text stringlengths 16 3.88k | source stringlengths 60 201 |
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3
total noise power of �2
after lowpass filtering. As long as no aliasing occurs, the total noise
12M
power at the output of a compressor-by-M is the same as that of its input signal, so we know
that the total noise power at the output of the system is
Total noise power at output of system in OSB Figure 4.57 =
�2... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2e04b01fe7072772ba39f0c8dfb58306_lec06.pdf |
ing a linear system to reduce the effects of quantization noise. Looking at OSB Figure
4.69 for further insight, the system works by first obtaining the error signal e[n], delaying it, and
then using it to pre-compensate the input signal so that as long as e[n] is changing very slowly,
it is mostly cancelled at y[n] b... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2e04b01fe7072772ba39f0c8dfb58306_lec06.pdf |
is shown in OSB Figure 4.65, and a number of implementations of this
type of cascaded system are discussed in OSB.
Note that in practice, higher-order noise shaping systems are often designed to provide more
detailed control of the resultant noise power spectrum. There are a number of practical issues
associated wi... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2e04b01fe7072772ba39f0c8dfb58306_lec06.pdf |
Table of Contents
Lec 1
Introduction to Nanotechnology and Nanoscale Transport Phenomena.
Microscopic Pictures of Heat Carriers
Lec 2
Characteristic Time and Length, Simple Kinetic Theory, Characteristic
Lec 3
Schrödinger Equation
Lec 4
Quantum Wells, Harmonic Oscillators, Rigid Rotors, and Hydrogen Atoms
Lec... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
1. Overview for nano sciences
1.1 Length scale
1.2 Examples in microtechnology
1.3 Examples in nanotechnology
1.4 Nano for energy (phonon, phonon, electron; wavelength, mean free path)
1.5 Nanoscale heat transfer in devices (e.g., CMOS)
1.6 Nano and microfabrication
1.7 Transport regimes
1.8 Overview of the boo... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
flux of radiation is evaluated as
q =σ (T − T2
4
1
4 ) .
T1
ε 1=1
T2
ε 2=1
2.2 Newton shear stress law
The shear stress for the sketched one-dimensional flow is:
τ = µ
xy
∂ u
∂ y
µ [N-s/m2] is dynamic viscosity.
where:
y
τ xy
x
2.57 Fall 2004 – Lecture 1
4 2.3 Fick’s diffusion law
j = −ρD
i
dmi
dx
... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
/ m , σ = 78mN / m into this equation, we get
3
2
r =1 m, γ = 8.4e4; r =1 mm, γ = 8.4e-2.
mg
2.57 Fall 2004 – Lecture 1
5
4. Microscopic pictures of energy carriers
4.1 Heat
4.1.1 Heat conduction
Gases: hotter air molecules (with larger kinetic energy) randomly pass their excess
energy to cooler molecules. He... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
You can see similarities
and why I said we can describe them in parallel.
2.57 Fall 2004 – Lecture 1
7 2.57 Nano-to-Macro Transport Processes
Fall 2004
Lecture 2
5.1 Heat conduction
Th
Tc
In last lecture, we describe electrons as free electron gas and lattice vibrations as
phonon gas. Basically they are both... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
Lecture 2
1
8 5.4 Pressure and shear stress
x
As is shown in the figure, the velocities of gas molecules distribute randomly in all
directions. Pressure is caused by their momentum changes normal to the wall. For one
molecule, we have
G
G
G
F ma
=
=
(
d mv )
dt
or F =
x
m vx>0 − v
(
x<0 =
) m∆v
∆t
x
.
∆... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
they travel?
— How do they interact with each other?
6.1 How much energy/momentum can a particle have?
Energy
EKinetic
Classical mechanics
E = EKinetic+EPotential
Quantum mechanics
E is the eigenvalue of the
Schrödinger equation
E
Translation =
2
mv
2
(for quantum case, I did not give answer but point out ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
π
=
G
| k | .
6.2 How many particles have the specified energy E?
For a monoatomic ideal gas system, the only energy of each atom is their kinetic
energy,
m (v + v y + vz ).
2
x
2
2
E =
2
Statistical thermodynamics gives the probability density f(E), defined as the probability
of finding the carriers at energ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
dvx ∫ dvy ∫
-∞
-∞
∞
2
m (vx + v + vz
2 )
-∞ 2
2
y
3/2
⎛
⎞
m
⎟ exp
⎜
2πκ T ⎠
⎝
B
⎡
⎢−
⎣⎢
2
2
( x
m v + v2 + vz ) ⎤
y
⎥ dv
2κ BT
⎦⎥
z
3
= k T
2 B
At room temperature (300 K), this average energy is 39 meV, or 6.21e-21 J. For He gas,
m=6.4e-27 kg. Using
2
mv
2
=
3
2 B
k T , we can calculate averag... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
ρ π D P
k T
B
2
πD PN A πD P
R T
u
2
Λ =
=
=
,
where the universal gas constant R = N k = 8.314 J mol K .
/
⋅
u
A B
The ideal gas law can also be derived as following:
In section 5.4, we derive P mn
=
2
v
3
. In section 6.3, we get
2mv
2
=
3
2 B
k T . Thus
P = mn
2
v
3
= nk T =
B
N
V B
k T =
... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
theory
In the figure, half of the carriers within vxτ can go across the interface before being
scattered. Here vx is the x component of the random velocity of the heat carriers and τ is
the relaxation time — the average time a heat carrier travels before it is scattered and
changes its direction. So the net heat fl... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
Λ > d for
thin films
Λ=100 nm
d=10~20 nm
Disk drive
Thin film k
In example 2, we can further reduce the film thickness to enhance the size effects. With
measured data for k and specific heat c, the mean free path in silicon can be estimated by
k =
cvΛ
3
,
where v is sound velocity. The approximated mean free ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
electron, D=1 mm, we have
E =
2
n
8× 9.1 10 −31
×
(
6.6 ×10 −34
10−3
)2 ~ 10
−34
2
n << k
T
B
= 4.14
×10
−21
J at room temperature.
2) For D=1e-8 m, we calculate E k
> B
T . Further reducing D results in more observable E.
8.3 Fast transport
For many materials, we have τ = 10−12 −10−11 s . Laser pulse can be ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
sine and cosine functions, e.g.
A e−i(ωt kx ) = A ⎡cos (ωt-kx ) − i sin (ωt kx )⎦⎤ .
−
⎣
−
2.1.3 Standing wave
x=0
x = π/k
We can create a standing wave by superimposing two traveling waves along the positive
and negative x-directions (assuming that the problem is linear such that the superposition
principle ap... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
2.57 Fall 2004 – Lecture 3
16 To explain the blackbody radiation, Max Planck (1858-1948) introduced a radical
hypothesis that the allowable energy of the electromagnetic field at a frequency ν is not
continuous, but is a multiple of the following basic energy unit
E = nhν ,
in which h is called the Planck constant... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
long wavelength. The first proof of the wave properties of particles came from the
electron diffraction experiment, in which peak signals are observed in specified incident
angles.
Transmission Electron Microscope (TEM)
A TEM uses thermal excitation or applying a high voltage to draw electrons from the tip
end. Th... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
t
= Ψ ∇
t
i=
−
,
2
which is similar to heat conduction equation k T = ρ c
∇ 2
∂ T
∂ t
but the magic imaginary unit
“i” really gives rise to wave behavior.
2.57 Fall 2004 – Lecture 3
18 Schrödinger himself did not come up with a correct explanation for the meaning of
wavefunction. The right explanation was g... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
� 2 + U −=
2
=
m2
+ U =
−=
+ U
⎜ ∂
2
⎝ ∂x
+
∂
∂y2
+
∂
∂z 2
+ U
⎞
⎟
⎟
⎠
Note: in equation < Ω >= ∫ Ψ ΩΨ dV , you cannot switch Ψt
gradient operator ∇ and the Laplace operator ∇ 2 .
t
*
t
* and Ψt if Ω contains the
Standard deviation:
Similar to
x
∆ =
n
1
− ∑
(
x
− < >
1
n
x
i
1
=
2
)
,
... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
2
⎢−
Ψ ⎢⎣ 2m
where E is a constant (eigenvalue) since Ψ depends on r only and Y depends on t only.
Its meaning will be explained later ( H = E ). Solving for Y leads to
⎤
UΨ⎥ = i=
⎥
⎦
1 dY
Y dt
2
+ Ψ ∇
= E ,
Y C1 exp −i
=
E
⎤
⎡
⎥ = C1 exp [−iωt] .
t
⎢⎣ = ⎦
And the governing equation for Ψ(r) is called the... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
principle states
< ∆ >< ∆ >≥
x
p
=
2
;
< ∆ >< ∆
t
E >≥
=
2
.
2.3 Example solutions:
G
G
( , )
( ) by the boundary conditions and will not consider the u r t
u r
Here we determine
case.
2.3.1 Free particles in 1D
In this case, there are no constraints for the particles. The potential energy u=0 so that
E ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
Same as the free particles, the solution for first equation is still
−ikx
Ψ = Ae + Be
ikx
where
k =
,
mE
2
2
=
=
2
mE
=
=
p
=
.
The general boundary conditions are the continuity of the wave functions and their first
derivatives at the boundaries. The latter derives from the continuity of particle flu... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
Finally we get A i=
1
2D
.
2.3.2 Particle in a 2D box
y
D
O
U=
0
U = ∞
x
D
We establish a coordinate system as shown in the schematic above. Clearly, outside the
potential well, we have Ψ = 0 because U = ∞ . We thus focus on the solution inside the
potential well. The Schrödinger equation inside the well ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
. Applying these boundary conditions, we see that
kx =
nπ
D
(n = 1, 2,3
") .
Similarly, k y =
lπ
D
Thus
(l = 1, 2,3
") .
2.57 Fall 2004 – Lecture 4
23
(A + n )π =
2 2
2
2
2
2mD
E An =
and
( A ,n=1,2,…)
⎛ Aπy ⎞
⎛ nπx ⎞
.
sin
Ψ = C sin
⎜
⎟
⎜
⎟
⎝ D ⎠
⎝ D ⎠
A... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
, however, does not tell the whole story on the quantum state of
a particle. For example it cannot distinguish the spin of particles. For electrons,
corresponding to each wavefunction obtained from the Schördinger equation, there are
two quantum states (or two relativistic wavefunctions), which are usually denoted b... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
interaction between the nucleus and the orbiting electron is governed by
the Coulomb force
F = −
1 e2
4πεo r
2 = −
du
dr
,
where εo=1.124x10-10/4π [C2/(m2N)] is the electrical permittivity of the vacuum. It yields
u r( ) = −
2e
4πε0r
.
Using separation of variables, we assume ΨnAm = RnA (r)YA
energy level... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
orbitals
1s
n=1
(- 13.6 eV)
Note: the energy gap between different n values is much larger than the thermal energy
(kBT~26 meV) and it is almost impossible to thermally excite electrons to a higher n level.
A stable element is obtained only if all the orbitals for the highest n are completely filled,
such as He.... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
obtain from solving Schrodinger
equation. Similarly, we can deal with electrons moving around the nucleus but changing
the potential energy to u =
e2
4πε0 xn
. Finally we get E ∝
1
n2
.
Rigid rotation
G
G
In classic mechanics, the angular momentum is expressed as r
p . However, because of
the uncertainty princ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
n=1
U=0
x
Energy has discrete levels, and we have one quantum number n.
E =
2h n2
2
8m D
(n=1,2,…)
For 2D constraints, we have two quantum numbers n and l. In the discussions, the
conception of “degeneracy” is introduced.
3. Spin
1
For electrons, we have talked about s = ±
2
, where s=1/2 is called spin up ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
. The energy of 3d is lifted up
above 4s because of the electron-electron interaction. For the element potassium (K),
it has 19 electrons but the n=3 energy levels are not totally filled and one electron
goes to the 4s orbit.
2.57 Fall 2004 – Lecture 5
29 2.3.5 Energy Quantization observation
Absorption or emissi... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
(n f − ni );ν =
k
1
2π m
,
where the reduced mass is m =
m m2 =
1
m m2
+1
m1 ( m = m2 ), and the selection rule states
2
1
ni
n f −
= ± 1. Positive for absorption and negative for emission. We can measure the
vibrational frequency from which, to deduce the spring constant. For hydrogen, the
2.57 Fall 2004 – ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
=0,1,2, …).
2=
I 2
hν = hB
[
where B=1.8e12 Hz. Similarly we have A f − A i = ±1, 0 . As an example, A f −
= 1
(absorption) gives νp = 2B l +1) . The normal wavelength for rotational energy is around
100 µm (far infrared regime), which is much larger than the emission wavelength at the
room temperature (10 µm).
A i
... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
of electrons.
( +
(
3.1.2 Possible approaches
First we will review the problems solved in Lecture 4.
(1) Free electron E p2 / 2m
=
2
(2) Quantum well E ∝ n
2.57 Fall 2004 – Lecture 5
32 ENERGY AND
WAVEFUNCTION
U ∞ =
n=3
n=2
n=1
U=0
x
(3) Electron-nucleus system (hydrogen atom) En
u
r
el = −
13.6 eV
2n
... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
let us go back to the quantum wells. Due to the finite depth, the wave functions will
not be zero at the boundary like standing waves. Instead, they will decay exponentially
from the interface. However, this indicates two wavefunctions overlap in the middle of
the barrier, which conflicts with the Pauli exclusion pr... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
Born-von
Karman periodic boundary condition. The Born-von Karman boundary condition deals
with the end points of a crystal. Ordinarily, we would think that the two end points are
different from the internal points. For many applications, however, distinguishing the
boundary points from the internal points is not ne... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
same quantum state and should be counted only once.
Thus, rather than plotting the energy eigenvalues for all the wavevectors, we can plot
them in one period, as shown in the subsequent figure. This way of representation is
called the reduced-zone representation. Often, only half of the band, [0,π /a], needs to be
... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
z
i
l
a
m
r
o
N
0
-1
0
k / ( π /a)
1
2.57 Fall 2004 – Lecture 5
37
2.57 Nano-to-Macro Transport Processes
Fall 2004
Lecture 6
Quick review of Lecture 5
(
,
[x
n
) ] = Ψ x
( ) ikn a +b)
e
In the last... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
condition is no longer valid when N becomes very small in
nanomaterials.
/
o
E
E
y
g
r
e
n
E
n
o
r
t
c
e
l
E
d
e
z
i
l
a
m
r
o
N
15
10
5
0
-
1
0
1
k / (π/a)
2.57 Fall 2004 – Lecture 6
38
3.1.3 Consequences of solid energy levels we just obtained:
(1) Electrons wave function extends through the whole... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
levels is small. Thus, these electrons can flow freely, making the
materials good electrical conductors, which is the case for metals.
2.57 Fall 2004 – Lecture 6
39
Energy
Energy
Energy
Energy
Energy
Energy
Energy
Energy
Energy
Energy
Energy
Energy
Energy
Energy
Energy
Energy
Energy
Energy
Energy
Conduction
Co... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
trinsic
Intrinsic
Intrinsic
Semiconductor
Semiconductor
Semiconductor
n-type
n-type
n-type
Semiconductor
Semiconductor
Semiconductor
p-type
p-type
p-type
Semiconductor
Semiconductor
Semiconductor
If the valence electrons exactly fill one or more bands, leaving others empty, the crystal
will be an insulator at zero tem... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
,
x
y
,
z
talk about the electron filling in a 3D structure. (2) The shape of bands can be affected by
heavily doping. (3) The lowest band starts from a nonzero energy, which is the
consequence of the uncertainty principle.
Impurities are added to most semiconductors and these impurities have energy levels
somewher... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
)
(
m
= E k
)
(
m
+
+
1 ∂ 2 E
2 ∂ k
1 =2
2 m *
2 |km
(k
−
k )2
m
(k
−
k )2 ,
m
where the effective mass is defined as m * =
2=
(∂ 2 E k ) |km
/ ∂ 2
. In differential geometry, the
/ ∂ 2
term 1/ ∂ 2 E k
is just the curvature. Thus, effective mass is proportional to local
curvature at band maxima or minim... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
Direct
Eg
k
Eg
k
In above figures, we demonstrate the idea of direct and indirect band gaps. In a direct
band structure, both the minimum in the conduction band and the maximum in the
valence band occur at the same location of k (=0 for the example given). A good example
for direct band gap is GaAs with Eg =1.4... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
between the nearest neighbors. Second, the interaction force between
atoms is a harmonic force (which obeys Hook's law). This can be justified as we have
done for harmonic oscillators. Now consider a typical atom j. The displacement of atom
j from its equilibrium position o
jx
is
2.57 Fall 2004 – Lecture 6
43
... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
2
K
m
sin
ka
2
.
Na
(j-1)a ja (j+1)a
Note: When k approaches zero for large wavelength, the frequency becomes a linear
K ka
m 2
function of the wavevector, i.e. ω≈ 2
. We can calculate the sound
K
m
= ka
velocity by v
sound
=
dω
dk
.
ω
k
π/a
In last two lectures, we have derived the allowed k val... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
,2,3,…),
in which ω = 2
K
m
sin
ka
2
.
The basic vibrational energy quanta, hν, is called a phonon. Comparison between
electrons, phonons, and photons:
(1) Electrons obey the Pauli exclusion principle, which says that each quantum state
can only have at most one electron. Photons and phonons are not limited ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
ons in 3D crystals
a. Each direction is different
b. Two transverse waves, one longitudinal wave
Transverse
Longitudinal
c. For m atoms per basis, we have 3 acoustic waves and 3(m-1) optical waves.
d. Each kx, ky, kz represents a normal mode.
e. The energy dispersion (E-k relationship) can be totally different ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
, i.e.
crystal = lattice + basis.
2.57 Fall 2004 – Lecture 7
48
From a mathematical point of view, the location of each point can be described by a
vector. Due to the periodic arrangement of lattice points, we can choose a basic set of
vectors ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
lattice points in such a rectangle---the center point plus the four
corners, each of the latter is shared by four cells. Because the choices of primitive lattice
vectors are not unique, there can be different ways to draw a primitive unit cell, as the
two examples in the figure. One method to construct a unit cell u... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
6 3
5e −10
9
) = e8
,
which indicates the material should be similar to bulk crystal.
(3) Miller index
The Miller indices of crystal planes (hkl) are obtained in accordance with the following
steps:
(a) Find the intercepts of the crystal plane with the axes formed by the lattice vectors a1,
a2, a3 in terms of the ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
y
x
2.57 Fall 2004 – Lecture 7
51
3.1.3 Bonding potential
Interatomic
potential Φ
Repulsion
x
Attraction
Recall the previous discussion about the interatomic forces. Here we will learn them in
more details.
The force interaction between atom... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
– Lecture 7
52
What makes a crystal structure a favorable structure is that the total potential energy of
the system
⎛
1
B
⎜
U = ∑
⎜ 12
2
i≠ j rij
⎝
−
⎞
⎟
1
A
6 ⎟ 2
rij
⎠
12
⎛
⎞
⎛
⎜
⎜ σ ⎟
= ∑ 4ε⎜
⎟
⎜
⎜
⎟
⎜⎜ r
i≠ j
⎝ ij ⎠
⎝
−
⎛
⎞
⎜ σ ⎟
⎟
⎜
⎜
⎟
r
⎝ ij ⎠
6
⎞... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
R (r) = U e
o
U
where B, ζ, and Uo are empirical constants determined from experimental data, such as
the interatomic spacing and the binding energy.
(Born-Mayer)
Interatomic
potential Φ
Repulsion
x
Attraction
2.57 Fall 2004 – Lecture 8
54
Combining this attractive potential (van d... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
silicon,
and germanium are all covalent crystals. Each atom has four electrons in the outer shell
and forms a tetrahedral system of covalent bonds with four neighboring atoms, as
indicated in the figure.
1 1 1
4 4 4
,
,
) atom is shared by neighboring atoms.
Note: For instance, in silicon the (
2.57 Fall 2004 – L... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
i.e.
t
expanded into a Fourier series as,
f
( + ) = f ( )t
t T
), its can be
∞ ⎛
⎛ 2πn ⎞⎞
⎛ 2πn ⎞
f (t) = ∑ ⎜⎜ a n sin⎜
t ⎟⎟⎟
t ⎟ + bn cos⎜
T
T
⎠
⎠⎠
⎝
⎝
n = −∞ ⎝
= ∑(
+ n '
b e
a n e
'
−inωt
inωt
∞
)
n =−∞
Here the angular frequency ω=2π/T is the Fourier conjugate of the time periodicity such
that eiωT... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
In the above figure, the electron energy dispersion shows a period of 2π/a because of the
periodic potential field ( + ) = u x ( )
(recall the Bloch theorem).
u x a
Note: (1) The discussed function f(x) can represent not only the charge density but also
other periodically distributed properties. (2) The Born-von ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
. For the
one-dimensional case, we have G =2πn/a, r =x.
With the above definitions, we can show that u(r) is indeed invariant with any
translational lattice vector in the real space, T(=n1a1+n2a2+n3a3), where n1, n2 n3 integers,
u( + ) =
u e • + •
ir G iT G
i(r T+
•) G
=
r T ∑ G
u e
G
∑ G
G
ir G π n m n m ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
an incident wave from the source along direction k. The incident wave is
, where r is any point in the sample and rs is the location of the
proportional to e
source relative to the origin of coordinates. The wave scattered into the detector is then
proportional to ( )r
, where kf is the propagation direction of t... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
of the source, the sample, and the detector. The
reciprocal lattice vectors G, and thus the crystal structure, can be determined from
diffraction experiments.
− f
i
θ
a
2.57 Fall 2004 – Lecture 8
58
Consider the special set of crystal planes s... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
,…)
For wavefunction Ψ
,n s
(2) Harmonic oscillator
The energy is
, we have degeneracy g(n)=2 due to the spin.
E = hν(n +1/ 2);ν =
n
1 K
2π m
(n=0,1,2…)
The wavefunction is Ψ , and the degeneracy is g(n)=1.
(3) Rigid rotation
The energy eigenvalues are
n
E l =
2=
I 2
A A + 1) = hB A A + 1)
(
(
(for |m|≤ A ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
10
5
0
-1
0
1
k / (π/a)
ω
Debye
k
π/a
In the Debye approximation, we have energy dispersion as
ω= vk = v
2
k
x
2
+
k
y
2
k
+
z
; E
n = hν(n +
1
2
)
,
where v is sound velocity.
Density of (quantum mechanical) states (DOS):
(a) Electron
ky
π4
L
π2
L
0
Volume of
One
Unit Cell
k+dk
dk
π2... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
2m *
k
=
(
2
2π dE 2π =
1
2
2
)3/ 2 (E E )
c
−
1/ 2
.
E
l
e
c
t
r
o
n
D
e
n
s
i
t
y
o
f
S
t
a
t
e
s
Ec
EF
E
(b) Phonon
According to the Debye model that assumes three modes (two transverse, one
longitudinal) are identical, we have
2.57 Fall 2004 – Lecture 9
62
D(
)
ω
= 3
3
∆N
=
2
V ∆... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
�
1 ⎛
3π2 ⎜
in which the electron density n=N/V.
E f
n = ∫
E c
(E − E c )
)
(
Ed E D
3 / 2
3 / 2
=
f
,
For nonzero temperatures, Ef is replaced by the chemical potential.
Nanostructures:
z
D
For a thin film with thickness d, we have the energy as
(
k E x k
, y n
, ) =
2
2
= k xy
m 2 *
+ n 2
= 2π2... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
just like
jumps at different Enl k .
− nl
(
kz (n)
D(E
)
kx
bulk (3D)
quantum well (2D)
n=3
n=2
n=1
E
ky
Chapter 4 Statistical thermal & energy storage
In Lecture 2, we have mentioned that matter tends to occupy the lowest energy levels.
For an energy level Ei, we have given the Boltzmann factor for its ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
applied to
closed systems, while for an open system exchanging energy with the outside, the
probability becomes
P E Ni ) = Aexp[ −(E − µN ) / k T ] ,
where Ni is the particle number, chemical potential µ is the criteria for the equilibrium
state of mass exchanging process with the outside, just as pressure for mech... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
ν
k T
B
) = 1
exp(−
hν
1
B2k T 1− exp( −
) A
A = exp(
)(1 − exp( −
hν
2k T
B
= 1.
)) .
)
hν
k TB
hν
k T
B
And
(P E) = exp( −
)[1 − exp( −
)] .
nhν
k T
B
hν
k T
B
Then the average number of the phonons, or the occupancy of the quantum state is
< >= ∑ nP E
(
) ,
n
∞
n
n=0
= − exp( −
[1
)]∑ n exp(... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
number of quantum states per unit interval of
energy and per unit volume
D E(
)
=
1 ∆N
V E
∆
=
k 2 dk
k 2 ∆k
=
2π2 ∆E 2π2 dE
.
A factor that considers polarization of waves may be added (electron, spin up and down,
thus a factor of 2, photon, two polarizations, phonons, 3 polarizations)
For an energy level Ei,... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
=
1
k T
( E −µ ) /
B
i
e
,
−1
where µ is again the chemical potential of the boson gas.
The Bose-Einstein distribution changes the “plus one” in the denominator of the Femi-
Dirac distribution into minus one. When E − µ k T , we can ignore the ±1 term in the
denominator. Both distributions reduce to the Boltzma... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
to the
temperature. For constrained electrons, we can find that the ground state (quantum
numbers n l
= = = ) has much larger occupancy than any other energy level. This is
more apparent at lower temperatures. Therefore, most molecules will go to the ground
state when the temperature approaches 0 K. The phenomenon ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
1
x2, P2
Quantum
State Ω
x3, P3
Microcanonical ensemble
U, V, N fixed
Isolated systems
Canonical ensemble
V, N, T fixed
In contact with a thermal
reservoir; isothermal.
equal
of
1/
iP = Ω
Principle
probability:
Boltzmann principle gives
S
The entropy S is additive.
S U V N
,
)
Ω =
ln
k
B
=
(
... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
ln
=
B
becomes
potential in this case.
the
thermal
Grand canonical
V, µ, T fixed
isothermal system.
Open,
Exchanging both energy
and particles with
the
reservoir.
P E N
)
,
i
i
(
=
(
−
E
i
e
−
µ
N
i
)
/
k T
B
Z
The numerator is the Gibbs
factor and
= ∑∑
(
−
Z
e
k T
B
.
N
i
E
i
µ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
z
)× exp
⎛ E − µ ⎞
⎜ −
⎟
k T
⎝
⎠
B
∑∑∑ ( x
z )
E k k k
,
exp
,
y
k
x
k y
k
z
=
∑∑∑ exp ⎜ −
⎝
k
z
k
k
y
x
⎛ E ⎞
⎜ −
⎟
k T
B ⎠
⎝
,
⎛ E ⎞
⎟
k T
B ⎠
If the energy separation is very small (quasi-continuous), we can evaluate Z by
integration instead of discrete summation, i.e.
Z = ∑∑∑ exp ⎜ −
⎝... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
κBT/2 to the average energy of the system.
The internal energy is
2.57 Fall 2004 – Lecture 10
71
u = k N T = R T ,
A
B
u
3
2
3
2
in which Ru is the universal gas constant. The specific heat is
CV =
∂u
∂T
3
= Ru .
2
In the following figure we draw the specific heat of hydrogen gas. At low ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
�
dω
2.57 Fall 2004 – Lecture 10
73
2.57 Nano-to-Macro Transport Processes
Fall 2004
Lecture 11
4.1.1 Photons (continue)
First let us continue the discussion of photons. We have
n = f (
ω T =
)
,
1
=ω/ k TB
e
.
−1
The internal energy is
u = ∑∑∑ ωf (
= ω
,
k
x
k
y
k
z
T =
)
∞
∫0
=ωf (
ω T D (
)
,
ω dω... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
, we
obtain uλ =
dω
dλ
uω =
2πc
λ
2 uω , which accounts for the power difference in above two
expressions. Therefore, we have obtained the exact Planck’s blackbody radiation law.
The radiation intensity, defined as energy flux per unit solid angle and normal area, is
calculated as
cu
λ
,
4π
Iλ =
where 4π i... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
3
2
2 π v
D
3 ∫
exp
0
= ω3dω
(=
) −
ω/κ T 1
B
.
Note: For one monoatomic chain with N atoms, we have N quantum states. Similarly, for
N atoms in a crystal, we have 3N quantum states (three acoustic branches) for phonons.
This yields
1
V kx
ω 3ω2
∫
2π vD
0
∑∑∑ ∫
1 =
ω
D D
)
d ω ω
(
ω 3
D
2
2π vD... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
k T , which is consistent with C = 3
B
N
V B
result is similar to the ideal gas case. A common Cv-T curve is drawn as following.
V
k . This
CV
Cv ~T3
CV → const
Temperature
In many sources, people use the specific heat per unit mass instead of per unit volume.
This normally causes a factor difference. In general... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
y ⎞ ⎛ dkz ⎞
µ,T )
⎟
⎜
⎝ 2 /π L ⎠ ⎝ 2 /π L ⎠ ⎝ 2 /π L ⎠
⎜
⎟
⎟ ⎜
∞
,
0
)
f E
(
,
,
)µ T D E dE
(
= ∫
where N is the total number of electrons, and we use density of state to rewrite the
summation in an integral form. The chemical potential µ is solved if N/V is given.
The specific heat is derived as
T
1 N
2 ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
77
)
.
K
g
k
/
J
(
T
A
E
H
C
F
C
E
P
S
I
I
103
102
101
100
10-1
10-2
10-3
1
PHONON
ELECTRON
100
10
TEMPERATURE (K)
1,000
Chapter 5 Energy transport by waves
Consider energy transported between two points. The net energy transfer rate is
(
q =
12
E f T E D E ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
.
Based on the Schrödinger equation
2∇ Ψ + (U − E)Ψ = 0 ,
−
=
2m
we have
Ψ = Ae− ω
(
i
i
t k )
− 1x
Ψ = Be− ω
(
i
t k 1 )
+ x
r
Ψ = Ce− ω
i
t k x
( − 2 )
t
(incoming wave),
(reflected wave), k =
1
(transmitted wave), k =
2
2mE
2=
2 (
m E u −
)
2
=
.
r
i
x=0+
The boundary conditions are applied ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
)
2
A k1 ,
where we use the fact that k1 is a real number in the last step.
2.57 Fall 2004 – Lecture 11
79
2.57 Nano-to-Macro Transport Processes
Fall 2004
Lecture 12
5.1 Plane waves & their interface reflection (continue)
Transmission wave
Reflection wave
Incoming
wave
Energy b... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
e
(ω − * )
t k x
1
|
⎞
x=0 ⎟
⎠
*
(
1 − 1 ) x
i k k
|
x=0
)
=
=
=
=
m
=
m
=
m
2
A
*
1
Re(k e
)
*
k1
A 2 Re(
2
A k1 ,
where we use the fact that k1 is a real number in the last step. Similarly, we have
2.57 Fall 2004 – Lecture 12
80
2B k1 (negative ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
occur for all the three types of carriers (phonons, electrons, photons) and
the descriptions of these phenomena are also similar.
JJG
An electromagnetic wave in vacuum is characterized by an electric field vector, E , and
a magnetic field vector, H . Consider a pair of charged particles placed in an electrical
fiel... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
which is
superimposed onto the external magnetic field. A measure of the total magnetic field
inside the medium is called magnetic induction, B (N.s m-1.C-1),
B=µH
where µ is the magnetic permeability.
The propagation of an electromagnetic wave is governed by the following Maxwell
equations:
(1) ∇ × E = −
∂B
... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
� 2E
∂t
∂E
∂t
denotes the current. Without µσ
∂E
∂t
, the
equation ∇2E=µε
∂ 2E
∂t 2
is just a regular wave function. The additional term µσ
∂E
∂t
corresponds to damping and is also called dissipation term.
By solving equations (1)-(5), we obtain the following results for EM waves
E(r,t) = Eoexp[-i(ωt- k • ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
2 = εµω can be rewritten as
2
2
2
2 ω ω c0 ω 2
N ,
2
=
=
k =
2
2
c
0
2
c
2
c
0
2
c
or
k =
Nω
co
.
One can prove that the electromagnetic wave is a transverse wave, and that the electrical
and magnetic fields are perpendicular to each other, i.e.,
E⊥H⊥k
2.57 Fall 2004 – Lecture 12
83
... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
along direction ki (wave vector direction) and meets an interface with norm nˆ .
The reflected wave and refracted wave propagates along the kr and kt directions,
respectively. We call the plane formed by ki and nˆ as the plane of incidence, and the
angle formed between nˆ and ki as the angle of incidence.
n1
n2
E... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
⎞⎤
⎟⎥
⎥
⎠⎦ ,
λ
o
⎛
E// t exp ⎢−i ⎜ωt − n2 2π
⎝
⎡
⎢
⎣
x sinθt + z cos θt ⎞⎤
⎟⎥ ,
⎠⎥
⎦
λo
where in the second equation the negative sign before z cos θr indicates different
propagation direction (upward in the z direction) from the incoming and refraction
waves. The subscript “//” means that the electric field... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
��⎦
o
r r
= cosθt E// t exp ⎢−iω⎜ t − 2
⎡
⎢
⎣
⎛
⎝
n x sinθ ⎞⎤
t ⎟⎥
⎠⎥
⎦
co
where the above equation is valid only when the exponents are equal because x can take
any value. Thus we have
2.57 Fall 2004 – Lecture 12
85
i
r
2
1
n sinθ = n sinθ = n sinθ
t
1
which ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
2θ + sin 2θ
t
t
t ,
Note: (1) For an incident light shown as below (e.g. from air to water), in this case the
refraction light will be bended and the object in the second media will looks higher if the
observer is in the first media. (2) If n2<0, n1>0, the refraction light will be bended as the
right figure. Howev... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
k1 + k 2
r
− =
t =
=
Ψ
r
Ψi
2k1
k1 + k 2
R = −J / J =
r
2
i
B
A
Re (k2 )
)k1
*
*
Jt
=
Ji Re (
T =
2
=
k
k2−1
k1 + k2
R//
=
2
t
T//
=
Normal incidence onto an interface
− n2 + n1
n2 + n1
E// r
E// i
r//
=
=
t //
=
2n1
n2 + n1
Sr z Sr
, =
Si z
Si
,
St z = Re ⎜
,
Si z
,
=
2
r//
⎛ n2 cos θ ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
angle, the Snell law gives,
sinθ =
t
n1 sinθi > 1,
n2
and thus,
⎛ n1 sin θi ⎟
⎞
cosθt = 1− sin θt = i ⎜
n2
⎠
⎝
2
2
−1 = ai .
In the wave function of the transmitted wave, the imaginary cos θt leads to an
exponential decay wave
⎡
⎛
E// t exp ⎢
−iω⎜
t −
⎢
⎝
⎣
2
n x sinθ + n z cos θt ⎞⎤
⎟⎥
co
⎠⎥⎦
2
t
E=... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
adding
layers that have a low refractive index than the core.
2.57 Fall 2004 – Lecture 13
89 Electrode
Cladding Layer
Core
Cladding Layer
Quantum
Wells
Electrode
2) Brewster angle
π
2
When θ θi+
=
t
, we have the electric wave r// = 0 , but the magnetic wave r⊥ ≠ 0 . This
incident angle θ is called the Bre... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
the phase
Refraction wave
factor. The energy flow is still
JK JJK *
S = Re ( E × H ).
1
2
Incident wave
Amplitude
contours
4) Acoustic waves
n
Recall the Newton’s law
K
JK
,
F ma
=
JK
K
K
= ⋅
where the force is F n
K
u
x= − x0 . We have
K
K
K d x du
v =
.
dt
dt
=
σ, acceleration is a =
. Denote u ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
wave
When the incident wave is polarized in the direction parallel to the plane of incidence,
one longitudinal wave and one transverse wave are excited for both reflection and
transmission waves. If the materials are anisotropic, one more transverse is excited for
both reflection and transmission waves.
Incident w... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
of the electric field E ( ) and y-component of
( ) on the interface are related by 2×2 matrix A, 2×2 matrix M1, 2
the magnetic field H y
×1 matrix B. And we obtain
⎛ Ei
⎞
⎜
⎟
⎝ Er ⎠
= A M BEt .
−1
1
x
z
2.57 Fall 2004 – Lecture 13
93 (0) ⎞
⎛ E
x
⎜
⎜ H
(0)
⎝
y
⎛ Ei ⎞
⎟
E
r ⎠
⎟⎟ = A⎜
⎝
⎠
x
(0) ⎞
⎛ E... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
2
12
r
23
r
23
cos ϕ2
r
2
12
2 2
r
r
cos ϕ2 +
2
23 ,
12
in which the cosine function indicates periodicity of R. This is just the interference effect.
Discussions:
(1) Periodic variation in R
R
φ
Note: In microfabrication, the color of a thin film will change periodically according to
the thickness, ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
4
(
U m 2
o − d E
)
/ =]
= τ
or
≈ τ
o − E)
exp[− 2
16
(
U E
2U o
(
U m 2
)
o − d E
16
/ =] =
(
o − E) − k 2 2
U E
e
2
Uo
d
.
2.57 Fall 2004 – Lecture 13
95
2.57 Nano-to-Macro Transport Processes
Fall 2004
Lecture 14
Review of last lecture
d
R
n1
n2
n3
φ
In lecture 13, we talked abou... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
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