# Efficient hot carrier dynamics in near-infrared photocatalytic metals

Cesar E. P. Villegas  Departamento de Ciencias, Universidad Privada del Norte,
Lima 15434, Peru.  Marina S. Leite  Department of Materials Science and
Engineering, University of California Davis, CA, 95616 USA.  Andrea Marini
Istituto di Struttura della Materia of the National Research Council, Via
Salaria Km 29.3,I-00016 Monterotondo Stazione, Italy.  A. R. Rocha  Instituto
de Física Teórica, Universidade Estadual Paulista (UNESP), Rua Dr. Bento T.
Ferraz, 271, São Paulo, SP 01140-070, Brazil.

###### Abstract

Photoexcited metals can produce highly-energetic hot carriers whose controlled
generation and extraction is a promising avenue for technological
applications. While hot carrier dynamics in Au-group metals have been widely
investigated, a microscopic description of the dynamics of photoexcited
carriers in the mid-infrared and near-infrared Pt-group metals range is still
scarce. Since these materials are widely used in catalysis and, more recently,
in plasmonic catalysis, their microscopic carrier dynamics characterization is
crucial. We employ _ab initio_ many-body perturbation theory to investigate
the hot carrier generation, relaxation times, and mean free path in bulk Pd
and Pt. We show that the direct optical transitions of photoexcited carriers
in this metals are mainly generated in the near-infrared range. We also find
that the electron-phonon mass enhancement parameter for Pt is 16 $\%$ higher
than Pd, a result that help explains several experimental results showing
diverse trends. Moreover, we predict that Pd (Pt) hot electrons possess total
relaxation times of up to 35 fs (24 fs), taking place at approximately 0.5 eV
(1.0 eV) above the Fermi energy. Finally, an efficient hot electron generation
and extraction can be achieved in nanofilms of Pd (110) and Pd (100) when
subject to excitation energies ranging from 0.4 to 1.6 eV.

##  I Introduction

Noble metals belong to a class of materials whose physical properties have
been extensively studied, both theoretically Xu  _et al._ (2017); Choi  _et
al._ (2006); Marini  _et al._ (2002); Tang and Fultz (2011); Wilson and Coh
(2020); Gall (2016) and experimentally Werner  _et al._ (2019); Olmon  _et
al._ (2012); Suemoto  _et al._ (2021); Garcia-Basabe  _et al._ (2019). This
has lead to a wide range of applications in photovoltaics Linic  _et al._
(2011); Atwater and Polman (2010), plasmonics Jiang  _et al._ (2017); Gong and
Leite (2016), and most recently in an emergent area called plasmonic catalysis
Aslam  _et al._ (2017, 2018); Cortés  _et al._ (2020). Some of these metals’
outstanding properties stem from the formation of large surface areas, high
electron densities and strong interaction of light with plasmons. This enables
the absorption of vast amounts of electromagnetic radiation near the metallic
surface, a phenomenon known as surface plasmon resonance Yu  _et al._ (2019);
Barnes  _et al._ (2003). Thus, photoexcitated metals can produce highly-
energetic hot carriers, via non-radiative surface plasmon decay Sönnichsen
_et al._ (2002); Maier (2007), whose subsequent dynamics strongly depends on
inelastic electron-electron ($e$-$e$), electron-phonon ($e$-$ph$), and defect
scattering processes Fatti  _et al._ (2000); Brown  _et al._ (2016), as
schematically shown in Figure 1a.

As demonstrated in recent experiments, the efficient generation and extraction
of photoexcitated carriers, which ultimately depends on their hot carrier
relaxation time and mean free path, may be used in a number of novel
applications Clavero (2014); Brongersma  _et al._ (2015); Ng  _et al._ (2016).
Therefore, having a precise knowledge of – and perhaps control over – the
microscopic processes that hot carriers undergo is of utmost importance for
efficient device development.

While hot carrier dynamics in Au-group metals (Au,Ag,Cu) covering the
ultraviolet and visible spectral range have been widely studied due to their
highly-energetic efficient generation via interband excitations,Gall (2016);
Bernardi  _et al._ (2015); Brown  _et al._ (2016) a microscopic understanding
of the carrier relaxation processes in Pt-group metals (Pt,Pd,Ni) have
received little attention Suemoto  _et al._ (2021). These carriers are
generated in the mid-infrared and near-infrared range, and subsequent
engineering of the electronic and optical properties by alloying them with
other noble metals has emerged as a route for the efficient generation of hot
carriers as this procedure brings the positions of the $d$-band closer to the
Fermi level, EF Gong  _et al._ (2020, 2018); Dias and Leite (2019); McClure
_et al._ (2019). Consequently it reduces the interband energy threshold,
permitting the efficient generation of long-lived hot carriers in the infrared
and near-infrared region Stofela  _et al._ (2020); Krayer  _et al._ (2020).

Figure 1:  a) Schematic representation of the formation of a surface plasmon
and its subsequent decay into hot carriers (hot electron and hot hole). The
inelastic phonon scattering processes of hot carriers and their corresponding
relaxation path is depicted by the arrows. b) Simplified representation of
density of states for near- infrared plasmonic metals. c) Quasiparticle
electronic bandstructure for Pd and Pt (blue lines) obtained within the real-
axis GW approximation compared with angle-resolved ultraviolet photoemission
measurements (orange dotted line) Himpsel and Eastman (1978); Eyers  _et al._
(1984). The gray shaded regions indicate the crystallographic directions where
the most relevant direct optical transitions occur. d) Comparison of the
calculated interband and intraband absorption with experimental measurements
for Pd Lafait  _et al._ (1978) and Pt Weaver (1974).

Accurate theoretical descriptions of catalytic metals, including their
electronic Östlin  _et al._ (2016); Ladstädter  _et al._ (2004); Windmiller
_et al._ (1969); Andersen (1970), vibrational Corso (2007); Shen  _et al._
(2016); Savrasov and Savrasov (1996); Xu  _et al._ (2020), and optical Weaver
(1974); Glantschnig and Ambrosch-Draxl (2010); Lafait  _et al._ (1978)
properties abound. Nonetheless, _ab initio_ calculations addressing the
scattering relaxation rates and mean free paths are still scarce. Indeed, the
lack of predictive studies concerning the microscopic e-ph scattering
mechanism become especially critical, in light of the available experimental
measurements indicating that Pd and Pt present large mass enhancement
parameters, $\lambda(E_{F})$, which play an important role in many physical
phenomena such as electronic heat capacity and superconductivity. This is even
more striking, since these metals do not exhibit superconductivity despite the
large value of $\lambda(E_{F})$– ranging from 0.6 to 0.7– which arises from
the contribution of the electron-phonon and electron-paramagnon interactions.
The later (the spin fluctuations) has been suggested to be the responsible for
the suppression of the superconductive state Allen (1987); Knapp and Jones
(1972). In addition, there is a long standing debate over the hierarchy of
enhancement factors between Pt and Pd. Available specific heat measurements
suggest that $\lambda_{Pd}$ > $\lambda_{Pt}$, which implies that the electron-
phonon lifetime at the Fermi energy for Pt is greater than that of Pd Knapp
and Jones (1972); Andersen (1970). This conclusion is, however, contrary to
what is estimated from semi-empirical electron-phonon lifetimes obtained from
resistivity measurements, in which the lifetimes of Pd are slightly higher
compared Pt.

We set out to elucidate this set of interesting phenomena, all of which,
closely related to the inelastic e-ph and e-e scattering rates. We do so by
carrying out _ab initio_ many-body perturbation theory Onida  _et al._ (2002)
to study the hot carrier generation, carrier lifetimes, and carrier transport
in bulk Pd and Pt. Our results show that the photoexcited carriers in these
metals are mainly generated in the near-infrared range by direct optical
transitions. We also found that the electron-phonon mass enhancement parameter
for Pt is $16\%$ higher than Pd, a result that corroborate recent experimental
trends obtained by resistivity Wilson and Coh (2020) and ultrafast
luminescence measurements Suemoto  _et al._ (2021). This is due to a higher
than previously obtained density of states for Pt at the Fermi level. Finally,
we show that thin films of Pd (110) and Pd (100) subject to excitation
energies ranging from 0.4 to 1.6 eV can be used for efficient hot carrier
generation and extraction as a consequence of its 30 nm mean free path,
comparable to that of noble metals.

##  II Results and discussion

Figure 2: a) Imaginary part of the $e$-$e$ and $e$-$ph$ self-energies for Pd
and Pt. The electron-phonon rates were computed at room temperature. The
insets show the temperature dependence of the electron-phonon linewidth at
Fermi energy. b) electron-phonon spectral function $\alpha^{2}_{tr}F(\omega)$
(solid lines) and phonon density of states (shaded curve). The phonon DOS
curve is normalized to enclose the same area as the spectral function. c)
comparison of the calculated phonon dispersion and experimental measurements
for Pd Miiller and Brockhouse (1968) and Pt Dutton  _et al._ (1972). Right
panels shows the generalized Eliashberg spectral function at the highest
occupied energy states for the high-symmetry points $G$, $X$, $L$ and $K$.

In metals, the dielectric response function, which gives the photon absorption
probability, is composed by two contributions. First i) the intraband one,
$\epsilon_{2}^{intra}(\omega)$: indirect electronic excitations between
partially filled $sp$ bands, that require a transfer of momentum provided by
either phonons or plasmons. This typically occurrs in the infrared energy
range. Second ii) due to the interband transitions
$\epsilon_{2}^{inter}(\omega)$: direct electronic excitations from occupied
$d$ bands to unoccupied $sp$ states, which causes a strong absorption peak.
These two contributions are schematically represented in Figure 1-b. Given
that in Pd and Pt, the most intense peaks of the dielectric function are
located in the infrared and near-infrared spectral range, the intraband and
interband contributions may overlap. Thus, separetely analyzing each
contributions is a requirement. In order to understand the microscopic
mechanism of photocarrier generation, governed by the electronic structure, in
Figure 1c we present the quasiparticle GW bandstructure, fully including spin-
orbit interactions, which are compared with experimental data from angle-
resolved ultraviolet photoelectron spectroscopy along the $\Gamma L$ path.
Clearly, our predicted QP energies show excellent agreement with
experimentally available data in the vicinity of the $L$ point, close to the
Fermi energy, whose electronic states are relevant for the carrier photo-
excitation in the near-infrared energy window. As shown, the Fermi level is
intersected by $d$ bands narrowing the energy window for allowed $d$-to-$sp$
transitions, when compared with gold group metals (see supplementary
information). A careful analysis of the oscillator strength for both metals
reveals that these optical transitions occur around the high symmetry points
$L$ and $X$, which are schematically represented by the shaded regions in
Figure 1c. In Figure 1d the interband contribution of the dielectric function
is obtained fully ab-initio while the intraband one is computed by
substituting the estimated experimental electron lifetime and plasma frequency
for Pd Lafait  _et al._ (1978) and Pt Weaver (1974), into the Drude model. Our
quasiparticle optical absorption for Pd shows two intense peaks located at
0.33 and 0.53 eV, which are in excellent agreement with the experimental
measurements obtained by Lafait _et al_ Lafait  _et al._ (1978). Accordingly,
the calculated absorption spectrum of Pt presents an intense peak at 0.76 eV
which is less than 100 meV larger than the one 0.67 eV found in experimental
measurements Weaver (1974). Furthermore, notice that the Pd intraband
contribution strongly overlaps with the interband one in the mid-infrared
region. This result clarify an old debate since it was not possible to
unambiguously distinguish in experiments the intraband and interband
contributions in Pd Lafait  _et al._ (1978); Weaver (1974); Vargas  _et al._
(2019), a task that became more challenging due to the lack of accurate band
structure calculations. In contrast, the intraband contribution of Pt weakly
overlaps with the interband one. It should be noted that the weak (strong)
overlap between intraband and interband contributions for Pt (Pd) is directly
ascribed to the position of $d$ states (around the high symmetry point $L$)
with respect to the Fermi energy.

Figure 3: a) hot carrier relaxation time as a function of energy for Pd (top
panel) and Pt (bottom panel). The scattering rates in green fully account for
$e$-$e$ and $e$-$ph$ contribution, while the red curves only accounts for
$e$-$ph$ contribution. The vertical blue dashed line indicates the Fermi
level. b) Mean free path as a function of energy along the [100], [110], and
[111] crystallographic directions.

Despite the experimental and theoretical works characterizing the carrier
scattering rates of Pt group metals Windmiller  _et al._ (1969); Ladstädter
_et al._ (2004); Gall (2016), the microscopic mechanism related to the
electron-phonon relaxation rates is still poorly understood. Following an
approach developed by Bernardi _et al_ in references Bernardi  _et al._ (2015,
2014), we employ a quantum mechanical microscopic approach based on _ab
initio_ many-body perturbation theory to calculate the hot carrier scattering
rates, taking into account both the $e$-$e$ and $e$-$ph$ contributions. These
are related to the imaginary part of the $e$-$e$
(Im$\Sigma^{e-e}_{n\textbf{k}}$) and $e$-$ph$
(Im$\Sigma^{e-ph}_{n\textbf{k}}$) self-energies. Figure 2-a shows the $e$-$e$
and $e$-$ph$ self-energy within a 5 eV energy window around $E_{F}$. For both
metals, the $e$-$ph$ self-energy resembles the energy dependence of the
electronic density of states (DOS). This is expected, since the DOS dictates
the number of available electronic states that can effectively couple with
lattice excitations. Clearly, the $e$-$ph$ contribution for Pd and Pt
dominates within a narrow energy window of 0.8 eV around EF, indicating that
the energy loss within this range of energy is dominated by electron-phonon
scattering processes. Concerning the $e$-$e$ self-energy, commonly related
with the impact ionization rate, the results show that for Pd, the energy loss
of hot carriers satisfying the condition $|E|>4$ eV, is dominated by inelastic
impact ionization. In contrast, for Pt, there is not a clear dominance of the
$e$-$e$ interactions. Overall, the energy loss of hot holes is greater that of
hot electrons by up to a factor of four. This feature has not been observed in
gold group metals Bernardi  _et al._ (2015) within this energy range and can
be ascribed to the proximity of $d$ states to the Fermi level Bauer  _et al._
(2015); Zhukov  _et al._ (2002). The $e$-$ph$ linewidth,
$\Gamma^{e-ph}(E_{n\textbf{k}})=2$Im$\Sigma^{e-ph}_{n\textbf{k}}$, can also be
used to estimate the electron-phonon mass enhancement parameter,
$\lambda(E_{n\textbf{k}})$. By considering, the $e$-$ph$ self-energy in the
regime at which $k_{B}T$ is greater than the maximum phonon energy, one can
write down a linear relation for the $e$-$ph$ linewidth with temperature
Grimvall (1981)

$\Gamma^{e-ph}(E_{n\textbf{k}})=2\pi\lambda(E_{n\textbf{k}})k_{B}T.$ (1)

In order to estimate the electron-phonon mass enhancement parameter at the
Fermi energy, $\lambda(E_{F})$, we computed the $e$-$ph$ linewidth, along the
same high-symmetry directions used for the quasiparticle bandstructure (see
Figure 1b) and then averaged over a narrow energy window of 10 meV around the
$E_{F}$. The results, presented in the insets of Figure 2a, clearly show an
increasing monotonic behavior of the linewidth with temperature. From equation
(1), we estimate the electron-phonon mass enhancement (the slope of the
curves) to be $\lambda=0.66$ and $\lambda=0.78$ for Pd and Pt, respectively.
Given that our calculations fully include the spin-orbit interaction, it can
be argued that they partially take into account some effects related to the
spin degree of freedom Andersen (1970). This implies that the carrier lifetime
at the Fermi level for Pt is greater than that of Pd (the greater
$\lambda\left(E_{n\mathbf{k}}\right)$, the shorter the relaxation time). In
this sense, the value obtained for Pd can be compared with the one estimated
from specific heat measurements by Knapp and Jones Knapp and Jones (1972)
($\lambda=0.7$), Allen Allen (1987) ($\lambda=0.66$) and Savrasov Savrasov and
Savrasov (1996) ($\lambda=0.69$), whose mass enhancement parameters include
the electron-phonon and spin fluctuation contributions. Note that even though
Pd and Pt present large mass enhancement parameters, superconductivity is
absent in these metals due to the large paramagnon contribution Hsiang  _et
al._ (1981). The mass enhancement parameter obtained for Pt, on the other
hand, over-estimates experimental measurements obtained from specific heat
that span from 0.33 to 0.63 Allen (1987); Knapp and Jones (1972); Andersen
(1970). It is important to point out that the experimental results rely on _ab
initio_ calculations for the density of states at the Fermi level. This
parameter is, in fact, extremely sensitive towards the electronic
bandstructure Allen (1987), which at the time most of these studies were
carried out, suffered from a computational limit on the number of k-grid
samples, yielding less accurate values for the electronic structure. Thus, we
argue here, that this issue is related to a required theoretical input (the
proper calculation of the density of states at Fermi level), which demands
very fine q-and k-grid samples, as those employed in this work. In fact, more
recent semi-empirical electron-phonon lifetimes estimated from resistivity
measurements, reported by Wilson and Coh Wilson and Coh (2020) point out in
the same direction.

We proceed with the analysis of the electron-phonon coupling strength. Thus,
in Figure 2b we present the calculated phonon dispersion at 0 K together with
previous experimental neutron scattering measurements for Pd Miiller and
Brockhouse (1968) and Pt Dutton  _et al._ (1972). We clearly observe a nice
agreement between the compared curves. For completeness, in Figure 2c we
present the phonon density of states in which one notes the presence of an
intense peak at higher energies. We then calculate the generalized Eliashberg
spectral function, $\alpha^{2}F_{n\textbf{k}}(\omega)$, which determines the
probability of specific phonon modes to decay into an electron-hole pair (see
methods section). The Eliashberg spectral functions related to the highest
occupied energy states of the high-symmetry points $\Gamma$, $X$, $L$ and $K$
(represented by green squares in Figure 1c) are shown in Figure 2d. In
general, the selected spectral functions show considerable differences with
the phonon density of states, especially in the intensity at low and high
energies. This effect can be attributed to the complex Fermi surface that
allows more effective Fermi nesting Mueller and Freeman (1970); Xu  _et al._
(2020); Andersen (1970). In fact, Pd and Pt, present an intense peak around 16
meV and 13 meV, respectively, which are related to the contribution of
transverse acoustic modes. The less intense peaks at higher frequencies of
approximately 28 meV and 23 meV for, Pd and Pt respectively, is directly
associated to the longitudinal acoustic mode.

The hot carrier relaxation times, presented in Figure 3a-b, show that for Pd
(Pt) hot electrons possess total relaxation times of up to 35 fs (24 fs),
taking place at approximately 0.5 eV (1.0 eV) above the Fermi energy. In
contrast, hot holes present faster relaxation times with a maximum of 6.8 and
5.6 fs for Pd and Pt, respectively, which are found close to the Fermi level.
This result suggest that the extraction of hot holes before thermalization is
a challenging task, since these carriers lose energy on the time scales below
5 fs. Note that the inclusion of $e$-$e$ scattering rates for Pt, drastically
changes the relaxation times of hot holes for energies 1.0 eV below the Fermi
level. A similar trend for hot electrons is observed at energies above 3 eV of
the Fermi level. This results clearly highlight that a complete description of
hot carriers in metals must include both the $e$-$e$ and $e$-$ph$
contributions. The carrier lifetimes at the Fermi level, calculated by
averaging the relaxation time over a 10 meV energy window around the Fermi
level, are 6.3 fs and 5.2 fs for Pd and Pt, respectively. These findings are
in agreement with the electron-phonon lifetimes reported by Wilson and Coh
Wilson and Coh (2020), which employ semi-empirical models based on resistivity
measurements yielding values of 8.6 fs for Pd and 6.1 fs for Pt.

Based on the results for the total carrier lifetime ($\tau_{n,\textbf{k}}$)
and the carrier velocity $v_{n,\textbf{k}}=\frac{1}{\hbar}\frac{\partial
E_{n,\textbf{k}}}{\partial\textbf{k}}$, obtained from the GW bandstructures,
we calculate the mean free path,
$\Lambda_{n,\textbf{k}}=\tau_{n,\textbf{k}}|v_{n,\textbf{k}}|$. Figure 3c-d
shows the carrier mean free path for hot holes and hot electrons along the
$\Gamma X$, $\Gamma K$, and $\Gamma L$ directions of the Brillouin zone, which
represent the [100], [110], and [111] crystallographic directions. As expected
from the relaxation time results, hot holes in Pd an Pt possess mean free
paths of less than 5 nm in all directions, making extraction challenging. In
contrast, hot electrons in Pd (Pt), present mean free paths of up to 30 nm (20
nm). Such longer values are obtained for energies up to 0.8 eV above the Fermi
level, and are ascribed to carrier velocities of hybridized $sp$ bands along
the $\Gamma K$ direction, combined with the relative long lifetimes associated
to the inherent low DOS of these bands. These results suggest that an
efficient extraction of hot electrons in thin films of Pd (100) and Pd (110)
may be achieved with excitation energies spanning from 0.4 to 1.6 eV.
Accordingly, we predict that the efficient way of extracting hot electrons in
Pt might be accomplished in thin films grown in the (110) direction for
excitation energies within the 0.7 to 1.5 eV range. Our predictions are
consistent with some available experimental measurements of optical
absorption, electron-phonon mass enhancement and electron-phonon lifetimes.
Nevertheless, this is not true for calculations in the absence of the spin-
orbit coupling, which seems to understimate the optical absorption peaks while
overstimates the carrier mean free path (see SI). Finally, we should mention
that our predicted scattering rates are in fact equilibrium rates based on
standard _ai_ -MBPT, which are expected to be reliable at low carrier
densities. These rates, however, can be employed as inputs in a complete
description of carrier dynamics based on the evolution of the non-equilibrium
Green’s functions, which has been recently devised and employed to describe
the real-time carrier dynamics in some paradigmatic materials Marini (2012);
Sangalli and Marini (2015); Molina-Sánchez  _et al._ (2017). The latter is,
however, beyond of the scope of this work.

##  III Conclusions

Based on _ab initio_ many body perturbation theory, we provided a
comprehensive theoretical description of the hot carrier generation, carrier
lifetimes, and carrier transport in bulk Pd and Pt. Our optical response
results showed that the photoexcited carriers in these metals are mainly
generated in the near-infrared range (from 0.3 to 0.78 eV) by direct optical
transitions, while intraband transitions are relevant at infrared frequencies.
We also found that the electron-phonon mass enhancement parameter for Pt is
slightly higher than Pd, a striking result that shed light into the
understanding of previous experimental measurements. Our findings indicate
that the extraction of hot holes in Pt and Pd is a challenging task as these
carriers possess sub-5-fs lifetimes. In addition, we showed that thin films of
Pd (110) and Pd (100) subject to excitation energies ranging from 0.4 to 1.6
eV can be used for efficient hot electron generation and extraction as a
consequence of its 30 nm mean free path, comparable to that of noble metals.

##  IV Acknowledgements

A.R.R. acknowledges support from ICTP-SAIRF (FAPESP project 2011/11973-4).
M.S.L. acknowledges the financial support from the National Science Foundation
(award number 20-16617). A.M. acknowledges the funding received from the
European Union projects: MaX Materials design at the eXascale
H2020-EINFRA-2015-1, Grant agreement n. 676598, and
H2020-INFRAEDI-2018-2020/H2020-INFRAEDI-2018-1, Grant agreement n. 824143;
Nanoscience Foundries and Fine Analysis - Europe H2020-INFRAIA-2014-2015,
Grant agreement n. 654360. This work uses the computational resources from
GRID-UNESP.

##  V APPENDIX

###  V.1 Theory and computational details

We consider pristine face centered cubic crystals of Pd and Pt with fully
relaxed lattice parameters of 3.86 and 3.91 Å, respectively. Our calculations
are conducted in two steps. First, plane-wave density functional theory is
used to obtain the electronic ground-state within the local density
approximation (LDA) for the exchange-correlation potential. We employed norm-
conserving pseudopotentials treating the semicore $s$ and $p$ states as
valence electrons, a 90 Ry kinetic energy cutoff for Pd/Pt and a $k$-sampling
grid in the Monkhorst-Pack scheme of 12$\times$12$\times$12 as implemented in
the Quantum-Espresso code Giannozzi and et. al. (2009). The structures were
fully optimized to their equilibrium position with pressures on the lattice
unit cell smaller than 0.02 kbar. The Marzari-Vanderbilt smearing scheme
including a width of 0.02 Ry is employed. Next, we use density functional
perturbation theory (DFPT) Baroni  _et al._ (2001) to compute the vibrational
frequencies $\omega_{\textbf{q}\lambda}$ and the derivatives of the self-
consistent Khon-Sham potential with respect to the atomic displacements,
inputs required to evaluate the electron-phonon coupling matrix elements,
$g^{\textbf{q}\lambda}_{nn^{\prime}\textbf{k}}$, which represent the
probability amplitude for an electron to be scattered due to emission or
absorption of phonons,

$\begin{split}g^{\textbf{q}\nu}_{nn^{\prime}\textbf{k}}=\sum_{s\alpha}[2M_{s}\omega_{\textbf{q}\nu}]^{-1/2}e^{i\textbf{q}.\tau_{s}}\xi_{\alpha}(\textbf{q}\nu|s)\\\
\times\langle n^{\prime}\textbf{k-q}|\frac{\partial
V_{scf}(\textbf{r})}{\partial R_{s\alpha}}|n\textbf{k}\rangle,\end{split}$ (2)

where, $M_{s}$ is the atomic mass of the s$-$th atom, $\tau_{s}$ is the
position of the atomic displacement in the unit cell,
$\xi_{\alpha}(\textbf{q}\nu|s)$ are the components of the phonon polarization
vectors, and $V_{scf}(\textbf{r})$ is the self-consistent DFT ionic potential.

Many-Body perturbation theory (MBPT) Onida  _et al._ (2002) is used to
describe the electron-phonon interaction which is treated perturbatively
Marini  _et al._ (2015) by considering the first order Taylor expansion term
in the nuclear displacement, commonly known as the Fan term. Given that the
Fan self-energy term is a complex function, it is responsible for providing
the imaginary part of the quasiparticle energies that correspond to the
quasiparticle linewidths. Thus, the Fan self-energy is given by

$\begin{split}\Sigma^{Fan}_{n\textbf{k}}(\omega,T)=\sum_{n^{\prime}\textbf{q}\nu}\frac{|g^{\textbf{q}\nu}_{nn^{\prime}\textbf{k}}|^{2}}{N_{q}}\Big{[}\frac{N_{\textbf{q}\nu}(T)+1-f_{n^{\prime}\textbf{k-q}}}{\omega-\epsilon_{n^{\prime}\textbf{k-q}}-\omega_{\textbf{q}\nu}-i\delta}\\\
+\frac{N_{\textbf{q}\nu}(T)+f_{n^{\prime}\textbf{k-q}}}{\omega-\epsilon_{n^{\prime}\textbf{k-q}}+\omega_{\textbf{q}\nu}-i\delta}\Big{]},\end{split}$
(3)

where $N_{\textbf{q}\lambda}$ and $f_{n^{\prime}\textbf{k-q}}$ represent the
Bose-Einstein and Fermi-Dirac distribution functions, while Nq is the number
of q vectors in the Brillouin zone. The electron-phonon relaxation time is
given by,

$\tau_{n,\textbf{k}}=\frac{\hbar}{2}Im[\Sigma^{Fan}_{n\textbf{k}}]^{-1}.$ (4)

We employ both, the EPW Poncé  _et al._ (2016) and YAMBO Sangalli  _et al._
(2019) code to compute the Fan self-energy and the generalized Eliashberg
spectral functions, respectively. The electron-phonon matrix elements required
for the EPW code are initially calculated on a course 4$\times$4$\times$4
q-grid, and then interpolated onto fine grids using maximally localized
Wannier functions (MLWFs) Mostofi  _et al._ (2016). The random projection
method, including eighteen (nine) Wannier functions in the presence (absence)
of spin-orbit coupling, was used to construct MLWFs.

We use a small Lorentzian broadening of 10 meV and a fine randomly distributed
q-vectors mesh to better map out the phonon transferred momentum Poncé  _et
al._ (2014). Thus, the electron phonon self-energy calculation obtained with
EPW (YAMBO) uses a 40$\times$40$\times$40 (26$\times$26$\times$26) k-grid and
2.7$\times$104 (5$\times$102) random q-points in the Brillouin zone. We found
that the converged parameters for both codes provide similar results for the
e-ph self-energy (See supplementary information).

The generalized Eliashberg spectral function is be obtained by the relation

$\alpha^{2}F_{n\textbf{k}}(\omega)=\frac{1}{N}\sum_{n^{{}^{\prime}}\textbf{q}\lambda}\Big{[}\frac{|g^{\textbf{q}\lambda}_{nn^{\prime}\textbf{k}}|^{2}}{\epsilon_{n\textbf{k}}-\epsilon_{n^{\prime}\textbf{k-q}}}\Big{]}\times\delta(\omega-\omega_{\textbf{q}\lambda}),$
(5)

which enables us to visualize the e-ph coupling strength for a given state
$|n\textbf{k}\rangle$.

The real and imaginary part of the electron-electron self-energy
(Im$\Sigma^{e-e}_{n\textbf{k}}$) is computed within the full frequency GW
approximation as implemented in the YAMBO code. An energy cutoff of 68 Ry for
the exchange potential of Pd/Pt, respectively. The energy cutoff for the
polarization function is set at 20 Ry for Pd/Pt. For both metals, a q-vectors
grid of 18 $\times$ 18 $\times$ 18 together with 240 unoccupied bands were
employed for the polarization and Green’s function. The frequency dependent
dielectric function is computed over 400 frequencies along the real axis for
its numerical evaluation. Finally, The optical absorption is calculated at the
Random Phase Approximation level in a grid of 56 $\times$ 56 $\times$ 56,
taking into account the electronic quasiparticle corrections computed within
full frequency GW approximation.

## References

  * Xu  _et al._ (2017) M. Xu, J.-Y. Yang, S. Zhang, and L. Liu, Phys. Rev. B 96, 115154 (2017).
  * Choi  _et al._ (2006) W. S. Choi, S. S. A. Seo, K. W. Kim, T. W. Noh, M. Y. Kim, and S. Shin, Phys. Rev. B 74, 205117 (2006).
  * Marini  _et al._ (2002) A. Marini, G. Onida, and R. D. Sole, Phys. Rev. B 66, 115101 (2002).
  * Tang and Fultz (2011) X. Tang and B. Fultz, Phys. Rev. B 84, 054303 (2011).
  * Wilson and Coh (2020) R. B. Wilson and S. Coh, Commun. Phys. 3, 179 (2020).
  * Gall (2016) D. Gall, J. Appl. Phys. 119, 085101 (2016).
  * Werner  _et al._ (2019) W. S. M. Werner, K. Glantschnig, and C. Ambrosch-Draxl, Phys. Chem. Ref. Data 38, 1013 (2019).
  * Olmon  _et al._ (2012) R. L. Olmon, B. Slovick, T. W. Johnson, D. Shelton, S.-H. Oh, G. D. Boreman, and M. B. Raschke, Phys. Rev. B 86, 235147 (2012).
  * Suemoto  _et al._ (2021) T. Suemoto, K. Yamanaka, N. Sugimoto, Y. Kobayashi, T. Otsu, S. Tani, and T. Koyama, J. Appl. Phys. 130, 025101 (2021).
  * Garcia-Basabe  _et al._ (2019) Y. Garcia-Basabe, G. F. Peixoto, D. Grasseschi, E. C. Romani, F. C. Vicentin, C. E. P. Villegas, A. R. Rocha, and D. G. Larrude, Nanotechnology 30, 475707 (2019).
  * Linic  _et al._ (2011) S. Linic, P. Christopher, and D. B. Ingram, Nature Mat. 10, 911–921 (2011).
  * Atwater and Polman (2010) H. A. Atwater and A. Polman, Nature Mater. 9, 205–213 (2010).
  * Jiang  _et al._ (2017) N. Jiang, X. Zhuo, and J. Wang, Chem. Rev. 118, 3054–3099 (2017).
  * Gong and Leite (2016) C. Gong and M. S. Leite, ACS Photonics 3, 507–513 (2016).
  * Aslam  _et al._ (2017) U. Aslam, S. Chavez, and S. Linic, Nature Nanotech. 12, 1000 (2017).
  * Aslam  _et al._ (2018) U. Aslam, V. G. Rao, S. Chavez, and S. Linic, Nat. Catal. 1, 656 (2018).
  * Cortés  _et al._ (2020) E. Cortés, L. V. Besteiro, A. Alabastri, A. Baldi, G. Tagliabue, A. Demetriadou, and P. Narang, ACS Nano 14, 16202 (2020).
  * Yu  _et al._ (2019) H. Yu, Y. Peng, Y. Yang, and Z.-Y. Li, npj Comput. Mater. 5, 45 (2019).
  * Barnes  _et al._ (2003) W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824–830 (2003).
  * Sönnichsen  _et al._ (2002) C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, Phys. Rev. Lett. 88, 077402 (2002).
  * Maier (2007) S. Maier,  _Plasmonics: fundamentals and applications_ (Springer Science and Business Media, 2007).
  * Fatti  _et al._ (2000) N. D. Fatti, C. Voisin, M. Achermann, S. Tzortzakis, D. Christofilos, and F. Vallée, Phys. Rev. B 61, 16956 (2000).
  * Brown  _et al._ (2016) A. M. Brown, R. Sundararaman, P. Narang, W. A. G. III, and H. A. Atwater, ACS Nano 10, 957–966 (2016).
  * Clavero (2014) C. Clavero, Nature Photon. 8, 95–103 (2014).
  * Brongersma  _et al._ (2015) M. L. Brongersma, N. J. Halas, and P. Nordlander, Nature Nanotech 10, 25–34 (2015).
  * Ng  _et al._ (2016) C. Ng, J. J. Cadusch, S. Dligatch, A. Roberts, T. J. Davis, P. Mulvaney, and D. E. Gómez, ACS Nano 10, 4704 (2016).
  * Bernardi  _et al._ (2015) M. Bernardi, J. Mustafa, J. B. Neaton, and S. G. Louie, Nature Commun. 6, 7044 (2015).
  * Gong  _et al._ (2020) T. Gong, P. Lyu, K. J. Palm, S. Memarzadeh, J. N. Munday, and M. S. Leite, Advanced Opt. Mat. 8, 2001082 (2020).
  * Gong  _et al._ (2018) C. Gong, A. Kaplan, Z. A. Benson, D. R. Baker, J. P. McClure, A. R. Rocha, and M. S. Leite, Advanced Opt. Mat. 6, 1800218 (2018).
  * Dias and Leite (2019) M. R. S. Dias and M. S. Leite, Acc. Chem. Res. 52, 2881–2891 (2019).
  * McClure  _et al._ (2019) J. P. McClure, J. Boltersdorf, D. R. Baker, T. G. Farinha, N. Dzuricky, C. E. P. Villegas, A. R. Rocha, and M. S. Leite, ACS Appl. Mater. Interfaces 11, 24919–24932 (2019).
  * Stofela  _et al._ (2020) S. K. F. Stofela, O. Kizilkaya, B. T. Diroll, T. R. Leite, M. M. Taheri, D. E. Willis, J. B. Baxter, W. A. Shelton, P. T. Sprunger, and K. M. McPeak, Advanced Mat. 32, 1906478 (2020).
  * Krayer  _et al._ (2020) L. J. Krayer, K. J. Palm, C. Gong, A. Torres, C. E. P. Villegas, A. R. Rocha, M. S. Leite, and J. N. Munday, ACS Photonics 7, 1689–1698 (2020).
  * Himpsel and Eastman (1978) F. J. Himpsel and D. E. Eastman, Phys. Rev. B 18, 5236 (1978).
  * Eyers  _et al._ (1984) A. Eyers, F. Schäfers, G. Schönhense, U. Heinzmann, H. P. Oepen, K. Hünlich, J. Kirschner, and G. Borstel, Phys. Rev. Lett. 52, 1559 (1984).
  * Lafait  _et al._ (1978) J. Lafait, F. Abeles, M. L. Theye, and G. Vuye, J. Phys. F: Met. Phys. 8, 1597 (1978).
  * Weaver (1974) J. H. Weaver, Phys. Rev. B 11, 4 (1974).
  * Östlin  _et al._ (2016) A. Östlin, W. H. Appelt, I. D. Marco, W. Sun, M. Radonjić, M. Sekania, L. Vitos, O. Tjernberg, and L. Chioncel, Phys. Rev. B 93, 155152 (2016).
  * Ladstädter  _et al._ (2004) F. Ladstädter, U. Hohenester, P. Puschnig, and C. Ambrosch-Draxl, Phys. Rev. B 70, 235125 (2004).
  * Windmiller  _et al._ (1969) L. R. Windmiller, J. B. Ketterson, and S. Hornfeldt, Jour. Appl. Phys 40, 1291 (1969).
  * Andersen (1970) O. K. Andersen, Phys. Rev. B 2, 883 (1970).
  * Corso (2007) A. D. Corso, Phys. Rev. B 76, 054308 (2007).
  * Shen  _et al._ (2016) Y. Shen, C. W. Li, X. Tang, H. L. Smith, and B. Fultz, Phys. Rev. B 93, 214303 (2016).
  * Savrasov and Savrasov (1996) S. Y. Savrasov and D. Y. Savrasov, Phys. Rev. B 54, 16487 (1996).
  * Xu  _et al._ (2020) B. Xu, M. D. Gennaro, and M. J. Verstraete, Phys. Rev. B 102, 155128 (2020).
  * Glantschnig and Ambrosch-Draxl (2010) K. Glantschnig and C. Ambrosch-Draxl, New J. Phys. 12, 103048 (2010).
  * Allen (1987) P. B. Allen, Phys. Rev. B 36, 2920(R) (1987).
  * Knapp and Jones (1972) G. S. Knapp and R. W. Jones, Phys. Rev. B 6, 1761 (1972).
  * Onida  _et al._ (2002) G. Onida, L. Reining, and A. Rubio, Rev. Mod. Phys. 74, 601 (2002).
  * Miiller and Brockhouse (1968) A. P. Miiller and B. N. Brockhouse, Phys. Rev. Lett. 20 (1968).
  * Dutton  _et al._ (1972) D. H. Dutton, B. N. Brockhouse, and A. P. Miiller, Can. J. Phys. 50, 2915 (1972).
  * Vargas  _et al._ (2019) W. E. Vargas, N. Clark, F. Muñoz-Rojas, D. E. Azofeifa, and G. A. Niklasson, Phys. Scr. 94, 055101 (2019).
  * Bernardi  _et al._ (2014) M. Bernardi, D. Vigil-Fowler, J. Lischner, J. B. Neaton, and S. G. Louie, Phys. Rev. Lett. 112, 257402 (2014).
  * Bauer  _et al._ (2015) M. Bauer, A. Marienfeld, and M. Aeschlimann, Progress Surf. Sci. 90, 319 (2015).
  * Zhukov  _et al._ (2002) V. P. Zhukov, F. Aryasetiawan, E. V. Chulkov, and P. M. Echenique, Phys. Rev. B 65, 115116 (2002).
  * Grimvall (1981) G. Grimvall,  _The electron–phonon interaction in metals (Selected Topics in Solid State Physics)_ (New York: North-Holland, 1981).
  * Hsiang  _et al._ (1981) T. Y. Hsiang, J. W. Reister, H. Weinstock, G. W. Crabtree, and J. J. Vuillemin, Phys. Rev. Lett. 47, 523 (1981).
  * Mueller and Freeman (1970) F. M. Mueller and A. J. Freeman, Phys. Rev. B 1, 12 (1970).
  * Marini (2012) A. Marini, J. Phys.: Conf. Ser. 427, 012003 (2012).
  * Sangalli and Marini (2015) D. Sangalli and A. Marini, Eur. Phys. Lett. 110, 47004 (2015).
  * Molina-Sánchez  _et al._ (2017) A. Molina-Sánchez, D. Sangalli, L. Wirtz, and A. Marini, Nano Lett. 17, 4549 (2017).
  * Giannozzi and et. al. (2009) P. Giannozzi and et. al., J. Phys. Condens. Matter 21, 395502 (2009).
  * Baroni  _et al._ (2001) S. Baroni, S. de Gironcoli, A. D. Corso, and P. Giannozzi, Rev. Mod. Phys. 73, 515 (2001).
  * Marini  _et al._ (2015) A. Marini, S. poncé, and X. Gonze, Phys. Rev. B. 91, 224310 (2015).
  * Poncé  _et al._ (2016) S. Poncé, E. Margine, C. Verdi, and F. Giustino, Computer Physics Communications 209, 116 (2016).
  * Sangalli  _et al._ (2019) D. Sangalli, A. Ferretti, H. Miranda, C. Attaccalite, I. Marri, E. Cannuccia, P. Melo, M. Marsili, F. Paleari, A. Marrazzo, G. Prandini, P. Bonfa, M. O. Atambo, F. Affinito, M. Palummo, A. Molina-Sánchez, C. Hogan, M. Grüning, D. Varsano, and A. Marini, J. Phys.: Condens. Matter 31, 325902 (2019).
  * Mostofi  _et al._ (2016) A. A. Mostofi, J. R. Yates, G. Pizzi, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, Comput. Phys. Commun. 209, 116 (2016).
  * Poncé  _et al._ (2014) S. Poncé, G. Antonius, P. Boulanger, E. Cannuccia, A. Marini, M. Côté, and X. Gonze, physica status solidi (c) 83, 341–348 (2014).