# Role of Coulomb Interaction in Valley Photogalvanic Effect

V. M. Kovalev  Novosibirsk State Technical University, Novosibirsk 630073,
Russia  A. V. Parafilo  Center for Theoretical Physics of Complex Systems,
Institute for Basic Science (IBS), Daejeon 34126, Korea  O. V. Kibis
Novosibirsk State Technical University, Novosibirsk 630073, Russia  I. G.
Savenko  Department of Physics, Guangdong Technion–Israel Institute of
Technology, 241 Daxue Road, Shantou, Guangdong, China, 515063  Technion –
Israel Institute of Technology, 32000 Haifa, Israel  Guangdong Provincial Key
Laboratory of Materials and Technologies for Energy Conversion, Guangdong
Technion–Israel Institute of Technology, Guangdong 515063, China

###### Abstract

We develop a theory of Coulomb interaction-related contribution to the
photogalvanic current of electrons in two-dimensional non-centrosymmetric
Dirac materials possessing a nontrivial structure of valleys and exposed to an
external electromagnetic field. The valley photogalvanic effect occurs here
due to the trigonal warping of electrons and holes’ dispersions in a given
valley of the monolayer. We study the low-frequency limit of the external
field: the field frequency is smaller than the temperature $T$, and the
electron-electron and electron-hole scattering times are much larger than the
electron-impurity scattering time. In this regime, we employ the Boltzmann
transport equations and show that electron-hole scattering dominates electron-
electron scattering in intrinsic semiconductors. Coulomb interaction-related
contribution to the valley photogalvanic current can reduce the value of the
bare photogalvanic current as these two currents flow in the opposite
directions.

##  I Introduction

The influence of the Coulomb scattering, namely, electron-electron (e-e) and
electron-hole (e-h) interaction, on the transport properties of solids is an
active research area. At low temperatures, when particle-phonon scattering is
frozen, the particle-particle scattering mechanisms may determine the
temperate behaviour of transport coefficients, in particular, the Drude
conductivity [1]. In Galilean-invariant systems with the parabolic spectrum,
particle collisions do not affect the conductivity since the velocity is
proportional to the particle momentum, and thus, the conservation of total
momentum under e-e and h-h collisions results in the conservation of velocity.
Nevertheless, in strongly disordered samples, elastic scattering on impurities
plays sufficient role: impurities break the Galilean invariance, and then, the
influence of particle collisions not only can be finite, but also it results
in strong Coulomb-induced corrections to conductivity [2]. In particular, weak
localization corrections and corrections to magneto-oscillation transport
phenomena [3], including the Shubnikov–de Haas effect, can emerge.

Figure 1: (a) System schematic: a 2D material exposed to a linearly polarized
electromagnetic (EM) field with the frequency $\Omega$ and a circularly
polarized light with the polarization $\sigma$. A circular-polarized EM field
changes electron population at a given valley due to photoinduced interband
transition, whereas a linearly polarized EM results in a valley photogalvanic
effect (vPGE). (b) Reciprocal space of the 2D system, reflecting the multi-
valley structure. Electrons scatter by holes in different valleys.

Interaction between charged particles can be repulsive or attractive, and it
reveals different behavior in different temperature ranges. Indeed, in
addition to the direct Coulomb repulsion between identical particles (e-e or
h-h) and attraction of different particles (e-h in semiconductors), the other
interaction channels may also strongly affect the conductivity of the
material. In particular, strong particle-particle attraction in the Cooper
channel leads to a renormalization of the Drude conductivity by
superconducting fluctuations at sufficiently low temperatures (in the vicinity
of the superconducting transition temperature). This constitutes the
phenomenon referred to as the paraconductivity [4]. Furthermore, recent
technological achievements open a way to creation of ultra-clean
nanostructures, where the electron (and hole) mean free path, which is usually
limited by the interaction with impurities, is comparable with or even exceeds
the sample width [5, 6]. The system is in the hydrodynamic regime of electron
transport [7, 8, 9]. In this regime, the particle momentum predominantly
changes due to electron scattering off the sample boundaries. In some
materials, the e-e and e-h interaction start to play a key role in the
particle transport determining the viscosity of electron liquid.

This subject is especially timely for novel two-dimensional (2D) materials
such as monolayers of graphene [10, 11] and transition-metal dichalcogenides
(TMD) [12, 13, 14]. A key specifics of these materials is the two-valley
structure of the dispersion of carriers of charge. Characteristic band
structure results in the emergence of specific valley transport phenomena,
such as the valley acoustoelectric effect and valley Hall effect [15, 16, 17,
18, 19, 20, 21]. Recently, both the diffusive and hydrodynamic regimes of
valley Hall and anomalous Hall effects have been intensively studied
theoretically [22, 23, 24]. One of the critical properties of electrons and
holes in the valleys of TMD materials is the trigonal warping of valleys
reflecting the trigonal symmetry of the crystal lattice. It results in the
modification of the particle energy: $\epsilon_{\mathbf{p}}\approx
p^{2}/2m+\eta w(p_{x}^{3}-3p_{x}p_{y}^{2})$, where $\mathbf{p}$ is the
momentum of an electron or a hole, $m$ is the effective mass, $\eta=\pm 1$ is
the valley index, and $w$ is the warping parameter. Thanks to that, in
addition to the valley Hall effect, these materials reveal fascinating
nonlinear transport phenomena such as the valley photogalvanic effect (vPGE)
[25, 26, 27, 28]. From the mathematical point of view, the vPGE represents the
second-order response of the stationary electric current density to the
external alternating EM field with the frequency much smaller than the
material bandgap.

There might exist various contributions to vPGE at small temperatures. For
instance, attractive e-e interaction (the Cooper channel) can influence the
magnitude of the vPGE [29]. Namely, the emergence of superconducting
fluctuations strongly affect the vPGE in the vicinity of the superconducting
transition temperature, obeying the law $(T-T_{c})^{-3}$ when $T\rightarrow
T_{c}$ from above [29]. The goal of this paper is to examine the vPGE in TMDs
due to particle-particle interaction in the Coulomb channel, when the
temperature is high enough (or a particle density is low), thus the
superconducting fluctuations do not play important role. In a previous work by
some of the authors of this paper [26], bare vPGE in a non-degenerate electron
gas was studied. Here, we will assume that the electron and hole gases are
non-degenerate, satisfying the Boltzmann statistics, and apply the two-band
model with the parabolic dispersion of electrons and holes with equal
effective masses and accounting for the warping-related corrections to the
parabolic dispersion. In an intrinsic MoS2 at finite temperature, there can
exist three channels of particle-particle interaction: e-e, h-h, and e-h ones.
Our calculations show, that e-e and h-h contributions do not play role,
despite small corrections to the warping and, consequently, to the parabolic
dispersions of electrons and holes. Thus, we focus on e-h scattering as a
dominating source of particle-particle interaction correction to the vPGE.

##  II General formalism

We will consider the diffusive regime, assuming that the temperature is low
enough that particle-impurity collision rate exceeds the particle-particle one
[1]. We will also assume that the frequency of external electromagnetic field
producing the vPGE (the probe) does not exceed the temperature which
determines the mean kinetic energies of electrons and holes. Such an
approximation allows us to use the Boltzmann equations, considering the effect
of particle-particle collision integral via successive approximations. The EM
field falls normally to the 2D monolayer (see Fig. 1). Furthermore, we will
use a two-band model with single valence and conduction bands disregarding the
spin-orbit-splitting effects for simplicity. The main advantages of this model
are that (i) it captures all the main physical effect and (ii) it can also be
applied to a gapped monolayer graphene.

The second-order transport phenomena in general and particularly the vPGE are
usually sensitive to the polarization of the EM field and the symmetry of the
system under study, namely, the time reversal symmetry and the spatial
inversion symmetry. The phenomenological relation connecting the photoinduced
rectified electric current and the amplitude of external EM field reads
$j_{\alpha}=\lambda_{\alpha\beta\gamma}E_{\beta}E^{*}_{\gamma}$, where
$\lambda_{\alpha\beta\gamma}$ is the third-order tensor acquiring non-zero
components in non-centrosymmetric materials. In non-gyrotropic semiconductor
materials, the (rectified) photoinduced electric current occurs as a second-
order response to linearly polarized external EM field. This constitutes the
PGE. This effect does not directly relate to either light pressure, the
photon-drag phenomena, or non-uniformity of either the sample or light field
intensity, like the photoinduced Dember effect. Within the D3h point symmetry
group, which MoS2 belongs to, the third-order conductivity (or transport
coefficient) tensor possesses only one nonzero component. Thus,
phenomenologically, the PGE current can be expressed as

$\displaystyle
j_{x}=\lambda(|E_{x}|^{2}-|E_{y}|^{2}),\,\,\,j_{y}=-\lambda(E_{x}E_{y}^{*}+E^{*}_{x}E_{y}).$
(1)

Therefore, our main task comes down to calculation of the coefficient
$\lambda$ taking into account the contribution of electron-hole interaction.

###  II.1 The Boltzmann equations

The Boltzmann equations, describing (i) the electron and hole scattering on
impurities and (ii) the inter-particle scattering in the framework of the two-
band model read:

$\displaystyle\frac{\partial f_{\bf p}}{\partial t}-{\bf
F}(t)\cdot\frac{\partial f_{\bf p}}{\partial{\bf p}}=Q_{ei}\\{f_{\bf
p}\\}+Q_{eh}\\{f_{\bf p}\\},$ (2) $\displaystyle\frac{\partial f_{\bf
k}}{\partial t}+{\bf F}(t)\cdot\frac{\partial f_{\bf k}}{\partial{\bf
k}}=Q_{hi}\\{f_{\bf k}\\}+Q_{he}\\{f_{\bf k}\\},$ (3)

where ${\bf F}(t)={\bf F}e^{-i\Omega t}+{\bf F}^{*}e^{i\Omega t}$ is the time-
dependent force acting on the mobility electrons and holes of the MoS2
monolayer exposed to an external linearly polarized EM field
$\textbf{E}=(E_{x},E_{y})$, thus ${\bf F}=e{\bf E}$, $e>0$. Throughout the
text, the momenta ${\bf k}$ and ${\bf k}^{\prime}$ refer to holes, and ${\bf
p}$ and ${\bf p}^{\prime}$ to electrons (for clarity and in order to avoid
mistakes). The right-hand sides of these Boltzmann equations contain the
collision integrals, describing the particle-impurity and inter-particle
interactions. In general, the r.h.s. of Eq. (2) should also contain e-e and
h-h interactions. However, an analysis shows, that in the framework of a two-
band model with equal masses of electrons and holes, these contributions
vanish [30]. Furthermore, let us limit ourselves to considering electron and
hole scattering on short-range impurities, and assume the corresponding
scattering times identical and independent on the electron and hole energies.
In this case, it is easier to implement the relaxation time approximation for
the particle-impurity scattering,

$\displaystyle Q_{e(h)i}\\{f_{{\bf p}({\bf k})}\\}=-\frac{f_{{\bf p}({\bf
k})}-n_{{\bf p}({\bf k})}}{\tau_{i}},$ (4) $\displaystyle Q_{eh}\\{f_{\bf
p}\\}=\frac{2\pi}{\hbar S^{3}}\sum_{{\bf p}^{\prime},{\bf k}^{\prime},{\bf
k}}|U_{{\bf p}^{\prime}-{\bf p}}|^{2}\delta_{{\bf k}^{\prime}+{\bf
p}^{\prime}-{\bf k}-{\bf p}}$ (5) $\displaystyle\times[(1-f_{\bf k})(1-f_{\bf
p})f_{{\bf k}^{\prime}}f_{{\bf p}^{\prime}}-(1-f_{{\bf k}^{\prime}})(1-f_{{\bf
p}^{\prime}})f_{{\bf k}}f_{{\bf p}}]$
$\displaystyle~{}~{}~{}~{}~{}\times\delta(\epsilon_{{\bf
k}^{\prime}}+\epsilon_{{\bf p}^{\prime}}-\epsilon_{{\bf k}}-\epsilon_{{\bf
p}}),$

where $n_{{\bf p}({\bf k})}$ are equilibrium distribution functions of
electrons (holes). Expression (5) determines the electron-hole collision
integral. Evidently, $Q_{eh}$ is a nonlinear function of the electron and hole
distribution functions. Here, $U_{\bf p}$ is a Fourier-transform of electron-
hole interaction potential. In what follows, we will set the sample area $S=1$
in front of the sums for brevity (in most of places). These factors cancel out
in the derivations and do not appear in the final formulas. We also use
$\hbar$$=$$k_{B}$$=$$1$ units and restore dimensionality in the final results.

In Eqs. (4) and (5), we account for trigonal warping of the electron and hole
dispersion: $\epsilon_{\mathbf{p}}=\epsilon_{p}^{0}+w_{\mathbf{p}}$ with
$\epsilon_{p}^{0}=p^{2}/(2m)$, and
$w_{\mathbf{p}}=w_{e}(p_{x}^{3}-3p_{x}p_{y}^{2})$. Note, we include the valley
index $\eta_{e}$ into the definition of the warping parameter:
$\eta_{e}w_{e}\rightarrow w_{e}$, for brevity. Thus, ${\bf v}^{e}({\bf
p})={\bf
p}/m+w_{e}\left[3(p_{x}^{2}-p_{y}^{2})\mathbf{i}-6p_{x}p_{y}\mathbf{j}\right]$
or ${\bf v}^{e}({\bf
p})=(p_{x}/m+3w_{e}(p_{x}^{2}-p_{y}^{2}),p_{y}/m-6w_{e}p_{x}p_{y})$. The
relation for the holes is similar with the replacement $w_{e}\rightarrow
w_{h}$.

The actual distribution functions entering Eqs. (4) and (5) can be expanded
into the powers of external EM field as
$f_{\textbf{p}(\textbf{k})}=n_{\textbf{p}(\textbf{k})}+\delta
f^{(1)}_{\textbf{p}(\textbf{k})}(t)+\delta f^{(2)}_{\textbf{p}(\textbf{k})}$
with $\delta f^{(1)}_{\textbf{p}(\textbf{k})}(t)=\delta
f^{(1)}_{\textbf{p}(\textbf{k})}e^{-i\Omega
t}+f^{(1)\ast}_{\textbf{p}(\textbf{k})}e^{i\Omega t}$, thus $\delta
f^{(1)}_{\textbf{p}(\textbf{k})}$ and $\delta
f^{(2)}_{\textbf{p}(\textbf{k})}$ are the alternating first-order and the
stationary second-order corrections to the equilibrium distribution function
with respect to external EM field amplitude. The first-order corrections (to
electron and hole distribution functions) satisfy the equations:

$\displaystyle\frac{\partial\delta f^{(1)}_{\bf p}(t)}{\partial t}-{\bf
F}(t)\cdot\frac{\partial n_{\bf p}}{\partial{\bf p}}+\frac{\delta f^{(1)}_{\bf
p}(t)}{\tau_{i}}=Q^{I}_{eh}\\{\delta f^{(1)}_{\bf p}(t)\\},~{}~{}~{}$ (6)
$\displaystyle\frac{\partial\delta f^{(1)}_{\bf k}(t)}{\partial t}+{\bf
F}(t)\cdot\frac{\partial n_{\bf k}}{\partial{\bf k}}+\frac{\delta f^{(1)}_{\bf
k}(t)}{\tau_{i}}=Q^{I}_{he}\\{\delta f^{(1)}_{\bf k}(t)\\},~{}~{}~{}$ (7)
$\displaystyle Q^{I}_{eh}\\{\delta f^{(1)}_{\bf p}\\}=-2\pi\sum_{{\bf
p}^{\prime},{\bf k}^{\prime},{\bf k}}|U_{{\bf p}^{\prime}-{\bf
p}}|^{2}\delta(\epsilon_{{\bf k}^{\prime}}+\epsilon_{{\bf
p}^{\prime}}-\epsilon_{{\bf k}}-\epsilon_{{\bf p}})\delta_{{\bf
k}^{\prime}+{\bf p}^{\prime}-{\bf k}-{\bf p}}$ (8)
$\displaystyle~{}~{}~{}~{}~{}\times\Bigl{[}\delta f^{(1)}_{\bf p}[(1-n_{\bf
k})n_{{\bf k}^{\prime}}n_{{\bf p}^{\prime}}+n_{\bf k}(1-n_{{\bf
k}^{\prime}})(1-n_{{\bf p}^{\prime}})]-\delta f^{(1)}_{{\bf
p}^{\prime}}[(1-n_{{\bf k}})(1-n_{{\bf p}})n_{{\bf k}^{\prime}}+n_{\bf
k}n_{\bf p}(1-n_{{\bf k}^{\prime}})]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\delta f^{(1)}_{\bf k}[(1-n_{\bf
p})n_{{\bf k}^{\prime}}n_{{\bf p}^{\prime}}+n_{\bf p}(1-n_{{\bf
k}^{\prime}})(1-n_{{\bf p}^{\prime}})]-\delta f^{(1)}_{{\bf
k}^{\prime}}[(1-n_{{\bf k}})(1-n_{{\bf p}})n_{{\bf p}^{\prime}}+n_{\bf
k}n_{\bf p}(1-n_{{\bf p}^{\prime}})]\Bigr{]},$

where $Q^{I}_{eh}\\{\delta f^{(1)}_{\bf p}\\}$ is a linearized collision
integral with respect to the first-order correction to the electron
distribution function; $Q^{I}_{he}\\{\delta f^{(1)}_{\bf k}\\}$ can be found
analogously (see Supplemental Material for details
[[SeeSupplementalMaterialat[URL],
whichgivesthedetailsofthederivationsandrefersto~\cite[cite]{[\@@bibref{Number}{Zaitsev:2014aa,
34, Durnev:2018}{}{}]}]SMBG]).

The stationary vPGE current density is determined by the stationary part of
the second-order correction to the particle distribution functions, which
satisfy the equations:

$\displaystyle-{\bf F}\cdot\frac{\partial\delta f^{(1)*}_{\bf p}}{\partial{\bf
p}}-{\bf F}^{*}\cdot\frac{\partial\delta f^{(1)}_{\bf p}}{\partial{\bf
p}}+\frac{\delta f^{(2)}_{\bf p}}{\tau_{i}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=Q^{I}_{eh}\\{\delta
f^{(2)}_{\bf p}\\}+Q^{II}_{eh}\\{\delta f^{(1)}_{\bf p}\\},$ (9)
$\displaystyle{\bf F}\cdot\frac{\partial\delta f^{(1)*}_{\bf k}}{\partial{\bf
k}}+{\bf F}^{*}\cdot\frac{\partial\delta f^{(1)}_{\bf k}}{\partial{\bf
k}}+\frac{\delta f^{(2)}_{\bf k}}{\tau_{i}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=Q^{I}_{eh}\\{\delta
f^{(2)}_{\bf k}\\}+Q^{II}_{eh}\\{\delta f^{(1)}_{\bf k}\\},$ (10)
$\displaystyle Q^{I}_{eh}\\{\delta f^{(2)}_{\bf p}\\}=-2\pi\sum_{{\bf
p}^{\prime},{\bf k}^{\prime},{\bf k}}|U_{{\bf p}^{\prime}-{\bf
p}}|^{2}\delta(\epsilon_{{\bf k}^{\prime}}+\epsilon_{{\bf
p}^{\prime}}-\epsilon_{{\bf k}}-\epsilon_{{\bf p}})\delta_{{\bf
k}^{\prime}+{\bf p}^{\prime}-{\bf k}-{\bf p}}$ (11)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\Bigl{[}\delta
f^{(2)}_{\bf p}[(1-n_{\bf k})n_{{\bf k}^{\prime}}n_{{\bf p}^{\prime}}+n_{\bf
k}(1-n_{{\bf k}^{\prime}})(1-n_{{\bf p}^{\prime}})]-\delta f^{(2)}_{{\bf
p}^{\prime}}[(1-n_{{\bf k}})(1-n_{{\bf p}})n_{{\bf k}^{\prime}}+n_{\bf
k}n_{\bf p}(1-n_{{\bf k}^{\prime}})]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\delta f^{(2)}_{\bf
k}[(1-n_{\bf p})n_{{\bf k}^{\prime}}n_{{\bf p}^{\prime}}+n_{\bf p}(1-n_{{\bf
k}^{\prime}})(1-n_{{\bf p}^{\prime}})]-\delta f^{(2)}_{{\bf
k}^{\prime}}[(1-n_{{\bf k}})(1-n_{{\bf p}})n_{{\bf p}^{\prime}}+n_{\bf
k}n_{\bf p}(1-n_{{\bf p}^{\prime}})]\Bigr{]},$ $\displaystyle
Q^{II}_{eh}\\{\delta f^{(1)}_{\bf p}\\}=-2\pi\sum_{{\bf p}^{\prime},{\bf
k}^{\prime},{\bf k}}|U_{{\bf p}^{\prime}-{\bf p}}|^{2}\delta(\epsilon_{{\bf
k}^{\prime}}+\epsilon_{{\bf p}^{\prime}}-\epsilon_{{\bf k}}-\epsilon_{{\bf
p}})\delta_{{\bf k}^{\prime}+{\bf p}^{\prime}-{\bf k}-{\bf p}}$ (12)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\Bigl{[}\overline{\delta
f^{(1)}_{\bf p}(t)\delta f^{(1)}_{\bf k}(t)}[(1-n_{{\bf
k}^{\prime}})(1-n_{{\bf p}^{\prime}})-n_{{\bf k}^{\prime}}n_{{\bf
p}^{\prime}}]-\overline{\delta f^{(1)}_{{\bf p}^{\prime}}(t)\delta
f^{(1)}_{{\bf k}^{\prime}}(t)}[(1-n_{{\bf k}})(1-n_{{\bf p}})-n_{{\bf
k}}n_{{\bf p}}]\Bigr{]},$

where $\overline{A(t)B(t)}$ stands for the time averaging; $Q^{II}$ conbines
the second-order terms after the linearization of e-h collision integral.
Solving Eqs. (6)-(II.1) by successive approximations with respect to the
particle collision integrals, we come up with the Coulomb-induced corrections
as $\delta f^{(2)}_{\bf p}=\delta_{0}f^{(2)}_{\bf p}+\delta_{C}f^{(2)}_{\bf
p}$ and $\delta f^{(2)}_{\bf k}=\delta_{0}f^{(2)}_{\bf
k}+\delta_{C}f^{(2)}_{\bf k}$, where

$\displaystyle\delta_{0}f^{(2)}_{\bf p}=\tau_{i}\left({\bf
F}\cdot\partial_{\bf p}\delta_{0}f^{(1)*}_{\bf p}+{\bf
F}^{*}\cdot\partial_{\bf p}\delta_{0}f^{(1)}_{\bf p}\right),$ (13)
$\displaystyle\delta_{0}f^{(2)}_{\bf k}=-\tau_{i}\left({\bf
F}\cdot\partial_{\bf k}\delta_{0}f^{(1)*}_{\bf k}+{\bf
F}^{*}\cdot\partial_{\bf k}\delta_{0}f^{(1)}_{\bf k}\right),$ (14)

and thus,

$\displaystyle\delta_{C}f^{(2)}_{\bf p}=\tau_{i}\left({\bf
F}\cdot\partial_{\bf p}\delta_{C}f^{(1)*}_{\bf p}+{\bf
F}^{*}\cdot\partial_{\bf p}\delta_{C}f^{(1)}_{\bf p}\right)$ (15)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+\tau_{i}\left(Q^{I}_{eh}\\{\delta_{0}f^{(2)}_{\bf
p}\\}+Q^{II}_{eh}\\{\delta_{0}f^{(1)}_{\bf p}\\}\right),$
$\displaystyle\delta_{C}f^{(2)}_{\bf k}=-\tau_{i}\left({\bf
F}\cdot\partial_{\bf k}\delta_{C}f^{(1)*}_{\bf k}+{\bf
F}^{*}\cdot\partial_{\bf p}\delta_{C}f^{(1)}_{\bf k}\right)$ (16)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+\tau_{i}\left(Q^{I}_{he}\\{\delta_{0}f^{(2)}_{\bf
k}\\}+Q^{II}_{he}\\{\delta_{0}f^{(1)}_{\bf k}\\}\right).$

Expressions (15) and (16) describe full e-h scattering corrections to the
electric current density,

$\displaystyle j_{\alpha}=e\sum_{\bf k}v^{h}_{\alpha}({\bf
k})\delta_{C}f^{(2)}_{\bf k}-e\sum_{\bf p}v^{e}_{\alpha}({\bf
p})\delta_{C}f^{(2)}_{\bf p},$ (17)

which has the units A/m (as a 2D current density). Without the loss of
generality, we can choose a particular direction of the force to make our
derivations clear: $\mathbf{F}=(F_{x},0)$, and then,

$\displaystyle j_{x}=e\sum_{\bf k}v^{h}_{x}({\bf k})\delta f^{(2)}_{\bf
k}-e\sum_{\bf p}v^{e}_{x}({\bf p})\delta f^{(2)}_{\bf p}.$ (18)

It is more convenient to split contributions (15) and (16) into several terms
and consider them separately (see Supplemental Material [30] for a detailed
derivation). Performing the calculations, we find

$\displaystyle
j_{x}\propto(w_{e}-w_{h})\frac{e}{\hbar}\frac{\tau_{i}^{3}F_{x}^{2}}{1+\Omega^{2}\tau_{i}^{2}}\frac{n_{e}n_{h}}{\sqrt{mT^{3}}}$
(19)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\frac{1}{\sqrt{\pi}}\int\limits_{0}^{\infty}dq|U_{q}|^{2}q^{2}e^{-\frac{q^{2}}{4mT}},$

where $n_{e}$ and $n_{h}$ are electron and hole equilibrium densities at a
given temperature and we restored the Planck’s and other constants. Formula
(19) is a general expression for the Coulomb interaction-induced vPGE current.
When applied to a TMD MoS2 monolayer, $U_{q}$ takes the form of the Keldysh-
Rytova potential [31, 32]: $U_{q}=2\pi e^{2}/q(1+2\pi\alpha_{\textrm{2D}}q)$,
where $\alpha_{\textrm{2D}}$ is a polarizability of the monolayer. Introducing
the dimensionless variable $x=q/\sqrt{4mT}$, we find electron-hole
interaction-induced contribution to $\lambda$, describing vPGE current
density:

$\displaystyle\lambda=(w_{e}-w_{h})\frac{30}{4}\frac{e^{3}}{\hbar}\frac{\tau_{i}^{3}}{1+\Omega^{2}\tau_{i}^{2}}\frac{n_{e}n_{h}}{T}(2\pi
e^{2})^{2}{\cal J},$ (20) $\displaystyle{\cal
J}=\frac{1}{\sqrt{\pi}}\int\limits_{0}^{\infty}dx\left[\frac{1}{x(1+x\,2\pi\alpha_{\textrm{2D}}\sqrt{4mT})}\right]^{2}x^{2}e^{-x^{2}}.$

Formula (20) is the central result of this paper. It should be mentioned, that
the dimensionless integral ${\cal J}$ can be estimated numerically or even
found analytically, but the resulting expression is rather complicated, and
thus, we do not present it here.

##  III Results and discussion

Formula (20) potentially has two important limiting cases due to the presence
of two effective distances: $\alpha_{\textrm{2D}}$ and $\hbar/\sqrt{4mT}$ (in
the power of the exponence). The estimation shows that
$\alpha_{\textrm{2D}}\sqrt{4mT}\ll 1$ up to the temperatures, when particle-
phonon scattering starts to play the dominating role. Thus, we can safely
disregard the term proportional to $\alpha_{\textrm{2D}}$. Then, the integral
over $x$ turns into the Euler-Poisson integral, which gives

$\displaystyle\lambda=(w_{e}-w_{h})\frac{15}{4}\frac{e^{3}}{\hbar}\frac{\tau_{i}^{3}}{1+\Omega^{2}\tau_{i}^{2}}(2\pi
e^{2})^{2}\frac{n_{e}n_{h}}{T}.$ (21)

Let us mention, that in (21), all the temperature dependence is in the
fraction containing the concentrations $n_{e}$, $n_{h}$, and $T$.

Furthermore, let us compare Eq. (21) with the bare vPGE effect, reported
elsewhere [26]. Combining together the bare vPGE current and the Coulomb-
induced correction to the current yields

$\displaystyle\tilde{\lambda}=12\frac{e^{3}\tau_{i}^{2}}{1+\Omega^{2}\tau_{i}^{2}}\left[(n_{h}w_{h}-n_{e}w_{e})\right.$
(22)
$\displaystyle~{}~{}~{}~{}\left.-(w_{h}-w_{e})\frac{15}{12}\frac{(2\pi)^{2}}{\hbar}\frac{e^{4}n_{e}n_{h}}{(2T)^{2}}(T\tau_{i})\right],$

where we use $\tilde{\lambda}$ instead of $\lambda$ to distinguish between the
Coulomb-related and full (bare plus Coulomb force-induced) vPGE currents.
Evidently, the dependence on the particles densities of the first and the
second terms here are different. The dependence on the product $n_{e}n_{h}$ of
the second term is an attribute of the Coulomb-related contribution.

In order to clearly understand the physics of the Coulomb-induced term, let us
take into account that the equilibrium densities in an intrinsic material
equal $n_{e}=n_{h}=n_{i}$, where $n_{i}=(mT/2\pi)\exp(-\Delta/2T)$ is an
intrinsic particle density with $\Delta$ the material bandgap. Thus, the
square brackets in Eq. (22) read as

$\displaystyle
n_{i}(w_{h}-w_{e})\left[1-\frac{15}{12}\frac{(2\pi)^{2}}{\hbar}\left(\frac{e^{2}\sqrt{n_{i}}}{2T}\right)^{2}(T\tau_{i})\right].$
(23)

We immediately conclude that the second term has an opposite to the bare vPGE
contribution sign. It results in suppression of the vPGE due to Coulomb
interaction between elecrtrons and holes. Qualitatively, such behavior can be
explained by the following argument: electrons and holes experience friction
(or a coulomb drag) since they move in the opposite directions. To
qualitatively estimate the Coulomb contribution, an estimation of intensity is
necessary. In the second term, the first factor, $e^{2}\sqrt{n_{i}}/2T\ll 1$,
is the relation of the Coulomb interaction energy of the electron and the hole
to their kinetic energy given by temperature for non-degenerate gas
statistics. Such behavior is expected because we considered the Coulomb-
related contribution to vPGE perturbatively.

The second factor, $T\tau_{i}$, reflects the relation between the kinetic
energy of particles with their broadening due to the scattering on impurities.
The latter should be weak in comparison with the particle energy, resulting in
large factor $T\tau_{i}\gg 1$. This factor compensates the smallness of the
first factor, and it can even result an essential suppression of the vPGE due
to the electron-hole scattering.

Furthermore, the general result (22) only holds in the equilibrium. The net
vPGE current should be summed over two non-equivalent valleys. Thus, it
vanishes due to the time-reversal symmetry. Indeed, restoring the valleys
indices $\eta_{e}=\pm 1$ and $\eta_{h}=\mp 1$ for electron and hole warping
amplitudes, $w_{e}\rightarrow\eta_{e}w_{e}$,
$\,w_{h}\rightarrow\eta_{h}w_{e}$, and summing up over $\eta_{e}=\pm 1$ and
$\eta_{h}=\mp 1$ at equilibrium densities
$n_{e}^{\eta_{e}}=n_{h}^{\eta_{h}}=n_{i}$ in both valleys, yields zero net
vPGE current. In order to achieve a nonzero net current, the time reversal
symmetry is to be destroyed. It can be done by sample illumination with
circularly-polarized pump EM field with frequency producing interband
transitions populating a given valley (at given circular polarization), see,
e.g., [14], and leaving the other valley with equilibrium populations,
$n_{i}$. Let the valley with $\eta_{e}=-\eta_{h}=1$ be pupulated with
densities $n_{e}=n_{h}=n_{i}+\delta n_{ph}$, where $\delta n_{ph}$ is the
photoinduced density correction. If electrons and holes in the other valley
$\eta_{e}=-\eta_{h}=-1$ have equilibrium densities $n_{i}$, the net vPGE
current is determined by the coefficient $\lambda$ in the form:

$\displaystyle\lambda=-12\frac{e^{3}\tau_{i}^{2}}{1+\Omega^{2}\tau_{i}^{2}}(w_{h}+w_{e})$
(24)
$\displaystyle~{}~{}~{}~{}~{}~{}\times\left[1-\frac{15}{12}\frac{(2\pi)^{2}}{\hbar}\frac{e^{4}(2n_{i}+\delta
n_{ph})}{T^{2}}(T\tau_{i})\right]\delta n_{ph}.$

At low intensity of photogeneration, $n_{i}\gg\delta n_{ph}$, the total
current increases linearly with the circular light intensity, $\delta
n_{ph}\sim I$. With increasing the intensity when the density of non-
equilibrium carries exceeds the density of equilibrium ones, $n_{i}\ll\delta
n_{ph}$, the Coulomb-induced term increases as $I^{2}$, and it can totally
suppress the vPGE effect.

## Conclusions

We developed a theory of the Coulomb interaction-related contribution to the
valley photogalvanic effet in two-dimensional non-centrosymmetric Dirac
materials possessing nontrivial structure of valleys and exposed to an
external electromagnetic field, taking MoS2 as an example. The valley
photogalvanic effect is the result of two factors: (i) the presence of the
trigonal warping of electrons and holes’ bands and (ii) the Coulomb
interaction between the carriers of charge. We considered the low-frequency
limit of the external linearly-polarized probe field: the field frequency is
smaller than the temperature $T$, and the electron-electron and electron-hole
scattering times are much larger than the electron-impurity scattering time.
using the Boltzmann transport equations, we demonstrated that the electron-
hole scattering dominates electron-electron scattering in intrinsic
semiconductors and it might be comparable with the bare valley photogalvanic
effect. We found, that the Coulomb interaction-related contribution to the
valley photogalvanic current is opposite to the bare valley photogalvanic
current.

###### Acknowledgements.

We were supported by the Institute for Basic Science in Korea (Project No.
IBS-R024-D1), Ministry of Science and Higher Education of the Russian
Federation (Project FSUN-2023-0006), and the Foundation for the Advancement of
Theoretical Physics and Mathematics “BASIS”.

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