# Ultrafast nonadiabatic phonon renormalization in photoexcited single-layer
MoS2

Nina Girotto  Centre for Advanced Laser Techniques, Institute of Physics,
10000 Zagreb, Croatia  Fabio Caruso  Institut für Theoretische Physik und
Astrophysik, Christian-Albrechts-Universität zu Kiel, 24098 Kiel, Germany
Dino Novko<EMAIL_ADDRESS>Centre for Advanced Laser Techniques,
Institute of Physics, 10000 Zagreb, Croatia

###### Abstract

Comprehending nonequilibrium electron-phonon dynamics at the microscopic level
and at the short time scales is one of the main goals in condensed matter
physics. Effective temperature models and time-dependent Boltzmann equations
are standard techniques for exploring and understanding nonequilibrium state
and the corresponding scattering channels. However, these methods consider
only the time evolution of carrier occupation function, while the self-
consistent phonon dressing in each time instant coming from the nonequilibrium
population is ignored, which makes them less suitable for studying ultrafast
phenomena where softening of the phonon modes plays an active role. Here, we
combine _ab-initio_ time-dependent Boltzmann equations and many-body phonon
self-energy calculations to investigate the full momentum- and mode-resolved
nonadiabatic phonon renormalization picture in the MoS2 monolayer under
nonequilibrium conditions. Our results show that the nonequilibrium state of
photoexcited MoS2 is governed by multi-valley topology of valence and
conduction bands that brings about characteristic anisotropic electron-phonon
thermalization paths and the corresponding phonon renormalization of strongly-
coupled modes around high-symmetry points of the Brillouin zone. As the
carrier population is thermalized towards its equilibrium state, we track in
time the evolution of the remarkable phonon anomalies induced by
nonequilibrium and the overall enhancement of the phonon relaxation rates.
This work shows potential guidelines to tailor the electron-phonon relaxation
channels and control the phonon dynamics under extreme photoexcited
conditions.

###### keywords:

ultrafast dynamics, phonon renormalization, electron-phonon coupling,
transition metal dichalcogenides, density functional theory

Kiel Nano, Surface and Interface Science KiNSIS, 24118 Kiel, Germany

##  1 Introduction

Recent advancements of ultrafast spectroscopy techniques have opened many
avenues for controlling and understanding fundamental interactions in quantum
materials 1, 2. Due to ultrashort duration, usually below the characteristic
thermalization timescale, the corresponding laser sources are not only able to
probe and disentangle the relaxation pathways of electrons, phonons, spin, and
other degrees of freedom 3, 4, but can reveal new physical phenomena and
phases of matter beyond the thermodynamical equilibrium 1. Namely, photo-
induced nonequilibrium carrier distribution and the accompanying modifications
of the potential energy landscape was shown to promote nonlinear lattice
control 5, 6, 7, elevate or quench existing superconducting phase 8, 9, 10,
11, 12, alter the transition temperature 13, 14 or dimensionality 15, 16 of
the known charge-density-wave (CDW) order or even induce new ordered phases of
matter 17, 18, 19, 20, 21, 22, 23, as well as switch ferroelectric 24, 25, 26,
27, 28, 29, 30 and ferromagnetic 31 properties.

Electron-phonon coupling (EPC) plays a crucial role in the aforesaid ultrafast
phenomena 32, 1, 33, 34 and thus it is of utmost importance to master and
comprehend microscopic channels ruling phonon dynamics in extreme
nonequilibrium conditions. Complementary to the time-resolved photoemission
methods that provide an important access to the electron-hole thermalization
process 3, 4, 35, 36, 37, 38 and electronic structure changes 39, 40, there
are several ultrafast techniques, such as ultrafast electron diffraction
scattering 41, 42, 43, 44, 45, 46, 47, 48, coherent phonon spectroscopy 49,
50, 51, 52, 53, and time-resolved Raman spectroscopy 54, 55, 56, 57, that can
precisely track the phonon relaxation channels following the photoexcitation
and corresponding EPC strength 50, 51. For instance, ultrafast electron
diffraction had uncovered highly anisotropic non-thermal phonon relaxation in
black phosphorus 46, and mapped momentum-resolved electron-phonon scattering
channels and strengths in various transition metal dichalcogenides (TMDs) 41,
47, 45, 48. Intriguingly, these methods are able to analyze photo-induced
phonon frequency modifications and uncover the relevant microscopic processes,
as it was done, for example, for zone-center strongly-coupled $E_{2g}$ optical
mode in graphite with coherent phonon 49 and time-resolved Raman
spectroscopies 54, as well as for the amplitude CDW mode in TiSe2 by means of
ultrafast electron diffraction 45. In combination with other time-resolved
spectroscopy approaches, the latter technique allowed to pinpoint the phonon
modes that play an active role in unconventional superconductivity of FeSe
thin films on SrTiO3, and to extract the correlation-induced EPC constants 50.
Further, recent study on graphite reported dynamics of coherent vibrations for
both zone-center and zone-edge phonon modes with unprecedented energy and time
resolution allowed by attosecond core-level spectroscopy 52.

In order to obtain a complete insight into the ultrafast phonon dynamics and
unveil the corresponding electron-phonon scattering paths, the above studies
need to be complemented with microscopic theoretical methods, preferably based
on quantitative description of nonequilibrium state beyond simple
phenomenological approaches. Probably the most widespread theoretical
description of electron-lattice energy flow is the two temperature model (TTM)
and its extensions 58, 59, 60, 61, 62, 63, 64, where it is assumed that
electron and lattice subsystems are already thermalized. It is commonly used
in its phenomenological form to supplement the experimental observations of
the heat transfer 59, 60, 61, 65, and was further useful to address the hot
phonon dynamics and phonon bottleneck in materials with highly anisotropic
EPC, such as graphene 61, 66, 64, graphite 67, MgB2 68, 69, MoS2 70, and lead
halid perovskites 71. It was also used as a basis to study laser-induced
energy renormalization of hot phonons in MgB2 as a function of time 68.

However, the TTM is likely to fail in describing femtosecond lattice dynamics
below the electron thermalization time, where full information of energy-
momentum phase space in nonequilibrium state is required 64. A considerable
improvement in describing nonequilibrium scattering events is met with time-
dependent Boltzmann equations (TDBE), especially with its first-principles
implementations 72, 73, 46, 74, 75, 48. When both electron-phonon and phonon-
phonon scattering channels are included, the TDBE can accurately describe
energy-, momentum-, and time-dependent modifications to electron and phonon
population in both sub-picosecond and picosecond regimes 76. Despite these
strengths, common TDBE studies do not account for the self-consistent
renormalization of phonon energies in each time instant coming from the
updated carrier populations 64, and are, therefore, not fully suited for
exploration of the photo-induced soft phonon dynamics, structural phase
transitions, ferroelectricity, and charge density waves.

An alternative _ab-initio_ method to track carrier dynamics upon laser
excitation is based on real-time time-dependent density functional theory (rt-
TDDFT) and Ehrenfest dynamics, and along the EPC allows to include many-body
electron-electron and electron-hole interactions 77, 78, 79, 80, 81, 82, 83.
In combination with molecular dynamics and real-space lattice distortions, the
latter method can account for time-resolved self-consistent renormalization of
phonon energies and EPC strengths 81, 82, providing a microscopic information
on laser-induced structural transitions 83 and enhanced superconducting
properties 81, 82. Since it relies on real-space distortions and supercell
approaches to account for electron-lattice interactions, it is numerically
challenging for the rt-TDDFT to provide full momentum-resolved analysis on
phonon dynamics and, in practice, usually considers only very few coherent
optical phonons, such as zone-center $E_{2g}$ and zone-edge $A_{1}^{\prime}$
modes in graphene 81, and zone-center $A_{1g}$ mode in MoS2 82.

On the other hand, the non-thermal renormalization of phonons in the whole
Brillouin zone can be acquired from the adiabatic density functional and
density functional perturbation theories by constraining the occupation of
electronic states to a nonequilibrium distribution (i.e., from cDFT and cDFPT)
84, 85, 86, simulating thus laser-excited nonequilibrium states fixed in time,
such as population inversion in semiconductors and semimetals. For instance,
these methods were used to study photo-induced phonon frequency modifications
in tellurium 84 and bismuth 85, 86, phase transition in ferroelectrics 87 and
MoTe2 88, as well as structural rippling of hBN monolayer 89. However, the
cDFT and cDFPT approaches are time independent and lack information on carrier
population dynamics and its temporal evolution towards thermal equilibrium
caused by various scattering events.

Here we combine the TDBE and many-body phonon self-energy calculations in
order to obtain first-principles information on the time-dependent phonon
renrmalization process and electron-phonon scattering channels following laser
excitation with a full momentum and frequency resolution. With this
methodology we investigate a photoexcited MoS2 monolayer, a prototypical 2D
semiconductor with exceptional electronic 90 and optoelectronic 91, 92, 39,
93, 94, 40 properties for which vibrational, electronic, valley, spin, and
other degrees of freedom play an active and important role. Thermalization of
carrier distribution function with band and momentum resolution
$f_{n\mathbf{k}}$ is obtained as a function of time by means of TDBE where
electron-phonon and phonon-phonon scatterings are included 74, 64, 48. Thus
acquired distribution functions $f_{n\mathbf{k}}$ are then utilized to
construct the time-resolved phonon self-energy $\pi_{\mathbf{q}\nu}(\omega;t)$
and the full nonadiabatic phonon spectral functions
$B_{\mathbf{q}\nu}(\omega;t)$, which enables analysis of phonon frequency and
linewidth (i.e., relaxation rate) modifications. Characteristic multi-valley
landscape of MoS2 electronic states in momentum space permits only selective
population dynamics and anisotropic electron-phonon scatterings. Namely,
photo-holes in $\mathbf{k}=\mathrm{\Gamma}$ and $\mathbf{k}=\mathrm{K}$
valance valleys promote specifically $\mathbf{q}=\mathrm{\Gamma}$ and
$\mathbf{q}=\mathrm{K}$, while photo-electrons in $\mathbf{k}=\mathrm{K}$ and
$\mathbf{k}=\mathrm{Q}$ conduction valleys promote dominantly
$\mathbf{q}=\mathrm{\Gamma}$ and $\mathbf{q}=\mathrm{M}$ electron-phonon
scatterings of optical and acoustic phonons. This in turn influences
considerably phonon frequency and linewidths, and results in remarkable
anisotropic nonequilibrium phonon softening and dynamical Kohn anomalies at
aforesaid phonon momenta. For instance, large dynamical Kohn anomaly of the
$A_{1g}$ optical mode appears close to $\mathbf{q}=\mathrm{\Gamma}$,
surpassing the strength of the corresponding phonon anomaly in equilibrium
state of doped MoS2 samples 95, 96, 97. Also, sizeable phonon softening is
induced for the longitudinal acoustic (LA) phonon at $\mathbf{q}=\mathrm{M}$,
which is considered relevant for the appearance of the superconductivity 98
and CDW 99 in doped MoS2. Importantly, we show that overall phonon scattering
rate is significantly increased in nonequilibrium, opening a possibility for
enhancing the total EPC strength. These findings demonstrate that photo-
induced nonequilibrium state is a promising route for tailoring vibrational
properties of quantum matter, especially for MoS2 where phonons play a primary
role in the emergence of novel quantum phenomena, such as in exciton dynamics
100, 101, 102, 103 as well as formation of Holstein polaron 104, CDW 99 and
superconductivity 105, 106, 98, 107.

##  2 Theoretical methods

The dynamics of non-thermal electron-lattice system in the TDBE is described
by modifications of electron and phonon occupation functions
$f_{n\mathbf{k}}(t)$ and $n_{\mathbf{q}\nu}(t)$, while electron and phonon
energies, as well as the corresponding coupling functions are unaltered and
fixed to their equilibrium values. The time evolution of $f_{n\mathbf{k}}(t)$
and $n_{\mathbf{q}\nu}(t)$ is dictated by electron-phonon and phonon-phonon
scattering processes and can be described with the following coupled integro-
differential equations 74, 64:

$\displaystyle{\partial_{t}f_{n\mathbf{k}}(t)}$ $\displaystyle=\Gamma^{\rm
ep}_{n\mathbf{k}}[f_{n\mathbf{k}}(t),n_{\mathbf{q}\nu}(t)],$ (1)
$\displaystyle{\partial_{t}n_{\mathbf{q}\nu}(t)}$ $\displaystyle=\Gamma^{\rm
pe}_{{\mathbf{q}\nu}}[f_{n\mathbf{k}}(t),n_{\mathbf{q}\nu}(t)]+\Gamma^{\rm
pp}_{\mathbf{q}\nu}[n_{\mathbf{q}\nu}(t)],$ (2)

where $\partial_{t}=\partial/\partial t$, while $\Gamma_{n\mathbf{k}}$ and
$\Gamma_{{\mathbf{q}\nu}}$ denote the collision integrals for electrons and
phonons, for which electron-phonon, phonon-electron, and anharmonic phonon-
phonon scatterings are accounted for. The corresponding expressions for the
collision rates are derived from the standard first-order Fermi’s golden rule
and the explicit forms can be found in Refs. 74, 48, 64. The time dependence
of the collision integrals is explicitly accounted for in the calculations by
reevaluating $\Gamma^{\rm ep}_{n\mathbf{k}}$ and $\Gamma^{\rm
pe}_{{\mathbf{q}\nu}}$ at each time step of the time propagation with new
occupation functions. The phonon-phonon collision integral is treated with the
relaxation-time approximation. Electron-electron scattering channel governs
the thermalization of photoexcited carriers when $f_{n\mathbf{k}}(t)$ deviates
significantly from the equilibrium distribution. Here we consider exclusively
electronic excited state characterized by a weak deviation from a Fermi-Dirac
function. In this regime, electron-electron scattering plays only a minor role
in the carrier dynamics and it is therefore neglected.

At thermal equilibrium, $f_{n\mathbf{k}}$ and $n_{\mathbf{q}\nu}$ are time
independent and they coincide with the Fermi-Dirac and the Bose-Einstein
occupations: $f^{\rm
FD}_{n\mathbf{k}}=\left[e^{{(\varepsilon_{n\mathbf{k}}-\varepsilon_{\rm
F})}/{k_{\rm B}T}}+1\right]^{-1},n^{\rm
BE}_{\mathbf{q}\nu}=\left[e^{\hbar\omega_{\mathbf{q}\nu}/{k_{\rm
B}T}}-1\right]^{-1}.$ Here, $\varepsilon_{\rm F}$ is the Fermi energy,
$\varepsilon_{n\mathbf{k}}$ is the single-particle energy of a Bloch electron,
and $\hbar\omega_{\mathbf{q}\nu}$ the phonon energy. The initial photoexcited
concentration of electrons is taken to be $n=10^{14}$ cm-2. The corresponding
initial photo-holes and photo-electrons are defined with two separate chemical
potentials and high carrier temperature, while keeping in mind the
conservation of carrier number. Namely, we define $f_{n\mathbf{k}}(t=0)=f^{\rm
FD}_{n\mathbf{k}}(\mu_{\rm e/h},T^{0}_{e})$, with $\mu_{\rm e}$ ($\mu_{\rm
h}$) being the electron (hole) chemical potential, $T^{0}_{\rm e}=2000$ K,
while phonon temperature is set to $T^{0}_{\rm p}=100$ K. Equations (1) and
(2) are solved by time-stepping the derivatives with small time step of 1 fs
up to 40 ps.

In order to have a full time-dependent information of electron-phonon dynamics
an important step forward is to update the phonon frequencies coming from the
modified electron occupation functions $f_{n\mathbf{k}}(t)$. This can be
calculated by means of the time-resolved phonon spectral function defined as

$B_{\mathbf{q}\nu}(\omega;t)=-\frac{1}{\pi}\mathrm{Im}\left[\frac{2\omega_{\mathbf{q}\nu}}{\omega^{2}-\omega_{\mathbf{q}\nu}^{2}-2\omega_{\mathbf{q}\nu}\widetilde{\pi}_{\mathbf{q}\nu}(\omega;t)}\right].$
(3)

The crucial ingredient to the above expression is the time-resolved NA phonon
self-energy 108 defined as
$\widetilde{\pi}_{\mathbf{q}\nu}(\omega;t)=\pi_{\mathbf{q}\nu}(\omega;t)-\pi_{\mathbf{q}\nu}(0)$,
where the adiabatic part at $t\rightarrow-\infty$, i.e.,
$\pi_{\mathbf{q}\nu}(0)$, is subtracted, and where

$\pi_{\mathbf{q}\nu}(\omega;t)=\sum_{\mathbf{k}nm}\left|g_{\nu}^{nm}(\mathbf{k},\mathbf{q})\right|^{2}\frac{f_{n\mathbf{k}}(t)-f_{m\mathbf{k+q}}(t)}{\omega+\varepsilon_{n\mathbf{k}}-\varepsilon_{m\mathbf{k+q}}+i\eta}.$
(4)

The electron-phonon matrix elements are denoted by
$g_{\nu}^{nm}(\mathbf{k},\mathbf{q})$ and $\eta$ is an infinitesimal
parameter. In our approach, the electron occupation functions
$f_{n\mathbf{k}}(t)$ entering the phonon spectral function (3) and phonon
self-energy (4) are no longer Fermi-Dirac distributions as in the standard
thermal case 108, but are extracted from TDBE Eq. (1) and, therefore,
represent nonequilibrium occupations and redistribution of charge carriers at
each time instant after the laser excitation. The corresponding photo-induced
renormalization of phonon frequency and modifications to the phonon linewidth
(relaxation rate) can be tracked as a function of time by means of the
following expressions 108

$\displaystyle\Omega^{2}_{\mathbf{q}\nu}(t)$
$\displaystyle=\omega^{2}_{\mathbf{q}\nu}+2\omega_{\mathbf{q}\nu}{\rm
Re}\,\widetilde{\pi}_{\mathbf{q}\nu}(\Omega_{\mathbf{q}\nu}(t);t),$ (5)
$\displaystyle\gamma_{\mathbf{q}\nu}(t)$ $\displaystyle=-{\rm
Im}\,\widetilde{\pi}_{\mathbf{q}\nu}(\Omega_{\mathbf{q}\nu}(t);t).$ (6)

A similar idea was adopted recently in Refs. 46, 48 where occupation functions
as obtained from TDBE were introduced into dynamic structure factors to
complement ultrafast electron diffraction experiments.

Figure 1: (a) Crystal structure of MoS2 single layer. (b) Corresponding
Brillouin zone and high-symmetry points. (c) Electronic band structure of MoS2
with time evolution of the photoexcited occupation functions as obtained from
the time-dependent Boltzmann equations. Figure 2: Phonon spectral functions
$B_{\nu}(\mathbf{q},\omega)$ of photoexcited MoS2 shown along high-symmetry
points in the first Brillouin zone and for several time frames after
excitation: (a) $t=0$ ps, (b) $t=0.5$ ps, (c) $t=1$ ps, and (d) $t=2$ ps. The
dashed lines are phonon dispersions for pristine equilibrium MoS2 as obtained
from the adiabatic DFPT. Significant photo-induced phonon broadening
enhancements and dynamical Kohn anomalies can be observed around the high-
symmetry points both for optical and acoustic branches.

We also define the spectral representation of the phonon scattering rate
$\gamma F(\omega)$ in order to quantify the modifications to the electron-
phonon scattering channels relevant for phonon dynamics and to discuss
possible enhancements to the total EPC strength upon the laser excitation

$\displaystyle\gamma
F(\omega;t)=\sum_{\mathbf{q}\nu}\gamma_{\mathbf{q}\nu}(t)\delta\left(\omega-\Omega_{\mathbf{q}\nu}(t)\right),$
(7)

where $\delta(x)$ is the Dirac delta function. Note that the above spectral
function is defined in a similar manner as the Eliashberg function or
electron-phonon spectral function $\alpha^{2}F(\omega)$ 109. The cumulative
scattering rate of phonons can then be written as

$\displaystyle\gamma(\omega;t)=\int_{0}^{\omega}d\omega^{\prime}\gamma
F(\omega^{\prime};t),$ (8)

while the total phonon scattering rate is $\gamma(\omega\rightarrow\infty;t)$.

All of the above equations and the corresponding input parameters are
calculated in this work by means of DFPT 110 and Wannier interpolation 111 of
EPC matrix elements $g_{\nu}^{nm}$ 112.

We use Quantum ESPRESSO 113, 114, 115 for the DFT calculations, and for EPC we
use the EPW code 116, 117, 118. All calculations are performed with the norm-
conserving Perdew-Burke-Ernzerhof pseudopotential with a kinetic energy cutoff
of 120 Ry. The lattice constant is set to the value of 3.188 Å, while the
neighboring MoS2 sheets are separated by 12.7 Å. The self-consistent electron
density calculation is done on a $20\times 20\times 1$ k-point grid and the
phonon calculation on a $6\times 6\times 1$ q-point grid. Both are done with
the equilibrium electron occupation functions for pristine MoS2, with the
valence band occupied and the conduction band unoccupied. To interpolate
electron-phonon quantities, we use 11 maximally localized Wannier functions
119 with the initial projections of d-orbitals on the Mo sites and p-orbitals
on the S atom sites. All electronic structure parameters for the self-
consistent cycle and the Wannierization match the ones used to solve the TDBE.
The fine sampling of the Brillouin zone for the electron-phonon interpolation
is done on a $200\times 200\times 1$ grid. The fine q-point grid is always
extracted among these 40000 points, whether it is a path in q-space or a full
Brillouin zone calculation. Smearing in the EPW calculation is set to 40 meV.

##  3 Results and discussion

Figure 1 depicts the crystal structure of single-layer MoS2, the corresponding
Brillouin zone and electronic band structure along high-symmetry points. MoS2
is a semiconducting TMD with a direct band gap at the K point of the Brillouin
zone, and interesting multi-valley topology of valence and conduction bands.
The latter is considered to be instrumental for various physical phenomena in
MoS2, such as enhanced EPC of the $A_{1g}$ phonon mode as observed with Raman
spectroscopy 95, exciton-phonon coupling 102, 103, anisotropic electron-phonon
scattering following laser excitation 48, and multi-valley superconductivity
105, 106, 98, 107. As it will be shown below, the rich phase space for
electron-phonon scatterings come from the $\Gamma$ and K valleys in the
valence band and the K and Q valleys in the conduction band.

Additionally, in Fig. 1(c) we show occupation functions $f_{n\mathbf{k}}(t)$
for several time instants up to $t=2$ ps as obtained from the solution of the
TDBE (1)-(2). Initial hot distribution of photo-holes and photo-electrons
(grey lines) thermalize to smaller effective electron temperatures (about 180
K) with different time scales. Namely, holes are almost thermalized at $t=0.5$
ps, while it takes around 2 ps for excited electrons to equilibrate. The
reason for this is the difference in the phase space of valence and conduction
bands, where the Q valley acts as a sort of bottleneck for electron-phonon
scattering in the conduction band and slows down the process (see the
accumulated charge for $t=0.5$ ps that forms non-Fermi-Dirac distribution). On
the other hand, the obained thermalization time of a nonequilibrium phonon
distribution is somewhere between 5 and 10 ps 74. Note that while at the
initial step and after thermalization time the population of electrons and
phonons are well described with quasi-equilibrium distribution functions
having corresponding temperatures, the TDBE allows for large deviations from
the equilibrium in the between, as it is for instance the case for the time
period between 0.1 and 1 ps for electrons, and between 0.1 and 5 ps for
phonons.

The impact of the nonequilibrium electron and hole distributions on phonons is
shown in Fig. 2 where we report the full phonon spectral functions for several
time delays. Considerable and time-varying nonequilibrium phonon
renormalizations are observed for both acoustic and optical branches,
especially around the high-symmetry points of the BZ. In addition, these
renormalizations are accompanied by a remarkable enhancement of the phonon
broadenings. Particularly strong modifications of phonon frequency is observed
for $A_{1g}$ optical phonon at $\mathbf{q}=\Gamma$, where a much larger
dynamical Kohn anomaly is formed by excited carriers compared to the case of
doped MoS2 in equilibrium state 96. Kohn anomalies are distinct softenings of
phonon dispersion coming from singularities in static phonon self-energy
$\pi_{\mathbf{q}\nu}(0)$, which in turn come from highly anisotropic electron-
phonon matrix elements $g_{\nu}^{nm}(\mathbf{k},\mathbf{q})$ as well as from
the intense and anisotropic density of states of electron-hole pair
excitations 108. If the Kohn anomaly calculated in nonadiabatic regime [i.e.,
with $\pi_{\mathbf{q}\nu}(\omega)$] is different from the static one, it is
usually dubbed dynamical. The dynamical Kohn anomaly of optical phonons at
$\mathbf{q}=\Gamma$ was reported for various single-layer and bulk materials
(such as graphene, hole-doped diamond, MgB2, and doped TMDs) in thermal
equilibrium 120, 76, 121, 96, 122, 123. Here we illustrate to which extent a
photoexcited carrier distribution can provide a route to trigger and control
the emergence of Kohn anomalies over transient timescales. The $A_{1g}$ mode
was shown to be significantly coupled to electrons once multiple valleys are
occupied 95, as it is the case here for photoexcited MoS2. The same phonon
mode is as well quite affected at the M point, while energetically lower
$E_{2g}$ optical mode is softened and broadened at the K point. Also, both
longitudinal and transverse acoustic modes are modified, especially around the
M and K points. Softening of the LA mode at M point particularly stands out,
where at time $t=2$ ps its frequency is decreased by about 10 meV compared to
the equilibrium value. Since the LA mode at $\mathbf{q}=\mathrm{M}$ is
instrumental for phonon-mediated superconductivity 98 as well as the CDW
formation 99, these results suggest a possibility to tune these ordered states
by laser excitations.

Figure 3: Nonequilibrium electron-phonon scattering channels coming from
valence- (VB) and conduction-band (CB) distributions. The VB- and CB-resolved
contributions are presented for two time instants (a) $t=0\,\mathrm{ps}$ and
(b) $t=2\,\mathrm{ps}$. Upper panels depict the momentum-resolved occupation
functions $f_{n\mathbf{k}}(t)$ for the CB and VB within the first Brillouin
zone. White arrows represent the dominating inter-valley scatterings channels.
The lower panels show the VB- and CB-resolved contributions to the phonon
dispersion and linewidth $\gamma_{\mathbf{q}\nu}$. In addition, the overall
result of dynamical phonon renormalization, coming from both VB and CB
nonequilibrium channels, is shown. The dashed lines again show the phonon
dispersions for pristine equilibrium MoS2 as obtained from the adiabatic DFPT
Figure 4: (a) Total phonon scattering rate (summed over all branches) along
the high-symmetry points and as a function of time. (b) Spectral
representation of phonon scattering rates $\gamma F(\omega)$ (phonon density
of states weighted with phonon linewidth contributions) as a function of
frequency $\omega$ shown for several time delays. Right axis shows the
cumulative scattering rate $\gamma(\omega)$.

In the following we study the phase space arguments and analyze the electron-
phonon scattering events out of equilibrium that lead to these remarkable
phonon renormalizations and phonon linewidth enhancements. In the top panels
of Figs. 3(a) and 3(b) we show the momentum-resolved occupation functions
$f_{n\mathbf{k}}(t)$ of valence and conduction bands for $t=0$ and $2$ ps,
respectively. The corresponding bottom panels show the contributions to the
phonon dispersions and linewidths coming only from the electron-phonon
scatterings in the valence and conduction bands. Both time frames are
characterized by the similar phase space for the valence and conduction bands,
i.e., depopulated $\mathbf{k}=\Gamma$ and $\mathbf{k}=\mathrm{K}$ valleys in
the valence band and populated $\mathbf{k}=\mathrm{K}$ and
$\mathbf{k}=\mathrm{Q}$ valleys in the conduction band. Such occupations
promote $\mathbf{q}=\Gamma$ and $\mathbf{q}=\mathrm{K}$ electron-phonon
scatterings in the former, while $\mathbf{q}=\Gamma$, $\mathbf{q}=\mathrm{K}$
and $\mathbf{q}=\mathrm{M}$ electron-phonon scatterings within the latter
band. Consequently, these intra- and inter-valley scatterings in the valence
band lead to nonequilibrium renormalization of the optical $A_{1g}$ and
$E_{2g}$ phonons at $\Gamma$ and K points, respectively, as well as of the LA
mode at the K point. Similarly, the scattering channels in the conduction band
produce phonon softenings of the optical $A_{1g}$ phonon at $\Gamma$ and M
points, and the LA mode at the K and M points. As the initially hot
distribution of electrons and holes at $t=0$ ps is distributed into more sharp
occupations at the top and bottom of valence and conduction bands at $t=2$ ps,
some alterations of the phonon bands and broadenings become more pronounced
while others are reduced. For instance, as the K valley becomes more
populated, while the occupation in the Q valley is reduced in the conduction
band, the corresponding modifications of the $\mathbf{q}=\mathrm{K}$
($\mathbf{q}=\mathrm{M}$) phonons coming only from conduction-band scatterings
are reduced (enhanced).

Femtosecond electron diffraction experiment revealed that scatterings in
multi-layer MoTe2 are dominated by the zone-center $A_{1g}$ and $E_{2g}$
optical phonons, as well as by the LA phonons at the M point of the BZ 47.
Momentum-resolved picture of the energy transfer between excited electrons and
phonons in thin bulk-like films of WSe2 reveals importance of inter-valley
scattering between two Q points followed by emission of the acoustic M-point
phonons 93, 41. On the other hand, coherent phonon dynamics extracted from
femtosecond pump-probe spectroscopy had shown that ultrafast intervalley
scattering in monolayer MoSe2 is dictated dominantly by the LA phonons, but at
K point of the BZ 53. Multi-layer films of MoTe2 and WSe2 are indirect-band-
gap semiconductors, and are therefore characterized by different scattering
phase space for excited carriers compared to monolayer MoSe2, which has a
direct band gap at the K point. Consequently, monolayer and thin films of TMDs
have distinct ultrafast phonon dynamics, coming from different shapes of
conduction and valence valleys. For instance, in bulk-like semiconducting TMDs
the Q conduction valley has lower energy than the K valley, and therefore the
$\mathrm{Q}\leftrightarrow\mathrm{Q^{\prime}}$ inter-valley scatterings are
dominant with emission of the $\mathbf{q}=\mathrm{M}$ phonons. For
corresponding single layers the situation is different, where the K conduction
valley has the lowest energy, and more dominant are the
$\mathrm{K}\leftrightarrow\mathrm{K^{\prime}}$ scatterings (actually in the
both conduction and valence bands) and the concomitant $\mathbf{q}=\mathrm{K}$
phonon emission. This opens many possibilities to tailor ultrafast phonon
scatterings, e.g., by strain, pressure, doping, and other techniques that can
significantly alter the energy positions of valleys 124.

To further demonstrate the implications of nonequilibrium carrier distribution
on phonon dynamics, we show in Fig. 4 the total phonon scattering rate (summed
over all branches) as a function of time along the high-symmetry points and
spectral representation of phonon scattering rates $\gamma F(\omega)$ with the
corresponding cumulative scattering rate $\gamma(\omega)$ [see Eqs. (7) and
(8)]. The momentum resolved scattering rate $\gamma_{\mathbf{q}}$ reveals
important role of the $\mathbf{q}=\Gamma$ and $\mathbf{q}=\mathrm{K}$ phonons
and their dominance in the overall phonon relaxation dynamics. Importance of
these specific phonon modes are in line with the ultrafast coherent phonon
dynamics in single-layer MoSe2 53, while anisotropic phonon response is in
accordance with recent results obtained with ultrafast electron diffraction
spectroscopy in MoS2 monolayer 48. Certain discrepancies in timescales between
experiments and theoretical results obtained here could be attributed to the
screening of the matrix element induced by the substrate 48. Note that the
recent theoretical study based on nonequilibrium Green’s functions also
obtained that the $\Gamma$\- and K-point phonons are dominantly involved in
nonequilibrium carrier dynamics in monolayer MoS2 125. Interestingly, they
also show that the optical phonons participate more in the relaxation dynamics
compared to the acoustic modes.

The obtained relaxation rates around $\Gamma$, K, as well as M points are also
significantly increased in time, where values at $t=2$ ps are $4-5$ times
larger compared to the corresponding values at the initial time. The
frequency-resolved scattering rate is presented in Fig. 4(b) via $\gamma
F(\omega)$, where it shown that nonequilibrium phonon scattering rates is
larger for optical modes above 40 meV. The results for cumulative rate
$\gamma(\omega)$ confirms the gradual increase of the total rate as a function
of time. This result seems surprising at first, since one expects that
vibrational energy exchange rate decreases towards the thermalization instant,
i.e., when the effective electronic temperature is decreased 41. Nevertheless,
in some cases (like nickel and platinum as well as photo-doped MoS2 as shown
here) due to specific Fermi surface and density of states, the opposite is
possible 126.

In addition, we want to explore the difference in phonon dynamics between the
photo-doped (i.e., photoexcited) scenario investigated here and the standard
case of the electron-doped material via field-effect techniques or atom
adsorption (i.e., via dopants). Figure 5 compares the total and mode-resolved
phonon scattering rates along the high-symmetry points for photoexcited MoS2
at $t=2$ ps [panel (a)] and MoS2 doped with electron carrier concentration of
$n_{\rm eff}=2\times 10^{14}$ cm-2, which corresponds to the effective photo-
induced carrier (both electron and hole) density at $t=2$ ps [panel (b)]. For
the electron-doped case, the active phase space for electron-phonon
scatterings consists only of the K and Q valleys in the conduction band. This
in turn promotes dominantly intra-valley $\mathbf{q}=\Gamma$ and inter-valley
$\mathbf{q}=\mathrm{M}$ scatterings and consequently phonon renormalizations
and broadening around these symmetry points. Furthermore, in Fig. 5(c) we show
spectral representation of phonon scattering rates $\gamma F(\omega)$ and the
cumulative scattering rate $\gamma(\omega)$ for these two cases, where it is
clear that photo-doping induces larger and richer phonon-electron scatterings,
since the corresponding scattering phase space includes both photo-holes and
photo-electrons, i.e., both valence and conduction valleys.

Overall, the present results clearly demonstrate a notable increase of phonon-
electron scattering rate with photoexcitation, which points to the potential
of enhancing the total EPC strength out of equilibrium 82 as well as inducing
and modifying the concomitant superconducting properties. Note that in order
to have a more conclusive answer to the intriguing physical problem of photo-
induced superconductivity, one would need to go beyond the present
consideration and adopt more rigorous time-dependent methodology of
superconductivity, such as nonequilibrium Green’s function techniques 1, 127,
128, 7, 33.

Figure 5: (a) Total and mode-resolved nonequilibrium phonon scattering rates
along the high-symmetry points for $t=2$ ps. (b) Same as (a) but for the case
of electron-doped equilibrium MoS2 for the effective carrier concentration
$n_{\rm eff}=2\times 10^{14}$ cm-2 that matches the photoexcited carrier
(i.e., both photo-hole and photo-electron) density at $t=2$ ps. (c) Spectral
representation of phonon scattering rates $\gamma F(\omega)$ as a function of
frequency $\omega$ and the cumulative scattering rate $\gamma(\omega)$ for the
photo-doped and doped cases presented in (a) and (b) panels.

As a final remark, we want to note that, besides on the phonon dynamics as
studied here, the laser-induced nonequilibrium carrier and phonon
distributions could potentially have a significant impact on the electron
dynamics, such as ultrafast band renormalizations. Within the present
theoretical framework, these ultrafast features could be captured by updating
the electronic energies and linewidths with electron self-energy due to EPC,
which are in turn based on the TDBE results. This future direction might
provide some interesting microscopic insights on the observed ultrafast band
gap renormalizations in TMDs 39, 40 and other layered materials 129, as well
as on the intriguing Floquet physics, such as phonon-driven Floquet matter 130
and ultrafast quasiparticle dressing by light 131.

##  4 Conclusions

We have explored the phonon relaxation pathways and the ensuing nonequilibrium
phonon renormalization in the photoexcited MoS2 monolayer by combining the
_ab-initio_ time-dependent Boltzmann equation and the phonon self-energy
calculations. Our findings show how population and depopulation of conduction
and valence valleys promote anisotropic electron-phonon scatterings and
triggers strong Kohn anomalies, softenings, and increase of relaxation rate
for strongly-coupled optical and acoustic phonon modes. Nonequilibrium of the
electronic energy levels induces strong Kohn anomaly of the $E_{2g}$ mode
close to the center and edge of the Brillouin zone, and strongly softens the
longitudinal acoustic phonon at the M point. In accordance to the recent
ultrafast experiments, our momentum-resolved analysis demonstrates that the
$\mathbf{q}=\Gamma$ and $\mathbf{q}=\mathrm{K}$ phonon modes play a key role
in intra- and inter-valley scattering channels and they are thus characterized
by large relaxation rates. It is also shown that as the effective electron
temperature decreases, i.e., as the photo-holes and photo-electrons are
scattered towards the top and bottom of valence and conduction valleys, the
overall phonon relaxation rate is significantly enhanced. The richness of the
phase space for the photo-carriers and the corresponding impact on phonon
dynamics was further demonstrated in comparison to the electron-doped MoS2 in
equilibrium, where, instead of K-point, M-point modes are ruling the phonon
relaxation and renormalization, and where phonon relaxation rates have less
intensity. In general, we believe that present results and methodology might
be instrumental for gaining crucial microscopic insights of photoexcited
states in multi-valley systems, such as transition metal dichalcogenides, as
well as discovering new photo-induced ordered phases (e.g., superconductivity
and charge density waves) and structural transformations in condensed matter.

Useful discussions with Jan Berges, Samuel Poncé, Yiming Pan are gratefully
acknowledged. We acknowledge financial support from the Croatian Science
Foundation (Grant no. UIP-2019-04-6869) and from the European Regional
Development Fund for the “Center of Excellence for Advanced Materials and
Sensing Devices” (Grant No. KK.01.1.1.01.0001). F.C. acknowledges funding from
the Deutsche Forschungsgemeinschaft Grant No. 443988403 and 499426961.

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