# Numerical extrapolation method for complex conductivity of disordered metals

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  D. Manca
Department of Experimental Physics, Comenius University, SK-84248 Bratislava,
Slovakia  J. Greguš  Department of Experimental Physics, Comenius University,
SK-84248 Bratislava, Slovakia  S. Volkov  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

Recently, quantum corrections to optical conductivity of disordered metals up
to the UV region were observed. Although this increase of conductivity with
frequency, also called anti-Drude behaviour, should disappear at the electron
collision frequency, such transition has never been observed, or described
theoretically. Thus, the knowledge of optical conductivity in a wide frequency
range is of great interest. It is well known that the extrapolation of complex
conductivity is ill-posed - a solution of the analytic continuation problem is
not unique for data with finite accuracy. However, we show that assuming
physically appropriate properties of the searched function $\sigma(\omega)$,
such as: symmetry, smoothness, and asymptotic solution for low and high
frequencies, one can significantly restrict the set of solutions. We present a
simple numerical method utilizing the radial basis function approximation and
simulated annealing, which reasonably extrapolates the optical conductivity
from visible frequency range down to the far infrared and up to the
ultraviolet region. The method was compared with two widely used analytic
continuation techniques and resulting extrapolation obtained on MoC and NbN
thin films was checked by transmission measurement across a wide frequency
range.

##  I Introduction

Optical properties of thin films, especially superconducting metals, are
currently of great interest. Knowledge of the electric response is essential
for the rapidly expanding field of superconducting devices e.g. photon
detectors,Day et al. (2003) as well as for purely academic purposes, due to
the presence of phenomena like weak localization. Therefore, this class of
materials attracts much investigation. Banerjee et al. (2018); Semenov et al.
(2009); Neilinger et al. (2019) Measurement of optical properties in a broad
frequency range is often demanding and the experimental window of devices is
naturally limited. However, as it is known, the knowledge of the response
function on a finite interval is sufficient to extend the domain because this
function must be analytical, i.e. satisfying the Kramers-Kronig relations
(KK). Unfortunately, this problem, known as analytic continuation (AC), is
ill-posed - its solution is unstable and not uniquely determined when a small
error is introduced into an input data set.Trefethen (2020) In order to solve
such a problem, regularization should be carried out by making use of
additional information.

For example, one can presume that the studied system can be described by a
certain physical model.Kuzmenko (2005); Tanner (2015) In the case of metals,
the complex conductivity is usually analysed in terms of the classical Drude-
Lorentz model. This model allows to fit the conductivity of clean metals well,
but it fails to describe the conductivity of disordered metals, where quantum
corrections play a significant roleAltshuler and Aronov (1985) and can persist
up to the ultraviolet range.Neilinger et al. (2019) The discussion on the
applicability of this model is in the Appendix C. In contrast, there are more
general, rather mathematical approaches to the AC, e.g. the Padé approximation
and the singular value decomposition (SVD). However, they exhibit instability
due to noise in the data, as we demonstrate in Appendix C.2, and the
regularization by physically motivated constraints is not
straightforward.Schött et al. (2016); Beach and Gooding and Marsiglio (2016);
Creffield (2001)

In this paper, we describe a method which allows to add constraints
systematically until a reasonable extrapolation is obtained. As we
demonstrate, our method is able to find the expected solution, presuming
physically well justified restrictions on the searched function. Namely, that
the complex conductivity i) cannot oscillate on scales smaller than the
relaxation rate $\Gamma$, ii) cannot diminish at frequencies higher than
$5\Gamma$, iii) has a non-negative real part and iv) the contribution from
high frequency ($\gg\Gamma$) transitions can be included in the ”infinite
frequency” dielectric constant $\epsilon_{\infty}$ . This allows a model-
independent determination of $\sigma(\omega)$, extending the experimental
window of spectroscopic ellipsometry by making use of DC sheet resistance,
both measured at room temperature. The approach is demonstrated on data
measured on highly disordered metals close to the metal-insulator transition.
Our simple illustrative procedure is compared to the aforementioned
techniques, namely to the Drude-Lorentz data fit, Padé approx. and SVD.

##  II Method description

Let a set of complex values
$\\{\sigma^{\prime}_{e}(\omega_{i}^{e})+i\sigma^{\prime\prime}_{e}(\omega_{i}^{e})\\}$
be the complex conductivity at discrete frequencies $\omega_{i}^{e}$, from
interval $[\omega^{e}_{min},\omega^{e}_{max}]$, as well as at zero frequency,
$\sigma^{\prime}_{e}(0)=\sigma_{dc}$, measured at temperature $T_{R}$. We
start with the discretization of the $\omega$ \- axis. Taking into account
that the error of analytical continuation of experimental data measured with
finite precision increases exponentially with distance from the known
interval,Trefethen (2020) we use a logarithmic scale. This also helps to avoid
problems with splining square-root and logarithmic singularities with
polynomial functions. We introduce a new discrete variable $x_{i}$ with values

$\displaystyle x_{i}=$ $\displaystyle 0,1,...\quad\quad$ (1)
$\displaystyle...k,\frac{1}{a}\ln(\hbar\omega_{0}^{e}/k_{B}T_{R}),\frac{1}{a}\ln(\hbar\omega_{1}^{e}/k_{B}T_{R}),...$
$\displaystyle...,\frac{1}{a}\ln(\hbar\omega_{N}^{e}/k_{B}T_{R}),l,l+1,...,n,$

where the number of points $k$ below the experimental window is set by a
parameter $a=\ln(\hbar\omega_{0}^{e}/k_{B}T_{R})/(k+1)$ and $l$ is the lowest
integer greater than $\ln(\hbar\omega_{max}^{e}/k_{B}T_{R})$. The
discretization is obtained simply by

$\omega_{i}=\frac{k_{B}T_{R}}{\hbar}e^{a\cdot x_{i}}.$ (2)

An example of such procedure is shown in Fig. 1, where the exponential spacing
of red points $\omega_{i}$ is interrupted by the experimental values (blue
points) at $\omega^{e}_{i}$. Values of the real part of conductivity at the
frequencies $\omega_{i}$ are denoted by $y_{i}$, i.e.
$y_{i}=\sigma^{\prime}(\omega_{i})$ and they will become the fitting
parameters to be optimized. The integer $n$, determining the number of fitted
points, can be estimated as the index of a reasonably high value of
$\omega_{n}$, where the conductivity can be safely fixed to zero, since
$\sigma(\omega\to\infty)\xrightarrow[]{}0$, whereas the contributions from
high energy transition, being far enough from the studied region, are included
in parameter $\epsilon_{\infty}$ introduced later on. According to Fermi
liquid theory, finite temperature can be taken into account by the
transformation

$\omega\mapsto\Omega=\sqrt{\omega^{2}+\gamma(T)^{2}},$ (3)

where $\gamma(T)$ is of the same order of magnitude as $k_{B}T/\hbar$, often
taken as $\gamma(T)=\pi k_{B}T/\hbar$ (e.g. Eq. 6.7 in Ref. Altshuler and
Aronov, 1985). Equation (3) implies that for $\omega\ll\gamma(T)$, the
conductivity $\sigma^{\prime}(\Omega)$ is constant and equals to
$\sigma_{dc}=\sigma^{\prime}(\gamma(T))$. Thus a smooth function
$y=f_{\\{y_{i}\\}}(x)$ can be constructed by a cubic spline using the Radial
Basis Function (RBF) methodDyn and Levin (1983); Weber et al. (2009), with two
boundary conditions, $y_{0}=\sigma_{dc}$ measured at room temperature and
$y_{n}=0$. The first condition is satisfied by fixing the first point of the
discretization to frequency $\approx\gamma(T_{R})$, which is the reason for
the presence of the temperature, at which the optical measurement was
conducted, in the discretization (1). The spline proceeds in the logarithmic
scale, where the distribution of centres $x_{i}$ given by equation (1) is
equidistant, which is optimal for RBF method. Iske (2000)

From known trial function $f$, the complex conductivity is calculated as
$\sigma_{\\{y_{i}\\}}^{\prime}(\omega)=f_{\\{y_{i}\\}}(x(\omega))$ and
$\sigma_{\\{y_{i}\\}}^{\prime\prime}(\omega)=\mathcal{H}[\sigma^{\prime}_{\\{y_{i}\\}}(\omega)]$.
Here, $\mathcal{H}[\sigma^{\prime}_{\\{y_{i}\\}}(\omega)]$ is the Hilbert
transform of $\sigma^{\prime}_{\\{y_{i}\\}}(\omega)$. The curve
$f_{\\{y_{i}\\}}(x)$ is found as the best fit of its Hilbert transform
$\sigma_{\\{y_{i}\\}}^{\prime\prime}(\omega)$ to experimental set of points
$\sigma^{\prime\prime}_{e}(\omega_{i}^{e})$ by least-squares method,
minimizing the functional

$\displaystyle\mathcal{F}[\sigma^{\prime}_{\\{y_{i}\\}}(\omega)]=\sum_{\omega_{j}^{e}}\Big{(}\sigma^{\prime\prime}_{e}(\omega_{j}^{e})-\sigma^{\prime\prime}_{\\{y_{i}\\}}(\omega_{j}^{e})\Big{)}^{2}$
(4)
$\displaystyle+\sum_{\omega_{j}^{e}}\Big{(}\sigma^{\prime}_{e}(\omega_{j}^{e})-\sigma^{\prime}_{\\{y_{i}\\}}(\omega_{j}^{e})\Big{)}^{2}.$

This approach leads to the following key observation: Suppose, for a moment,
that the values of $\sigma^{\prime}_{e}(\omega_{i}^{e})$ and
$\sigma^{\prime\prime}_{e}(\omega_{i}^{e})$ were not obtained by measurement,
but were generated instead by restricting the domain of a dimensionless model
function $g^{\prime}(\omega)$ and its Hilbert image
$g^{\prime\prime}(\omega)$, i.e.
$\sigma^{\prime}_{e}(\omega_{i}^{e})=\sigma_{0}g^{\prime}(\omega_{i}^{e})$ and
$\sigma^{\prime\prime}_{e}(\omega_{i}^{e})=\sigma_{0}g^{\prime\prime}(\omega_{i}^{e})$.
It means that the solution corresponding to the input data
$\sigma^{\prime}_{e}(\omega_{i}^{e})$ and
$\sigma^{\prime\prime}_{e}(\omega_{i}^{e})$ is already known (the whole
function $g^{\prime}(\omega)$). It is clear that the spline of finite number
of points $f_{\\{y_{i}\\}}(x)$ can not perfectly recover the function, even if
$y_{i}=\sigma_{0}g^{\prime}(\omega_{i})$. Slight differences yield small, but
non-zero values of the functional (4), denoted by $\mathcal{F}_{0}$.
However,by applying the corresponding optimization method, one can find curves
with an even lower value of the functional (4), while these curves are
significantly different from $g^{\prime}(\omega)$.

Figure 1: Reconstruction of Drude complex conductivity by the described method
from precisely known values, shown as blue points. The real part of Drude
conductivity is shown by the red dashed line together with three curves, which
have considerably different values $y_{i}$ (red points) at higher frequencies.
Inset shows the deviation
$\Delta\sigma^{\prime\prime}/\sigma_{0}=\mathcal{H}[\sigma^{\prime}_{\\{y_{i}\\}}]/\sigma_{0}-g^{\prime\prime}$
of the imaginary part of conductivity calculated from these curves by Hilbert
transform, from the Drude one. The deviation is surprisingly larger for the
light pink curve, which has all the red points lying on the real part of Drude
conductivity, than for the brown and orange curves oscillating around it. The
values of the functional (4) for the light pink curve are
$\mathcal{F}_{0}=1.2\times 10^{-5}\sigma^{2}_{0}$ and for the orange and brown
curves are $2.0\times 10^{-7}\sigma^{2}_{0}$. Nevertheless, the average of
brown and orange curves is very close to the Drude one.

This is interpreted as a practical consequence of the ill-posed nature of the
problem and in order to regularize it, we add the assumption that the searched
function is slowly varying. This can be confirmed by the fact that the
relaxation rate $\Gamma$ in highly disordered metals has a large value (for
estimation of $\Gamma$ see Appendix D) and therefore, their conductivity
varies on large scales.Neilinger et al. (2019) Utilization of this requirement
is based on the following idea: For the sake of simplicity, let
$g^{\prime}(\omega)$ be the Lorentzian

$g^{\prime}(\omega)=\frac{1}{1+(\omega/\Gamma)^{2}}$ (5)

shown in Fig. 1 as red dashed line. The choice of input data
$\sigma^{\prime}_{e}(\omega_{i}^{e})$ and the discretization of the problem
are depicted in Fig. 1 as blue and red dots respectively. Let us investigate
different solutions with the value of functional (4) smaller or equal to the
$\mathcal{F}_{0}$. Such solutions have the following property: if the red
point at $\omega_{11}$ is lifted and the red point at $\omega_{12}$ lowered,
or vice versa, the deviation of the imaginary conductivity from the Drude one,
shown in inset of Fig. 1, changes only very slightly and can even decrease the
deviation caused by the spline error. Thus, a solution with the least
structure can be obtained by averaging of all generated curves.Gunnarsson et
al. (2010); Schött et al. (2016) Curves with energy (4) lower than
$\mathcal{F}_{0}$ can be found, utilizing the simulated annealing technique by
repeated melting of the system. We suggest inverse logarithmic temperature
decay and rather small step in the form of $n$-dimensional Gaussian random
variable, recommended for continuous optimization.Spall (2005) Optimized
parameters are forced to be non-negative and the upper bound is chosen high
enough to include the shape of an expected solution but not too high to make
optimization time-consuming. Points in the measured interval, i.e. values
$y_{i}=\sigma^{\prime}(\omega_{i})$ where
$\omega_{i}\in[\omega^{e}_{min},\omega^{e}_{max}]$ are optimized within
intervals
$[\sigma^{\prime}_{e}(\omega_{i}^{e})-\chi,\sigma^{\prime}_{e}(\omega_{i}^{e})+\chi]$,
where $\chi$ is the noise level of the data.

The generated curves $\sigma_{\\{y_{i}\\}}^{\prime}(\omega)$ which satisfy
$\mathcal{F}(\sigma_{\\{y_{i}\\}}^{\prime}(\omega))\leq\mathcal{F}_{0}$ are
used to compute the averaged curve
$\overline{\sigma^{\prime}}(\omega)=\sigma^{\prime}_{\\{\overline{y_{i}}\\}}(\omega)$.
Although the Hilbert transformation is linear, the averaged curve
$\overline{\sigma^{\prime}}(\omega)$ gives a slightly higher value of the
functional $\mathcal{F}$. Therefore we search a curve varying similarly to
$\overline{\sigma^{\prime}}(\omega)$, while minimizing the functional (4).
Such a multi-objective optimization is performed by linear
scalarization,Emmerich and Deutz (2018) i.e. minimizing functional:

$\tilde{\mathcal{F}}_{\lambda}[{\scriptstyle\sigma^{\prime}_{\\{y_{i}\\}}(\omega)}]=\mathcal{F}[{\scriptstyle\sigma^{\prime}_{\\{y_{i}\\}}(\omega)}]+\lambda\int\displaylimits_{\omega_{0}}^{\omega_{n}}{\scriptstyle\Big{(}\frac{d^{2}\sigma^{\prime}(\omega)}{d\omega^{2}}-\frac{d^{2}\overline{\sigma^{\prime}}(\omega)}{d\omega^{2}}\Big{)}^{2}}\mathrm{d}\omega,$
(6)

where $\omega_{0}$ and $\omega_{n}$ are the first and the last point of the
spline defined by equation (2). The parameter $\lambda$ is not a priori known
and must be carefully chosen for the particular problem. One can start with a
high value of $\lambda$, such that $\lambda\gg\mathcal{F}/I$, where $I$ is the
integral in (6), estimated with an arbitrarily chosen curve from the ensemble.
The found solution is very similar to $\overline{\sigma^{\prime}}(\omega)$ and
with a similar value of the first error functional (4). One can lower the
value of $\lambda$ further, until the error (4) is lower then
$\mathcal{F}_{0}$. In real data analysis, the precision limit
$\mathcal{F}_{0}$ is set by the measurement error in such a way, that the
accepted curve lies within the estimated measurement error.

Figure 2: (a) Top: Red dashed line is a Gaussian model of optical conductivity
with square-root quantum corrections (MoC). Blue points laying on the curve
are input data for the extrapolation. Silver lines are curves from the
ensemble. Black line is a normalized averaged curve
$\overline{\sigma^{\prime}}(\omega)=\sigma^{\prime}_{\\{\overline{y_{i}}\\}}(\omega)$.
The final curve is green. Bottom: Histogram showing occupancy of deviations
around the searched function by curves from ensemble and deviation of final
curve shown as green line. (b) Top: Red dashed line is the optical
conductivity as a sum of two Lorentzians, with square-root quantum corrections
(NbN). Blue points are input data, ensemble curves are silver, the average
curve is black and the final curve is green. Bottom: histogram of deviation of
curves from ensemble and deviation of final green line.

##  III Numerical results

The extrapolation range and accuracy of the described procedure, i. e. finding
reasonable complex conductivity curves to extrapolate the experimental curves
with respect to KK relations, naturally depends on the degree of complexity of
the extrapolated functions. Thus, to demonstrate the feasibility of this
method, we tested the extrapolation on sets of data created from two
functions, which qualitatively describe the real part of the optical
conductivity of MoC and NbN thin films, respectively. The conductivity of the
former is modeled by a simple function motivated by Ref. Neilinger et al.,
2019, which contains the observed square-root quantum corrections to the
optical conductivity of these films:

$g_{1}(\omega,T)=e^{-\Omega^{2}/\Gamma_{1}^{2}}+Q^{2}(\sqrt{\Omega/\Gamma_{1}}-1)e^{-4\Omega^{2}/\Gamma_{1}^{2}}.$
(7)

Here, $\Omega$ is defined by equation (3), $\Gamma_{1}$ is the scattering
rate, and the quantumness $Q^{2}$ characterizes the strength of the square-
root corrections, which are significant up to a certain crossover frequency,
chosen to be half of the scattering rate. This function is shown as the red
dashed line in square-root-scale in top plot of Fig. 2(a) for $Q^{2}=0.66$.
The input data - depicted as blue dots - were obtained by sampling the above
function (7) at $20$ regularly placed frequencies. Points of discretization
are depicted as red dots. Next, to test the extrapolation of a slightly more
complex function, we study a model function with two peaks motivated by Refs.
Semenov et al., 2009; Neilinger et al., 2019

$\displaystyle g_{2}(\omega,T)=\frac{1}{1+(\Omega/\Gamma_{2,1})^{2}}+$
$\displaystyle
Q^{2}(\sqrt{\Omega/\Gamma_{2,1}}-1)e^{-4\Omega^{2}/\Gamma_{2,1}^{2}}$ (8)
$\displaystyle+$
$\displaystyle\frac{r_{2}}{1+((\Omega-\Omega_{2})/\Gamma_{2,2})^{2}},$

where $\Gamma_{2,1}/2$ is the cut-off for quantum corrections and $r_{2}$ is
the peak height ratio. Here, the cut-off frequency was not determined by the
$\Gamma_{2,1}$ but by the appropriate scale describing the ultimate decrease
of the function to zero, which we assumed to be $\Omega_{2}+\Gamma_{2,2}$. As
we show in the Appendix D, the estimated relaxation rate $\Gamma$ for the NbN
film corresponds rather to this scale. The searched function (8) is shown in
the top plot of Fig. 2(b) as the red dashed line, the input data are blue dots
and the optimized points are red. A few curves from the averaged ensemble are
also shown as silver lines. The average curves are depicted as black lines and
the final conductivity is green. As shown in section (IV), function (8) based
on Drude-Lorentz modelSemenov et al. (2009) qualitatively describes the
measured conductivity of our NbN films, which indicates the presence of two
peaks in optical conductivity.

The results presented in Fig. 2 show the usability of the procedure. For the
$g_{1}(\omega)$ model, there is good agreement of the final green curve with
the original curve (7) at both higher and lower frequencies. The deviation
$\Delta\sigma^{\prime}=g_{1}-\sigma_{y_{i}}^{\prime}$ shown in bottom plot of
Fig. 2(a) is below $5\%$. Shown is also a histogram generated by the ensemble
of found curves. The conductivity model $g_{2}(\omega)$ is more complex,
therefore, the function (8) is accurately reconstructed in the vicinity of the
extrapolated region, whereas further from this region, the deviation
increases. The deviation of the final curve from the searched function (8),
shown in bottom plot of Fig. 2(b), reaches $20\%$ and variance of the curves
is also larger than for the simpler model (7). This is visible at higher
frequencies in the slightly underestimated second peak’s height, whereas at
low frequencies, the function is recovered with lower deviation. In both
cases, we performed both the Padé approx. and SVD. The results are presented
and discussed in Appendix C.2.

##  IV Extrapolation of experimental data

The procedure was applied to extrapolate the conductivity of disordered thin
films, as shown in Fig. 3(a) for MoC and Fig. 3(b) for NbN, respectively. For
the details of sample preparation, see Appendix B and Refs. Neilinger et al.,
2019; Volkov et al., 2019. Complex conductivity was determined from
spectroscopic ellipsometry, and DC conductivity was measured by the Van der
Pauw method. The contribution of interband transitions at higher energies must
be taken into account in the imaginary part of the conductivity. Following the
KK relations, the contribution from the bound electrons can be expressed as
$\sigma^{\prime\prime}_{bound}=-\epsilon_{0}(\epsilon_{\infty}-1)\omega$,
where $\epsilon_{0}$ is the permittivity of vacuum and $\epsilon_{\infty}$ is
the bound-electron contribution to the static dielectric constant. For MoC,
the value of $\epsilon_{\infty}$ was estimated as 1.4 in Ref. Neilinger et
al., 2019 and for NbN, we calculated $\epsilon_{\infty}=2.6$ utilizing the
same procedure, while a similar value was estimated in Ref. Semenov et al.,
2009. Subsequently, the corresponding contribution to the imaginary part of
the conductivity $\sigma^{\prime\prime}_{bound}$ was subtracted from measured
data. Since the relaxation rate $\Gamma$ can not be easily used as a fitting
parameter, we estimated the value from the sheet resistance measured on our
thin films and the density of carriers. The details on both estimations of
$\epsilon_{\infty}$ and $\Gamma$ are given in the Appendix D. For MoC
conductivity, the extrapolation fits between two theoretical curves, a
Lorentzian (red) and a Gaussian (pink), proposed in Ref. Neilinger et al.,
2019. Taking $\gamma(T)$ in the form $\pi k_{B}T/\hbar$ (same as in Ref.
Neilinger et al., 2019), the inverse transformation given by the equation (3)
allows to compare the extrapolation (dotted lines) with the terahertz-
frequency real part of conductivity (light blue thick lines), obtained from
the temperature dependent DC transport measurement. The agreement for MoC is
very good, especially taking into account the large distance between optical
frequencies (500 THz) and the frequency ($\pi kT/h\approx 50$ THz) relevant
for transport measurements. The agreement for NbN is less satisfying. However,
even for these more complicated conductivity spectra, the extrapolation can
match the transport measurements reasonably.

Figure 3: Real (solid green line) and imaginary (dashed green line) parts of
the extrapolation of normalized complex sheet conductance
$g=g^{\prime}+ig^{\prime\prime}=(\sigma^{\prime}+i\sigma^{\prime\prime})dZ_{0}$,
obtained from spectroscopic ellipsometry (blue data) for MoC (a) and NbN (b)
thin films with thicknesses $d=5$ nm and $3.5$ nm, respectively, measured at
room temperature. All found complex conductivities with functional (4)
$\leq\mathcal{F}_{0}$ are shown as silver (real part) and gold lines
(imaginary part). Since Eq. (3) implies
$g^{\prime}(\omega=0,T)=g^{\prime}(\omega=\pi k_{B}T,0)$, the real part of
complex conductivity at zero temperature (dotted line), obtained from
extrapolation procedure, can be compared to temperature-dependent DC transport
measurement shown as light blue thick lines. Plot (a) also shows two
theoretical curves, a Lorentzian (red) and a Gaussian (pink), proposed in Ref.
Neilinger et al., 2019.

In order to verify the extrapolation procedure experimentally, the real part
of conductivity, determined from the extrapolation of ellipsometric data, was
compared to the conductivity calculated from the transmission of MoC and NbN
films (see Appendix A), directly measured in the much wider frequency range
accessible by our spectrometer, as shown in Fig. 4. The good agreement of our
extrapolation with the data from transmission measurement for MoC is the
encouraging result. The mismatch in the NbN’s transmission above 5.0 eV could
be caused by interdiffusion between sapphire and NbN. The transmission of
sapphire is nearly frequency independent between 0.3-5.0 eV (inset in Fig. 4
(b)), but above 5.0 eV, the transmission strongly depends on impurities in the
sapphire.Dobrovinskaya et al. (2009) To verify the reliability of transmission
data above 5.0 eV, we used them to compute the real part of conductivity,
taking the imaginary part from ellipsometry for the same sample, and used both
as the basis for extrapolating a complex conductivity curve, plotting its real
part (Figure 4). Such extrapolation failed if the entire transmission dataset
(6.2 eV) was used, indicating that such conductivity violates the Kramers-
Kronig relations. By the bisection method, we found the cut-off, for which the
transmission data still process into valid complex conductivity, to be 5.4 eV
(dotted line, brown); if a data set with cut-off at 5.0 eV was used, the real
part of the generated conductivity (orange) matches that produced from
ellipsometry alone (green).

Figure 4: Green curves are extrapolations of normalized sheet conductance for
MoC (a) and NbN (b) obtained from spectroscopic ellipsometry data (blue
points) (the same as in Fig. 3) and the experimental values obtained from
transport (light blue) and transmission (red) measurements. In plot (b) brown
and orange curves are extrapolations of data obtained from transmission
measurement with upper cut-off 5.4 eV and 5.0 eV, respectively. The value 5.4
eV of the energy cut-off (dotted line) is the largest, where the extrapolation
can be done. The highest value of the cut-off energy, where the influence of
the substrate can be neglected, is 5.0 eV. The rapid change in the brown curve
above 5.0 eV is probably due to the influence of the substrate, whose
transmission changes rapidly above this value, as shown in the inset. Data are
shown in linear scale.

##  V Conclusion

A numerical procedure of extrapolating the complex conductivity of disordered
metals based on the Kramers-Kronig relations was introduced. In general, this
task has many solutions with standard methods finding a single particular one.
The proposed method finds a single solution by averaging many of them which is
reasonable, if the requirement of slow variation of their conductivity on
energy scale $\lesssim\Gamma$ is met. A slow variation of $\sigma(\omega)$ is
reasonably satisfied in highly disordered metals, where experimental data do
not indicate rapid oscillations in conductivity and the described method
offers a robust and reliable extrapolation procedure of the measured optical
conductivity. The range and precision of the extrapolation depend on the
complexity of the function. For the simple single-peak model of conductivity,
which is the case for MoC thin films, even the DC transport measurements can
be reconstructed from the optical measurements, despite the presence of strong
quantum corrections. For the more complex, double-peaked conductivity model,
which is the case of NbN films, the reliability range of the extrapolation is
decreased. However, the extrapolation still reasonably matches the DC
measurements and predicts the second peak.

###  V.1 Acknowledgments

This work was supported by the Slovak Research and Development Agency under
the contract APVV-16-0372, APVV-18-0358 and by the QuantERA grant SiUCs, by
SAS-MTVS.

##  Appendix A Optical properties calculation

Both the ellipsometry data and the extrapolated conductivity were used to
obtain the optical properties of our thin films. The real and imaginary part
of the complex refractive index are defined in Ref. Hecht, 2002 as

$\displaystyle
n(\omega)=\sqrt{\frac{1}{2}(\Re\\{\epsilon_{R}(\omega)\\}+\lvert\epsilon_{R}(\omega)\rvert)}$
(9) $\displaystyle
k(\omega)=\textrm{sgn}(\omega)\sqrt{\frac{1}{2}(-\Re\\{\epsilon_{R}(\omega)\\}+\lvert\epsilon_{R}(\omega)\rvert)},$

where $\epsilon_{R}$ is the (complex) relative permittivity given by

$\epsilon_{R}(\omega)=1+i\sigma(\omega)/\omega\epsilon_{0}.$ (10)

The transmission $T_{vfs}$ of a thin film with the complex refractive index
$\tilde{n}(\omega)=n(\omega)+ik(\omega)$, placed upon a semi-infinite
substrate with index $\tilde{n}_{s}=n_{s}$, i.e. vacuum-film-substrate, whose
imaginary part is neglected is Li and Heinz (2018)

$T_{vfs}=\Big{\lvert}\frac{2}{(1+\tilde{n}_{s})\textrm{cos}\phi-i(\tilde{n}_{s}+\tilde{n}^{2})\phi_{0}\frac{\textrm{sin}\phi}{\phi}}e^{-i\phi_{0}}\Big{\rvert}^{2}.$
(11)

Here $\phi=\tilde{n}\frac{\omega}{c}d$ and $\phi_{0}=\frac{\omega}{c}d$ are
phase shifts in the film and in vacuum of thicknesses $d$, respectively. The
transmission $T_{vfsv}$ through a system with finite substrate thickness is

$T_{vfsv}=\frac{T_{vfs}T_{sv}}{1-R_{vfs}R_{sv}},$ (12)

where $T_{sv}=4n_{s}/(1+n_{s})^{2}$ is the transmission of the substrate-
vacuum interface, and the reflection $R$ is obtained from the corresponding
transmission simply by $1-T$. Next, the transmission of the vacuum-substrate-
vacuum system is

$T_{vsv}=\frac{T_{sv}^{2}}{1-R_{sv}^{2}}.$ (13)

Finally, the equations (12) and (13) are used to express the transmission of
the system vacuum-film-substrate-vacuum normalized to the transmission of the
substrate itself as

$\frac{T_{vfsv}}{T_{vsv}}=\frac{T_{vfs}}{T_{sv}}\frac{1-R_{vs}^{2}}{1-R_{vfs}R_{sv}},$
(14)

which was also a measured quantity. The right-hand side of equation (14) is a
function of complex conductivity marked as
$T_{n}(\sigma^{\prime},\sigma^{\prime\prime})$ and can be utilized to
calculate one of its parts, real or imaginary, from another part and from
normalized transmission. Instead of inverting equations (9) - (14) one can
simply minimize the difference between
$T_{n}(\sigma^{\prime},\sigma^{\prime\prime})$ and measured
$T_{vfsv}/T_{vsv}$, namely

$\sigma^{\prime}=\arg\min\Big{(}(T_{n}(\sigma^{\prime},\sigma^{\prime\prime})-T_{vfsv}/T_{vsv})^{2}\Big{)},$
(15)

which is, in fact, plotted in Fig. (4) as red symbols.

##  Appendix B Sample preparation

The MoC thin film was prepared by means of reactive magnetron sputtering from
Mo target in argon-acetylene atmosphere (both gases used of purity 5.0) on
c-cut sapphire wafer heated to 200 degrees Celsius. The flow rates of argon
and acetylene were set by flow meters. During deposition, the magnetron
current was held constant at 200 mA, corresponding to a deposition rate
$\approx 11$ nm/min. The deposition time, and thus the thickness, was
regulated by means of a programmable shutter control interface with a
precision of 1 s. The chamber was evacuated to approximately $5\times 10^{-5}$
Pa. For details on the preparation of the MoC films and their characterization
see Ref. Trgala et al., 2014. The sheet resistance of the studied MoC sample
with a nominal thickness $d=5$ nm was $R_{\Box}=720~{}\Omega$.

The thin NbN film was prepared by pulsed laser ablation from an Nb-target
(purity $99.99~{}\%$) in atmosphere of $\textrm{N}_{2}$ with added $1~{}\%$
$H_{2}$ (purity of the gas mixture is $5.0$). The NbN thin film was deposited
on c-cut sapphire wafer, heated to the $600\ ^{\circ}\textrm{C}$. The used
laser fluency of the KrF laser of $4.9~{}\textrm{Jcm}^{–2}$ corresponded to
the deposition rate of $2.4~{}\textrm{nm}/\textrm{min}$. The vacuum chamber
was evacuated to $2\cdot 10^{–6}~{}\textrm{Pa}$ before deposition. For more
details of the preparation of NbN film and its growing features see.Volkov et
al. (2019) The sheet resistance of our $3.5$ nm thick NbN films was
$R_{\Box}=655~{}\Omega$.

##  Appendix C Standard methods of extrapolation

In the following section, the standard methods of the complex conductivity
extrapolation are applied to the MoC and NbN noisy experimental data and
compared to our method.

###  C.1 Fitting to model function

The presumed model determines a direct physical interpretation of the
parameters of the fit, which have a direct physical interpretation, are
determined by . The ill-posed nature of the analytic continuation enables one
to fit the finite interval data with reasonable accuracy by a variety of
models. Therefore, the final solution should be verified by an independent
measurements. For example, the conductivity of metallic thin films is commonly
analysed by the Drude-Lorentz model.Ketterson (2016), as presented by Semenov
et al. in Ref. Semenov et al., 2009 for NbN thin film:

$\epsilon_{r}=\epsilon_{\infty}-\frac{\omega_{p}^{2}}{\omega(\omega+i\Gamma)}+\sum_{n=1}^{2}\frac{\Omega_{Sn}^{2}}{\Omega_{0n}^{2}-\omega^{2}-i\Omega_{Dn}}$
(16)

Here, $\epsilon_{r}$ is the relative permittivity from which the conductivity
is obtained by equation (10). This function does not take into consideration
the presence of quantum corrections to the optical conductivity, which are
clearly present in highly disordered NbN films. These corrections can strongly
affects the optical conductivity of these films, depending on the degree of
disorder. The contribution of quantum corrections to optical conductivity
should be present up to UV frequencies, as assumed in theory and recently
experimentally demonstrated in Ref. Neilinger et al., 2019. Still, such
contributions are usually neglected above infra-red frequencies and the
complex conductivity is commonly analysed by the Drude-Lorentz model. This
leads to either unrealistic physical parameters or to the conclusion that the
complex conductivity cannot be fitted by the standard model.Archer (2017) The
fit of our NbN data by the Drude-Lorentz model (16) is shown in Fig. 5. The
first Lorentz peak, $i=1$ in (16), which should corresponds to a band
transition, is misused to fit the conductivity suppressed by quantum
corrections at lower frequencies and the normal regime at higher frequencies.
As a consequence, the Drude contribution is suppressed to create a crossover
to the low frequency conductivity and the fit provides an extremely small
value of the relaxation rate $\hbar\Gamma\approx 0.26~{}eV$ in comparison with
rough estimation $\hbar\Gamma\approx 10~{}eV$ presented in next section. In
fact, all quantities the listed in Tab I are unphysical, as was explicitly
pointed out in Ref. Semenov et al., 2009, too. Moreover, fitting data by an
inappropriate physical model can lead to the appearance of additional features
that do not have to be real. On the contrary, our ”unbiased” method has
predictive strength and can uncover phenomena present at frequencies outside
the measured interval, just assuming that the optical conductivity cannot
rapidly oscillate on a scale determined by the relaxation rate $\Gamma$,
whereas rapid changes are not excluded. This is shown in Fig. 5 by arrows
which mark frequencies, where optical conductivity changes its monotonicity.
Due to quantum corrections, the change can be very sharp, e.g. thermally
smeared square-root or logarithmic singularity at zero frequency.

Figure 5: Fit of the normalized sheet conductance (blue dots), by the Drude-Lorentz model (red line). The fit does not match the increase of the conductivity indicated by the temperature-dependent DC transport measurement (light blue thick line) in contrast to the prolongation (green line), see Fig. 4 (b) in the main text. The pink, brown and violet dashed lines show the contributions from the Drude and the two Lorentz terms in (16), respectively. $\omega_{p}\hbar$ | $\Gamma\hbar$ | $\hbar\Omega_{S1}$ | $\hbar\Omega_{01}$ | $\hbar\Omega_{D1}$ | $\hbar\Omega_{S2}$ | $\hbar\Omega_{02}$ | $\hbar\Omega_{D2}$
---|---|---|---|---|---|---|---
(eV)
2.88 | 0.25 | 9.42 | 0.86 | 2.12 | 11.71 | 5.54 | 3.31
Table 1: Parameters given by the fit of the real and imaginary part of the
complex conductivity obtained from spectroscopic ellipsometry on our NbN
sample by model (16). We assumed $\epsilon_{\infty}=2.6$, coming from
calculation as mentioned in the main text.

###  C.2 Padé approximation and SVD

In the Padé approximation, the data are fitted by the ratio of two general
complex polynomials, $\sigma(\omega)=P(\omega)/Q(\omega)$ with orders $N$ and
$M$, respectively. The first boundary condition is satisfied by including the
point at zero frequency and the asymptotic behaviour at higher frequencies is
ensured by setting $M=N+1$.Schött et al. (2016) Results of such constrained
Padé approx. strongly depends on the orders $N$ and $M$, which were chosen to
avoid reaching negative, unphysical values. This approach offered poor
precision of extrapolation (dash-dotted line in Fig. 6) even for the best fit,
which was obtained for the orders $N=5$ and $M=6$ for model (7) and $N=4$ and
$M=5$ for model (8). Surprisingly, fitting data in the interval
$[\omega^{e}_{min},\omega^{e}_{max}]$ only (solid orange curve), without
condition on the orders, offered the best result. For the single peak model
(7) the maximum deviation reached was $~{}1\%$. For the double peak function
(8), the result was similar but only in higher frequencies. Non-negativity was
again ensured by the orders $N$ and $M$, which were optimized to reach the
best match to the value at zero frequency. For the unconstrained Padé, they
were both equal to $6$. The Padé approx. which worked reliably for the
noiseless model data, was applied to the MoC and NbN noisy experimental data
as well (Fig. 7). Although the noise was carefully filtered out, the
unconstrained Padé approx. (solid orange curve) rapidly varies and diverges
for the NbN data. The orders of polynomials were chosen to reach the best
match to the static conductivity, being $N=5$, $M=6$ and $N=8$, $M=9$ for MoC
and NbN, respectively. The constrained Padé (dash-dotted line) gave a solution
which was more stable, but poorly fitted the data, especially at the edges of
the experimental window.

In SVD, the behaviour of the searched function is prescribed at both
boundaries. Following Ref. Dienstfrey and Greengard, 2001, the problem of
finding a function satisfying the KK relations, while fitting the input data,
is turned into solving a linear system by expanding the known and searched
functions into a proper series. Regularization is done by truncating the
series and the resulting system is solved by the SVD, which means that the
solution is one with the least squares error. For the model function (7), a
result similar to Padé was achieved by SVD, with the rank of the decomposed
matrix being equal to 7. The rank was chosen to minimize the rapid
oscillations of the resulting curve. For the double peak model (8), SVD
recovers the searched function at low frequencies very well, but at high
frequencies gives additional structure. To obtain this result the matrix rank
was equal to 8. The results obtained from SVD are shown in Fig. 6 as black
lines.

When applied to the experimental data, the SVD solution oscillates, its real
part reaching negative values, which can not be forbidden in the SVD approach.
The rank of the matrix of which we calculated the pseudo-inverse by SVD was
again chosen to minimize the oscillations, even though they are still present
in the final curves (black lines in Fig. 7). This rank was 7 and 6 for MoC and
NbN, respectively.

We believe, that the local variability of the curves generated by our method
allows us to suppress the effects of the noise and averaging inside the
measured interval provides additional smoothing of data, but with respect to
KK.

Figure 6:  Application of the Padé and SVD prolongation method on the model
data (blue dots, same as in Fig. 2). The red dashed line in panel (a) and (b)
corresponds to the model function (7) and (8), respectively. Black line is
real part of the solution obtained by SVD, orange dash-dotted line is Padé
approx. with both boundary conditions included and orange solid line is Padé
approx. without them.

Figure 7: Comparison of the real parts of normalized sheet conductance
obtained by our method (green line) extrapolating ellipsometric (blue) data to
the solutions obtained by the Padé approx. (orange solid), constrained Padé
(orange dash-dotted) and SVD (black).

##  Appendix D Estimation of dielectric constant $\epsilon_{\infty}$ and
relaxation rate $\Gamma$

Extrapolating the complex conductivity as described requires two crucial
parameters, electron relaxation rate $\Gamma$ and an ”infinity frequency”
dielectric constant $\epsilon_{\infty}$. Our estimation of $\epsilon_{\infty}$
closely follows calculation presented in Ref. Neilinger et al., 2019:

$\epsilon_{\infty}\approx 1+\sum\frac{\Omega_{j}^{2}}{\omega_{j}^{2}}.$ (17)

Here, approximate equality means that the high frequency permittivity consists
of resonant contributions of transitions between bands modeled by atomic-like
levels and conduction band at the Fermi energy. Estimation of weights of the
oscillator was based on the $f-$sum rule as
$\Omega_{j}^{2}=\frac{Z_{j}n_{at}e^{2}}{m\epsilon_{0}}$, where $Z_{j}$ is the
number of electrons at the atomic level $j$ and $n_{at}$ is the density of the
particular atom type. The excitation energies of the atomic levels
$\hbar\omega_{j}$ were taken from Ref. Williams, 2013 and corresponding
densities were calculated from volumes of elementary cells, which can be found
in Ref. Persson, 2016a, b. The calculation for MoC yields
$\epsilon_{\infty}=1.38$.Neilinger et al. (2019) Relevant levels of niobium
are 4p (6 electrons, 31 eV), 4s (2 electrons. 56 eV) and 3d (10 electrons, 203
eV) and levels of nitrogen are 2s (2electrons, 20 eV) and 1s (2 electrons, 410
eV) which yields $\epsilon_{\infty}=2.58$ for NbN.

Our rough estimation of the relaxation rate $\Gamma$ is based on the Drude
formula for static conductivity

$\Gamma=\frac{e^{2}n}{m\sigma_{0}}.$ (18)

Here, the density of carriers $n\approx 4.2\times 10^{23}~{}cm^{-3}$ is taken
from the Hall measurement in Ref. Destraz et al., 2017. The quantum
corrections to the static conductivity can be included via the quantumness
$Q^{2}\approx 1/{(k_{F}l)^{2}}$, which relates the classical value
$\sigma_{0}$ to the measured one as
$\sigma_{dc}=\sigma_{0}(1-Q^{2})$.Neilinger et al. (2019) For a reasonable
quantumnessNeilinger et al. (2019) $Q^{2}=0.7$ we obtained $\hbar\Gamma\approx
16~{}eV$ and $\approx 10~{}eV$ for MoC and NbN thin film, respectively. Here
we should emphasize, that our method requires a rough estimation of the
relaxation time (order of magnitude) only. For example, if $\Gamma$ is
overestimated, the method sets more high frequency points close to zero
conductivity. On the other hand, if $\Gamma$ is considerably underestimated,
the simulated annealing finds no curve. In each case, one can properly adjust
$\Gamma$ to obtain reasonable curves found by the simulated annealing.

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