# An adaptive model for the optical properties of excited gold

P. D. Ndione<EMAIL_ADDRESS>Department of Physics and OPTIMAS Research
Center, RPTU Kaiserslautern-Landau, Erwin-Schrödinger-Straße 46, 67663
Kaiserslautern, Germany  S. T. Weber  Department of Physics and OPTIMAS
Research Center, RPTU Kaiserslautern-Landau, Erwin-Schrödinger-Straße 46,
67663 Kaiserslautern, Germany  D. O. Gericke  Centre for Fusion, Space and
Astrophysics, Department of Physics, University of Warwick, Coventry CV4 7AL,
United Kingdom  B. Rethfeld  Department of Physics and OPTIMAS Research
Center, RPTU Kaiserslautern-Landau, Erwin-Schrödinger-Straße 46, 67663
Kaiserslautern, Germany

###### Abstract

We study the temperature-dependent optical properties of gold over a broad
energy spectrum covering photon energies below and above the interband
threshold. We apply a semi-analytical Drude-Lorentz model with temperature-
dependent oscillator parameters. Our approximations are based on the
distribution of electrons over the active bands with a density of states
provided by density functional theory. This model can be easily adapted to
other materials with similar band structures and can also be applied to the
case of occupational nonequilibrium. Our calculations show a strong
enhancement of the intraband response with increasing electron temperature
while the interband component decreases. Moreover, our model compares well
with density functional theory-based calculations for the reflectivity of
highly excited gold and reproduces many of its key features. Applying our
methods to thin films shows a sensitive nonlinear dependence of the reflection
and absorption on the electron temperature. These features are more prominent
at small photon energies and can be highlighted with polarized light. Our
findings offer valuable insights for modeling ultrafast processes, in
particular, the pathways of energy deposition in laser-excited samples.

##  I Introduction

The optical response of materials has been subject to extensive studies over
the last decades due to its potential application in various fields, including
plasmonics [1, 2], material processing [3], and optoelectronics [4]. Optical
properties are also often used as probes and thus play a crucial role in
understanding the fundamental properties of matter. For example,
nonequilibrium electron kinetics can be studied in condensed matter using
reflectivity and/or transmissivity measurements [5, 6, 7]. Similarly, the
combination of optical probing [8, 9] and theoretical models [10, 11] has been
used to provide valuable quantitative benchmarks of important properties such
as the electron-phonon coupling strength in warm dense matter.

Laser excitation of matter can strongly modify the energy distribution of its
electrons, changing its optical properties, which creates a feedback mechanism
to the absorption process. Thus, an accurate description of the transient
energy deposition requires an appropriate evaluation of the time-dependent
dielectric function. Near equilibrium, it can be provided by _ab-initio_
methods such as density functional theory (DFT) via the Kubo-Greenwood formula
[12, 13, 14, 11, 15]. Unfortunately, these methods are not always applicable
due to their high computational costs. Moreover, often only one component of
the response function is directly calculated [12, 11], while the other
component is obtained with the Kramers-Kronig relations, which may lead to
increasing inaccuracies.

For practical reasons, simplified approaches such as the Drude [16] or the
Drude-Lorentz (DL) description are often used. The DL model provides a
comprehensive framework for modeling the optical response of various materials
as it incorporates interband transitions. It has been widely used [17, 18, 19,
14, 20, 21], is easy to implement and flexible, _i.e_ , it is easy to adapt to
different materials, thus, allowing for fast predictions and a quick analysis
of experiments. Recently, the DL model has been fitted to data for gold
obtained by DFT calculations [14].

Here, we study the temperature-dependent optical properties of gold across a
broad range of probe energies by applying a modified DL model. In contrast to
Ref. [14], we start with a fit to the measured dielectric function at room
temperature [22]. To obtain the optical response at elevated temperatures, we
apply temperature-dependent collision frequencies in the Drude part [23],
i.e., the intraband response is fully determined by the temperature-dependent
occupation of the conduction band and the changing Drude damping. For the
interband response, we introduce an additional temperature dependence of the
Lorentz oscillators. In our approach, we connect the amplitudes of the Lorentz
oscillators with the density of oscillating electrons at constant oscillator
positions. The resonance energies are modified according to the shift of the
chemical potential and the broadening of the Fermi edge at elevated
temperatures. The remaining freedom for the Lorentz parameters can be fixed by
comparison with DFT-based calculations. Finally, the damping of the Lorentz
oscillators is described in the same ways as the Drude damping taking into
account electron-phonon and electron-electron scattering. In this way, our
model allows for a direct transfer of the parameters describing Lorentz
oscillators at room temperature to a description applicable to gold with
highly excited electrons.

Our results show a strong enhancement of the intraband response with
increasing electron temperature while the interband component decreases. Most
importantly, our model reproduces features in the reflectivity of highly
excited gold that have been found with extensive DFT-based calculations [24].
Applying our methods to thin gold films, we find a strong nonlinear dependence
of the optical properties on the electron temperature. We find that the
changes are more pronounced at photon energies below the interband threshold
and are particularly sensitive to $p$-polarized light, making such a setup
ideal for probing.

##  II Theoretical framework

We consider the excitation of matter by a short pulse of infrared or optical
light. For noble metals like gold, such excitations involve only the two upper
electron bands, i.e., the $5d$ valence and the $6sp$ conduction electrons. In
general, both types of electrons contribute to the optical properties, but the
low-frequency behavior is dominated by the $6sp$ electrons [22]. The density
of states (DOS) can be calculated by density functional theory (DFT). Fig. 1
shows results for gold as obtained by the ELK code [25] as well as the
occupation of these states for a finite temperature. By projection, partial
density of states (PDOS) for $d$\- and $sp$-electrons can be obtained as well
(see, _e.g._ , Ref. [26]).

The model we present here focuses on time scales when the electron
distribution is relaxed and can be described with a temperature. This
assumption is often satisfied for excitations with sufficiently high energy,
where the kinetic stage establishing a Fermi distribution lasts only a few
femtoseconds [27, 28]. Here, the $sp$ and $d$-electrons are assumed to share
the same temperature $T_{e}$ and chemical potential $\mu$, i.e., we have

$f\\!\left(E,\mu,T_{e}\right)=\frac{1}{\exp\\!\left[\left(E-\mu\right)/k_{B}T_{e}\right]+1}$
(1)

over both bands. Note that this assumption may not hold on time scales when a
common electron temperature has been reached but the band occupation has not
fully equilibrated [10, 29]. The equilibration of the band occupation is,
however, much faster than the electron-phonon relaxation, thus there is a
stage with a joint Fermi distribution (1) over both bands where most of the
absorbed laser energy is contained in the electronic system. This stage with
electron-phonon nonequilibrium is the main focus of interest for our current
investigation.

The complex-valued dielectric function is a central and intrinsic quantity to
describe how the material responds to an external electromagnetic field, such
as the radiation field of a laser pulse. Once known, material properties such
as reflection, conductivity, or absorption can be calculated directly [30].
The dielectric function is determined by the electron states specific to the
material, i.e., the band structure. For metals like gold with a $d$-band
character, the total dielectric function is a combination of intra- and
interband contributions: $\varepsilon_{\rm tot}=\varepsilon_{\rm
intra}+\varepsilon_{\rm inter}$. While the intraband response is dominated by
the $sp$-electrons, the interband response depends on the occupation within
the $d$-band.

Figure 1: Total density of states of gold. The Fermi energy $E_{F}$ is plotted
as a blue dashed line and sets the zero point of the energy scale. A Fermi
distribution for $T_{e}=$2\text{\times}{10}^{4}\text{\,}\mathrm{K}$$ and the
chemical potential at this temperature are plotted with red solid lines. The
resonance energies $E_{j}$ (Lorentz oscillators) considered in this work are
highlighted with blue solid lines.

###  A Intraband response

In the optical limit with negligible momentum transfer from the photons, the
intraband response for the electrons can be modeled with the Drude theory [16]

$\varepsilon_{\rm
intra}\\!\left(\omega,T_{i},T_{e}\right)=\varepsilon_{\infty}-\frac{\omega_{p}^{2}\\!\left(T_{e}\right)}{\omega^{2}+i\omega\nu\\!\left(T_{i},T_{e}\right)}\,,$
(2)

where $\omega$ is the frequency probed, $\omega_{p}$ denotes the plasma
frequency, and $\nu$ is the collision or damping frequency, which depends on
the phonon and electron temperatures $T_{i}$ and $T_{e}$, respectively. The
background dielectric constant $\varepsilon_{\infty}$ differs from unity in
noble metals and is $\varepsilon_{\infty}=3.104$ for gold [20].

The plasma frequency sets the frequency of collective electron excitations and
is defined by the density of $sp$-electrons via
$\omega_{p}^{2}\\!\left(T_{e}\right)=e^{2}\,n_{sp}\\!\left(T_{e}\right)/\epsilon_{0}\,m_{sp}^{*}$
with $e$ being the electron charge, $\epsilon_{0}$ the vacuum permittivity,
and $m_{sp}^{*}$ the effective mass of the conduction electrons. For the
latter, the free-electron mass is a good approximation in gold [22]. In our
case, the plasma frequency becomes temperature-dependent as thermal
excitations increase the $sp$-density

$n_{sp}\\!\left(T_{e}\right)=\int\mathrm{d}E\;f\\!\left(E,T_{e},\mu\right)\;D_{sp}\\!\left(E\right)\,,$
(3)

where $D_{sp}\\!\left(E\right)$ is the PDOS of the $6sp$-band.

The Drude damping in Eq. (2) is a key parameter when modeling the optical and
transport properties of highly excited metals [23, 24, 26]. Its main
contributions in gold are electron-phonon and electron-electron scattering
rates. Thus, we have $\nu_{\rm
tot}=\nu_{ei}\\!\left(T_{i}\right)+\nu_{ee}\\!\left(T_{e}\right)$. At room
temperature, it is dominated by the scattering of the $sp$-electrons with the
lattice phonons, $\nu_{ei}$, as Pauli-blocking prohibits scattering with
$d$-electrons. This rate is set in this work to match the experimental data of
Johnson & Christy [22]. For a heated lattice, we assume $\nu_{ei}$ to increase
linearly with $T_{i}$. For the description of $\nu_{ee}$, we follow the model
presented in Ref. [26] which takes into account the temperature dependence of
the $d$-band hole and $sp$-band electron densities. At elevated $T_{e}$, the
scattering of $sp$-electrons by $d$-electrons is the dominant damping
mechanism and depends strongly on the number of conduction electrons [23],
given by Eq. (3). Details for the description of the Drude damping are given
in Ref. [26].

###  B Interband response - room temperature

Photons with energies larger than the interband threshold can excite $d$-band
electrons into the empty states of the conduction band. Strictly speaking,
such interband processes should be treated using the matrix elements that
describe the effect of light on the electrons as well as the density of states
[31]. However, the Lorentz model provides a simplified way to describe the
interband response and it has been applied frequently to materials at room
temperature [17, 20, 32].

In the Lorentz model, valence electrons are assumed to be bound to the nucleus
similar to a spring. The model mimics the motion of electrons around their
equilibrium positions with a damped harmonic oscillator driven by an electric
field. These oscillators have specific resonance frequency, amplitude, and
damping. The inclusion of damping in the Lorentz model implies that the
electrons change their state, e.g., via collisions. The damping broadens the
absorption line to a finite width [31], which also avoids singularities at the
resonance frequency. Most materials have multiple characteristic resonances
and the total response is a sum over all transitions possible in the optical
medium. In the framework of this approach, the interband response can be
written as [20]

$\varepsilon_{\rm
inter}\\!\left(\omega\right)=\sum_{j=1}^{\ell}\frac{a_{j,0}}{\omega_{j,0}^{2}-\omega^{2}-i\omega\Gamma_{j,0}}\,,$
(4)

where $a_{j,0}$, $\omega_{j,0}$ and $\Gamma_{j,0}$ are the amplitude, the
resonance frequency, and the damping of the Lorentz oscillators, respectively.
Here, we choose to apply five Lorentz oscillators. This number is
phenomenological and may be optimized to the energy spectrum of interest. An
important property of the Lorentz dielectric function is the fact that it
fulfills the Kramers-Kronig relations as long as the dampings $\Gamma_{j,0}$
are positive [19, 33].

For gold at room temperature, we obtain the Lorentz parameters $a_{j,0}$ and
$\Gamma_{j,0}$ through a fit to experimental data [22]. The resulting
parameters are summarized in Tab. 1. The five resonance frequencies used here
were taken from Ref. [20]. Fig. 1 shows the position of the resonance energies
$E_{j}=-\hbar\omega_{j,0}$ (blue vertical lines) in relation to the density of
states. Here, and for the values given in Tab. 1, we make the Fermi energy
$E_{F}$ the zero point of our energy scale.

Table 1: Fit parameters of the Lorentz oscillators at room temperature. $\hbar\Gamma_{j,0}$, $\hbar\omega_{j,0}$, and $\hbar a_{j,0}$ are given in eV.  $j$ | $1$ |  $2$ |  $3$ |  $4$ | $\quad 5$
---|---|---|---|---|---
$\hbar a_{j,0}$ | 4.577 |  4.999 |  12.281 |  24.999 |  34.999
$\hbar\Gamma_{j,0}$ | 0.499 |  0.699 |  0.999 |  1.651 |  2.190
$\hbar\omega_{j,0}$ | 2.784 |  3.183 |  3.799 |  4.695 |  6.349

###  C Interband response - excited matter

The application of the Lorentz model to excited matter is not straightforward.
As for the intraband response, we have to account for modifications due to the
electron and phonon temperatures [34]. In particular, new transitions become
possible as an increasing number of free states are created below the Fermi
energy when the temperature rises. We propose here to keep the form of the
interband response described by Eq. (4), but introduce temperature-dependent
Lorentz parameters. Recently this approach has been applied to electron-phonon
nonequilibrium by fitting the Lorentz model to results obtained by DFT at
distinct elevated electron temperatures [14]. Our approach aims to introduce a
temperature dependence to the strength and damping of the Lorentz oscillators
in a smooth way by taking into account the changing occupation numbers in the
two optical active bands. The oscillator positions will be kept constant with
respect to the density of states. In this way, we provide an easy-to-use
description that is smoothly connected to the values at room temperature and
might also be applied in nonequilibrium.

We start by considering the amplitudes of the Lorentz oscillators $a_{j}$.
This parameter can be connected to the number of oscillating $d$-band
electrons [35]. Thus, we scale $a_{j}$ with the electron occupation at the
oscillator position in the $d$-band normalized to the value at
$300\text{\,}\mathrm{K}$,

$a_{j}\\!\left(T_{e}\right)=a_{j,0}\;\frac{n_{d}^{E_{j}}\\!\left(T_{e}\right)}{n_{d}^{E_{j}}\\!\left($300\text{\,}\mathrm{K}$\right)}\,.$
(5)

Here, $n_{d}^{E_{j}}\\!\left(T_{e}\right)$ is the density of electrons in the
$d$-band within an energy interval $\left[E_{j}-\Delta,E_{j}+\Delta\right]$
that contains the energy $E_{j}$ of the oscillator $j$

$n_{d}^{E_{j}}\left(T_{e}\right)=\int_{E_{j}-\Delta}^{E_{j}+\Delta}\mathrm{d}E\;f\\!\left(E,T_{e},\mu\right)\;D_{d}\\!\left(E\right)\,,$
(6)

where $D_{d}$ is the PDOS of the $d$-band. The parameter $\Delta$ represents a
small energy interval. We apply the mean value theorem for integrals to Eq.
(6) and obtain that the amplitude is proportional to the distribution function
at the oscillator position $E_{j}$

$a_{j}\\!\left(T_{e}\right)=a_{j,0}\;\frac{f\\!\left(E_{j},T_{e}\right)}{f\\!\left(E_{j},$300\text{\,}\mathrm{K}$\right)}\,.$
(7)

Next, we focus on the temperature dependence of the resonance frequencies.
When the electron temperature rises, holes are created in the $d$-band below
$E_{F}$ as more and more $d$-band electrons are thermally excited into the
conduction band, which becomes increasingly populated. Moreover, the chemical
potential $\mu$ also shifts at elevated temperatures, which reflects the
conservation of the total number of particles. The details of the shift depend
on the shape of the DOS around $E_{F}$. In the case of gold, the DOS exhibits
high $d$-band peaks below $E_{F}$ whereas much fewer states exist above the
Fermi energy (see Fig. 1). Thus, the chemical potential for gold rises with
increasing electron temperature [24]. The chemical potential at a temperature
of $T_{e}=$2\text{\times}{10}^{4}\text{\,}\mathrm{K}$$ and the corresponding
Fermi distribution are plotted in Fig. 1 (red lines).

This shift of the chemical potential renders interband transitions more
difficult as the $d$-electrons need to overcome a broader energy gap. However,
the broadening of the Fermi edge facilitates the transitions from $d$\- into
$sp$-states below the chemical potential as not all of these are fully
occupied anymore, i.e., we see a softening of the Pauli blocking with rising
electron temperature. To account for both effects, the resonance frequencies
are modified according to

$\hbar\omega_{j}\\!\left(T_{e}\right)=\hbar\omega_{j,0}+\Delta\chi\\!\left(T_{e}\right)\,,$
(8)

where
$\Delta\chi\\!\left(T_{e}\right)=\chi\\!\left(T_{e}\right)-\chi\\!\left($300\text{\,}\mathrm{K}$\right)$
is the temperature-dependent shift from the fixed basis points of the Lorentz
oscillators to the lowest available state for the conduction band. The
function $\chi$ is set to be

$\chi\\!\left(T_{e}\right)=\mu\\!\left(T_{e}\right)-c\,k_{B}T_{e}\,.$ (9)

The first term accounts for the shifting chemical potential and the second
term for the temperature broadening of the Fermi edge. The constant $c$ sets
the strength of the second effect. It can be fixed by comparison to
experimental data or first-principles simulations.

The last quantities to consider are the damping parameters. We describe their
temperature dependence similarly to the Drude damping, i.e., including
electron-phonon and electron-electron scatterings

$\Gamma_{j}\\!\left(T_{i},T_{e}\right)=\Gamma^{j}_{ei}\\!\left(T_{i}\right)+\Gamma_{ee}\\!\left(T_{e}\right)\,.$
(10)

At room temperature, the electron-phonon damping $\Gamma_{ei}^{j}$ dominates
and their value is identical to $\Gamma_{j,0}$ presented in Tab. 1. For
elevated temperatures, one can assume a linear increase with the lattice
temperature. However, we consider cases where the probe time is fast and the
lattice remains cold, i.e., the damping due to scattering with phonons remains
unchanged here. In contrast, the damping due to electron-electron scattering
$\Gamma_{ee}$ is strongly varying with electron temperature via changes in the
occupation of the $d$\- and $sp$-bands. Here, we propose to use the same form
as for the intraband part, since the damping of the interband transitions
relies on the same scattering process [23, 26].

To summarize our model, we propose a description of the dielectric function
employing the Drude-Lorentz form with temperature-dependent oscillator
frequencies, amplitudes, and damping terms, thus, describing intraband and
interband contributions. For the Drude part, that is for the intraband
contributions, this extension has been shown to work well and to agree with
experimental data [23, 10]. The Lorentz part, describing transitions between
bands, will be tested against DFT-based data in the following section. Despite
its simplicity, we will demonstrate that the model is able to reproduce many
features of the optical properties of excited gold with very low computational
effort.

##  III Results and discussion

We present the results of our analysis of the dielectric function, and derived
optical properties, of gold under electron-phonon nonequilibrium. We also
compare our results for the reflectivity of gold with calculations based on
DFT. Applying our methods to thin films reveals more detailed insights into
the optical properties and points toward potential applications in various
fields.

###  A Temperature-dependent dielectric function

Figure 2: Broadband dielectric function of gold for various electron
temperatures. Imaginary part (left panel) and real part (right panel). (a) and
(b) intraband Drude response, (c) and (d) interband Lorentz part, and (e) and
(f) total response. The experimental data at room temperature are taken from
Ref. [22].

We examine the dielectric function of gold across a broad spectral range to
cover absorption of and probing by photons of different energy. Fig. 2 shows
the dielectric function in dependence on the photon energy for various
electron temperatures while keeping the phonons at $300\text{\,}\mathrm{K}$.
The individual intra- and interband contributions are shown in Figs. 2 to 2.
The intraband part is calculated with Eq. (2) and the interband response is
determined by Eq. (4). The total dielectric function, i.e. the sum of both
contributions, is shown in Figs. 2 and 2.

At room temperature and photon energies in the infrared regime, the imaginary
part of the dielectric function is dominated by intraband contributions, i.e.,
the Drude response of free carriers as interband transitions are very
unlikely, compare Figs. 2 and 2. However, this is not true for the real part
that has positive and non-negligible interband contributions at low photon
energy, as depicted in Fig. 2 and in qualitative agreement with the findings
of Ref. [36]. That is, the Drude model may not be sufficient to describe the
real part of the dielectric function of gold even for the small photon
energies.

With the onset of interband transitions, i.e., for photon energies around
$1.9\text{\,}\mathrm{eV}$, the imaginary part of the interband (Lorentz) part
increases significantly as shown in Fig. 2. The real part, plotted in Fig. 2,
shows a maximum at a photon energy equal to the first resonance energy
$\hbar\omega_{1,0}$, before decreasing at higher photon energies. In contrast,
the imaginary part of the intraband (Drude) contribution decreases
monotonically, while its real part increases, see Figs. 2 and 2.

The total dielectric response function, Figs. 2 and 2, shows a Drude-like
behavior for energies below the onset of interband transitions and is
relatively flat above $\sim$$3\text{\,}\mathrm{eV}$. We take the experimental
data of Ref. [22] (gray circles) as a reference at $300\text{\,}\mathrm{K}$
and match these data reasonably well with the fit parameters in Table 1.

At elevated electron temperatures, we observe a strong enhancement of
intraband contributions in the imaginary part over the entire spectrum ,
plotted in Fig. 2 for $10\,000\text{\,}\mathrm{K}$ and
$25\,000\text{\,}\mathrm{K}$. We attribute this enhancement to a strong rise
of both the Drude damping and the plasma frequency [26]. In contrast, the
temperature effects on the real part of the Drude response are small.

The interband (Lorentz) response also varies with the electron temperatures.
It is calculated using Eqs. (4) to (10) and by setting $c=1/2$ in Eq. (9), we
include the temperature broadening of the Fermi edge in the $sp$-band. The
imaginary part becomes smaller for increasing electron temperatures, as can be
seen in Fig. 2. This behavior is expected: With rising $T_{e}$ the chemical
potential shifts to higher energies, increasing the resonance frequencies and
thus decreasing the probability of transitions. The real part of the interband
response decreases at low energies as well. Moreover, the main resonance peak
shifts to higher energies (see Fig. 2). Due to the softer decrease of the
distribution function at the chemical potential, the transition peaks are more
and more washed out at elevated $T_{e}$, leading to smoother interband curves,
where the changes in the individual contributions are stronger at photon
energies below the interband transition threshold. Figures 2 and 2 show that
the total dielectric function is considerably modified at these smaller photon
energies, whereas the total response changes just slightly above the
threshold. Such behavior is in qualitative agreement with previous
observations for copper obtained by the Kubo-Greenwood formalism [12].

Figure 3: Imaginary part of the dielectric function of Au in dependence of the
electron temperature at laser wavelengths of $800\text{\,}\mathrm{nm}$ (red)
and $400\text{\,}\mathrm{nm}$ (blue). Intraband parts (dashed lines),
interband contributions (dots), and total (solid lines). Note the logarithmic
scale on the vertical axis.

Figure 4: Surface reflectivity of bulk gold versus electron temperature and
photon energy. Panel (a) uses $c=1/2$ in Eq. (9), i.e., including a
temperature broadening of the Fermi edge for the lowest available state. In
Panel (b) we consider $c=0$, i.e., the lowest available state in the $sp$-band
is determined only by $\mu\\!\left(T_{e}\right)$. The last panel (c) shows
DFT-based calculations from Ref. [24].

Figure 3 shows the important imaginary part of the dielectric function in
dependence on the electron temperature $T_{e}$ for two fixed laser
wavelengths: $800\text{\,}\mathrm{nm}$ ($1.55\text{\,}\mathrm{eV}$) and
$400\text{\,}\mathrm{nm}$ ($3.1\text{\,}\mathrm{eV}$). At room temperature,
the Drude part dominates the total dielectric function $\varepsilon_{\rm
total}^{i}$ for the long wavelength, _i.e._ low photon energy. The Lorentz
contribution is very small (but finite) in that case. Driven by the Drude
contribution, $\varepsilon_{\rm total}^{i}$ increases nonlinearly for
increasing electron temperatures. The behavior for the short wavelength of
$400\text{\,}\mathrm{nm}$ is quite different. For low electron temperatures,
the Lorentz part dominates and the value of $\varepsilon_{\rm total}^{i}$ is
relatively constant, before it starts a minor decline toward a minimum around
$20\,000\text{\,}\mathrm{K}$, where the Drude and Lorentz curves intersect.
Above this temperature, $\varepsilon_{\rm total}^{i}$ increases again, which
is not well visible due to the logarithmic scale of Fig. 3. This rise is
attributed to a strong enhancement of the Drude part.

To understand this behavior, it is essential to assess individual
contributions. According to Eq. (2), $\varepsilon_{\rm intra}^{i}$ is
proportional to $\nu$ and $\omega_{p}^{2}$. Thus, its trend is fully defined
by changes in the collision frequency $\nu$ and the plasma frequency
$\omega_{p}$. At sufficiently high $T_{e}$, a significant number of
$d$-electrons are thermally excited into free states of the $sp$-band and
create holes below $E_{F}$. These holes allow for efficient electron-electron
scattering and, thus, considerably increase the Drude damping $\nu$ [23, 26].
Combined with an increase in the plasma frequency, the intraband response
rises. The interband response, on the other hand, shows an overall decrease
with increasing temperature. This results from a lower probability of electron
transitions as the chemical potential and, thus, the interband excitation
threshold rises with temperature. Here, we note that the resonance frequencies
are the most critical parameters in the Lorentz approach.

###  B Broadband reflectivity of bulk gold

The reflectivity of materials is a measurable quantity and can serve as a
benchmark for models. It can be directly calculated from the dielectric
function discussed above. Under normal incidence, the surface reflectivity is
determined as

$R\\!\left(\omega,T_{i},T_{e}\right)=\left\lvert\frac{\tilde{n}\\!\left(\omega,T_{i},T_{e}\right)-1}{\tilde{n}\\!\left(\omega,T_{i},T_{e}\right)+1}\right\rvert^{2}\,,$
(11)

where $\tilde{n}$ is the complex refractive index and is related to the
dielectric function via $\tilde{n}=\sqrt{\varepsilon}$.

For the following results, we use the Drude-Lorentz form of the dielectric
function developed in Sec. II and, as before, keep the phonons at room
temperature. Fig. 4 displays the reflectivity of bulk gold as a function of
electron temperature and photon energy. For Fig. 4, we set $c=1/2$ in Eq. (9),
whereas for the results in Fig. 4 the value $c=0$ is used. This means that the
temperature broadening of the Fermi edge is applied in the left panel, whereas
it is neglected in the middle panel, where only the shift of the chemical
potential is applied. Accordingly, transitions of $d$-electrons are easier in
the former case as additional free states are available around the Fermi edge.
We compare both results with data from DFT-based calculations [24] plotted in
Fig. 4.

Figure 5: Reflectivity (top row), transmissivity (middle row), and absorption
(bottom row) of a $25\text{\,}\mathrm{nm}$ thin free-standing gold film as a
function of electron temperature and photon energy. The results shown in the
left vertical column are calculated for light at normal incidence. The panels
in the central and right vertical columns are calculated for $s$\- and
$p$-polarized light, respectively, with an angle of incidence of
$\theta_{1}=$45\text{\,}\mathrm{\SIUnitSymbolDegree}$$.

At room temperature, bulk gold is highly reflective below the onset of
interband transition energy of $\sim$1.9\text{\,}\mathrm{eV}$$. Above this
energy, absorption of photons by $d$-electrons starts to increase strongly
and, thus, decrease the reflectivity. With increasing electron temperature,
the reflectivity decreases in the infrared regime. Our results depicted in
Fig. 4 show a redshift of the reflectivity edge at temperatures below
$\sim$5000\text{\,}\mathrm{K}$$. This results from the term $\Delta\chi$ in
Eq. (8) being negative for $c=1/2$ at these low temperatures, _i.e._ , the
softening of the Fermi edge dominates the increase in the chemical potential.
For higher electron temperatures, the chemical potential increases
significantly and shifts the lowest available $sp$-state towards higher
energies. Thus, the reflectivity edge experiences a blueshift. In Fig. 4, we
observe such a blueshift of the reflectivity edge at all temperatures. The
blueshift is much more pronounced in Fig. 4 for the elevated $T_{e}$ because
no broadening of the Fermi edge is considered and, thus, the full energy gap
is accounted for. A similar blueshift of the reflectivity edge is also
observed in the DFT-based results plotted in 4. Moreover, we see that the edge
in the reflectivity is smeared out at elevated $T_{e}$ both in Figs. 4 and 4
and also visible in the DFT-based results [24]. This effect is mainly
attributed to the softening of the distribution function. Comparing to the DFT
data in panel 4, we find better agreement of the reflectivity shown in Fig. 4
for the range of temperatures considered here. Thus, we take $c=1/2$ in Eq.
(9) as the best option in future calculations.

Overall, our model shows similar features in the reflectivity as the DFT-based
calculations of Ref. [24] but requires significantly less computational
effort. We conclude therefore, that our model provides a reasonably accurate
and easily applicable approximation for the behavior of the optical response
in highly excited gold including intra- and interband transitions.

###  C Application to thin films

Many experiments applying laser excitation have been performed on thin films.
This approach avoids temperature gradients as the foil can be homogeneously
heated by ballistic electrons [37, 8, 38, 39]. To calculate the reflectivity,
transmissivity, and absorption of a free-standing thin gold film, we apply
multiple reflections theory [30]. We assume a single gold layer with two
parallel surfaces and vacuum at both interfaces of the thin film, like, e.g.,
used in Refs. [37, 8, 40]. We label the vacuum parts with indices $1$ and $3$,
respectively, and the gold film is indexed with $2$ leading to [30]

$\displaystyle R\\!\left(\omega,T_{i},T_{e}\right)$
$\displaystyle=\left|\frac{r_{12}+r_{23}e^{i2\beta}}{1+r_{12}r_{23}e^{i2\beta}}\right|^{2}\,,$
(12a) $\displaystyle T\\!\left(\omega,T_{i},T_{e}\right)$
$\displaystyle=\left|\frac{t_{12}t_{23}e^{i\beta}}{1+r_{12}r_{23}e^{i2\beta}}\right|^{2}\frac{n_{3}\cos\\!\left(\theta_{3}\right)}{n_{1}\cos\\!\left(\theta_{1}\right)}\,,$
(12b)

where $r_{km}$ and $t_{km}$ are the reflection and transmission coefficients
of light propagating from medium $k$ to medium $m$. These coefficients are
determined by the Fresnel equations [30]. In our case, $n_{1}$ and $n_{3}$ are
the refractive indices of the vacuum and are thus equal. $\theta_{1}$ is the
angle of incidence, and $\theta_{3}$ is the angle between the transmitted
light and the surface normal ($\theta_{1}=\theta_{3}$ because media $1$ and
$3$ are both vacua). The quantity $\beta=2\pi h\cos(\theta_{2})n_{2}/\lambda$
denotes the phase change of light resulting from multiple internal
reflections. Here, $\lambda$ is the wavelength of the probe pulse, $h$ is the
film thickness, and $\theta_{2}$ is the refracted angle of light within the
gold film (medium 2) after entering from vacuum (medium 1). $n_{2}$ is the
complex refractive index of gold calculated with the temperature-dependent
dielectric function described above.

Figure 5 shows the reflectivity $R$ (top row), transmissivity $T$ (middle
row), and absorption $A=1-R-T$ (bottom row) of a free-standing gold film with
$25\text{\,}\mathrm{n}\mathrm{m}\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{k}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{a}\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$
of the probed photon energy and the electron temperature. We present the
results for normal incidence in the left columns. For the results shown in the
central and right columns, we consider an angle of incidence of
$\theta_{1}=$45\text{\,}\mathrm{\SIUnitSymbolDegree}$$, where Figs. 5(b-e-h)
are for $s$-polarized light and Figs. 5(c-f-i) assume $p$-polarized light.

At $300\text{\,}\mathrm{K}$, the thin film reflects less light than the bulk
material (compare the top panels to Fig. 4) for all considered cases.
Moreover, the reflectivity edge is lower and less pronounced for the thin
film, especially for the $p$-polarized light. As $T_{e}$ increases, the
reflectivity decreases in the infrared regime. The $s$-polarized light is more
reflective over the whole spectrum compared to the $p$-polarized light, whose
reflectivity shows enhanced features. Similar to bulk, we also observe a
blueshift of the reflectivity edge and its broadening at elevated electron
temperatures.

The effect of elevated electron temperature on the transmissivity is
relatively low, except for photon energies between $\sim$1\text{\,}$$ and
$\sim$3\text{\,}\mathrm{eV}$$. The transmissivity maps all show a peak around
$\sim$2.4\text{\,}\mathrm{eV}$$, which decreases and shifts towards higher
energies with increasing $T_{e}$. These features appear at normal incidence
and for both polarizations considered but are more pronounced for the
$p$-polarized light.

Since the transmissivity is low, the absorption is more or less the inversion
of the reflectivity. For small photon energies below the absorption edge, the
absorption at room temperature is very low (less than $6\%$), but increases
significantly with increasing temperature. In contrast, the absorption is
large for higher photon energies, due to the possibility of transferring
$d$-electrons into free states of the $sp$-band, and appears to be nearly
independent of temperature. Similar to the reflectivity, we also observe a
shift of the absorption edge and its broadening.

Our results for thin films show a similar qualitative behavior of the optical
parameters for normal and for oblique incidence, but with large differences in
magnitude. Moreover, some of the features observed for $R$, $T$, and $A$ are
much more pronounced for $p$-polarized light than for $s$-polarized light. The
latter is thus more sensitive and seems to be ideal for probing changes in the
optical properties. Figs. 5 also reveal that photon energies below the
interband threshold are better suited to detect changes in material properties
than higher photon energies.

Figure 6: Reflectivity and absorption of a $25\text{\,}\mathrm{nm}$ thin free-
standing gold film for $p$-polarized light as a function of the electron
temperature. (a) for a wavelength of $800\text{\,}\mathrm{nm}$ and (b) for
$400\text{\,}\mathrm{nm}$.

The dependence of the optical properties on electron temperature is difficult
to assess quantitatively in Fig. 5. We thus highlight it for two fixed probe
wavelengths, one below and one above the interband threshold. Fig. 6 shows the
reflectivity and absorption for $p$-polarized light as a function of the
electron temperature for a gold film of $25\text{\,}\mathrm{nm}$ thickness. We
consider probe wavelengths of $800\text{\,}\mathrm{nm}$
($1.55\text{\,}\mathrm{eV}$) and $400\text{\,}\mathrm{nm}$
($3.1\text{\,}\mathrm{eV}$). Depending on wavelength, we observe a different
qualitative behavior of reflectivity and absorption. For
$800\text{\,}\mathrm{nm}$, the reflectivity decreases with increasing $T_{e}$.
This is due to an increase in the number of free states below $E_{F}$. In
contrast, for $400\text{\,}\mathrm{nm}$, Fig. 6 shows a small increase in $R$
when gold is heated. This is followed by a small decrease reaching a minimum
at about $\sim$15\,000\text{\,}\mathrm{K}$$ before $R$ experiences a larger
increase. The increase in reflection is due to a decrease in the number of
free states above the Fermi level with increasing $T_{e}$.

Equally interesting is the absorption shown in Fig. 6. For the
$800\text{\,}\mathrm{nm}$ laser, the absorption increases nonlinearly with the
electron temperature. At the highest temperature shown, the absorption is more
than five times larger than its initial value. That is, the absorption follows
a trend similar to that of $\varepsilon_{\rm total}^{i}$ shown in Fig. 3. At
sufficiently high $T_{e}$, more $d$-electrons, which serve as a reservoir, are
excited above $E_{F}$ and are responsible for the observed nonlinearity. For
$400\text{\,}\mathrm{nm}$, the absorption is nearly constant. The absorption
only reduces slightly at very high $T_{e}$ when many $d$-electrons are
thermally excited.

Overall, our calculations show the importance of considering temperature-
dependent optical properties when modeling the optical response of thin films
after ultrafast excitation. This behavior is particularly important when
estimating the time-dependent absorption during transient energy deposition,
especially for thin films. Probing with different polarizations can also help
to highlight features arising in heated matter and, thus, can be used for
probing the states created.

##  IV Conclusion

In summary, we have studied the optical properties of excited gold for a broad
spectrum of probe frequencies and a temperature range from room temperature up
to high electron temperatures. We develop our approach starting with a Drude-
Lorentz model fitted to experimental data for room temperature. For excited
gold, we generalize the description by introducing temperature-dependent
parameters describing interband transition in the Lorentz approach. The main
quantities here are resonance frequencies, amplitudes, and scattering
frequencies, which depend on electron temperature as well as on the occupation
of electron states around the Fermi edge. The temperature dependence of the
intraband response, _i.e._ , the Drude part, has been studied previously [26].
Our model shows a good agreement with reflectivity data for highly excited
gold obtained by DFT-based calculations [24]. In particular, it reproduces its
key features well but requires much less computational effort.

When applied to a thin gold film, our method reveals a high sensitivity of the
reflectivity and the absorption on the electron temperature. For both
quantities, we observe a qualitatively different behavior for photon energies
below and above the threshold for interband transitions. There is a larger
sensitivity to increasing electron temperature for the photon energies below
the interband threshold. Moreover, we find pronounced features for
$p$-polarized light which, thus, can be considered an ideal tool for probing
temperature effects.

The model described here can be easily implemented in numerical simulations
and offers valuable insights for modeling ultrafast processes during and
shortly after the energy deposition with short laser pulses. The model can be
applied to situations with Fermi-distributed electrons as well as to
nonequilibrium situations. Although only results for a cold lattice are shown,
the effect of an excited phonon system can also be included, _e.g._ , by
incorporating additional electron-phonon scattering processes. Extensions of
the present work towards materials with a similar band structure, such as
copper and silver, are straightforward and require only the adjustment of a
few parameters.

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