# Electron recoil effect in electrically tunable MoSe2 monolayers

Jonas Zipfel  Department of Physics, University of Regensburg, Regensburg
D-93053, Germany  Molecular Foundry, Lawrence Berkeley National Laboratory,
Berkeley, California 94720, USA  Koloman Wagner  Department of Physics,
University of Regensburg, Regensburg D-93053, Germany  Dresden Integrated
Center for Applied Physics and Photonic Materials (IAPP) and Würzburg-Dresden
Cluster of Excellence ct.qmat, Technische Universität Dresden, 01062 Dresden,
Germany  Marina A. Semina  Ioffe Institute, Saint Petersburg, Russian
Federation  Jonas D. Ziegler  Department of Physics, University of Regensburg,
Regensburg D-93053, Germany  Dresden Integrated Center for Applied Physics and
Photonic Materials (IAPP) and Würzburg-Dresden Cluster of Excellence ct.qmat,
Technische Universität Dresden, 01062 Dresden, Germany  Takashi Taniguchi
International Center for Materials Nanoarchitectonics, National Institute for
Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan  Kenji Watanabe
Research Center for Functional Materials, National Institute for Materials
Science, 1-1 Namiki, Tsukuba 305-0044, Japan  Mikhail M. Glazov  Ioffe
Institute, Saint Petersburg, Russian Federation  Alexey Chernikov
<EMAIL_ADDRESS>Department of Physics, University of Regensburg,
Regensburg D-93053, Germany  Dresden Integrated Center for Applied Physics and
Photonic Materials (IAPP) and Würzburg-Dresden Cluster of Excellence ct.qmat,
Technische Universität Dresden, 01062 Dresden, Germany

###### Abstract

Radiative recombination of excitons dressed by the interactions with free
charge carriers often occurs under simultaneous excitation of either electrons
or holes to unbound states. This phenomenon, known as the electron recoil
effect, manifests itself in pronounced, asymmetric spectral lineshapes of the
resulting emission. We study the electron recoil effect experimentally in
electrically-tunable monolayer semiconductors and derive it theoretically
using both trion and Fermi-polaron pictures. Time-resolved analysis of the
recoil lineshapes is employed to access transient, non-equilibrium states of
the exciton-carrier complexes. We demonstrate cooling of the initially
overheated populations on the picosecond timescales and reveal the impact of
lattice temperature and free carrier density. Both thermally activated phonons
and the presence of free charges are shown to accelerate equilibration.
Finally, we find strong correlations between relaxation times from recoil
analysis and luminescence rise times, providing a consistent interpretation
for the initial dynamics of trion/Fermi-polaron states.

two-dimensional materials, light-matter interaction, trions, Fermi-polarons,
dynamics

##  I Introduction

Light-matter coupling of bound electron-hole pairs, known as excitons, plays a
fundamental role in the optical properties of semiconductors [1]. The nature
of this coupling, however, can change substantially when additional
quasiparticles are introduced [2]. Among frequently encountered cases of
interacting many-particle systems are mixtures of excitons with free charge
carriers. It leads to the formation of composite excitonic states, commonly
labeled as trions [3, 4] or Fermi-polarons in the low-density limit [5, 6, 7].
When these complexes recombine radiatively, their center-of-mass momenta are
almost entirely transferred to the free carriers that remain in the system.
Broadly known as electron recoil effect for trions [8], it expands the number
of optically accessible states and alters both spectral shape and the dynamics
of the emission. This behavior strongly contrasts recombination of single
excitons that is typically restricted to near-zero momenta within the light-
cone [9].

Experimentally, electron recoil has been demonstrated in two-dimensional
nanosystems such as quantum wells [8, 10] and, more recently, in monolayers of
transition-metal dichalcogenides (TMDCs) [11]. The latter are particularly
well suited to explore exciton-electron phenomena due to strong Coulomb
interactions [12, 13] that stabilize excitonic complexes even at elevated
temperatures and high carrier densities. Interestingly, while the spectral
asymmetry due to recoil is frequently observed in TMDCs [11, 14, 15, 16, 17],
it has been studied exclusively under steady-state conditions so far. Time-
resolved detection of the recoil flanks, however, would provide a direct
access to non-equilibrium distributions and trion cooling dynamics. These are
typically challenging to access, often requiring advanced experimental
approaches in combination with microscopic theory, as it has been recently
demonstrated for the case of free electrons [18]. Accurate examination of the
recoil effect should further profit from recent advances in material
preparation using state-of-the-art encapsulation in hBN. In addition, recent
analysis highlighted an alternative description of the excitons interacting
with free carriers as attractive and repulsive Fermi-polarons [5, 6, 7, 19,
20, 21, 22]. A consistent theoretical description of the electron-assisted
recombination of dressed exciton states in both trion and Fermi polaron
pictures is thus highly desirable 111Please also see a recent preprint
discussing spectral lineshape asymmetry for Fermi polarons: T. Wasak et al.
arXiv:2103.14040.

Here, we address the above by studying electron recoil effect in two-
dimensional TMDCs from both experimental and theoretical perspectives. Optical
studies of field-effect transistors based on high-quality hBN-encapsulated
MoSe2 monolayers with thin-layer graphite gates are combined with analytical
theory of trions and Fermi-polarons. It enables us to develop a consistent
physical description of the interactions of excitons with resident charge
carriers. The manuscript is organized as follows: in Sect. II we summarize the
employed techniques of material preparation and time-resolved microscopy.
Section III provides an overview of the main experimental observables and
their quantitative analysis within a phenomenological approach. The theory of
the recoil effect is then discussed in Sect. IV justifying the description of
the experimental data. Implications for light-matter coupling are derived in
both trion and Fermi-polaron pictures. Time-resolved measurements of the
recoil effect are presented in Sect. V. They provide access to the transient
excess temperatures of trion/Fermi-polaron complexes and reveal accelerated
cooling at elevated lattice temperatures on picosecond time-scales. Similar
behavior is observed for increased free carrier densities that facilitate
relaxation, as discussed in Sect. VI. The main results and conclusions are
then summarized in Sec. VII.

##  II Experimental details

The field-effect device under study was fabricated by mechanical exfoliation
and dry-stamping [24] of bulk crystals on SiO2/Si substrates with prepatterned
gold electrodes. A MoSe2 monolayer was encapsulated between 10’s of nm thick
hBN layers. Exfoliated thin-layer graphite flakes were used for top- and
bottom-gates as well as to contact MoSe2. The quality of encapsulation was
confirmed by observing spectrally narrow luminescence across sufficiently
large areas of many $\mu$m2 on the sample. The device was fixed onto an
electrically-contacted sample holder and placed into a microscopy cryostat. It
was cooled down to heat-sink temperatures between 5 and 50 K. Free charge
carrier densities were tuned by changing the gate voltage and calibrated by
evaluating relative spectral shifts between neutral excitons and trions
(repulsive and attractive Fermi polarons, respectively), see Appendix D. Their
values were varied across the range between $4\times 10^{10}$ and $2\times
10^{12}$ cm-2, consistent with those estimated from a parallel-plate capacitor
model at higher voltages.

For optical measurements we used a 140 fs pulsed Ti:sapphire laser with a
repetition rate of 80 MHz as excitation source. The laser was tuned to a
photon energy of 1.657 eV, corresponding to near-resonant excitation into the
high-energy flank of the ground-state exciton of MoSe2 at 1.642 eV with an
effective absorption of a few percent. The incident light was focused onto the
sample by a 60x microscope objective to a spot of 1 $\mu$m diameter. The
excitation energy density was set to 6 $\mu$Jcm-2, resulting in the estimated
injected electron-hole pair-density per pulse of several 1011 cm-2. The
emitted photoluminescence (PL) signal was dispersed in a spectrometer and
detected by a picosecond streak camera in synchroscan, photon-counting mode.
Additional details on sample preparation and experimental procedures are given
in the supplementary of Ref. 25.

##  III Electron recoil in $\mbox{MoSe}_{2}$

In this section we introduce the experimental observations of the recoil
effect and the data analysis. Time-integrated PL spectra of the gate-tunable
MoSe2 monolayer at 5 K are presented in Fig. 1 (a). The corresponding full set
of PL measurements is shown in Fig. 1 (b) as a function of gate voltage and
emission energy. Free carrier densities are estimated to be in the range of
1011 cm2. The data illustrate a typical response of MoSe2 [11, 26], dominated
by a single exciton resonance at charge neutrality (i). Upon doping, an
additional peak emerges that is shifted to lower energies by 27 meV in the
electron-doped regime (n) and by a slightly smaller value of 25 meV in the
case of hole-doping (p), consistent with the literature [27, 26]. It stems
from the radiative recombination of a three-particle state known as trion [3,
4], involving an exciton that binds an additional electron or a hole in an
atom-like picture [28]. These quasiparticles can be also understood as
attractive Fermi-polarons [29, 6, 7], – excitons dressed by a cloud of free
carriers. We note, that the two descriptions are conceptually equivalent at
sufficiently low charge densities [22] studied here, and thus use the term
trion throughout most parts of the manuscript.

While the neutral excitons in MoSe2 emit as a narrow, symmetric peak with
about 2 meV linewidth, the trion PL is asymmetric with a broader low-energy
flank. The asymmetry becomes increasingly pronounced at higher temperatures,
as observed in time-integrated spectra presented in Fig. 1 (c) for the hole-
doped regime. Such lineshapes are characteristic for trion emission [10] and
are the consequence of the energy and momentum conservation during radiative
recombination. The process is schematically illustrated in Fig. 1 (d) adopting
the picture of a n-doped case (it applies equally for p-doping). A trion with
the energy $E_{\bm{k}}^{tr}$ recombines by emitting a photon and leaving a
free electron. Due to the vanishing momentum of the photon almost the entire
trion momentum $\bm{k}$ is transferred to the remaining electron. As a
consequence, the electron gains the kinetic energy
${E_{\bm{k}}^{e}}\propto\bm{k}^{2}$, motivating the labeling of the effect as
electron recoil in the literature [8, 11]. Energy conservation then requires
the photon being emitted at a lower energy than that of the trion by
$E_{\bm{k}}^{e}$. It leads to an asymmetric emission peak of the trions with a
low-energy flank following their thermal distribution.

Figure 1: (a) Representative, time-integrated PL spectra of hBN-encapsulated
MoSe2 monolayer under electrical gating in the p-doped, neutral, and n-doped
regimes at 5 K. (b) False-color plot of the PL intensity as function of
emission energy and gate voltage. Dashed lines represent spectra presented in
panel (a). The gate voltage of $\pm$1 V corresponds to the electron and hole
density of about $1.6\times 10^{11}$ cm2 and $2.9\times 10^{11}$ cm2,
respectively, estimated from the trion-exciton energy separation (at higher
voltages the scaling is 2x1011 cm2/V for holes and 4.8x1011 cm2/V for
electrons). (c) Time-integrated PL spectra at different lattice temperatures
in the p-doped regime. The hole density is set to $1\times 10^{11}$ cm-2 for
all temperatures. The spectra are presented as function of the emission energy
relative to the zero-momentum trion peak energy $E_{0}^{tr}$ at each
temperature. Fit functions are convolutions of a Lorentzian peak with an
exponential low-energy flank as well as a Gaussian with a fixed linewidth of 1
meV. (d) Schematic illustration of the electron recoil effect during trion
recombination. (e) Exponential low-energy constant $\epsilon^{*}$ extracted
from the fits in panel (c) (filled triangles), presented in the form of
$k_{B}/\epsilon^{*}$ as function of the inverse lattice temperature for $T\geq
15$ K. Filled circles correspond to a second measurement for temperatures up
to 70 K at a different position. Fit of both data sets using Eq. (2) is shown
in red. Corresponding values obtained from time-resolved measurements at 50 ps
are shown by open circles.

Assuming Boltzmann statistics, the resulting PL lineshape $I_{PL}$ can be thus
described by a convolution of a symmetric peak function $I_{0}$ and an
exponential function on the low-energy side:

$I_{PL}(\epsilon)=I_{0}(\epsilon)\otimes\exp({\epsilon/\epsilon^{*}})\Theta({-\epsilon}),$
(1)

with $\epsilon={\hbar\omega-E_{0}^{tr}}$ denoting the photon energy
($\hbar\omega$) relative to that of the trions at rest ($E_{0}^{tr}$) and
$\Theta(\epsilon)$ representing the Heaviside step function; $\otimes$ denotes
the convolution. For the symmetric component $I_{0}(\epsilon)$ we use a
convolution of a Lorentzian peak and a Gaussian with a fixed linewidth of 1
meV to account for homogeneous and inhomogeneous broadening, respectively. The
value $\epsilon^{*}$ in the low-energy exponent combines two contributions
[10]:

$\frac{1}{\epsilon^{*}}=\frac{1}{\epsilon_{1}}+\frac{1}{k_{B}T}\frac{m_{e}}{m_{x}}.$
(2)

The first term comes from an exponential approximation for the energy-
dependent light-matter coupling strength of the trions. The constant
$\epsilon_{1}$ is usually of similar magnitude as the trion binding energy
[10]. The second term accounts for electron recoil and depends on the trion
temperature $T$ as well as the ratio of electron ($m_{e}$) and exciton
($m_{x}$) total translational masses. The origin of Eq. (2) is discussed in
detail in Sect. IV.

The model described by Eq. (1) is used to fit time-integrated experimental
spectra presented in Figs. 1 (a) and (c). Setting symmetric broadening and the
low-energy exponential $\epsilon^{*}$ as free parameters we obtain an overall
good match to the measured lineshapes with only small deviations. The
extracted values for $\epsilon^{*}$ are presented in Fig. 1 (e) in the form of
$k_{B}/\epsilon^{*}$ as function of the inverse lattice temperature for $T>$10
K. The plot thus directly represents the relation in Eq. (2) with the slope
being proportional to the mass ratio $m_{h}/m_{x}$ and the y-axis offset
corresponding to $1/\epsilon_{1}$. Only at the lowest temperatures of
$T\leq$15 K we find deviations from the linear behavior of
$k_{B}/\epsilon^{*}$ as function of $T^{-1}$. These mainly originate from the
increased influence of overheated trion populations, as discussed further
below in Sect. V and in Appendix A.

From the analysis presented in Fig. 1 (e) we obtain the parameters of
$m_{h}/m_{x}=0.44$ and $\epsilon_{1}=7$ meV. Both values are reasonable in
view of the nearly equal electron and hole masses in the monolayer MoSe2 with
$m_{x}=m_{e}+m_{h}\approx 2m_{h}$ as well as the trion binding energies of a
few 10’s of meV. The presented quantitative description of the temperature-
dependent spectral asymmetry of the trion PL strongly supports the
interpretation of this observation as the recoil effect. It provides a solid
basis for using time-resolved measurements of the recoil flanks to access
transient trion temperatures and cooling dynamics presented in Sect. V and
Sect.VI. In the following Sect. IV we start with the theoretical discussion of
the electron recoil to outline the consequences for light-matter coupling and
rationalize the applied lineshape analysis.

##  IV Theory of electron recoil

In this section we derive the analytical expressions describing the PL spectra
and recoil effect in a two-dimensional system. We use both trion and Fermi-
polaron approaches and demonstrate that at sufficiently low densities of
resident electrons they give essentially the same result, as it is the case
for their spectral characteristics [30]. We recall that the trion picture [3,
4] is a few-body approach that considers an exciton, a bound state of an
electron and a hole, and a trion, a bound state of two electrons and a hole
(or, in the case of p-type structure, of two holes and an electron). By
contrast, in the Fermi-polaron approach we consider the exciton as a rigid
particle immersed in the Fermi-sea of electrons (or holes) [31, 32, 33, 5, 6,
7, 34]. The exciton is then “dressed” by the Fermi-sea excitations resulting
in the formation of the attractive and repulsive Fermi-polaron states.

###  IV.1 Trion picture

It is instructive to start the analysis of the photoluminescence spectra in
terms of trions providing a convenient physical picture of the effect, as
illustrated in Fig. 1(d). In the trion language the rate of the photon
emission with the frequency $\hbar\omega$ by the trion gas can be expressed
using the Fermi golden rule in the form

$W^{tr}(\omega)=\frac{2\pi}{\hbar}\sum_{\bm{k}}f_{\bm{k}}|M^{tr,opt}(\bm{k})|^{2}\delta(\hbar\omega+E^{e}_{\bm{k}}-E^{tr}_{\bm{k}}).$
(3)

Here $E_{\bm{k}}^{e}=\hbar^{2}k^{2}/2m_{e}$ and
$E_{\bm{k}}^{tr}=E_{0}^{tr}+\hbar^{2}k^{2}/2m_{tr}$ are the electron and trion
kinetic energies with $m_{e}$ and $m_{tr}=m_{x}+m_{e}$ being the electron and
trion effective masses,

$f_{\bm{k}}=\mathcal{N}\exp{\left(-\frac{\hbar^{2}k^{2}}{2m_{tr}k_{B}T_{*}}\right)}$
(4)

is the trion distribution function in the Boltzmann approximation. Here,
$T_{*}$ is the effective temperature of the trion gas which can, in the
general case, deviate from that of the lattice and electrons, $\mathcal{N}$ is
the normalization constant, and $M^{tr,opt}_{\bm{k}}$ is the matrix element of
the trion radiative decay [10, 30] 222Here we correct a typo in Eqs. (24),
(26) of Ref. [30] where the factor $m_{x}/m_{tr}$ was missing in the exponent.

$M^{tr,opt}(\bm{k})=\mathfrak{M}_{r}\int\varphi(0,\bm{\rho})e^{-\mathrm{i}\bm{k}\bm{\rho}(m_{x}/m_{tr})}d\bm{\rho}.$
(5)

Here $\varphi(\bm{\rho}_{1},\bm{\rho}_{2})$ is the trion envelope
wavefunction, $\bm{\rho}_{1,2}$ are the relative coordinates of the electrons
with respect to the hole. In derivation of Eq. (3) we further assumed that the
electron density $N_{e}$ is sufficiently low such that
$\hbar^{2}N_{e}/m_{e}\ll E_{b,tr},k_{B}T_{*}$ where $E_{b,tr}$ is the trion
binding energy. In this case one can neglect modifications of the trion state
by the resident carriers [30]. It also allows us to neglect the occupancy of
the final electron state after the trion recombination.

Figure 2: The top panels schematically illustrate overlap between trion/Fermi-
polaron wavefunction with that of the final state of the free electron.
Presented are two cases of high and low kinetic energies resulting in small
and large overlap integrals, respectively. The main panels shows calculated
dipole matrix elements as function of the kinetic energy of the quasiparticle,
using either trion or Fermi-polaron wavefunctions. Dashed black line is
calculated after Eq. (8) and dashed magenta line is calculated after Eq. (11).
Single-exponential approximation is presented by the solid line for direct
comparison. For the trion we set $a_{tr}=3a_{x}$ and obtain the binding energy
of $17$ meV, close to the experimental values (we note that such deviations
are common for variational models). For the Fermi-polaron approach we set
$E_{b,tr}$ to the experimental value of $25$ meV and the exponential
approximation corresponds to $\epsilon_{1}=7$ meV.

Altogether, Eq. (3) describes the radiative transition where the trion with
the in-plane wavevector $\bm{k}$ recombines emitting a photon with negligible
momentum and leaving behind an electron with the wavevector $\bm{k}$. As a
consequence, the trion PL spectrum has a typical recoil low-energy wing in the
form

$W^{tr}(\omega)\propto\exp{\left(\frac{\hbar\omega-E^{tr}_{0}}{k_{B}T_{*}}{\frac{m_{e}}{m_{x}}}\right)}\\\
\times\left|M^{tr,opt}\left(\sqrt{\frac{2m_{e}m_{tr}(E^{tr}_{0}-\hbar\omega)}{m_{x}\hbar^{2}}}\right)\right|^{2}\Theta(E^{tr}_{0}-\hbar\omega).$
(6)

This expression agrees with Eqs. (1) and (2), where, following Ref. [10], the
energy dependence of $|M^{tr,opt}|^{2}$ has been approximated by the
exponential function. To justify this assumption we use the trion envelope
function in the simple form [36, 37, 38]

$\psi(\bm{\rho}_{1},\bm{\rho}_{2})=e^{-\rho_{1}/a_{x}}e^{-\rho_{2}/a_{tr}}+e^{-\rho_{2}/a_{x}}e^{-\rho_{1}/a_{tr}},$
(7)

with $a_{e}$ and $a_{tr}$ being the effective “exciton” and “trion” Bohr
radii, the matrix element is evaluated as

$M^{tr,opt}(\bm{k})=\frac{\mathfrak{M}_{r}}{\sqrt{a_{tr}^{2}a_{x}^{2}/8+2a_{tr}^{4}a_{x}^{4}/(a_{tr}+a_{x})^{4}}}\\\
\times\left(\frac{a_{tr}^{2}}{[1+a_{tr}^{2}k^{2}({m_{x}}/m_{tr})^{2}]^{3/2}}\right.\\\
\left.+\frac{a_{x}^{2}}{[1+a_{x}^{2}k^{2}({m_{x}}/m_{tr})^{2}]^{3/2}}\right).$
(8)

Figure 2 shows the matrix element calculated after Eq. (8) (dashed black
curve) and exponential approximation (solid magenta curve) as functions of the
trion kinetic energy. It demonstrates a good accuracy of the exponential
approximation to describe the form of the function, especially in the energy
range of the $\epsilon_{1}$ constant.

###  IV.2 Fermi-polaron picture

Within the Fermi-polaron approach the PL can be described as a process of the
polaron recombination. After recombination, the Fermi-sea can either remain in
the unperturbed state or acquire an excitation in the form of the Fermi-sea
electron-hole pair (with the hole being an unoccupied state under the Fermi
level) depending on the momentum of the polaron. As a result, a low-energy
flank in the PL is formed.

Fermi-polaron effects can be conveniently described using the diagram
technique [32, 7, 34, 30]. We consider the low electron density regime where,
as above, $\hbar^{2}N_{e}/m_{e}\ll E_{b,tr}\ll E_{b,x}$, evaluate the diagram
in Fig. 3(a) and derive the following expression for the PL spectrum (using
the non-equilibrium Keldysh technique, see Appendix F for technical details):

$W^{FP}(\omega)=-\sum_{\bm{k}}Z_{\bm{k}}\mathrm{\Im}\\{\Sigma_{PL}(\hbar\omega)\\}\\\
=\frac{2\pi}{\hbar}\sum_{\bm{k}}f_{\bm{k}}\frac{E_{b,tr}}{\mathcal{D}}\frac{|\mathfrak{M}_{r}|^{2}}{(\hbar\omega-
E_{0}^{x})^{2}}\delta(\hbar\omega+E^{e}_{\bm{k}}-E_{\bm{k}}^{tr}).$ (9)

Here $\mathcal{D}={\mu_{tr}}/(2\pi\hbar^{2})$ is the exciton-electron reduced
density of states with $\mu_{tr}=m_{e}{m_{x}}/(m_{e}+{m_{x}})$ and
$Z_{\bm{k}}\approx N_{e}/(\mathcal{D}E_{b,tr})$ is the renormalization factor.
The explicit expressions for the self-energy $\Sigma_{PL}(\hbar\omega)$
obtained via the Fermi-sea polarization loop $\Pi(\hbar\Omega,\bm{k})$,
scattering amplitude $T(\varepsilon,\bm{k})$, and Greens functions are
presented in the Appendix F. Summing over the $\bm{k}$ we arrive at

$W^{FP}(\omega)\propto\exp{\left(\frac{\hbar\omega-E^{tr}_{0}}{k_{B}T_{*}}\frac{m_{e}}{m_{x}}\right)}|M^{fp,opt}|^{2}\Theta(E^{tr}_{0}-\hbar\omega).$
(10)

The factor

$|M^{fp,opt}|^{2}=\frac{\mathfrak{M}_{r}^{2}}{\left(\hbar\omega-
E_{0}^{x}\right)^{2}}=\frac{1}{(E_{\bm{k}}^{tr}-E_{\bm{k}}^{e}-E_{0}^{x})^{2}},$
(11)

can be directly associated, in the model of the short-range exciton-electron
interaction, with the squared absolute value of the matrix element in Eqs. (5)
and (6).

Consequently, Eqs. (6) for trions and (10) for attractive Fermi-polarons are
essentially equivalent. To illustrate it quantitatively, we show
$|M^{fp,opt}|^{2}$ calculated after Eq. (11) in Fig. 2 by dashed magenta
curve. The result of the Fermi-polaron model is thus in good agreement with
the optical matrix element found with the trion wavefunction in Sec. IV.2 and
can also be quite accurately approximated by the exponential function.

Figure 3: Fermi-polaron interaction with photons. (a) The full diagram
describing the Fermi-polaron PL in the Keldysh technique. (b) Self-energy
responsible for the radiative decay. Wavy line shows the photon Greens
function. Exciton Greens function $\mathcal{G}_{x}$ renormalized by the Fermi-
sea is shown by the thick green line. Blue boxes show the electron-exciton
scattering amplitude $T$, and $\Pi$ denotes the Fermi-sea polarization loop
composed of the bare electron Greens functions (black lines with arrows).
Signs $+$ and $-$ denote the components of the Greens functions in Keldysh
technique.

To make our description of the trion/attractive Fermi-polaron PL complete, we
also analyze the radiative decay rate of the trion (attractive Fermi polaron)
ensemble. To that end, we calculate the self-energy of the attractive polaron
related to the light-matter coupling, Fig. 3(b), see Appendix G for details:

$\Sigma_{phot}(\varepsilon,\bm{k})=-\mathrm{i}\pi\frac{|\mathfrak{M}_{r}|^{2}}{(\varepsilon-
E_{\bm{k}}^{e}-E_{0}^{x})^{2}}N_{e}T^{2}(\varepsilon,\bm{k})\mathcal{D}_{phot},$
(12)

where $\mathcal{D}_{phot}$ is the density of photon states with the frequency
$\omega=\varepsilon/\hbar$. Using the Dyson equation
$\mathbb{G}_{x}=\mathcal{G}+\mathcal{G}\Sigma\mathbb{G}$ we obtain the
radiative damping rate of the Fermi-polaron with the energy
$\varepsilon=E^{tr}_{\bm{k}}-N_{e}/\mathcal{D}$ in the form

$\gamma_{r,\bm{k}}=\frac{\pi}{\hbar}E_{b,tr}\frac{\mathcal{D}_{phot}}{\mathcal{D}}\frac{|\mathfrak{M}_{r}|^{2}}{(\varepsilon-
E_{\bm{k}}^{e}-E_{0}^{x})^{2}}.$ (13)

In derivation of Eqs. (12) and (13) we disregarded polarization-dependent
factors, cf. Ref. [10] which do not significantly affect the radiative decay
rate of the quasiparticles ensemble. Equation (13) can be recast as

$\gamma_{r,\bm{k}}=C(q_{phot}a_{tr})^{2}\gamma_{x}\frac{E_{b,tr}}{(\hbar\omega-
E_{\bm{k}}^{e}-E_{0}^{x})^{2}},$ (14)

where $C\sim 1$ is a numerical constant, $\gamma_{x}$ is the light-cone
exciton radiative decay rate, and $q_{phot}$ is the photon wavevector at the
frequency $\omega$. Equations (12) and (13) thus take into account the
radiative decay processes accompanied by the recoil effect.

However, for the Fermi-polarions with $k<q_{phot}$, direct recombination is
also possible. In this case, the “exciton” component of the polaron has a
small momentum corresponding to the light-cone and can decay radiatively
without the necessity to transfer its momentum to the remaining electron.
Corresponding contribution to the decay rate is directly proportional to the
Fermi-polaron oscillator strength [30, 39] and given by (with polarization-
dependent factors being omitted):

$\gamma_{phot,\bm{k}}=Z_{\bm{k}}\gamma_{x}=\frac{N_{e}}{\mathcal{D}E_{b,tr}}\gamma_{x},\quad
k<q_{phot}.$ (15)

The resulting total radiative recombination rate is then obtained for the
thermalized ensemble of the exciton-polarons by averaging Eqs. (14) and (15)
with the quasiparticle distribution function $f_{\bm{k}}$, Eq. (4). Omitting
numerical coefficients, the resulting total rate can be estimated as:

$\gamma_{tot}\sim\left[\frac{E_{F}}{k_{B}T_{*}}+1\right](q_{phot}a_{tr})^{2}\gamma_{x}.$
(16)

This expression illustrates the interplay of two distinct Fermi-polaron decay
mechanisms: At sufficiently low doping densities and elevated temperatures of
the trion/Fermi-polaron ensemble, where $E_{F}\ll k_{B}T_{*}$, the radiative
decay via the recoil effect is more important. This is the regime relevant for
our experiments.

In contrast, for the high doping/low temperature conditions of Fermi-polaron
ensemble, $E_{F}\gg k_{B}T_{*}$, the decay via the exciton component is
dominant and takes place via the Fermi-polarons within the light-cone. For the
charge carrier masses of MoSe2 on the order of $0.5\,m_{0}$ and trion
temperatures of 10’s of K, this high doping regime should roughly correspond
to carrier densities in the $10^{12}$ cm-2 range. Then, one would need to
modify the spectral lineshape analysis of Eq.(1) by introducing an additional
symmetric peak to account for direct recombination of Fermi-polarons.

Note that for the repulsive Fermi-polarons (excitons) the lineshape asymmetry
is expected to be much weaker. Indeed, the repulsive polaron recombination
takes place mainly via its excitonic component (which is significant as
compared with the attractive polaron) making the effect of the recoil smaller.
The PL asymmetry may arise in this case mainly due to the exciton-phonon
interaction instead, see Ref. [40, 41].

##  V Trion/Fermi-polaron cooling

In this section we discuss time-resolved analysis of the recoil effect as well
as temperature-dependent cooling dynamics of the trions. We briefly illustrate
theoretical considerations of relaxation time-scales followed by presenting
experimental results and discussion.

###  V.1 Model

A theoretical description of the cooling effects can be performed within
either the trion or Fermi-polaron approach. Provided that the resident
electron density is low enough and the quasiparticles temperature $T^{*}$ is
sufficiently large such that ${E_{F}}\ll k_{B}T^{*}$ both approaches yield the
same results. Here we use the trion picture; the same result can be derived in
the Fermi-polaron approach, see Appendix H.

In particular, we consider cooling of trions due to emission of longitudinal
acoustic phonons. Their interaction occurs via the deformation-potential
mechanism with the bare matrix element of the trion-phonon interaction
represented in the form [42]

$\left|M_{\bm{k}\to\bm{k}^{\prime}}^{tr,\pm\bm{q}}\right|^{2}={\delta_{\bm{k},\bm{k}\mp\bm{q}}}\mathcal{B}_{tr}(q),$
(17)

where the top (bottom) sign refers to the phonon emission (absorption), the
normalization area, as above, is set to unity, and the $\mathcal{B}(q)$ is
reduced rate. It reads (cf. [43, 41, 22])

$\mathcal{B}_{tr}(q)=\frac{\hbar}{2\varrho
s}\times\begin{cases}(2\Xi_{c}-\Xi_{v})^{2},\quad\mbox{n-type,}\\\
(2\Xi_{v}-\Xi_{c})^{2},\quad\mbox{p-type,}\end{cases}$ (18)

where $\Xi_{c}$, $\Xi_{v}$ are the conduction and valence band deformation
potentials, $\varrho$ is the mass density of the monolayer, $s$ is the
longitudinal sound speed, and we consider both n- and p-type structures. In
derivation of Eq. (18) we disregarded the form-factor arising from the overlap
of the phonon and trion wavefunctions assuming that $qa_{tr}\ll 1$.

Figure 4: (a) Series of time-resolved PL spectra of the X+ resonance at 5 K
for different times after near-resonant excitation of the neutral exciton at
1.663 eV. Estimated pump density of the injected electron-hole pairs is
$4.5\times 10^{11}$ cm-2, considering absorption of 2% at the pump energy. The
density of the free holes is set to about $7\times 10^{10}$ cm-2. (b)
Corresponding streak camera image of the PL intensity in a false-color plot,
as function of time and emission energy. Dotted and dashed lines represent the
time-dependent exponent $\epsilon^{*}$ of the low-energy flank (x4) and the
trion peak energy $E^{tr}_{0}$, respectively. They are obtained by an
automated fit procedure of spectra in intervals of 1 ps using Eq. (1). (c)
Extracted, transient temperatures of the trion measured for different lattice
temperatures between 5 and 50 K. (d) Characteristic 1/e cooling times of the
trions as function of lattice temperature. Rise-time of the PL transients is
presented for direct comparison. (e) Corresponding PL transients of the trion
emission. Laser scattering signal representing instrument response function
with 4 ps width is shown by the gray area. An exponential rise-decay model was
used for fitting, convoluted with a Gaussian to account for the instrument
response.

The energy loss rate by the non-degenerate trion gas characterized by the
distribution function $f_{\bm{k}}$ reads

$Q=\frac{1}{\sum_{\bm{k}}f_{\bm{k}}}\sum_{\bm{k}\bm{q}}\hbar\omega^{ph}_{\bm{q}}\left[f_{\bm{k}}W_{\bm{k}\to\bm{k}-\bm{q}}^{+\bm{q}}-f_{\bm{k}}W^{-\bm{q}}_{\bm{k}\to\bm{k}+\bm{q}}\right],$
(19)

where $\hbar\omega^{ph}_{\bm{q}}=\hbar sq$ is the phonon dispersion,

$W_{\bm{k}\to\bm{k}^{\prime}}^{\pm\bm{q}}=\frac{2\pi}{\hbar}\left|M_{\bm{k}\to\bm{k}^{\prime}}^{tr,\pm\bm{q}}\right|^{2}\left(n^{ph}_{\bm{q}}+\frac{1}{2}\pm\frac{1}{2}\right)\delta(E^{tr}_{\bm{k}}-E^{tr}_{\bm{k}^{\prime}}\pm\hbar\omega^{ph}_{\bm{q}}),$
(20)

$n^{ph}_{\bm{q}}$ is the phonon distribution function. Making the replacement
$\bm{k}\to\bm{k}-\bm{q}$ in the second term in Eq. (19) and using Eq. (17) we
arrive at

$Q=\frac{1}{\int d\bm{k}f_{\bm{k}}}\int
d\bm{k}\int\frac{d\bm{q}}{2\pi\hbar}\hbar\omega_{\bm{q}}\mathcal{B}(q)\delta(E^{tr}_{\bm{k}}-E^{tr}_{\bm{k}-\bm{q}}-\hbar\omega^{ph}_{\bm{q}})\\\
\times\left[f_{\bm{k}}(1+n^{ph}_{\bm{q}})-f_{\bm{k}-\bm{q}}n^{ph}_{\bm{q}}\right].$
(21)

Making use of the explicit form of the distribution function Eq. (4) and
assuming that phonons are characterized by the lattice temperature $T$, Eq.
(21) can be recast as

$Q=\frac{k_{B}(T_{*}-T)}{\tau_{tr,T}},$ (22)

where the cooling time $\tau_{T}$ is defined as

$\frac{1}{\tau_{tr,T}}=\frac{2m_{tr}^{2}}{\hbar^{3}\varrho}\times\begin{cases}(2\Xi_{c}-\Xi_{v})^{2},\quad\mbox{n-type,}\\\
(2\Xi_{v}-\Xi_{c})^{2},\quad\mbox{p-type.}\end{cases}$ (23)

Note, that the cooling time of excitons is described by the same expression
with the replacements $m_{tr}\to M_{X}$ and
$(2\Xi_{c}-\Xi_{v})^{2}\to(\Xi_{c}-\Xi_{v})^{2}$ [22]. We should thus expect
the trions to cool at least as efficiently as the excitons and much faster
than free charge carriers due to the larger mass.

###  V.2 Experiment

In the experiment we focus on the p-doped regime due to the overall narrower
linewidths that allow for a more accurate quantitative analysis (similar
observations for n-doping are presented in the Appendix C). Fig. 4 (a)
presents PL spectra of the trion emission obtained at different times after
near-resonant excitation of the neutral exciton with the photon energy of
1.663 eV. Corresponding streak camera image of the emission is shown in Fig. 4
(b). The spectra are normalized at each time step for better lineshape
comparison.

Directly after the excitation, the PL exhibits a broad low-energy recoil flank
that subsequently narrows at later times. It shows that trions are initially
overheated with comparatively high kinetic energies. Over time, they cool down
towards thermal equilibrium determined by the lattice temperature. In
addition, the emission maximum shifts slightly in energy during the first few
picoseconds, following the shift of the neutral exciton resonance. The latter
is a common observation upon optical excitation of semiconductors and is often
related to the interplay of bandgap and exciton binding energy renormalization
at finite excitation densities [1, 44, 45].

For quantitative analysis of the equilibration dynamics we follow the
procedure outlined in Sect. III. We use the model from Eq. (1) to fit the
spectra at each time step. The resulting fit curves are shown in Fig. 4 (a)
and the corresponding low-energy exponents $\epsilon^{*}$, multiplied by
factor of 4 for better illustration, are indicated in Fig. 4 (b). From
$\epsilon^{*}$ we extract the effective, transient temperature of the trions
by using Eq. (2). We fix the exponential parameter $\epsilon_{1}$ representing
light-matter coupling dependence on kinetic energy to the previously
determined value of 7 meV, see Fig. 1 (e). The presented time-resolved
measurements of the electron recoil and their analysis are then repeated for a
series of lattice temperatures up to 50 K.

The obtained effective temperatures of the trions are presented in Fig. 4 (c)
as function of time after the excitation. For all studied lattice
temperatures, the initial, non-equilibrium trion temperature is higher than
that of the lattice by about 20 K. This value does not depend strongly on the
excess energy of the pump laser, as we obtain a similar excess temperature
also for non-resonant excitation conditions (see Appendix B). It is thus
likely to represent the residual excess energy of the trions after their
formation on a picosecond timescale. At later times the effective trion
temperature decreases and reaches values that are indeed very close to the
respective lattice temperatures. Only at 5 K the trions do not seem to cool
down sufficiently, equilibrating above 10 K. The latter stems likely from
additional heating of the system by the laser illumination, discussed in the
Appendix A.

As illustrated in Fig. 4 (c), the trion temperature decays over time and is
well described by a single-exponential law. Corresponding cooling times are
summarized in Fig. 4 (d) and are on the order of 10 to 15 ps at the lowest
studied temperatures. These values are consistent with the expectations from
the phonon-assisted cooling involving long-range acoustic phonon modes. They
roughly correspond to the relaxation time-scales previously determined for
excitons in TMDCs [46, 47] at liquid helium temperature in agreement with Eq.
(23) which predicts about the same order of magnitude for the exciton and
trion cooling times. In contrast, trion cooling seems indeed to be faster than
that of free electrons in MoSe2, recently determined to occur within 70 ps at
5 K [18]. Similar to electrons, however, trions are found to equilibrate more
rapidly at higher temperatures indicating additional contributions from
thermally activated optical and zone-edge phonons [48, 18].

The independent access to cooling times from transient recoil analysis further
allows for a consistent interpretation of the PL intensity dynamics of the
trions, presented in Fig. 4 (e). In general, it is a priori unclear whether
the PL rise time, for example, corresponds to relaxation or recombination,
being determined by the faster one of the two processes. For bright excitons
in MoSe2 it was indeed shown that a much shorter radiative lifetime governs
the rise time [46] and the slower relaxation time on the order of 20 ps
accounts for the PL decay. In the present case of the trions, however, the
extracted rise time closely follows the values of the cooling time, as
illustrated in Fig. 4 (d). This result seems reasonable in view of the low
doping densities $<10^{11}$ cm-2 used in our study. In this regime, the
transfer of the oscillator strength from the neutral exciton to the
trion/attractive Fermi-polaron is rather small and should thus lead to
comparatively long radiative lifetimes. Consequently, the recombination of the
trions, whether radiative or non-radiative, then determines the subsequent
decay of the PL.

##  VI Impact of free carriers on cooling

In this section we address the trion cooling mediated by free carriers that
can contribute or even dominate relaxation dynamics at finite densities in
addition to phonons. Experimentally, we elucidate the role of free carriers
for trion cooling from time-resolved analysis of the recoil effect as function
of charge carrier density in the p-doped regime.

Figure 5: (a) Representative, time-integrated PL spectra of X+ at different
doping densities up to $5\times 10^{11}$ cm-2 (using the hole mass of 0.6 m0,
1 meV energy shift corresponds to change of doping by $2.5\times 10^{11}$
cm-2). All experimental parameters are kept the same as for the measurements
presented in Fig. 4: photon energy of 1.663 eV, pump density of 6 $\mu$J/cm-2,
estimated injected electron-hole density of $4.5\times 10^{11}$ cm-2, and T =
5 K. (b) Transient excess temperatures of the X+ trion for different densities
of the free holes with respect to the equilibrium temperature at 50 ps. (c)
Extracted cooling time from the exponential fit as function of hole density.
Rise time of the trion PL is shown for comparison. (d) Corresponding PL
transients of the trion emission together with exponential rise-decay fits
$\propto(\tau_{decay}-\tau_{rise})^{-1}[\exp(-t/\tau_{decay})-\exp(-t/\tau_{rise})]$.

Typical, time-integrated PL spectra of the X+ trion emission at 5 K are
presented in Fig. 5 (a) for hole doping densities between $5\times 10^{10}$
cm-2 and $5\times 10^{11}$ cm-2. While at the lowest doping density the
spectral asymmetry from recoil effect is rather pronounced, the spectra
acquire a more symmetric lineshape at higher doping densities. This
observation is partially related to a slightly increased symmetric broadening
due to additional density-induced scattering [7, 49, 25] as well as potential
contributions from direct recombination discussed in the context of Eqs. (15)
and (16). However, it is also associated with increasingly faster trion
cooling that leads to a less pronounced asymmetry in the time-integrated
response.

The doping-induced increase in the trion cooling rates is illustrated in Fig.
5 (b) by evaluating the change of the excess temperature as function of time
for a series of free hole densities. Here, we used the same procedure as
outlined in Sect. V for the time-resolved analysis of the recoil effect. The
resulting density-dependent cooling times obtained from single-exponential
fitting of the temperature transients are summarized in Fig. 5 (c). In the
low-density doping regime, below $3\times 10^{11}$ cm-2, the trion cooling
does not depend on the density of free carriers and is mediated by scattering
with phonons. At elevated hole densities, the cooling times decrease below 5
ps at $2\times 10^{12}$ cm-2. The corresponding, hole-mediated cooling rate
coefficient obtained from the inverse slope of the time constants is 0.042
cm2/s. It means that at the free carrier density of $10^{11}$ cm-2 the
corresponding contribution to the trion cooling time is above 200 ps and
decreases to the range of 10’s of ps for densities above $10^{12}$ cm-2. At
these high densities and above relaxation of the trions is almost exclusively
governed by scattering with free carriers.

Similar to the temperature-dependent analysis, we also find very similar
cooling and PL rise times, as illustrated in Fig. 5 (c). Corresponding
transients are plotted in Fig. 5 (d) together with exponential rise-decay
fits. These results show that while direct radiative recombination of the
trions / attractive Fermi-polarons is expected to occur at elevated doping
levels, cooling processes largely determine the rise time over a broad range
of free carrier densities.

##  VII Conclusion

To summarize, we have studied electron recoil effect in charge-tunable MoSe2
monolayers employing time-resolved photoluminescence spectroscopy (Sect. II)
and analytical theory (Sect. IV). Characteristic spectral asymmetry of the
bound states between excitons and free charge carriers was identified to
originate from carrier-assisted radiative recombination: a photon is emitted
and the free carrier system is left in an excited state (Sect. III).
Theoretically, this process was shown to occur using either a three-particle
trion or a many-particle Fermi polaron description of the exciton-carrier
mixtures (Sect. IV). In both formalisms the low-energy flank of the emission
is determined by the thermal distribution of the quasiparticles and momentum-
dependent light-matter coupling strength at sufficiently low carrier densities
with $E_{F}<k_{B}T$. The resulting spectral lineshapes of trion/Fermi-polaron
luminescence was shown to provide an accurate quantitative description of
temperature-dependent spectra (Sect. III).

Based on this analysis, time-resolved measurements of the recoil effect
allowed for a direct evaluation of the transient trion temperature and
relaxation dynamics (Sect. V). Trions were shown to cool down on a timescale
of about 15 ps at $T=5K$, determined by scattering with long-wavelength
acoustic phonons. The cooling process became increasingly faster at elevated
temperatures, attributed to thermal activation of additional phonon modes.
Similarly, free carriers were found to mediate trion/Fermi-polaron cooling and
dominate it at doping densities close to $10^{12}$ cm-2 and above (Sect. VI).
Finally, we observed a close correspondence between relaxation timescales
obtained from recoil analysis and rise times of the PL transients, providing a
consistent interpretation of the PL dynamics.

Overall, direct experimental access to the non-equilibrium states of bound
exciton-carrier complexes and their relaxation dynamics should be very useful
for the broader community working on exciton-carrier mixtures in monolayer
materials and their heterostructures. Theoretically demonstrated agreement of
trion and Fermi-polaron pictures for the appropriate description of the recoil
effect in the low-density regime should be particularly helpful to motivate
and support future work in this field. Open questions include the interplay of
direct and carrier-assisted recombination, radiative coupling in the high-
density regime, as well as relaxation of higher-particle complexes such as
neutral and charged biexcitons.

##  VIII Acknowledgments

We thank Andrea Bergscheider, Martin Kroner, and Malte Selig for fruitful
discussions as well as Christian Bäuml and Nicola Paradiso for their
assistance with pre-patterned substrate preparation. Financial support by the
DFG via Emmy Noether Initiative (CH 1672/1, Project-ID: 287022282), Walter
Benjamin Programme (Project-ID: 462503440, J.Z.), SFB 1277 (project B05), as
well as the Würzburg-Dresden Cluster of Excellence on Complexity and Topology
in Quantum Matter ct.qmat (EXC 2147, Project-ID 390858490) is gratefully
acknowledged. K.Watanabe and T.T. acknowledge support from the Elemental
Strategy Initiative conducted by the MEXT, Japan (Grant Number
JPMXP0112101001) and JSPS KAKENHI (Grant Numbers 19H05790, 20H00354 and
21H05233). The development of the analytical theory by M.M.G. has been
supported by RSF Project No. 19-12-00051; numerical calculations of the trion
and Fermi-polaron states by M.A.S. have been supported by RSF Project No.
19-12-00273. J. Z. and K. Wagner contributed equally to this manuscript.

## References

  * Haug and Koch [2009] H. Haug and S. W. Koch,  _Quantum theory of the optical and electronic properties of semiconductors_ , 5th ed. (World Scientific, Singapore, 2009).
  * Klingshirn [2007] C. Klingshirn,  _Semiconductor Optics_ , 3rd ed. (Springer, Berlin Heidelberg New York, 2007).
  * Lampert [1958] M. A. Lampert, Mobile and Immobile Effective-Mass-Particle Complexes in Nonmetallic Solids, Phys. Rev. Lett. 1, 450 (1958).
  * Kheng  _et al._ [1993] K. Kheng, R. Cox, M. d’ Aubigné, F. Bassani, K. Saminadayar, and S. Tatarenko, Observation of negatively charged excitons X- in semiconductor quantum wells, Phys. Rev. Lett. 71, 1752 (1993).
  * Koudinov  _et al._ [2014] A. V. Koudinov, C. Kehl, A. V. Rodina, J. Geurts, D. Wolverson, and G. Karczewski, Suris Tetrons: Possible Spectroscopic Evidence for Four-Particle Optical Excitations of a Two-Dimensional Electron Gas, Phys. Rev. Lett. 112, 147402 (2014).
  * Sidler  _et al._ [2017] M. Sidler, P. Back, O. Cotlet, A. Srivastava, T. Fink, M. Kroner, E. Demler, and A. Imamoglu, Fermi polaron-polaritons in charge-tunable atomically thin semiconductors, Nat. Phys. 13, 255 (2017).
  * Efimkin and MacDonald [2017] D. K. Efimkin and A. H. MacDonald, Many-body theory of trion absorption features in two-dimensional semiconductors, Phys. Rev. B 95, 035417 (2017), arXiv:1609.06329 .
  * Esser  _et al._ [2001] A. Esser, R. Zimmermann, and E. Runge, Theory of Trion Spectra in Semiconductor Nanostructures, Phys. status solidi 227, 317 (2001).
  * Ivchenko [2005] E. L. Ivchenko,  _Optical spectroscopy of semiconductor nanostructures_ (Alpha Science, Harrow England, 2005).
  * Esser  _et al._ [2000] A. Esser, E. Runge, R. Zimmermann, and W. Langbein, Photoluminescence and radiative lifetime of trions in GaAs quantum wells, Phys. Rev. B 62, 8232 (2000).
  * Ross  _et al._ [2013] J. S. Ross, S. Wu, H. Yu, N. J. Ghimire, A. M. Jones, G. Aivazian, J. Yan, D. G. Mandrus, D. Xiao, W. Yao, and X. Xu, Electrical control of neutral and charged excitons in a monolayer semiconductor., Nat. Commun. 4, 1474 (2013).
  * Yu  _et al._ [2015] H. Yu, X. Cui, X. Xu, and W. Yao, Valley excitons in two-dimensional semiconductors, Natl. Sci. Rev. 2, 57 (2015).
  * Wang  _et al._ [2018] G. Wang, A. Chernikov, M. M. Glazov, T. F. Heinz, X. Marie, T. Amand, and B. Urbaszek, Colloquium : Excitons in atomically thin transition metal dichalcogenides, Rev. Mod. Phys. 90, 021001 (2018).
  * Christopher  _et al._ [2017] J. W. Christopher, B. B. Goldberg, and A. K. Swan, Long tailed trions in monolayer MoS2: Temperature dependent asymmetry and resulting red-shift of trion photoluminescence spectra, Sci. Rep. 7, 14062 (2017).
  * Lyons  _et al._ [2019] T. P. Lyons, S. Dufferwiel, M. Brooks, F. Withers, T. Taniguchi, K. Watanabe, K. S. Novoselov, G. Burkard, and A. I. Tartakovskii, The valley Zeeman effect in inter- and intra-valley trions in monolayer WSe2, Nat. Commun. 10, 2330 (2019).
  * Zhumagulov  _et al._ [2020] Y. V. Zhumagulov, A. Vagov, D. R. Gulevich, P. E. Faria Junior, and V. Perebeinos, Trion induced photoluminescence of a doped MoS2monolayer, J. Chem. Phys. 153, 1 (2020), arXiv:2005.09306 .
  * Park  _et al._ [2021] S. Park, B. Han, C. Boule, D. Paget, A. C. H. Rowe, F. Sirotti, T. Taniguchi, K. Watanabe, C. Robert, L. Lombez, B. Urbaszek, X. Marie, and F. Cadiz, Imaging Seebeck drift of excitons and trions in MoSe 2 monolayers, 2D Mater. 8, 045014 (2021).
  * Venanzi  _et al._ [2021] T. Venanzi, M. Selig, S. Winnerl, A. Pashkin, A. Knorr, M. Helm, and H. Schneider, Terahertz-Induced Energy Transfer from Hot Carriers to Trions in a MoSe 2 Monolayer, ACS Photonics , acsphotonics.1c00394 (2021).
  * Chang  _et al._ [2018] Y.-C. Chang, S.-Y. Shiau, and M. Combescot, Crossover from trion-hole complex to exciton-polaron in n -doped two-dimensional semiconductor quantum wells, Phys. Rev. B 98, 235203 (2018).
  * Fey  _et al._ [2020] C. Fey, P. Schmelcher, A. Imamoglu, and R. Schmidt, Theory of exciton-electron scattering in atomically thin semiconductors, Phys. Rev. B 101, 195417 (2020).
  * Cotlet  _et al._ [2020] O. Cotlet, D. S. Wild, M. D. Lukin, and A. Imamoglu, Rotons in optical excitation spectra of monolayer semiconductors, Phys. Rev. B 101, 205409 (2020).
  * Glazov [2020a] M. M. Glazov, Quantum Interference Effect on Exciton Transport in Monolayer Semiconductors, Phys. Rev. Lett. 124, 166802 (2020a).
  * Note [1] Please also see a recent preprint discussing spectral lineshape asymmetry for Fermi polarons: T. Wasak et al. arXiv:2103.14040.
  * Castellanos-Gomez  _et al._ [2014] A. Castellanos-Gomez, L. Vicarelli, E. Prada, J. O. Island, K. L. Narasimha-Acharya, S. I. Blanter, D. J. Groenendijk, M. Buscema, G. a. Steele, J. V. Alvarez, H. W. Zandbergen, J. J. Palacios, and H. S. J. van der Zant, Isolation and characterization of few-layer black phosphorus, 2D Mater. 1, 025001 (2014).
  * Wagner  _et al._ [2020a] K. Wagner, E. Wietek, J. D. Ziegler, M. A. Semina, T. Taniguchi, K. Watanabe, J. Zipfel, M. M. Glazov, and A. Chernikov, Autoionization and Dressing of Excited Excitons by Free Carriers in Monolayer WSe2, Phys. Rev. Lett. 125, 267401 (2020a).
  * Smoleński  _et al._ [2019] T. Smoleński, O. Cotlet, A. Popert, P. Back, Y. Shimazaki, P. Knüppel, N. Dietler, T. Taniguchi, K. Watanabe, M. Kroner, and A. Imamoglu, Interaction-Induced Shubnikov–de Haas Oscillations in Optical Conductivity of Monolayer MoSe2, Phys. Rev. Lett. 123, 097403 (2019).
  * Shepard  _et al._ [2017] G. D. Shepard, J. V. Ardelean, O. A. Ajayi, D. Rhodes, X. Zhu, J. C. Hone, and S. Strauf, Trion-Species-Resolved Quantum Beats in MoSe 2, ACS Nano 11, 11550 (2017).
  * Rau [1996] A. R. P. Rau, The Negative Ion of Hydrogen, J. Astrophys. Astr. 17, 113 (1996).
  * Ganchev  _et al._ [2015] B. Ganchev, N. Drummond, I. Aleiner, and V. Fal’ko, Three-Particle Complexes in Two-Dimensional Semiconductors, Phys. Rev. Lett. 114, 107401 (2015).
  * Glazov [2020b] M. M. Glazov, Optical properties of charged excitons in two-dimensional semiconductors, J. Chem. Phys. 153, 034703 (2020b), arXiv:2005.05829 .
  * Suris  _et al._ [2001] R. Suris, V. Kochereshko, G. Astakhov, D. Yakovlev, W. Ossau, J. Nurnberger, W. Faschinger, G. Landwehr, T. Wojtowicz, G. Karczewski, and J. Kossut, Excitons and trions modified by interaction with a two-dimensional electron gas, Phys. status solidi 227, 343 (2001).
  * Suris [2003] R. A. Suris, Optical properties of 2d systems with interacting electrons (NATO ASI, 2003) Chap. Correlation between trion and hole in fermi distribution in process of trion photo-excitation in doped QWs.
  * Schmidt  _et al._ [2012] R. Schmidt, T. Enss, V. Pietilä, and E. Demler, Fermi polarons in two dimensions, Phys. Rev. A 85, 021602 (2012).
  * Cotlet  _et al._ [2019] O. Cotlet, F. Pientka, R. Schmidt, G. Zarand, E. Demler, and A. Imamoglu, Transport of Neutral Optical Excitations Using Electric Fields, Phys. Rev. X 9, 041019 (2019).
  * Note [2] Here we correct a typo in Eqs. (24), (26) of Ref. [30] where the factor $m_{x}/m_{tr}$ was missing in the exponent.
  * Sergeev and Suris [2001] R. A. Sergeev and R. A. Suris, Ground-state energy of $x^{-}$ and $x^{+}$ trions in a two-dimensional quantum well at an arbitrary mass ratio, Phys. Solid State 43, 746 (2001).
  * Berkelbach  _et al._ [2013] T. C. Berkelbach, M. S. Hybertsen, and D. R. Reichman, Theory of neutral and charged excitons in monolayer transition metal dichalcogenides, Phys. Rev. B 88, 045318 (2013).
  * Courtade  _et al._ [2017] E. Courtade, M. Semina, M. Manca, M. M. Glazov, C. Robert, F. Cadiz, G. Wang, T. Taniguchi, K. Watanabe, M. Pierre, W. Escoffier, E. L. Ivchenko, P. Renucci, X. Marie, T. Amand, and B. Urbaszek, Charged excitons in monolayer WSe2 : Experiment and theory, Phys. Rev. B 96, 085302 (2017), arXiv:1705.02110 .
  * Imamoglu  _et al._ [2020] A. Imamoglu, O. Cotlet, and R. Schmidt, Exciton-polarons in two-dimensional semiconductors and the tavis-cummings model (2020), arXiv:2006.15963 [cond-mat.mes-hall] .
  * Christiansen  _et al._ [2017] D. Christiansen, M. Selig, G. Berghäuser, R. Schmidt, I. Niehues, R. Schneider, A. Arora, S. M. de Vasconcellos, R. Bratschitsch, E. Malic, and A. Knorr, Phonon Sidebands in Monolayer Transition Metal Dichalcogenides, Phys. Rev. Lett. 119, 187402 (2017).
  * Shree  _et al._ [2018] S. Shree, M. Semina, C. Robert, B. Han, T. Amand, A. Balocchi, M. Manca, E. Courtade, X. Marie, T. Taniguchi, K. Watanabe, M. M. Glazov, and B. Urbaszek, Observation of exciton-phonon coupling in MoSe2 monolayers, Phys. Rev. B 98, 035302 (2018).
  * Gantmakher and Levinson [1987] V. F. Gantmakher and Y. B. Levinson,  _Carrier Scattering in Metals and Semiconductors_ (North-Holland Publishing Company, 1987).
  * Kaasbjerg  _et al._ [2012] K. Kaasbjerg, K. S. Thygesen, and K. W. Jacobsen, Phonon-limited mobility in n-type single-layer MoS2 from first principles, Phys. Rev. B 85, 115317 (2012).
  * Steinhoff  _et al._ [2014] A. Steinhoff, M. Rösner, F. Jahnke, T. O. Wehling, and C. Gies, Influence of excited carriers on the optical and electronic properties of MoS2., Nano Lett. 14, 3743 (2014).
  * Schmidt  _et al._ [2016] R. Schmidt, G. Berghäuser, R. Schneider, M. Selig, P. Tonndorf, E. Malić, A. Knorr, S. Michaelis de Vasconcellos, and R. Bratschitsch, Ultrafast Coulomb-Induced Intervalley Coupling in Atomically Thin WS 2, Nano Lett. 16, 2945 (2016).
  * Fang  _et al._ [2019] H. H. Fang, B. Han, C. Robert, M. A. Semina, D. Lagarde, E. Courtade, T. Taniguchi, K. Watanabe, T. Amand, B. Urbaszek, M. M. Glazov, and X. Marie, Control of the Exciton Radiative Lifetime in van der Waals Heterostructures, Phys. Rev. Lett. 123, 067401 (2019).
  * Rosati  _et al._ [2020] R. Rosati, K. Wagner, S. Brem, R. Perea-Causín, E. Wietek, J. Zipfel, J. D. Ziegler, M. Selig, T. Taniguchi, K. Watanabe, A. Knorr, A. Chernikov, and E. Malic, Temporal Evolution of Low-Temperature Phonon Sidebands in Transition Metal Dichalcogenides, ACS Photonics 7, 2756 (2020).
  * Selig  _et al._ [2016] M. Selig, G. Berghäuser, A. Raja, P. Nagler, C. Schüller, T. F. Heinz, T. Korn, A. Chernikov, E. Malic, and A. Knorr, Excitonic linewidth and coherence lifetime in monolayer transition metal dichalcogenides, Nat. Commun. 7, 13279 (2016).
  * Carbone  _et al._ [2020] M. R. Carbone, M. Z. Mayers, and D. R. Reichman, Microscopic model of the doping dependence of linewidths in monolayer transition metal dichalcogenides, J. Chem. Phys. 152, 194705 (2020).
  * Leo  _et al._ [1988] K. Leo, W. W. Rühle, and K. Ploog, Hot-carrier energy-loss rates in GaAs/AlxGa1-xAs quantum wells, Phys. Rev. B 38, 1947 (1988).
  * Mak  _et al._ [2013] K. F. Mak, K. He, C. Lee, G. H. Lee, J. Hone, T. F. Heinz, and J. Shan, Tightly bound trions in monolayer MoS2., Nat. Mater. 12, 207 (2013).
  * Kormányos  _et al._ [2015] A. Kormányos, G. Burkard, M. Gmitra, J. Fabian, V. Zólyomi, N. D. Drummond, and V. Fal’ko, K · P Theory for Two-Dimensional Transition Metal Dichalcogenide Semiconductors, 2D Mater. 2, 022001 (2015).
  * Larentis  _et al._ [2018] S. Larentis, H. C. P. Movva, B. Fallahazad, K. Kim, A. Behroozi, T. Taniguchi, K. Watanabe, S. K. Banerjee, and E. Tutuc, Large effective mass and interaction-enhanced zeeman splitting of $k$-valley electrons in ${\mathrm{mose}}_{2}$, Phys. Rev. B 97, 201407 (2018).
  * Wagner  _et al._ [2020b] K. Wagner, E. Wietek, J. D. Ziegler, M. A. Semina, T. Taniguchi, K. Watanabe, J. Zipfel, M. M. Glazov, and A. Chernikov, Autoionization and dressing of excited excitons by free carriers in monolayer ${\mathrm{wse}}_{2}$, Phys. Rev. Lett. 125, 267401 (2020b).
  * Chevy [2006] F. Chevy, Universal phase diagram of a strongly interacting fermi gas with unbalanced spin populations, Phys. Rev. A 74, 063628 (2006).
  * Wasak  _et al._ [2021] T. Wasak, F. Pientka, and F. Piazza, Fermi polaron laser in two-dimensional semiconductors,  (2021), arXiv:2103.14040 .

##  Appendix A Pump power dependence of trion/Fermi-polaron cooling

In this section we present the dependence of the measured cooling dynamics on
the pump power of the excitation laser. The experimental approach to determine
transient trion/Fermi-polaron temperatures is described in Sect. V.2 of the
main manuscript. It is based on the time-resolved analysis of the low-energy
recoil flank. For the measurements presented here, the lattice temperature was
set to 5 K and the free hole density was fixed to about $n_{h}=1\times
10^{11}$ cm-2. The effect of the photo-excitation on the free carrier doping
was compensated by readjusting the gate voltage during the measurement (see
Appendix D). The injected electron-hole pair density is obtained as
$n_{eh}=4\alpha P/(E_{ph}f_{rep}\pi w^{2})$, with the pump power P, repetition
rate $f_{rep}=80$ MHz, full-width-at-half-maximum of the spot of $w=1\mu$m,
excitation energy $E_{ph}=1.663$ eV and an effective absorption of
$\alpha=2\%$. The latter is estimated from absorption type reflectance
measurements.

Figure 6:  (a) Transient trion temperature for different injected electron-
hole pair densities $n_{eh}$ at a fixed hole density of $n_{h}\approx 1\times
10^{11}$ cm-2, near-resonant excitation at $1.663$ eV and lattice temperature
T $=5$ K. Solid lines correspond to mono-exponential fits of the data. The
extrapolated zero-density trion temperature after 50 ps is 7.6 K. The inset
shows the corresponding X+ peak energy shift $\Delta$E${}_{0}^{tr}$ as a
function of pump density. (b) Extracted 1/e cooling times from (a) as function
of pump density $n_{eh}$. Rise time of the trion PL is shown for comparison.

Fig. 6 (a) shows the transient trion temperature for several different pump
densities $n_{eh}$ in the range of $4\times 10^{10}$ to $8\times 10^{11}$
cm-2. We obtain similar cooling transients for the lowest pump densities,
indicating linear excitation regime. At these conditions, the extracted
effective temperature after 50 ps reaches 8 K for the lowest pump density.
This value is very close to the nominal heat sink temperature of 5 K.

At higher electron-hole pair densities, however, both excess and equilibrium
temperatures increase, while the cooling time remains almost unchanged within
experimental error, as illustrated in Fig. 6 (b). The observed increase of the
excess and equilibrium temperatures is consistent with the observed shift of
the trion emission peak to lower energies, shown in the inset of Fig. 6 (a).
We also note that the shift of the neutral exciton peak energy is almost
identical (not shown here). Transient increase of the trion/Fermi-polaron
temperature may be associated with additional heating due to reabsorption of
non-equilibrium phonons, created during initial relaxation, known as “hot-
phonon effect” in the literature [50]. In addition, laser-induced heating of
the lattice could contribute to the increase of the equilibrium temperature
[17]. The latter is consistent with the shift of all resonances to lower
energies following temperature-induced decrease of the bandgap.

We note that in the temperature-dependent experiments presented in Fig. 1 and
Fig. 4 of the main manuscript we used an excitation density of $4$x$10^{11}$
cm-2 for increased signal-to-noise ratio. We would thus expect additional
contributions from laser heating leading to slightly larger absolute values of
the extracted trion/Fermi-polaron temperatures. These should be particularly
pronounced for measurements at nominal heat sink temperatures T $\leq 15$ K.
Most importantly, however, the cooling times are robust with respect to the
pump power. As demonstrated in the main manuscript, they also match the values
of the PL rise time with small deviations at lowest pump densities.

##  Appendix B Trion / Fermi-polaron cooling at different excitation energies

Figure 7:  Transient trion / Fermi-polaron temperature for near-resonant and
non-resonant excitation of the neutral exciton transition. The pump density is
set to a few $10^{11}$ cm-2 and the hole density to $n_{h}\approx 1\times
10^{11}$ cm-2. Mono-exponential fit (solid line) corresponds to a cooling time
of 9.5 ps.

The PL studies reported in the main manuscript are performed using pulsed
excitation at the photon energy of 1.657 eV, near-resonant to the A-exciton
transition at $1.642$ eV. In this section, we present the comparison of these
results with the case of non-resonant excitation with about 0.13 eV excess
energy by tuning the laser to 1.774 eV. The lattice temperature is set to 5 K
and we consider the regime of low and comparable densities of the free holes
and optically injected electron-hole pairs in the range of $10^{11}$ cm-2. As
illustrated in Fig. 7, we find essentially the same behavior of the transient
trion / Fermi-polaron temperature for near-resonant and non-resonant
excitation conditions. Both the initial temperatures and the cooling times are
the same. This result strongly suggests that the detected dynamics are not
determined by the initial relaxation of the optically injected electron-hole
pairs. The latter should thus occur on much faster, sub-picosecond time-scales
below the resolution of the setup.

##  Appendix C Trion / Fermi-polaron cooling in the n-doped regime

As we illustrate in Figs. 1 (a) and (b) both n- and p-type trions / attractive
Fermi-polarons exhibit similar low-energy flanks in their PL due to the recoil
effect. For the time-resolved studies, however, discussed in the main
manuscript we focus on the p-doped region due to the overall smaller symmetric
broadening allowing for more accurate quantitative analysis. In this section,
we address the n-doped regime for direct comparison.

In Fig. 8 (a) we compare the extracted transient trion temperature at free
carrier densities $n_{e}\approx n_{h}\approx 7\times 10^{11}$cm-2 at the
lattice temperature of 5 K. Here, we follow the same procedure as discussed in
the main manuscript the trion temperature. We further consider the light-
matter coupling exponent $\epsilon^{*}$ to be same for the p- and n-doped
species due to matching binding energies. Both $X^{+}$ and $X^{-}$ features
exhibit very similar initial excess temperatures on the order of 10 to 20 K
and, more importantly, almost equivalent cooling times. The latter roughly
correspond to the rise times of the PL intensity transients, presented in Fig.
8 (b). We note, that only the decay times of the trions / Fermi-polarons are
substantially shorter in the n-doped regime. Overall, these results
demonstrate that the cooling dynamics do not depend strongly on the charge of
the free carriers.

Figure 8:  (a) Transient excess trion temperature for estimated free carrier
densities of $n_{e}\approx n_{h}\approx 7\times 10^{11}$ cm-2. The excess
temperature is plotted with respect to the equilibrium temperature at 40 ps.
(b) Corresponding normalized trion PL transients in the p- and n- doped
regime. Extracted exponential cooling times as well as PL rise and decay times
are illustrated.

##  Appendix D Estimation of free charge carrier concentration

In the studied devices the free charge carrier density is controlled by
applying a variable gate voltage in a plate capacitor geometry: thin-layer
graphite flake and MoSe2 monolayer separated by an thin hBN spacer (additional
details regarding fabrication of the devices are presented in the
supplementary of Ref. [25]). While a capacitor model can be used to estimate
free carrier density, it tends to be less reliable in the low-doping regime
motivating the use of established spectroscopic observables instead.

Figure 9:  (a) Relative exciton-trion energy separation as function of
temperature corresponding to the measurements presented in Fig. 1 (c). (b)
Relative exciton-trion energy separation as function of pump density $n_{eh}$
corresponding to the measurements presented in Fig. 6. The remaining
experimental fluctuation of the doping level is estimated by setting the
exciton-trion energy variations equal to the Fermi energy shift using a hole
mass of $0.6$ m0. In both (a) and (b) the average hole density is $1\times
10^{11}$ cm.2.

In our study we use the relative energy separation between exciton and trion
peaks (or repulsive and attractive Fermi polarons). This quantity scales
linearly with the free carrier density in both trion and Fermi polaron
formalisms [30]. While there are small differences in the prefactors [30],
resulting in systematic deviations on the order of 30%, we follow the approach
in Ref. [51] and set it equal to the Fermi energy shift $\Delta E_{f}$. The
energy separation is extracted from fitting gate voltage-dependent PL spectra
with a multi-Lorentzian model in the low-density regime. From this analysis,
we also determine the onset for zero-doping at -0.2 and 0.4 V for p- and
n-doping, respectively.

Considering that free charge carriers reside in the energetically lowest
conduction and valence bands at the K/K’ valleys, the doping level is
calculated as $n_{e,h}=\Delta E_{f}m_{e,h}/(\hbar^{2}\pi)$ with the effective
masses of either $m_{e}=0.5$ m0 or $m_{h}=0.6$ m0 [52]. From the dependence of
exciton-trion energy separation on gate voltage we obtain scaling factors for
the charge carrier densities $2\times 10^{11}$ cm${}^{-2}/V$ and $4.8\times
10^{11}$ cm${}^{-2}/V$ for p- and n-doping, respectively. We note, however,
that the extracted ratio $m_{h}/m_{x}=0.44$ from the analysis of temperature-
dependent measurements in Fig. 1 (e) with Eq. (2) indicates either a lower
hole or a higher electron effective mass. The latter would be consistent with
transport experiments at in monolayer MoSe2 [53] that demonstrate an electron
mass as large as 0.8 m0. In that case, electron densities would be
underestimated in our analysis, while the hole densities should remain largely
correct.

We also note that that the doping density can slowly change over time as well
as under optical illumination. To compensate for that, we manually controlled
the gate voltage to ensure constant doping density throughout all temperature-
and pump-density dependent experiments presented in Figs. 1, Fig. 5 and Fig.
6. The gate voltage was slightly adjusted such that the exciton-trion energy
separation remained fixed within a small experimental uncertainty far below 1
meV. Fig. 9 (a) and (b) show the corresponding relative exciton-trion energy
separations as function of temperature and injected electron-hole pair density
$n_{eh}$, respectively. The data illustrates residual fluctuations of the free
charge carrier on the order of several 1010 cm-2.

##  Appendix E Basics of the Fermi-polaron approach

We use a simplified approach to the description of the exciton interaction
with the resident charge carriers presented in Ref. [30]. For definiteness we
consider an electron-doped sample (the modification of the results for the
case of the hole doping are evident) and, for simplicity, we assume that the
electrons are degenerate. We consider the regime where

$E_{F}\ll E_{b,tr}\ll E_{b,x}.$ (24)

Here $E_{F}$ is the electron Fermi energy, $E_{b,tr}$ is the trion binding
energy and $E_{b,x}$ is the exciton binding energy, respectively. The exciton
Greens function in the presence of the resident electrons is given by

$\mathcal{G}_{x}(\varepsilon,\bm{k})=\frac{1}{\varepsilon-E^{x}_{\bm{k}}-\Sigma_{x}(\varepsilon,\bm{k})}\\\
\approx\frac{Z_{\bm{k}}}{\varepsilon-E^{tr}_{\bm{k}}-N_{e}\mathcal{D}^{-1}}+\frac{1-Z_{\bm{k}}}{\varepsilon-E^{x}_{\bm{k}}}.$
(25)

The approximate equality holds for the energies in the vicinity of the trion
(attractive Fermi-polaron) and exciton (repulsive Fermi-polaron) resonances.
Here $E^{x}_{\bm{k}}$ is the exciton energy dispersion, $E^{tr}_{\bm{k}}$ is
the trion energy dispersion, $\Sigma_{x}(\varepsilon,\bm{k})$ is the exciton
self-energy,

$Z_{\bm{k}}\approx N_{e}/(\mathcal{D}E_{b,tr}),$ (26)

is the renormalization factor proportional to the electron density $N_{e}$
which describes, in particular, the transfer of the oscillator strength
between the repulsive polaron (exciton) and attractive polaron (trion).

Generally, in agreement with [31, 32, 33, 34, 30, 54] the exciton self-energy
and the electron-exciton scattering amplitude obey the set of the self-
consistent equations

$\Sigma_{x}(\varepsilon,\bm{k})=\int\frac{d\xi}{2\pi\mathrm{i}}\sum_{\bm{p}}G_{e}(\xi,\bm{p})T\left(\varepsilon{+}\xi,\bm{k}+\bm{p}\right)\\\
=\sum_{\bm{p}}n_{{\bm{p}}}T\left(\varepsilon+\frac{\hbar^{2}p^{2}}{2m_{e}},\bm{k}+\bm{p}\right),$
(27a)
$1/T(\varepsilon,\bm{k})=-\mathcal{D}\ln{\left(\frac{\bar{E}}{E_{b,tr}}\right)}\\\
{+}\sum_{\bm{p}}(1-n_{\bm{p}})\mathcal{G}_{x}\left(\varepsilon-\frac{\hbar^{2}p^{2}}{2m_{e}};\bm{k}-\bm{p}\right),$
(27b)

where $G_{e}(\xi,\bm{p})$ is the electron Greens function, $n_{\bm{p}}$ is the
electron distribution function, $T$ is the scattering amplitude, $\mathcal{D}$
is the exciton-electron reduced density of states, $\bar{E}$ is the cut-off
energy and $E_{b,tr}>0$ is the trion binding energy. Hereafter we set the
normalization area to unity. At negligible electron density where Eq. (24)
holds electron occupation numbers in Eq. (27b) can be set to zero.

In the non-self-consistent approach where in Eq. (27b) $\mathcal{G}_{x}$ is
replaced by the bare exciton Greens function $G_{x}$ and $n_{\bm{p}}$ is
neglected, we arrive at the second line of Eq. (25) with $Z_{\bm{k}}$ given by
Eq. (26), see Ref. [30] for details. Such an approach is equivalent to the
Chevy representation of the Fermi-polaron state. Following Refs. [55, 6], see
also [32], we present the wavefunction of the system “exciton+Fermi-gas” in
the form

$\Psi_{\bm{k}}=\varphi(\bm{k})X_{\bm{k}}^{\dagger}|0\rangle+\sum_{\bm{p},\bm{q}}F_{\bm{p},\bm{q}}(\bm{k})X^{\dagger}_{\bm{k}+\bm{q}-\bm{p}}e^{\dagger}_{\bm{p}}e_{\bm{q}}|0\rangle.$
(28)

Here $|0\rangle$ is the ground state of the (doped) semiconductor with the
filled valence band and partially filled (up to $k=k_{F}$) conduction band,
$e^{\dagger}$, $e$ are the conduction electron creation and annihilation
operators, $X^{\dagger}$, $X$ is the exciton creation and annihilation
operators. In Eq. (28) the summation over $\bm{q}$ is limited by $q<k_{F}$. At
small doping, where the product to the trion radius, $a_{tr}$, and the Fermi
wavevector is small, $a_{tr}k_{F}\ll 1$ [cf. Eq. (24)], the dependence of
$F_{\bm{p},\bm{q}}$ on $\bm{q}$ can be disregarded. In this limit, the
function $F_{\bm{p},\bm{q}}(\bm{k})$ has a meaning of the “trion”
wavefunction, it accounts for the correlation between the exciton and
electron. Under condition of Eq. (24) and replacing the summation over
$\bm{q}$ by the multiplication by the electron density $N_{e}$ we arrive at
the following equations for the functions $\varphi(\bm{k})$ and
$F_{\bm{p},\bm{q}}$ and the energy of the Fermi-polaron $\varepsilon$:

$E^{x}_{\bm{k}}\varphi(\bm{k})+V_{0}N_{e}\varphi(\bm{k})\\\
+V_{0}N_{e}\sum_{\bm{p}}F_{\bm{p},\bm{q}}(\bm{k})=\varepsilon\varphi(\bm{k}),$
(29a)
$\left(E^{x}_{\bm{k}-\bm{p}}+E^{e}_{\bm{p}}\right)F_{\bm{p},\bm{q}}(\bm{k})\\\
+V_{0}\varphi(\bm{k})+V_{0}\sum_{\bm{p}^{\prime}}F_{\bm{p}^{\prime},\bm{q}}(\bm{k})=\varepsilon
F_{\bm{p},\bm{q}}(\bm{k}),$ (29b)

where $V_{0}$ is the bare constant of the electron-exciton scattering.
Assuming $V_{0}$ to be small and working in the linear order in $N_{e}$ we
obtain

$\displaystyle\varphi(\bm{k})\approx\frac{V_{0}N_{e}}{-E_{b,tr}}\mathcal{F}(\bm{k}),$
(30a) $\displaystyle
F_{\bm{p},\bm{q}}(\bm{k})\approx\frac{V_{0}}{E_{\bm{k}}^{tr}-E_{\bm{k}-\bm{p}}^{x}-E_{\bm{p}}^{e}}\mathcal{F}(\bm{k}),$
(30b)

where

$\mathcal{F}(\bm{k})=\sum_{\bm{p}}F_{\bm{p},\bm{q}}(\bm{k}).$ (31)

The self-consistency condition, Eq. (31), yields the trion dispersion
$E^{tr}_{\bm{k}}=E^{tr}_{0}+\hbar^{2}k^{2}/2m_{tr}$. The function
$\mathcal{F}(\bm{k})$ can be found from the normalization condition

$1=|\varphi(\bm{k})|^{2}+\sum_{\bm{p},\bm{q}}|F_{\bm{p},\bm{q}}|^{2}\approx\sum_{\bm{p},\bm{q}}|F_{\bm{p},\bm{q}}|^{2}.$
(32)

At not too large $k$ ($\hbar^{2}k^{2}/2\mu_{eX}\ll E_{b,tr}$) we find in the
lowest order in $N_{e}$ that

$|\mathcal{F}(\bm{k})|^{2}\approx\frac{E_{b,tr}}{N_{e}\mathcal{D}V_{0}^{2}}.$
(33)

##  Appendix F Photoluminescence of Fermi-polarons

In order to calculate the PL we evaluate the photon self-energy related to the
photon generation as a result of the Fermi-polaron decay [Fig. 3(a)]:

$\Sigma_{PL}(\hbar\omega)=|\mathfrak{M}_{r}|^{2}\\\
\times\int\frac{d\Omega}{2\pi\mathrm{i}}\int\frac{d\xi}{2\mathrm{i}\pi}\sum_{\bm{q},\bm{k}}\mathcal{G}_{x}^{--}(\hbar\omega,0)\mathcal{G}_{x}^{++}(\hbar\omega,0)\\\
\times
T^{2}(\hbar\omega+\hbar\Omega+\xi,\bm{k}+\bm{q})\mathcal{G}_{x}^{-+}(\hbar\omega+\hbar\Omega,\bm{k})\\\
\times G_{e}^{-+}(\xi,\bm{q})G_{e}^{+-}(\xi+\hbar\Omega,\bm{q}+\bm{k}).$ (34)

Here $G_{x}(\varepsilon,\bm{k})$ is the dressed exciton Keldysh Greens
function, $G_{e}(\xi,\bm{q})$ is the resident electron Greens function,
superscripts $+$ and $-$ denote the branch of the Keldysh contour (cf. Ref.
[30]), and $T(\varepsilon,\bm{k})$ is the exciton-electron scattering
amplitude. Equation (34) can be simplified taking into account the fact that
the summation over $\bm{q}$ takes place over the occupied states: Hence, for
degenerate carriers, $\xi$ and $E^{e}_{\bm{q}}\lesssim E_{F}$ that is much
smaller than the trion binding energy. As a result, we can omit the dependence
on $\bm{q}$ and $\xi$ in $T^{2}$. As a result, we obtain

$\Sigma_{PL}(\hbar\omega)=\frac{|\mathfrak{M}_{r}|^{2}}{(\hbar\omega-
E_{0}^{x})^{2}}\int\frac{d\Omega}{2\pi\mathrm{i}}\sum_{\bm{k}}T^{2}(\hbar\omega+\hbar\Omega,\bm{k})\\\
\times\Pi(\hbar\Omega,\bm{k})\mathcal{G}_{x}^{-+}(\hbar\omega+\hbar\Omega,\bm{k}),$
(35)

with

$\Pi(\hbar\Omega,\bm{k})=\sum_{\bm{q}}\int\frac{d\xi}{2\pi\mathrm{i}}G_{e}^{+-}(\xi+\hbar\Omega,\bm{k}+\bm{q})G_{e}^{-+}(\xi,\bm{q})\\\
\approx-2\pi\mathrm{i}N_{e}\delta(E^{e}_{\bm{q}}-\hbar\Omega),$ (36)

being the electron polarizability. The last approximate equality holds at
$k\gg k_{F}$. This condition enables us to neglect the dynamics of the hole in
the Fermi sea, cf. Ref. [32], and the Pauli blocking effects for the recoil
electron.

Using Eq. (36) (last approximate equality), assuming thermal equilibrium of
Fermi-polarons with

$\mathcal{G}_{x}^{-+}(\varepsilon,\bm{k})=2\pi\mathrm{i}f(\varepsilon)Z_{\bm{k}}\delta(\varepsilon-
E_{\bm{k}}^{tr}-N_{e}/\mathcal{D}),$ (37)

with quasi-equilibrium distribution
$f(\varepsilon)\propto\exp{(-\varepsilon/k_{B}T_{*})}$, cf. Eq. (4), using the
expression

$T(\varepsilon,\bm{k})=\frac{\mathcal{D}^{-1}E_{b,tr}}{\varepsilon-
E_{\bm{k}}^{tr}+\mathrm{i}\delta},$ (38)

valid in the vicinity of the attractive Fermi-polaron resonance, and retaining
small energy shift $N_{e}/\mathcal{D}$ in the argument of $T$ only (which is
crucial to avoid the divergence and properly account for the renormalization
factors) we arrive at Eq. (9) of the main text.

##  Appendix G Radiative decay rate

Self-energy is shown by Fig. 3(b) and reads

$\Sigma_{phot}(\varepsilon,\bm{k})=\int
d\omega\sum_{\bm{q},\bm{p}}\int\frac{d\xi}{2\pi\mathrm{i}}\int\frac{d\xi^{\prime}}{2\pi\mathrm{i}}\\\
\times
T(\varepsilon+\xi,\bm{k}+\bm{q})G_{e}(\xi,\bm{q})G_{e}(\xi^{\prime},\bm{p})T(\varepsilon+\xi,\bm{p})\\\
\times\delta_{\bm{p},\bm{k}+\bm{q}}|\mathfrak{M}_{r}|^{2}\mathcal{G}_{x}^{2}(\varepsilon+\xi-\xi^{\prime},0)G_{phot}(\varepsilon+\xi-\xi^{\prime},\hbar\omega).$
(39)

Here
$G_{phot}(\varepsilon,\hbar\omega)=\pi\mathrm{i}\mathcal{D}_{phot}\delta(\varepsilon-\hbar\omega)$
is the imaginary part of the photon Greens function, $\mathcal{D}_{phot}$ is
the photon density of states at the energy $\varepsilon$. For simplicity we
omitted the polarization-dependent factors in Eq. (39). As in Sec. F the
summation over $\bm{q}$ is restricted by the occupied states, thus
$\xi,E_{\bm{q}}^{e}\lesssim E_{F}$ are much smaller than other relevant
energies, including $E_{b,tr}$ and $E^{e}_{\bm{k}}$, $E^{tr}_{\bm{k}}$. As a
result, Eq. (39) simplifies to

$\Sigma_{phot}(\varepsilon,\bm{k})=\int
d\omega|\mathfrak{M}_{r}|^{2}T^{2}(\varepsilon,\bm{k})\int\frac{d\xi^{\prime}}{2\pi\mathrm{i}}\\\
\times\Pi(\xi^{\prime},\bm{k})\mathcal{G}_{x}^{2}(\varepsilon-\xi^{\prime},0)G_{phot}(\varepsilon-\xi^{\prime}).$
(40)

The remaining integrations are trivial and yield Eq. (12) of the main text.

##  Appendix H Fermi-polaron cooling

Figure 10: Diagrams describing the Fermi-polaron interaction with phonons. We
consider a $n$-type system with resident electrons. Dash-dotted line is the
phonon Greens function. (a-c) Basic diagrams showing individual contributions
resulting in $(\Xi_{c}-\Xi_{v})^{2}$, $\Xi_{c}^{2}$ and
$2\Xi_{c}(\Xi_{c}-\Xi_{v})$ terms in $\mathcal{B}_{tr}(q)$, Eq. (18). (d-f)
Fermi-polaron self-energies. Note that the diagrams with at least one of the
phonon vertices describing the Fermi sea hole interaction with phonons (not
shown) are negligible at small electron densities. The diagrams with the
phonon line encompassing $\mathcal{G}_{x}$ (not shown) have an additional
smallness $\propto N_{e}/(\mathcal{D}E_{b,tr})$.

Relevant diagrams describing Fermi-polaron coupling with phonons are shown in
Fig. 10. In what follows we consider the situation where the typical
wavevector transferred in the course of the Fermi-polaron-phonon scattering
exceeds the Fermi wavevector of the resident charge carriers $k_{F}$. This
assumption allows us to neglect the interaction of the Fermi-sea hole with the
phonon because the momentum of the Fermi-sea-hole is limited by $k_{F}$. As a
result we are left with the essentially exciton- and electron-phonon
interaction, or, in terms of constituent quasi-particles, the phonon
interaction with the electron-in-exciton, outer electron in trion and hole-in-
exciton.

The Keldysh technique allows to derive the kinetic equation for the quasi-
particles distribution function. Correspondingly, the expressions for the
cooling rate in the form of Eqs. (19) and (20) can be derived, cf. [30, 56].
The key issue is to calculate the corresponding self-energies or matrix
elements of the scattering. The relevant diagrams for the self energies are
shown in Fig. 10(d-e). It is illustrative, however, to derive the transition
matrix elements

$M_{\bm{k}\to\bm{k}^{\prime}}^{FP,\pm\bm{q}}=\langle\Psi_{\bm{k}^{\prime}}|V_{ph}^{\pm}(\bm{q})|\langle\Psi_{\bm{k}}\rangle,$
(41)

directly using the Chevy ansatz wavefunctions, Eq. (28). Here $V_{ph}^{\pm}$
are the operators of the electron (and exciton) interaction with longitudinal
acoustic phonons [41]:

$V_{ph}^{+}(\bm{q})=\sqrt{1+n^{ph}_{\bm{q}}}\sqrt{\frac{\hbar}{2\varrho\omega_{\bm{q}}^{ph}}}q\\\
\times\left[\sum_{\bm{k}}(\Xi_{c}-\Xi_{v})X^{{\dagger}}_{\bm{k}-\bm{q}}X_{\bm{k}}+\sum_{\bm{p}}\Xi_{c}e^{{\dagger}}_{\bm{p}-\bm{q}}e_{\bm{p}}\right],$
(42a)
$V_{ph}^{-}(\bm{q})=\sqrt{n^{ph}_{\bm{q}}}\sqrt{\frac{\hbar}{2\varrho\omega_{\bm{q}}^{ph}}}q\\\
\times\left[\sum_{\bm{k}}(\Xi_{c}-\Xi_{v})X^{{\dagger}}_{\bm{k}+\bm{q}}X_{\bm{k}}+\sum_{\bm{p}}\Xi_{c}e^{{\dagger}}_{\bm{p}+\bm{q}}e_{\bm{p}}\right].$
(42b)

Hre $\Xi_{c}$ and $\Xi_{v}$ are the deformation potential constants for the
conduction and valence bands. The extension of the approach to account for
other phonon modes and other types of interaction is straightforward.

In the lowest in the resident electron density $N_{e}$ order, the matrix
elements $M_{\bm{k}\to\bm{k}^{\prime}}^{FP,\pm\bm{q}}$ are contributed by the
transitions with the excited Fermi sea [second term in Eq. (28)]. For example,

$M_{\bm{k}\to\bm{k}^{\prime}}^{FP,-\bm{q}}=\sqrt{n^{ph}_{\bm{q}}}\sqrt{\frac{\hbar}{2\varrho\omega_{\bm{q}}^{ph}}}q\\\
\times\sum_{\bm{p},\bm{q}^{\prime}}\left[(\Xi_{c}-\Xi_{v})F^{*}_{\bm{p},\bm{q}^{\prime}}(\bm{k}+\bm{q})F_{\bm{p},\bm{q}^{\prime}}(\bm{k})\right.\\\
+\Xi_{c}F^{*}_{\bm{p}+\bm{q},\bm{q}^{\prime}}(\bm{k}+\bm{q})F_{\bm{p},\bm{q}^{\prime}}(\bm{k})\\\
\left.-\Xi_{c}F^{*}_{\bm{p},\bm{q}^{\prime}+\bm{q}}(\bm{k}+\bm{q})F_{\bm{p},\bm{q}^{\prime}}(\bm{k})\right].$
(43)

Under assumptions $k_{F}\ll q\ll a_{tr}^{-1}$ we can neglect the last term in
Eq. (43) because the transition of a Fermi sea hole with large wavevector
transfer is impossible, and summations over $\bm{p}$ and $\bm{q}$ can be
reduced to $1$ by virtue of Eq. (32). As a result, we arrive at

$M^{FP,\pm\bm{q}}_{\bm{k}\to\bm{k}^{\prime}}=M^{tr,\pm\bm{q}}_{\bm{k}\to\bm{k}^{\prime}},$

where $M^{tr}$ is given by Eq. (17) demonstrating the equivalence of the trion
and Fermi-polaron approaches.