# Enhancement of Hot Carrier Effects and Signatures of Confinement in Terms of
Thermalization Power in Quantum Well Solar Cells

I. Makhfudz  Aix Marseille Université, CNRS, Université de Toulon, IM2NP UMR
7334, 13397, Marseille, France  N. Cavassilas  Aix Marseille Université, CNRS,
Université de Toulon, IM2NP UMR 7334, 13397, Marseille, France  M. Giteau
Research Center for Advanced Science and Technology, The University of Tokyo,
Komaba 4-6-1, Meguro-ku, Tokyo 153-8904, Japan  NextPV, LIA RCAST-CNRS, The
University of Tokyo, Komaba 4-6-1, Meguro-ku, Tokyo 153-8904, Japan  H.
Esmaielpour  NextPV, LIA RCAST-CNRS, The University of Tokyo, Komaba 4-6-1,
Meguro-ku, Tokyo 153-8904, Japan  CNRS, Ecole Polytechnique-IP Paris, Institut
Photovoltaique d’Ile de France (IPVF), UMR 9006, 18 Boulevard Thomas Gobert,
91120 Palaiseau, France  D. Suchet  NextPV, LIA RCAST-CNRS, The University of
Tokyo, Komaba 4-6-1, Meguro-ku, Tokyo 153-8904, Japan  CNRS, Ecole
Polytechnique-IP Paris, Institut Photovoltaique d’Ile de France (IPVF), UMR
9006, 18 Boulevard Thomas Gobert, 91120 Palaiseau, France  A.-M. Daré  Aix
Marseille Université, CNRS, Université de Toulon, IM2NP UMR 7334, 13397,
Marseille, France  F. Michelini  Aix Marseille Université, CNRS, Université de
Toulon, IM2NP UMR 7334, 13397, Marseille, France

###### Abstract

A theoretical model using electron-phonon scattering rate equations is
developed for assessing carrier thermalization under steady-state conditions
in two-dimensional systems. The model is applied to investigate the hot
carrier effect in III-V hot-carrier solar cells with a quantum well absorber.
The question underlying the proposed investigation is: what is the power
required to maintain two populations of electron and hole carriers in a quasi-
equilibrium state at fixed temperatures and quasi-Fermi level splitting? The
obtained answer is that the thermalization power density is reduced in two-
dimensional systems compared to their bulk counterpart, which demonstrates a
confinement-induced enhancement of the hot carrier effect in quantum wells.
This power overall increases with the well thickness, and it is moreover shown
that the intra-subband contribution dominates at small thicknesses while the
inter-subband contribution increases with thickness and dominates in the bulk
limit. Finally, the effects of the thermodynamic state of phonons and
screening are clarified. In particular, the two-dimensional thermalization
power density exhibits a non-monotonic dependence on the thickness of the
quantum well layer, when both out-of-equilibrium longitudinal optical phonons
and screening effects are taken into account. Our theoretical and numerical
results provide tracks to interpret intriguing experimental observations in
quantum well physics. They will also offer guidelines to increase the yield of
photovoltaic effect based on the hot carrier effect using quantum well
heterostructures, a result critical to the research toward high-efficiency
solar cell devices.

##  I Introduction

In the contemporary solar cell research progress, it has been suggested that
if the charge carriers of the semiconductor constituting the solar cell can be
maintained at high effective temperature, or eventually in an ”hot” a-thermal
distribution, the yield of the solar cell can be improved Conibeer1 . Major
efforts and proposals have thus been put forward in order to achieve hot
carrier effect Conibeer2 . Hot carrier phenomenon has actually been known in
semiconductor physics research since early in the 60’s JShah . Its relevance
to solar cell study was shown by Ross and Nozik RossNozik  who first
demonstrated the hot carrier effect as a principle to beat the efficiency
limit imposed by Shockley-Queisser theory ShockleyQueisser .

A key mechanism for achieving hot carrier effect is impeding the carrier
thermalization process, that is, preventing the charge carriers from cooling
down via their interaction with phonons Mahan . There are various ways in
which this can happen which include reducing carrier-phonon coupling strength
JShahSolidStateElectronics , reducing longitudinal optical (LO) phonon density
MahyarAPL , and reducing LO phonon decay to acoustic phonons RidleyBook .
Overall, electrons and holes under continuous illumination can reach a
temperature higher than the one of lattice if the rate at which the power is
dissipated via carrier-phonon scattering can be reduced.

One of the most promising tracks to promote a hot carrier effect is to confine
electrons in a small region, which restricts their interaction with phonons
and, hence, prevents electrons from thermalizing. This can be achieved by
employing semiconductor heterostructures with reduced dimensionality (D), in
which carriers experience confinement in one (2D), two (1D) or three (0D)
directions Hirst0 , compared to the bulk case (3D). In such low dimensional
systems, the band structure splits into subbands whose energy separation
depends on semiconductor properties, as energy gap and effective mass in a
simple empirical modeling. In optoelectronics and photovoltaics, quantum wells
(2D) based on polar semiconductors of III-V family are widely used due to
their good absorption properties. In these materials, the predominant electron
thermalization channel is the scattering between electrons and longitudinal
optical (LO) phonons JShahSolidStateElectronics , while optical phonons may
themselves decay into acoustic phonons ShahLeite . While several experiments
have indicated that the approach does give rise to a hot carrier effect
Pelouch ; Ryan ; ShahPRL ; Lyon ; Balkan ; Rosenwaks ; Hirst ; ConibeerIEEE ,
only a few theoretical investigations estimates the contribution of quantum
confinement to the hot carrier effect. Furthermore, understanding the
intricate mechanisms of this contribution is key to improve the design of
solar cells integrating nanostructured absorbers ConibeerIEEE .

Numerous theoretical studies on charge transport in quantum wells exist from
works in the 80’s, but mostly focused on the calculation of scattering rates
without specific application to hot carrier phenomenon nor relevance to solar
cells Ferry1978 ; Hess ; Price1981 ; Ridley1 ; Ridley2 . These works reported
that a quantum well structure increases the electron-phonon scattering rate
relatively to its value in the bulk material. Other works applying Boltzmann
transport equation on ultrafast dynamics of electrons and phonons in a quantum
well LugliGoodnickPRL  however reported that, when the LO phonons are assumed
to be in a non-thermal distribution, the electron-phonon scattering rate is of
the same order GoodnickLugliPRB  or even reduced compared to the 3D case Ge ;
Sarma . The apparent opposite behaviors in terms of electron-phonon scattering
rate in quantum wells with respect to bulk points out the critical role of LO
phonons throughout the type of equilibrium or non-equilibrium distributions
they follow. Finally, related theoretical studies demonstrated high effective
electron and hole temperatures on short time scales upon ultrafast excitation
and estimated the relaxation time from the transient decay of the effective
temperature JoshiFerry ; GoodnickReview .

In the present work, we demonstrate that confinement in a quantum well
suppresses the electron-phonon scattering rate, a phenomenon referred to as
the phonon bottleneck Bastard . This phenomenon has been observed in quantum
dots where the discretization of energy levels prevents state relaxation
QDphononbottleneck . As a consequence, the hot carrier effect is shown to be
enhanced from quantum confinement. For that, we discuss a quantity which can
be deduced from experiment, the so-called thermalization power. Our work
relies on a semi-classical kinetic theory developed recently for bulk solar
cells Tsai , which we have extended to include 2D confinement for assessing
carrier thermalization in quantum well solar cells. Moreover, simulations are
performed for InGaAsP/InGaAs/InGaAsP quantum wells varying the well thickness,
and using input parameters from experiment performed under steady-state, i.e.,
non-transient excitation, thus close to realistic situations in which solar
cells normally operate.

This article begins with a description of the main lines of the
phenomenological model of carrier cooling in homogeneous bulk solar cells.
Then, a specific model of electron-phonon interaction is built taking into
account the confinement of electrons in a finite quantum well, and then used
to derive the electron-phonon scattering rate, detailing the distinction
between intra and inter-subband processes. The unconfined LO phonons in the
model are characterized by a non-thermal distribution. The role of a non-
equilibrium state of LO phonons on carrier cooling is clarified from the
scattering rate. The thermalization power needed to achieve electron and hole
populations at given temperatures is computed and analyzed, with an emphasis
on the relative importance of a non-equilibrium phonon distribution and
screening effect in the case of confining heterostructures. Finally, our
theoretical findings are discussed with respect to a number of experimental
observations reported in the literature, old and recent, in the context of
seeking for the hot carrier effect in quantum wells.

##  II On Steady-State Carrier Thermalization and Thermalization Power

###  II.1 Principle

Absorption of a photon generates a pair of electron and hole in the conduction
and valence bands respectively of a semiconductor. For photon energy larger
than the band gap, the electron (hole) may be generated with energy well above
(below) the conduction (valence) band edge. The ensemble of photo-generated
carriers redistribute the absorbed energy among themselves, and may form an
effective high-temperature thermal distribution, referred as hot carrier
effect. But, this excess kinetic energy is mainly dissipated through carrier-
phonon scattering, in a process known as thermalization, until the carriers
reach the band edges and fulfill an effective equilibrium distribution at the
lattice temperature. The tendency of electrons to cool down can however be
impeded if the electron-LO phonon scattering rate is strongly suppressed, so
that electrons can hardly release their energy by emitting LO phonons. In
other words, the net phonon emission rate needs to be reduced, either by
reducing phonon emission or by increasing phonon absorption.

The rate of energy transfer from electrons to LO phonons thus measures the
tendency of electrons to cool down. The corresponding power is called
thermalization power. It is the quantity of central interest in our work
because of two important reasons; it can be deduced from experiment and it is
meaningful: a higher thermalization power for a given carrier temperature
means that it is necessary to pump more energy into the system to achieve the
same carrier temperature.

Figure 1: (Color online) Illustration of a steady-state hot carrier effect
involving non-equilibrium hot LO phonons.

###  II.2 Main lines of 3D theoretical model

A semi-classical model of carrier cooling was derived by Tsai in Ref. Tsai
for homogeneous 3D systems of charge carriers, and longitudinal and acoustic
phonons in bulk semiconductors used in solar cells. The fundamental assumption
of the model is that acoustic phonons remain at lattice temperature $T_{L}$
while electrons form a thermalized population at a carrier temperature $T_{c}$
that can be significantly higher than $T_{L}$, thus becoming hot carriers. The
first part of the assumption is justified by the fact that acoustic phonons do
not interact directly with electrons because in polar semiconductors,
electrons only couple strongly to polar optical phonons
JShahSolidStateElectronics ; ShahLeite . The thermalization power can be
computed from the rate of change of the number of LO phonons due to their
interaction with electrons, which represents the only path of thermalization
for carrier in that model. More precisely, the thermalization power in units
of W/m2 is given by

$P^{c-LO}_{\mathrm{th}}=L^{ph}_{z}\int\frac{d^{3}\mathbf{q}}{(2\pi)^{3}}\hbar\omega_{\mathbf{q}}\frac{dN_{\mathbf{q}}}{dt}\Bigr{|}_{c-LO},$
(1)

where $L^{ph}_{z}$ is the dimension of the LO phonon bath along the well
confinement direction $\mathbf{q}$ is the LO phonon wave vector,
$\hbar\omega_{\mathbf{q}}$ its energy, while $N_{\mathbf{q}}$ is the number of
phonons at wave vector $\mathbf{q}$ which is not necessarily equal to its
equilibrium value. The equilibrium population at a given temperature $T$ is
given by the Bose-Einstein distribution:
$N_{\mathbf{q}}(T)=(\exp(\hbar\omega_{\mathbf{q}}/(k_{B}T))-1)^{-1}$ where
$\hbar$ is the reduced Planck constant, and $k_{B}$ the Boltzmann constant.

A key insight put forward in Ref. Tsai  argues that the total rate
$dN_{\mathbf{q}}/dt$ for LO phonons receives two principal contributions: one
from electrons which emit or absorb LO phonons and the other from the acoustic
phonons from 3-phonon scattering process where an LO phonon transforms into
two acoustic phonons or vice versa RidleyBook . In other words, we can write

$\frac{dN_{\mathbf{q}}}{dt}=\frac{dN_{\mathbf{q}}}{dt}\Bigr{|}_{c-LO}+\frac{dN_{\mathbf{q}}}{dt}\Bigr{|}_{LO-
ac},$ (2)

where

$\frac{dN_{\mathbf{q}}}{dt}\Bigr{|}_{c-LO}=\frac{N_{\mathbf{q}}(T_{c})-N_{\mathbf{q}}}{\tau^{c-LO}_{\mathbf{q}}}\
,$ (3) $\frac{dN_{\mathbf{q}}}{dt}\Bigr{|}_{LO-
ac}=\frac{N_{\mathbf{q}}(T_{L})-N_{\mathbf{q}}}{\tau^{LO-ac}_{\mathbf{q}}}\ ,$
(4)

and the subscripts $c$, $LO$, $ac$ stand for charge carrier, longitudinal
optical phonon, and acoustic phonon respectively, to be used in the rest of
this article.

A steady-state is achieved when the rate of LO phonon emission by electrons in
Eq. (3) equals the rate of the LO phonon conversion into acoustic phonons in
Eq. (4). Under this condition, the LO phonons themselves will remain out of
equilibrium, with a population number that can be expressed as a thermodynamic
average between Bose-Einstein distributions at $T_{c}$ and $T_{L}$, with no
well-defined effective temperature. Setting $dN_{\mathbf{q}}/dt=0$, one
arrives at Tsai

$N_{\mathbf{q}}=\frac{N_{\mathbf{q}}(T_{c})\tau^{LO-
ac}_{\mathbf{q}}+N_{\mathbf{q}}(T_{L})\tau^{c-LO}_{\mathbf{q}}}{\tau^{c-LO}_{\mathbf{q}}+\tau^{LO-
ac}_{\mathbf{q}}}\ ,$ (5)

which defines an athermal population of LO phonons. Substituting Eq. (5) into
Eqs. (1) and (3), one obtains Tsai

$P^{c-LO}_{\mathrm{th}}=L^{ph}_{z}\int\frac{d^{3}q}{(2\pi)^{3}}\hbar\omega_{\mathbf{q}}\frac{N_{\mathbf{q}}(T_{c})-N_{\mathbf{q}}(T_{L})}{\tau^{c-LO}_{\mathbf{q}}+\tau^{LO-
ac}_{\mathbf{q}}}$ (6)

for the thermalization power under steady-state condition. The thermalization
power reflects the deexcitation of hot carriers via interaction with LO
phonons. Thus a reduced thermalization power represents a lower cost to
maintain a carrier population hot, it is hence beneficial for technological
purposes.

For 3D bulk system, the electron-LO phonon scattering rate is given by Tsai

$\frac{1}{\tau^{c-LO}_{\mathbf{q}}}=\frac{m^{2}_{c}k_{B}T_{c}|M_{\mathbf{q}}|^{2}V_{c}}{\pi\hbar^{5}q}\mathrm{ln}\left[\frac{1+\exp(\eta_{c}-\varepsilon_{\mathrm{min}}+\varepsilon_{\mathrm{LO}})}{1+\exp(\eta_{c}-\varepsilon_{\mathrm{min}})}\right]\
,$ (7)

where $m_{c}$ is the carrier mass, $M_{q}$ the electron-phonon coupling
function, $V_{c}$ the carrier spatial volume, $q$ the LO phonon wave vector.
Furthermore $\eta_{c}=\mu_{c}/(k_{B}T_{c})$, where $\mu_{c}$ is the chemical
potential, $\varepsilon_{\mathrm{LO}}=\hbar\omega_{\mathrm{LO}}/(k_{B}T_{c})$,
$\omega_{\mathrm{LO}}$ the LO phonon frequency (assumed constant independent
of $q$),
$\varepsilon_{\mathrm{min}}=\hbar^{2}k^{2}_{\mathrm{min}}/(2m_{c}k_{B}T_{c})$
where $k_{\mathrm{min}}=q/2+m_{c}\omega_{\mathrm{LO}}/(\hbar q)$.

To show that non-equilibrium phonons do contribute to hot carrier effect, we
compare Eq. (1) with the corresponding expression assuming LO phonons to be at
equilibrium at lattice temperature $T_{L}$. Then
$N_{\mathbf{q}}=N_{\mathbf{q}}(T_{L})$. Substituting it into Eq. (3) gives

$P^{c-LO}_{\mathrm{th}}=L^{ph}_{z}\int\frac{d^{3}\mathbf{q}}{(2\pi)^{3}}\hbar\omega_{\mathbf{q}}\frac{N_{\mathbf{q}}(T_{c})-N_{\mathbf{q}}(T_{L})}{\tau^{c-LO}_{\mathbf{q}}}\
,$ (8)

which is always larger than the thermalization power with non-equilibrium
phonons expressed as in Eq. (6). This thus demonstrates explicitly that non-
equilibrium hot phonons help to maintain a hot carrier effect in bulk systems.
It is emphasized here that the hot carrier effect studied in our work is a
steady-state property, thus distinguishing itself from the transient hot
carrier effect known in the literature GoodnickReview .

The analytical expression for the electron-LO phonon scattering time
$\tau^{c-LO}_{\mathbf{q}}$ (Eq. (7)) was derived in Ref. Tsai  for bulk
homogeneous semiconductor solar cells. In the following section, we will
derive the expression for electron-LO phonon scattering time for 2D systems,
and we will analyze the effect of confinement on the hot carrier effect.

##  III 2D Theoretical Model of Confined Carrier Thermalization

In this part, we propose to derive expressions for the electron-LO phonon
scattering rate $\tau^{c-LO}_{\mathbf{q}_{\perp},m}$ and the resulting
thermalization power $P^{c-LO}_{\mathrm{th}}$ for two-dimensional
semiconductor heterostructures. The general framework of the derivation
assumes that electron-electron scattering is faster than any other process,
such that the carriers follow a Fermi-Dirac distribution given by a
temperature $T_{c}$ hotter than the lattice one, and a quasi Fermi level
$\mu_{c}$. The expression for 2D systems starts from Eq. (1)

$\displaystyle P^{c-LO}_{\mathrm{th}}$ $\displaystyle=$
$\displaystyle\sum_{m}\int\frac{d^{2}\mathbf{q}_{\perp}}{(2\pi)^{2}}\hbar\omega_{\mathbf{q}_{\perp},m}\frac{dN_{\mathbf{q}_{\perp},m}}{dt}|_{c-LO}$
(9) $\displaystyle=$
$\displaystyle\sum_{m}\int\frac{d^{2}\mathbf{q}_{\perp}}{(2\pi)^{2}}\hbar\omega_{\mathbf{q}_{\perp},m}$
$\displaystyle\times\frac{N_{\mathbf{q}_{\perp},m}(T_{c})-N_{\mathbf{q}_{\perp},m}(T_{L})}{\tau^{c-LO}_{\mathbf{q}_{\perp},m}+\tau^{LO-
ac}_{\mathbf{q}_{\perp},m}}\ ,$

which involves a sum over the number of phonon modes $m$, and an integration
over the transverse wave vector $\mathbf{q}_{\perp}$ of the LO phonon. In the
following treatment, the LO phonons are supposed dispersionless, such that we
shall take $\hbar\omega_{\mathbf{q}_{\perp},m}=\hbar\omega_{LO}$, hence the
only quantity in the integral which depends on $({\bf q}_{\perp},m)$ is
$\tau^{c-LO}_{{\bf q}_{\perp},m}$.

To be concrete and perform illustrative calculations, we focus on a single
quantum well of width $L_{z}$, made of an InGaAs layer sandwiched between two
InGaAsP layers, as illustrated in Fig. 2. In the analytical derivation, we
will rely on the approximation of infinite barrier height for the wave
functions that will be used to determine interaction matrix elements. In the
numerical calculations, we will consider finite barrier height for the
confinement energies implemented to account for subband positions and
fillings. This impacts the number of subbands existing in the well, as well as
their minima (subband edge energy).

For the scattering between LO and acoustic phonons, we will use the following
result RidleyBook

$\frac{1}{\tau^{LO-
ac}_{\mathbf{q}}}=\frac{\Gamma^{2}\hbar\omega^{3}_{\mathbf{q}}(N_{\mathbf{q}_{1}}+N_{\mathbf{q}_{2}}+1)}{32\pi\rho
v^{3}_{s}}=\frac{N_{\mathbf{q}_{1}}+N_{\mathbf{q}_{2}}+1}{\tau^{LO-ac}_{0}}\
.$ (10)

In this expression, $N_{\mathbf{q}_{1(2)}}$ is the population of acoustic
phonons at $T_{L}$ at wave vector $\mathbf{q}_{1(2)}$ satisfying
$\mathbf{q}=\mathbf{q}_{1}+\mathbf{q}_{2}$. $\rho$ is the mass density of the
material, $v_{s}$ the speed of sound wave, $\Gamma$ is a deformation potential
coefficient related to the material’s Gruneisen parameter by RidleyBook

$\Gamma=\sqrt{\frac{4}{3}}\frac{\gamma\omega_{q_{\perp},m}}{v_{s}}\ ,$ (11)

giving the zero temperature LO phonon-acoustic phonon scattering time

$\tau^{LO-ac}_{0}=\frac{32\pi\rho
v^{3}_{s}}{\Gamma^{2}\hbar\omega^{3}_{q_{\perp},m}}.$ (12)

In the 3-phonon scattering process that converts an LO phonon into two
acoustic phonons, the energy conservation law mandates that
$\hbar\omega_{\mathbf{q}}=\hbar\omega_{\mathbf{q}_{1}}+\hbar\omega_{\mathbf{q}_{2}}$.
While there are virtually infinitely many possible pairs of
($\mathbf{q}_{1},\mathbf{q}_{2}$) that satisfy the above energy conservation
equation for each given $\omega_{\mathbf{q}}$, we consider a simplifying
assumption of equal energy partition normally employed in literature:
$\hbar\omega_{\mathbf{q}_{1}}=\hbar\omega_{\mathbf{q}_{2}}=\hbar\omega_{\mathbf{q}}/2$
Klemens . It gives the following population numbers of acoustic phonons

$N_{\mathbf{q}_{1}}=N_{\mathbf{q}_{2}}=\frac{1}{\exp(\frac{\hbar\omega_{\mathbf{q}}}{2k_{B}T_{L}})-1}.$
(13)

Acoustic phonons are at equilibrium at $T_{L}$, in addition, they are not
confined. Thus, Eqs. (10-13) will be used in the forthcoming 2D and 3D
numerical calculations with

$\tau^{LO-ac}_{\bf q}=\tau^{LO-
ac}_{0}\frac{e^{\frac{\hbar\omega_{LO}}{2k_{B}T_{L}}}-1}{e^{\frac{\hbar\omega_{LO}}{2k_{B}T_{L}}}+1}.$
(14)

For InGaAs quantum well, $\Gamma$ of Eq. (11) can be estimated from
experimental values for InAs and GaAs Sparks . For $T_{L}=300$ K, we thus have
$\tau^{LO-ac}_{0}=2.2\ 10^{-11}$ s from Eq. (12), and $\tau^{LO-ac}_{\bf
q}\simeq 0.74\ 10^{-11}$ s from Eq. (14) (see Table I of Supp. Mat. for
parameter values).

Figure 2:  Illustration of the band diagram of a finite quantum well potential
of InGaAs layer hosting a few confined states of electron and holes. For the
calculations, $E_{g}=0.8$ eV,$V_{\mathrm{con}}=0.235$ eV, and
$V_{\mathrm{val}}=0.185$ eV for the energy gap, the conduction band well
depth, and valence band well depth respectively. All other parameters used in
our calculations are given in Supp. Mat.

.

###  III.1 Confined electron and unconfined phonon states

In the proposed derivation, we will differ from the models developed for
quantum well heterostructures in the 80’s Rucker91 ; Babiker ; RidleyPRB ;
RidleyJPC ; Huang  which considered the limit where both electrons and phonons
are perfectly confined in the quantum well, assumed to be an infinite
potential well in first approximation. Instead, we propose to consider
unconfined phonon states with several arguments. First of all, we rely on the
conclusion put foward in Ref. Rucker91  that the impact of dimensionality in
electron-LO phonon scattering rate so dramatically depends on the model used
to account for phonon confinement that it is safer to use bulk phonon states.
Moreover, such a simplified treatment has been widely adopted in theoretical
literature Price1981 ; dasSarmaMason . Experimentally, it is reasonably
realized in lattice-matched systems, where phonons are generally assumed to
keep their bulk properties. Finally, recent theoretical works ZhangDysonRidley
which compared model based on bulk-like phonon and more sophisticated approach
based on dielectric continuum model RidleyBook2  where phonons can be confined
also show that the difference in the result is relatively insignificant.

The corner stone of the thermalization power defined in Eq. (1) is the rate
equation of the LO phonon number. Using the Fermi golden rule, it can be
written for bulk systems Tsai

$\frac{dN_{\mathbf{q}}}{dt}\Bigr{|}_{c-LO}=2\frac{2\pi}{\hbar}|M_{\mathbf{q}}|^{2}\sum_{\mathbf{k}}I^{2}_{\mathbf{k},\mathbf{k}-\mathbf{q}}[(N_{\mathbf{q}}+1)f_{\mathbf{k}}(1-f_{\mathbf{k}-\mathbf{q}})$
$-N_{\mathbf{q}}f_{\mathbf{k}-\mathbf{q}}(1-f_{\mathbf{k}})]\delta(E_{\mathbf{k}}-E_{\mathbf{k}-\mathbf{q}}-\hbar\omega_{\mathbf{q}})$
(15)

where $M_{\mathbf{q}}$ is the electron-phonon scattering matrix element at
wave vector $q$, $I^{2}_{\mathbf{k},\mathbf{k}-\mathbf{q}}$ represents the
overlap of the wave functions of electron and phonon before and after the
electron-LO phonon scattering (LO phonon emission or absorption process),
$N_{\mathbf{q}}$ is the number of phonon at wave vector $\mathbf{q}$ and
$f_{\mathbf{k}}$ is the Fermi-Dirac distribution of the electron at wave
vector $\mathbf{k}$. The extra overall factor 2 in front of the right hand
side counts for the spin degeneracy. It is emphasized that Eq. (15) is valid
only for 3D theory because the momentum conservation applies to all the three
orthogonal directions during the electron-phonon scattering process. In the 3D
theory derived in Ref. Tsai , the thermalization power finally takes the form
of Eq. (6) with the scattering rate given Eq. (7).

In the presence of confinement, the above scattering rate has to be modified
appropriately. The wave vectors for carriers will be noted ${\bf k}=({\bf
k}_{\perp},k_{z})$. The longitudinal wave vector component $k_{z}$ takes
discrete values of index $n$, that are related to the well width $L_{z}$
TannoudjiQM . Only transverse components of the wave vectors
($\mathbf{k}_{\perp},\mathbf{k}^{\prime}_{\perp}=\mathbf{k}_{\perp}-\mathbf{q}_{\perp}$)
satisfy momentum conservation. The rate of change of phonon number becomes

$\frac{dN_{\mathbf{q}_{\perp},q_{z}}}{dt}\Bigr{|}_{c-LO}=2\frac{2\pi}{\hbar}|M_{\mathbf{q}_{\perp},q_{z}}|^{2}\sum_{\mathbf{k}_{\perp}}\sum_{n,n^{\prime}}I^{2}_{\mathbf{k}_{\perp},n;\mathbf{k}_{\perp}-\mathbf{q}_{\perp},n^{\prime}}$
$\times\Bigl{(}(N_{\mathbf{q}_{\perp},q_{z}}+1)f_{\mathbf{k}_{\perp},n}(1-f_{\mathbf{k}_{\perp}-\mathbf{q}_{\perp},n})-N_{\mathbf{q}_{\perp},q_{z}}f_{\mathbf{k}_{\perp}-\mathbf{q}_{\perp},n^{\prime}}(1-f_{\mathbf{k}_{\perp},n})\Bigr{)}$
$\times\delta(E_{\mathbf{k}_{\perp},n}-E_{\mathbf{k}_{\perp}-\mathbf{q}_{\perp},n^{\prime}}-\hbar\omega_{\mathbf{q}_{\perp},q_{z}})$
(16)

where the electron-phonon scattering matrix element is given by Tsai

$|M_{\mathbf{q}}|^{2}=\frac{e^{2}\hbar\omega_{q_{\perp},q_{z}}}{2\varepsilon_{0}q^{2}V^{ph}}\left(\frac{1}{K_{\infty}}-\frac{1}{K_{s}}\right)\
,$ (17)

with $q^{2}=(q_{\perp}^{2}+q_{z}^{2})$ for the 2D formulation,
$\omega_{q_{\perp},q_{z}}=\omega_{LO}$ as we have supposed LO phonons to be
dispersionless, and $V^{ph}=SL^{ph}_{z}$ the volume of the LO phonon bath (not
necessarily equal to the carrier volume) for which $S=L^{2}_{\perp}$ is the
cross-sectional area of the device. Moreover, $e$ is the electron charge,
$\varepsilon_{0}$ the vacuum permittivity, $K_{\infty}$ the dielectric
constant at infinite frequency and $K_{s}$ the static dielectric constant.

The wave function overlap $I^{2}_{\mathbf{k},\mathbf{k}-\mathbf{q}}$ occurring
in Eq. (16) is given in terms of the following factor Price1981 ;
dasSarmaMason

$I^{2}_{\mathbf{k}_{\perp},n,\mathbf{k}_{\perp}-\mathbf{q}_{\perp},n^{\prime}}=|G_{n,n^{\prime}}(q_{z})|^{2}\
,$ (18)

where

$G_{n,n^{\prime}}(q_{z})=\int^{L_{z}}_{0}dz\psi_{n}(z)\exp(iq_{z}z)\psi^{*}_{n^{\prime}}(z)\
,$ (19)

with

$\psi_{n}(z)=\sqrt{\frac{2}{L_{z}}}\sin\left(\frac{n\pi z}{L_{z}}\right)\ ,$
(20)

relying on the approximation of infinite barrier height. The calculation leads
to

$G_{n,n^{\prime}}(q_{z})=\frac{iL_{z}q_{z}}{(n\pi-n^{\prime}\pi+L_{z}q_{z})(-n\pi+n^{\prime}\pi+L_{z}q_{z})}$
$+\frac{iL_{z}q_{z}}{(n\pi+n^{\prime}\pi-
L_{z}q_{z})(n\pi+n^{\prime}\pi+L_{z}q_{z})}$
$+\frac{-iL_{z}q_{z}e^{iL_{z}q_{z}}\cos[(n-n^{\prime})\pi]}{(n\pi-n^{\prime}\pi+L_{z}q_{z})(-n\pi+n^{\prime}\pi+L_{z}q_{z})}$
$-\frac{iL_{z}q_{z}e^{iL_{z}q_{z}}\cos[(n+n^{\prime})\pi]}{(n\pi+n^{\prime}\pi-
L_{z}q_{z})(n\pi+n^{\prime}\pi+L_{z}q_{z})}\,.$ (21)

The profile of $G_{n,n^{\prime}}(q_{z})$ is illustrated in Fig. 3, reflecting
a distribution maximum at $q_{z}=0$ which sharpens with increasing the quantum
well thickness. The overlap factor has a symmetry property
$G_{n,n^{\prime}}(q_{z})=G_{n^{\prime},n}(q_{z})$.

Figure 3:  Profile of the overlap factor $|G_{11}(q_{z})|^{2}$ from Eq. (21),
for thicknesses $L_{z}=5.5$, $10$ and $15$ nm.

###  III.2 Intra-subband and inter-subband scattering rates

The next step is the derivation of the electron-phonon scattering rate
$dN_{\mathbf{q}_{\perp},q_{z}}/dt$, which was done with the hypothesis that
thermalization occurs only in the conduction band, due to the large effective
mass values of holes inside the valence band. Such an hypothesis helps to
decipher the thermalization power behavior, but it remains easy to include all
band contributions.

We evaluate the expression for the rate of change of the number of LO phonons
Eq. (16) using the electron-phonon scattering matrix element Eq. (17) and the
overlap factor Eq. (18). The details of the derivation are given in Supp. Mat.
Defining handy notations, for the scattering time,
$\tau^{c-LO}_{\mathbf{q}_{\perp},q_{z}}$, and its inverse the scattering rate
$R_{\mathbf{q}_{\perp},q_{z}}$, it can be written as

$\frac{dN_{\mathbf{q}_{\perp},q_{z}}}{dt}\Bigr{|}_{c-LO}=\frac{N_{\mathbf{q}}(T_{c})-N_{\mathbf{q}_{\perp},q_{z}}}{\tau^{c-LO}_{\mathbf{q}_{\perp},q_{z}}}$
$=R_{\mathbf{q}_{\perp},q_{z}}(N_{\mathbf{q}}(T_{c})-N_{\mathbf{q}_{\perp},q_{z}})$
(22)

where

$\frac{1}{\tau^{c-LO}_{\mathbf{q}_{\perp},q_{z}}}=\frac{m_{c}e^{2}\omega_{LO}}{\pi\varepsilon_{0}\hbar^{2}q_{\perp}}\left[\frac{1}{K_{\infty}}-\frac{1}{K_{s}}\right]\frac{1}{(q^{2}_{\perp}+q^{2}_{z})L^{\mathrm{phon}}_{z}}\
\ \times$
$\sum_{n,n^{\prime}}|G_{n,n^{\prime}}(q_{z})|^{2}\int^{+\infty}_{k^{\mathrm{min}}_{\perp}(n,n^{\prime})}dk_{\perp}\frac{k_{\perp}}{\sqrt{k^{2}_{\perp}-(k^{\mathrm{min}}_{\perp}(n,n^{\prime}))^{2}}}$
$\times\left[f(E_{\mathbf{k}_{\perp},n}-\hbar\omega_{LO})-f(E_{\mathbf{k}_{\perp},n})\right]\
.$ (23)

We have defined

$k^{\mathrm{min}}_{\perp}(n,n^{\prime})=\Bigl{|}\frac{q_{\perp}}{2}+\frac{m_{c}\omega_{LO}}{\hbar
q_{\perp}}+\frac{({n^{\prime}}^{2}-n^{2})}{2q_{\perp}}\frac{\pi^{2}}{L^{2}_{z}}\Bigr{|}$
(24)

the minimum transverse wave vector for an electron to emit an LO phonon.

Upon emitting a phonon of energy $\hbar\omega_{LO}$, an electron may stay in
the same subband, which constitutes an intra-subband scattering process, or
move to another subband, which represents an inter-subband scattering process.
Splitting the intra-subband and inter-subband contributions, the LO phonon
emission rate for each phonon wave vector
$\mathbf{q}=(\mathbf{q}_{\perp},q_{z})$ can be written as

$\frac{1}{\tau^{c-LO}_{\mathbf{q}_{\perp},q_{z}}}=\sum^{N_{s}}_{n=1}\frac{1}{\tau^{c-LO}_{\mathbf{q}_{\perp},q_{z}}(n,n)}+\sum^{N_{s}}_{n,n^{\prime}=1(n\neq
n^{\prime})}\frac{1}{\tau^{c-LO}_{\mathbf{q}_{\perp},q_{z}}(n,n^{\prime})},\ $
(25)

with $N_{s}$ the number of subbands inside the well. We have thus introduced
$1/\tau^{c-LO}_{{\bf q}_{\perp},qz}(n,n^{\prime})$ as the net phonon emission
rate from subband $n$ to subband $n^{\prime}$. It is not symmetric upon
$n-n^{\prime}$ permutation. It is important to note that our theory only takes
into account the scattering between electron states confined in the quantum
well and the LO phonons. That is, we do not consider scattering processes
involving electron states whose energy are higher than the InGaAsP barrier
height in Fig. 2.

The numerical results for the summed scattering rates

$R_{nn^{\prime}}=L^{ph}_{z}\int\frac{dq_{z}}{2\pi}\frac{1}{\tau^{c-LO}_{{\bf
q}_{\perp},qz}(n,n^{\prime})}$ (26)

are shown in Fig. 4 both for intra-subband $(n^{\prime}=n)$ and inter-subband
$(n^{\prime}\neq n)$ processes. We observe that he intra-subband contribution
of the first subband, $R_{11}$, gives the largest contribution to the
electron-phonon scattering rate, and that intra-subband contributions of
higher subbands decrease with the subband index, a conclusion which is in
agreement with that of the classic work by Riddoch-Ridley Ridley2 . Moreover,
for small thicknesses where only few subbands exist in the well ($N_{s}=2$ for
$L_{z}=10$ nm and $N_{s}=3$ for $L_{z}=15$ nm), the $R_{21}$ inter-subband
term is much larger than other inter-subband contributions to the scattering
rate. Eventually, as the thickness is increased further, the number of
subbands increases, and the inter-subband scattering rate (summing over all
inter-subband processes) overtakes that of the first intra-subband scattering
rate.

Figure 4:  Intra (grey curves) and inter-subband (colored curves) scattering
rates $R_{n,n^{\prime}}$ (see Eq. (24) for definition) in 10 nm a) and b), and
15 nm c) and d) quantum wells, as functions of $q_{\perp}$ for $T_{c}=450$K
and $T_{L}=300$K. Other parameters are $m_{c}=0.041m_{0}$ with $m_{0}$ the
bare electron mass, $\hbar\omega_{LO}=36$meV, $K_{\infty}=10.9$, $K_{s}=12.9$,
while the chemical potential $\mu_{c}=-35.9074$ meV for $10$ nm and
$\mu_{c}=-37.7681$ meV for $15$ nm quantum wells.

###  III.3 Impact of non-equilibrium phonon distribution hypothesis

A key mechanism that is believed to drive hot carrier effect is the occurrence
of a non-equilibrium distribution of LO phonons. This hypothesis means that
the LO phonon occupation number do not match a Bose-Einstein function defined
at the lattice temperature $T_{L}$ with which acoustic phonons stay in
equilibrium. The effect of a non-equilibrium LO phonon distribution seems to
be significant in bulk 3D systems Tsai , but this is not necessarily the case
in quantum well heterostructures, due to the quantum confinement which
constrains the interaction between electrons and phonons.

In this section, we therefore investigate the relative strengths of the
effects of confinement and non-equilibrium phonon distribution on the
electron-LO phonon scattering rate. With LO phonons characterized by an
a-thermal distribution specified in Eq. (5), the thermalization power is given
by Eq. (6), while, with LO phonons at the lattice temperature $T_{L}$,
henceforth called equilibrium phonons, the thermalization power is given by
Eq. (8). To this end, using Eqs. (23) and (14), we plot in Fig. 5 the two
integrated scattering rates

$R^{\rm{NonEq}(\rm{Eq})}({\bf{q}_{\perp}})=L^{ph}_{z}\int\frac{dq_{z}}{2\pi}\frac{1}{\tau^{c-LO}_{\bf{q}_{\perp},q_{z}}+(\tau^{LO-
ac}_{\bf q})}\,,$ (27)

where $\tau^{LO-ac}_{\bf q}$ in parenthesis is present or absent respectively
in the non-equilibrium (NonEq) and equilibrium (Eq) cases. With the same
convention, the 3D results, $1/(\tau_{{\bf q}}^{c-LO}+(\tau^{LO-ac}_{\bf q}))$
from Eqs. (7) and (14) are also represented. In the figure, the solid curves
encompass a screening effect that will be discussed in the following section.
Focussing on the dashed curves of Fig. 5, the results show that the expected
reduction of the scattering rate due the non-equilibrium phonon distribution
hypothesis is suppressed for thinner wells, while it remains strong in the 3D
case.

The ratio $\tau^{LO-ac}_{\bf q}/\tau_{{\bf q}}^{c-LO}$ provides a simple tool
to explain why the LO phonons have a weaker tendency to go out-of-equilibrium
in quantum wells. On the one hand, the scattering rate
$1/\tau^{c-LO}_{\mathbf{q}}$ is strongly reduced due to confinement relatively
to bulk, as shown in section III.2. On the other hand, the LO phonon-acoustic
phonon scattering rate remains unchanged, as phonons are assumed to be
unconfined, $1/\tau^{LO-ac}_{\mathbf{q}}\simeq 1.35\,10^{11}$ s-1. Starting
from the 3D case, the characteristic times are in ratio $\tau^{LO-ac}_{\bf
q}/\tau_{{\bf q}}^{c-LO}\simeq 30$ for the wave vector corresponding to the
maximum of the scattering rate, the corresponding phonon number $N_{\bf q}$
from Eq.(5) is then leaning towards $N_{\bf q}(T_{c})$: the a-thermal
distribution of LO phonons approaches a hot equilibrium distribution. In
contrast, in the quasi 2D case, and more particularly for narrow quantum
wells, the scattering rate $1/\tau^{c-LO}_{\mathbf{q}}$ is strongly reduced
due to confinement, the reduction can be as large as 2 orders of magnitude
examining Fig. 5. The ratio $\tau^{LO-ac}_{\bf q}/\tau_{{\bf q}}^{c-LO}$ thus
becomes smaller than 1, which steers the corresponding phonon number towards
$N_{\bf q}(T_{L})$: the a-thermal distribution of LO phonons approaches the
lattice temperature equilibrium distribution of acoustic phonons. This
analysis may be interpreted as follows: in quantum well solar cells, that once
an LO phonon is emitted by an electron, it never gets re-absorbed back by the
electron. Instead, the LO phonon quickly transforms into acoustic phonons
which stay in equilibrium at the lattice temperature. In other words, LO
phonons quickly cool down to the lattice temperature, suppressing the non-
equilibrium LO phonon effect. Hot carrier effect is thus controlled by
confinement in reduced dimensionality.

Figure 5:  Electron-LO phonon scattering rates as defined in Eq. (27) for a
non-equilibrium (NonEq, red curves) and an equilibrium (Eq, blue curves) LO
phonon distribution, as functions of $q_{\perp}$. Solid and dashed lines
correspond to the cases with and without screening respectively. Results are
shown for three quantum wells of thicknesses $L_{z}=5.5$, $10$, and $15$nm, as
well as for bulk 3D model. For the sake of clarity, 3D scattering rates for
non-equilibrium phonons have been multiplied by a factor 5 (lower right
quadrant). The parameter values are the same as in Fig. 4.

###  III.4 Dependence on Electron-Electron Screening Effect

So far, we did not consider the screening of the electron-phonon interaction
in Eq.(17). However, screening ultimately renormalizes the scattering matrix
element $|M_{\mathbf{q}_{\perp},q_{z}}|^{2}$ of Eq. (16) as

$|M^{\mathrm{screen}}_{\mathbf{q}_{\perp},q_{z}}|^{2}=|M_{\mathbf{q}_{\perp},q_{z}}|^{2}\left(\frac{\varepsilon_{0}}{\varepsilon_{\mathrm{screen}}(\mathbf{q}_{\perp},q_{z},\omega_{LO})}\right)^{2}.$
(28)

Making a static and strictly 2D approximation, we have to evaluate the
following screening function RidleyBook2

$(\varepsilon_{\mathrm{screen}})^{n_{4},n_{3}}_{n_{2},n_{1}}(\mathbf{q}_{\perp},0)=\varepsilon_{0}\delta_{n_{2},n_{4}}\delta_{n_{1},n_{3}}$
$-\frac{e^{2}}{2q_{\perp}S}F^{n_{4}n_{3}}_{n_{2}n_{1}}(\mathbf{q}_{\perp})\sum_{{\mathbf{k}_{\perp}},\sigma}\frac{f(E_{n_{2},\mathbf{k}_{\perp}+\mathbf{q}_{\perp}})-f(E_{n_{1},\mathbf{k}_{\perp}})}{E_{n_{2},\mathbf{k}+\mathbf{q}_{\perp}}-E_{n_{1},\mathbf{k}_{\perp}}}$
(29)

where $F^{n_{4},n_{3}}_{n_{2},n_{1}}(\mathbf{q})$ is the form factor of the
electron-electron interaction involving four indices for the confined electron
subbands $n_{1},n_{2},n_{3},n_{4}$. $\sum_{\sigma}$ indicates the sum over the
spin.

The number of form factor tensor elements involved in the calculation
increases with the number of subbands, going as $N^{4}_{s}$ where $N_{s}$ is
the number of subbands. For the sake of simplicity, we make the approximation
of accounting for only the screening of the first intra-subband process, with

$F^{11}_{11}=\frac{1}{2}\frac{\eta(\eta^{2}+\pi^{2})(3\eta^{2}+2\pi^{2})-\pi^{4}(1-e^{-2\eta})}{[\eta(\eta^{2}+\pi^{2})]^{2}}$
(30)

where $\eta=q_{\perp}L_{z}/2$ RidleyBook2 . We thus ignore the screening of
all other scattering processes. This is justified by the fact that the first
intra-subband scattering rate $R_{11}$ strongly dominates over all other
scattering rates $R_{nn^{\prime}}$, as can be seen from Fig. 4. As a
consequence, the present approximation which underestimates the screening
effect, overestimates most scattering rates.

Figure 5 gathers the effects of screening on the electron-phonon scattering
rates in the bulk and in quantum wells, for LO phonons out-of equilibrium or
at equilibrium with the lattice. As a general trend, screening renormalizes
downward the scattering rate, whatever the hypothesis made for the LO phonon
distribution. The reduction is more pronounced for wider quantum wells up to
the 3D limit. However, at the smallest thickness where the confinement effect
is strongest, it appears that the hypothesis of out-of equilibrium LO phonons
has only a small influence on scattering rates due to an increasing impact of
$\tau^{LO-ac}_{\bf q}$, whatever screening is included, which complies with
the analysis we have developed section III.3.

##  IV Signatures of Quantum Confinement and Enhancement of Hot Carrier Effect
in terms of Thermalization Power

###  IV.1 Calculation protocol

The objective of this section is to discuss the power needed to maintain a
carrier population at fixed temperature $T_{c}$ and quasi-Fermi level
splitting $\Delta\mu$ PWurfel , when confinement increases in
InGaAsP/InGaAs/InGaAsP quantum well heterostructures shown Fig. 2. These two
thermodynamic parameters have been extracted from photoluminescence
experiments done for a particular quantum well, which makes the obtained
conclusions more reasonable and exploitable to interpret real situations. We
thus have $T_{c}=450$ K and $\Delta\mu=0.653$ eV, and we keep it constants all
along the study. For each quantum well thickness, these values are used to
numerically estimate the chemical potential $\mu_{c}$ from the
electroneutrality equation. Details of the calculation protocol are given in
Supp. Mat. For LO phonons, we take
$\hbar\omega_{\mathbf{q}}\equiv\hbar\omega_{\mathbf{q}_{\perp},m}=\hbar\omega_{LO}=36$
meV in the calculations, independent of wave vector. Additionnally, acoustic
phonons are assumed to be in an equilibrium distribution at the lattice
temperature $T_{L}=300$ K. The 2D thermalization power is finally computed
within our theoretical model combining Eqs. (23), (9) and (14).

###  IV.2 Findings: 2D thermalization power

We compare the thermalization power resulting from the proposed 2D theoretical
model with the one obtained from the 3D model of Ref. Tsai , within different
frameworks of hypothesis shown Fig. 6: a) without (in blue) and b) with (in
red) the assumption that LO phonon distribution is non-equilibrium following
the derivations of section III.3, and including or not the screening effects
as described in section III.4.

The first important observation that can be made is that the thermalization
power density is lower in 2D than in 3D systems, whatever the used
assumptions. It means that the amount of power needed to achieve a hot carrier
population at given temperature and Fermi level splitting is lower in quantum
well heterostructures than in the bulk material. The result suggests that
reducing the system’s dimensionality reduces the system’s propensity for
thermalization. Confinement thus enhances the hot carrier effect, which is
expected to make quantum well based solar cells more efficient, due to the
fact that the quantification of electronic states increases the amount of
carrier energy to be converted into electrical power because it reduces the
thermalization power. Actually, the reduction of the scattering rate in
quantum wells was also pointed out in more sophisticated models of electron-LO
phonon interaction Lassnig ; RiddochRidley ; Sawaki . Moreover, the
thermalization power reduction of about one order of magnitude is consistent
with existing works in the literature, e.g. MonteCarlo .

Figure 6: Thermalization power per unit volume as a function of the quantum
well thickness in InGaAs heterostructure, considering a) an equilibrium (in
blue) b) a non-equilibrium (in red) LO phonon distribution, with (filled
circles) and without (empty circles) screening effect of electrons computed
from the 2D model using Eq. (9), and from the 3D model using Eq. (6). Input
parameters are $T_{c}=450$ K, $\Delta\mu=0.653$ eV and $T_{L}=300$ K.

In the 3D bulk case, the thermalization power density is constant, as can be
seen in Fig. 6. This behavior is in agreement with experimental results where
the thermalization power is found to vary linearly with the thickness of the
absorber for relatively thick samples of 100-200 nm, where the physics
approaches well that of a bulk system GiteauJAP . In contrast, the 2D
thermalization power density increases with the quantum well thickness, but
remains always lower than the value computed in the 3D model under the same
conditions: LO phonons in equilibrium or not, with screening effect taken into
account or not. As a general trend, the assumption of a non-equilibrium LO
phonon distribution always reduces the thermalization power, by an amount
decreasing with the quantum well thickness for the 2D model whereas it is
independent of thickness for the 3D model.

Moreover, a non-trivial dependence on the quantum well thickness clearly
appears in the insets of Figs. 6 a) and b). Ignoring screening (empty
circles), the thermalization power exhibits a staircase shape with a quasi-
plateau for thicknesses between 8 and 16 nm, which reflects the evolution of
subband filling as the quantum well thickness increases. At small well
thicknesses, each subband starts to fill significantly when the energy
separation between the subband minimum and the Fermi level goes bellow
$4k_{B}T_{c}\approx 4\times 39$ meV (at which Fermi occupation goes beyond
0.01). This evolution is represented for the two first subbands in Fig. 7. As
the well thickness increases, the filling increases. However in 2D systems,
the density of states is proportional to $1/L_{z}$ inside each subband (for a
fixed Fermi energy), so the carrier density must finally decrease at larger
well thickness.

Figure 7: First and second conduction subband minima and quasi Fermi level, as
a function of the quantum well thickness. The energy origin is at the bulk
conduction band minimum.Thermal energy, $k_{B}T=0.039$ eV, is also
represented.

The thermalization increase by plateau is thus interpreted by distinguishing
the intra- and inter-subband contributions to the 2D thermalization power of
the first and second subbands, as represented Fig. 9.

Figure 8: Detailed subband contributions to the 2D thermalization power per
unit volume as a function of the quantum well thickness, in the simplest
framework of an equilibrium LO phonon distribution, without screening effect.

At first, the separation energy between the first subband minimum and the
quasi Fermi level decreases below $4k_{B}T_{c}$, while the energy separation
between the second subband and the quasi Fermi level remains above this value,
as can be seen Fig. 7. Consequently, the intraband contribution of the first
subband increases then decreases when the $1/L_{z}$ term dominates. In a
second stage, above 12 nm, the second subband approaches the quasi Fermi level
$\mu_{c}$, it starts to fill and participates to thermalization while the
first subband depopulates. The first intra-subband contribution to power
decreases while both the inter- and intra- second subband contributions
increase. In total, the thermalization power plotted Fig. 6 increases from
smaller well thicknesses, then saturates when the contributions of two first
subbands compensate, next starts to increase again when intra- and inter-
second subband components increase faster than the intra- first subband
component decreases. In these confining heterostructures, the thermalization
process thus follows several channels that are activated one after the other.
The occurence of a plateau transition depends on temperature, but also on
confinement energies, that are directly related to material nature throughout
effective mass and barrier potential. The proposed analysis also explains why
the 2D thermalization power remains far from the 3D value at 25 nm, because
only two subbands are at play for this thickness value, while there are
already five subbands inside the well at 25 nm.

Finally, screening in general also reduces thermalization power, by an amount
that varies with thickness in the 2D model, while it is independent of
thickness in the 3D model. In particular, the screening-induced reduction of
power is non-monotonic in quantum well heterostructures. As a result, the
thermalization power density passes through a minimum in the plateau region,
before increasing again as one keeps increasing the thickness of the quantum
well layer. This non-monotonic behavior thus emerges as a signature of energy
quantization in 2D systems.

##  V Comparison with Experiments

Real systems employed in experiments are far more complex than what we model
in the present study. We describe a simple quantum well, for which interface
and barrier contributions to the scattering rate are ignored. Moreover, the
screening is only partially accounted for, and the form factors and overlap
integrals that shape the matrix elements of the electron-phonon interaction
and screening potential are estimated in the approximation of infinite well
barriers. Finally, electron-LO phonon interaction is assumed to be the only
mechanism of carrier energy loss, while the crucial role of radiative
recombination have been advanced Hirst . Despite these limitations, the
scattering theoretical model we developed for carrier thermalization in 2D
systems can provide means of interpretation of why quantum well solar cell has
been observed to give rise to a slower cooling rate for carriers than a bulk
solar cell in several experimental works reported in the literature. We
discuss in this section comparison between our theoretical predictions and
available experimental data, which is summarized in terms of quantities that
measure hot carrier effect enhancement in quantum well heterostructure in
Table 1.

One of the first works comparing quantum well and bulk solar cells did observe
a significant reduction of the cooling rate in quantum well solar cells by
more than one order of magnitude, by a factor of 60 Ryan , which is comparable
to our results presented Fig. 6, where a reduction factor of about 10 is
overall obtained. Many later works suggested that such a reduction of the
cooling rate occurs only at a high enough charge density Rosenwaks . In these
earlier works, the relative role of confinement, screening, and non-
equilibrium hot LO phonon effects have not been addressed in detail.

A pioneering work toward a detailed investigation of the different factors
that drive the hot carrier effect in 2D systems was given by Shah and co-
workers in Ref. ShahPRL . They found that the energy loss rate is about 25
times smaller for electrons than for holes in semiconductor quantum wells
(segment on the ratio of thermalization powers for (heavy) hole and electron
in Table 1). This observation can easily be explained within the framework of
kinetic scattering theory in terms of the effective mass dependence of the
electron-LO phonon scattering rate. As can be seen from Eq. (19) of Supp.
Mat., the scattering rate depends on the $3/2$ power of the carrier effective
mass $m_{c}$ in the leading order. For GaAs, the heavy hole effective mass
$m_{h}=0.5m_{0}$ is about 7 times larger than the electron effective mass
$m_{e}=0.07m_{0}$, where $m_{0}$ is the electron mass, implying that the
cooling rate of holes must be about $7\sqrt{7}\simeq 19$ times larger than
that of electrons, providing a reasonable explanation for the observation
reported in Ref. ShahPRL . Furthermore, the measured energy-loss rate was
found to be approximately 8 times smaller than expected for 2D electrons based
on existing theoretical calculations at that time Ridley2 . This reduction
factor is off only by $20\%$ relative to the value obtained from our
theoretical results of Fig. 6 b), in which a reduction factor of about 10 is
obtained at the minimum of the thermalization power density, where both non-
equilibrium LO phonon and screening effects are included (segment on reduction
factor in Table 1). In Ref. ShahPRL , this reduction was argued to be caused
by non-equilibrium LO phonons whose population exceeds the one expected from
equilibrium phonons at lattice temperature, rather than by confinement ShahPRL
.

In contrast, our findings gathered in Fig. 6 show that confinement, non-
equilibrium LO phonons, and screening effects, all contribute to reducing the
electron-phonon scattering rate in quantum well relative to bulk solar cells.
However, confinement has been concluded to be the principal factor responsible
for reducing the scattering rate for the smallest quantum well thicknesses.
Reducing the system dimensionality exhibits a stronger impact than the one of
non-equilibrium LO phonon or electrostatic screening. Within the 2D semi-
classical framework that we have built, the effect of an a-thermal LO phonon
distribution is suppressed due to the confinement-induced reduction of the
electron-LO phonon scattering rate which annihilates the acoustic phonon
contribution in the LO phonon dynamics, as implied by Eq. 27. Remarkably, this
relative feeble role of the non-equilibrium hot LO phonons effect in the
carrier thermalization process seems to be in agreement with the result of a
classic experiment on GaAs quantum well Lyon .

The use of a multi-quantum well or super-lattice system to enhance efficiency
over a bulk GaAs solar cell was first proposed in Ref. BarnhamDuggan . The
suppression of the radiative recombination in the quantum well compared to
that in the bulk regions of the cell was first demonstrated in Ref. Nelson .
Ref. EkinsDaukes  demonstrated the excellent material quality of a strain-
balanced GaAsP/InGaAs multi-quantum well. Such material quality enabled the
observation of hot carrier effects in these wells in Ref. Hirst0 . Ref. Mazzer
proposed that the suppressed radiative recombination in the quantum wells,
observed in Ref. Nelson , was due to the thermoelectric (Seebeck) effect
resulting from the presence of hot carriers in the wells.

The hot carrier effect has in fact been confirmed in the strain-balanced
quantum wells, especially in the so-called “deep well” samples, for which it
was shown that the hot carrier effect is stronger in deep than shallow well
samples Hirst0 . The impact of the quantum well depth can be explained
following the arguments pointed out in section IV.2. Indeed, the deeper the
well is, the larger the inter-subband separation energy is, which supports the
interpretation that a fewer number of subbands channels thermalization in the
case of “deep well” samples, due to the fact that only a fewer number of
subbands are able to drive the thermalization process, although a deep well
accommodates more subbands. This interpretation is also in agreement with
recent experimental studies on strain-balanced multiple InGaAs/GaAsP quantum
wells solar cells, in which it was shown that a deeper quantum well gives rise
to significantly higher effective carrier temperature, attributed to hot
carrier dynamics Hirst . Indeed, a higher carrier temperature means a weaker
system’s propensity to loss its energy harvested from light, and thus a lower
thermalization power. Overall, considering deeper wells augments the degree of
confinement inside the solar cell absorbers, which is concluded to enhance the
hot carrier effect.

A quantitative measure of this enhancement of the hot carrier effect with a
deeper well is given in Table 1 in the segment on well depth dependence.
There, we compare a quantity $\eta_{PV}$ which measures how much
thermalization power changes relative to the change in the potential barrier
height (or equivalently, well depth); $\eta_{PV}<0(>0)$ means lower(higher)
thermalization power with deeper well or, equivalently, stronger (weaker) hot
carrier effect respectively. The experimental value is estimated from
thermalization powers at $T_{c}=450$K extracted from Fig. 2(a) of Hirst0  for
samples with 20% In content ($V^{(2)}_{c}=0.25$eV) and 11% In content
($V^{(1)}_{c}=0.17$eV) respectively. The theoretical counterpart values are
from our calculation. The thermalization power is found to decrease with
deeper well, giving $\eta_{PV}<0$ for all the four theoretical hypotheses, in
agreement with the experiment. The experimental reduction factor is about
-0.98, closest to our theoretical result with non-equilibrium phonon no
screening hypothesis.

Hot carrier effect assessment parameter | Hypothesis | Our 2D Theory | Experiment
---|---|---|---
| (2D Theory) | (InGaAs/InGaAsP) |
Reduction factor | Eq. NS | 11 | 8(GaAs/AlGaAs)ShahPRL
of electron thermalization power $\frac{P^{3D}_{th}}{P^{2D}_{th}}$ | Eq. S | 20 |
(evaluated at $L_{z}=11$ nm as example) | NonEq. NS | 5 |
| NonEq. S | 9 |
Ratio of (heavy) hole thermalization power | Eq. NS | 35 | 25(GaAs/AlGaAs)ShahPRL
to electron thermalization power | Eq. S | 18 |
Leading-order value: 19 (2D theory) | NonEq. NS | 35 |
| NonEq. S | 24 |
Well depth-dependent enhancement factor | Eq. NS | -0.66 | -0.98
$\eta_{PV}=\left(\frac{\left(P^{c-LO}_{\mathrm{th}}\right)_{2}}{\left(P^{c-LO}_{\mathrm{th}}\right)_{1}}-1\right)/\left(\frac{V^{(2)}_{c}}{V^{(1)}_{c}}-1\right)$ | Eq. S | -4.03 | (GaAsP/InGaAs)Hirst0
(evaluated at $L_{z}=11$ nm as example) | NonEq. NS | -0.85 |
($V^{(1)}_{c}=0.235$eV$<V^{(2)}_{c}=0.4$eV for theory) | NonEq. S | -3.69 |
Optimal well thickness $L^{*}_{z}$ (nm) | Eq. NS | - | 8.0 (GaAs/AlAs)ConibeerIEEE
| Eq. S | 11.0 |
| NonEq. NS | - |
| NonEq. S | 11.5 |
Table 1: Comparison of quantities characterizing hot carrier effect in our
theory and available experimental values from slightly different semiconductor
materials but still of III-V family. Eq.(NonEq.) and S (NS) respectively
represent hypothesis with equilibrium (non-equilibrium) LO phonons and with
(without) screening effect. The first segment shows the amount of hot carrier
effect enhancement induced by carrier confinement, relative to bulk system.
The second segment gives an approximate value of the ratio between
thermalization powers of (heavy) hole and electron, first from the leading-
order dependence of thermalization power on carriers (electron and (heavy)
hole) masses in the first column while giving the exact numerical values at
$L_{z}=11$nm in the third column for various hypotheses. In the third segment,
we have defined $\eta_{PV}$ as a measure of the change in thermalization power
relative to the change in well depth; $\eta_{PV}<0(>0)$ means a deeper well
gives a stronger (weaker) hot carrier effect. In the last segment, an optimal
well thickness is defined as the thickness that yields maximum effective
carrier (electron) temperature or, equivalently, minimum thermalization power.
It is to be noted that this table is not intended to reproduce each available
experimental data but to illustrate an overall qualitative and order-of-
magnitude quantitative agreement between our theory and experiments. Figure 9:
Comparison between experimental curve for carrier effective temperature (black
circles) reproduced with permission from Fig. 5 in ConibeerIEEE  with linear
interpolation and our theoretical curve for carrier thermalization power with
non-equilibrium phonon and screening hypothesis (orange circles). The arrow in
each curve marks the optimum well layer thickness for which the hot carrier
effect is maximized. The difference in arrow positions is partly due to
different materials used; GaAs/AlAsConibeerIEEE  and InGaAs/InGaAsP(ours).

Finally, in the investigation conducted by Conibeer et al. ConibeerIEEE ;
ConibeerJJAP , a non-monotonic dependence of the carrier temperature on the
thickness of the quantum well layer was reported, as illustrated in Fig. 9.
Phonon emission was considered there as one of the possible mechanisms in
explaining the observed hot carrier effect. While our calculations are based
on the parameters of InGaAsP-InGaAs-InGaAsP quantum well layers, we can
provide a direct qualitative analysis. We argue that the non-monotonic
dependence of the carrier temperature on quantum well layer thickness in Ref.
ConibeerIEEE  corresponds directly to the non-monotonic dependence of the
thermalization power density plotted Fig. 6 d) (red solid circles in the
inset), within the most realistic framework in which both non-equilibrium LO
phonon and screening effects are taken into account, shown in Fig. 9. Based on
the principle that a lower thermalization power density means a stronger hot
carrier effect, the hot carrier effect first increases from 6.5 nm to 10.5 nm,
but then decreases from 10.5 nm to 25 nm (and beyond, eventually). It suggests
that the carrier temperature should increase when going from 6.5 nm to 10.5
nm, and then decreases in going from 10.5 nm to 25 nm (and beyond). This is in
accordance with the thickness dependence of the carrier temperature displayed
in Fig. 5 of Ref. ConibeerIEEE  in going from 6 nm to 30 nm for the quantum
well layer thickness, while the barrier thickness is fixed to 40 nm. An
optimal thickness of about 11.5 nm in our result and 8 nm in ConibeerIEEE
gives the strongest hot carrier effect, in terms of lowest thermalization
power and highest effective carrier temperature respectively, as summarized in
the last segment in Table 1. Some of the sources of incomplete analysis of
this work of Conibeer and co-workers include the facts that we use different
materials and that our simple model assumes infinitely thick barriers on both
sides compared to the 40 nm thick barrier. Overall, bulk 3D theory gives a
constant power density, which means that there is no dependence of carrier
temperature on the quantum well thickness at all, and thus cannot explain the
investigation of Ref. ConibeerIEEE .

##  VI Conclusion.

In this work, we have derived a kinetic scattering model for determining the
thermalization power in 2D systems, which was used to understand the
confinement-induced thermalization reduction in InGaAsP/InGaAs/InGaAsP quantum
wells. In the proposed theoretical investigation, electron and holes are
assumed to be in two fixed thermal distributions with fixed temperatures and
Fermi level splitting. Moreover, we consider that electron-LO phonon
scattering is the unique channel for carrier cooling. Our main findings are
listed below.

1\. The thermalization power in quantum wells is reduced by carrier
confinement relatively to bulk theory prediction, which occurs because
confinement decreases electron-phonon scattering rate. Reducing dimensionality
is thus demonstrated to enhance the hot carrier effect, and it is finally
expected to improve the efficiency of quantum well solar cells relatively to
solar cells with bulk absorbers.

2\. Intra-subband contribution to electron-phonon scattering rate initially
dominates over inter-subband contribution at relatively small thicknesses
where only few confined electron states exist. As one increases thickness,
inter-subband contribution overtakes that of intra-subband scattering process,
as inter-subband terms outnumber intra-subband ones.

3\. The effects of confinement, a non-equilibrium LO phonon distribution and
screening altogether combine to reduce the electron-phonon scattering and
energy loss rates in quantum well heterostructure. In the 3D case, the
scattering rate reduction due to non-equilibrium phonons is more significant
than in the 2D case, where the impact of the LO phonon disintegration in two
acoustic phonons is indirectly weakened due to the confinement-induced
reduction of thermalization.

4\. In the samples studied, it exists a cross-over between the reduction of
the first intra-subband contribution and the emergence of both intra and
inter-subband contributions of the second subband. It results in a plateau
region, where the thermalization power adopts a 3D-like behavior paradoxically
originating from electronic state quantization. Including the screening effect
then gives rise to a non-monotonic dependence on the well thickness. There is
a finite thickness that minimizes the thermalization power density and thus
maximizes hot carrier effect. For the InGaAsP-InGaAs-InGaAsP quantum wells
under consideration, the optimal thickness is about 10.5 nm. We argue that
this optimal thickness corresponds to the optimal thickness that would drive
to a maximum value of carrier temperature, as observed experimentally
ConibeerIEEE .

Finally, in experiments, there are various other factors that may come into
play and contribute to the scattering process or carrier thermalization, such
as impurity, defect, interface scattering, thermionic emission, and exciton
effects. We leave these complexities as open problems for future works. It is
to be noted that the various additional complexities not considered in our
work tend to provide additional channels to carrier thermalization and thus
tend to increase the resulting thermalization power. The numerical value of
thermalization estimated from our simple model should thus be interpreted as
the lower limit of what one should expect by considering more realistic
situations. Furthermore, our semi-classical approach does not discuss
interface-type LO phonon modes, which requires more rigorous treatment, such
as the one based on dielectric continuum model AndoMori . It is expected that
the inclusion of interface-type LO phonon modes would not change qualitatively
the conclusions of the present work ZhangDysonRidley . Additionally, surface
modes manifest experimentally as a constant additive term on the
thermalization power density per surface unit GiteauJAP . Surface modes are
thus not expected to change qualitatively the dependence of thermalization
power on the thickness of the quantum well (or volume of a bulk sample).

To conclude, this work provides a proof of principle that 2D confinement of
charge carriers in a semiconductor quantum well does indeed give rise to hot
carrier effect and precise the details of how this is accomplished, a finding
that is expected to be critical and timely to the effort in achieving high
solar cell efficiency.

Acknowledgements.— I.M. acknowledges funding from the French Agence Nationale
de la Recherche (ANR) through project ANR-ICEMAN under Grant
Number:19-CE05-0019-01. The research is involved within the framework of
program 6: PROOF (ANR-IEED-002-02) at IPVF.

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Supplementary Materials: Enhancement of Hot Carrier Effect and Signatures of
Confinement in Terms of Thermalization Power in Quantum Well Solar Cell

I. Makhfudz1, N. Cavassilas1, M. Giteau,2, 3H. Esmaielpour,3, 4D. Suchet,3,
4A.-M. Daré ,1 and F. Michelini 1

1 Aix Marseille Université, CNRS, Université de Toulon, IM2NP UMR 7334, 13397,
Marseille, France

2Research Center for Advanced Science and Technology, The University of Tokyo,
Komaba 4-6-1, Meguro-ku, Tokyo 153-8904, Japan

3NextPV, LIA RCAST-CNRS, The University of Tokyo, Komaba 4-6-1, Meguro-ku,
Tokyo 153-8904, Japan

4CNRS, Ecole Polytechnique-IP Paris, Institut Photovoltaique d’Ile de France
(IPVF), UMR 9006, 18 Boulevard Thomas Gobert, 91120 Palaiseau, France

In these Supplementary Materials, we first give the derivation of the generic
form of electron-phonon scattering rate. Then, we give the resulting
analytical expression for electron-phonon scattering rate when Dirac-delta
function approximation is used in place of the exact overlap. The
determination of chemical potential starting from quasi-Fermi level splitting
and charge neutrality condition in intrinsic semiconductors is then described.
Finally, we provide the material parameters used in the calculations.

##  VII Derivation of the Generic Expression for Electron-Phonon Scattering
Rate

The generic expression for the electron-phonon scattering rate reads
$\frac{dN_{{\bf
q}_{\perp},q_{z}}}{dt}\Bigr{|}_{c-LO}=\frac{2\pi}{\hbar}|M_{{\bf
q}_{\perp},q_{z}}|^{2}A_{\mathbf{q}_{\perp},q_{z}}$ where

$A_{\mathbf{q}_{\perp},q_{z}}=2\sum_{\mathbf{k}_{\perp}}\sum_{k_{z},k^{\prime}_{z}}I^{2}_{\mathbf{k},\mathbf{k}-\mathbf{q}}((N_{\mathbf{q}_{\perp},q_{z}}+1)f_{\mathbf{k}_{\perp},k_{z}}(1-f_{\mathbf{k}_{\perp}-\mathbf{q}_{\perp},k^{\prime}_{z}})-N_{\mathbf{q}_{\perp},q_{z}}f_{\mathbf{k}_{\perp}-\mathbf{q}_{\perp},k^{\prime}_{z}}(1-f_{\mathbf{k}_{\perp},k_{z}}))$
$\times\delta(E_{\mathbf{k}_{\perp},k_{z}}-E_{\mathbf{k}_{\perp}-\mathbf{q}_{\perp},k^{\prime}_{z}}-\hbar\omega_{\mathbf{q}_{\perp},q_{z}})\
.$ (31)

We adopt the notation ${\bf k}=({\bf k}_{\perp},k_{z})$ and the same for ${\bf
q}$. Henceforth we suppose that there is no dispersion in the LO phonon
spectrum, thus $\omega_{\mathbf{q}_{\perp},q_{z}}=\omega_{LO}$. The difference
of carrier energies reads

$E_{\mathbf{k}_{\perp},k_{z}}-E_{\mathbf{k}_{\perp}-\mathbf{q}_{\perp},k^{\prime}_{z}}=\frac{\hbar^{2}}{2m_{c}}\left[(k^{2}_{z}-{k^{\prime}_{z}}^{2})-q^{2}_{\perp}+2k_{\perp}q_{\perp}\cos\phi\right]\
,$ (32)

where $\phi$ is the angle between ${\bf k_{\perp}}$ and ${\bf q}_{\perp}$.
Using

$\sum_{\mathbf{k}_{\perp}}...\rightarrow\frac{S}{(2\pi)^{2}}\int_{0}^{2\pi}d\phi\int_{0}^{+\infty}d{k}_{\perp}k_{\perp}\
...\ ,$ (33)

(the upper bound can be safely rejected to infinity), and expressing the
discrete nature of the confined electron longitudinal wave vector
$k_{z}=n\pi/L_{z},k^{\prime}_{z}=n^{\prime}\pi/L_{z}$, while leaving the LO
phonon longitudinal wave vector $q_{z}$ as it is, we get

$A_{\mathbf{q}_{\perp},q_{z}}=\frac{2m_{c}}{\hbar^{2}q_{\perp}}\sum_{n,n^{\prime}}\frac{S}{(2\pi)^{2}}\int_{0}^{2\pi}d\phi\int_{0}^{+\infty}d{k}_{\perp}I^{2}_{\mathbf{k}_{\perp},n,\mathbf{k}_{\perp}-\mathbf{q}_{\perp},n^{\prime}}\left[(N_{\mathbf{q}_{\perp},q_{z}}+1)f_{\mathbf{k}_{\perp},n}(1-f_{\mathbf{k}_{\perp}-\mathbf{q}_{\perp},n^{\prime}})-N_{\mathbf{q}_{\perp},q_{z}}f_{\mathbf{k}_{\perp}-\mathbf{q}_{\perp},n^{\prime}}(1-f_{\mathbf{k}_{\perp},n})\right]$
$\times\delta\left(\cos\phi+\frac{(n^{2}-{n^{\prime}}^{2})}{2k_{\perp}q_{\perp}}\frac{\pi^{2}}{L_{z}^{2}}-\frac{q_{\perp}}{2k_{\perp}}-\frac{m_{c}\omega_{LO}}{\hbar
k_{\perp}q_{\perp}}\right)$ (34)

The overlap integral for unconfined phonons
$I^{2}_{\mathbf{k}_{\perp},n,\mathbf{k}_{\perp}-\mathbf{q}_{\perp},n^{\prime}}=|G_{n,n^{\prime}}(q_{z})|^{2}$
was detailed in the main text (Eq. (21)). Note that due to the
$\delta$-function in Eq. (31), the arguments of the Fermi functions do not
depend on $\phi$ and integration over angle can be readily made. Indeed

$\int^{2\pi}_{0}d\phi\delta(\cos\phi-\cos\phi_{0})=\frac{2}{\sqrt{1-\cos^{2}\phi_{0}}}\
,$ (35)

with

$\cos\phi_{0}=\frac{q_{\perp}}{2k_{\perp}}+\frac{m_{c}\omega_{LO}}{\hbar
k_{\perp}q_{\perp}}-\frac{(n^{2}-{n^{\prime}}^{2})}{2k_{\perp}q_{\perp}}\frac{\pi^{2}}{L_{z}^{2}}\
.$ (36)

This enforces the following constraint, as $k_{\perp}$ and $q_{\perp}$ are
radial wave vectors and so positive,

$k_{\perp}\geq
k_{\perp}^{\mathrm{min}}(n,n^{\prime})=\Bigl{|}\frac{q_{\perp}}{2}+\frac{m_{c}\omega_{LO}}{\hbar
q_{\perp}}-\frac{(n^{2}-{n^{\prime}}^{2})}{2q_{\perp}}\frac{\pi^{2}}{L_{z}^{2}}\Bigr{|}\
,$ (37)

meaning that only electrons from the $n$ subband whose transverse wave vector
is greater than $k^{\mathrm{min}}_{\perp}(n,n^{\prime})$ can emit LO phonons
before joining the $n^{\prime}$ subband. We have

$\cos\phi_{0}=\pm\frac{k^{\mathrm{min}}_{\perp}(n,n^{\prime})}{k_{\perp}}$
(38)

giving

$\int^{2\pi}_{0}d\phi\delta(\cos\phi-\cos\phi_{0})=\frac{2k_{\perp}}{\sqrt{k^{2}_{\perp}-(k^{\mathrm{min}}_{\perp}(n,n^{\prime}))^{2}}}$
(39)

Let us turn to the square bracket in Eq. (34). The relation imposed by energy
conservation ($\delta$-function argument in Eq. (31)) enables to use the nice
identity Wurfel ,

$f_{\mathbf{k}_{\perp},n}(1-f_{\mathbf{k}_{\perp}-\mathbf{q}_{\perp},n^{\prime}})=N_{\mathbf{q}}(T_{c})(f_{\mathbf{k}_{\perp}-\mathbf{q}_{\perp},n^{\prime}}-f_{\mathbf{k}_{\perp},n})\
,$ (40)

where $N_{\mathbf{q}}(T_{c})=1/(\exp(\hbar\omega_{LO}/(k_{B}T_{c}))-1)$ is the
equilibrium phonon distribution at the carrier temperature $T_{c}$. Gathering
the previous results, we finally get

$A_{\mathbf{q}_{\perp},q_{z}}=(N_{\mathbf{q}}(T_{c})-N_{\mathbf{q}_{\perp},q_{z}})\frac{m_{c}}{\hbar^{2}q_{\perp}}\frac{S}{\pi^{2}}\sum_{n,n^{\prime}}|G_{n,n^{\prime}}(q_{z})|^{2}\int^{+\infty}_{k^{\mathrm{min}}_{\perp}(n,n^{\prime})}dk_{\perp}\frac{k_{\perp}}{\sqrt{k^{2}_{\perp}-(k^{\mathrm{min}}_{\perp}(n,n^{\prime}))^{2}}}\left[f(E_{\mathbf{k}_{\perp},n}-\hbar\omega_{LO})-f(E_{\mathbf{k}_{\perp},n})\right]\,.$
(41)

Finally, defining the scattering time $\tau^{c-LO}_{\bf{q}_{\perp},q_{z}}$ by

$\frac{2\pi}{\hbar}|M_{{\bf
q}_{\perp},q_{z}}|^{2}A_{\mathbf{q}_{\perp},q_{z}}=\frac{(N_{\bf{q}}(T_{c})-N_{\bf{q}_{\perp},q_{z}})}{\tau^{c-LO}_{\bf{q}_{\perp},q_{z}}}$
(42)

and using Eq. (16) of the main text, we obtain the Eq. (23) of the body text.

To evaluate the integral on the right hand side in the Eq. (41), namely

$I=\int^{\infty}_{k^{\mathrm{min}}_{\perp}}dk_{\perp}\frac{k_{\perp}}{\sqrt{k^{2}_{\perp}-(k^{\mathrm{min}}_{\perp})^{2}}}(f_{1}-f_{2})\
,$ (43)

where $f_{i}=\frac{1}{e^{x_{i}}+1}$, with respectively
$x_{1}=x_{2}-\beta\hbar\omega_{LO}$ and
$x_{2}=\beta(\frac{\hbar^{2}}{2m_{c}}(k_{\perp}^{2}+\frac{n^{2}\pi^{2}}{L_{z}^{2}})-\mu_{c})$,
with $\mu_{c}$ the carrier chemical potential and $\beta=1/k_{B}T_{c}$, we can
use an obvious change of variable which leads to

$I=\frac{1}{2}\frac{\sqrt{2m_{c}k_{B}T_{c}}}{\hbar}\int_{0}^{+\infty}dx\frac{1}{\sqrt{x}}\Bigl{[}\frac{1}{e^{(x-a)}+1}-\frac{1}{e^{(x-b)}+1}\Bigr{]}\
,$ (44)

with

$\displaystyle a$ $\displaystyle=$ $\displaystyle
b+\beta\hbar\omega_{LO}\,,\rm{and}$ (45) $\displaystyle b$ $\displaystyle=$
$\displaystyle-\beta\Bigl{[}\frac{\hbar^{2}}{2m_{c}}((k^{min}_{\perp}(n,n^{\prime}))^{2}+\frac{n^{2}\pi^{2}}{L_{z}^{2}})-\mu_{c}\Big{]}\,.$
(46)

Eqaution (14) can be expressed in terms of a polylogarithmic function using
Ref. nist, ,

$\int_{0}^{+\infty}\frac{dt}{\sqrt{t}}\frac{1}{e^{t-u}+1}=-\Gamma\Bigl{(}\frac{1}{2}\Bigr{)}Li_{1/2}(-e^{u})\
.$ (47)

This leads to

$I=\frac{1}{2}\frac{\sqrt{2m_{c}k_{B}T_{c}}}{\hbar}\Gamma\Bigl{(}\frac{1}{2}\Bigr{)}(Li_{1/2}(-e^{b})-Li_{1/2}(-e^{a}))\,,$
(48)

and finally, the expression for the electron-phonon scattering rate takes the
form

$\displaystyle\frac{1}{\tau^{c-LO}_{\bf{q}_{\perp},q_{z}}}$ $\displaystyle=$
$\displaystyle\frac{2S}{h}\frac{|M_{{\bf
q}_{\perp},q_{z}}|^{2}}{q_{\perp}}\Big{[}\frac{2m_{c}}{\hbar^{2}}\Big{]}^{3/2}\sqrt{k_{B}T_{c}}\sum_{n,n^{\prime}}|G_{n,n^{\prime}}(q_{z})|^{2}$
(49)
$\displaystyle\times\Gamma\Bigl{(}\frac{1}{2}\Bigr{)}(Li_{1/2}(-e^{b})-Li_{1/2}(-e^{a}))\,.$

##  VIII Scattering Rate for Confined Phonons

There is a substantial literature concerning the electron-LO-phonon
interaction in quantum wells, particularly about the confrontation between
macroscopic dielectric models with various boundary conditions and microscopic
models Rudin91 ; Rucker91 . The former are easier to manipulate, while the
latter maybe more quantitative. In Ref. Rucker91  it was shown that crude bulk
models may provide acceptable results. In the main part of the paper we have
considered unconfined phonons, by arguing that barriers and quantum well
lattice parameters are quite close. In the present section we consider the
opposite case of confined phonons Ridley89 , however ignoring any interface
phonon. In this case, the overlap (Eq. (21) of the main text) will be replaced
by

$G_{n,n^{\prime}}(m)=\int^{L_{z}}_{0}dz\psi_{n}(z)\cos(q_{z}z)\psi^{*}_{n^{\prime}}(z)=\frac{1}{2}\Bigl{(}-\delta_{m,n+n^{\prime}}+\delta_{m,n^{\prime}-n}+\delta_{m,n-n^{\prime}}\Bigr{)}\
,$ (50)

where the longitudinal phonon wave number satisfies
$q_{z}=\frac{m\pi}{L_{z}}$. Aside from the values of
$|G_{n,n^{\prime}}(q_{z})|^{2}$, the calculations are the same, and we can
reuse present Eq. (41) and Eq. (23) of the main text.

Figure 10:  2D Electron-LO confined phonon scattering rates with screening
(solid lines) and without screening (dashed curves), summed over $q_{z}$, as
functions of transverse wave vector $q_{\perp}$ for different well widths, as
well as replicated 3D results (lower right quadrant) from Fig. 5 of the main
text.

We plot in Fig. 10, for the quantum well, the quantities

$\sum_{m}\frac{1}{\tau^{c-LO}_{\bf{q}_{\perp},m}+(\tau^{LO-ac}_{\bf q})}$ (51)

$\tau^{LO-ac}_{\bf q}$ in parenthesis being present or absent respectively in
the non-equilibrium and equilibrium cases. With the same convention, the 3D
results out-of and at equilibrium, $1/(\tau_{{\bf q}}^{c-LO}+(\tau^{LO-
ac}_{\bf q}))$ are also presented. Results with and without screening are also
shown. Contrarily to the unconfined phonon case, (Fig. 5 of the body text), in
quasi 2D, for all three thicknesses, the resulting confined LO phonon
scattering rate is lower than $1/\tau^{LO-ac}_{\mathbf{q}}\simeq 1.36\times
10^{11}\mathrm{s}^{-1}$, in such a way that the effect of non-equilibrium
state of LO phonons does not make any visible difference for $L_{z}=5.5$ nm,
and is still faint for $L_{z}=10$ nm. In this case, LO phonons are close to
equilibrium with their acoustic counterparts at lattice temperature. Finally,
we notice that for $L_{z}=10$ nm, phonon confinement causes two peaks to
appear on the curve of the scattering rate. These peaks are actually related
to the phonon modes involved by the thermalization selection rules. For
$L_{z}=5.5$ nm, there is only one band and therefore only one peak, while for
$L_{z}=15$ nm, the overlapping of the peaks does not allow to separate them.

Figure 11:  Thermalization power per unit volume as a function of quantum well
thickness, considering all possible combinations of assumptions: with and
without non-equilibrium LO phonons (upper two figures, while the screening is
included or not) and with and without screening effect (lower two figures,
while the LO phonons are assumed in non-equilibrium or equilibrium states),
computed with $\Delta\mu=0.653$ eV, $T_{c}=450$ K, and $T_{L}=300$ K.

The resulting thermalization power for confined phonons is presented in Fig.
11, where the results for various combination of the state of LO phonons
(equilibrium or non-equilibrium) and screening (included or not) are
displayed. The curves show that 2D thermalization power always falls below the
3D results for the same state of LO phonon and screening. The reduction in
thermalization power is rather strong for the first two quantum well layer
thicknesses (5.5 and 6 nm) where there is only one confined electron state, by
about two orders of magnitude. However, once we increase the quantum well
layer thickness to accomodate two or more confined electron states, the degree
of reduction becomes only about one order of magnitude, closer to those found
in the existing works in the literature, e.g. MonteCarloS . The 2D curves also
exhibit well defined jumps in the thermalization power density as one
increases the thickness, reflecting the quantization of energy levels of
confined electrons.

##  IX Determination of the chemical potential

The quantity which connects our theory to photoluminescence experiment, is the
chemical potential $\mu_{c}$. Indeed the quasi-Fermi level splitting, that is
the difference between electron and hole chemical potentials, can be measured
experimentally. To determine chemical potentials, we assume that electrons
become hot at temperature $T_{c}$, and so do the light holes, while the heavy
holes remain cold at lattice temperature $T_{L}$, in tune with the finding of
transient study JoshiFerry , due to a cooling rate which is slower for light
holes than for heavy holes. We have verified that considering the same
temperature for heavy and light holes does not change our results
qualitatively. We substitute these effective temperatures $T_{L}$ and $T_{c}$
into charge neutrality equation when solving for the carrier chemical
potential as described in detail below.

The chemical potential $\mu_{c}$ used in the calculations is determined from
solving the charge neutrality equation for intrinsic semiconductors, taking
into account light and heavy holes which share the same chemical potential but
not the same temperature:

$n_{c}=p_{vl}+p_{vh}\ ,$ (52)

where $n_{c}$ is the electron density, and $p_{vl}(p_{vh})$ are respectively
light (heavy) hole densities. In 3D, the electron density is defined as

$n_{c}=\int_{\varepsilon_{c}}^{+\infty}d\epsilon
g_{c}(\epsilon)\frac{1}{e^{(\epsilon-\mu_{c})/k_{B}T_{c}}+1}\ ,$ (53)

where $\varepsilon_{c}$ is the conduction bande edge. Using well known results
for 3D density of states, we have

$n_{c}=\frac{1}{2\pi^{2}}\Bigl{(}\frac{2m_{c}k_{B}T_{c}}{\hbar^{2}}\Bigr{)}^{3/2}\int_{0}^{+\infty}dx\frac{\sqrt{x}}{e^{x-\eta_{c}}+1}\
,$ (54)

with $\eta_{c}=(\mu_{c}-\varepsilon_{c})/k_{B}T_{c}$. While, for holes, from
the definition

$p_{v}=\int_{-\infty}^{\varepsilon_{v}}d\epsilon
g_{h}(\epsilon)\frac{1}{e^{(\mu_{h}-\epsilon)/k_{B}T_{h}}+1}\ ,$ (55)

with $\varepsilon_{v}$ the valence band edge, we find

$p_{v}=\frac{1}{2\pi^{2}}\Bigl{(}\frac{2m_{v}k_{B}T_{h}}{\hbar^{2}}\Bigr{)}^{3/2}\int_{0}^{+\infty}dx\frac{\sqrt{x}}{e^{x+\eta_{h}}+1}\
.$ (56)

$m_{v}$, $T_{h}$ and $\eta_{h}$ need to be specified for both hole types: For
light holes (mass $m_{hl}$, temperature $T_{c}$), we have
$\eta_{hl}=\eta_{c}+(E_{g}-\Delta\mu)/k_{B}T_{c}$ and the corresponding
density will be noted $p_{vl}$, while, for heavy holes (mass $m_{hh}$,
temperature $T_{L}$),
$\eta_{hh}=\frac{T_{c}}{T_{L}}\eta_{c}+(E_{g}-\Delta\mu)/k_{B}T_{L}$, and the
density will be noted $p_{vh}$. We have defined
$E_{g}=\varepsilon_{c}-\varepsilon_{v}$, the band gap, and
$\Delta\mu=\mu_{c}-\mu_{h}$ the quasi Fermi level splitting. The implicit Eq.
(52) enables to determine $\eta_{c}$, hence $\mu_{c}-\varepsilon_{c}$.

In quasi 2D, for finite depth quantum well of width $L_{z}$, the electron
density of state is expressed as a sum over the subbands given by

$g_{c,Q2D}(\epsilon)=\frac{m_{c}}{\pi\hbar^{2}}\frac{1}{L_{z}}\sum_{i=1}^{\infty}\Theta(\epsilon-(E_{ic}+\varepsilon_{c}))\
,$ (57)

with $E_{ic}$ the edges of the different subbands. It leads to

$n_{c}=\frac{m_{c}k_{B}T_{c}}{\pi\hbar^{2}L_{z}}\sum_{i}\int_{0}^{+\infty}dx\frac{1}{e^{x+E_{ic}/k_{B}T_{c}-\eta_{c}}+1}\
,$ (58)

where $\eta_{c}$ has been previously defined. The integral can be evaluated,
this gives

$n_{c}=\frac{m_{c}k_{B}T_{c}}{\pi\hbar^{2}L_{z}}\sum_{i}\ln{(1+e^{\eta_{c}-E_{ic}/k_{B}T_{c}})}\
.$ (59)

For holes, we use the following DOS

$g_{v,Q2D}(\epsilon)=\frac{m_{h}}{\pi\hbar^{2}}\frac{1}{L_{z}}\sum_{i}\Theta\Bigl{(}\varepsilon_{v}-E_{ih}-\epsilon\Bigr{)}$
(60)

with $E_{ih}$ the edges of the different hole subbands. For light holes (mass
$m_{hl}$, temperature $T_{c}$), the calculation gives

$p_{vl}=\frac{m_{hl}}{\pi\hbar^{2}}\frac{k_{B}T_{c}}{L_{z}}\sum_{i}\ln{\Bigl{(}1+e^{-\eta_{hl}-E_{ihl}/k_{B}T_{c}}\Bigr{)}}\
.$ (61)

On the other hand, for the heavy holes (mass $m_{hh}$, temperature $T_{L}$),
we get

$p_{vh}=\frac{m_{hh}}{\pi\hbar^{2}}\frac{k_{B}T_{L}}{L_{z}}\sum_{i}\ln{\Bigl{(}1+e^{-\eta_{hh}-E_{ihh}/k_{B}T_{L}}\Bigr{)}}\
.$ (62)

$\eta_{hh}$ and $\eta_{hl}$ have been formerly defined.

The quasi-Fermi level splitting $\Delta\mu$ and the carrier temperature are
determined from the modeling of photoluminescence spectrum, which measures the
photon current density and can be described by the generalized Kirchoff law
Wurfel :

$j_{\gamma}(\hbar\omega)=a(\hbar\omega)\frac{(\hbar\omega)^{2}}{4\pi^{2}\hbar^{3}c^{2}}\frac{1}{\exp(\frac{\hbar\omega-\Delta\mu}{k_{B}T_{c}})-1}$
(63)

where the absorptivity is given by

$a(\hbar\omega)=\frac{\left[1-R(\hbar\omega)\right](1-\exp(-\alpha(\hbar\omega)L_{z}))}{1-R(\hbar\omega)\exp(-\alpha(\hbar\omega)L_{z})}\,,$
(64)

$R$ being the reflectivity and $\alpha$ the absorption coefficient. In the
proposed investigation, we assume the quasi-Fermi level splitting $\Delta\mu$
and the electron temperature to be fixed as we vary the quantum well
thickness, even in the limit of the 3D case. Translated into experimental
conditions, it would mean that we have discussed the thermalization power in
different quantum well heterostructures, tuning the excitation conditions to
provide the same photoluminescence spectrum.

In the calculations, we have used the data from experiment using 405 nm laser
irradiating multiple quantum well absorber consisting of 5 InGaAsP-InGaAs-
InGaAsP layers with thickness 10nm-5.5nm-30nm and transverse dimension
$L_{\perp}=3$ cm: $\Delta\mu=0.653$ eV and $T_{c}=450$ K. From these data, we
have estimated the chemical potentials from the 3D model and the proposed 2D
model, together shown in Fig. 12, where the asymptotic convergence towards the
3D case is nearly reached in the quasi-2D case beyond 15 nm.

Figure 12: Profiles of the chemical potential for our 2D model (circle) and 3D
model (+), for quantum well layer thicknesses $L_{z}=5.5-25$ nm.

##  X Material Parameters

Material parameters that were used to produce the figures of the main text and
the Suppl. Mat. are gathered in Tab. 2.

Quantity | Symbol | Value
---|---|---
LO phonon energy | $\hbar\omega_{LO}$ |  $36$ meV
Energy gap (InGaAs) | $E_{g}$ | 0.8 eV
Well depth for electrons | $V_{con}$ | 0.235 eV
Well depth for holes | $V_{val}$ | 0.185 eV
Electron effective mass (InGaAs) | $m_{c}$ | 0.041 $m_{0}$
Heavy hole effective mass | $m_{hh}$ | 0.363 $m_{0}$
Light hole effective mass | $m_{hl}$ | 0.051 $m_{0}$
Sound velocity (InGaAs) | $v_{s}$ |  $3.5\times 10^{3}$ ms-1
Infinite-frequency susceptibility | $K_{\infty}$ | 12.9
Static susceptibility | $K_{s}$ | 10.9
Deformation potential coefficient | $\Gamma$ | 0.43
LO phonon decay time (InGaAsP) | $\tau^{LO-ac}_{0}$ |  $2.2\times 10^{-11}$ s
(0 K) |  |
Lattice constant (InGaAsP) | $a$ | 5.75 Å
Material density (InGaAsP) | $\rho$ | 5500 kg/m3
Table 2:  Material parameters needed to compute the electron-LO phonon
scattering rate and thermalization power in InGaAsP/InGaAs/InGaAsP quantum
well heterostructures of varying thickness.

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