SI_v3

# Theory of non-equilibrium “hot” carriers in direct band-gap semiconductors
under continuous illumination

Subhajit Sarkar<EMAIL_ADDRESS>Department of Chemistry, Ben-Gurion
University of the Negev  School of Electrical and Computer Engineering, Ben-
Gurion University of the Negev  Ieng-Wai Un and Yonatan Sivan  School of
Electrical and Computer Engineering, Ben-Gurion University of the Negev
Yonatan Dubi<EMAIL_ADDRESS>Department of Chemistry, Ben-Gurion University
of the Negev  Ilse Katz Center for Nanoscale Science and Technology, Ben-
Gurion University of the Negev

###### Abstract

The interplay between the illuminated excitation of carriers and subsequent
thermalization and recombination leads to the formation of non-equilibrium
distributions for the “hot” carriers and to heating of both electrons, holes
and phonons. In spite of the fundamental and practical importance of these
processes, there is no theoretical framework which encompasses all of them and
provides a clear prediction for the non-equilibrium carrier distributions.
Here, a self-consistent theory accounting for the interplay between
excitation, thermalization, and recombination in continuously-illuminated
semiconductors is presented, enabling the calculation of non-equilibrium
carrier distributions. We show that counter-intuitively, distributions deviate
more from equilibrium under weak illumination than at high intensities. We
mimic two experimental procedures to extract the carrier temperatures and show
that they yield different dependence on illumination. Finally, we provide an
accurate way to evaluate photoluminescence efficiency, which, unlike
conventional models, predicts correctly the experimental results. These
results provide a starting point towards examining how non-equilibrium
features will affect properties hot-carrier based application.

##  I Introduction

Illumination of semiconductors causes the generation of high energy non-
thermal (aka “hot”) carriers (HCs) [1] which critically impacts the
performance of semiconductor-based electronic and opto-electronic devices [2,
3, 4]. Although the “hot” carrier (HC) dynamics in semiconductors have been
studied for the past few decades [1, 2], the topic has seen a growing interest
in recent times due to technological relevance [3, 5, 6, 7, 8]. Much
experimental and theoretical efforts have been devoted to studying the
transient dynamics of HCs in semiconductors under ultrafast pulsed
illumination [9, 10, 11, 12, 13, 14, 15, 7, 16, 17, 18, 19, 8, 20] with the
aim of understanding the HC thermalization time scales and role of HC-phonon
interaction that can affect performance of HC based solar energy conversion
[21, 22] and lighting applications (e.g., photo-detection, HC based lasers)
[23, 24].

Various key applications of semiconductors, such as solar-cells [3, 5, 6, 25,
26, 27, 28] and lighting applications [29, 30] involve instead continuous-wave
(CW) illumination, under which the system is necessarily in a non-equilibrium
steady-state (NESS). The NESS consists of a carrier distribution that deviates
from the thermal (i.e., Fermi-Dirac) distribution, and may also involve
carrier temperatures differing from each other and from the lattice
temperature.

Despite the extensive work on the topic, most studies involved only a
macroscopic description of the problem, while the fewer existing state-of-the-
art modelling of the NESS have several limitations. First, the tenet that the
photo-excited HCs ultimately achieve the lattice temperature in the steady-
state [31] has been challenged in a recent experiment on group III-V
semiconductors; it unveils that in the NESS carrier temperatures can be much
higher than lattice temperature [32, 33, 34, 35]. A theoretical understanding
of the existence of such a high carrier-temperature in NESS is still lacking
[32, 33, 34, 35], especially in light of the different techniques available
for measuring the various temperatures. Second, the existing theoretical
studies of the steady-state HC dynamics have considered the HC relaxation and
interband recombination within the relaxation time approximation (RTA) where
the scattering rate is assumed to be independent of the NESS distributions
[36, 37, 38, 39]. Although the RTA makes the calculation of NESS distribution
simpler, it neither conserves particle number nor energy within each band.
Moreover, a constant (distribution-independent) relaxation rate may
overestimate/underestimate the scattering rates, such that consequently, the
results become inaccurate. In addition, previous studies neglected the heating
of the lattice [40, 41, 42], thus, they cannot treat correctly the amount of
energy dissipated to the environment from the lattice, overall making it
impossible to assess the difference between the carrier, phonon and
environment temperatures. Last but not least, theoretical modelling of
semiconductor photoluminescence (PL) and cooling of SCs has been based on
carrier-only (macroscopic) rate equations of the total number density (rather
than the Fermi golden rule expression which relies on the energy distributions
of electrons and holes) [43, 44, 45, 46, 30, 47, 48], thus, neglecting the
role of carrier-phonon interactions and system-environment coupling [49, 50,
51, 52]; this, again, leads to an inaccurate assessment of the lattice
temperatures and even led to some debate in temperature measurements in
optical cooling experiments [53]. The above list of approximations indicate
the lack of a comprehensive theoretical approach to understand the NESS
properties of semiconductors under CW illumination that incorporates full non-
equilibrium distributions in determining carrier excitation, recombination,
carrier-phonon and carrier-carrier scatterings.

Motivated by the semiconductor applications perspective, the lack of a
complete theory incorporating the non-equilibrium nature of the photo-excited
carriers, and the advance in the theoretical formulation in metallic systems
[54, 55, 56, 57], we put forward a theoretical formalism for the non-
equilibrium dynamics of HCs under CW illumination. Our primary aim is to
describe the steady-state properties. In particular, we present a set of
coupled Boltzmann-heat equations for the non-equilibrium photo-generated
carrier distributions, whereby photo-excitation, carrier-carrier and carrier-
phonon scattering rates, and recombination of electrons and holes are
evaluated self-consistently within the Fermi golden rule, thus introducing
(semi-) quantum behavior into the Boltzmann equation (BE). The macroscopic
properties (e.g., the power flows and the phonon temperature) are obtained by
integrating over the distributions and employing the conservation of particle
number and energy for each subsystem [40, 41, 42].

Incorporation of the full non-equilibrium distributions in a self-consistent
way along with the energy and number conservation distinguishes our work from
existing formalisms that treat the non-equilibrium in semiconductors only
within RTA [40, 41, 42, 36, 37, 38, 39]. Our central results can be summarized
as follows:

  * •

Counter-intuitively, we find the NESS carrier distribution is more out-of-
equilibrium (i.e., less resembles an equilibrium distribution), indicating
higher density of non-thermal carriers at low illumination intensities than at
high intensities [58] due to inefficient thermalization at low intensities. An
equivalent interpretation commonly adopted in semiconductor textbooks [59, 60]
is to interpret the distribution as being thermal with yet a higher effective
chemical potential.

  * •

Consequently, the temperatures of the carriers become ill-defined at low
illumination, and depend on the way the temperatures are measured. We show
this by extracting carrier temperatures using two generic experimental
procedures frequently used, namely (i) extracting a temperature from the heat
transfer between carriers and phonons (measured via, e.g., a floating thermal
probe or a thermocouple [61, 62]), and (ii) by fitting the photoluminescence
(PL) spectra [63, 64, 65]. These two temperatures are shown to be very
different from one another.

  * •

Finally, our formalism allows us to evaluate the energy partition for an
illuminated semiconductor, i.e., how much of the power pumped into the system
goes to heating-up the lattice, and how much remains in the “hot” electron
sub-system and subsequently (partially) dissipates through radiative
recombination. We show that this partition depends on the energy of the single
photon but not on the total number of photons, and explain experimental
observations of the PL efficiency in CW-illuminated Galium Arsenide (GaAs).

The paper is organized as follows. in Sec. II we describe our microscopic
formulation and clearly state the simplifying assumptions as a first step,
e.g., we consider parabolic band structure. We also describe why the
assumption made are well suited in the context of many III-V semiconductors,
in particular, for GaAs. Our formulation of photo-excitation and recombination
are distinctive due to the fact, unlike the existing formulations, e.g., Refs.
[40, 41, 42, 36, 37, 38, 39], ours incorporate the full non-equilibrium
distribution explicitly. Our formulation of carrier-phonon interaction,
although incorporating acoustic phonons only, can easily be extended to other
phonon modes such as optical phonons or even piezoelectric phonons [66]. We
emphasize that a comparative study of the role of different phonon modes in
the steady-state properties is beyond the scope of the present paper. We
incorporate carrier-carrier interaction within hard-sphere approximation using
Fermi golden rule which is a good approximation in most of the undoped III-V
semiconductors due to low carrier densities. In Sec. III we describe the
macroscopic quantities that are essential for defining a steady-state. In Sec.
IV we describe our results, and conclude and indicate possible improvements to
our formalism in Sec V. In Appendices we describe the necessary mathematical
details of our formulation.

Figure 1: Schematic picture of the band structure of a representative
semiconductor (GaAs) and the different processes participating in determining
the non-equilibrium steady-state distributions. $\mathcal{E}_{c}$ and
$\mathcal{E}_{v}$ are the energies of the conduction and valence band edges,
respectively, and the chemical potential is placed at the middle of the band-
gap. Orange and red arrows denote photo-excitation and recombination (that
leads to PL), respectively. e-e (h-h) solid black (white) arrow denotes
carrier-carrier scattering and e-ph (h-ph) black (white) arrow denotes the
carrier-phonon scattering.

##  II Microscopic Formulation

To calculate the NESS carrier distributions $f_{e}(\mathcal{E})$ (for
electrons) and $f_{h}(\mathcal{E})$ (for holes), while keeping the phonon
subsystem in its own thermal equilibrium [54], we use the semi-quantum
Boltzmann equations (BEs),

$\displaystyle\frac{\partial f_{c}(\mathcal{E};T_{ph})}{\partial t}$
$\displaystyle=$ $\displaystyle\left(\frac{\partial f_{c}}{\partial
t}\right)_{exc}+\left(\frac{\partial f_{c}}{\partial t}\right)_{rec}$ (1)
$\displaystyle+$ $\displaystyle\left(\frac{\partial f_{c}}{\partial
t}\right)_{c-ph}+\left(\frac{\partial f_{c}}{\partial t}\right)_{c-c},$

where $f_{c}$ be the electron distribution for $c=e$ in the conduction band,
and hole distribution for $c=h$ in the valence band at an energy
$\mathcal{E}$; $T_{ph}$ is the phonon temperature.

The right-hand-side of the BEs includes four central processes. The first is
the photo-excitation process, which occurs upon absorption of a photon of
energy $\hbar\omega$. The next process is the recombination of excited
carriers by which photo-excited electrons and holes lose energy by emitting
photons, a process known as PL. Third, the carrier-phonon ($c-ph$) interaction
is responsible for transferring energy from the electrons and holes to the
lattice. Last but not least, carrier-carrier scattering is responsible for
thermalization of the electrons and holes in their respective bands. Going
forward, we explain and formulate these four processes in Sections II.1-II.4.
We neglect re-absorption of spontaneously emitted photons, Auger recombination
and carrier-carrier Umklapp scattering, the reasons being explained in
Sections II.2, and II.4, respectively.

###  II.1 Photo-excitation

The first term in Eq. (1), $\left(\frac{\partial f_{c}}{\partial
t}\right)_{exc}$, denotes the change in the population of carriers in the
conduction and valence bands due to absorption of a photon of energy
$\hbar\omega$, see Fig. 1. In semiconductors, following the absorption of a
photon with energy $\hbar\omega>\mathcal{E}_{g}$, the band-gap, an electron-
hole pair is created either across the band-gap or inside a specific band. We
adapt the formalism introduced in 67 to incorporates the band-gap
corresponding to semiconductors. The formalism of 67 is simple enough, and
correctly incorporate the quantum-like term ($\sim|E|^{2}$, incoherent) in the
photo-excitation term of the BE, as has been argued in a previous study in
connection to the metal nanoparticles [54]. It further bypasses the need for
$a~{}priori$ exact determination of the electron-photon interaction matrix
element by using physical conditions, such as the normalization of the total
probability of all the electronic processes and the particle number
conservation, with an input of the absorption lineshape [67]. The change in
carrier distribution at energy $\mathcal{E}$, due to photon absorption is
given by

$\left(\frac{\partial f_{c}(\mathcal{E})}{\partial
t}\right)_{exc}=\frac{N_{exc}\phi_{exc}^{(c)}(\mathcal{E})}{\rho_{c}(\mathcal{E})},$
(2)

where $\phi_{exc}^{(c)}(\mathcal{E})$ is the net change in the carrier
population at $\mathcal{E}$ in the conduction band (for $c=e$) and valence
band (for $c=v$). See Appendices A.1 and A.2 for detailed derivation of the
explicit formulae. The rate of number of electrons excited due to the
absorption of photon of energy $\hbar\omega$ is then given by
$\frac{dn_{c}}{dt}=\int\rho_{c}(\mathcal{E})\left(\frac{\partial
f_{e}(\mathcal{E})}{\partial t}\right)_{exc}$, which is equal to $N_{exc}$
electrons per unit time per unit volume. A positive value of $N_{exc}$
signifies an increase in the particle number, i.e., an increase in the number
of electrons in the conduction band and holes in the valence band. When the
semi-conductor is continuously illuminated by light, an ever increasing
particle number indicates that the system shall never reach a steady-state
until all the electrons (holes) are moved to the conduction (valence) band,
thereby losing the semiconducting property. A balancing process that comes to
rescue this catastrophe is the spontaneous recombination which we formulate in
Sec. II.2.

In our formalism of the photo-excitation we do not explicitly incorporate
momentum conservation, because in the presence of interactions of electrons
and holes with phonons, defects and impurities present in the system, as well
as for systems with finite size, the momentum is no longer a good quantum
number. In this regard, for direct band-gap SCs, our formulation of the photo-
excitation process provides an upper bound for the number of excitation events
(i.e., the maximum number of photo-excited carriers).

For indirect band-gap SCs, the interband transition due to photon absorption
may require intraband carrier-phonon interaction to provide the necessary
momentum transfer. This is because the conduction band minimum and the valence
band maximum do not appear at the same $\mathbf{k}$-point in the Brillouin
zone while an optical transition is associated with near-zero momentum
transfer. Moreover, any inter-valley (intraband optical) transition (for
example, from the $\Gamma$-valley to $L(\text{or~{}}X)$-valley) requires a
direct intraband optical transition accompanied by a carrier-phonon and/or a
carrier-carrier Umklapp scattering. Therefore, due to the absence of explicit
momentum conservation our formulation of photo-excitation process can
incorporate both indirect interband transitions and inter-valley (intraband)
optical transitions only in the sense that it provides the upper bound for the
number of excitation events. Nevertheless, for the intensities considered in
this study, we explicitly show in Appendix A.1 (equations (20) and (21)) that
intraband optical transitions are negligible compared to their interband
counterpart. We show this to be true irrespective of whether the transition is
direct or indirect. However, for even higher intensities and
$\hbar\omega\sim\mathcal{E}_{g}$ intra-band optical transitions should be
considered explicitly. Moreover, how much the explicit incorporation of inter-
valley scattering would change the NESS quantitatively at the steady state in
our CW illumination case is beyond the scope of present study.

###  II.2 Recombination

The second term in Eq. (1), $\left(\frac{\partial f_{c}}{\partial
t}\right)_{rec}$, denotes the interband recombination of excited carriers, by
which excited electrons and holes lose energy by spontaneously emitting
photons leading to PL, see Fig. 1. We adopt the standard formulation, see
Appendices B.1 and B.2, for PL, [68, 69, 36] which is also consistent with the
formulation of the photo-excitation (Section II.1) considered above. The rate
of change of electron population due to the recombination of carriers with
energy $\mathcal{E}$ is given by

$\left(\frac{\partial f_{c}}{\partial
t}\right)_{rec}=-\frac{N_{rec}\phi_{rec}^{(c)}(\mathcal{E})}{\rho_{c}(\mathcal{E})},$
(3)

and the rate of change of particles due to recombination is given by
$\frac{dn_{rec}}{dt}=\int
d\mathcal{E}\rho_{c}(\mathcal{E})\left(\frac{\partial f_{e}}{\partial
t}\right)_{rec}$ $=-N_{rec}$, a negative sign signifying a loss of particles.
Here, $\phi_{rec}^{(c)}(\mathcal{E})$ is the net change in the carrier
population at energy $\mathcal{E}$ due to the recombination. See Appendix B
for details of the formulation. At the steady-state, the rate of photo-
excitation of electrons would become the same as that of recombination of
electrons, and this fixes $N_{rec}=\frac{dn_{e}}{dt}$ so that
$\frac{dn_{e}}{dt}+\frac{dn_{rec}}{dt}=0$. It is worthwhile to point out the
recombination rate, therefore, indirectly depends on the photo-excitation rate
via $\frac{dn_{e}}{dt}$.

To this end, we relax the requirement of the momentum conservation for the
same set of reasons corresponding to the photo-excitation process. Therefore,
our formulation of recombination provides only an upper limit of the number of
recombination events for indirect band-gap SCs.

Furthermore, we assume absorption of spontaneously emitted photons is a weak
effect. Such an assumption is valid for systems with size smaller than the
optical skin depth, such as nanoparticles and thin films. Most semiconductor
applications, such as solar cells, lighting applications [36, 3, 5, 6, 29, 30]
etc. meet this assumption. At this system size, the effective local electric
field corresponding to the illumination is homogeneous throughout the system,
and almost all the spontaneously emitted photons from inside the bulk of the
system are radiated out of the system with a negligible fraction getting
reabsorbed thus, justifying our assumption.

###  II.3 Carrier-phonon interaction

The electron-phonon ($e-ph$) and hole-phonon ($h-ph$) interactions (third term
in Eq. (1)) are responsible for transferring energy from the electrons and
holes to the lattice. These are therefore, responsible for the heating of the
lattice and cooling of the carriers and shown schematically shown in Fig. 1.
They occur within an energy window comparable to the Debye energy near the
band edge (thus, it is typically narrow with respect to the photon energy).
For simplicity, we consider the deformation potential approximation for the
interaction between the carriers (both electrons and holes) and the phonons
[18], where only the longitudinal acoustic phonons are taken into account;
nevertheless, our formulation can be extended to incorporate optical phonons,
both polar and longitudinal [66]. In general, a more accurate quantitative
estimation of the energy transfer from carriers to phonon in semiconductors
would require explicit inclusion of other phonon modes, such as, transverse
acoustic, polar optical phonons [66] (arguably, known to be the dominant one
in III-V semiconductors [19, 13]), inter-valley carrier-phonon scatterings
[66]. However, such a detailed analysis is not the primary motivation of the
present study. Our calculations can be considered as an order of magnitude
estimation of the carrier-phonon energy transfer and in Sec. IV.3 we show that
our formalism agrees well with the experimental predictions in the appropriate
regime.

The energy of an electron in the conduction band is measured from the
conduction band edge, and
$\mathcal{E}=\mathcal{E}_{c}+\frac{\hbar^{2}k^{2}}{2m_{e}^{*}}$ where
$m_{e}^{*}$ is the conduction band effective mass of the electron and
$\mathcal{E}_{c}$ is the energy of the conduction band edge. Following Refs.
[66, 18], the Fermi golden rule expression for the scattering between the
conduction band electrons and phonons is obtained to be

$\displaystyle\left(\frac{\partial f_{e}(\mathcal{E})}{\partial
t}\right)_{e-ph}$ $\displaystyle=$
$\displaystyle\frac{\Xi_{c}^{2}}{4\pi\rho_{c}}\frac{\sqrt{m_{e}^{*}}}{\sqrt{2(\mathcal{E}-\mathcal{E}_{c})}}\int_{0}^{\hbar\omega_{D}}\frac{d\mathcal{E}_{q}~{}\mathcal{E}_{q}^{2}}{(\hbar
v_{ph})^{4}}\Bigg{[}(n_{B}(\mathcal{E}_{q},T_{ph})+1)\Big{(}f_{e}(\mathcal{E}+\mathcal{E}_{q})[1-f_{e}(\mathcal{E})]$
(4) $\displaystyle-$ $\displaystyle
f_{e}(\mathcal{E})[1-f_{e}(\mathcal{E}-\mathcal{E}_{q})]\Big{)}+n_{B}(\mathcal{E}_{q},T_{ph})\Big{(}f_{e}(\mathcal{E}-\mathcal{E}_{q})[1-f_{e}(\mathcal{E})]$
$\displaystyle-$ $\displaystyle
f_{e}(\mathcal{E})[1-f_{e}(\mathcal{E}+\mathcal{E}_{q})]\Big{)}\Bigg{]},$

where we have assumed that a phonons are in thermal equilibrium characterized
by a Bose-Einstein distribution $n_{B}(\mathcal{E}_{q},T_{ph})$ at the lattice
temperature $T_{ph}$, with
$n_{B}(\mathcal{E}_{q},T_{ph})=(e^{\mathcal{E}_{q}/k_{B}T_{ph}}-1)^{-1}$,
$k_{B}$ being the Boltzmann constant expressed in eV and $\hbar\omega_{D}$ is
the ‘Debye’ energy. Within the deformation potential approximation the
acoustic phonons exhibit a linear dispersion, viz.,
$\omega_{\mathbf{q}}=v_{ph}|\mathbf{q}|$, $v_{ph}$ being the speed of sound
(see Table 1 for the numerical values used) [70]. In (4), $\Xi_{c}$ is the
deformation potential constant, $\rho_{c}$ is the material density, $V$ is the
volume of the system [70].

Analogously, the energy of a hole in the valence band is measured from the
valence band edge, and
$\mathcal{E}=\mathcal{E}_{v}-\frac{\hbar^{2}k^{2}}{2m_{h}^{*}}$, where
$m_{h}^{*}$ is the valence band effective mass of the hole and
$\mathcal{E}_{v}$ is the energy of the valence band edge. Therefore, energy of
a hole with momentum $\mathbf{k}$ is
$\mathcal{E}_{h}=\mathcal{E}_{v}-\mathcal{E}$, i.e., the lower the value of
$\mathcal{E}$ the more energy the hole exhibits. A Fermi golden rule
expression corresponding to the hole-phonon interaction, analogous to the
electron-phonon one, is given by,

$\displaystyle\left(\frac{\partial f_{h}(\mathcal{E})}{\partial
t}\right)_{h-ph}$ $\displaystyle=$
$\displaystyle\frac{\Xi_{v}^{2}}{4\pi\rho_{m}}\frac{\sqrt{m_{h}^{*}}}{\sqrt{2(\mathcal{E}_{v}-\mathcal{E})}}\int_{0}^{\hbar\omega_{D}}\frac{d\mathcal{E}_{q}~{}\mathcal{E}_{q}^{2}}{(\hbar
v_{ph})^{4}}\Bigg{[}(n_{B}(\mathcal{E}_{q},T_{ph})+1)\Big{(}f_{h}(\mathcal{E}-\mathcal{E}_{q})[1-f_{h}(\mathcal{E})]$
(5) $\displaystyle-$ $\displaystyle
f_{h}(\mathcal{E})[1-f_{h}(\mathcal{E}+\mathcal{E}_{q})]\Big{)}+n_{B}(\mathcal{E}_{q},T_{ph})\Big{(}f_{h}(\mathcal{E}+\mathcal{E}_{q})[1-f_{h}(\mathcal{E})]$
$\displaystyle-$ $\displaystyle
f_{h}(\mathcal{E})[1-f_{e}(\mathcal{E}-\mathcal{E}_{q})]\Big{)}\Bigg{]},$

where $\Xi_{v}$ is the deformation potential for the $h-ph$ interaction
corresponding to the holes in the valence band (see Table 1).

###  II.4 Carrier-carrier interaction

The electron-electron interaction in the conduction band and hole-hole
interaction in the valence band (fourth term in Eq. (1), altogether carrier-
carrier scattering) are responsible for thermalization of the electrons and
holes in their respective bands. This enables carriers to reach a quasi-
equilibrium where a Fermi-Dirac distribution with a carrier temperature
[$T_{e(h)}$ for electrons (holes)] can be associated to the carriers.

Following the standard notations from the 18, we obtain the net scattering
rate corresponding to the electrons with energy within $\mathcal{E}$ and
$\mathcal{E}+d\mathcal{E}$ is given by

$\displaystyle\left(\frac{\partial f_{e}(\mathcal{E})}{\partial
t}\right)_{e-e}=\frac{4\pi}{\hbar}\frac{V^{2}}{16\pi^{4}}\int
d\mathcal{E}_{1}d\mathcal{E}_{2}d\mathcal{E}_{3}\left[\int_{k_{low}}^{k_{high}}d\kappa|M(\kappa)|^{2}\right]\delta(\mathcal{E}_{1}-\mathcal{E}_{2}+\mathcal{E}-\mathcal{E}_{3})\times$
$\displaystyle\frac{\hbar}{\sqrt{(\mathcal{E}-\mathcal{E}_{c})}}\left([1-f_{e}(\mathcal{E})][1-f_{e}(\mathcal{E}_{1})]f_{e}(\mathcal{E}_{2})f_{e}(\mathcal{E}_{3})-f_{e}(\mathcal{E})f_{e}(\mathcal{E}_{1})[1-f_{e}(\mathcal{E}_{2})][1-f_{e}(\mathcal{E}_{3})]\right),$
(6)

where $V$ is the volume of the system under consideration and $M(\kappa)$ is
the matrix element of the electron-electron interaction Hamiltonian within the
first-order perturbation theory; it depends on the momentum exchange,
$\kappa=\mathbf{k}-\mathbf{k}_{2}=\mathbf{k}_{1}-\mathbf{k}_{3}$. The limits
of the integration over the matrix element are
$k_{low}=\text{max}(|k-k_{2}|,|k_{1}-k_{3}|)$ and
$k_{high}=\text{min}(k+k_{2},k_{1}+k_{3})$ where the electron wave-number
$k_{j}$ corresponds to $\mathcal{E}_{j}$.

We consider the short-ranged, hard sphere interaction between the electrons
for which
$\int_{k_{low}}^{k_{high}}d\kappa|M(\kappa)|^{2}=\frac{\sigma_{t}(m_{e}^{*})^{2}}{4\pi^{2}\hbar^{3}V^{2}}(k_{\text{high}}-k_{\text{low}})$,
where $\sigma_{t}=6\times 10^{-16}~{}m^{2}$ is the total scattering cross-
section corresponding to electrons in GaAs [18]. Notice the fact that the
matrix element $M^{2}(\kappa)$ contains a $V^{-2}$ factor which cancels the
$V^{2}$ appearing in Eq. (6), thereby making the expression for electron-
electron scattering volume normalized. We point out in passing that our choice
of the hard-sphere scattering is a trade-off between the computation time and
a more realistic model of screened Coulomb interaction, the Debye screening
[59, 66]. In 18, 71, 72 it has been pointed out that at low densities (
$n_{e}<10^{22}m^{-3}$) the “Debye” screening formula [59, 66] breaks down and
electrons act as unscreened particles. Therefore, following 18 we take hard-
sphere interaction with the total scattering cross-section $\sigma_{t}$
representing the value of the scattering cross-section of unscreened
electrons.

An expression for the hole-hole interaction in the valence band, analogous to
the electron-electron interaction term, is given by

$\displaystyle\left(\frac{\partial f_{h}(\mathcal{E})}{\partial
t}\right)_{h-h}=\frac{4\pi}{\hbar}\frac{V^{2}}{16\pi^{4}}\int
d\mathcal{E}_{1}d\mathcal{E}_{2}d\mathcal{E}_{3}\left[\int_{k_{low}}^{k_{high}}d\kappa|M(\kappa)|^{2}\right]\delta(\mathcal{E}_{1}-\mathcal{E}_{2}+\mathcal{E}-\mathcal{E}_{3})\times$
$\displaystyle\frac{\hbar}{\sqrt{(\mathcal{E}_{v}-\mathcal{E})}}\Big{(}[1-f_{h}(\mathcal{E})][1-f_{h}(\mathcal{E}_{1})]f_{h}(\mathcal{E}_{2})f_{h}(\mathcal{E}_{3})-f_{h}(\mathcal{E})f_{h}(\mathcal{E}_{1})[1-f_{h}(\mathcal{E}_{2})][1-f_{h}(\mathcal{E}_{3})]\Big{)}.$
(7)

The hard sphere interaction between the holes is given by
$\int_{k_{low}}^{k_{high}}d\kappa|M(\kappa)|^{2}=\frac{\sigma_{t}(m_{h}^{*})^{2}}{4\pi^{2}\hbar^{3}V^{2}}(k_{high}-k_{low})$,
where $\sigma_{t}=6\times 10^{-16}~{}m^{2}$, being equal to that of the
electrons, is the total scattering cross-section corresponding to holes in
GaAs.

The quantum nature of the carrier-carrier scattering comes from the Pauli
exclusion principle incorporated into the distribution-dependent term. The
carrier-carrier scattering rate depends on the density of (excited) carriers,
and therefore, on the intensity of the incident light and the carrier
temperature [71, 73]. For carrier densities $n_{e}<10^{24}~{}m^{-3}$, we
replace the the screened Coulomb interaction by the hard sphere interaction
involving a total carrier-carrier scattering cross-section that determines the
strength of carrier-carrier scattering [71, 72, 74]. In this carrier density
regime screening energy,
$\mathcal{E}_{TF}=\frac{\hbar^{2}\kappa_{TF}^{2}}{2m_{e}^{*}}$ with
$\kappa_{TF}$ being the Thomas-Fermi wave-vector [59], always remain much
smaller than the average energy of the electrons which validates the use of
hard-sphere interaction (this applies to the holes too). However, in this
situation carrier-carrier exchange interaction also become important [75, 76]
but within hard-sphere interaction it can be incorporated within an adjustable
parameter in the total carrier-carrier scattering cross-section [77]. In most
of the undoped III-V binary SCs with wider band-gaps [with exceptions being
InSb ($\mathcal{E}_{g}=0.235~{}eV$), InAs ($\mathcal{E}_{g}=0.417~{}eV$) at
room temperature] carrier density does not reach such high values for the
illumination intensities considered in our study [78, 71, 72]. Therefore, our
formulation of carrier-carrier interaction applies to a large class of III-V
binary SCs. For Ternary and Quaternary alloys energy band-gaps and room
temperature carrier density strongly depend on the alloy composition [79], and
therefore, these are needed to be studied case by case. Importantly, we
discard Auger (non-radiative) recombination here, as it is negligible for the
intensities we consider [80, 45, 81], and become appreciable only under
extremely high carrier densities with spatial confinement (such as small nano-
structures with system size even much less than that we already assumed in
connection to absorption of the spontaneously emitted photons in Section II.2)
where spatial confinement can lead to a large overlap between electron and
hole wave functions [82]. Moreover, we consider only pristine SCs, such that
carrier-impurity scattering is neglected [60].

##  III Macroscopic formulation - Energy and number conservation

Crucially, in order to account correctly for the non-equilibrium properties,
energy conservation must be considered for each sub-system (electrons, holes
and phonons) separately [54]. The rate of change of energy of the electronic
subsystem (and analogously for holes) is obtained by integrating over the
distribution functions (1) by $\int
d\mathcal{E}(\mathcal{E}-\mathcal{E}_{C})\left(\frac{\partial f_{e}}{\partial
t}\right)\rho_{e}(\mathcal{E})$, $\mathcal{E}_{C}$ being the energy of the
conduction band edge $\Big{(}\int
d\mathcal{E}(\mathcal{E}_{V}-\mathcal{E})\left(\frac{\partial f_{h}}{\partial
t}\right)\rho_{h}(\mathcal{E})$, $\mathcal{E}_{V}$ being the energy of the
valence band edge$\Big{)}$ [see Fig. 1] , resulting in

$\frac{d\mathcal{U}_{c}}{dt}=W_{c-exc}-W_{c-rec}-W_{c-ph},$ (8)

where $\rho_{c}(\mathcal{E})$ is the carrier density of states (cDOS). We
interpret equation (8) as the balance between the rate of gain of energy of
the electronic (hole) subsystem due to the photo-excitation, viz.,
$W_{c-exc}$, the rate of loss of excess energy the electrons (holes) due to
they recombination back to the conduction (valence) band, viz., $W_{c-rec}$,
and the rate of energy flow to the lattice, $W_{c-ph}$. The elastic carrier-
carrier scattering processes do not cause any change in the energy of the
carrier subsystems, respectively. At the steady-state
$\frac{d\mathcal{U}_{c}}{dt}=0$ signifies the conservation of energy. See
Appendix D for the detailed expressions.

Similarly, the total energy of the lattice, $\mathcal{U}_{ph}$, is balanced by
the heat flowing in from the electron and hole subsystems and flowing out to
the environment, viz.,

$\frac{d\mathcal{U}_{ph}}{dt}=(W_{e-ph}+W_{h-ph})-G_{ph-
env}(T_{ph}-T_{\text{amb}}),$ (9)

where $G_{ph-env}$ is the coupling between the lattice and the environment,
and is phenomenologically introduced in the formulation.See Appendix D for the
detailed expressions. Moreover, value of $G_{ph-env}$ may strongly depend on
the geometry of the sample and on the thermal conductivity of the host [54,
83].

We search numerically for a NESS based on the energy conservation, i.e.,
$\frac{d\mathcal{U}_{e}}{dt}=\frac{d\mathcal{U}_{h}}{dt}=\frac{d\mathcal{U}_{ph}}{dt}=0$,
when the semiconductor is maintained under CW illumination. When a steady-
state is obtained, equations (1)-(9) provide us with the electron and hole
distributions corresponding to the NESS, and the temperature of the lattice,
$T_{ph}$. We then extract the steady-state electron and hole temperatures,
$T_{e}$ and $T_{h}$, respectively, by adapting two generic experimental
procedures frequently used, (i) from the exchange of power between the
electron (hole) sub-systems and the phonons [61, 62], and (ii) from the
steady-state PL spectra [63, 64, 65].

The physical picture of the power transfers corresponding to the energy
conservation is the following. The incident radiation excites electrons in the
conduction band and holes in the valence band, thereby generating high energy
(non-thermal) carriers in the respective bands. To reach a steady-state,
electrons in the conduction band then lose their energy chiefly by (i)
emitting phonons (via electron-phonon ($e-ph$) interactions), and (ii) by
recombining with the holes in the valence band. Holes too follow the same
processes (but recombine with electrons in the conduction band). Under
continuous illumination, given $e(h)-ph$ interactions are intra-band
processes, these alone cannot force the system to reach a steady-state, so
that recombination turns out to be important as well. There are other
processes, such as intra-band Auger recombination, impact ionization etc.,
however, the rates of these are known to be much slower compared to the $e-ph$
interactions and the interband recombinations [8, 84]. The intraband carrier-
carrier scattering is even slower a process in the case of low density of
electrons (and holes) in the conduction (and valence) band. For GaAs carrier-
carrier interaction plays an active role only at carrier densities
$n_{e}=n_{h}=10^{24}~{}m^{-3}$ [8, 84]. A steady-state in the electron and
hole gases is reached when $\frac{d\mathcal{U}_{e}}{dt}=0$ and
$\frac{d\mathcal{U}_{h}}{dt}=0$, respectively. Energy transferred to the
lattice then dissipates to the environment maintained at the ambient
temperature $\approx 297$ K in our case, and the lattice phonons reach a
steady-state when $\frac{d\mathcal{U}_{ph}}{dt}=0$.

##  IV Results

We take intrinsic (undoped) Gallium Arsenide (GaAs) as a specific material to
understand and determine the non-equilibrium carrier distributions under CW
illumination with varying intensities. However, our formulation can well be
applied to pulsed-illumination with a suitable modification in the photo-
excitation term $\left(\frac{\partial f_{c}}{\partial t}\right)_{exc}$. The
valence band edge, conduction band edge and the chemical potential are at
$\mathcal{E}_{V}=3.49$ eV, $\mathcal{E}_{C}=4.91$ eV and $\mu=4.2$ eV,
respectively. The illumination photons have energy of $\hbar\omega=1.65$ eV
and the GaAs permittivity is set to
$\epsilon^{\prime\prime}(\omega)=3.3\epsilon_{0}$ at $\hbar\omega$ [85], where
$\epsilon_{0}$ is the free-space permittivity. The intrinsic carrier
concentration in the GaAs at room temperature is
$n_{e}^{\text{amb}}=1.317\times 10^{12}~{}/m^{3}$, which is considered to be
quite low compared to many other semiconductors [86, 87]. Other parameters
used in this study are given in Table 1.

parameter | parameter’s symbol | value
---|---|---
effective mass of electron | $m_{e}^{*}/m_{e}$ | 0.063 [88]
effective mass of hole | $m_{h}^{*}/m_{e}$ | 0.49 [88]
band-gap | $\mathcal{E}_{g}$ | 1.42 eV [88]
chemical potential | $\mu$ | 4.2 eV
bottom of valence band | $\mathcal{E}_{b}$ | $-\mu/10$
top of conduction band | $\mathcal{E}_{t}$ | $2\mu+\mu/10$
imaginary part of the dielectric function | $\epsilon^{\prime\prime}(\omega)$ | 3.3$\epsilon_{0}$ [85]
Debye frequency | $\omega_{D}$ | 0.0243 eV
deformation potential in conduction band | $\Xi_{c}$ | 7.04 [79]
deformation potential in valence band | $\Xi_{v}$ | 3.6 [89]
sound velocity | $v_{ph}$ | 3650 m/s
ambient temperature | $T_{amb}$ | 297 K
mass density | $\rho_{m}$ | 5320 $\text{kg}/\text{m}^{3}$
free space permittivity | $\epsilon_{0}$ | $8.8542\times 10^{-12}~{}F/m$
lattice-environment coupling | $G_{ph-env}$ | $5\times 10^{14}~{}W/m^{3}K$
Table 1: Values of parameters corresponding to intrinsic (undoped) GaAs. The
chemical potential $\mu$ is completely arbitrary and is used in fixing the
centre of the band-gap only.

We reiterate that we intend to calculate steady-state distributions and as a
first step we consider the parabolic band dispersion with the corresponding
carrier density of stats [cDOS] given by,

$\displaystyle\rho_{e}(\mathcal{E})$ $\displaystyle=$
$\displaystyle\frac{(2m_{e}^{*})^{3/2}}{2\pi^{2}\hbar^{3}}\sqrt{\mathcal{E}-\mathcal{E}_{c}},~{}~{}\text{for}~{}\mathcal{E}>\mathcal{E}_{c},$
$\displaystyle\rho_{h}(\mathcal{E})$ $\displaystyle=$
$\displaystyle\frac{(2m_{h}^{*})^{3/2}}{2\pi^{2}\hbar^{3}}\sqrt{\mathcal{E}_{v}-\mathcal{E}},~{}~{}\text{for}~{}\mathcal{E}<\mathcal{E}_{v},$
(10)

where $m_{e}^{*}$ and $m_{h}^{*}$ represent the effective band masses for
electrons and holes, respectively. In general realistic band structures of
almost all the SCs are far from being parabolic, however for suitable
illumination frequencies, for example for GaAs
$\hbar\omega\leq\mathcal{E}_{g}+0.3~{}eV=1.72~{}eV$ our results are exact.

However, we point out that our formulation is general enough to accommodate
any dispersion relation as long as the cDOS is a function of energy. It is
indeed the case with cDOS, be it experimentally obtained or calculated from
ab-initio density functional theory. Moreover, our formulation can well be
generalized to a more complicated band dispersion by incorporating a
$\mathbf{k}\cdot\mathbf{p}$ perturbation theory [66].

###  IV.1 Non-equilibrium distribution:

FIG. 2(a) shows the steady-state distributions of holes $f_{h}$ and electrons
$f_{e}$ as a function of the energy $\mathcal{E}$ in the valence and
conduction bands, respectively, for electric field levels ranging from
$|E|^{2}=4$(V/m)2 [$1.976\times 10^{-6}$ W/cm2] to $|E|^{2}=4\times
10^{9}$(V/m)2 [$1.976\times 10^{3}$ W/cm2]. For comparison, we plot the
thermal (Fermi-Dirac) distributions for electrons and holes $f_{e}^{T}$ and
$f_{h}^{T}$ at ambient temperature.

Figure 2: Hole and the electron distributions: (a) Plots of the hole
distribution $f_{h}(\mathcal{E})$ in the valence band, the electron
distributions $f_{e}(\mathcal{E})$ in the conduction band at different values
of the local field. The valence band edge at $\mathcal{E}_{V}$, the conduction
band edge at $\mathcal{E}_{C}$ and the chemical potential $\mu$ are shown by
green dashed vertical lines. The superscript $T$ denotes the Fermi-Dirac
distribution. (b) Logarithmic slope of the non-equilibrium distribution of
electrons $f_{e}(\mathcal{E})$ at electronic energy $\mathcal{E}=4.914$ eV (an
energy near the conduction band edge) as a function of local field $|E|^{2}$.

The population of the carriers increases with the illumination intensity while
the (logarithmic) slopes of the steady-state distribution are nearly the same
as that of the ambient thermal distribution at lower intensities. The common
textbook interpretation is that the carrier temperatures do not change at all,
and only the effective chemical potential for each of the carriers are shifted
to their respective band edges, viz., for electrons towards the conduction
band edges and for holes towards the valence band edges [60, 59]. However, in
the CW illuminated SC the distribution can only be thermalized via carrier-
carrier scattering. Due to the negligible carrier-carrier scattering the
distributions are indeed not thermalized. Therefore, one can interpret the
distributions in the following alternative way. Since for a thermalized (i.e.,
Fermi-Dirac) distribution the slope is inversely proportional to the carrier
temperature, an increase of the distribution without a change in slope is
indicative of a non-thermal distribution. It is only at higher intensities,
when $e-e$ and $h-h$ interactions become comparable to the $e(h)-ph$ coupling,
that the slopes of $f_{e(h)}$ start to deviate from
$f_{e(h)}^{T}(\mathcal{E},T_{amb})$. To demonstrate this, in FIG. 2(b) we plot
the slopes of $f_{e}(\mathcal{E})$ at $\mathcal{E}=4.914$ eV (near the
conduction band edge, where the strength of the $e-e$ interaction is the
maximum and therefore, can affect $f_{e}(\mathcal{E})$ the most), as a
function of the local field $|E|^{2}$. The slope of $f_{e}(\mathcal{E})$
remains unchanged, until the intensity reaches a critical value such that the
$e-e$ interaction becomes comparable to the $e-ph$ interaction. This will also
be reflected in the carrier temperature (see discussion below and Fig. 3(b)).
Thus, at low intensities due to inefficient carrier-carrier interaction, the
distributions are truly non-thermal, and only start to thermalize at higher
illumination intensities. This conclusion is independent of incident photon
energies, as demonstrated in Appendix C.

###  IV.2 Carrier temperatures

We now use the NESS distributions to extract macroscopic characteristics of
interest. First, we extract (“effective”) electron and hole temperatures from
the carrier-phonon power transfers. See Appendix E for the details of the
formalism. We denote $\Delta T_{e}$ and $\Delta T_{h}$ as the deviation of
electron and hole temperatures, respectively, from the ambient temperature.
These are plotted as a function of the local field in FIG. 3(a), showing a
nonlinear increase with the illumination intensity by several hundreds of
degrees. Such a nonlinear dependence stems from the fact that $W_{e(h)-ph}$
exhibits a nonlinear dependence on $T_{e(h)}-T_{ph}$. The difference between
the values of $\Delta T_{e}$ and $\Delta T_{h}$ is due to the different
effective band masses of electrons and holes, and the different deformation
potentials in the conduction and valence bands, see Table 1.

Figure 3: Temperatures and energy partitions: (a) The plot of $\Delta
T_{e}=T_{e}-T_{\text{amb}}$ and $\Delta T_{h}=T_{h}-T_{\text{amb}}$
corresponding to the rise in the electron and hole temperatures, respectively,
from the ambient temperature as a function of the local field $|E|^{2}$, in
the units of $(V/m)^{2}$, on a log-log scale obtained from $e-ph$ interaction.
(b) The plot of $\Delta T_{c}=T_{c}-T_{\text{amb}}$ carrier temperatures from
the ambient temperature as a function of $|E|^{2}$, in the units of
$(V/m)^{2}$, on a log-log scale obtained from fitting the PL spectra, with a
$95\%$ fitting accuracy. (c) The plot of $\Delta T_{ph}=T_{ph}-T_{\text{amb}}$
the rise in phonon temperature from the ambient temperature, indicating the
lattice heating, as a function of $|E|^{2}$ on a log-log scale. The values of
$\Delta T_{ph}$ indicates that the lattice does not heat up at all.

FIG. 3(b) shows the rise in the carrier temperature above ambient, denoted by
$\Delta T_{c}$, extracted from the PL (see Appendix F), as a function of the
local field. It is worth emphasizing that the rise in electron and hole
temperatures, obtained from the PL spectra is the same, because the
recombination process that gives rise to PL is symmetric with respect to
electrons and holes. $\Delta T_{c}$ exhibits a nonlinear increase with the
intensity, and differs considerably from $\Delta T_{e(h)}$ both qualitatively
and quantitatively throughout the entire range of the local fields. In
particular, $T_{c}$ obtained from the PL spectra deviate $5-15$ K from
$T_{amb}$ which is much smaller compared to the deviation of $T_{e(h)}$ from
$T_{amb}$ obtained from carrier-phonon power transfers. However, as explained
above, with increasing intensity, thermalization becomes more important,
especially above a critical intensity (corresponding to a carrier density
$n_{e}\approx 5.2\times 10^{19}~{}m^{-3}$ here). The value of $\Delta T_{c}$
starts increasing above this critical intensity indicating the role of
thermalization in a PL-based temperature measurement. This is also evident
from the similarity between FIG. 3(b) and FIG. 2(b).

In FIG. 3(c) we plot the increase in phonon (lattice) temperature from the
ambient, $\Delta T_{ph}$, as a function of the local field, showing a linear
dependence on the square of the local electric field. The total power
transferred to the lattice from both electron and hole sub-systems exhibits
the same linear dependence on the square of the local field (not shown). At
the steady-state we find that $\Delta T_{ph}=(W_{e-ph}+W_{h-ph})/G_{ph-env}$,
explaining the linear dependence of $\Delta T_{ph}$ on $|E|^{2}$, see Eq. (9).
The tiny increase in $T_{ph}$ with respect to the ambient temperature implies
that the lattice hardly heats up at all. However, since $\Delta T_{ph}\propto
G_{ph-env}^{-1}$, weaker lattice-environment coupling would lead to more
heating of the lattice and vice versa. Interestingly, in the case of metals,
electron-phonon coupling $G_{e-ph}$ was orders of magnitude higher due to the
larger number of electrons, and even exceeded $G_{ph-env}$ [54]. In that
sense, while the bottleneck of the energy flow in metals was the heat transfer
from the phonons to the environment, for semiconductors, the bottleneck is the
$e(h)-ph$ coupling.

###  IV.3 Energy partition and efficiency

Our formulation provides a unique prediction for the partition between the two
channels by which the electron/hole sub-system dissipates its energy, viz.,
interband recombination and energy transfer to the lattice via phonons. This
is crucial for the correct prediction of semiconductor heating and associated
photo-thermal nonlinearlity in SC nanostructures (for example, recently
studied in Silicon nanostructures; notably an indirect band-gap SC) [90, 91],
quantification of the strength of PL, applications relying on these quantities
such as thermometry and imaging, and most importantly, lighting applications
of semiconductors. [92, 93, 29].

Figure 4: Energy partitions: The ratios $\eta_{rec}^{(e)}$ (blue circles and
red stars) and $\eta_{ph}^{(e)}$ (yellow square and magenta cross) (see text
definitions) corresponding to the photon energies $\hbar\omega=1.65$ eV
(yellow and blue solid lines, respectively) and $2.05$ eV (magenta and red
dashed lines, respectively), respectively, as a function of $|E|^{2}$ on a
semi-log scale.

Thus, we define $\eta_{rec}^{(c)}=W_{c-rec}/W_{c-exc}$ as the ratio of the
power dissipated from the electron/hole sub-systems through recombination
$W_{c-rec}$ to the power absorbed by the electron/hole sub-systems,
$W_{c-exc}$; similarly,
$\eta_{ph}^{(c)}=W_{c-ph}/W_{c-exc}[\approx(1-\eta_{rec}^{(c)})]$ as the ratio
between the power transferred from the electron/hole sub-systems to the
phonons, $W_{c-ph}$, and the power absorbed by the electron/hole sub-systems.
FIG. 4 shows $\eta_{rec}^{(e)}$ and $\eta_{ph}^{(e)}$ for two different photon
energies 1.65 and 2.05 eV ($\eta_{rec}^{(h)}$ and $\eta_{ph}^{(h)}$ are
quantitatively the same as that of the electrons). We note that the energy
partition is inversely proportional to the photon energy, i.e.,
$\eta_{rec}^{(c)}\propto(\hbar\omega-\mathcal{E}_{g})^{-1}$, but does not
depend on intensity, see Fig. 4. To further clarify, we plot
$\eta_{rec}^{(e)}$ for four different values photon energy in Fig. 8 (a) and
(b) in Appendix G, which corroborate these conclusions. The reason is rather
straightforward; for higher photon energies, electrons cross the energy gap to
higher energies in the conduction band. They, thus, have more energy to lose
to phonons before reaching the band edge and recombining.

Most importantly, the energy partitions can also be used to evaluate the
efficiency of the PL process. The total power absorbed $W_{abs}$ by the system
is $\hbar\omega N_{exc}$, where $N_{exc}$ is the rate of photon absorption per
unit volume, and the total power lost due to PL is
$W_{PL}=W_{e-rec}+W_{h-rec}+\mathcal{E}_{g}N_{exc}$ (note that
$N_{exc}=N_{rec}$, the rate of recombination being equal to the rate of
excitation at the steady-state, as required by the particle number
conservation). From Eq. (D) and Eq. (D) we find
$W_{e-exc}=\frac{\hbar\omega-\mathcal{E}_{g}}{2}N_{exc}$ and
$W_{h-exc}=\frac{\hbar\omega-\mathcal{E}_{g}}{2}N_{exc}$; these are the powers
absorbed by the electron and hole subsystems measured from the conduction and
valence band edges, respectively. This signifies that out of the total power
absorbed by the semiconductor, viz. $\hbar\omega N_{exc}$, a power of
$(\hbar\omega-\mathcal{E}_{g})N_{exc}$ is absorbed together by electron and
hole subsystems and is equally distributed between them; it also means that a
power of $\mathcal{E}_{g}N_{exc}$ is lost in overcoming the band-gap.
$W_{c-rec}$ and $W_{h-rec}$ defined in Eq. (D) and Eq. (D), respectively,
measure the power lost by the electrons and holee, respectively. Therefore,
the total power lost due to PL is
$W_{PL}=W_{e-rec}+W_{h-rec}+\mathcal{E}_{g}N_{exc}$ which incorporates all
possible recombination events from the electrons and holes away from the band
edges. Then, the efficiency of PL is defined by
$\eta_{PL}^{QE}=\frac{W_{PL}}{\hbar\omega N_{exc}}$, which leads to Eq. (4).
In deriving the final form of Eq. (4) we define
$\eta_{rec}^{(c)}=\frac{W_{c-rec}}{W_{c-exc}}$ [$c=e$ for electrons and $c=h$
for holes] as the ratio of the power dissipated from the electron/hole sub-
systems through the recombination $W_{c-rec}$ to the power absorbed by the
electron/hole sub-systems, $W_{c-exc}$.

$\displaystyle\eta_{PL}^{QE}$ $\displaystyle=$
$\displaystyle\frac{W_{PL}}{W_{abs}}=\frac{\eta_{rec}^{(e)}+\eta_{rec}^{(h)}}{2}+\frac{\mathcal{E}_{g}}{\hbar\omega}\left[1-\frac{\eta_{rec}^{(e)}+\eta_{rec}^{(h)}}{2}\right].$

For the example we study here, we find the efficiency $\eta_{PL}^{QE}$ to be
90.6% and 73% at photon energies 1.65 and 2.05 eV, respectively. We compare
this to experimental observations [45], where the PL efficiency was measured
as a function of illumination intensity (under laser light of 633nm). In these
experiments, the low-intensity regime is dominated by trap-assisted
recombination (e.g., Shockley-Read-Hall recombination), an effect which we do
not account for. At high intensities interband recombination is dominant, and
the PL efficiency saturates at around 71% (obtained by interpolating the
experimental data), very close to our theoretical result, showing that the
saturation of efficiency comes due to the phonon-mediated dissipation of
carrier energy.

It is worth emphasizing again that although existing formulations for
evaluation of PL efficiency account for trap-assisted and Auger recombination
[43, 44, 45, 46, 30], these are macroscopic formulations based on (rate
equations of) carrier density instead of carrier distribution and ignore the
role of phonons. This prevents these formulations from being able to explain
the high intensity PL efficiency. Put simply, without considering carrier-
phonon interactions, PL efficiency should reach 100% at high intensities,
which is in contradiction to experimental observations. Our theory thus
provides a much improved theoretical prediction for PL efficiency. Moreover,
our theory may also serve as a basis for a more rigorous microscopic
formulation (in terms of carrier distribution $f_{c}$) of laser cooling of
semiconductors [52], and can allow the quantification of the possible role of
carrier-phonon scattering as a heating path-way that hinders cooling [53].

##  V Conclusions and discussion

In conclusion, we employed a semi-quantum BE formulation (including,
importantly, energy and particle number conservation), to study the full non-
equilibrium carrier distributions and temperatures in an SC under continuous
illumination, taking GaAs as a specific example. Our formalism can easily be
applied to all the direct band-gap semiconductors, and for indirect
semiconductors, it can provide an upper limit of the full non-equilibrium
carrier distributions and temperatures. Under low intensity illumination, we
find that thermalization processes are inefficient, and the system remains at
strong non-equilibrium. Somewhat surprisingly, for high intensities the SC
tends to thermalize more efficiently due to increased carrier-carrier
interaction at increased particle number densities. Although the lack of
thermalization of the “hot” (non-thermal) carriers leaves room for the use of
“hot” carrier- based SC electronics, how much the (more efficient)
thermalization (at even higher intensities than considered here) can limit the
use of “hot” carriers remains to be determined. Our theoretical formulation
serves as a first step towards such a quantitative estimation.

Our formulation also allowed us to evaluate the steady-state carrier
temperatures in two ways, corresponding to two different experimental
techniques, namely (i) through carrier-phonon energy transfer (measured via,
e.g., a floating thermal probe or a thermocouple [61, 62]), and (ii)
photoluminescence-based carrier temperature. We find that these two values
deviate substantially from each other, again indicating strong non-
equilibrium, and providing direct experimental predictions to test our theory.

Finally, our formulation provides a simple way to evaluate the the steady-
state PL efficiency, which is found to compare remarkably well with
experimental observations and goes beyond existing theoretical models [43, 44,
45, 46, 30]. Our model can be used as a theoretical platform to study
outstanding open questions in the field, for instance, the persistent long-
lived non-thermal carriers and the increase in carrier temperature recently
observed in GaAs and other III-V semiconductors under CW illumination [32, 33,
34, 35], and may enable the clarification of the controversial issues related
to optical cooling and temperature of SCs [53]. Our formalism can be extended
to other non-equilibrium situations, such as current-carrying junctions, doped
semiconductors, and be used to evaluate and optimize the PL efficiency of,
e.g., HC solar cells [3, 5, 6], optical cooling [52, 45, 94], and many other
light emitting devices [29, 30]. Our formalism can further be fine tuned by
incorporating the explicit band structure, inter-valley scatterings, different
phonon modes, Auger and impurity scatterings [66] for a more quantitative
estimation of carrier NESS, temperatures and PL efficiency, which we leave for
future studies.

## Acknowledgements:

IWU and YS were supported by Israel Science Foundation (ISF) grant (340/2020)
and by Lower Saxony - Israel cooperation grant no. 76251-99-7/20 (ZN 3637).
Authors acknowledge David W. Snoke for valuable suggestions and discussions.

##  Appendix A Carrier-photon interaction: photo-excitation

###  A.1 Photo-excitation of electrons

#### Interband transition:

Upon absorption of a photon of energy $\hbar\omega$, an electron-hole pair is
created across the band-gap, i.e., if the electron is created in the
conduction band at energy $\mathcal{E}$, then the corresponding hole is
created in the valence band at energy $\mathcal{E}-\hbar\omega$. The joint
probability of absorption of a photon of frequency between $\omega$ and
$\omega+d\omega$ and the excitation of an electron in the conduction band at
energy $\mathcal{E}$ is given by

$A_{exc}^{v\rightarrow
c}(\mathcal{E}_{f}=\mathcal{E},\hbar\omega)=K_{exc}^{v\rightarrow
c}D_{J}(\mathcal{E},\mathcal{E}-\hbar\omega)\rho_{J,exc}^{v\rightarrow
c}(\mathcal{E},\mathcal{E}-\hbar\omega),$ (12)

where $D_{J}(\mathcal{E}_{final},\mathcal{E}_{initial})$ is the square of the
matrix element corresponding to the electron-photon interaction leading to the
electronic process $\mathcal{E}_{final}\rightarrow\mathcal{E}_{initial}$,
$K_{exc}^{v\rightarrow c}$ is the proportionality constant to be defined
later, and the superscript ‘$v\rightarrow c$’ denotes the excitation (denoted
by the subscript ‘$exc$’) process from the valence band to the conduction
band. In Eq. (12), $\rho_{J,exc}^{v\rightarrow
c}(\mathcal{E},\mathcal{E}-\hbar\omega)$ is the joint density of states of the
excitation of an electron at $\mathcal{E}$ in the conduction band leaving a
hole at $\mathcal{E}-\hbar\omega$ in the valence band, and is given by

$\displaystyle\rho_{J,exc}^{v\rightarrow
c}(\mathcal{E},\mathcal{E}-\hbar\omega)$ $\displaystyle=$
$\displaystyle[1-f_{h}(\mathcal{E}-\hbar\omega)]\rho_{h}(\mathcal{E}-\hbar\omega)\left[1-f_{e}(\mathcal{E})\right]\rho_{e}(\mathcal{E}),\quad\mathcal{E}>\mathcal{E}_{c},$
(13)

where $[1-f_{h}(\mathcal{E}-\hbar\omega)]\rho_{h}(\mathcal{E}-\hbar\omega)$ is
the electron distribution in the valence band and
$[1-f_{e}(\mathcal{E})]\rho_{e}(\mathcal{E})$ is the hole distribution in the
conduction band, and the density of electron and hole states
$\rho_{e}(\mathcal{E})$ and $\rho_{h}(\mathcal{E})$, respectively, are plotted
in Fig. 5 (considering the parameters of GaAs). Eq. (13) is schematically
shown by the black arrow in Fig. 5.

#### Intraband transition:

The intraband transition, due to the absorption of a photon, can lead to the
excitation of an electron-hole pair both within the conduction band and within
the valence band with the corresponding probability to be defined below. The
joint probability of absorption of a photon of frequency between $\omega$ and
$\omega+d\omega$ and the excitation of an electron-hole pair within the
conduction band (denoted by superscript $c\rightarrow c$), with the electron
at energy $\mathcal{E}+\hbar\omega$ and hole at energy $\mathcal{E}$, is given
by

$\displaystyle A_{exc}^{c\rightarrow
c;out}(\mathcal{E}_{f}=\mathcal{E}+\hbar\omega,\hbar\omega)=K_{exc}^{c\rightarrow
c;out}D_{J}(\mathcal{E}+\hbar\omega,\mathcal{E})\rho_{J,exc}^{c\rightarrow
c;out}(\mathcal{E}+\hbar\omega,\mathcal{E}),\quad\mathcal{E}>\mathcal{E}_{c},$
(14)

where $K_{exc}^{c\rightarrow c;out}$ is the proportionality constant, and the
corresponding joint density of state is given by

$\rho_{J,exc}^{c\rightarrow
c;out}(\mathcal{E}+\hbar\omega,\mathcal{E})=f_{e}(\mathcal{E})\rho_{e}(\mathcal{E})[1-f_{e}(\mathcal{E}+\hbar\omega)]\rho_{e}(\mathcal{E}+\hbar\omega),$
(15)

and the superscript ‘$out$’ represents the fact that the electron is going out
of the state with energy $\mathcal{E}$. The equation (15) is schematically
shown by the red arrow in Fig. 5.

Similarly, there exists a finite joint probability for absorption of a photon
of frequency between $\omega$ and $\omega+d\omega$, and the excitation of an
electron-hole pair within the valence band (denoted by superscript
$v\rightarrow v$), with the electron at energy $\mathcal{E}-\hbar\omega$ and
hole at energy $\mathcal{E}-2\hbar\omega$, is given by

$A_{exc}^{v\rightarrow
v;in}(\mathcal{E}_{f}=\mathcal{E}-\hbar\omega,\hbar\omega)=K_{exc}^{v\rightarrow
v;in}D_{J}(\mathcal{E}-\hbar\omega,\mathcal{E}-2\hbar\omega)\rho_{J,exc}^{v\rightarrow
v;in}(\mathcal{E}-\hbar\omega,\mathcal{E}-2\hbar\omega),\quad\mathcal{E}>\mathcal{E}_{c},$
(16)

where $K_{exc}^{v\rightarrow v;in}$ is the proportionality constant, and the
corresponding joint density of state is given by

$\rho_{J,exc}^{v\rightarrow
v;in}(\mathcal{E}-\hbar\omega,\mathcal{E}-2\hbar\omega)=[1-f_{h}(\mathcal{E}-2\hbar\omega)]\rho_{h}(\mathcal{E}-2\hbar\omega)f_{h}(\mathcal{E}-\hbar\omega)\rho_{h}(\mathcal{E}-\hbar\omega),$
(17)

and the superscript ‘$in$’ represents the fact that the electron is going in
to the state with energy $\mathcal{E}-\hbar\omega$ by creating a hole at
energy $\mathcal{E}-2\hbar\omega$ in the valence band. Equation (17) is
schematically shown by the blue arrow in Fig. 5. The above-mentioned joint
probability densities corresponding to Eqs. (12), (14), and (16) satisfy the
following property,

$\int_{-\infty}^{\infty}d\mathcal{E}[A_{exc}^{v\rightarrow
c}(\mathcal{E},\hbar\omega)+A_{exc}^{c\rightarrow
c;out}(\mathcal{E}+\hbar\omega,\hbar\omega)+A_{exc}^{v\rightarrow
v;in}(\mathcal{E}-\hbar\omega,\hbar\omega)]=\frac{n_{exc}(\omega)}{N_{exc}},$
(18)

where $n_{exc}(\omega)$ is the number density of absorbed $\hbar\omega$
photons per unit time between $\omega$ and $\omega+d\omega$, and

$N_{exc}=\int d\omega
n_{exc}(\omega)=\hbar^{-1}\epsilon^{\prime\prime}(\omega,T_{e},T_{h},T_{ph})\langle\mathbf{E}(t)\cdot\mathbf{E}(t)\rangle_{t},$
(19)

$\langle\cdot\rangle_{t}$ being the temporal average over a single optical
cycle. In the calculation we use
$N_{exc}=\frac{2\epsilon_{0}\epsilon^{\prime\prime}(\omega)}{\hbar}|E|^{2}$,
$\hbar$ being in the units of $(J\cdot s)$, which is the total number density
of absorbed photons per unit time (in units of $m^{-3}s^{-1}$). Its value is
known from electromagnetic simulations. For simplicity, we consider the
effective local electric field corresponding to the illumination to be
homogeneous throughout the sample. In that sense, we implicitly assume the
system size to be smaller than that of the optical skin-depth of the
illumination.

Figure 5: Schematic diagram for the excitation and recombination processes.
Left $y$-axis plots the hole density of states $\rho_{h}(\mathcal{E})$ and
right $y$-axis plots the electron density of states $\rho_{e}(\mathcal{E})$as
a function of energy $\mathcal{E}$, where $\mathcal{E}_{c}$ denotes the
conduction band edge and $\mathcal{E}_{v}$ denotes the valence band edge. The
expression for the joint density of states corresponding to different
transitions are explained in the text. Parameters chosen here are for GaAs.

To simplify Eq. (18) we approximate
$D_{J}(\mathcal{E}_{final},\mathcal{E}_{initial})$ by a constant, and consider
all the processes corresponding to Eqs. (12), (14), and (16) to be equally
probable [67, 54] such that $K_{exc}^{v\rightarrow
v;in}D_{J}(\mathcal{E}-\hbar\omega,\mathcal{E}-2\hbar\omega)=K_{exc}^{c\rightarrow
c;out}D_{J}(\mathcal{E}+\hbar\omega,\mathcal{E})=K_{exc}^{v\rightarrow
c}D_{J}(\mathcal{E},\mathcal{E}-\hbar\omega)=K_{exc}^{(e)}$; the normalization
constant (with the superscript ‘$(e)$’ denoting that this normalization
corresponds to the excitation of electrons), is determined via the following
condition,

$K_{exc}^{(e)}\int_{-\infty}^{\infty}d\mathcal{E}[\rho_{J,exc}^{v\rightarrow
c}(\mathcal{E},\mathcal{E}-\hbar\omega)+\rho_{J,exc}^{c\rightarrow
c;out}(\mathcal{E}+\hbar\omega,\mathcal{E})+\rho_{J,exc}^{v\rightarrow
v;in}(\mathcal{E}-\hbar\omega,\mathcal{E}-2\hbar\omega)]=\frac{n_{exc}(\omega)}{N_{exc}}.$
(20)

To simplify further, we can consider all the intraband photo-excitations
probabilities to be negligible, and we show that
$\frac{\rho_{J,exc}^{c\rightarrow
c;out}(\mathcal{E}+\hbar\omega,\mathcal{E})}{\rho_{J,exc}^{v\rightarrow
c}(\mathcal{E},\mathcal{E}-\hbar\omega)}\ll 1$ and
$\frac{\rho_{J,exc}^{v\rightarrow
v;in}(\mathcal{E}-\hbar\omega,\mathcal{E}-2\hbar\omega)}{\rho_{J,exc}^{v\rightarrow
c}(\mathcal{E},\mathcal{E}-\hbar\omega)}\ll 1$.

####

Firstly, fro Eqs. (13) and (15), it is easy to see,

$\displaystyle\frac{\rho_{J,exc}^{c\rightarrow
c;out}(\mathcal{E}+\hbar\omega,\mathcal{E})}{\rho_{J,exc}^{v\rightarrow
c}(\mathcal{E},\mathcal{E}-\hbar\omega)}=\frac{f_{e}(\mathcal{E})[1-f_{e}(\mathcal{E}+\hbar\omega)]\rho_{e}(\mathcal{E}+\hbar\omega)}{[1-f_{h}(\mathcal{E}-\hbar\omega)]\rho_{h}(\mathcal{E}-\hbar\omega)[1-f_{e}(\mathcal{E})]}~{}~{}\ll
1,$ (21)

given the fact that for an electron at energy $\mathcal{E}$ in the conduction
band, $[1-f_{e}(\mathcal{E})]\approx 1$,
$[1-f_{e}(\mathcal{E}+\hbar\omega)]\approx 1$, $f_{e}(\mathcal{E})\ll 1$, and
$[1-f_{h}(\mathcal{E}-\hbar\omega)]\approx 1$, these are ensured by the
presence of the gap because $f_{e(h)}(\mathcal{E})\ll 1$ far away from the
chemical potential, a condition that is mostly satisfied for intensities
considered in the present study. Moreover, the ratio of the density of states
in the above equation brings in a factor $(m_{e}^{*}/m_{h}^{*})^{3/2}\ll
1~{}(0.046$ for GaAs, for example). Similar arguments lead to
$\rho_{J,exc}^{v\rightarrow
v;in}(\mathcal{E}-\hbar\omega,\mathcal{E}-2\hbar\omega)/\rho_{J,exc}^{v\rightarrow
c}(\mathcal{E},\mathcal{E}-\hbar\omega)\ll 1$. Therefore, we can rewrite Eq.
(20) as,

$K_{exc}^{(e)}\int_{-\infty}^{\infty}d\mathcal{E}\rho_{J,exc}^{v\rightarrow
c}(\mathcal{E},\mathcal{E}-\hbar\omega)=\frac{n_{exc}(\omega)}{N_{exc}}.$ (22)

We have explicitly verified these arguments in the numerical calculations for
intensities we have considered and found the interband transition to be
dominant so that using either Eqs. (20), or (22) does not change the final
result. Finally, the net change in the electronic population at $\mathcal{E}$
in the conduction band is, therefore,

$\displaystyle\phi_{exc}^{(e)}(\mathcal{E})=\int_{0}^{\infty}d\omega\frac{n_{exc}(\omega)}{N_{exc}}\left[\frac{\rho_{J,exc}^{v\rightarrow
c}(\mathcal{E},\mathcal{E}-\hbar\omega)}{\int_{-\infty}^{\infty}d\mathcal{E}\rho_{J,exc}^{v\rightarrow
c}(\mathcal{E},\mathcal{E}-\hbar\omega)}\right],~{}~{}\mathcal{E}\geq\mathcal{E}_{c},$
(23)

where the explicit expression for the normalization constant $K_{exc}^{(e)}$,
given by Eq. (22), has been used. Therefore, the change in electron
distribution at energy $\mathcal{E}$, in the conduction band due to photon
absorption is given by

$\left(\frac{\partial f_{e}(\mathcal{E})}{\partial
t}\right)_{exc}=\frac{N_{exc}\phi_{exc}^{(e)}(\mathcal{E})}{\rho_{e}(\mathcal{E})},$
(24)

which is (2) for $c=e$ in the conduction band.

###  A.2 Photo-excitation of holes

We aim to formulate the rate of change of hole population $f_{h}(\mathcal{E})$
in the valence band due to photon absorption by adapting the formalism
introduced in 67. The population probability of holes in the valence band
changes both due to the interband and intra-band transitions. However,
following the explanations given in the section A.1 (see equations Eqs. (20),
(21), and (22), and nearby discussions), we neglect any intra-band transition
due to the absorption of a photon. Therefore, the joint probability of
absorption of a photon of frequency between $\omega$ and $\omega+d\omega$ and
excitation of a hole in the valence band is given by

$A_{exc}^{c\rightarrow
v}(\hbar\omega,\mathcal{E}_{f}=\mathcal{E})=K_{exc}^{(h)}D_{J}(\mathcal{E},\mathcal{E}+\hbar\omega)\rho_{J,exc}^{c\rightarrow
v}(\mathcal{E},\mathcal{E}+\hbar\omega),$ (25)

where $K_{exc}^{(h)}$ is the normalization to be defined later and the
superscript ‘$(h)$’ denotes that the normalization constant corresponds to the
excitation of hole. The joint density states corresponding to Eq. (25) is

$\rho_{J,exc}^{c\rightarrow
v}(\mathcal{E},\mathcal{E}+\hbar\omega)=[1-f_{h}(\mathcal{E})]\rho_{h}(\mathcal{E})[1-f_{e}(\mathcal{E}+\hbar\omega)]\rho_{e}(\mathcal{E}+\hbar\omega),$
(26)

where a hole at $\mathcal{E}+\hbar\omega$ in the conduction band moves to the
valence band at $\mathcal{E}$ upon absorption of a photon of energy
$\hbar\omega$, $[1-f_{h}(\mathcal{E})]\rho_{h}(\mathcal{E})$ being the
electron distribution in the valence band and
$[1-f_{e}(\mathcal{E}+\hbar\omega)]\rho_{e}(\mathcal{E}+\hbar\omega)$ being
the hole distribution in the conduction band. Assuming
$D_{J}(\mathcal{E},\mathcal{E}+\hbar\omega)$ to be a constant, i.e., all hole
transitions are equally probable, and absorbing them into the normalization
constant $K_{exc}^{(h)}$, we find the joint density of states in Eq. (13)
satisfies the following condition,

$K_{exc}^{(h)}\int_{-\infty}^{\infty}d\mathcal{E}\rho_{J,exc}^{v\rightarrow
c}(\mathcal{E},\mathcal{E}+\hbar\omega)=\frac{n_{exc}(\omega)}{N_{exc}}.$ (27)

Then the net change in the hole population at $\mathcal{E}$ in the valence
band is given by

$\displaystyle\phi_{exc}^{(h)}(\mathcal{E})=\int_{0}^{\infty}d\omega\frac{n_{exc}(\omega)}{N_{exc}}\left[\frac{\rho_{J,exc}^{c\rightarrow
v}(\mathcal{E},\mathcal{E}+\hbar\omega)}{\int_{-\infty}^{\infty}d\mathcal{E}\rho_{J,exc}^{v\rightarrow
c}(\mathcal{E},\mathcal{E}+\hbar\omega)}\right],~{}~{}\mathcal{E}\leq\mathcal{E}_{v}.$
(28)

Therefore, the rate of change in the hole population in the valence band due
to the absorption of a photon is given by

$\left(\frac{\partial f_{h}(\mathcal{E})}{\partial
t}\right)_{exc}=\frac{N_{exc}\phi_{exc}^{(h)}(\mathcal{E})}{\rho_{h}(\mathcal{E})}$
(29)

where $\rho_{h}(\mathcal{E})$ the eDOS in the valence band. The number of
holes excited due to photon absorption is $\frac{dn_{h}}{dt}=\int
d\mathcal{E}\rho_{h}(\mathcal{E})\left(\frac{\partial f_{h}}{\partial
t}\right)_{exc}$ which is equal to $N_{exc}$ holes per unit time. Our
formulation thus ensures that the number of photo-excited electrons and holes
are the same.

##  Appendix B Recombination: spontaneous emission

###  B.1 Recombination of electrons

We adapt the theoretical formalism for the spontaneous emission obtained using
the usual Fermi golden rule, as explained in 68, to make it well-matched with
the formulation of the photo-absorption. The rate of change of electron
population due to the recombination of electrons with energy $\mathcal{E}$ in
the conduction band is given by

$\left(\frac{\partial f_{e}}{\partial
t}\right)_{rec}=-\frac{N_{rec}\phi_{rec}^{(e)}(\mathcal{E})}{\rho_{e}(\mathcal{E})},$
(30)

and the rate of change of particles due to recombination is given by
$\frac{dn_{rec}}{dt}=\int
d\mathcal{E}\rho_{c}(\mathcal{E})\left(\frac{\partial f_{e}}{\partial
t}\right)_{rec}$ $=-N_{rec}$, a negative sign signifying a loss of particles.
Here, $\phi_{rec}^{(e)}(\mathcal{E})$ is the net change in the electronic
population at energy $\mathcal{E}$ due to the recombination in the conduction
band which is given by

$\phi_{rec}^{(e)}(\mathcal{E})=\int_{0}^{\infty}d\omega^{\prime}\frac{n_{rec}^{(e)}(\hbar\omega^{\prime})}{N_{rec}}\left[\frac{\rho_{J,rec}^{c\rightarrow
v}(\mathcal{E},\mathcal{E}-\hbar\omega^{\prime})}{\int_{-\infty}^{\infty}d\mathcal{E}\rho_{J,rec}^{c\rightarrow
v}(\mathcal{E}-\hbar\omega^{\prime},\mathcal{E})}\right],~{}~{}\mathcal{E}\geq\mathcal{E}_{c},$
(31)

where the joint density of states are given by

$\rho_{J,rec}^{c\rightarrow
v}(\mathcal{E}-\hbar\omega^{\prime},\mathcal{E})=f_{e}(\mathcal{E})\rho_{e}(\mathcal{E})f_{h}(\mathcal{E}-\hbar\omega^{\prime})\rho_{h}(\mathcal{E}-\hbar\omega^{\prime}),~{}~{}\mathcal{E}\geq\mathcal{E}_{c},$
(32)

and is normalized such that

$\displaystyle\int_{-\infty}^{\infty}d\mathcal{E}K_{rec}^{(e)}\rho_{J,rec}^{c\rightarrow
v}(\mathcal{E}-\hbar\omega^{\prime},\mathcal{E})=\frac{n_{rec}^{(e)}(\omega^{\prime})}{N_{rec}},$
(33)

where $n_{rec}^{(e)}(\omega^{\prime})$ is the number density of emitted
photons (per unit time, unit volume, and per unit frequency) of energy
$\hbar\omega^{\prime}$ within the range $\omega^{\prime}$ and
$\omega^{\prime}+d\omega^{\prime}$. The total number of emitted photons is
given by $N_{rec}=\int d\omega^{\prime}n_{rec}^{(e)}(\hbar\omega^{\prime})$
which equals the number of electrons recombining per unit volume. The
intraband downward transition induced by photon emission, i.e., the intraband
recombination is assumed to be negligible [67].

We initially choose $n_{rec}^{(e)}(\omega^{\prime})$ by the analytic result of
the LHS of Eq. (33) for thermal distributions [68, 69], namely,

$n_{rec}^{(e)}(\omega^{\prime})=\text{const.}~{}\rho_{phot}(\hbar\omega^{\prime})\sqrt{\hbar\omega^{\prime}-\mathcal{E}_{g}}e^{-\beta_{e}^{*}(\hbar\omega^{\prime}-\mathcal{E}_{g})}.$
(34)

where $\rho_{phot}(\hbar\omega^{\prime})$ is the photonic density of states.
In vacuum
$\rho_{phot}(\hbar\omega^{\prime})=\frac{\sqrt{\epsilon(\omega^{\prime})}\hbar^{2}\omega^{\prime
2}}{\pi^{2}(\hbar c)^{3}}$ (in units of $(eV\cdot m^{3})^{-1}$) where $c$ is
the speed of light and $\sqrt{\epsilon(\omega^{\prime})}$ is the frequency
dependent refractive index corresponding to the bulk SC. Note the
$\beta_{e}^{*}$ appearing in the above equation represents (the inverse of) an
energy scale coming from the energy conservation and aids the convergence in
the self-consistent calculation, thereby serving as a parameter for
convergence.

It is worthwhile to point out that an alternative choice of
$n_{rec}^{(e)}(\omega^{\prime})=\Theta(\hbar\omega^{\prime}-\epsilon_{g})\Theta(2\hbar\omega-\epsilon_{g}-\hbar\omega^{\prime})$,
$\hbar\omega$ being the energy of the incident light, which doesn’t use any
convergence parameter like $\beta_{e}^{*}$, also brings in the energy
conservation. This indicates that the matrix element $K_{rec}^{(e)}$ is quite
insensitive to the choice of $n_{rec}^{(e)}(\omega^{\prime})$ which can be
attributed to the condition of the particle number conservation, viz., at the
steady-state the number of electrons excited must be equal to the number of
electrons recombine. The same holds for the holes too.

###  B.2 Recombination of holes

Analogous to the recombination of electrons, the joint probability of
recombination of a hole from the valence band with an electron in the
conduction band and emission of a photon of energy in the range
$\omega^{\prime}$ and $\omega^{\prime}+d\omega^{\prime}$, similar to that of
the photo-excitation, is given by

$\displaystyle A_{rec}^{v\rightarrow
c}(\mathcal{E}_{f}=\mathcal{E}+\hbar\omega^{\prime},\hbar\omega^{\prime})=K_{rec}^{c\rightarrow
v}D_{J}(\mathcal{E}+\hbar\omega^{\prime},\mathcal{E})\rho_{J,rec}^{v\rightarrow
c}(\mathcal{E}+\hbar\omega^{\prime},\mathcal{E}),$ (35)

with the corresponding joint density of states,

$\displaystyle\rho_{J,rec}^{v\rightarrow
c}(\mathcal{E}+\hbar\omega^{\prime},\mathcal{E})=f_{h}(\mathcal{E})\rho_{h}(\mathcal{E})f_{e}(\mathcal{E}+\hbar\omega^{\prime})\rho_{e}(\mathcal{E}+\hbar\omega^{\prime}),~{}~{}\mathcal{E}\leq\mathcal{E}_{v},$
(36)

where $K^{c\rightarrow v}_{rec}$ is the normalization. Assuming all the
electronic (in this case hole) transitions corresponding to the recombination
to be equally probable, i.e.,
$D_{J}(\mathcal{E},\mathcal{E}+\hbar\omega^{\prime})K_{rec}^{v\rightarrow
c}=K_{rec}^{(h)}$, (the superscript ‘$(h)$’ denote that this normalization
corresponds to the recombination of a hole), Eq. (35) satisfy the following
condition,

$\displaystyle\int_{-\infty}^{\infty}d\mathcal{E}A_{rec}^{v\rightarrow
c}(\mathcal{E}+\hbar\omega^{\prime},\hbar\omega^{\prime})=\int_{-\infty}^{\infty}d\mathcal{E}K_{rec}^{(h)}\rho_{J,rec}^{v\rightarrow
c}(\mathcal{E}+\hbar\omega^{\prime},\mathcal{E})=\frac{n_{rec}^{(h)}(\omega^{\prime})}{N_{rec}},$
(37)

which define the normalization constant $K_{rec}^{(h)}$. The total number of
emitted photons is given by $N_{rec}=\int
d\omega^{\prime}n_{rec}^{(h)}(\omega^{\prime})$ which is equal to the number
of re-combinations happening, where $n_{rec}^{(h)}(\omega^{\prime})$ is the
number density of emitted photons of energy $\hbar\omega^{\prime}$ per unit
time per unit frequency of emitted photons. We take the expression for
$n_{rec}^{(h)}(\omega^{\prime})$ to be the same as that of the electrons but
with $\beta_{h}^{*}$ in place of $\beta_{e}^{*}$.

Then, the rate of change of electron population due to the recombination of a
hole at energy $\mathcal{E}$ in the valence band is given by

$\left(\frac{\partial f_{h}(\mathcal{E})}{\partial
t}\right)_{rec}=-\frac{N_{rec}\phi_{rec}^{(h)}(\mathcal{E})}{\rho_{h}(\mathcal{E})},$
(38)

where the net change in the electronic population at energy $\mathcal{E}$ in
the valence band due to the recombination of holes with the electrons in the
conduction band is given by

$\displaystyle\phi_{rec}^{(h)}(\mathcal{E})=\int_{0}^{\infty}d\omega^{\prime}\frac{n_{rec}^{(h)}(\hbar\omega^{\prime})}{N_{rec}}\left[\frac{\rho_{J,rec}^{v\rightarrow
c}(\mathcal{E}+\hbar\omega^{\prime},\mathcal{E})}{\int_{-\infty}^{\infty}d\mathcal{E}\rho_{J,rec}^{v\rightarrow
c}(\mathcal{E}+\hbar\omega^{\prime},\mathcal{E})}\right],~{}~{}\mathcal{E}\leq\mathcal{E}_{v}.$
(39)

##  Appendix C Logarithmic slope of the distribution for different photon
frequency

In Sec. IV.1 we demonstrate that due to negligible carrier-carrier scattering
the distributions are indeed more non-thermal at low illumination intensities
and start thermalizing only at higher intensities due to increased carrier-
carrier scattering. Here we demonstrate that this conclusion is independent of
the energy of the incident photons, viz., $\hbar\omega$. To demonstrate we
plot in Fig. 6, the logarithmic slope of $f_{e}(\mathcal{E})$ at electronic
energy $\mathcal{E}=4.914$ eV as a function of local field $|E|^{2}$ for three
more photon energies $\hbar\omega=1.55$ eV, $1.90$ eV, and $2.05$ eV,
respectively. For each $\hbar\omega$ the slope of $f_{e}(\mathcal{E})$ remains
unchanged, until the intensity reaches a critical value such that the $e-e$
interaction becomes comparable to the $e-ph$ interaction.

Figure 6: Logarithmic slope of the non-equilibrium distribution of electrons
$f_{e}(\mathcal{E})$ at electronic energy $\mathcal{E}=4.914$ eV (an energy
near the conduction band edge) as a function of local field $|E|^{2}$ for
different photon energies.

##  Appendix D Conservation of energy and number of particles

The rates of change of particle numbers, due to the different scattering
mechanisms, are given by the following definitions

$\displaystyle\left(\frac{dn_{e(h)}}{dt}\right)_{exc}$ $\displaystyle=$
$\displaystyle\int d\mathcal{E}\left(\frac{\partial f_{e(h)}}{\partial
t}\right)_{exc}\rho_{e(h)}(\mathcal{E})$
$\displaystyle\left(\frac{dn_{e(h)}}{dt}\right)_{rec}$ $\displaystyle=$
$\displaystyle\int d\mathcal{E}\left(\frac{\partial f_{e(h)}}{\partial
t}\right)_{rec}\rho_{e(h)}(\mathcal{E})$
$\displaystyle\left(\frac{dn_{e(h)}}{dt}\right)_{e(h)-ph}$ $\displaystyle=$
$\displaystyle\int d\mathcal{E}\left(\frac{\partial f_{e(h)}}{\partial
t}\right)_{e(h)-ph}\rho_{e(h)}(\mathcal{E})=0$
$\displaystyle\left(\frac{dn_{e(h)}}{dt}\right)_{e(h)-e(h)}$ $\displaystyle=$
$\displaystyle\int d\mathcal{E}\left(\frac{\partial f_{e(h)}}{\partial
t}\right)_{e-e}\rho_{e(h)}(\mathcal{E})=0.$

Here, the subscripts ‘$e$’ and ‘$h$’ stand for electrons and holes,
respectively. At the steady-state, particle number conservation requires
$\left(\frac{dn_{e(h)}}{dt}\right)_{exc}+\left(\frac{dn_{e(h)}}{dt}\right)_{rec}=0$
for both electrons and holes separately. Both the electron (hole)-phonon and
electron-electron (hole-hole) interactions are number conservingThe powers
transferred between the different carriers in the conduction band are

$\displaystyle W_{e-exc}$ $\displaystyle=$ $\displaystyle\int
d\mathcal{E}~{}\rho_{e}(\mathcal{E})(\mathcal{E}-\mathcal{E}_{c})\left(\frac{\partial
f_{e}}{\partial t}\right)_{exc},$ $\displaystyle W_{e-rec}$ $\displaystyle=$
$\displaystyle-\int
d\mathcal{E}\rho_{e}(\mathcal{E})(\mathcal{E}-\mathcal{E}_{c})\left(\frac{\partial
f_{e}}{\partial t}\right)_{rec},$ $\displaystyle W_{e-ph}$ $\displaystyle=$
$\displaystyle\int
d\mathcal{E}\rho_{e}(\mathcal{E})(\mathcal{E}-\mathcal{E}_{c})\left(\frac{\partial
f_{e}}{\partial t}\right)_{e-ph},$ $\displaystyle W_{e-e}$ $\displaystyle=$
$\displaystyle\int
d\mathcal{E}\rho_{e}(\mathcal{E})(\mathcal{E}-\mathcal{E}_{c})\left(\frac{\partial
f_{e}}{\partial t}\right)_{e-e}=0.$ (41)

In the above, $W_{e-exc}$ is the power that electrons receive from the
electromagnetic field due to photon absorption, and $W_{e-rec}$ is the power
that electrons lose due to the recombination. Note that Eq. (30) is negative
and therefore, the definition of $W_{e-rec}$ in Eq. (D) makes $W_{e-rec}$ a
positive quantity. The superscript ‘$(e)$’ denotes that these powers
correspond to the electrons in the conduction band, and the powers defined in
(D) correspond to the rates of change of excess energy of electrons measured
from the conduction band edge. Then, part of the net power stored in the
electronic bath gets transferred to the phonon bath (i.e., the lattice) via
electron-phonon scattering, and is given by $W_{e-ph}$. The electron-electron
interaction, being an elastic scattering mechanism, does not induce any power
transfer. The power balance equation corresponding to the electrons is given
by

$\frac{d\mathcal{U}_{e}}{dt}=c_{e}\frac{dT_{e}}{dt}=W_{e-exc}-W_{e-rec}-W_{e-ph},$
(42)

where $c_{e}$ is the electronic heat capacity. In the non-equilibrium steady-
state (NESS), we have $\frac{d\mathcal{U}_{e}}{dt}=0$ which determines the
steady-state electron temperature.

In analogy to the powers defined for the electrons in Eq. (D), for the holes
we have

$\displaystyle W_{h-exc}$ $\displaystyle=$ $\displaystyle\int
d\mathcal{E}~{}\rho_{h}(\mathcal{E})(\mathcal{E}_{v}-\mathcal{E})\left(\frac{\partial
f_{h}}{\partial t}\right)_{exc},$ $\displaystyle W_{h-rec}$ $\displaystyle=$
$\displaystyle-\int
d\mathcal{E}~{}\rho_{h}(\mathcal{E})(\mathcal{E}_{v}-\mathcal{E})\left(\frac{\partial
f_{h}}{\partial t}\right)_{rec},$ $\displaystyle W_{h-ph}$ $\displaystyle=$
$\displaystyle\int
d\mathcal{E}~{}\rho_{h}(\mathcal{E})(\mathcal{E}_{v}-\mathcal{E})\left(\frac{\partial
f_{h}}{\partial t}\right)_{h-ph},$ $\displaystyle W_{h-h}$ $\displaystyle=$
$\displaystyle\int
d\mathcal{E}~{}\rho_{h}(\mathcal{E})(\mathcal{E}_{v}-\mathcal{E})\left(\frac{\partial
f_{h}}{\partial t}\right)_{h-h}=0,$ (43)

calculated with respect to the valence band edge. The power balance equation
corresponding to the holes is given by

$\frac{d\mathcal{U}_{h}}{dt}=c_{h}\frac{dT_{h}}{dt}=W_{h-exc}-W_{h-rec}-W_{h-ph},$
(44)

where $c_{h}$ is the hole heat capacity.

Finally, we introduce a phenomenological description for the phonon
temperature consisting of the total power transferred to the phonon sub-system
(or the lattice) and the power transferred to the surrounding from the
lattice. The power balance equation corresponding to the phonons is given by

$\displaystyle\frac{d\mathcal{U}_{ph}}{dt}=c_{ph}\frac{dT_{ph}}{dt}=(W_{e-ph}+W_{h-ph})-G_{ph-
env}(T_{ph}-T_{env}),$ (45)

where $c_{ph}$ is the heat capacity of phonons and $G_{ph-env}$ coupling
between the lattice and the environment. In the above equation,
$\frac{d\mathcal{U}_{ph}}{dt}$ is the rate of change of energy of the phonon
sub-system.

##  Appendix E Extraction of $T_{e}$ and $T_{h}$ from $W_{c-ph}^{T}$

The carrier-phonon power transfers at the thermal equilibrium, $W_{c-ph}^{T}$
are defined as

$\displaystyle W_{e-ph}^{T}$ $\displaystyle=$ $\displaystyle\int
d\mathcal{E}(\mathcal{E}-\mathcal{E}_{c})\left(\frac{\partial
f_{e}^{T}}{\partial t}\right)_{e-ph}\rho_{e}(\mathcal{E})$ $\displaystyle
W_{h-ph}^{T}$ $\displaystyle=$ $\displaystyle\int
d\mathcal{E}(\mathcal{E}_{v}-\mathcal{E})\left(\frac{\partial
f_{h}^{T}}{\partial t}\right)_{h-ph}\rho_{h}(\mathcal{E}),$ (46)

respectively, for electrons and holes, where $\left(\frac{\partial
f_{e}^{T}}{\partial t}\right)_{e-ph}$ is given by Eq. (4) and
$\left(\frac{\partial f_{h}^{T}}{\partial t}\right)_{h-ph}$ is given by Eq.
(5).

To extract $T_{e}$ and $T_{h}$ from $W_{e-ph}^{T}$ and $W_{h-ph}^{T}$,
respectively, we first obtain $W_{c-ph}^{T}$ as a function of $T_{c}-T_{ph}$
by taking $T_{c}$ as a variable, and then, we invert the $W_{c-ph}^{T}$ vs.
$(T_{c}-T_{ph})$ relation to obtain
$(T_{c}-T_{ph})=\mathcal{F}(W_{c-ph}^{T})$. Next, we find from $\mathcal{F}$
for which value of $(T_{e}-T_{ph})$ and $(T_{h}-T_{ph})$ we have
$W_{e-ph}^{T}=W_{e-ph}$ and $W_{h-ph}^{T}=W_{h-ph}^{T}$, respectively, where
$W_{e-ph}$ and $W_{h-ph}$ are obtained from our self-consistent calculation.
From $T_{e}-T_{ph}$ and $T_{h}-T_{ph}$ we can further extract the rise in
electron temperature, $\Delta T_{e}$, and the rise in hole temperature $\Delta
T_{h}$ above that of the ambient. $\Delta T_{e}$ and $\Delta T_{h}$ are
plotted as a function of $|E|^{2}$ in Fig. 3(a) of the main paper and exhibit
a non-linear dependence on the intensity of the incident light.

The following method can be used to measure the temperature extracted from
$W_{c-ph}$. At the steady-state we have $W_{c-exc}-W_{c-rec}-W_{c-ph}=0$,
where we approximate $W_{c-ph}=G_{c-ph}(T_{c},T_{ph})(T_{c}-T_{ph})$ for both
electrons and holes. Now, if we add a floating thermocouple (TC) for e.g., as
considered in 61, 62, which measures the tunnelling of the electrons and
holes, then, the energy flow equations for both electrons and holes get
modified to

$\displaystyle W_{c-exc}$ $\displaystyle-$ $\displaystyle
W_{c-rec}-G_{c-ph}(T_{c},T_{ph})(T_{c}-T_{ph})$ (47) $\displaystyle-$
$\displaystyle G_{TC}(T_{c}-T_{TC})=0,$

at the steady-state, where $c=e(h)$ for electrons (holes) and $G_{TC}$ is the
coupling between the electronic subsystem of the SC and the TC. As a measuring
probe we keep $T_{TC}$ a floating parameter and at the steady-state the flow
equations reach at a fixed point where $T_{TC}=T_{c}$, giving us a unique
value of carrier temperatures from the measurement.

##  Appendix F Steady-state Photoluminescence

Figure 7 plots the steady-state photoluminescence (PL) spectra for a specific
intensity corresponding to $|E|^{2}=40~{}(V/m)^{2}$. The non-equilibrium PL
spectrum $n_{PL}(\omega^{\prime})$ is defined by integrating the right-hand-
side of Eq. (32). Here, we plot a normalized $n_{PL}(\omega^{\prime})$ (in the
sense that $\int_{0}^{\infty}d\omega^{\prime}n_{PL}(\omega^{\prime})=1$). We
normalize the PL spectra because we are interested only in its shape. We fit
the steady-state PL spectrum with the thermal PL spectra defined by the right-
hand-side of Eq. (33) without the photonic density of states $\rho_{phot}$,
whose functional form given by
$n_{PL}(\omega^{\prime},T_{c})\propto\sqrt{\hbar\omega^{\prime}-\mathcal{E}_{g}}e^{-\beta_{c}(\hbar\omega^{\prime}-\mathcal{E}_{g})}$
where $\beta_{c}=1/k_{B}T_{c}$ [68, 69].

Figure 7: Steady-state photoluminescence at $|E|^{2}=40~{}(V/m)^{2}$ Plot of
the PL spectra as a function of the frequency of the emitted photons (in eV)
where blue square data corresponds to the steady-state
$n_{PL}(\omega^{\prime})$ PL spectra, and the red solid line corresponds to a
fit with $n_{PL}(\omega^{\prime},T_{c})$ where is the thermal PL spectra.

#### Estimation of electron and hole from steady-state photoluminescence
spectra:

The fitted data are plotted, after the due normalization, in Fig. 7 and
provides us with a value of the carrier temperature $T_{c}$. The temperatures
obtained from such a fitting for each values the intensity are plotted as a
function of $|E|^{2}$ in Figure 3(b) of the main paper. For all energies
$\mathcal{E}>\mathcal{E}_{C}$, $\rho_{J,rec}^{c\rightarrow
v}(\mathcal{E}-\hbar\omega^{\prime},\mathcal{E})=\rho_{J,rec}^{v\rightarrow
c}(\mathcal{E}-\hbar\omega^{\prime},\mathcal{E})$, i.e., the joint density of
states for an electron at $\mathcal{E}$ recombining with a hole at
$\mathcal{E}-\hbar\omega^{\prime}$ is the same as the joint density of states
for a hole at $\mathcal{E}-\hbar\omega^{\prime}$ recombining with an electron
at $\mathcal{E}$, as seen from (32) and (36). This is also true for all
energies $\mathcal{E}<\mathcal{E}_{V}$. Therefore, the PL spectrum is
symmetric with respect to electrons and holes, making the electron and hole
temperatures obtained from the PL spectra the same.

##  Appendix G Ratio of power transfer for different photon energies

In deriving the final form of Eq. (IV.3) we define
$\eta_{rec}^{(c)}=\frac{W_{c-rec}}{W_{c-exc}}$ [$c=e$ for electrons and $c=h$
for holes] as the ratio of the power dissipated from the electron/hole sub-
systems through the recombination $W_{c-rec}$ to the power absorbed by the
electron/hole sub-systems, $W_{c-exc}$. These ratios are plotted in Fig. 4 of
the main text for $\hbar\omega=1.65$ eV. Here in Fig. 8 left panel, we plot
$\eta_{rec}^{(e)}$ as a function of illumination intensities for four
different values of photon energies $\hbar\omega$. Fig. 8 right panel shows
$\eta_{rec(ph)}^{(e)}$ vs. $(\hbar\omega-\mathcal{E}_{g})$ on a log-log scale
at $|E|^{2}=4\times 10^{7}~{}(\frac{V}{m})^{2}$, where $\eta_{ph}$ is defined
in Sec. IV.3.

Figure 8: Left Panel: $\eta_{rec}^{(e)}$ as a function of illumination
intensities for four different values of photon energies $\hbar\omega$. Right
panel: $\eta_{rec(ph)}^{(e)}$ vs. $(\hbar\omega-\mathcal{E}_{g})$ on a log-log
scale at $|E|^{2}=4\times 10^{7}~{}(\frac{V}{m})^{2}$.

Fig. 8 right panel further shows that
$\eta_{rec}\propto(\hbar\omega-\mathcal{E}_{g})^{-1}$ which is expected from
the fact that $\eta_{rec}^{(c)}=W_{c-rec}/W_{c-exc}$ and
$W_{c-exc}\propto(\hbar\omega-\mathcal{E}_{g})$. This further indicates that
$W_{c-rec}$ is independent of the illumination photon energies. Moreover,
$\eta_{ph}^{(c)}=\left(1-\frac{\text{const.}}{(\hbar\omega-\mathcal{E}_{g})}\right)$
due to $\eta_{rec}^{(c)}+\eta_{ph}^{(c)}=1$

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