2408/2408.00623.md

Title: 1 INTRODUCTION

URL Source: https://arxiv.org/html/2408.00623

Markdown Content: Numerical modeling of SNSPD absorption utilizing optical conductivity with quantum corrections

M.Baránek 1, P.Neilinger 1,2, S.Kern 1, and M.Grajcar 1,2

1 Department of Experimental Physics, Comenius University, SK-842 48, Bratislava, Slovakia

2 Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, SK-845 11, Bratislava, Slovakia

ABSTRACT. Superconducting nanowire single-photon detectors are widely used in various fields of physics and technology, due to their high efficiency and timing precision. Although, in principle, their detection mechanism offers broadband operation, their wavelength range has to be optimized by the optical cavity parameters for a specific task. We present a study of the optical absorption of a superconducting nanowire single photon detector (SNSPD) with an optical cavity. The optical properties of the niobium nitride films, measured by spectroscopic ellipsometry, were modelled using the Drude-Lorentz model with quantum corrections. The numerical simulations of the optical response of the detectors show that the wavelength range of the detector is not solely determined by its geometry, but the optical conductivity of the disordered thin metallic films contributes considerably. This contribution can be conveniently expressed by the ratio of imaginary and real parts of the optical conductivity. This knowledge can be utilized in detector design.

Strengthening the security of digital communication channels by building Quantum Key Distribution (QKD) infrastructure is an ongoing effort worldwide[1, 2]. Most QKD systems are based on entangled photon pairs as quantum bits and utilize the existing, low-loss fiber optical infrastructure [3]. To implement an efficient algorithm, a single-photon detector with a high overall detection efficiency, low dark counts, and high speed is required[4]. The State-of-the-art detectors that fulfill these requirements are the superconducting nanowire single-photon detectors (SNSPD)[5]. They are superior in terms of detection efficiency and dark count rates to other types[6], such as the Avalanche Photodiodes (APD)[7], and do not require sub-kelvin temperatures, as the Transition Edge Sensors (TES)[8] do. Moreover, their superior properties make them beneficial in a broad range of applications in different areas of physics, for example, physical chemistry and spectroscopy[9], fluorescent luminescence[10], or fast space-to-ground communication[11] often requiring a lower wavelength range. The detector’s wavelength can be optimized by its design depending on the required spectral range. This is also important in the implementation of modern quantum networks[3, 12] as not just the detection efficiency of the detector, but its bandwidth is a relevant parameter.

Figure 1: Scheme of the SNSPD with coupled optical fiber. Nanowire has thickness t 𝑡 t italic_t, width w 𝑤 w italic_w, and spacing s 𝑠 s italic_s. Optical λ 𝜆\lambda italic_λ/4 resonator consists of a dielectric spacer with thickness d 1 subscript 𝑑 1 d_{1}italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and a gold reflector with d 2 subscript 𝑑 2 d_{2}italic_d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. The gap between optical fiber and nanowire is g 𝑔 g italic_g. Right: Top-down view of the nanowire of diameter 2⁢r 2 𝑟 2r 2 italic_r, with red line showing the position of the cross-sectional view.

The working principle of SNSPD is straightforward: a nanowire in the superconducting state is current-biased close to the edge of the normal metal transition. The absorption of a single photon results in the destruction of superconductivity at some place along the nanowire. Due to the current bias, the normal state region expands, which results in a voltage pulse across the nanowire. These nanowires are fabricated on 6-15 nm thick, highly disordered superconductors, such as Niobium Nitride[13], Titanium Niobium Nitride[14], Tungsten Silicide[15] and their width is in the range of 30-200 nm.

The overall system detection efficiency (SDE) of detectors is determined by the efficiency of the photon coupling to the active area of the detector η coupling subscript 𝜂 coupling\eta_{\mathrm{coupling}}italic_η start_POSTSUBSCRIPT roman_coupling end_POSTSUBSCRIPT, the photon absorption efficiency of the superconductor η abs subscript 𝜂 abs\eta_{\mathrm{abs}}italic_η start_POSTSUBSCRIPT roman_abs end_POSTSUBSCRIPT, and by the above-described efficiency of transformation of the absorbed photon to electric signal η intrinsic subscript 𝜂 intrinsic\eta_{\mathrm{intrinsic}}italic_η start_POSTSUBSCRIPT roman_intrinsic end_POSTSUBSCRIPT and is given as[16]:

SDE=η coupling⋅η abs⋅η intrinsic,SDE⋅subscript 𝜂 coupling subscript 𝜂 abs subscript 𝜂 intrinsic\mathrm{SDE}=\eta_{\mathrm{coupling}}\cdot\eta_{\mathrm{abs}}\cdot\eta_{% \mathrm{intrinsic}},roman_SDE = italic_η start_POSTSUBSCRIPT roman_coupling end_POSTSUBSCRIPT ⋅ italic_η start_POSTSUBSCRIPT roman_abs end_POSTSUBSCRIPT ⋅ italic_η start_POSTSUBSCRIPT roman_intrinsic end_POSTSUBSCRIPT ,(1)

The absorption of a nanowire (thin film) η abs subscript 𝜂 abs\eta_{\mathrm{abs}}italic_η start_POSTSUBSCRIPT roman_abs end_POSTSUBSCRIPT on bare substrate is insufficient (below 30%percent%%, Fig. 2). To increase the absorption at the desired wavelength (λ=𝜆 absent\lambda=italic_λ =1550 nm) the nanowires are commonly fabricated in an optical resonator, at the cost of limiting its bandwidth. A common choice is a λ/4 𝜆 4\lambda/4 italic_λ / 4 resonant cavity [17].

This resonator consists of a thin dielectric layer with thickness d 1 subscript 𝑑 1 d_{1}italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, matching its optical length Λ Λ\Lambda roman_Λ with λ/4 𝜆 4\lambda/4 italic_λ / 4 of the desired light’s wavelength. The nanowire is fabricated on one side of the dielectric and the other side is covered with a low-loss mirror metallic layer (as shown in Fig. 1). The resonator enhances the absorption of the photons in the nanowire close to 100%percent%% (Fig. 2) by creating an antinode of the electric field of TE mode in the superconductor. The detectable light is guided by an optical fiber to this structure.

To properly model the optical properties of this structure, and thus to maximize the absorption η abs subscript 𝜂 abs\eta_{\mathrm{abs}}italic_η start_POSTSUBSCRIPT roman_abs end_POSTSUBSCRIPT in the nanowire, not only has the dielectric thickness d 1 subscript 𝑑 1 d_{1}italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT to be optimized, but the precise optical properties of the disordered metal have to be considered. However, their optical properties in the metallic state are not trivial due to the presence of quantum corrections, which the standard Drude model fails to describe [18]. This means that, to model their optical absorption, the optical properties of the specific thin film have to be either measured in the infrared spectra, or the right optical model - or an extrapolation method - has to be chosen, if measurements in the required range are not accessible.

In this paper, we investigate the dependence of the optical absorption of NbN nanowires in λ/4 𝜆 4\lambda/4 italic_λ / 4 resonant cavity on film thickness and nanowire design. The optical conductivity of a set of NbN films was determined from spectroscopic ellipsometry in the visible range, and they were fitted by the modified Drude-Lorentz model, which takes the quantum corrections into account [19] and provides an excellent fit of these conductivities. The presence of quantum corrections in the IR range results in a strong wavelength and thickness dependence of the optical properties of NbN films. The obtained optical fits and thickness dependencies of its parameters are used to model the absorption spectra of nanowires. The presented approach can be applied to other disordered films, such as MoC and NbTiN. The optical properties of the latter[20] are almost identical to the properties of NbN.

Figure 2: Nanowire optical absorption η abs subscript 𝜂 abs\eta_{\mathrm{abs}}italic_η start_POSTSUBSCRIPT roman_abs end_POSTSUBSCRIPT on bare sapphire substrate and on λ/4 𝜆 4\lambda/4 italic_λ / 4 resonator.

2 Optical properties of NbN films

Niobium nitride samples were deposited by means of Pulsed Laser Deposition[21] from a 99% pure niobium target on top of single-side polished sapphire substrate in a nitrogen atmosphere with 1% of hydrogen. The film thickness varied from 8 up to 22 nm. The optical conductivities of our films were determined from spectroscopic ellipsometry measured in wavelength range from 400 to 1000 nm at room temperature.

In Ref.[19] the quantum corrections to the optical conductivity of these NbN films were thoroughly studied. These corrections are arising from localization and interaction effects both having a square root energy dependence[22]. The optical conductivity σ=σ 1+i⁢σ 2𝜎 subscript 𝜎 1 𝑖 subscript 𝜎 2\tilde{\sigma}=\sigma_{1}+i\sigma_{2}over~ start_ARG italic_σ end_ARG = italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_i italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is described by the modified Drude-Lorentz model in the form:

σ 1⁢(ω)=subscript 𝜎 1 𝜔 absent\displaystyle\sigma_{1}(\omega)=italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_ω ) =σ D 1+(Ω Γ)2⁢(1−𝒬 2⁢(1−Ω Γ)⁢e−1 2⁢(Ω Γ)2)+limit-from subscript 𝜎 𝐷 1 superscript Ω Γ 2 1 superscript 𝒬 2 1 Ω Γ superscript 𝑒 1 2 superscript Ω Γ 2\displaystyle\frac{\sigma_{D}}{1+(\frac{\Omega}{\Gamma})^{2}}\Big{(}1-\mathcal% {Q}^{2}(1-\sqrt{\frac{\Omega}{\Gamma}})e^{-\frac{1}{2}(\frac{\Omega}{\Gamma})^% {2}}\Big{)}+divide start_ARG italic_σ start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT end_ARG start_ARG 1 + ( divide start_ARG roman_Ω end_ARG start_ARG roman_Γ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( 1 - caligraphic_Q start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 1 - square-root start_ARG divide start_ARG roman_Ω end_ARG start_ARG roman_Γ end_ARG end_ARG ) italic_e start_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( divide start_ARG roman_Ω end_ARG start_ARG roman_Γ end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ) +(2) σ L 1+(Ω L 2−Ω 2 Ω⁢Γ L)2,subscript 𝜎 𝐿 1 superscript superscript subscript Ω 𝐿 2 superscript Ω 2 Ω subscript Γ 𝐿 2\displaystyle\frac{\sigma_{L}}{1+\left(\frac{\Omega_{L}^{2}-\Omega^{2}}{\Omega% \Gamma_{L}}\right)^{2}},divide start_ARG italic_σ start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_ARG start_ARG 1 + ( divide start_ARG roman_Ω start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - roman_Ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG roman_Ω roman_Γ start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ,

σ 2⁢(ω)=ℋ⁢[σ 1⁢(ω)]−(ε∞−1)⁢ε 0⁢ω subscript 𝜎 2 𝜔 ℋ delimited-[]subscript 𝜎 1 𝜔 subscript 𝜀 1 subscript 𝜀 0 𝜔\displaystyle\sigma_{2}(\omega)=\mathcal{H}[\sigma_{1}(\omega)]-(\varepsilon_{% \infty}-1)\varepsilon_{0}\omega italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_ω ) = caligraphic_H [ italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_ω ) ] - ( italic_ε start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT - 1 ) italic_ε start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_ω(3)

respectively. Here, σ D subscript 𝜎 𝐷\sigma_{D}italic_σ start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT is the Drude conductivity, Γ Γ\Gamma roman_Γ is the electron relaxation rate and 𝒬 𝒬\mathcal{Q}caligraphic_Q is the strength of the quantum corrections, also referred to as quantumness. The second term in eq. 2 describes the transition peak at Ω L subscript Ω 𝐿\Omega_{L}roman_Ω start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT, with strength σ L subscript 𝜎 𝐿\sigma_{L}italic_σ start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT, and width Γ L subscript Γ 𝐿\Gamma_{L}roman_Γ start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT. This formula describes the anomalous suppression in the IR range and below, which is characteristic of these films. The optical conductivity for films with different thicknesses is shown in Fig.3 The same result can be represented by σ 1 subscript 𝜎 1\sigma_{1}italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT vs. σ 2 subscript 𝜎 2\sigma_{2}italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT plots in Fig.4, where the wavelength dependence is color-coded. The solid spirals are the fits with eqs. 2,3. The fitted parameters are listed in Tab. 1.

As can be seen in Fig. 3a, the Drude conductivity σ 0 subscript 𝜎 0\sigma_{0}italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT varies weakly with thickness, but the relaxation rate Γ Γ\Gamma roman_Γ and the quantumness 𝒬 𝒬\mathcal{Q}caligraphic_Q are strongly thickness dependent. This reflects the commonly observed suppression of DC conductivity and, most importantly, the significant changes in the optical properties of films, especially in the IR range. By lowering the thickness, the real part of the IR range conductivity is suppressed and the imaginary part changes its sign. The dielectric function

ε r⁢(ω)=ε 1+i⁢ε 2=1+i⁢σ⁢(ω)/(ϵ 0⁢ω)subscript 𝜀 𝑟 𝜔 subscript 𝜀 1 𝑖 subscript 𝜀 2 1 𝑖𝜎 𝜔 subscript italic-ϵ 0 𝜔\varepsilon_{r}(\omega)=\varepsilon_{1}+i\varepsilon_{2}=1+i\tilde{\sigma}(% \omega)/(\epsilon_{0}\omega)italic_ε start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( italic_ω ) = italic_ε start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_i italic_ε start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 1 + italic_i over~ start_ARG italic_σ end_ARG ( italic_ω ) / ( italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_ω )(4)

for various film thicknesses is plotted in Fig.3b. The known thickness dependence of these parameters allows us to conveniently model the optical properties of SNSPDs. The complex refraction index can be expressed in terms of permittivity, as:

n=ε r⁢(ω),𝑛 subscript 𝜀 𝑟 𝜔\tilde{n}=\sqrt{\varepsilon_{r}(\omega)},over~ start_ARG italic_n end_ARG = square-root start_ARG italic_ε start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ( italic_ω ) end_ARG ,(5)

Table 1: Parameters of optical model eq.(2) obtained from the ellipsometric data fit.

Figure 3: Specific conductivity (top) and permittivity (bottom) of thin NbN films, consisting of real (solid line) and imaginary (dashed line) parts. Thick lines are the result of spectroscopic ellipsometry measurement, and thin lines result from the model described above. Dots at zero frequency are the room temperature DC conductivities.

Figure 4: Wavelength-dependent real and imaginary part of conductivity for films with various thicknesses. The arrows show the mismatch in the simple rescaling of the conductivities, easily visible by the change in the angle θ 0 subscript 𝜃 0\theta_{0}italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT of arrows, which point at the conductivity at 1550 nm.

It is important to note, that not just the magnitude of the conductivity changes with the thickness, but the ratio of real to imaginary parts changes as well. This is easily visible by the change of the angle of the arrow pointing to the conductivity at λ 0=subscript 𝜆 0 absent\lambda_{0}=italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT =1550 nm in Fig. 4. This angle is defined by the tangent of real and imaginary parts of the optical conductivity as tan⁡(θ 0)=σ 2⁢(λ 0)/σ 1⁢(λ 0)subscript 𝜃 0 subscript 𝜎 2 subscript 𝜆 0 subscript 𝜎 1 subscript 𝜆 0\tan{\theta_{0}}=\sigma_{2}(\lambda_{0})/\sigma_{1}(\lambda_{0})roman_tan ( start_ARG italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG ) = italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) / italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) and is a characteristic of the films for a given thickness - it does not depend on the fill factor of the nanowire, since scaling the conductivity scales both real and imaginary part.

3 Optical absorption simulations

We studied the effect of the film’s conductivity on the absorption spectra of SNSPD with a similar detector design to [14]. The numerical simulations were carried out in a commercial frequency-domain EM solver[23]. The air gap between the fiber and the surface of the detector leads to additional Fabry-Perót resonances superposing the absorption spectra of the λ/4 𝜆 4\lambda/4 italic_λ / 4 resonator. This could be expressed in the wavelength dependence of the coupling parameter η coupling⁢(λ)subscript 𝜂 coupling 𝜆\eta_{\mathrm{coupling}}(\lambda)italic_η start_POSTSUBSCRIPT roman_coupling end_POSTSUBSCRIPT ( italic_λ ). In our 2D simulations, the optical fiber output is modelled as a plane wave coupled to air and we assume infinite size of the detector in the plane perpendicular to the light propagation, which corresponds to the ideal coupling η coupling=1 subscript 𝜂 coupling 1\eta_{\mathrm{coupling}}=1 italic_η start_POSTSUBSCRIPT roman_coupling end_POSTSUBSCRIPT = 1 and allows us to focus on the absorption in the nanowire η abs subscript 𝜂 abs\eta_{\mathrm{abs}}italic_η start_POSTSUBSCRIPT roman_abs end_POSTSUBSCRIPT.

The optical absorption of the SNSPD can be described by the effective impedance model[24], which approximates the thin superconducting meander as a lumped element effective homogeneous medium, with effective sheet conductance G=G 1+i⁢G 2𝐺 subscript 𝐺 1 𝑖 subscript 𝐺 2\tilde{G}=G_{1}+iG_{2}over~ start_ARG italic_G end_ARG = italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_i italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT:

G=σ⁢w w+s⁢t⁢[S/□],𝐺𝜎 𝑤 𝑤 𝑠 𝑡 delimited-[]S□\tilde{G}=\tilde{\sigma}\frac{w}{w+s}t\mathrm{\ [S/\square]},over~ start_ARG italic_G end_ARG = over~ start_ARG italic_σ end_ARG divide start_ARG italic_w end_ARG start_ARG italic_w + italic_s end_ARG italic_t [ roman_S / □ ] ,(6)

where σ⁢(t)𝜎 𝑡\tilde{\sigma}(t)over~ start_ARG italic_σ end_ARG ( italic_t ) is the optical conductivity (in units of S/m) of the NbN film with thickness t 𝑡 t italic_t, and the ratio f=w/(w+s)𝑓 𝑤 𝑤 𝑠{f={w}/{(w+s)}}italic_f = italic_w / ( italic_w + italic_s ) is the fill factor. This approximation is valid for the width and the thickness of the nanowires w,t≪λ much-less-than 𝑤 𝑡 𝜆 w,t\ll\lambda italic_w , italic_t ≪ italic_λ. As these parameters are usually w∼similar-to 𝑤 absent w\sim italic_w ∼30-200 nm, t∼similar-to 𝑡 absent t\sim italic_t ∼5-22 nm, and s∼similar-to 𝑠 absent s\sim italic_s ∼50-300 nm, this approximation holds well for λ>1⁢μ⁢m 𝜆 1 𝜇 m\lambda>1\mu\mathrm{m}italic_λ > 1 italic_μ roman_m. The validity of this approximation is demonstrated by numerical modeling of the absorption spectra of a detector with different nanowire widths and constant fill factor f=0.5 𝑓 0.5 f=0.5 italic_f = 0.5, shown in Fig. 5. The absorption spectra are almost identical for w≲400⁢n⁢m less-than-or-similar-to 𝑤 400 n m w\lesssim 400\mathrm{nm}italic_w ≲ 400 roman_n roman_m. The negligible differences in the absorption maximum (<0.5%absent percent 0.5<0.5%< 0.5 %) and its wavelength (<50 absent 50<50< 50 nm) may originate in simulation mesh discretization. The ≈\approx≈ 1% losses originate in the Au layer with d 2=100 subscript 𝑑 2 100 d_{2}=~{}100 italic_d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 100 nm.

Figure 5: Simulated absorption spectra of the detector for different nanowire widths and constant fill factor f=0.5 𝑓 0.5 f=0.5 italic_f = 0.5

In Fig. 6, we present the absorption spectra of a detector with 100 nm width NbN nanowire of different thicknesses t=8−22 𝑡 8 22 t=8-22 italic_t = 8 - 22 nm on SiO 2 layer with thickness d 1 subscript 𝑑 1 d_{1}italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT=250 nm. The absorption maximum corresponding to the optical length of an ideal λ/4 𝜆 4\lambda/4 italic_λ / 4 resonator is 4⁢Λ=4⁢n⁢d 1≈1600 4 Λ 4 𝑛 subscript 𝑑 1 1600 4\Lambda=4nd_{1}\approx 1600 4 roman_Λ = 4 italic_n italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≈ 1600 nm. For each thickness, the fill factor f 𝑓 f italic_f was tuned to maximize the absorption at λ 0 subscript 𝜆 0\lambda_{0}italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT= 1550 nm. The maximal absorptions at λ 0 subscript 𝜆 0\lambda_{0}italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and the wavelength of the absolute maximal absorption λ 𝜆\lambda italic_λ are listed in Table 2. Note, that for this analysis, the optical properties of NbN films were extracted from ellipsometric measurements on films deposited on c-cut sapphire substrate. The optical properties of films deposited on different substrates, for example, SiO 2 or Si 3 N 4, can differ. However, the presented results can be easily generalized for any combination of film and dielectric material, as we will show below.

Figure 6: Simulated absorption spectra of nanowires with different thicknesses for fill factor maximizing absorption at 1550 nm.

Table 2: Simulated maximum absorption at 1550 nm and the optimized fill factor for 1550 nm, wavelength of maximum absorption, and maximum absorption value for different thickness

The absorption spectra for t=𝑡 absent t=italic_t = 8, 9, 14, 22 nm are shown in Fig.6. As it is visible, due to the thickness dependence of the optical conductivity (refraction index), the thickness of the film significantly shifts the position of the absorption maxima and changes the shape of the absorption spectra. The maximum of the absorption value max(η a⁢b⁢s subscript 𝜂 𝑎 𝑏 𝑠\eta_{abs}italic_η start_POSTSUBSCRIPT italic_a italic_b italic_s end_POSTSUBSCRIPT), at its peak wavelength λ m⁢a⁢x subscript 𝜆 𝑚 𝑎 𝑥\lambda_{max}italic_λ start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT, is also lowered for thicker films due to the increased reflection, but it is negligible compared to the drop of absorption at the desired wavelength λ 0 subscript 𝜆 0\lambda_{0}italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 1550 nm. To study how G 1 subscript 𝐺 1 G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G 2 subscript 𝐺 2 G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT separately affect the absorption spectra, we simulated the absorption of the nanowire with w=100 𝑤 100 w=100 italic_w = 100 nm, s=100 𝑠 100 s=100 italic_s = 100 nm (f=0.5 𝑓 0.5 f=0.5 italic_f = 0.5), t=10 𝑡 10 t=10 italic_t = 10 nm on d 1=250 subscript 𝑑 1 250 d_{1}=250 italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 250 nm SiO 2 dielectric and d 2=100 subscript 𝑑 2 100 d_{2}=100 italic_d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 100 nm Au reflector.

The absorption spectra for varying G 1 subscript 𝐺 1 G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and constant G 2 subscript 𝐺 2 G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT=0 reveal that the real part of the effective sheet conductance G 1 subscript 𝐺 1 G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT (σ 1 subscript 𝜎 1\sigma_{1}italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, respectively) mainly affects the value of the absorption maxima, but not their position λ m⁢a⁢x subscript 𝜆 𝑚 𝑎 𝑥\lambda_{max}italic_λ start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT, shown in Fig.7a. On the other hand, variation of G 2 subscript 𝐺 2 G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT(σ 2 subscript 𝜎 2\sigma_{2}italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, respectively) for constant G 1 subscript 𝐺 1 G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT results in the shift of absorption peak λ m⁢a⁢x subscript 𝜆 𝑚 𝑎 𝑥\lambda_{max}italic_λ start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT, shown in Fig.7b. The simulated range of tan⁡(θ)𝜃\tan{\theta}roman_tan ( start_ARG italic_θ end_ARG ) is approx. 0 - 0.4, which is comparable to the values of our films (0 - 0.32). Thus, the resonant absorption peak of the detector is shifted by the imaginary part of the conductance towards lower values and thus, its wavelength λ m⁢a⁢x subscript 𝜆 𝑚 𝑎 𝑥\lambda_{max}italic_λ start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT is not solely determined by the thickness of the dielectric layer, as one could simply assume. This effect, governed by tan⁡(θ)𝜃\tan{\theta}roman_tan ( start_ARG italic_θ end_ARG ), is universal and should be present in any thin film with similar values of tan⁡(θ)𝜃\tan{\theta}roman_tan ( start_ARG italic_θ end_ARG ).

Figure 7: Optical absorption in nanowire with fixed G 2=0⁢m⁢S subscript 𝐺 2 0 𝑚 𝑆 G_{2}=0\ mS italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0 italic_m italic_S (left) and varying G 1 subscript 𝐺 1 G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, and with fixed real effective conductance G 1=2.5⁢m⁢S subscript 𝐺 1 2.5 𝑚 𝑆 G_{1}=2.5\ mS italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 2.5 italic_m italic_S (right).

The dependence of 1/λ m⁢a⁢x 1 subscript 𝜆 𝑚 𝑎 𝑥 1/\lambda_{max}1 / italic_λ start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT on the conductivity of the NbN films, represented by tan⁡(θ)𝜃\tan{\theta}roman_tan ( start_ARG italic_θ end_ARG ), is linear for a broad range of tan⁡(θ)𝜃\tan{\theta}roman_tan ( start_ARG italic_θ end_ARG ), as is visible in Fig.8, and can be approximated as:

1 λ m⁢a⁢x⁢(σ 2)=1 λ m⁢a⁢x⁢(σ 2=0)+1 λ c⁢tan⁡θ,1 subscript 𝜆 𝑚 𝑎 𝑥 subscript 𝜎 2 1 subscript 𝜆 𝑚 𝑎 𝑥 subscript 𝜎 2 0 1 subscript 𝜆 𝑐 𝜃\frac{1}{\lambda_{max}(\sigma_{2})}={\frac{1}{\lambda_{max}(\sigma_{2}=0)}+% \frac{1}{\lambda_{c}}\tan\theta},divide start_ARG 1 end_ARG start_ARG italic_λ start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT ( italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG = divide start_ARG 1 end_ARG start_ARG italic_λ start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT ( italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0 ) end_ARG + divide start_ARG 1 end_ARG start_ARG italic_λ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG roman_tan italic_θ ,(7)

where λ c subscript 𝜆 𝑐\lambda_{c}italic_λ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT is a fitting parameter. The value λ c subscript 𝜆 𝑐\lambda_{c}italic_λ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT is governed by the optical properties of the dielectric layer and its thickness. For the studied geometry of the detector on SiO 2, λ c∼4.2 similar-to subscript 𝜆 𝑐 4.2\lambda_{c}\sim 4.2 italic_λ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ∼ 4.2 μ 𝜇\mu italic_μ m. Here, we assumed tan⁡θ 𝜃\tan\theta roman_tan italic_θ to be constant and equal to tan⁡θ 0 subscript 𝜃 0\tan\theta_{0}roman_tan italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, which is a feasible approximation in the vicinity of λ 0 subscript 𝜆 0\lambda_{0}italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, as is shown in Fig. 10.

Since tan⁡(θ)𝜃\tan{\theta}roman_tan ( start_ARG italic_θ end_ARG ) increases with decreasing wavelength, the positive feedback further lowers the resonance wavelength.

Figure 8: Dependance of wavelengths of the simulated absorption maxima on the conductivity ratio tan⁡(θ)=σ 2/σ 1=G 2/G 1 𝜃 subscript 𝜎 2 subscript 𝜎 1 subscript 𝐺 2 subscript 𝐺 1\tan{\theta}=\sigma_{2}/\sigma_{1}=G_{2}/G_{1}roman_tan ( start_ARG italic_θ end_ARG ) = italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT / italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT / italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, given by samples with different thicknesses. Left: On top of the 250nm SiO 2 substrate, Right: On top of the 200 nm Si 3 N 4 membrane

This simple relation can be directly implemented in the detector design. The absorption maximum shift due to the imaginary part of the conductivity (contribution of the thin film), for a given dielectric layer, limits the possible range of film thicknesses.

However, using e.g. a commercially available 200 nm thick Si 3 N 4 membrane, the resonance wavelength determined by the membrane is 4⁢Λ≈1800 4 Λ 1800 4\Lambda\approx 1800 4 roman_Λ ≈ 1800 nm and the corresponding absorption at λ 0 subscript 𝜆 0\lambda_{0}italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is thus lower, as can be seen in Fig. 8. Using thicker NbN films, the resonance wavelength can be lowered and the absorption can be significantly increased, see Tab.3. For each film thickness, the fill factor was optimized to obtain the maximum absorption at λ 0 subscript 𝜆 0\lambda_{0}italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Thus, fine-tuning the absorption peak position up to 200 nm is possible, without significant loss of total optical absorption η a⁢b⁢s subscript 𝜂 𝑎 𝑏 𝑠\eta_{abs}italic_η start_POSTSUBSCRIPT italic_a italic_b italic_s end_POSTSUBSCRIPT.

Table 3: Table of sample parameters with simulated maximum absorption at 1550 nm, optimized fill factor for 1550 nm, the wavelength of maximum absorption, and maximum absorption value, respectively.

The fit parameter for the 200 nm Si 3 N 4 membrane is λ c=6.3⁢μ subscript 𝜆 𝑐 6.3 𝜇\lambda_{c}=6.3\mu italic_λ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = 6.3 italic_μ m, estimated from the linear fit in Fig. 8.

3.1 Discussion

In similar studies, the properties of a single deposited film are often measured (for example by spectroscopic ellipsometry), and the determined specific conductivity is considered to be thickness-independent in the simulations. Consequently, the nanowire’s thickness and width are then tuned to maximize the optical absorption in the nanowire at the aimed wavelength λ 0 subscript 𝜆 0\lambda_{0}italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. For example, in Ref.[25], or Ref.[26] the same refraction index was used for samples with thicknesses ranging from 4 to 10 nm, in case of Ref.[13] even from 1 to 16 nm. Rescaling the thickness of a specific film to a requested value results in the inaccuracy of the simulated absorption. This fact is acknowledged in [24], however, no further analysis is present. Rarely, the refractive index is also considered to be wavelength independent[25].

To illustrate the error arising from this approximation, we simulated the optical absorption in the above-described nanowire for the respective conductivity σ⁢(t)𝜎 𝑡\tilde{\sigma}(t)over~ start_ARG italic_σ end_ARG ( italic_t ), and the corresponding optical indices, given by eq. 5. For example, using the conductivity of the film with t m⁢e⁢a⁢s=subscript 𝑡 𝑚 𝑒 𝑎 𝑠 absent t_{meas}=italic_t start_POSTSUBSCRIPT italic_m italic_e italic_a italic_s end_POSTSUBSCRIPT =14 nm and simply rescaling it to t=9 𝑡 9 t=9 italic_t = 9 nm for the simulation would result in a decrease of the simulated absorption maxima by approximately 200 nm, as can be seen in Fig.9.

Figure 9: Simulated optical absorption of 9 nm thick sample on top of the resonant cavity. Other curves show the change in optical absorption of the optical conductivity (refraction index) taken from a sample with different thicknesses.

If one assumes the optical conductivity (refraction index) of these films to be wavelength-independent, the error in the absorption spectra will accumulate with the distance from the reference wavelength, as is shown in Fig.10.

Figure 10: Simulated absorption of a sample with wavelength dependent n⁢(λ)𝑛 𝜆 n(\lambda)italic_n ( italic_λ ), constant n⁢(λ=1000,1300,1550⁢n⁢m)𝑛 𝜆 1000 1300 1550 n m n(\lambda=1000,1300,1550\mathrm{nm})italic_n ( italic_λ = 1000 , 1300 , 1550 roman_n roman_m ) and n 𝑛 n italic_n corresponding to constant σ⁢(λ=1550⁢n⁢m)𝜎 𝜆 1550 n m\sigma(\lambda=1550\mathrm{nm})italic_σ ( italic_λ = 1550 roman_n roman_m ), respectively.

4 CONCLUSION

We modelled the absorption spectra of NbN nanowires in a λ/4 𝜆 4\lambda/4 italic_λ / 4 resonator for different film thicknesses and showed that the optical conductivity of thin films significantly affects the absorption spectra. The optical conductivity of disordered metallic films is strongly affected by the presence of quantum corrections, resulting in nontrivial wavelength and thickness dependence of the optical conductivity. The real part of the conductivity determines the amplitude of absorption of the nanowire in the λ/4 𝜆 4\lambda/4 italic_λ / 4 optical resonator, whereas the imaginary part contributes to the wavelength shift of the maxima. The wavelength shift of the absorption maxima can be expressed in terms of the ratio of imaginary and real parts of the optical conductivity, which is a characteristic of a film with a particular thickness. This knowledge can be utilized in detector design and optimization of its optical properties. Thus, to properly simulate and optimize the absorption of the detectors at a required wavelength, it is necessary to work with the precise optical conductivity of the particular layers. These can be obtained by direct measurements of the samples in the required range, or by utilizing a model that describes the optical conductivity of the disordered samples well, as presented in Ref. [19].

5 ACKNOWLEDGMENT

This work was supported by the Slovak Research and Development Agency under the contract APVV-20-0425, and by the project skQCI (101091548), founded by the European Union (DIGITAL) and the Recovery and Resilience Plan of the Slovak Republic.

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