the growth of grid penetration of PV. They're trying to guess in the future, how much PV is going to come onto the grid by a certain date? And hence, what the price will be, as well. And thus, reduce their feed-in tariff accordingly. And since it's impossible to look into a crystal ball and nail it-- especially since this is a nonlinear system-- the price depends on the feed-in tariff, the feed-in tariff depends on the price, right? So there's a little bit of that interaction going on. They have to reassess, from time to time, what the new rates are going to be. And that's why you have these various-- well, aside from the initial-- you have these various reevaluations from time to time, looking at the feed-in tariff. Now, what has happened more recently-- there was a reevaluation in January 2009. Another mid-2010 that decreased it even further. So more recently-- this, unfortunately, only goes to 2010-- but more recently, there have been more significant, stronger cuts to the feed-in tariff in Germany in response to a few things. So let me go over this real quick. I'll get back to that in a second. First off, this is the payment of fees in millions of s versus time that the rate payers are paying in total. So that winds up being something in the order of 135 s per head in Germany. That's not spread equally amongst everybody-- per year-- that's not spread equally amongst everybody. That's, as well, their industry bears more, obviously than the residential customer would. But it's a line item of a few s on your utility bill per month as a customer. And that begins to add up. So Germany has begun putting on the brakes on the incentives. And further, if they look at how much they've installed versus other countries-- again, this is the same chart we showed last class-- their portion of all new installations is very large. And so they began looking around to the rest of the world, saying, hey folks, we don't have a lot of sun here. Why aren't you doing your part to put solar on your grids? Compounded by the fact that now they have a growing percentage of manufacturing that's not in Germany. The percentage of German manufacturing of the PV modules themselves has stayed more or less flat. And so Germany's sitting here thinking, OK, we're in a financial crisis right now. Something has to give. Let's put a damper on this feed-in tariff for a little bit until the situation straightens itself out and until there's more growth in other markets besides just Germany. Let's try to reduce the incentive that we give to put PV on our grid and maybe increase the share of PV going onto US, China, and so forth-- other big markets around the world. So that we're not bearing the sole burden of trying to reduce the cost of PV to grid parity. As a result of this-- the decline in the feed-in tariff in Germany-- and as a result of a massive amount of new production capacity coming online in China and Taiwan, we're now in an oversupply condition. What means an oversupply condition? What it means is that there are more PV modules available today than there are customers to buy them at the given prices that are available. And the price is dictated, in part, by the feed-in tariff. And so what you've seen is-- sorry, I'm just going to drive through these slides right over here till I get to that one. So what you see is that chart that Secretary Chu presented yesterday during his talk. This was in first quarter of 2008. This was when the market began softening. Right around here, the German feed-in tariffs really started going down tremendously. Chinese manufacturing and Taiwanese manufacturing really started ramping up around here, 2007, 2008. And so what happened? This here is price, not cost. Price. So this is being driven by market dynamics, not solely by the costs of manufacturing. So we have what people are willing to pay for their hours of watt. So if the feed-in tariff is going down in Germany, which is acquiring 50% of the modules in the market, that means that the price has to come down, as well. If you're going to be able to sell your modules, you're going to have to sell them at a lower price because the feed-in tariff is now lower. At the same time, you have now more supply on the market and you have companies competing against each other to get their modules on the market. And so prices are going to come down by that, as well. So what this chart is telling you-- let's look at the blues, for instance. Let's start here. These are estimates made in 2008. And this blue line extending forward is the estimated price-- not cost-- the estimated price of what a PV module would sell for, projected forward to the end of 2010. Then, we enter 2009-- the reds here. And the real prices continue to drop precipitously. Again, here we have the analysts' estimates for what the price is going to be, moving forward to end of 2011. And you can see that the analysts' estimates are always above the actual prices over the last three years. Which means that the actual prices have fallen faster than anybody-- or the analysts-- expected to. Maybe there were some smart people in the actual industry who saw this coming in quite the same way. But what this means is the prices have come down a lot faster than what people expected. I don't think people expected that Germany would cut the feed-in tariff rate quite as large as it did in '09 and in '10. And some people, who haven't been paying attention to the market, might not have expected as much supply to be available from China as there is today. Those people should have been paying better attention. But that combination of factors resulted in a much faster price decline than what people saw coming. And as a result, companies that were formed in, say, 2007, 2008, and got venture capital-- and saw one of these lines right here and said, oh, we're going to be able to intersect them in 2010-- are now looking at these sorts of prices here. And saying, oh, gee, we're not to be able to intersect them in 2010, it's going to be more like 2015 before we get our production costs low enough to compete at those prices. And so the venture catalysts are now sitting back thinking, so let me get this straight. You came to us three years ago and told us that you'd be cost competitive by 2010. But now the story is you're going to be cost competitive by 2015. This kind of smells fishy. I don't know if I want to lend you an additional round here, especially if I have to wait another five years before you're profitable. Let me just cut my losses and pick up shop and leave. And your company will go under. So that's happened a few times. That's happened in a high profile way, which we all know about-- Solyndra. It's also happened to a few other companies-- SpectraWatt. Even earlier ones at the beginning of the financial crisis-- OPTI-Solar and others. So the companies that are surviving right now-- there are still more than 100 startup companies in the United States. Dozens and dozens of startup companies in solar. Those smart ones that are surviving are usually in pre-production stage. They don't have big manufacturing lines, hundreds, thousands of people to pay, supply chains to pay for, and customers evaporating, so they're not caught in that situation. That was Solyndra. They had a big production line. They had 1,100 people employed in that line. And they had customers lined up. They had suppliers shipping in materials that they were converting into product. And some of the customers disappearing were not willing to pay as high prices anymore. And that leads to a very difficult situation. You don't have a cash cushion, you don't have any reserves in the bank, you have to sell your product. And you're not able to compete at these prices. It's a recipe for disaster if you're a mid-size company. So small companies can survive in what I call spore mode. They're like a spore. They don't have that big manufacturing line to pay for. They can survive off grants, they can survive off venture capital. And their cash burn rate is very low, they're developing technologies. The big companies have cash cushions already, they have cash reserves. They might even be able to access low interest rate loans from banks that borrow money at ridiculously low rates from the treasury right now. They may even have financial branches within their own company, like GE Finance, that can do that sort of thing. So big companies are, so far, surviving, the really tiny companies are, so far, surviving, but the market dynamic is really hitting those mid-size companies that already have a manufacturing line. And so you hear about layoffs, you hear about job losses. These are often the mid-size companies just trying to go back to spore mode so they can survive this difficult period until prices equilibrate. And when we look at price is equilibrating-- let's look at the price now. We're headed towards Q4 2011. Let's put a data point right here for Q4 2011. Let's do it right now. So I'll go back here and-- sorry about that. That was me registering a website. That's for the project that Doug was working on. We'll go to PVinsights. And we're going to add the latest data point here. So solar wafer-- this is silicon, this is wafer, this is cell, and this is module. All right. This is our low, this is our high. These are all prices in dollars per watt peak. Prices. Our low and our high and our average for-- I guess, last update was yesterday. The low-- I can tell you this particular low number right here came from a tier three manufacturer in China. What is tier one, tier two, tier three? So tier one are brand names. Yingli-- they advertised during the World Cup. Suntech, mentioned in Secretary Chu's presentation yesterday. Trina Solar, LDK, and so forth. These are tier one manufacturers, the big dogs. Tier three are companies you've never heard of but are employing thousands of people and manufacturing modules in the hundreds of megawatts range. Perhaps even reaching a gigawatt scale. And because banks have never heard of them either-- maybe that's an overstatement, but I'm making a point here. They're not as well known, they're not as reliable from the bank's perspective. Maybe they haven't been around as long and it's questionable whether they're going to survive this difficult economic climate. They have difficulty to sell their product. The installers don't want to take their product. And so they have to undercut the competition. They have to leave money on the table. And selling at prices at $0.75 per watt. Now, what Doug's calculations are indicating-- as you saw from the very beginning-- the cost of manufacturing in the US is around $1.30. The cost of manufacturing in China is around $1.00. And then you have to ship it over to the US. So if somebody is willing to sell you a module at 0.75 dollars per watt, that means that the price is below the cost. That means that that company is desperate to get rid of inventory. They must have modules stacking up in their shipment yard. They're unable to move them. And so the chief financial officer walks over and says, we got to get rid of this stuff. It's costing us money. It costs us money to keep this product on the books. Sell it for whatever you have to sell it for to get rid of it. And as a consequence, they sell below cost, put it out into the market. And then you have SolarWorld, or a US company, coming to the Department of Commerce saying, they're dumping product. It's an unfair competition. According to the World Trade Organization, you can't sell at a price below your cost in order to squeeze out and gain market share. So it's a complex situation right now. I've described for you, in as much detail as I can, my impression of what's happening in the world today. This low price right here, of 0.75 from a Chinese tier three manufacturer. And the high price here coming from, most likely, a German supplier or US supplier. Selling what is known to be a very high quality, reputable, product, has been selling for the last 10 years. Very reliable, very few incidences of consumers returning the product. And banks like that product. So they're able to extract a premium for their product. They're able to sell and move those modules at a higher price because they're more bankable. This average right here is more representative of what Chinese tier one manufacturers are currently selling for, and what many of the US and European average module makers are having to compete against with costs on the order of $1.30. That's the constriction right now. So if we add that one data point onto this chart right here, where we have Q4 2011-- we're solidly in Q4 right now-- and we're at 0.98 with an error bar somewhere around here. So we're at 0.98. You see the prices are still coming down. And will likely come down for maybe another quarter before they start to stabilize. And as companies fail-- as more companies leave the market-- you'll have consolidation of market share, you'll have the few remaining companies that had the large cash cushion, that had the lowest cost structure, survive. And increase their market share and reduce the number of players out on the market. And probably, prices will come back up afterwards. Because you can't continue selling below cost for very long before everybody goes out of business. I saw a hand going up over there. AUDIENCE: I guess, those prices-- a transaction happened at that price, or that was just the asked price of the the manufacturer? PROFESSOR: So these right here-- I know that the 0.75, that was an offer price. AUDIENCE: Oh. PROFESSOR: Yeah. So it was 0.85 during Solar Power International in Texas about a month ago. And it made a big splash and everybody was really awed by it. The 0.75 is news to me. It probably came up over the last week or so. And in response, most likely, to that company selling at 0.85-- unable to move their product-- going even lower in a desperate attempt just to get rid of their inventory. AUDIENCE: But these are offered prices? The transaction hasn't happened at that price as of yet? PROFESSOR: So these prices, most likely, are coming from a few different routes. So you, as an individual, can send an email or fax over a request for a quote from any one of 100 module manufacturers around the world. And you will get a quote back or an offer sheet back. And most likely, what PVinsights is doing is some combination of that-- a guerrilla tactic, let's gather information. And as well, information gleaned from their installer base. So they have contacts to various installer companies. They keep the finger on the pulse over there, talking to their friends saying, how much are they offering you the modulus for? AUDIENCE: It's not like the stock market, where the price sold at is subjective? PROFESSOR: Yeah. The sell and the buy price are a little different. No-- or currency exchange markets. No. These are individual companies trying to assess what the market is willing to pay for their product. In the case on the low end, these are desperate producers trying to move product. And on the high end, these are companies with high cost structures, typically, in the west-- typically US and Germany-- that have a reputation. And they are clinging for as long as they can on to the high prices. For as long as they can do it, before the market finally says, you know what? I'm sorry. We've been good friends. We've worked together for the last 10 years. But honestly, I'm not going to buy it $1.45. Not when Suntech is offering me a module at $1.05 and it's comparable in quality. They've proven themselves. The days when all Chinese modules were inferior are over. Now we have several tier one Chinese manufacturers that have proven their modules out in the open market. There hasn't been a large number of recalls. So I'm willing to take the risk with them. And at that point, you'll start to see the higher priced US and European products begin to soften. Yeah. AUDIENCE: If the oversupply condition doesn't get resolved in enough time, does it have the potential to, basically, stall the entire industry? And if so, how long would that take? PROFESSOR: Well, keep in mind that this is just the module. There's a whole other dynamic happening on the installation side. And the reason I'm not getting into that in too much detail is because it varies so much from country to country. Although I will say a few comments before the end of class. If the oversupply condition continues and if the feed-in tariffs continue to be low, what you'll see is a continued softening of the module price into the point where almost all manufacturers are selling below manufacturing costs. They're all desperately trying to reduce manufacturing costs, reduce overhead. It's forcing them to innovate-- at least in manufacturing innovation-- faster. Not on product innovation, not on, how do we design this cell differently? But more on the process innovation on the line of saying, gee, how do we mix the silver with the cheaper metal like aluminum in a ratio so that we eek out half a cent per watt peak? Because any small fraction counts at this point. We're desperate. And you can think of this oversupply condition as a bunch of horses running nose to nose. And which ones will begin falling out? In the beginning, you could point to easy candidates. Those that have already begun to go by the wayside. But now, companies are burning through the cash cushion, reporting negative profits. You see right and left, even the Chinese tier one manufacturers are reporting negative earnings this last quarter. So that's a reflection not only of the oversupply condition, but also the fact that they're continuing to expand, despite the oversupply condition. The idea being, well, we can withstand one, two, three, four quarters of losses as long as we consolidate market share. Once we emerge from this oversupply condition, we'll be able to increase prices a bit more. And then return to a more sustainable market. But we'll be the big dogs and everyone else will be out. So I think all companies right now are trying to play that game of survive. Survive this oversupply condition, make it through. Some are able to continue growing and other ones are just stagnant. So the stagnant ones are going to become niche players. They won't be the major players in the market. The ones who continue growing will be 80%, 90% of the market. AUDIENCE: Has there been talk of increasing feed-in tariffs to try to keep the American companies alive? PROFESSOR: So US doesn't do many feed-in tariffs. This goes back to the US case right here. So due to lack of leadership at the national level, there are a variety of state level incentives put forth. So these are the policies-- we kind of talked about this last class, but to dive into a bit more detail. The rebate programs for the renewables-- these are state programs plus the utility and/or nonprofit programs. Utility, local, and/or nonprofit programs only and state programs only. So you see, for example, in the state of Massachusetts, we have what used to be the Massachusetts Technology Collaborative. There was a bit of a power struggle within Mass State and it got incorporated into a more centralized organization downtown. So that used to be in a nonprofit organization, which is now more affiliated with the state. The state also has a rebate. We have a very different set of ways of doing things than, say, California, which has a clean energy commission that rates each module and gives you a rebate depending on what they rate the module as performing in California. So every state has its own way of doing things and it becomes very complicated very quickly. To enact a national incentive, beyond the tax rebate that is already offered today, is challenging, in this political climate especially. Because anything that you would do to raise the cost of something, which most likely would come from some federal program-- right now, the Republican Party is demanding that there be an offset, a reduction of spending elsewhere within the government. Many of the places that are easy to cut have already been cut. And so you've gone through the fat. You're now hitting muscle, and pretty soon you're going to be hitting bone. So it's difficult to enact something that the government would pay for. It is even more difficult to enact something that the utilities and rate payers would pay for at a national level because, oftentimes, those powers are delegated to the states. And you would enter a federal versus state fight over that, which would go to the courts, most likely, and be held up there for several years. There are about 18,000 different independent jurisdictions within the United States governing how solar is added to the grid. And so what the Department of Energy has done, which I think is the wisest thing to do-- given this situation that we have-- is to say, OK, we're not going to force anybody to change, but we're going to give an incentive for people to change. And like they had in the Department of Education-- the race to the top, where states competed against each other to implement best practices in education-- they're having a similar program on the installation side of solar, trying to get various states to adopt a best practices. To streamline the permitting process to get solar onto the grid in the most efficient way possible, and hopefully reduce the installation costs associated with that. So let's turn our attention, quickly, to installations, since I want to give some time to have people ask questions about their class projects. Since time is running very short, if you're hitting up against a roadblock, I want to make sure that we resolve that. I'll say a couple of words about the installation. So a funny thing happened on Tuesday night. I met with a colleague from Wisconsin. And he said, well, I met the editor of-- I think it was an editor from one of the big journals. We'll protect the innocent. And this was a very high-impact scientific journal. And he said, well, the majority of the cost right now is in the installation side, not in the module, in solar. So I'm not going to be interested in any papers to my journal that describe new concepts for PV modules. I'm interested in the installation side because that's where the majority of the cost is. And I went, oh my goodness, here's another one who can't distinguish between cost and price. So the installation cost right now-- let's focus on cost first and then we'll get to price. So the installation cost in the United States is partially reflective of the fact that we have those 18,000 different ways of doing it here, versus in Germany, there is one way to do it nationally. The federal government said, we are going to demand that everybody in Germany install in this protocol. And the paperwork is very, very brief. It's a couple of pages to get it installed. And the inspection is one. And so Germany has a much more efficient system. And they've installed roughly six to seven times more solar than we have. And if we remember experience learning curves, when you have the reduction of cost the more you do it, that means you have three doublings. Germany has three doublings over us. And even if you assume a very leisurely 20% reduction for each doubling, that means Germany is about the half of the cost as we are, because they've done more. They know how to do it more efficiently than we do. OK. So that's on the cost side. On the price side, if you look around the US, you have some states like New Jersey, which had an amazing incentive program for a while. And California's as well, quite generous. So there are certain states where it's almost like a gold mine. And so there's no incentive for the installer to reduce their price. Their costs can be going down, but their price can be maintained high. And usually-- this isn't always the case, but in most industries-- when prices are high, the industry becomes lazy. Now this isn't always the case, but is often the case. It is not atypical that when the prices are high, the industry says, oh, prices are high, that's pretty good. I'm enjoying myself right now. I'm not going to be focused on cost reduction. It seems that, at least, the lower level managers suddenly become fixated on cost reduction when it's too late. When margins have already begun shrinking, when prices are collapsing, and when it's do or die for the company. Then, all of sudden, cost becomes imperative. So there are very few companies. The ones that usually become leaders are the ones who recognize, gee, prices can't remain high forever. We have to reduce our costs now. And hey, it'll be even to our benefit, because our margins will be greater. We'll be able to take advantage of this right now and build up some cash, so that when the prices do collapse, we have some buffer. And we can survive the oversupply condition. So installers, right now, I have to say I'm disappointed in our installers in the United States for not doing more amongst themselves to reduce the amount of paperwork. For not taking a leadership role in reducing the paperwork burden. And the true cost of installing PV on the balances system, [INAUDIBLE] installation side. I think that's a very resolvable problem. You look to Germany and you have residential systems going in for below three euros per watt peak price taking advantage of the low module prices right now. And the fact that the installation costs in Germany are low, as well. Because they've learned how to do it well, they've learned how to do it efficiently. And they've reduced their cost structure. So I personally-- I think there is innovation to be done on the installation side. I think there's a lot that can be done with pre-fabrication. Maybe moving a robot out there that can assemble things in a big rack and then a crane that puts it on the roof. As opposed to having 10 people going out to a house and spending a couple days putting panels on the roof. It's not that bad. Maybe it's more like a day. But it's still a very labor-intensive industry right now in the US. So more innovation can be done. But the lion's share of the installation price right now is driven by what I would call inefficiencies in the way installation is done and inefficiencies in the way the permitting process is done, the paperwork is done. So that's my soapbox speech on installation. By all means, there is innovation to be done. So don't give up on that side either. It's important. It's where the rubber hits the road. But I think you can't just look at the price of modules today and the price of installing the system today in the United States-- and realize that 80% of the price right now is wrapped up in the installation-- and say, oh, well, there isn't any more innovation to be done on the module side. You have to look at cost. And you have to recognize that the module cost is driven by the efficiency of the module. And the installation cost, as well, will be driven by the efficiency the module. And so the module being the engine of the system is still very, very, very important. And so the work that folks are doing-- on new PV materials, especially-- is important. |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | 2_The_Solar_Resource.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: And I think we're about ready to get started. So welcome, folks, for lecture number two of Fundamentals of Photovoltaics focused on the solar resource. What I wanted to do to get everybody in the mood of thinking about the sun is pass around a few balls. So this is really a, to limber you all up, but b-- whoop. [LAUGHTER] There we go. All right. Don't let it fall. There you go. I know Ashley's got solid hands. All right. Here's one. Pass that around as well. We'll get a few more there, there, and lastly, right up the middle. There you go. OK. So today's lecture is really about the solar resource. And as we go through, it kind of helps to have a sphere in your hands since oftentimes we perceive the world as being flat-- no fault of our own. Locally, one can approximate it as a flat body. That's certainly a good possibility. You wouldn't mind passing these out to your friends as well? AUDIENCE: Sure. PROFESSOR: Thanks. But in reality, if we really want to understand the solar resource, we really have to begin understanding or thinking in terms of spheres and in terms of circles or, in most cases, ellipses. And so we're going to dive into the solar resource. Before we really dive in in detail into the solar resource, I wanted to give you feedback to your surveys. So you did a background assessment survey, a census, if you will, and filled out a number of questions last class about your backgrounds. And I wanted to provide you the feedback, the consolidated information, because it's really telling about who your colleagues are. This right here is a little bit of a snapshot of expertise and current career trajectory. So the self-defined expertise in the bottom left is really, I think, the most telling parameter. For the undergrads here, you may define yourself by your major today, but when you graduate and go on to grad school, you might, say, for example, do your undergrad in physics and then do your graduate school in mechanical engineering, but still consider yourself a physicist at heart. And so that's why I asked this question here-- what is your self-defined expertise-- because there are several graduate students who have changed fields, if you will, from undergrad to graduate school. Most people in the audience, by and large, consider themselves engineers-- either material science engineers or mechanical engineers. Chemistry is strong as well. And then we have about 10 different departments here represented. And that's really cool. It's going to manifest itself in the class projects. And you'll see the diversity of different inputs and perspectives from your colleagues. The degree in progress-- undergrad/grad is about split 1/3 2/3 undergrad and grad. ASP is Advanced Studies Program, so these are folks coming in from industry who are actually here in the classroom. Some of your colleagues in the class are folks who are in industry and perhaps have real world PV experience. Several of the people here in the class, as well, have gained-- how do we say it-- have gained expertise in solar with their hobbies, with their work. Some have installed solar panels. Other ones have done research or are doing research in solar. So it's a pretty diverse group. And some are, as well, members of the solar car team, which is rolling out its new model in a few days' time. In terms of learning methods, it was pretty well split between hands-on labs, field trips, and guest lectures. And I'll get back to that in a few slides. In terms of class project interest, there was a strong interest in working with pre-established projects, so we've listened to that. This is pretty consistent with previous years. And so we have several pre-prepared class projects ready for you. And a few of you had an interest in the self-design project. I'd like to talk to you. I'd like to begin developing those ideas as soon as possible so that when we start assembling teams, if you do have a strong idea for class project, we can begin crafting that and molding that starting now. So please come up and have a chat with me after class or during office hours or during recitation. These are your learning objectives defined by you. And they range-- I tried to give it some continuum spectrum from natural sciences to social sciences and engineering in the middle. And obviously, this is more of a loop than a linear line, but bear with me. There was a strong interest in fundamentals. And that certainly, I think, what the core of the class is about or at least the first third of the class. Going in terms of size of the bubble, these are the number of people who listed a particular topic as of great interest to them. Economics and market, systems and grid, current technologies, and emerging technologies. And so listening to all of this, we have, or we are in the process of preparing for you some guest lectures and field trips based on this feedback right here. We have already lined up a field trip to a local PV research laboratory that produces modules like this one right here only much, much bigger and has strong collaboration with existing companies, startup companies, in the area as well as more established companies. So that'll be a lot of fun. And we're currently in the process of arranging other field trips and guest lectures to match this feedback right here, so thank you. We'll mold the course, shape it, craft it to fit your interests. So to hop into the solar resource and without further ado, the subject of today and the motivation for wearing this tie is really the solar resource, the sun. This is where it all starts. If we're to understand PV, photovoltaics, the conversion of sunlight into electricity, it starts from the sun. And so spending some good time thinking about the sun is really, really important. And it will avoid the embarrassing situation-- how many of you have been at a shopping center, walking along, and a little child is asking his parent, Dad, why is the sky blue? Or why is the such and such? And the answers you'll hear just make you want to tear your ears out, say, my goodness. And so part of this is just general knowledge. It's getting a feel for the world and the universe and asking those questions again that the little children will ask but we forget to ask as we move on with our lives, right? OK. So moving forward, the learning objectives for today are these right here. By the end of the lecture-- and hopefully, you already have a good sense of this based on your readings already. I'll quiz you on that second-- verbally. We want to be able to quantify the available solar resource relative to human energy needs and other fuel sources. We want to recognize and plot air mass zero and air mass 1.5 solar spectra and describe the physical origins. We want to describe how solar insulation maps-- these are solar resource maps, in other words, how much sunlight is available. And we want to be able to estimate a solar resource amount locally at a specific spot on the planet. We want to list the causes of variation and intermittency of the solar resource and quantify their time constant in magnitude. In other words, we want to be able to discern what are the big effects and what are the ones that don't really matter. We want to be able to estimate the land area needed to provide sufficient solar resource for a project, whether it's a house, a car, a village, a country, a world. And a lot of this will be on your homework assignment, so we'll give you the tools here, but then ask you to address those questions. And for those of you who have already picked up your p-set number one, you'll see relevant questions. Where is this? Does anybody recognize this right here? If I start rambling off names, what city has Pennsylvania Avenue, Independence Ave? AUDIENCE: Washington, DC. PROFESSOR: Washington, DC. This is right outside of the National Air and Space Museum. What this little girl here is pointing to is the sun. And then we have Mercury, Venus, Earth, Mars, and so forth. So it's essentially a solar system to scale. As you walk out of the Air and Space Museum and walk down the street, you'll be passing the different bodies in our solar system. And so just as a quick little review to kind of get us situated and to ask the questions that little kiddies might ask us, how far is it from the earth to the sun? AUDIENCE: 93 million miles. PROFESSOR: About 100 million miles. Yeah. 93 million miles. Plus or minus somewhere in the range of maybe a percent or so, a few percent depending on what time of year we are since we're in a little bit of an elliptical orbit. Good. So that's the distance to the sun. It would be about 150 million kilometers. How long does it take light to travel that distance? AUDIENCE: Eight minutes. PROFESSOR: About eight minutes, eight and a third minutes, right? So it takes a little bit for the light to reach us. Good. Some more questions-- how far are the other planets in our solar system to the sun, going in order from Mercury out? It's to get us situated here. If we are, at any point, planning on throwing up satellites and sending them with other planets, this is a good thing to kind of keep in the back of our minds. So if we define an astronomical unit-- not in terms of our national debt, but in terms of the distance from the earth to the sun-- that's an astronomical unit. Mercury would be somewhere around 0.4. Venus 0.7. Mars 1.5. So that's all kind of in our neighborhood, right? And then from Mars to Jupiter is a bit of a jump. It goes from 1.5 to 5. Then from Jupiter to Saturn is 10-- well, sorry. 10 is the distance from Saturn to the Sun. And then 30 and then 40. Sorry, 10, 20, 30, 40. So it goes-- Jupiter's 5, Saturn 10. Uranus would be 20. Neptune 30. Pluto 40. Pluto, planet, sort of. So it's easy to remember those numbers because it goes 5, 10, 20-- that's just essentially a sequence of doubling-- and then 10, 20, 30, 40. I'm giving you approximate numbers here, but that's something just to keep in mind all in terms of astronomical units. So in case a little kid comes up and asks you, you can spit out the answer. Let's talk about the sun. This is just a review of our readings right here. This was a representation of the sun and the Earth moving around. And what is solstice and equinox? What do those refer to? What is the equinox? AUDIENCE: Equal amounts of light and dark. PROFESSOR: Yeah, equal amounts of light and dark throughout pretty much all the world except if you're really standing on the tippy top and the bottom. So equal amount of light and dark throughout the world on that particular day. So the day and the night have the same amount of length. If you move over to this region right here, this would be a region of our northern hemisphere summer, southern hemisphere, winter. Over here, vice versa. And the solstice would be? AUDIENCE: The shortest day of the year. PROFESSOR: Yeah. So the shortest or the longest day of the year, depending on what side you're on. So in the northern hemisphere, the June solstice would be the summer solstice. For us, it would be the longest day. And if you're in the southern hemisphere, it would be the shortest day. Depending on what time zone you're in, there might be a variation of one day hither to. Good. OK. And what is this? What are the seasons caused by? What is this kind of tilt right here? What is that called? AUDIENCE: Declination angle. PROFESSOR: Declination angle. And approximately how much is that? AUDIENCE: 23 and 1/2. PROFESSOR: 23 and 1/2. 23.45. Yeah, so 23 1/2 degrees. Good. OK. And we can visualize all of this on the PV CD-ROM on the website. So this was part of your assigned reading. And here's the earth going around the sun in this representation. Likewise, if you want to take one of your balls and just kind of imagine being on one of those surfaces, you see the diurnal rotation here. It's spinning around its axis. And as well, the seasonal variation as it spins around the sun. That's important for a number of reasons, right? That will determine how much sunlight is normally incident on the planet. If you are normally incident, if you're at this exact spot right here, you're receiving the sunlight full on. But if you're up here somewhere, your surface normal is some vector pointing out like that. You're only going to be receiving the cosine theta of that amount of sun. So if you're an extreme example, if you're right here, you're not going to get any. But if you're in this part right here, you're going to get cosine of 0, which would be 1. So you get the full amount of sun. And likewise, as you move through the angles here. So it's important to understand what the relative angle is between our surface normal and the vector pointing at the sun. That varies as a function of season. It varies as a function of time of day. And obviously, over the entire earth, you can define the precise amount of sunlight coming in, the precise solar resource, by a series of trig formula. It gets pretty complex. And this website will actually walk you through it if you're so interested. Now this is all review since folks have all done the background ready, right? All done the background reading. I expect you to before class. Let me ask you a few trickier questions just to see if our creative juices are really moving at this early time of day. When would be the shortest day of the year? Let's start there. The shortest day of the year is approximately December 22, right? In the solstice. When is the latest sunrise and when is the earliest sunset? You might need to pick up your little ball and rotate it around. Does anybody have even just a gut sense? Would it be on the solstice? How many people think it's going to be exactly on the solstice? Let's see. Earliest sunset-- how many people think that the earliest sunset's going to be a little bit before the solstice? How many people think a little bit after the solstice? I know people don't really know. The reality is that the earliest sunset would be a little bit before the solstice here, and the latest sunrise would be a little bit after the solstice. And it flips in summertime. It would be, let's see, the earliest sunrise and the latest sunset before and after the solstice respectively. OK. And you can think about that in terms of what is the solar noon. The solar noon is when the sun this is directly overhead relative to our chronological noon, which is, I would say, less dependent on the specific angle of the earth relative to the sun as it moves around this trajectory. Let's get more into that in recitation. I sense since there wasn't much traction there I don't want to dwell. OK. Good, good, good. OK. Let's think a little bit more in terms of the trajectory of the sun later on as we move through some of the introductory material. I don't want to dwell too much. I want to give a little bit of review since I'm sensing that not everybody did the readings. I expect you to do the readings, folks. So let's keep with me here. So a touch of history, since that was asked for. It was requested. I decided to launch a little bit into a history of the study of the sun. Philosophers, going back to what I suppose most would consider early India, studied the sun. There were some writings of some of the earlier philosophers that were recorded. Some have interpreted these writings as being indicative of, perhaps, heliocentric models. These are poetry, folks. It's a very different style of communication than what we have today of technical writing, so it's difficult for us to discern, or difficult for me, at least, to discern when I read the lines verbatim if this is really somebody thinking about the heliocentric model or whether this is somebody just describing the universe in the best of their abilities. I would say the real beginnings of heliocentric models began in the third century Before the Common Era and the notions of interstellar distances estimated in a similar manner to what we just walked through today, right? Estimating the distance between the earth and the sun and then using that as a measure or yardstick with which to measure the distances to the other planets began somewhere during that time. Likewise, these writings of old made their way to the Middle East. And in the 10th and 11th centuries of the Common Era, the Middle East, the Arab world is really where science and technology was at. And these days, I mean, Europe was still largely mired in the Middle Ages, starting to emerge in a few places. But by and large, the carriers of civilization in the Western world were really centered and in some of the Arab cities in the Middle East. And al-Biruni, in particular, was very avid at applying methods of astronomical observation, in particular to aid travel, but in the process, discovering a thing or two about our known universe. And of course, finally, Johannes Kepler in Europe, once it starts to emerge from the Middle Ages, with really taking observation some other scientists, who very carefully plotted out the position of the different bodies, he came up with some of the mathematical models that describe the motion of the planets through the skies and is largely credited with developing a series of laws that define interplanetary motion. So a couple of interesting things to note is that international collaboration was really essential. These ideas didn't develop in isolation. They were flowing throughout the world. It's important to know that many of the scientists were well-traveled polyglots, meaning they spoke different languages. And that's how they were able to interpret the texts and readings of other people. And it's also to note that parallel astronomical developments were happening in other regions of the world-- the Far East, Mesoamerica, and so forth, right? And so obviously, there may have been some communication-- I would say rather sparse-- between especially with the Far East. But there was a fair amount of communication between these regions here. And you can imagine writings or ideas traveling from word of mouth along trade routes. So there was some communication. So back to our learning objectives. Today, we're about to quantify the available solar resource relative to human energies and other fuel sources. So let's do that. We'll jump right into one of the slides that I showed you last time. This is in terms of terawatts av. Terawatts is a unit of power, a very big one. "Tera" is 10 to the 12. And you can see here the resource of the sun relative to the wind energy resource base relative to human energy needs. If we just consider the resource falling on the earth's surface, as opposed to that falling on the outer atmosphere, there's a little bit of a discount. But we're still very large compared to human energy use. And if we redefine our units from terawatts into HECs, which are Human Energy Consumptions, defined in, say, 2050, where one HEC is the average, let's say, energy burn rate of 2050, you can see here that these numbers are a few orders of magnitude larger than what our human needs are. So if you were able to capture only 1% of all of the solar resource falling on the earth's crust, we would be actually in pretty good shape. And 1% of the total land area on the earth's crust-- I believe somewhere between 1% and 2% of the United States is covered in asphalt right now for roads. So this could be on houses, on rooftops, on buildings, and so forth. We wouldn't necessarily have to exclusively repave virgin farmland with solar panels. So going back to quantifying the solar power, we have our sun. We're going to start by quantifying the solar resource by assuming that the sun is a black body. The same way that hot objects emit light-- say, for example, when you turn up your stove and you have a very warm glow coming out of it-- the sun is as well a hot body, a black body, sorry. Very hot as well, somewhere around 6,000 Kelvin. And the total radiated power is given by Stefan Boltzmann's law here in the following way where we have temperature to the fourth and the temperature is somewhere around 6,000 Kelvin. And so we have this power being radiated out at the surface of the sun. And then, as it travels outward, it becomes, you could say, diluted in effect. Because the total surface area of that sphere is increasing, obviously as r squared. And so by the time that that light reaches the earth, it's only a very small cross section, or very small solid angle to be more precise, of the sun's surface that is radiating directly at the earth right here because this, in the absence of scattering centers in the universe that might reflect or bounce the light back toward the earth. And you can calculate the total power incident on the earth by that simple formula right there. What is the radius of the earth? Again, one of these simple numbers you should kind of having your head. 6,370 kilometers, somewhere in that order, right? And so you can begin estimating here the total power and the order of magnitude. It's going to be tiny compared to the total power that the sun is radiating thankfully, or else we'd be pretty hot right now. So the average power coming from the sun on the surface of the outer atmosphere is around 1,366 watts per square meter. Who's heard of 1,366 before? AUDIENCE: Yeah. PROFESSOR: Yeah? You've heard about it? All right. You know where the number comes from now. It's pretty wonky. 1366 is a startup company, a spin-off of MIT focused on solar energy. OK. And so that's at the equinox, which we just learned is occurring somewhere around March 21, September 21-- coming up soon. Celebration. The ratio of the surface areas of the spheres is really something to keep in mind right there. OK. So we've quantified the available resource relative to human energy needs. Now we have to come up with some language that we use to describe the sunlight moving through the atmosphere of the earth and reaching the surface of the earth, right? So we're going to use what's called the air mass convention or AM convention. AM stands for Air Mass. Even without knowing much about how the light is absorbed or how to quantify it mathematically, we can assume that our atmosphere contains molecules. It contains particulate matter. And that's going to interact with the light in some ways. Probably either going to absorb or scatter it. And so as the light passes through our atmosphere, there's going to be some absorption. And the greater the distance, the greater the optical path length through the atmosphere, the more absorption and more scattering there will be. And so we use air mass, or AM convention, to define the path length or the path distance through the atmosphere. AM0 would mean the outer atmosphere. AM1 would be essentially just going straight through, normal incidence so that the direction of the trajectory of the light is parallel to the surface normal of the earth at that location. And then air mass 1.5 and so forth is as we increase the angle of the entry of light relative to the surface normal. So in other words, as we go further from the equator to northern latitudes that air mass number is going to go up, up, up. OK. We'll explain it with a few graphs and figures and a few slides. So we have our atmospheric absorption. When we just glance at the earth in these beautiful pictures taken from outer space, we can see very obviously the clouds present. And more importantly, if we were to zoom in on one of these regions right here, we would see this bluish hue coming from our atmosphere, which is scattering preferentially the shorter wavelengths of light. And more importantly, this radius right here is somewhere on the order of 6,370 kilometers. And this thin atmospheric shell in on the order of 30. Right? So that's why you don't really see too much of a ring around the planet from this distance. So the atmospheric effects, let's try to bend them into discrete buckets. This is a simplification, but it helps us gain a foothold in understanding. And then from there, we can make our understanding a bit more complex. So we have incoming solar radiation coming from here. We have a number now that's 342 watts per square meter. Why is that number so much lower than the 1,366 that we were just talking about? AUDIENCE: Particles in the air and pollution in the atmosphere. PROFESSOR: Yeah. So this is meant to be an average over the entire day at one fixed point along the ground, right? So as we're rotating around, we have at least half of the day normally when we don't have sunlight. And then there's, I would say, the cosine theta angle is not 1 at all times. It's very rarely 1. And so this is a bit of a discounted incoming solar radiation, essentially a time-averaged solar radiation for a given patch of the planet over a typical day. And so we have a variety of processes here. We have reflection off of clouds. That's pretty clear to see from here. It looks nice and white. We have some absorbed by the atmosphere, typically in a rotational or vibrational modes of molecules up in the outer atmosphere, sometimes by particulate matter as well. And then we have the amount that's absorbed here by the surface, of course, reflected as well. And so the amount that's incident on the surface or coming down to us is what we can actually use to make solar energy. And this is an average, right? Because sometimes the cloud cover is a lot greater. And for that particular day, we will have a lot less resource striking the ground at a given time. This over here, this is all mostly infrared, right? Where this is visible coming in, once the light gets absorbed and then gets re-emitted, it usually gets re-emitted in the longer wavelength light or the infrared light. And this is the stuff that gets blocked by or absorbed by greenhouse gases and then re-emitted equiangularly. And some of it makes its way back to the earth, right? OK. So air mass, let's define that so that we have a common language that we can use to describe the solar resource from place to place. So again, this is the sun. This is the surface right here. And let's imagine that the angle between the incident sunlight and the surface normal is 0 such that the cosine theta term is 1. Air mass here at this point, at this point on the surface of the earth, is going to be AM 1. So we call it air mass 1 if the sun is literally directly overhead. What did we learn about the declination angle of the earth? It's about 23 and 1/2 degrees, right? How far north are we? What is our latitude here in Boston? AUDIENCE: 41. PROFESSOR: 41, 42ish, right? So let's for simplicity say that here in Boston, our latitude of Boston Logan Airport is approximately 41 degrees north. And let's say that the declination of our planet is approximately 23 and 1/2 degrees. So what would be the angle of the sun in the sky if you were to lie on your back in the middle of the summer solstice and the middle of the winter solstice and you're lying straight on your back looking up at solar noon, what would the angle of the sun in the sky be relative to the surface normal? How far south would the sun be? Imagine that this is 0 degrees and that's 90, right? So relative to this angle right here, where would the sun be in the sky? Why don't you turn to your neighbor right now and discuss? On the summer solstice, the winter solstice, come up with some set of angles there. All right, folks. What do we come up with? This is our little human being, you, in Boston. This is south, and that's north. And we're at the solar noon in the winter solstice in the summer solstice. So here's you. This is directly above. So I would say if you're lying on your back and looking straight up, that's the surface normal of the earth. In the summer, at solar noon, the sun would be at what angle relative to the surface normal of the earth? AUDIENCE: 17 and 1/2. PROFESSOR: 17 and 1/2, somewhere around there. And how did you get that number? Subtract those two, right? So you get 41 minus 23 and 1/2. You're somewhere around 17, 18, somewhere around there. So we'll call it 18 degrees in summer, again, relative to the surface normal. And in wintertime, what does that work out to be? AUDIENCE: 64 and 1/2. PROFESSOR: 64 and 1/2. Similar logic, right? So we'll call it 65 degrees in winter. Good. So what does that work out to be in terms of air mass for winter and summer? Quick engineering approximation, I'm going to say that, in summer, it's approximate air mass 1, but you can calculate it real quick. Somebody with a calculator want to plug those in? AUDIENCE: And roughly 2 in winter. PROFESSOR: And roughly 2 in winter. All right? OK. We're going to get into scattering of light in next lecture actually. And we'll see why that matters in terms of especially of getting sunburned since the shorter wavelengths, the ultraviolet, are more sensitive to the path length. OK. Good. Very good. So we have pretty much a gut sense now of where the sun is in the sky. And as it moves from summer to winter, from our perspective, it follows a little bit of a, I would say, sinusoidal path, right? It stays in summer for a long period of time up here. And then it moves pretty quickly down here and stays here. So versus time, the angle of the sun in the sky is following a sine curve, or cosine curve if you will. Right? And so right now, the sun is actually close to the middle. We're in September 13. The solstice is coming up in a week's time. And so the slope of that sine curve is at a maximum right about now. And so that the amount of time that the day will change in length is changing at its greatest point right now in the year. And so you really begin to notice it if you start paying attention or if you go to weather.com and start looking up how long is today's day going to last, when is the sunset tomorrow. I don't know about you folks, but I like to cycle. And when I'm doing my evening rides, I'm noticing it now that I have to start earlier and earlier if I want to put in, say, 30 or 40 miles. I'm not going to be able to make it home in time. So that's the sun and how it relates to you in your daily lives. We're going to get back to this in a minute, so keep this in mind. Don't let it out of your RAM. Let's talk about the actual solar spectrum for a minute. This is the sunlight intensity in some very real units. We'll get to that in a minute. But think of this in terms sort of like the total amount of power in a given bandwidth. So the wavelength right here-- or the power density in the bandwidth. The wavelength is the wavelength of light. Shown for your convenience here is the visible spectrum. That's what our eye-- mostly what our eye-- is able to detect in this wavelength range right here. And the sun is emitting over a much broader range of wavelengths. It's emitting following a black body emission source at 6,000 Kelvin. And that's in this very difficult to see green line right there. Air mass 0 spectrum looks like this, this red line right here. And again, let me remind you that the air mass 0 is the light that's falling on the outer atmosphere. There is no earth atmospheric absorption yet. Why do we have these little lines here? Do you see it's not a perfect black body. We have some-- I'll give you a hint-- absorption lines. Where is that light being absorbed? Is it the ether between the earth and the sun? No. There's no ether between the earth and sun. Where is that light being absorbed? AUDIENCE: Hydrogen ions and stuff? PROFESSOR: In the sun itself, right? So these are absorption events occurring in the solar atmosphere. And now, if we do air mass 1 or 1.5-- let's push it up a little bit from air mass 0-- this is now passing through an angle of somewhere around 60 degrees. Now what do we have? We have several absorption lines occurring, right? And these correspond to absorption events where? AUDIENCE: In the earth's atmosphere. PROFESSOR: In the earth's atmosphere. Exactly. And so we can attribute each of these little absorption lines here to a particular-- usually it's a molecule in the earth's atmosphere. Note the sensitivity of the human in black right here and how well-matched it is to the air mass 1, air mass 1.5 spectrum. That's pretty cool. That's pretty cool. OK. So there you have the spectrum. Let's get to these units of power density per bandwidth for a second. The way to think about those units is as follows-- kilowatts per meter squared. OK, I get that. It's the amount of power falling on a unit area. Per micron, the reason it's normalized per micron is because the wavelength units right here is in microns. And if you take the product of the two, it makes it pretty easy to calculate the total power density, right? So if you want to calculate the power density, the total watts per square meter, falling on the earth, say, between 0.5 and 1 micron, you can calculate the total amount of power by multiplying one versus the other. So that's why they're in this weird unit right here. It's to help you perform calculations like the ones you'll do in your homework, like the ones you'll do for your class project and so forth. And it strikes a little bit odd the first time you look at it, but it begins making sense. And you're appreciative of it after a while. There are standard spectra. They're standard reference spectra. Sure, you can go outside and using some form of spectrophotometer. You can measure the incident solar radiation and map out the spectral irradiance as a function of wavelength. In terms of the planning or communicating with other scientists, we typically refer to standards. We use common yardsticks, common metrics. And that facilitates communication, avoids ambiguity, avoids misunderstanding. And so these standards right here, these ASTMs, refer to the particular standards that are used for those solar spectra. And in your supporting online material, at the very end of the lecture slides online, we go into some more detail regarding that for those who are interested. So again, these little absorption lines here correspond to specific atmospheric events, interactions of that particular wavelength of light with some molecule usually in the atmosphere. We can, as well, have generalized attenuation due to other scattering mechanisms. Notice right here in the short wavelengths what's happening. From the red to the blue, this light is particularly effective. We have a sharp drop in the shorter wavelengths. And it grows sharper the shorter in wavelength you go. So the attenuation due to passing through the atmosphere grows or increases the shorter in wavelength you go. And this is a process generally called Rayleigh scattering. And there's a wavelength to the fourth dependence. So as you go shorter and shorter in wavelength, the likelihood or probability of scattering will increase. Why is that pertinent to us? Well, now you can answer that little child who walks up to you in the shopping mall and says, why is the sky blue? You say, well, there is this elastic scattering mechanism of electromagnetic radiation whereby in a broad spectral event, such as the sun, black body emission, we have the shorter wavelengths that are scattered more. And that's why when we look away from the sun, in the other direction, we see those shorter wavelengths that are scattered back to us. That is pertinent for two reasons. A, it makes the sky look blue. Secondly, it's not only the short wavelengths in blue light that we're worried about, but also the ultraviolet radiation. Right? So even on a cloudy day, if there are a few open patches and you can get scattered light coming in, you can still get sunburned. And secondly, there is a very strong dependence on the path length, the optical path length, the air mass, right? So if you go further north in latitudes, where your air mass increases, right-- because now, if you think about the atmosphere as being kind of a flat. Just giving you an approximation for minute. If you think of the earth being flat and the atmosphere of being flat, now the path length in winter is much, much greater than the path length in summer. Same sun, just different path length. You're much more likely to get sunburned in the summer than you are in winter because the path length is a lot shorter and the amount of short wavelength ultraviolet radiation that will be scattered away is going to be less in the summer than in winter. That's also why if you go south latitude, for example, from here to, say, Miami, your incidence of getting sunburned is a lot greater, a lot more than the total increase of the visible portion of the spectrum. So the sun might not look that different to you, but your incidence of sunburn events goes up quite a bit. And that is due to Rayleigh scattering. And now, by this point, the little child has already run crying back to the parent. But you have a full satisfaction of knowing how the universe around you is put together. And that actually was a pretty deep problem. It took a long time for at least European scientists to crack that nut and figure out what was going on. Describe how solar insulation maps are made and use them to estimate the local solar resource. So we have metrology. We have techniques that we can use to measure the amount of sunlight that is out there. And now we want to apply those in some systematic fashion to measure the average solar resource around the planet, including the oceans, and then use that information to estimate later on the land area needed, or the size of the array, or how many cells we're going to have to string together based on the solar resource locally. So how are these insulation maps made, these maps that we'll use as engineers to size our systems? First off, let me define insolation. This is not insulation, as in stuff you put around the house to keep the heat from going out. This is insolation with an "o," a shorthand for incoming solar radiation. Insolation at the top there. It's typically given in units of energy per unit area per unit time, so kilowatt hours-- that's energy-- per meter squared per day. And it's helpful when designing or projecting these PV systems. And it's affected by a bunch of stuff, which we'll get to over the next few slides. So we can measure insolation from the ground. That's a surefire way to do it. This right here is a pyranometer. Pyro, fire, sun. Ano, on top of. So anode/cathode. Cata, under. Ano, above. Catatonic, under. Right? OK. So pyranometer. So it's basically measuring the sun above, right? Measuring the sunlight above. This is a full hemisphere measuring the sunlight coming in from all angles. There's a small little sensor right here. It's lying flat. And that glass is essentially allowing the light from different angles to get into the sensor. And this is a very narrow, solid angle of the sky. It's probably just looking at the sun, or in a particular direction rather, plus or minus 2 and 1/2 degrees in either direction. So it's a very limited solid angle of the sky. This one over here would be more appropriate, say, for a flat panel that's receiving scattered light coming in at all angles. This one over here would be more appropriate for a tracking system, especially a concentrator system that has optics that only accept light in from a very limited solid angle. So imagine you have a lens that has to have like incident to it to focus it on the right spot. And if the sun moves in the wrong spot or the-- put it another way-- if the lens is in the wrong position relative to the incident solar radiation, the light is being focused off of the solar cell. And it doesn't produce any power. So this is a system that's used for measuring the direct solar spectrum, which will be useful for calculating the total output from concentrator systems. And this pyranometer over here is useful for flat panel systems. And we can also measure the total amount of incident solar radiation, total amount of insolation, from the sky using satellite imagery. This is an example of a measurement. And this right here is insolation, average insolation, from 0 to 550 watts per square meter taken from a NASA satellite with the NASA Earth Observatory. Very cool website, great place to spend a Friday night if you don't have plans. Just log on here, and bunches of maps from snow cover, to population density, to CO2 being emitted, to wildfires around the planet-- anything that a satellite can measure, they're measuring. And the insolation value is one of them. And so we have data from various points. This is the insolation in January. So in January, it's the Southern Hemisphere's summer, the Northern Hemisphere's winter. And as a result, we have less insolation up north. We're in the blues. And the Southern Hemisphere is more in the reds. And of course, the tide turns in July. We have our summer and the Southern Hemisphere has their winter. And the poor folks here in Antarctica have nothing. So a couple of things to note just already straight off the bat, we're noticing that there's, in general, higher insolation near the equator, the equator passing right through here approximately. Fun fact-- small city up there in the north of Brazil, there's a soccer field that is half in the north and half in the south. That's neither here nor there. We have rainforests up here in the north of Brazil, Central Africa, and here in Southeast Asia. And even when the sun is directly overhead, those clouds are preventing some of the sunlight from getting in. And that's why right at the equator itself, we typically have less insolation than we do in the tropics, say, Tropic of Capricorn, Tropic of Cancer. Tropic of Capricorn running straight through Sao Paulo, Brazil, Tropic of Cancer running through Key West. Just to situate yourselves. And the Tropics are how far away from the equator? Right. OK. 23 and 1/2. Good guess. Good. So what we're going to do is now launch into our next learning objective, which is to list the causes of variation and intermittency of the solar resource and quantify the time constants in magnitudes. This is really, really, really, really important. The other stuff is very useful from an engineering point of view from answering certain questions in your homework. This right here is the singular reason-- one of the singular reasons-- why solar doesn't behave like a regular fossil fuel source, why solar does not produce power all the time. It is variable in terms of its power output. Variability generally refers to the fact that we can predict it's coming. It's going to vary, but at least we can predict it's coming. Intermittency, while not a strict definition, the understanding when somebody says "intermittency" or "intermittent power source," the impression that it gives is that it's unpredictable in terms of its variability and its variation. So we've talked a little bit about the variation so far and about the predictable nature of the sun. We've talked about how the sunlight, the solar resource, varies from summer to winter. We've talked about how the solar resource varies as a function of latitude, right? But now, we're going to talk not only in a little bit more depth about that and have a few fun in-class exercises to get us really grasping that concept in its entirety, but also talk about some sources of intermittency, which if you have a large amount of solar contributing to the grid and it is intermittent, and you have no way of dealing with that, you're going to have fluctuations of energy level on the grid or power levels here as a function of time. And that's not going to be good. So in terms of the seasonal variations, in terms of predicting the amount coming from the sun at a given point-- I told you it looks a little bit like a sine, a cosine wave. And indeed, it does. You can calculate those values based on this website right here. Just to show you how nifty and cool it is, our friends at Arizona State University, Stuart Bowden and Christiana Honsberg, really put in a lot of time to make this. You can vary the time or the day of year right here for instance. Right? And you can see how the solar resource-- this is the direct radiation, kilowatts per meter squared. And this is the time. So if you take the integral of the curve right here, you're going to get what? Units of-- AUDIENCE: Power. PROFESSOR: Power. Power times time is? AUDIENCE: Energy. PROFESSOR: Energy. Energy per unit area, right? So you're going to be able to calculate the total amount of energy falling on a given area per day, let's say, right? So if we look at the size of this little curve in winter, the total area under this is going to be very small. And that's because the solar resource is very small. And the sun rises late and sets early in winter. And as we move towards summer, obviously, the total amount of the solar resource increases. Not only it increases because of this that we have at solar noon. We have less of a path through the atmosphere. We have more sunlight reaching the earth. We have a total increase of the amount reaching the earth. We also have that cosine theta term here dictating the cross-section incident to that sunlight coming in increasing. And so that's driving this going up. And we also have a second fact that the time of the day, the total duration of the day, increases, at least in northern latitudes here at around 40, let's say, 41 degrees north, here in Boston. And we have because these two effects a much larger area underneath that curve. And so as we go through summer and now finally to September 13 and back to winter, our solar resource goes back down again. So you can calculate it. You can visualize it. That's cool. And we can plot the total amount of energy per unit area per day, essentially the integral under that curve, as a function of location around the US, around the world per month let's say, right? So this is January. This is kilowatt hours per meter squared per day. So it's just taking the integral of the curves measured. So it's accounting for cloudy days, which kind of has a depressive effect. This is an envelope function, if you will, the maximum you could get. And then, of course, local weather patterns will suppress that, drive it down. So this is the real map of the United States. And you can see in sunnier areas that are less cloudy, over here, for example, in Arizona and New Mexico, there's a large solar resource even in January. Atlanta, which has half of the number of sunny days per year as Phoenix does, even as it's at the same latitude, is getting about half the solar resource. They got a short end of the stick. Again, this curve right here is the envelope function, right? And off of that, you can only go down. You can only decrease the amount of solar resource actually arriving at our feet here. And so this is in January. And this is all in the same color scale here as we move through the months. So we'll move from January, to February, to March, April, May, June, July, August, September, October, November, December. So you can see across the United States how the resource is distributed geographically. The general trend that as you go from south to north you have a decreasing solar resource holds. You can also see the influence of local weather patterns as well for the same latitude. So that's pretty nifty. And another nifty fact, if you look at the year average value, annual-- this is the annual average value-- here in Boston, we're averaging around 4.5 kilowatt hours per meter squared per day. Phoenix, Arizona can be upwards of 6 somewhere in the outskirts. It's not that bad. It's only a few tenths of percent. It's not that bad, I tell myself. I don't believe it myself either, but I try to convince myself of that during winter. All right. Let me show you the seasonal and diurnal variations. We're increasing the level of sophistication as we go along, right? We've assumed you've done your readings. We've started with some simple examples, and now we're really taking it one step further, which is to introduce the full 3D model. And I'm going to do that by use of this really cool app that's available here. Right here. This is you standing on the earth. And you can drag and pull this around. You can see there's north, south, east, and west. So I'm going to pull it up a little bit just to give us a little bit of perspective. Still, south is facing toward us. North is away. The sun will rise in the east and set in the west. Now, let's say I pull the date back to September. So this little tool is so cool because it recognizes your IP address and situates you at the proper latitude, so we don't have to touch that at all. It's approximately right. We're at 40, yeah, about 41 degrees. We're right here in September. And in terms of time of day, we can pretty much just cycle through the time of day if we like. We could, for example, start animation. Let's see, this is going very fast right now. I'm going to slow it down so you can see the time of day moving right over here. And you can see that relative to our vantage point on the surface of the earth, this little yellow dot here and this yellow line is tracing through the path of the sun in the sky from our perspective. And so as we go through the seasons, I'm going to speed it up just a little bit so that we pay more attention to the position of this yellow line and less attention to the diurnal variations. We're paying more attention to the seasonal variations. I'm going to vary the seasons by force here. I'm going to go back to July or June if I may. Here we go. AUDIENCE: What's the blue line? So you can see it? PROFESSOR: Yeah. So there's a number of other lines right here, and they're all explained very carefully. There is the hour of ascension, which would be prime hour circle. Yes. I'd have to go back and double-check all of this, but I believe they relate to would be the sunrise and sunset of that given day. Let's see if our hypothesis is correct. No, it is not. That would have to be, since it is varying in a systematic way from January through the summer and then back to the winter, I'm imagining this has something to do with the direction of the sun relative to the earth, right, as it traces that ellipse through the sky. So let's pay attention to that yellow line for a minute. That's the one I want to attract everybody's attention to. Now we're in June, so in the height of summer. And relative to this observer right here, the sun is further up in the sky just like we traced out right there. And now, as we go to winter, that line drops close to the horizon. So a couple of things happen. If we look like this for instance, now we're looking straight down on the observer. In wintertime, the sun will rise in the southeast, and it will trace this arc through the sky and set in the southwest over here. In the summertime, the sun will rise almost in the northeast, slightly north of east, just slightly north of east. And that's why if you have a north-facing window and you put your little plant on the window sill, it'll get a little bit of direct sunlight early in the morning and late at night. Because when the sun is tracing this part or that part through, it's tracing the sky. So it's worth sitting down with one of these plots, toying around with it, getting accustomed to it, and understanding really how the sun traces its arc throughout the sky relative to our position right here on the earth. If we shift this further up north, really interesting things begin to happen. So for example, my wife is in Sweden. If we go to her hometown right here in the middle summertime at the solstice, you can see the sun traces this awesome route from north to north barely leaving the horizon. If we look at, again, from the perspective of the little creature here, that yellow arc is really close to the horizon. Maybe it goes up about that high, but it continues going all the way to the north. And if you keep going north, it will never set during the middle of summer. It'll just be light all the time. And you can see here it just traces that orbit right around there. It's all trig, folks. We can do it. We can sit down, and we can work through the equations by hand. I did that once. It took me a long time. I didn't learn that much. I would instead advise you to go to one of these simulations right here, but to understand all the inputs into it, all the different components, the fact that the earth is moving around the sun as a declination angle, seasonal variations, and so forth. Very useful tool. You have the website link right here. And yeah. So from this tool-- actually, one last tiny, tiny thing. From this tool right here, we can understand why-- OK, this is a real stretch. And forgive me, social scientists in the room, for doing this, but I have to project a little bit of science onto human behavior. How far west is Madrid from GMT? Madrid, Spain? It's 3 degrees west of the Great Meridian. So the line that divides the East and the West Hemispheres is 3 degrees west. But it is one time zone earlier than London, so it's in the same time zone as Germany and all the other cities that are east of London. And this is just for convenience factor. If you're traveling from one continental European country to the other, it just makes sense to have everything be on the same time zone. You get to work at the same time sort of. Pick up the phone, call somebody, you're doing business. Now relative to everybody else in Europe, though, is the sun setting later or earlier if you're that far west in your time zone? AUDIENCE: Later. PROFESSOR: Later, right? So the sun is setting later if you're there. So if you're eating according to the sun, not according to what your watch is saying, but if you're choosing to eat dinner when the sun is setting, when will your watch say, oh my goodness, it's really late when you're in Berlin or when you're in Madrid? AUDIENCE: Madrid. PROFESSOR: When you're in Madrid, right? So again, I'm not saying that this is the sole reason for social behavior being a little different on the Iberian Peninsula, since Portugal also eats very late and they're in the same time zone as London, but it could be a contributing factor. The sun is still up in the sky when it's 5:00 PM in winter, let's say, where in Germany, it's set a long time ago. So these are just little things to keep in mind. An easy way to calculate, when is the solar noon, you look at the earth more or less like we're looking at this right now. We have 360 degrees. We divide that into 24 time zones. And then we say, OK, about 15 degrees each. And then we can begin counting from there. If in Boston, were 41 degrees north, but we're 71 degrees west, we can say, OK we should be for GMT minus 5. We should be at around 75 degrees. And so we're a little earlier, so we do things a little earlier around here than what the solar noon should be telling us to do things-- wake up a little earlier, go to bed a little earlier. And that's why students are like, dang, there's no night life around here. I'm not saying that's the only reason, but it could be a contributing factor. Whereas the opposite happens when you're far west in your time zone. OK. So again, just trying to wrap our heads around the solar resource and around the world around us so that we can answer that little child in the shopping mall when they come with questions. What are these? AUDIENCE: [INAUDIBLE] PROFESSOR: Solar trash compactors, right? AUDIENCE: At the Student Center. PROFESSOR: At the Student Center. Anna's Taqueria is right over there. Dunkin' Donuts is there, right? So these are solar panels mounted on the tops of those. And what the solar is doing is charging a battery inside. Once the trash reaches a critical lever, a sensor is triggered. It stops you from opening this bin, and it compacts the trash and then releases and allows you to open and put more stuff in. And what it does is it minimizes the number of times between trash pickups. If labor is a large portion of the cost of trash management, of refuse management, then it eliminates some of the labor, transferring it instead to the technology. And so installing these at the Student Center, I had a little bit of a pet peeve. [LAUGHTER] So you have angle of the sun right here, which we just walked through, 18 degrees in summer. And it just so happened to work out that-- and these numbers were approximate. This was me going with my kind of engineering sense. That's about 45 degrees. Count the number of paces. Equilateral triangle. Estimate the height of there. But in the middle of summertime, when you are at the solstice, there is no so direct sunlight hitting this trash can because of that overhang way up there. And it's not a problem for this particular trash collector, since those panels are way oversized for the amount of energy that the trash collector actually needs. And there is a fair amount of what we call diffuse sunlight, meaning sunlight being scattered off of other things. That's why this portion of the image looks white. It wouldn't look white if there was no diffuse scatter. It'd look black, pitch black, if there was nothing to scatter the light off. Like an outer space, there's nothing to scatter this way. It looks like a black night. But instead, there's a large amount of diffused light. There is some sunlight reaching it. And since the panels are way oversized and the system is over-engineered, it still manages to acquire enough energy to compact the trash. And it doesn't have a catastrophic stop. But it was an example of somebody not really thinking much about the direction or the angle of the sun in the sky. They probably installed it sometime around March when the snow started to melt and the sun was right around here. And the angle was around there, and it made it into the trash collectors. But as the summer came along, it got shaded. So it's something to keep in mind when doing a solar installation. It is important to calculate these things. And I just pick out once more example. I'd encourage you to walk through. I was going to have that be a small little in-class example, but since we're running short on time, I'll just give it to you like that. OK. Fixed versus tracking systems. So if the sun is moving as a function of season and as a function of time of day throughout the sky-- and we can see that very nicely, again, through our demo right here-- so if the sun is actually moving, one embodiment would say, OK, I know more or less what the sun is going to do as a function of season. Forget the diurnal variations, but just the seasonal variations. I know that the sun is going to be on average somewhere around here, somewhere around my latitude. So if I point my panel at latitude tilt, since this angle was-- what was it-- 41 plus 23. This angle was 41 minus 23, so this would be right around 41 latitude. So if I aim my solar panels at latitude tilt, then I'm going to get, on average, some pretty decent power throughout the year. I'll have a little lesson in winter, a little more in summer because, well, just because of the amount of solar resources available. But all in all, I'll be all right. At most, I'll be off by 23 and 1/2 degrees. You can do that. And that's called fixed latitude tilt. And typically, you'll face the panels south approximately. We'll get to that in a minute. It depends on local weather patterns. If you have fog in the morning, for instance, you want to face them a little to the west. But we'll face them south and at latitude tilt. Or we can decide, no, let's actually track the sun throughout the sky throughout the day. And so we'll start in the east in the morning and have it rotating through on one-axis tracker throughout the day. Or we can have a two-axis tracker where it rotates to follow the sun throughout the seasons as well, a kind of a north/south tilt. And that's what's plotted right here. This is a quick approximation of the fixed one-axis and two-axis trackers for a given system in Boston using a simulation tool called PVWatts. There's a link to that. It's based on the National Renewable Energy Laboratory website. There's a link to that at the end of the slides. But this shows you the total system output in terms of kilowatt hours in terms of hours per day. I think, yeah, it's a little bit of an odd units there on the one-axis. But it shows you the relative gain that you would get by going to a one-axis and then, finally, a two-axis tracker. So in many places, it makes sense to go to one-axis tracker, since especially you broaden out the peak near the peak hours of the day. And that's really good. But not always does it make sense financially to go to a two-axis tracker. You're adding another motor onto that thing. It's really not gaining you that much. Obviously, you have to calculate it out yourself for that specific location. But by and large, a generality, one-axis tracker makes sense for a flat panel. Now if you have a concentrator lens in the front and it has to be looking directly at the sun, then you're kind of forced to go to two-axis tracker. OK. And there you have your total system output, which is just the integral under the curve over the entire year. So definitions-- I should have done this before. But direct sunlight, looking directly at the sun, and then diffuse sunlight or scattered light coming off of other things. So diffuse sunlight would be coming from the other angles in other directions in the sky. The direct sunlight would pretty much be at the sun plus or minus a few degrees. And we have different ways of measuring the-- this would be flat plate. Up there on the upper left it says, flat plate facing south latitude tilt. Just like we said, it's facing south. In the United States, that's good. Latitude tilt, meaning it's tilted at our latitude. So if we go from the southern tip of Florida and Texas up to the northern tip of Minnesota, we'll be tilting it more and more toward the south like this, going from Texas to Minnesota. From Texas to Minnesota. And so latitude tilt facing south flat plate. This is essentially accumulating all of the sunlight, the direct and the scattered light, because it's all being collected by the flat plate there. This map, on the other hand, is for a two-axis tracker. And it's looking at the sun plus or minus 2 and 1/2 degrees. And so it's really only picking up this right here out of the sky. And when it's sunny all the time, you're golden. You're tracking the sun. You're actually getting more energy than you would if you just at a flat plate because the cosine theta angle is changing throughout the day if you have a flat plate. When the sun is in the morning time, you have a flat plate like this. So if my sunlight is coming in like this and I do cosine theta of the angle, I'm only getting a very small amount of the incident sunlight projected onto this plate right here. If I was in the middle of the day, now cosine theta is 1, I get the full sunlight. But if I have, for instance, a tracker, I would be able to face this panel due east and then track it throughout the day. And that's where I'd get this additional energy boost here in the mornings right there from the tracking system. So if I have a tracker, I get a big gain in the places that have a lot of direct sun. And if I'm only looking at a very small solid angle of the sky, if I'm only looking at, say, this little portion the sun, on cloudy days, most of the sunlight is coming off of the diffused light, not from the sun. If I have a flat plate, I win. If I have a concentrator, I lose on a cloudy day. And so that's why, in some of the regions of the United States that are notoriously cloudy-- I won't point to any one in particular-- it's actually better for non-concentrating solar in terms of total energy output. And in other regions of the United States where we have a lot of sunny days, you start having two-axis trackers making sense. And it's just really the ratio of these two maps-- whoopsie-- this one, which is flat plate, and that one, which is concentrator. You see over here, we win if we go for the concentrator. And over here, we get more energy out-- oopsie-- more energy out if we use the flat plate. Now that's just energy. Obviously, cost and economics factors what it takes to install it. OK. And weather patterns, I promised I'd get back to this. Interesting to note that for right near the equator, we have this drop of the insolation. And there are these beautiful maps put out by, again, NASA Earth Observatory that show you the cloud fraction coverage of particular spots around the planet. And you can see that right near the equators, typically we have these beautiful tropical forests. The high cloud cover in those regions is blocking out some of the sun. And likewise, you can see the dichotomy between Phoenix and Atlanta. Same latitude again, but Atlanta having a higher cloud coverage than Phoenix and, hence, a lower solar resource, a lower insolation. So you how all this kind of ties together. That's all predictable in a sense. This is unpredictable at a local level for one system. If the sun is tracing its route through the sky, and you have that envelope function that you've just spent so much effort calculating out with all your trig functions in the computer simulation, the code that you've just been given here. And now, a cloud comes over, some random cloud that was very hard to predict. And you just have one panel, one tiny little thing like this. And the cloud goes over it. Boom, all of a sudden, you get a drop in your instantaneous power output. Boom, drops again. Another cloud, drops again. And then a thunderstorm in the afternoon. So a meteorologist could have told you that this thunderstorm was coming, but a meteorologist would be hard pressed to be able to predict the evolution of a tiny little cloud over your system. And so the question is, to what degree are these local weather patterns predictable and unpredictable? And hopefully, some of you, over this class, will be able to help answer that question by analyzing data from tens of thousands of systems that have been installed throughout California in local geographical systems, for example, in the Los Angeles region, San Francisco region, and so forth. If you start averaging the curves, the energy outputs as a function of time, from a variety of systems throughout a neighborhood, you can probably average out the small tiny clouds. You probably can't average out the thunderstorm. But you probably could have predicted that the thunderstorm would've come along. And so this is a hot area of solar research at the systems level. It's trying to understand, to what degree are PV systems predictable? To what degree can the power output of a PV system or an ensemble of PV systems be predicted in advance so we don't wind up in a situation where all of a sudden we have this catastrophic drop of a cumulative PV system output of, say, 30% and meanwhile, it's a hot summer day, everybody's air conditions are going, and we cause failure of the power grid? Kind of worst case scenario. So this boils down to intermittency with a short time constants tend to be less predictable. Cloud cover. And it's relevant for predicting power supply reliability. And the longer time constants tend to be more predictable. The diurnal or seasonal variations-- and these are relevant to calculating total energy output annually. And oftentimes, when we just work off of these long time constant variability issues, we're assuming we have access to easy storage. Right now, the solar panels on top of my roof, they're producing in excess of what we're consuming right now because neither my wife or I are at home. And they're injecting that power into the grid, and the grid is serving as our big battery, as our storage unit. And I'm riding free. I'm a free rider right now. I don't pay for that service necessarily. I do to NSTAR. But they're not charging extra for the service of using the grid as my big battery. Another thing to keep in mind is that we're calculating all this on the basis of engineering and scientific principles. We're calculating the solar resource, which is a good, important first step. Now if we look at the actual solar installations by year and by country-- this is a big table, but follow me on the two underlined red lines here, DEU, that stands for Deutschland, Germany, and USA. We'll see that Germany has about seven times more solar installed cumulatively than the United States does. And yet, if we look at the solar resource in Germany relative to the US-- this is average annual, same scale, going from 900 to 1,200 kilowatt hours per kilowatt per year-- we can see that there's a lot more sun in the United States than there is in Germany. So there's something else going on than just the solar resource. And that's economics. That's why we talk about the policy and economics later on in class. That's why we dedicate a sizable portion later on talking about that. Lastly, our final learning objectives before we halt for questions and for comments, we need to estimate the land area needed to provide sufficient solar resource for a project. And this is really where your homework picks up, and we'll spend some time in recitation walking through this. But it's important to get your units right. And so I want to do a quick quiz right now. Think in your minds which of these properties corresponds to which units or are described by which units over here on the right. So do a kind of a linkage in your mind one to one, connect, connect, connect, connect, without looking at your slides. And then we'll do it in three, two, one. Those are your answers. So the ones that usually get confused are power and energy, kilowatts and kilowatt hours. I'm not saying it's easy, but the way to remember it, the easiest way to remember it, would be to remember that the power time product is equal to energy. So if you have power instantaneous energy burn rate versus time, some plot that looks like this as you take the integral of that curve, that's your total energy. So in terms of units, current voltage power and energy, a hair dryer versus a fridge, which is more likely to blow a fuse and which is more likely to blow your budget? AUDIENCE: A hair dryer is more likely to blow a fuse. PROFESSOR: A hair dryer is more likely to blow a fuse. Why is that? AUDIENCE: High voltage. PROFESSOR: Yeah, high voltage. Well, high current, really. Because everything's running 120. It's a higher current. It's pushing the wattage up to around 1.5, 1.7, 1,500, 1,700 watts. And so it's running close to the 15 amp limit. And the fridge, on the other hand, is probably an order of magnitude lower. But the fridge is running all the time, and your hair dryer is only running a few minutes a day. So in terms of blowing a fuse, it uses a large amount of power for a very short amount of time. Whereas, the fridge uses a small amount of power for a long amount of time. And the integral under that curve winds up being more, typically, then your hair dryer. That's the total amount of energy. And you pay by the energy. You pay by the kilowatt hour to your utility company, and that's why it would blow your budget. Yeah. Oopsie. So the numbers work out to somewhere around 1 kilowatt hour per day for the fridge and about 1/2 a kilowatt hour for the hair dryer. A kilowatt hour is how much in here? $0.18? Most people pay about $0.18 per kilowatt hour. And so you can calculate how much it costs to keep things running in your house. Most of us don't usually think about that. And last, last, last, last point, in your homework assignments, you're going to be asked to size out systems, PV systems, photovoltaic systems, in different parts of the world. And the easiest way to do that is if we take a panel and we say that this panel right here is rated at a certain amount of power, so under peak illumination conditions. When the panel is seated at incident sunlight, so the cosine theta term is 1, and the incident solar resource is 1,000 watts per square meter, so around AM 1.5 conditions, this panel right here would be producing-- this one is tiny. It would be producing 6 watts peak. But most of these panels over here are producing somewhere around a few hundred watts peak. And so the panels are rated in terms of watt peak because the panel manufacture doesn't know where you're going to install them. You could decide that you're going to install it in Alaska, in which case, it's going to produce about half the energy than if you installed it down south in the continental US in, say, Arizona. Maybe a third of the energy. And so it doesn't want to rate the panel in terms of the energy output. It's not going to guarantee that you're going to get a certain energy output off of the panel. But it will guarantee that the panel was rated at maximum power of such and such under standard testing conditions. Now that's good because you can generalize it, which you'll do for your homework. But it also has a downside because standard testing conditions aren't real world conditions. The panel isn't always operating at 25 degrees Celsius. The panel isn't operating always at incident sunlight normal, right? And so the gap between actually knowing how much the panel is going to output in terms of its energy for a specific location and what our calculations, the back of the envelope calculations, will give us, that gap is, from an economics point of view, there's a lot of money to be made there. If you understand really well how much of a given panel will output, you can then predict to a much finer degree what you should be charging your customer for installing panels in a particular location. So there's a lot of work being done right now, more on the business side of pulling the systems engineers along and trying to increase the accuracy of predictions. And so with that, I will leave off on this slide. We can get to it during recitation tomorrow at 4:00. I welcome your questions here at the front if you have any. Otherwise, I'll see you tomorrow. |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | Tutorial_Texturing.txt | [MUSIC PLAYING] PROFESSOR: Hello, everyone. Today we'll be taking a look at how light interacts with the surface of a solar cell. Right now I'm standing next to a solar module made up of individual silicon solar cells. If you look closely, these cells actually appear black. And they appear black for a very important reason. Solar engineers work very hard to make their solar cells as efficient as possible. Reflected light is lost energy, so good engineers will want to minimize the total amount of reflected light. To make solar cells absorb as much light as possible, and appear black, solar engineers do two things. First, they grow this very thin film of a dielectric layer on the surface. This layer is aptly called an anti-reflection coating. Second, they texture the wafer. And today we'll demonstrate how texturing is performed, and quantify its enhancement for reducing light reflection. Silicon wafers don't start out black. In fact, they appear gray. Polished wafers even look mirror-like, and reflect quite a bit of light. Here we see a polished wafer, which reflects around 1/3 of the light off its surface. And to create the rough surface that reflects less light, solar engineers immerse their silicon wafers into a hot, wet chemical bath, which helps create tiny surface features. In this example, we use a solution of potassium hydroxide, or KOH, which is heated to around 80 degrees Celsius. This violent reaction is actually etching into the silicon, and carving out little tiny pyramids on the surface. And the result is a wafer that loses its shiny appearance, and appears to have a dull finish. The textured wafer is left with a surface that is covered with microscopic pyramids, whose base is around a micron, or about 1/50 of the width of a human hair. It turns out that this wafer only reflects about 1/3 as much light as it previously did. The reason this KOH which bath work so well at texturing the silicon surface is due to the fact that these silicon wafers are large crystals, which in this case means that the atoms are formed in an ordered, repeating pattern. I have a model of the silicon crystal structure right here. On our model, I've highlighted the surface in red. Note that each silicon atom below the surface is actually bonded to four other silicon atoms, with four covalent bonds. Note that the surface atoms are only bonded to two other silicon atoms, and it has two bonds that are unbonded. The alkaline etch is able to remove silicon atoms more rapidly, when they have fewer bonds holding to the lattice. Hence, a solution quickly removes atoms on the surface. Now if I were acting as the KOH solution, I would remove all the atoms that only have two covalent bonds. Let's remove a few atoms. So this one only has two covalent bonds. It gets removed. These three atoms on the surface only have two covalent bonds holding them, so I'll remove them as well. Let's go ahead. Now we can see that after we have removed a few atoms, we have created some atoms below our original surface that only have two covalent bonds holding to the lattice. These atoms will also get removed by the KOH. Now if we continue this process, it would look something like this. So what we're going to do is measure the reflectivity of both a flat and a textured wafer. First, we'll measure the flat wafer, and use a laser pointer as a light source to simulate sunlight coming from very far away point. We'll shine light down on the surface, and measure the amount of light that gets reflected or bounced back away from the surface, which we'll label as R. So I'm standing next to our first experiment, which I just outlined in the previous sketch. We'll be using this laser pointer, shining it onto this silicon wafer, and into our photodiode. For those of you who don't know what a photodiode is, it is a tool that can measure the amount of light hitting its surface. The current that's read off of this ammeter will be proportional to the amount of light hitting our photodiode. To help visualize the beam path, we're going to use some steam. Now if we turn on our laser pointer, hits our photodiode, and we can get a good reading. And right now we see that it's reading around 0.9 milliamps. But how does this compare to a textured wafer? Let's find out. Now it's hard to measure the reflectivity of a textured wafer with a laser pointer, because it bounces the light off at several different angles, and we can't measure the entire beam with a photodiode alone. However, we can simulate what this would be like. Let's go to our sketch board to show how we can approximate this measurement. To measure the reflectivity of a textured surface, we'll create a 10,000 to one scale model. We'll approximate our textured surface using two pieces of silicon. Again, we use a laser pointer as our light source, and shine it on one side of the pyramid. It'll bounce off that next surface, and off the adjacent side, and then we'll measure the amount of light reflected. Let's go to our experimental setup. So let's clarify our set up. The two angled lines in our drawing, the adjacent sides, on our atomic models, and the two angled, non-textured wafers in our set up, are all representing the pyramid structure of a textured wafer, just like the one visible in our scanning electron microscope image. Let's use some steam to visualize the beam path. Now that the steam has settled, we can get an accurate reading of the reflectivity of our modeled silicon surface. And I turn on our laser pointer. So now we can see that our photodiode is reading around 0.33 milliamps. This corresponds to around a 9% reflectivity, which is quite a huge reduction from what we had before, by about a factor of 3. So in summary, today we learned how silicon is etched using a KOH solution, and how the resultant pyramids increase the efficiency of our solar cells, by reducing the amount of reflective losses by a factor of 3. If you found this interesting, please watch our other solar demos to learn more about how these exciting devices work. I'm Joe Sullivan from MIT, and thanks for watching. I'm out of here. [MUSIC PLAYING] |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | Student_Project_Presentations_Part_1.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Final presentations should be very exciting-- fruits of your labor over the entire semester in reality. The fundamentals in the first third, the technologies in the second, gearing up toward the cross cutting themes in the third. I understand that we've had an accelerated project schedule this semester. We've completed the entire projects over the duration of about a month and a half, so you are to be congratulated for your hard work in a very intense period of time McKenzie tiger team style. For that, I reward you with doughnuts and coffee over there. I understand many of you were up late last night, so you're welcome to ingest some shortchange carbohydrates and some caffeine. If you would like to get some, get it now. We're going to have another minute of blah, blah, blah before we dive into the presentations and the real fun begins. I'd like to introduce our panelists up here in the front who will be the evaluation criteria a la American Idol style, except that you're, of course, a lot smarter and equally well dressed. Starting from right to left in front of me we have Dr. Jasmin Hofstetter, who comes from IES, Spain. That's the Institute for Solar Energy in Spain. That's where she did her PhD with Antonio Luque. Those who have been studying intermediate band solar cell materials may know the name as one of the fathers of the field. She studied under Antonio Luque's organization with Carlos de Canizo and is the winner of presentation awards at scientific conferences, among others, so Jasmin is welcome here. We have thought that Dr. Mark Winkler, as well, a PhD in Eric Mazur's laboratory at Harvard. Those who are familiar with femtosecond laser characterization may be familiar with Eric Mazur, also one of the fathers of the field. Mark started the Harvard Journal Energy Club, which is Harvard's version of the MIT Energy Club-- a lot smaller and a lot less dynamic than MIT's version, but nevertheless to be congratulated. And of course, a very good organization. I kid. There's a little bit of MIT Harvard rivalry. And of course, our very own Joe Sullivan, who has been with you the entire semester. For those who might not be familiar as much with this research as you are with his teaching, Joe is studying intermediate band solar cell materials here at MIT in the Media Lab and has been working for the last-- what is it now? JOE SULLIVAN: Three and a half. PROFESSOR: Three and a half years-- focused on intermediate band solar cell materials, coming from a very broad background in energy from climate science. So with that, I'd like to welcome our first team down, and the floor is yours. STUDENT 1: Good morning, we are the PV smart retrofit group, and our project goal was to assess whether or not the there was an electrical benefit or loss from retrofitting an old home with a PV system. Now, in less lofty terms, it essentially means from an on site energy perspective, does it make sense to put PV panels on my house? You'd think that that's a kind of an obvious question. We'd all say, well, yeah. We produce energy for free, except that you have to consider other things, such as shading from trees that you'd have to cut down or the color of the roof that you might be changing by adding a black panel. These would both reduce the thermal load of your house in normal situation. So by adding a PV panel and increasing the thermal load, you actually add the energy to cool your house during the summer. We considered multiple variables in this project. One was location, which has an effect on the amount of sunlight you're receiving. The PV panel's presence and its size. We had a couple of situations where there was no panel as a kind of a baseline. And then also different sizes to figure out if a bigger panel had a bigger effect. We looked at two different colors, black and white. Those are the ends of the spectrum, and they give us endpoints to look at, and the color matters because a darker color will absorb more heat. We looked at roof pitch. The reason for this-- well, first off, roof pitch is the angle of your roof. And we looked at this because we assumed that our panels were fixed and parallel to the roof, so this kind of controlled what angle your solar panel was facing towards the sun. And finally you had your house footprint, which is the area that the house covers, and when you combine that with the roof pitch, you get the area of the roof, which is really what we're concerned with. We had five scenarios, and as you can see from our cute little diagrams, we have a black roof, a white roof, a white roof with the tree, a white roof with a solar panel, and a white roof with a solar panel and a cut tree so the solar panel is getting plenty of sunlight. In evaluating the five scenarios, we had three models of increasing complexity from left to right, which you can note from the fact that model one only covers three of the five, and these models all had common assumptions. The first was that you had a single story house in a suburban locations, so you didn't have shading from other buildings, for example. We had a common house size of about 2000 square feet, or 186 square meters. The roof pitch, which in construction is set usually at 5/12, which means 5 inches of rise for 12 inches of travel. We had an unfinished attic space, which means that it's sealed to the outside, but you didn't make it livable, a five kilowatt PV system covering 36 square meters, and we chose reflectance values of 0.08 for black and 0.35 for white, and this, again, reflects the effect that color has on heat absorption. With that, I will turn you over to, Jordan. JORDAN: So the first one that we looked at was basically using most readily available and simple models there can be. So this is from the Department of Energy to measure-- well, to get a gauge of how the color of the roof and the roof properties affect the thermal loads in the house. This is a simple one dimensional model where you've got an inside cavity of 65 Fahrenheit, and you basically input the location, and from that it has a lookup table of the average insulation on the house, as well as the number of heating degree days and cooling degree days relative to that 65 inside temperature. The other parameters are just at the roof, so we insert the reflectance, and we're using values that represent real tiles for an average house, black and white, as well as the thermal resistance and the heat absorbance of the tiles-- from this model, the outputs and the thermal loads in terms of the heating you need to put in, as well as the cooling energy load. So with the thermal model assessed, we can assess the photovoltaic output. And for this, we're just using a simple model PVWatts. This basically takes in the location and the angle of the panels, as well as a derate factor from converting to AC to DC. It's quite a simple model, and output from this is the amount of kilowatt hours per year that you get from the panels. STUDENT 2: So I'll be talking about a couple thermal electric model. This is the model that we built in our group, and we developed this model based on two sets of individual parameters and individual models. I would say that one comprises of the thermal model, and there is the electric model. So what we need to note from this model here is it's a further step in complexity when compared to the module one that Jordan just discussed. And it takes into account various input parameters that the model one doesn't take into account. So the basic structure of this model is as follows. We have a thermal model which takes into account several input conditions, such as the insulation and shading, and it outputs a living space temperature, which is this temperature of the living room in our house. And then this temperature is fed into an electric model, which calculates the cost and energy values, and thus we can compare an energy production and energy consumption. Going to the thermal model in detail, I've just shown a picture here. So it considers basically when we start from the top of the house, we use insulation and shading as the input parameters. We then calculate the temperature of the PV panel, and then the temperature of the roof using two different energy balance models. And these two energy balance models are pretty robust in the sense that they consider all these physical phenomenon which are realistic, such as the convection, radiation, and PV electrical output. And based on these energy balance equations, we can calculate the PV panel temperature and then the roof temperature. And once we get these two temperatures, we get the heat flux that goes into the roof and that enters the attic. And once you get this, we find out the attic temperature, which then determines what is the ceiling temperature. And then we consider the convection via ceiling, and then finally we end up with the living space temperature. And then the couple this living space temperature with another electric model, which I'll discuss now. So the electric model is basically based on an ideal gas assumption. So it basically-- what it does is it calculates the energy needed by the AC, which could be the heating or cooling, in order to maintain the living space at a particular temperature. And we use this formula, m dot Cp delta T, which is an ideal gas formula, which gives out the energy needed by the cooler. And then so the electrical model uses the power consumption, and then we know the power production through PV output. So comparing these two, we can really assess whether PV installation is favorable or not. And this is just the model in making. What we want to signify here is we actually made this model on our own, and this is the MATLAB code we wrote. And the model not only just predicts energy values, but it also can do a lot more things, such as predicting temperature. And what I've shown here is the PV output, and then the cooling load that is required, so it can do a lot of other things, as well. This is the temperature of the roof in terms of direct sunlight and diffuse sunlight. So if anyone is interested, I'd be happy to discuss with them more. Thank you. And I'll now pass it on to Heidi. HEIDI: So I take over from here talking about the third and most complex model that we used. For this model, we used two different softwares-- one called BEopt from NREL, and the second, called EnergyPlus, which you'll see later, developed by the Department of Energy. So what we did in this model was we took into account the 3D effects of these thermal and electric loads that have been talked about in the other two models. The first thing we did was to actually model the 3D house, as you can see right over here. BEopt allows for a very nice interface, where you can easily model the house and easily and put a whole bunch of input parameters for the house. And for these input parameters, we consulted with an experienced building inspector for the construction inputs and used BEopt default values for the rest of the inputs. Once this model of the house was done, we had to actually export it into EnergyPlus because EnergyPlus gives us a much more detailed look into all these different parameters, I guess. And you can actually go in and modify different things. For the materials, we can modify every single material property-- conductivity, density, specific heat-- and we actually did that for the roof. And we also removed a whole bunch of other miscellaneous loads that BEopt had included. So this is how we modeled the trees and the panels that we talked about in our scenarios before. For the trees, we modelled them as these really large 5 by 20 meter rectangles that act as shading for the house. So the trees are located on the south side of the house, and we modelled them as deciduous trees, so we set up a transmitting schedule so that they have a higher transmittance in the winter when the leaves have fallen and a low one in the summer. And for the case of scenario E in which the trees are cut, we modelled the trees as being five meters tall. On the other hand, for the PV panel, we modeled it as being fixed on the roof. We actually had to completely change the model from BEopt, because they modeled it as being decoupled from the entire system, placed 30 meters away from the house. So we completely changed that, and we read extensively into EnergyPlus literature and found this particular object-- I guess you could call it-- called the integrated exterior vented cavity object. And what this does is it models a surface as being, in our case, 0.5 meters away from the roof, and it models the convection and radiation between these two surfaces. We also considered the solar panel to have a solar absorbance of 0.92 and thermal emissivity of 0.9. STUDENT 3: And so we actually got a lot of results, as you might imagine, from all those different models, but just for the purposes of comparison for the presentation, we're just going to show you-- just summarize results. And basically what we're showing you here is specifically for Boston, and this figure that we're showing is the y-axis is the relative energy gain. So in order to compare them, we decided within each model compare it to a common situation. So we decided to say that, if we're in Boston, let's say we start with a white roof and a tree, so that's scenario C there, so that's why it's 0. So everything is in comparison to that. And right off, we could see that, as we might expect, putting a PV on makes sense energetically. And specifically scenario D, which is where you completely remove the tree instead of just cutting it, in Boston at least, is what makes the most sense. And this is for both models-- actually, for all the models-- although model one can't really model a tree necessarily. So that's why it basically doesn't apply for that. In terms of comparing the results between the models, model one actually does a pretty decent job in Boston in getting close to model three, which is impressive, because model one is significantly more crude, much simpler than model three, which required many, many inputs. And the reason we believe the discrepancy is between models two and three is that model two-- the way that we basically treated the solar insulation-- it doesn't treat diffuse sunlight differently, whereas basically the models PVWatts and EnergyPlus will take all that into account, so we think that's a larger reason for that. And then similar thing in Phoenix. And basically we see a slightly different case here. Actually scenario E, where you just-- you install the PV, but you cut the tree instead of completely removing it-- gives you a slightly better increase in net energy gain. In terms of-- and this is, again, just in terms of energies. There could be rounding errors. This is ignoring the fact that, if you cut a tree down, the net greenhouse gas emissions would be changed and altered, and this is ignoring all those other effects. This is just in terms of net energy gain of the house. And just to point out the discrepancy between model one here, we think that model one is actually, again-- that's using PVWatts to get your energy output. And that's basically assuming peak solar insulation, whereas our models basically use more empirical formulas to get the estimates for your PV output. And then the last thing we did to do the sensitivity-- or to compare the models, was do a sort of sensitivity analysis. So basically on your y-axis, what you have is your percent change in your relative energy gain-- relative, again, to that scenario C-- divided by the percent change in your parameters. So we just considered four parameters. The x-axis is a rough estimate of how difficult it would be to actually change that parameter in your house. So the far left one is the PV size. We just assumed if you went-- instead of a five kilowatt to a 5.5 kilowatt, so that's why the price is roughly $3,000. We assumed about a $5 to $6 per watt installation cost for that. So that's basically the easiest one to do, and you get various significant change in your thermal energy gains because of that. And then just because of time I'm going to rush through these, but basically a lot of the models follow similar trends in terms of the sensitivities. The last one is the roof pitch. You obviously wouldn't really want to change that. It's very expensive to do. Luckily, for most of the models, it's not actually that sensitive to it, at least within a close amount to where you start with. So in conclusion, in all three models, it makes sense to install a PV system. It's kind of what we expected from the beginning, but it's nice to get that sort of conclusion. For models one and two, we actually were able to get pretty reasonable results, but they're limited in terms of what you can consider within those models. The advantage of looking at this basically is that, if you have a user that isn't as familiar with EnergyPlus in model three, which is very sophisticated-- it requires a lot of inputs-- they can still get a rough estimate, which is relatively close, using these much simpler models. And model three-- again, we were taking that to be the more realistic case, but you'd have to compare it to real life data and do an empirical analysis to see how close it actually does correlate. And then just, again, to summarize the results we got from model three-- in Boston, it makes sense to install the PV, but completely remove the tree. And your payback period is about 24 years. In Phoenix, the best scenario is to install the PV, cut the tree, and it's about 51 years. And interestingly, the maximum of that relative energy gain was essentially the same in both, even though the scenarios were different. So we'd just like to acknowledge Professor Buonassisi for helping assist us and guiding the direction of the project, and Bryan Urban at Fraunhofer and other members at Fraunhofer for giving us guidance. And with that, we would like to ask you for questions. [APPLAUSE] AUDIENCE: My question is I grew up in a neighborhood that has a lot of trees, and so cutting down all the trees wouldn't be very practical, but do you at all consider PV systems that could handle shading at different times of the day? So somehow decoupling different parts of it knowing that some of them will be in sunlight for part of the day, some of them will be shaded, and that will change? STUDENT 3: So you could make the model-- especially in EnergyPlus, you could make it as complex as you want. We just did this for simplicity, just to put boundaries around what are problem is that we were considering, but you could definitely-- yeah, you could definitely add that to the model if you wanted to. STUDENT 1: Cutting the tree was not worrying about the output of the actual cell in terms of whether or not some shading was going to bring down the rest of the cell. The reason we would cut the trees is to increase the maximum amount of sunlight per day hitting the panel. And it's because we were looking mostly at endpoints, trying to get the spectrum ends. That was why we went to such extremes. Cutting down selective trees and just parts of trees would be kind of in between. It's a little bit more difficult to assess. STUDENT 2: Cutting down trees is actually [INAUDIBLE]. It's not [INAUDIBLE] because we found that the shading factor doesn't play a much bigger role if you look at the relative [INAUDIBLE]. So you would as well have increased the-- there will be objectively small loss in the energy gain, but that shouldn't matter much. AUDIENCE: Sort of a philosophical question. If the payback period in Phoenix is 51 years, is it worth it? That's a long time period for-- I guess economically you could say that you could do other things with that capital instead that would have a shorter payback period. JORDAN: Well, 51 years-- then the answer is probably not if you're looking at the benefit of cost money-wise. We did analysis about the energy. So we found an estimate for the embedded energy of the panel. This is from-- I can't remember the source, but they change quite a bit. But this example says it's 1,500 kilowatt hours per meter squared, so that equivalents to nine years payback. Pretty much nine years in Boston. I think it turned out to be eight years in Phoenix. So there is a benefit energy-wise, but in this example, perhaps not cost-wise-- perhaps not the most advantageous to do per dollar. MARK WINKLER: A related question, actually. Can you back to your two slides back maybe? JORDAN: Yeah. MARK WINKLER: Your look at the net energy gain was quite similar. So why the large difference in yearly savings and payback period. STUDENT 3: Just the cost of electricity in each location. We estimated it as about $0.07 per kilowatt hour in Phoenix and $0.17 in Boston for residential. STUDENT 1: Which is why you would get a shorter payback period for Boston-- is because the cost of the electricity that you're [INAUDIBLE]. MARK WINKLER: So I would have assumed that the generation mix is sort of similar. Is that regulatory, or-- I would assume they're coal/gas centric generation mixes. STUDENT 3: Yeah. You mean in terms of how the houses are-- MARK WINKLER: This is a little outside the scope of what you guys did. I was just curious if you guys had any sense of why the big difference in wholesale electricity price is between Boston-- STUDENT 3: I think part of it is just how plentiful energy is. I guess Boston is at the very end in the corner of the US. It's more difficult to get fuel, oil, gas shipped over here. I think Arizona-- I think they're relatively close to a nuclear power plant over there. Oil-- I think it's just location--wise. JOE SULLIVAN: There's a lot of coal there. They actually ship a lot of the electricity to California because they can't [INAUDIBLE] in California. One quick question, though. The relative energy gains are the same for both. Do you have different sized panels, or is heating that much? That's a big deal. STUDENT 3: Heating, yeah. JOE SULLIVAN: OK. So if you-- HEIDI: Also for Phoenix. For Phoenix, you can see this is the cooling over here and heating on the right over there, and for Phoenix, you can just look at the values and see that there's a lot more cooling than heating compared to the Boston case, where there's a lot more heating by many orders of magnitude more. And so that kind of balances it out. JOE SULLIVAN: And so that's only in a cutting down a tree case if you were already well shaded or not shaded at all? HEIDI: These are actually all the curves for all the scenarios, and-- JOE SULLIVAN: I see. OK. HEIDI: So it does matter just because of location. So if you're not cutting down a tree, then there's no decrease in shading. Or is it just the panel itself that's heating up more? STUDENT 3: Well, actually if you go back to-- JOE SULLIVAN: Sorry, I think I missed something. STUDENT 3: Actually, in Boston it actually makes more sense to have a black roof than a white roof. So you actually want-- shading isn't necessarily good in Boston, just because there's so much heating that you need in the winter. It seems to be the dominant effect in Boston, and in Phoenix, it's the opposite. The cooling is the dominant effect. PROFESSOR: Did you consider the possibility that snow also insulates the house once it falls on the roof? STUDENT 3: No. We did not. I don't know if-- is that built into-- I don't know what would happen. No, I don't think we did, but that's a good point. AUDIENCE: Along those lines, do you have any intuition as to why in one case it's better to cut down a tree other than remove it, and then the other is better to remove it rather than just cut it down? Are you expecting it to grow back and then have to incur more costs because you're going to have to cut it down again, or-- what's going on? STUDENT 3: Well, just in terms of pure energy, it was very, very slightly better in this case to have the tree just cut just in terms of the balance between heating and cooling [INAUDIBLE]. STUDENT 1: I may be able to help clarify that. The idea is in Boston we're relatively cold most of the year, so the more sunlight that hits your house is going to add more heat to your house, and that's less energy that you have to pay for. So the reason that it's beneficial to cut down the tree completely in Boston is because it allows more sunlight to hit your house, whereas in Phoenix, you don't want the sunlight to hit your house. If you cut down the tree completely, that's more you have to pay for AC in the summer. So the-- AUDIENCE: [INAUDIBLE] the difference between cutting down the tree and removing it? STUDENT 1: So-- STUDENT 3: So-- go ahead. STUDENT 1: Cutting the tree is assuming that you're going to maintain it at that certain level. So by cutting the tree, you keep a certain amount of shading on the lower part of your house, but you still allow sunlight to hit your solar panel. Cutting down the tree completely means there's no shading on your house at all. AUDIENCE: I guess I have a philosophical question. So I think there are a lot of people-- motivation for the solar panels is not just the [INAUDIBLE], but rather the desire to do something good for the environment, and to lower carbon emissions, et cetera. But when you cut down trees, that increases your carbon emission because you're reducing the plant, that reduces your carbon output. So given that you won't have a tree there for like 50 years, does that offset the carbon emission gains that you get by-- STUDENT 1: Just my two cents. If you really want to go in depth, you can look at how much carbon is going to be produced by the coal power plant to give you the energy that you're going to be using for 50 years and compare that to the amount of carbon that one tree was going to save you, or you could ask yourself, am I planning to have a child during those 50 years, which will produce so much more CO2 than that tree will take out? Either way, it's relatively a small value. However, we do acknowledge that there was a lot of philosophical questions that we argued amongst ourselves, but we realized we didn't have the time to try to evaluate, or the materials, and scope. PROFESSOR: One more question, and then we're going to have to switch groups. Jasmin? JASMIN HOFSTETTER: Do you have any real data to compare your model results to. From your results, it seems that it doesn't make any sense to install solar panels ins Phoenix. Is that right? That's the impression? STUDENT 3: Financially. Just purely financially, yeah. In terms of the PV output, it seemed to be pretty close in comparison to what we got from other sources. So it seems like the net gain in energy is roughly right, but obviously people still install panels there. So either, I'm guessing, subsidies, or larger installations, or something else, or just the desire to install it just for installing it-- not necessarily for financial reason-- in Phoenix. JASMIN HOFSTETTER: What was the temperature that you assumed? I suppose you assumed a constant temperature in the house that was like the-- STUDENT 3: Yes, that would also change it. JASMIN HOFSTETTER: What was this temperature? STUDENT 3: The set points for our model was-- the cooling set point was 71. HEIDI: 76. Yeah, 76. And then the heating set point was 71. STUDENT 3: Yes. JASMIN HOFSTETTER: Can you say that again, please? HEIDI: The cooling set point was 76 degrees Fahrenheit, and the heating was 71. STUDENT 3: For both locations. HEIDI: For both locations. JASMIN HOFSTETTER: Thank you. AUDIENCE: [INAUDIBLE]. JOE SULLIVAN: Are we out of-- PROFESSOR: No, that's it. You guys are done. Congratulations. [APPLAUSE] [INAUDIBLE] coming up, PV grid. What happens when you install loads, and oodles, and oodles, and oodles of solar onto the grid? We're going to hear all about. And take it away. Knock it out of park, guys. IBRAHIM: So as [? Tony ?] mentioned, we're the PV grid project. I'm Ibrahim. MARY: I'm Mary. RITA: I'm Rita. ASHLEY: I'm Ashley. JARED: I'm Jared. IBRAHIM: All right, so I'm just going to start with the motivation behind our project. So as we discussed in class, PV installations have witnessed very significant growth rates over the last few years. Last year alone PV installation growth rates were around 17%. Around 18 gigawatts globally were installed. As the cost of PV approaches grid parity, more investors and consumers are going to want to adopt PV systems. However, one lingering or major obstacle preventing the further or high penetration levels of PV systems is intermittency. So as we discussed in class, there's variability in terms of the solar resource, both on a long-term scale and a short-term scale seconds to minutes. So on a long-term scale, we're talking about the position of the sun relative to the Earth and so on. So in that respect, that's predictable and can be planned for. When we define or talk about intermittency, it's the short-term unpredictable effects that change the power output significantly. So what we have here is the fractional change in power output over the course of one day. So as you can see, between the two consecutive seconds, the power output can almost double, and it can at other times drop by half. So from a system operator perspective, that's obviously a major challenge because demand should match supply at all times. So again, these effects, or these intermittency issues, arise due to regional weather patterns that can be predicted and also due to local weather patterns that are less predictable. So in our project, what we tried to address is, can the weather report be used to predict the power output from an ensemble of smaller distributive PV systems? That is, can we average out these local less predictable intermittency effects? I'll give it to Mary to discuss our approach. MARY: So our goal of this project was to design a model that could quantitatively analyze a PV grid and determine its robustness in terms of variability. And our main components were meantime between failure-- which is the average time between two system failures, which Rita will define and discuss later-- number of systems in the grid, and geographic dispersion, which we measured through geometric mean distance. Our data set was from the Oahu airport, which is part of the National Renewable Energy Laboratory. There are 17 systems all within about a kilometer of each other, so it's a very small, very dense system, but there was second interval data for a year, which we used. So there's a fair amount of data to give us an estimate of how intermittency varies over the course of a system and the number of systems and density. RITA: So our first step was to define what was a PV system failure. In order to do so, we accessed the CAISO website-- that is, the California Independent System Operator-- and we took that data from one week of the actual demand and hour hand demand forecast. They give this value for every hour, so we took the value for every hour of the week, and then we plotted in this graph that we have a line for each day of the week, and can see that both the magnitudes and the shapes throughout the week are almost the same. We can also see that the values are almost all positive. This means that they usually underestimate. They usually think that the demand is going to be under what it really happens. And so what we defined was that, if this estimation is OK for CAISO, if they can manage that the grid with this variation, then they could also manage the grid with this variation in a PV output. And so we looked at 5:00 PM. That is the hour that we have the biggest variation between the two, and we averaged the value, and we got to 6%. So this means that, if our intermittency is above 6%, we are going to have a PV system failure. If the variation is below 6%, then the intermittency is not going to be a failure. Then we could define the mean time between failures-- that is, the mean time between two intermittencies higher than 6%. JARED: OK, now that we have some context of what the problem is, and we have an idea of what variability is, and we have a data set to work with, I'm going to talk about how we actually solve the problem. We use coding in MATLAB to handle this huge data set. NREL had 17 systems out there for every second of an entire year. And so we took all the files from NREL and put them all into one huge matrix. You can imagine it was-- it ended up being about 23 fields by several million, and it's about 677 megabytes. So actually handling the data was an issue in itself. I don't recommend it with an old computer. And we also the GPS coordinates for each of those locations and the variability from the California ISO, so with that data we could begin to build our code to figure out a quantitative description of mean time between failure and our idea of density. So once everything was loaded into one big matrix that we could work with, we moved on to use the GPS coordinates. And of those 17 systems, we found every single combination of 17 choose 2, 17 choose 3-- every possible way that you could connect these systems-- and came up with something like 60,000 different ways of connecting these, and then for each possible connection, we had a function that would calculate this geometric mean distance that would give you an idea of the density of that particular connection. And so to compare our mean time between failure for these systems while holding the density constant, we then searched through those possible combinations and found this magic number that kind of existed for each of those possible combinations of two, of three, or four, all the way through. And it kind of lined up for geometric mean distance of 400 meters. So using that set, we could then go on and see how increasing the number of systems helped the mean time between failure. And then for a given set, we ended up using eight. 17 choose 8 gave us like 24,000 possible ways to connect them. We searched through and found varying densities for one set number. Then finally we wrote a function that calculated the mean time between failure that went through our data from NREL and said-- looked at the fractional difference and said, OK, each time it's about 6%, that's a failure, and then measured that distance, took the average of that, and that was our mean time between failure. And then finally, once we had all that together, we crunched all the numbers, took a long time on my computer. We were able to plot it together and get some very nice trends. One of the great things, I think, about our code is that it was only 525 lines, and if you've ever built a programmer, a big application, that's really small. It's very easy just to go in and see exactly what's going on. So it's very flexible. We could hand it off to another company, to another research group, and they go in and adapt it to just about any data set. If you are able to get data in California, or from Germany, or from somewhere else, and bring it into our format, it's very easy just to plug it in and run the data. Very, very minimal changes within our code. And then you could also build on our code to look at other problems. So we have the change in the-- we have the variability data as a function of time. We also have the solar output as a function of time. So you could conceivably go in and figure out how your meantime between failure changes based on the time of day and change your critical percentage based on the time of day, and there are several other problems that you could go, and launch off of our code, and continue on. And if you're interested at all, I actually put the code of my public space. There's the link there. Check it out. It's pretty cool. And then Ashley is going to talk about our results. ASHLEY: Cool, so the first thing that we did in order to try to see the trends in these huge fields of data was just to plot the data. It was actually a much bigger task than I thought it was going to be. The plot on the left is for one day's worth of data, and the plot on the right is one week's worth of data. The y-axis is power density in watts per square meter, and the x-axis is the time in seconds. The blue is all 17 of our systems together, and the red is just for one system. So as you would assume, the power output for all 17 together is clearly a lot greater than the output from just one system, but this give us a sense of being able to see fluctuations within one day, and also were able to see when the sun rose, and peaked, and also fell each day. And in order to quantify all those different fluctuations, we did the fractional change in power density versus time, once again, for one day, and then for one week. And red is the one system. Blue is all 17 systems together, and we can already see just from plotting the data that having all 17 systems together does start to average out the fluctuations of individual systems by a significant amount. So then Rita earlier mentioned that we use 6% as our cut off for failure. We actually went ahead and did 6%, 12%, and 18% just to see how sensitive our analysis was to that threshold value. So here we have plotted on the left the meantime between failure versus the number of systems, and on the right, meantime between failure versus the geometric mean distance. I also calculated these values for using a week's worth of data, a month's worth of data, and a year's worth of data. So the week would give you more fluctuations, but the year would give you the more long-term overall system behavior. Relationship between the mean time between failure and number of systems is quadratic, and we found a linear relationship between the mean time between failure and the GMD. So this is four 6% cut off. This is for 12% cut off, and this is for 18% cut off. And the mean time between failure increases dramatically as you go from the 6% cut off to the 18% cut off. So a lot of this makes sense, but it was really cool to quantify that. RITA: So after applying those graphs we could take our conclusions and answer our question. And so the first thing that we noticed, but we were expecting, is that a big data sample should be used if conclusions are going to be used as a design tool. As Ashley said, we used for a week, a month, and a year. And so we know that the bigger the data set, it's going to be-- it's not going to be influenced by abnormal things that can happen in a given day. And we also saw that there is a linear relation between mean time between failure and GMD. When GMD increases-- that is, when density decreases-- we are going to have an increase in mean time between failure. This was also what we were expecting because the local effects will not affect systems that are further apart. We also saw that there was a quadratic relation between mean time between failure and number of systems. Number of systems increased. Mean time between failures also increased. This was also according to what we expected because we know that the percentage and the total output is going to be lower. We also saw that the mean time between failure is very low, even when we can see the 17 systems together, we have about 900 seconds between failures. This means that some backup systems should be used in order to take over the load when we have a failure. And so now we're running conditions to ask our first question. And so we conclude that localized predictable intermittency do average out and that this effect decreases as the number of systems and the GMD increase. The data that we used was for 17 systems, and the biggest distance between them was one kilometer. So we believe that it's important to run our code for a bigger set of data, because only in this way we can confirm our conclusions and guidelines for the design of PV systems can be defined. Thank you, and we'll be happy to answer your questions. [APPLAUSE] JOE SULLIVAN: So a couple things. First of all, you ended at exactly 15 minutes. I find that remarkable. Additionally, just-- sorry. Can you repeat what exactly a failure mode is defined as? Are you looking at 6% intermittency varying from second to second? So if you look at the output from one second to the next, does that change by over 6%? RITA: Mm-hm. JOE SULLIVAN: It wasn't average out over an hour. RITA: No, no. It was second by second. JOE SULLIVAN: You got the 6% from Cal ISO. RITA: We said that if there-- in a given hour, we measured-- let me just-- JARED: They only had an hour of data [INAUDIBLE]. RITA: Yeah, they only gave hourly data. So the difference between the actual demand and the hour-ahead demand forecast. So this is what they are expecting, but the difference between what they are expecting and what the grid is really asking them. So if they can manage this difference on a second base, they can also manage this difference on the PV grid. JOE SULLIVAN: So you took the worst case. Is that how you got 6%? RITA: Yeah, we took the average of the worst case. It's the 5:00 PM. The 5:00 PM is always the worst hour. It's always when they have that peak. And in fact, all of the base-- almost all of the base were around 6%. Our peak was like 6.8%, and we averaged, and it was 6%. MARK WINKLER: So that's essentially their peaking capacity? RITA: Yeah. ASHLEY: Also, so I actually wrote down the numbers for 6% function, 18%-- like our mean time between failure. For 6%, we had up to 15 minutes between failure. So it's a pretty low amount of time between failures. And if you allow 12% as your intermittency, you can get up to about half a day. And then for 18% as your cut off, you get about nine days. So it is still very intermittent, and you would pretty often have to have backup systems if you had the small of a system. So if you had a much wider spread system and a lot more systems in your grid, then you could definitely significantly increase the mean time between failure. Yeah, Joe? JOE SULLIVAN: So you have this awesome graph. So if you go back to the time between failure number of systems. The interesting takeaway is how large of an area do you have average over, right? So 300 seconds on a grid perspective is unacceptable for widespread PV developed point. We need to be on the order of years. And so do you have an idea of what that distance is? ASHLEY: If we just extrapolate it out? JOE SULLIVAN: If you extrapolate-- this is obviously like we're taking the very, very edge of that function and then extrapolating [INAUDIBLE]. ASHLEY: So looking at the numbers-- JOE SULLIVAN: But it looks like it's going up exponentially, or do you have an idea of what that trend is? ASHLEY: For 6% for the one year, it was almost exactly x squared. It was like x squared plus 50, or 100, or whatever that would be. So if you want, you could say, mean time how many seconds are in a year equals number squared. So the square root of however many number of seconds there are in a year would give you your number of systems required for a year between failures. IBRAHIM: But this is for a given geometric mean distance, so you have two factors. If you sort of spread them out more, probably going to require less systems. JARED: And if you looked that map, that's all at the end of a runway at the Honolulu airport. So if you have a huge field in Arizona, thousands of systems, your mean time between failure is going to be a lot better. MARK WINKLER: So I'm actually really surprised that there's such a huge effect from adding systems, just because it seems as though the relevant length scales for weather should be very large. Do you guys say anything about that? JARED: I think the idea was that, for long-term weather, you can predict that. So if you know there's going to be a storm front coming through, you can add natural gas. You can add coal to the system. Back up-- ASHLEY: And that would cover the entire system. JARED: Our kind of variability we're talking about is say, if one cloud goes over, or a flock of birds, or something. So we were thinking that would be on a few seconds for a single module for a cloud just go over shade it for a short distance. So if you add thousands of modules, the other modules wouldn't be shaded while that one is. MARK WINKLER: But these fields-- I mean, 100 meters on the scale of cloud cover, this still seems like a somewhat small length scale. Let me rephrase the question. Do you think that the graph on the right would be a smooth function of distance, or do you think there's some length scale at which the behavior on that plot changes significantly? JARED: That would be interesting if we could find-- ASHLEY: The assumption is definitely evenly dispersed in an area. JARED: That would be something that, if we had another data set that had wider distances, it would be very easy to plug it in. I think our code's really flexible. It would show us that relationship. MARK WINKLER: What do you guys think, though? JARED: It's a good question. ASHLEY: I wouldn't be surprised if it was linear still. I guess another complexity we could do would be you would have-- right now we just have one big field of systems, but if you had one set of systems that was spaced x distances apart, and then you had some number of kilometers away from another one space-- I'm not sure how exactly would model that, but I think that at that point I'm not sure what the curve would look like, but a continuing linear trend seems reasonable to me. IBRAHIM: So I guess another thing to keep in mind is we did not take into account transmission costs, so I guess you'd have to weigh the cost of failure versus, I guess, the added incurred cost for transmission lines and so on, so there's sort of an optimum point where you want to space them and have a certain number of systems where I guess, after a certain point, your returns diminish and are not equal to, I guess, the cost of failure. So that's something where, I guess, future people can come in and expand on. AUDIENCE: So all this is data from Hawaii, which has a very notable climate and weather. I've never been there, but-- [LAUGHTER] Do you think that this is really-- your code is flexible, so I understand that, but do you think the conclusions are really extensible to other parts of the world with different weather patterns or climate? RITA: That's why we think that the future mark is really to do it for a different place and for a bigger set of date because we really want to be sure that the conclusions are going to be applicable, because we had that same question. We were talking just about a small place. We said that it's one kilometer apart for the distance that we have. So we also want to run for a bigger set of data and for another place just to be sure that our conclusions are applicable everywhere. JARED: And I would say the relationship would probably hold because if you have-- say, if your regional weather is very different, that wouldn't show up in the fractional second to second difference that we had. And so the timescale that we measured it on I think would be, say, small clouds or intermittent events that would occur over a wide range of different climates. The general regional weather is predictable, and it isn't investigated in our study at all. So I would say I think the relation would hold. ASHLEY: I think that the big change between different regions would just be the total output power that you can get, but I think-- I wouldn't be surprised if the fluctuation is still the same or is similar. And I think certainly that, as you increase a number systems and as you decrease the density with which they're packed, you should be able to have a more robust grid. I would be very surprised if that weren't the case. AUDIENCE: Do you see shading for planes at the airport? ASHLEY: There's no way for us to determine what causes the shading. The raw data we have is just output. JOE SULLIVAN: Can you see how they move? [LAUGHTER] ASHLEY: It's like there is this line-- IBRAHIM: We actually did that for one plot. You could see the cloud moving around the plot. JOE SULLIVAN: That's cool. IBRAHIM: And you see the power output for [INAUDIBLE]. ASHLEY: Yeah, it was on the order of-- we had like two billion data points, I think, which was overwhelming. But yeah, it was really cool. Any more questions? Yeah? AUDIENCE: Can you describe a little more what these PV system failures entail? And what happens, and how long does it take to get them back up and running? What has to be done to do that? ASHLEY: You wanna get that one? JARED: Sure. So basically there is a certain capacity that the grid would have. Say, you can compensate for a 6% drop in this case, or a 20% drop, or something like that. So if your system is completely powered by PV, which is not realistic, and you have, say, a 20% drop and nothing to compensate that, you have a blackout. And so we investigated 18%, for instance. So that would be, say, if your grid is a certain percentage of PV and then has natural gas, or coal, or something that you can bring online quickly to compensate a drop in PV. That would be an idea of what a failure is-- if you aren't able to compensate that fluctuation AUDIENCE: And how long [INAUDIBLE]? JARED: How long would a failure last? It depends. If you can't meet the demand-- AUDIENCE: [INAUDIBLE]. JARED: For a PV system, I think the problem is the PV system would come back up right after the cloud was over, but if you can't meet power demand, you've got all kind of protection systems that would trip off, and it would be mess. So I don't think would come back very quickly. ASHLEY: That's a good question, though. JARED: That's a good question. PROFESSOR: It's relevant because you can envision back up power that could kick in really quick, but exhaust itself within the period of the delta t necessary. AUDIENCE: For your definition of intermittency, did you look at the absolute value or just the drop? Because the grid can't deal with excess power as well, and so I was just wondering if you had insight on that. Like if you dumped 60% more power in the demand, there's no way for you to-- JARED: We did the absolute value. So 6% more, 6% less. JASMIN HOFSTETTER: So I'm going to ask you for real data. So do you know where more or less data points would lie for, let's say, PV systems on houses that are like-- with a typical distance in some kind of neighborhood. JARED: I think that would just be you would adjust your geometric mean distance to whatever the distance from the houses are. I don't think our data has to be a solar farm, for instance. I think it could be houses in a neighborhood, for instance. So if they're perfectly connected to the grid, I think that our code would account for that. ASHLEY: This was for eight, right? JARED: Uh-huh. ASHLEY: The right-hand graphic held the number of systems at eight. And so if you had eight houses spread apart by an average of 150 meters, then you would-- and if you considered a year's worth of data-- is it like 250? I just can't see it. So you'd have meantime between failure of 250 seconds, which is four minutes? Doing math under pressure. JARED: Right, but if you have a grid to back that up, it's not big of a deal. AUDIENCE: I'm confused about the plot on the right here. What it's suggesting is that one week you picked was significantly below the year average [INAUDIBLE], and you could have equally picked another week that was significantly above. JARED: Right. This was, I think, just to give the trend. The relationship between the day, and a week, and a year is just the day that we-- I'm sorry. A week, and a month, and a year is just the week we picked, the month we picked. I think you see on some of the other plots that the week and the month actually shift. It's just the year was kind of the average of those. MARK WINKLER: I would assume that areas, or specifically countries, that made large investments in solar would have studied this question in a detailed fashion. Do you know if, for example, Germany or Spain have looked at this problem when it's spread across hundreds of kilometers. ASHLEY: Ibrahim, do you know that one? I think you might be-- IBRAHIM: I was actually very-- we didn't find a lot of literature actually. For wind, there was a lot of data out there, I guess, because the high penetration levels with PV. There were very few studies. Most of them actually were addressing the US. I didn't find any, actually, on Germany or Spain. Probably maybe they're in Spanish or German, so I don't know. JOE SULLIVAN: So what I find really startling is that, for a given system, the time between failure of the 6% intermittency is on the order of a minute. Do you have any the idea-- is that vastly different for wind and what that number is? And this is outside of your-- I'm just wondering if in your literature searching. JARED: You probably know the most about it. ASHLEY: You would know from it. JOE SULLIVAN: It seems like you have this big rotor. There's some momentum, and that to slow that thing down requires more time, but I don't-- as opposed to electrons. JARED: I would say wind would definitely have a much longer time scale than solar, I think. There's a lot of momentum there. RITA: But when wind stops, the times that you have intermittency is going to be much bigger. And there'd be backup systems you need to have to take over for a long period of time. JARED: And maybe in high winds you would have more of an issue, because if the turbine is spinning too fast, you actually have to stop it. So maybe there you'd run into issues of variability on the order of minutes. AUDIENCE: So I think-- and this is kind of going back to location data set-- comes from Hawaii, which I would imagine has mostly direct sunlight. For locations such as Boston, would the data set change for, say, diffuse light and would that generally bring in panels closer together or require more panels at the same geometric distance to get the same results? ASHLEY: Well, the raw data that we have doesn't separate direct and diffuse, so I think that the first thing would be we'd want to look at a data set and from whatever other location you wanted to know about and look at how diffuse and direct differs. I don't think we have a sense here of that effect. Does that answer your question to some degree? AUDIENCE: Some degree. I don't know if someone else wants to add more. JOE SULLIVAN: [INAUDIBLE] after you respond. JARED: I think it's interesting-- I was just thinking about this. Something that might be interesting to investigate is concentrated solar. If it's easier to shade, it would look like a denser system. So maybe that would be-- maybe a concentrated solar farm might be a bad idea if you have lots of little clouds. So that's something I think that you could expand into from this project. IBRAHIM: And another thing, I guess, to add to your point, if you look at, I guess, solar thermal systems probably because of the diffuse sunlight, the intermittency I would expect is going to be probably less. You're going to have less, or the mean time to failure is going to be longer, so you could maybe add a solar thermal system, sort of balance the power output, and decrease your intermittency even further. JOE SULLIVAN: Any last questions? No? All right, let's thank our group. [APPLAUSE] |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | 14_PV_Efficiency_Measurement_and_Theoretical_Limits.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Good. Well, why don't we go ahead and get started. We're going to be discussing photovoltaic efficiency, measurement, and theoretical limits. And there will be plenty of natural breaks over the course of today's presentation for us to have our debate. This is a fun lecture because we start out by talking about how to measure solar cell device efficiency. Later, we will discuss the theoretical efficiency limits of solar cells. Why do we focus an entire lecture on efficiency? Well first, as we discussed previously, efficiency is a very strong determining factor for cost. The rationale, again, is that if you have low efficiency, you're going to need more commodity materials to make a given watt peak. That means you'll need a larger area of solar module to make a certain amount of power, which means you'll need more glass, encapsulance, and so forth. So efficiency is a strong lever determining cost of all downstream components except for the area independent factors like the inverter. Secondly, efficiency is tricky to measure accurately. That is why there are only a few laboratories around the world, a handful, that are certified to measure solar cell efficiencies. These are the efficiencies that could be reported in, say, the efficiency compendiums, an example of which you've just picked up here as one of the readings. The reality is that we can measure efficiency or get a pretty close value for an efficiency of a device within our own laboratories. But there are a number of possible errors that can creep up and nip us in the heel if we're not careful. And that's why we spend some time in today's lecture discussing those potential pitfalls. And thirdly, there are new technologies that are being promised right and left to overcome some of the fundamental limits of traditional solar cell devices, like this one right here. And we have to understand what those limits are so that we can design better ways to overcome them. So learning objectives. Bit of a small font here. But the idea is-- our very first point is to identify the sources of record solar cell efficiencies to understand where one goes to look them up. How do you find the record efficiency of say, a silicon device or a [INAUDIBLE] device? Eventually, we'll talk about measurement of solar cell efficiencies. And finally, the theoretical or fundamental limits to solar cell efficiencies. So the key concepts for-- learning objective number one, to identify a source for record efficiencies. My go to place is a Progress in Photovoltaics, it's a journal and in the PB field. And every six months, PiP comes out with solar cell efficiency tables led by one of their editors, Martin Green, professor at University of New South Wales in Australia. The latest addition that I could find was version 38. I believe this was from July or June this summer. And every six months or so, they come out with a new version. And what you'll find inside of that paper-- this is one of the four handouts that you have today. One of the three articles that you have. What you'll find inside of a typical Martin Green solar cell efficiency table is a listing of-- in table one, a listing of individual cells. In table two of modules. And within the cells, any new record efficiency will be shown in bold. Over the six months preceding the release of the latest version, there were indeed four record efficiencies that had been made. And that's pretty impressive advance. Note one thing which we'll come back to later. Note the plus minus appearing here after the efficiency number. Is anybody surprised at that number? Ashley, did you expect it to be that big? AUDIENCE: No, not that big. PROFESSOR: Not that big, right? AUDIENCE: Yeah. But say the record efficiency of crystal silicon device, 25% plus or minus 0.5. Pretty large delta. We'll explain some of the reasons why that error bar is so large. Another thing to keep in mind. Look at gallium arsenide at 28.1%, just achieved by Alta Devices in March. Keep that number, 28.1 plus or minus 20% in your mind, at least the first one, 28.1 We'll march on to the module efficiency tables right here. So now the gallium arsenide module efficiency record for the efficiency tables right here is 21%, or 21.1%. Crystalline silicon has dropped from 25 to 23 and so forth. And this is fairly typical that record module efficiencies are in the order of 2% to 7% lower than record efficiency cells. Can anybody guess why that might be? AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah. You have a mix of different performers. And when you connect them in series and in parallel, you're going to be limited by the lowest voltage or current, respectively. Yes. OK. So certainly there are mismatches between the individual devices inside of a module. That's where the majority of that comes from for, say, crystalline silicon or discrete monolithic wafers. But how about for some of these thin film devices? They are deposited using these large chemical vapor deposition reactors, for example, or PVD reactors. And you deposit a uniform thin film over a large area, use lasers to cut little trenches in the films and discretize the devices that way. So how come there are differences between record cell and record module for thin film? PROFESSOR: Exactly. So if you have inhomogeneities in thickness or in composition, or even in surface quality from region to region in that large area, you're likely to reduce your performance. One analogy, since I know many of you are mechanical engineers, one analogy to this is when you're doing tensile tests with ceramics or brittle specimens at room temperature, and you pull on your specimen, and you obtain a certain fracture stress, you then take that smaller piece, pull again. Now the fracture stress is higher. Pull on that smaller piece that broke off. Yet again, another fracture stress is even higher. In other words, in that large specimen, there was one point that was extremely weak, another point that was sort of weak, and another point that was mildly weak. And as you increase the size of your specimen, the likelihood of having one of these failure points increases. That's an analogy, let's say, to a large area module, as well, if we could have pinholes or other manufacturing defects inside of a large area module that could reduce the performance locally, and everything is interconnected, it tends to drop the performance overall. So we have the record laboratory efficiencies. This is another reference source. I am not personally aware where Larry Kazmerski publishes this on a regular basis. I know he maintains this table. And if you email him very nicely, he'll email you back with the most updated version. But I'm not aware of any publication outlet where this is regularly appearing. But nevertheless, it captures the record efficiencies versus time the same way that you, too, could do if you went to the Martin Green records. And I believe we're at, was it, number 39 now? 38. We're at Version 38. And if you went back in time to all of the different tables and tabulated the results versus time, you would get a plot that looked very similar to this for each technology. So next up we're going to identify the sources of standard solar spectrum. So we know the record efficiencies. We're taking their word for it right now that they did everything right. Now we're going to learn what that everything is and how to do it right, or at least some of the pieces of doing it right. We're going to identify, first of all, the sources of standard solar spectrum. So in one scenario you could say, well, let me just go outside and measure the solar spectrum. If you go outside today, it's kind of cloudy. The spectrum would look very different than tomorrow. And so it would be very difficult for you to compare the performance of your device against somebody who might be in Germany or somebody who might be in Finland, or somebody who might be in Brazil. So that's why we come up with the standard reference solar spectrum. And it's shown right here. The ASTM standard reference spectrum is shown on the nice NREL website for air mass zero. That's in the outer reaches of our earth's atmosphere and air mass 1.5, which is assumed to be a standard in, say, temperate climates. So there have been numerous revisions to the standard. You might want to use the latest one just to make sure that everything is up to proper spec. The latest standard here, essentially they, in very pedantic detail, walk through all of the possible scattering mechanisms in the atmosphere. And this is their justification for the specific solar spectrum that they're measuring. And so you have a nice explanation in much detail. Let me just show you once again the standard spectrum. You've seen this before. You've worked with it in homework number, I believe it was one or two. And we see AM0, AM1.5 direct, and AM1.5 global, AM1.5 global being the capture of sunlight from a full hemisphere, direct being looking very closely, in a very small solid angle, directly at the sun on a sunny day. Given these atmospheric conditions with a small amount of atmospheric scattering, obtaining an integrated power density, watts per square meter if you integrate over all wavelengths, it's about 90% of the global full hemisphere measurement. Any questions so far about this? OK. So that's pretty straightforward and the measurement devices for the global and direct measurement today. Obviously we didn't have those fancy contraptions 60, 70 years ago in quite the same way, with the same materials and the same design and the same quality of glass. But we had other measurement devices. And you'll see in a few slides how the solar constant has been varying as a function of time. Granted, the solar output also varies, but our ability to measure the solar output varies as well. So if you look at the evolution of the solar constant versus time, this is the number of watts per meter squared in the outer reaches of our atmosphere, so AM0. And you can see that our values have changed. So if you were to ask, is this because of the change of our measurement capabilities or because of the change of the solar output, the people who study the sun will tell you that the variation in solar output is expected to be rather small. So the likely origin of this large fluctuation, pre-1960, is most likely due to our ability to measure the spectrum accurately. So next we'll describe how to simulate the solar spectrum in the laboratory, and we'll describe how a solar simulator works. So great. We have an idealized solar spectrum right here. How do we recreate this in a laboratory environment? How do we obtain a light source that follows this profile exactly? Well, barring the ability to recreate a small fusion source in the laboratory and the ability to introduce exactly the right Fraunhofer lines into our spectrum, we are not going to be able to reproduce that exactly. But we have several techniques that come fairly close. So the solar simulator. This is a schematic coming off of the Newport website. Many of the solar simulators in the laboratories here at MIT, as you'll see as you walk around, be Newport Oriole or related brands, we have the light source back here. Note the type of the lamp right here. It's a xenon arc lamp. Mirrors, essentially a series of optics to create the right form factor of the light and ultimately work our way toward a planar incoming beam. And the what's called spectral correction filter, which is right in the middle of the optics train. And that's essentially to correct for variances between the emission of the xenon arc clamp and the ASTM solar spectrum as defined right here. And we have a shutter, as well, to block the light. For example, if your device or cell or material is photosensitive, you might want to not expose it for long periods of time. There's also-- when I say photosensitive I don't mean that the device will stop working under light, but that the performance will change. There are materials, for example, amorphous silicon that we discussed, where the performance does change as a function of illumination, cumulative illumination intensity. So another interesting thing to note here is that we have quasi-planar light coming in at the end of this optics train. But if we're going to be measuring, for example, concentrating solar cell apparatus, it's much more important to have a higher degree of planarity of the incoming light, and then other optics would be used. One might envision, for instance, increasing the distance between the light source and the actual sample, or the optical path length between the collimating mirror and the sample. So now that we have light approaching our sample right here, so light is incoming on the sample, we have three things, broadly, that we have to worry about. We have to worry about uniformity of the light, the uniformity from small region to region of the illuminated area. We need to worry about spectral fidelity. That means, how closely does the spectrum of our lamp match the ASTM standard solar spectrum? And thirdly, temporal stability. That means if I turn on the lamp this morning and want to take in a measurement immediately, is it stable yet? OK. Let me wait a half hour for things to-- for example, the thermal loads inside of the system to reach equilibrium with environment. Now I'm going to measure it. What if I come back in an hour and a half? Will I still get the same result? What if I come back tomorrow or next month? Temporal stability is another major concern for solar simulators. On the right-hand side is just an example of the radiance versus wavelength of a given light source versus time. And you can see there's a new lamp after a certain working period of 1,200 hours. So we have non-ideal matches, several examples of light sources that don't quite get it right or have several spikes in the output spectrum. So in all cases, here the AM1 direct spectrum is shown in this dash dot line. And different light sources are shown either solid or dashed lines. So not great. And then finally, we reach our xenon arc lamp with air mass filters. And the filters are to suppress certain peaks and certain general portions of the spectrum so that we have an approximation of our ASTM standard reference spectrum. And you notice that it's not perfect. You notice that there are spikes in the output of the xenon arc lamp, as you might expect from the physics involved in the light source. This is of some concern if your device is particularly sensitive to a region, a spectral region where those might be present. So it's not a bad idea to take a measurement of your light source and actually understand how it compares, how it matches up against the ASTM standard spectrum. In terms of standards, or ranking different types of solar simulators, there are three classes according to the IEC standards. There are, as well, other common standards. There's a standard used in Japan, and the ASTM standards as well. But let's focus on the IEC 904-9. These are the requirements for solar simulators measuring crystalline silicon single junction devices. So it's a very specific standard. And in a few slides we'll explain what the potential differences are when you're measuring other types of solar cell materials. We have the spectral match, or spectral fidelity. We have the non-uniformity and the temporal instability in this case, since we're defining a relatively small parameter. Class A solar simulators have relatively tight specs. But you'll still notice here that the non-uniformity of the irradiance plus or minus 2%, the temporal instability plus or minus 2%. This is where you start to see some of those error bars on the record efficiency measurements, right? So if a laboratory has a very good handle on its reference solar simulator, it will be able to calculate these effects and estimate what their impact is on the actual solar cell efficiency measure. Note that the temporal instability for the Japanese standard is a little bit more stringent than the IEC test. Minor detail, but depending on who your collaborators are, where they are in the world, they might be using a different standard than you. Just keep that in mind. The solar simulator downstairs in Building 35, in the laboratory that several of you have seen yesterday when you did the phosphorus diffusions-- I poked my head in and saw everybody there. So about a dozen of you might have walked past the solar simulator in the laboratory. That's a large area, as in the illuminated area at the working plane is around 20 by 20 centimeter squared. And it is a Class AAB solar simulator. So I believe that would be spectral match, non-uniformity, and then temporal stability. So you'll typically see solar simulators rated in this way, AAA, Triple A, or Class B solar simulator, or AAB and so forth. So again, note the significant figures. That's where that comes from. Pretty straightforward. Next we're going to describe how to accurately measure and report cell efficiency and some common pitfalls to avoid when actually measuring the cells. So this is really directed toward people who are doing active research right now in the field of PV. For those who aren't, enjoy. And we'll come back as soon as this little section is over to the debate. And finally some topics of general interest. So this is just a small subset of things to keep in mind when you're measuring an actual solar cell. This is by no means a comprehensive list. There are, indeed, people who spend their entire lives optimizing and perfecting the art of measuring solar cells. So the very first thing that you might want to do is have a reference solar cell encapsulated, and then mailed to you after a measurement is performed at a certification laboratory. So this is an example of a very small crystalline silicon solar cell device inside of an encapsulated frame. And the encapsulation is meant to prevent any degradation to the solar cell, as well as damage that might incur during accidental use. And this cell is called a reference cell. It was tested at NREL. The current voltage, the short circuit current, open circuit voltage, and fill factor of that device is well-known and reported, and is essentially sent with that device back to our laboratory. Now whenever we want to make a new measurement on any cell that we want to measure inside the laboratory, we'll measure the reference cell first just to make sure that our solar simulator is well-behaved. When I say well-behaved, what could happen? Well, one of the many things that could happen is that the lamp intensity decreases. That's probably the most common thing that can happen. You'll just have an overall reduction in the output of your lamp, in which case you'll notice a reduction in what cell parameter? Current, voltage, or fill factor? Current, right? So the light intensity decreases. You'll have a reduction primarily in your current, logarithmic reduction in voltage. So you have your standard calibrated reference cell. This is definitely something that each laboratory that is going to be serious about measuring efficiency should have. This is also an apparatus that doesn't leave our laboratory except for class purposes. It's something we treat very carefully, since I think the cost ran in the few hundreds or thousands of dollars. Avoid extraneous-- basically, avoid light from coming in from outside of your solar simulator. It is often shown, or the solar simulators are often shown in this manner right here, where you have everything out in the open. That's to show you what's going on inside. When you actually take the measurement, typically you have a small black curtain that's light tight around your apparatus, or maybe even a box. Ensure 25 degrees C measurement conditions. Remember that the open circuit voltage can change, depending on the band gap. The larger the band gap, the smaller this effect. The smaller the band gap, the larger this effect. The VOC can change as the temperature changes. And so it's important to have a good handle of your temperature. That means that you might, for instance, have active heating and cooling on your chuck. And you'll also, at the minimum, be measuring the temperature on top of the chuck, where the actual solar cell will sit, not far away removed through some layers of insulation away from the chuck. Even if your temperature is not precisely at 25 degrees C, if you know the temperature dependence of your solar cell, you can correct for it or account for it to pull it back to 25 degrees C. Next, choose your probe locations judiciously to avoid series resistance losses. Let's think that through for a second. So if we have our solar cell device right here and we're going to be measuring its cell efficiency, if I put one little probe right here in the corner, then the current that's being generated over here has to travel a very large distance through a lot of metal to reach that point. But if I have probes that essentially will consist of many different individual metal points, and they come down on either bus bar like shown right here-- where you see the green right here, and off of the green are many little probe tips, individual probe tips that will make contact with the bus bar, so you have essentially one coming down right here, another coming down right here-- your series resistance losses will be much less. But one thing to keep in mind is that when you do have these probes sticking on top of your solar cell device, they will scatter some of the light as well. So there are best practices in terms of what color they should be and how tall they should be, as well. So choosing the probe location is a large step toward achieving high efficiencies. I don't under emphasize that point. It really does make a huge difference where you put your probe tips, especially for cells with high series resistance, which can be several of the new materials that are being developed. Account for spectral mismatch between calibration cell and your cell. Let me drive that point home. So I have a crystal and silicon calibration cell right here. I know its spectral response. I know at what wavelengths it responds the strongest. And so it will detect any mismatch between the ASTM standard and the actual light source there where it responds most strongly. Now let's imagine that I'm designing a new organic material that responds really well. Pick something, the infrared. And so now silicon will stop responding, let's say, at around 1,100 nanometers. So anything beyond here, silicon won't be able to detect. But let's say my device is very sensitive in that spectral region. My silicon calibration cell says go ahead, take your measurement, everything's fine. It matches your ASTM standard. But then when I stick my cell in there, all of a sudden I'm getting a super high efficiency. I just made an organic device, and my efficiency's 11%, which would be a world record. I'm ecstatic. I'm really happy until I do the quantum efficiency of both the standard calibration cell and my new cell, and I realize, wait a second. They're nowhere near each other in terms of their responsivity, what region of the solar spectrum they can respond well at. And the reason I'm bringing that up is, again, to emphasize we have, say, for example, an amorphous silicon device, a gallium arsenide device over here, a SiGs device showing the different regions of the solar spectrum, the solar spectrum shown in this dark orange in the background right here, peaking at around 550. We can see that the different materials are responding to different regions of the solar spectrum. And crystalline silicon would be in between the amorphous silicon-- actually, it would be a little further over, be starting up at around between 1,200 and 1,100 nanometers. So we can see how different materials are more sensitive to different regions of the solar spectrum. And there's even a standard test method for determining the spectral mismatch parameter between your device and the reference cell, an ASTM standard for it. And the typical way to account for this is to measure the spectral irradiance as a function of wavelength of your light source, measure the quantum efficiency, meaning the responsivity versus wavelength of your device, and then measure your calibration cell. And using that math, you can really begin to normalize for these extraneous effects. Why do I bring that up? I bring that up because there was an example, several examples, of folks in the literature-- I've taken off their names, so protecting the innocent here-- folks in the literature who report very high efficiency devices, or at the time was a near-record efficiency device. What they showed were the QE curves of the devices, and then the IV curves of their devices. And you guys, in your homeworks, calculated the short circuit current, which is shown here-- essentially the intercept with the y-axis-- you calculated the short circuit current from the QE. So you know how to perform that calculation. Well, guess what? Folks at NREL also know how to do that calculation, and a whole lot more. So the folks at NREL did that and said, well, wait a second. When I do the integration of your QE, I'm not getting these short circuit currents over here. I suspect what happened was, in your solar simulator, you were using a silicon reference cell. But your cells were more sensitive to a different region of the solar spectrum-- in this particular case they were more sensitive to the shorter wavelengths-- and you didn't have a properly calibrated solar simulator. So you're over reporting your current outputs. And of course that's very embarrassing for a group. In this particular case there were merits on both sides. There was a rebuttal to the rebuttal. So it's not a simple black and white case for this particular story, although the logic does fall more strongly on one side. So this is to say, avoid this sort of controversy. Perform your measurements properly, don't over report your efficiencies, and when in doubt, you can always ship your cells to NREL or [INAUDIBLE] or another certified testing center and get a certified cell efficiency. Then you can place your IV curve inside of your publication, and in the little corner over here it'll have the figure of NREL or [INAUDIBLE] and the properly certified information. Now that can be rather complicated if you're, for example, growing organic materials that degrade quickly. Somehow you have to transport it over there without it degrading. And that's where arranging in advance the transfer of the materials, and perhaps even looking into what sort of transfer chamber you're going to encapsulate your device in could be of interest. But at the very least, please, please, please remember spectral response mismatch when you're doing efficiency measurements. Any questions so far? AUDIENCE: So then does NREL also make non-silicon reference cells? PROFESSOR: So NREL actually doesn't make these cells. They're local companies that live right around NREL that are manufacturing these and putting them together. NREL will test those cells. Since it takes a few months, typically, to get turnaround on a cell efficiency measurement-- unless you're fast tracked in because your device is degrading or you've arranged in advance and kind of put your place in line-- because it takes a few months turnaround, there's actually a premium on inventory. And so these companies will manufacture the devices, send them in, get them tested and calibrated, or get the calibrated measurements performed at NREL, and then bring them back to the company and put them on the shelf until you put pick up the phone and call them and say, I'd like a certified cell. You can actually make your own, too, if you follow a set of standard protocols that you can receive from the folks at NREL. I believe Keith Emery might be a good point contact at first. There are a very specific set of protocols that you should follow if you want to make your own in the lab. In other words, you have to design the contacts a certain way, the encapsule in a certain way, make sure that the materials comprising the remainder are black so that they don't reflect the light back in, little details that are only gathered by experience. If you follow all of those parameters, and the folks over there, Keith Emery or Paul [INAUDIBLE] look at it, and they inspect it and say, yeah, yeah, looks good, then you can get that tested and serve as a calibration cell as well. So you can circumvent some of the cost associated with buying a certified cell from a company. But there is a premium to them. They don't tend to be particularly cheap. So yeah, if you have another material and you'd like to have it calibrated, you can make your own calibration reference standard as long as it doesn't degrade. Question. AUDIENCE: Putting the multiple contact in yours while you're measuring the efficiency. Is that very representative of the devices that actually work in the field? PROFESSOR: So there are a lot of things-- let me spend a minute waxing poetic about the discrepancies between these cell efficiency measurements and what actual cells experience in the field. We understand the logic behind cell efficiency measurements. We understand we have to have a universal way of comparing a cell in Japan, in Boston, in Freiburg, Germany. So we understand that we need some standard method of cell measurement. This cell measurement proves very useful when you measure the output in terms of peak watts, because then you can multiply by the number of peak hours of sunlight per day and estimate the energy output as a function of location on the earth. So it has its uses. Now as to its drawbacks. We're measuring at 25 degrees Celsius. These cells are typically operating around 60. We know that there's a voltage drop, primarily a voltage drop with increasing temperature. That's point one. Point number two, the contacting scheme. Typically on these devices in a module, you'll have soldered contacts in the front, and so they make contact with the entire bus bar, but really, depending on your soldering machine, only in a few locations where there's really a good electrical contact, and on the back in three locations. So yes, there are differences in how the cell is contacted in real life. Could be one of the reasons for discrepancies between module and cell efficiency, probably one of the minor ones compared to homogeneities. Why don't we take a quick little pause right here before we dive into describing the efficiency limitations of a typical cell and get some cool demos at the very end. What we're going to do is have ourselves a quick debate. So I'd like to call forward at the front of the room the representatives of the two teams, one of which is going to debate in favor of the development of novel materials for solar cells. Let me give you an example of one material that's attracted quite a bit of attention, which is pyrite. I can pass this around as folks are coming up to the front. Here's an example of an iron sulfide-based mineral which is purported to have a very high degree of manufacturability because of the large resource abundance, and also the large refining capacity for the respective elemental constituents. There is a gap between the performance of current pyrite cells and their theoretical record efficiencies, quite substantial, the record efficiency being up closer to 20%, the actual efficiency being down at around two. That is not unlike, for example, tin sulfide, or other related materials. I'd say copper zinc tin sulfide is probably the most advanced at around 10%, but still about half of its theoretical limit. And so the big question that we're going to debate is, does it make sense to invest a lot of funds to come up with these earth abundant alternatives for our existing solar cell materials, our cadtel, our copper indium gallium selenide, cognizant that the supply of some of these heavier elements, tellurium and indium, is limited. And we may not have enough of these elements in the earth's crust to scale up to the terawatts level. And so we'll hear two points of view, one in favor of development of new materials, and one against. And just to situate ourselves in a position or a location where these debates actually do happen, you can imagine yourself, for example, in the Office of Science and Technology Policy, which reports directly to Barack Obama. These are about 50 PhDs who are all in an office under the direction of a director, thinking deeply about some of the scientific challenges that our nation faces and the proper scientific response, coordinating amongst many agencies, including the DOD, DOE, NSF, and so forth. And so you can picture yourselves in a debate, in a lively discussion, all friendly, but really with the potential to influence national policy. Do we direct resource funds to develop these novel, earth abundant alternatives that we might need in 10 years' time? Or do we focus and allocate resources elsewhere? So I'll welcome the two participants up to the front. So folks want some insight into where the Office of Science and Technology Policy actually decided to go. A gentleman who took a version of this course in Berkeley in 2003, I think it was, his name is Cyrus Wadia. Graduated with his PhD from Berkeley and actually went off to join Office of Science and Technology Policy. He took his class project for the PB course, which was analyzing alternative materials and published that paper that you read by Cyrus Wadia on resource abundance that was published in 2009, for that work earned himself a TR 35 reward from Tech Review, and then joined the Obama administration's OSTP, and has been developing the Materials Genome Project within OSTP. The first solicitation for proposals was issued, I believe, a week and a half ago. So that's funneled through the NSF. But it's a larger effort to develop some of these materials involving NSFD, DOD, DOE, and so forth. So it actually did come to fruition through OSTP. Budgeting is always the big question, though, because that, of course, gets done through the committees in Congress and ultimately reconciled between the House and the Senate, and has to make it through OMB and finally to the individual directorates. So that's how things actually get done. But it takes the vision of OSTP, sometimes, to drive these larger projects forward. So we'll wish him the best. We're going to describe the efficiency limitations of a typical solar cell now. And what I'll do is I'll pass around some of these books just to, again, situate ourselves. Did I grab Jenny Nelson's on the way out? Thought I did. Hm. OK. Well. Peter Wurfel and-- oh, there it is. Yes, absolutely. So there's two books that are entitled Physics of Solar Cells. You can see it's a very popular topic. And one is The Photovoltaic Handbook. All refer to some aspect of efficiency limits. And as you look through your sheets, you'll see at the bottom, typically the different pages of each of the books are listed here. So the very first thing that we should consider before we get into PV technology, we just have some solar device looking at the sun. And the sun is radiating at it at 6,000 Kelvin, 5,800 Kelvin. And it, the solar contraption on the surface the earth, is radiating back at the sun with black body radiation at 300 Kelvin. So much lower power from Stefan Boltzmann's Law we know exactly the amount of power being emitted by that device. So the two are radiating at each other, and they're in equilibrium. And that results in the very first, we'll call it blackbody efficiency, or maximum solar heat engine efficiency, which would be around 86%. So that's our first fundamental cutoff. Now that 86% is averaged over all wavelengths. There tends to be a wavelength dependence to this as well. So the next step was to say, OK, well, we know that we can't do better than the theoretical thermodynamic limit of one object looking at another. But what is the actual limit of a solar cell? And this was a question that the first developers of the solar cell asked themselves. So Prince was one of the three of the team that in 1954 published on the crystal silicon solar cell device, and very quickly followed it up with another article here in JP focused on the theoretical efficiency limit. And a curve was proposed, looking much like this, with two data points for germanium and silicon, theoretical limits, that is, and power density. So not a conversion efficiency per se, but a power density. And of course, assuming a certain input power density, one can calculate an efficiency from there. And along came-- well, at this time, H-J Queisser had just moved over from Germany to Palo Alto, to The Apricot Barn, and working with Shockley, the esteemed Shockley at the time, to develop a detailed balance model for describing how a solar cell performs, or what its ultimate theoretical efficiency limit could be. Now what a detailed balance model does is just basically accounting, accounting for all of the photons coming in and out of the device. Through the photons, the electron hole pairs are generated. This is assuming a very high quality material that does not have any form of recombination other than radiative recombination. So we talked about the different methods of limiting lifetime. Radiative recombination is one of the methods of limiting performance that occurs in high quality materials. If you have a poor quality material, you'll have non-radiative recombination, say, Shockley-Read-Hall recombination. But in the detailed balance model, only radiative recombination was assumed. That way you can count number of photons coming in and number of photons going out of your device. Furthermore, they assumed that the mobility of carriers was infinite inside of their material. A few laughs over here coming from the folks who are working on organic materials. But it's true. They assumed that the mobility was infinite, such that the separation of the quasi Fermi energies throughout the entire device was equal. So if you had a limited mobility, if you had a certain higher density of carriers in the front than toward the back, you'd have a difference in the chemical potential through the thickness of your device. They assumed infinite mobility. So it was a very simple, yet elegantly insightful model that, upon first submission, was rejected. So the first time they submitted this model for publication it was rejected outright. And it took them another several months of edits, probably about a year, and they resubmitted it. And then it became a sleeper paper. It wasn't cited that much. If you go to the Web of Science, for example, and look at this manuscript, you'll see that the number of citations in the early years was rather limited. Nowadays there isn't a talk about the fundamental efficiency of a solar cell without mentioning the Shockley-Queisser efficiency limit. That's coming directly from this paper right here. And that serves as a motivational story for you. If your paper is rejected, just remember that now you're along with some several esteemed individuals who have set precedence in the field of photovoltaics and are Nobel worthy. So that's my personal opinion. If you fall into that category, don't feel discouraged. Regroup and find a way to make your paper better. You have a copy of this manuscript right here, by the way, in your handouts for today. And interestingly, through their detailed balance model, they obtained an ultimate efficiency versus wavelength, or energy in this case, curve that looks very similar to the curve that you derived in your earlier homeworks, where you just assumed two loss mechanisms. One was non-absorption of light and the other was thermalization of carriers. So the detailed balance model, the way they reached this point, is different, fundamentally different. The physics that was assumed is basically listed out right here. But the end result is fairly similar to the rough back of the envelope calculation that you performed in an earlier homework. So again, just to go up, they assumed that the photons with energies greater than the band gap are absorbed, create one electron hole pair. They assume thermalization. So these were your two assumptions that you did in the homework. Then they do a few additional things. They assume that radiative losses occur within the material, that not every carrier is collected-- that's important-- so you have radiative losses in your device, so that reduces the performance. And then they assume further that there is a thermodynamic loss in their device, in other words, that you don't extract the full band gap of energy, but that there's a thermodynamic loss as you go from band to band, which would be the band gap energy, to the difference or separation of the quasi Fermi energies, which is this delta mu here, which is the change in the chemical potential from the front side to the back side of the device. So there are some additional loss terms that were included. For a full description of the detailed balance limit, and all the math, I definitely encourage you go to this website. As well, Peter Wurfel's text does a wonderful job of describing this. And of course, you have your paper here. You can read through the original paper yourself, or perhaps suggest it at an upcoming journal meeting for your group. Here is the calculation of the detailed balance limit for AM0 and AM1.5, essentially as a function of band gap, coming from the PVCDROM. And you can see how silicon and gallium arsenide are pretty close to the theoretical maximum in terms of the theoretical maximum efficiency for a single band gap semiconductor material. So this curve right here, again, is just representing one band gap material, a single band gap material. So let's take it from here and venture forward into some more realistic performance reduction effects. We can, for instance, take into account recombination mechanisms that aren't only-- for example, that aren't only radiative. We can take Auger recombination. We can take Shockley-Read-Hall recombination into account. We can also take what's called photon recycling into account. So what is photon recycling? Photon recycling is when you have a radiative recombination event, and that photon gets trapped within the material. It's not allowed to escape. But it gets trapped because, for example, off the top, there's an index of refraction mismatch. Or the angle at which it tries to exit is too oblique, and so you have total internal reflection within your device. And you eventually have reabsorption. So that's called photon recycling because you have a radiative recombination event. It emits a photon inside of your solar cell device, and that photon bounces around a few more times until it's reabsorbed, generating another electron hole pair. And that is, essentially, the major boost in some of these ultra thin, high performance solar cell materials such as the Alta Devices record efficiency 28% gallium arsenide cell has the ability to do this photon recycling. Now how do you modify the detailed balance limit to account for finite mobility? I suspect this corner of the room is going to want to hear this. There is a beautiful piece of work done by [INAUDIBLE]. The PhD thesis is even more insightful than the manuscript in terms of actually breaking down each individual component. A great deal of modeling went into this, including photon recycling, including some things that are very difficult to model. And the effect of finite mobility was calculated on a modified detailed balance limit model. And you can see number one, that curve that's starting here at the 20. That is for a mobility that is close to the optimal. If you drop by an order of magnitude, or two orders of magnitude, relative to the maximum mobility inside of a material, your performance begins to degrade considerably. And now you can understand why silicon, or crystalline silicon, which has, say, hole mobilities somewhere in the range of a few hundreds of centimeters squared per volt second. So keep that number in your mind, on the order of hundreds of centimeters squared per volt hole mobility. Now you compare to amorphous silicon, which has something in the range of 10 to the minus 3 to 10 to the minus 1 centimeters per volt second. And you begin to see why those materials with poor mobilities are really impacted in terms of their performance. There is a method to calculate the impact of limited mobility on device performance. And if that intrigues you, I would definitely refer you to this manuscript. One very simple way to think about it is, at least for a band conductor, you would have a certain diffusion length. At least a first order would be the diffusivity times the lifetime and the diffusivity would be related to the mobility inside of your material. So your mobility is factoring into your diffusion length, and the diffusion length is affecting how many charge carriers are collected. So that's a simple way to think about the problem. A similar related approach that kind of pulls all this together in a graphical form is at the very last page-- some of the very last pages of Peter Wurfel's text. And this is going through the efficiency calculation that we just saw, starting from a certain amount of light that's not absorbed. The light that is absorbed-- so that's efficiency loss mechanism number one-- for a thin device, let's say light trapping isn't perfect. We're at around 74% efficiency of light trapping, of absorption of light. Let's say now that we have a thermalization event that results in about a 33% loss, so we're down to 67% efficiency for the thermalization of charge carriers, just a step. So now our combined efficiency is going to be the multiplicative product of those two. So just the thermalization of charge carriers results in a 33% drop of an efficiency. Now there's a delta between, say, the band to band and the actual chemical potential inside of our material once we consider the ensemble of carriers, not just those free carriers, but the ensemble of carriers, which is going to dictate the ultimate potential. So we have some carriers that are excited and others that aren't. These thermodynamic losses results in another 36% drop. And finally, fill factor losses, which represent the solar cell and practical operation. When we have series resistance and so forth, another 11%. So if we multiply these four numbers together, we drop down to 28. And the beauty of doing breaking things out like this-- and what Peter Wurfel does very nicely is dives into each of these in great detail and explains to you exactly how those numbers are derived. The beauty of doing something like this is you can pick the lowest number, say, 64 and 67, and say, I want to work on those. I want to make my PhD thesis about those parameters because that has the biggest impact. And that's what you can do with an analysis like this. Now you can take this further. You can say, well, these are only four of the many parameters that impact performance. I would like to look at many, many more. And that's what one has done for crystalline silicon, which is arguably the most researched and, from the point of view of physical understanding, advanced solar cell technology. We can see a variety of performance loss mechanisms that have been taken into account inside of the crystal silicon devices. This was an invited talk by Dick Swanson, the founder of Sun Power, former professor at Stanford, who pulled this together, a very nice presentation. He speaks with authority because Sun Power produces the highest efficiency crystalline silicon solar cell device commercially. And interestingly, right here, this is the thermodynamic limit and the calculated limit efficiency versus time. So we start from Prince. That was the very first one. And we have Shockley-Queisser up there, and so forth. So the theoretical limit of solar cell performance has also changed with time. And their calculations vary. So it's important to be able to understand this as well as you go into it. And here is the actual best laboratory performance. So if we had, for example, a 25% efficient cell over there, it would already be higher than some of the earlier efficiency calculations for crystalline silicon. What Dick Swanson thinks is the practical limit is right there. That's the Swanson prediction, that we won't get much above 26% for crystalline silicon. So loss mechanisms visualized right here, and several good readings on efficiency limits. What I'm going to do is pause here. There is still some material in your text, and some really cool demos that we're going to have to wait until next week to see. |
MIT_2627_Fundamentals_of_Photovoltaics_Fall_2011 | 18_Cost_Price_Markets_Support_Mechanisms_Part_I.txt | The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Today we're going to dive into Cost, Price, Markets, and Support Mechanisms. The support mechanisms otherwise known as subsidies. So this is Lecture 18. We're approaching the end of our course, actually. We have about, we have a handful of lectures left, and then we go our separate ways. This particular lecture will be followed up not this Thursday but the following Thursday. We'll have a guest speaker come in and talk about the cost model that he's developed for PV into a very high level of detail, so it'll be a lot of fun. And you'll be able to use that cost model to model your own PV devices, apparatus, and so forth. Next Tuesday, a week from today, we'll be touring a PV facility. It'll be here on campus to make it easy for everybody. We'll go over to the student center and tour the PV system up on the roof there, as well the balance of system components. So we'll be able to have a close-up look at how that works. But today, cost, price, markets, and subsidies. We want to talk about those items because, at the end of the day, PV is a product that is competing against bulk electricity. And if we can't compete against the bulk electricity, then our on-grid applications are going to be rather limited. So we want to understand how all this works and fits together. I'll be providing you several snapshots and several pieces of the puzzle with a lot of discussion back and forth over the course of today's lecture. First off, let's dive into PV cost and price. What is the difference between cost and price? They're used interchangeably in colloquial language, but there's a big difference. Jessica? AUDIENCE: Cost might be what it costs the manufacturer, and price is what it's sold at [INAUDIBLE]. PROFESSOR: Absolutely. So cost is what it actually costs to make, to manufacture, and price is what people are willing to pay for it, what the market is demanding. So sometimes price can be above costs-- you're hopefully in that situation most of the time-- and sometimes price can actually be low cost. What is an example of when price could be below cost? AUDIENCE: The Amazon Kindle? PROFESSOR: The Amazon Kindle? Why would it be doing that? AUDIENCE: Because they want get people to adopt the device and then make money on subscriptions to books or [INAUDIBLE]. PROFESSOR: Hm. A loss leader, right? Something like, for example, if you buy your razor handle, and that's really cheap, but then they gouge you on blades. Or other examples include the cheap items in the front of a store. You walk inside the store, and then you're barraged by all the more expensive ones right inside. So loss leading can be one example of price below cost. Another? Other examples? What if I start making a gizmo, and I pocket an enormous profit. And all of you start watching me make that gizmo and say, hey, I can do that. It's pretty simple. It doesn't take a rocket scientist to manufacturer that gizmo. I can do it too. And everybody starts manufacturing gizmos. Pretty soon, we overwhelm the demand, at least at that given price point, and the price is depressed as we enter a price war. We enter what is called an oversupply condition. That's another example where price can fall below cost. And another reason why price can fall below cost is simply the price, or the market you're trying to address, simply won't buy your product at that cost. And that's the case with substitution economics. If we're competing against fossil fuel-based electricity, let's say, and we want to compete against that, we might not be able to manufacture solar panels cheap enough to address certain markets. For example, Wyoming, which has $0.05 per kilowatt hour electricity due to cheap fossil fuel. The southeast of the United States as well, where the TVA, the Tennessee Valley Authority, has very low-priced nuclear and coal power. So these are examples of where price might be below cost. We're going to get more into that over the course of today's lecture, because there are some very interesting geopolitical debates occurring right now. Oftentimes the two sides are very staunch in their positions and there isn't much nuance, there isn't much shade of gray, there aren't many rational arguments presented. And instead, we're going to be diving into some of that, discussing the nuance over today's lecture. Let's dive into cost first up. This is a paper that I presented already in class. I've also steered some of the project groups toward it. This is a proceeding back in-- whoa. This wasn't 2009. My apologies. This is 2003. This was presented at the 3rd World Conference of Photovoltaic Energy Conversion by Tom Surek, presenting a very simple cost model, if you will, for PV, more specifically the impact of efficiency on cost. And by no means was this the first time that something like this had ever been presented, but it was a nice summary of the work to date, highlighting several, I would say, key levers, cost levers. Efficiency-- that's the solar conversion efficiency. Processing costs-- that's the materials and processing costs for the module in dollars per meter squared. The manufacturing yield-- that means out of, say, 100 cells into your manufacturing line, how many make it through to the other side without breaking or being discarded due to manufacturing defects? Capital equipment cost-- that's depreciated over several years, meaning you buy equipment up front, but then due to financial gimmicks, you're allowed to allow that cost to hit your books over an extended period of time, not all at once upfront. Overhead, and so forth-- overhead being the health insurance, if it is paid to the workers, and, of course, R&D and the CEO's salary, and so forth could be lumped in. So this is a very simple way of estimating cost. It's a linear equation, a direct relationship. What is, I would say, the economics, or a more sophisticated way of looking at cost? Other than just saying it's the dollars per watt-peak, you would look at it in terms of cents per kilowatt hour. Right? You would look at it in terms of, how much do you pay for your electricity coming out of the wall? Or in this case, out of the panels? What would factor in to what is called the levelized cost of electricity, when you're actually calculating cents per kilowatt hour? How would you convert dollars per watt-peak-- OK, I know how many dollars it took to manufacture this. I can also depreciate my equipment costs over several years to manufacture this. How would I go from dollars per watt-peak into cents per kilowatt hour? We all agree that cents per kilowatt hours is the metric of importance, right? That's what we pay on our electricity bills, or at least some of us do. So when we pay our electricity bills, we're paying in cents per kilowatt hour from the grid. And when we manufacture our solar panels, we pay in dollars per watt-peak. Let's start simple. Why is dollars per watt-peak at all useful? It's so far removed from cents per kilowatt hour that it almost seems an artificial metric. Why, again, do we use dollars per watt-peak? Jessie? AUDIENCE: Because most coal-based or most fossil fuel-based electricity is based on a capacity factor, which is measured in kilowatts. PROFESSOR: That's a good way of looking at it. And then the capacity factor of solar would be based on what? On the solar resource locally, right? And that might vary from location to location. OK. So what dollars per watt-peak allows you to do is, given a rated nameplate capacity, you can calculate, based on the solar resource locally, how much energy will be produced over a certain period of time. And then from that, you can calculate your cents per kilowatt hour, because now we're converting from power, or rated nameplate power, into energy, which we can use, and which has economic value. So there is a rationale, then, for giving nameplate capacity in terms of watt-peak. In other words, rating a factory in terms of megawatts per year or gigawatts per year produced. That means that each module that goes on to the cell tester is rated, and there's an estimate based on the cumulative module production what the total watt-peak output of that factory was. And then depending on where those modules go in the world, they might produce different amounts of energy. If you take those same models and install them in Alaska or Arizona, you're going to get widely varying energy outputs. OK. So then, how do we transition? We have the dollars per watt-peak. We have to know the local solar insulation that would allow us to calculate what the cents per kilowatt hour would be, assuming a certain cost of capital. We have to buy those panels up front. You have to buy from me a huge number of panels, which are going to last for 20, 25 years. I'll guarantee it. But you have to front that money up front, which means that you need to lend that money from a bank or from a financial institution, and then you'll be paying a certain amount of interest every year. And it's the spread, it's the difference between the interest payments and the money saved that's going to turn your profit. And that's what's called the rate of return of your investment. And there's also a payback period, or a monetary payback period, over which you're not going to be making money on average, but after the payback period is finished, then your solar array will be a money press. It'll be printing money. And so over the entire lifetime of the investment, you can calculate an average rate of return. OK. So given that setup, we're going to be talking about some incentive mechanisms and, today, really, really simple cost model. More on Thursday when we get back to this, how we calculate cost in a more sophisticated manner. I want to introduce this concept very simply at first, just because there are some parameters here in the sensitivity analysis that overwhelm all the others. And they can be very simply seen an equation like that right up there. And one of them is efficiency. The processing costs also matter quite a bit. OK. Inside of processing costs, you also have labor and commodity materials. And those can vary significantly from region to region. There are a number of assessments out there. I'm going to be leading us into the current big debate which has exploded in Washington DC, the case of Solyndra, and then SolarWorld filing the complaint with the US Commerce Department. I'll be easing us into this whole question of US and foreign manufacturing, using that more as a hook to get us interested in the overall topics of the day. And then toward the end, hopefully you'll be able to see, with more shades of gray, what exactly is going on and perhaps formulate opinions of your own. So where is PV cost actually at? Where do the current cost numbers currently stand? It is extremely difficult to get at true cost. Price? Price is easy. You go out there, probe the market, see what people are willing to sell panels at, maybe send a few emails to a few companies saying, hey, I want to install 100 of your panels. How much are you willing to sell them to me? This is what people do. They probe the market. They test it. So price is fairly easy to gauge. But manufacturing cost? If I go up to you and say, hey, can you tell me how much it costs to manufacture panels in your factory? Would you be willing to give me that information? They'd probably say no. Probably say no. Definitely to me, as a professor. Maybe you as students, they might show you a little bit more of how it actually works under the hood, since many people believe in the educational mission. But, by and large, people are fairly secretive about this, because they realize that their stock price is heavily dependent on market perception. And if I go out there and write an article and say, hey, your company produces panels at twice the cost of yours, investors are going to flee your company, especially if it comes out of a university like MIT. And so there are big repercussions associated with the divulgation of cost numbers, and that's why it's very hard to get to the bottom of it. What you find most often are aggregators. These are consultancies that work with several companies. They may be, for example, photon consulting across the river in Boston. It might be Greentech Media consulting branch, which is located here in Cambridge, Massachusetts. There are many consulting companies, and several of them based here in the Boston area. And these consulting companies work with several PV manufacturers and, over time, begin aggregating data and presenting market trends, generalized market trends. A few of them single out specific companies, but most of them just aggregate the data and present general trends. I presented to you this, a few articles as of late. These are relatively recent-- within the last three to four years-- and because there has been so much change over the last three to four years within the PV market, even an article three years ago is highly outdated. So you'll want information that's within the last year if you're going to be using cost and price information for your projects. Even within the last few months. And if any of you have any doubts in that regard, you can come talk to me. So the fully loaded module manufacturing cost is shown here. An estimate, again. An aggregation based on access to multiple companies' data. Polysilicon-- this, of course, being crystalline silicon technology, which accounts for about 85% of the current market. Polysilicon in blue, depreciation in red. What's depreciation? AUDIENCE: Is that the discounting of the capital that we just talked about? PROFESSOR: Exactly. Exactly. So what that means is, I borrow a lot of money to manufacture these panels, to buy the equipment, to buy the factory and set everything up. And then I have to pay interest on my loans. I have to pay interest on the loans, and what I do is, I start writing off the equipment as a loss to the company. So that equipment is going to be useful over a certain period of time, and then we can assume that it's outdated and that I'll need to buy the next generation of manufacturing equipment. So I can begin writing off the value of that equipment over a certain schedule, over a certain period of years. Five, seven, depending on the company, depending on the way they're-- I don't want to say cooking their books, but the way they're manipulating the numbers in the accounting sector. So the depreciation varies. And it varies because the interest rate at which you acquire the loan, and the interest rate-- or the inflation rate of the particular currency in question, is different in different regions. So when you're calculating depreciation both of those things matter, and that's why those numbers can vary from place to place. Materials. Materials cost. These are the materials used to manufacture the module, typically called commodity materials. You would look at them as the extruded aluminum components, the glass in the front side, the encapsulant materials, and so forth. All the fancy stuff that we saw when we visited Fraunhofer last week. Let's see what else. We have labor. That's pretty straightforward to see. If we're manufacturing in China, we are looking at a labor rate, could be as low-- base labor without adding housing and so forth-- the base labor rate could be as low as $2.75 an hour in US dollars. And in US dollars in the United States, we could be looking at labor rates of somewhere in the range of $16 an hour before you start adding in Social Security and benefits and so forth. And there may be different levels of automation in the two different places, which shifts costs from labor into capital equipment. So if you realize that you have a much higher labor cost, you might want to buy more robots to do the manufacturing. And vice versa, if you're in China and you realize, oh my goodness, our labor rates are increasing almost exponentially, definitely super linearly with time. As the country takes off and there is inflation, wage inflation, a company that is trying to project forward five or seven years might say, well goodness, it doesn't make sense for me to flood my manufacturing line with people right now, because that manufacturing line still has to make a profit in five years. And so I'm going to buy more robots now. Even though I might not need it today, I might need in five years. And so there's a bit of risk calculation that gets thrown into this as well. Utilities and overhead. That's pretty straightforward. For looking at utilities, that means the electricity, mostly, to run the lines. It could also be water which is used in the manufacturing process. Note the caveats in several of these studies, especially the one out of Lawrence Berkeley National Laboratory. Manufacturing cost, you can do a number of things with that. You can assume that you're buying your polysilicon, you're buying the polysilicon, manufacturing cells, modules, and systems. Or you can assume that you're buying the cells, and the cells are commodity products equal in price throughout the world, and that you're just manufacturing the modules. So you can cut costs in many different ways, depending on what assumptions you make. The price of the cell on the international market may be very different than the cost to manufacture that cell in your particular country. And cost is almost one of those things that you can torture until it tells you the story you want it to tell. It's one of those things where, if you did these numbers a little bit differently and said, OK, we're buying cells and just making modules, and then adding shipment fees-- let's say transport fees from China to the United States-- China and US might look almost equal. Whereas if you integrate over the entire supply chain and say, OK, I'm going to be manufacturing my polysilicon, then my wafers, then my cells, and finally the modules, then you might start seeing some disparity based on these other parameters shown here. The point being, be careful when you see a cost assessment. Probe their base assumptions and try to understand what their biases and motivations were for presenting that particular comparison. Especially nowadays, where you have more and more parties with vested interests in presenting one story or another. We'll get to that in a few slides. Experience learning curve. Now we're getting into price. We're venturing beyond cost and into the regime of price, which is definitely more easily measurable. And that's why we have some fairly good data going back several decades. And you can see here, this is cumulative sales in gigawatt peak. That means the cumulative number of widgets, in, this case, watt-peaks, produced by solar industries worldwide. And this is a global average module selling price. Not the system, the module. So not the balance of system installation, labor, and so forth, but just the manufacturing of the module. And what we can see here is a general trend over time with a decreasing price. That's good. Sorry, decreasing price with time. That's excellent. What is that called? This curve here, plotted in a log-log scale, where you have a line through the cumulative manufacturing production versus price. What is that curve called? AUDIENCE: Is it an experience learning curve? PROFESSOR: It's an experience learning curve. And you'll get something similar for any high-tech product-- computers, toasters-- as long as the product isn't changing significantly with time. And even sometimes if they are, as is the case with the solar panel. Solar panels back in the 1960s, or the 1970s, rather, looked very different than the solar panels today in terms of the materials and the processes used. And it all falls along this very interesting experience learning curve. So it's extremely tempting to say, oh well, what price do we need to reach to be cost competitive-- or competitive, let's say, with bulk electricity? We need to reach a price of, say, $0.50 per watt-peak? Oh, easy. We'll just project forward and we'll see how much cumulative production is needed, and then we'll subsidize until we get to that cumulative production. And bingo, voila, it'll happen by magic. Well, the reality is that each little bump here along the learning curve was some-- if you look closely, you can kind of see these little bumps here. There were many traumatic events within the industry that forced people to innovate, to produce better technology, whether it's the technology itself, something designed here in the laboratory at MIT or Harvard, or whether it's something innovated on the manufacturing line where they realized, oh, this is a more efficient way of manufacturing the solar panels. It's cheaper. This little bump right here, boom. This represents a period in which prices actually went back up year in and year out. What could cause prices to go up? What are some of the motivations for that? I mean, scale's increasing. The market's growing. It's not shrinking at all. It's not like the points went back here as they went up. So the market continued to grow, the manufacturing capacity continued to grow, but the price went back up. What could have caused that? AUDIENCE: Demand outpaces supply? PROFESSOR: Demand outpaces supply. Exactly. So in this specific case, what happened was the polysilicon feedstock, which is the input material into this entire process, was in short supply. It takes about, in those days it took about 24 to 36 months to get a new plant online. Long lead time and billions of dollars of investment. And so the polysilicon suppliers didn't really want to invest unless they knew photovoltaics was for real. And the PV industry had exhausted the elasticity of the supply market in the polysilicon business. And polysilicon suppliers looked at the situation and said, well, let's let prices go up a little bit. It can't hurt us too bad. We've been starved for several years because of low polysilicon feedstock prices. Let's let this increased demand kind of push prices up a little bit before we really decide what to do. And by the time they decided what to do, there was a lot of them getting in the market all at once, which had this effect. Boom. Now, it's not only the polysilicon, but also the cell manufacturers, the wafer manufacturers, that expanded their capacity during this time between 2007 and 2010. Now today's most recent price point, as I saw it on one of our more trusted websites, put us down at around $1.03, $1.05 per watt-peak. So we're down here. We're well below the historical average trend. We're in an oversupply condition right now. Why did the oversupply condition come about? Well, partly because of this undersupply condition, many people saw an opportunity and said, well, we can address that demand. We can grow in this industry right here. To grow in the industry, you need capital. You need access to finances to expand. What happened in 2008 in Western countries-- and 2008 was more or less when it really hit the fan, if you will, in the United States in particular. What happened in the capital markets? AUDIENCE: Capital was severely restricted becase we're not providing a lot bones. PROFESSOR: Exactly. We had the financial crisis here in the US. So in the United States, there was a pull back of lending. The government stepped in shortly thereafter, realizing that this was going to be an issue. There was the ARRA. Does anybody know what that is? It's not the American Association of Retired People. This is very different. ARRA is the American Recovery and Reinvestment Act, right? This was the Stimulus Act. And this act actually did inject a lot of money, or a lot of capital, into renewable energy projects that began hitting in sometime between 2009, let's say, at the beginning, until 2011. In this period right over here. Anybody hear of the word Solyndra? Yeah? You've heard about it? Solyndra was one of the companies that received funding under the ARRA, or "arra." Under the Stimulus Act. So a lot of interesting things were happening during this brief little period right here. During the mid 2000s, China was getting a lot of bad rap by environmental groups, and the United States as well, for its growing greenhouse gas emissions. There was a growing concern over greenhouse gases culminating in the Copenhagen Discussions, that countries, especially developing countries, had to do more to reduce their greenhouse gas production. And this, of course, invited a wonderful tug of war between a developing country bloc-- the most progressive of the developing countries, let's say, normally described as BRICO, so Brazil, Russia, India, China, and other, including South Africa-- and the developed countries. And you can see perspectives from both, and it's wonderful. The sophistication of that debate in terms of poli-sci arguments was just beautiful to watch. On one hand you had the developing countries that said, well, wait a second. You folks in the developed world, you used a lot of fossil fuels. And the majority of the CO2 emissions that have occurred to date were from developed countries, today. So it's a bit unfair that you're asking us to reduce our CO2 intensity, because that might hamper our own development. Why don't you pay reparations for the CO2 that you've already emitted and help us decrease our CO2 intensity? So there were these beautiful arguments being constructed on both sides of the debate. What China decided to do-- so on the international spectrum, not much happened. And that's fairly-- sadly, it's fairly typical of most of today's-- I'd say the larger the body happens to be, whether it's the federal government or the world institutions, it seems that things happen at a much slower pace the larger the entity is. But at a smaller entity, for example, the state level, which we'll see in a few slides, in the United States, a lot's happening right now. And within China's central government, a lot happened in response to some of that criticism. They said well, there's a point. More a point that fossil fuel emissions really decreases our quality of life. If you look at some of the pollution in our cities, that's not very becoming. We can do better. And furthermore, we can create jobs for the people who are flooding our cities from the countryside looking for economic opportunity. We can create new jobs in this industry. And we'll do it by making capital available to this new nascent industry at a time when it's very difficult to achieve capital or acquire capital in the West, in Europe and the United States. And so that's again happening really in the mid 2000s. And what we'll see in a few slides is the massive growth of the Chinese manufacturing market in response to the availability of capital in those countries. Let me go back to a slide that I presented, I think it was lecture number 1, where we looked at the cumulative production of PV as a function of year. And if we plot this on the log-linear plot, we can read the growth rate off of the slope of that curve. So this was somewhere around 10%, 40%, maybe upwards to 60%, depending on what data points you include in those lines. Interestingly, back in 1990, if we look at the distribution of different technologies, multicrystalline silicon was 1/3, single-crystalline silicon about 1/3, and thin-film technology, namely, pushed by amorphous silicon, was about 1/3 of all manufacturing production. As the market evolved-- sorry about this. There we go. So again, the different technologies, just to situate ourselves, single-crystalline, multicrystalline, and thin films. As the market evolved going into the 2000s, we saw this type of breakdown occur. We have silicon, or crystalline silicon comprising around 85% to 90% of the market, and thin films not growing as fast as crystalline silicon. It was still growing. The market overall is going gangbusters, but crystalline silicon technology was going faster than the others. And in part, this was due to the fact that crystalline silicon technology was a bit cheaper than thin films at the time, largely driven by the efficiency parameter that we've just seen in the previous slides. And turnkey equipment was available. So if you had capital someplace in the world, anywhere in the world, and the labor and commodities were such that you could compete in the market, and the shipping costs weren't extremely prohibitive to get your product to the most interesting markets in the world, you could compete because you could buy turnkey equipment. Even if you knew nothing about how to manufacture a solar cell, knew nothing about semiconductors or solid-state physics, what we've discussed here in this class, you could still go out and buy a turnkey manufacturing line and get technicians to come in and teach you how to manufacture solar cells. That was the beauty of these equipment companies. And so the equipment companies specialized in crystalline silicon technologies. And as such, many of the turnkey lines that grew up out of Greenfield factories, especially in China, Taiwan, and other places around the world, leveraged these turnkey manufacturers to a great extent. So price, markets, and subsidies. We're going to be looking at-- it's a bit of a hodgepodge in the sense, if we're addressing cost, price, and manufacturing all together in one big stew, I think that's useful because the three are interrelated. A price is difficult to set without a cost, and the manufacturing is part of that story and can help us tease apart what exactly is going on right now with Solyndra, with SolarWorld, and so forth. OK. So let's start with customer needs. Just to acquaint ourselves, if we're talking about price, we can't divorce price from our customer. And in terms of what our customer needs are, we have on-grid applications. That's represented by, for example, a solar system on my house, on the student center. These are systems that are tied to the grid and using the grid as a battery to store the excess energy. Off-grid, this represents, for example, the Lighting Africa project here within our class. These are folks who don't have access to an electrical grid and who need to have that electricity there. So while people on the grid are worried about cents per kilowatt hour, people off the grid might be worried about the dollars per hour of light. That might be their metric of merit. So from the customer's perspective, what is their value? What do they get from the solar PV system on the roof? These parameters under here, underneath each of the pictures, represent the value parameter. And again, it's a very cartoonish way of thinking about it. It's a lot more complex when you start getting into the weeds and figure out what the customer actually wants. Reliability factors in, access to the product, repair, reliability, and so forth, factor in as well. There are many factors that add in to value. But what we've done here is emphasize the biggest levers, if you will, and some of the biggest differences between different applications of PV. If you're looking at the solar panels on top of the Toyota Prius, you might not really care about the cents per kilowatt hour, because you're paying $20,000 for your car, so what's a few extra dollars here or there? But you might care about the watts per meter squared, which is really an efficiency parameter. You might want it to satisfy a certain function that it could not do otherwise if the panel was too low efficiency. If you're sending something into outer space, again, you might not care about the manufacturing cost of the panel, , because the shipment costs, in other words, putting it onto the rocket and sending it up in outer space, is $10,000 per kilogram. You might be more worried about, how much does it weigh? What is the specific power of the solar panels? Likewise, if you're installing them in a big-box company, for example at Walmart or Kmart, and you have a big flat roof that isn't very strong, isn't well reinforced, you might care about that parameter as well, or the grams per meter squared. I'm sure that factors in somewhere. Actually, it doesn't. So there are many other parameters that could matter for a particular application. Dollars per meter squared for aesthetics for a building integrated system. I've had architects come to me and say, can you make a yellow solar panel? I'd really like a yellow solar panel. And here I am thinking of the solar spectrum and seeing the biggest, the peak of the solar spectrum being reflected away from the panel into the observer's eye, and I'm thinking, well, I can make it kind of dark yellow, yellowish. OK. How much are you willing to pay? And there are many different parameters here that matter for the customer. This one right here is a concentrator system, so you have these optics that concentrate the light into a small little spot, and they're looking at the watts per millimeter squared. A very small device, how much power does it output? The cost of the actual device is almost irrelevant. It's the power output that matters, because the majority of the cost is sunk in all the commodity materials around it. All right. So we're getting into this because we're going to focus largely on on-grid applications, and I'm going to explain to you why. If you're in the Lighting Africa project, you probably care mostly about that. But the on-grid applications currently comprise 95% of the current market. Substitution economics. We're substituting PV with, or we're substituting fossil fuel-based power, on average in the United States, somewhere around 600 grams of CO2 per kilowatt hour. We're substituting that with PV which is on the order of a factor of 5 to 20 lower. So what types of grid electricity will PV substitute? What does this mean for traditional generation companies, also called gencos? And what is a fair selling price for PV electricity? Very interesting questions. So in terms of markets, this is just a breakdown of off-grid consumer applications and on-grid. When I first got into this, somewhere around here, it was broken down about 50-50, PV on-grid and off-grid. Today, much further down, off-grid applications, probably below 10%. I don't know the precise numbers for you, but it's stayed relatively flat compared to the growth rate of the on-grid. And the reasons for this, we'll see in a couple of slides. This is the value of PV electricity, per a 2008 report. If you look at what is easily monetized-- easily monetized depending on the policy in a particular case-- you have the cost of fuel, the cost of capital, typically for a power plant, the CO2 emissions offset. This is if you have a price of carbon. Notice how small it actually is, leading a lot of people to conclude that maybe carbon pricing isn't the biggest lever for bringing PV onto the grid. That's a whole other discussion. We can have that later. Grid losses. These are the transmission and distribution losses, the difference between having the PV mounted or distributed, the power generation source mounted right on the site of use, as opposed to a centralized location that has to distribute it. These are much more difficult to monetize. And you can add in here health impacts related to emissions, and so forth. You're looking at the security, the reduced risk of having a product producing electricity at a certain known price for 20 years. You're also looking at tax bonuses. These are tricky because tax policy can change. You're looking at uncertainty in your raw feedstock material, the fuel. And the need for backup is something that reduces the value of PV electricity. If you, at some point in the future, will need to back up PV power with something that is more dispatchable and it's still an uncertainty in the power grid, that's something that is reducing the value of installing the PV today. Some might argue that that $0.01 negative is actually a drastic underestimation of the risk associated with backup power needs. But this is a fair look at what the so-called true value of PV electricity might be, which would lead us into pricing. If we know the value, we can enter pricing. And not many people can argue with this, because this is substitution economics. This is just saying, OK, how much does it cost to produce fossil fuel electricity? And let's substitute that. Anything over here in the yellow is what people might say, well gee, if we really look at the true price, or the true cost of fossil fuels, this is what it really costs if we take away all of the subsidies and add in all of the externalities. What's an externality? What's an externality mean? Yeah? AUDIENCE: Is it when the consumer isn't aware of something that the [INAUDIBLE] is doing? PROFESSOR: Yeah. So when the consumer-- or let's put it this way. When the true impact, price impact, is not factored into the selling price. Let's say-- hm. Let's say that I sell you a miracle drug. And it allows you to be 5 times more productive than you are right now. But then every time you have your needs, down the toilet goes a bunch of chemicals that the water plant now needs to filter out, and the water plant begins failing some of its standard tests when they measure water quality. So they add in some more filters into their system. They figure out how to get rid of this compound. And now all of the water treatment facilities around the country begin adding in these new filters, and it costs somewhere on the order of $1 billion. That is an externality. That's something that wasn't factored into the price of selling you that miracle drug that allowed you to be 5 times more productive. It's an example of how everybody pays for the acts or the purchases of a few. And that's the case with energy production as well. So it's difficult to argue in economic terms-- or it's difficult to put a specific price on externalities, because these could be things like premature deaths. And there are statisticians who will calculate the number of people who die prematurely as a result of exposure to mercury or cadmium or some other emission coming off of a fossil fuel-burning plant. But then how do you monetize that? You have to assign a value to a human life and say that a certain economic value was lost, both in terms of productivity cost, but also in terms of the investment due to that person dying. And they do that. In government, there is a value to a human life, and it can vary somewhat from one group to another. But it's very difficult to price in externalities. And that's why oftentimes these things in yellow here are neglected or not considered. For the purposes of today's discussion, we're just going to factor in mostly these parameters right here because we have a greater hold on them. So in terms of PV installations worldwide cumulative, I want to compare and contrast where our customers are versus where our manufacturing is. And we'll get to some really interesting questions associated with that. So look at that. Where would you expect PV to be installed? You'd expect it to be installed in the sunniest places. Why? Because the amount of energy produced is related, proportional, to the amount of solar resource that's available in that spot. But from a customer's point of view, solar resource isn't everything. They're also looking at the cost of displacement. What is the price of electricity I'm paying? And how much of that can I displace with my PV electricity? So there's some more sophisticated concern here. And further, the price of electricity might not only be dictated by the true cost of production of the fossil fuel power plants. There may be a few governments out there that say, well, look at all these externalities. We want to begin factoring those in. We want to provide an incentive for people to produce PV electricity. That's commonly referred to as a subsidy. So if we look at the total installation, EU is really leading the charge here. And we saw on those installation maps, there's not a heck of a lot of sun there in the EU. Germany has a solar radiance similar to what we have here in the northeast of the country. And we know that most of the PV installed in the United States is going into the southwest. Japan, rest of world, USA, a relatively small fraction up there in China. Tiny, tiny little bleep. China's the peach one right up there at the top. So I would say less than 5% of total installed worldwide. Actually this is 1.5%. You can read it off right here, the division. OK. So we have a very interesting perspective here about our customers. This is a breakdown, again, in terms of customers again. A little bit better detail. Instead of just seeing EU, we're looking at a variety of different customers. And this is new installations worldwide. So if, for example, this blue bar goes down, it just means we installed less PV this year than we did last year. It doesn't mean that the total amount on the grid went down. It's just new PV installs. So the blue down here is now Germany. We have these countries that are like flashes in the pan. Italy, for example-- sorry, Spain. This one, Spain, grew up really quick and then disappeared. Italy grew up really quick and then is shrinking. If we extended out to 2011, it would be back to a very small amount. We have USA that's consistently growing. That's a nice healthy market. In Germany, that's growing as well. Wow. What an interesting dynamic. Why did Spain have this flash in the pan and then shrink suddenly? What did they do differently than Germany did? We'll start asking those questions and answering them in a few slides. If you really want to get detailed information about the US, where PV is installed in the United States, one of your MIT colleagues and a bunch of NREL folks got together and put up this beautiful archive, if you will, compendex of as many PV installations as they possibly could in the United States. So that's the website. Unfortunately, I didn't include it in your slides, so you might want to write that down if you're curious about it. Wonderful resource. Again, NREL, National Renewable Energy Laboratory based in Golden, Colorado. And you can see the install distribution throughout the United States. Again, a bias toward states that have high electricity prices, like New York and Massachusetts, and lots of sun, like Arizona. The price of electricity is rather low in Arizona. It's below $0.10 per kilowatt hour, but the amount of sun available is very high. Whereas California has both. OK. So we talked about the customer. Now here we're talking about the manufacturers. So we have the customers, the demand; the manufacturers, the supply. Obviously, the manufacturers are growing at the same rate as our customers are installing. Even faster, mind you. Inventory rates are almost, somewhere between 25% and 50% nowadays. So the production has grown faster than the demand has, at least at current prices. What we've done right here, or what is done right here is a normalization by market share, just to demonstrate how the market dynamics are changing as the industry grows. So from 2005 to 2009, this doesn't mean that the industry stayed flat. It continued growing at a breakneck pace, but the countries which comprise the manufacturers have changed significantly. So let's look at Europe first off. Europe pretty much started really feeling it during the financial crisis. They couldn't keep expanding in 2008, and then when prices really started to drop, 2009, as we saw, they didn't continue expanding. Were a bit uncompetitive. US, shown here in the green, again, dropping. India, I guess sort of growing now. It's definitely on the upswing. Japan decreased considerably, considerably. During the 1990s and the early 2000s, Japan had the largest solar company in the world, Sharp, better known for microelectronics that you might find around your house. Somehow, some way, the executives at Sharp saw this coming. Saw a huge rise in demand coming from Europe, and ramped up capacity in Japan. And when they did that, they were able to address large portions of that. They made a healthy profit for several years, and then they just stopped expanding in PV. Part of it might have been they saw the market dynamics changing. They saw their costs, their manufacturing costs, relative to, for example, China-- this is the orange right here-- and Taiwan, above China, above the orange-- and comprised today-- this is 2009 numbers, if we fast-forward to today in 2011, China and Taiwan comprise 55% to 60% of the PV industry worldwide. Production, manufacturing, production. So what does that mean? Let's explore together some of the interesting results of that. Let's look at some of the good things. What are some of the good things about production going to China and Taiwan from, say, Europe and the United States, of solar panels? Let's look at that for a second. So we'll look at the glass half full perspective. What are some of the good things? AUDIENCE: It's more likely that these developing countries will use solar panels? PROFESSOR: More likely that the developing countries will use solar panels? Was that the case over here? Not really. It's going. It's going. OK. I suppose, if the technology's available there. Maybe the technology isn't quite cost competitive with local electricity yet, so there's isn't that demand pull locally. Maybe the realization is that, well, goodness, we can address this European market. They're willing to pay a lot more for the panels than we are, so we might as well export right now. But at some point in the future we might be able to satisfy internal demand. Yeah. And there are gigawatt plants, PV installs going up in China right now. In part due to increased electricity demand, in part due to weakened demand elsewhere in the world, and a very large manufacturing base in China that has to put their panels out somewhere. So there is, yes, adoption. I'll cede that point. Yeah? AUDIENCE: Lower cost to the consumer? PROFESSOR: Lower cost to the consumer, yeah. So in the United States, if you look at the price of installing a PV system in the United States, the price of installing the PV system is around $5.20 on your roof today. The cost of buying the module is around $1.03, $1.05, from China. And so that means that 80% of the profit margin right now is being gobbled up by a US company. The price came down-- let's see, when I installed the panels on my roof, I got a little bit of a better deal. But I would say the average price for a PV system in 2007 must've been somewhere around $8 per watt-peak. And now the price is down at around $5.20. That said, in Europe, in Germany, which has 10 times more installed PV than the US does, the price of installing a PV system on your roof could be below 3 euros per watt-peak. So the price is lower because the profit margin that the installers are getting right now is smaller. So yes, absolutely. Lower cost product in US markets. That isn't the whole story about what you're going to pay, because there's still the installer that's stuck in between the Chinese manufacturer of the module and you serving as the middle person. Providing you value, still, but still extracting a very large profit right now, a disproportionately large profit, shall I say. What are some of the other good things. Yeah? AUDIENCE: If cost in big markets is reduced, then that could lead to more market penetration, and then the [INAUDIBLE] adopted more, and even if prices go up, it might maintain [INAUDIBLE]. PROFESSOR: Yeah. And so what you're leading to here is really what the demand curve of PV looks like. And in the past, if you look at what people are willing to pay for PV, let's say-- let's convert it into dollars per watt-peak, since that's the universal unit of PV cost. And this is the, let's call it total market size in terms of watt-peak, and this being a very, very large number. What you can do to-- first order. Just how would we construct the demand curve for the United States for PV. How would we go about doing that? How much are people willing to pay for their PV modules to offset the cost of electricity? Well, to do it right, we need the levelized cost of electricity analysis. We'd assume we're borrowing money from the bank and do those fancy economics that involve something to the power of something, and that being the interest rate, and then we calculate it through. Just a first order. Hand-wavy Mickey-Mousey. We might take into consideration the manufacturing cost of the module. Sorry. Back up one step. We might take into consideration the cents per kilowatt hour of the electricity that we're getting from the grid and the insulation, the solar resource, that is available at that particular location. So if we have data granular to the state level of the price of electricity on average throughout the state, and the solar resource availability on average throughout the state, we can immediately comprise 150 markets in the US. Residential, commercial, industrial for 50 states. And then we'll have a demand curve, a very simple one, for how much people are willing to pay for their PV electricity. And it looks more or less like this, if you start working it out. Rough sketch. Rough sketch. This is Hawaii. Those poor critters over there, although they have beautiful sun and enjoy a wonderful life, they're paying a lot for their energy, because they're on a few rocks out there. They have no natural resource under the ground to speak of, except geothermal, I suppose. But they're shipping in a lot of their fuel. That's why if you've ever gone to Hawaii and rented a car, you are surprised at the sticker shock when you go off to the gas station and try to refuel. Their price of electricity is about $0.30 per kilowatt hour residential. But it's a very, very tiny market. So you're not going to be able to satisfy much demand. You're not going to be able to produce too many panels there. You start having interesting things happen when you start hitting the bigger markets, like the tiers four and five of California, Texas, New York. These are big markets that have a lot of people in them, and they have larger electricity prices and/or large solar resource available there. And so at some point, the demand curve reaches these plateaus, where if you decrease the price even a little bit, of a sudden, voom, the amount of market you can address for this amount of price decrease, the amount of market that you can address is huge. And so the slope of this line, the slope of the demand curve, is indicative of what happens when you reduce your price just a little bit. And so yes, producing cheaper panels, when you start hitting these plateaus, can result in massive, massive demand pull, or market pull. So that's another good reason to have cheap panels. Let's look at the flip side. What would be some of the downsides, let's say, to module manufacturing going into China? You can assume anything is on the table. CO2 emissions, jobs, et cetera. Let's start teasing into some of those questions and looking at some data. I might flip back and forth during the presentation if we have to address specific topics. AUDIENCE: Bad politics if American photovoltaic manufacturers fail, and it reflects poorly on the industry and the American political scene and decreases support for that. PROFESSOR: Yeah. So bad politics. In DC, this is often referred to as the optics of the situation. How people observe, or how people perceive something. The litmus test is not DC itself. I was down there on Thursday, and everybody in the solar space was-- I think it was Wednesday and Thursday, yeah. Everybody in solar space was freaking out about a particular event that was going on in Capitol Hill. I was telling folks, don't worry. It's not going to-- I mean, in terms of the rest of the country and the perception of the rest of the country, it won't be that significant. And I'm sure Secretary Chu will do a phenomenal job up there in front of the congressional panel. He did phenomenal. And so people were really, really worried about something that didn't have too big of an effect outside. Granted, what people should be worried about is the impact, for example, that desequestration of the discretionary funds will have on R&D once they start kicking in in 2013. If you mandate a reduced funding level for funding agencies, you will have an impact, and a long-term one, for that matter. So the optics of the situation. There has been a slight decrease in public support, I believe on the order of 5% to 10% since the beginning of the whole Solyndra affair, support for solar renewables. But the support remains high, and still continues to be high among Republicans and Democrats alike. Especially when compared to, say, congressional approval ratings. But let's not look down. Let's compare ourselves to higher. What else? What are some of the potential negatives? Let me start guiding you, because you hear so much about bad this-- jobs, whatever. Let me guide you into some of the questions you might not have thought of. There is a colleague of ours here at MIT, Tim Gutowski's group, that is looking into matters concerning manufacturing of PV and installation of PV from a CO2 perspective. So if you manufacture your PV in a coal-rich country that is spewing out emissions into the air, you are going to have a greater CO2 intensity per kilowatt hour of energy that the panels will produce over their lifetime than if the panels were produced, say, in a country that has a large abundance of hydropower or geothermal. Or, hey, even produces solar panels using other solar panels. That'd be nice. But in the growth situation where solar is rising, or the manufacturing capacity is rising at 60% a year, it's a little bit unfeasible to think that solar will power itself forward. It almost violates one of the laws of thermodynamics. So we look at the CO2 balance, right? We think about, gee, what if we were to be smart about where we manufacture and install PV? Then it might make more sense to manufacture PV, say, in Norway, which is based on hydropower, and install them in China, which is largely running on coal. So from the CO2, from the global carbon perspective, again, these are externalities that aren't being priced in. OK? CO2 and-- yeah. Go ahead, please. AUDIENCE: But wouldn't they actually be priced in Norway, because they are in the new emissions trading scene? I mean, they're not priced in China, but the cost of electricity in Norway doesn't [INAUDIBLE]. PROFESSOR: Yeah. So point conceded. In certain countries, some of the externalities are factored in. Not, perhaps, at the levels they should be, but some are factored in, whereas in other countries, they're not at all. So we have what is referred to in some terms as an uneven playing field in the sense that in some places there are regulations in place that increase the price of electricity, whereas they aren't present in other places. And in some cases, they work to your advantage, like in Norway, for instance, or Switzerland. Big hydro countries. What are some other potential disadvantages? You hear a lot about jobs, jobs moving overseas. Let's start looking at some of the numbers next Thursday, when we have Doug coming in here and talking about the cost model. A good number to keep in mind, a good number to keep in mind for the manufacturing of modules-- so back to this part right over here-- for the module itself, typical numbers would be 2 to 12 people, depending on what region of the world you're in, 2 to 12 people per megawatt. So if you have a total market of, say, 10 gigawatts-- it's larger than that. We'll do quick math. So if it's-- the megawatt would be 20,000 people per gigawatt, about 10 gigawatts, somewhere in that range, on the low end. And on the high end it would be somewhere around 200,000 people per gigawatt, on the high end of the labor intensity. So the high end of the labor intensity represents a company like Trina Solar in China that is extremely labor intensive. It's located in Changzhou. It's on the high-speed rail between Shanghai and Nanjing. The low end of that scale would be more representative of, say, Suntech, which is almost next door in Wuxi, which has adopted a different approach, saying, we're going to invest in robots because we're uncertain about where the whole labor thing is going to go in China, where the prices are going to go. Or REC, which is a Norwegian company, but they opened a factory in Singapore, which has very, very low labor rates, because they invested a lot of money in capital equipment, in robots to move modules around their factories. US, if you can look at turnkey manufacturing lines, are typically round 3, 3.5 people per megawatt. Those are some good numbers to keep in mind. So when anybody comes to you and says, jobs, solar jobs, with these sorts of numbers you can begin estimating, OK, give me a number. How big is the solar market going to grow to? What is the level of automation? What is the labor intensity of manufacturing of the modules? And you can make an estimate on the total number of jobs, direct jobs, direct jobs associated with the module manufacturing. Then there are all the indirect jobs as well. There are all the people supplying the commodity materials into your manufacturing line. There all the people who are working as administrative assistants and R&D staff and so forth. So it's not quite as simple as that, but it gives you a number to really start grabbing. And that, of course, doesn't include all the installations, installer jobs. All right. So we're talking about grid-tied electricity. I'd like to move on to our next few topics here. We're talking about grid-tied electricity. This is-- ooh, it's in German. Sorry about that. This is over a period-- "tag der woche" means "day of the week." And this is basically the peak power in gigawatts, instantaneous at any given point. So we have morning, middle of the day, night, middle of the day, night. There's a little bump right here during prime time TV. And we have the output of PV going from 5 gigawatt peak to 30 gigawatt peak, you can see, beginning to eat into the profits, essentially the peak hours. So let me differentiate between base load and peak. Base load power would be voom, right? Something providing a constant power output as a function of time, something like a nuclear power plant, coal fired power plant. Peakers would be natural gas fired power plants that receive the call. It's like, Jessica, you're in charge of the peaker over there. Based on my weather report today, I'm going to need so many gigawatts tomorrow between the hours of 10:00 and 12:00. You think you can turn yourself on then? I'll pay this amount. Yep. Jessica's going to battle. Yes, I can do that. PV, on the other hand, is coming on if the sun's coming on. And to date, there isn't a great degree of predictability. We can see if that's going to change, based on one of our class projects here. But there wasn't a great degree of predictability. You can see here, for example, and here varying amounts of sunlight from one day to the other as a result of changing insulation. There's, of course, changing demand as a day of the week goes by. On Saturday and Sunday there's less demand for electricity. So what PV is doing is eating into some of the highest-use periods, which is good from the market's perspective-- from the, shall we say, the "person in charge of maintaining grid stabilities" perspective, because they want power to be produced then. They don't want to run out of generation capacity. And PV is avoiding the situation, or at least prolonging the situation a few more years, until we run out of power generation capacity on the grid. As our population grows and our energy intensity grows, we're needing more and more power, but we're not building new power plants at that rate, generally, in Germany and the United States. And so PV is helping to defray some of that deferred investment in new generation capacity. So that's good. On the bad side, somebody who has invested-- let's say Mary's invested in a natural gas generation plant. She put the money down for Jessica's plant. Jessica's running it. And now Mary realizes, well goodness, I thought I was going to get a steady rate of return over 20 years from my natural gas fired power plant. That rate of return was factored into my economic analysis when they borrowed money at a certain rate from a bank, and now you're telling me that this upstart technology, this new PV stuff, is coming online and is going to take away some of my profit. That doesn't work for me. As a matter of fact, on peak days in Bavaria, which is the southeastern part of Germany, today PV can comprise as much as 40% of electricity on the grid during peak hours during the summer. And what that does sometimes is force conventional electricity producers to either turn off or go into negative pricing, meaning they shunt the power to ground so that they don't have to pay to put their electricity on the grid. So that's an interesting market dynamic that's evolving as more and more PV electricity enters the grid in these regions of high penetration, which include California, Hawaii-- the island of Lanai, I believe, has up to 40% as well on weekdays. Bavaria in Germany, which is the southeastern part of Germany. Let me get to this point right here. Predicting where the market will go, we talked about the demand curve for PV, and we collapsed both insulation and price of electricity from the grid into this dollars per watt-peak figure, which is how much people are willing to pay. We can also break it out into annual solar energy yield. This is the solar resource available. It's using the units of kilowatt hours per kilowatt peak, meaning the number of kilowatt hours, the amount of energy, that a unit of rated power, kilowatt peak, of the solar panel is going to produce. So how much energy will a unit of solar panel produce? That's down here. And of course, larger amounts mean more sun, more solar resource. On the ordinate, we have average power price of household. This is US dollars. It should be average electricity price, US dollars per kilowatt hour. So that's how much people are paying for the electricity, what you're displacing. And you can see that these little bubbles here represent the size of the market. So as the price of PV electricity comes down, we are able to address more and more markets. As the subsidization of PV electricity increases, you move these little bubble up. And as the subsidization comes back down, the market pull-- not the manufacturing subsidies but the use or installation subsidies-- as they come back down, the little bubbles move down, and eventually rest at their so-called true market price without the externalities factored in. And so you can see how we're beginning to enter a regime where PV is starting to look interesting in a lot of places. And as we begin addressing these markets, or hitting these markets, you can imagine two Gaussian curves intersecting, one being the price of PV electricity, the other being the price of grid electricity, as they intersect and overlap. You expect an exponential growth, and that's what we're seeing in the market today. Question? AUDIENCE: What is the California [INAUDIBLE] for tier 4 and tier 5? PROFESSOR: Yeah. So California decided to implement more of a-- well, it is a little bit regressive in that sense. But what they did is, they said let's, instead of paying the same price for electricity for everybody-- which actually it would be even the opposite way. You know how residential, we typically pay higher rates than industrial? That's because industrial buys in bulk. They buy electricity Costco size. We just buy it corner-store size. And so they pay less for their electricity. And California realized this and said, well, this is a perverse incentive. What this does is it gives an incentive to people to use more electricity. Because if you buy in Costco size, you pay less per kilowatt hour than if you buy in the corner store. Instead, what we're going to do is reverse that incentive by penalizing the people who buy more electricity, so your rate is going to be higher then if you just bought a small amount. So what they did is, the government bureaucrats got together with some of the scholars and decided, these are reasonable amounts for a residential house to consume. This is the next tier up, this is the next tier up, this is next tier up, and this is completely unreasonable. And so what they did is they tiered their electricity. And you have California tiers 1, 2, and 3 represented by the big bubble, and tier 4 and tier 5 as represented at higher pricing. So if you happen to fall into the tier 5 category, you could be paying upward of $0.30 per kilowatt hour for your electricity. And it's a way of penalizing you for powering two swimming pools in your backyard along with, I don't know, something in the range of 10 kilowatts of lighting inside of your house. Whereas, if you're running on a meager budget, you might do much better. Do you have a question over here? AUDIENCE: Is it only residential? PROFESSOR: California's tier 1, 2, 3, and 4, and 5? That a great question. Joe, would you happen to know if commercial and industrial fall into tiers as well, or if it's just residential? AUDIENCE: My guess would be no, because that would be political suicide. PROFESSOR: Well, even the residential is political suicide. AUDIENCE: Yeah. The residential system is actually kind of a bad one, because if you have like eight people living in a house versus two, it doesn't take that into account. So it's not a per capita thing. [INAUDIBLE]. PROFESSOR: So many states and countries have tried to adjust incentive mechanisms. And I'm not sure specifically about the commercial and industrial. You might want to jot it down and give a quick Google search after. Germany. Growing gangbusters. The market versus time. This is installed capacity annually versus year. And again, subsidies, support mechanisms, tax breaks, incentives. What we wanted to emphasize here was that Germany is not one of the sunnier places in the world. Look at this. It's almost half of the modulus installed, and yet it has less insulation than even here in the US northeast. Even the sunniest places in Germany. This is Bavaria. This is the place that hits 40% of electricity on peak summer days coming from PV. And this is where we are, right up there. Same scale. So something's going on in Germany, and it's not sun. The German government instituted a number of-- this little D represents Deutschland, Germany-- implemented a number of incentive programs to increase the amount of PV installed on the grid, under the assumption that if they had demand pull, they would have a supply push. Meaning, they would start up local industries. Initially it didn't quite take off as fast as they hoped, so they began providing direct manufacturing support, direct subsidies to install PV manufacturing plants, to the point where they were willing to pay $0.50 on the euro for each piece of capital equipment that moved in on German soil to produce plants in the mid 2000s. So that combination of events left them with a booming market, upwards of 100,000 jobs, I believe, in Germany related to PV manufacturing, and a lot of PV installed on their grid as a result of the market pull. So really, when it comes to subsidies and support, you have to decouple what is market pull, which means we want PV installed on our grid to offset our country's carbon emissions, versus manufacturing push, which is to say, you want to set up a factory in my country? Fine. Come right in. I'll provide you the finances. I'll provide you a lower cost of capital on your loan, or help you by providing a direct grant or loan. And these are two different support mechanisms, and that's why you see that dichotomy between Germany, which installs a lot of PV and has a decent amount of manufacturing; the US, which has a lot of installation but not a whole lot of manufacturing traditionally, although it's changed, the support has changed over the last three years; and China, which has a heck of a lot of manufacturing but not a whole lot of local demand. It's because each country has adopted a different approach based on optimization of different functions. In terms of where a lot has happened on the state level, these are RPS, renewable portfolio standard, policies with solar and distributed generation provisions distributed throughout the United States. And you can see that the states have picked up where the federal government lacked in terms of leadership. In terms of getting PV on the grid, the states really pulled hard. In, specifically, New Jersey-- some really oddball states, like New Jersey here, installing about 5 gigawatts of solar. California had another more of a market pull mechanism. Rebate programs for renewables, this is if you install solar panels you might get a few thousand dollars back from the state government. And it just gives you a sense of how the states have filled in that void where the federal government didn't step in. And this has come, in part, to our detriment. It's nice, because we can allow-- the federalist approach allows individual states to say, hey, we prioritize green jobs. All right. Let's support it. Let's make it happen. And that's certainly Massachusetts' take. California, Texas, certain other states throughout the US, including Mississippi most recently. But the downside of that is that you have each state doing its own thing, each municipality, each local electric utility grid doing its own thing. And what can happen is diversification of support mechanisms and incentives to the point now that if you want to install panels on your roof in the United States, there over 18,000 ways you could do it within the US alone. 18,000 unique types of paperwork that you have to fill out, depending on where you are in the United States. In Germany, you have one unified paperwork for the entire country, and it's very short. I believe it's two pages. And in the United States, you can fill out a few reams of paper. One of our friends in California once estimated it takes about $1 a watt to fill out paperwork and to get certifications and so forth. Maybe it was a little bit of an exaggeration, but the price is high. And so recently the DOE has attempted to address this as well by fostering a "race to the top" type incentive mechanism for states to facilitate installation. But still, in Massachusetts, we had-- remember for the system on my roof, we had at least two inspectors come by from different agencies. That could have been consolidated into one. So while state independence is good, and local utility independence is great, from the perspective that the federal government is lacking in terms of moving forward, the states can push forward at their comfortable speed. On the downside, it makes it very difficult for an installer company to reduce its costs because it has to address a multitude of local markets throughout the US. And that adds to cost. In terms of projections, I really hesitate. Because in 2006, there were a number projections of where we'd be in 2010, and many of them-- you can see the range here, going from 4 gigawatts up to, actually, I think Photon Consulting had us close to 30 gigawatts back in 2010. Photon Consulting was closer to right than most of the other groups, interestingly. The 4 gigawatts was obviously pretty far off. So projections are very dangerous because you could be easily wrong. This is just to show you an example. These are projections of where the solar market is going to go. From the EPIA, which is the European Photovoltaics Industry Association, it's one of the better sources that I could find with the various scenarios giving you an indication, a tornado plot, if you will, of where the market is going to head over the future. And the bottom line is that most likely it will continue to grow. Even though it's going to enter a difficult year, a difficult two years as the oversupply condition works itself out of the market, we're headed for some pretty massive growth. And just keep those numbers in mind. If this grows up to be, say, a 100-gigawatt industry, and there's two people per megawatt, that's 200,000 people employed just in the manufacturing of modules. Not counting all the other people on top of that, maybe another order of magnitude, if we use the car industry as an example. So we could be looking at two million jobs worldwide. And where they are, who works on them, how the value chain is distributed between R&D, manufacturing, this is all open. The next decade will decide that. You'll help decide that. So with that message, I let you go. |
World_History_TED_Talks | The_incredible_history_of_Chinas_terracotta_warriors_Megan_Campisi_and_PenPen_Chen.txt | What happens after death? Is there a restful paradise? An eternal torment? A rebirth? Or maybe just nothingness? Well, one Chinese emperor thought that whatever the hereafter was, he better bring an army. We know that because in 1974, farmers digging a well near their small village stumbled upon one of the most important finds in archeological history: vast underground chambers surrounding that emperor's tomb, and containing more than 8,000 life-size clay soldiers ready for battle. The story of the subterranean army begins with Ying Zheng, who came to power as the king of the Qin state at the age of 13 in 246 BCE. Ambitious and ruthless, he would go on to become Qin Shi Huangdi, the first emperor of China after uniting its seven warring kingdoms. His 36 year reign saw many historic accomplishments, including a universal system of weights and measures, a single standardized writing script for all of China, and a defensive barrier that would later come to be known as the Great Wall. But perhaps Qin Shi Huangdi dedicated so much effort to securing his historical legacy because he was obsessed with his mortality. He spent his last years desperately employing alchemists and deploying expeditions in search of elixirs of life that would help him achieve immortality. And as early as the first year of his reign, he began the construction of a massive underground necropolis filled with monuments, artifacts, and an army to accompany him into the next world and continue his rule. This magnificent army is still standing in precise battle formation and is split across several pits. One contains a main force of 6,000 soldiers, each weighing several hundred pounds, a second has more than 130 war chariots and over 600 horses, and a third houses the high command. An empty fourth pit suggests that the grand project could not be finished before the emperor's death. In addition, nearby chambers contain figures of musicians and acrobats, workers and government officials, and various exotic animals, indicating that Emperor Qin had more plans for the afterlife than simply waging war. All the figurines are sculpted from terracotta, or baked earth, a type of reddish brown clay. To construct them, multiple workshops and reportedly over 720,000 laborers were commandeered by the emperor, including groups of artisans who molded each body part separately to construct statues as individual as the real warriors in the emperor's army. They stand according to rank and feature different weapons and uniforms, distinct hairstyles and expressions, and even unique ears. Originally, each warrior was painted in bright colors, but their exposure to air caused the paint to dry and flake, leaving only the terracotta base. It is for this very reason that another chamber less than a mile away has not been excavated. This is the actual tomb of Qin Shi Huangdi, reported to contain palaces, precious stones and artifacts, and even rivers of mercury flowing through mountains of bronze. But until a way can be found to expose it without damaging the treasures inside, the tomb remains sealed. Emperor Qin was not alone in wanting company for his final destination. Ancient Egyptian tombs contain clay models representing the ideal afterlife, the dead of Japan's Kofun period were buried with sculptures of horses and houses, and the graves of the Jaina island off the Mexican coast are full of ceramic figurines. Fortunately, as ruthless as he was, Emperor Qin chose to have servants and soldiers built for this purpose, rather than sacrificing living ones to accompany him, as had been practiced in Egypt, West Africa, Anatolia, parts of North America and even China during the previous Shang and Zhou dynasties. And today, people travel from all over the world to see these stoic soldiers silently awaiting their battle orders for centuries to come. |
World_History_TED_Talks | 히틀러는_어떻게_힘을_얻을_수_있었을까_알렉스_겐들러_안토니_하자드Alex_Gendler_Anthony_Hazard.txt | How did Adolf Hitler, a tyrant who orchestrated one of the largest genocides in human history, rise to power in a democratic country? The story begins at the end of World War I. With the successful Allied advance in 1918, Germany realized the war was unwinnable and signed an armistice ending the fighting. As its imperial government collapsed, civil unrest and worker strikes spread across the nation. Fearing a Communist revolution, major parties joined to suppress the uprisings, establishing the parliamentary Weimar Republic. One of the new government's first tasks was implementing the peace treaty imposed by the Allies. In addition to losing over a tenth of its territory and dismantling its army, Germany had to accept full responsibility for the war and pay reparations, debilitating its already weakened economy. All this was seen as a humiliation by many nationalists and veterans. They wrongly believed the war could have been won if the army hadn't been betrayed by politicians and protesters. For Hitler, these views became obsession, and his bigotry and paranoid delusions led him to pin the blame on Jews. His words found resonance in a society with many anti-Semitic people. By this time, hundreds of thousands of Jews had integrated into German society, but many Germans continued to perceive them as outsiders. After World War I, Jewish success led to ungrounded accusations of subversion and war profiteering. It can not be stressed enough that these conspiracy theories were born out of fear, anger, and bigotry, not fact. Nonetheless, Hitler found success with them. When he joined a small nationalist political party, his manipulative public speaking launched him into its leadership and drew increasingly larger crowds. Combining anti-Semitism with populist resentment, the Nazis denounced both Communism and Capitalism as international Jewish conspiracies to destroy Germany. The Nazi party was not initially popular. After they made an unsuccessful attempt at overthrowing the government, the party was banned, and Hitler jailed for treason. But upon his release about a year later, he immediately began to rebuild the movement. And then, in 1929, the Great Depression happened. It led to American banks withdrawing their loans from Germany, and the already struggling German economy collapsed overnight. Hitler took advantage of the people's anger, offering them convenient scapegoats and a promise to restore Germany's former greatness. Mainstream parties proved unable to handle the crisis while left-wing opposition was too fragmented by internal squabbles. And so some of the frustrated public flocked to the Nazis, increasing their parliamentary votes from under 3% to over 18% in just two years. In 1932, Hitler ran for president, losing the election to decorated war hero General von Hindenburg. But with 36% of the vote, Hitler had demonstrated the extent of his support. The following year, advisors and business leaders convinced Hindenburg to appoint Hitler as Chancellor, hoping to channel his popularity for their own goals. Though the Chancellor was only the administrative head of parliament, Hitler steadily expanded the power of his position. While his supporters formed paramilitary groups and fought protestors in streets. Hitler raised fears of a Communist uprising and argued that only he could restore law and order. Then in 1933, a young worker was convicted of setting fire to the parliament building. Hitler used the event to convince the government to grant him emergency powers. Within a matter of months, freedom of the press was abolished, other parties were disbanded, and anti-Jewish laws were passed. Many of Hitler's early radical supporters were arrested and executed, along with potential rivals, and when President Hindenburg died in August 1934, it was clear there would be no new election. Disturbingly, many of Hitler's early measures didn't require mass repression. His speeches exploited people's fear and ire to drive their support behind him and the Nazi party. Meanwhile, businessmen and intellectuals, wanting to be on the right side of public opinion, endorsed Hitler. They assured themselves and each other that his more extreme rhetoric was only for show. Decades later, Hitler's rise remains a warning of how fragile democratic institutions can be in the face of angry crowds and a leader willing to feed their anger and exploit their fears. |
World_History_TED_Talks | 책의_진화ㅣ쥴리_드레이프스Julie_Dreyfuss.txt | What makes a book a book? Is it just anything that stores and communicates information? Or does it have to do with paper, binding, font, ink, its weight in your hands, the smell of the pages? Is this a book? Probably not. But is this? To answer these questions, we need to go back to the start of the book as we know it and understand how these elements came together to make something more than the sum of their parts. The earliest object that we think of as a book is the codex: a stack of pages bound along one edge. But the real turning point in book history was Johannes Gutenberg's printing press in the mid-15th century. The concept of moveable type had been invented much earlier in Eastern culture, but the introduction of Gutenberg's press had a profound effect. Suddenly, an elite class of monks and the ruling class no longer controlled the production of texts. Messages could spread more easily, and copies could constantly be produced, so printing houses popped up all over Europe. The product of this bibliographic boom is familiar to us in some respects, but markedly different in others. The skeleton of the book is paper, type, and cover. More than 2,000 years ago, China invented paper as a writing surface, which was itself predated by Egyptian papyrus. However, until the 16th century, Europeans mainly wrote on thin sheets of wood and durable parchment made of stretched animal skins. Eventually, the popularity of paper spread throughout Europe, replacing parchment for most printings because it was less expensive in bulk. Inks had been made by combining organic plant and animal dyes with water or wine, but since water doesn't stick to metal type, use of the printing press required a change to oil-based ink. Printers used black ink made of a mixture of lamp soot, turpentine, and walnut oil. And what about font size and type? The earliest movable type pieces consisted of reversed letters cast in relief on the ends of lead alloy stocks. They were handmade and expensive, and the designs were as different as the people who carved their molds. Standardization was not really possible until mass manufacturing and the creation of an accessible word processing system. As for style, we can thank Nicolas Jenson for developing two types of Roman font that led to thousands of others, including the familiar Times Roman. Something had to hold all this together, and until the late 15th century, covers consisted of either wood, or sheets of paper pasted together. These would eventually be replaced by rope fiber millboard, originally intended for high quality bindings in the late 17th century, but later as a less expensive option. And while today's mass produced cover illustrations are marketing tools, the cover designs of early books were made to order. Even spines have a history. Initially, they were not considered aesthetically important, and the earliest ones were flat, rather than rounded. The flat form made the books easier to read by allowing the book to rest easily on a table. But those spines were damaged easily from the stresses of normal use. A rounded form solved that issue, although new problems arose, like having the book close in on itself. But flexibility was more important, especially for the on-the-go reader. As the book evolves and we replace bound texts with flat screens and electronic ink, are these objects and files really books? Does the feel of the cover or the smell of the paper add something crucial to the experience? Or does the magic live only within the words, no matter what their presentation? |
World_History_TED_Talks | 북아메리카_탄생_배경_피터_J_하프로프_Peter_J_Haproff.txt | The geography of our planet is in flux. Each continent has ricocheted around the globe on one or more tectonic plates, changing quite dramatically with time. Today, we'll focus on North America and how its familiar landscape and features emerged over hundreds of millions of years. Our story begins about 750 million years ago. As the super continent Rodinia becomes unstable, it rifts along what's now the west coast of North America to create the Panthalassa Ocean. You're seeing an ancestral continent called Laurentia, which grows over the next few hundred million years as island chains collide with it and add land mass. We're now at 400 million years ago. Off today's east coast, the massive African plate inches westward, closing the ancient Iapetus Ocean. It finally collides with Laurentia at 250 million years to form another supercontinent Pangea. The immense pressure causes faulting and folding, stacking up rock to form the Appalachian Mountains. Let's fast forward a bit. About 100 million years later, Pangea breaks apart, opening the Southern Atlantic Ocean between the new North American Plate and the African Plate. We forge ahead, and now the eastward-moving Farallon Plate converges with the present-day west coast. The Farallon Plate's greater density makes it sink beneath North America. This is called subduction, and it diffuses water into the magma-filled mantle. That lowers the magma's melting point and makes it rise into the overlying North American plate. From a subterranean chamber, the magma travels upwards and erupts along a chain of volcanos. Magma still deep underground slowly cools, crystallizing to form solid rock, including the granite now found in Yosemite National Park and the Sierra Nevada Mountains. We'll come back to that later. Now, it's 85 million years ago. The Farallon Plate becomes less steep, causing volcanism to stretch eastward and eventually cease. As the Farallon Plate subducts, it compresses North America, thrusting up mountain ranges like the Rockies, which extend over 3,000 miles. Soon after, the Eurasian Plate rifts from North America, opening the North Atlantic Ocean. We'll fast forward again. The Colorado Plateau now uplifts, likely due to a combination of upward mantle flow and a thickened North American Plate. In future millennia, the Colorado River will eventually sculpt the plateau into the epic Grand Canyon. 30 million years ago, the majority of the Farallon Plate sinks into the mantle, leaving behind only small corners still subducting. The Pacific and North American plates converge and a new boundary called the San Andreas Fault forms. Here, North America moves to the south, sliding against the Pacific Plate, which shifts to the north. This plate boundary still exists today, and moves about 30 millimeters per year capable of causing devastating earthquakes. The San Andreas also pulls apart western North America across a wide rift zone. This extensional region is called the Basin and Range Province, and through uplift and erosion, is responsible for exposing the once deep granite of Yosemite and the Sierra Nevada. Another 15 million years off the clock, and magma from the mantle burns a giant hole into western North America, periodically erupting onto the surface. Today, this hotspot feeds an active supervolcano beneath Yellowstone National Park. It hasn't erupted in the last 174,000 years, but if it did, its sheer force could blanket most of the continent with ash that would blacken the skies and threaten humanity. The Yellowstone supervolcano is just one reminder that the Earth continues to seethe below our feet. Its mobile plates put the planet in a state of constant flux. In another few hundred million years, who knows how the landscape of North America will have changed. As the continent slowly morphs into something unfamiliar, only geological time will tell. |
World_History_TED_Talks | 알리야_부터_제이지_까지_힙합_역사에서_잡혀진_순간들_조나산_매니언_Jonathan_Mannion.txt | This is The Notorious B.I.G., 1995, the Palladium nightclub, New York City. What really I want to talk about is my dedication, my 100% focus, and finding something that I love, my passion point. I fought to be on this stage, to be able to stand next to Lil' Kim and take all these pictures to create one definitive photo. A photo that was the most important of anything that I took. And I really wanted it for future generations to be able to feel the energy that was in that room, and certaintly you can feel that right there. I'm a photographer, this is what I do for a living. I chase these moments, these fractions of seconds, that will never be the same again. You can't take this picture again because I took it, and he's no longer with us, sadly, rest in peace. The definitive portrait of that person, in that moment, is what I strive for every single time out, like when you close your eyes and think of a picture of Jay-Z, I want it to be my picture. So far, so good. Eight album covers for Jay-Z later, I'm not doing too bad. Aaliyah. Gorgeous and amazing, and I really spent proper time with her, conversing, having a conversation, connecting to her, which I find is a big part of my work. That connection, to be able to have a conversation, to say you're doing great, to say why I want a certain picture and a certain attitude. In this moment, we talked about shooting in the Caribbean, and it's sort of a passion point for me, Caribbean culture, Trinidad, Barbados, Jamaica, and I said, "We should do a photo shoot there because I think it would be incredible." And she said, "Yeah, you know, let's do it." You know, as we're taking these beautiful pictures, all this is happening, I'm continuing to converse with her and really connect. So, she believed in my vision at that moment, you know, even while all the work was happening there was a human connection that happened. At the end of the day, I have people sign Polaroids just as kind of a diary for myself. And she wrote, "I can't wait for the Caribbean. I'll see you then." (Exhales) You've got to take a breath. You've got to take a breath. She's missed incredibly. And I think what the reminder is that these moments are really precious, and you really have to take the time to connect to them, to be part of that process, to make a difference in people's lives in that moment. You never know how you're affected by somebody, clearly I am, you can see it, but you never know how you affect that person, how you make that moment a little more important for them. This is my good friend Drake. I had the opportunity to work with him for the FADER magazine. He is a beast and one of my favorites. He's my friend. I shot him for three different days, two in New York, and he had just signed his record deal. He said, "What's most important to me is going home to Toronto to celebrate with my family." So, we went home. We flew home. And he said, "I've got to do a pit stop. I've got to go see my grandmother in the afternoon, and so you probably don't want to bother with that." I said, "It would be an honor to meet your grandmother." I asked for what I wanted. I asked for access that nobody else had because this is what makes a photographer sort of greater, to have a picture that somebody else doesn't have. We want these unique moments. I asked for these moments, right? And I'm reminded of a story. He said, "Grandma, I just got millions of dollars. I just signed a record deal. I've got millions of dollars from Cash Money Universal and Young Money." And she said, "A million dollars?" He said, "No Grandma, millions of dollars. What do you want?" And she said, "I want a kiss, and I want a hug." Again, the reminder of, like, why we do this. It also revealed for me another layer of Drake as a character, and how important this guy is, you know, his message. He's not just about the limelight and self-serving. He's connected to these moments as much as I was connected to this moment as a photographer. I know everybody does this everyday, they ask DMX to get in a pool of blood. This was my challenge this day. I needed to get him to see my vision. I was very clear in what I wanted for "Flesh of My Flesh, Blood of My Blood," which was his album cover. You know, I envisioned photographs of him in this pool of blood, just making these things happen, you know? And what was difficult was he didn't see it the same way I saw it, right? He walked in, and he had these pants on. Brand new pants. Everybody feels fresh when they have brand new pants on, right? He said, "I'm not getting in that pool of blood." I said, "Oh yes, you're going to get in that pool of blood." You know? And he said, "No, my dogs, I'm not getting in that pool of blood." And I said, "My dude, you're going to get in that pool of blood." And in a bold statement, using all the psychology knowledge that I had from my schooling at Kenyon College, I dropped my pants in front of 40 people on a set that all were, like, stunned. DMX laughed. He said, "Alright, my dude, put your pants back on, and I'll get in the blood." One of the most epic photos that we've ever created in this hip-hop movement. I'm on home soil. I'm in New Orleans. We've got to give love to Lil Wayne. Lil Wayne is an incredible MC, and the biggest point that I want to drive home here is about the opportunity to create somebody's legacy with them, to take pictures, to see them grow. It was about trust in the moment and understanding that you could make a difference, have a communication, see somebody elevate and move forward, to take a variety of different pictures, you know? Him in a spacesuit. He's a Martian, you know, what can I say, you know? But it was about seeing his growth, and having an important role and communication with this guy in the moment. DJ Quick once said, "Isn't it incredible how you make people see your vision in hip-hop? The way that people look at hip-hop is through your eyes." I'm really, really proud of what I've created, what I've done, and passionately working to make quality work constantly. It's not about taking a photo. For me, it's about giving a photo to people that believe in it so much. Thank you. |
World_History_TED_Talks | 비단길_역사_최초의_세계교역망_샤논_헤리스_카스텔로Shannon_Harris_Castelo.txt | A banker in London sends the latest stock info to his colleagues in Hong Kong in less than a second. With a single click, a customer in New York orders electronics made in Beijing, transported across the ocean within days by cargo plane or container ship. The speed and volume at which goods and information move across the world today is unprecedented in history. But global exchange itself is older than we think, reaching back over 2,000 years along a 5,000 mile stretch known as the Silk Road. The Silk Road wasn't actually a single road, but a network of multiple routes that gradually emerged over centuries, connecting to various settlements and to each other thread by thread. The first agricultural civilizations were isolated places in fertile river valleys, their travel impeded by surrounding geography and fear of the unknown. But as they grew, they found that the arid deserts and steps on their borders were inhabited, not by the demons of folklore, but nomadic tribes on horseback. The Scythians, who ranged from Hungary to Mongolia, had come in contact with the civilizations of Greece, Egypt, India and China. These encounters were often less than peaceful. But even through raids and warfare, as well as trade and protection of traveling merchants in exchange for tariffs, the nomads began to spread goods, ideas and technologies between cultures with no direct contact. One of the most important strands of this growing web was the Persian Royal Road, completed by Darius the First in the 5th century BCE. Stretching nearly 2,000 miles from the Tigris River to the Aegean Sea, its regular relay points allowed goods and messages to travel at nearly 1/10 the time it would take a single traveler. With Alexander the Great's conquest of Persia, and expansion into Central Asia through capturing cities like Samarkand, and establishing new ones like Alexandria Eschate, the network of Greek, Egyptian, Persian and Indian culture and trade extended farther east than ever before, laying the foundations for a bridge between China and the West. This was realized in the 2nd century BCE, when an ambassador named Zhang Qian, sent to negotiate with nomads in the West, returned to the Han Emperor with tales of sophisticated civilizations, prosperous trade and exotic goods beyond the western borders. Ambassadors and merchants were sent towards Persia and India to trade silk and jade for horses and cotton, along with armies to secure their passage. Eastern and western routes gradually linked together into an integrated system spanning Eurasia, enabling cultural and commercial exhange farther than ever before. Chinese goods made their way to Rome, causing an outflow of gold that led to a ban on silk, while Roman glassware was highly prized in China. Military expeditions in Central Asia also saw encounters between Chinese and Roman soldiers. Possibly even transmitting crossbow technology to the Western world. Demand for exotic and foreign goods and the profits they brought, kept the strands of the Silk Road in tact, even as the Roman Empire disintegrated and Chinese dynasties rose and fell. Even Mongolian hoards, known for pillage and plunder, actively protected the trade routes, rather than disrupting them. But along with commodities, these routes also enabled the movement of traditions, innovations, ideologies and languages. Originating in India, Buddhism migrated to China and Japan to become the dominant religion there. Islam spread from the Arabian Penninsula into South Asia, blending with native beliefs and leading to new faiths, like Sikhism. And gunpowder made its way from China to the Middle East forging the futures of the Ottoman, Safavid and Mughul Empires. In a way, the Silk Road's success led to its own demise as new maritime technologies, like the magnetic compass, found their way to Europe, making long land routes obsolete. Meanwhile, the collapse of Mongol rule was followed by China's withdrawal from international trade. But even though the old routes and networks did not last, they had changed the world forever and there was no going back. Europeans seeking new maritime routes to the riches they knew awaited in East Asia led to the Age of Exploration and expansion into Africa and the Americas. Today, global interconnectedness shapes our lives like never before. Canadian shoppers buy t-shirts made in Bangladesh, Japanese audiences watch British television shows, and Tunisians use American software to launch a revolution. The impact of globalization on culture and economy is indisputable. But whatever its benefits and drawbacks, it is far from a new phenomenon. And though the mountains, deserts and oceans that once separated us are now circumvented through super sonic vehicles, cross-continental communication cables, and signals beamed through space rather than caravans traveling for months, none of it would have been possible without the pioneering cultures whose efforts created the Silk Road: history's first world wide web. |
World_History_TED_Talks | How_inventions_change_history_for_better_and_for_worse_Kenneth_C_Davis.txt | Transcriber: tom carter Reviewer: Bedirhan Cinar This is the story of an invention that changed the world. Imagine a machine that could cut 10 hours of work down to one. A machine so efficient that it would free up people to do other things, kind of like the personal computer. But the machine I'm going to tell you about did none of this. In fact, it accomplished just the opposite. In the late 1700s, just as America was getting on its feet as a republic under the new U.S Constitution, slavery was a tragic American fact of life. George Washington and Thomas Jefferson both became President while owning slaves, knowing that this peculiar institution contradicted the ideals and principles for which they fought a revolution. But both men believed that slavery was going to die out as the 19th century dawned, They were, of course, tragically mistaken. The reason was an invention, a machine they probably told you about in elementary school: Mr. Eli Whitney's cotton gin. A Yale graduate, 28-year-old Whitney had come to South Carolina to work as a tutor in 1793. Supposedly he was told by some local planters about the difficulty of cleaning cotton. Separating the seeds from the cotton lint was tedious and time consuming. Working by hand, a slave could clean about a pound of cotton a day. But the Industrial Revolution was underway, and the demand was increasing. Large mills in Great Britain and New England were hungry for cotton to mass produce cloth. As the story was told, Whitney had a "eureka moment" and invented the gin, short for engine. The truth is that the cotton gin already existed for centuries in small but inefficient forms. In 1794, Whitney simply improved upon the existing gins and then patented his "invention": a small machine that employed a set of cones that could separate seeds from lint mechanically, as a crank was turned. With it, a single worker could eventually clean from 300 to one thousand pounds of cotton a day. In 1790, about 3,000 bales of cotton were produced in America each year. A bale was equal to about 500 pounds. By 1801, with the spread of the cotton gin, cotton production grew to 100 thousand bales a year. After the destructions of the War of 1812, production reached 400 thousand bales a year. As America was expanding through the land acquired in the Louisiana Purchase of 1803, yearly production exploded to four million bales. Cotton was king. It exceeded the value of all other American products combined, about three fifths of America's economic output. But instead of reducing the need for labor, the cotton gin propelled it, as more slaves were needed to plant and harvest king cotton. The cotton gin and the demand of Northern and English factories re-charted the course of American slavery. In 1790, America's first official census counted nearly 700 thousand slaves. By 1810, two years after the slave trade was banned in America, the number had shot up to more than one million. During the next 50 years, that number exploded to nearly four million slaves in 1860, the eve of the Civil War. As for Whitney, he suffered the fate of many an inventor. Despite his patent, other planters easily built copies of his machine, or made improvements of their own. You might say his design was pirated. Whitney made very little money from the device that transformed America. But to the bigger picture, and the larger questions. What should we make of the cotton gin? History has proven that inventions can be double-edged swords. They often carry unintended consequences. The factories of the Industrial Revolution spurred innovation and an economic boom in America. But they also depended on child labor, and led to tragedies like the Triangle Shirtwaist fire that killed more than 100 women in 1911. Disposable diapers made life easy for parents, but they killed off diaper delivery services. And do we want landfills overwhelmed by dirty diapers? And of course, Einstein's extraordinary equation opened a world of possibilities. But what if one of them is Hiroshima? |
World_History_TED_Talks | The_history_of_our_world_in_18_minutes_David_Christian_TED.txt | First, a video. Yes, it is a scrambled egg. But as you look at it, I hope you'll begin to feel just slightly uneasy. Because you may notice that what's actually happening is that the egg is unscrambling itself. And you'll now see the yolk and the white have separated. And now they're going to be poured back into the egg. And we all know in our heart of hearts that this is not the way the universe works. A scrambled egg is mush -- tasty mush -- but it's mush. An egg is a beautiful, sophisticated thing that can create even more sophisticated things, such as chickens. And we know in our heart of hearts that the universe does not travel from mush to complexity. In fact, this gut instinct is reflected in one of the most fundamental laws of physics, the second law of thermodynamics, or the law of entropy. What that says basically is that the general tendency of the universe is to move from order and structure to lack of order, lack of structure -- in fact, to mush. And that's why that video feels a bit strange. And yet, look around us. What we see around us is staggering complexity. Eric Beinhocker estimates that in New York City alone, there are some 10 billion SKUs, or distinct commodities, being traded. That's hundreds of times as many species as there are on Earth. And they're being traded by a species of almost seven billion individuals, who are linked by trade, travel, and the Internet into a global system of stupendous complexity. So here's a great puzzle: in a universe ruled by the second law of thermodynamics, how is it possible to generate the sort of complexity I've described, the sort of complexity represented by you and me and the convention center? Well, the answer seems to be, the universe can create complexity, but with great difficulty. In pockets, there appear what my colleague, Fred Spier, calls "Goldilocks conditions" -- not too hot, not too cold, just right for the creation of complexity. And slightly more complex things appear. And where you have slightly more complex things, you can get slightly more complex things. And in this way, complexity builds stage by stage. Each stage is magical because it creates the impression of something utterly new appearing almost out of nowhere in the universe. We refer in big history to these moments as threshold moments. And at each threshold, the going gets tougher. The complex things get more fragile, more vulnerable; the Goldilocks conditions get more stringent, and it's more difficult to create complexity. Now, we, as extremely complex creatures, desperately need to know this story of how the universe creates complexity despite the second law, and why complexity means vulnerability and fragility. And that's the story that we tell in big history. But to do it, you have do something that may, at first sight, seem completely impossible. You have to survey the whole history of the universe. So let's do it. (Laughter) Let's begin by winding the timeline back 13.7 billion years, to the beginning of time. Around us, there's nothing. There's not even time or space. Imagine the darkest, emptiest thing you can and cube it a gazillion times and that's where we are. And then suddenly, bang! A universe appears, an entire universe. And we've crossed our first threshold. The universe is tiny; it's smaller than an atom. It's incredibly hot. It contains everything that's in today's universe, so you can imagine, it's busting. And it's expanding at incredible speed. And at first, it's just a blur, but very quickly distinct things begin to appear in that blur. Within the first second, energy itself shatters into distinct forces including electromagnetism and gravity. And energy does something else quite magical: it congeals to form matter -- quarks that will create protons and leptons that include electrons. And all of that happens in the first second. Now we move forward 380,000 years. That's twice as long as humans have been on this planet. And now simple atoms appear of hydrogen and helium. Now I want to pause for a moment, 380,000 years after the origins of the universe, because we actually know quite a lot about the universe at this stage. We know above all that it was extremely simple. It consisted of huge clouds of hydrogen and helium atoms, and they have no structure. They're really a sort of cosmic mush. But that's not completely true. Recent studies by satellites such as the WMAP satellite have shown that, in fact, there are just tiny differences in that background. What you see here, the blue areas are about a thousandth of a degree cooler than the red areas. These are tiny differences, but it was enough for the universe to move on to the next stage of building complexity. And this is how it works. Gravity is more powerful where there's more stuff. So where you get slightly denser areas, gravity starts compacting clouds of hydrogen and helium atoms. So we can imagine the early universe breaking up into a billion clouds. And each cloud is compacted, gravity gets more powerful as density increases, the temperature begins to rise at the center of each cloud, and then, at the center, the temperature crosses the threshold temperature of 10 million degrees, protons start to fuse, there's a huge release of energy, and -- bam! We have our first stars. From about 200 million years after the Big Bang, stars begin to appear all through the universe, billions of them. And the universe is now significantly more interesting and more complex. Stars will create the Goldilocks conditions for crossing two new thresholds. When very large stars die, they create temperatures so high that protons begin to fuse in all sorts of exotic combinations, to form all the elements of the periodic table. If, like me, you're wearing a gold ring, it was forged in a supernova explosion. So now the universe is chemically more complex. And in a chemically more complex universe, it's possible to make more things. And what starts happening is that, around young suns, young stars, all these elements combine, they swirl around, the energy of the star stirs them around, they form particles, they form snowflakes, they form little dust motes, they form rocks, they form asteroids, and eventually, they form planets and moons. And that is how our solar system was formed, four and a half billion years ago. Rocky planets like our Earth are significantly more complex than stars because they contain a much greater diversity of materials. So we've crossed a fourth threshold of complexity. Now, the going gets tougher. The next stage introduces entities that are significantly more fragile, significantly more vulnerable, but they're also much more creative and much more capable of generating further complexity. I'm talking, of course, about living organisms. Living organisms are created by chemistry. We are huge packages of chemicals. So, chemistry is dominated by the electromagnetic force. That operates over smaller scales than gravity, which explains why you and I are smaller than stars or planets. Now, what are the ideal conditions for chemistry? What are the Goldilocks conditions? Well, first, you need energy, but not too much. In the center of a star, there's so much energy that any atoms that combine will just get busted apart again. But not too little. In intergalactic space, there's so little energy that atoms can't combine. What you want is just the right amount, and planets, it turns out, are just right, because they're close to stars, but not too close. You also need a great diversity of chemical elements, and you need liquids, such as water. Why? Well, in gases, atoms move past each other so fast that they can't hitch up. In solids, atoms are stuck together, they can't move. In liquids, they can cruise and cuddle and link up to form molecules. Now, where do you find such Goldilocks conditions? Well, planets are great, and our early Earth was almost perfect. It was just the right distance from its star to contain huge oceans of liquid water. And deep beneath those oceans, at cracks in the Earth's crust, you've got heat seeping up from inside the Earth, and you've got a great diversity of elements. So at those deep oceanic vents, fantastic chemistry began to happen, and atoms combined in all sorts of exotic combinations. But of course, life is more than just exotic chemistry. How do you stabilize those huge molecules that seem to be viable? Well, it's here that life introduces an entirely new trick. You don't stabilize the individual; you stabilize the template, the thing that carries information, and you allow the template to copy itself. And DNA, of course, is the beautiful molecule that contains that information. You'll be familiar with the double helix of DNA. Each rung contains information. So, DNA contains information about how to make living organisms. And DNA also copies itself. So, it copies itself and scatters the templates through the ocean. So the information spreads. Notice that information has become part of our story. The real beauty of DNA though is in its imperfections. As it copies itself, once in every billion rungs, there tends to be an error. And what that means is that DNA is, in effect, learning. It's accumulating new ways of making living organisms because some of those errors work. So DNA's learning and it's building greater diversity and greater complexity. And we can see this happening over the last four billion years. For most of that time of life on Earth, living organisms have been relatively simple -- single cells. But they had great diversity, and, inside, great complexity. Then from about 600 to 800 million years ago, multi-celled organisms appear. You get fungi, you get fish, you get plants, you get amphibia, you get reptiles, and then, of course, you get the dinosaurs. And occasionally, there are disasters. Sixty-five million years ago, an asteroid landed on Earth near the Yucatan Peninsula, creating conditions equivalent to those of a nuclear war, and the dinosaurs were wiped out. Terrible news for the dinosaurs, but great news for our mammalian ancestors, who flourished in the niches left empty by the dinosaurs. And we human beings are part of that creative evolutionary pulse that began 65 million years ago with the landing of an asteroid. Humans appeared about 200,000 years ago. And I believe we count as a threshold in this great story. Let me explain why. We've seen that DNA learns in a sense, it accumulates information. But it is so slow. DNA accumulates information through random errors, some of which just happen to work. But DNA had actually generated a faster way of learning: it had produced organisms with brains, and those organisms can learn in real time. They accumulate information, they learn. The sad thing is, when they die, the information dies with them. Now what makes humans different is human language. We are blessed with a language, a system of communication, so powerful and so precise that we can share what we've learned with such precision that it can accumulate in the collective memory. And that means it can outlast the individuals who learned that information, and it can accumulate from generation to generation. And that's why, as a species, we're so creative and so powerful, and that's why we have a history. We seem to be the only species in four billion years to have this gift. I call this ability collective learning. It's what makes us different. We can see it at work in the earliest stages of human history. We evolved as a species in the savanna lands of Africa, but then you see humans migrating into new environments, into desert lands, into jungles, into the Ice Age tundra of Siberia -- tough, tough environment -- into the Americas, into Australasia. Each migration involved learning -- learning new ways of exploiting the environment, new ways of dealing with their surroundings. Then 10,000 years ago, exploiting a sudden change in global climate with the end of the last ice age, humans learned to farm. Farming was an energy bonanza. And exploiting that energy, human populations multiplied. Human societies got larger, denser, more interconnected. And then from about 500 years ago, humans began to link up globally through shipping, through trains, through telegraph, through the Internet, until now we seem to form a single global brain of almost seven billion individuals. And that brain is learning at warp speed. And in the last 200 years, something else has happened. We've stumbled on another energy bonanza in fossil fuels. So fossil fuels and collective learning together explain the staggering complexity we see around us. So -- Here we are, back at the convention center. We've been on a journey, a return journey, of 13.7 billion years. I hope you agree this is a powerful story. And it's a story in which humans play an astonishing and creative role. But it also contains warnings. Collective learning is a very, very powerful force, and it's not clear that we humans are in charge of it. I remember very vividly as a child growing up in England, living through the Cuban Missile Crisis. For a few days, the entire biosphere seemed to be on the verge of destruction. And the same weapons are still here, and they are still armed. If we avoid that trap, others are waiting for us. We're burning fossil fuels at such a rate that we seem to be undermining the Goldilocks conditions that made it possible for human civilizations to flourish over the last 10,000 years. So what big history can do is show us the nature of our complexity and fragility and the dangers that face us, but it can also show us our power with collective learning. And now, finally -- this is what I want. I want my grandson, Daniel, and his friends and his generation, throughout the world, to know the story of big history, and to know it so well that they understand both the challenges that face us and the opportunities that face us. And that's why a group of us are building a free, online syllabus in big history for high-school students throughout the world. We believe that big history will be a vital intellectual tool for them, as Daniel and his generation face the huge challenges and also the huge opportunities ahead of them at this threshold moment in the history of our beautiful planet. I thank you for your attention. (Applause) |
World_History_TED_Talks | Where_did_English_come_from_Claire_Bowern.txt | When we talk about English, we often think of it as a single language but what do the dialects spoken in dozens of countries around the world have in common with each other, or with the writings of Chaucer? And how are any of them related to the strange words in Beowulf? The answer is that like most languages, English has evolved through generations of speakers, undergoing major changes over time. By undoing these changes, we can trace the language from the present day back to its ancient roots. While modern English shares many similar words with Latin-derived romance languages, like French and Spanish, most of those words were not originally part of it. Instead, they started coming into the language with the Norman invasion of England in 1066. When the French-speaking Normans conquered England and became its ruling class, they brought their speech with them, adding a massive amount of French and Latin vocabulary to the English language previously spoken there. Today, we call that language Old English. This is the language of Beowulf. It probably doesn't look very familiar, but it might be more recognizable if you know some German. That's because Old English belongs to the Germanic language family, first brought to the British Isles in the 5th and 6th centuries by the Angles, Saxons, and Jutes. The Germanic dialects they spoke would become known as Anglo-Saxon. Viking invaders in the 8th to 11th centuries added more borrowings from Old Norse into the mix. It may be hard to see the roots of modern English underneath all the words borrowed from French, Latin, Old Norse and other languages. But comparative linguistics can help us by focusing on grammatical structure, patterns of sound changes, and certain core vocabulary. For example, after the 6th century, German words starting with "p," systematically shifted to a "pf" sound while their Old English counterparts kept the "p" unchanged. In another split, words that have "sk" sounds in Swedish developed an "sh" sound in English. There are still some English words with "sk," like "skirt," and "skull," but they're direct borrowings from Old Norse that came after the "sk" to "sh" shift. These examples show us that just as the various Romance languages descended from Latin, English, Swedish, German, and many other languages descended from their own common ancestor known as Proto-Germanic spoken around 500 B.C.E. Because this historical language was never written down, we can only reconstruct it by comparing its descendants, which is possible thanks to the consistency of the changes. We can even use the same process to go back one step further, and trace the origins of Proto-Germanic to a language called Proto-Indo-European, spoken about 6000 years ago on the Pontic steppe in modern day Ukraine and Russia. This is the reconstructed ancestor of the Indo-European family that includes nearly all languages historically spoken in Europe, as well as large parts of Southern and Western Asia. And though it requires a bit more work, we can find the same systematic similarities, or correspondences, between related words in different Indo-European branches. Comparing English with Latin, we see that English has "t" where Latin has "d", and "f" where latin has "p" at the start of words. Some of English's more distant relatives include Hindi, Persian and the Celtic languages it displaced in what is now Britain. Proto-Indo-European itself descended from an even more ancient language, but unfortunately, this is as far back as historical and archeological evidence will allow us to go. Many mysteries remain just out of reach, such as whether there might be a link between Indo-European and other major language families, and the nature of the languages spoken in Europe prior to its arrival. But the amazing fact remains that nearly 3 billion people around the world, many of whom cannot understand each other, are nevertheless speaking the same words shaped by 6000 years of history. |
World_History_TED_Talks | When_will_the_next_mass_extinction_occur_Borths_DEmic_and_Pritchard.txt | About 66 million years ago, something terrible happened to life on our planet. Ecosystems were hit with a double blow as massive volcanic eruptions filled the atmosphere with carbon dioxide and an asteroid roughly the size of Manhattan struck the Earth. The dust from the impact reduced or stopped photosynthesis from many plants, starving herbivores and the carnivores that preyed on them. Within a short time span, three-quarters of the world's species disappeared forever, and the giant dinosaurs, flying pterosaurs, shelled squids, and marine reptiles that had flourished for ages faded into prehistory. It may seem like the dinosaurs were especially unlucky, but extinctions of various severities have occurred throughout the Earth's history, and are still happening all around us today. Environments change, pushing some species out of their comfort zones while creating new opportunities for others. Invasive species arrive in new habitats, outcompeting the natives. And in some cases, entire species are wiped out as a result of activity by better adapted organisms. Sometimes, however, massive changes in the environment occur too quickly for most living creatures to adapt, causing thousands of species to die off in a geological instant. We call this a mass extinction event, and although such events may be rare, paleontologists have been able to identify several of them through dramatic changes in the fossil record, where lineages that persisted through several geological layers suddenly disappear. In fact, these mass extinctions are used to divide the Earth's history into distinct periods. Although the disappearance of the dinosaurs is the best known mass extinction event, the largest occurred long before dinosaurs ever existed. 252 million years ago, between the Permian and Triassic periods, the Earth's land masses gathered together into the single supercontinent Pangaea. As it coalesced, its interior was filled with deserts, while the single coastline eliminated many of the shallow tropical seas where biodiversity thrived. Huge volcanic eruptions occurred across Siberia, coinciding with very high temperatures, suggesting a massive greenhouse effect. These catastrophes contributed to the extinction of 95% of species in the ocean, and on land, the strange reptiles of the Permian gave way to the ancestors of the far more familiar dinosaurs we know today. But mass extinctions are not just a thing of the distant past. Over the last few million years, the fluctuation of massive ice sheets at our planet's poles has caused sea levels to rise and fall, changing weather patterns and ocean currents along the way. As the ice sheets spread, retreated, and returned, some animals were either able to adapt to the changes, or migrate to a more suitable environment. Others, however, such as giant ground sloths, giant hyenas, and mammoths went extinct. The extinction of these large mammals coincides with changes in the climate and ecosystem due to the melting ice caps. But there is also an uncomfortable overlap with the rise of a certain hominid species originating in Africa 150,000 years ago. In the course of their adaptation to the new environment, creating new tools and methods for gathering food and hunting prey, humans may not have single-handedly caused the extinction of these large animals, as some were able to coexist with us for thousands of years. But it's clear that today, our tools and methods have become so effective that humans are no longer reacting to the environment, but are actively changing it. The extinction of species is a normal occurrence in the background of ecosystems. But studies suggest that rates of extinction today for many organisms are hundreds to thousands of times higher than the normal background. But the same unique ability that makes humans capable of driving mass extinctions can also enable us to prevent them. By learning about past extinction events, recognizing what is happening today as environments change, and using this knowledge to lessen our effect on other species, we can transform humanity's impact on the world from something as destructive as a massive asteroid into a collaborative part of a biologically diverse future. |
World_History_TED_Talks | A_brief_history_of_rhyme_Baba_Brinkman_TEDxNavesink.txt | Translator: Tanya Cushman Reviewer: Peter van de Ven Yes, indeed, I am a rap artist, but I release records with subject matters that are a little bit different. One of my albums is a hip-hop adaptation of Chaucer's "Canterbury Tales." (Laughter) Another one is a hip-hop interpretation of Charles Darwin's "Theory of Evolution by Natural Selection." And natural selection is a pretty potent idea; it's used to understand design and complexity in the natural world or the appearance of design. But I think we can also use it to understand how culture evolves, including hip-hop culture and the word play through history - going back a lot further than hip-hop. Let's take it to the Stone Age of rap; where did it start out? I mean, hip-hop is a culture that is based on word play; rappers are known for their word play. They use metaphor; they use simile; they use alliteration; they use irony; they use humor, double entendre. But I want to focus really on the rhymes, and see where the rhymes in rap come from and how they've changed over the years. Here's the first rap record that came out in 1979, "Rapper's Delight." So this was pretty much the first rap lyrics that anyone outside of the New York area would have heard because it was the first one on any mass media. And here's what it sounded like: (Music) "Now what you hear is not a test, I'm rappin' to the beat. And me, the groove, and my friends, are gonna try to move your feet." (Music ends) Now, familiar stuff, right? But let's just look at the rhyme schemes and the word play. The basic rhyme is "beat" and "feet," the same kind of rhyme as you might find in Shakespeare, Chaucer, William Blake. It's not really an innovation. Then there's an off-rhyme, maybe, on the other line: "test" and "friends," and there are internal rhymes on "groove" and "move," but they're all monosyllabic single rhymes. If you go forward a few years, 1982, rap starts to become more complex in its message. This song is called "The Message": (Music) "Rats in the front room, roaches in the back. Junkies in the alley with the baseball bat. I tried to get away, but I couldn't get far 'cause a man with a tow truck repossessed my car.'" (Music ends) Getting rhythmically a bit more interesting, the subject matter is more conscious, but the rhymes are still quite simple: "back," "bat," "far," "car." A few years later, an innovation happened; it was like a mutation. It was a mutation that happened in the mind of this guy on the right, William Griffin, aka Rakim, and he did something innovative with rap that no one had done before. So, see if you can see what it is in these lyrics that constitutes an innovation in rap. (Music) "Write a rhyme in graffiti in every show you see me in, deep concentration, 'cause I'm no comedian. Jokers are wild, if you wanna be tame, I'll treat you like a child, and you're gonna be named." (Music ends) These are called "mosaic rhymes." A mosaic rhyme takes the constituent parts of a multisyllable word and assembles those parts from smaller words to recreate the pattern. "Comedian," "see me in," "graffiti in." A polysyllabic rhyme, or multisyllable rhyme - rappers call them "multis" - takes words and combines them together into a pattern. So "wild," "wanna be tame," "child," "gonna be named." This caught like wildfire through hip-hop culture. After that, it was the game that rappers had to play because rap is so competitive; there's too many emcees and not enough mics, which means rappers have to innovate in order to get their prestige and their recognition. So a few years later, hip-hop culture had decided that an idea worth spreading was this concept of multisyllable rhymes. They proliferated, not instantly but fairly quickly, so that if you go forward about seven years, to where Nas was in the early '90s, you get all rappers doing this, but I think Nas was one of the best. Here's how he played with multisyllable rhyme schemes: (Music) "It's like that. You know it's like that. I got at him, now you'll never get the mic back. When I attack, there ain't a army that could strike back, so I react, never calmly, on a hype track." (Music ends) You can see how more and more of the line is being used as part of the rhyme scheme. It's not just "like that," and "mic back"; it's "attack, there ain't an army that could strike back," "react never calmly on a hype track." The word play is sort of proliferating through the lyrics. And then a few years later, Eminem did something kind of novel as well, where he took a small pattern of rhymes and just tried to see how many variations of it he could flip in the same lyric. So here's four bars that uses the "work a sweat," or "circus net," rhyme pattern: (Music) "I feel like I'm walking a tight rope without a circus net poppin' Percocet; I'm a nervous wreck. I deserve respect, but I work a sweat for this worthless check. I'm about to burst this tech in somebody to reverse this debt." (Music ends) "Circus net," "Percocet," "nervous wreck," "deserve respect," "work a sweat" "worthless check," "burst this tech," "reverse this debt." That's only four lines. So when I first discovered these rhyming patterns, I was 19. I was first starting out as a rap artist myself, but I was also studying English literature at the time. So I thought if I go back to the oldest poetry in our language, if I go back 1,000 years, will I find these rhyme patterns? So I started with Geoffrey Chaucer in the late 1300s because he was sort of the founder of what we call English literature today. And here's the opening lines from Chaucer's "Pardoner's Tale": "In Flanders whilom was a company Of youngè folk that haunteden folly, As riot, hazard, stewès, and taverns Where, as with harpès, lutès and gitterns. They dance, and play at dice both day and night, And eat also and drink over their might Through which they do the devil sacrifice Within that devil's temple in cursèd wise." (Applause) Clearly, there's word play going on here, but are there multisyllable rhyme schemes? Not really - it's rhymes like "company," "folly," "taverns" "gitterns," one syllable and two syllable rhymes. There is a strange little possible multi here on "stewès, and taverns" and "lutès and gitterns." And if you look through Chaucer, you find these occasionally sprinkled in, but he seems to just do it to spice up the playfulness of the lyricism; he never uses a sustained pattern of polysyllabic rhymes. To find this pattern of polysyllabic rhymes in English literary history, you have to go forward to 1663, where Samuel Butler's poem "Hudibras" was published. And it had these lines: "Beside, he was a shrewd philosopher, And had read ev'ry text and gloss over; Profound in all the Nominal And Real ways, beyond them all." Now, these are rap-style rhymes: "philosopher," "gloss over," right? This was an innovation for Samuel Butler, as far as we know. But this poem, Hudibras, it was a mock epic. It was meant to be silly, absurd, a comedy piece. So henceforth, this kind of rhyming became known as "Hudibrastic rhyme." And Hudibrastic signified both the multisyllabled patterns and the absurdist, comedy context. And all poets, from then on, seemed to never divest these two different aspects of Hudibrastic rhyme. So if you go 150 years forward, you get Lord Byron using the technique in his poem "Don Juan," where he writes the classic line "But—Oh! ye lords of ladies intellectual, Inform us truly, have they not hen-peck'd you all?" Now, that is a classic hip-hop-style rhyme: "intellectual" and "hen-peck'd you all." But once again, it's used in an absurdist comedy context. I made a bit of a study when I was in grad school of the English literary canon to see if I could find any poets in the history of the English language using multisyllable rhymes outside of a comedy context, and I couldn't really find any. If anybody knows of a counterexample of this, of serious polysyllabic rhyme in English literary history, I'd love to know of it; I couldn't find any. Actually, the only one I could find was, strangely enough, in "Lord of the Rings," by JRR Tolkein. Now, in "Lord of the Rings," Bilbo, he's kind of done with the quest; he's hanging at Riverrun, chilling with the elves, smoking a pipe, sitting by the fire, writing poetry. And when Frodo comes through on his quest with the Fellowship, Bilbo takes him aside and goes, "I've worked on this poem; I want to read it to you. It's kind of a funny little experiment. See what you think." And he reads him the "Song of Eärendil," and here's how that one sounds: "Through Evernight he back was borne on black and roaring waves that ran o'er leagues unlit and foundered shores that drowned before the Days began, until he heard on strands of pearl where ends the world the music long, where ever-foaming billows roll the yellow gold and jewels wan." These are rap-style polysyllabic rhymes: "waves that ran," "days began," "foundered shores," "drowned before." So Tolkein discovered this mutation that Rakim discovered, the possibility of using mutisyllable rhymes to express a noncomedic emotional tone, but Tolkein puts it in a meta-context of humor because he doesn't give the poem to someone serious, he gives it to old, senile Bilbo sitting by the fire, just goofing off in his retirement, right? So we still can't escape from this comedy context. Rappers manage to do this; they managed to escape from this comedy context, and it may help that Rakim's early lyric in this, the first thing people heard in multisyllable rhyme, was "I ain't no comedian," right? And the song title was "I ain't no joke." So in case you were tempted to take it as a joke, Rakim did not give you that option. And henceforth, rappers could use the multisyllabic rhyme to express all possible emotional tones: aggression, anger, frustration, grief, melancholy, and yes, also humor, but there was not that same limitation. It became open, and this redeemed the possibility of rhyme, because if you look through the last 100 years of poetry, you'll notice that poets pretty much gave up on rhyme. Rhyme used to be definitive, and then it became free verse; it just became too obvious: moon, June, soon. It's, like, not creative anymore. But my analogy is that - imagine poets throughout history were like painters who only knew how to paint in primary colors. All they had was red, yellow and blue. And then rappers came along, and they were like, "Yo, you can mix them together." And suddenly, there's aquamarine and mauve and turquoise and fuchsia, and they're all appearing, and the color potential emerges. So, next time you listen to rap, I want you to just listen to the word play that the rappers are using because you'll hear this legacy of polysyllabic rhyme. You'll hear that what they say is part of the word play, but the structure, the intricacy of the rhymes, is another. And this is a historically significant literary innovation that rappers deserve credit for, for creating that potential space. Now, I started out saying that I was a rap artist, so I'm going to have to show you how I play with these rhymes. I'll take the one pattern "broken glass" and see how many times I can flip it. It's a 16-bar rap I'm going to do, and I use the pattern four times in each of the sixteen bars. Don't be tempted to read ahead; we'll do it line by line. (Music) Like this. I been cracking jokes and having loads of plans, but no cash. I'm known for passing notes in class and going back to old Casanova tactics, holding hands and poems, and that's my only passion, so my rap is open to action. No relaxin'. No, I'm smashing holes; fasten open caskets, slow dancing over scattered broken glass and cold molasses; no fashion moguls have my clothes - I'm happy sewing patches. I don't have a home; I'd rather roam the map of no attachment, so I'm a laughin' nomad though I travel roads and pass it, no average Joe can; I won't collapse and go flaccid. No chance, it won't happen, only after no challenge known to man goes unmastered, only after more than half my bones are fractured, only passive, goes to pasture, no one asks for pros, they ask for smokin' raps like solar flashes flowing past the globe's axis, so I task my overactive dome with matching no unstandards and only hatchin' poems that have the motions of explosive gases. (Music ends) And that's what rhymes with broken glasses. Thank you. (Cheers) (Applause) |
World_History_TED_Talks | 만사_무사_가장_부유하게_살았던_사람들_중_한_사람_제시카_스미스.txt | If someone asked you who the richest people in history were, who would you name? Perhaps a billionaire banker or corporate mogul, like Bill Gates or John D. Rockefeller. How about African King Musa Keita I? Ruling the Mali Empire in the 14th century CE, Mansa Musa, or the King of Kings, amassed a fortune that possibly made him one of the wealthiest people who ever lived. But his vast wealth was only one piece of his rich legacy. When Mansa Musa came to power in 1312, much of Europe was racked by famine and civil wars. But many African kingdoms and the Islamic world were flourishing, and Mansa Musa played a great role in bringing the fruits of this flourishing to his own realm. By strategically annexing the city of Timbuktu, and reestablishing power over the city of Gao, he gained control over important trade routes between the Mediterranean and the West African Coast, continuing a period of expansion, which dramatically increased Mali's size. The territory of the Mali Empire was rich in natural resources, such as gold and salt. The world first witnessed the extent of Mansa Musa's wealth in 1324 when he took his pilgrimage to Mecca. Not one to travel on a budget, he brought a caravan stretching as far as the eye could see. Accounts of this journey are mostly based on an oral testimony and differing written records, so it's difficult to determine the exact details. But what most agree on is the extravagant scale of the excursion. Chroniclers describe an entourage of tens of thousands of soldiers, civilians, and slaves, 500 heralds bearing gold staffs and dressed in fine silks, and many camels and horses bearing an abundance of gold bars. Stopping in cities such as Cairo, Mansa Musa is said to have spent massive quantities of gold, giving to the poor, buying souvenirs, and even having mosques built along the way. In fact, his spending may have destabilized the regional economy, causing mass inflation. This journey reportedly took over a year, and by the time Mansa Musa returned, tales of his amazing wealth had spread to the ports of the Mediterranean. Mali and its king were elevated to near legendary status, cemented by their inclusion on the 1375 Catalan Atlas. One of the most important world maps of Medieval Europe, it depicted the King holding a scepter and a gleaming gold nugget. Mansa Musa had literally put his empire and himself on the map. But material riches weren't the king's only concern. As a devout Muslim, he took a particular interest in Timbuktu, already a center of religion and learning prior to its annexation. Upon returning from his pilgrimage, he had the great Djinguereber Mosque built there with the help of an Andalusian architect. He also established a major university, further elevating the city's reputation, and attracting scholars and students from all over the Islamic world. Under Mansa Musa, the Empire became urbanized, with schools and mosques in hundreds of densely populated towns. The king's rich legacy persisted for generations and to this day, there are mausoleums, libraries and mosques that stand as a testament to this golden age of Mali's history. |
World_History_TED_Talks | Hunting_for_Perus_lost_civilizations_with_satellites_Sarah_Parcak.txt | In July of 1911, a 35-year-old Yale graduate and professor set out from his rainforest camp with his team. After climbing a steep hill and wiping the sweat from his brow, he described what he saw beneath him. He saw rising from the dense rainforest foliage this incredible interlocking maze of structures built of granite, beautifully put together. What's amazing about this project is that it was the first funded by National Geographic, and it graced the front cover of its magazine in 1912. This professor used state-of-the-art photography equipment to record the site, forever changing the face of exploration. The site was Machu Picchu, discovered and explored by Hiram Bingham. When he saw the site, he asked, "This is an impossible dream. What could it be?" So today, 100 years later, I invite you all on an incredible journey with me, a 37-year-old Yale graduate and professor. (Cheers) We will do nothing less than use state-of-the-art technology to map an entire country. This is a dream started by Hiram Bingham, but we are expanding it to the world, making archaeological exploration more open, inclusive, and at a scale simply not previously possible. This is why I am so excited to share with you all today that we will begin the 2016 TED Prize platform in Latin America, more specifically Peru. (Applause) Thank you. We will be taking Hiram Bingham's impossible dream and turning it into an amazing future that we can all share in together. So Peru doesn't just have Machu Picchu. It has absolutely stunning jewelry, like what you can see here. It has amazing Moche pottery of human figures. It has the Nazca Lines and amazing textiles. So as part of the TED Prize platform, we are going to partnering with some incredible organizations, first of all with DigitalGlobe, the world's largest provider of high-resolution commercial satellite imagery. They're going to be helping us build out this amazing crowdsourcing platform they have. Maybe some of you used it with the MH370 crash and search for the airplane. Of course, they'll also be providing us with the satellite imagery. National Geographic will be helping us with education and of course exploration. As well, they'll be providing us with rich content for the platform, including some of the archival imagery like you saw at the beginning of this talk and some of their documentary footage. We've already begun to build and plan the platform, and I'm just so excited. So here's the cool part. My team, headed up by Chase Childs, is already beginning to look at some of the satellite imagery. Of course, what you can see here is 0.3-meter data. This is site called Chan Chan in northern Peru. It dates to 850 AD. It's a really amazing city, but let's zoom in. This is the type and quality of data that you all will get to see. You can see individual structures, individual buildings. And we've already begun to find previously unknown sites. What we can say already is that as part of the platform, you will all help discover thousands of previously unknown sites, like this one here, and this potentially large one here. Unfortunately, we've also begun to uncover large-scale looting at sites, like what you see here. So many sites in Peru are threatened, but the great part is that all of this data is going to be shared with archaeologists on the front lines of protecting these sites. So I was just in Peru, meeting with their Minister of Culture as well as UNESCO. We'll be collaborating closely with them. Just so you all know, the site is going to be in both English and Spanish, which is absolutely essential to make sure that people in Peru and across Latin America can participate. Our main project coprincipal investigator is the gentleman you see here, Dr. Luis Jaime Castillo, professor at Catholic University. As a respected Peruvian archaeologist and former vice-minister, Dr. Castillo will be helping us coordinate and share the data with archaeologists so they can explore these sites on the ground. He also runs this amazing drone mapping program, some of the images of which you can see behind me here and here. And this data will be incorporated into the platform, and also he'll be helping to image some of the new sites you help find. Our on-the-ground partner who will be helping us with education, outreach, as well as site preservation components, is the Sustainable Preservation Initiative, led by Dr. Larry Coben. Some of you may not be aware that some of the world's poorest communities coexist with some of the world's most well-known archaeological sites. What SPI does is it helps to empower these communities, in particular women, with new economic approaches and business training. So it helps to teach them to create beautiful handicrafts which are then sold on to tourists. This empowers the women to treasure their cultural heritage and take ownership of it. I had the opportunity to spend some time with 24 of these women at a well-known archaeological site called Pachacamac, just outside Lima. These women were unbelievably inspiring, and what's great is that SPI will help us transform communities near some of the sites that you help to discover. Peru is just the beginning. We're going to be expanding this platform to the world, but already I've gotten thousands of emails from people all across the world -- professors, educators, students, and other archaeologists -- who are so excited to help participate. In fact, they're already suggesting amazing places for us to help discover, including Atlantis. I don't know if we're going to be looking for Atlantis, but you never know. So I'm just so excited to launch this platform. It's going to be launched formally by the end of the year. And I have to say, if what my team has already discovered in the past few weeks are any indication, what the world discovers is just going to be beyond imagination. Make sure to hold on to your alpacas. Thank you very much. (Applause) Thank you. (Applause) |
World_History_TED_Talks | 역사_대_블리디미르_레닌알렉스_젠들러_Alex_Gendler.txt | He was one of the most influential figures of the 20th century, forever changing the course of one of the world's largest countries. But was he a hero who toppled an oppressive tyranny or a villain who replaced it with another? It's time to put Lenin on the stand in History vs. Lenin. "Order, order, hmm. Now, wasn't it your fault that the band broke up?" "Your honor, this is Vladimir Ilyich Ulyanov, AKA Lenin, the rabblerouser who helped overthrow the Russian tsar Nicholas II in 1917 and founded the Soviet Union, one of the worst dictatorships of the 20th century." "Ohh." "The tsar was a bloody tyrant under whom the masses toiled in slavery." "This is rubbish. Serfdom had already been abolished in 1861." "And replaced by something worse. The factory bosses treated the people far worse than their former feudal landlords. And unlike the landlords, they were always there. Russian workers toiled for eleven hours a day and were the lowest paid in all of Europe." "But Tsar Nicholas made laws to protect the workers." "He reluctantly did the bare minimum to avert revolution, and even there, he failed. Remember what happened in 1905 after his troops fired on peaceful petitioners?" "Yes, and the tsar ended the rebellion by introducing a constitution and an elected parliament, the Duma." "While retaining absolute power and dissolving them whenever he wanted." "Perhaps there would've been more reforms in due time if radicals, like Lenin, weren't always stirring up trouble." "Your Honor, Lenin had seen his older brother Aleksandr executed by the previous tsar for revolutionary activity, and even after the reforms, Nicholas continued the same mass repression and executions, as well as the unpopular involvement in World War I, that cost Russia so many lives and resources." "Hm, this tsar doesn't sound like such a capital fellow." "Your Honor, maybe Nicholas II did doom himself with bad decisions, but Lenin deserves no credit for this. When the February 1917 uprisings finally forced the tsar to abdicate, Lenin was still exiled in Switzerland." "Hm, so who came to power?" "The Duma formed a provisional government, led by Alexander Kerensky, an incompetent bourgeois failure. He even launched another failed offensive in the war, where Russia had already lost so much, instead of ending it like the people wanted." "It was a constitutional social democratic government, the most progressive of its time. And it could have succeeded eventually if Lenin hadn't returned in April, sent by the Germans to undermine the Russian war effort and instigate riots." "Such slander! The July Days were a spontaneous and justified reaction against the government's failures. And Kerensky showed his true colors when he blamed Lenin and arrested and outlawed his Bolshevik party, forcing him to flee into exile again. Some democracy! It's a good thing the government collapsed under their own incompetence and greed when they tried to stage a military coup then had to ask the Bolsheviks for help when it backfired. After that, all Lenin had to do was return in October and take charge. The government was peacefully overthrown overnight." "But what the Bolsheviks did after gaining power wasn't very peaceful. How many people did they execute without trial? And was it really necessary to murder the tsar's entire family, even the children?" "Russia was being attacked by foreign imperialists, trying to restore the tsar. Any royal heir that was rescued would be recognized as ruler by foreign governments. It would've been the end of everything the people had fought so hard to achieve. Besides, Lenin may not have given the order." "But it was not only imperialists that the Bolsheviks killed. What about the purges and executions of other socialist and anarchist parties, their old allies? What about the Tambov Rebellion, where peasants, resisting grain confiscation, were killed with poison gas? Or sending the army to crush the workers in Kronstadt, who were demanding democratic self-management? Was this still fighting for the people?" "Yes! The measures were difficult, but it was a difficult time. The new government needed to secure itself while being attacked from all sides, so that the socialist order could be established." "And what good came of this socialist order? Even after the civil war was won, there were famines, repression and millions executed or sent to die in camps, while Lenin's successor Stalin established a cult of personality and absolute power." "That wasn't the plan. Lenin never cared for personal gains, even his enemies admitted that he fully believed in his cause, living modestly and working tirelessly from his student days until his too early death. He saw how power-hungry Stalin was and tried to warn the party, but it was too late." "And the decades of totalitarianism that followed after?" "You could call it that, but it was Lenin's efforts that changed Russia in a few decades from a backward and undeveloped monarchy full of illiterate peasants to a modern, industrial superpower, with one of the world's best educated populations, unprecedented opportunities for women, and some of the most important scientific advancements of the century. Life may not have been luxurious, but nearly everyone had a roof over their head and food on their plate, which few countries have achieved." "But these advances could still have happened, even without Lenin and the repressive regime he established." "Yes, and I could've been a famous rock and roll singer. But how would I have sounded?" We can never be sure how things could've unfolded if different people were in power or different decisions were made, but to avoid the mistakes of the past, we must always be willing to put historical figures on trial. |
World_History_TED_Talks | History_vs_Napoleon_Bonaparte_Alex_Gendler.txt | After the French Revolution erupted in 1789, Europe was thrown into chaos. Neighboring countries' monarchs feared they would share the fate of Louis XVI, and attacked the New Republic, while at home, extremism and mistrust between factions lead to bloodshed. In the midst of all this conflict, a powerful figure emerged to take charge of France. But did he save the revolution or destroy it? "Order, order, who's the defendant today? I don't see anyone." "Your Honor, this is Napoléon Bonaparte, the tyrant who invaded nearly all of Europe to compensate for his personal stature-based insecurities." "Actually, Napoléon was at least average height for his time. The idea that he was short comes only from British wartime propaganda. And he was no tyrant. He was safeguarding the young Republic from being crushed by the European monarchies." "By overthrowing its government and seizing power himself?" "Your Honor, as a young and successful military officer, Napoléon fully supported the French Revolution, and its ideals of liberty, equality, and fraternity. But the revolutionaries were incapable of real leadership. Robespierre and the Jacobins who first came to power unleashed a reign of terror on the population, with their anti-Catholic extremism and nonstop executions of everyone who disagreed with them. And The Directory that replaced them was an unstable and incompetent oligarchy. They needed a strong leader who could govern wisely and justly." "So, France went through that whole revolution just to end up with another all-powerful ruler?" "Not quite. Napoléon's new powers were derived from the constitution that was approved by a popular vote in the Consulate." "Ha! The constitution was practically dictated at gunpoint in a military coup, and the public only accepted the tyrant because they were tired of constant civil war." "Be that as it may, Napoléon introduced a new constitution and a legal code that kept some of the most important achievements of the revolution in tact: freedom of religion abolition of hereditary privilege, and equality before the law for all men." "All men, indeed. He deprived women of the rights that the revolution had given them and even reinstated slavery in the French colonies. Haiti is still recovering from the consequences centuries later. What kind of equality is that?" "The only kind that could be stably maintained at the time, and still far ahead of France's neighbors." "Speaking of neighbors, what was with all the invasions?" "Great question, Your Honor." "Which invasions are we talking about? It was the neighboring empires who had invaded France trying to restore the monarchy, and prevent the spread of liberty across Europe, twice by the time Napoléon took charge. Having defended France as a soldier and a general in those wars, he knew that the best defense is a good offense." "An offense against the entire continent? Peace was secured by 1802, and other European powers recognized the new French Regime. But Bonaparte couldn't rest unless he had control of the whole continent, and all he knew was fighting. He tried to enforce a European-wide blockade of Britain, invaded any country that didn't comply, and launched more wars to hold onto his gains. And what was the result? Millions dead all over the continent, and the whole international order shattered." "You forgot the other result: the spread of democratic and liberal ideals across Europe. It was thanks to Napoléon that the continent was reshaped from a chaotic patchwork of fragmented feudal and religious territories into efficient, modern, and secular nation states where the people held more power and rights than ever before." "Should we also thank him for the rise of nationalism and the massive increase in army sizes? You can see how well that turned out a century later." "So what would European history have been like if it weren't for Napoléon?" "Unimaginably better/worse." Napoléon seemingly unstoppable momentum would die in the Russian winter snows, along with most of his army. But even after being deposed and exiled, he refused to give up, escaping from his prison and launching a bold attempt at restoring his empire before being defeated for the second and final time. Bonaparte was a ruler full of contradictions, defending a popular revolution by imposing absolute dictatorship, and spreading liberal ideals through imperial wars, and though he never achieved his dream of conquering Europe, he undoubtedly left his mark on it, for better or for worse. |
World_History_TED_Talks | Is_graffiti_art_Or_vandalism_Kelly_Wall.txt | Spray-painted subway cars, tagged bridges, mural-covered walls. Graffiti pops up boldly throughout our cities. It can make statements about identity, art, empowerment, and politics, while simultaneously being associated with destruction. And, it turns out, it's nothing new. Graffiti, or the act of writing or scribbling on public property, has been around for thousands of years. And across that span of time, it's raised the same questions we debate now: Is it art? Is it vandalism? In the 1st century BCE, Romans regularly inscribed messages on public walls, while oceans away, Mayans were prolifically scratching drawings onto their surfaces. And it wasn't always a subversive act. In Pompeii, ordinary citizens regularly marked public walls with magic spells, prose about unrequited love, political campaign slogans, and even messages to champion their favorite gladiators. Some, including the Greek philosopher Plutarch, pushed back, deeming graffiti ridiculous and pointless. But it wasn't until the 5th century that the roots of the modern concept of vandalism were planted. At that time, a barbaric tribe known as the Vandals swept through Rome, pillaging and destroying the city. But it wasn't until centuries later that the term vandalism was actually coined in an outcry against the defacing of art during the French Revolution. And as graffiti became increasingly associated with deliberate rebellion and provocativeness, it took on its vandalist label. That's part of the reason why, today, many graffiti artists stay underground. Some assume alternate identities to avoid retribution, while others do so to establish comradery and make claim to territory. Beginning with the tags of the 1960s, a novel overlap of celebrity and anonymity hit the streets of New York City and Philadelphia. Taggers used coded labels to trace their movements around cities while often alluding to their origins. And the very illegality of graffiti-making that forced it into the shadows also added to its intrigue and growing base of followers. The question of space and ownership is central to graffiti's history. Its contemporary evolution has gone hand in hand with counterculture scenes. While these movements raised their anti-establishment voices, graffiti artists likewise challenged established boundaries of public property. They reclaimed subway cars, billboards, and even once went so far as to paint an elephant in the city zoo. Political movements, too, have used wall writing to visually spread their messages. During World War II, both the Nazi Party and resistance groups covered walls with propaganda. And the Berlin Wall's one-sided graffiti can be seen as a striking symbol of repression versus relatively unrestricted public access. As the counterculture movements associated with graffiti become mainstream, does graffiti, too, become accepted? Since the creation of so-called graffiti unions in the 1970s and the admission of select graffiti artists into art galleries a decade later, graffiti has straddled the line between being outside and inside the mainstream. And the appropriation of graffiti styles by marketers and typographers has made this definition even more unclear. The once unlikely partnerships of graffiti artists with traditional museums and brands, have brought these artists out of the underground and into the spotlight. Although graffiti is linked to destruction, it's also a medium of unrestricted artistic expression. Today, the debate about the boundary between defacing and beautifying continues. Meanwhile, graffiti artists challenge common consensus about the value of art and the degree to which any space can be owned. Whether spraying, scrawling, or scratching, graffiti brings these questions of ownership, art, and acceptability to the surface. |
World_History_TED_Talks | 역사_대_크리스토퍼_콜롬버스_알렉스_젠들러.txt | Many people in the United States and Latin America have grown up celebrating the anniversary of Christopher Columbus's voyage, but was he an intrepid explorer who brought two worlds together or a ruthless exploiter who brought colonialism and slavery? And did he even discover America at all? It's time to put Columbus on the stand in History vs. Christopher Columbus. "Order, order in the court. Wait, am I even supposed to be at work today?" Cough "Yes, your Honor. From 1792, Columbus Day was celebrated in many parts of the United States on October 12th, the actual anniversary date. But although it was declared an official holiday in 1934, individual states aren't required to observe it. Only 23 states close public services, and more states are moving away from it completely." Cough "What a pity. In the 70s, we even moved it to the second Monday in October so people could get a nice three-day weekend, but I guess you folks just hate celebrations." "Uh, what are we celebrating again?" "Come on, Your Honor, we all learned it in school. Christopher Columbus convinced the King of Spain to send him on a mission to find a better trade route to India, not by going East over land but sailing West around the globe. Everyone said it was crazy because they still thought the world was flat, but he knew better. And when in 1492 he sailed the ocean blue, he found something better than India: a whole new continent." "What rubbish. First of all, educated people knew the world was round since Aristotle. Secondly, Columbus didn't discover anything. There were already people living here for millennia. And he wasn't even the first European to visit. The Norse had settled Newfoundland almost 500 years before." "You don't say, so how come we're not all wearing those cow helmets?" "Actually, they didn't really wear those either." Cough "Who cares what some Vikings did way back when? Those settlements didn't last, but Columbus's did. And the news he brought back to Europe spread far and wide, inspiring all the explorers and settlers who came after. Without him, none of us would be here today." "And because of him, millions of Native Americans aren't here today. Do you know what Columbus did in the colonies he founded? He took the very first natives he met prisoner and wrote in his journal about how easily he could conquer and enslave all of them." "Oh, come on. Everyone was fighting each other back then. Didn't the natives even tell Columbus about other tribes raiding and taking captives?" "Yes, but tribal warfare was sporadic and limited. It certainly didn't wipe out 90% of the population." "Hmm. Why is celebrating this Columbus so important to you, anyway?" "Your Honor, Columbus's voyage was an inspiration to struggling people all across Europe, symbolizing freedom and new beginnings. And his discovery gave our grandparents and great-grandparents the chance to come here and build better lives for their children. Don't we deserve a hero to remind everyone that our country was build on the struggles of immigrants?" "And what about the struggles of Native Americans who were nearly wiped out and forced into reservations and whose descendants still suffer from poverty and discrimination? How can you make a hero out of a man who caused so much suffering?" "That's history. You can't judge a guy in the 15th century by modern standards. People back then even thought spreading Christianity and civilization across the world was a moral duty." "Actually, he was pretty bad, even by old standards. While governing Hispaniola, he tortured and mutilated natives who didn't bring him enough gold and sold girls as young as nine into sexual slavery, and he was brutal even to the other colonists he ruled, to the point that he was removed from power and thrown in jail. When the missionary, Bartolomé de las Casas, visited the island, he wrote, 'From 1494 to 1508, over 3,000,000 people had perished from war, slavery and the mines. Who in future generations will believe this?'" "Well, I'm not sure I believe those numbers." "Say, aren't there other ways the holiday is celebrated?" "In some Latin American countries, they celebrate the same date under different names, such as Día de la Raza. In these places, it's more a celebration of the native and mixed cultures that survived through the colonial period. Some places in the U.S. have also renamed the holiday, as Native American Day or Indigenous People's Day and changed the celebrations accordingly." "So, why not just change the name if it's such a problem?" "Because it's tradition. Ordinary people need their heroes and their founding myths. Can't we just keep celebrating the way we've been doing for a century, without having to delve into all this serious research? It's not like anyone is actually celebrating genocide." "Traditions change, and the way we choose to keep them alive says a lot about our values." "Well, it looks like giving tired judges a day off isn't one of those values, anyway." Traditions and holidays are important to all cultures, but a hero in one era may become a villain in the next as our historical knowledge expands and our values evolve. And deciding what these traditions should mean today is a major part of putting history on trial. |
World_History_TED_Talks | 징기스칸_대_역사_알렉스_젠들러.txt | He was one of the most fearsome warlords who ever lived, waging an unstoppable conquest across the Eurasian continent. But was Genghis Khan a vicious barbarian or a unifier who paved the way for the modern world? We'll see in "History vs. Genghis Khan." "Order, order. Now who's the defendant today? Khan!" "I see Your Honor is familiar with Genghis Khan, the 13th century warlord whose military campaigns killed millions and left nothing but destruction in their wake." "Objection. First of all, it's pronounced Genghis Kahn." "Really?" "In Mongolia, yes. Regardless, he was one of the greatest leaders in human history. Born Temüjin, he was left fatherless and destitute as a child but went on to overcome constant strife to unite warring Mongol clans and forge the greatest empire the world had seen, eventually stretching from the Pacific to Europe's heartland." "And what was so great about invasion and slaughter? Northern China lost 2/3 of its population." "The Jin Dynasty had long harassed the northern tribes, paying them off to fight each other and periodically attacking them. Genghis Khan wasn't about to suffer the same fate as the last Khan who tried to unite the Mongols, and the demographic change may reflect poor census keeping, not to mention that many peasants were brought into the Khan's army." "You can pick apart numbers all you want, but they wiped out entire cities, along with their inhabitants." "The Khan preferred enemies to surrender and pay tribute, but he firmly believed in loyalty and diplomatic law. The cities that were massacred were ones that rebelled after surrendering, or killed as ambassadors. His was a strict understanding of justice." "Multiple accounts show his army's brutality going beyond justice: ripping unborn children from mothers' wombs, using prisoners as human shields, or moat fillers to support siege engines, taking all women from conquered towns--" "Enough! How barbaric!" "Is that really so much worse than other medieval armies?" "That doesn't excuse Genghis Khan's atrocities." "But it does make Genghis Khan unexceptional for his time rather than some bloodthirsty savage. In fact, after his unification of the tribes abolished bride kidnapping, women in the Mongol ranks had it better than most. They controlled domestic affairs, could divorce their husbands, and were trusted advisors. Temüjin remained with his first bride all his life, even raising her possibly illegitimate son as his own." "Regardless, Genghis Khan's legacy was a disaster: up to 40 million killed across Eurasia during his descendents' conquests. 10% of the world population. That's not even counting casualties from the Black Plague brought to Europe by the Golden Horde's Siege of Kaffa." "Surely that wasn't intentional." "Actually, when they saw their own troops dying of the Plague, they catapulted infected bodies over the city walls." "Blech." "The accounts you're referencing were written over a hundred years after the fact. How reliable do you think they are? Plus, the survivors reaped the benefits of the empire Genghis Khan founded." "Benefits?" "The Mongol Empire practiced religious tolerance among all subjects, they treated their soldiers well, promoted based on merit, rather than birth, established a vast postal system, and inforced universal rule of law, not to mention their contribution to culture." "You mean like Hulagu Khan's annihilation of Baghdad, the era's cultural capital? Libraries, hospitals and palaces burned, irrigation canals buried?" "Baghdad was unfortunate, but its Kalif refused to surrender, and Hulagu was later punished by Berke Khan for the wanton destruction. It wasn't Mongol policy to destroy culture. Usually they saved doctors, scholars and artisans from conquered places, and transferred them throughout their realm, spreading knowledge across the world." "What about the devastation of Kievan Rus, leaving its people in the Dark Ages even as the Renaissance spread across Western Europe?" "Western Europe was hardly peaceful at the time. The stability of Mongol rule made the Silk Road flourish once more, allowing trade and cultural exchange between East and West, and its legacy forged Russia and China from warring princedoms into unified states. In fact, long after the Empire, Genghis Khan's descendants could be found among the ruling nobility all over Eurasia." "Not surprising that a tyrant would inspire further tyrants." "Careful what you call him. You may be related." "What?" "16 million men today are descended from Genghis Khan. That's one in ever 200." For every great conqueror, there are millions of conquered. Whose stories will survive? And can a leader's historical or cultural signifigance outweigh the deaths they caused along the way? These are the questions that arise when we put history on trial. |
World_History_TED_Talks | 흑사병의_과거_현재_그리고_미래.txt | Imagine if half the people in your neighborhood, your city, or even your whole country were wiped out. It might sound like something out of an apocalyptic horror film, but it actually happened in the 14th century during a disease outbreak known as the Black Death. Spreading from China through Asia, the Middle East, Africa, and Europe, the devastating epidemic destroyed as much as 1/5 of the world's population, killing nearly 50% of Europeans in just four years. One of the most fascinating and puzzling things abut the Black Death is that the illness itself was not a new phenomenon but one that has affected humans for centuries. DNA analysis of bone and tooth samples from this period, as well as an earlier epidemic known as the Plague of Justinian in 541 CE, has revealed that both were caused by Yersinia pestis, the same bacterium that causes bubonic plague today. What this means is that the same disease caused by the same pathogen can behave and spread very differently throughout history. Even before the use of antibiotics, the deadliest oubreaks in modern times, such as the ones that occurred in early 20th century India, killed no more than 3% of the population. Modern instances of plague also tend to remain localized, or travel slowly, as they are spread by rodent fleas. But the medieval Black Death, which spread like wildfire, was most likely communicated directly from one person to another. And because genetic comparisons of ancient to modern strains of Yersinia pestis have not revealed any significantly functional genetic differences, the key to why the earlier outbreak was so much deadlier must lie not in the parasite but the host. For about 300 years during the High Middle Ages, a warmer climate and agricultural improvements had led to explosive population growth throughout Europe. But with so many new mouths to feed, the end of this warm period spelled disaster. High fertility rates combined with reduced harvest, meant the land could no longer support its population, while the abundant supply of labor kept wages low. As a result, most Europeans in the early 14th century experienced a steady decline in living standards, marked by famine, poverty and poor health, leaving them vulnerable to infection. And indeed, the skeletal remains of Black Death victims found in London show telltale signs of malnutrition and prior illness. The destruction caused by the Black Death changed humanity in two important ways. On a societal level, the rapid loss of population led to important changes in Europe’s economic conditions. With more food to go around, as well as more land and better pay for the surviving farmers and workers, people began to eat better and live longer as studies of London cemeteries have shown. Higher living standards also brought an increase in social mobility, weakening feudalism and eventually leading to political reforms. But the plague also had an important biological impact. The sudden death of so many of the most frail and vulnerable people left behind a population with a significantly different gene pool, including genes that may have helped survivors resist the disease. And because such mutations often confer immunities to multiple pathogens that work in similar ways, research to discover the genetic consequences of the Black Death has the potential to be hugely beneficial. Today, the threat of an epidemic on the scale of the Black Death has been largely eliminated thanks to antibiotics. But the bubonic plague continues to kill a few thousand people worldwide every year, and the recent emergence of a drug-resistant strain threatens the return of darker times. Learning more about the causes and effects of the Black Death is important, not just for understanding how our world has been shaped by the past. It may also help save us from a similar nightmare in the future. |
World_History_TED_Talks | 타투의_역사_애디슨_앤더슨Addison_Anderson.txt | Thinking of getting a tattoo? Decorating your birthday suit would add another personal story to a history of tattoos stretching back at least 8000 years. Tattooed mummies from around the world attest to the universality of body modification across the millennia, and to the fact that you really were stuck with it forever if your civilization never got around to inventing laser removal. A mummy from the Chinchorro culture in pre-Incan Peru has a mustache tattooed on his upper lip. Ötzi, mummified iceman of the Alps, has patterned charcoal tats along his spine, behind his knee and around his ankles, which might be from an early sort of acupuncture. The mummy of Amunet, a priestess in Middle Kingdom Egypt, features tattoos thought to symbolize sexuality and fertility. Even older than the mummies, figurines of seemingly tattooed people, and tools possibly used for tattooing date back tens of thousands of years. Tattoos don't have one historical origin point that we know of, but why do we English speakers call them all tattoos? The word is an anglophonic modification of "tatao," a Polynesian word used in Tahiti, where English captain James Cook landed in 1769 and encountered heavily tattooed men and women. Stories of Cook's findings and the tattoos his crew acquired cemented our usage of "tattoo" over previous words like "scarring," "painting," and "staining," and sparked a craze in Victorian English high society. We might think of Victorians having Victorian attitudes about such a risque thing, and you can find such sentiments, and even bans, on tattooing throughout history. But while publicly some Brits looked down their noses at tattoos, behind closed doors and away from their noses, lots of people had them. Reputedly, Queen Victoria had a tiger fighting a python, and tattoos became very popular among Cook's fellow soldiers, who used them to note their travels. You crossed the Atlantic? Get an anchor. Been south of the Equator? Time for your turtle tat. But Westerners sported tattoos long before meeting the Samoans and Maori of the South Pacific. Crusaders got the Jerusalem Cross so if they died in battle, they'd get a Christian burial. Roman soldiers on Hadrian's Wall had military tattoos and called the Picts beyond it "Picts," for the pictures painted on them. There's also a long tradition of people being tattooed unwillingly. Greeks and Romans tattooed slaves and mercenaries to discourage escape and desertion. Criminals in Japan were tattooed as such as far back as the 7th century. Most infamously, the Nazis tattooed numbers on the chest or arms of Jews and other prisoners at the Auschwitz concentration camp in order to identify stripped corpses. But tattoos forced on prisoners and outcasts can be redefined as people take ownership of that status or history. Primo Levi survived Auschwitz and wore short sleeves to Germany after the war to remind people of the crime his number represented. Today, some Holocaust survivors' descendants have their relatives numbers' tattooed on their arms. The Torah has rules against tattoos, but what if you want to make indelible what you feel should never be forgotten? And those criminals and outcasts of Japan, where tattooing was eventually outlawed from the mid-19th century to just after World War II, added decoration to their penal tattoos, with designs borrowed from woodblock prints, popular literature and mythical spirtual iconography. Yakuza gangs viewed their outsider tattoos as signs of lifelong loyalty and courage. After all, they lasted forever and it really hurt to get them. For the Maori, those tattoos were an accepted mainstream tradition. If you shied away from the excruciating chiseling in of your moko design, your unfinished tattoo marked your cowardice. Today, unless you go the traditional route, your tattoo artist will probably use a tattoo machine based on the one patented by Samuel O'Reilly in 1891, itself based on Thomas Edison's stencil machine from 1876. But with the incredibly broad history of tattoos giving you so many options, what are you going to get? This is a bold-lined expression of who you are, or you want to appear to be. As the naturalist aboard Cook's ship said of the tataoed Tahitians, "Everyone is marked, thus in different parts of his body, according maybe to his humor or different circumstances of his life." Maybe your particular humor and circumstances suggest getting a symbol of cultural heritage, a sign of spirituality, sexual energy, or good old-fashioned avant-garde defiance. A reminder of a great accomplishment, or of how you think it would look cool if Hulk Hogan rode a Rhino. It's your expression, your body, so it's your call. Just two rules: you have to find a tattooist who won't be ashamed to draw your idea, and when in doubt, you can never go wrong with "Mom." |
World_History_TED_Talks | 아테네에서_민주주의의_진실한_의미는_무엇인가_Melissa_Schwartzberg.txt | Hey, congratulations! You've just won the lottery, only the prize isn't cash or a luxury cruise. It's a position in your country's national legislature. And you aren't the only lucky winner. All of your fellow lawmakers were chosen in the same way. This might strike you as a strange way to run a government, let alone a democracy. Elections are the epitome of democracy, right? Well, the ancient Athenians who coined the word had another view. In fact, elections only played a small role in Athenian democracy, with most offices filled by random lottery from a pool of citizen volunteers. Unlike the representative democracies common today, where voters elect leaders to make laws and decisions on their behalf, 5th Century BC Athens was a direct democracy that encouraged wide participation through the principle of ho boulomenos, or anyone who wishes. This meant that any of its approximately 30,000 eligible citizens could attend the ecclesia, a general assembly meeting several times a month. In principle, any of the 6,000 or so who showed up at each session had the right to address their fellow citizens, propose a law, or bring a public lawsuit. Of course, a crowd of 6,000 people trying to speak at the same time would not have made for effective government. So the Athenian system also relied on a 500 member governing council called the Boule, to set the agenda and evaluate proposals, in addition to hundreds of jurors and magistrates to handle legal matters. Rather than being elected or appointed, the people in these positions were chosen by lot. This process of randomized selection is know as sortition. The only positions filled by elections were those recognized as requiring expertise, such as generals. But these were considered aristocratic, meaning rule by the best, as opposed to democracies, rule by the many. How did this system come to be? Well, democracy arose in Athens after long periods of social and political tension marked by conflict among nobles. Powers once restricted to elites, such as speaking in the assembly and having their votes counted, were expanded to ordinary citizens. And the ability of ordinary citizens to perform these tasks adequately became a central feature of the democratice ideology of Athens. Rather than a privilege, civic participation was the duty of all citizens, with sortition and strict term limits preventing governing classes or political parties from forming. By 21st century standards, Athenian rule by the many excluded an awful lot of people. Women, slaves and foreigners were denied full citizenship, and when we filter out those too young to serve, the pool of eligible Athenians drops to only 10-20% of the overall population. Some ancient philosophers, including Plato, disparaged this form of democracy as being anarchic and run by fools. But today the word has such positive associations, that vastly different regimes claim to embody it. At the same time, some share Plato's skepticism about the wisdom of crowds. Many modern democracies reconcile this conflict by having citizens elect those they consider qualified to legislate on their behalf. But this poses its own problems, including the influence of wealth, and the emergence of professional politicians with different interests than their constituents. Could reviving election by lottery lead to more effective government through a more diverse and representative group of legislatures? Or does modern political office, like Athenian military command, require specialized knowledge and skills? You probably shouldn't hold your breath to win a spot in your country's government. But depending on where you live, you may still be selected to participate in a jury, a citizens' assembly, or a deliberative poll, all examples of how the democratic principle behind sortition still survives today. |
World_History_TED_Talks | 화약의_치명적인_역설_에릭_로사도_Eric_Rosado.txt | Everybody loves fireworks -- the lights, the colors, and, of course, the big boom. But the history of fireworks isn't all hugs and celebrations. Long before epic fireworks displays, chemists in China invented the key ingredient that propels those bright lights into the sky. That invention was what we now call gunpowder. Our story begins back in ancient China in the mid-ninth century where early Chinese alchemists were trying to create a potion for immortality. Instead, what they created was a flammable powder that burned down many of their homes. They quickly realized that this black powder, which they called fire medicine, was precisely the opposite of something that would make you live forever. In these early days, the Chinese hadn't yet figured out how to make the powder explode; it was simply very flammable, and their armies used it to make flaming arrows and even a flamethrower. But once they figured out the right proportions of ingredients to create a blast, they began using the powder even more, creating fireworks to keep evil spirits away and bombs to defend themselves against Mongol invaders. It was these Mongols, most likely, who spread the invention of gunpowder across the world. After fielding Chinese attacks, they learned how to produce the powder themselves and brought it with them on their conquests in Persia and India. William of Rubruck, a European ambassador to the Mongols, was likely responsible for bringing gunpowder back to Europe around 1254. From there, engineers and military inventors created all kinds of destructive weapons. From bombs to guns to cannons, gunpowder left its mark on the world in some pretty terrible ways, in contrast to the beautiful marks it can leave in the air. So, how does black powder propel fireworks into the sky? You might have seen old Westerns or cartoons where a trail of gunpowder is lit and it leads to a large and obviously explosive barrel. Once the fire gets to the barrel, a large boom occurs. But why doesn't the trail itself explode? The reason is that burning the powder releases energy and gases. While the trail is burning, these are easily released into the surrounding air. But when the gunpowder is contained within the barrel, the energy and gases cannot easily escape and build up until BOOM! Firework canisters provide a single, upward-facing outlet to channel this explosive energy. The wick ignites the gunpowder and the energy takes the easiest exit from the canister, launching the firework high into the sky. The flame then makes its way through the firework's encasing and the same reaction occurs high above our heads. So, while the Chinese alchemists never found the compound for eternal life, they did find something that would go on to shape all of civilization, something that has caused many tragic moments in human history, and yet still gives us hope when we look up in celebration at the colorful night sky. |
World_History_TED_Talks | 햇빛은_당신의_생각보다_나이가_많습니다_스텐_오덴왈드.txt | You may know that it takes light a zippy eight minutes to reach us from the surface of the Sun, so how long do you think it takes light to travel from the Sun's core to its surface? A few seconds or a minute at most? Well, oddly enough, the answer is many thousands of years. Here's why. Photons are produced by the nuclear reactions deep in the core of our Sun. As the photons flow out of the core, they interact with matter and lose energy, becoming longer wavelength forms of light. They start out as gamma rays in the core, but end up as x-rays, ultraviolet or visible light as they near the surface. However, that journey is neither simple nor direct. Upon being born, each photon travels at a speed of 300,000 kilometers per second until it collides with a proton and is diverted in another direction, acting like a bullet ricocheting off of every charged particle it strikes. The question of how far this photon gets from the center of the Sun after each collision is known as the random walk problem. The answer is given by this formula: distance equals step size times the square root of the number of steps. So if you were taking a random walk from your front door with a one meter stride each second, it would take you a million steps and eleven days just to travel one kilometer. So then how long does it take for a photon generated in the center of the sun to reach you? We know the mass of the Sun and can use that to calculate the number of protons within it. Let's assume for a second that all the Sun's protons are evenly spread out, making the average distance between them about 1.0 x 10^-10 meters. To random walk the 690,000 kilometers from the core to the solar surface would then require 3.9 x 10^37 steps, giving a total travel time of 400 billion years. Hmm, that can't be right. The Sun is only 4.6 billion years old, so what went wrong? Two things: The Sun isn't actually of uniform density and photons will miss quite a few protons between every collision. In actuality, a photon's energy, which changes over the course of its journey, determines how likely it is to interact with a proton. On the density question, our models show that the Sun has a hot core, where the fusion reactions occur. Surrounding that is the radiative zone, followed by the convective zone, which extends all the way to the surface. The material in the core is much denser than lead, while the hot plasma near the surface is a million times less dense with a continuum of densities in between. And here's the photon-energy relationship. For a photon that carries a small amount of energy, a proton is effectively huge, and it's much more likely to cause the photon to ricochet. And for a high-energy photon, the opposite is true. Protons are effectively tiny. Photons start off at very high energies compared to when they're finally radiated from the Sun's surface. Now when we use a computer and a sophisticated solar interior model to calculate the random walk equation with these changing quantities, it spits out the following number: 170,000 years. Future discoveries about the Sun may refine this number further, but for now, to the best of our understanding, the light that's hitting your eyes today spent 170,000 years pinballing its way towards the Sun's surface, plus eight miniscule minutes in space. In other words, that photon began its journey two ice ages ago, around the same time when humans first started wearing clothes. |
World_History_TED_Talks | 고대_로마의_4자매_레이_로렌스_Ray_Laurence.txt | Translator: Andrea McDonough Reviewer: Jessica Ruby Today, we're going to look at the world of Rome through the eyes of a young girl. Here she is, drawing a picture of herself in the atrium of her father's enormous house. Her name is Domitia, and she is just 5 years old. She has an older brother who is fourteen, Lucius Domitius Ahenobarbus, named after her dad. Girls don't get these long names that boys have. What is worse is that Dad insists on calling all his daughters Domitia. "Domitia!" His call to Domitia drawing on the column, Domitia III. She has an older sister, Domitia II, who is 7 years old. And then there's Domitia I, who is ten. There would have been a Domitia IV, but mom died trying to give birth to her three years ago. Confused? The Romans were too. They could work out ancestry through the male line with the nice, tripartite names such as Lucius Domitius Ahenobarbus. But they got in a real mess over which Domitia was married to whom and was either the great aunt or the great stepmother and so on to whom when they came to write it down. Domitia III is not just drawing on the pillar, she's also watching the action. You see, it's early, in the time of day when all her dad's clients and friends come to see him at home to pay their respects. Lucius Popidius Secundus, a 17 year old, he wants to marry Domitia II within the next five to seven years, has come as well. He seems to be wooing not his future wife, but her dad. Poor Lucius, he does not know that Domitia's dad thinks he and his family are wealthy but still scumbags from the Subura. Afterall, it is the part of Rome full of barbers and prostitutes. Suddenly, all the men are leaving with Dad. It's the second hour and time for him to be in court with a sturdy audience of clients to applaud his rhetoric and hiss at his opponent. The house is now quieter. The men won't return for seven hours, not until dinner time. But what happens in the house for those seven hours? What do Domitia, Domitia, and Domitia do all day? Not an easy question! Everything written down by the Romans that we have today was written by men. This makes constructing the lives of women difficult. However, we can't have a history of just Roman men, so here it goes. We can begin in the atrium. There is a massive loom, on which Dad's latest wife is working on a new toga. Domitia, Domitia, and Domitia are tasked with spinning the wool that will be used to weave this mighty garment, 30 or more feet long and elliptical in shape. Romans loved the idea that their wives work wool. We know that because it's written on the gravestones of so many Roman women. Unlike women in Greece, Roman women go out the house and move about the city. They go to the baths in the morning to avoid the men or to separate baths that are for women only. Some do go in for the latest fad of the AD 70s: nude bathing with men present. Where they have no place is where the men are: in the Forum, in the Law Court, or in the Senate House. Their place in public is in the porticos with gardens, with sculpture, and with pathways for walking in. When Domitia, Domitia, and Domitia want to leave the house to go somewhere, like the Portico of Livia, they must get ready. Domitia II and Domitia III are ready, but Domitia I, who is betrothed to be married in two years to darling Philatus, isn't ready. She's not slow, she just has more to do. Being betrothed means she wears the insignia of betrothal: engagement rings and all the gifts Pilatus has given her - jewels, earrings, necklaces, and the pendants. She may even wear her myrtle crown. All this bling shouts, "I'm getting married to that 19 year old who gave me all this stuff I'm wearing!" While as they wait, Domitia II and Domitia III play with their dolls that mirror the image of their sister decked out to be married. One day, these dolls will be dedicated to the household gods on the day of their wedding. Okay, we're ready. The girls step into litters carried by some burly slaves. They also have a chaperone with them and will be meeting an aunt at the Porticus of Livia. Carried high on the shoulders of these slaves, the girls look out through the curtains to see the crowded streets below them. They traverse the city, pass the Coliseum, but then turn off to climb up the hill to the Porticus of Livia. It was built by Livia, the wife of the first emperor Augustus, on the site of the house of Vedius Pollio. He wasn't such a great guy. He once tried to feed a slave to the eels in his fish pond for simply dropping a dish. Luckily, the emperor was at the dinner and tamed his temper. The litters are placed on the ground and the girls get out and arm in arm, two by two, they ascend the steps into the enclosed garden with many columns. Domitia III shot off and is drawing on a column. Domitia II joins her but seeks to read the graffiti higher up on the column. She spots a drawing of gladiators and tries to imagine seeing them fighting, something she will never be permitted to do, except from the very rear of the Coliseum. From there, she will have a good view of the 50,000 spectators but will see little by way of blood and gore. If she really wanted a decent view, she could become a vestal virgin and would sit right down the front. But a career tending the sacred flame of Vesta is not to everybody's taste. Domitia I has met another ten year old also decked out in the insignia of betrothal. Home time. When they get there after the eighth hour, something is up. A smashed dish lies on the floor. All the slaves are being gathered together in the atrium and await the arrival of their master. Dad is going to go mad. He will not hit his children, but like many other Romans, he believes that slaves have to be punished. The whip lies ready for his arrival. No one knows who smashed the dish, but Dad will call the undertaker to torture it out of them, if he must. The doorkeeper opens the front door to the house. A hush comes over the anxious slaves. In walks not their master but, instead, a pregnant teenager. It is the master's eldest daughter, age 15, who is already a veteran of marriage and child birth. Guess what her name is. There is a five to ten percent chance she won't survive giving birth to her child, but, for now, she has come to dinner with her family. As a teenage mother, she has proved that she is a successful wife by bringing children and descendants for her husband, who will carry on his name in the future. The family head off to the dining room and are served dinner. It would seem Dad has had an invite to dinner elsewhere. With dinner concluded, the girls crossed the atrium to bid farewell to their older sister who is carried home in a litter, escorted by some of Dad's bodyguards. Returning to the house, the girls cross the atrium. The slaves, young and old, male and female, await the return of their owner. When he returns, he may exact vengeance, ensuring his power over the slaves is maintained through violence and terror, to which any slave could be subjected. But, for the girls, they head upstairs for the night, ready for bed. |
World_History_TED_Talks | 예술에서의_종교의_짧은_역사_TEDEd.txt | It's only been the last few hundreds years or so that Western civilization has been putting art in museums, at least museums resembling the public institutions we know today. Before this, for most, art served other purposes. What we call fine art today was, in fact, primarily how people experienced an aesthetic dimension of religion. Paintings, sculpture, textiles and illuminations were the media of their time, supplying vivid imagery to accompany the stories of the day. In this sense, Western art shared a utilitarian purpose with other cultures around the world, some of whose languages incidentally have no word for art. So how do we define what we call art? Generally speaking, what we're talking about here is work that visually communicates meaning beyond language, either through representation or the arrangement of visual elements in space. Evidence of this power of iconography, or ability of images to convey meaning, can be found in abundance if we look at art from the histories of our major world religions. Almost all have, at one time or another in their history, gone through some sort of aniconic phase. Aniconism prohibits any visual depiction of the divine. This is done in order to avoid idolatry, or confusion between the representation of divinity and divinity itself. Keeping it real, so to speak, in the relationship between the individual and the divine. However, this can be a challenge to maintain, given that the urge to visually represent and interpret the world around us is a compulsion difficult to suppress. For example, even today, where the depiction of Allah or the Prophet Muhammad is prohibited, an abstract celebration of the divine can still be found in arabesque patterns of Islamic textile design, with masterful flourishes of brushwork and Arabic calligraphy, where the words of the prophet assume a dual role as both literature and visual art. Likewise, in art from the early periods of Christianity and Buddhism, the divine presence of the Christ and the Buddha do not appear in human form but are represented by symbols. In each case, iconographic reference is employed as a form of reverence. Anthropomorphic representation, or depiction in human form, eventually became widespread in these religions only centuries later, under the influence of the cultural traditions surrounding them. Historically speaking, the public appreciation of visual art in terms other than traditional, religious or social function is a relatively new concept. Today, we fetishize the fetish, so to speak. We go to museums to see art from the ages, but our experience of it there is drastically removed from the context in which it was originally intended to be seen. It might be said that the modern viewer lacks the richness of engagement that she has with contemporary art, which has been created relevant to her time and speaks her cultural language. It might also be said that the history of what we call art is a conversation that continues on, as our contemporary present passes into what will be some future generation's classical past. It's a conversation that reflects the ideologies, mythologies, belief systems and taboos and so much more of the world in which it was made. But this is not to say that work from another age made to serve a particular function in that time is dead or has nothing to offer the modern viewer. Even though in a museum setting works of art from different places and times are presented alongside each other, isolated from their original settings, their juxtaposition has benefits. Exhibits are organized by curators, or people who've made a career out of their ability to recontextualize or remix cultural artifacts in a collective presentation. As viewers, we're then able to consider the art in terms of a common theme that might not be apparent in a particular work until you see it alongside another, and new meanings can be derived and reflected upon. If we're so inclined, we might even start to see every work of art as a complementary part of some undefined, unified whole of past human experience, a trail that leads right to our doorstep and continues on with us, open to anyone who wants to explore it. |
World_History_TED_Talks | This_is_Sparta_Fierce_warriors_of_the_ancient_world_Craig_Zimmer.txt | In ancient Greece, violent internal conflict between bordering neighbors and war with foreign invaders was a way of life, and Greeks were considered premier warriors. Most Greek city-states surrounded themselves with massive defensive walls for added protection. Sparta in its prime was a different story, finding walls unnecessary when it had an army of the most feared warriors in the ancient world. So what was Sparta doing differently than everyone else to produce such fierce soldiers? To answer that question, we turn to the written accounts of that time. There are no surviving written accounts from Spartans themselves, as it was forbidden for Spartans to keep records, so we have to rely on those of non-Spartan ancient historians, like Herodotus, Thucydides, and Plutarch. These stories may be embellished and depict Sparta at the apex of its power, so take them with a grain of salt. For Spartans, the purpose for their existence was simple: to serve Sparta. On the day of their birth, elder Spartan leaders examined every newborn. The strong healthy babies were considered capable of fulfilling this purpose, and the others may have been left on Mount Taygetus to die. Every Spartan, boy or girl, was expected to be physically strong, mentally sharp, and emotionally resilient. And it was their absolute duty to defend and promote Sparta at all costs. So in the first years of their lives, children were raised to understand that their loyalty belonged first to Sparta, and then to family. This mindset probably made it easier for the Spartan boys, who upon turning seven, were sent to the agoge, a place with one main purpose: to turn a boy into a Spartan warrior through thirteen years of relentless, harsh, and often brutal training. The Spartans prized physical perfection above all else, and so the students spent a great deal of their time learning how to fight. To ensure resilience in battle, boys were encouraged to fight among themselves, and bullying, unlike today, was acceptable. In order to better prepare the boys for the conditions of war, the boys were poorly fed, sometimes even going days without eating. They also were given little in the way of clothing so that they could learn to deal with different temperatures. Spartan boys were encouraged to steal in order to survive, but if they were caught, they would be disciplined, not because they stole, but because they were caught in the act. During the annual contest of endurance in a religious ritual known as the diamastigosis, teenage boys were whipped in front of an altar at the Sanctuary of Artemis Orthia. It was common for boys to die on the altar of the goddess. Fortunately, not everything was as brutal as that. Young Spartans were also taught how to read, write, and dance, which taught them graceful control of their movements and helped them in combat. While the responsibilities for the girls of Sparta were different, the high standards of excellence and expectation to serve Sparta with their lives remained the same. Spartan girls lived at home with their mothers as they attended school. Their curriculum included the arts, music, dance, reading, and writing. And to stay in peak physical condition, they learned a variety of sports, such as discus, javelin, and horseback riding. In Sparta, it was believed that only strong and capable women could bear children that would one day become strong and capable warriors. To all Spartans, men and women, perhaps the most important lesson from Spartan school was allegiance to Sparta. To die for their city-state was seen as the completion of one's duty to Sparta. Upon their death, only men who died in battle and women who died in childbirth were given tombstones. In the eyes of their countrymen, both died so that Sparta could live. |
World_History_TED_Talks | 고대_올림픽의_기원_아마드_디안고어_Armand_DAngour.txt | Thousands of years in the making, what began as part of a religious festival honoring the Greek god Zeus in the rural Greek town of Olympia has today become the greatest show of sporting excellence on Earth. The inception date in 776 BC became the basis for the Greek's earliest calendar, where time was marked in four-year increments called olympiads. What could it be? Why, it's the Olympic games, of course. Competition fosters excellence, or so thought the Ancient Greeks. In addition to sporting events, contests were held for music, singing, and poetry. You can read about them all yourself in classical literary works, like Homer's "Iliad" and Virgil's "Aeneid." Even mythical heroes appreciate a good contest every now and then, wouldn't you say? For the first thirteen games, the Ancient Greek Olympics featured just one event, the two hundred yard dash. But over time, new exciting contests, like boxing, chariot and mule racing, and even a footrace where the competitors wore a full suit of armor enticed many hopeful champions into the Olympic stadium. The combined running, jumping, wrestling, javelin throwing, and hurling the discus events known as the pentathlon inspired world-class competition, and the pankration, a no holds barred fight where only biting and eye-gouging were prohibited, ensured the toughest men were victorious. And victorious they were. Nobody tops the local baker Coroebus, who 776 BC became the very first Olympic champion. And we'll never forget Orsippus of Megara, the 720 BC Olympic victor tore away his loincloth so he could race unimpeded, inaugurating the Ancient Greek tradition of competing in the nude. Now there's a winning streak, if ever we've seen one. But all good things must end. In 391 AD, the Christian Roman Emperor Theodosius banned pagan practices, so the world soon bid a fond farewell to the Olympic games. But just like those early pankration athletes, you can't keep a good one down, and 1500 years later in 1896, the modern Olympic games kicked off in Athens, Greece. Today, the Summer and Winter Olympics bring international world-class athletes together by the thousands, uniting fans by the billions for the world's foremost sporting competition. Citius, Altius, Fortius. Three cheers for the Olympics. |
World_History_TED_Talks | A_TED_Talk_on_the_History_of_Everything_Big_History_Project.txt | [Music] [Music] first a video yes it is a scrambled egg but as you look at it I hope you'll begin to feel just slightly uneasy because you may notice that what's actually happening is that the egg is on scrambling itself and you'll now see the yolk and the white have separated and now they're going to be poured back into the egg and we all know in our heart of hearts that this is not the way the universe works a scrambled egg is mush tasty mush but it's mush an egg is a beautiful sophisticated thing that can create even more sophisticated things such as chickens and we know in our heart of hearts that the universe does not travel from mush to complexity in fact this gut instinct is reflected in one of the most fundamental laws of physics the second law of thermodynamics or the law of entropy what that says basically is that the general tendency of the universe is to move from order and structure to lack of order lack of structure in fact to mush and that's why that video feels a bit strange and yet look around us what we see around us is staggering complexity Eric pine hotter estimates that in New York City alone there are some 10 billion skews or distinct commodities being traded that's hundreds of times as many species as there are on earth and they're being traded by a species of almost 7 billion individuals who are linked by trade travel and the internet into a global system of stupendous complexity so here's a great puzzle in a universe ruled by the second law of thermodynamics how is it possible to generate the sort of complexity I've described the sort of complexity represented by you and me and the convention center well the answer seems to be the universe can create complexity but with great difficulty in pockets there appear what my colleague Fred's vehicle's Goldilocks conditions not too hot not too cold just right for the creation of complexity and slightly more complex things appear and where you have slightly more complex things you can get slightly more complex things and in this way complexity builds stage by stage each stage is magical because it creates the impression of something utterly new appearing almost out of nowhere in the universe we refer in big history to these moments as threshold moments and at each threshold the going gets tougher the complex things get more fragile more vulnerable the Goldilocks conditions get more stringent and it's more difficult to create complexity now we as extremely complex creatures desperately need to know this story of how the universe creates complexity despite the second law and why complexity means vulnerability and fragility and that's the story that we tell in big history but to do it you have to do something that may have first sight seemed completely impossible you have to survey the whole history of the universe so let's do it let's begin by winding the timeline back 13.7 billion years to the beginning of time around us there's nothing there's not even time or space imagine the darkest emptiest thing you can and cube it a gazillion times and that's where we are and then suddenly a universe appears and the entire universe and we've crossed our first threshold the universe is tiny it's smaller than an atom it's incredibly hot it contains everything that's in today's universe so we can imagine it's busting and it's expanding at incredible speed and at first it's just a blur but very quickly distinct things begin to appear in that blur within the first second energy itself shatters into distinct forces including electromagnetism and gravity and energy does something else quite magical it congeals to form matter quarks that will create protons and leptons that include electrons and all of that happens in the first second now we move forward 380,000 years that's twice as long as humans have been on this planet and now simple atoms appear of hydrogen and helium now I want to pause for a moment 380,000 years after the origins of the universe because we actually know quite a lot about the universe at this stage we know above all that it was extremely simple it consisted of huge clouds of hydrogen and helium atoms and they have no structure they're really a sort of cosmic mush but that's not completely true recent studies by satellites such as the W map satellite have shown that in fact there are just tiny differences in that background what you see here the blue areas are about a thousandth of a degree cooler than the red areas these are tiny differences but it was enough for the universe to move on to the next stage of building complexity and this how it works gravity is more powerful where there's more stuff so where you get slightly denser areas gravity starts compacting clouds of hydrogen and helium atoms so we can imagine the early universe breaking up into a billion clouds and each cloud is compacted gravity gets more powerful as density increases the temperature begins to rise at the center of each cloud and then at the center of each cloud the temperature crosses the threshold temperature of 10 million degrees protons start to fuse there's a huge release of energy and bam we have our first stars from about 200 million years after the Big Bang stars begin to appear all through the universe billions of them and the universe is now significantly more interesting and more complex stars will create the Goldilocks conditions for crossing to new thresholds when very large stars die they create temperatures so high that protons begin to fuse in all sorts of exotic combinations to form all the elements of the periodic table if like me you're wearing a gold ring it was forged in a supernova explosion so now the universe is chemically more complex and in a chemically more complex universe it's possible to make more things and earth starts happening is that around young sons young stars all these elements combine they swirl around the energy of the star stirs them around they form the particles they form snowflakes they form little dust motes they form rocks they form asteroids and eventually they form planets and moons and that is how our solar system was formed four-and-a-half billion years ago rocky planets like our earth are significantly more complex than stars because they contain a much greater diversity of materials so we've crossed a fourth threshold of complexity now the going gets tougher the next stage introduces entities that are significantly more fragile significantly more but they also much more creative and much more capable of generating further complexity I'm talking of course about living organisms living organisms are created by chemistry we are huge packages of chemicals so chemistry is dominated by the electromagnetic force that operates over smaller scales and gravity which explains why you and I are smaller than stars or planets now what are the ideal conditions for chemistry what are the Goldilocks conditions well the first you need energy but not too much in the center of a star there's so much energy that any atoms that combine will just get busted apart again but not too little in intergalactic space there's so little energy that atoms can't combine what you want is just the right amount and planets it turns out are just right because they're close to stars but not too close you also need a great diversity of chemical elements and you need liquids such as water why well in gases atoms move past each other so fast that they can't hitch up in solids atoms stuck together they can't move in liquids they can cruise and cuddle and link up to form molecules now where do you find such Goldilocks conditions well planets are great and our early Earth was almost perfect it was just the right distance from its star to contain huge oceans of liquid water and deep beneath those oceans that cracks in the Earth's crust you've got heat seeping up from inside the earth and you've got a great diversity of elements so at those deep oceanic vents fantastic chemistry began to happen and atoms combined in all sorts of exotic combinations but of course life is more than just exotic chemistry how do you stabilize those huge molecules that seem to be viable well it's here that life introduces an entirely new trick you don't stabilize the individual you stabilize the template the thing that carries information and allow the template to copy itself and DNA of course is the beautiful molecule that contains that information you'll be familiar with the double helix of DNA each rung contains information so DNA contains information about how to make living organisms and DNA also copies itself so it copies itself and scatters the templates through the ocean so the information spreads notice that information has become part of our story the real beauty of DNA though is in its imperfections as it copies itself once in every billion runs there tends to be an error and what that means is that DNA is in effect learning it's accumulating new ways of making living organisms because some of those errors work so DNA is learning and it's building greater diversity and greater complexity and we can see this happening over the last 4 billion years for most of that time of life on earth living organisms have been relatively simple single cells but they had great diversity and inside great complexity then from about 600 to 800 million years ago multi-celled organisms appear you get fungi you get fish you get plants you get amphibia you get reptiles and then of course you get the dinosaurs and occasionally there are disasters 65 million years ago an asteroid landed on earth near the Yucatan Peninsula creating conditions equivalent to those of a nuclear war and the dinosaurs are wiped out terrible news for the dinosaurs but great news for our mammalian ancestors who flourished in the niches left empty by the dinosaurs and we human beings are part of that creative evolutionary pulse that began 65 million years ago with the landing of an asteroid humans appeared about 200,000 years ago and I believe we count as a threshold in this great story let me explain why we've seen that DNA learns in a sense it accumulates information but is so slow DNA accumulates information through random errors that just some of which just happened to work but DNA had actually generated a faster way of learning it had produced organisms with brains and those organisms can learn in real time they accumulate information they learn the sad thing is when they die the information dies with them now what makes humans different is human language we are blessed with a language a system of communication so powerful and so precise that we can share what we've learned with such precision that it can accumulate in the collective memory and that means it can outlast the individuals who learnt that information and it can accumulate from generation to generation and that's why as a species we're so creative and so powerful and that's why we have a history we seem to be the only species in four billion years to have this gift I call this ability collective learning it's what makes us different we can see that work in the earliest stages of human history we evolved as a species in the savanna lands of Africa but then you see humans migrating into new environments into desert lands into jungles into the Ice Age tundra of Siberia tough tough environment into the Americas into Australasia each migration involved learning learning new ways of exploiting the environment new ways of dealing with their surroundings then 10,000 years ago exploiting a sudden change in global climates with the end of the last ice age humans learnt to farm farming was an energy bonanza and exploiting that energy human populations multiplied human societies got larger denser more interconnected and then from about 500 years ago humans began to link up globally through shipping through trains through telegraph through the internet until now we seem to form a single global brain of almost 7 billion individuals and that brain is learning at warp speed and that the last 200 years something else has happened we've stumbled on another energy bonanza in fossil fuels so fossil fuels and collective learning together explain the staggering complexity we see around us so here we are back at the Convention Center we've been on a journey a return journey of 13.7 billion years I hope you agree this is a powerful story and it's a story in which humans play an astonishing and creative role but it also contains warnings collective learning is a very very powerful force and it's not clear that we humans are in charge of it I remember very vividly as a child growing up in England living through the Cuban Missile Crisis for a few days the entire biosphere seemed to be on the verge of destruction and the same weapons are still here and they're still armed if we avoid that trap others are waiting for us we're burning fossil fuels at such a rate that we seemed to be undermining the Goldilocks conditions that made it possible for human civilizations to flourish over the last 10,000 years so what big history can do is show us the nature of our complexity and fragility and the dangers that face us but it can also show us our power with collective learning and now finally this is what I want I want my grandson Daniel and his friends and his generation throughout the world to know the story of big history and to know it so well that they understand both the challenges that face us and the opportunities that face us and that's why a group of us are building a free online syllabus in big history for high school students throughout the world we believe that big history will be a vital intellectual tool for them as Daniel and his generation face the huge challenges and also the huge opportunities ahead of them at this threshold moment in the history of our beautiful planet I thank you for your attention [Applause] [Music] the precision of a watch is a function of its movement for Rolex and for hands whilst off to guarantee the precision of a timepiece the pressing question was how to protect the movement itself from the elements not only water but also tiny particles of dust in 1926 a major step was taken with the creation of the world's first waterproof and dustproof wristwatch the Rolex Oyster was born over the years subtle changes in the design continued to improve the oyster adding more comfort while keeping the style contemporary and along with style more functions have been added a Rolex wristwatch was the first to show the date through a small aperture on the face it was also the first wristwatch to spell out the day of the week in foam [Music] in the early 1950s Rolex developed professional watches whose functions went far beyond telling the time launched in 1953 the Submariner was the first Rolex watch guaranteed waterproof to a depth of 100 meters already on an incredible journey of innovation and design Rolex decided to push the boundaries even further in 1960 the bathyscaphe Trieste and Rolex made history the submersible successfully dived to ten thousand nine hundred and sixteen metres below the surface of the ocean a Rolex deep-sea special was strapped to the outside the development of undersea exploration led to the launching in 1967 of the sea dweller mm water proved to a depth of 610 metres in 2008 fitted with the patented rolex rim lock system the Rolex deep-sea safely descends to 3,900 metres the Submariner continues to evolve in 2008 the model in gold is redesigned and features a new unidirectional rotatable bezel with a sarah chrome disc and two years later the steel Submariner is introduced with a green color combination Rolex has incorporated countless hours and more than a century of experience years of research innovation and development into every one of its models [Music] and the benefits arising from this work including waterproofness precision and durability are the result of groel X's continuous pursuit of perfection [Music] from the most elegant and prestigious models to the professional timepieces or our exquisitely confident [Music] piece by piece we design and manufacture every single watch [Applause] [Music] [Applause] [Music] and the story continues [Music] |
World_History_TED_Talks | The_Akune_brothers_Siblings_on_opposite_sides_of_war_Wendell_Oshiro.txt | There are many stories that can be told about World War II, from the tragic to the inspring. But perhaps one of the most heartrending experiences was that of the Akune family, divided by the war against each other and against their own identities. Ichiro Akune and his wife Yukiye immigrated to America from Japan in 1918 in search of opportunity, opening a small grocery store in central California and raising nine children. But when Mrs. Akune died in 1933, the children were sent to live with relatives in Japan, their father following soon after. Though the move was a difficult adjustment after having been born and raised in America, the oldest son, Harry, formed a close bond with his grand uncle, who taught him the Japanese language, culture, and values. Nevertheless, as soon as Harry and his brother Ken were old enough to work, they returned to the country they considered home, settling near Los Angeles. But then, December 7, 1941, the attack on Pearl Harbor. Now at war with Japan, the United States government did not trust the loyalty of those citizens who had family or ancestral ties to the enemy country. In 1942, about 120,000 Japanese Americans living on the West Coast were stripped of their civil rights and forcibly relocated to internment camps, even though most of them, like Harry and Ken, were Nisei, American or dual citizens who had been born in the US to Japanese immigrant parents. The brothers not only had very limited contact with their family in Japan, but found themselves confined to a camp in a remote part of Colorado. But their story took another twist when recruiters from the US Army's military intelligence service arrived at the camp looking for Japanese-speaking volunteers. Despite their treatment by the government, Harry and Ken jumped at the chance to leave the camp and prove their loyalty as American citizens. Having been schooled in Japan, they soon began their service, translating captured documents, interrogating Japanese soldiers, and producing Japanese language propaganda aimed at persuading enemy forces to surrender. The brothers' work was invaluable to the war effort, providing vital strategic information about the size and location of Japanese forces. But they still faced discrimination and mistrust from their fellow soldiers. Harry recalled an instance where his combat gear was mysteriously misplaced just prior to parachuting into enemy territory, with the white officer reluctant to give him a weapon. Nevertheless, both brothers continued to serve loyally through the end of the war. But Harry and Ken were not the only Akune brothers fighting in the Pacific. Unbeknownst to them, two younger brothers, the third and fourth of the five Akune boys, were serving dutifully in the Imperial Japanese Navy. Saburo in the Naval Airforce, and 15-year-old Shiro as an orientation trainer for new recruits. When the war ended, Harry and Ken served in the allied occupational forces and were seen as traitors by the locals. When all the Akune brothers gathered at a family reunion in Kagoshima for the first time in a decade, it was revealed that the two pairs had fought on opposing sides. Tempers flared and a fight almost broke out until their father stepped in. The brothers managed to make peace and Saburo and Shiro joined Harry and Ken in California, and later fought for the US Army in Korea. It took until 1988 for the US government to acknowledge the injustice of its internment camps and approve reparations payments to survivors. For Harry, though, his greatest regret was not having the courage to thank his Japanese grand uncle who had taught him so much. The story of the Akune brothers is many things: a family divided by circumstance, the unjust treatment of Japanese Americans, and the personal struggle of reconciling two national identities. But it also reveals a larger story about American history: the oppression faced by immigrant groups and their perseverance in overcoming it. |
World_History_TED_Talks | Neil_MacGregor_2600_years_of_history_in_one_object.txt | The things we make have one supreme quality -- they live longer than us. We perish, they survive; we have one life, they have many lives, and in each life they can mean different things. Which means that, while we all have one biography, they have many. I want this morning to talk about the story, the biography -- or rather the biographies -- of one particular object, one remarkable thing. It doesn't, I agree, look very much. It's about the size of a rugby ball. It's made of clay, and it's been fashioned into a cylinder shape, covered with close writing and then baked dry in the sun. And as you can see, it's been knocked about a bit, which is not surprising because it was made two and a half thousand years ago and was dug up in 1879. But today, this thing is, I believe, a major player in the politics of the Middle East. And it's an object with fascinating stories and stories that are by no means over yet. The story begins in the Iran-Iraq war and that series of events that culminated in the invasion of Iraq by foreign forces, the removal of a despotic ruler and instant regime change. And I want to begin with one episode from that sequence of events that most of you would be very familiar with, Belshazzar's feast -- because we're talking about the Iran-Iraq war of 539 BC. And the parallels between the events of 539 BC and 2003 and in between are startling. What you're looking at is Rembrandt's painting, now in the National Gallery in London, illustrating the text from the prophet Daniel in the Hebrew scriptures. And you all know roughly the story. Belshazzar, the son of Nebuchadnezzar, Nebuchadnezzar who'd conquered Israel, sacked Jerusalem and captured the people and taken the Jews back to Babylon. Not only the Jews, he'd taken the temple vessels. He'd ransacked, desecrated the temple. And the great gold vessels of the temple in Jerusalem had been taken to Babylon. Belshazzar, his son, decides to have a feast. And in order to make it even more exciting, he added a bit of sacrilege to the rest of the fun, and he brings out the temple vessels. He's already at war with the Iranians, with the king of Persia. And that night, Daniel tells us, at the height of the festivities a hand appeared and wrote on the wall, "You are weighed in the balance and found wanting, and your kingdom is handed over to the Medes and the Persians." And that very night Cyrus, king of the Persians, entered Babylon and the whole regime of Belshazzar fell. It is, of course, a great moment in the history of the Jewish people. It's a great story. It's story we all know. "The writing on the wall" is part of our everyday language. What happened next was remarkable, and it's where our cylinder enters the story. Cyrus, king of the Persians, has entered Babylon without a fight -- the great empire of Babylon, which ran from central southern Iraq to the Mediterranean, falls to Cyrus. And Cyrus makes a declaration. And that is what this cylinder is, the declaration made by the ruler guided by God who had toppled the Iraqi despot and was going to bring freedom to the people. In ringing Babylonian -- it was written in Babylonian -- he says, "I am Cyrus, king of all the universe, the great king, the powerful king, king of Babylon, king of the four quarters of the world." They're not shy of hyperbole as you can see. This is probably the first real press release by a victorious army that we've got. And it's written, as we'll see in due course, by very skilled P.R. consultants. So the hyperbole is not actually surprising. And what is the great king, the powerful king, the king of the four quarters of the world going to do? He goes on to say that, having conquered Babylon, he will at once let all the peoples that the Babylonians -- Nebuchadnezzar and Belshazzar -- have captured and enslaved go free. He'll let them return to their countries. And more important, he will let them all recover the gods, the statues, the temple vessels that had been confiscated. All the peoples that the Babylonians had repressed and removed will go home, and they'll take with them their gods. And they'll be able to restore their altars and to worship their gods in their own way, in their own place. This is the decree, this object is the evidence for the fact that the Jews, after the exile in Babylon, the years they'd spent sitting by the waters of Babylon, weeping when they remembered Jerusalem, those Jews were allowed to go home. They were allowed to return to Jerusalem and to rebuild the temple. It's a central document in Jewish history. And the Book of Chronicles, the Book of Ezra in the Hebrew scriptures reported in ringing terms. This is the Jewish version of the same story. "Thus said Cyrus, king of Persia, 'All the kingdoms of the earth have the Lord God of heaven given thee, and he has charged me to build him a house in Jerusalem. Who is there among you of his people? The Lord God be with him, and let him go up.'" "Go up" -- aaleh. The central element, still, of the notion of return, a central part of the life of Judaism. As you all know, that return from exile, the second temple, reshaped Judaism. And that change, that great historic moment, was made possible by Cyrus, the king of Persia, reported for us in Hebrew in scripture and in Babylonian in clay. Two great texts, what about the politics? What was going on was the fundamental shift in Middle Eastern history. The empire of Iran, the Medes and the Persians, united under Cyrus, became the first great world empire. Cyrus begins in the 530s BC. And by the time of his son Darius, the whole of the eastern Mediterranean is under Persian control. This empire is, in fact, the Middle East as we now know it, and it's what shapes the Middle East as we now know it. It was the largest empire the world had known until then. Much more important, it was the first multicultural, multifaith state on a huge scale. And it had to be run in a quite new way. It had to be run in different languages. The fact that this decree is in Babylonian says one thing. And it had to recognize their different habits, different peoples, different religions, different faiths. All of those are respected by Cyrus. Cyrus sets up a model of how you run a great multinational, multifaith, multicultural society. And the result of that was an empire that included the areas you see on the screen, and which survived for 200 years of stability until it was shattered by Alexander. It left a dream of the Middle East as a unit, and a unit where people of different faiths could live together. The Greek invasions ended that. And of course, Alexander couldn't sustain a government and it fragmented. But what Cyrus represented remained absolutely central. The Greek historian Xenophon wrote his book "Cyropaedia" promoting Cyrus as the great ruler. And throughout European culture afterward, Cyrus remained the model. This is a 16th century image to show you how widespread his veneration actually was. And Xenophon's book on Cyrus on how you ran a diverse society was one of the great textbooks that inspired the Founding Fathers of the American Revolution. Jefferson was a great admirer -- the ideals of Cyrus obviously speaking to those 18th century ideals of how you create religious tolerance in a new state. Meanwhile, back in Babylon, things had not been going well. After Alexander, the other empires, Babylon declines, falls into ruins, and all the traces of the great Babylonian empire are lost -- until 1879 when the cylinder is discovered by a British Museum exhibition digging in Babylon. And it enters now another story. It enters that great debate in the middle of the 19th century: Are the scriptures reliable? Can we trust them? We only knew about the return of the Jews and the decree of Cyrus from the Hebrew scriptures. No other evidence. Suddenly, this appeared. And great excitement to a world where those who believed in the scriptures had had their faith in creation shaken by evolution, by geology, here was evidence that the scriptures were historically true. It's a great 19th century moment. But -- and this, of course, is where it becomes complicated -- the facts were true, hurrah for archeology, but the interpretation was rather more complicated. Because the cylinder account and the Hebrew Bible account differ in one key respect. The Babylonian cylinder is written by the priests of the great god of Bablyon, Marduk. And, not surprisingly, they tell you that all this was done by Marduk. "Marduk, we hold, called Cyrus by his name." Marduk takes Cyrus by the hand, calls him to shepherd his people and gives him the rule of Babylon. Marduk tells Cyrus that he will do these great, generous things of setting the people free. And this is why we should all be grateful to and worship Marduk. The Hebrew writers in the Old Testament, you will not be surprised to learn, take a rather different view of this. For them, of course, it can't possibly by Marduk that made all this happen. It can only be Jehovah. And so in Isaiah, we have the wonderful texts giving all the credit of this, not to Marduk but to the Lord God of Israel -- the Lord God of Israel who also called Cyrus by name, also takes Cyrus by the hand and talks of him shepherding his people. It's a remarkable example of two different priestly appropriations of the same event, two different religious takeovers of a political fact. God, we know, is usually on the side of the big battalions. The question is, which god was it? And the debate unsettles everybody in the 19th century to realize that the Hebrew scriptures are part of a much wider world of religion. And it's quite clear the cylinder is older than the text of Isaiah, and yet, Jehovah is speaking in words very similar to those used by Marduk. And there's a slight sense that Isaiah knows this, because he says, this is God speaking, of course, "I have called thee by thy name though thou hast not known me." I think it's recognized that Cyrus doesn't realize that he's acting under orders from Jehovah. And equally, he'd have been surprised that he was acting under orders from Marduk. Because interestingly, of course, Cyrus is a good Iranian with a totally different set of gods who are not mentioned in any of these texts. (Laughter) That's 1879. 40 years on and we're in 1917, and the cylinder enters a different world. This time, the real politics of the contemporary world -- the year of the Balfour Declaration, the year when the new imperial power in the Middle East, Britain, decides that it will declare a Jewish national home, it will allow the Jews to return. And the response to this by the Jewish population in Eastern Europe is rhapsodic. And across Eastern Europe, Jews display pictures of Cyrus and of George V side by side -- the two great rulers who have allowed the return to Jerusalem. And the Cyrus cylinder comes back into public view and the text of this as a demonstration of why what is going to happen after the war is over in 1918 is part of a divine plan. You all know what happened. The state of Israel is setup, and 50 years later, in the late 60s, it's clear that Britain's role as the imperial power is over. And another story of the cylinder begins. The region, the U.K. and the U.S. decide, has to be kept safe from communism, and the superpower that will be created to do this would be Iran, the Shah. And so the Shah invents an Iranian history, or a return to Iranian history, that puts him in the center of a great tradition and produces coins showing himself with the Cyrus cylinder. When he has his great celebrations in Persepolis, he summons the cylinder and the cylinder is lent by the British Museum, goes to Tehran, and is part of those great celebrations of the Pahlavi dynasty. Cyrus cylinder: guarantor of the Shah. 10 years later, another story: Iranian Revolution, 1979. Islamic revolution, no more Cyrus; we're not interested in that history, we're interested in Islamic Iran -- until Iraq, the new superpower that we've all decided should be in the region, attacks. Then another Iran-Iraq war. And it becomes critical for the Iranians to remember their great past, their great past when they fought Iraq and won. It becomes critical to find a symbol that will pull together all Iranians -- Muslims and non-Muslims, Christians, Zoroastrians, Jews living in Iran, people who are devout, not devout. And the obvious emblem is Cyrus. So when the British Museum and Tehran National Musuem cooperate and work together, as we've been doing, the Iranians ask for one thing only as a loan. It's the only object they want. They want to borrow the Cyrus cylinder. And last year, the Cyrus cylinder went to Tehran for the second time. It's shown being presented here, put into its case by the director of the National Museum of Tehran, one of the many women in Iran in very senior positions, Mrs. Ardakani. It was a huge event. This is the other side of that same picture. It's seen in Tehran by between one and two million people in the space of a few months. This is beyond any blockbuster exhibition in the West. And it's the subject of a huge debate about what this cylinder means, what Cyrus means, but above all, Cyrus as articulated through this cylinder -- Cyrus as the defender of the homeland, the champion, of course, of Iranian identity and of the Iranian peoples, tolerant of all faiths. And in the current Iran, Zoroastrians and Christians have guaranteed places in the Iranian parliament, something to be very, very proud of. To see this object in Tehran, thousands of Jews living in Iran came to Tehran to see it. It became a great emblem, a great subject of debate about what Iran is at home and abroad. Is Iran still to be the defender of the oppressed? Will Iran set free the people that the tyrants have enslaved and expropriated? This is heady national rhetoric, and it was all put together in a great pageant launching the return. Here you see this out-sized Cyrus cylinder on the stage with great figures from Iranian history gathering to take their place in the heritage of Iran. It was a narrative presented by the president himself. And for me, to take this object to Iran, to be allowed to take this object to Iran was to be allowed to be part of an extraordinary debate led at the highest levels about what Iran is, what different Irans there are and how the different histories of Iran might shape the world today. It's a debate that's still continuing, and it will continue to rumble, because this object is one of the great declarations of a human aspiration. It stands with the American constitution. It certainly says far more about real freedoms than Magna Carta. It is a document that can mean so many things, for Iran and for the region. A replica of this is at the United Nations. In New York this autumn, it will be present when the great debates about the future of the Middle East take place. And I want to finish by asking you what the next story will be in which this object figures. It will appear, certainly, in many more Middle Eastern stories. And what story of the Middle East, what story of the world, do you want to see reflecting what is said, what is expressed in this cylinder? The right of peoples to live together in the same state, worshiping differently, freely -- a Middle East, a world, in which religion is not the subject of division or of debate. In the world of the Middle East at the moment, the debates are, as you know, shrill. But I think it's possible that the most powerful and the wisest voice of all of them may well be the voice of this mute thing, the Cyrus cylinder. Thank you. (Applause) |
MIT_Learn_Differential_Equations | Impulse_Response_and_Step_Response.txt | GILBERT STRANG: OK. So this is a video in which we go for second-order equations, constant coefficients. We look for the impulse response, the key function in this whole business, and the step response, too. So those are the responses. So I'm going to call g-- that will be the impulse response, where the right-hand side is a delta function, an impulse, a sudden force at the moment t equals 0. So that's the equation. That's the impulse. And g is the response, and we want a formula for it. Then the other possibility, very interesting possibility, is when the right-hand side is a step function. And then we want the response to that function. I click a switch. The machine starts working, and it approaches a steady response. The solution rises from 0. So it starts at r of 0 equals r prime of 0 equals 0. The step response starts from rest. The action happens when I click a switch at t equals 0, and then r of t will rise to a constant. Very, very important solutions. But we'll focus especially on this one. OK. So that's our equation with the right-hand side delta. And of course, that right-hand side is not totally familiar, not as nice as e to the st. But there is something that-- there is another way to approach it-- that's a key idea here-- that gives us this all-important function from solving a null equation. How's that? I start with a null equation, but now this had no initial condition. So this one started from g of 0 and g prime of 0 both 0. And everything happened, boom, from the delta function. This is the same function. Except when I look to see what happens at t equals 0, what happens is g prime immediately jumps to 1. So another way I can approach g, the computation of g, is to think of it-- I'm just looking for a null solution. I'm looking for the null solution that starts from 0. But it starts with an initial derivative, slope equal 1. So I know that g is a combination. So I know how to solve equations like that, null equation. You remember s1 and s2? I look at s squared-- I've made this coefficient 1-- so s squared plus Bs plus C equals 0. That gives us s1 and s2. And now I'll tell you what the g is. So that gives us the s1 and the s2 in the null solution, and we're looking for a null solution. So our g of t is some combination of e to the s1t and e to the s2t. OK? It's some combination of those. And we want it to be 0. So no surprise, if I subtract those-- I'm starting at t equals 0. I'm starting-- this is 1 minus 1. It's 0. And now I just have to fix the initial slope, the first derivative, to be 1. Well, what's the derivative of this? This brings down an s1. This brings down an s2. At t equals 0, I'm getting an s1 minus s2. So I'll just divide by that, s1 minus s2. There you go. That's the impulse response-- a null solution that satisfies these special initial conditions. So that's the function in mathematics that's sometimes called the fundamental solution. It's a solution from which you can create all solutions. It's really the mother of solutions to this second-order differential equation. Because if I have another forcing function, this tells me that growth rate. It's just like e to the at for a first-order equation. Remember the growth rate e to the at for the simple first-order equation with interest rate coefficient a? Now we have two. Instead of a, we have an s1 and an s2, and that's the special function. OK. We need to get more insight on that for particular cases. So let me show you the same function when I have no damping. Start with that case, always the easiest case. When B is 0-- B was the damping coefficient, the first derivative in our differential equation. Can I just bring down the differential equation? When B is 0 here, that's no damping. I just have the second derivative and the function. That's when things oscillate forever. So that's what will happen. With B equals 0, I have pure oscillation. The s1 and s2 are cosines and sines that oscillate. Or it's neater to stay with exponentials, the i omega and minus i omega, where that's omega n, the natural frequency. Now, if I just plug in that s1 and s2-- the plus is s1 and the minus is s2-- I plug it in there, I get a nice formula for g of t. So that's what g of t looks like for no damping. It just oscillates. OK. The next case is underdamping. It's good to see all these cases each time. So that's a small value of B. Underdamping means there is some damping, but it's small enough so there's now a real part, but there's still an imaginary part. So this is, in a way, the trick is the case when s is complex. If I go higher with the dampening, increase B further, then I'll hit a point where there are two real solutions equal. And if I push B beyond that, I've got overdamping, and those two real solutions separate. They're different, but they're real. And then my formula, in that case, overdamping, that would be the best formula. But with underdamping, I can see the oscillation. If I just plug in those two solutions for s1 and s2, you'll see that I have the e to the minus B over 2t appears throughout. But then I have the sine of omega over omega, same as I had before, except now the damping frequency is a bit slower than the natural frequency. Damping slows that frequency down. And in a different video, we had a formula for omega damping, omega damping. And then increase the damping some more, then this part-- this omega damping goes to 0. We don't see any imaginary part in the solution. We see two equal real values. They're simple. They have to be just minus B/2. So that's a case of two s's coming together. And when two things come together, we're used to seeing a factor t appear. So I have that they came together at minus B/2. So I have the exponential of that. But I have a factor t from the merge of the two. And then if B increases beyond that, that's my formula. The two s's are real. I don't think one memorizes all this. I had to look them up and write them on the board before starting this video. But I hope you see that they're extremely nice. The no-damping case with [? pure ?] frequencies and the underdamping case with a real-- a decay. The critical damping when you increase B further, you just have that and no oscillation. And then beyond that is overdamping. OK. So we're good for the impulse response. And now I just have to say, what's the step response? So can I end this video by going back to my equation? I have to bring the board down to show it to you. So now I'm going to deal with the step response. So the equation is the same. I'm calling the solution r for response. And the point is, the right-hand side is now a step instead of a delta. So we'd like to solve that equation starting from rest. So a switch went on, and I want a formula for r of t. That's all that remains. And actually, that's just-- well, you can see what the particular solution is. We look at a particular solution. Well, this right-hand side is 1. This right-hand side is 1 beyond t equals 0. So I'm looking for a way to get 1 out of that. Well, or it can be just a constant. The particular solution is the steady state that we're approaching. And there's one little cool thing to do. Sometimes people who have the dimensions and units of things clearly in their mind will put a C in there. This is really the good thing to do is to have that C in there because now the units for r are the same as-- r is going to go to the 1. The steady state will be 1 now because I have Cr equals C times 1. And out in infinity, the simple solution is r equal 1. If when r is 1, its derivative is 0. Second derivative's 0. r equal 1 is a solution. It's a particular solution. It's the steady-state solution. Good. But that r of t equal 1 does not start correctly. We want to start at 0, with a slope of 0. So I have to subtract off one of these particular solutions with e to the s1. And now I have to get it so that I have to subtract it off so that this thing starts from 0. Let me see if I can do it. I think maybe if I do an s2 e to the s1t and subtract off an s1 e to the s2t. Do you see what that one has achieved? At t equals 0, I have-- well, at least at t equals 0, I've made the derivative 0 because the derivative will bring down an s1 there. That derivative will bring down an s2. And when I put t equals 0, I get s1, s2 minus s1, s2. Good, good, good. OK. Now I think that together they're all correct. I need to divide by s1 minus s2. Let me just say, I think that's it. I think that's it. It wouldn't be a bad idea if I just checked. And having checked, I've learned that that's a plus sign. OK. So the graph of r. This is a graph of r of t. It starts from 0, and it rises to 1. Asymptotically it's 1. This is a graph of r of t. And in practice, that's a very important number. What is the rise time? How far do you have to go in time before it rises up to, let's say, 95% of 1? All these questions are extremely practical questions for an engineer. What's the rise time? And you're playing with this formula. So let me just make another comment on this r of t step response. My other comment is I've emphasized g of t, the impulse response is like responsible for everything. It's always with us. And how are those connected? That's my final question in this video. How is r of t connected to g of t? Well, let me ask about the right-hand sides. How is the step function connected to the delta function? Answer, the step function is the integral of the delta function. The integral of the delta function is 0 as long as you're integrating off where the delta function is 0. But as soon as you pass the big spike, then the integral jumps to 1, and you have a step function. So the step function is the integral of the delta function. So the step response is the integral of the delta response. It's the integral of the impulse response. r is the integral of g. r is the integral of g with the correct initial conditions that gave us this and eventually [INAUDIBLE] approach 1. So that is the two key solutions, you could say. The impulse response important in theory and in practice. The step response extremely important in practice because turning on a switch is so basic an operation in engineering. Good. Thank you. |
MIT_Learn_Differential_Equations | Laplace_Transforms_and_Convolution.txt | PROFESSOR: OK. This is one more thing to tell you about Laplace transforms, and introducing a new word, convolution. And so we're going to find our old formula in new language, a new way. But the formula is familiar. And the problem is our basic problem, second order, linear, constant coefficient with a forcing term. And we know that the Laplace-- and I'll take zero boundary conditions. So that the Laplace transform is just s squared y, sy, and that's the transform of our equation. No problem. OK, now I'll divide by that. So I move that as 1 over, and I call it G. So this G is 1 over s squared, plus Bs plus C. And that has the name transfer function. And then this is the transform of the forcing term. OK. So here we have a nice formula for y of s, after I do that division. It's a product. The transform of the solution that we want is that transform times that transform. This is the transform of the impulse response. This is the transform of the right-hand side. Now I just have a Laplace transform question. Suppose my transform is one function of s times another function of s, what is the inverse transform? What is the inverse transform? What function y of t gives me G times F? And I'm just going to answer that. The answer is the g and the f, those are the ones that give that. But I do not just multiply those. The new operation that gives the right answer is called convolution. And I'll use a star. So right now I'm going to say what does that convolution mean. So this is a general question. If I have two functions multiplied together, then I want the inverse transform, then I take the separate inverse transforms, little g and little f, and I convolve them, I do convolution. And let me tell you what convolution is. So convolution is-- here is the formula for convolution. It's an integral from 0 to t of one function-- maybe I better use capital T, better-- times the other function, integrated. That's what convolution is. So what have I achieved here? The same old formula. The formula which we described way back at the beginning as inputs f, growth factors over the remaining time, g. Put all those together by integration. Put all the inputs with their growth factors. Integrate to put them all together. And that is y. So it's a familiar formula, with only a new word. But you see that I could jump to the answer, once I knew about the convolution formula, and I knew that this is the function whose transform its-- let me say again. Its transform is GF. So if I multiply transforms, I convolve functions. And looking at it the other way, if I multiply functions I would convolve their transforms. So convolution grows the number of functions that we can deal with on Laplace transform. Because it tells us what to do with products, capital G capital F. Or it tells us what to do with little g little f. So I'm almost through, because I don't plan to check. I could. But this isn't the right place. The book does it accurately. I don't plan to check that this statement is true that the transform of that one is that one. But it is true. But I do plan to do an example. Now second degree gets a little messy. So let me do a first degree example. Example, I'll take the equation dy dt minus ay. That's our usual first degree differential equation. And I'll take e to the ct on the right-hand side. OK. I'm doing those, because I can take the transforms and check everything. So let me transform both of those starting from 0. So the transform of that is s y of s, minus a y of s, equals, well I know the transform f of s. I know the transform of that is 1 over s minus c. So this is just, s minus a factors out. So well y of s is 1 over s minus a, and s minus c. Again, this is the simplest differential equation with a forcing term that I could use as an example. So now I'm looking for what is y of T. I'm looking for y of T. And I'm now going to use the language of convolution. This is the transform of e to the at. This is the transform of-- so you see I'm thinking of that as the transform of e to at, and the transform of e to the st. So there is one factor. And there's the other factor. So according to the convolution formula, I can write down the inverse transform, the y of t I want as the integral. I'm just going to copy the convolution. But I know the functions for that. So it's an integral from 0 to t. What do I have? g of t minus t. What is the inverse transform of 1 over s minus a? It's e to the a t minus t. And what is the inverse transform of 1 over s minus c? e to the cT dT. So I have used the-- I've just put in what I know in the convolution formula. And this should be the correct answer. And I can do this integral. And what do I get? Well, I'm pretty sure that I get e to the-- down below there will be a-- you see I'm going to combine those exponentials. So I'll have a c minus a. It comes out perfectly. e to the ct, minus e to the at. That's the right answer. It's not only what the convolution formula tells me, it's what I know. So that example is a good one to show that when-- so I didn't use partial fractions. Normally I would separate this into partial fractions, and then I would recognize those two pieces of the answer. I didn't do that this time. Instead of using partial fractions, the algebra, I used the convolution formula, and did the integral or almost did it. We can do it. And we get that answer. OK. So the point of this video is simply to introduce the idea of a convolution, which is the quantity we need, the function we need, when the transform is a product of two transforms. OK. Thank you. |
MIT_Learn_Differential_Equations | Similar_Matrices.txt | GILBERT STRANG: OK, thanks. Here's a second video that involves the matrix exponential. But it has a new idea in it, a basic new idea. And that idea is two matrices being called "similar." So that word "similar" has a specific meaning, that a matrix A, is similar to another matrix B, if B comes from A this way. Notice this way. It means there's some matrix M-- could be any invertible matrix. So that I take A, multiply on the right by M and on the left by M inverse. That'd probably give me a new matrix. Call it B. That matrix is called "similar" to B. I'll show you examples of matrices that are similar. But first is to get this definition in mind. So in general, a lot of matrices are similar to-- if I have a certain matrix A, I can take any M, and I'll get a similar matrix B. So there are lots of similar matrices. And the point is all those similar matrices have the same eigenvalues. So there's a little family of matrices there, all similar to each other and all with the same eigenvalues. Why do they have the same eigenvalues? I'll just show you, one line. Suppose B has an eigenvalue of lambda. So B is M inverse AM. So I have this. M inverse AMx is lambda x. That's Bx. B has an eigenvalue of lambda. I want to show that A has an eigenvalue of lambda. OK. So I look at this. I multiply both sides by M. That cancels this. So when I multiply by M, this is gone, and I have AMx. But the M shows up on the right-hand side, I have lambda Mx. Now I would just look at that, and I say, yes. A has an eigenvector-- Mx with eigenvalue lambda. A times that vector is lambda times that vector. So lambda is an eigenvalue of A. It has a different eigenvector, of course. If matrices have the same eigenvalues and the same eigenvectors, that's the same matrix. But if I do this, allow an M matrix to get in there, that changes the eigenvectors. Here they were originally x for B. And now for A, they're M times x. It does not change the eigenvalues because of this M on both sides allowed me to bring M over to the right-hand side and make that work. OK. Here are some similar matrices. Let me take some. So these will be all similar. Say 2, 3, 0, 4. OK? That's a matrix A. I can see its eigenvalues are 2 and 4. Well, I know that it will be similar to the diagonal matrix. So there is some matrix M that connects this one with this one, connects this A with that B. Well, that B is really capital lambda. And we know what matrix connects the original A to its eigenvalue matrix. What is the M that does that? It's the eigenvector matrix. So to get this particular-- to get this guy, starting from here, I use M is V for this example to produce that. Then B is lambda. But there are other possibilities. So let me see. I think probably a matrix is-- there is the matrix, A transpose. Is that similar to A? Is A transpose similar to A? Well, answer-- yes. The transpose matrix has those same eigenvalues, 2 and 4, and different eigenvectors. And those eigenvectors would connect the original A and this A or that A transpose. So the transpose of a matrix is similar to the matrix. What about if I change the order? 4, 0, 0, 2. So I've just flipped the 2 and the 4, but of course I haven't changed the eigenvalues. You could find the M that does that. You can find an M so that if I multiply on the right by M and on the left by M inverse, it flips those. So there's another matrix similar. Oh, there could be plenty more. All I want to do is have the eigenvalues be 4 and 2. Shall I just create some more? Here is a 0, 6. I wanted to get the trace right. 4 plus 2 matches 0 plus 6. Now I have to get the determinant right. That has a determinant of 8. What about a 2 and a minus 4 there? I think I've got the trace correct-- 6. And I've got the determinant correct-- 8. And there the determinant is 8. So that would be a similar matrix. All similar matrices. A family of similar matrices with the eigenvalues 4 and 2. So I want to do another example of similar matrices. What will be different in this example is there'll be missing eigenvectors. So let me say, 2, 2, 0, 1. So that has eigenvalues 2 and 2 but only one eigenvector. Here is another matrix like that. Say, so the trace should be 4. The determinant should be 4. So maybe I put a 2 and a minus 2 there. I think that has the correct trace, 4, and the great determinant, also 4. So that will have eigenvalues 2 and 2 and only one eigenvector, so it's similar to this. Now here's the point. You might say, what about 2, 2, 0, 0. That has the correct eigenvalues, but it's not similar. There's no matrix M that connects that diagonal matrix with these other matrices. That matrix has no missing eigenvectors. These matrices have one missing eigenvector. What's called the Jordan form. The Jordan form. So that didn't belong. That's not in that family. The Jordan form is-- you could say-- well, that'll be the Jordan form. The most beautiful member of the family is the Jordan form. So I have a whole lot of matrices that are similar. That is the most beautiful, but it's not in the family. It's related but not in the family. It's not similar to those. And the best one would be this one. So the Jordan form would be that one with the eigenvalues on the diagonal. But because there's a missing eigenvector, there has to be a reason for that. And it's in the 1 there, and I can't have a 0 there. OK. So that's the idea of similar matrices. And now I do have one more important note, a caution about matrix exponentials. Can I just tell you this caution, this caution? If I look at e to the A times e to the B. The exponential of A times the exponential of B. My caution is that usually that is not e to the B, e to the A. If I put B and A in the opposite order, I get something different. And it's also not e to the A plus B. Those are all different. Which, if I had 1 by 1, just numbers here, of course, that's the great rule for exponentials. But for matrix exponentials, that rule doesn't work. That is not the same as e to the A plus B. And I can show you why. e to the A is I plus A plus 1/2 A squared and so on. e to the B is I plus B plus 1/2 B squared and so on. And I do that multiplication. And I get I. And I get an A. And I get a B times I. And now I get 1/2 B squared and an AB and 1/2 A squared. Can I put those down? 1/2 A squared, and there's an A times a B. And there's a 1/2 B squared. OK. This makes the point. If I multiply the exponentials in this order, I get A times B. What if I multiply them in the other order, in that order? If I multiply e to the B times e to the A, then the B's will be out in front of the A's. And this would become a BA, which can be different. So already I see that the two are different. Here is e to the A, e to the B. It has A before B. If I do it this way, it'll have B before A. If I do it this way, it'll have a mixture. So e to the A plus B will have a I and an A and a B and a 1/2 A plus B squared. So that'll be 1/2 of A squared plus AB plus BA plus B squared. Different again. Now I have a sort of symmetric mixture of A and B. In this case, I had A before B. In this case, I had B on the left side of A. So all three of those are different, even in this term of the series that defines those exponentials. And that means that systems of equations, if the coefficients change over time, are definitely harder. We were able to solve dy dt equals, say, cosine of t times y. Do you remember how-- that this was solvable for a 1 by 1. We put the exponent-- the solution was y is e to the-- we integrated cosine t and got sine t times y of 0. e to the sine t-- Can I just think of putting that into the differential equation-- its derivative. The derivative of e to the sine t will be e to the sine t. I'm using the chain rule. The derivative of e to the sine t will be e to the sine t again, times the derivative of sine t, which is cos t, so it works. That's fine as a solution. But if I have matrices here-- if I have matrices, then the whole thing goes wrong. You could say that the chain rule goes wrong. You can't put the integral up there and then take the derivative and expect it to come back down. The chain rule will not work for matrix exponentials, the simple chain rule. And the fact is that we don't have nice formulas for the solutions to linear systems with time-varying coefficients. That has become a harder problem when we went from one equation to a system of an equation. So this is the caution slide about matrix exponentials. They're beautiful. They work perfectly if you just have one matrix A. But if somehow two matrices are in there or a bunch of different matrices, then you lose the good rules, and you lose the solution. OK. Thank you. |
MIT_Learn_Differential_Equations | Midpoint_Method_ODE2.txt | PROFESSOR: The cost of a numerical method for solving ordinary differential equations is measured by the number of times it evaluates the function f per step. Euler's method evaluates f once per step. Here's a new method that evaluates it twice per step. If f is evaluated once at the beginning of the step to give a slope s1, and then s1 is used to take Euler's step halfway across the interval, the function is evaluated in the middle of the interval to give the slope s2. And then s2 is used to take the step. For obvious reasons, this is called the midpoint method. Here's ode2. It implements the midpoint method, evaluates the function twice per step. The structure is the same as ode1. Same arguments, same for loop, but now we have s1 at the beginning of the step, s2 in the middle of the step, and then the step is actually taken with s2. Here's an example involving a trig function. Dy dt is the square root of 1, minus y squared. Starting at the origin on the interval from 0 to pi over 2. Now, because I've called it a trig example, you might just-- this is a separable equation-- do the integral, or you can just guess at the-- guess that the answer is sine t. Because the derivative of sine t is the cosine of t, and that's the square root of 1 minus y squared. Let's set it up. F is the anonymous function square root of 1 minus y squared. T0 is 0. I'm going to take h to be pi over 32. And tfinal is pi over 2. And y0 is 0. And here's my call to ode2, with these five arguments, and it produces this output. Now I want to plot it. Let's get t to go along with it. There is the t values as a column-- vector-- and let's plot. And do some annotation on the plot. Here's our plot. So there's the graph of our, there's the graph of sine t, the points generated by ode2. Now I can't help but go look at these answers. This is supposed to be the values of sine t. This should be getting to 1 at pi over 2. We've got 0.997. That gives you a rough idea of what kind of accuracy we're getting out of this crude numerical method. Let's take a look at an animation of the midpoint method. The differential equation is y prime is 2y, starting at t0 equals 0 with a step size of 1, going up to 3, and starting with y0 equals 10, and using ode2. Here is the animation. Here's t0 and y0. Evaluate the function at y0. 2 times y0 is 20, step halfway across the interval with that slope, that gets us to 20. Evaluate the function there, the slope is 40, so we take a step with slope 40 all the way across the interval to get up to 50. That's the first step. Now we'll rescale the plot window. Here we are at 50. Evaluate the function there. The slope is 100, step halfway with that slope, get to the middle of the interval, evaluate the function there. The slope is 200, so we take a step with slope 200 to get up to 250. That's the second step. Rescale the plot window. Evaluate the function there. The slope is 500. Take that step halfway across the interval, evaluate the slope there. The slope is 1,000, so we take a step with slope of 1,000 to get up to 1,250 as our final value. Since this is a rapidly increasing function of y, the values we generate here with the midpoint method are far larger than the values generated with the Euler method that we saw with ode1. Here's an exercise. Modify ode2, creating ode2t, which implements the companion method, the trapezoid method. Evaluate the function at the beginning of the interval to get s1. Use s1 to go all the way across the interval. Evaluate the function at the right-hand endpoint of the interval to get s2. And then, use the average of s1 and s2 to take the step. That's the trapezoid method. |
MIT_Learn_Differential_Equations | The_Tumbling_Box_in_3D.txt | PROFESSOR: OK. Here's an example that's more or less for fun. Because you'll see me try to do it. You can do it better. I call the problem the tumbling blocks. Only in this example, in my demonstration, it's going to be a tumbling book. I'm going to take a book, the sacred book, and throw it in the air. And I'll throw it three different ways. And the question is, is the spinning book stable or not? And let me tell you the three ways and then give you the three equations that came from Euler. So those are the three equations. You see that they're not linear. And those are for the angular momentum. So there's a little physics behind the equations. But for us, those are the three equations. So the first throw will spin around the very short axis, just the thickness of the book, maybe an inch. So when I toss that, as I'll do now, you will see if I can toss it not too nervously I hope. It came-- it was stable. The book came back to me without wobbling. Of course, my nerves would give it a little wobble, and that wobble would continue. It will be only neutrally stable. The wobble doesn't disappear. But it doesn't grow into a tumble. OK. So that's one axis, the short axis. Then I'll throw it also around the long axis, flipped like this. I think that will be stable too. And then, finally, on the intermediate axis, is middle length axis. Notice the rubber band that's holding the book together. Holding so the pages don't open. And this, we'll see, I think, will be unstable. And similarly, throwing a football, throwing other Frisbees, whatever your throw. Any 3D object has got these three axes: a short one, a medium one, and a long axis. And the equations will tell us short and long axes should give a stable turning. And the in between axis is unstable. Well, how do we decide for our differential equation whether the fixed point, a fixed point, that's a critical point, a steady state-- we have to find this steady state, and then for each steady state we linearize. We find the derivatives at that steady state. And that gives us a constant matrix at that steady state. And then the eigenvalue is decided. So first, find the critical points. Second, find the derivatives at the critical points. Third, for that matrix of derivatives, find the eigenvalues and decide stability. That's the sequence of steps. OK. The first time we've ever done a three by three matrix. Maybe the last time. OK. Let me, before I start-- before I find the critical points-- notice some nice properties. If I multiply this equation by x, this one by y, this one by z, and add, those will add to 0. When there's an x there, a y there, and a z there, I get a 1 minus 2 and a 1 they add to 0. So x times dx dt. y times dy dt. z time dz dt adds to 0. That's an important fact. That's telling me that the derivative of something is 0. That something will be a constant. So I'm seeing here the derivative of that whole business would be the derivative of a half probably. x squared, because the derivative of x squared will be with a half. The derivative will be x dx dt. And y squared and z squared is the derivative is 0. The derivative of that line is just this line. It's 0. So this is a constant. No doubt, that's probably telling me that the total energy, the kinetic energy, is constant. After I've tossed that book up in the air, I'm not touching it. It's doing its thing. And it's not going to change energy because nothing is happening to it. It's just out there. Now there are other-- so that's a rather nice thing. This is a constant. Now there's another way. If I multiply this one by 2x, and I multiply this one by y, and add just those two, that cancels. So 2x dx dt-- 2x times the first one-- and y times the second one gives 0. Again, I'm seeing something is constant. The derivative of something, and that something is x squared plus 1/2 y squared is a constant. Another nice fact. Another quantity that's conserved. And as I'm flying around in space, this quantity x squared plus 1/2 y squared does not change. This sort of-- that involved all of xyz. And of course that's the equation of a sphere. So in energy space, or in an xyz space, our solution is wandering around a sphere. And this is the equation for, I guess, it's an ellipse. So there's an ellipse on that's sphere that it's actually staying on that ellipse. And in fact there's another ellipse because I could've multiplied this one by 2z and this one by y and added. And then those would have canceled. Minus 2 xyz plus 2xyz. So that also tells me that it would be probably z squared plus 1/2 y squared equals a constant. That's another ellipse. z squared plus 1/2 y squared. You see this? If I take the derivative of that, I have 2z times dz dt plus y times dy dt. Adding give 0. The derivative is 0. The thing is a constant. But! But, but, but! If I subtract this one from this one, take the difference of these two. Suppose I take this one minus this one. The 1/2 y squared will go. So that will tell me that x squared minus z squared is a constant. Oh, boy! I haven't solved my three equations. But I found out a whole lot about the solution. The solution stays on the sphere, wanders around somehow. It also at the same time stays on that ellipse. And it stays on that ellipse. But this is not an ellipse, not an ellipse. That's the equation of a hyperbola. And that's why-- which, of course, goes off to infinity. And that's why the-- well, it goes off to infinity, but it has to stay on the sphere. It wanders. This will be responsible for the unstable motion. Professor [INAUDIBLE], who would do this far better than me, his great lecture in 1803, Differential Equations, was exactly this. The full hour to tell you everything about the tumbling box. So I'm going to do the demonstration and write down the main facts and understand the stability, the discussion of stability. I'm ready to move on to the discussion of stability. Again, here are my three equations. We're up to three equation, so we're going have a three by three matrix. And first I have to find out the critical points, the steady states of this motion. How could I toss it so that if I toss it perfectly it stays exactly as tossed? And the answer is, around the axis. If I toss this perfectly, with no nerves, it'll just spin exactly as I'm throwing it. The x, y, and z will all be constant. Now, when I toss it on that axis. I'm looking for-- here are my right hand side. YZ, minus 2XZ, and XY. And I wrote those in capital letters because those are going to be my steady states. Now I'm looking for are points where nothing's happened. If those three right hand sides of the equation are 0, I'm not going to move. xyz will stay where they are. So can you see solutions of those three equations? Well, they're pretty special equations. I get a solution when, for example, solutions could be 1, 0, 0/ If two of the three-- if y and z are 0. y is 0, z is 0, y and z are 0, I get 0. So that is a certainly steady state. x equal 1, y and z equal 0 and 0. And that steady state is spinning around one axis. And, actually, I could have also a minus 1 would also be. So I've found, actually, two steady states with y and z 0. Then there'll be two more with x and z 0. And this could be-- that'll be spinning around the middle axis. And then 0, 0, 1 or minus 1, that would be spinning around the third axis, the long axis. So those are my steady states. And I guess, come to think of it, 0, 0, 0 would also be a steady state. I think I found them all. These are the xy's. These are the x, y, z steady states. OK. So now once you know the steady states, that's usually fun, as it was here. Now the slightly less fun step is find all the derivatives, find that Jacobian matrix of derivative. So I've got three equations. Three unknowns, xyz. Three right hand sides. And I have to find-- I'm going to have a three by three matrix of derivatives. This Jacobian matrix. So J for the Jacobian, the matrix of first derivatives. So what goes into the matrix of first derivative? Let me write Jacobian. It is named after Jacoby. It's the matrix of first derivatives. On the top row are the derivatives of the first function with respect to x. Well, the derivative with respect to x is 0. The derivative with respect to y is z. The derivative with respect to z is y. Those were partial derivatives. They tell me how much the first unknown x moves. They tell me what's happening with the first unknown x around the critical point whichever it is. OK. What about the partial derivatives from the second equation? it's partial derivatives will go into this row. So x has a minus 2z. y derivative is 0. z derivative is minus 2x. And the third one, the z derivative is 0 here. The y derivative in x. And the x derivative is y. I've found the 3 by 3 matrix with the nine partial first derivatives. OK. It's the eigenvalues of that matrix at these points that decide stability. So I write that down. Eigenvalues of J at the critical points x, y, z that's what I need. That's what decides stability. Let me just take the first critical point. What is my matrix? I have to figure out what is the matrix at that point? And I'll just take 1, 0, 0. 1, 0, 0. If x is 1-- so I'm getting, this is at the point x equal 1. y and z are 0. So if x is 1, then that that's a minus 2 and a 1. And I think everything else is 0. So it'll be the eigenvalues of that matrix that decide the stability 1, 0, 0 of that fixed point. And remember, that's the toss around the narrow axis. That's the toss around the short axis. OK. What about the eigenvalues of that matrix? Well, I can see here that really it's three by three. But really, with all those 0s, that gives me an eigenvalues of 0. So I'm going to have an eigenvalue of 0 here. And then I'm going to have eigenvalues from the part of that matrix, which is two by two. So I'll have a lambda equals 0 here. And two eigenvalues from here. And I look at that, and what do I see? Now this is a two by two problem. I see the trace is 0. 0 plus 0. My eigenvalues are a plus and minus pair because they add to 0. They multiply to give the determinant. The determinant of that matrix is 2. The determinant of that matrix is 2. OK. So it has a positive determinant. That's good for stability. But the trace is only 0. It's not quite negative. It's not positive. It's just at 0. So this is going to be a case of neutral stability. The eigenvalues will be-- I'll have a 0 eigenvalue from there. The eigenvalues from this two by two will be-- there'll be a square root of 2 times i and a minus the square root of 2 times i. I think those are the eigenvalues. And what I see there is they're all imaginary. This is a pure oscillation. The wobbling keeps wobbling. Doesn't get worse. Doesn't go away. It's neutral stability at this point. So neutral stability is what we hopefully will see again. Yes. And I think, also, if I flip on the long axis. Good. Did you see that brilliant throw? It's neutral stability. It came back without doing anything too bad. And I finally have to do the axis that we're all intensely waiting for, the middle axis. And the middle axis is when the book starts tumbling, and it's going to be a question of whether I can catch it or not. May I try? And then may I find-- what am I expecting on the neutral axis? I'm expecting instability. I think actually it will be a saddle point. But there'll be a positive eigenvalues. There will be a positive eigenvalue. And it is responsible for the tumbling, the wild tumbling that you will see. And it's connected with the point staying on this hyperbola that wonders away from-- so it's this one now that I'm doing. This guy is the-- I'll put a box around-- a double box around it. That's the unstable one, which I'm about to demonstrate. Ready? OK. Whoops. OK. It took two hands to catch it. Let me try it again. The point is it starts tumbling, and it goes in all directions. It's like a football, a really badly thrown football. It's like a football being thrown that goes end to end. The whole flight breaks up, and the ball is a mess. Catching it is ridiculous. And I'm doing it with a book. Yes. You saw that by watching really closely. OK. Better if you do it. I'll end with the eigenvalues at this point. So the eigenvalues at that point-- can I just erase my matrix? So this was a neutrally stable one, a center in the language of stability. That's a center which you just go around and round and round. But now I'm going to just take x and z to be 0 and y to be 1. So can I erase that matrix and take-- If x and z are 0, and y is 1-- so I get a 1 down here. And I get a 1 up there. And nothing else. Everything else is 0. OK. That's my three by three matrix. What are its eigenvalues? What are the eigenvalues of that three by three very special matrix? This is now the-- this was the first derivative matrix, the Jacobian matrix, at this point, corresponding to the middle axis. OK. Again, I'm seeing some 0s. I'll reduce this to that two by two matrix and this matrix. Really, I have this two by two matrix in the xz, and this one in the y. How about that guy? You recognize what we're looking at with this matrix. So with that matrix, I can tell you the eigenvalues. We can see the trace is 0. The eigenvalues add to 0. They multiply to the determinant. And the determinant is minus 1. So the eigenvalues here are 1 and minus 1. And then this guy gives 0. And it's that eigenvalue of 1 that's unstable. That eigenvalue of 1 is unstable. OK. So mathematics shows what the experiment shows: an unstable rotation tumbling around that middle axis. Thank you. |
MIT_Learn_Differential_Equations | Boundary_Conditions_Replace_Initial_Conditions.txt | GILBERT STRANG: OK. Well my problem today is a little different. Because I don't have two initial conditions, as we normally have for a second-order differential equation. Instead, I have two boundary conditions. So let me show you the equation. So I'm changing t to x because I'm thinking of this as a problem in space rather than in time. So there's the second derivative. The minus sign is for convenience. This is the load. But here's the new thing. I'm on an interval 0 to 1. And at 0-- let me take 0 for the two boundary conditions. So my solution somehow does something like this. Maybe up and back down. So it's 0 there, 0 there, and in between it solves the differential equation. Not a big difference, but you'll see that it's an entirely new type of problem. OK. As far as the solution to the equation goes, there is nothing enormously new. I still have a y particular. A particular solution that solves the equation. And then I still have the y null, the homogeneous solution, any solution that solves the equation with 0 on the right hand side. And in this example-- this is especially simple-- the null equation would be second derivative equal 0. And those are the functions, linear functions, that have second derivative equal zero. So there's the general solution. And now I have to put in, not the initial conditions, but the boundary conditions. OK. So I substitute x equal 0. And I substitute x equal 1 into this. I have to find y particular. I'll do two examples. I'll do two examples. But the general principle is to get these numbers, these constants like C1 and C2, from the boundary conditions. I'll put in x equal zero. And then I'll have y of 0, which is y particular at 0, still defined, plus C times 0, plus d. That's the solution at the left end, which is supposed to be 0. And then at the right end, end, I have whatever this particular solution is at 1, plus now I'm putting in x equal 1. I'm just plugging in x equals 0, and then x equal 1. And x equal 1, I have C plus D. C plus D. And that gives me 0. The two 0's come from there and there. OK. Two equations. They give me C and D. So I'm all solved. Once I know how-- I know how to proceed once I find a particular solution. So I'll just do two examples. They'll have two particular solutions. And they are the most important examples in applications. So let me start with the first example. So my first example is going to be the equation minus D second y, Dx squared, equal 1. That will be my load. f of x is going to be 1. So I'm looking for a particular solution to that equation. And of course I can find a function whose second derivative is 1, or maybe minus 1. My function will be-- well, if I want the second derivative to be 1, then probably 1/2 x squared is the right thing. And that would give me a minus. So I think I have a minus 1/2 x squared. That solves the equation. And now I have the Cx plus D. The homogeneous, the null solution. And now I plug in. And again, I'm always taking y of 0 to be 0, and also y of 1 to be 0. Boundary conditions. Again, boundary condition, not initial condition. OK. Plug in x equal 0. At x equals 0, what do I learn? x equal 0. That's 0, that's 0, so I learn that D is 0. At x equal 1, what do I learn? This is minus 1/2. D is 0 now. And x is 1. So I think we learn C is plus 1/2. OK with that? At x equal 1, I'm supposed to get 0 from the boundary condition. So I have minus 1/2, plus 1/2, plus 0. I do get 0. This is good. So this answer is-- Cx, then, is 1/2 x minus 1/2 x squared. That's it. That's my solution. That function is 0 at both ends, and it solves the differential equation. So that's a simple example. And maybe I can give you an application. Suppose I have a rod. Here's a bar. And those lines that I put at the top and bottom are the ones that give me the boundary conditions. And I have a weight. A weight of 1. Maybe the bar itself. It gives-- elastic force. Gravity will pull, displace, the bar downwards because of its weight. It's elastic. And this function gives me the solution, gives me the distribution. If I go down a distance x, then that tells me that this part of the bar, originally at x, will move down by an additional y. Moves. So this is now at x plus y of x. And that's the y. And that is 0 at the bottom, 0 at the top, and positive in between. OK. That was a pretty quick description of an application. And more important, a pretty quick solution to the problem. Can I do a second example that won't be quite as easy? OK. So again, my equation is going to be minus the second derivative equals a load. But now it will be a point load. A point load. That's a point load at x equal A. This is my friend, the delta function. The delta function, you remember, is 0, except at that one point where this is 0. This is 0 at the point x equal A. In my little picture of a physical problem, now I don't have any weight in the bar. The bar is thin. Weightless. But I'm putting on, at the point x equal A, right at this point, I'm attaching a weight. So this distance is x equal A. Here's my weight, my load, hanging at this point. So I can see what will happen. That load hanging down there will stretch the part above the bar, above the load, and compress the part below the load. So it's a point load. Very important application. OK. Now I have this equation to solve. OK. I can solve it on the one side of A, x equal A. And I can solve it on the other side of x equal A. Let me do that. For x less than A, I have minus the second derivative. And what's the delta function for x below A, on the left side of the spike? 0. And x on the right side of the load, again, 0. And what are the solutions to the null equation? y is Cx plus D on the left side of the load, there. And now here it may have some different constants. y equals, what shall I say, E x plus F, on the right side of the load. And now I've got four numbers to find. C, D, E, and F. And what do I know? I know two boundary conditions. Always I know that y of 0 is 0, from fixing the top of the bar. So y of 0 equal 0. And when I put in x equal 0, that will tell me D is 0. And then also y of 1 is 0. And that will be on this side of the load. So when I put in x equal 1, that will tell me that E plus F is 0, at x equal 1. So it tells me that F is minus E, right? So what do I know now? D is gone. 0. F is minus E. So can I just change this to F is minus E. So I had E x minus E. E times x minus 1 takes care of that boundary condition. At x equal 1, it's gone. OK. But I still have two, C and E, to find. So what are my two further conditions at the jump? So far I'm on the left of the jump, the spike, the impulse, the delta function. And on the right of it. But now I've got to say, what's happening at the impulse? At the delta function. Or at the point load. OK. Well, what's happening there? I need two equations. I've still got C and E to find. So my first equation is that at that load, the bar is not going to break apart. It's just going to be stretched above and compressed below. But it is not going to break apart. So at the load, at which is x equal A. So now I'm ready for x equal A. OK. What happens at x equal A? That's the same as that. Let me draw a picture of the solution, here. Here is x. This is x. Here's y. Here is x equal 0. Here is x equal 1. I see a linear function. Cx up to the point x equal A. And here I have a linear function coming back to 0. You see? That's the picture of the solution. The graph of the solution. It has this 0 at the left boundary. It has 0 at the right boundary. It has, in between, it is Cx in the x minus E. And I have made it continuous at x equal A. The bar is not coming apart. So that this solution runs into that solution. That's good. That's one more condition. But I need one further, one final, condition. And somehow I have to use the delta function. And what does the delta function tell me? I'm just going to go give you the answer here, rather than a theory of delta functions. That equation. So you see what my solution is. It's a broken line with a change of slope. It's a ramp. It has a corner. All those words describe functions like this. So I have some slope going up here, and some slope-- and let me tell you. I'll tell you what those slopes are. I'll tell you what those slopes are in this. So I'll tell you the answer and then we'll check. So that C turns out to be 1 minus A. So in this region, I have 1 minus A times x. In that region. And in this region, below, so that's stretching. The fact that it's positive displacement means it's stretching. Now this part is going to be in compression, with that negative slope. And I think in this region it's 1 minus x times A, which will be coming from there. So there is my solution. Because of the delta function, I need a two part solution. To the left of the delta function, the point load. And to the right of the point load. And then we could check that at the load, x equal A. This is 1 minus A times A. This is 1 minus A times A. They do meet. And now comes this mysterious fourth condition about the slopes. The slope drops by 1. Here the slope is 1 minus A. That's 1 minus A is the slope there. And here the slope is minus A. You see minus x times A, so the derivative is minus A. So it was 1 minus A. The 1 dropped away and left me with minus A. That's what the solution looks like. And now I have to say one word about why did the slope drop by 1. The slope dropped by 1, from 1 minus A to minus A. And that has to come from this delta function. And of course you remember about the delta function. The key point is if when you integrate the delta function, you get 1. So when I integrate this equation, I get a 1 on the right hand side from the delta. And on the left hand side, I'm integrating the second derivative, so I get the first derivative. Great. The first derivative at the end point, minus the first derivative at the start point, should be the 1. And that's the drop of 1. I'll do a full-scale job with delta functions in another video. I want to keep this one under control. We're seeing the new idea is boundary conditions, and here we're seeing a delta function equation in this boundary value problem. Thank you. |
MIT_Learn_Differential_Equations | Examples_of_Fourier_Series.txt | This video is to give you more examples of Fourier series. I'll start with a function that's odd. My odd function means that on the left side of 0, I get the negative of what I have on the right side of 0. F at minus x is minus f of x. And it's the sine function that's odd. The cosine function is even, and we will have no cosines here. All the integrals that involve cosines will tell us 0 for the coefficients AN. What we'll get is the B coefficients, the sine functions. So you see that I chose a simple odd function, minus 1 or 1, which would give a square wave if I continue it on. It will go down, up, down, up in a square wave pattern. And I'm going to express that as a combination of sine functions, smooth waves. And here was the formula from last time for the coefficients bk, except now I'm only integrating over half, over the zero to pi part of the interval, so I double it. So that's an odd function, that's an odd function. When I multiply them, I have an even function. And the integral from minus pi to 0 is just the same as the integral from 0 to pi. So I'll do only 0 to pi and multiply by 2. But my function on 0 to pi is 1. My nice square wave is just plus 1 there, so I'm just integrating sine kx dx. We can do this. It's minus cosine kx divided by k, right? That's the integral with the 2 over pi factor. Now I have to put in pi and 0 and put in the limits of integration and get the answer. So what do I get? I get 2 over pi. For k equal 1, I think I get-- so k is 1, the denominator will be 1, and I think the numerator is 2. Yes, when k is 0, I get yeah. When k is 1, I get 2. When k is 2, so this is 4 over pi, I figured out as the first coefficient. The coefficient b1 is 4 over pi. The coefficient b2, now if I take k equal to 2, I have a 2 down below. But above, I have a 0 because the cosine of 2 pi is the same as the cosine of 0. When I subtract I get nothing, so that's 0. Now I go to k equals 3. So the k equals 3 will come down here. And now when k is 3, it turns out I get-- they don't cancel, they reinforce. I get another 2. Good if you do these. And when k is 4, I get a 0 again. You see the pattern? The pattern for the integrals is the k is going 1, 2, 3, 4, 5. This part gives me a 2 or a 0 or a 2 or a 0 in order. If you check that, you'll get it. So I see that now for this function, which is better than the delta function also. It's not very smooth. It has jumps. It's a jump function, a step function. I see some decay, some slow decay, in the Fourier coefficients. This factor k is growing so the numbers are going to 0, but not very fast. Not very fast. Because my function is not very smooth. So now you see-- so if I use those numbers, I'm saying that the square wave, this function, the minus 1 to 1 function, is equal to, let's see. I might as well take that 4 over pi times 1. So that's 1 sine x, 0, sine 2x's then 4 over pi sine 3x's, but with this guy there's a 3, 0 sine 4x's, sine 5x comes in over 5, and so on. That's a kind of nice example. It turns out that we have just the odd frequencies 1, 3, 5 in the square wave and they're multiplied by 4 over pi and they're divided by the frequency, so that's the decay. There is an odd function. Why don't I integrate that function? If I want to get an even function to show you an even example, I'll just integrate that square wave. When I integrate it square wave, it'll be even. Maybe I'll start the integral at 0, then it goes up at 1. And here the integral is negative, so it's coming down. So you see it's a-- what am I going to call this function? Sort of a repeating ramp function. It's a ramp down and then up, down and then up. But of course from minus pi to pi, that's where I'm looking. I'm looking between minus pi and pi. And I see that function is even. And what does even mean? That means that my function at minus-- there is minus x-- is the same as the value at x. And what that means for a Fourier series is cosine. Even functions only have cosine terms. And of course, since I've just integrated, I might as well just integrate that series. So this is this ramp, this repeating ramp function, is going to be 4 over pi. I could figure out the cosine coefficients, the a's, patiently. But why should I do that when I can just integrate? So the integral of sine x will be minus is the integral of sine x, is minus cosine x, so I'll put the minus there, cosine x over 1 I guess. Now what's the integral of this? The integral of sine 3x is a cosine 3x over 3. And there's another 3 and there's a minus sign, which I've got. So I think it's cosine of 3x over 3 squared, because I have one 3 there and I get another 3 from the integration. And similarly here, when I integrate sine 5x I get cos 5x with a 5. And then I already had one 5, so 5 squared. So there you go. [LAUGHTER] There's something in freshman calculus which I totally forgot, the constant term. So there is a constant term, the average value, that a0. I've only found the a1, 2, 3, 4, 5. I haven't found the a0, and that would be the average of that. I don't know, what's the average of this function? Its goes from 0 up to pi and it seems like it's pretty-- I didn't draw it well, but half way. I think probably its average is about pi over 2, right? Let's hope that's right. So let me sneak in the constant term here. The ramp is, I think I have a constant term is pi over 2. That's the average value. It would come from the formula and those-- well, what do you see now? That's the other example I wanted you to see. You see a faster drop off. 1, 9, 25, 49, whatever. It's dropping off with k squared. And the reason it drops off faster than this one is that it's smoother. This function has corners. This function has jumps. So a jump is one level more rough, more word noisy than a ramp function. The smoother function has faster decay. Smooth-- let me write those words-- smooth function connects with faster decay. Faster drop off of the Fourier coefficient. It means that the Fourier series is much more useful. Fourier series is really terrific for functions that are smooth because then you only need to keep a few terms. For functions that have jumps or delta functions, you have to keep many, many terms and the Fourier series calculation is much more difficult. So that's the second example. Let's see, what more shall I say? We learned something about integrating and taking the derivative so let me end with just two basic rules. Two basic rules. So the rule for derivatives. What's the Fourier series of df dx? And the second will be the rule for shift. What's the Fourier series for f of x minus a shift? You know that when I change x to x minus d, all that does is shift the graph by a distance d. That should do something nice to its Fourier coefficient. So I'm starting with-- oh, I haven't given you any practice with a complex case. This would be a good time. Suppose start is f of x equals the sum of ck, a complex coefficient e to the ikx, the complex exponential. And you'll remember that sum went from minus infinity to infinity. So I have a Fourier series. I'm imagining I know the coefficients and I want to say, what happens if I take the derivative? Well, just take the derivative. You'll have a sum of the derivative brings down a factor ik. So that's the rule. Simple, but important. That's why Fourier series is so great because you have orthogonality and then you have this simple rule with derivatives. it just brings a factor ik so the derivative make sure function noisier and you have larger coefficients. And if I do f of x minus d, so I'll change x to x minus d, so I'll see the sum of ck e to the ikx, e to the minus ikd, right? I've put in x minus d instead of x. And here I see that the Fourier coefficient for a shifted function-- so the ck was a Fourier coefficient for f. When I shift f, it multiplies that coefficient by a phase change. The magnitude stayed the same because that's a number-- everybody recognizes that as a number of magnitude 1 and just has a phase shift. Those are two would rules that show why you can use Fourier series in differential equations and in difference equations. Thank you. |
MIT_Learn_Differential_Equations | Lorenz_Attractor_and_Chaos.txt | CLEVE MOLER: The Lorenz strange attractor, perhaps the world's most famous and extensively studied ordinary differential equations. They were discovered in 1963 by an MIT mathematician and meteorologist, Edward Lorenz. They started the field of chaos. They're famous because they are sensitive to their initial conditions. Small changes in the initial conditions have a big effect on the solution. Lorenz is famous for talking about the butterfly effect. How flapping of butterflies' wings can affect the weather. A butterfly flying in Brazil can cause a tornado and Texas is a flamboyant version of a talk he gave. The equations are almost linear. There's two quadratic terms here. The equations come out of a model of fluid flow. The Earth's atmosphere is a fluid. But this range of parameters, the three parameters, sigma, rho, and beta, these are outside the range that actually represents the Earth's atmosphere. We're going to take a look at these parameters. These are the most commonly used parameters. But we're going to be interested in other values of rho as well. But I'm a matrix guy, so I like to write the equations in this form. Y dot equals Ay. It looks linear except A depends upon y. And so there's y2, the second component of y, appears in the matrix A. This helps me study the differential equations in this form. This matrix form is convenient for finding the critical points. Put a parameter eta in place of y2. Try to make the matrix singular. That happens when eta is beta times the square root of rho minus 1. And then the null vector is the critical point. If we take this vector as the starting value of the solution, then the solution stays there. Y prime is 0. This is an unstable critical point. And values near this solution deviate the solution. Won't stay near the solution. In May of 2014, I wrote a series and blog post in Cleve's Corner about the MATLAB ordinary differential equations suite. And I used the Lorenz attractor as an example. And I included a program called Lorenz plot that I'd like to use here. Here's Lorenz plot. Set the parameters. Set the initial value of the matrix A. Here is the critical point. Here is an initial value near the critical point. Integrate from 0 to 30. Use ODE 23. Give it a function called the Lorenz equation. Capture the values t and y and then plot the solution. I'm going to do a plot with the three components offset from each other. And here's an internal function Lorenz equation that is called by ODE 23. And it continuously, every time it called, it modifies the matrix A updates it with the new values of y2. So let's run that function. And here's the output. Here is the three components of the Lorenz attractor. Time series is functions of t. It's pretty hard to see what's going on here except to say they start out with their initial values, oscillate around them, close them through for a little while, and then begin to deviate. And it's hard to see what they're doing. They're just oscillating in an unpredictable fashion. We need another graphic to see what's really going on here. I want to write a program called Lorenz GUI. Lorenz Graphic User Interface. That's out of my old book calle this one is really out of Numerical Computing with MATLAB, NCM. OK, I hit the Start button. Here are the two critical points in green. We started near the critical point. We oscillate around the critical point. And here is the orbit. This is just going back and forth. It oscillates around one critical point then decides to go over and oscillate around the other for a while. It continues around like this forever. This is not periodic. It never repeats. Now, the butterfly is associated with Lorenz in two ways. One is the butterfly effect on the weather. Also, this plot looks like a butterfly. I can grab this with my mouse and rotate it in three dimensions. So I can get different views of the orbit. It's still being computed. We're adding more and more to the plot. And I can look at it from different points of view to get some notion of how this is proceeding in three dimensions. It almost lives in two dimensions, but not quite. Earlier, we've seen solutions, differential equations with periodic solutions. Here, this isn't periodic. Just going like this [? forever. ?] Now, this is perfectly-- this isn't random. This is completely determined by the initial conditions. If I were to start it over again with those exact conditions, with those exact initial conditions, I'd get exactly this behavior. But it's unpredictable. It's hard to say where this is going. I can clear this out and see the orbit continue to develop. Press Stop. Now I have a choice. This pull down menu here allows me to choose other values of rho. 28 is the value of rho that is almost always studied, but there's a book by a Colin Sparrow that I've referenced in my in my blog about periodic solutions to Lorenz equations. And let's take another value. Let me choose rho equal to 160 and clear and restart. Now, after an initial transient, this is now periodic. So this is not chaos. This is a periodic solution. And these other values of rho, not rho equals 28, that's chaotic, but these other values of rho give periodic solutions with different character. That's a long, interesting story that I talk about in my blog following the work of Sparrow. |
MIT_Learn_Differential_Equations | Unforced_Damped_Motion.txt | GILBERT STRANG: OK. So today is unforced-- that means zero on the right-hand side, looking for null solutions-- damped-- that means there is a coefficient B in the first derivative. And what's the solution? This is really a basic, basic equation. In many applications, A would be the mass. In a spring, for example, A would be a mass. B is the damping, the friction. And C is the spring constant, the force that pulls the mass back. Or in electronics, B would be the resistance. It's giving some friction, giving some heat. So that's our equation. Just we have to be able to solve it. And we want to look for exponentials. A pure exponential is just right for a constant coefficient equation like that. So I'll substitute y equals e to the st. And what happens? Well, so there's a C e to the st. That's a Cy. The derivative brings down an s. Two derivatives bring down s squared. So I simply have As squared, Bs, and C all together, multiplying e to the st. And how'd I get 0? Well the 0 is not going to come from e to the st, so it has to come from this. So fundamentally, the whole video and more is about a quadratic equation. As squared plus Bs plus C equals 0. We have to solve it. We have to understand how does the answer depend on these numbers, A and B and C, the constants? OK? So we know the route. We know the solutions. The quadratic formula tells us that the two solutions-- there are always two, but they could be equal-- have this-- do you recognize this expression? You see the damping coefficient coming in. You see this all important square root, which tells us, depending on whether B squared is bigger than 4AC, B squared is equal 4AC, B squared is smaller. So smaller B means less damping. And it would be called underdamping.. So here are the possibilities. B could be 0. That puts us back in a previous video when the solution was a pure complex exponential, pure sine cosine. There was no damping. It just oscillated forever. That's the one we know. Now, the new ones with a B are B squared could be smaller. So that's only a little damping. And in that case, what does the solution look like? If B squared is smaller than 4AC, I have to always look back here. So I have something negative. Something negative, a minus B over 2A. And then plus or minus-- and what's important point about this guy? B squared is smarter than 4AC. This is negative inside. So its square root is an imaginary number. So the plus or minus, there's an imaginary number. A little different from the natural frequency. It's called the damp frequency. Not quite as frequent. The damping slows the oscillation down a little bit, but it brings in exponential decay. I'll draw a picture this time or next time of the solution, e to the minus st. But you see, e to the minus st is decaying to 0. It's like a spring that's slowly winding down to 0. But it's oscillating as it does it. OK. Now, we've got two more possibilities here. And these are just fun to get straight. There's the next one. When B squared is exactly 4AC. Exactly 4AC. B squared over 4AC, that's going to be the critical ratio here. And here that ratio is 1. And in that case, that square root is nothing. The square root of B squared minus 4AC is 0. So we get s1 and s2 equal. Two equal frequencies. Real numbers. They're the minus B over 2A. So s1 and s2 are both minus B over 2A. That's just the edge between the decay with oscillation from underdamping. And the opposite extreme is overdamping. Overdamping, when B squared is bigger than 4AC, then our formula-- we're always looking back at that quadratic formula. B squared bigger than 4AC, this is real. Square root of a positive number. It's perfectly real. So in that case, we have two, s1 and s2. Oh, let me draw you some pictures. I'll draw you the parabola, the As squared. So this is a graph of As squared. Let me plot As squared plus Bs plus C. So in this overdamping case, we might have a lot of damping. B is pretty large. And that would be-- this direction is s. And I'm graphing here s squared plus-- well, let me see. I think probably plus 2s. And now I'm going to choose this-- oh, actually, plus 0 to get that one. So this has a root at-- this is overdamping. Overdamping. A big number there. C is 0. No mass at all. No stiffness at all. So this parabola has a root at 0 and a root at minus 2. That would be an extreme case with no stiffness whatever. Now, let me add some stiffness. So what happens if I change the constant term? It lifts the graph. It lifts the graph. So let me make it 1. So I move the whole graph up by 1. So you see that as I move it up, these roots will come closer and closer. And at the right point, when I move it up to 1, what's going on there? Critical damping. Critical damping. This is the case s squared plus 2s plus 1. Which is exactly s plus 1 twice. And it has that repeated root, which we always recognized as the marginal case when it doesn't go below, but it doesn't stay above either. It hits here twice. So the root is minus 1 twice. That's critical. And now what? Increase this constant C even further. You'll lift the graph, lift the parabola up here. Lift it by one more, let's say. And what's going on there? So this would be s squared plus 2s plus 2. And you see what that graph is telling us. It's telling us there are no roots, no real roots. This is a real graph in the real plane, and it's never 0. This is s plus 1. This is the case when I have s plus 1 squared plus 1. Can't be 0. Never 0. So this has-- sorry. With the 2 there, it has damping. It has damping, but it's underdamping. The roots are complex. So this s plus 1 squared is 0. What are the roots of that? s is minus 1 plus or minus i. That's the case of underdamping. So let me write that. We see so much from these graphs. So this was a graph in which I changed the constant term, the capital C, the stiffness, k. So that lifted the graph. And eventually the roots were complex. They weren't real anymore, because the parabola didn't cross the axis anymore. The roots are where it crosses the axis. So that's one. I'll draw in the next video another picture. Maybe change B. Change the damping. So what would happen if I change the damping? If there's no damping, the roots are pure imaginary. Yeah. Let's just go back through those four possibilities, because that's what you have to learn. If B is 0, there's no damping at all. And we're back in the pure oscillation, where is pure imaginary. We're just seeing cosines and sines. Then if we add a little damping, there appears a decay turn. If we add-- I could even draw these possibilities. Let me draw the solution y of t, against t. OK. So underdamping-- the solutions say it starts there-- will oscillate, but decay. So this is underdamping. And you remember when that happens? That's when B is present but smaller. B squared is smaller than 4AC. Overdamping, if it starts, let's say, here, it will going to 0, probably with no oscillation at all. It could have one oscillation. Depends how it starts. So ooh, let me make this picture. So that one is the overdamping. This guy is the underdamping. Underdamping still has oscillations. Overdamping has, at most, once. It could cross once. But overdamping is B squared bigger than 4AC. OK. There's more to do. This is so central that you have to think about the ratio of B squared to 4AC. That has its own name. And it's a damping ratio, or the square of a damping ratio. And we identify all these solutions. Does everybody know where the solution will be when B is 0? No damping at all. In that case, it will just oscillate. So my picture is getting, you could say, a little messed up. But this pure oscillation would be the B equals 0 with undamped. OK. That's a first look at quadratic equations. As squared plus Bs plus C. It has the big name characteristic equation, but you could see, it's the fundamental equation for a second order differential equation. So we'll see more of it. Thanks. |
MIT_Learn_Differential_Equations | Positive_Definite_Matrices.txt | GILBERT STRANG: OK. This is positive definite matrix day. Our application was the second-order equation with a symmetric matrix, S. And we solved this equation. Second derivative, plus S times y, equals 0. And you maybe remember how we solved it. We looked for an exponential solution. e to the I omega t, times a vector x. We substituted, and we discovered x had to be an eigenvector of S, as usual. And lambda, which was omega squared, is the eigenvalue. But I didn't stop to point out that if we want lambda to be omega squared, we need to know lambda greater or equal to 0. So that is the best of the best matrices. Symmetric and positive definite, or positive semidefinite, which means the eigenvalues are not only real, they're real for symmetric matrices. They're also positive. Positive definite matrices-- automatically symmetric, I'm only talking about symmetric matrices-- and positive eigenvalues. OK. There it is. Positive definite matrix. All the eigenvalues are positive. Positive semidefinite. That word semi allows lambda equal 0. The matrix could be singular, but all the eigenvalues have to be greater or equal to 0. And let me show you exactly where those matrices come from. Those matrices come from A transpose A. If I take any matrix A, could be rectangular. And A transpose A will be square. A transpose A will be symmetric. And it will be at least positive semidefinite. Why is that? This is the simple step that is worth remembering. What's special about A transpose A x equal lambda x? The good idea? Multiply both sides by x transpose. Take the inner product of both sides with x. Then I have x transpose times the left side, is x transpose times the right side. I'm OK with equation 2. When my S is A transpose A, that's my S. OK. But now I look at this. That is A x in a product with itself. That's the length of A x squared. Any time I have y transpose y, I'm getting the length of y squared. Here y is A x, so I'm getting the length of A x squared. Over here, y is x, so I'm getting the length of x squared. And you see that number lambda is, in this equation, I have a number that can't be negative. A number that's positive, for sure. Because x is not the 0 vector. So lambda is never negative. A x could be the 0 vector. If we were in a singular case, A x could be the 0 vector. In that case, I would only learn lambda equals 0, and I'd be in this semidefinite case. So if I want to move from semidefinite to definite, then I must rule out A x equals 0 there. Because that's certainly a possibility. If I took the 0 matrix, all 0's, as A, A transpose A would be the 0 matrix. That would be symmetric. All its eigenvalues would be 0. Would it be positive semidefinite? Yes. Yes. All its eigenvalues would actually be 0. Of course, that's not a case that we are really thinking about. More often we're in this good case where all the eigenvalues are above 0. OK. So that's the meaning. And now the next job. How do we recognize a positive definite matrix? It has to be symmetric. That's easy to see. But how can we tell if its eigenvalues are positive? That's not fun because computing eigenvalues is a big job. For a large matrix, we take time on that. We didn't know how to do it a little while ago. Now there are good ways to do it, but it's not for paper and pencil, and not for I. So how can we tell that all the eigenvalues are positive? Well, we only want to know their sign. We don't have to know what they are. We don't know that we need the number, we just want to know is it a positive number. And there are several neat tests. Can I show you them? I'm going to have five tests. Five equivalent tests. Any one of these tests is sufficient to make the matrix S positive definite. There is a particular S there that I'll use as a test matrix. So there is a symmetric matrix S. And I know it is positive definite. But how do I know? OK. Well. So can you take five things here? They connect all of linear algebra. It's really beautiful. That the eigenvalues, that's one chapter of linear algebra. The pivots are another chapter of linear algebra. Do you remember about pivots? That's when you do elimination. So 4 is the first pivot. The first pivot. Pivot number 1 is the 4. And then when I take a multiple of that away from that, I get a second pivot. And I'd see that that was positive. So what's that? Maybe I take 1 and 1/2 away of this. I multiply that by 1 and 1/2, 6, 9. Subtract from 6, 10. So I actually get a 1 down there. So pivot number 2 is a 1 in that case. Right? 6, 9 taken away from 6, 10 leaves me 0, 1. OK. It passed the pivot test. Notice I didn't compute the eigenvalues. I'm just doing other tests. Now here's another beautiful test. It involves determinants. Now, I have to say upper. Upper left. Upper left determinants greater than 0. What do I mean by an upper left determinant? I look at my matrix. That's a 1 by 1 determinant. Certainly positive. That determinant is 4. Here is a 2 by 2 determinant. And that determinant is 40 minus 36, so happened to be 4 again. So the determinant of the matrix is 4. But I also needed the ones on the way. I can't just find the determinant of the whole matrix. That's the last part of this test, but I have to do all the others as I get there. So it passes that test. Check. It works. So that test is passed. I'm doing more work than I need to do because one test would have done the job. Now here comes another one. S is A transpose A. That's what we looked at a minute ago. If S has this form A transpose A. Oh, what did we convince ourselves? We said that this was sure to be semidefinite. And I needed some condition to avoid A x equals 0. There was the possibility of A x equals 0. I'll just bring that down. You remember we started there and ended up here. And if A x was 0 then lambda was 0. We were in the semidefinite case. So I have to avoid that. So I have to say when A has independent columns. And I think I could factor that matrix S into A transpose A. I'm sure I could. And get independent columns. And it would pass test 4. I want to go on to test 5. Which really, in a way, is the definition, the best definition, of positive definite. So if I took number 5, it's the energy definition. So can I tell you what that means? I mean that x transpose Sx. If I take my matrix S that I'm testing for positive definite, I multiply on the right by any vector x, any x, and on the left by x transpose. Well, I get a number. S is a matrix. Sx is a vector. Inner product with a vector. I get a number. And that number should be positive for all x. Oh, I have to make one exception. If x is the 0 vector, then of course that answer is 0. All x except the 0 vector. OK. So that would be a way to-- another test. And this is associated in applications with energy. So I call this the energy test, or really the energy definition, of positive definite. x transpose Sx. I'd like to apply that test. So you'll see what does it mean. Now we're looking at all x to this particular example. But I won't throw away this board here. You see eigenvalues, pivots, determinants, A transpose A, and energy. Really all the pieces of linear algebra. A transpose A. We'll see it more and more. It comes up in least squares. If I have a general matrix A, it's not even square. It doesn't have great properties. But when I compute A transpose A, then I get a symmetric matrix. And now I know also a positive semidefinite. And with a little bit more positive definite matrix. OK. By the way, are there five tests for semidefinite matrices? Yes. There are five similar tests. All eigenvalues greater or equal to 0. All pivots greater or equal to 0. I can go down this and just allow that borderline case that brings in semidefinite. I won't do that. Let me take my matrix S. That small, example matrix. And apply the energy test. OK. So I'm looking at energy. So I'm looking at x. That's x1 x2, times my matrix 4, 6, 6, 10, times x1 x2. That's the energy. That's my x transpose Sx. x transpose Sx. Is that what we wanted to compute? Yes. x transpose Sx. Now, can I compute that? Yes. It's a matrix multiplication. Nothing magical here. But when I do, I'll show you the shortcut. When I do that, a 4 x1 is going to appear, and it'll be multiplied by that x1 over there. I'll get a 4 x1 squared. And then I'll have a 6 x2 that's multiplying that x1. So there's a 6 x1 x2. And now from this. That was the first component. And now I have 6 x1 and 10 x2. Multiply an x2. Well, that's another 6 x1 x2. And the 10, we'll multiply x2 and x2. x2 squared. I did that quickly. But the result is just easy to see. The 4, 6, 6, 10 show up in the squares. The diagonal 4 and 10 show up in the squares. And the off diagonal 6, which doubles to 12, shows up in the x1 x2, the cross term. OK. Now why should that-- so that's a number for every x1 and x2. Suppose x1 is 1 and x2 is 1. Then the number I get is 4, plus 6, plus 6, plus 10. That's probably 26. It's positive. What if x1 is 1-- let me try this. x1 is 1 and x2 is minus 1. Do I still get a positive energy? So my vector is 1 minus 1. So I get 4. Now, because of that, I have minus 6, and minus 6, and 10, from the x2 squared. And that's 14 minus 12. That's 2. It's positive. Well, I tested two vectors. I tested the 1, 1 vector and the 1, minus 1 vector. But you have to know that for every vector x, this does turn out to be positive. And I can show you that by something called completing the square. It's not what I plan to do. But the beauty is we now understand this energy test. What it means to take x transpose Sx, write it out, and ask is it always positive. Is it always positive? OK. So that's the fifth, number 5, test. But I think of it really as the definition. And it means-- can I draw a picture? Here is x1. Here's x2. And now I'm going to-- this is my function. x transpose A x. My energy. If I graph that, I have an x, and a y, and a function z. That function of x and y. What kind of a graph does it have? When x1 and x2 are 0, it's there. When x1 and x2 move away from 0, it goes positive. That graph is like that. It's sort of a bowl. And I have a minimum. One of the main application of derivatives in calculus is to find the test for a minimum, and decide minimum or maximum. Minimum or maximum. And you remember the second derivative decides a minimum or maximum. Positive second derivative, minimum. Negative second derivative, maximum. It tells you about the bending of the curve. Well, we're in two dimensions now, with a function of two variables. This is multivariable calculus. So what becomes positive second derivative, becomes positive definite matrix. A matrix of second derivatives. This is the whole subject of optimization. Maximizing, minimizing, comes here. OK. That's for another day. I just would like to tell you one more thing about positive definite matrices. I got a book in the mail which could be quite valuable. It's a little paperback, and the title is Frequently Asked Questions in Interviews for Financial Math. Being a Quant. Going to Wall Street. Becoming rich. So they don't give you all the money right away. They make you show that you know something. And so they ask a few math questions. And the first question was-- I was happy to see this. The first question asked, when is this matrix positive definite? OK. Can you see that matrix? 1 is on the diagonal. Those are correlation. This is a correlation matrix. That's why it's important in finance. It might be the three correlations of bonds, and stocks, and foreign exchange. So each one is correlated to itself with a full correlation of 1. But there'll be a correlation between bonds and stocks going up together, but not perfectly together, by some number a. And bonds and foreign exchange with some number b. Stocks and foreign exchange, some number c. So that's the matrix of correlations. And the key point is, it is positive definite. So the question when you go to Wall Street to apply for the money. If you're asked what's the test on those numbers a, b, c, to have a positive definite, proper correlation matrix? I would suggest the determinant test. The determinant test, if I'm given a small matrix, I'll just do the determinants. So that determinant is 1. No problem. This determinant, what's the 2 by 2 determinant? 1 minus a squared. So 1 minus a squared has to be positive. I'm doing the determinant test. And what's the 3 by 3 determinant? 1 from the diagonal. And I have an acb and an acb. I think I have two acb's from the three terms. Now, those terms have the plus signs. And now I have some with a minus sign, which better not be too big. That's the whole point on positive definite matrices. The off diagonal is not allowed to overrun the diagonal. The diagonal should be the biggest numbers. OK. So I saw that a squared had to be below 1. But now what's the determinant test? I think this has to be bigger than what I'm getting from this direction, which is a b squared, and a c squared, and an a squared. Oh, look at that. a squared, b squared, and c squared. That would be the answer. That first test there. Second test there. Well, the easy test was just 1, is positive. So really, that's what they're looking for. That would be the test, those numbers. So abc can't be too large or that would begin to fail. Good. So positive definite matrices have lots of applications. Here was minimum. Here was correlation matrices and finance. Many, many other places. Let me just bring down the five tests. Eigenvalues, pivots, determinants, A transpose A, and energy. And I'll stop there. Thank you. |
MIT_Learn_Differential_Equations | Classical_RungeKutta_ODE4.txt | PROFESSOR: Here is the classical Runge-Kutta method. This was, by far and away, the world's most popular numerical method for over 100 years for hand computation in the first half of the 20th century, and then for computation on digital computers in the latter half of the 20th century. I suspect it's still in use today. You evaluate the function four times per step, first in the beginning of the interval. And then use that to step into the middle of the interval, to get s2. Then you use s2 to step into the middle of the interval again. And evaluate the function there again to get s3. And then use s3 to step clear across the interval, and get s4. And then take a combination of those four slopes, weighting the two in the middle more heavily, to take your final step. That's the classical Runge-Kutta method. Here's our MATLAB implementation. And we will call it ODE4, because it evaluates to function four times per step. Same arguments, vector y out. Now we have four slopes-- s1 at the beginning, s2 halfway in the middle, s3 again in the middle, and then s4 at the right hand. 1/6 of s1, 1/3 of s2, 1/3 of s3, and 1/6 of s4 give you your final step. That's the classical Runge-Kutta method. Carl Runge was a fairly prominent German mathematician and physicist, who published this method, along with several others, in 1895. He produced a number of other mathematical papers and was fairly well known. Martin Kutta discovered this method independently and published it in 1901. He is not so nearly well known for anything else. I'd like to pursue a simple model of combustion. Because the model has some important numerical properties. If you light a match, the ball of flame grows rapidly until it reaches a critical size. Then the remains at that size, because the amount of oxygen being consumed by the combustion in the interior of the ball balances the amount available through the surface. Here's the dimensionless model. The match is a sphere, and y is its radius. The y cubed term is the volume of the sphere. And the y cubed accounts for the combustion in the interior. The surface of the sphere is proportional y squared. And the y squared term accounts for the oxygen that's available through the surface. The critical parameter, the important parameter, is the initial radius, y0, y naught. The radius starts at y0 and grows until the y cubed and y squared terms balance each other, at which point the rate of growth is 0. And the radius doesn't grow anymore. We integrate over a long time. We integrate over a time that's inversely proportional to the initial radius. That's the model. Here's an animation. We're starting with a small flame here, a small spherical flame. You'll just see a small radius there. The time and the radius are shown at the top of the figure. It's beginning to grow. When time gets to 50, we're halfway through. The flame sort of explodes, and then gets up the radius 1, at which time the two terms balance each other. And the flame stops growing. It's still growing slightly here, although you can't see it on this this scale. Let's set this up for Runge-Kutta. The differential equation is y prime is y squared minus y cubed. Starting at zero, with the critical initial radius, I'm going to take to be 0.01. That means we're going to integrate out to two over y0 out to time 200. I'm going to choose the step size to take 500 steps. I'm just going to pick that somewhat arbitrarily. OK, now I'm ready to use ODE4. And I'll store the results in y. And it goes up to 1. I'm going to plot the results. So here's the values of t I need. And here's the plot. Now, you can see the flame starts to grow. It grows rather slowly. And then halfway through the time interval, it's sort of explodes and goes up quickly, until it reaches a radius of 1, and then stays here. Now this transition period is fairly narrow. And we're going to continue to study this problem. And is this transition area which is going to provide a challenge for the numerical methods. Now here, we just went through it. We had a step size h, that we picked pretty arbitrarily. And we just generated these values. We have really little idea how accurate these numbers are. They look OK. But how accurate are they? This is the critical question about the about the classical Runge-Kutta method. How reliable are the values we have here in our graph? I have four exercises for your consideration. If the differential equation does not involve y, then this solution is just an integral. And the Runge-Kutta method becomes a classic method of numerical integration. If you've studied such methods, then you should be able to recognize this method. Number. two-- find the exact solution of y prime equals 1 plus y squared, with y of 0 equals zero. And then see what happens with ODE4, when you try and solve it on the interval from t from 0 to 2. Number three- what happens if the length of the interval is not exactly divisible by the step size? For example, if t final is pi, and the step size is 0.1. Don't try and fix this. It's just one of the hazards of a fixed step size. And finally, exercise four-- investigate the flame problem with an initial radius of 1/1,000. For what value of t does the radius reach 90% of its final value? |
MIT_Learn_Differential_Equations | Linearization_of_two_nonlinear_equations.txt | GILBERT STRANG: OK. Two equations, the question of stability for two equations, stability around a critical point. OK. So the idea will be to linearize, to look very near that critical point, that point. But now we're in two dimensions. So that's a little more to do. So here's the general picture, and then here is an example. So here's the general setup. We have an equation for the changes in y. But z is involved. And we have an equation for the rate of change of z. But y is involved. So they're coupled together. It's that coupling that's going to be new. So what's a critical point? Critical point is when those right-hand sides are 0. Because then y and z are both constant. So they stay at that point. Wherever they are at this critical point is steady state. They stay steady. They stay steady. They stay at that constant value. So we want that to be 0. And we want this to be 0. We have two equations, f equals 0, and g equals 0, two equations. But we have two unknowns, y and z. So we expect some solutions. And each solution has to be looked at separately. Each solution is a critical point. It's like well, you could think of a golf course, with a surface going up. So critical points will be points where the maximum point maybe, or the minimum point, or we'll see something called a saddle point. Actually, let's do the example. The example is a famous one. Predator-prey it's known as. Predator like foxes, prey like rabbits. So the foxes eat the rabbits. And the question is, what are the steady states where foxes' and rabbits' constant values could stay? So here this is the equation for what happens to the prey. So if the rabbits are left alone, the prey is the rabbits. If they're left alone they multiply, plenty of grass. Go for it. But if there are foxes, and z counts the number of foxes, then foxes eat, shall I say, those rabbits. And we lose rabbits. So we see that with a minus sign. And the amount of preying that goes on is proportional to the number of foxes times the number of rabbits. Because that gives the number of possible meetings. And what about the foxes, the predator? The predator increases. So this is from encountering the rabbits. That tends to make the number of predators increase. But if there were no rabbits, the foxes don't eat grass. They're out of luck, and they decay. So I see a minus z there. So you see the pattern? So starting from 0 and 0, at that point, so I've defined critical points. So there's my f. And that should be 0. And there is my g, and that should be 0. And it turns out there are just two possibilities. This is one. If y and z are both 0, then certainly I would get 0's. So that's like starting with a very small number of foxes and rabbits. Or if y is 1 and z equal 1, do you see that that would be, they would be in perfect balance? If y is 1, and z is 1, then that's 0 and that's 0. So the equation is satisfied. We can stay at-- y can stay at 1, and z can stay at 1. It's a steady state. And the question is, is that steady state where rabbits are staying-- their population is staying at 1, because they're eating grass, good. But they're getting eaten by the foxes, bad. And those two balance, and give a rate of change of zero. And the foxes similarly, the foxes get a positive push from eating rabbits. But natural causes cut them back. And they balance at z equal to 1. OK. So what I have to do is linearize. And that's the real point of the lecture. That's the real point for linear-- how do linearize for two functions? And how do you linearize for these two functions? So let me-- I have to write the general formula first, so you'll see it. And then I'll apply it to those two functions. OK. So here is the idea of linearize. So I'm linearizing. So my first function is whatever its value is at this point, it's like a tangent line. But now I've got two derivatives. Because the function depends on two variables. So I have a y minus capital Y, times now I have to do partial derivative. So that's the slope in the y direction, multiplied by the movement. And then similar term, z minus capital Z times the movement in the z direction. And I have to, because I stopped, this is the linear part of the function. I have to put an approximate symbol. Because I've ignored higher derivatives. And of course this is 0. So that's why we have linear in y and linear in z, times some numbers, the slopes. But we have two more slopes. Because we have another function g of y and z. And that again will just be approximately g at the critical point, which is 0 plus y minus capital Y, times dg dy plus z minus capital Z times dg dz. So altogether, the linear stuff and four numbers, the derivatives of f in the y and z directions, the derivatives of g in the y and the z directions. OK. Now we had an example going. Let me bring that example down again. There was my f. There was my g. Can easily find those partial derivatives. So let me do it. So the partial derivative with respect to y will be 1 minus z, z held constant for that partial derivative. And the partial derivative with respect to z will be minus y. Let me write all those things down. So here is in the example. So can I create a little matrix, df dy? It's the nice way, if I've got four things, 2 by 2 matrix is great. df, dz; that number, dg, dy; and dg dz. You could say this is the first derivative matrix. It's the matrix of first derivatives and it's always named after Jacobi who studied these first. So it's called the Jacobian matrix. Maybe I'll put his name to give him credit, Jacobi. And the matrix is the Jacobian matrix. And its determinant is important. It's a very important matrix, important in economics. We're doing things-- I'm speaking here about predator-prey, little animals running around. But serious stuff is the economy. Is the economy stable? If it's a running along at some steady state and we move it a little bit, does it return to that steady state, or does it get totally out of hand? So there is the Jacobian matrix. And what are those derivatives? Remember again, what functions. There's my functions. So the y derivative is 1 minus z. And the z derivative is minus y. The y derivative is z, and the z derivative is y minus 1. Is that all right? This is what we need to know from the functions. I've forgotten the functions in the board that went up. Here are their derivatives. Now that's my Jacobian matrix. That's my matrix of the-- that matrix has these four coefficients, those four numbers. And really, so the linearization, let me call that matrix. I should call it J for Jacobian. I will call it J for Jacobian. OK. That's the Jacobian matrix. So then my approximate-- what's my linearized equation? My linearized equation is the time derivative of y and z. So this left-hand side is just dy dt, and dz dt. So I'm using vector notations, putting y and z together, instead of separately, no big deal. So then I have this matrix J, this 2 by 2 matrix, times you notice here is a y minus capital Y and a z minus capital Z. There is the linearized problem. Linearized because this is constant, and this is linear, single y, single z, and we have a matrix J. So I have to find-- so now my little job is find the critical points. I've got everything ready. I have to find the critical points. Now remember, the critical points are where f and g are 0. And let me remember what those are. So there is the f. There's the g. One critical point was that one. Everything is 0. Another critical point is that one. Again, everything is 0. So I have two critical points, two Jacobian matrices, one at the first point, one at the second point. So what are those matrices? At y and z equals 0, I have the matrix-- I'll put it here, and then I'll copy it. If y and z are 0, I have a 1 and a 0 and a 0 and a minus 1. I just took y and z to be 0. That was the first critical point. The second critical point gives me the Jacobian at that second point. The second point was when z was 1. So that's 0 now. And y was 1, so that's minus 1. z is a 1, y minus 1, y is a 1. So that's a 0. That's the second Jacobian. So we're seeing something interesting here. We're seeing how 2 by 2 matrices will work by really nice examples, 1, 0, minus 1. What's that telling me? That's telling me that the rabbits grow. Because the rabbits are the first, the y. And the foxes decay. And that's what's happening when the two populations are really small. When the two populations are really small, multiplying them together is extremely small. So when the two populations are really small, forget the eating. There aren't enough people around, enough foxes and rabbits around to make a decent meal. So I just have dy dt equal y. Rabbits are growing from eating grass. dz dt is minus z, foxes are decaying from natural causes. So that's what kind of a stationary point will 0, 0 be? Rabbits are growing. It's an unstable point. We're leaving at 0, 0. Rabbits are increasing. Now how about the second point? The second point was when they were both 1. when they're both 1, then we got this as the Jacobian matrix. Oh, this is the interesting one. Can I just stay with this one to finish? So I'm interested in the-- and I'll put these facts on a new board. So again, y prime dy dt is y minus yz. z prime is yz, rabbits are getting eaten, minus z. And I'm interested in the point y equal 1, z equal 1. And my matrix of this is the Jacobian matrix. The Jacobian matrix had the derivatives, which were 1 minus z, minus y. The y derivative of that is z. The z derivative is y minus 1. And at this point y and z are 1. So that became 0, minus 1, 1 and 0. And what kind of a problem do I have here? So my linearized equation, so linearized, linearized around the point 1,1. My equation is y minus 1 prime, sorry. It's the distance to the critical points. the derivative of y minus 1 is, I see here a minus 1. I see a minus z minus 1. And here a plus y minus 1. You've got to understand this pair of linearized equations. If I use some other variable, the derivative of the first guy is minus the second. The derivative of the second guy is plus the first. What will happen? Initially if I'm a little bit-- if I have extra foxes, the rabbit population will drop. The rabbit population-- this will be negative. If z, the number of foxes, is a little higher than 1, then the rabbit population drops. And when the rabbit population drops, z starts dropping. As z starts dropping below 1, the rabbit population starts increasing. I get, what shall we say, sort of exchange of rabbits and foxes, oscillation between rabbits and foxes? So this is the-- right in the center there, that would be the point where y is 1 and z equal 1, the critical point. And if I start out with some extra rabbits, then the number of rabbits will drop. Because foxes are eating them. The number of foxes will increase. I'll go up. So there I have a little bit later, I have foxes now, but no rabbits to eat. So the foxes start dropping, and what happens? I think, yeah. The number of rabbit starts increasing. And this is what happens. I'll go around and around in a circle. If you remember the pictures of the paths for 2 by 2 equations, there were saddle points. That's what this is. y equals 0, z equals 0 was a saddle point. So no, it's a saddle. And what I'm discovering now for y equal 1 is an oscillation between foxes and rabbits. So again, I could say it'll be our center that was the very special picture where it didn't spiral out. It didn't spiral in. The special numbers, the eigenvalues of that matrix are-- well, better leave eigenvalues for the future. Because they happen to be i and minus i here. It's motion in a circle. It comes from this, the equation here. Motion in a circle as in y double prime plus y equals 0. That's motion in a circle. And that's what we've got. So this is a center. Now what about-- do I call a center stable? Not quite, because the rabbits and foxes don't approach 1. They stay on a circle around 1. Either I've got extra rabbits or extra foxes. But the total energy or the total stays a constant on that circle. And I would call that neutrally neutral. Neutral stability, because it doesn't blow up. I don't leave the area around. I stay close to the critical point. But I don't approach it either. OK. So that's a case where we could see the stability, based on the linearization. OK. One more example to come in another lecture. Thanks. |
MIT_Learn_Differential_Equations | The_Column_Space_of_a_Matrix.txt | GILBERT STRANG: OK. We're coming to the point where we need matrices. That's the point when we have several equations, several differential equations instead of just one. And it's a matrix that does that coupling. So can I-- this won't be a full course in linear algebra. That would be available, you may know on, open courseware for 18.06. That's the linear algebra course. But [INAUDIBLE] facts, and why not just say them here in a few minutes? So I have a matrix. Well there's a matrix. That's a 3 by 3 matrix. And first I want to ask how does it multiply a vector. So there it is multiplying a vector, v1, v2, v3. And what's the result, key idea? It takes the answer on the right-hand side is this number v1, times that column, plus this number, that number times the second column, plus the third number, the third number times the third column, combination of the columns of a. That's what a times v is. That's what the notation of matrix multiplication produces. That's really basic to see it as a combination of columns. Now I want to build on that. That's one particular, if you give me v1, v2, and v3, I know how to multiply it. I take the combination. Now I would like you to think about the result from all v1, v2, and v3. If I take all those numbers, and I get a whole lot of answers. They're all vectors, the result of A times v is another vector, Av, And I want to think about Av, those outputs, for all inputs v. So I take v1, v2, v3 to be [AUDIO OUT] numbers. And I get all combinations of those three columns. And usually I would get the whole 3-dimensional space. Usually I can produce any vector, any output b1, b2, b3 from A times v. But not for this matrix, not for this matrix. Because this matrix is, you could say, deficient. That third column there, 2, 3, 3, is obviously the sum of columns one and column two. So this v3 times that third column just produces something that I could already get from column one and column two. That v3 times that column three, I could x out. That's the same as column one, plus column two for this matrix, not usually. And then so I only really have a combination of two columns. It's a combination of three. But the third one was dependent on the others. And it's really a combination of two columns. So combinations of two columns, two vectors in 3-dimensional space produce a plane. I only get a plane. I don't get all of 3-dimensional space, only a plane. And I call that plane the column space, so the column space of the matrix. So if you gave me a different matrix, if you change this 3 to an 11, probably the column space now changes to-- for that matrix I think the column space would be the whole 3-dimensional space. I get everything. But when this third column is this the sum of the first two columns, it's not giving me anything new. And the column space is only a plane. And you can think of a matrix where the column space is only a line, just one independent column. OK. So that, we thought about this. [AUDIO OUT] is all combinations of the columns. In other words, it's all the results, all the outputs from A times v. It's all the outputs from A times v. Those are the combinations of the columns. So we can answer the most basic question of linear algebra. When does Av equal b? Have [AUDIO OUT]. When is there a v so that I can solve this? When is there a v that solves this equation? So it's a question about b. What is it about b that must be true if this can be solved? Well this says that equation is saying b is a combination of the columns of a. So this has a solution when b must be-- shall I say must be in the column space. For that example, only b's that where we can get a solution on b's that are combinations of the first two columns. Because having the third column at our disposal gives us no help. It doesn't give us anything new. [AUDIO OUT] It will be solvable if b equalled 1, 1, 1. That's a combination of the column, or if b equals 1, 2, 2. That's another simple combination of the columns. Or if b equals 2, 3, 3. But I'm only, I'm staying on a plane there. And most b's are off that plane. Now when there is a solution. All right. Now a second key idea of linear algebra, can we do it in this short video? I want to know about the equation Av equals 0. So now I'm setting the right-hand side to be 0. That's the 0 vector, 0, 0, 0. Does it have a solution? Does it have a solution? Let's take this example. 1, 1, 1; 1, 2, 2; 2, 3, 3; now I'm looking at the solutions when the right side is all 0. Does that have a solution? Is there a combination of those three columns that gives 0? Well there is always one combination. I could take 0, 0, and 0. I could take nothing, 0 of everything. 0 of this column, 0 of that column, 0 of the third column, would give me to the 0 [AUDIO OUT]. That solution is always available. The big question is, is there another solution. And here for this deficient, singular, non-invertible matrix, there is. There is another solution. Let me just write it down. Let me put it in there. Do you see what the solution is? The third column is the sum of those two. So if I want one of that column, I should take minus 1 in other column. So this is minus this column, minus this column, plus this column gives me the 0 column. That is a vector in the null space. That's a solution to Avn equals [AUDIO OUT]. So the null space is all solutions to Av equals 0. It's all the v's. The null space is a bunch of v's. The column space was a bunch of b's. It's just going to just emphasize that difference. I was looking at which b [AUDIO OUT]. I wasn't paying attention to what that solution was, just is there a solution. Then that b is in the column space. I take b equals 0. I fixed that all important b. And now I'm looking at the solutions. And here I find one. Can you find any more solutions? I think minus 10, minus 10, and 10 would be another solution. It's 10 times as much. And 0, 0, 0 is solution. [AUDIO OUT] line of solutions. We had a plane for the column space. But we have a line for the null space. Isn't that neat? One's a plane, one's a line, dimension two plus dimension one. Two for the plane, one for the line, adds to dimension three, the dimension of the whole space. OK. That's a little going at in. All right. Now I ask, what our all solutions? Complete solution to Av equals, well let me choose some right-hand side where there is a solution. Let me choose a right-hand side, say if I add that column and that column, I'll get Av-- maybe I'll take two of that column plus one of that column. Two of the first column with one of the second would be 3, 2 plus that would be a 4, 2 plus that would be another 4. OK. That's my b. It's a combination of the columns. You saw me create it from the first two columns. So now I ask, what are all the solutions? It's in the column space. It's 2 times the first column, plus the second column. But there may be other solutions. So all solutions, a complete solution, v complete is here's the key idea. And the point is that it's the same that we know from differential equations. It's particular solution plus any null solution. Plus all, you can say all v null. Particular plus null solution. It's such an important concept we just want to see it again. One particular solution with that thing would be particular, v particular could be-- 2-- how did we produce that? Out of two these, plus one of these, plus zero of that. So v particular could be 2, 1, 0. It works for that particular b, two of the first column, one of the second. Now then we could add in anything in the null solution. So we have infinitely many solutions here. We've got one solution plus added to that, a whole line of solutions. This, all the null space, would be all vectors like that. OK. That's the picture that we've seen for differential equations. And I just want to bring it out again for matrix equations, using the language of linear algebra. That's what I'm introducing here. I have one particular solution, plus anything in the null [AUDIO OUT] space of vectors that is the heart of linear algebra. Thank you. |
MIT_Learn_Differential_Equations | The_Stability_and_Instability_of_Steady_States.txt | GILBERT STRANG: This is a topic I think is interesting. I like this one. It's about stability or instability of a steady state. So let me show you the differential equation. It could be linear, but might be non linear. dy dt is f of y. I'm going to-- I keep it that right hand side not depending on t, so just a function of y. And when do I have a steady state? There's a steady state when the derivative is 0. So if the derivative is 0 when f of y equals 0, let me call those special y's by a capital letter. So capital Y is a number, a starting value, where the right hand side of the equation is 0. And if the right hand side of the equation is 0, the left side of the equation is 0, and dy dt is 0, and we don't go anywhere. So the solution-- if f or y is equal to 0, then we have y stays at y. It's a constant for all time, and my question is, if we start near capital Y, do we approach capital Y as time goes on? It's, in that case, I would say, stable-- or does the solution when we start near y go far away from Y? From capital Y? Leave the steady state? In that case, I would call the steady state unstable. So stable or unstable, and it's very important to know which it is. And let me just do some examples, and you'll see the whole point. So here is first starting with a linear equation. So what is capital Y in this case? Well, this is f of y here. So if I set that to 0, the steady state is capital Y-- capital-- equals 0 in this case. That is 0. So if I start at 0, I stay at 0. Here is a second example, the logistic equation, where I've taken the coefficients to be 1. What are the steady states for the logistic equation? Again, I set the right hand side to 0. I find two possible steady states-- capital Y equals 0 or 1. That right hand side is 0 for both of those, so in both cases, those are both constant solutions, steady states. If the solution starts at 0, it stays there because the derivative is 0. Has no reason to move. And finally, now I let y minus y cubed equals 0. I solve y equals y cubed, and I find three solutions, three steady states. Y could be 0 again. It could be 1 again, or it could be minus 1. y equals y cubed. Then y can be any of those three groups, and of course, these are examples. The actual problem could have sines, and cosines, and exponentials, but these are three clear cases, and of course, the linear case is always the good guide. So in the linear case, when does a solution stay near 0? If I start small, when do I go to 0, and when do I leave? So I'm ready for the answer here. So, stable or not. In this example, y equals 0. That's stable if-- well, do you see what's coming? The solution is e to the at if I start-- or constant times e to the at. When does that go to 0? When does it approach the steady state? I need a to be negative. That's going to be the key to everything. That number a should be negative. Now, over here, we don't have an a. The key point will be to see what is that that should be negative in these examples. And can I tell you the answer? So the thing to look at, negative or positive, stable or unstable, is the derivative. Look at the derivative of that right hand side at y equals y at the steady state. And if the derivative df dy is negative, then stable. That was correct in this linear case. The derivative of ay was just a, we know that we get stable when a is negative because the solution has an e to the at. a is negative. We go to 0. What about examples two and three? So with those two examples you'll see the whole idea. So look at the second example, y minus y squared. f is y minus y squared. We look at its derivative. Its derivative is 1 minus 2y. The derivative of-- so I'm looking at 1 minus 2y. That's df dy. So what's the story on that? If y is 0, then that derivative is 1 plus 1 unstable. So y equals 0 is now unstable, and the other possibility, y equals 1, I think, will be stable, because when y is 1, 1 minus 2y-- that derivative that we check-- 1 minus 2y comes out minus 1 now-- negative-- and that's the test for stability. So capital Y equals 1-- you remember how those S curves went up and approached the horizontal line, the steady state capital Y equals 1? So OK with two different steady states there-- one unstable and one stable. And now here we have three steady states, and in other examples, we could have many, or they might be hard to find, but here we can see exactly what's happening. Now, I look at the derivative df dy. It's the derivative of y minus y cubed. So that's 1 minus 3y squared. So again, y equals 0 is bad news. y equals 0 I get 1-- positive number, unstable. So y equals 0, unstable. Whereas y equals 1 or minus 1-- those are the other two steady states-- then 1 minus 3y squared. y squared will be 1 in those cases. So I have 1 minus 3 minus 2. [INAUDIBLE] it's negative. So those are stable. Do you see how easy the test is? Compute the derivative df dy at the steady state, and just see is it stable or is it not stable, and that gives the-- see whether is it negative or is it positive. That decides stable or unstable. Now, I just want to show why briefly and then show you an example by throwing the book, and this would be an example in three dimensions that we will get to when we're doing a system of equations. So for something flying in three dimensions, we'll need three differential equations, and all this discussion, which is coming to the end of first order-- one first order equation. So this stability is one of the nice topics. Now, what's the reasoning behind it? Behind this test? Here's our test. If df dy is 0, that's our test, and why is that our test? Can I explain it here? I want to look at the difference between y and the steady state. And my question is, if that goes to 0, I have something stable. If that blows up, if y goes further away from this steady state, it's unstable. So dy dt is f of y. d capital Y dt-- well, that's actually 0. Capital Y is that constant steady state, and at the same time, f of Y is 0. So I've just put a 0 on the left side and a 0 on the right side, remembering that capital Y solves the equation with no movement at all. It's just steady. Now I have f of y minus f at capital Y. I'm going to use calculus. The difference between the function at a point and the function at a nearby point is approximately-- and the mean value theorem tells me that it really is-- is approximately the derivative df dy times y minus 1. That's the whole point of calculus actually-- to be able to estimate the difference between f at two points. This is delta f, if you like, and this is delta Y. And delta f divided by delta Y is approximately df dy, and approximately means more and more approximately, closer and closer, as these points-- as little y and capital Y come close. So in other words, what I have is approximately the linear equation-- the linear equation where the test is this is my a. Here is the a. Well, it's only approximate because this isn't a truly linear equation. We are allowing more terms, but calculus says it's better and better when you're close, and so our question is, do we get closer or do we not get close? And the answer is that when that a is negative, then it's just like the linear equation. The exponential of at goes to 0. y minus capital Y goes to 0-- stable-- when this thing is positive or maybe even 0. 0 is kind of a marginal case. I don't know whether I shoot off or go to 0, so I'm only going to say if that is negative, it's stable, and if it's positive, then my e to the at blows up. And that e to the at is y minus capital Y. It gets bigger and bigger-- unstable. So that's the reasoning behind the beautiful, simple, easy to apply test, which is, if the derivative is negative, then stable. That's good, and now I'm ready to show you the example of a tumbling box. That is, I'm more or less ready. I'm going to take a copy of the book, and I'm going to throw it in the air. Well, I have put on a rubber band to hold it together, because the book is rather precious to be throwing around. And let me say here I learned about this experiment from Professor Alar Toomre and just today I've asked him would he like to do the experiment on a video? If yes, then he will do it properly. If no, I will do more with it when we get to three equations, because we're in three dimensions, but let me just show you the point. So the point is I'm going to throw this book up. Does it wobble all over the place-- unstable-- or does it turn nicely on its axis? I'm going to throw it up on the narrow axis here, the thin axis, half an inch or so. To me, that's stable. I'm not as good as Professor Toomre at catching it, but with a rubber band on it I caught. Now, that's one axis, but I'm in 3D. Here's another way to throw it-- is this way. Now, you can try this on somebody else's book. I'm going to throw it now this way. I'm going to start it this way, and the question is, does it turn steadily this way or not? No. Absolutely not. It went all over the place. Shall I do that one again? You see how it tumbles? Not so easy to catch. So that is unstable, and then there is a third direction. Let's see. I've done the very narrow one, the middle one. Probably the third direction is this one, and if I do it-- I'm going to leave well enough alone-- it will come out stable. So two directions-- stable, one unstable-- for a tumbling box, and the website and the book have lots more details, and we'll do more. One more thing I want to add to this board for these three examples-- can I do that? One more thing. I want a picture that shows-- so here's a line off to plus infinity, and this way to minus infinity. And if I took this example, 0 in example one, say, for dy dt equals minus y a negative is stable. It is stable. So the solutions here are approaching 0. This is approaching-- so my first example is dy dt equals minus y. I draw a line of y's. There's y equals 0, and the solution, wherever it starts, approaches 0. Now I'm ready to do the logistic equation. dy dt equals y minus y squared. Now I have y equals 0 is unstable now. y equals 0 is now unstable. y doesn't approach 0 anymore. It goes away from 0. And what does it go to? The other steady state, if you remember, was y equals 1. 1 minus 1 is 0. The derivative is 0. That's a steady state. Let me put it in here-- 1-- and that was a stable steady state. So that arrow is correct, and it goes to 1, and it's also going to 1 from above. So there is the stability line. Let me call this the stability line of y's that shows in the simplest possible picture what direction what direction the solution moves, which is the same as showing me the sine of dy dt. The sine of dy dt is positive, and this y minus y squared is positive for y between 0 and 1. Between 0 and 1, 1/2 would be a 1/2 minus 1/4 positive. So it increases, but it approaches 1. And now finally can I add in-- can I create the stability line for y minus y cubed? This is still correct. y equals 1 is still a stable point. 0 is an unstable point, but now I have 0, 1, or that other possibility, minus 1. So let me put that into the picture-- minus 1. That is a stable one. So for the y minus y cubed example, I'm stable, unstable-- you see the arrow is going away and going into 1 and minus 1, and then they go in from both sides. Isn't that a simple picture to put together what we discovered from the derivatives of this thing, the 1 minus 3y squared at those three points? At this point, the derivative was negative. We go to it. At this point, the derivative, one minus 3y squared, was positive. We leave it. At this point, this is another stable one. The derivative df dy is negative there, and the solution approaches y equals 1. We didn't have any formula for the solution. That's the nice thing. We're getting this essential information by just taking the derivative of that simple function and looking to see is it negative or positive and getting that picture without a formula. So tumbling book, stability, and instability, and more to do in higher dimensions. Thank you. |
MIT_Learn_Differential_Equations | Eigenvalues_and_Stability_2_by_2_Matrix_A.txt | GILBERT STRANG: This is a good time to do two by two matrices, their eigenvalues, and their stability. Two by two eigenvalues are the easiest to do, easiest to understand. Good to separate out the two by two case from the later n by n eigenvalue problem. And of course, let me remember the basic dogma of eigenvalues and eigenvectors. We're looking for a vector, x, and a number, lambda, the eigenvalue, so that Ax is lambda x. In other words, when I multiply by A, that special vector x does not change direction. It just changes length by a factor lambda, which could be positive. It could be zero. Could be negative. Could be complex number. It's a number, though. So that's the key equation. Let me go toward its solution. So I want to move that onto the left hand side. So I just write the same equation this way. And now I see that this matrix times the vector gives me 0. Now, when is that possible? That matrix can't be invertible. If it was invertible, the only solution would be x equals 0. No good. So this matrix must be singular. It's determined it must be 0. And now we have an equation for the eigenvalue lambda. So lambda is how much we shift the matrix to make the determinant 0. We shift by lambda times the identity to subtract that from the diagonal. So can I begin with very easy two by two matrix, the kind that we met first, called a companion matrix. So we met this matrix when we had a second order equation. So I started with the equation y double prime plus By prime plus Cy equals, say, 0. So I started with one second order equation. And then I introduced y prime as a second unknown. So now I have a vector unknown, y and y prime. And then, when I wrote the equation down-- I won't repeat that-- it led us to a two by two matrix. Two equations for two unknowns, y and y prime. So there is a two by two matrix that we're interested in. But we really are going to be interested in all two by twos. So let me take that to be my matrix A, my companion matrix. So I just want to go through the steps of finding its eigenvalues. What are the eigenvalues of that matrix? We just take the matrix, subtract lambda from the diagonal, and take the determinant. And when I take the determinant of a two by two matrix, it's just that times that, which is minus lambda times minus lambda is lambda squared. This gives me a B lambda. And the other part of the determinant is this product, minus C. But it comes with a minus sign, so it's plus C. So there's my equation for the eigenvalues of a companion matrix. And of course you see that's exactly the same equation that we had for the exponent s. So lambda for the matrix case is the same as s, s1 and s2 for the single second order equation. So this equation has solutions e to the st when the matrix has the eigenvalues lambda equal s. Those same s1 and s2. But now I move on to a general two by two matrix. What are its eigenvalues? What does that equation looks like for its two eigenvalues? So this will be a special case of this. Here, I have a general matrix, a, b, c, d. I've subtracted lambda from the diagonal. I'm taking the determinant. That'll give me the two eigenvalues. Let's do it. Minus lambda times minus lambda is lambda squared. Then I have a minus lambda d and a minus lambda a. So I have an a plus a d lambda. And then I have the part that doesn't involve lambda. The part that doesn't involve lambda is just the determinant of a, b, c, d. It's just the ad and the minus bc. So there's an ad and a minus bc, and all that is 0. It's a quadratic equation, second degree. A two by two matrix has two eigenvalues, the two roots of that equation. I just want to understand more and more and more about the connection of the roots, lambda 1 lambda 2, to the matrix a, b, c, d. If I know the two by two matrix, this tells me the eigenvalues. So this will, being a quadratic equation, have two roots. So if I factor this, this will factor into lambda minus lambda 1 times lambda minus lambda 2. And of course, if the numbers are nice, then I can see what lambda 1 and lambda 2 are. In that case, I find the eigenvalues. If the numbers are not nice, then lambda 1 and lambda 2 come from the quadratic formula, the minus b plus or minus square root of b squared minus 4ac. The quadratic formula will solve this equation, will tell me these two numbers. And if I multiply it out this way, I see lambda squared. I see minus lambda times lambda 1 and lambda 2. And then I see plus lambda 1 times lambda 2 equals 0. Here, I've written the equation for the two lambdas. Here, I've written the equation when I know the two lambdas. Why did I do this? I want to match this with this and see that this number, whatever it is, is the same as that number. They show up there, the coefficient of minus lambda. So that's the first step, that lambda 1 plus lambda 2 is the same as a plus d. Just matching those two equations. This is just like a general fact about a quadratic equation. The sum of the roots is the minus coefficient of lambda. And then the constant term is the constant term. So lambda 1 times lambda 2 is ad minus bc. These are facts about a two by two matrix, a, b, c, d. The sum of the eigenvalues. So this is the sum of the eigenvalues-- so I'll put s-u-m to indicate that I'm looking at the sum-- is that a plus d. A plus d are the numbers on the diagonal. So that's a little special. When I add the diagonal numbers, I get something called the trace of the matrix. I'm introducing a word, trace. Trace is the add up down the diagonal. And that matches a plus d. And this one is the product of the eigenvalues lambda 1 times lambda 2. So that's the product. And that's equal to the determinant of a. I'm just making all the neat connections that are special for a two by two. So that if I write down some matrices, we could look at them immediately. Let me write down a matrix. Suppose I write down that matrix. Oh, let me make them 0, 1-- well, 0, 4-- ah, let me improve this a little. 2, 4, 4, 9. 2, 4, 4, 2 would be even easier. Sorry. I look at that matrix. I see immediately the two eigenvalues of that matrix add to 4. 2 plus 2 is 4. I took the trace. The two eigenvalues of that matrix multiply to the determinant, which is 2 times 2 is 4 minus 16 minus 12. So the sum here for that matrix would be 4. The determinant of that matrix would be 4 minus 16 is minus 12. And maybe I can come up with the two numbers that have add to 4 and multiply to minus 12. I think, actually, that they are six and minus 2. I think that the eigenvalues here are 6 and minus 2 because those add up to 4, the trace, and they multiply 6 times minus 2 is minus 12. That's the determinant. Two by two matrices, you have a good chance at seeing exactly what happens. Now, my interest today for this video is to use all this, use the eigenvalues, to decide stability. Stability means that the differential equation has solutions that go to 0. And we remember the solutions are e to the st, which is the same as e to the lambda t. The s and the lambda both come from that same equation in the case of a second order equation reduced to a companion matrix. So I'm interested in when are the eigenvalues negative. When are the eigenvalues negative? Or if they're complex numbers, when are their real parts negative. So can we remember trace, the sum, product, the determinant. And answer the stability questions. So I'm ready for stability. So stability means either lambda 1 negative and lambda 2 negative. This is in the real case. Or in the complex case, lambda equals some real part plus and minus some imaginary part. Then we want the real part to be negative. Real part of a lambda, which is a, should be 0. So that's our requirement. If the eigenvalues are complex, we get a pair of them and the real part should be 0 so that e to the-- the point about this negative a is that e to the at will go to 0. The point about these negative lambdas is that e to the lambda t will go to 0. This is stability. So my question is, what's the test on the matrix that decides this about the eigenvalues? Can we look at the matrix-- maybe we don't have to find those eigenvalues. Maybe we can use the fact. Again, the fact is that lambda 1 plus lambda 2 is the trace and lambda 1 times lambda 2 is the determinant. And we can read those numbers off from the matrix. Then there's a quadratic equation. But if we only want to know information like are the eigenvalues negative? Are their real parts negative? We can get that information from these numbers without going to finding the eigenvalues from that quadratic equation. Wouldn't be that hard to do, but we don't have to do it. So suppose we have two negative eigenvalues. Then certainly, this would mean the trace would be negative. Because the trace is the sum of the eigenvalues. If those are both negative, trace is negative. So we can check about the trace just right away. What about the determinant? If that's negative and that's negative, then multiplying those will give a positive number. So the determinant should be positive. So trace less than 0. Determinant greater than 0. That is the stability test. That's the stability test. Stable. The two by two matrix A, B, C, D, if its trace is negative and its determinant is positive, is stable. That's the test. And actually, it works also if lambda comes out complex because lambda 1 plus lambda 2-- lambda 1 is a plus i omega. Lambda 2 is a minus omega. The sum is just 2a. And we want that to be negative. So again, trace negative. Trace negative even if the roots are real or if they're complex. That still tells us that the sum of the roots is negative and the determinant also works. If a plus i omega times a minus i omega-- in this case, lambda 1 times lambda 2-- if I multiply those numbers, I get a squared plus omega squared. With a plus. So that would be positive. And we're good. So my conclusion is this is the test for stability. And I can apply it to a few matrices. I wrote down a few matrices. Can I just look at that test-- can you look at that test-- and just apply it to see. So here's an example. Say minus 2, minus 1, 3, and 4. Is that any good? The trace is minus 3. That's good. The determinant is 2 minus 12 minus 10. That's bad. That's bad. So that would be unstable. That has a negative determinant. Unstable. So I'll put an x through that. Unstable. Let me take a stable one. Stable one, I'm going to want like minus 5, and 1, let's say. That's OK. The trace is negative. Minus 4. And now I want to make the determinant positive. So maybe I better put like 6 and minus 7. Just picking numbers. So now the determinant is minus 5 plus 42. A big positive number. And the determinant test is passed. So that is OK. That one would be stable. If this was my matrix A, then the solutions to dy dt equal Ay, y prime equal Ay is my differential equation. The two solutions which would track the eigenvectors would have negative lambdas. Negative lambdas because the trace is negative and the determinant is positive. Passes the stability test and the solutions would go to minus infinity. That's two by twos. Thank you. |
MIT_Learn_Differential_Equations | Fourier_Series.txt | GILBERT STRANG: OK, I'm going to explain Fourier series, and that I can't do in 10 minutes. It'll take two, maybe three, sessions to see enough examples to really use the idea. Let me start with what we're looking for. We have a function. And we want to write it as a combination of cosines and sines. So those our basis functions-- the cosines and the sine. And a n's and the b n's are the coefficients that we have to look for. That tells us how much of cosine nx is in the big function f of x. Notice that the cosines start at n equals 0, because cosine of 0 is 1. So there's an a0 in our sum. But there isn't a b0, because n equals zero of the sine would be zero, and we don't get anything there. So we're looking for the a n's and b n's. And, really, I want to show you, at the same time, the complex form with coefficient cn. And now n goes from minus infinity to infinity. That's really the more beautiful form because that one formula for cn does the job, whereas here I will need a separate formula for a n and for bn. OK. So this is natural when the function is real, but in the end, and for the discrete Fourier transform, and for the fast Fourier transform, the complex case will win. And, of course, everybody sees that e to the inx, by Euler's great formula, is a combination of cosine nx and sine nx. So, I can use those, or I can use cosine and sine. OK. So, how do you find these numbers? The key is orthogonality. So that's the first central idea here in Fourier series, is the idea of orthogonality. Now what does that mean? That means perpendicular. And for a vector, and a second vector, we have an idea of what perpendicular means. The 90 degree angle between them. And we check that by the dot product-- or inner product, whichever name you like-- between the two vectors should be 0. OK. But here we have functions-- like cosine functions. So here's one cosine, and here's a different cosine. So those are two different basis functions-- say, cosine of 7x and cosine of 12 x. The coefficients a7 and a12 would tell us how much of cosine 7x is in the function. You see, we're separating the function into frequencies. We're looking into pure oscillations, pure harmonics. And we expect, probably, that's the lower harmonics the smoother ones cos x, cos 2x, cos 3x, have most of the energy. And the high harmonics, cosine 12x, cosine 100x, probably those are quickly alternating, those contain noise, and high frequency. Quick changes in the function will show up in the high frequencies. OK. So what's the answer to this integral-- cosine of 7x times cosine of 12x dx, over the range minus pi to pi? Orthogonality comes in, the answer is 0. That's the crucial fact. That's what makes it possible to separate out a7 and a12 and get hold of them. So let me show you how to do that. So I'm going to use this fact, which is the function version of 90 degree angle. So, you see, it's a little like a dot product. Well, let me remember, a dot product would be something like c1 d1 plus c2 d2 equals 0, if I had a vector c1 c2 and a vector d1 d2. That would be the dot product, and it would be 0 if the vectors are orthogonal. Here, instead of adding, I'm integrating because I have functions. So just that's the meaning of dot product-- the integral of one function times the other function gives 0. OK. I'll use that now. OK, how will I use this? I will look what I want. This is my goal. I'll multiply both sides of this equation by cosine kx. And then I'll integrate. And the beauty is, that when I multiply by cosine kx, and I integrate, everything goes to zero except what I want. By the way, all the sines times cosine kx integrate to 0. All the sines are orthogonal to all the cosines. And all the cosines will be orthogonal to all the other cosines. So let me show you what I get. So I multiply my f of x by cosine kx, and I integrate from minus pi to pi. OK? Now, on the right-hand side, this is my integral from minus pi to pi, of my big sum of all these terms, 0 to infinity, a n cos nx, etcetera-- including the sines but I'm not even put them in because they're going to get killed by this integration-- times cosine kx dx. All I did was take the f of x equal that formula, multiplied both sides by cosine kx, and integrated. And, now the orthogonality pays off, because this times this, when I integrate gives 0, with one exception. When n equals k, then I do get the integral. The only term I get is ak, cosine kx, twice dx. Only k equal n survives this process. And then that integral of cosine squared happens to be pi, so this is just ak times pi. Look, I've discovered what ak is. I've discovered the k Fourier cosine coefficient. I just divide by pi. So can I just divide by pi to get this formula for ak? Ak is 1 over pi. The integral from minus pi to pi of my function, times cosine kx dx. That's the formula. That tells me the coefficient. And I could only do that with orthogonality to knock out all but one term. And now, if I wanted the sine coefficients, bk, it would be the same formula except that would be a sine. And if I wanted the complex coefficient, ck, it turns out it'd be the same formula expect-- well maybe it's 2 pi there, 1 over 2 pi-- and this becomes an e to the minus ikx. In a complex case, the complex conjugate e to the minus ikx shows up. So this is really the dot product, the inner product, of the function with the cosine. OK. So let me do some examples. Maybe I should write up the sine formula that I just mentioned. So bk is the integral 1 over pi, the integral of my function, times sine kx dx. And there's one exception. A0 has a little bit different formula, the pi changes to 2 pi. I'm sorry about that. When k is 0 or it's the integral of 1, from minus pi to pi, and I get 2 pi. So, a0 is 1 over 2 pi-- the integral of f of x times when k is zero cosine-- this is 1 dx. That has a simple meaning. That's the average of f of x. OK. So the basis function was just 1 when k was zero. When k is 0, the function of my cosine is just one, and I get the integral of the function times 1 divided by 2 pi. Could we just do an example? So I want to take a function. And in this video why don't I take an easy, but very important, function-- the delta function. So I plan to use these formulas on the delta function. Let me draw a little picture of the delta function. I'm only going between minus pi and pi, and the delta function, as we know, is 0, it's infinite, at the spike, and 0 again. The reason I wanted to draw it is, that's an even function. That's a function which is symmetric between x and minus x. And in that case, there will be no sines. Sine functions are odd. The integral from minus pi to pi of an odd function gives 0. The odd means that when you cross x equals 0 you get minus the result for x greater than 0. So my point is, this is an even function-- delta of x is the same as delta of minus x, and only cosines. Good. The sine coefficients automatically dropped our 0 so, of course, the integral would show it. But we see it even before we integrate. OK I'm ready for the delta function. So I'm going to write delta of x, and we remember what the delta function is-- a combination of cosines. OK. That's the delta function between minus pi and pi. OK. And what's our formula for the a n? Well, you remember we had a special formula for a0, which was 1/2 pi times the integral, from minus pi to pi, of our function, which is delta, times the basis function, which n equals 0, the basis function is 1 dx. OK, we know the answer to that. We can integrate the delta function. The one key thing about the integral of the delta function is, it's always 1-- if we cross x equals 0, which we will. So that integral is 1 so I'm getting 1/2 pi. What about the other for a coefficient? So that's 1/pi, now. The integral from minus pi to pi of all of my function times cosine kxdx. You know what I'm doing. I'm using my formula to find the coefficients. My formula says take the function, whatever it is-- and in this example, it's the delta function-- multiply by the cosine, integrate, and divide by the factor pi. OK. Well, of course, we can do that integral. Because when you integrate a delta function, times some other function, all the action is at x equals 0. At x equals 0, this function is 1. And I don't care what it is elsewhere, it's just 1. So this is the same as integrating delta of x times 1, which gives us-- well, the interval the delta function 1. So that integral is one, so I'm getting 1/pi. Good. OK. So now, do you want me to write out the series for the delta function? It looks kind of unusual. This is telling us something quite remarkable. It's telling us that all these coefficients are the same. All the frequencies, all the harmonics, are in the delta function in equal amounts. Usually, we would see a big drop off of the coefficients ak, but for the delta function, which is so singular, all a big spike at one point, there's no drop off and no decay in the coefficients, they just constant. OK. So I'm saying that the delta function is the constant term, 1/2pi, and then 1/pi times cosine of x, and cosine of 2x, and so on. OK. All frequencies there are the same. And I'll stop with that one example here. So the key points were orthogonality, the formulas for the the coefficients, and this example. Thank you. |
MIT_Learn_Differential_Equations | Incidence_Matrices_of_Graphs.txt | GILBERT STRANG: OK. I want to continue the last video, which was about incidence matrices, and graphs, and networks, and flows in the network. So that was 5.6. This is 5.6b. And I'll remember the same graph. You remember a graph is some nodes, four nodes here, and some edges, and in this case five edges. So I have a 5 by 4 matrix, and that's what it was. And I'll remember how it was created. Every row corresponds to an edge. So the first edge there goes from node 1 to node 2, so I put a minus 1 and a 1 in columns one and two. That tells me what that first edge is doing and it gives me one row of the incidence matrix. Five edges give me five rows. There is the matrix. And here I'll multiply by v, thinking of a vector v as voltages at the four nodes, and I get that answer. The 1 and minus 1 produce this kind of answer. OK. Now I'm ready for questions about the matrix A, the 5 by 4 matrix. These matrices, these incidence matrices, are beautiful examples of rectangular matrices where we can ask all the key questions about a matrix and get a nice answer. And the key questions that I have in mind are what are their solutions to Av equals 0? Are there-- That says, are there are combinations of the columns that give the zero column? So it's asking, are the columns dependent? If the columns were dependent, then I'll find some solutions, and here I will. If the columns are independent, the only solution I will find will be v equals 0. But those columns are dependent. Now, how can we see that? Well, in this case, we can find a solution to Av equals 0, because I can see how do I get all those differences to be 0? Well, not hard. v could be the vector of all 1's. Then the differences would all be 1 minus 1, would all be 0. I would be solving Av equals 0. And of course, I can multiply by any constant. The voltage is-- So all I'm saying is if all the voltages are equal, there won't be any flow. If all the voltages are equal and I don't have any batteries or other sources in the network, there will be no flow. And those are all the solutions. But the only way I could make all those 0 would be for all the v's to be the same. So all the v's have to be the same. v is C, C, C, C. And I learn something important. Av equals 0 has some solutions. And I'll just jump ahead one electrical moment. That's not good if we want an invertible matrix. In the end we would have A transpose A and it won't be invertible unless we do something. And what do we do? We want to get rid of that last column. We can have three columns. Those will be independent, but that fourth column is a combination of the others. And what we do, in reality, is we ground a node, which means we set one of the v's, maybe v4, if we set that to 0, it's like we're fixing the temperature, we're fixing the voltage, we often have to do this on a sliding scale. If we only know differences in temperature, we have to say, where is 0? And if we make that point 0, then we have only three unknown voltages and a 5 by 3 matrix and all well. OK. So that's the discussion of Av equals 0. Now, what about A transpose w equals 0? So now I'm asking about the transpose of that matrix. Now, this is a 4 by 5 matrix. Again, a beautiful example, 4 by 5 matrix. w, of course. It's a 4 by 5 matrix multiplying w, which is 5 by 1. So 4 by 5 times 5 by 1. And I want to get all zeros, four zeros this time. Right. So first of all, if I have a 4 by 5 matrix, so when I transpose this is you could say short and wide, I think there are automatically solutions. There will be solutions in a 4 by 5 matrix. With five unknowns and only four equations, I'm going to have some solutions to that system. So there will be some solutions. Well, the question is how many different w's could I find, how many different solutions, and what do they mean. And that's the beauty of this example, that it's not just a bunch of 20 numbers in the matrix. The matrix has a meaning. The incidence matrix takes differences A to Av is differences in v, but what's the meaning of A transpose? That's the key question here. Why is this equation very important? OK. So I have to tell you the meaning of A transpose. And maybe I have to copy down what A transpose is. So let me go to the next board and copy down A transpose. So I'm looking now at A transpose w. So now it will be 4 by 5. So that first row becomes a column. The second row becomes another column in the transpose. The third row, another column, the fourth row, is that column. And the fifth row is that one. And that will multiply w1, 2, 3, 4, and 5 to give 0, 0, 0, 0. And that's called the current law, Kirchhoff's current law. And what is that law? What does it mean? It means that in the network at a typical node, so at node 1, you remember, there was an edge out. Edge 1 went out. Three edges went out actually. This was to node 2. This was to node 3, and that was to node 4. At node 1, three edges are going out. And what does the current law tell me? It tells me that the total flow out is 0. The net flow, any flow in, which would be negative w's, and any flows out, which would be positive w's-- w, that came from the first edge. This was maybe the second edge. And I think that happened to be the fourth edge-- flows out of w. And that's what I see here. A 1, a 2, and a 4 are multiplying w1, w2, and w4 are the-- The first equation there is minus. w1 plus w2 plus w4 equals 0. So that came from the first row of A transpose w equals 0. Right? I just took those numbers from the first row. I wrote down that first equation. And you see it says exactly the sum of those three flows has to be 0. So if there's some positive flows going out, there must be some negative w's coming in to balance. OK. And that was at node 1, and similarly at nodes 2, and 3, and 4, currents balance. It's the balance equation. Kirchhoff's law, it's the balance equation. It's conservation. A fundamental equation in modeling applied mathematics is if a body is sitting there in equilibrium, then the forces on it are in balance. If I have steady flow around the network, the currents are in balance. Always there's a balance equation, so that things are not collecting up at a node. It's stable. OK. So that's the meaning of Kirchhoff's current law. That's the meaning of A transpose w equals 0. And what about solutions? Solutions w. Now, so now we're getting down into the details. Can we actually find the w's? Well, there will be some. There will be some. As I said, we've got five unknowns here and only four equations. So we're certainly going to find a solution. And let me suggest one good way to look for it. Suppose the flow-- Let me put in the other two edges-- suppose the flow goes around a loop. Loops are the key here. The key to the solution is a loop. So that's a flow that sends a flow of 1 along that edge, a flow of 1 going that way along that edge, which I think was w5, and a flow of 1 going that way. Pay attention. It's going to send 1 Amp around the loop. I go with the arrow, with the flow, this way and this way, but this one is against the arrow. So I'm thinking that a solution is w1 equals 1. You see I'm writing down a solution without doing any elimination or other linear algebra. I'm just understanding the picture. w1 is 1. w5 is 1. w5 is 1. And what is w4? Negative 1, because it's going against the arrow. And the other two w's are 0, w2 and w3. This was w3 here. Those are not involved in this loop. So there is a solution with w2 and w3 equal to 0. And I think that how could it fail on Kirchhoff's current law? Nothing is piling up at a node. We're just sending it around a loop. And of course, I put in that's a 1. w2 is a 0. w4 is a minus 1. I have a 1 and a minus 1. I get 0, just right. And all the equations would be solved. In other words, conclusion, the solutions w come from loops in the network. Every loop in the network gives me a w. Here's another loop. I could send flow down there. Now that would be a w4 plus 1. This way. This way. Do you see that second loop? Let me draw my little loopy symbol. Flow going around that loop. That loop happens to have four edges on it. So I'd have four w's. 1 minus 1, 1, and minus 1, and no flow on edge 1, and I would have another solution. And it would be a different solution. So I'm going from-- Can I insert here two loops? In that graph I see two loops, two small loops. And each of those small loops gives me a flow, a w, that solves the current law, because it's just continuously running around and around. Now, there's another question to ask you, and that is what about the big loop, w1, w3-- I think that is-- and minus w2? What if I send flow around the big loop? No problem. That gives me another set of w's . Those satisfy Kirchhoff's current law. They satisfy these equations. They satisfy A transpose w equals 0. But I don't want that big loop. I don't want to include that in my list of w's, because I was only looking for two w's. I was only looking for two w's. And linear algebra told me that was the number to look for. And here you're suggesting-- I'll blame you-- a third around the big loop. So what's up? Well, do you see it? The flow around that big loop does solve A transpose w equals 0, but it's not new. It's the sum of a flow around that loop plus a flow around that. Do you see? If I add together the flow vector, the loop vector for w for that loop and for that loop, they will cancel on the edges that are in both loops, and I'll just be left with the flow there, the flow there, and the flow there, and that's the big loop. In other words, that big loop doesn't give me a new vec-- It doesn't give me-- It gives me a vector w that's a combination of what I already have. And in linear algebra, that's always the question. You want the number of independent w's, and this big loop is a dependent w, because it's a combination of the other two. OK. So that's the picture for one particular example. I'll just end with linear algebra facts, linear algebra facts. OK. So how many-- So if I have an m by n matrix, and suppose Av equals 0 has how many independent solutions shall I say? k independent solutions. And in my example, the incidence matrix, the answer was, for A equal incidence matrix, k was 1. So if I know the number of solutions to that equation, then how many solutions do I expect to-- This has-- So how many solutions do I expect there? The difference between m and n comes in it, and then plus k. So independent solutions. That's a basic fact of linear algebra that I never wrote down before. I never wrote it in this notation. I'll make that a question on a future linear algebra exam. What I'm saying is that if I know how many solutions Av has, how many combinations, these are combinations of the columns of A that give 0, then I know how many combinations of the rows of A. Let's just check that this counting theorem was correct. This was k equals 1, right? The only solution to Av equals 0 was the constants, 1, 1, 1, 1. Then m was 5. n was 4. k was 1. 5 minus 4 plus 1 is 2. And that's the number of loop solutions to Kirchhoff's current law. OK. We have voltages. We have currents. And there's a lot of beautiful linear algebra involved with these matrices. I'll also include a video about RLC circuits, which are totally an application of this. And there I'll begin with just one loop, one RLC loop. But the reality of modern electronics is thousands of nodes, thousands of edges, maybe tens of thousands of edges, and many, many loops. Good. Thank you. |
MIT_Learn_Differential_Equations | Systems_of_Equations.txt | PROFESSOR: Many mathematical models involve high order derivatives. But the MATLAB ODE solvers only work with systems of first order ordinary differential equations. So we have to rewrite the models to just involve first order derivatives. Let's see how to do that with a very simple model, the harmonic oscillator. x double prime plus x equals 0. This involves a second order derivative. So to write it as a first order system, we introduced the vector y. This is a vector with two components, x, and the derivative of x. We're just changing notation to let y have two components, x and x prime. Then the derivative of y is the vector with x and x double prime. So the differential equation now becomes y2 prime plus y1 equals zero. Do you see how we've just rewritten this differential equation. so y2 prime is playing of x double prime? Once you've done that, everything else is easy. The vector system is now y1, y2 prime is y2 minus y1. The first components says y1 prime is y2. That's just saying that the derivative of the first component is the second. Here's the differential equation itself. Y2 prime is minus y1 is the actual harmonic oscillator differential equation. When we write this as an autonomous function for MATLAB, here's the autonomous function. f is an autonomous function of t and y, that doesn't depend upon t. First it's a vector now, a column vector. The first component of f is y2. And the second component is -y1. The first component here is just a matter of notation. All the content is in the second component, which expresses the differential equation. Now for some initial conditions-- suppose the initial conditions are that x of 0 is 0, and x prime of 0 is 1. In terms of the vector y, that's y1 of 0, the first component of y is 0. And the second component is 1. Or in vector terms, the initial vector is 0, 1. That implies they solution is sine t and cosine t. When we write the initial condition in the MATLAB, it's the column vector 0, 1. Let's bring up the MATLAB command window. Here's the differential equation. y1 prime is y2. And y2 prime is -y1. Here's the harmonic oscillator. We're going to integrate from 0 to 2pi, because they're trig functions. And I'm going to ask for output in steps of 2 pi over 36, which corresponds to every 10 degrees like the runways at an airport. I'm going to need an initial condition. y0 not is 0. I need a column vector, 0, 1, for the two components. I'm going to use ODE45, and if I call it with no output arguments, ODE45 of the differential equation f, t span the time interval, and y0 the initial condition. If I call ODE45 with no output arguments, it just plots the solution automatically. And here we get a graph of cosine t starting at 1, and sine t starting at 0. Now if I go back to the command window, and ask to capture the output in t and y, I then get vectors of output. 37 steps, vector t, and two components y, the two columns containing sine and cosine. Now I can plot them in the phase plane. Plot ya against y2. If I regularize the axes, I get a nice plot of a circle with small circles every 10 degrees, as I said, like the runways at an airport. The Van der Pol oscillator was introduced in 1927 by Dutch electrical engineer, to model oscillations in a circuit involving vacuum tubes. It has a nonlinear damping term. It's since been used to model phenomena in all kinds of fields, including geology and neurology. It exhibits chaotic behavior. We're interested in it for numerical analysis because, as the parameter mu increases, the problem becomes increasingly stiff. To write it as a first order system for use with the MATLAB ODE solvers, we introduce the vector y, containing x and x prime. So y prime is x prime and x double prime. And then the differential equation is written so that the first component of y prime is y2. And then the differential equation is written in the second component of y. Here's the nonlinear damping term minus y1. When mu is 0, this becomes the harmonic oscillator. And here it is as the anonymous function. Let's run some experiments with the Van der Pol oscillator. First of all, I have to define the value of mu. And I'll pick a modest value of mu. Mu equals 100. And now with mu set, I can define the anonymous function. It involves a value of mu. And here is the Van der Pol equation in the second component of f. I'm going to gather statistics about how hard the ODE solver works. And for that, I'm going to use ODE set, and tell it I want to turn on stats. I need an initial condition. Now I'm going to run ODE45 on this problem. And I'm specifying just a starting value of t, and a final value of t. ODE45 is going to pick its own time steps. And I know with t final equals 460, it's going to integrate over it about two periods of the solution. Now we can watch it go for a while. It's taking lots of steps. And it's beginning to slow down as it takes more and more steps. Now this begins to get painfully slow as it runs into stiffness. I'll turn off the camera for a while here, so you don't have to watch all these steps. We're trying to get up here to 460. And I'll turn it back on as we get close to the end. OK, well, the camera's been off about three minutes. And you can see how far we've gotten. We're nowhere near the end. So I'm going to press the stop button here. And we'll go back to the command window. And oh, so we didn't get to the end here. Let me back off on the time interval and try this value here. See how that works. So this is going to still take lots of steps. But we'll be able to-- This will go over about one period. We'll actually get to the end here. I'll leave the camera on until we finish. OK so that took a little under a minute. And it took nearly 15,000 steps. So let's run it with a stiff solver. There. So it took half a second, and only 500 steps. So there's a modest example of stiffness here. So let's examine the Van der Pol equation using the stiff solver. Let's capture the output and plot it. Because that plot wasn't very interesting. I want to plot it a couple of different ways. And again, I want to go back up to the-- capture a couple periods. Let's plot one of the current components as a function of time. And here it is. Here's the Van der Pol equation. And you can see it has an initial transient, and then it settles into this periodic oscillation with these really steep spikes here. And even this stiff solver is working hard at these rapid changes. We've got a fair number of points in here, as it is it chooses the step size. And now, let's go back to the command line and do a phase plane plot. So here's the phase plane plot of this oscillator with damping. And it's nowhere near a circle, which it would be if mu was 0. And this is the characteristic behavior of the Van der Pol oscillator. |
MIT_Learn_Differential_Equations | Tumbling_Box.txt | PROFESSOR: Here are the differential equations for the angular momentum of a tumbling box. Try throwing a book, or a box, or any rectilinear object whose three dimensions are all different, into the air with a twist, to make a tumble. You could go to rotate about its longest axis, or about its shortest axis. But you can't get to rotate about its middle axis. Let's examine that phenomena numerically. Here's the anonymous function defining those system of three first order differential equations. Now I'm going to start with an initial condition that's near the first critical point. 1, 0, 0 is a critical point. And I'm going to take 0.2 times a random number, to sort of be near the critical point, and then normalize it, so that it has length 1. So the largest component is the first component. And the other two are small, but not too small. This is an easy problem numerically. There's no stiffness involved here. So I'm going to use ODE 23, integrate from 0 to 10, and here's the solution. The blue component is the first one, and it stays near 1. And the other two are periodic, rotating around the 0. Let's go back and take another starting condition. Here it is again. Now the other two components are really small. And when we integrate that, the blue component stays flat near 1. And the other two guys hardly move at all. Now I'm going to go to the third critical point, 0, 0, 1. Do the same thing. Take a random number near there. Use ODE 23. Now the yellow component stays near one. And the other two move periodically around 0. Run that again. The third component is near 1. The other two are not too big. And run the ODE 23. The other component stays near 1. And the other two rotate periodically around 0. Now we're going to go to the middle critical point. We're going to try and get the box to rotate around its middle axis. The second component is the one near 1. And now we see completely different behavior. This sienna component doesn't stay near 1. It goes down near -1, and comes back up. Let's integrate over a longer period, so we can see that behavior. So it's periodic. But it goes down to -1 and comes back to 1. And the other two move in large amplitudes around 0. So this is the instability of that middle critical value. Let's doing again. Same thing. 1 down to -1, and back up. It's periodic. These solutions are all periodic. But that middle critical point is unstable. Now I want to view these in a different way, graphically. The differential equations have these three critical points. Any solutions to start exactly in these initial conditions stay there. But what happens if you start near these initial conditions? Well it turns out, that x and z are stable critical points. But y is an unstable critical point. If the angular momentum is near x or near z, it stays near there. But if it starts near y, it moves away quickly. You can think of x as the short axis, and z is the long axis. Rotation near the short axis is stable. And rotation near the long axis a stable. But rotation near the middle axis is unstable. We can see that in the following graphic. It turns out that if a solution starts with an initial condition that has norm 1, it stays with norm 1. So the solution lives on the unit sphere. Here's our unit share with our three critical points, x, y, and z. If this were the Earth, z will be the North Pole. Axis where the 0-th meridian crosses the equator. That's in the east Atlantic, a little off of West Africa. y would be where the 90th meridian crosses the equator. That's in the Indian Ocean, west of Sumatra. If we start with an initial condition near x, the solution orbits around x. That's stable rotation around the short axis. If we start with an initial condition near z, the solution orbits around z. That's stable rotation around the long axis. But if we start near y, the solution takes off, goes over to near -y, turns around, and comes back to y. Periodic, but goes clear around the globe. That turns out, that's a circle actually, an orbit around x. If we come up a little above y, we get an orbit around z. Go down a little below y, we get an orbit around -z. Go to the right of y, we get an orbit around -x. Let's zoom in a little bit. And we can see that y is a classic unstable critical point. Let's conclude by drawing a few orbits. |
MIT_Learn_Differential_Equations | Powers_of_Matrices_and_Markov_Matrices.txt | GILBERT STRANG: OK. So this video is about using eigenvectors and eigenvalues to take powers of a matrix, and I'll show you why we want to take powers of a matrix. And then the next video would be using eigenvalues and eigenvectors to solve differential equations. The two big applications. So here's the first application. Let me remember the main facts. That if A-- if. This is an important point. Not every matrix has n independent eigenvectors that would go into matrix V. You remember V is the eigenvector matrix, and I need n independent eigenvectors in order to have a V inverse, to make that formula correct. So that's the key formula for using eigenvalues and eigenvectors. And the case where we might run short of eigenvectors is when maybe one eigenvalue is repeated. It's a double eigenvalue, and maybe there's only one eigenvector to go with it. Every eigenvalue's got at least one line of eigenvectors. But we might not have two when the eigenvalue is repeated or we might. So there are cases when this formula doesn't apply. Because I must be able to take V inverse, I need n independent columns there. OK. But when it works, it really works. So the n-th power, just remembering, is V lambda V inverse, V lambda V inverse, n times. But every time I have a V inverse and a V, that's the identity. So I move V out at the beginning. I have lambda, lambda, lambda, n of those, and a V inverse at the very end. So that's the nice result for the n-th power of a matrix. Now I have to show you how to use that formula, how to use eigenvalues and eigenvectors. OK. So we know we can take powers of a matrix. So first of all, what kind of equation? There's an equation. That's called a difference equation. It goes from step k to step k plus 1 to step k plus 2. It steps one at a time and every time multiplies by A. So after k steps, I've multiplied by A k times from the original u0. So instead of a differential equation, it's a step difference equation with u0 given. And there's the solution. That's the quickest form of the solution. A to the k-th power, that's what we want. But just writing A to the k, if we had a big matrix, to take its hundredth power would be ridiculous. But with eigenvalues and eigenvectors, we have that formula. OK. But now I want to think. Let me try to turn that formula into something that you just naturally see. And we know what happens. If u0 is an eigenvector, if u0 is an eigenvector, that probably won't happen because there are just n eigenvector directions. But if it happened to be an eigenvector, then every step we'd multiply by lambda, and we'd have the answer, lambda k times. But what do we do for all the initial vectors u0 which are maybe not an eigenvector? How do I proceed? How do I use eigenvectors when my original starting vector is not an eigenvector? And the answer is, it will be a combination of eigenvectors. So making this formula real starts with this. So I write u0 as a combination of the eigenvectors. And I can do it because if I have n independent eigenvectors, that will be a basis. Every vector can be written in the basis. So I'm looking there at a combination of eigenvectors. And now the point is that as I take these steps to u1-- what will u1 be? u1 will be Au0. So I'm multiplying by A. So when I multiply this by A, what happens? That's the whole point. c1, A times x1 is lambda 1 times x1. It's an eigenvector. c2 tells me how much of the second eigenvector I have. When I multiply by A, that multiplies by lambda 2, and so on, cn lambda n xn. And that's the thing. Each eigenvector goes its own way, and I just add them together. OK. And what about A to the k-th power? Now, that will give me uk. And what happens if I do this k times? You've seen what I got after doing it one time. If I do it k times, that lambda 1 that multiplies its eigenvector will happen k times. So I'll have lambda 1 to the k-th power. Do you see that? Every step brings another factor lambda 1. Every step brings another factor lambda 2. Every step brings-- that's the answer. That is-- well, that answer must be the same as this answer. And I'll do an example in a minute. Right now, I'm just getting the formulas straight. So I have the quickest possible formula, but it doesn't help me much. I have the using the eigenvectors and eigenvalue formula. And here I have it that, really, it's the same thing written out as a combination of eigenvectors. And then this is my answer. That's my answer to the-- that's my solution uk. That's it. So that must be the same as that. Do you want to just think for one minute why this answer is the same as that answer? Well, we need to know what are the c's? Well, the c's came from u0. And if I write that equation for the c's-- do you see what I have as an equation for the c's? u0 is this combination of eigenvectors. That's a matrix multiplication. That's the eigenvector matrix multiplied by the vector c of coefficients, right? That's how a matrix multiplies a vector. The columns, which are the x's, multiply the numbers c1, c2, cn. There it is. That's the same as that. So u0 is Vc. So c is V inverse u0. Oh, that's nice. That's telling us what are the coefficients, what are the numbers, what amount of each eigenvector is present in u0. This is the equation. But look, you see there that V inverse u0, that's the first part there of the formula. I'm trying to match this formula with that one. And I'm taking one step to recognize that this part of the formula is exactly c. You might want to think about that. Run this video once more just to see that step. Now what do we do? We've got the lambdas. So I'm taking care of the c's, you could say. Now I need the lambda to the k-th power-- lambda 1 to the k-th, lambda 2 to the k-th, lambda n to the k-th. That's exactly what goes in here. So that factor is producing the lambdas to the k-th power. And finally, this factor has-- everybody's remembering here. V is the eigenvector matrix x1, x2, to xn. And when I multiply by V, it's a matrix times a vector. This is a matrix. This is a vector. And I get the combination-- I'm adding up. I'm reconstructing the solution. So first I break up u0 into the x's. I multiply them by the lambdas, and then I put them all together. I reconstruct uk. I hope you like that. This formula, which it's like common sense formula, is exactly what that algebra formula, matrix formula, says. OK. I have to do an example. Let me finish with an example. OK. Here's a matrix example. A equals-- this'll be a special matrix. I'm going to make the first column add up to 1, and I'm going to make the second column add up to 1. And I'm using positive numbers. They're adding to 1. And that's called a Markov matrix. So it's nice to know that name-- Markov matrix. One of the beauties of linear algebra is the variety of matrices-- orthogonal matrices, symmetric matrices. We'll see more and more kinds of matrices. And sometimes they're named after somebody who understood that they were important and found their special properties. So a Markov matrix is a matrix with the columns adding up to 1 and no negative numbers involved, no negative numbers. OK. That's just by the way. But it tells us something about the eigenvalues here. Well, we could find those two eigenvalues. We could do the determinant. You remember how to find eigenvalues. The determinant of lambda I minus A will be something. Could easily figure it out. There's always a lambda squared, because it's two by two, minus the trace. 0.8 and 0.7 is 1.5 lambda, plus the determinant. 0.56 minus 0.06 is 0.50, 0.5. And you set that to 0. And you get a result that one of the eigenvalues is-- this factors into lambda minus 1, lambda minus 1/2. And the cool fact about Markov matrices is lambda equal 1 is always an eigenvalue. So lambda equal 1 is an eigenvalue. Let's call that lambda 1. And lambda 2 is an eigenvalue, and that depends on the numbers, and it's 1/2, 0.5, 0.5. Those are the eigenvalues. 1 plus 1/2 is 1.5. The trace is 0.8 plus 0.7, 1.5. Are we good for those two eigenvalues? Yes. And then we find the eigenvectors that go with them. I think that this eigenvector turns out to be 0.6, 0.4. I could check. If I multiply, I get 0.48 plus 0.12 is 0.60, and that's the same as 0.6. And that goes with eigenvalue 1. And I think that this eigenvector is 1, minus 1. Maybe that's always for a two-by-two Markov matrix. Maybe that's always the second eigenvector. I think that's probably good. Right. OK. Yeah. All right. What now? What now? I want to use the eigenvalues and eigenvectors, and I'm going to write out now uk. So if I apply A k times to u0, I get uk. And that's c1 1 to the k-- this lambda 1 is 1-- times its eigenvector 0.6, 0.4 plus c2, however much of the second eigenvector is in there, times its eigenvalue, 1/2 to the k-th power times its eigenvector, the second eigenvector, 1, negative 1. That is a formula. c1 lambda 1 to the k-th power x1 plus c2 lambda 2 to the k-th power x2. And c1 and c2 would be determined by u0, which I haven't picked a u0. I could. But I can make the point, because the point I want to make is true for every u0, every example. And here's the point. What happens as k gets large? What happens if Markov multiplies his matrix over and over again, which is what happens in a Markov process, a Markov process? This is like-- actually, the whole Google algorithm for page rank is based on a Markov matrix. So that's like a multi-billion-dollar company that's based on the properties of a Markov matrix. And you repeat it and repeat it. That just means that Google is looping through the web, and if it sees a website more often, the ranking goes up. And if it never sees my website, then for that, when it was googling some special subject, it never came to your website and mine, we didn't get ranked. OK. So this goes to 0. 1/2 to the-- it goes fast to 0, quickly to 0. So that goes to 0. And of course, that stays exactly where it is. So there's a steady state. What happens if page rank had only two websites to rank, if Google was just ranking two websites? Then its initial ranking, they don't know what it is. But by repeating the Markov matrix and this part going to 0, right, goes to 0 because of 1/2 to the k-th power, there is the ranking, 0.6, 0.4. That's where Google-- so this first website would be ranked above the second one. OK. There's an example of a process that's repeated and repeated, and so a Markov matrix comes in. This business of adding up to 1 means that nothing is lost. Nothing is created. You're just moving. At every step, you take a Markov step. And the question is, where do you end up? Well, you keep moving, but this vector tells you how much of the time you're spending in the two possible locations. And this one goes to 0. OK. Powers of a matrix, powers of a Markov matrix. Thank you. |
MIT_Learn_Differential_Equations | Electrical_Networks_Voltages_and_Currents.txt | GILBERT STRANG: This video is about one of the key applications of ordinary differential equations to electrical flow, flow of currents in a network. And so I drew a network, a very simple network. It's just called an RLC loop. It's only got one loop, so it's a really simple network. The R stands for resistance to the flow. The L stands for an inductance. And the C is the. Capacitance those are the three elements of a simple linear constant coefficient problem associated with one loop. And then there is a switch, which I'll close, and the flow will begin. And there is a voltage source, so like a battery, or maybe let's make this alternating current. So the voltage source will be some voltage times an e to the i omega t. So we're going to have alternating current. And the question is, what is the current? We have to find the current, I. So the current is I of t going around the loop. And we saw our differential equation will have that unknown I of t, rather than my usual y. I'm going to use I for current. Again, this is an RLC loop that everybody has to understand, as in electrical engineering. So I'm going to have a second order differential equation. Well, you'll see what that equation is. So you'll remember Ohm's law. That the voltage is the current times the resistance. So this gives me a voltage across the resistor. If the current is I and the resistance is R, then the voltage drop from here to here is I times R. So that's that term. But now I have also my current is changing with time. This is alternating current. It's going up and down. So the current is also going through the inductance. And there, the voltage drop across the inductance has this form. The derivative of the current comes into it. And in the capacitance, which is building up charge, the integral of the current comes in. So there, that's the physical equation that expresses this voltage law, which says that around a closed loop-- this is a closed, loops are closed-- add to 0. So I have four terms, and they combine to give us 0. So there's an equation I'd like to solve. And how am I going to solve that equation? By the standard idea which applies when I have constant coefficients and I have a pure exponential forcing term. I look for a solution that is a multiple of that exponential, right? The solution to differential equations with constant coefficients, if they have an exponential forcing, then the solution is I equals some, shall I say W e to the i omega t. Some multiple of the source gives me the solution to that differential equation. Well, it's actually a differential integral equation. I can make it a more familiar looking differential equation by taking the derivative of every term. Suppose I do that. Suppose I take the derivative of every term, just to make it look really familiar. That would be L times I double prime. Taking the derivative of the derivative. This would be RI prime. The derivative of the integral would be just I itself. So I'd have 1 over C I. And I would have the derivative here, I omega V e to the I omega t. So it's just a standard second order constant coefficient linear differential equation. And in fact, if you are a mechanical engineer, you would look at that and say, well, I don't know what L, R, and 1 over C stand for. But I know that I should see the mass, the damping, and the stiffness there. So we have a complete parallel between two important fields of engineering, the electric engineering with L, R, and 1 over C, mechanical engineering with M, B for damping, and K for stiffness. And actually, that parallel allowed analog computers-- which came before digital computers and lost out in that competition. An analog computer was just solving this linear equation by actually imposing the voltage and measuring the current. So an analog computer actually solved the equation by creating the model and measuring the answer. But we're not creating an analog computer here. We're just doing differential equations. So why don't I figure out what that W is. So what am I going to do? As always, I have this equation. I have a pure exponential. I look for a solution of that same form. I plug it in. And I get an equation for W. That's exactly what I'll do on the next board. I'll put W e to the I omega t into this equation and find W. Let's do it. Maybe I'll bring that down just a hair and I'll do it here where you can watch me do it. So I have L times the derivative. So I have L. The derivative will bring down an I omega L. Everything is going to multiply W and match V. When I put this into the equation, the derivative is an I omega L W e to the I omega t, and it's matching V e to the I omega t. Now, what happens when I put I in for that second term, R. I just get an R. R times W times e to the I omega t. No problem. And now finally, a 1 over C. The integral. The integral of the exponential brings down-- let me put it in the denominator neatly-- I divide by I omega when I integrate e to the I omega t. I have a division by I omega. That's it. That's it. Those are the three terms that come-- times W, the unknown. This is to find. And of course, we find it right away. We find W is V over-- and now we're seeing this I omega L plus R. Oh, let me combine the I omegas. Combine the real part and the imaginary part. The real part is R. And the imaginary part is I omega L minus 1 over I omega C. Straightforward. And that has a name. That is the resistance. But when there's also terms coming from an inductance and a capacitance, then the whole thing is called the impedance. So this whole thing, this whole denominator, is called the complex impedance. Believe me, all these ideas are so important. There's a whole vocabulary here. But you see, we've done exactly the same thing for other constant coefficient equations. We just called the coefficient A, B, C. Or maybe M, B, K. And now we have slightly different letters, but we don't have a new idea here. The idea is this 1 over, that 1 over the impedance, that will be the transfer function, which multiplies the source to give the complex number W. And W is a complex number. I have to now think about that. And that impedance is always called Z. So we now have a new letter for the important quantity that shows up in the denominator there. And again, its real part is the resistance. Its imaginary part comes from L and C. So we can easily see how large-- what's the size of that impedance? What's going to be the magnitude of this current? We want the size of that number. V is the size of the voltage. Here is the size of the impedance. And the answer will give us the size of W. I'm using size or magnitude to say that when I only do magnitudes, you won't be seeing the phase lag. So complex numbers, like this complex number has a magnitude which we're about to write down. And also it has a phase lag that tells us how much is in the imaginary part and how much is in the real part. But the magnitude is easy. What's the magnitude of a complex number? It's the real part squared and the imaginary part squared. Oh, that should have been a plus there, I think. I don't know how it became a minus. It will become a minus, so I was thinking if I put the I up there. Let me show you what I'm saying. The imaginary part is omega L minus 1 over omega C. What I'm saying is that if I put the I up there, then 1 over I is minus I. That's the brilliant step I just took there. So all that squared. Are you OK with that? It's the real part squared, which is the resistance. And this combination gives the imaginary part. We square that. That's maybe called the reactants. And the sum of those squares is the impedance squared, the magnitude. So we have essentially successfully solved a second order constant coefficient single equation for the current. What to do now. Just let me add a little bit more. Maybe just a comment. That video was about one loop. When I told Dr. Mohler that one of the applications, one of the real applications in this series of videos would be an RLC circuit, his reply was an RLC circuit is not an application, not a realistic application. One loop. So how do we proceed with a full scale circuit with many nodes, many resistors, many conductors, many edges? Well, we have a big decision to make. And that's the comment I want to make. They have a choice. They can use Kirchoff's current law at the nodes and solve for the voltages at those nodes. Or they can do as we did for one loop, use Kirchoff's voltage law around that one loop which said that the currents in the loop gave a total voltage drop adding to 0. So we solve the current equation for the unknown I. This is what we did for one loop. My message is just for a big system, this is the winner. So writing down the equations in terms of Kirchoff's current law, that the currents-- we get the nodal picture, the picture with an equation for every node instead of the picture for an equation for every loop. Because it's not so easy to see which are the loops to consider and which loops are combinations of other loops. The linear algebra is the question. And the linear algebra, to get the loop picture independent and clear, is more difficult than the node picture. The node picture with the unknown voltages, V at the nodes, is the good one. And the matrix that comes into that is the incidence matrix. It connects nodes and edges. It says how the network is put together. And that matrix, I'll study with a little bit of linear algebra. So that comes in a later video. If you look for incidence matrices, you'll see probably two videos about those very, very important and beautiful matrices. Thank you. |
MIT_Learn_Differential_Equations | Fourier_Series_Solution_of_Laplaces_Equation.txt | GILBERT STRANG: OK. So this is using Fourier series. So I had to pick an equation where we were given a function, and not just a couple of initial values. So I made the equation a partial differential equation. The most famous one, Laplace's equation. So this is the setup. And you'll see how Fourier series comes in. We're in a circle. I'm going to make this a nice model problem. So inside this circle we're solving Laplace's equation. Laplace's equation was the second derivative of u in the x direction, plus the second derivative of u in the y direction, is 0. That's the way heat, temperature, distributes itself when you leave it alone. In this problem I'm going to put a source of heat at that point. So it'll be a point source. A delta function. And on the rest of the boundary, temperature 0. So the boundary function is a delta function with a spike at that one point, and 0 elsewhere. And our problem is to solve the Laplace's equation inside the circle. And we use polar coordinates because we've got a circle. So there is the equation with x and y, but we really are thinking r and theta. And the reason is, you get beautiful solutions to this equation using r and theta. And that was a family of solutions. r to the n-th cos n theta just works. And so does r to the n-th sine n theta. And that's for every n. So we have-- we can combine. We have a linear equation. We can take combinations of solutions with coefficients a n in the cosines, and bn in the sines. And now here's the key step. Put in r equal 1. Put r equal 1. And then this solution, u at 1 and theta-- r equal 1-- is the boundary. It's the circle. And that's where we're given u of 1 to be the delta function. The point source. The delta function. Delta of theta. The point source at theta equals 0. So you see our job. That function, that boundary condition, is supposed to tell us the a's and the b's. And then we have our solution. So by putting r equal to 1 in this formula, we're supposed to get the delta function. So let me put r equal 1. Easy to do. It's the sum of a n, 1 to the n-th, cos n theta, plus the sum of bn, 1 to the n-th, sine n theta, is supposed to match the delta function. So that's the Fourier series for the delta function. That's the whole point. That we use a Fourier series expression for the boundary function, whatever that boundary function is. Here it's a particularly nice neat one. And actually, the delta function is an even function. It's 0 at theta and it's 0 at minus theta. So changing theta to minus theta still leaves me the spike at 0. So because it's an even function, I won't see any signs. I won't see any odd functions. The sine theta. And I have an easy time to find the coefficients a n of the cosines. Actually, we did that directly from the formula for the a n's. Let me just remember that formula. The formula was a0 was 1 over 2 pi times the average. a0 is the average value of the temperature. And the temperature on the boundary is delta theta. And that integrates to 1, and we get the answer 1 over 2 pi. That's the average temperature. Isn't that a little weird? The temperature 0 except at one point. At that point it's a delta function with the coefficient 1 outside it. And then we get 1 over 2 pi as the average. The other a n's, the coefficients of the cosines, are 1 over pi, times the integral of our delta function, times cos n theta d theta. And the delta function, that point source, picks out that number at theta equals 0. And that number is 1. So I'm getting 1 over pi. So finally I now know the a's and b's. When I put those in, that tells me the solution. The solution-- now I can put r back in the picture-- it's a sum. Well, let me take the a0 term. The a0 is 1 over 2 pi-- that's the constant, that's the average-- plus the sum of 1 over pi cos n theta, from n equals 1 to infinity. And r to the n-th. Sorry. r to the n-th. So you see what happens. When r is 1, we have the Fourier series for the delta function. That's the very exceptional function that's given on the boundary. As soon as r is less than 1, these r to the n-th's get small, and we have a series that adds up to a reasonable sum. And we can actually-- it's possible to add up that series. It's possible to add up that series. It's a geometric series if you switch from cosines to exponentials. That's usually the good way to get good formulas. And here, so you can add it up. And I think there's a 1 over 2 pi still there. And I think it's 1 minus r squared, over 1 plus r squared, minus 2r cos theta. Let me just be sure I got that right. Yep, looks good. Looks good. And we could check if it's good. Let's take theta equal 0. So if we take theta equal 0. Let me draw that circle again. Theta equal 0. We're coming out on that ray. And we're expecting to see infinity when we get there, at r equal 1. So theta equal 0. So let me just put that. On the ray, theta equal 0. This is what you should do. We have a formula for all r and theta, but let's look at some particular points to see what's happening. So along that ray, where theta is 0, I have 1 over 2 pi, 1 minus r squared, over 1 plus r squared, minus 2r. Because cos theta is 1. And 1 plus r squared, minus 2r, is 1 minus r squared. Right? Because cos theta is 1 on this ray. Theta is 0. Cos theta is 1. And now 1 minus r will factor out of this. And I think we get 1 plus r. And we still have a 1 minus r down below. I like that. You don't often, for partial differential equations, get some nice expression for the solution. So that's the solution. And as r goes to 1, this solution blows up. Right. The temperature is infinite on the boundary, but the temperature is something reasonable inside. And at r equals 0, I have 1 over 2 pi. Well, of course. It's the average value. Right at the center that temperature is going to be the average on the boundary. That's a natural key property of Laplace's equation. It averages everything. Actually, if I take a little circle in anywhere, those temperature in the center of that circle would be the average of the temperatures around the little circle. For all the circles. It's just the Laplace's equation. Solving Laplace's equation averages everything. And the result is that the temperature function sort of smoothes out as I come in. Around the boundary it's far from smooth. There's a big jolt at theta equals 0. But if I look on that circle, or that circle, or this circle, the temperature is a nice smooth function. And it's never going to be above the maximum on the boundary. And it's never going to be below the minimum on the boundary. Everything's being averaged. So that's, you see, a use of the Fourier series. For one particular function. I could do another function, but I don't think I will. I could take the function that's 1 on the top of the circle and minus 1 on the bottom half of the circle. OK, that's a function with a jump, but not a delta function. So we would see a Fourier series that would give us the a's and the b's. There would probably only be b's in that case. Sine. Sine terms. And we'd get an answer. May I just, while I'm talking about averages, add one final comment. Usually, for a complicated region, we can't solve Laplace's equation with formulas. It's not possible. We can't find sines and cosines that match some crazy boundary. So we have to replace Laplace's equation. So I'll write Laplace's equation again. That goes into u-- we have a region. And we carve it out with a grid. And then at each point on the grid, we have an equation connecting the value of u at that point. Say u0 at the center. With u east, maybe, u west, u north, and u south. So we have an equation, and I want to write that equation down. u center is just going to be the average. It's just going to be 1/4 of u east, u west, u north, and u south. So that'll be true at-- that equation will hold. The unknowns are all these u's. The u's of all the mesh points. And I have an equation at every mesh point. So I have the same number of equations from the mesh points as unknowns at the mesh points. I solve that big system, and that gives me a solution u. An approximate solution u to Laplace's equation. So this would be called Laplace's difference equation, or Laplace's five-point scheme, because it uses five points in that average. OK. That's an important problem in numerical analysis. Thank you. |
MIT_Learn_Differential_Equations | Pictures_of_Solutions.txt | PROFESSOR: OK. So we've moved on into Chapter 3. Chapter 1 and 2 were about equations we could solve, first order equations, chapter one; second order equations in chapter 2, often linear, constant coefficient sometimes. Now we take any equation. And I'll start with first order. First derivative is some function and not a linear function, so I don't expect a formula. A solution will exist. But I won't have a formula for the solution. But I can make a picture of the solution. You see what's happening as time goes on. And so that's today's lecture, is a picture. So this function, whatever it is, gives the slope of y. That's the slope. And it will be the slope of the arrows that I will draw in this picture. So here's a picture that started, y, t. And the slope of the arrows is f. And here is my example. Well, you will see. I chose a constant coefficient linear equation. Because I could find a solution. So 2 minus y, I know from that minus sign that I'm going to have exponential decay in the null solution. And then y equal to 2 is a very special particular solution, a constant. And in my picture, y equal to 2, it jumps out. Because when y is 2, when y is 2 the slope is 0. So all my arrows on the y equal to 2 line, have slope 0. So that's a very special line. And since the solution follows the arrows that's the whole point. The solution follows the arrows. Because the arrows tell the slope. So if I'm on that line, the solution just follows those arrows, and stays on the line. y equal to 2 is a fixed point, fixed point of the solution, a fixed point for the equation. And the question is, if I don't start at y equal to 2, do I move toward 2 or away from it? OK. So I can see from the formula what the answer is going to be. If I start with some other value, some other value of c not 0, then there will be a null solution part. But as t gets large that goes to 0. So I move toward 2. Now let's see that in the picture. So let me-- I'm drawing the arrows first. So this is all time starting at if y is 0, then if y is 0 then dy dt is 2. So I draw arrows with slope 2, along the y equals 0 line. This is the y equals 0 line. All my arrows have slope 2. Now what else? So that's a few arrows that show what will-- so the solution if it starts there, will start in the direction of that arrow. But then I have to see what the other arrows are for other values of y. Because right away the solution y will change. And the slope will change. And that's it needs more arrows. Well, actually it needs way more arrows than I can possibly draw. Let me draw another line of arrows when y is 1, along that line, along the line y equal 1. When y is 1, 2 minus 1 is 1. The slope is 1. f is 1. And my arrows have slope 1. So all along here, the arrows go up. Those went up steeply with slope 2. Now the arrows will go up. So I'll have arrows that are going a 45-degree angle, slope 1. Do you see? I hope you begin to see the picture here. The solution might start there. It would start with that slope. But it will curve down. Because the arrows are not so steep. As I go upward, the arrows are getting flat. And so the curve that follows the arrows has to flatten out, flatten out, flatten out. The arrows are still, at that point the arrows are still slope 1. But it's flattening out. And it's never going to cross this line. And it will run closer and closer to that line. And wherever it starts, if it starts at time t equals to 1 there, it'll do the same thing. And it will stay just below the other one. Do you see what the pictures are looking like? If it starts at different times, so these are different times. These are different starts. Yeah, really we're used to, at t equals 0, we're used to giving y of 0. So this is starting at 0. This is starting at 2. Starting at 1 would be a higher start. What about starting at 4? Suppose y of 0 is 4. That point is t equals 0, y equal 4. So that point is y of 0 equal 4. What's the graph of the solution with that start? Actually, I could figure out what the solution would be if y of 0 was 4, I'd have 2 plus 2 e to the minus t. At t equals 0, that's 4. And it fits. It solves the equation. And it's going to be its graph. I should be able learn that from the arrows. So along this line of y equal 4, all the arrows when y is 4, the slope is minus 2. So these arrows from these points, go down with slope minus 2. But the solution starts down. So it starts like that. But then it has to follow the new arrows. And the new arrows are not so steep. So the new arrows are I have slope minus 1. I hope my picture is showing the steeper slope 2 along this line, and the flatter slope 1, or rather minus 1 downwards, along this line. So it just follows along here. Well of course it's just a mirror image of that one. It's a mirror image of that one. I'm trying to show the graph of all solutions from all starts, the whole plane. Actually I could go, t could go to minus infinity. And y could go all the way from minus infinity up, all the way up. I could fill the whole board here with arrows, and then with solutions. And the solutions would follow the arrows, the arrows changing slope and actually in this case, all solutions wherever you started, would approach 2. And that's what the formula says. But we get that information from the arrows with no formula. Let me show you a next example. And here's our next example. The logistic equation, it's not linear. So it's going to be more interesting. And do you remember the solution? You remember maybe the trick with the logistic equation was 1 over the solution, gave a linear equation and expression like that. OK. Time to draw arrows. OK. When y is 0-- so here's y-- when y is 0, the slope is 0. So I have a whole line of flat. I have a flat horizontal line. That's the solution, y equals 0 fixed point. Also we have another fixed point. When y is 1, 1 minus 1 is 0. Slope is 0. Slope stays 0. The arrows all have zero slope along the line y equal 1. So there is another solution, which doesn't do anything exciting. It just stays at 1. y equal 1 is another fixed point, steady state, whatever words we want to use. But again, the real picture is what about other starts. What about a start at 1/2? Well if it starts at-- if y is 1/2 half at the starting time, what is the slope? 1/2 minus 1/4 is 1/4. So the slope is upwards, but not very steep. The slope along the-- and it doesn't depend on t in these examples. So that slope is the same as long as y is 1/2, doesn't matter what the time is. y equals 1/2 gives me a 1/2 minus 1/4, which is a 1/4. It gives me that slope. What about the slope 1/4? So 1/4, I have 1/4 minus 1/16. I think that's 3/16. So it's beginning to climb upward. So it's upwards again. But 3/16 is a little-- I don't know if I'm going to get the picture too brilliantly. The slope, as soon it-- if it just starts a little above 0, what happens to the solution that starts a little bit above 0? It climbs. Because if y is above 0, say if it starts between 0 and 1, if y is between 0 and 1, then y is bigger than y squared. And the slope is positive. And it goes up. So do you see what it's doing? The slope will just, if it starts a little bit above, it'll have a small slope. But that slope will gradually increase. But then actually at this point, the slope is 3/4 minus whatever it is. It slows down, still going upwards. y is still bigger than y squared. You recognize what the curve is going to look like. So there is an S curve. It's an S curve. Which we saw for the logistic equation, and here we have a formula for it. Well, the whole point of today's video was we don't need a formula. So you don't need that. The arrows will tell you that it starts up slowly. It gets only-- that's the biggest slope it gets. And then it starts down. The slope goes down again. But it's still a positive slope. Still climbing, climbing, climbing, and approaching 1. Now that's sort of a sandwich in the picture. But it could start with a negative. So what happens if it starts at y equal minus 1? The slope, if y is minus 1, we have minus 1, minus 1, a slope of minus 2. That's a steeper serious downward slope. So the solution that starts here has-- that's tangent. You see that it's tangent to the arrow, because it has the same slope as the arrow. And it comes down. But as it goes down, the slopes are getting steeper. Whoops, not flatter, but steeper. For example, if y is minus 2, I have minus 2, minus 4 is minus 6. So as soon as it gets down to minus 2, the slope has jumped way down to minus 6. So here is the-- it falls right off. It's a drop-off curve, a drop-off curve. It falls right off actually to infinity. It never makes it out to-- it falls down to y equal minus infinity in a fixed time, in a definite time. And so here's a whole region of curves going down to minus infinity. Here is a whole region. What happens in this region? Suppose y starts at plus 2? Well, I have 2 minus 4. So the slope is negative. The slope is negative up here. Yeah. And this is the big picture. The slope, the arrows are positive below this line. They're upward. They were downward here. They're slowly upward in this sandwich. And then up above, they're downward again. So if slopes are coming down, and they drop into actually it's a symmetric picture. Really-- no reason not to go backwards in time. Where are these coming from? They're all coming from curves. The whole plane is full of curves. And these start at plus infinity. They drop into 2. These start below 0, and they drop off to minus infinity. And then the real interest in studying population was these. Can you do one more example? Let me take a third example that has a t in the function. So the arrows won't be the same along the whole line. In fact, the arrows will be the same. So if I have 1 plus t minus y, that's the f, equal a constant. Then that's a curve-- well, it's actually a straight line. It's actually a 45-degree line in this plane. And along that line the f, this is the f, the f of t and y, the arrow slope. The arrows slopes are the same along that line. That line is called an isocline. This is called an I-S-O, meaning the same, cline, meaning slope. So that's an isocline. Here's an isocline. It's a 45-degree line. That's the 45-degree line, 1 plus t minus y equal 1. Let me draw the 45-degree line 1 plus t minus y equals 0. So it's a little bit higher. OK. Now arrows, and then put in the curves, the solution curves that match the arrows. So the arrows have this slope along that line. Along this line, 1 plus t minus y, they have slope 0. Oh, interesting. At every point on the line the slope is 0. Because this is the slope of the arrows. At every point on this line, the slope is 1, also very interesting. Because that's right along the line. So here we have a solution line. That must be a solution line. That's the line where y is t. That's a very big 45-degree important line. Because if y equals t, if y equals t then dy dt should be 1. And it is 1 for y equals t. So that's a solution line with that solution. Now what about a line with 1 plus t minus y equal minus 1, a line? If 1 plus t minus y is minus 1, if f is minus 1, the slope is negative. So what does that mean? If 1 plus t minus y is minus 1, the slope is negative. So at points on this line, the slope is going downwards. Oh, interesting. I wasn't quite expecting that. Let me just see if I got a suitable picture. Why is it not right? If 1 plus t minus y is-- oh, I'm sorry. This is the line, y equal 1 plus t. I think what I'm expecting to see is I'm expecting to see it from the formula too that as time goes on, this part goes to 0, and y goes to t. I believe that all the solutions will approach this y equal to t. I think their slopes, their slopes here-- darn, that's not right. Their slopes should be coming upwards. Yeah, let me-- I can figure that out. If t is let's say 1, and y is 0. OK. If t is 1, and y is 0, I have a slope of 2. Good. OK. There's a point t equal to 1. Here is 0, 0. Here's the point t equal to 1, y equals 0. The slope came out to be 2. It went up that way. So along that line, the slopes are going up. Along this line, the slopes are right on the line. On this line the slopes are flat, and the curve is moving toward the line. I'll just draw the beautiful picture now of the solution. So the solutions look like this. They are-- this is the big line. You've got to keep your eye on that line. Because that's the steady state line that all solutions are approaching. So if you have the idea of arrows to show the slope, fitting solution curves through tangent to the arrows, and sometimes having a formula to confirm that you did it right, you get a picture like this. So that's the idea of first order equations, which are graphed in the y-t plane. And the arrows tell you the derivative. Thanks. |
MIT_Learn_Differential_Equations | Laplace_Equation.txt | Today I'm speaking about the first of the three great partial differential equations. So this one is called Laplace's equation, named after Laplace. And you see partial derivatives. So we have-- I don't have time. This equation is in steady state. I have x and y, I'm in the xy plane. And I have second derivatives in x and then y. So I'm looking for solutions to that equation. And of course I'm given some boundary values. So time is not here. The boundary values, the boundary is in the xy plane, maybe a circle. Think about a circle in the xy plane. And on the circle, I know the solution u. So the boundary values around the circle are given. And I have to find the temperature u inside the circle. So I know the temperature on the boundary. I let it settle down and I want to know the temperature inside. And the beauty is, it solves that basic partial differential equation. So let's find some solutions. They might not match the boundary values, but we can use them. So u equal constant certainly solves the equation. U equal x, the second derivatives will be 0. U equal y. Here is a better one, x squared minus y squared. So the second derivative in the x direction is 2. The second derivative in the y direction is minus 2. So I have 2, minus 2, it solves the equation. Or this one, the second derivative in x is 0. Second derivative in y is 0, those are simple solutions. But those are only a few solutions and we need an infinite sequence because we're going to match boundary conditions. So is there a pattern here? So this is degree 0, constant. These are degree 1, linear. These are degree 2, quadratic. So I hope for two cubic ones. And then I hope for two fourth degree ones. And that's the pattern, that's the pattern. Let me find-- let me spot the cubic ones. X cubed, if I start with x cubed, of course the second x derivative is probably 6x. So I need the second y derivative to be minus 6x. And I think minus 3xy squared does it. The second derivative in y is 2 times the minus 3x is minus 6x, cancels the 6x from the second derivative there, and it works. So that fits the pattern, but what is the pattern? Here it is. It's fantastic. I get these crazy polynomials from taking x plus iy to the different powers. Here to the first power, if n is 1, and I just have x plus iy and I take the real part, that's x. So I'll take the real part of this. The real part of this when n is 1, the real part is x. What about when n is 2? Can you square that in your head? So we have x squared and we have i squared y squared, i squared being minus 1. So I have x squared and I have minus y squared. Look, the real part of this when n is 2, the real part of x plus iy squared, the real part is x squared minus y squared. And the imaginary part was the 2ixy. So the imaginary part that multiplies i is the 2xy. This is our pattern when n is 2. And when n is 3, I take x plus iy cubed, and that begins with x cubed like that. And then I think that the other real part would be a minus 3xy squared. I think you should check that. And then there will be an imaginary part. Well, I think I could figure out the imaginary part as I think. Maybe something like minus-- maybe it's 3yx squared minus y cubed, something like that. That would be the real part and that would be the imaginary part when n is 3. And wonderfully, wonderfully, it works for all powers, exponents n. So I have now sort of a pretty big family of solutions. A list, a double list, really, the real parts and the imaginary parts for every n. So I can use those to find the solution u, which I'm looking for, the temperature inside the circle. Now of course, I have a linear equation. So if I have several solutions, I can combine them and I still have a solution. X plus 7y will be a solution. Plus 11x squared minus y squared, no problem. Plus 56 times 2xy. Those are all solutions. So I'm going to find a solution, my final solution u will be a combination of this, this, this, this, this, this, this, and all the others for higher n. That's going to be my solution. And I will need that infinite family. See, partial differential equations, we move up to infinite family of solutions instead of just a couple of null solutions. So let me take an example. Let me take an example. We're taking the region to be a circle. So in that circle, I'm looking for the solution u of x and y. And actually in a circle, it's pretty natural to use polar coordinates. Instead of x and y inside a circle that's inconvenient in the xy plane, its equation involves x equals square root of 1 minus y squared or something, I'll switch to polar coordinates r and theta. Well, you might say you remember we had these nice family of solutions. Is it still good in polar coordinates? Well the fact is, it's even better. So the solution of u will be the real part and the imaginary part. Now what is x plus iy in r and theta? Well, we all know x is r Cos theta plus ir sine theta. And that's r times Cos theta plus i sine theta, the one unforgettable complex Euler's formula, e to the I theta. Now, I need its nth power. The nth power of this is wonderful. The real part and imaginary part of the nth power is r to the nth e to the in theta. That's my x plus iy to the nth. Much nicer in polar coordinates, because I can take the real part and the imaginary part right away. It's r to the nth Cos n theta and r to the nth sine n theta. These are my solutions, my long list of solutions, to Laplace's equation. And it's some combination of those, my final thing is going to be some combination of those, some combination. Maybe coefficients a sub n. I can use these and I can use these. So maybe b sub n r to the nth sine n theta. You may wonder what I'm doing, but what I'm achieved, it's done now, is to find the general solution of Laplace's equation. Instead of two constants that we had for an ordinary differential equation, a C1 and a C2, here I have these guys go from up to infinity. N goes up to infinity. So I have many solutions. And any combination working, so that's the general solution. That's the general solution. And I would have to match that-- now here's the final step and not simple, not always simple-- I have to match this to the boundary conditions. That's what will tell me the constants, of course. As usual, c1 and c2 came from the matching the conditions. Now I don't have just c1 and c2, I have this infinite family of a's, infinite family of b's. And I have a lot more to match because on the boundary, here I have to match u0, which is given. So I might be given, suppose I was given u0 equal to the temperature was equal 1 on the top half. And on the bottom half, say the temperature is minus 1. That's a typical problem. I have a circular region. The top half is held at one temperature, the lower half is held at a different temperature. I reach equilibrium. Everybody knows that along that line, probably the temperature would be 0 by symmetry. But once the temperature there halfway up, not so easy, or anywhere in there. Well, the answer is u in the middle, u of r and theta inside is given by that formula. And again, the ANs and the BNs come by matching the-- getting the right answer on the boundary. Well, there's a big theory there how do I match these? That's called a Fourier series. That's called a Fourier series. So I'm finding the coefficients for a Fourier series, the A's and B's, that match a function around the boundary. And I could match any function, and Fourier series is another entirely separate video. We've done the job with Laplace's equation in a circle. We've reduced the problem to a Fourier series problem. We have found the general solution. And then to match it to a specific given boundary value, that's a Fourier series problem. So I'll have to put that off to the Fourier series video. Thank you. |
MIT_Learn_Differential_Equations | Heat_Equation.txt | GILBERT STRANG: This is heat equation video. So this is the second of the three basic partial differential equations. We had Laplace's equation, that was-- time was not there. Now time comes into the heat equation. We have a time derivative, and two-- matching with two space derivatives. So I have my function. My solution depends on t and on x, and I hope I can separate those two parts. This is exactly like the way we solved the ordinary systems of differential equations. We pulled out an e to the lambda t, where lambda was the eigenvalue, and then we had the eigenvector. Here, it's an eigenfunction because it depends on x. We didn't have x before, but now we have partial differential equations. X is also a coordinate here. So I look for solutions like that. And just as always, I substitute that into the differential equation to discover what-- what determines S, S of x. The time derivative brings down a lambda. The space derivative brings down-- has two space derivatives. So that's what I get when I substitute e to the lambda t S eigenvalue, times eigenfunction, into the equation. And always, I cancel e to the lambda t, that's the beauty of it, and I have an eigenvalue equation, or I'm looking again for a function. So the second derivative of my function is lambda, some number, times my function. OK. What I'm looking for for functions S, they'll be sine functions. S will be, S of x, will be the sine of K pi x, the sine of x, the sign of something times x. What's the eigenvalue? Take two derivatives of this, I get back sine k pi x, which is, that's great, it's an eigenfunction. And out comes the eigenvalue, lambda is, well when I take two derivatives, k pi comes out twice with a minus sign. So it's minus k squared, pi squared. So I have found a bunch of eigenvectors, eigenfunctions, and the eigenvalues. So this was a simple pair, but now the general solution, the general solution, u of t x, will be? If I know several solutions, and I have a linear equation, I just take combinations. That's always our way. Take combinations of those basic solutions, so I'll have a sum, k going from 1 to infinity. In differential equation, partial differential equations, I need a big sum here, of some coefficients, let me call them B k, times this solution, e to the lambda-- what was lambda-- minus k squared, pi squared t, times S, S number k. I should have given this eigenfunction it's number, k. So there is the, this is my family of eigenfunctions and eigenvalues. And then, this is the combination of these solutions. There is the general solution. And for S k, I should have written in sine k pi x. So that's the dependence on t. Let's have a look at this. So the dependence on t is fast decay. And if K, as K gets larger, later terms in this sum, the decay is really fast. So the term that decays most slowly, k equal 1, there'll be a B1, an e to the minus, pi squared t. That's decaying already pretty fast. When I'm talking about decay, what's happening here? I have a bar, a material bar. The ends of the bar are kept at temperature zero, they're frozen, and heat is in the bar. Heat is flowing around in the bar. And where is it going? It's flowing out the ends. The bar is approaching freezing. The ends are kept at freezing, and the inside of the bar, whatever heat is there at the beginning, is going to flow out the ends. So, you see, that I have these sine functions. When x is 0 the sine is 0 so that's frozen at one end. When x is 1, I have the sine of k pi, which is, again, 0, so it's frozen at the other end. So I'm freezing it at both ends. The temperature is escaping out of the center, so let me graph the solution. So maybe I start with-- here is my bar from 0 to 1, and I'm keeping it frozen. F for frozen, f for frozen at that end. Maybe I, maybe I start off with a warm bar. So u at 0 and x, I'll say it'd be 1. This way is x. So I have an ordinary heated bar, and I put it in the freezer. So I insulate the sides so the heat is escaping out the two ends, out the end x equals 0 and out the end x equal 1. And the solution will be, let me remember what the general solution looked like, and I have to find these numbers. OK. And those numbers, of course, the numbers are always found by matching the initial conditions. This is the initial, this is an initial picture. OK. So I have to match that by-- so this is t equals 0, I have to match the sum of B k. This is k going from 1 to infinity. When t is, when t is zero, this is 1, and I just have the Sk. The sine of k pi x, that has to match the 1. And from that, I find the Bk's, and then the final solution. T greater than 0 uses those Bks. And we're again faced with a Fourier series problem. Anytime I have to find these coefficients, this is a Fourier sine series, I have only sines, not cosines here. And I'm finding the coefficients, so that this will match 1, the initial condition. And then, for t greater than 0, solution u will be, as we said, the sum of these Bk's, which come from the initial conditions, come from this-- Fourier coefficients, we still have to do that video on Fourier series to know what these numbers will come out to be-- times the e, to the minus k squared, pi squared t times the sine of k pi x. You may think, well it's a pretty messy solution, because it's an infinite sum. But it's not bad for a partial differential equation. We have numbers, we have something depending on time and decaying rapidly, and something depending on x. So at time 1, if I drew a picture, suppose the heat is, the temperature starts out through the whole bar at 1. But with this kind of time decay, a little later in time, the temperature's going to be something like that. It'll be way down at the ends, pretty low in the middle. And so at some time t, the temperature will look like that, and then soon after that, the temperature will go down here, and the steady state, of course, is the whole thing is at temperature 0. So that's what solutions to the heat equation look like. And this is the step of finding the-- which I didn't take, it's the Fourier series step-- of finding the coefficients in our infinite series of solutions. Once again, we have infinitely many solutions. We're talking about a partial differential equation. We have a whole function to match, so we need all of those. And Fourier series tells us how to do that matching, how to find these Bk's. So that's a separate and important question, Fourier series. Thank you. |
MIT_Learn_Differential_Equations | Phase_Plane_Pictures_Spirals_and_Centers.txt | GILBERT STRANG: OK. So this is the second lecture about these pictures, in the phase plane that's with axes y and y prime, for a second order constant coefficient linear, good problem. Good problem. And you remember that we study that equation by looking for special solutions y equals e to the st. When we plug that into the equation, we get this simple quadratic equation. And everything depends on that. So today this video is about the case when the roots are complex. You remember, so the roots, complex roots, you have a real part, plus or minus an imaginary part. And this happens when b squared is smaller than 4ac. Because you remember, there's a square root in the formula for the solution of a quadratic equation. There's a square root of b squared minus 4ac, the usual formula from school. And if b squared is smaller, we have a negative number under the square root. And we get complex roots. So last time the roots were real. The pictures in the phase plane set off to infinity, or came in to 0, more or less almost on straight lines. Now we're going to have curves and spirals, because of the complex part. So here are the three possibilities now. We had three last time. Here are the other three with complex roots. So the complex, the real part, everything depends on this real part that the stability going in, going out, staying on a circle depends on that real part. If the real part is positive, then we go out. We have an exponential e to the a plus i omega t. And if a is positive that e to the at would blow up, unstable. So that's unstable. Here is a center. When a is zero, then we just have e to the i omega t. That's the nicest example. I do that one first. So in that case we're just going around in a circle or around in an ellipse. And finally, the physical problem where we have damping, but not too much damping. So the roots are still complex. But they're going in. Because if a is negative, e to the at is going to 0. So that's a stable case. That's a physical case. We hope to have a little damping in our system, and be stable. This one we could say neutrally stable. This one is certainly unstable. Let me start with that, the neutrally stable. Because that's the most famous equation in second order equation in mechanics. It's pure oscillation, a spring going up and down, an LC circuit going back and forth, pure oscillation. And we see the solutions. So I've written-- I've taken this particular equation. You notice no damping. There's no y prime term. OK. So here is the solution, famous, famous solution. And y prime it will be c1, I guess the derivative of the cosine is minus omega times sine omega t, plus c2. The derivative of that is omega cos omega t. So that's the y and y prime. So for every t, it's going to be an easy figure. Here is y. Here is y prime. And that's the phase plane, phase plane. So at each time t, I have a y and a y prime. And it gives me a point. So let me put it in there. As time moves on that point moves. And it's the picture in the phase plane, the orbit sometimes you could say, it's kind of like a planet or a moon. So for that, what is the orbit for that one? Well it goes around in an ellipse. It would be a circle with-- let me draw it. This is the case omega equal 1. In that case, in that most famous case, we simply go around a circle. There's y. There's y prime. We have cosine and sine and cos squared plus sine squared is 1 squared. And we're going around a circle of radius 1, or another circle depending on the initial condition. Here there's a factor omega, giving an extra push to y prime. So if omega was 2, for example, then we'd have a 2 in y prime from the omega, which is not in the y. And that would make y prime a little larger. And it would be twice as-- it would go up to twice as-- that's meant to be, meant to be an ellipse with height 2 up there, or in general omega, and 1 there. So in the y direction there is no factor omega. And we just have cosine and sine. And that would be a typical picture in the phase plane. But if we started with smaller initial conditions, we would travel on another ellipse. But the point is-- and these are called, this picture is called a center. So that's one of the six possibilities, and in some way, kind of the most beautiful. You get ellipsis in the phase plane. They close off. Because the solution just repeats itself every period. It's periodic. y is periodic. y prime is periodic. They come around again and again and again. No energy is lost, conservation of energy, perfection. And I would say neutrally stable, neutral stability. The solution doesn't go into 0. Because there's no damping. It doesn't go out to infinity. Because there's constant energy. And that's the picture in the phase plane. OK. So that's the center. And now I'll draw one with a source, or a sink. I just have to change the sign on damping to get source or sink. So let me do that. So now I'm going to do a spiral source or sink. This is the unstable one, going out to infinity. This is the stable one coming in to 0. And let me do y double prime, plus or maybe minus 4y prime, plus 4y equals 0. Suppose I take that equation. Then I have s squared plus 4s, oh maybe-- maybe 2 is a nicer number. 2 is nicer than 4. Let me change this to a 2, and a 2. And so I have s squared plus 2s plus 2 or minus 2s plus 2 equals 0. So those are my-- positive damping would be with a plus. So with a plus sign, the roots are s squared plus 2s plus 2. The roots are 1, or rather a minus 1 plus or minus i. Plus sign, and then the minus sign, with a minus 2. Then all the roots have a plus, plus or minus i. Everything is depending on these roots, these exponents, which are the solutions of the special characteristic equation, the simple quadratic equation. And you see that depending on positive damping or a negative damping, I get stability or instability. And let me draw a picture. I don't if I can try two pictures in the same thing, probably not. That wouldn't be smart. So what's happening then? Let's take this one. So this solution y is e to the minus t. That's what's making it stable coming into 0, times-- and from here we have c1 cos t and c2 sine t. That's what we get from the usual, as in the case of a center that carries us around the circle. So what's happening in this picture, in this phase plane? Here's a phase plane again, y and y prime. Without the minus 1, we have a center. We just go around in a circle. But now because of the minus 1, which is the factor e to the minus t in the solution, as we go around we come in. And the word for that curve is a spiral. So this would be the center, going around in a circle. But now suppose we start here. Suppose we start at y equal 1, and y prime equal 0, start there at time 0. Let time go. Plot where we go. Where does this y and the y prime, where is the point, y, y prime? OK. I'm starting it at-- so I'm probably taking c1 as 1, and c2 as 0. So I'm starting it right there. And then I'll travel along, depending on sines. I would go, I think, probably this way. So it will travel on a-- it comes in pretty fast, of course. Because that exponential is a powerful guy that e to the minus t. So this is the solution, damping out to 0. That's with the plus sign, plus damping, which gives the minus sign in the s, in the exponent. And then so that is a spiral sink. Sink meaning just as water in a bathtub flows in, that's what happens. Now what happens in a spiral source that's what we have with a minus sign. Now we have a 1. Now we have an e to the plus t. Everything is growing. So instead of decaying, we're going around but growing-- OK. I'm off the board, way off the board with that spiral. Which is going to keep going around, but explode out to infinity. So those are the three possibilities for complex roots, centers, spiral source, and spiral sink. For real roots we had ordinary source, and ordinary sink, no spiral. And then the other possibility was a saddle point, where almost surely we go out. But there was one direction that came into the saddle point. OK. Those six pictures are going to control the whole problem of stability, which is our next subject. Thank you. |
MIT_Learn_Differential_Equations | Integrating_Factor_for_Constant_Rate.txt | GILBERT STRANG: OK. This is our last look at the first order linear differential equation that you see up here. The dy dt is ay, that's the interest rate growing in the bank example. y is our total balance. And q of t is our deposits or withdrawals. Only one change. We allow the interest rate a to change with time. This we didn't see before. Now we will get a formula. It will be a formula we had before when a was constant. And now we'll see it. It looks a little messier, but the point is, it can be done. We can solve that equation by a new way. So that's really the other point. Everybody in the end likes these integrating factors. And I will call it m. And let me show you what it is and how it works. What it is is the solution to the null equation, with a minus sign. With a minus sign. dM dt equals minus a of tM. No source term. We can solve that equation. If a is constant-- and I'll keep that case going because that's the one with simple, recognizable formulas. If a is a constant, we're looking for the function M whose derivative is minus aM. And that function is e to the minus at. The derivative brings down the minus a that we want. In case a is varying, we can still to solve this equation. It will still be an exponential of minus something. But what we have to put here when I take the derivative of M, the derivative of that will come down. So I want the integral of a here. And then the derivative of the integral is a minus a, coming down as it should. So I want minus the integral of a. And can I introduce dummy variables, say a of T dT, just to make the notation look right. OK. You see that, again, the derivative of M is always with an exponential. It's always the exponential times the derivative of the exponent. And the derivative of that exponent is minus a. Because by the fundamental theorem of calculus, if I integrate a and take its derivative, I get a again. And it's that a that I want. Now, why do I want this M? How does it work? Here's the reason M succeeds. Look at the derivative of M times y. That's a product. So I'll use the product rule. I get the derivative of y times M, and then I get the derivative of M times y. But the derivative of M is minus a of tM, so I better put the derivative of M is minus a of tM times y. But what have I got here? Factor out an M and that's just dy dt minus ay, dy dt minus ay is q. So when I factor out the M, I just have q. All together, this is M times q. Look, my differential equation couldn't look nicer. Multiplying by M made it just tell us that a derivative is a right-hand side. To solve that equation, we just integrate both sides. So if you'll allow me to take that step, integrate both sides and see what I've got, that will give us the formula we know when we're in the constant case, and the formula we've never seen when t is varying. And then I'll do an example. Let me do an example right away. Suppose a of t, instead of being constant, is growing. The economy is really in hyperinflation. Take that example if a of t is, let's say, 2t. Interest rate started low and moves up, then growth is going to be faster and faster as time goes on. And what will be the integral of 2t? The integral of 2t is t squared, so M, in that case, will be e to the minus a t squared. Sorry, there's no a anymore. a is just the 2t. e to the minus t squared. With a minus sign, it's dropping fast. In a minute, we'll have a plus sign there and we'll see the growth. Do you see that this is the integrating factor when a of t happens to be 2t? OK. Now I come back to this equation and integrate both sides to get the answer. OK. All right. The integral of My, of the derivative, the integral of the derivative is just M of t y of t minus M of 0 y of 0. That's the integral on the left side. And on the right side, I have the integral of M times q from 0 to t. And again, I'm going to put in an integration variable different from t just to keep things straight. OK. So now I've got a formula for y. It involves the M. Actually, the y is multiplied by M, I better divide by-- first of all, do we remember what M of 0 is? That's the growth factor at 0. It's just 1. Nothing's happened. It's the exponential of 0 in our formulas for M. M of 0 is 1. That's where M starts. So M of 0 is 1. I can remove that. OK. And now-- oh, let me put that on the other side so this will be equals y of 0 plus that. OK. And now if I divide by M, I have my answer. So those are the steps. Find the integrating factor. Do the integration, which is now made easy because I have a perfect derivative whose integral I just have to integrate. And then put in what M is, and divide by it so that I get y. OK. So I'm dividing by M. So what is 1 over M? Well, M has this minus sign in the exponent. 1 over M will have a plus sign. M here has e to the minus t squared. 1 over M will be e to the plus t squared. So when I divide by M, I get y of t. This will be 1 over M. That will be e to the plus the integral of a of t dt y of 0. That's the null solution. That's the solution that's growing out of y of 0. And now I have plus the integral from 0 to t of-- remember, I'm dividing by M. And that's e to the plus the integral from 0 to s of a times q of s ds. OK. Oh, just a moment. I'm dividing by M and I had an M there. Oh, wait a minute. I haven't got it right here. So I want to know what is M at time s divided by M at time t? So this was the integral from 0 to s. This is an integral from 0 to t. And both are in the exponent. This is-- can I say it here? This is a e to the-- divided by M is the integral from 0 to t. And then I'm multiplying by e to the minus the integral from 0 to s. The rule for exponents is if I have a product of two exponentials, I add the exponents. When I add this to this, this knocks off the lower half of the integral. I'm left with the integral from s to t of a. So this was an integral of a minus an integral of a. Let me do our example. Our example up here. Example-- M of t will be-- when a is equal to 2t, this was the example a equal to 2t. The first time we've been able to deal with a varying interest rate. So the integral of 2t is t squared. From the-- is e to the t squared. And I subtract the lower limit, s squared. That's the growth factor. That's the growth factor from time s to time t. When a was constant, that exponent was just a times t minus s. That told us the time. But now, a is varying and the growth factor between s and t is e to the t squared minus s squared. So that's what goes in here. Let me-- that's the growth factor. May I just put it in here? In this example, it's e to the t squared minus s squared. Instead of e to the a t minus s, I now have t squared minus s squared, because I had an integral of a of t, and a is not constant anymore. This is my example. And I don't know if you like this formula. Can I just describe it again? This was an integral from 0 to t, so that would be-- this part would be e to the t squared. That's the growth factor that multiplies the initial deposit. The growth factor that multiplies the later deposit is e to the t squared minus s squared. And we allow deposits all the way from s equals 0 to t. So when we add those up, we get that sum. We've solved an equation that we hadn't been able to solve before. That's a small triumph in differential equations. Small, admittedly. I'd rather move next to non-linear equations, which we have not touched. And that's a big deal. Thank you. |
MIT_Learn_Differential_Equations | Gilbert_and_Cleve_Introduction.txt | GILBERT STRANG: Hi. I'm Gilbert Strang. I'm a math professor at MIT. CLEVE MOLER: And I'm Cleve Moler. I'm one of the founders of MathWorks, and I'm chief mathematician. Gil and I have made a video series about ordinary differential equations. GILBERT STRANG: This developed really from my experience of the linear algebra class. It's videotaped and shown on OpenCourseWare. And it was my first time to discover that millions of people were learning, watching it. And it just seemed possible to tackle the other major undergraduate courses after calculus. The other lead in to engineering and science. Differential equations with the two sides of the formulas and the computations. CLEVE MOLER: This is not only for students that are currently in school, after calculus. But it's also for lifelong learners. People who want to come back to this material, maybe after years out of school. We want to help people who are using MATLAB to solve differential equations. To understand the MATLAB ODE suite. To turn the black boxes into grey boxes, where you have some understanding of the mathematics that underlies them. GILBERT STRANG: Really, differential equations is the expression, the mathematical expression of change. So engineering, finance, economics, life sciences, medical sciences where you're seeing things change over time. Those are modeled by differential equations. So we want to understand what does solutions look like for formulas? And what do they look like for numbers? For actual quantities. CLEVE MOLER: Gil and I are matrix guys. Our professional lives have been involved with matrices. And MATLAB started life as Matrix Laboratory. Here we are doing differential equations. Why is that? That's because matrices are fundamental in understanding modern differential equations. Systems of ordinary differential equations are the key to understanding modern applications of differential equations. GILBERT STRANG: We hope you enjoy it. |
MIT_Learn_Differential_Equations | Solving_Linear_Systems.txt | GILBERT STRANG: So this is the key video about solving a system of n linear constant coefficient equations. So how do I write those equations? Y is now a vector, a vector with n components. Instead of one scalar, just a single number y-- do you want me to put an arrow on y? No, I won't repeat it again. But that's to emphasize that y is a vector. Its first derivative, it's a first order system. System meaning that there can be and will be more than one unknown, y1, y2, to yn. So how do we solve such a system? Then the matrix is multiplying that y and they equate. The y's are coupled together by that matrix. They're coupled together, and how do we uncouple them? That is the magic of eigenvalues and eigenvectors. Eigenvectors are vectors that go in their own way. So when you have an eigenvector, it's like you have a one by one problem and the a becomes just a number, lambda. So for a general vector, everything is a mixed together. But for an eigenvector, everything stays one dimensional. The a changes just to a lambda for that special direction. And of course, as always, we need n of those eigenvectors because we want to take the starting value. Just as we did for powers, we're doing it now for differential equations. I take my starting vector, which is probably not an eigenvector. I'd make it a combination of eigenvectors. And I'm OK because I'm assuming that I have n independent eigenvectors. And now I follow each starting value c1 x1-- what does that have? What happens if I'm in the direction of x1, then all the messiness of A disappears. It acts just like lambda 1 on that vector x1. Here's what you get. You get c1, that's just a number, times e to the lambda 1t x1. You see there, instead of powers, which we had-- that we had lambda 1 to the kth power when we were doing powers of a matrix, now we're solving differential equations. So we get an e to the lambda 1t. And of course, next by superposition, I can add on the solution for that one, which is e to the lambda 2t x2 plus so on, plus cne to the lambda nt xn. You can see when, I could ask, when is this stable? When do the solutions go to 0? Well, as t gets large, this number will go to 0, provided lambda 1 is negative. Or provided its real part is negative. We can understand everything from this piece by piece formula. Let me just do an example. Take a matrix A. In the powers of a matrix-- in that video I took a Markov matrix-- let me take here the equivalent for differential equations. So this will give us a Markov differential equation. So let me take A now. The columns of a Markov matrix add to 1 but in the differential equation situation, they'll add to 0. Like minus 1 and 1, or like minus 2 and 2. So there is the eigenvalue of 1 for our powers is like the eigenvalue 0 for differential equations. Because e to the 0t is 1. So anyway, let's find the eigenvalues of that. The first eigenvalue is 0. That's what I'm interested in. That column adds to 0, that column adds to 0. That tells me there's an eigenvalue of 0. And what's its eigenvector? Probably 2, 1 because if I multiply that matrix by that vector, I get 0. So lambda 1 is 0. And my second eigenvalue, well the trace is minus 3 and the lambda 1 plus lambda 2 must give minus 3. And its eigenvector is-- it's probably 1 minus 1 again. So I've done the preliminary work. Given this matrix, we've got the eigenvalues and eigenvectors. Now I take u0-- what should we say for u0? U0-- y0, say y of 0 to start. Y of 0 as some number c1 times x1 plus c2 times x2. Yes, no problem, no problem. Whatever we have, we take this-- some combination of that vector and that eigenvector will give us y of 0. And now the y of t is c1 e to the 0t-- e to the lambda 1t times x1, right? You see, we started with c1x1 but after a time t, it's either the lambda t and here's c2. E to the lambda 2 is minus 3t x2. That's the evolution of a Markov process, a continuous Markov process. Compared to the powers of a matrix, this is a continuous evolving evolution of this vector. Now, I'm interested in steady state. Steady state is what happens as t gets large. As t gets large, that number goes quickly to 0. In our Markov matrix example, we had 1/2 to a power, and that went quickly to 0. Now we have the exponential with a minus 3, that goes to zero. E to the 0t is the 1. This e to the 0t equals 1. So that 1 is the signal of a steady state. Nothing changing, nothing really depending on time, just sits there. So I have c1x1 is the steady state. And x1 was this. So what am I thinking? I'm thinking that no matter how you start, no matter what y of 0 is, as time goes on, the x2 part is going to disappear. And if you only have the x1 part in that ratio 2:1. So again, if we had movement between Y1 Y2 or we have things evolving in time, the steady state is-- this is the steady state. There is an example of a differential equation, happen to have a Markov matrix. And with a Markov matrix, then we know that we'll have an eigenvalue of - in the continuous case and a negative eigenvalue that will disappear as time goes forward. E to the minus 3t goes to 0. Good. I guess I might just add a little bit to this video, which is to explain why is 0 an eigenvalue when whenever-- if the columns added to 0, minus 1 plus 1 is 0. 2 minus 2 is zero. That tells me 0 is an eigenvalue. For a Markov matrix empowers the columns added to 1 and 1 was an eigenvalue. So I guess I have now two examples of the following fact. That if all columns add to some-- what shall I say for the sum, s for the sum-- then lambda equal s is an eigenvalue. That was the point from Markov matrices, s was 1. Every column added to 1 and then lambda equal 1 was an eigenvalue. And for this video, every column added to 0 and then lambda equal 0 was an eigenvalue. And also, this is another point about eigenvalues, good to make. The eigenvalues of a transpose are the same as the eigenvalues of A. So I could also say if all rows of A add to s, then lambda equal s is an eigenvalue. I'm saying that the eigenvalues of a matrix and the eigenvalues of the transpose are the same. And maybe you would like to just see why that's true. If I want the eigenvalues of a matrix, I look at the determinant of lambda I minus A. That gives me eigenvalues of A. If I want the eigenvalues of a transpose, I would look at this equals 0, right? This equaling 0. That equation would give me the eigenvalues of a transpose just the way this one gives me the eigenvalues of A. But why are they the same? Because the determinant of a matrix and the determinant of its transpose are equal. A matrix and its transpose have the same determinant. Let me just check that with A, B, C, D. And the transpose would be A, C, B, D. And the determinant in both cases is AD minus BC, AD minus BC. Transposing doesn't affect. So this, that is the same as that. And the lambdas are the same. And therefore we can look at the columns adding to s or the rows added to s. So this explains why those two statements are both true together because I could look at the rows or the columns and reach this conclusion. That if all columns add to s-- now why is that, or all rows add to s? Let me just-- I'll just show you the eigenvector. In this case, A times the eigenvector would be all ones. Suppose the matrix is 4 by 4. If I multiply A by all ones, when you multiply a matrix by a vector of ones, then the dot product of this row with that is the sum, is that plus that plus that plus that, would be we s. This would be s because this first row-- here is A-- first row of A adds to s. So these numbers add to s, I get s. These numbers add to s, I get s again. These numbers add to s. And these, finally those numbers add to s. And I have s times 1, 1, 1, 1. Are you OK with this? When all the rows add to s, I can tell you what the eigenvector is, 1, 1, 1, 1. And then the eigenvalue, I can see that that's the sum s. So again, for special matrices, in this case named after Markov, we are able to identify important fact about their eigenvalue, which is that it's that common row sum s equal 1 in the case of powers and s equal 0 in this video's case with-- let me bring down A again. So here, every column added to 0. It didn't happen that the rows added to 0. I'm not requiring that. I'm just saying either way, A or A transpose has the same eigenvalues and one of them is 0 and the other is whatever the trace tells us, that one. These collection of useful fact about eigenvalues show up when you have a particular matrix and you need to know something about its eigenvalues. Good, thank you. |
MIT_Learn_Differential_Equations | ODE45.txt | PROFESSOR: The most frequently used ODE solver in MATLAB and Simulink is ODE45. It is based on method published by British mathematicians JR Dormand and PJ Prince in 1980. The basic method is order five. The error correction uses a companion order four method. The slope of tn is, first same as last left over from the previous successful step. Then there are five more slopes from function values at 1/5 h, 3/10h, 4/5h, 8/9h and then at tn plus 1. These six slopes, linear combinations of them, are used to produce yn plus 1. The function is evaluated at tn plus 1 and yn plus 1 to get a seventh slope. And then linear combinations of these are used to produce the error estimate. Again, if the error estimate is less than the specified accuracy requirements the step is successful. And then that error estimate is used to get the next step size. If the error is too big, the step is unsuccessful and that error estimate is used to get the step size to do the step over again. If we want to see the actual coefficients that are used, you can go into the code for ODE45. There's a table with the coefficients. Or you go to the Wikipedia page for the Dormand-Prince Method and there is the same coefficients. As an aside, here is an interesting fact about higher order Runge-Kutta methods. Classical Runge-Kutta required four function evaluations per step to get order four. Dormand-Prince requires six function evaluations per step to get order five. You can't get order five with just five function evaluations. And then, if we were to try and achieve higher order, it would take even more function evaluations per step. Let's use ODE45 to compute e to the t. y prime is equal to y. We can ask for output by supplying an argument called tspan. Zero and steps of 0.1 to 1. If we supply that as the input argument to solve this differential equation and get the output at those points, we get that back as the output. And now here's the approximations to the solution to that differential equation at those points. If we plot it, here's the solution at those points. And to see how accurate it is, we see that we're actually getting this result to nine digits. ODE45 is very accurate. Let's look at step size choice on our problem with near singularity, is a quarter. y0 is close to 16. The differential equation is y prime is 2(a-t) y squared. We let ODE45 choose its own step size by indicating we just want to integrate from 0 to 1. We capture the output in t and y and plot it. Now, here, there's a lot of points here, but this is misleading because ODE45, by default, is using the refine option. It's only actually evaluating the function at every fourth one of these points and then using the interpolant to fill in in between. So we actually need a little different plot here. This plot shows a little better what's going on. The big dots are the points that ODE45 chose to evaluate the differential equation. And the little dots are filled in with the interpolant. So the big dots are every fourth point. And the refine option says that the big dots are too far apart and we need to fill it in with the interpolant. And so this is the continuous interpolant in action. The big dots are more closely concentrated as we have to go around the curve. And then, as we get farther away from the singularity the step size increases. So this shows the high accuracy of ODE45 and the automatic step size choice in action. Here's an exercise. Compare ODE23 and ODE45 by using each of them to compute pi. The integral 4 over 1 plus t squared from 0 to 1 is pi. You can express that as a differential equation, use each of the routines to integrate that differential equation and see how close they get to computing pi. |
MIT_Learn_Differential_Equations | Step_Function_and_Delta_Function.txt | GILBERT STRANG: OK, this is the video about two neat functions-- the step function and its derivative the delta function. So if I can just introduce you to those functions and show you that they're very natural inputs to a differential equation. They happen all the time in real life. And so we need to understand how to compute these formulas and compute with them. OK, so the first one is the step function and it's-- I'll call it h after its inventor who was an engineer named Heaviside, started with an H. And the step function, let me write the formula. h of t is 0 for t negative and 1 for t greater or equal to 0. OK. So, that's the step function. It just has two values and it has a jump. You could say jump function also. Jump function, step function. All right. And notice I've also graphed the shifted step function. What happens to any function including this one if I change from t, which jumps at 0, to h of t minus t? If I put in t minus some fixed number t as the variable, then the jump happens. So the jump will happen when this is 0. Step functions jump when that's 0. And that's 0 at t equal to t. So the jump in dotted line. So the shifted step function will just shift over. That's the complete effect of changing from t to t minus a capital T, is just to shift the whole thing by capital T. OK. So you keep your eye on the standard step function, which jumps at t equals 0. It jumps by 1. And take its derivative. So what's the derivative of this step function? Well, the function is 0 along there, so the derivative is 0. The function is constant along here, so the derivative is again 0. It's just at this one point everything happens. So now this is the delta function. The delta function runs along at 0, continues at 0, but at t at 0, the whole thing explodes. The derivative is infinite. You see an infinite slope there. And the point is infinity is not a sufficiently precise word to tell you exactly what's happening. So we don't have really-- this graph of a delta function is not fully satisfactory. It's perfect for all the uninteresting boring part. But at the moment of truth, when something happens in an instant, we need to say more. We need to say more, not just its infinite. And again, if it's shifted, then the infinite slope happens at t equal a capital T. So the infinity is just shifted over. And that'd be the delta function there. So this is what I would use. If that was the source term in my differential equation, what would that mean? If this was the q of t in the differential equation reflecting input at different times, that function would say no input except at one moment and one instant, capital T. At that instant of time, you put 1 in, over in an instant. And remember, that otherwise q of t has been a continuous input. Put in $1.00 per year over the whole year. This one puts in $1.00 at one moment. But of course, you see that that's really what we do. So, you see that that's a function we need to do things in an instant. And as I took the example of a golf club hitting a golf ball, well, it's not quite 0 time. But it's so close to zero time that the two are connected. And then the ball takes off. And so a simple model, a workable model is to say it happens in 0 time with a delta function. So I really want to use delta functions. And they're not difficult to use. They're just not quite perfect for calculus because the derivative of the step function is not quite legitimate at the jump. OK. But what you can do, the part of calculus that works correctly is integration. Integration tends to make things smoother. The delta function-- sorry, the step function is the integral of the delta function. Right? We're going in the opposite direction. We take derivatives, we get craziness. If we take integrals to go from delta-- so the integral of the delta is the step function. And that's really how you know a delta function. That's the math way to describe more exactly than this arrow that just fires off what the delta function is doing. So the key property of the delta function is to know what it's integral is. The integral of the delta function is the total deposit over, let's say, it started-- time could have started even at minus infinity, and it could go on forever to plus infinity. So that's the total deposit, the total input coming from this source term delta of t. And what is the answer? Well, the integral of delta should be the step function. The step function out in infinity is 1. Back at minus infinity it's 0. Do you see what I'm saying here? This would be h of t evaluated between t equal minus infinity and plus infinity because those are the limits of integration. And what do I get? At plus infinity the step function is 1. This is 0. So I get 1. And everybody catches on to that key fact that the total integral of the delta function is 1. Again, you only made the deposit at one moment, but that deposit was a full dollar. And that, adding up all deposits is just that $1.00. So, that's the integral of the delta function. Now actually, to use delta functions I need to give you a slight generalization of that. So as I say, delta functions are really known-- we don't like to take their derivative. The derivative of a delta function is a truly crazy function. It shoots up to infinity and then it shoots down the minus infinity, the slope of that arrow. But it's integrals that we want. So now let me integrate from minus infinity to infinity my delta function times any other function, say f of t dt. That's something we'll need to be able to compute. What's the right integral for that? And again, delta is doing everything at one moment at t equals 0. At that moment t equals 0, at that moment when t is 0 and that's the only place any action is happening, f of t is f of 0. It's whatever value it has at that point t equals 0. And that's the answer. f of 0. So if f of t was the constant function 1, then we're back to our integral up there. If that's just 1, I'm integrating delta of t. My function is 1, I get 1. But if that function is, suppose that function is sine t. What's the integral of delta of t times sine t dt? Well, sine t happens to disappear just at the moment when the delta function is ready to turn on at t equals 0. So the integral of delta of t sine t is sine of 0 is 0. You have one term turned on, but the other term turned off. So nothing happened altogether. Whereas the integral of delta t e to the t-- yeah, tell me that one. The integral of delta t e to the t dt is-- well e to the t is doing all sorts of stuff for all time. But the delta function is 0 all that time, except at t equals 0. So, the integral of delta t e to the t dt would be 1 because at that moment, t equals 0, the only important moment would be e to the t function is e to the 0, and it's just 1. Let me ask you for another example. The integral of minus infinity to infinity of delta-- let me use the shifted delta e to the t dt. Can you compute that integral. Well again, that function is 0 almost all the time. The only time that impulse, the moment that impulse hits is t equals capital T. At that moment, this is equal to e to the capital T. And that's all that matters. OK. So now, let me use a delta function as the source term in our differential equation. So we are seeing one last time one more-- I still call it a nice function, even though it's not legitimately a function at all, the delta. But let me solve the equation dy dt equals ay plus the delta function turned on at capital T. And let me start it from 0. So I don't make an initial deposit to my account. I don't make any deposit at all, except at one moment t equal capital T. And in that moment, I deposit $1.00 because delta-- this is the unit delta. If I was depositing $10, I would make it 10 delta. OK. So we know what the solution is from a deposit of $1.00 made at one time, t equal to capital T. What is the solution? y of t, we have 0 up to t equal to T. Nothing whatever has happened. And at capital T time t, in goes the $1.00 and it grows. It grows so that it grows over the remaining time e to the t minus capital T. This is for t larger than T. t larger than or equal I could say. When t and capital T are equal, that's e to the 0. That's our $1.00 just gone in. When t minus capital T is a year later, our dollar is worth e. When t minus capital T, when it's been in there for a year, that $1.00 has increased to-- well, that was if the interest rate was 100% you may feel. You'd be fortunate to get that. But let's suppose you do. At 100% interest, after one year, you might say, well, my money just doubled because I got the interest equaled the original. So I got twice it. But not true because that money went in-- was growing. Interest was being added, compounded through the whole year so that after one year, starting with 1, you have e at a is 100%. Oh, OK. My formula isn't incorrect here because I had an a here and it belongs here. So let me fix that. It's e to the a t minus T. That's the growth factor. That's the growth factor up to time t starting from the earlier time capital T. So you see that we were able just to write down the solution to the differential equation even though it's entirely new or different or non-standard input. The step function input-- so we're finding here the impulse response. That's a very, very important concept in engineering, the impulse response, the response to an impulse. And for second order differential equations, this is going to be-- it's really a crucial function in the subject. So this is the response to an impulse. It's the impulse response from our standard first order equation that we've been dealing with. Now we've got just to remember one more step is still linear would be to allow the interest rate to change. That's one lecture, the next one. And then we get non-linear equations. So that's what's coming. But here is delta functions for the first time and not for the last time. Thank you. |
MIT_Learn_Differential_Equations | Singular_Value_Decomposition_the_SVD.txt | PROFESSOR: The previous video was about positive definite matrices. This video is also linear algebra, a very interesting way to break up a matrix called the singular value decomposition. And everybody says SVD for singular value decomposition. And what is that factoring? What are the three pieces of the SVD? So this is the fact is every matrix, rectangular, every matrix factors into-- these are the three pieces. U sigma V transpose. People use those letters for the three factors. The factor U is an orthogonal matrix, an orthogonal matrix. The factor sigma in the middle is a diagonal matrix. The factor V transpose on the right is also an orthogonal matrix. So I have orthogonal, diagonal, orthogonal, or physically, rotation, stretching, rotation. Now we have seen three factors for a matrix, V, lambda, V inverse. What's the difference? What's the difference between this SVD, this, and the V, lambda, V transpose, V inverse, V lambda, V inverse for diagonalizing other matrices? So the lambda is diagonal and the sigma is diagonal, but they're different. The key point is I now have two different matrices, not just V and V inverse, but two different matrices. But the new great advantage is they are orthogonal matrices, both of them. So by going to-- and I can do it for rectangular matrices also. Eigenvalues really worked for square matrices. Now we really are-- we have two. We have an input matrix and an output matrix. In those spaces m and n can have different dimensions. So by allowing two separate bases, we get rectangular matrices, and we get orthogonal factors with, again, a diagonal. And this is called-- these numbers sigma instead of eigenvalues, are called singular values. So these are the singular values. These are the singular vectors, the right singular vectors and the left singular vectors. That's the statement of the factorization. But we have to think a little bit, what are those factors? What are the-- can we see why this works? So I want that. And let me do, as you see this coming, I'll look at A transpose A. I like A transpose A. So A transpose will be, I transpose this. V sigma transpose U transpose, right? That's A transpose. Then I multiply by A U sigma V transpose. And what do I have? Well, I've got six matrices. But U transpose U in here is the identity, because U is an orthogonal matrix. So I really have just the V on one side, a sigma transpose sigma, that'll be diagonal, and a V transpose the right. This I recognize. This I recognize. Here is a single V, a diagonal matrix, a V transpose. What I'm showing you here, what we reached is the eigenvalue, the diagonalization, the usual eigenvalues are in here and the eigenvectors are in here. But the matrix is A transpose A. Once again, A was rectangular and completely general and we couldn't see perfect results. But when we went to A transpose A, that gave us a positive semidefinite matrix, symmetric for sure. Its eigenvectors will be orthogonal. That's how I know this V matrix, the eigenvectors for this symmetric matrix, are orthogonal and the eigenvalues are positive. And they're the squares of the singular value. So this is telling me the lambdas for A transpose A are the sigma squareds for s-- for A. For A itself. Lambda is the same. Lambda for A transpose A is sigma squared for the matrix A. Well that tells me V, that tells me sigma, and U disappeared here because U transpose U was the identity. It just went away. How would I get hold of U? Well, here's one way to see it. I multiply A times A transpose in that order, in that order. So now I have U sigma V transpose times the transpose, which is the V sigma transpose U transpose-- I'm having a lot of fun here with transposes. But V transpose V is now the identity in the middle. So what do I learn here? I learn that U is the eigenvector matrix for AA transpose. So these have the same eigenvalues, A times B has the same eigenvalues as B times A in this case, it comes out here. Same eigenvalues. This has eigenvectors V, this has eigenvectors U, and those are the V and the U in the singular value decomposition. Well, I have to show you an example I have to show you an example and an application, and that's it. So here's an example. Suppose A, I'll make it a square matrix, 2, 2, minus 1, 1, not symmetric. Certainly not positive definite. I wouldn't use the word because that matrix is not symmetric. But it's got an SVD, three factors. And I work them out. This is the orthogonal matrix. I have to divide by square root of 5 to make it unit vectors. Oops, that's not going to work. How about that? The two columns are orthogonal and that's a perfectly good U. And then in the sigma, I got, well that's a-- oh, I did want 1 and 1. I did want 1 and 1, yes. So I have a singular matrix, determinant 0, singular matrix. So my eigenvalues will be 0 and it turns out square root of 10 is the other eigenvalue for that-- other singular value for this guy. And now I'll put in the V transpose matrix, which is 1, 1, and 1, minus 1 is it? And those have length square root of 2, which I have to divide by. Well, I didn't do that so smoothly, but the result is clear. U, sigma, V transpose, so here's the sigma. And the singular values of this matrix are square root of 10 and then 0 because it's a singular matrix. And the eigenvectors, well the singular vectors of the matrix are the left singular vectors and the right singular vectors. That looks good to me. And now the application to finish. A first application is, well, very important. All the time in this century, we're getting matrices with data in them. Maybe in life sciences, we test a bunch of sample people for genes. So I have a-- my data comes somehoe-- I have a gene expression matrix. I have samples, people, people 1, 2, 3 in those columns. And I have in the rows, let me say four rows, I have genes, gene expressions. That would be completely normal. A rectangular matrix, because the number of people and the number of genes is not the same. And in reality, those are both very, very big numbers, so I have a large matrix. And out of it, I want to-- and each number in the matrix is telling me how much the gene is expressed by that person. We may be searching for genes causing some disease. So we take several people, some well, some with the disease, we check on the genes. We get a big matrix and we look to understand something about of it. What can we understand? What are we looking for? We're looking for the correlation, the connection, between some combination maybe of genes and some-- we're looking for a gene people connection here. But it's not going to be person number one. We're not looking for one person. We're going to find a mixture of those people, so we're going to have sort of an eigensample, eigenpeople. Oh, that's a terrible-- eigenperson would be better. So I think I'm seeing an eigenperson. Let me see where I'm going to put this. So yeah, I think my matrix would be written-- oh, here is the main point. That just as I see in this example, it's the first vector and the first vector and the biggest sigma that are all important. Well, in that example the other sigma was 0, nothing. But in this example, I'll probably have three different sigmas. But the largest sigma, the first, the U1 and the V1, it's that combination that I want. I want U1 sigma 1 V1 transpose, the first eigenvector of A transpose A and of AA transpose. And the first singular, the biggest singular value, that's the information. That's the best sort of put together person, eigenperson, combination of these people and the best combination of genes. It has the-- in statistics, I would say the greatest variance. In ordinary English, I would say the most information. The most information in this big matrix is in this very special matrix with only rank one, only a single column repeated. A single row repeated, and a number sigma 1, the number that tells me that. Because remember, U is a unit vector. V is a unit vector. It's that number sigma 1 that's selling me. So it's like that unit vector times that number, key number, times that unit vector, that's this. I'm talking here about principle component analysis. I'm looking for the principle component, this part. Principle component analysis. A big application in applied statistics. You know, in large scale drug tests, statisticians really have a central place here. And this is on the research side, to find the-- get the information out of a big sample. So U1 is sort of a combination of people. V1 is a combination of genes. Sigma 1 is the biggest number I can get. So that's PCA, all coming from the singular value decomposition. Thank you. |
MIT_Learn_Differential_Equations | Graphs.txt | PROFESSOR: OK. This video is a different direction. It will be about linear equations and not differential equations. A matrix is at the center of this video and it's called the incidence matrix. And that incidence matrix tells me everything about a graph. Now, what do I mean by the word graph? I don't mean a graph of sine x or cosine x. The word graph is used in another way completely for some edges and some nodes. So I have some nodes. In this case 1, 2, 3, 4 nodes. That's my number n. The number m is the number of edges that connect the nodes. So I have edge 1 connecting those nodes, edge 2, edge 3, 4, and 5. And I didn't put in an edge 6. A complete graph would have all possible edges, but a general graph can have some edges. Some pairs of nodes are connected others are not connected. So now I want to create the matrix that shows me everything that's in that picture. Then I can work with the matrix and graphs. And their matrices are the number one application, number one model for so many applications, like the world wide web. The web might have-- every website would be a node and there would be an edge between two nodes if those websites are linked. So the world wide web is a giant graph. Or the telephone company has a giant graph in which the nodes are the telephones, and there is an edge when a call is made from one phone to another, between two phones. So, nodes and edges. And our brain-- which is the great problem of the 21st century is to understand the graph that represents our brain, the connections of neurons in our thinking-- well, that's a tougher problem than we'll solve today. Let me work with that graph and create the matrix. So the matrix has five rows coming from the five edges. Let me take the first edge. So the first edge, there's edge number 1, goes from node 1 to node 2. The nodes correspond to columns. So if I want an edge from node 1 to node 2, that edge 1 will go in row 1. So edge 1. First edge is connected to row 1. So that edge goes from node 1 to node 2, so I put a minus 1 and a 1. And it doesn't touch nodes 3 and 4. That's edge 1. That's row 1. Now that tells me everything I see about edge 1. Edge 2 goes from 1 to 3. So I'll put a minus 1, a 0, and a 1 in row 2 because row 2 comes from edge 2 and it goes from 1 to 3. Edge 3 will give me row 3, from 2 to 3. So edge 3 giving me row 3, 2 to 3. Edge 4 went from 1 to 4. So minus 1, nothing, nothing, 1. That tells me that edge 4 is going from node 1 to node 4. And finally, from node 2 to node 4 is the final row. Do you see there the graph? Everything, all the information in this picture is now captured in that matrix. So we can work with the matrix. And what does a matrix do? It multiplies vectors. That's what a matrix does, it acts on vectors. So what happens if I multiply that matrix by a vector? So now let me take out these edge numbers and do a multiplication. That matrix has four columns, it's a 5 by 4 matrix, m by n. 5 by 4. So it multiplies a vector with four components and those four components will come from the four nodes. And maybe they represent voltages at the nodes. Let me think like an electrical engineer for a moment. So if there's my matrix, I imagine I have voltages, v1, v2, v3, v4, at the nodes. So there's a v1 voltage here, v2, a v3, and a v4, and where those voltages' currents will flow. So my unknowns are the voltages, the four voltages, and the five currents. That's what the engineer needs to know. So first of all, when I multiply A times v, what do I get? Let me just do that multiplication. So that first row times that gives me v2 minus v1, right? The dot product of the row with the vector. The next one is v3 minus v1. Then I have a minus 1 there. It's a v3 minus a v2. Then I have a minus 1 and a 1. I think that's v4 minus v1. And finally, this dot product of that will give me a v4 minus v2. So what am I seeing here? This is now A times v. I've done a multiplication by a vector of voltages. And what have I found? I found the differences in voltages, the voltage difference between one end of the edge and the other one. I have five edges and now I have five results and those are the voltage differences. And what does a difference in voltage do if these are at different voltages, different potentials? Current flows. If they're at the same potential, no current flows, right? That's the fundamental driving equation of currents from voltages is the difference in the voltage. The difference in the potential drives the flow. And now, how much flow? So now I'm looking for the flows. So can I call those w, for the flows. So I have a w2 is the flow on that edge. A w1 is a flow there. A w5, a w3, and a w4. My pair of unknowns-- and that's the beauty of this picture-- is the voltages v1 to v4 four at the nodes, and the currents, the flows, w1 to w5 on the five edges. And I've seen that Av gives me the voltage differences. I'm going to briefly, briefly approach the fundamental laws of flow, of current flow, of flow in any network. We're talking about the most basic equation, I would almost say, of applied mathematics. Maybe I should say of discrete applied mathematics. By discrete I mean a graph without derivatives. I'm not seeing derivatives here, I'm just seeing matrices and vectors. So I have to remember that incidence matrix, A-- let me write it down again. Av gave the voltage differences. And that's one part of my picture. Another part is what is the equation that finally brings it together? That if I have the currents-- so the v's were the voltages. Now, there's going to be an equation involving w, the currents. This, what I'm going to write here, is going to be really important. It's going to be Kirchhoff's Current Law, KCL. And I just emphasized that there are two Hs in Kirchhoff's name. So Kirchhoff's Current Law says-- and pay attention-- it says that the total flow into a node equals the flow out. We're talking about equilibrium here. So if current is traveling around my graph, my network, and it's a stable equilibrium here so that flow into node 1 equals flow out of node 1. And let me tell you what that equation is in terms of the matrix A. This voltage difference is involved A and, beautifully, the Kirchhoff's Current Law involves A transpose. So A transpose now is 4 by 5. These are the flows, a vector with five components because I have five edges. And Kirchhoff's Current Law would say that's 0. So between A and A transpose, the incidence matrix is leading me to the fundamental equilibrium condition for flow in a network. Now, one more law is needed. It has to connect voltage differences to flows, potentials to currents. Do you know who created that law in electrical engineering? It was Ohm. So Ohm's Law, finally, Ohm's Law is edge by edge that the potential difference, the drop in potential, the potential forcing current is proportional to the current. So voltage difference-- let me write it in words. Voltage difference-- voltage drop I could say-- between the ends or across a resistor is proportional to, and there is some resistance, some physical number comes in here. This is where the material we're working with comes in. In Kirchhoff's Laws, those laws hold for a network before we even say what the network is made of. But now if our network is made of resistors or pipes or whatever we have, then this will be some conductance. So E equal IR, some resistance, times the flow, times the current flow, w. So a difference in v's is some number, this is the physical constant that we have to measure in a lab to know how many ohms our resistor is. That equation is on each edge. So we have a bunch of equations and together they tell us the four voltages and the five currents. And maybe I'll just make the main point here. The main point is that this matrix is crucial. A is crucial. A transpose is crucial. A gives voltage differences, it makes something happen. A transpose is the balance law, the balance or current balance at each node. And you won't be surprised that when the whole thing is put together and we have a final equation to solve, we end up with A transpose and A. And that magic combination, A transpose A, is central to graph theory. It's called the graph Laplacian and has a name and a fame of its own. Thank you. |
MIT_Learn_Differential_Equations | Response_to_Exponential_Input.txt | GILBERT STRANG: OK. We're still talking about first order differential equations with a dy dt. And there is still a gross term proportional to the balance. This might be the interest that's added on to y. And then there is input, a source term, of deposits being made all the time. So I'm looking to solve that differential equation. And this is the best possible function for differential equations. Exponentials. Because the derivative of exponential is exponential. It's just easiest to work with. And the output, the solution is called the exponential response. That word response says what comes out when e to the st goes in. And as before, we have some starting deposit, y of 0. Initial condition at time 0. OK. Here is the key point, that with this nice source function, the solution, or one solution, a particular solution, will be just a multiple of e to the st. So all I have to do is find that number, capital Y, and I've got a solution to that equation. How do I do it? Substitute this into the equation and solve for Y. So let's do that. The derivative of this, the derivative of exponential will bring down a factor s. So there'll be a ys e to the st from the derivative. And that will have to equal a times Y e to the st, plus the source term e to the st. good? I just substituted it in. Now, the nice thing, I cancel, I divide by e to the st, which is never 0. So I divide by-- factor out e to the st, factor that out. It just leaves me with a 1. So I have Y times s, Y times a, plus 1. So let me write that equation so you see it. That's just s minus a times Y is 1. Right? I took a times Y and put it on the left side of the equation. So I've discovered the exponential response. 1 over s minus a. OK. So I have a solution to the equation. That's not the end, because that solution won't match the initial condition. So how do I match an initial condition? What I've found is a particular solution, and I need also the null solution, the homogeneous solution. So the full solution, y of t, is this Y particular. So capital Y, I now know is that. So I have an e to the st over-- I'm putting in Y, the right value of Y. So that's the particular solution that I've found. Plus any null solution. So remember, the null solutions, that term is gone. So the source is 0. That's why the word null. So I'm looking for the solution to dy dt equal ay. And the solutions to dy dt equal ay are e to the at, times any number. Because the right-hand side is 0 now. This is y particular. And let me write that. This is y particular, and this is y null, or y homogeneous. So that's the general solution. The complete solution has that form. And now I can match y equal y of 0 at t equal 0. I put in t equal 0, I get y of 0 equals-- t equals 0, this is 1. So 1 over s minus a. And when t is 0, this is 1, so plus C. So now I know what C is. And notice, C is not just y of 0, as sometimes in the past. C is y of 0 minus this. Are you ready now for the complete solution with satisfying the initial conditions? So now I'm going to-- this in the correct form. This tells me what C has to be. So I put it in and I have this solution. y of t is e the st over s minus a, the easy one, plus C. Now, C is y of 0 minus 1 over s minus a. That's what we needed. Times e to the at. That's our answer. That's our answer. I can make it look a little nicer. I want to. I want to separate out the y of 0 part, the part that's just growing from the initial condition, from the part that is coming from the source term. So I just want to put that together with this. So I have the same s minus a below. Here is an e to the st above. And I have a minus, that 1 over s minus a, times e to the at. And then I have this term, which is growing. Well, this is really good. This is the part growing out of the initial deposit. I'm using, again, money in a bank with additional deposits, e to the st. And this is the part coming from those later deposits. Initial part, and the part coming from there. So this is, again, a null solution. A multiple of e to the at. This is another particular solution. Remember, there isn't just one particular solution. Any solution is a particular solution. And this is, I call that the very particular solution. Because it has the nice property that it starts from 0. So a t equals 0, that's 1 minus 1, I'm getting 0. So I would call that y vp. I'll introduce those letters, not standard, for that part. And then y homogeneous, or y null, is this part. OK. Problem solved. The exponential grows in this way, and the initial condition grows directly that way. OK. The problem is solved with one exception. And now I have to take a minute with that exception. The exception is the formula breaks down if s equals a. If s equals a, I'm dividing by 0. My formula is falling apart. And that's the case of resonance. So let me put over here, s equal a. That resonance. And we always have to expect that that's a possibility, that we're putting money with the same exponential as the natural growth of the money, or whatever we're growing. And our formula has to change. You what you might say it's infinite, because I'm dividing by 0. But notice the part above is also 0. If s equals a, this is e to the st minus e to the at. Those are the same. So I have a 0/0 situation. My formula is breaking down, but it isn't dying. It needs more thinking. The case of resonance needs to-- I have to understand what this is when s equals a. Let me tell you what it is. And then show you why. So this is the case s equal a. So if s equal a, then y very particular plus y null space. So it's this very particular solution that has to have a different form. And here's the form it gets. A factor t appears. You just learn to recognize resonance by that factor t. So it'll be a t e to the at. a is the same as s now, so s doesn't appear. So that's the solution which starts from 0, and it comes from the input. And this is the part that starts from y of 0 and grows. So you see, eventually, this is going to be the bigger one. The resonant case, it grows like e to the at, with that little bit extra growth of t. OK. So now I have a solution in that special case also, when s equals a. All right. Do you want to know how this comes out of this as s approaches a? Let me take three minutes to tell you about that. It's L'Hopital's rule. Do you remember from calculus, 0/0, the way to deal with that was called by this guy's name, L'Hopital? Hospital, I guess. Probably hospital in French. And you this 0/0 expression. It's a ratio of two things. The top going to 0 when s goes to a, because these become the same. The bottom going to 0 when s goes to a. And the right-- L'Hopital's cool idea was you get the same answer if you take the ratio of the derivatives. So L'Hopital says, take the ratio of the derivatives of the top minus the derivative-- oh, divided by the derivative of the bottom. And then let s go to a in the end. OK. So I have to take this derivative, I have to take that derivative with respect to s. Often in calculus, it was an x. Here it's an s. No big deal. So that derivative is-- You take the derivative with respect to s. We'll bring down-- ah, here comes the t. The derivative of s is t e to the st. And the derivative of this thing is 1. And now I let s go to a. Well, it's easy to let s go to a now. This thing approaches that. I have the t e of the at in the limit. So L'Hopital's rule was the reason behind this formula for resonance. But again, I emphasize that expect a factor t when you have this resonance. OK. So that's the solution for the best possible right-hand side e to the st. Well, maybe the best is a constant. Second simplest is an exponential. Next, will come sines and cosigns. That's the next lecture. Thanks. |
MIT_Learn_Differential_Equations | An_Example_of_Undetermined_Coefficients.txt | GILBERT STRANG: OK. So can I begin with a few words about the big picture of solving differential equations? So if that was a nonlinear equation, we would go to computer solutions. And Cleve Moler is making a parallel video series about the Matlab suite of codes for solving differential equations. Then when that equation is linear, as it is here, with constant coefficients, as like the 1, minus 3, and the 2, we can always get a formula for the answer. Involves an integral. There's still one integral to do involving the impulse response. And you'll see that. But there are few, the most beautiful, the most simple equations when the right-hand side has a special form. And that's one possibility-- t, or t squared we could deal with, or e to the t. We'll see all those. Then we know what the solution looks like. For example-- first of all, we know the null solutions, of course. That's when this is 0. And then I just wrote for e to the st, and I find s could be 1 or s could be 2. Those are two solutions with right side 0. So we can match initial conditions. But we need a particular solution. And that's where this one is especially simple. So the idea is to try the form we know the solution will have. So I'm going to try a particular solution. When I see a t there, I'll want a t in the particular solution. But I also need the constant term. So I'll try a plus bt. What I mean by try is put that into the equation, match the left side and the right side, and we have a solution. And I'll just go ahead and do that. So there's several-- this video is mostly about the list of possible nice functions. And that's one of them. OK. So if I put that into the equation, the second derivative is 0 for that. So I have minus 3. The first derivative is just b. Then I have plus 2 times y itself, which is a plus bt. So the left side of the equation is just this much. And it has to equal 4t. And that we can make happen. I see 2bt and 4t here, so I want b equal to 2. And so when b is 2, then I have 4t matching 4t. And then I have minus 3b plus 2a should be 0. Minus 3b plus 2a, the constant part there we don't want, so that should be 0. We already know b is 2, so that's minus 6 plus 2a. a is 3. a is 3. Now, that's perfectly satisfied by b equal 2a equal 3, and that's the answer. Correct answer is b was 2, a was 3, and we don't have to say try anymore. We got it. Done. One other right-hand side. Once you get a nice one like this, you look for more. If it was t squared, we would assume-- what would we assume if it was a 4t squared there? We would assume a plus bt plus ct squared. We want to match the right-hand side. Now, a different type of right-hand side we know already. What if this was e to the-- say, e to the 3t? Or e to the st. Let me put any exponent there for the moment. So now we have a different right-hand side. A very nice function. The best function of differential equations. And now what we will try with this right-hand side, e to the st. We've seen it before. The particular solution is just y e to the st. The undetermined coefficients were a and b in the first time. Here the undetermined coefficient is capital Y. I'm just going to plug that into the equation and match the left side and right side. And I'll determine this coefficient, capital Y. So what happens when I put that into the equation? I get second derivative brings down an s squared, and first derivative, we have a minus 3. The first derivative brings down an s, and I have plus 2. All that is Y e to the st, right? I put in Y e to the st. The derivatives brought down this familiar polynomial. And it's all supposed to match e to the st. We can do that. That's a perfect match. The left side has the same form as the right side. I can cancel the e to the st's, and I discover that Y is 1 over s squared minus 3s plus 2. Good. That's the coefficient, 1 over that. Let's see. I have to make two comments. One comment is that s-- we're totally golden if that s is imaginary. If s is i omega, well, minus i omega, or both, both possibilities, those will give us sine and cosine. So we'll add those to the nice function. So the nice functions are t, polynomials. E to the st, exponentials. Sines and cosines. And you see this worked perfectly. Well, perfectly except we have to be sure that we don't have 1/0. When would we have 1/0? If s is 1, this would be 1 minus 3 plus 2, that would be 1/0. I can't deal with s equal 1 with that assumption. Doesn't work. Also, s equal 2. What's special about s equal 1 and s equal 2? Those make that 0. They give the null solutions. And I know that if this happened to be e to the t, and a null solution was also e to the t, I have to fix the particular solution by giving it an extra factor t. Let me do that at the end of the lecture. That's the resonance idea, where my null solution is the same as what I hope for the particular solution. So I have to change that particular solution with an extra factor t. Let me keep going here to be sure we get all the possibilities. Again, we're getting a very small set of nice functions. But fortunately, they appear quite often in applications. Constants, linear like 4t, exponentials like e to the 5t, oscillation like e to the i omega t. So let me make a list. So polynomials, like the forcing function could be t, could be 1. Constant, that would be like a ramp. That would be like a step function. They could be 3 to the st, but not s equals 1 and 2 in this problem. They could be e to the i omega t, and e to the minus i omega t, which leads me directly to cosine of omega t and sine of omega t. I'm creating here a list of the nice functions-- polynomials, exponentials, cosine and sine, because those come from exponentials, and finally, I can multiply these. I could have, for example, t e to the st. I could allow that. And now, all I have to do is tell you, what form do I try to plug-in? The form will have undetermined coefficients. I'll substitute in the equation, and I'll determine the coefficients. So here we saw a plus bt. That was good. That was the Y. Here we saw Y e to the st. That was good. Here, what will I have? If I have cosine, I need the sine there also. So I'll have to allow a combination of those. Say M on this plus N of that. If I tried to do cosines alone, I would be in danger of taking the derivative, getting a sine, and having nothing to match. So I'll take that. Now, final question, or next question is, if I multiply two of these, the product rule is still going to tell me that derivatives of that have the same form as that. Derivatives of this t e to the st have the same-- they also involve t or e to the st, with factors s, from the product rule. So what do I assume here? Well, when I see t there, I have to include, as I did up there, also constants. And if I saw a t squared, I would go up to t squared. I'd have three coefficients. Now, that e to the st, I can keep. So what I've put on the right-hand side is the right form to assume. It's just like good advice. Put that into the differential equation when this is the right-hand side. Match left side with the right side. That will tell you a and b. And you have the answer. You have the particular solution. You have a particular solution, and that's what you wanted. OK. And if I had t cosine omega t, do you want to see in the whole business? If I had t cosine omega t-- oh dear, it's getting a little messy. I'd need an a plus bt to deal with the t. And I'd need a cosine to deal with the cosine. And then, just to make the problem a little messier, can't be helped, I'd need the sine. So I need a sine omega t. And I'm going to need a c plus dt there. OK. I'm up to four coefficients to determine by plugging in. It uses more ink, but it doesn't use more thinking. You just put it in, match all terms, and you discover A, B, C, and D. Finally, I have to say that word about resonance that I mentioned earlier. The possibility that if s happened to be 1 or 2 on the right-hand side, that's also a null solution. This method would give me 1/0, infinite answer, no good. And we know what to do. We know how to deal with resonance. So with resonance-- so could I just finally complete the whole story. Resonance. In this example, s equal 1 or 2. What would I do with f of t equals e to the t? If that was the right-hand side, that would give me resonance. It's got the exponent s equal 1, which is also in the null solution, e to the t. So what do I try? So I try-- everybody knows what happens when there's resonance. When you have this doubling up. You need an extra factor t to rescue. So you would try y of t. y, this is the particular solution I'm looking for. You would try t e to the t, with a Y. And you would plug that in. You would find the right number for capital Y. And you'd have the particular solution. Only, I think, doing a few examples of this, you get the knack of assume the right form, put it into the equation, match left side with right side, and that reveals the undetermined coefficients. It tells you what a and b and capital Y and c and d, tells you what they all have to be. So this is a really good method that applies to really nice right-hand sides. Good. Thanks. |
MIT_Learn_Differential_Equations | Separable_Equations.txt | GILBERT STRANG: OK. So speaking today about separable equations. These are, in principle, the easiest to solve. They include nonlinear equations but they have a special feature that makes them easy, makes them approachable. And that special feature is that the right hand side of the equation separates into some function of t divided by or multiplied by some function of y. The t and the y have separated on the right hand side. And for example, dy/dt equal y plus t would not be separable. They'd be very simple but not separable. Separable means that we can keep those two separately and do an integral of f and an integral of g and we're in business. OK. Examples. Suppose that f of y is 1. Then we have this simplest differential equation of all, dy/dt is some function of t. That's what calculus is for. y is the integral of g. Suppose there was no t. Just a 1 over f of y, with g of t equal one. Then I bring the f of y up. I integrate the [? f ?] dy. And moving the dt there, I'm just integrating dt. So the right hand side would just be t. And the left hand side is an integral we have to do. That's the minimum amount of work to solve a differential equation. But the point is, with y and t separate, we just have integration to do. And here is the case when there is both a g of t and an f of y. Then let me just emphasize what's happening here. The f of y I am moving up with a dy. The dt I'm moving up with a g of t dt. So I g of t dt equals f of y dy and I integrate both sides. The left side is an integral of y with respect to y. The right hand side is an integral with respect to t or a dummy variable s. The integral going from 0 to t. This integral going from y of 0 to y of t. Those are the two integrations to be done. And you will get examples of seperable equations. And what you have to do is two integrals. And then there's this one little catch at the end. This is some function of y when I integrate. But I usually like to have the solution to a differential equation just y equal something. And you'll see in the examples. I have to solve it for y because this isn't going to give me just y. It's going to give me some expression involving y. So let me do examples. Let me do examples. You see why it's correct. OK. So here are examples. So let me take, what about the equation dy/dt equals t over y. Clearly separable. The function's f. g of t is just t. f of y is just y. I combine those to y dy equalling t dt. You see I've picked a pretty straightforward example. Now I'm integrating both sides from y of 0 to y of t on the left. And from 0 to t on the right. And of course that is 1/2 t squared. And the left hand side is 1/2 y squared between these two limits. So I'm getting the integral of that is 1/2 y squared. So up top I have 1/2 y of t squared minus, at the bottom end, 1/2 y of 0 squared equalling the right hand side 1/2 t squared. So you see, we got a function of y equal to a function of t. And the equation is solved, really. That differential equation is solved. But I haven't found it in the form y of t equal something. But I can do that. I just move this to the other side. So that will go to the other side with a plus. And then I'll cancel the 1/2. And then I'll take the square root. So the solution y of t is the square root of y of 0 squared plus t squared. That's the solution to the differential equation. Maybe I make a small comment on this equation. Because it's essential to begin to look for dangerous points. Singular points where things are not quite right. Here the dangerous point is clearly y equal zero. If I start at y of 0 equals zero then I'm not sure what. What's the solution to that equation if I start at y of 0 equals 0? I'm starting with a 0 over 0. What a way to begin your life, starting with a 0 over 0. This, well, actually the solution would still be correct. If y of 0 is 0, I would get the square root of t squared. I would get t. So y of 0 equals 0 allows the solution y equals t. And that is a solution. That if y is equal to t then dy/dt is 1. And on the right hand side t over y is t over t is 1. So the equation is solved. But my point was, there's got to be something going a little strange when y of 0 is 0. And what happens strangely is there are other solutions. I like, I think, y equal negative t. And more, probably. But if y is equal to negative t, then its derivative is minus 1. And on the right hand side, I have t over negative t minus 1 again. So the equation is solved. That's a perfectly good solution. That's a second solution. It's an equation with more than one solution. And we'll have to think, when can we guarantee there is just one solution, which is of course what we want. OK. I'd better do another example going beyond this. And maybe the logistic equation is a good one. So that's separable. And it's going to be a little harder. So let me do that one. dy/dt is y minus y squared, let's say. The logistic equation. Linear term minus a quadratic term. That's separable because the g of t part is 1. And what's the f of y? Remember f of y-- I want to put that on the y side. But it's going to show up in the denominator. So I have dy over y minus y squared equaling dt. And I have to integrate both sides to get the solution y. Now, integrating the right hand side is of course a picnic. I get t. But integrating the left hand side, I have to either know how, or look up, or figure out the integral of 1 over y minus y squared. So let me just make a little comment about integrating, because examples often have this problem. Integrating when there is a polynomial a quadratic in the denominator. There are different ways to do it. And the time that we'll really see this type of problem is when we discuss Laplace transforms. So I'm going to save the details of the method until then. But let me give the name of the method. The name is partial fractions, which is a method of integration. Partial fractions. And I'll just say here what it means. It means that I want to write this 1 over y minus y squared in a nicer way. What over y minus y squared can be split up into two fractions? Those are the partial fractions. One fraction is-- so I'm going to factor that y minus y squared factors into y and 1 minus y. The partial fractions will be some number over the y and some other number over the 1 minus y. This is just algebra now. Partial fractions is just algebra. It's not calculus. So I factored the y minus y squared into these two terms. You see that if I come to a common denominator, if I put these two fractions together, then the denominator is going to be that. And the numerator, if I choose a and b correctly, will be 1. So, integrating this, I can separately integrate a over y dy and b dy over 1 minus y. And those are easy. So partial fractions, after you go to the effort of finding the fractions, then you have separate integrations that you can do. That integral is just a times the log of y. And this is maybe b times-- maybe it's minus b times the log of 1 minus y. So we've integrated. Just remember though. That with this particular equation, the logistic equation, we didn't have to use partial fractions. We could have done-- we've just seen how, thinking of it as a separable equation. But that logistic equation had the very neat approach. Much quicker, much nicer. We just introduced z equal 1 over y. We looked at the unknown 1 over y, called it z, found the equation for z, and it was linear. And we can write down its solution. So when we can do that it wins. But if we don't see how to do that, partial fractions is the systematic way. One fraction, another fraction. Integrate those fractions. Put the answer together. And then, and then, at the end, this is some integral depending on y equal to t. And to finish the problem perfectly, I would have to solve for y as a function of t. And that was what came out so beautifully by letting 1 over y bz. We got an easy formula for z and then we had the formula for y. This we would integrate easily enough. But then we have to solve to find that formula for y. OK. That's a more serious example. This example was a very simple one. You can do other examples of separable equations. y and t integrated separately. Good. Thank you. |
MIT_Learn_Differential_Equations | The_Big_Picture_of_Linear_Algebra.txt | GILBERT STRANG: I would like you to see the big picture of linear algebra. We're not doing, in this set of videos, a full course on linear algebra. That's already on OpenCourseWare 1806. And now I'm concentrating on differential equations, but you got to see linear algebra this way. And this way means subspaces. And there are four of them in the big picture. And we-- previous video described the column space and the null space. Now, we've got two more, making four. And let me look at this matrix-- it's for subspaces-- and put them into the big picture. So the first space I'll look at is the row space. Now, the row space has these rows-- has the vector 1, 2, 3 and the vector 4, 5, 6, two vectors there, and all their combinations. That's the key idea in linear algebra, linear combinations. So 1, 2, 3 is a vector in three dimensional space. 4, 5, 6 is another one. Now, if I take all their combinations, do you visualize that if I have two vectors, and I add them, and I get another vector that's in the same plane? Or if I subtract them, I'm still in that plane. Or if I take five of one and three of another, I'm still in that plane. And I fill the plane when I take all the combinations. So the row space-- can I try to draw a picture here? It's a plane. This is the row space. I'll just put row. And in that plane are the vectors 1, 2, 3 and the vectors 4, 5, 6, those two rows. And the plane fills our combinations. Well, I can't draw am infinite plane on this MIT blackboard. But you get the idea. It's a plane. And we're sitting in three dimensions. Now, the other-- so there's more. We've only got one-- a plane here, a flat part, like a tabletop, extending to infinity, but not filling 3D because we've got another direction. And in that other direction is the null space. That's the nice thing. So I would like to know the null space of that matrix. I'd like to solve so the null space, N of A-- I'm solving Av equals all 0s. So some combination of those three columns will give me the 0 column. Let me write it in as a 0 column. What could v be? What combination of that column, that column, and that column give 0, 0? Now, I know there are some interesting combinations because I-- only amounts to two equations with three unknowns, v1, v2, v3. I want to multiply that by v1, that by v2, that by v3. So I have three unknowns, but I've only two 0s to get, only two equations. And if I have three unknowns and two equations, there will be lots of solutions. And I can see one. Do you see that if I add that and that, I get 4, 10 And that's the same-- 4, 10 is the same as 2 times 2, 5. In other words, I believe v equal-- if I took 1 of the first and 1 of the third, and if I subtracted 2 of the second column-- so Av will give me 1 of the first column, 1 of the third column, and subtracting 2 of the second column will give me 0, 0. So here's my null space. My null space heads off in this direction, in the direction of 1, minus 2, 1. But, of course, I get more solutions by multiplying v by any number. 10 times that vector would still give me 0s and still be in the null space. So I really have-- the null space is a whole line of vectors. It's that vector and any multiple of that vector. So it's a whole infinite line, which is a one dimensional subspace, the null space. So the null space in my picture-- here is the null space. Well, it is not very thick is it because it's just a line. So I'll call this N of A, this line. Well, you see I'm trying to draw a three dimensional space. That line goes both ways. But it's perpendicular to the plane. That's the fabulous part. That's wonderful. This line, the null space, is perpendicular to this plane, the row space. You want to know why? You want to just see it? Because if I take A times v, that would be 1, 2, 3 times v. 1, 2, 3 is perpendicular to that. How do I check perpendicular for two vectors? 1, 2, 3 dot product. 1, minus 2, 1, the dot product is 1 times 1, minus 2 times 2-- that's 4-- plus 3 times 1, that's 3. 1 minus 4 plus 3 is 0. And, similarly, 4 minus 10 plus 6 is 0. So this is a right angle here. It's a right angle, 90 degrees between those two subspaces. And again, in this example, one space is two dimensional, a plane. The other space is one dimensional, a perpendicular line. I can show with my hands, but I can't draw on this flat blackboard. I have the plane going infinitely far, and I have the line going perpendicular to it and meeting, of course, at 0-- at the 0 vector. That solves Av equals 0, and it also-- it's a combination, a 0 combination of the rows. That's half of the big picture, the row space and the null space. Now, I'm ready for the other half, which is a second side of the other-- the right-hand side of the big picture contains the column space first of all. So what's the column space of that matrix? So the column space of a matrix, we take all combinations of those three columns. And that will fill out a space. Now, I have-- so I take the vector 1, 4. And I take the vector 2, 5, maybe there. And then I'm going to also take the vector 3, 6. Well, I've got three columns. So I'm counting out 3, up 6. Good. Take those combinations of those vectors, and what do you get? This is a picture in two dimensional space because these columns are in two dimensions, 1,4; 2, 5; 3, 6. When I take the combinations of 1, 4 and 2, 5, those are in different directions. The combinations already give me all of two dimensional space, so the column space is the whole space, including 0, 0 because I could take 0 of 1 plus 0 of the other vector. And that third column can't contribute anything new. It's sitting in the column space. It's a combination of those two. But the first two are independent. Their combinations give the whole plane. So the column space is the whole plane. Column space. There's not much room for our fourth subspace. But the fourth subspace, in this example, is quite small. Let me tell you about the fourth subspaces then. So we know the null space, N of A. And we know the column space, C of A. The null space was in this picture. The column space was in that picture. Now, what about-- what's the name for the row space? Well, if I transpose the matrix, the row space turns into the column space. Transpose rows into columns of the matrix A transpose. So by transposing a matrix, it turns these two rows into two columns. And that's what I have here. The row space is the-- this is the column space of the transpose matrix. I like it. I don't want to introduce a new letter for the row space. I like having just column space and null space. So I-- and I'm OK to go to A transpose. Now, what's that fourth guy? Oh, just by beautifulness, general principles of elegance here. If I have columns space and null space of A, and if I have column space of A transpose, the fourth guy has to be the null space of A transpose. Sorry, I wrote that so small, so small. But I did write this a little larger. The null space of A transpose, all the w's that solve that equation. A transpose w equals 0. The null space of A transpose is all the w's that solve that equation. What does that equation looks like? Ha! Well, that equation-- A transpose will have two columns. So A transpose-- this will be w1 of the first column. 1, 2, 3, when I transpose. And w2 of the second column, 4, 5, 6 equaling 0, 0, 0. Well, now I've got, for this null space-- because my matrix here is 2 by 3, for this fourth subspace, I have three equations and only two unknowns, w1 and w2. And, in fact, the only solutions are w1 equals 0, w2 equals 0, because that's the only way I can get combination-- that's the only combination of that vector and that vector that gives me 0 is to take 0 of that and zero of that. Do you see that the-- in this example, the null space of A transpose is just-- null space of A transpose-- is just what I call the 0 subspace. The subspace that has only one puny vector in it, the 0 vector. But that's OK. It follows the rule for subspaces. And it completes the picture of four subspaces. In other examples, we could have all four subspaces nonzero. But we would have two over here that, together, complete the full N dimensional space. And over here, we have two that together complete the full M dimensional space. And here, for this matrix, M was 2, so that this is completed. The column space was all of R2 in this case. All of two dimensional space was the column space, and that didn't leave any room for the left null space, the null space of A transpose. So do you see that picture? Let we may be just sketch it once more with a clean board. So I have the row space. Let me draw it maybe going this way, the row space. And perpendicular to that is the null space. That's in-- we're here in N dimensions, and they are perpendicular, those spaces. And then, over here, I have the column space. And perpendicular to that is the left null space. And we're here in M dimensions. Those are our four subspaces. And they have-- they sit in N dimensional space, two of them, two of them in M dimensional space, perpendicular. And I could tell you something about their dimensions. So this row space, in that example, was two dimensional. It was a plane. In general, the dimension equals-- let's say R. That's an important number, the rank of A. Oh, that's a key number. Maybe, I better speak separately about the rank of a matrix. But I'll complete the idea here. So the dimension of the row space is the number of independent rows. And I call that number R. And the beauty is that this has the same dimension. Dimension Is also the rank R. Can I say that wonderful fact in a sentence? The column space and the row space have the same dimension. The number of independent rows equals the number of independent columns. That's like a miracle for a giant matrix, say 57 by 212, there might be 40 independent rows. Then, there would be 40 independent columns. And then the null space and the left null space have the remaining dimension. So the null space has dimension N minus R because, altogether-- together they have dimension N. And this has dimension M minus R because, together, they have dimension M. That's the picture with the dimensions put in. And let me say a little more about the idea of dimension in a separate video. Thank you. |
MIT_Learn_Differential_Equations | Independence_Basis_and_Dimension.txt | PROFESSOR: So as long as I'm introducing the idea of a vector space, I better introduce the things that go with it. The idea of its dimension and, all important, the idea of a basis for that space. That space could be all of three dimensional space, the space we live in. In that, case the dimension is three, but what's the meaning of a basis-- a basis for three dimensional space. Or a basis for other spaces. OK, so I have to explain independence, basis, and dimension. Dimension's easy if you get the first two. OK, independence. Are those vectors independent? Well, if I draw them, in three dimensional space, I can imagine 2, 1, 5 going in some direction. Let me draw it. How's that? 2, 1, 5, whatever! Goes there. That's a1. OK. Now is a2 on the same line? If a2 is on the same line then it would be dependent. The two vectors would be dependent if they're on the same line. But this one is not on that line. A 4, 2, 0. So it doesn't go up and all. It's somewhere in this plane, 4, 2, 0. I'll say there. Whatever. a2. So those are independent. So their combinations give me a space. The combinations of a1 and a2 give me a plane, a flat plane, in three dimensional space. That plane is, I would say, they span the plane. a1 and a2 span a plane. And here's the key word: span. So there are two vectors. They're in three dimensional space. And the plane they span is all their combinations. That's what we're always doing: taking all the combinations of these vectors. OK. So there-- and actually, a1 and a2 are a basis for that pane. a1 and a2 are a basis for that plane because their combinations fill the plane. And also, they're independent. I need them both. If I threw away one, I would only have one vector left, and it would only span a line. OK. Now let me bring in a third vector in three dimensions. Well, what shall I take for that third vector? Ha! Suppose I take a1 plus a2 as my third vector. So 6, 3, 5. What about the vector 6, 3, 5? Well, what do I know? It's obviously special. It's a1 plus a2. It's in the same plane. So if I took a3 equal 6, 3, 5, that would be dependent. The three vectors would be dependent with that a3. They would span the plane still. Their combinations would still give the plane, but they wouldn't be a basis for the plane. a1 and 12 and a3 together, that's too much, too many vectors for a single plane. The vectors are dependent. And we don't-- a basis has to be independent vectors. You have to need them all. We don't need all three here. So that's a dependent one. It can't go into a basis with a1 and a2 because the three vectors are dependent. Now let me make a difference choice. So that one's dead. That did not do it. All right. Let me take a3 equal to some other, not a combination of these, but headed off in some new direction. Well, I don't know what that new direction is. Maybe 1, 0, 0. What the heck? I believe-- I hope I'm right-- that 1, 0, 0 is not a combination here. I say 1, 0, 0 goes off. It's pretty short. Here's a3. Better a3 then that loser 6, 3, 5. 1 0, 0 is a winner. These three vectors-- So now a1, a2, and let me add in a3, all three of them span a-- what do they span? What are all the combinations of a1, a2, a3? It's three dimensional? It's the whole three dimensional space. They span all of 3D, the whole three dimensional space. They're a basis for the whole three dimensional space. They're independent. So let me-- you see that picture before I move it? a1, a2, a3 are independent. None of them is a combination of the others. They fill a three dimensional space. They're are a basis for that three dimensional space. And that space is, in this example, is the whole of R^3. So let me just write down on the next blackboard what I mean. Independent. Independent. So independent columns of a matrix. Independent columns of a matrix A means the only solution to Av equals 0 is v equals 0. So if I have independent columns, then I haven't got any null space. If I have independent columns, then the null space of the matrix is just the 0 vector. So let me write down that example again. A was the matrix 2, 1, 5, 4, 2, 0, 1, 0, 0. So I believe that matrix has independent columns. So its column space is the full three dimensional space. It's null space only contains-- let me put it, make that clear that that's a vector. And now I'm ready to write down the idea of a basis. So what is a basis for the space? A basis for a space, a subspace. Independent vectors. That's the key. Independent vectors that span the space, the subspace. Whatever it is. By the way, if the column space is all a three dimensional space, as it is here, that's a subspace too. It's the whole space, but the whole space counts as a subspace of itself. And the 0 vector alone counts as the smallest possible. So if we're in three dimensions, the idea of subspaces has-- we have just the 0 vector. Just one point. That's a smallest. We have the whole three dimensional space. That's the biggest. And then we have all the lines through 0. Those are on the small side. We have all the planes through 0. Those are a bit bigger. And those dimensions are 0, 1, 2, 3. The possible dimensions is told to us by how many basis vectors we need. So let me look at that and then come to dimension. OK. So independent means that the only-- that no combination, no other combination of the vectors, no combination of these vectors gives the 0 vector except to take 0 of that, 0 of that, and 0 of that. So those are a basis for the column space because they're independent and their combinations give the whole column space. OK. And now I wanted to say something about dimensions. OK. Dimension. It's a number. It's the number of basis vectors for the subspace. Oh! But you might say, that the subspace has other bases, not just the one you happen to think of first. And I agree. Many different bases. For this example, all I need to get a basis for, in this case, for three dimensional space is I need three independent vectors. Any three. But the point is, the point about dimension is that I need exactly three. I can never get two vectors that span all of R^3. And I can never get four vectors that are independent in R^3. If I have fewer than the dimension number, I don't have enough. They don't span. If I have too many, than the dimension, they're dependent. They won't be independent. They can't be a basis. Every basis has the same number. And that number is the dimension of the subspace. All right, let's just take an example, just with a picture. I'll stay in three dimensional space, but my subspace will just be a plane. So here I'm in three dimensional space. Good. Now I have my subspace is a plane. So it goes through the origin, but it's only a plane. So I'm expecting that I could take a vector in the plane, and I could take another vector in the plane, and they could be independent. They are. They're different directions. I couldn't find a third independent vector in the plane. Every basis for the plane-- So here every basis for this plane contains two vectors. Always two. And that number two is the dimension of a plane. Well, I'm just saying the plane there is two dimensional. It's not the same as r2. it's not the same. That plane is a plane in r3. It's not ordinary two dimensional space. But its dimension is two because it takes any vector. And if I didn't like the looks of this one, well, that's no problem. Let me go that way. That's just as good. Those two vectors are independent. They span the plane. They're a basis for the plane. The plane is two dimensional. That's the set of key ideas. Independent. Span. Basis. Basis is fundamental. Basis is a bunch of vectors. And dimension is how many vectors. OK. Those are key ideas in linear algebra. And you'll see them come into the big picture of linear algebra. Thank you. |
MIT_Learn_Differential_Equations | Integrating_Factor_for_a_Varying_Rate.txt | GILBERT STRANG: OK? I want to talk about a slightly different way to solve a linear first-order equation. And if you look at the equation-- I'll do an example. That's the best. Do you notice what's different from our favorite equation? The change is 2t. The interest rate a is increasing with time, changing with time. So we still have a linear equation, still just y. But the coefficient is varying. We have a variable coefficient 2t. And if we think here of applications to economy, to banks, that would be rampant inflation, the interest rate 2t climbing and climbing forever. But we want to see that this is a class of equations that we can solve. OK. And the new method is called an integrating factor. It's a magic factor that makes the equation simple. So that's another nice way to solve all the problems that we've dealt with so far, plus this new one. So what is this factor? Well, for this 2t problem, the right factor is e to the minus t squared. And why is that the right factor? This is the factor that I'm going to multiply the equation by and make it simple. And the reason that's the right choice is that the derivative of this-- you remember how to take the derivative by the chain rule? The derivative will be the same e to the minus t squared, the same I, times the derivative of the exponent. And the derivative of that exponent is minus 2t. Minus t squared becomes minus 2t. So it's that little device that gives us an integrating factor that makes the equation simple. And now I'm going to look at the equation. What I want to look at is the derivative of I times y. Instead of just dy dt, let me look at the derivative of I times y. So I have a product here. Got to use the product rule. So that will be I-- so I dy dt. OK. dy dt is-- we can take dy dt from the equation, I times 2ty plus q of t. And now I have to add on dI dt y. Good. So it's the product rule-- I times the derivative of y plus the derivative of I times y. But now look. dI dt we know is minus 2tI. So that dI dt, now I'm using the key fact about I, that that's minus 2tIy. Look, minus 2tIy cancels 2tIy. So I have a nice equation now. The derivative of Iy is Iq. The derivative of Iy is Iq. I can just integrate both sides. And that's the key. That's the key. If I integrate the left-hand side-- so I'll just move this up-- integrate the derivative-- of course, the integral of the derivative is the function-- at time t, Iy at time t minus Iy at time 0, y of 0. Because notice that I at t equals 0-- can I just mention that-- I at 0 is 1. When t is 0, I of e to the 0 power, which is 1. So that I of 0 is 1. So that's the integral of the derivative. And on the right-hand side, I have the integral from 0 to t of I times q. So I'll put in-- yeah, e to the minus s squared q of s ds. I've introduced a variable of integration s going from 0 to t. You remember this type of formula? The input is continuous over time, and I'm looking at the resulting output at time t. So all the inputs go in. They're all multiplied by some factor and integrated gives the total result from those inputs. OK. I'm almost here. I just want to remember I want to divide by I of t so I have a formula for y. OK. So my formula for y. When I divide by I of t-- don't forget what I of t is. Let me put it again here. Let me remind myself. I of t is e to the minus t squared. That was the magic integrating factor. OK. So I'm going to divide by that, which means I'll multiply by e to the t squared. So that will knock out the I here. I'll put this on the other side of the equation, y of 0, y of 0, and it will be multiplied by the e to the t squared. And this thing will be multiplied by e to the t squared. The integral from 0 to t of e to the t squared minus s squared q of s ds. That's my answer. Well, let's look at it. I have y of t. This is what comes out of y of 0. You see that the growth factor has changed from our old e to the at-- that was the growth at constant rate, interest rate a-- to e to the t squared. That's our growth from an increasing interest rate. And over here, I'm seeing the result, the output, from the input q, from all the inputs between 0 and t. Each input is multiplied by now that factor is the growth not from 0 to t. This is the growth from 0 to t. This is the growth from s to t, because the input went in at time s, and it had the shorter time, t minus s, to grow. So that's the formula for the answer. If you give me any particular q of s, I just do the integral, and I find the solution to the differential equation. So that integrating factor has made things work. Maybe I should say what the integrating factor would be in general. So let me take a moment to see-- this was an example. This was an example with a of t equal to 2t. What's the general integrating factor? So we always want the integrating factor. Our construction rule is that it should give us-- the derivative should be minus a of t times I itself. That's how we chose the e to the minus t squared. Then a of t was 2t. Now I want to give the general rule. The general rule for the integrating factor is the solution to that equation. The solution to that equation is giving us the e to the t squared in the example. This was the example. But now I want a formula just to close off the entire case of varying interest rate. I want to find the solution to that equation. And it is-- so here's the integrating factor. It's e to the minus because of that minus sign. Now I'm wanting a to come down when I take a derivative. So what I'll put up here, the integral of a of t dt, say, from 0 to t. Now, let me just do again this example just to see. I have e to the minus the integral of 2t, which is e to minus t squared. That's how I get t squared as the right choice for our example. And the general rule is there. That's the integrating factor. And finally, finally, if a is a constant, which is the most common case-- the only case we've had until this video-- if a is a constant, then the integral of a from 0 to t is just a times t. So number one example, number zero example, would be e to the minus at. That would be the correct integrating factor if we had constant a. And I'll create some examples, some problems, just to go through the steps in that best case of all with constant integrating factor. But now we can solve it with a varying interest rate. Good. Thank you. |
MIT_Learn_Differential_Equations | Second_Order_Equations_with_Damping.txt | GILBERT STRANG: I'm coming back to the number one example, but not the easiest example, of a second order equation with an oscillating forcing term, cosine omega t. We have to know the answer to this problem. And it's a little messy, but the method is not messy. The method is straightforward. So let me begin by looking for the rectangular form. I call this the rectangular form. It separates the cosine with its amplitude and the sine with its amplitude into two separate pieces. So if I'm looking for that solution, and m and n are the numbers I want to find, how do I proceed? It's a case of undetermined coefficients, M and N. And the way to determine them is substitute this into the equation and match the cosine term and find M and N. And the way we find M and N, we need two equations for two quantities, M and N. And imagine this substituted in there. I'll get some cosines. So the cosines on one side will match the cosine on the other side. And also from the derivative, I'll get some sines and they should match 0 because I have no sine omega t on the right hand side. So I have two equations, matching the sines, matching the cosines. And I solve those. Two equations, two unknowns. And I just write the answer down. M involves a C minus omega squared. M is coming from the cosines. And we get cosines from that term and that term. Divided by some number, D, that I'll write down. And N is just B omega divided by that same D. And now I'll write down D. That's C minus A omega squared squared plus B omega squared. This is what comes out from the two equations for M and N. I just solve those equations. This D here is the two by two determinant if we think about the linear algebra behind two equations. And that's what it is. And so the answer now is in terms of A, C, B, and D, which is a mixture of all of A, B, and C. That's the solution. Only I always want to show you a different form of the solution. And in this case, a better form. Because the most important physical quantity is the magnitude. How large does y get? What is the amplitude of this? This is a sinusoid. And we remember that every sinusoid can be written in a polar form. Says that y of t is some amplitude of G, the gain, times a cosine of omega t with a shift, with a lag, with an angle alpha. So I have two numbers now. That's the gain. And this is the phase shift alpha. And that's an attractive form because it has only one term. The two numbers, G and alpha, get put into a single term where we can see the magnitude of the oscillation. And what does that come out to be? I won't go through all the steps. I'll just write down what G turns out to be. G turns out to be-- it comes from there-- and it's 1 over the square root of D. Well, G is the square root of M squared plus N squared. The square root of M squared plus N squared. And if I put M squared and N squared, then I have D over D squared. I get that answer. That's the gain. Let me write that word, gain, again. Because you got it there. Here it is again. And as always, the tangent of alpha is the N over the M, which is just B omega over C minus A omega squared. I like that polar form. And I feel I should just do an example. I didn't do any of the algebra in this video. But you know where the algebra came from. It came from substituting the form we expect for the solution. And of course, that form that we expect is the form we get provided omega, the driving frequency, is different from omega N. Well, no. I guess we're all right even if omega is omega N, because we have a damping term. So that's the answer. So an example. Why not an example? y double prime plus y prime plus 2y equals cosine of t. That's a simple example. I took omega to be 1, you see. And there is omega. And then A is 1, B is 1, C is 2. We can evaluate everything. In fact, I think M and N are 1/2. D, by the way, will be 1 squared plus 1 squared. That's 2 square root. Sorry. D will be 2. 1 squared plus 1 squared. So what do I know? Do I know the rectangular form? Yes. Rectangular form is 1/2. 1/2 for both the cosine and the sine. 1/2 of cosine t plus sine t. That's the rectangular form. Two simple things, but I have to add them. And in my mind, I don't necessarily see how the cosine adds to the sine. But the sinusoidal identity, the polar form, gives it to me. So what is it in polar form? So G, the gain, is going to be 1 over the square root of 2. At the highest point, the cosine and the sine are the same. They're both 1 over the square root of 2. I have two of them. So I get 1 over the square root of 2 cosine of t minus pi over 4 is the angle, the phase lag. When I add the cosine and the sine, I get a sinusoid that's sitting over pi over 4, 45 degrees. So those are the two forms. So in a nice example, we certainly got a nice answer. We certainly did. Yes. So that is the-- worked out, more or less worked out, in principle, worked out-- is the solution to what I think of as the most important application when the forcing term is a cosine. So it gives oscillating motion. It gives a phase shift. And it gives these formulas. The only thing I would add is that I need to comment on better notation. So I have used in these formulas A, B, and C. But those have meaning as mass, damping constant, spring constant. M, B, and K. And it's combinations of those that come in. So let me just take this moment to say better notation. Or maybe I should say engineering notation instead of A, B, C, which are mass, damping, spring constant. Well, that's already better to use letters that have a meaning. But the small but very important point is that two combinations of A, B, C, M, B, K are especially good. One is the natural frequency that we've seen already, square root of C over A. Square root of K over M. So that gives us one important combination of A and C. And the other one is the damping ratio. And it's called zeta. And that damping ratio is B over the square root of 4ac. Ha! You'll say, where does that come from? Or I can use these letters, B over the square root of 4mk. That damping ratio is, so to speak, it's the right dimensionless quantity. The dimensions of this ratio are just numbers. Those two quantities have the same dimension. And we can see that because in the quadratic formula comes-- you remember that in a quadratic formula comes the square root of b squared minus 4ac? Now if you see a formula that has b squared minus 4ac in it, you know that these must have the same units. Otherwise, subtraction would be a crime. So they have the same ratio and the same units and therefore the ratio is dimensionless. Let me write that word. Dimensionless. So conclusion. I could rewrite the answer in terms of these quantities omega n and zeta. I won't do that here. That can wait for another time. But just to say since we've found a solution to the most important application with cosine omega t there, since we found the solution, appropriate to comment that we could write the answer in terms of omega n, the natural frequency, and z, zeta, the damping ratio. Thank you. |
MIT_Learn_Differential_Equations | Forced_Harmonic_Motion.txt | GILBERT STRONG: This is the second video on second order differential equations, constant coefficients, but now we have a right hand side. And the first one was free harmonic motion with a zero, but now I'm making this motion, I'm pushing this motion, but at a frequency omega. This is my forcing term. So I think I'm having a forcing frequency, omega, and remember that for this one, for the no solution, there was a natural frequency omega n. It's very important are those close, are those well separated? That governs whether the bridge that you're walking over oscillates too much and eventually falls. Or in the extreme case, are they equal? If omega n is equal to omega that's called resonance. Let me put that word in. Resonance. When omega equals omega n. And we're not going to deal with today, but you should know that always the formula has an omega minus omega n dividing by that. So if that is 0, if omega equals omega n our formula has to change. Today, this won't happen. No. So what's the formula? What is yp? I'm looking for a particular solution. That's a nice function and also important in practice. So I would like to hope that the particular solution could be some multiple of that cosine omega t. And in this problem that's possible. Because if I have a cosine, I've got a cosine on the right hand side, and if that cosine comes here, it's on the left side, and the second derivative of the cosine is, again, a cosine, I'm going to have a match of cosine omega t terms. And then I'll just choose the right number capital Y. I won't be able to do that when there's a first derivative in there, because the first derivative of cosine will bring in signs. I'll have a mixture of cosines and sines and then I better allow for that mixture. But here I don't have to. There's the forcing function. Response, this is the forced response. I'd like to get used to that word, response, for the solution. Here's the input, the response is the output. So let me just plug that into the equation and find capital Y. So here I have m, second derivative is going to be a Y, and second derivative will bring out a minus omega squared times the cosine. And here I have kY is Y times the cosine equal the cosine. I could have a constant there, but the whole thing would be no more interesting, no more difficult than with a 1. So what do I do? The nice thing is here I have all cosines, so I'm just going to have minus omega squared m and a k. So it's k minus m omega squared. Can I write it that way? Times Y. I'm going to cancel the cosines. That's just a 1. On the side is a 1. I've canceled the cosine, so I've kept kY. I've kept the 1, and I've kept a minus omega squared mY. So that tells me Y right away. It's just like plugging in an exponential and canceling exponentials all the way along. Here, I'm canceling cosines all the way because every term was a cosine. So I know Y. So I know the answer. So the final answer is Y(t) is Yn. Well, let me put Y particular first plus Yn. So I've just found Y particular. Y particular is this capital Y cosine omega t. So it's cosine omega t times Y and Y is 1 over this. Here's goes Y. Down below I have k minus m omega squared. Right? That's what we just found, that particular solution. The capital Y, the multiplying constant, was 1 over that constant. And now comes the C1 cosine of omega nt and the C2 sine of omega nt. Remember, omega n is different from omega. Actually, this is pretty nice here. I could write that another way so you would see the important here. So remember, what is omega n squared? Can I just remember that omega n squared is k over m. Right? Yup. k is the same as-- I'm going to put that m up here-- k is the same as m omega n squared. k is the same as m omega n squared and here I'm subtracting m omega squared. You'll see the whole point of resonance or near resonance when the bridge is getting forced buy a frequency close to its resonant frequency. This difference, omega squared, the difference between the two frequencies squared is in the denominator and will be small and then the effect is large. And if we get those too close, the effect is too large. So we'll see this cosine omega t over this is, I would call, the frequency response is this factor. 1 over m omega n squared minus omega squared. That's the key multiplier for when the forcing term is a pure frequency, that frequency gets exploded. And now, of course, what are capital C1 and capital C2? We find those from the initial condition. At t equals 0, we put in t equals 0, and that tells us what C1 has to be. And we put in t equals 0 again to match the velocity Y prime at 0, and that tells us C2. Are you OK with that? Just look at the beauty of that solution. This is null part. This is the forced part, the particular part, the cosine divided by that constant. There's one more equation, one more forcing term I'd like often and always and now to discuss. And that is a delta function, an impulse. So I'm going to add one more example. my double prime plus ky equal the delta function. Delta function. It's called an impulse. So I'd like to solve that equation also. When the forcing term just happens at one second, at the initial second. At t equals 0, the delta function, I'm hitting the spring. So the spring is sitting or the pendulum is sitting there. Actually, let's set it at rest. Here's my pendulum. I'll try to draw a pendulum. I don't know. That's not much of a pendulum. But it's good enough. This equation says what happens if I hit it with a point source? At t equals 0, I hit it but I give it a finite velocity. It doesn't move in that instant second. This is where delta functions come in so let me give you the result of what happens and then we'll see them again. So what am I doing? I want to solve this equation when the forcing function is a delta function. So I'm going to call y the impulse response. It's the solution that comes when the forcing function is an impulse. So y is the impulse response. In fact, it's so important, I'm going to give it its own letter. g. Now, can I turn that y into a g? So that g is g of t is the impulse response. If I can solve that equation. You might say, not so easy. With a delta function, it's not even a genuine function. It's a little bit crazy. It all happens in one second. I'm sorry, in one instant. Not over one second, but one moment. But I can solve it. I can solve it for this reason. I can think of it as an impulse here or I have an option, another way which clearly I can think of it as solving it with no force mg double prime. Same problem, same solution is 0. But I start from rest. Nothing's happening. y of 0 is 0. And it starts from an initial velocity, y prime of 0. The impulse starts it out like a golf ball. Just go. And there's a 1 over m there. I'll discuss that another time. What I want to see now is that I have either this somewhat mysterious equation or this totally normal equation, even a no equation starting from y of 0 equals 0. But with an initial velocity that the impulse gave to the system. And I should be calling this g. This is the g. We'll see impulse responses again, but let's see it this time by solving this equation. So I plan to solve that equation and actually we solved it last time. You remember the solution to this one? When it starts from 0, there's no cosine. But when the initial velocity is 1 over m, there is a sign. So I'm going to just write down the g of t, which is just sine of omega nt. And why is it the natural frequency? Because I'm solving the no. I'm looking for a no solution. But the previous video on no solutions gets me this. Only I have to divide by, get 1 over m as the initial velocity. You'll see that that will solve the no equation. This is what happens to the pendulum or the golf ball. Well, pendulum much better. Actually, golf ball is poor example. Sorry about that. Golf balls don't swing back and forth. They tend to go. I'm looking at pendulums, springs going up and down. So the spring starts out, has an initial velocity of 1 over m and then after that nothing happens. So that is the impulse response. The response to an impulse. And why do I like that? First of all, its beautiful. Simple answer. Secondly, every forcing function, and the output comes from this one. We'll see that point. So we've introduced forcing functions, cos omega t, where the particular solution was a multiple of cos omega t. And now, we've introduced a forcing function delta, the delta function where the response is a sine function. Thank you. |
MIT_Learn_Differential_Equations | Eigenvalues_and_Eigenvectors.txt | GILBERT STRANG: So today begins eigenvalues and eigenvectors. And the reason we want those, need those is to solve systems of linear equations. Systems meaning more than one equation, n equations. n equal 2 in the examples here. So eigenvalue is a number, eigenvector is a vector. They're both hiding in the matrix. Once we find them, we can use them. Let me show you the reason eigenvalues were created, invented, discovered was solving differential equations, which is our purpose. So why is now a vector-- so this is a system of equations. I'll do an example in a minute. A is a matrix. So we have n equations, n components of y. And A is an n by n matrix, n rows, n columns. Good. And now I can tell you right away where eigenvalues and eigenvectors pay off. They come into the solution. We look for solutions of that kind. When we had one equation, we looked for solutions just e to the st, and we found that number s. Now we have e to the lambda t-- we changed s to lambda, no problem-- but multiplied by a vector because our unknown is a vector. This is a vector, but that does not depend on time. That's the beauty of it. All the time dependence is in the exponential, as always. And x is just multiples of that exponential, as you'll see. So I look for solutions like that. I plug that into the differential equation and what happens? So here's my equation. I'm plugging in what e to the lambda tx, that's y. That's A times y there. Now, the derivative of y, the time derivative, brings down a lambda. To get the derivative I include the lambda. So do you see that substituting into the equation with this nice notation is just this has to be true. My equation changed to that form. OK Now I cancel either the lambda t, just the way I was always canceling e to the st. So I cancel e to the lambda t because it's never zero. And I have the big equation, Ax, the matrix times my eigenvector, is equal to lambda x-- the number, the eigenvalue, times the eigenvector. Not linear, notice. Two unknowns here that are multiplied. A number, lambda, times a vector, x. So what am I looking for? I'm looking for vectors x, the eigenvectors, so that multiplying by A-- multiplying A times x gives a number times x. It's in the same direction as x just the length is changed. Well, if lambda was 1, I would have Ax equal x. That's allowed. If lambda is 0, I would have Ax equals 0. That's all right. I don't want x to be 0. That's useless. That's no help to know that 0 is a solution. So x should be not 0. Lambda can be any number. It can be real, it could be complex number, as you will see. Even if the matrix is real, lambda could be complex. Anyway, Ax equal lambda x. That's the big equation. It got a box around it. So now I'm ready to do an example. And in this example, first of all, I'm going to spot the eigenvalues and eigenvectors without a system, just go for it in the 2 by 2 case. So I'll give a 2 by 2 matrix A. We'll find the lambdas and the x's, and then we'll have the solution to the system of differential equations. Good. There's the system. There's the first equation for y1-- prime meaning derivative, d by dt, time derivative-- is linear, a constant coefficient. Second one, linear, constant coefficient, 3 and 3. Those numbers, 5, 1, 3, 3, go into the matrix. Then that problem is exactly y prime, the vector, derivative of the vector, equal A times y. That's my problem. Now eigenvalues and eigenvectors will solve it. So I just look at that matrix. Matrix question. What are the eigenvalues, what are the eigenvectors of that matrix? And remember, I want Ax equals lambda x. I've spotted the first eigenvector. 1, 1. We could just check does it work. If I multiply A by that eigenvector, 1, 1, do you see what happens when I multiply by 1? That gives me a 6. That gives me a 6. So A times that vector is 6, 6. And that is 6 times 1, 1. So there you go. Found the first eigenvalue. If I multiply A by x, I get 6 by x. I get the vector 6, 6. Now, the second one. Again, I've worked in advance, produced this eigenvector, and I think it's 1 minus 3. So let's multiply by A. Try the second eigenvector. I should call this first one maybe x1 and lambda 1. And I should call this one x2 and lambda 2. And we can find out what lambda 2 is, once I find the eigenvectors of course. I just do A times x to recognize the lambda, the eigenvalue. So 5, 1 times this is 5 minus 3 is a 2. It's a 2. So here I got a 2. And from 3, 3 it's 3 minus 9 is minus 6. That's what I got for Ax. There was the x. When I did the multiplication, Ax came out to be 2 minus 6. Good. That output is two times the input. The eigenvalue is 2. Right? I'm looking for inputs, the eigenvector, so that the output is a number times that eigenvector, and that number is lambda, the eigenvalue. So I've now found the two. And I expect two for a 2 by 2 matrix. You will soon see why I expect two eigenvalues, and each eigenvalue should have an eigenvector. So here they are for this matrix. So I've got the answers now. y of t, which stands for y1 and y2 of t. Those are-- it's e to the lambda tx. Remember, that's the picture that we're looking for. So the first one is e to the 6t times x, which is 1, 1. If I put that into the equation, it will solve the equation. Also, I have another one. e to the lambda 2 was 2t. e to the lambda t times its eigenvector, 1 minus 3. That's a solution also. One solution, another solution. And what do I do with linear equations? I take combinations. Any number c1 of that, plus any number c2 of that is still a solution. That's superposition, adding solutions to linear equations. These are null equations. There's no force term in these equations. I'm not dealing with a force term. I'm looking for the null solutions, the solutions of the equations themselves. And there I have two solutions, two coefficients to choose. How do I choose them? Of course, I match the initial condition, so at t equals 0. At t equals 0. At t equals 0, I would have y of 0. That's my given initial condition, my y1 and y2. So I'm setting t equals 0, so that's one of course. When t is 0, that's one. So I just have c1 times 1, 1. And c2-- that's one again at t equals o-- times 1 minus 3. That's what determines c1 and c2. c1 and c2 come from the initial conditions just the way they always did. So I'm solving two first order linear constant coefficient equations, homogeneous, meaning no force term. So I get a null solution with constants to choose and, as always, those constants come from matching the initial conditions. So the initial condition here is a vector. So if, for example, y of 0 was 2 minus 2, then I would want one of those and one of those. OK. I've used eigenvalues and eigenvectors to solve a linear system, their first and primary purpose. OK. But how do I find those eigenvalues and eigenvectors? What about other properties? What's going on with eigenvalues and eigenvectors? May I begin on this just a couple more minutes about eigenvalues and eigenvectors? Basic facts and then I'll come next video of how to find them. OK, basic facts. Basic facts. So start from Ax equals lambda x. Let's suppose we found those. Could you tell me the eigenvalues and eigenvectors of A squared? I would like to know what the eigenvalues and eigenvectors of A squared are. Are they connected with these? So suppose I know the x and I know the lambda for A. What about for A squared? Well, the good thing is that the eigenvectors are the same for A squared. So let me show you. I say that same x, so this is the same x, same vector, same eigenvector. The eigenvalue would be different, of course, for A squared, but the eigenvector is the same. And let's see what happens for A squared. So that's A times Ax, right? One A, another Ax. But Ax is lambda x. Are you good with that? That's just A times Ax. So that's OK. Now lambda is a number. I like to bring it out front where I can see it. So I didn't do anything there. This number lambda was multiplying everything so I put it in front. Now Ax. I have, again, the Ax. That's, again, the lambda x because I'm looking at the same x. Same x, so I get the same lambda. So that's a lambda x, another lambda. I have lambda squared x. That's what I wanted. A squared x is lambda squared x. Conclusion. The eigenvectors stay the same, lambda goes to lambda squared. The eigenvalues are squared. So if I had my example again-- oh, let me find that matrix. Suppose I had that same matrix and I was interested in A squared, then the eigenvalues would be 36 and 4, the squares. I suppose I'm looking at the n-th power of a matrix. You may say why look at the n-th power? But there are many examples to look at the n-th power of a matrix, the thousandth power. So let's just write down the conclusion. Same reasoning, A to the n-th x is lambda. It's the same x. And every time I multiply by A, I multiply by a lambda. So I get lambda n times. So there is the handy rule. And that really tells us something about what eigenvalues are good for. Eigenvalues are good for things that move in time. Differential equations, that is really moving in time. n equal 1 is this first time, or n equals 0 is the start. Take one step to n equal 1, take another step to n equal 2. Keep going. Every time step brings a multiplication by lambda. So that is a very useful rule. Another handy rule is what about A plus the identity? Suppose I add the identity matrix to my original matrix. What happens to the eigenvalues? What happens to the eigenvectors? Basic question. Or I could multiply a constant times the identity, 2 times the identity, 7 times the identity. And I want to know what about its eigenvectors. And the answer is same, same x's. Same x. I show that by figuring out what I have here. This is Ax, which is lambda x. And this is c times the identity times x. The identity doesn't do anything so that's just cx. So what do I have now? I've seen that the eigenvalue is lambda plus c. So there is the eigenvalues. I think about this as shifting A by a multiple of the identity. Shifting A, adding 5 times the identity to it. If I add 5 times the identity to any matrix, the eigenvalues of that matrix go up by 5. And the eigenvectors stay the same. So as long as I keep working with that one matrix A. Taking powers, adding multiples of the identity, later taking exponentials, whatever I do I keep the same eigenvectors and everything is easy. If I had two matrices, A and B, with different eigenvectors, then I don't know what the eigenvectors of A plus B would be. I don't know those. I can't tell the eigenvectors of A times B because A has its own little eigenvectors and B has its eigenvectors. Unless they're the same, I can't easily combine A and B. But as always I'm staying with one A and its powers and steps like that, no problem. OK. I'll stop there for a first look at eigenvalues and eigenvectors. |
MIT_Learn_Differential_Equations | Stiffness_ODE23s_ODE15s.txt | INSTRUCTOR: I want to illustrate the important notion of stiffness by running ode45, the primary MATLAB ODE solver, on our flame example. The differential equation is y prime is y squared minus y cubed, and I'm going to choose a fairly-- an extremely small initial condition, 10 to the minus sixth. The final value of t is 2 over y naught, and I'm going to impose a modest accuracy requirement, 10 to the minus fifth. Now let's run ode45 with its default output. Now, see it's taking-- it's moving very slowly here. It's taking lots of steps. So I'm take- pressing the stop button here. It's working very hard. Let's zoom in and see why it's taking so many steps, very densely packed steps here. This is stiffness. It's satisfying the accuracy requirements we imposed. All these steps are within 10 to the minus sixth of one, but it's taken very small steps to do it. These steps are so small that the graphics can't even discern the step size. This is stiffness. It's an efficiency issue. It's doing what we asked for. It's meeting the accuracy requirements, but it's having to take very small steps to do it. Let's try another ODE solver-- ode23. Just change this to 23 and see what it does. It's also taking very small steps for the same reason. If we zoom in on here, we'll see the same kind of behavior. But it's taking very small steps in order to achieve the desired accuracy. Now let me introduce a new solver, ode23s. The s for stiffness. This was designed to solve stiff problems. And boom, it goes up, turns the corner, and it takes just a few steps to get to the final result. There it turns the corner very quickly. We'll see how ode23s works in a minute, but first let's try to define stiffness. It's a qualitative notion that doesn't have a precise mathematical definition. It depends upon the problem, but also on the solver and the accuracy requirements. But it's an important notion. We say that a problem is stiff if the solution being sought very slowly, but there are nearby solutions that very rapidly. So the numerical method must take small steps to obtain satisfactory results. Stiff methods for ordinary differential equations must be implicit. They must involve formulas that involve looking backward from the forward timestep. The prototype of these methods is the backward Euler method, or the implicit Euler method. This formula, it involves-- defines y n plus 1, but doesn't tell us how to compute it. We have to solve this equation for y n plus 1. And I'm not going to go into detail about how we actually do it. It involves something like a Newton method that would-- requires knowing the derivative, or an approximation to the derivative of f. But this gives you an idea of what you can expect in stiff methods. I like to make an analogy with taking a hike in one of the slot canyons we have here in the Southwest. Explicit methods like ode23 and 45 take steps on the walls of the canyon and go back and forth across the sides of the canyon, make very slow progress down the canyon. Whereas implicit methods, like ode15s, look ahead down the canyon and look ahead to where you want to go and make rapid progress of the canyon. The stiff solver, ode23s, uses an implicit second-order formula and an associated third-order error estimator. It evaluates the partial derivatives of f with respect to both t and f at each step, so that's expensive. It's efficient at crude error tolerances, like graphic accuracy. And it has relatively low overhead. By way of comparison, the stiff solver ode15s, can be configured to use either the variable order numerical differentiation formula, NDF, or the related to backward differentiation formula BDF. Neither case it saves several values of the function over previous steps. The order varies automatically between one and five, it evaluates the partial derivatives less frequently, and did see efficient at higher tolerances then 23s. |
MIT_Learn_Differential_Equations | Exponential_Response_Possible_Resonance.txt | GILBERT STRONG: Well, you see spring has finally come to Boston. My sweater is gone and it's April the 16th, I think. It's getting late spring. So today, this video is the nice case, constant coefficient, linear equations, and the right hand side is an exponential. Those are the best. And we've seen that before. In fact, let me extend, we saw it for first order equations, here it is for second order equations, and it could be an nth order equation. We could have the nth derivative and all lower derivatives. The first derivative, the function itself, 0-th derivative with coefficients, constant coefficients, equalling e to the st. That's what makes it easy. And what do we do when the right hand side is e to the st? We look for a solution, a multiple of e to the st. Capital Y times e to the st, that's going to work. We just plug into the equation to find that transfer function capital Y. Can I just do that? I'll do it for the nth degree. Why not? So will I plug this in for y, every derivative brings an s. Capital Y is still there, the exponential is going to be still there, and then there are all the s's that come down from the derivatives, and s's from the nth derivative. One s from the first derivative, no s from the constant term. Do you see that equation is exactly like what we had before with as squared plus Bs plus C. We have quadratic equation, the most important case. Now, I'm including that with any degree equation, nth degree equation. And what's the solution for Y? Because e to the st cancels e to the st. That whole thing equals 1. I divide by this and I get Y equal 1 over that key polynomial. It's a nth degree polynomial. And the 1 over it is called the transfer function. And that transfer function transfers the input-- e to the st-- to the output-- Ye to the st. It gives the exponential response. Very nice formula. Couldn't be better. And you remember for second degree equations, our most important case is as squared Bs plus C. That's the solution almost every time. But one thing can go wrong. One thing can go wrong. Suppose for the particular s, the particular exponent, in the forcing function, suppose that s in the forcing function is also one of the s's in the no solutions. You remember the no solutions, there are two s's-- s1 and s2-- that make this 0, that make that 0. Those are the s1 and s2 that go into the no solutions. Now, if the forcing s is one of those no solution s's, we have a problem. Because this is 1 over 0 and we haven't got an answer yet. 1 over 0 has no meaning. So I have to, this is called resonance. Resonance is when the forcing exponent is one of the no exponents that make this 0. And there are two of those for second order equations and there will be n different s's-- s1, s2, up to sn-- for nth degree equations. Those special s's, I could also call them poles of the transfer function. The transfer function has this in the denominator and when this is 0, that identifies a pole. So the s1, s2, to sn are the poles and we hope that this s is not one of those, but it could be. And if it is, we need a new formula. So that's the only remaining case. This is a completely nice picture. We just need this last case with some resonance when s equals say s1. I'll just pick s1 and when A, I know that As 1 squared plus Bs 1 plus C is 0. So Y would be 1 over 0. And we can't live with that. I've written here for the second degree equation same possibility for the nth degree, An s1 to the nth plus A0 equals 0. That would be a problem of resonance. In the nth degree equation, this gives us resonance, you see, because remember, the no solutions were e to the s1t was a no solution. If I plug-in e to the s1t, the left side will give 0. So I can't get for equal to a forcing term on the right side. I need a new solution. I need a new y of t. Can I tell you what it is? It's a typical case of L'Hopital's rule from calculus when we approach this bad situation, and we are getting a 1 over 0. Well, you'll see a 0 over 0 and that's L'Hopital's aim for. So where do we start? A Y particular solution is this e to the st over this As squared plus Bs plus C. Right. That's our particular solution. If it works. Resonance is the case when this doesn't work because that's 0. Now, that's a particular solution. I'll subtract off a no solution. I could do that. I still have a solution. So I subtract off e to the s1t. So S1 is, e to the s1t is a no solution. This is what I would call a very particular solution. It's very particular because a t equals 0, it's 0. Do you see that, you see what's happening here. The question, resonance happens when s approaches s1. Resonance is s equal to, resonance itself is at the thing. Now, we let s sneak up on s1 and we ask what happens to that formula. You see, we're sneaking up on resonance. At resonance, when s equals s1, that will be 0 and that will be 0. That's our problem. So approach it and you end up with the derivative of this divided by the derivative of this. Do you remember L'Hopital? It was a crazy rule in calculus, but here it's actually needed. So as s goes to s1 this goes to 0 over 0, so I have to take derivative over derivative. So let me write the answer. Y resonant. Can I call this the resonant solution when s equals s1. And what does it equal? Well, I take the s derivative of this and divide it by the s derivative of that. Derivative over derivative. The s derivative of that is te to the st. And the s derivative of this is 2As plus B. And now, I have derivative over derivative, I can let s go to s1. So s goes to s1, this goes to s1, and I get an answer. The right answer. This is the correct solution and you notice everybody spots this t factor. That t factor is a signal to everyone that we're in a special case when two things happen to be equal. Here the two things are the s and the s1. So that will work. So do you see the general picture? It's always this te to the st, t above. And down below we have the derivative of this polynomial at s equal s1. You know it's theoretically possible that we could have double resonance. We would have double resonance if that thing is 0. If s1 was a double root, if s1 was a double root, then, well, that's just absurd, but it could happen. Then not only is that denominator 0, but after one use of L'Hopital, so we have to drag L'Hopital back from the hospital and say do it again. So we would have a second derivative. I won't write down that solution because it's pretty rare. So what has this video done? Simply put on record the simplest case possible with a forcing function, e to the st. And above all, we've identified this transfer function. And let me just anticipate that if we need another way to solve these equations, instead of in the t domain, we could go to the Laplace transform in the s domain. We could solve it in the s domain. And this is exactly what we'll meet when we take the Laplace transform. That will be the Laplace transform, which we have to deal with. So that transfer function is a fundamental, this polynomial tells us, its roots tell us the frequencies, s1, s2, and the no solutions. And then 1 over that tells us the right multiplier in the force solution. So constant coefficients, exponential forcing, the best case possible. Thank you. |
MIT_Learn_Differential_Equations | Linearization_at_Critical_Points.txt | GILBERT STRANG: OK. I'm concentrating now on the key question of stability. Do the solutions approach 0 in the case of linear equations? Do they approach some constant, some steady state in the case of non-linear equations? So today is the beginning of non-linear. I'll start with one equation. dy dt is some function of y, not a linear function probably. And first question, what is a steady state or critical point? Easy question. I'm looking at special points capital Y, where the right-hand side is 0, special points where the function is 0. And I'll call those critical points or steady states. What's the point? At a critical point, here is the solution. It's a constant. It's steady. I'm just checking here that the equation is satisfied. The derivative is 0 because it's constant, and f is 0 because it's a critical point. So I have 0 equals 0. The differential equation is perfectly good. So if I start at a critical point, I stay there. That's not our central question. Our key question is, if I start at other points, do I approach a critical point, or do I go away from it? Is the critical point stable and attractive, or is it unstable and repulsive? So the way to answer that question is to look at the equation when you're very near the critical point. Very near the critical point, we could make the equation linear. We can linearize the equation, and that's the whole trick. And I've spoken before, and I'll do it again now for one equation. But the real message, the real content comes with two or three equations. That's what we see in nature very often, and we want to know, is the problem stable? OK. So what does linearize mean? Every function is linear if you look at it through a microscope. Maybe I should say if you blow it up near y equal Y, every function is linear. Here is f of y. Here it's coming through-- it's a graph of f of y, whatever it is. If this we recognize as the point capital Y, right, that's where the function is 0. And near that point, my function is almost a straight line. And the slope of that tangent is the coefficient, and everything depends on that. Everything depends on whether the slope is going up like that-- probably that's going to be unstable-- or coming down. If it were coming down, then the slope would be negative at the critical point, and probably that will be stable. OK. So I just have to do a little calculus. The whole idea of linearizing is the central idea of calculus. That we have curves, but near a point, we can pretend-- they are essentially straight if we focus in, if we zoom in. So this is a zooming-in problem, linearization. OK. So if I zoom in the function at some y. I'm zooming in around the point capital Y. But you remember the tangent line stuff is the function at Y. So little y is some point close by. Capital Y is the crossing point. And this is the y minus Y times the slope-- that's the slope-- the slope at the critical point there is all that's-- you see that the right-hand side is linear. And actually, f of Y is 0. That's the point. So that I have just a linear approximation with that slope and a simple function. OK. So I'll use this approximation. I'll put that into the equation, and then I'll have a linear equation, which I can easily solve. Can I do that? So my plan is, take my differential equation, look, focus near the steady state, near the critical point capital Y. Near that point, this is my good approximation to f, and I'll just use it. So I plan to use that right away. So now here's the linearized. So d by dt of y equals f of y. But I'm going to do approximately equals this y minus capital Y times the slope. So the slope is my coefficient little a in my first-order linear equation. So I'm going back to chapter 1 for this linearization for one equation. But then the next video is the real thing by allowing two equations or even three equations. So we'll make a small start on that, but it's really the next video. OK. So that's the equation. Now, notice that I could put dy dt as-- the derivative of that constant is 0, so I could safely put it there. So what does this tell me? Let me call that number a. So I can solve that equation, and the solution will be y minus capital Y. It's just linear. The derivative is the thing itself times a. It's the pure model of steady growth or steady decay. y minus Y is, let's say, some e to the at. Right? When I have a coefficient in the linear equation ay, I see it in the exponential. So a less than 0 is stable. Because a less than 0, that's negative, and the exponential drops to 0. And that tells me that y approaches capital Y. It goes to the critical point, to the steady state, and not away. Example, example. Let me just take an example that you've seen before, the logistic equation, where the right side is, say, 3y minus y squared. OK. Not linear. So I plan to linearize after I find the critical points. Critical points, this is 0. That equals 0 at-- I guess there will be two critical points because I have a second-degree equation. When that is 0, it could be 0 at y equals 0 or at y equals 3. So two critical points, and each critical point has its own linearization, its slope at that critical point. So you see, if I graph f of y here, this 3y minus y squared has-- there is 3y minus y squared. There is one critical point, 0. There is the other critical point at 3. Here the slope is positive-- unstable. Here the slope is negative-- stable. So this is stable, unstable. And let me just push through the numbers here. So the df dy, that's the slope. So I have to take the derivative of that. Notice this is not my differential equation. There is my differential equation. Here is my linearization step, my computation of the derivative, the slope. So the derivative of that is 3 minus 2y, and I've got two critical points. At capital Y equal 0, that's 3. And at capital Y equals 3, it's 3 minus 6, it's minus 3. Those are the slopes we saw on the picture. Slope up, the parabola is going up. Slope down. So this will correspond to unstable. So what does it mean for this to be unstable? It means that the solution Y equals 0, constant 0, solves the equation, no problem. If Y stays at 0, it's a perfectly OK solution. The derivative is 0. Everything's 0. But if I move a little away from 0, if I move a little way from 0, then the 3y minus y squared, what does it look like? If I'm moving just a little away from Y equals 0, away from this unstable point, y squared will be extremely small. So it's really 3y. The y squared will be small near Y equals 0. Forget that. We have exponential growth, e to the 3t. We leave the 0 steady state, and we move on. Now, eventually we'll move somewhere near the other steady state. At capital Y equals 3, the slope of this thing is minus 3, and the negative one will be the stable point. So where y minus 3, the distance to the steady state, the critical point will grow like e to the mi-- well, will decay, sorry, I said grow, I meant decay-- will decay like e to the minus 3t because the minus 3 in the slope is the minus 3 in the exponent. OK. That's not rocket science, although it's pretty important for rockets. Let me just say what's coming next and then do it in the follow-up video. So what's coming next will be two equations, dy dt and dz dt. I have two things. y and z, they depend on each other. So the growth or decay of y is given by some function f, and this is given by some different function g, so f and g. Now, when do I have steady state? When this is 0. When they're both 0. They both have to be 0. And then dy dt is 0, so y is steady. dz dt is 0, so z is steady. So I'm looking for-- I've got two numbers to look for. And I've got two equations, f of y-- oh, let me call that capital Y, capital Z-- so those are numbers now-- equals 0. So I want to solve-- equals 0, and g of capital Y, capital Z equals 0. Yeah, yeah. So both right-hand sides should be 0, and then I'm in a steady state. But this is going to be like more interesting to linearize. That's really the next video, is how do you linearize? What does the linearized thing look like when you have two functions depending on two variables Y and Z? You're going to have, we'll see, [? for ?] slopes-- well, you'll see it. So this is what's coming. And we end up with a two-by-two matrix because we have two equations, two unknowns, and a little more excitement than the classical single equation, like a logistic equation. OK. Onward to two. |
MIT_Learn_Differential_Equations | Second_Order_Systems.txt | GILBERT STRANG: OK. Now, I'm going to have differential equations, systems of equations, so there'll be matrices and vectors, using symmetric matrix. They'll be second order. So second order, second derivative, that y is the vector. And S is the symmetric matrix. And that's the first time we've been prepared for the most fundamental equation of physics, of mechanics, oscillating springs-- so many applications-- rotating torques. It's very important in applications. The finite element, giant finite element codes, are solving equations like that all the time. And we don't have a damping term here, so it-- or a forcing term, so it's the null solutions that I'm going to look for to match initial conditions. I don't have a forcing term. OK. So the real central equation always looks like that. This is-- Newton's law, is what this is-- mass times acceleration. So M will be a matrix, often a diagonal matrix, telling me the masses. Remember, I have n equations here, so I have n masses, as you'll see. And I have, let's say, a bunch of springs connecting those masses. And then there's a matrix K in multiplying y itself, and that's always called the stiffness matrix. So, actually, in applications, the first job is to take the problem and create these matrices. I'll give you an example, but let's suppose we've got them, and how do we solve them? We look for, as we always do, solutions where time is separate from the vector x. I substitute that into the equation, So I get M, second derivative will bring down the i omega, twice. E to the i omega t x, right? Plus, this term, K times e to the i omega t x, should be 0. So I'm just substituting the expected form for the solution. In that form, that exponential factor can cancel. And, I see, I have an eigenvalue problem. Let me just look at that eigenvalue problem. I'm going to put that on the opposite side. But what i squared is giving me minus one. I'm just going to be left with Kx. Well, let me put this on the other side, because it's got a minus. And then when I put it over there, will be a plus. M omega squared x. That's an eigenvalue problem. Here is the eigenvector. There is the eigenvalue. Oh, but we have two matrices. That's something a little new. Not new to MATLAB however. The MATLAB command to find these eigenvalues, let me call those eigenvalues lambda, so lambda will now be omega squared, because two derivatives brought down omega twice. But we have our two matrices, so the MATLAB command would be i of K and M. If you define the matrices, K and M, and you call that command, it will produce the eigenvalues and the eigenvectors x for this, you could say, generalized eigenvalue problem, two matrix eigenvalue problem. It's got a K, as usual, and then it has an M. But many, many times, M will be a multiple of the identity and present no problem. OK, so that-- this is the eigenvalue problem that we reached. And that's the command that would solve it. OK. That's the first step, is to look for solutions of that special form. Now let's do a little count. How many solutions are we expecting? How many initial conditions do we have? So we initially, we give y at 0 of course, the initial condition, the position. But we also give, in a second order equation, we also give the initial velocity, y prime of 0. And those are vectors, because those tell us the initial condition of n masses. And so I have n numbers from y of 0 and n more numbers. Two n, all together, initial conditions. I'm going to need two n solutions. I'm going to need two n solutions to match two n initial conditions and solve the-- solve the equation. OK, so what do they look like? All right. Here, I've gone ahead to put an application up. So, again, I'm taking the masses to be equal. Here are the masses, M, M, and M. And so I end up with a 3 by 3 matrix. I have three unknowns, n is 3 for this problem. And I've got 4 springs. These are springs-- maybe I make them look a little springier. OK. And they're connected at the top to a fixed support and at the bottom to a fixed support and they're connected to each other. Do you see what will happen? As I, maybe I, the initial condition, I drag all those masses down? As my initial condition, I let go? Then they will go up and down, up and down, just the way springs always do. And they will solve, their position will be the solution to my differential equation. M y double prime plus Ky equals 0. So I'm not, I'm just starting the motion, and then backing off. So I'm not forcing-- it's not forced motion. It's pure oscillation, pure oscillation. Right. But coupled, several oscillators are a couple, that's what's new. We know all about this equation when y is just a scalar, just one equation. We know that, and that led us to the square root of K over M. That was just a 1 by 1 eigenvalue problem, and now we will have a 3 by 3 eigenvalue problem. The mass matrix is simple. Here's what the stiffness matrix would look like, if all those springs were the same. Just, I wanted to see what kind of a matrix shows up in the problem. And time to write down solutions. OK, so what are we remembering from solutions? We're remembering that solutions look like this. I have, but I'll have three possible eigenvectors, because I have 3 by 3 matrices. So that will give me three solutions. But I want six because I have six initial conditions all together. Let me write those six solutions down. Y, the solution is, sub constant times the cosine of omega t, times the first eigenvector. And another constant times the sine of omega t, times the second eigenvector-- times the first eigenvector, sorry. The first eigenvector, I have three eigenvectors, and for each of them, I get two solutions, one a cosine and one a sine. And the frequencies, there would be omega 1, and you are remembering that lambda, lambda is omega squared. So if I write omega 1 there, it's the square root of lambda 1. That's the square root of lambda 1. So I've got two solutions so far, coming from the first eigenvector at its eigenvalue, and with the cosine and a sine. And then, I'll also have for the A2 and a B2, and an A3 and a B3. So I just, briefly, A2, B2 with omega, using omega 2 and x2. And then they'll be an A3 and a B3 using omega 3 and the eigenvector x3. That's pretty stupid looking. This is what I meant to represent. I don't want to rewrite all that with 2's, and then rewrite it again with 3's. That's what the solution looks like. OK. How do we-- when we match the initial condition, what happens? I said t equals 0, right? When I said t equals 0, the sines disappear. So it's the A's that match y of 0. So A's match y of 0. And when I look at the initial velocity, the derivative, the derivative of the cosine is the sine, and it's 0. But the derivative of the sine is the cosine and it's 1, so I'll see that the B's match y prime of 0. I'm trying to get the total count to be six. So there are three, three initial positions, and three A's. There are three initial velocities, cause there are three masses, and there are three B's. I get a perfect match, a total of six. Six constants, six numbers to match. It all works. OK. Now, do I want to try an example? Sure. Let me end with a particular example. I better go to 2 by 2 for an example. So can I do the same problem 2 by 2? So my problem is going to be y double prime plus S-- there's a K, there's a division by m, and there's a 2, 2 minus 1, minus 1 y equals 0. That's my equation. Now, I'm speaking now, about the problem with a spring, a first mass m, a spring, the second mass m, and a spring. So two masses, two equal masses, three equal springs. That's my equation. So what's a solution? How do I solve it? I need the eigenvalues and the eigenvectors of my matrix. So here's my matrix S. That's my matrix S. It's symmetric, it's got physical constants there. The stiffness of the springs divided by the masses, k over m, we're expecting that same k over m that always shows up. And we need the eigenvectors there. And what are they? The eigenvalues of that are 1 times k over m, And 3 times k over m. Cause the trace is 4. 2 plus 2 is 4. The determinant is 3. 4 minus 1 is 3. Those are the two eigenvalues. And these are the omega squareds, remember. OK. That's how I start the system from rest. So physically I pull down these masses, or maybe I push that one up and pull that one down, whatever I like. I hold them for a moment and I let go. I don't give them an initial velocity. They start from rest, so the B's will be 0. So y prime of 0, y prime of 0 is going to be 0 from rest. And that will give me B's are 0. So my solution will be, my A cosine of, so the eigenvalues cosine, I take the square root of k over m t times the first eigenvector-- one of the eigenvectors. The eigenvectors of this, probably are 1, 1. And then the other eigenvector, and that will have a cosine of the square root of omega, I have to take the square root of that, so that's the square root of 3, k over m t, times its eigenvector, which I think is 1 minus 1. It's going to be perpendicular to that one. I've solved the problem. The A1 and the A2 are determined by the initial condition. Now do you see what's happening in the motion? That's the last thing, last point for this video. This motion with a 1, 1 eigenvector, the two masses are in sync. They're growing together, up and down. That's one eigenvector of the problem. And it has a certain frequency that they go up and down-- a square root of k over m, our old friend. But also, with two masses, they can go against each other, like this motion. That's coming from this eigenvector, and it happens at a higher frequency. So those are going-- the final solution is a combination of the masses moving together at a little slower oscillation, and the masses moving opposite each other at a faster oscillation. Some combination of those two is the solution. And then if we had three masses, there would be three oscillations. One where all three are going together, one where the outside ones are opposite, and one where all three are, I see opposite signs. It's a beautiful subject. Highly developed, and highly important applications. But that's the nicest solution you could hope for. OK that's a second order system, solved. Good. |
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