# Theory of photoluminescence by metallic structures

Aurelian Loirette-Pelous<EMAIL_ADDRESS>Jean-Jacques
Greffet<EMAIL_ADDRESS>[

###### Abstract

Light emission by metals at room temperature is quenched by fast relaxation
processes. Nevertheless, Mooradian reported in 1969 the observation of
photoluminescence by metals pumped by a laser. Strikingly, while it is
currently at the heart of many promising applications, this phenomenon is
still poorly understood. In this work, we report a theory which reproduces
quantitatively previously published experimental data. We first provide a
general formula that relates the emitted power for a frequency, direction and
polarization state to a sum over all transitions involving matrix elements,
electronic distribution of all bands and the Green tensor. We then consider
the case of intraband recombination and derive a closed-form expression of the
emitted power depending only on macroscopic quantities. This formula, which is
a generalization of Kirchhoff’s law, answers many of the open questions
related to intraband photoluminescence.

###### keywords:

Plasmonics, Photoluminescence, Hot electrons, Nonequilibrium electrons,
Metallic nanoparticles, Kirchhoff’s law

Current address: Materials Physics Center, CSIC-UPV/EHU, Donostia-San
Sebastián, Spain Unknown University]Université Paris-Saclay, Institut
d’Optique Graduate School, CNRS, Laboratoire Charles Fabry IR,NMR,UV

##  1 Introduction

Photoluminescence, broadly defined as the process of light emission by a
material following the absorption of an incident radiation, is a major process
for light generation in physics. It is easily observable in materials with a
gap such as semiconductors or dye molecules. Light emission is then due to
electron-hole recombination and called fluorescence. Incandescence is an
alternative light emission mechanism, that arises due to heating of the
emitting material under strong illumination. An incident radiation can also
produce Raman scattering, which is an ultrafast coherent process that does not
involve electronic population modification.

Metals are known to emit light by incandescence but fluorescence is usually
quenched. Indeed, electron-electron and electron-phonon interactions provides
a fast relaxation path for excited electrons. Owing to the absence of a gap,
nothing prevents a full relaxation so that no light emission is observed.
However, observation of photoluminescence in the visible and near infrared was
reported by Mooradian in 1969 1. This phenomenon was surprising given that
metal heating was negligible in the experimental conditions. This effect is
characterized by a very low quantum yield on the order of $\sim 10^{-10}$ 1,
that has hindered its study for decades. Nevertheless, observation of
luminescence by metals is nowadays a simple task thanks to recent advances in
the fabrication and characterization of single metallic nanoparticles 2, 3, 4.
Due to a strong plasmonic resonance, quantum yields can be enhanced up to
$\sim 10^{-5}-10^{-6}$ 5, so that photoluminescence is now a source of
detrimental background signal when performing Raman scattering measurements 6.
By contrast, photoluminescence from metallic nanoparticles has become an
useful tool. For example, it enables to create new luminescent markers in
biomedecine 7, 8, to supervise chemical reactions 9, 10, 11 or to measure the
temperature at the nanoscale 12, 13, 14, 15, 16, 17, 18, 19, among other
applications.

Despite a large number of studies and these practical applications, the
physical origin of photoluminescence by metallic nanostructures is still under
debate 19, 20. On the one hand, the long standing explanation in term of
interband transitions 1, 21, 22, 23 is still widely accepted for emission at
frequencies beyond the interband threshold. On the other hand, many evidences
of emission at frequencies lower than the interband threshold have been
observed in the past twenty years 2, 24, 4, 12, 6, 25, 26, 27, revealing the
necessity to consider other mechanisms. Therefore, current alternative
theories focus on processes occurring within the metal conduction band.
Intraband recombinations have been first proposed as the origin of the
luminescence 2, later followed by other works 25, 28, 16, 29. However, the
peculiar temperature dependence of the anti-Stokes spectrum used for
thermometry was first explained by a mechanism based on electronic Raman
scattering 12, 6, 26. Nevertheless, later theoretical work showed that the
temperature dependence of the anti-Stokes spectrum can also been explained
with an intraband recombination process, considering that a non-equilibrium
electronic distribution is induced by the pumping 30. Finally, let us note
that this theory has predicted a peculiar scaling of the emitted power with
the pumping strength 30. Interestingly, recent experiments have observed this
scaling 31, giving support to the hypothesis of an intraband recombination
mechanism.

Leaving aside the controversy on the physical mechanism involved in
photoluminescence by metals, there is still a lack of models able to predict
quantitatively the emitted power. Such a model must account for two effects: a
nonequilibrium electronic distribution and the emission in a metallic
environment. The first theory enabling comparisons with experiments has been
proposed in Ref. 27, considering a non-equilibrium hot carriers distribution
and including emission from both interband and intraband electron-hole
recombinations. A good agreement with experimental measurements was obtained
for the Stokes part of the spectrum but the anti-Stokes part of the spectrum
could not be modeled due the assumption of zero electronic temperature. This
issue has been tackled in the case of intraband recombinations in Ref. 30.
Yet, in both theories the metallic environment is included by inserting a
photonic density of states. To date, the most elaborated theory has been
proposed in Ref. 23 to model the PL of planar gold flakes. It features a
calculation of the nonequilibrium electronic distribution for both intraband
and interband transitions, combined with a calculation of the corresponding
transition matrix elements and the calculation of the Green tensor. Yet, while
a good agreement is found with experimental data for emission from interband
transitions, it fails to reproduce the emission from intraband transitions.
This indicates that the intraband matrix elements are not correctly taken into
account or that an alternative mechanism should be introduced.

Here, we introduce a quantitative theory of photoluminescence by metals that
covers both intraband and interband transitions. In the particular case of
intraband photoluminescence, we circumvent the calculation of matrix elements
by deriving a closed-form expression which depends on the absorption cross-
section. To this end, we use a statistical physics approach of light emission.
The emitted power is computed from the knowledge of fluctuating currents which
are included in Maxwell equations in the spirit of a Langevin model. We derive
a fluctuation relation connecting the current density correlation function
with the non-equilibrium electronic distribution. This formulation is exact
and general but difficult to implement. In the case of intraband
photoluminescence, we show that it can be drastically simplified by inserting
further approximations. We manage to cast the result in the form of a
generalized Kirchhoff’s law so that the emitted power can be expressed as the
product of the absorption cross section of the metallic nanostructure and a
generalized blackbody radiance. When using non-equilibrium conduction band
distributions evaluated by solving the Boltzmann equation under pumping by a
continuous wave (CW) laser, we obtain analytical formulas that enable to solve
many debates related to intraband CW photoluminescence. Finally, we proceed to
quantitative comparisons with experimental data.

##  2 Results and discussion

###  2.1 General theory

modeling light emission by a metallic nanostructure is a subtle issue. On the
one hand, a quantum point of view is implicitly taken when referring to
electron-hole recombinations. On the other hand, radiation problems associated
to emitters embedded in a specific (possibly resonant) electromagnetic
environment are naturally tackled in the classical framework of Maxwell
equations, with electric dipoles as source terms. Hence, a quantitative theory
of photoluminescence by metals must connect the classical and quantum points
of view on light emission. In this section, we bridge this gap using a
statistical physics theory of light emission 32, 33. The cornerstone of this
framework is the cross spectral density
$W_{j^{\dagger}j}(\boldsymbol{r}_{1},\boldsymbol{r}_{2},\omega)$, that
quantifies the correlation of the fluctuating current density operators
$\hat{\boldsymbol{j}}$ at positions $\boldsymbol{r}_{1}$ and
$\boldsymbol{r}_{2}$ and at frequency $\omega$. We will show how this quantity
is connected to the two points of view underlying the emitted light. This will
eventually provide a general relation connecting the emitted power with the
non-equilibrium electronic distribution.

First, let us remark that $W_{j^{\dagger}j}$ is the source term of the
classical radiation problem involved in the light emission. Indeed, Maxwell
equations connect the radiated field $\hat{\boldsymbol{E}}$ and the current
density $\hat{\boldsymbol{j}}$, that can be written using the Green tensor
$\boldsymbol{G}$ as
$\hat{\boldsymbol{E}}(\boldsymbol{r})~{}=~{}i\omega\mu_{0}\int\boldsymbol{G}(\boldsymbol{r},\boldsymbol{r}^{\prime},\omega)\hat{\boldsymbol{j}}(\boldsymbol{r}^{\prime},\omega)d\boldsymbol{r}^{\prime}$.
We note that the Green tensor accounts for the electromagnetic environment
including field enhancement and resonances. Hence, we show in Section A. of
the Supporting Information that the power $dP_{e}$ emitted at a frequency
$\omega$, in a solid angle $d\Omega_{r}$ centered around a position
$\boldsymbol{r}$ in the far-field of a metallic structure and in a
polarisation state $p$ is related to $W_{j^{\dagger}j}$ through the equation:

$\frac{dP_{e}}{d\Omega_{r}}(\omega,\boldsymbol{u}_{r},p)=\sigma_{abs}(\omega,-\boldsymbol{u}_{r},p)\frac{\omega^{2}}{8\pi^{3}c^{2}}\frac{W_{j^{\dagger}j}(\omega)}{2\omega\epsilon_{0}\text{Im}[\epsilon(\omega)]},$
(1)

where $\boldsymbol{u}_{r}=\boldsymbol{r}/|\boldsymbol{r}|$ is the direction of
emission defined by the position $\boldsymbol{r}$, and $c$ is the vacuum speed
of light. $\sigma_{abs}(\omega,-\boldsymbol{u}_{r},p^{*})$ is the absorption
cross section of the metallic structure, calculated for an incident plane wave
of same frequency but reverse direction and polarization handedness compared
to the far field plane wave emitted by photoluminescence. This quantity
contains the information on the Green tensor, as shown in Section A. of the
Supporting Information. Let us also point out that the presence of the
absorption cross section in a calculation of light emission is a consequence
of Lorentz reciprocity 34. The dimensionless relative permittivity of the
emitting metal is noted $\epsilon(\omega,\boldsymbol{r})$. Finally, it is
assumed that the metal is isotropic and homogeneous, so that
$W_{j^{\dagger}j}$ and $\epsilon$ are scalars and their spatial dependence can
be dropped.

Second, $W_{j^{\dagger}j}$ describes the statistical behavior of the
fluctuating current densities in the metal. In Section B.1. of the Supporting
Information, we show that for a homogeneous and isotropic electron gas with
arbitrary mean occupation number of the electronic states, the cross spectral
density of the current density can be cast into the form:

$W_{j^{\dagger}j}(\omega)=\frac{\epsilon_{0}\omega_{p}^{2}}{m}\frac{2\pi}{N}\sum_{n,n^{\prime}=1}^{N}|p_{n,n^{\prime}}|^{2}f_{n^{\prime}}[1-f_{n}]\delta(\omega-[\omega_{n^{\prime}}-\omega_{n}]),$
(2)

where $\epsilon_{0}$ is the vacuum permittivity, $\omega_{p}$ is the metal
plasma frequency, $m$ is the electron mass, $N$ is the total electron number
in the metal, $n=1,2,...,N$ labels the electronic states without specifying a
band, $\hbar\omega_{n}$ labels their corresponding energy and $f_{n}$ their
mean occupation number, while $p_{n,n^{\prime}}$ is the transition matrix
element between state $n$ and state $n^{\prime}$. Importantly, let us
emphasize that Eq. (2) describes the current density fluctuations in any
nonequilibrium condition, i.e. the $f_{n}$ can take any value. Remarkably, in
the specific case of thermodynamic equilibrium, the mean occupation number
follows the Fermi-Dirac distribution, that is
$f_{n}=1/[\exp(\frac{\hbar\omega_{n}-\mu_{F}}{k_{B}T})+1]$ where $k_{B}$ is
the Boltzmann constant, $T$ is the temperature of the equilibrium electron gas
and $\mu_{F}$ its Fermi level. Eq. (2) then reduces to (see Sections B.2. and
B.3. of the Supporting Information):

$W_{j^{\dagger}j,eq}(\omega)=2\omega\epsilon_{0}\text{Im}[\epsilon_{eq}(\omega)]\frac{\hbar\omega}{\exp\big{(}\frac{\hbar\omega}{k_{B}T}\big{)}-1},$
(3)

where $\epsilon_{eq}$ is the metal relative permittivity at equilibrium.
Equation (3) is known as the Fluctuation-Dissipation Theorem (FDT) 35, 36.
Indeed, it connects the current fluctuations to the imaginary part of the
permittivity, that quantifies how the electrons dissipate the energy provided
by an external radiation. Hence, we note that the fluctuation relation Eq. (2)
generalizes the FDT Eq. (3) to nonequilibrium situations. Yet, we also note
that it cannot be related, in general, to $\text{Im}[\epsilon]$ so that it is
not a fluctuation-dissipation relation.

In summary, the fluctuation relation Eq. (2) enables to establish the expected
link between the quantum matter and classical radiation pictures of light
emission: On the one hand, $W_{j^{\dagger}j}$ is the source term of the
classical radiation problem involved in the light emission, as shown in Eq.
(1). On the other hand, the right hand side of Eq. (2) clearly displays the
quantum point of view through the product
$f_{n^{\prime}}[1-f_{n}]\delta(\omega-[\omega_{n^{\prime}}-\omega_{n}])$, that
corresponds to the recombination probability of an electron on a state with
energy $\hbar\omega_{n}+\hbar\omega$ with a hole on a state with energy
$\hbar\omega_{n}$.

Lastly, we insert Eq. (2) into Eq. (1), which shows that the photoluminescence
power can be cast into the form:

$\frac{dP_{e}}{d\Omega_{r}}(\omega,\boldsymbol{u}_{r},p)=\sigma_{abs}(\omega,-\boldsymbol{u}_{r},p^{*})\frac{\omega^{2}}{8\pi^{3}c^{2}}\frac{\omega_{p}^{2}}{2\omega\text{Im}[\epsilon(\omega)]}\frac{2\pi}{N}\sum_{n,n^{\prime}=1}^{N}|p_{n,n^{\prime}}|^{2}f_{n^{\prime}}[1-f_{n}]\delta(\omega-[\omega_{n^{\prime}}-\omega_{n}]).$
(4)

Eq. (4) is the first main result of this work. It provides a framework to
calculate quantitatively the emitted power at each frequency, direction and
polarization state by a metallic structure. Remarkably, Eq. (4) is of very
broad generality: (i) the absorption cross section is not the absorption by a
volume of the bare metal but the absorption of a system. It can be, e.g. the
absorption by a metallic nanoparticle in vacuum, or the same metallic
nanoparticle placed in a cavity. Hence, it takes into account the local
photonic density of states in the metal. (ii) Eq. (4) was derived for a
stationary excitation. Yet, it can also be used for incident pulses under a
quasistationary approximation using a time-dependent electronic population and
permittivity. (iii) Similarly, metal inhomogeneities can be accounted for
using a position-dependent absorption cross section density 37, electronic
population and permittivity. (iv) Light emission can be calculated with Eq.
(4) for nonequilibrium electrons. Hence, it entails light emission by metallic
bodies possibly driven out-of-equilibrium by an external (pulsed or
continuous) pumping. (v) Eq. (4) encompasses both intraband and interband
photoluminescence, as no hypothesis on the type of transitions has been made
at this point.

###  2.2 Generalized Kirchhoff’s law for emission from intraband transitions

Modeling photoluminescence by metallic structures with Eq. (4) requires the
knowledge of the band structure and of the transition matrix elements of the
metal. While the derivation of these microscopic quantities has been performed
in several publications 38, 39, 40, 41, 23, it demands complex and cumbersome
numerical investigations. In this section, we specifically focus on emission
arising from intraband transitions and introduce an approximation that
circumvent this difficulty, enabling to compute the emitted power knowing the
absorption cross section.

We start by assuming that the density of states of the conduction electrons,
as well as the transition matrix elements corresponding to intraband
transitions are constant over the electronic states, and equal to their value
at the metal Fermi level. Let us note that this approximation is usually
justified in calculations of the absorption by metals in the mid/far IR range
42, 43, but requires more investigations in the visible/near-IR range that are
out of the scope of this work. Here, we will instead evaluate the impact of
the approximation in the next section through comparison with experiments. We
also assume that the mean occupation numbers $f_{n}$ of the conduction band
electronic states follow a well defined distribution $f(E)$ that only depends
on energy $E$. These hypotheses eventually enable to cast the fluctuation
relation Eq. (2) into the form (see Section C. of the Supporting Information
for a detailed derivation):

$W_{j^{\dagger}j}^{intra}(\omega)=2\epsilon_{0}\omega\text{Im}[\epsilon_{eq}^{intra}(\omega)]\times\Theta^{intra}(\omega),$
(5)

where $\epsilon_{eq}^{intra}$ is the relative permittivity of the metal at
equilibrium and limited to intraband transitions, and where:

$\Theta^{intra}(\omega)=\int f(E+\hbar\omega)[1-f(E)]dE,$ (6)

corresponds to the integral of the electron-hole recombination probabilities
associated to indirect transitions of frequency $\omega$ in the conduction
band. Let us emphasize that $f(E)$ can be any arbitrary distribution. Hence,
Eqs.(5),(6) appears as a generalization of the Fluctuation-Dissipation theorem
Eq. (3) to out-of-equilibrium statistics of conduction band electrons.

Finally, as long as nonequilibrium effects yield a small deviation of the
imaginary part of the metal permittivity from the equilibrium one, inserting
Eq. (5) into Eq. (1) enables to cast the expression of the emitted power into
the form:

$\frac{dP_{e}}{d\Omega_{r}}(\omega,\boldsymbol{u}_{r},p)=\sigma_{abs}^{intra}(\omega,-\boldsymbol{u}_{r},p^{*})\frac{\omega^{2}}{8\pi^{3}c^{2}}\Theta^{intra}(\omega),$
(7)

where
$\sigma_{abs}^{intra}=(\text{Im}[\epsilon_{eq}^{intra}]/\text{Im}[\epsilon_{eq}])\times\sigma_{abs}$
is the part of the total absorption that is due to intraband transitions.

Eqs. (5),(7) constitute the second main result of this work. We now discuss
further their properties. Firstly, it is remarkable that at thermodynamic
equilibrium, Eq. (5) coincides with the equilibrium Fluctuation-Dissipation
theorem Eq. (3), and $\omega^{2}/(8\pi^{3}c^{2})\times\Theta^{intra}(\omega)$
yields the Planck’s law. Eq. (7) then reproduces the so-called Kirchhoff’s law
44, 37, that is the basic tool to model thermal radiation. Hence, Eqs. (7),
(6) identifies as a generalization of Kirchhoff’s law to metals featuring an
arbitrary conduction band electron distribution.

Secondly, as for the equilibrium Kirchhoff’s law, let us stress that our
approach is based on statistical physics. Hence, the detailed knowledge about
the microscopic transition processes (phonon-assisted transitions, electron-
assisted transitions etc…), as well as their respective weighting in the
emission is not necessary. Instead, it is fully taken into account within
macroscopic quantities: (i) the permittivity of the emitting metal, that
underlies the absorption cross section; (ii) the distribution of the electrons
in the conduction band.

Thirdly, as it only requires the knowledge of macroscopic quantities, the
generalized Kirchhoff’s law Eqs. (7),(6) turns out to be way simpler to use
than Eq. (4): (i) The electronic distributions can be routinely calculated
with satisfying accuracy solving the Boltzmann equation in the relaxation time
approximation 43, 45, 46, 47. (ii) The absorption cross sections
$\sigma_{abs}$ of the metallic nanostructure can be computed with Maxwell’s
solvers. Further, the absorption cross section contains all the information on
the plasmon resonance that shapes the emission spectrum, directivity and
polarization; the electronic distribution contains all the information on the
excitation frequency and, if defined, on the temperature.

###  2.3 Application to emission arising from intraband transitions in the CW
pumping regime

####  2.3.1 Generalized Kirchhoff’s law under CW pumping

We now focus on the intraband photoluminescence of single metallic
nanoparticules illuminated by a CW pumping laser. This pumping scheme has been
experimentally studied by many groups, so that it represents a convenient
testbed for our generalized Kirchhoff’s law. In particular, these efforts have
resulted in an empirical guess for the emitted power 19:

$\frac{dP_{e}}{d\Omega_{r}}(\omega)\propto
S_{LPR}(\omega)F(\hbar\omega-\hbar\omega_{L},T)I_{L},$ (8)

where $S_{LPR}$ is the localized plasmon resonance lineshape,
$F(\hbar\omega,T)$ is a function of the frequency and temperature,
$\omega_{L}$ is the frequency of the pump laser and $I_{L}$ its intensity at
the sample position. However, several aspects of this empirical expression are
still under debate. First, we note that $S_{LPR}$ has been modeled so far by
the scattering cross section 16 or by a quantity related to the local density
of photonic states 27. Instead, Ref. 19 suggested that the absorption cross
section should be rather used. Similarly, $F(\hbar\omega,T)$ is the Bose-
Einstein distribution for some authors 12, 14, 17, 18 and the Fermi-Dirac
distribution reduced to its Boltzmann component for others 48, 16.
Accordingly, $T$ has been proposed to be the temperature of the metal lattice
(that is of phonons) in the former case and that of the electron gas in the
latter. In this section, we compare Kirchhoff’s law Eqs. (7),(6) to Eq. (8)
and address the open questions discussed above.

Figure 1: Nonequilibrium electron-hole recombination (left) and nonequilibrium
electron distribution under CW pumping (right). In both panels, the Fermi
level is depicted by the horizontal, black dashed line at energy $\mu_{F}$.
Right panel: low energy electrons are promoted to states over the Fermi level
by absorption of photons with frequency $\omega_{L}$ from the pumping beam.
Accumulation of these hot electrons and their corresponding hot holes is
limited by relaxation toward equilibrium, mainly due to electron-electron
interaction. In the stationary regime, the competition between excitation and
relaxation yields a steady population of hot electrons and holes (red line)
with respective occupation probability $K_{n}$ and $1-K_{n}$. Left panel:
scheme of two electron-hole (e-h) recombinations events, one above the Fermi
level and one below, that are enabled by the nonequilibrium distribution of
the electrons. The corresponding state occupation probabilities are indicated
by the blue dotted lines. Note that an exaggerated value of $K_{n}\sim 0.2$
has been used to simplify the illustration, when realistic values are
$K_{n}\sim I_{L}/(10^{11}\text{W.cm}^{-2})$ with $I_{L}$ the pumping intensity
at sample. See Methods for the analytical form of the nonequilibrium
distribution and for the interpretation of $K_{n}$.

To start, let us note that CW pumping induces a stationary nonequilibrium
electron distribution, whose shape is schematically depicted on the right side
of Fig. 1. It is seen that the nonequilibrium pattern consists in the creation
of hot electron and hot hole populations, respectively above and below the
Fermi level, with occupation probabilities orders of magnitudes higher
compared to equilibrium (though its actual value is exaggerated on the figure
for the sake of clarity). Interestingly, an analytical expression of this
distribution has been derived in Refs. 47, 30, using the Boltzmann equation in
the relaxation time approximation (see Methods). In Section D. of the
Supporting Information, we use this expression to perform the integral
$\Theta^{intra}$ in Eq. (6). Inserting the result into Eq. (7), the emitted
power under continuous laser illumination at frequency $\omega_{L}$ can be
cast into the form:

$\frac{dP_{e}}{d\Omega_{r}}(\omega,\boldsymbol{u}_{r},p)=\sigma_{abs}^{intra}(\omega,-\boldsymbol{u}_{r},p^{*})\frac{\omega^{2}}{8\pi^{3}c^{2}}\frac{2(\hbar\omega-\hbar\omega_{L})}{\exp\big{(}\frac{\hbar\omega-\hbar\omega_{L}}{k_{B}T_{e}}\big{)}-1}\frac{\hbar\omega_{L}}{\hbar\omega}K_{n}^{eff},$
(9)

where $k_{B}$ is the Boltzmann constant, $T_{e}$ is the conduction electron
temperature, that can be defined rigorously due to the weakness of the
nonequilibrium effects (see Methods and Ref. 47), and where the dimensionless
parameter $K_{n}^{eff}$ is proportional to the rate of absorbed photons:

$K_{n}^{eff}\propto\frac{\sigma_{abs}^{intra}(\omega_{L},\boldsymbol{u}_{L},p_{L})I_{L}}{\hbar\omega_{L}},$
(10)

where $\boldsymbol{u}_{L}$, $p_{L}$ and $I_{L}$ are, respectively, the
direction, the polarization and the intensity at sample of the pumping beam.
The full expression of $K_{n}^{eff}$, given in Methods, shows that it can be
identified as an effective Knudsen number and that it does not depend
significantly on the emission frequency and electronic temperature.

We can now compare Kirchhoff’s law Eq. (9) to the empirical guess Eq. (8).
Remarkably, we observe that the three terms in the empirical guess are
recovered and unambiguously identified in Kirchhoff’s law. First, Kirchhoff’s
law states that the localized plasmon resonance lineshape is given by the
product of the absorption cross section by the vacuum mode density, that is
$S_{LPR}(\omega)=\sigma_{abs}^{intra,\,eq}(\omega)\times\omega^{2}/8\pi^{3}c^{2}$.
This is an important result, as the shape of the absorption cross section
usually differs from the one of the easily measurable scattering cross
section, especially for large particles when the electrostatic approximation
fails or for complex structures involving several resonances. In particular,
we note that our finding is corroborated by Ref. 49 where the PL spectra of Au
sphere dimers were shown to resemble more closely the absorption spectra than
the scattering spectra. Moreover, let us note that a resonance of the
absorption cross section is blueshifted by the $\omega^{2}$ dependence of the
vacuum mode density. This may explain the reproducible blueshift observed,
e.g., in Ref. 50.

Figure 2: Comparison between the function
$F(\hbar\omega-\hbar\omega_{L},T_{e})/2$ (red line) and a Bose-Einstein
distribution with enforced positive frequency $|\omega-\omega_{L}|$ (blue
line). Parameters: $T_{e}=$ 293 K, $\hbar\omega_{L}=$ 1.58 eV (i.e.
$\lambda_{L}=$ 785 nm).

Second, Kirchhoff’s law states that
$F(\hbar\omega-\hbar\omega_{L},T_{e})=2(\hbar\omega-\hbar\omega_{L})/[\exp\big{(}\frac{\hbar\omega-\hbar\omega_{L}}{k_{B}T_{e}}\big{)}-1]$,
as previously identified in Ref. 30. Interestingly, there is a Bose-Einstein
function in $F(\hbar\omega-\hbar\omega_{L})$. In the anti-Stokes part of the
spectrum, this function enables to recover the exponential scaling
$dP_{e}/d\Omega_{r}(\omega)\propto\exp\big{(}-\hbar\omega/k_{B}T_{e}\big{)}$
observed in experiments and used in the anti-Stokes thermometry technique 12,
13, 14, 15, 16, 17, 18, 19. By contrast, close to
$\hbar\omega=\hbar\omega_{L}$, the shape of the Bose-Einstein distribution is
deeply modified by the factor $\hbar\omega-\hbar\omega_{L}$, that turns the
divergence of the distribution into a smooth concave curve, as shown on Fig.
2. This behavior is in agreement with experiments, see, e.g., Ref. 51 in which
the spectrum around $\hbar\omega_{L}$ is finely probed using an ultra-narrow
bandwidth notch filter.

Further, we stress that the Bose-Einstein function does not originates from a
distribution of phonons. On the contrary, it only stems from the electronic
distributions product in Eq. (6), as shown in Section D. of the Supporting
Information. This is the reason why the Bose-Einstein distribution is ruled by
the temperature of the conduction electrons. This means that the temperature
measured through anti-Stokes thermometry is the electronic temperature, that
may differ from the targeted lattice temperature.

Third, the linear scaling of the Stokes spectrum with the pump intensity is
recovered in Kirchhoff’s law through the term $K_{n}^{eff}$ (see Eq. (10)). In
our theory, this term arises from the scheme depicted on Fig. 1: plasmons
and/or photons are absorbed by the metal, that creates "hot" electrons and
"hot" holes populations, as shown on the nonequilibrium distribution on the
right panel of the figure. Above the Fermi level, a hot electron then
recombines with a hole, that emits a plasmon or a photon. The reverse process
involving a hot hole is also possible under the Fermi level. Hence, the
overall scheme is based on a one-photon absorption process.

In summary, the considerations above show that Kirchhoff’s law Eq. (9) enables
to recover the empirical guess Eq. (8) and to solve most of the open questions
related to this empirical form.

####  2.3.2 Quantitative comparisons between theory and experiments

Figure 3: Comparisons between experimental photoluminescence spectra (PL) of
gold nanorods (red lines) and simulations with Kirchhoff’s law (black dotted
lines), under CW pumping at 785 nm. Kirchhoff’s law Eq. (9) is used in panels
(a-d), while Eq. (11) is used in panels (e-f). Panel (a): nanorod with
longitudinal plasmon resonance at 871 nm (measured in dark-field scattering)
and dimensions 18.2 nm $\times$ 85 nm (measured with SEM). Reprinted with
permission from Ref. 27. Panels (b-f): nanorod with longitudinal plasmon
resonance at 734 nm and dimensions 25 nm $\times$ 79 nm, illuminated with an
intensity at sample $I_{L}=$ 0.37 MW.cm-2 for panels (b-e), and $I_{L}=$ 1.30
MW.cm-2 for panels (d) and (f). The electronic temperature is the only fit
parameter and is indicated on the panels. Panel (b) shows same data as panels
(c) and (e), but focuses only on the anti-Stokes part of the spectrum and is
plotted on semilogarithmic scale. Reprinted with permission from Ref. 16.
Details on the fitting procedure are provided in Methods.

We now turn to a quantitative comparison of the theory with experiments.
First, we show on Fig. 3 (a-d) comparisons between gold nanorods emission
spectra measured experimentally (red lines) and simulated with Kirchhoff’s law
(black dotted lines). The position of the pumping wavelength (785 nm) with
respect to the longitudinal plasmon resonance enables to focus either on the
Stokes spectrum (panel (a)) or on the exponentially decaying part of the anti-
Stokes spectrum (panel (b)). In both panels, it is seen that the agreement
between theory and experimental data is satisfying. Next, panels (c) and (d)
show emission spectra spanning both the Stokes and anti-Stokes sides of the
spectrum. Also, these spectra are obtained from the same gold nanorod for two
different pumping intensities. It is seen that it induces substantial spectral
changes. Adjusting the electronic temperature $T_{e}$, these spectral changes
are well reproduced by the simulations at short wavelengths. Interestingly, it
suggests that these changes originate from an increase of the electronic
temperature due to the higher pumping intensity. However, it is also seen in
both panels that an agreement between theory and experiment cannot be found on
the whole spectrum. This points out a limitation of Kirchhoff’s law in the
form of Eq. (9) that will be addressed in the following subsection.

Second, let us stress that the theory presented in this work enables for the
first time to calculate quantitatively the PL quantum yield. For a gold
nanorod pumped continuously at 785 nm and with longitudinal plasmon resonance
around 860 nm, corresponding to the experimental conditions of Fig. 3 (a), we
calculate in Methods a theoretical quantum yield $\eta_{sim}\sim 1.3\times
10^{-7}$. Experimentally, the corresponding quantum yield has been estimated
to be $\eta_{exp}\sim 1.5\times 10^{-6}$ in Ref. 27. While these values do not
show a perfect agreement, they remain relatively close to each other. Hence,
this shows that light emission through intraband transitions has a high
quantum yield and is likely to be a dominant process in PL from metals. This
also calls for further refinements of the theory in order to reach a better
agreement.

####  2.3.3 Phenomenological Kirchhoff’s law

In the previous section, we observed that Kirchhoff’s law Eq. (9) reproduces
the emission spectrum for short wavelengths but fails for larger wavelengths.
Here, we introduce a phenomenological correction to Eq. (9) that enables to
solve this issue.

Let us consider here the phenomenological generalization of Kirchhoff’s law in
the CW pumping regime:

$\frac{dP_{e}}{d\Omega_{r}}(\omega,\boldsymbol{u}_{r},p)=\sigma_{abs}^{intra}(\omega,-\boldsymbol{u}_{r},p^{*})\frac{\omega^{2}}{8\pi^{3}c^{2}}\times
C\frac{\arctan(\frac{\hbar\omega-\hbar\omega_{L}}{k_{B}T_{e}})}{\exp\big{(}\frac{\hbar\omega-\hbar\omega_{L}}{k_{B}T_{e}}\big{)}-1},$
(11)

where $C$ is a constant. We show on Fig. 3 (e),(f) a comparison between this
model and the experimental data. The agreement is now satisfying on the whole
spectrum in both cases. Remarkably, these fits are achieved using the same
simulated absorption cross section, only moderately increasing the electronic
temperature $T_{e}$ as the pumping intensity is raised. We provide further
support to the validity of Eq. (11) in Section E. of the Supporting
Information. In addition, although our derivation of Eq. (9) and then Eq. (11)
has been restricted to intraband transitions, we have used it to compare with
PL spectra obtained using pumping above the interband threshold, so that
interband transitions are expected to play a dominant role. It is seen on
Figs. S.2 and S.3 of the Supporting Information that the theory is in good
agreement with the experimental measurements. This suggests that a theory for
the emission from interband transitions can be cast into a formula resembling
to Eq. (11).

In summary, the comparisons presented in this section suggest that Eq. (11) is
a powerful tool with only one free parameter ($T_{e}$) to predict the emission
spectrum of metallic structures pumped in the CW regime. In addition, let us
note that Eq. (11) has the same overall form as the generalized Kirchhoff’s
law Eq. (7). Interestingly, this could mean that the missing ingredients to
explain the origin of the $arctan$ function in Eq. (11) may be related to the
nonequilibrium electronic function $f$ rather than to the approximations
underlying Kirchhoff’s law Eq. (7). This calls for more detailed
investigations of the non-equilibrium CW electronic distribution.

##  3 Conclusion

In summary, we have introduced a model of photoluminescence by metallic
structures based on a statistical physics approach circumventing the
calculation of microscopic quantities.

We first derived a nonequilibrium fluctuation relation Eq. (2) which relates
the current density correlation function with the nonequilibrium electronic
distribution. This relation is instrumental in connecting the electron-hole
recombination picture of light emission with the Maxwell point of view based
on radiation by time-dependent currents. Equipped with this fluctuation
relation, we derived the general relation Eq. (4) between the power radiated
and the nonequilibrium electronic distribution. This relation shows that the
absorption cross section can be used to capture the spectral and directional
shaping of the emitted light. The full calculation requires a sum over all the
transitions including the corresponding populations and matrix elements.

To bypass this cumbersome microscopic approach, we derived in Eq. (5) a
nonequilibrium fluctuation-dissipation relation for intraband transitions. It
enabled to establish in Eq. (7) a closed-form expression of the emitted power.
This form requires only the knowledge of the nonequilibrium macroscopic
absorption cross section and the electronic distribution.

When applied to continuous wave pumping, this theoretical approach recovers
the main features of an ansatz used in the literature to analyze the
experiments. It enables to resolve several open questions. It also gives a
quantitative model of many different published experiments.

This approach has the structure of a generalized Kirchhoff law. We have found
that while it is has been derived for intraband transitions, it also enables
to reproduce experimental data attributed to interband transitions. This
suggests that our derivation of the fluctuation relation and Kirchhoff’s law
could be extended to interband transitions. This is left for future work.
Another natural extension of this work is photoluminescence induced by short
pulses with high intensities. In this case, the strong time dependence of the
absorptivity due to band filling effects should be included in the generalized
Kirchhoff law. We hope that this analytical model of photoluminescence will
prove useful to optimize the different applications already demonstrated.

##  4 Methods

###  4.1 Nonequilibrium electron distribution under CW pumping and Knudsen
number

We plotted the shape of the nonequilibrium electron distribution under CW
pumping on Fig. 1 (right panel, red line). To this end, we used the analytical
expression of the conduction electrons distribution under continuous wave
pumping $f_{cw}$ that has been derived in Refs. 47, 30, solving the steady-
state Boltzmann equation in the relaxation time approximation for the
electron-electron collision rate:

$\begin{split}f_{cw}(E,T_{e})=f_{FD}(E,T_{e})+K_{n}(E,T_{e})\times[f_{FD}(E-\hbar\omega_{L},T_{e})+f_{FD}(E+\hbar\omega_{L},T_{e})-2f_{FD}(E,T_{e})],\end{split}$
(12)

with the shorthand:

$K_{n}(E,T_{e})=\frac{4I_{L}\sigma_{abs}(\hbar\omega_{L},p_{L})\tau_{ee}(E,T_{e})}{[\hbar\omega_{L}]^{2}\rho_{c}(E)},$
(13)

where $E$ stands for the electron energy, $T_{e}$ the electronic temperature,
$f_{FD}(E,T_{e})=1/[\exp(\frac{E-\mu_{F}}{k_{B}T_{e}}+1]$ is the Fermi-Dirac
distribution with $\mu_{F}$ the metal Fermi level and $k_{B}$ the Boltzmann
constant, $I_{L}$, $\omega_{L}$ and $p_{L}$ are respectively the intensity,
the frequency and the polarization state of the pump beam, $\tau_{ee}$ is the
relaxation time associated to electron-electron collisions and $\rho_{c}$ is
the density of electronic states in the conduction band. In transport theory,
the dimensionless parameter $K_{n}$ is called Knudsen number, and quantifies
the competition between equilibrium and out-of-equilibrium driving mechanisms.
In particular, $K_{n}$ is the ratio of two characteristic times: (i)
$\tau_{ee}(E)$ which is the lifetime of an electron with energy $E$ before it
relaxes due to electron-electron collisions, and (ii)
$[\hbar\omega_{L}]^{2}\rho_{c}(E)/4I_{L}\sigma_{abs}(\hbar\omega_{L},p_{L})$
which is the time interval between two consecutive photon absorption events
promoting an electron on a final state with energy $E$. Further, it was shown
in Refs. 52, 30 that $K_{n}\sim I_{L}/(10^{11}\text{W.cm}^{-2})$, so that
$K_{n}\ll 1$ in experimental conditions. Hence, the nonequilibrium correction
to the equilibrium distribution is weak and perturbative, so that the
electronic temperature $T_{e}$ can still be rigorously defined 47. In
addition, let us note that electron-phonon collisions are neglected in Eq.
(12), which is valid provided $|E-\mu_{F}|\gg k_{B}T_{e}$ 30. Also note that
in Fig. 1 we neglected the dependence of $K_{n}$ on energy $E$ and used an
exaggerated value $K_{n}\sim 0.2$ to simplify the illustration. Finally, we
used in Eq. (10) an effective Knudsen number, that is related to the above
Knudsen number as (see Section D. of the Supporting Information):

$K_{n}^{eff}=K_{n}(E=\mu_{F},T_{e})\times\bigg{[}\frac{\pi
k_{B}T_{e}}{\hbar\omega_{L}}\bigg{]}^{2}=\frac{4I_{L}\sigma_{abs}(\hbar\omega_{L},p_{L})}{W_{ee}\rho_{c}(\mu_{F})[\hbar\omega_{L}]^{4}},$
(14)

where $W_{ee}$ is a characteristic electron-electron scattering constant (see
Section D. of the Supporting Information). The right hand side of this
equation further shows that $K_{n}^{eff}$ is independent on the electron
temperature $T_{e}$.

###  4.2 Numerical procedure for simulations with Kirchhoff’s law

Figure 4: Comparison between the experimental dark-field scattering spectra
(red lines) and the simulated scattering cross sections (dark dashed lines) of
the gold nanorods studied in Fig. 3 in the main text. Panel (a): nanorod with
longitudinal plasmon resonance at 871 nm (Ref. 27). Dimensions measured with
SEM are 18.2 nm $\times$ 85 nm. Adjusted dimensions 18.1 nm $\times$ 85.7 nm
are used in the simulation, as well as an absorption enhancement factor
$C_{abs}=$ 1.2. Panel (b): nanorod with longitudinal plasmon resonance at 734
nm (Ref. 16). Dimensions measured with SEM are 25 nm $\times$ 79 nm. Adjusted
dimensions 25 nm $\times$ 78.3 nm are used in the simulation, as well as an
absorption enhancement factor $C_{abs}=$ 1.0. A Quartz substrate with
refractive index 1.4536 has been taken into account.

We first simulate the scattering cross section corresponding to the metallic
structures under study, using the structure dimensions measured experimentally
by scanning electron microscopy (SEM). All simulations are carried out with a
home-build Maxwell solver based on the Finite Element Method. When possible,
we compare the scattering cross section to the experimental dark-field
scattering spectra, and adjust the structure dimensions in order to improve
their matching. As shown in Fig. 4, a nearly perfect agreement can be found
within a sub-nanometer variation of the dimensions for the nanorods studied in
Fig. 3 of the main text. We note that our simulations account for the presence
of a substrate, so that the experimental conditions are accurately reproduced.
In addition, in the simulations illumination is modeled by a plane wave at
normal incidence with respect to the substrate. While experimental platforms
may use a condenser so that the beam impinges on the nanoparticle with some
angle, Fig. 4 shows that it does not introduce any flaw. We eventually note
that the (normalized) dark-field scattering spectrum and the scattering cross
section are defined a bit differently (since the total light collected in the
former is reduced compared to the latter due to a limited objective aperture
in experiments), but this also does not introduce discrepancies. Finally, we
used the values of Ref. 53 for the gold permittivity. In order to improve the
fit quality, the imaginary part of the gold permittivity has been multiplied
by a dimensionless factor $C_{abs}>1$ in some simulations. Indeed, the
presence of defects and edges in the narrow nanorods modeled in this work may
enhance the absorption processes in the metal, and hence the imaginary part of
the permittivity 33.

Figure 5: Numerical calculation of the absorption cross section normalized to
maximum. The red line shows a calculation taking into account the full
numerical aperture of the microscope objective used to collect the emitted
light in the experiments (N.A.= 0.8). The dark dotted line shows calculation
only at normal incidence with respect to the substrate. The metallic structure
used in the calculations correspond to the one whose scattering cross section
is shown on Fig. 4 (a) and PL is shown on Fig. 3 (a).

Once the above adjustments are made, the absorption cross section (ACS) of the
metallic structure is computed with the same Maxwell solver. On Fig. 5, we
show an example of comparison between the normalized ACS calculated at normal
incidence with respect to the substrate (dark dotted line) and the normalized
ACS calculated taking into account the numerical aperture of the microscope
objective used to collect the emitted light in the experiments (red line). No
difference can be seen between both quantities, so that the former has been
used in the simulations presented in this work (except to calculate the
quantum yield in the next section where the full measured signal is needed).
We further repeat the calculation of the ACS for orthogonal polarizations.
Nevertheless, let us note that absorption in the polarization perpendicular to
the nanorods investigated here is negligible. Finally, we stress that only the
absorption due to intraband recombinations is to be considered in Fig. 3 of
the main text and Fig. S.1 of the Supporting Information. We note that this
condition is fulfilled since the contribution from interband recombinations to
the imaginary part of the permittivity of gold and silver is negligible in the
wavelength ranges studied here.

###  4.3 Calculation of the quantum yield in the CW pumping regime with
Kirchhoff’s law Eq. (9)

Here, we use Kirchhoff’s law Eq. (9) to calculate the PL quantum yield of the
gold nanorod whose PL is shown on Fig. 3 (a) of the main text. The theoretical
quantum yield $\eta_{sim}$ is evaluated from the power spectrum
$dP/d\Omega_{r}(\omega,\boldsymbol{u}_{r},p)$ as:

$\eta_{sim}=\frac{\sum_{p}\int_{\omega_{min}}^{\omega_{max}}d\omega\int_{\text{N.A.}}d\Omega_{r}(\boldsymbol{u}_{r})\frac{dP}{d\Omega_{r}}(\omega,\boldsymbol{u}_{r},p)}{\sigma_{abs}(\omega_{L},\boldsymbol{u}_{L},p_{L})I_{L}},$
(15)

where $\omega_{min/max}$ indicates the minimal and maximal frequencies over
which the PL signal is integrated, N.A. indicates that the integration runs
over the solid angle within the Numerical Aperture of the microscope
collecting the PL signal and $\omega_{L}$, $\boldsymbol{u}_{L}$, $p_{L}$,
$I_{L}$ are respectively the frequency, the direction, the polarization and
the intensity at sample of the pumping laser.

In the CW regime, inserting Eq. (9) and Eq. (14) into the above equation
yields:

$\begin{split}\eta_{sim}=\frac{4}{W_{ee}\rho_{c}(\mu_{F})[\hbar\omega_{L}]^{4}}\int_{\omega_{min}}^{\omega_{max}}d\omega&\Bigg{[}\sum_{p}\int_{0}^{\theta_{\text{N.A.}}}sin(\theta)d\theta\int_{0}^{2\pi}d\phi\,\,\sigma_{abs}^{intra}(\omega,\theta,\phi,p^{*})\Bigg{]}\\\
&\times\frac{\omega^{2}}{8\pi^{3}c^{2}}\frac{2(\hbar\omega-\hbar\omega_{L})}{\exp\big{(}\frac{\hbar\omega-\hbar\omega_{L}}{k_{B}T_{e}}\big{)}-1}\frac{\hbar\omega_{L}}{\hbar\omega},\end{split}$
(16)

where we used the spherical coordinate angles $\theta$ and $\phi$ to
parameterize the emission angle ($\theta$ being defined from the normal
direction with respect to the substrate). We then use the following numerical
values that are representative of the experimental conditions of Ref. 27: (i)
$W_{ee}=$ 0.01135 eV-2.fs-1, 111This value enables the best fit with Eq.
(S.39) of the function used in Ref. 30:
$\tau_{ee}(E)=e^{\frac{1}{a+b(E-\mu_{F})}}$ with $a=$ 0.08585 and $b=$ 0.1278
eV-1 in the range 1.24-1.57 eV.  (ii)
$\rho_{c}(\mu_{F})=[V/2\pi^{2}]\times[2m_{e}/\hbar^{2}]^{3/2}\times\sqrt{\mu_{F}}$
in the parabolic approximation for the conduction band, with $V=\pi\times
9.1^{2}\times 85$ nm3 27, $m_{e}$ the electron mass and $\mu_{F}=$ 5.5 eV 42,
(iii) $\hbar\omega_{L}=$ 1.58 eV, $\omega_{min}=$ 1.24 eV and $\omega_{max}=$
1.57 eV 27, (iv) $\theta_{\text{N.A.}}=\arcsin(0.8)$ 27, (v) $T_{e}$= 293 K.
Using these values and calculating numerically the absorption cross section
according to the procedure described in the last section, we obtain the
theoretical estimate of the quantum yield:

$\eta_{sim}\sim 1.3\times 10^{-7}.$ (17)

We thanks M. Besbes for assistance with finite element simulations, as well as
Y. Cai, J. Tauzin and S. Link for sharing with us their experimental data.
This work was supported by Agence Nationale de la Recherche (GYN project
ANR-17-CE24-0046).

##  5 A. Statistical physics model of light emission by a reciprocal medium

In this section, we introduce the key elements of a statistical physics model
of light emission by a reciprocal medium. In particular, we link the
electromagnetic power radiated by a metallic structure to the cross spectral
density of its electronic current density. We also show that accounting for
Lorentz reciprocity enables to reformulate the radiation problem into a
practical form using the absorption cross section.

###  5.1 A.1. Linking the emitted power spectrum to the cross spectral density
of the electronic current density

Let us consider an emitting metallic structure, with a basis centered
somewhere in the structure. We focus on an observation point $\boldsymbol{r}$
in the far field. The corresponding emission direction is defined by the unit
vector $\boldsymbol{u}_{r}=\boldsymbol{r}/|\boldsymbol{r}|$. The power
$dP_{e}(t,\boldsymbol{r},p)$ emitted by the metallic structure at time $t$, in
a polarization state $p$ and measured by a detector with surface $dA$ placed
at the point $\boldsymbol{r}$ is given by the Poynting vector 54:

$dP_{e}(t,\boldsymbol{r},p)=2\epsilon_{0}c\langle\hat{{\mathcal{E}}}^{(-)}(t,\boldsymbol{r},p)\hat{{\mathcal{E}}}^{(+)}(t,\boldsymbol{r},p)\rangle\,dA,$
(S.1)

where
$\hat{{\mathcal{E}}}^{(+/-)}(t,\boldsymbol{r},p)=\boldsymbol{e}^{(p)}\cdot\hat{\boldsymbol{\mathcal{E}}}^{(+/-)}(t,\boldsymbol{r})$
is the amplitude of the electric field operator along the unit vector
$\boldsymbol{e}^{(p)}$ defining the polarization state $p$, with
$\hat{\boldsymbol{\mathcal{E}}}^{(+)}(t,\boldsymbol{r})=\int_{0}^{+\infty}\hat{\boldsymbol{\mathcal{E}}}(\omega,\boldsymbol{r})e^{-i\omega
t}d\omega/2\pi\,\,$ the positive frequency part of the electric field operator
$\hat{\boldsymbol{\mathcal{E}}}(t,\boldsymbol{r})$, and
$\hat{\boldsymbol{\mathcal{E}}}^{(-)}$ the negative frequency part, that
satisfies
$\hat{\boldsymbol{\mathcal{E}}}^{(-)}=\hat{\boldsymbol{\mathcal{E}}}^{(+)\,\dagger}$.
In this work, $\langle...\rangle$ stands for the canonical ensemble average.
We note that the order $(-),(+)$ of the electric field operators corresponds
to normal ordering, that stems from the absorption process occurring in the
detector.

First, let us remark that the electric field operators appearing in the right
hand side of Eq. (S.1) are connected to the current density operator
$\hat{\boldsymbol{j}}(\omega)$ associated to a homogeneous metal through:

$\hat{{\mathcal{E}}}(\omega,\boldsymbol{r},p)=i\mu_{0}\omega\int_{V}d^{3}\boldsymbol{r}^{\prime}e^{(p)}_{x}G_{xy}(\omega,\boldsymbol{r},\boldsymbol{r}\,^{\prime})\hat{j}_{y}(\omega)$
(S.2)

where $\mu_{0}$ is the vacuum permeability, $V$ is the metal volume and
$G_{xy}$ is the component of the Green tensor $\boldsymbol{G}$ associated to
the vector components $x$ and $y$. Throughout this work, we use the Einstein
notation so that there is a sum over repeated indices.

Second, we remind that the cross spectral density $W_{j^{\dagger}j}(\omega)$
of a homogeneous and stationary system is defined from the correlation of its
current density 55:

$\langle\hat{j}_{x}^{\dagger}(\omega)\hat{j}_{y}(\omega^{\prime})\rangle=2\pi\delta(\omega-\omega^{\prime})\frac{1}{V}W_{j^{\dagger}_{x}j_{y}}(\omega),$
(S.3)

where $\delta$ is the Dirac function.

Hence, inserting Eq. (S.2) into the Fourier transform of Eq. (S.1) and using
Eq. (S.3), the power spectrum can be cast into the form:

$\frac{dP_{e}}{d\Omega_{r}}(\omega,\boldsymbol{r},p)=\frac{|\boldsymbol{r}|^{2}\omega^{2}}{\pi\epsilon_{0}c^{3}}\int_{V}d^{3}\boldsymbol{r}^{\prime}e^{(p)}_{u}G_{ux}(\omega,\boldsymbol{r},\boldsymbol{r}^{\prime})e^{(p)}_{v}G_{vy}(\omega,\boldsymbol{r},\boldsymbol{r}^{\prime})W_{j^{\dagger}_{x}j_{y}}(\omega),$
(S.4)

where we have used $d\Omega_{r}=dA/|\boldsymbol{r}|^{2}$ to denote the solid
angle subtended by the surface $dA$.

Finally, noting that the permittivity of common metals is isotropic, the cross
spectral density tensor reduces to
$W_{j^{\dagger}_{x}j_{y}}=\delta_{x,y}W_{j^{\dagger}j}$, so that Eq. (S.4) can
be simplified into:

$\frac{dP_{e}}{d\Omega_{r}}(\omega,\boldsymbol{r},p)=\frac{|\boldsymbol{r}|^{2}\omega^{2}}{\pi\epsilon_{0}c^{3}}W_{j^{\dagger}j}(\omega)\times\sum_{x}\int_{V}d^{3}\boldsymbol{r}^{\prime}|e^{(p)}_{u}G_{ux}(\omega,\boldsymbol{r},\boldsymbol{r}^{\prime})|^{2}.$
(S.5)

###  5.2 A.2. Role of Lorentz reciprocity

The emitted power in Eq. (1) of the main text is expressed as a function of
the absorption cross section of the metallic body. We now show how to relate
Eq. (S.5) to this quantity 37. To start, we recall the definition of the
absorption cross section of an isotropic and homogeneous metallic body of
volume $V$:

$\sigma_{abs}(\omega,-\boldsymbol{u}_{r},p)=\frac{\omega}{c}\text{Im}[\epsilon(\omega)]\int_{V}d\boldsymbol{r}^{\prime}\frac{|\boldsymbol{{\mathcal{E}}}(\boldsymbol{r}^{\prime},\omega,-\boldsymbol{u}_{r},p)|^{2}}{|\mathcal{E}_{L}(\omega,-\boldsymbol{u}_{r},p)|^{2}},$
(S.6)

where $\epsilon$ is the permittivity of the metal,
${{\mathcal{E}}}_{L}(\omega,-\boldsymbol{u}_{r},p)$ is the electric field
amplitude corresponding to an incident plane wave and
$\boldsymbol{{\mathcal{E}}}(\boldsymbol{r}^{\prime},\omega,-\boldsymbol{u}_{r},p)$
is the electric field induced by this plane wave at a position
$\boldsymbol{r}^{\prime}$ in the metal. It will be convenient to assume that
the incident plane wave is emitted by an electric dipole with frequency
$\omega$ and polarization $p$ placed at the point $\boldsymbol{r}$ in the far
field of the metallic body. This emission point defines the plane wave
direction $-\boldsymbol{u}_{r}=-\boldsymbol{r}/|\boldsymbol{r}|$.

In this context, the vector component $u$ of
$\boldsymbol{\mathcal{E}}(\boldsymbol{r}^{\prime},\omega,-\boldsymbol{u}_{r},p)$
can be related to $\mathcal{E}_{L}(\omega,-\boldsymbol{u}_{r},p)$ using the
Green tensor:

$\mathcal{E}_{u}(\boldsymbol{r}^{\prime},\omega,-\boldsymbol{u}_{r},p)=4\pi|\boldsymbol{r}|e^{-\frac{i\omega|\boldsymbol{r}|}{c}}G_{ux}(\omega,\boldsymbol{r}^{\prime},\boldsymbol{r})e^{(p)}_{x}\mathcal{E}_{L}(\omega,-\boldsymbol{u}_{r},p).$
(S.7)

Hence, inserting Eq. (S.7) into Eq. (S.6) yields:

$\sigma_{abs}(\omega,-\boldsymbol{u}_{r},p)=\frac{\omega}{c}\text{Im}[\epsilon(\omega)](4\pi|\boldsymbol{r}|)^{2}\times\sum_{u}\int_{V}d^{3}\boldsymbol{r}^{\prime}|G_{ux}(\omega,\boldsymbol{r}^{\prime},\boldsymbol{r})e^{(p)}_{x}|^{2}.$
(S.8)

Since the metal has a scalar permittivity, Lorentz reciprocity imposes 34:

$G_{xu}(\omega,\boldsymbol{r},\boldsymbol{r}\,^{\prime})=G_{ux}(\omega,\boldsymbol{r}\,^{\prime},\boldsymbol{r}).$
(S.9)

Using Eq. (S.9), the absorption cross section Eq. (S.8) can be inserted into
Eq. (S.5), which yields the desired connection between the emitted power, the
cross spectral density of $\hat{\boldsymbol{j}}$ and the absorption cross
section:

$\frac{dP_{e}}{d\Omega_{r}}(\omega,\boldsymbol{u}_{r},p)=\sigma_{abs}(\omega,-\boldsymbol{u}_{r},p)\frac{\omega^{2}}{8\pi^{3}c^{2}}\frac{W_{j^{\dagger}j}(\omega)}{2\omega\epsilon_{0}\text{Im}[\epsilon(\omega)]}.$
(S.10)

This last equation corresponds to the Equation (1) reported in the main text.

##  6 B. Fluctuation and dissipation of a stationary nonequilibrium electron
gas

In this section, we derive nonequilibrium forms of $W_{j^{\dagger}j}$ and
$\text{Im}[\epsilon]$. We start by calculating the cross spectral density of
the current density $W_{j^{\dagger}j}(\omega)$, that quantifies the system
fluctuations. We then derive $\text{Im}[\epsilon]$, through a calculation of
the linear response under optical excitation. Finally, we show that when the
gas is in thermodynamic equilibrium, $W_{j^{\dagger}j}(\omega)$ can be related
to the linear response, yielding the so-called Fluctuation-Dissipation
theorem.

###  6.1 B.1. Cross spectral density of the current density

The current density operator of a homogeneous metal at a time $t$ in the
Heisenberg picture can be expressed generally as a function of the electron
field operator $\hat{\psi}$ 56:

$\hat{\boldsymbol{j}}(t)=\frac{e}{2mV}\int_{V}d^{3}\boldsymbol{r}\Big{[}\hat{\psi}^{\dagger}(\boldsymbol{r},t)\textbf{p}\hat{\psi}(\boldsymbol{r},t)-\hat{\psi}(\boldsymbol{r},t)\textbf{p}\hat{\psi}^{\dagger}(\boldsymbol{r},t)\Big{]},$
(S.11)

where $e$ is the electron charge, $m$ is the electron mass, $V$ is the metal
volume, $\boldsymbol{r}$ is a position in the metal, † is the Hermitian
conjugate and $\textbf{p}=-i\hbar\nabla$ is the momentum operator with $\hbar$
the reduced Planck constant and $\nabla$ the gradient operator.

First, we focus on the electron field operator. In the following, we will
assume that the electron gas can be described as a non-interacting gas, and
will neglect the electron spin degree of freedom. Accordingly, the electronic
states are given by the solutions of the single-electron Schrödinger equation
in the metal (with a mean-potential accounting for the lattice). We index
these states by $n=1...N$ where $N$ is the total electron number, and note
their corresponding energy by $\hbar\omega_{n}$ and wavefunction by
$|\phi_{n}\rangle$. Hence, the second-quantization Hamiltonian describing the
electron gas writes:

$\hat{\mathcal{H}}=\sum_{n=1}^{N}\hbar\omega_{n}\hat{c}_{n}^{\dagger}\hat{c}_{n},$
(S.12)

where $\hat{c}_{n}^{\dagger}$, $\hat{c}_{n}$ are respectively the electron
creation and annihilation operators of the state $n$, that obey the Fermionic
commutation relations
$\\{\hat{c}_{n}^{\dagger},\hat{c}_{n^{\prime}}^{\dagger}\\}=0$;
$\\{\hat{c}_{n},\hat{c}_{n^{\prime}}\\}=0$ and
$\\{\hat{c}_{n}^{\dagger},\hat{c}_{n^{\prime}}\\}=\delta_{n,n^{\prime}}$,
where $\\{\hat{a},\hat{b}\\}=\hat{a}\hat{b}+\hat{b}\hat{a}$. In this context,
the electron field operator in the Heisenberg picture is defined as 56:

$\hat{\psi}(\boldsymbol{r},t)=\sum_{n}\phi_{n}(\boldsymbol{r})e^{-i\omega_{n}t}\hat{c}_{n},$
(S.13)

where $\phi_{n}(\boldsymbol{r})=\langle\boldsymbol{r}|\phi_{n}\rangle$.
Inserting the electron field operator definition Eq. (S.13) into Eq. (S.11)
and carrying out a time Fourier transform, we obtain the current density
operator in the frequency domain:

$\hat{j}_{x}(\omega)=\frac{e}{mV}\sum_{n,n^{\prime}}2\pi\delta(\omega-[\omega_{n^{\prime}}-\omega_{n}])p^{x}_{n,n^{\prime}}\hat{c}_{n}^{\dagger}\hat{c}_{n^{\prime}},$
(S.14)

where $x$ indicates a vector component in a given basis and where we
introduced the shorthand
$p^{x}_{n,n^{\prime}}=\langle\phi_{n}|p^{x}|\phi_{n^{\prime}}\rangle$.

Then inserting Eq. (S.14) into Eq. (S.3) defined in the last section, we can
calculate the correlation of the current densities. To this end, the main
difficulty to overcome is the evaluation of the 4-operators correlations
$\langle\hat{c}_{n^{\prime}}^{\dagger}\hat{c}_{n}\hat{c}_{m}^{\dagger}\hat{c}_{m^{\prime}}\rangle$.
Yet, let us remind that for a non-interacting electron gas, the electronic
states can be described as Fock states, that we note by
$|\\{n_{l}\\}\rangle=|n_{1}...n_{l}...n_{N}\rangle$ where $n_{l}$ denotes the
number of particles in the state $l$. The explicit form of the average
correlation is
$\langle\hat{c}_{n^{\prime}}^{\dagger}\hat{c}_{n}\hat{c}_{m}^{\dagger}\hat{c}_{m^{\prime}}\rangle=\sum_{\\{n_{l}\\}}P_{\\{n_{l}\\}}\langle\\{n_{l}\\}|\hat{c}_{n^{\prime}}^{\dagger}\hat{c}_{n}\hat{c}_{m}^{\dagger}\hat{c}_{m^{\prime}}|\\{n_{l}\\}\rangle$,
where $P_{\\{n_{l}\\}}$ is the canonical probability that the system is in the
Fock state $|\\{n_{l}\\}\rangle$. The elements in the sum can be evaluated
given that
$\langle\\{n_{l}\\}|\hat{c}_{n}^{\dagger}|\\{n_{l}\\}\rangle=\langle\\{n_{l}\\}|\hat{c}_{n}|\\{n_{l}\\}\rangle=0$
and
$\langle\\{n_{l}\\}|\hat{c}_{n}^{\dagger}\hat{c}_{n^{\prime}}|\\{n_{l}\\}\rangle=\delta_{n,n^{\prime}}f_{n}$,
where $f_{n}$ is the mean occupation number of the electronic state $n$. These
considerations enables to simplify the above correlations:

$\langle\hat{c}_{n^{\prime}}^{\dagger}\hat{c}_{n}\hat{c}_{m}^{\dagger}\hat{c}_{m^{\prime}}\rangle=\delta_{n,n^{\prime}}\delta_{m,m^{\prime}}\Big{[}\delta_{n,m}f_{n}+[1-\delta_{n,m}]f_{n}f_{m}\Big{]}+\delta_{n^{\prime},m^{\prime}}\delta_{n,m}[1-\delta_{n,n^{\prime}}]f_{n^{\prime}}[1-f_{n}].$
(S.15)

Importantly, let us stress that at this point the mean occupation numbers can
take any value, that is the $f_{n}$ may differ from the equilibrium Fermi-
Dirac distribution. We also note that the first term in the right hand side of
Eq. (S.15) only implies emission at zero frequency so that it is not relevant
to the present study. Conversely, the second term in the right hand side of
Eq. (S.15) involves mixing of the $n$ and $m$ electronic states, namely time-
dependent correlation between the two current densities. Using this relation
finally enables to derive the fluctuation relation reported in Eq.(2) of the
main text:

$W_{j^{\dagger}j}(\omega)=\frac{\epsilon_{0}\omega_{p}^{2}}{m}\frac{2\pi}{N}\sum_{n,n^{\prime}=1}^{N}|p_{n,n^{\prime}}|^{2}f_{n^{\prime}}[1-f_{n}]\delta(\omega-[\omega_{n^{\prime}}-\omega_{n}]),$
(S.16)

where $\epsilon_{0}$ is the vacuum permittivity and
$\omega_{p}~{}=~{}\sqrt{Ne^{2}/mV\epsilon_{0}}$ is the plasma frequency.

In summary, we have provided a derivation of the nonequilibrium cross spectral
density of $\hat{\boldsymbol{j}}$. We note that a similar expression was used
in Ref. 57 in the context of scintillation-induced light emission. To finish,
let us discuss further the transition matrix elements
$|p_{n,n^{\prime}}|=|\langle\phi_{n}|\textbf{p}|\phi_{n^{\prime}}\rangle|$
that appears in Eq. (S.16). We note that for a bulk, non-interacting electron
gas, the electronic wavefunctions reduce to Bloch waves so that
$|\langle\phi_{n}|\textbf{p}|\phi_{n^{\prime}}\rangle|=0$ if $n\neq
n^{\prime}$. In reality, these transition matrix elements are obviously non-
zero, otherwise the existence of thermal emission would be precluded.
Actually, it is well-known that intraband absorption processes are assisted by
phonons or electrons. While taking into account these microscopic processes is
beyond the scope of the present paper, we note that this issue has been
tackled in Ref. 58 (Chapter 15. Section 3.) for the contribution of static
impurities and in Ref. 59 (Chapter 5. Appendix A.) for the contribution of
phonon-assisted transitions.

###  6.2 B.2. Linear response

We now calculate the imaginary part of the permittivity of a nonequilibrium
electron gas in the linear response approximation. We start by writing the
semiclassical coupling Hamiltonian in the Schrödinger picture between an
incident radiation described by the vector potential $\boldsymbol{A}$ and a
homogeneous electron gas described by the electron current density
$\hat{\boldsymbol{j}}$:

$\hat{\mathcal{H}}_{int}(t)=-\hat{\boldsymbol{j}}.\boldsymbol{A}(t)V,$ (S.17)

where we have neglected spatial variations of the vector potential inside the
metallic structure, in accordance with the homogeneous gas description.

We then treat the Hamiltonian $\hat{\mathcal{H}}_{int}(t)$ as a time-dependent
perturbation of the electron gas. In the linear response approximation, the
so-called Kubo formula 60, 61 expresses elegantly the expectation value of
$\hat{\boldsymbol{j}}^{ind}$, the electron current density induced by the
perturbation:

$\langle\hat{j}^{ind}_{x}(t)\rangle=\langle\hat{j}_{x}(t_{0})\rangle-\frac{iV}{\hbar}\int_{t_{0}}^{t}dt^{\prime}\langle[\hat{j}_{x}(t),\hat{j}_{y}(t^{\prime})]\rangle{A}_{y}(t^{\prime}),$
(S.18)

where it should be noted that $\hat{\boldsymbol{j}}_{ind}(t)$ is written in
the interaction picture while $\hat{\boldsymbol{j}}(t)$ is now written in the
Heisenberg picture.

From Eq. (S.18), it is possible to define the electric susceptibility tensor
of the electron gas in the vector potential-current density gauge (noted by
$\boldsymbol{j}.\boldsymbol{A}$):

$\chi_{x,y}^{\boldsymbol{j}.\boldsymbol{A}}(t,t^{\prime})=-\frac{iV}{\hbar}\langle[\hat{j}_{x}(t),\hat{j}_{y}(t^{\prime})]\rangle
H(t-t^{\prime}),$ (S.19)

where $H$ is the Heaveside step function. Now, resorting again to the
assumption that the electrons are non-interacting, the current density
operator in the Heisenberg picture can be expressed by inserting Eq. (S.13)
into Eq. (S.11). Then inserting the result into Eq. (S.19) and using the
relation Eq. (S.15) to simplify the 4-operator correlation, we obtain
$\chi^{\boldsymbol{j}.\boldsymbol{A}}$ into the form:

$\chi_{x,y}^{\boldsymbol{j}.\boldsymbol{A}}(t-t^{\prime})=\frac{i}{\hbar}\frac{\epsilon_{0}\omega_{p}^{2}}{m}\frac{1}{N}\sum_{n,n^{\prime}}p^{x}_{n,n^{\prime}}p^{y}_{n^{\prime},n}[f_{n^{\prime}}-f_{n}]e^{-i(\omega_{n^{\prime}}-\omega_{n})(t-t^{\prime})}H(t-t^{\prime}).$
(S.20)

Finally, moving to the electric field-dipole moment gauge and noting that Eq.
(S.20) is stationary in time, the imaginary part of the permittivity tensor
(that is the imaginary part of the electric susceptibility tensor) associated
to metal writes in the frequency domain:

$\text{Im}[\epsilon_{x,y}](\omega)=\frac{\omega_{p}^{2}}{2\hbar
m\omega^{2}}\frac{2\pi}{N}\sum_{n,n^{\prime}=1}^{N}p^{x}_{n,n^{\prime}}p^{y}_{n^{\prime},n}[f_{n}-f_{n^{\prime}}]\delta(\omega-[\omega_{n^{\prime}}-\omega_{n}]).$
(S.21)

In the assumption that the metal is isotropic, this equation reduces to:

$\text{Im}[\epsilon](\omega)=\frac{\omega_{p}^{2}}{2\hbar
m\omega^{2}}\frac{2\pi}{N}\sum_{n,n^{\prime}=1}^{N}|p_{n,n^{\prime}}|^{2}[f_{n}-f_{n^{\prime}}]\delta(\omega-[\omega_{n^{\prime}}-\omega_{n}]),$
(S.22)

where it is stressed again that the $f_{n}$ can take any value.

###  6.3 B.3. Fluctuation-Dissipation theorem

To finish, let us focus on the peculiar but common case in which the electron
gas has relaxed to thermodynamic equilibrium. The mean occupation numbers
$f_{n}$ of the electrons then follow the Fermi-Dirac distribution:

$f_{n}=f_{FD}(\omega_{n},T_{e},\mu_{F})\equiv\frac{1}{\exp\Big{(}\frac{\hbar\omega_{n}-\mu_{F}}{k_{B}T_{e}}\Big{)}+1},$
(S.23)

where $T_{e}$ is the temperature of the electron gas and $\mu_{F}$ its Fermi
level. Using Eq. (S.23), it is straightforward to derive the identity:

$\frac{f_{FD}(\hbar\omega_{n})-f_{FD}(\hbar\omega_{n}+\hbar\omega)}{\exp\Big{(}\frac{\hbar\omega}{k_{B}T}\Big{)}-1}=f_{FD}(\hbar\omega_{n}+\hbar\omega)[1-f_{FD}(\hbar\omega_{n})].$
(S.24)

The right hand side of Eq. (S.24) can be viewed as the probability of
spontaneous emission in a photonic mode with energy $\hbar\omega$ whereas the
left hand side is the product of the probability of absorption by the Bose-
Einstein distribution. This equation is actually equivalent to the so-called
Van Roosbroeck-Shockley relation, which connects the emission and absorption
rates in a homogeneous semiconductor medium 62, 63.

Using Eq. (S.24), a connection can now be made between the cross spectral
density Eq. (S.16) and the linear response Eq. (S.22) of the electron gas at
thermodynamic equilibrium, that is called the Fluctuation-Dissipation theorem
35, 36:

$W_{j^{\dagger}j}^{eq}(\omega,T_{e})=2\omega\epsilon_{0}\text{Im}[\epsilon_{eq}(\omega,T_{e})]\frac{\hbar\omega}{\exp\big{(}\frac{\hbar\omega}{k_{B}T_{e}}\big{)}-1},$
(S.25)

which corresponds to equation (3) of the main text.

##  7 C. Generalized Kirchhoff’s law for emission from intraband transitions

In this section, we detail the approximations and calculation steps used to
derive Eq. (7) of the main text. Hence, we focus on the conduction band
electronic states and intraband transitions only. We want to calculate the
cross spectral density corresponding to these intraband recombination
processes, that we note $W_{j^{\dagger}j}^{intra}$.

We start by assuming that the transition matrix elements of intraband
transitions are constant over the electronic states, and equal to their value
at Fermi level. Building on the notations of section B., this amounts to write
that ${p}_{n,n^{\prime}}={p}_{F}$. This assumption enables to cast Eq. (S.16)
into:

$W_{j^{\dagger}j}^{intra}(\omega)=\frac{\epsilon_{0}\omega_{p,c}^{2}}{m}\frac{2\pi}{N_{c}}|p_{F}|^{2}\sum_{n,n^{\prime}=1}^{N_{c}}f_{n^{\prime}}[1-f_{n}]\delta(\omega-[\omega_{n^{\prime}}-\omega_{n}]),$
(S.26)

where $\omega_{p,c}=\sqrt{N_{c}e^{2}/mV\epsilon_{0}}$ with $N_{c}$ the number
of electrons in the conduction band. We stress that now the sums over the
electronic states is restricted to the conduction band.

We now convert the sum over electronic states into integrals. To this end, we
assume that the mean occupation numbers $f_{n}$ of the conduction band
electronic states follow a distribution $f$ that only depends on the state
energy. Hence, Eq. (S.26) can be written as:

$W_{j^{\dagger}j}^{intra}(\omega)=\frac{\epsilon_{0}\omega_{p,c}^{2}}{m}\frac{2\pi}{N_{c}}|p_{F}|^{2}D_{J}(\mu_{F},\hbar\omega)\int_{E_{c,0}}^{\infty}dE\,f(E+\hbar\omega)[1-f(E)],$
(S.27)

where $E$ is an electron state energy, $E_{c,0}$ the energy minimum of the
conduction band, and $D_{J}(E,\hbar\omega)$ is the energy distribution of the
joint density of states (defined by volume unit):

$D_{J}(\omega,E)=V^{2}\int_{E_{c,0}}^{\infty}dE^{\prime}\,\rho_{c}(E^{\prime})\rho_{c}(E)\delta(\hbar\omega-[E^{\prime}-E]),$
(S.28)

with $\rho_{c}(E)$ the density of states of the conduction band (defined by
volume unit). In addition, let us note that in Eq. (S.27) we assumed
$D_{J}(\omega,E)$ to be constant over the electronic states and equal to its
value at Fermi level, so that it can be taken out of the integral.

We now focus on the prefactor of the integral in Eq. (S.27). Performing the
same approximations as above on Eq. (S.22), the imaginary part of the metal
permittivity corresponding to absorption through intraband processes can be
cast into the form:

$\text{Im}[\epsilon^{intra}](\omega)=\frac{\omega_{p,c}^{2}}{2\hbar
m\omega^{2}}\frac{2\pi}{N_{c}}|p_{F}|^{2}D_{J}(\mu_{F},\omega)\int_{E_{c,0}}^{\infty}dE\,[f(E)-f(E+\hbar\omega)].$
(S.29)

At thermodynamic equilibrium, it is possible to simplify further Eq. (S.29).
Indeed, the distributions $f$ can then be replaced by Fermi-Dirac
distributions. Interestingly, we further note that the ambient-temperature
thermal energy ($\sim$ 25 meV) is negligible compared to the energy of visible
or NIR photons ($\sim$ 1 eV). Therefore, the Fermi-Dirac distribution can be
approximated by a step function, that is $f_{FD}(E)\approx 1-H(E-\mu_{F})$.
Inserting this approximation into Eq. (S.29), we find that at thermodynamic
equilibrium $\text{Im}[\epsilon^{intra}]$ reduces to:

$\text{Im}[\epsilon^{intra}_{eq}](\omega)=\frac{\omega_{p,c}^{2}}{2\hbar
m\omega^{2}}\frac{2\pi}{N_{c}}|p_{F}|^{2}D_{J}(\mu_{F},\hbar\omega)V\hbar\omega.$
(S.30)

Finally, inserting Eq. (S.30) into Eq. (S.27) yields a generalization of the
Fluctuation-Dissipation theorem, valid for any out-of-equilibrium electronic
distribution $f$:

$W_{j^{\dagger}j}^{intra}(\omega)=2\epsilon_{0}\omega\text{Im}[\epsilon^{intra}_{eq}(\omega)]\int_{E_{c,0}}^{\infty}dE\,f(E+\hbar\omega)[1-f(E)],$
(S.31)

that is the equation reported in Eqs. (5),(6) of the main text.

Now, inserting Eq. (S.31) into Eq. (S.10) yields:

$\frac{dP_{e}}{d\Omega_{r}}(\omega,\boldsymbol{u}_{r},p)=\sigma_{abs}(\omega,-\boldsymbol{u}_{r},p^{*})\frac{\omega^{2}}{8\pi^{3}c^{2}}\frac{\text{Im}[\epsilon^{intra}_{eq}(\omega)]}{\text{Im}[\epsilon(\omega)]}\int
f(E+\hbar\omega)[1-f(E)]dE.$ (S.32)

We finally define the absorption cross section due to intraband transitions
only as:

$\sigma_{abs}^{intra}(\omega,-\boldsymbol{u}_{r},p^{*})=\frac{\text{Im}[\epsilon^{intra}(\omega)]}{\text{Im}[\epsilon(\omega)]}\times\sigma_{abs}(\omega,-\boldsymbol{u}_{r},p^{*}),$
(S.33)

where we note that the imaginary part of the metal permittivity due to
intraband transitions $\text{Im}[\epsilon^{intra}(\omega)]$ is not necessarily
evaluated at equilibrium here. Inserting Eq. (S.33) into Eq. (S.32), we get:

$\frac{dP_{e}}{d\Omega_{r}}(\omega,\boldsymbol{u}_{r},p)=\sigma_{abs}^{intra}(\omega,-\boldsymbol{u}_{r},p^{*})\frac{\omega^{2}}{8\pi^{3}c^{2}}\frac{\text{Im}[\epsilon^{intra}_{eq}(\omega)]}{\text{Im}[\epsilon^{intra}(\omega)]}\int
f(E+\hbar\omega)[1-f(E)]dE.$ (S.34)

Finally, assuming that
$\text{Im}[\epsilon^{intra}_{eq}(\omega)]/\text{Im}[\epsilon^{intra}(\omega)]\approx
1$, namely that possible nonequilibrium effects are sufficiently weak so that
the imaginary part of the metal permittivity stays close to the equilibrium
one, we obtain the equation (7) reported in the main text. We note that this
last approximation is expected to hold in most pumping conditions, in
particular through continuous wave pumping. However, it may be inaccurate for
excitation through ultrashort laser pulses with high peak intensity 64.

##  8 D. Generalized Kirchhoff’s law for emission from intraband transitions
in the CW regime

In this section, we detail the derivation of the generalized Kirchhoff’s law
for emission from intraband transitions in the CW pumping regime Eq. (9) from
the generalized Kirchhoff’s law Eq. (7).

To start, we insert Eqs. (12), (13) of Methods into Eq. (6) of the main text.
Keeping only the components to first order in $K_{n}$, the integral
$\Theta^{intra}$ can be cast into the form:

$\begin{split}\Theta^{intra}_{cw}(\omega)=&\int
f_{FD}(E+\hbar\omega-\hbar\omega_{L})[1-f_{FD}(E)]K_{n}(E+\hbar\omega)dE\\\
&+\int
f_{FD}(E+\hbar\omega)[1-f_{FD}(E+\hbar\omega_{L})]K_{n}(E)dE.\end{split}$
(S.35)

Using the remarkable relation between Fermi-Dirac distributions:

$\frac{f_{FD}(E+E_{1})[1-f_{FD}(E+E_{2})]}{f_{FD}(E+E_{2})-f_{FD}(E+E_{1})}=\frac{1}{\exp(\frac{E_{1}-E_{2}}{k_{B}T_{e}})-1},$
(S.36)

we get:

$\begin{split}\Theta^{intra}_{cw}(\omega)=\frac{1}{\exp\big{(}\frac{\hbar\omega-\hbar\omega_{L}}{k_{B}T_{e}}\big{)}-1}\times\Bigg{[}&\int[f_{FD}(E)-f_{FD}(E+\hbar\omega-\hbar\omega_{L})]K_{n}(E+\hbar\omega)dE\\\
&+\int[f_{FD}(E+\hbar\omega_{L})-f_{FD}(E+\hbar\omega)]K_{n}(E)dE\Bigg{]}.\end{split}$
(S.37)

As in Section C., we now use the fact that thermal energy $k_{B}T$ at ambient
temperature is negligible compared to
$\hbar\omega,\,\hbar\omega_{L},\,\mu_{F}$. Therefore, we approximate the
Fermi-Dirac distribution by a step function $f_{FD}(E)\approx 1-H(E-\mu_{F})$.
Hence, Eq. (S.37) simplifies into:

$\Theta^{intra}_{cw}(\omega)=\frac{1}{\exp\big{(}\frac{\hbar\omega-\hbar\omega_{L}}{k_{B}T_{e}}\big{)}-1}\times\Bigg{[}\int_{\mu_{F}+\hbar\omega_{L}}^{\mu_{F}+\hbar\omega}K_{n}(E)dE+\int_{\mu_{F}-\hbar\omega}^{\mu_{F}-\hbar\omega_{L}}K_{n}(E)dE\Bigg{]}.$
(S.38)

We then need to perform an integral over the Knudsen number. To simplify this
integral, we assume that the electronic density of state $\rho_{c}$ can be
evaluated at Fermi level, as in the last section when we derived the
generalized Kirchhoff’s law. Hence, according to Eq. (13) in Methods, the only
quantity that depends on energy in $K_{n}$ is the relaxation time $\tau_{ee}$.
In Landau’s Fermi liquid Theory 65, this dependence is a Lorentzian:

$\tau_{ee}(E,T_{e})=\frac{1}{W_{ee}[\pi
k_{B}T_{e}]^{2}}\frac{1}{1+\Big{[}\frac{E-\mu_{F}}{\pi
k_{B}T_{e}}\Big{]}^{2}},$ (S.39)

where $W_{ee}$ is a characteristic electron-electron scattering constant.

Using Eq. (S.39) into Eq. (S.38), one finds an analytic result involving
$\arctan$ functions. Using the remarkable identity
$\arctan(x)-\arctan(y)=\arctan(\frac{x-y}{1-xy})$ with $x,y>0$ eventually
enables to cast $\Theta^{intra}(\omega)$ under CW pumping into the form:

$\Theta^{intra}_{cw}(\omega)=\frac{2(\hbar\omega-\hbar\omega_{L})}{\exp\big{(}\frac{\hbar\omega-\hbar\omega_{L}}{k_{B}T_{e}}\big{)}-1}\frac{\hbar\omega_{L}}{\hbar\omega}K_{n}^{eff},$
(S.40)

where we defined the effective Knudsen number also given in Methods:

$K_{n}^{eff}=K_{n}(E=\mu_{F},T_{e})\times\bigg{[}\frac{\pi
k_{B}T_{e}}{\hbar\omega_{L}}\bigg{]}^{2}=\frac{4I_{L}\sigma_{abs}(\hbar\omega_{L},p_{L})}{W_{ee}\rho_{c}(\mu_{F})[\hbar\omega_{L}]^{4}}$
(S.41)

that is independent on the electron temperature $T_{e}$.

##  9 E. Phenomenological Kirchhoff’s law: additional simulation-experiment
comparisons

In this appendix, we provide additional evidence that the phenomenological
Kirchhoff’s law Eq. (11) is an accurate tool to model PL from metals in the CW
pumping regime. We reproduce data obtained with silver and data measured with
gold pumped at 532 nm that exhibit PL in the 500-600 nm range.

Figure S.1: Comparison between experimental 25 photoluminescence spectra (PL)
from silver nanorods of various lengths (plain colored lines) and simulations
with the phenomenological Kirchhoff’s law Eq. (11) (black and blue triangles).
The colored dotted lines correspond to experimental dark-field scattering
spectra and evidence the longitudinal plasmon resonance of the nanorods. The
width of the nanorods was measured to 23 nm 25. All nanorods are pumped at 633
nm. The theory-experiment comparison lies on the nanorods with respectively
lower (exp: green line; sim: black triangles) and higher (exp: brown line;
sim: blue triangles) resonance wavelength. Their length is estimated to 72 nm
and 92 nm respectively through adjustments of the dark-field scattering
spectra with the scattering cross section (not shown). These adjustments also
required to use $C_{abs}=1.8$ and $C_{abs}=3.5$ respectively. Figure adapted
from Ref. 25. Details on the fitting procedure are provided in Methods.

First, we show on Fig. S.1 experimental PL spectra of silver nanorods with
different length under CW pumping at 633 nm (plain colored lines) 25. This
material is interesting as it does not feature interband transitions in the
visible range, which enables to study intraband PL at shorter excitation
wavelengths than we did previously for gold. For readability reasons, we only
show comparisons with simulations (colored triangles) for the nanorods with
respectively lower (exp: green line, sim: black triangles) and higher (exp:
brown line, sim: blue triangles) resonance wavelength. Again, the agreement
between theory and experiments is very good for both nanorods. In particular,
let us notice that the green curve features a small anti-Stokes emission
around 620 nm that is nicely reproduced by the simulation. Hence, this
provides support that calculations with Eq. (11) are robust whatever type of
emitting metal.

Figure S.2: Comparison between the experimental 16 photoluminescence spectrum
(PL) of a gold nanorod (red line) and a simulation with the phenomenological
Kirchhoff’s law Eq. (11) (black dashed line), under pumping at 633 nm. The
nanorod has a longitudinal plasmon resonance at 619 nm (measured in dark-field
scattering) and dimensions 30 nm $\times$ 63 nm (measured with SEM). Reprinted
with permission from Ref. 16. The electron temperature is the only fit
parameter and is indicated on the figure. Details on the fitting procedure are
provided in Methods.  Figure S.3: Comparison between the experimental 18
photoluminescence spectra (PL) of a gold nanosphere in water (colored lines)
and simulations with the phenomenological Kirchhoff’s law Eq. (11) (colored
triangles), under pumping at 532 nm and for several illumination intensities.
The nanosphere has a longitudinal plasmon resonance at 540 nm and diameter 80
nm. The theory-experiment comparison is made for the illumination irradiances
0.2 mW.$\mu$m2 (exp: purple line; sim: green triangles), 0.9 mW.$\mu$m2 (exp:
pink line; sim: blue triangles) and 1.6 mW.$\mu$m2 (exp: yellow line; sim:
black triangles). The electron temperature is the only fit parameter, and is
293 K for the green and blue triangles and 320 K for the black triangles. The
absorption cross section of the nanosphere in water has been calculated with a
Mie theory solver. The influence of the substrate has been neglected. Figure
adapted from Ref. 18.

Second, we show on Figs. S.2 and S.3 experimental PL spectra for metallic
nanostructures pumped in CW over the interband threshold (Gold nanorod pumped
at 633 nm in Fig. S.2; Gold nanosphere pumped at 532 nm in Fig. S.3). While
Kirchhoff’s law Eq. (11) has been developed for CW emission under the
interband threshold, we try to fit with it the experimental data, only
replacing $\sigma_{abs}^{intra}$ by $\sigma_{abs}$. It is seen on both figures
that a quite good agreement can be reached. We stress that interband
transitions are expected to provide the leading contribution in this frequency
range. Hence, this agreement suggests that the contribution of the interband
transitions can also be cast in a form resembling Eq. (11).

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