# Optical and transport properties of NbN thin films revisited

S. Kern  Department of Experimental Physics, Comenius University, SK-84248
Bratislava, Slovakia  P. Neilinger  Department of Experimental Physics,
Comenius University, SK-84248 Bratislava, Slovakia  Institute of Physics,
Slovak Academy of Sciences, Dúbravská cesta, Bratislava, Slovakia  M.
Poláčková  Department of Experimental Physics, Comenius University, SK-84248
Bratislava, Slovakia  M. Baránek  Department of Experimental Physics, Comenius
University, SK-84248 Bratislava, Slovakia  T. Plecenik  Department of
Experimental Physics, Comenius University, SK-84248 Bratislava, Slovakia  T.
Roch  Department of Experimental Physics, Comenius University, SK-84248
Bratislava, Slovakia  M. Grajcar  Department of Experimental Physics, Comenius
University, SK-84248 Bratislava, Slovakia  Institute of Physics, Slovak
Academy of Sciences, Dúbravská cesta, Bratislava, Slovakia

###### Abstract

Highly disordered NbN thin films exhibit promising superconducting and optical
properties. Despite extensive study, discrepancies in its basic electronic
properties persist. Analysis of the optical conductivity of disordered ultra-
thin NbN films, obtained from spectroscopic ellipsometry by standard Drude-
Lorentz model, provides inconsistent parameters. We argue that this
discrepancy arise from neglecting the presence of quantum corrections to
conductivity in the IR range. To resolve this matter, we propose a
modification to the Drude-Lorentz model, incorporating quantum corrections.
The parameters obtained from the modified model are consistent not only with
transport and superconducting measurements but also with ab initio
calculations. The revisited values describing conduction electrons, which
differ significantly from commonly adopted ones, are the electron relaxation
rate $\Gamma\approx 1.8~{}\textrm{eV}/\hbar$, the Fermi velocity $v_{F}\approx
0.7\times 10^{6}~{}\textrm{ms}^{-1}$ and the electron density of states
$N(E_{F})=2~{}$states of both spins/eV/$V_{\textrm{f.u.}}$.

##  I Introduction

Niobium nitride (NbN) stands out for its excellent properties, including
chemical stability, hardness, optical and superconducting characteristics.Toth
(2014); Pogrebnjak et al. (2016); Banerjee et al. (2018); Hazra et al. (2016)
Its relatively high superconducting critical temperature and large sheet
resistance make it suitable for applications such as superconducting nanowire
single photon detectors (SNSPDs) Gol’Tsman et al. (2001) and kinetic
inductance travelling wave parametric amplifiers (KITWPAs). Adamyan et al.
(2016) Transition metal nitrides, including NbN, nowadays garner interest as
plasmonic materials,Karl et al. (2020) exhibiting double epsilon-near-zero
(ENZ) behaviour. Bower et al. (2021) This behaviour means that the real part
of their dielectric function, $\epsilon_{r}(\omega)$, becomes zero at two
frequencies below the UV range. ENZ materials enable strong interaction of
light with plasma oscillations, offering a wide range of possibilities in
photonics.Wu et al. (2021) NbN has drawn attention as an ENZ material due to
its tunable plasma frequency through composition adjustments.Ran et al. (2021)
Moreover, the optical response directly influences the efficiency of SNSPDs,
emphasizing the importance of optical characterization of NbN thin films, as
highlighted in Refs. Semenov et al., 2009.

Despite more than 50 years of extensive study of NbN films, significant
disagreement persists with regards to some of their fundamental properties.
The primary source of mismatch arises from seemingly contradictory results
obtained through different measurement pathways.Semenov et al. (2009);
Sidorova et al. (2020) First, the characterisation of the disorder in NbN is
routinely obtained from transport and Hall effect measurements. This is done
via the Ioffe-Regel parameter $k_{F}l$, where $k_{F}$ is the Fermi wavevector
and $l$ is the electron mean free path. It is well known that $k_{F}l$ close
to unity can be obtained in thin NbN films, i.e. $l$ is comparable to electron
wavelength, and such highly disordered films are approaching metal-insulator
transition (MIT).Chockalingam et al. (2008); Pellan et al. (1990); Chand et
al. (2012) Alternatively, this criterion can be expressed via electron
scattering rate in energy units $\hbar\Gamma$, which is comparable to the
Fermi energy. Therefore, in highly disordered metals, $\hbar\Gamma$ is
expected to be a couple eV. Second, $\hbar\Gamma$ can be obtained from optical
response as well, namely as a parameter of the Drude model for the dielectric
function of conducting electrons. However, these measurements suggest
$\hbar\Gamma\approx 0.33~{}$ as obtained from fitting a Drude-Lorentz model to
ellipsometric data in the visible range,Semenov et al. (2009); Kuz’michev and
Motulevich (1983) which is an order of magnitude smaller than expected.

Thin NbN films, especially its $\delta-$phase, have superior superconducting
properties,Chand (2012) and can be deposited by various methods,Kafizas et al.
(2013); Volkov et al. (2019); Ziegler et al. (2012). They typically exhibit a
polycrystalline structure with grains of various sizes, each possessing a
relatively well-defined cubic lattice interrupted by vacancy defects. Lin et
al. (2013); Toth (2014) The grain boundaries consist of disordered NbN alloy,
often containing oxygen, as well.Lengauer and Ettmayer (1986); Cabanel et al.
(1990) Despite their granularity, NbN thin films can be considered as
homogeneously disordered metal, especially regarding their optical response.
This is natural in the case of high intergrain conductivityBeloborodov et al.
(2007) or for mean free path that is small in comparison to the grain size.
Reiss et al. (1986); Pellan et al. (1990)

It is known that the presence of disorder in metals, either granular or
homogenous, leads to quantum corrections (QCs) to the Drude conductivity. In
highly disordered metals, the density of states (DOS) at the Fermi level is
suppressed, suppressing their conductivity, as well. Altshuler and Aronov
(1985) The correction to the real part of conductivity in 3D homogeneously
disordered films, as obtained by Altshuler and Aronov in Ref. Altshuler and
Aronov, 1985 as well as from scaling arguments,McMillan (1981) can be
expressed in the following unified form Neilinger et al. (2019); Kaveh and
Mott (1982)

$\delta\sigma(\Omega)=-\mathcal{Q}^{2}\sigma_{0}\left(1-\sqrt{\frac{\Omega}{\Gamma}}\right).$
(1)

Here, $\sigma_{0}$ is the Drude conductivity, $\mathcal{Q}$ is the strength of
the correction, also called quantumness. Neilinger et al. (2019) The electron
relaxation rate relates is defined as reciprocal relaxation time
$\Gamma=1/\tau$. The energy $\Omega$ is determined by various energy scales,
such as incident photon energy $\hbar\omega$, temperature $\propto k_{B}T$, or
magnetic field $\propto\mu_{B}B$. Although this behaviour is routinely
observed at energies of the order of meV, it is rarely taken into account in
optical response analysis. Lee et al. (1995) Notably, in the study by
Neilinger et al.,Neilinger et al. (2019) the square root corrections (1) were
observed up to optical frequencies in MoC thin films. Although not directly
measured, a numerical study proposed a similar square root behavior in
NbN.Kern et al. (2021)

In the following, we argue that these corrections dramatically alter the
optical properties of NbN films and explain the ENZ phenomenon. Modeling the
optical conductivity by a quantum-corrected Drude-Lorentz model, we can
determine various quantities, including the diffusion coefficient and the
superconducting coherence length in agreement with the transport measurements.
Moreover, the determined carrier density and agrees with the DFT.
Additionally, we compare this model to commonly used models of the NbN
dielectric function. The experimental data used to establish a consistent set
of parameters, were obtained from spectroscopic ellipsometry (SE) on
$\delta-$NbN films of various thicknesses. Independent confirmation was
provided by transport and magneto-resistance measurements.

##  II Optical properties

Figure 1: Thick lines: real (solid) and imaginary (dashed) part of optical
conductivity for NbN films of various thickness, determined by SE. Thin lines
are fitted to Eq. (2). Circles: room-temperature DC conductivities measured by
van der Pauw method.

The optical properties of NbN thin films were investigated utilizing SE in the
range from 300 to 800 nm. Typically, SE is employed to determine the
parameters (commonly layer thickness) of a chosen optical model for the
studied sample. In our examination, where the goal is to select an appropriate
optical model, we evaluate the optical constants directly from the SE data for
each wavelength separately via a model-independent way (for details see Ref.
Neilinger et al., 2021).

The complex conductivity in the visible spectral range, determined from SE
(thick lines in Fig. 1) is smooth and exhibits a clear dependence on the
sample thickness. The real part of $\sigma(\omega)$ decreases with frequency
for all samples which is attributed to the Drude-like $1/\omega^{2}$ behaviour
typical for metals with finite relaxation rate $\Gamma$.

In Ref. Kern et al., 2021, a similar NbN film was studied, and a conductivity
peak in the UV range was determined by means of numerical extrapolation. This
spectral weight was attributed to inter-band transition at $\hbar\omega\approx
5-7~{}$eV. The presence of the inter-band transition, was observed through
optical measurements Sanjinés et al. (2006); Ran et al. (2021) and various ab-
initio simulation, as well.Fong and Cohen (1972); Pflüger et al. (1985); Ran
et al. (2021)

Furthermore, the numerical study presented in our previous work suggests that
thin NbN films exhibit suppression of $\sigma_{r}(\omega)$ in the IR range,
accompanied by a peak at $\approx 1$ eV. This ”anti-Drude” behaviour is also
known as anomalous or displaced Drude peak. In Ref. Bower et al., 2021, the
peak in the optical conductivity of NbN films was explained in terms of an
effective medium emerging from granular NbN dissolved in an insulating NbO
matrix. We discuss this approach in Appendix A, where we argue that it is not
appropriate for NbN, as it leads to unphysical conclusions.Cocker et al.
(2017) In Ref. Cocker et al., 2017, the peak was obtained within the Drude-
Smith model, which we analyze in Appendix B. In Ref. Lee et al., 1995, the
$\sigma_{r}(\omega)$ of metallic films close to MIT exhibited a square root
dependence at frequencies below IR range, which was ascribed to quantum
corrections to the Drude conductivity due to disorder. In order to describe
the $\sigma_{r}(\omega)$ in the optical range, a ”localization-modified Drude
model” was suggested as a simple multiplication of the Drude formula with a
square root term similar to Eq. (1).Kaveh and Mott (1982) However, this model
leads to incorrect behaviour of the conductivity at high frequencies, which
should drop as $\omega^{-2}$. Moreover, this model poorly fits our data. The
same square root behaviour of $\sigma_{r}(\omega)$ up to the visible range was
observed in Ref. Neilinger et al., 2019. But, it was suggested here that the
influence of disorder should disappear at frequencies of the order of the
relaxation rate. A square root correction was smoothly joined to the bare
Drude conductivity at a crossover frequency, and an excellent agreement with
the SE data was further confirmed by independent transmission measurements.
Therefore, we analyse our SE data utilizing the quantum corrected Drude model,
extended by a Lorentzian peak centred at $\omega_{1}$ with strength
$\sigma_{1}$ and width $\Gamma_{1}$, describing the inter-band transition. The
quantum correction exponentially vanishes at a scale $\Gamma/2$. Finally, we
obtained following model for the complex conductivity

$\displaystyle\sigma_{r}(\omega)=$
$\displaystyle\frac{\sigma_{0}}{1+(\nicefrac{{\omega}}{{\Gamma}})^{2}}\Big{(}1-\mathcal{Q}^{2}(1-\sqrt{\nicefrac{{\Omega}}{{\Gamma}}})e^{-\frac{1}{2}(\nicefrac{{\omega}}{{\Gamma}})^{2}}\Big{)}+$
(2)
$\displaystyle\frac{\sigma_{1}}{1+\left(\frac{\omega_{1}^{2}-\omega^{2}}{\omega\Gamma_{1}}\right)^{2}},$
$\displaystyle\sigma_{i}(\omega)=\mathcal{H}[\sigma_{r}(\omega)]-(\epsilon_{\infty}-1)\epsilon_{0}\omega.$
(3)

To take into account the temperature smearing of the steep square root
correction at low photon energies, following Ref. Neilinger et al., 2019, an
energy scale entering the correction was introduced as
$\Omega=\sqrt{\omega^{2}+(\pi k_{B}T/\hbar)^{2}}$.
$\mathcal{H}[\sigma_{r}(\omega)]$ denotes the Hilbert transform of
$\sigma_{r}(\omega)$, mathematically manifesting the Kramers-Kronig relations.
The Hilbert transform can be performed numerically, or by approximative
analytical formula for $\mathcal{H}[\sigma_{r}(\omega)]$, which we derived in
Appendix C. This model of $\sigma_{r}(\omega)$ does not consider the inter-
band transitions of bound electrons at high energies, which affects the
imaginary conductivity even at small frequencies. Their effect is therefore
expressed by the second term in Eq. (3) via the parameter
$\epsilon_{\infty}=1.6$ estimated in Appendix D.

In the Fig. 1, it is shown that the proposed model (thin lines) produces an
excellent fit to both the real and the imaginary parts of the conductivity.
The parameters of the best fit are listed in Table 1. The Drude conductivity
$\sigma_{0}$ and the parameters of the inter-band transition peak exhibit no
clear dependence on thickness. On the contrary, the scattering rate $\Gamma$
slightly increases with decreasing thickness, as expected. As $\Gamma$ rises,
$\mathcal{Q}$ increases, too. This comes from the expression for $\mathcal{Q}$
being $\approx 1/k_{F}l$, where $l=v_{F}/\Gamma$ and $v_{F}=\hbar k_{F}/m_{e}$
is the Fermi velocity, which we estimated as
$v_{F}=\sqrt{\hbar\Gamma/(\mathcal{Q}m_{e})}\approx 0.7-0.8\times
10^{6}~{}$ms-1.Altshuler and Aronov (1985); Kaveh and Mott (1982)

Figure 2: The real part of the dielectric function $\epsilon(\omega)$
corresponding to the conductivities in Fig. 1. The inset shows the lower
plasma frequencies (frequencies at which $\epsilon(\omega)=0$) dependent on
quantumness $\mathcal{Q}$. The solid line is a plot according to Eq. (5)

The peak at 5-7 eV agrees with the predictions from the joint density of
states calculation. Pflüger et al. (1985) Similarly, the joint density of
states calculation predicted transitions between the three highest occupied
bands, leading to a peak in the dielectric function at approx. 1 eV.
Therefore, it is tempting to assign the anomalous Drude peak to the inter-band
transitions, as was done in Refs. Semenov et al., 2009; Kuz’michev and
Motulevich, 1983. However, as can be seen from the angle-resolved
photoemission spectroscopy (ARPES) measurement, the disorder and/or thickness-
enhanced scattering smears the electron structure at the scale
$\hbar\Gamma\approx 1.5~{}$eV.Yu et al. (2021) Furthermore, the DFT
simulations suggest that such smearing is necessary to stabilize the
crystalline structure of $\delta-$NbN.Babu and Guo (2019) Therefore, these
inter-band transitions should not be present in the spectra. Also, modelling
the displaced Drude peak as an inter-band transition leads to a puzzling shift
of its central frequency to higher energies with decreasing thickness, whereas
for our model it is explained via the increase of the quantumness
$\mathcal{Q}$.

The corresponding real part of the dielectric function (Fig. 2) given as
$\epsilon_{r}(\omega)=1-\sigma_{i}(\omega)/\epsilon_{0}\omega$, exhibits the
discussed double ENZ feature. For $\mathcal{Q}=0$, the model gives (for
$\epsilon_{r}(\omega)$ and for the ordinary screened plasma frequency
$\omega_{p}$) the well-known results

$\epsilon_{r}(\omega)\approx\epsilon^{\prime}_{\infty}-\frac{\sigma_{0}/\epsilon_{0}\Gamma}{(1+(\omega/\Gamma)^{2})}\quad\rightarrow\quad\omega_{p}^{2}=\frac{\sigma_{0}\Gamma}{\epsilon_{0}\epsilon_{\infty}^{\prime}},$
(4)

where
$\epsilon_{\infty}^{\prime}=\epsilon_{\infty}+\sigma_{1}\Gamma_{1}/(\epsilon_{0}\Omega_{1}^{2})$
contains bound electrons contribution $\epsilon_{\infty}$ evaluated in
Appendix D and the second term is contribution from the inter-band transition
at $\approx 6~{}$eV. From Eq. (21), it can be shown, that for low energies
$\hbar\omega$, the imaginary part is dominated by the square root term. This
gives, for epsilon and for the newly unveiled second plasma frequency
$\omega_{p2}$

$\displaystyle\epsilon_{r}(\omega)$
$\displaystyle\approx\epsilon^{\prime}_{\infty}-\frac{\sigma_{0}}{\epsilon_{0}\Gamma}\left(1-\mathcal{Q}^{2}\sqrt{\frac{\Gamma}{\omega}}\right)\
\rightarrow\ $ (5) $\displaystyle\omega_{p2}$
$\displaystyle\propto\Gamma\mathcal{Q}^{4},$

clearly showing that, for non-negligible $\mathcal{Q}$, the second, lower
plasma frequency appears. Its value increases as $\mathcal{Q}^{4}$, whereas
the regular plasma frequency slightly decreases with $\mathcal{Q}$. In the
inset of Fig. 2, the second plasma frequencies of the samples are plotted. The
thickest sample with the lowest $\mathcal{Q}$ does not reach zero, as the
temperature smearing sufficiently suppresses the square root behaviour, which
is not taken into account in Eq. (5).

$d$ |  $\sigma_{0}$ |  $\hbar\Gamma$ |  $\mathcal{Q}$ |  $\sigma_{1}$ |  $\hbar\Gamma_{1}$ |  $\hbar\omega_{1}$
---|---|---|---|---|---|---
$(\textrm{nm})$ |  $(S\mu\textrm{m}^{-1})$ |  $(\textrm{eV})$ |  $(1)$ |  $(S\mu\textrm{m}^{-1})$ |  $(\textrm{eV})$ |  $(\textrm{eV})$
6.0 |  0.88 |  1.86 |  0.75 |  0.72 |  2.38 |  5.64
10.4 |  0.94 |  1.89 |  0.56 |  0.77 |  2.04 |  5.51
13.6 |  1.01 |  1.80 |  0.64 |  0.81 |  1.80 |  5.36
22.1 |  0.95 |  1.84 |  0.45 |  0.77 |  2.23 |  5.74
33.0 |  0.95 |  1.73 |  0.53 |  0.68 |  3.10 |  6.20
Table 1: Paremeters of optical model (2) and (3) providing the best fit to the
experimental data.

##  III Transport and superconducting properties

$d$ |  $L$ |  $\rho$ |  $R_{\square}$ |  $RRR$ |  $T_{c}$ |  $2k_{B}T_{c}$
---|---|---|---|---|---|---
$(\textrm{nm})$ |  $(\textrm{nm})$ |  $(\textrm{gcm}^{-3})$ |  $(\Omega)$ |  $(1)$ |  $(K)$ |  $(\textrm{meV})$
6.0 |  9 |  7.8 |  340 |  0.76 |  11.68 |  2.0
10.4 |  12 |  7.8 |  136 |  0.88 |  13.15 |  2.3
13.6 |  12 |  7.8 |  107 |  0.81 |  13.64 |  2.3
22.1 |  11 |  7.9 |  58 |  0.89 |  14.26 |  2.5
33.0 |  12 |  7.8 |  41 |  0.83 |  13.71 |  2.4
Table 2: Properties of thin NbN obtained from X-ray measurements and temperature-dependent transport measurements. RRR is obtained as $R_{\square}/R_{\square}(20~{}\textrm{K})$, where $R_{\square}$ is the room temperature sheet resistance. $d$ |  $k_{F}l$ |  $D_{opt}$ |  $D_{B_{c2}}$ |  $l$ |  $v_{F}$ |  $\xi_{0}$ |  $\xi_{GL}$ |  $B_{c2}$ |  $N(E_{F})$ |  $E_{F}-E_{c}$ |  $n$
---|---|---|---|---|---|---|---|---|---|---|---
$(\textrm{nm})$ |  (1) |  $(\textrm{cm}^{2}\textrm{s}^{-1})$ |  $(\textrm{cm}^{2}\textrm{s}^{-1})$ |  $(\textrm{\AA})$ |  $(10^{6}\textrm{ms}^{-1})$ |  $(\textrm{nm})$ |  $(\textrm{nm})$ |  $(\textrm{T})$ |  $(\textrm{eV}^{-1}V_{\textrm{f.u.}}^{-1})$ |  $(\textrm{eV})$ |  $(10^{28}\textrm{m}^{-3})$
|  $1/\mathcal{Q}$ |  $\frac{\hbar}{3\mathcal{Q}m_{e}}$ |  $-\frac{4k_{B}}{\pi e}\left(\frac{\partial B_{c2}}{\partial T}\right)^{-1}_{T_{c}}$ |  $\sqrt{\frac{\hbar}{\mathcal{Q}\Gamma m_{e}}}$ |  $\sqrt{\frac{\hbar\Gamma}{\mathcal{Q}m_{e}}}$ |  $\frac{\hbar v_{F}}{\pi\Delta}$ |  $0.855\sqrt{\xi_{BCS}l}$ |  $\Phi_{0}/2\pi\xi_{GL}^{2}$ |  $\frac{3\sigma_{0}\mathcal{Q}m_{e}}{\hbar e^{2}}$ |  $\frac{\hbar\Gamma}{2\mathcal{Q}}$ |  $\frac{\sigma_{0}\Gamma m_{e}}{e^{2}}$
6.0 |  1.33 |  0.51 |  0.73 |  2.34 |  0.66 |  69.2 |  3.44 |  27.0 |  2.27 |  1.24 |  8.82
10.4 |  1.78 |  0.69 |  0.68 |  2.68 |  0.77 |  70.2 |  3.71 |  23.1 |  1.81 |  1.69 |  9.57
13.6 |  1.56 |  0.60 |  0.60 |  2.57 |  0.70 |  64.1 |  3.47 |  26.5 |  2.23 |  1.41 |  9.80
22.1 |  2.22 |  0.86 |  0.63 |  3.03 |  0.85 |  71.1 |  3.97 |  20.2 |  1.47 |  2.04 |  9.42
33.0 |  1.88 |  0.73 |  0.57 |  2.88 |  0.76 |  66.1 |  3.73 |  22.9 |  1.73 |  1.63 |  8.86
Table 3: Electronic properties calculated from the fit parameters of the
proposed optical model to optical conductivity. For comparison, besides the
calculated diffusivity $D_{opt}$, diffusivity $D_{B_{c}2}$ obtained from the
temperature dependence of critical field $B_{c2}(T)$ is listed, too. For the
BCS coherence length $\xi_{0}$ we used the estimate of superconducting gap
from Table 2: $\Delta\approx 2k_{B}T_{c}$.

The introduced optical model, fitted to ellipsometric data, predicts the DC
conductivity that was independently evaluated as
$\sigma_{DC}(T=300~{}\textrm{K})=1/(R_{\square}d)$. Here, $R_{\square}$ is the
sheet resistance measured by the van der Pauw method at room temperature, and
$d$ is the thickness of the sample determined by the X-Ray reflection (XRR)
measurements. Both values are listed in table 2. The resulting DC conductivity
is plotted in Fig. 1 as dots, and they are perfectly recovered by the low-
frequency part of the optical conductivity fit. Here, we emphasize, that the
DC conductivity was not utilized during the fitting procedure.

In literature, it is common to compute the electronic parameters (i.e.
$k_{F}l$) from the measured DC conductivity $\sigma_{DC}=1/(R_{\square}d)$,
assuming it is equal to the Drude conductivity $\sigma_{0}$. However, within
our approach, the measured DC conductivity is
$\sigma_{DC}=\sigma_{0}(1-\mathcal{Q}^{2})$, therefore it can not be
interchanged with the Drude conductivity $\sigma_{0}$. Following Refs.
Altshuler and Aronov, 1985; Lee et al., 1995, we equate the quantumness
$\mathcal{Q}$ to $1/k_{F}l$, which enables us to easily estimate the Ioffe-
Regel parameter as well as the diffusivity $D=v_{F}l/3=\hbar
k_{F}l/(3m_{e})=\hbar/(3\mathcal{Q}m_{e})$.

To verify the diffusivity estimated from the optical measurements, the
magneto-resistance at low temperatures was measured (see Fig. 3), and the
temperature-dependence of the upper critical field $B_{c2}(T)$ was determined.
We start with the Ginzburg-Landau (GL) result for the upper critical magnetic
field

$B_{c2}=\frac{\Phi_{0}}{2\pi\xi_{GL}^{2}(T)},$ (6)

where $\Phi_{0}$ is the magnetic flux quantum and $\xi_{GL}$ is the GL
coherence length. In the dirty limit, $\xi_{GL}(T)$ satisfies

$\frac{1}{\xi_{GL}^{2}(T)}=\frac{1}{0.855^{2}\xi_{0}l}\frac{T_{c}-T}{T_{c}},$
(7)

where $\xi_{0}=\hbar v_{F}/(\pi\Delta)$ is the BCS coherence length, $T_{c}$
is the superconducting critical temperature, and $\Delta$ is the
superconducting gap. Recalling the BCS relation $\Delta=1.764\ k_{B}T_{c}$,
one can express the diffusivity via the temperature derivative of $B_{c2}$
asBartolf (2015)

$D_{B_{c2}}=-\frac{4k_{B}}{\pi e}\left(\frac{\partial B_{c2}}{\partial
T}\right)^{-1}_{T_{c}}.$ (8)

The diffusivities were estimated from the slope of $B_{c2}(T)$ curves showed
in Fig. 4. Comparison in the inset of Fig. 4 shows that the diffusivity
$D_{B_{c2}}$ estimated from the magneto-resistance is comparable to $D_{opt}$,
calculated from the optical model. The expected decrease of the diffusivity at
low thicknesses is present in $D_{opt}$, but $D_{B_{C2}}$ is increasing with
lowering the film thickness. This paradoxical behaviour was likewise observed
in Ref. Ezaki et al., 2012. In Ref. Shoji et al., 1992 relation (8) was
corrected to take into account that NbN is supposed to be a strong coupling
superconductor. However, this would lead to further increase of $D_{B_{c2}}$.
Alternatively, in Ref. Herman and Hlubina, 2016 the authors showed that the
broadened tunneling spectra of dirty superconductors, also known as Dynes
superconductors, such as NbNNoat et al. (2013); Chaudhuri et al. (2013), can
be explained by the presence of two types of scattering processes, namely the
pair-conserving and the pair-breaking scattering. The rate of these
scatterings is $\Gamma$ and $\Gamma_{D}$, respectively. They also calculated
the thermodynamic properties of these superconductor, expressing the GL
coherence length asHerman and Hlubina (2018)

$\frac{1}{\xi_{GL}^{2}(T)}=\frac{12[1-\zeta(2,\frac{1}{2}+\alpha)]}{\pi\zeta(2,\frac{1}{2}+\alpha)}\frac{\hbar\Gamma}{\Delta}\frac{k_{B}(T_{c}-T)}{\Delta}\frac{1}{\xi_{0}^{2}},$
(9)

where $\alpha=\hbar\Gamma_{D}/(2\pi k_{B}T_{c})$ and $\zeta(s,x)$ is the
Hurwitz zeta function. Then, diffusivity can be expressed as

$D^{\prime}_{B_{c2}}=-K(\alpha)\frac{k_{B}}{e}\left(\frac{\partial
B_{c2}}{\partial T}\right)^{-1}_{T_{c}}\\!\\!\\!,$ (10)

where $K=2\pi[1-\zeta(2,\frac{1}{2}+\alpha)]/\zeta(2,\frac{1}{2}+\alpha)$ and
can be approximated by the expression

$K(\alpha)\approx\frac{1.273+1.155\alpha}{1+2.432\alpha+2.206\alpha^{2}}.$
(11)

For $\alpha=0$, the function $K(\alpha)$ reduces to the BCS value $4/\pi$,
however, for thin films, $\alpha$ is non-zero and thus decreases the
diffusivity. Comparing the diffusivity values $D_{B_{c2}}$ for the 6 nm sample
and 30 nm samples, $\alpha=0.15$ was estimated. This yields an estimate of
$\hbar\Gamma_{\textrm{D}}=0.9$ meV, which is reasonable for thin NbN films.
$\hbar\Gamma_{\textrm{D}}$ estimated from tunneling spectra varies in the
range from negligibly small values up to tenths of meV, and even as large as
$\Delta/2$ for strongly disordered samples with suppressed $T_{c}$.Šindler et
al. (2014); Noat et al. (2013); Henrich (2014); Carbillet et al. (2020); Chand
(2012); Kamlapure et al. (2013)

The electron mean free path was estimated as $l=\sqrt{\hbar/\mathcal{Q}\Gamma
m_{e}}\approx 2-3~{}$Å is slightly above half of the lattice parameter
$a=4.4~{}$Å, which indicates that the samples are close to the Ioffe-Regel
limit $k_{F}l\to 1$. The Ginzburg-Landau coherence length $\xi_{GL}(0)$
estimated by both Eqs. (7) and (9) ranges from $3~{}$nm to $4~{}$nm, which
agrees with a commonly measured value in thin NbN films.Engel et al. (2006);
Jesudasan et al. (2011); Shoji et al. (1992)

Figure 3: Magnetic field variation of the temperature-dependent sheet
resistance $R_{\square}(T)$ for the 10 nm sample. Black lines are given by the
maximal slope of $R_{\square}(T)$ curves and the temperature of the
superconducting transition is determined by the intersect of the maximal slope
line (black solid lines) and the zero resistance line (black dotted line).
Figure 4: Temperature dependence of the upper critical field $B_{c2}$. The
solid lines are linear fits to the $B_{c2}(T)$ data. The color-coding is same
as in Fig. 1. The inset shows a comparison of the diffusivity obtained from
the slope of $B_{c2}(T)$ (green) and that estimated from the proposed optical
model (blue).

##  IV Discussion

It is now convenient to compare the revisited parameters of NbN with DFT
simulations. We estimated the density of carriers in NbN as
$n=\sigma_{0}\Gamma m_{e}/e^{2}=9\times
10^{28}~{}m^{-3}=2~{}V^{-1}_{\textrm{f.u.}}$, where $V_{\textrm{f.u.}}$ is the
formula unit volume, for $\delta$-NbN $V_{\textrm{f.u.}}=a^{3}/4$. This result
is in excellent agreement with the value obtained by integrating the DFT DOS
from the threshold of the peak responsible for the modeled inter-band
transition ($\approx E_{F}-4~{}$eV) up to $E_{F}$.Babu and Guo (2019); Amriou
et al. (2003); Sanjinés et al. (2006); Papaconstantopoulos et al. (1985);
Schwarz (1977); Palanivel et al. (1993) This also contradicts the common
assumption, that all 4 Nb d-orbital electrons are conducting.Chand (2012) We
should emphasize here, that the agreement of our estimation of the number of
electrons in the conductive band with the DFT result was obtained despite the
fact that the DFT DOS itself is two times smaller than the ”Drude” DOS, which
we determined from the Einstein relation $N(E_{F})=\sigma_{0}/(e^{2}D_{opt})$.
This can be explained by the fact that the determined bottom of the conductive
band $E_{c}=E_{F}-2~{}$eV is much closer to $E_{F}$ compared to DFT result
$4~{}$eV. The value of $E_{F}-E_{c}$ was estimated from free electron relation
between the density of electrons and DOS, given as
$n=2N(E_{F})(E_{F}-E_{c})/3$. Similar effect was observed in ARPES measurement
where the conductive band was significantly flattened in comparison to the DFT
calculation, naturally leading to higher DOS at the Fermi level.Yu et al.
(2021) The $1~{}$eV scale of smearing of the ARPES bandstructure also suggest
a high value of scattering rate $\Gamma$. Nevertheless, ARPES agrees with DFT
on the value of the Fermi momentum being approximately half of the $\Gamma$L
path with length $\sqrt{3}\pi/a=1.24$ Å-1. This agrees with the estimated
$k_{F}\approx 0.6$ Å-1.

For $N(E_{F})$, we obtained 2 states of both spins/eV/$V_{\textrm{f.u.}}$.
Omitting the corrections, i.e., utilizing $\sigma_{DC}$ instead of
$\sigma_{0}$, would lead to suppressed DOS
$N^{\prime}=(1-\mathcal{Q}^{2})N(E_{F})$ which is thickness-dependent.Sidorova
et al. (2020); Chockalingam et al. (2008) Here we point out that even though
in Ref. Chockalingam et al., 2008 this suppressed DOS $N^{\prime}$ from
transport measurements was successfully compared to DFT value, these are by
nature different. Moreover, it should be noted that the DFT calculations only
partially account for electron-electron or electron-phonon interactions, which
are significant in NbN,Geballe et al. (1966). This implies that even the
comparison of uncorrected Drude DOS $N(E_{F})$ with the DFT value 0.8 - 1.2
states of both spins/eV/Vf.u. is questionable.

##  V Conclusion

In conclusion, we argue that the quantum corrections to the conductivity of
NbN films are present at optical frequencies and significantly alter their
dielectric function. Therefore, we analyze their optical conductivities,
utilizing the quantum-corrected Drude-Lorentz model. The proposed model yields
to an excellent fit to the $\sigma(\omega)$ and provides parameters of the
electronic fluid such as: the electron concentration $n$, the diffusion
coefficient $D$, the Ioffe-Regel parameter $k_{F}l$, and the electronic
density of states $N(E_{F})$. The obtained diffusion coefficient agrees with
the magneto-transport measurement, moreover, the estimated $n$ is consistent
with ab initio simulations. The determined electron relaxation rate
$\hbar\Gamma\approx 1.8~{}$eV, consistent with the presence of high disorder
in NbN films, is an order of magnitude higher than the commonly considered
value obtained from standard optical models. This emphasizes the importance of
quantum corrections in the analysis. Moreover, various puzzling phenomena like
the double ENZRan et al. (2021), the increase of sheet resistance at lower
thicknesses, and inconsistencies in electron relaxation rates are explained by
this model. For other reported effects, such as increasing diffusivity with
lowering of the thickness, and high electron density of states, we have
suggested explanations, which could be verified by further experiments.

###  V.1 Acknowledgments

We thank to R. Hlubina and R. Martoňák for useful discussions. This work was
supported by the Slovak Research and Development Agency under the contracts
APVV-20-0425 and by the QuantERA grant SiUCs. M. Poláčková was supported by
the Slovak Research and Development Agency under Contract No. APVV-19-0303.

It is also the result of support under the Operational Program Integrated
Infrastructure for the projects: Advancing University Capacity and Competence
in Research, Development and Innovation (ACCORD, ITMS2014+:313021X329) and
UpScale of Comenius University Capacities and Competence in Research,
Development and Innovation (USCCCORD, ITMS 2014+:313021BUZ3), co-financed by
the European Regional Development Fund.

##  Appendix A Effective medium theory

In Ref. Bower et al., 2021, the double ENZ was explained via Maxwell-Garnet
(MG) effective medium theory, which describes the optical properties of
composite materials consisting of polarizable inclusions in an insulating
matrix. In the case of NbN, NbN nanoparticles are immersed in a matrix of
insulating niobium oxides. Taking the dielectric function of the oxide from
Ref. O’Hara et al., 2014, the resulting MG formula provides $\epsilon(\omega)$
with two zeros. However, such effective medium with an insulating matrix is
inevitably an insulator. Moreover, the presence of oxygen in the NbN films was
associated with the degree of porosity and it manifests itself in a
significant increase of the resistance and/or the residual-resistance ration
(RRR).Cabanel et al. (1990) Considering the sheet resistance and RRR (see
Table 2), our samples have negligible oxygen content. The presence of oxygen
should affect the density of the films as well,Cabanel et al. (1990) which was
estimated from XRR, to have a constant value of $7.8~{}$gcm$-3$, which is
close the ideal cubic NbN value.

If the matrix would be a bad conductor, with a conductivity peak at 5-7 eV,
the MG model indeed reproduces both the suppression of the DC conductivity and
the double ENZ behaviour. But, as can be seen in Fig. 5a, by varying the
volume fraction of the inclusion, the model interpolates between the Drude
conductivity and the conductivity of the poorly conducting matrix, i.e., the
conductivity varies in the whole frequency range. Besides the DC conductivity
suppression, $\sigma(\omega)$ would vary at energy $\approx\hbar\Gamma$, too,
and the optical peak would change rapidly its weight. None of this behaviour
was observed in the SE measurements.

##  Appendix B Drude-Smith model

Figure 5: a) Prediction of Maxwell-Garnet theory for a metallic inclusion in a
niobium oxide matrix for various volume fractions of the inclusion. b) Thin
lines are the Drude-Smith model curves obtained as best fit to data from
ellipsometry depicted by thick lines. Points at zero frequency are the
measured DC conductivities.

Another approach which yields to the anomalous Drude peak is the Drude-Smith
model, which is based on a material formed by granules whose boundaries cause
reflection of electrons. This model, as derived in Ref. Cocker et al., 2017,
leads to the following corrections to the Drude formula

$\sigma(\omega)=\frac{\sigma_{0}}{1-i\omega/\Gamma_{\textrm{diff}}}\left(1-\frac{c}{1-i\omega/a}\right).$
(12)

Here, the parameters $\Gamma_{\textrm{diff}}$ and $a$ are determined by the
granule size $L$, utilizing the following relations

$\Gamma_{\textrm{diff}}=\frac{1}{\tau}+\frac{2v_{th}}{L},\quad
a=\frac{12v_{th}}{L}\frac{\tau}{L/v_{th}+2\tau},$ (13)

where $v_{th}=\sqrt{k_{B}T/m_{e}}$ is the thermal velocity. This model fits
our experimental data very well (see Fig. 5b). However, the resulting granule
size is $5~{}$Å, which is at least one order of magnitude smaller than the
grain size estimated by X-Ray diffraction (XRD) measurement. For reasonable
parameters (the measured grain size $L\approx 10~{}$nm and relaxation rate
$1/\tau=\Gamma\approx 2~{}\textrm{eV}/\hbar$), this model leads to a displaced
Drude peak in THz frequency range (meV). Thus, the fit of this model to our
data produces unreasonable parameters. We do not claim that this effect is not
present as it still can play a role at much smaller energies, where our
measurements are not sensitive.

$d$ |  $\sigma_{0}$ |  $\hbar\Gamma_{\textrm{diff}}$ |  $c$ |  $a$ |  $\sigma_{1}$ |  $\hbar\Gamma_{1}$ |  $\hbar\Omega_{1}$
---|---|---|---|---|---|---|---
$(\textrm{nm})$ |  $(S\mu\textrm{m}^{-1})$ |  $(\textrm{eV})$ |  $(1)$ |  $(1)$ |  $(S\mu\textrm{m}^{-1})$ |  $(\textrm{eV})$ |  $(\textrm{eV})$
6.0 |  1.00 |  1.38 |  0.48 |  1.08 |  1.00 |  0.76 |  5.52
10.4 |  1.17 |  1.36 |  0.38 |  1.31 |  1.01 |  0.72 |  5.50
13.6 |  1.18 |  1.35 |  0.40 |  1.11 |  1.08 |  0.68 |  5.50
22.1 |  1.17 |  1.32 |  0.34 |  1.36 |  1.05 |  0.76 |  5.70
33.0 |  1.17 |  1.24 |  0.37 |  1.20 |  0.97 |  0.91 |  6.00
Table 4: Paremeters of Drude-Smith model (12) providing best fit to the
experimental data.

##  Appendix C Approximative formula for imaginary part of the modelled
conductivity

The Drude-Lorentz and Drude-Smith models are convenient because both the real
and the imaginary part are accessible in a simple closed formula. Therefore,
they can be easily implemented in a fitting procedure, which are
computationally less demanding. More complex models, typically expressing one
part of the dielectric function, usually require to compute the other one
numerically, as we have indeed done for the proposed model (2). However, we
derived a simple approximative analytical formula for the KK image of
$\sigma_{r}(\omega)$, too. We start by introducing the dimensionless frequency
$x=\omega/\Gamma$ and expanding the exponential function

$e^{-2x^{2}}=\Big{(}1+2x^{2}+\frac{1}{2}(2x^{2})^{2}+...\Big{)}^{-1}.$ (14)

Taking the first two terms, we obtained the approximative form of
$\sigma_{r}(x)$

$\tilde{\sigma}_{r}(x)=\frac{\sigma_{0}}{1+x^{2}}\Big{(}1-\mathcal{Q}^{2}\frac{1-\sqrt{x}}{1+2x^{2}}\Big{)}.$
(15)

Utilizing the fact that the real part of the conductivity is an even function
of $x$, the Hilbert transform of Eq. (15) is

$\mathcal{H}[\sigma_{r}(x)]=\frac{2x}{\pi}\mathrm{P}.\mathrm{V}.\int_{0}^{\infty}\frac{\sigma_{r}(s)}{x^{2}-s^{2}}dx.$
(16)

For the imaginary part $\tilde{\sigma}_{i}(x)$ we obtained

$\tilde{\sigma}_{i}(x)=\frac{\sigma_{0}x}{1+x^{2}}\Big{(}1-\mathcal{Q}^{2}\frac{ax^{2}-b+1/\sqrt{x}}{1+2x^{2}}\Big{)},$
(17)

where

$\displaystyle a$ $\displaystyle=2(2\sqrt{2}-2^{3/4}-1)\approx 0.293,$ (18)
$\displaystyle b$ $\displaystyle=(1-3\sqrt{2}+2^{7/4})\approx 0.121.$

The comparison of Eq. (17) (red dashed lines) with the numerical result (green
dashed lines) for various values of quantumness $\mathcal{Q}=0,\ 0.5,\ 0.75,\
1$ is in Fig. 6a. One can see a excellent match, except for the value
$\mathcal{Q}=1$, where a slight disagreement can be seen. This can be treated
by taking the next term in the expansion (14). We calculated the Hilbert
transform of the function

$\tilde{\tilde{\sigma}}_{r}(x)=\frac{\sigma_{0}}{1+x^{2}}\Big{(}1-\mathcal{Q}^{2}\frac{1-\sqrt{x}}{1+2x^{2}+2x^{4}}\Big{)},$
(19)

with the result

$\displaystyle\tilde{\tilde{\sigma}}_{i}(x)$
$\displaystyle=\frac{\sigma_{0}x}{1+x^{2}}\Big{(}1+\mathcal{Q}^{2}\Big{(}\sqrt{2}-$
(20) $\displaystyle\frac{(\alpha
x^{2}+\beta)(x^{2}+1)+1+1/\sqrt{x}}{1+2x^{2}+2x^{4}}\Big{)}\Big{)},$

where

$\displaystyle\alpha=$ $\displaystyle
2-2^{7/4}\sin\Big{(}\frac{\pi}{8}\Big{)}+\frac{8}{\pi}\int_{0}^{\infty}\\!\\!\\!\frac{x^{5/2}dx}{1+2x^{2}+2x^{4}}\approx
3.136,$ (21) $\displaystyle\beta$
$\displaystyle=2^{5/4}\sin\Big{(}\frac{\pi}{8}\Big{)}-\frac{4}{\pi}\int_{0}^{\infty}\\!\\!\\!\frac{x^{1/2}dx}{1+2x^{2}+2x^{4}}\approx
0.308.$

In Fig. 6b we compare numerical transformations of Eq. (2) (green dashed
lines) to the approximative formula (19) (red dashed lines), indicating
deviations smaller than $1~{}\%$. Finally, fitting the experimental data with
the approximate formula for the imaginary part Eq. (21) produces identical
results as the numerical transformation does.

Figure 6: a) Black lines are the modified Drude conductivity $\sigma_{r}(x)$
with quantum corrections of various strength $\mathcal{Q}=0,\ 0.5,\ 0.75,\ 1$.
Green dashed lines are their corresponding Kramers-Kronig images obtained
numerically. Red dashed lines are given by the formula (17) for the Hilbert
transforms of $\tilde{\sigma}_{r}(x)$. b) Green dashed lines are the same as
in a), red dashed lines are plots of the formula (19) for
$\mathcal{H}[\tilde{\tilde{\sigma}}_{r}(x)]$.

##  Appendix D Electron structure and $\epsilon_{\infty}$ estimation

The picture of bonding and the configuration for 10 niobium and nitrogen
valence electrons in NbN were proposed by many authors, Geballe et al. (1966);
Schwarz (1977, 1987) and it was later largely confirmed by the partial DOS
from DFT calculations and X-ray photoelectron spectroscopy.Babu and Guo
(2019); Sanjinés et al. (2006) Namely, there is a complicated Fermi surface
created by Nb’s 4d orbitals occupied by two electrons.Babu and Guo (2019);
Pflüger et al. (1985). This agrees with the optical estimation of the Drude
weight mentioned in the main text. Next, there is strong hybridization of Nb
4d and N 2p orbitals containing approx. 6 electrons forming a peak in DOS,
approx. 6 eV below $E_{F}$, providing electrons for the modelled inter-band
transition. A naive approach utilizing the strength and the width of the
optical peak $Z=\sigma_{1}\Gamma_{1}m_{e}/(n_{NbN}e^{2})$ gives 3-5 electrons.
Here, $n_{NbN}=1/V_{\textrm{f.u.}}$ is the concentration of formula units. The
estimated number of electrons matches well with the band structure value,
considering neglected joint DOS influence. Finally, the calculations indicate
that the remaining 2 electrons occupy bands low in energy (10-20 eV below
$E_{F}$), corresponding to N 2s orbitals. Contribution to the optical response
due to the transition of these electrons to Fermi level is included via the
parameter $\epsilon_{\infty}$ together with transitions of relevant remaining
core electrons.

Similarly to the estimation of the electron number in the DOS peak from the
weight of the inter-band transition peak, we estimated the contribution
$\epsilon_{\infty}$ to the dielectric function from high-energy transitions.
In Ref. Neilinger et al., 2019, the $\epsilon_{\infty}$ was expressed via the
number of core electrons in their respective atomic level $k$, i.e. $Z_{k}$,
as follows

$\epsilon_{\infty}\approx
1+\sum_{k}\frac{Z_{k}\Omega_{k}^{2}}{\omega_{k}^{2}},\quad\Omega_{k}^{2}=\frac{n_{\scriptscriptstyle{NbN}}e^{2}}{m_{e}\epsilon_{0}}.$
(22)

Here $\hbar\omega_{k}$ is the energy of the atomic level $k$ with respect to
the Fermi level. The values of $\hbar\omega_{k}$ are listed in Ref. ene, . The
relevant orbitals which are not too low in energy for niobium are 3s2, 3p6,
3d10, and 4s2, with energies 467 eV, 370 eV, 203 eV, and 56 eV, respectively.
Nitrogen contributes with 1s2 and 2s2, with energies 410 eV and 37 eV,
respectively. The energy of the nitrogen’s 2s orbital is not taken from Ref.
ene, , but instead, the value 15 eV was taken, which is suggested by the
predictions on the NbN electronic band structure, summarized in the previous
paragraph. Finally, we obtained $\epsilon_{\infty}=1.62$.

##  Appendix E Sample preparation

The thin NbN films were prepared by pulsed laser deposition (PLD, Omicron
system with Coherent Compex Pro 201 F laser) by means of a KrF laser with
wavelength of 248 nm and pulse duration of 35 ns. The films were grown on
c-cut sapphire substrates cleaned in ultrasonic bath in acetone, isopropyl
alcohol, and deionized water in succession. The deposition was performed in
high-vacuum chamber with the residual atmosphere pressure of $10^{-7}$ Pa. The
ablation was carried out from a niobium target in N${}_{2}+1\%$H2 reactive
atmosphere. The pressure of the atmosphere was 9.3 Pa and the substrate was
heated up to 600°C. For more details see Ref. Volkov et al., 2019.

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