Nonequilibrium theory of the photoinduced valley Hall effect

I. Vakulchyk,1, 2 V. M. Kovalev,3 and I. G. Savenko1, 2

1Center for Theoretical Physics of Complex Systems,Institute for Basic Science (IBS), Daejeon 34126, Korea

2Basic Science Program, Korea University of Science and Technology (UST), Daejeon 34113, Korea3Rzhanov Institute of Semiconductor Physics, Siberian Branch of Russian Academy of Sciences, Novosibirsk 630090, Russia

(Dated: February 1, 2021)

A recent scientific debate has arisen: Which processes underlie the actual ground of the valleyHall effect (VHE) in two-dimensional materials? The original VHE emerges in samples with ballistictransport of electrons due to the anomalous velocity terms resulting from the Berry phase effect. Indisordered samples though, alternative mechanisms associated with electron scattering off impuritieshave been suggested: (i) asymmetric electron scattering, called skew scattering, and (ii) a shift of theelectron wave packet in real space, called a side-jump. It has been claimed that the side-jump notonly contributes to the VHE but fully offsets the anomalous terms regardless of the drag force forfundamental reasons, and thus, the side-jump together with skew scattering become the dominantmechanisms. However, this claim is based on equilibrium theories without any external valley-selective optical pumping, which makes the results fundamentally interesting but incomplete andimpracticable. We develop in this paper microscopic theory of the photoinduced VHE using theKeldysh nonequilibrium diagrammatic technique, and find out that the asymmetric skew scatteringmechanism is dominant in the vicinity of the interband absorption edge. This allows us to explainthe operation of optical transistors based on the VHE.

INTRODUCTION

The concept of the Hall effect is the emergence of anelectric current or other flux of particles in a sample inthe direction transverse to both the dragging force andthe external magnetic field, which should be finite forthe effect to take place. If similar phenomena happenin the absence of a magnetic field, they are referred toas anomalous Hall effects (AHEs) [1]. The eminent ex-amples of the AHE include the Hall effect in magneticmaterials (with their built-in sample magnetization), thespin Hall effect, where the role of the magnetic field isplayed by spin-orbit interaction, and the valley Hall ef-fect (VHE) [2–5], which emerges in two-dimensional (2D)Dirac materials possessing nonequivalent valleys in recip-rocal space, like transition metal dichalcogenide (TMDC)monolayers [6–8]. There, electrons and holes occupy twovalleys, K and K′, that are connected by time-reversalsymmetry. TMDCs also represent a promising platformand testing ground for optoelectronics [9, 10] and spinvalleytronics [11] as direct band gap materials that obeyvalley-dependent optical selection rules [12, 13]. Theseproperties make them fundamentally interesting and ap-pealing for device design [14].

It is commonly accepted that there exist three princi-pal mechanisms behind the AHE in non-magnetic mate-rials [15]: (i) the Berry phase stipulated anomalous veloc-ity term (also called the intrinsic contribution) [16], (ii)the side-jump contribution, and (iii) the skew scattering(asymmetric) contribution. These three terms interplayand can partially compensate each other, as has beenreported in recent works on electron [17, 18] and exci-ton [19] transport in semiconductors. In particular, one

important recent work [18] shows that the side jump andskew scattering should not be disregarded under photonor phonon drag conditions, as is usually done when con-sidering the VHE [2, 4, 16, 20]. More precisely, it hasbeen demonstrated that the side jump compensates forthe intrinsic contribution to conductivity, and moreover,some terms in the side jump survive.

These fundamental conclusions undoubtedly play animportant role in our understanding of the microscopicprocesses underlying the VHE. However, the existing the-ories only consider equilibrium electrons initially occupy-ing two nonequivalent valleys. The valley Hall currentsresulting from these electrons flow in opposite directionsand, being of the same magnitude, the currents canceleach other out, leaving zero-net VHE current in the sam-ple. To observe a nonzero valley Hall current in actual ex-periments [3], the sample should be illuminated by an ex-ternal circularly polarized electromagnetic field of light.This destroys the time-reversal symmetry and predomi-nantly populates only one of the valleys due to the valley-dependent interband optical selection rules. As a result,the current contributions from nonequivalent valleys donot annul each other. It is important, then, to considernonequilibrium photo-excited electrons since they are theones actually contributing to the VHE. While this ideahas been briefly mentioned in literature [21], it has notbeen rigorously studied. In the meantime though, thelight-induced AHE has generally become an active fieldof research [22]. Analysis [3] of experimental VHE ob-servations is based on a phenomenological expression ofthe form σH ∝ δn, where σH is the valley Hall conduc-tivity and δn is the electron density imbalance betweenthe valleys due to interband photogeneration. The stan-dard derivation of this dependence is based on the Berry-

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FIG. 1. System schematic of a 2D Dirac material exposed tocircularly polarized light A and static drag field A. The lightcouples to the K or K′ valley depending on its polarization σ.

phase-related expression applicable to ballistic samples.

In this paper, we analyze the applicability of this ap-proach in the presence of all relevant electron-impurityscattering processes. We pose an intriguing question: Dothese statements (regarding the partial compensation ofthe intrinsic contribution) remain valid in the case of op-tically driven systems based on Dirac materials when cir-cularly polarized light pumps one of the valleys? Theanswer to this question is of utmost importance not onlyfrom a fundamental viewpoint (since 2D Dirac materialsare prone to interact with light) but also from the per-spective of optoelectronic applications, in particular, innovel van der Waals heterostructures. We consider theintrinsic, side-jump, and skew scattering contributions tovalley Hall photoconductivity using the nonequilibriumKeldysh diagram technique. Thus, we build a micro-scopic theory of the photoinduced VHE.

GENERAL THEORY

We consider a 2D system (Fig. 1) exposed to a cir-cularly polarized light (which results in interband tran-sitions),

A(t) = Ae−iωt + A∗eiωt, (1)

and thus A = A(1, iσ) with σ = ±1, and the in-planealternating drag electric field is

A(t) = A(e−iΩt + eiΩt

), (2)

where we assume that the drag field is linearly polarizedso that A is real-valued. At the end of the calculations,we will put Ω → 0 to find the static limit, which cor-responds to the drag effect. We define the coordinatessuch that the drag field is directed along the y axis, andtherefore our goal is to consider the valley Hall current

along x. The full system Hamiltonian reads (e < 0)

H =∆

2ŝz + V · p− eV ·A(t)− eV yA(t), (3)

where ∆ is the monolayer material bandgap, p =p(cosφ, sinφ) is the electron momentum, V = v0(ηŝx, ŝy)is the velocity, η = ±1 is the valley index, and ŝα are thePauli matrices with α = x, y, z. The Hamiltonian (3) iswritten in sub-lattice basis since the honeycomb latticeof a TMDC monolayer can be looked at as two trian-gle sub-lattices inserted into each other. However, it isinstructive and physically transparent to work in the c-and v-band basis (the cv basis in what follows). In ourcase, the external fields in (3) are uniform in space, thusconserving the electron momentum (which, hence, can beconsidered as a complex number). To transform into thecv basis, we use a unitary operator that depends only onthe electron momentum [23],

U =

(cos(θ/2) sin(θ/2)

sin(θ/2)eiηφ − cos(θ/2)eiηφ), (4)

where cos θ = ∆/2�p, sin θ = ηv0p/�p, and �p =√(∆/2)

2+ v20p

2 ≈ ∆/2 + p2/2m, where the electron ef-fective mass is m = ∆/(2v20) at small electron momenta,v0p� ∆. Using H = U+HU , we find

H = H0 − ev ·A(t)− evyA(t), (5)

where

H0 =(�c(p) 0

0 �v(p)

), v =

(vcc vcvvvc vvv

)(6)

are the bare Hamiltonian and the velocity operator in thecv-basis, with �c(p) ≡ �p and �v(p) = −�p (we will justwrite �c and �v in what follows, keeping in mind that theyboth depend on the absolute value of the momentum; wewill also omit ~ in the expressions below but restore it inthe final formulas).

The valley Hall current, being the linear response toexternal drag field A, reads [24],

jx(t) =

∫C

dt′Qxy(t, t′)A(t′), (7)

Qxy(t, t′) = ie2Tr [vxG(t, t′)vyG(t′, t)] , (8)

where C stands for the Keldysh contour, Tr is the traceoperator that should be taken over the bands, Q(t, t′)is the generalized conductivity representing a linear re-sponse function, and[

i∂t −H0 + ev ·A(t)]G(t, t′) = δ(t− t′) (9)

defines the matrix Green’s function in the cv-basis. Itshould be stressed that this (matrix) Green’s function

3

accounts exactly for the external pumping field. We canalso write Eq. (7) as

jx(t) = j(1)x (Ω)e

−iΩt + j(1)x (−Ω)eiΩt. (10)

The in-plane electric field is Ey(t) = −∂tA(t), andthe current can be found as jx(Ω) = A[Qxy(Ω, ω) +Qxy(−Ω, ω)]. Also, we define jx(ω) = σH(ω)Ey. It isthen possible to express the static (with respect to in-plane electric field Ey) valley Hall photoconductivity bythe standard formula [24]:

σH(ω) = limΩ→0

Qxy(Ω, ω)−Qxy(−Ω, ω)2iΩ

. (11)

As electron conductivity is due to particles from the vband being excited by the external field, to find Eq. (11),we have to consider the Green’s function of the electronsin the c band while accounting for interband pumping.This Green’s function G is the solution of Eq. (9) withthe retarded (advanced) component GR,A and the lessercomponent G

4

where ne =∑

p f0[εc(p)] is the density of the photo-excited electrons in the c-band of a given valley. Thiscontribution has the same structure as the one in theequilibrium case; the principal difference is that ne rep-resents here the density of photo-excited electrons insteadof the density of thermal-equilibrium electrons.

SIDE-JUMP CONTRIBUTION

The diagrams representing the side-jump contribu-tion contain the interband matrix element of electron-impurity scattering as depicted in Fig. 2(d). To calculatethe conductivity due to the side-jump impurity process,let us first introduce the impurity potential in the cv ba-sis. We assume an elastic scattering approximation andshort-range impurities, and using the matrix element ofthe disorder matrix in the cv basis we find [17]

u(p,p′) = u0(p,p′)

{[1− sin2

(θ

2

)(1− eiη(φ

′−φ))]

× ŝ0 + ŝz2

+ [1− cos2(θ

2

)(1− eiη(φ

′−φ))]ŝ0 − ŝz

2

+1

2sin θ(1− eiη(φ

′−φ))ŝx

}, (13)

where ŝ0 is the unity matrix, φ and φ′ are the angles

corresponding to the momenta p and p′, respectively,and 〈|u0(p,p′)|2〉 = niu20 with ni being the density ofimpurities and niu

20 = (mτi)

−1. Here 〈...〉 stands for theaveraging over the positions of impurities.

In the case of near-resonant pumping, renormalizationof the vertices is negligible [25]. Then, the impurity linestake the form of free Green’s functions with an additionalform-factor and give averages of the kind

ucc(p,p′)Gα(p′, δt)ucv(p′,p) ≈

≈ 1mτi

∫dp′

(2π~)2

{sin(θ)

1− eiη(φ−φ′)

2

}Gα(p

′, δt).(14)

Performing the diagram calculations taking into accountthe mass operator renormalization (see Fig. 2(b) and theSupplemental Material [25]) we find

σ(SJ)H = −4η

e2

~

(~v0∆

)2ne

2τiγ, (15)

where γ = (τi + τr)/2τiτr includes the recombinationtime τr reflecting the nonequilibrium nature of the ef-fect. Again, the contribution of the side-jump process isdetermined by the density of the photo-excited electrons,ne.

COHERENT SKEW SCATTERING

The skew mechanism is associated with asymmetricelectron scattering by impurities. It should be described

beyond the standard Born approximation in electron-impurity scattering probability. Coherent skew scatter-ing involves pairs of closely located impurities, and canbe illustrated by a diagram with crossed impurity lines[Fig. 2(e)]. In the framework of standard Drude the-ory, the diagrams possessing crossing impurity lines areparametrically small and usually do not play any role (ex-cept for the theory of weak localization, where maximallycrossed diagrams are responsible for the effect). Never-theless, the so-called X- and Ψ-type diagrams play anessential role in the AHE [26, 27]. For a delta-correlateddisorder, the contribution of the Ψ diagram vanishes,leaving only the X diagram for calculation (see Ref. [25]):

σ(X)H = 2η

e2

~

(~v0∆

)2ne

(2τiγ)2. (16)

Summing Eqs. (12), (15), and (16), we find

σ(I)H + σ

(SJ)H + σ

(X)H = 2η

e2

~

(~v0∆

)2ne

(1− 1

2τiγ

)2= 2η

e2

~

(~v0∆

)2ne

(1− τr

τr + τi

)2. (17)

In the limit τr → ∞ and assuming ne to be the equilib-rium electron density, we recover the known result of theequilibrium VHE, when these three contributions canceleach other out [18]. The typical values of the times areτr ∼ µs and τi ∼ ps, i.e., τr � τi, allowing us to concludethat the mutual impact of these three contributions onthe photoinduced VHE is negligibly small.

ASYMMETRIC SKEW SCATTERING

The last principal mechanism is associated with asym-metric electron skew scattering by impurities. It shouldalso be described beyond the Born approximation [18]and requires a non-vanishing impurity potential correla-tor of the third order. The corresponding Y-type Feyn-man diagrams, as in Fig. 2(f), have one-to-one corre-spondence with the Boltzmann equation result with anaccount of the asymmetric contribution to the electron-impurity collision integral [17, 18, 28]. The calculationgives

σ(Y)H = −η

e2

~

(u0ne∆

) 〈ε〉τi(2γτi)2~

, (18)

where 〈ε〉 = n−1e∑

p(εp−∆/2)f0[εc(p)] = (~ω−∆)/2 isthe mean energy of the photo-excited electrons in the c-band. Evidently, the asymmetric skew scattering givesthe dominant contribution to the photoinduced VHE,since the other contributions vanish, as we have shownabove.

5

DISCUSSION

Let us consider the approximations employed herein.First, photoinduced VHE conductivity contains the den-sity of the photo-excited electrons,

ne =mτr|Mcv(0)|2

2~3Θ[ω −∆], (19)

where Θ[ω] is the Heaviside step-function. This formulahas been derived by treating the external circularly po-larized pump field as a perturbation. This is only validas long as τr|Mcv(0)|/~ � 1. In the opposite regime,τr|Mcv(0)|/~ � 1, the pumping field cannot be con-sidered as a perturbation. While a full theory of thephotoinduced VHE in the strong-coupling regime is stillmissing, the intrinsic contribution (which dominates inballistic samples) has been studied [23].

The second limitation concerns the electron-impurityscattering: ~/τi should be small in comparison with themean value of electron energy. In the case of the station-ary nonequilibirum VHE, the characteristic energy of thephoto-excited electrons is 〈ε〉, and thus (~ω−∆)τi/~� 1should be fulfilled.

Furthermore, we have accounted for one principalmechanism resulting in the establishment of a station-ary nonequilibrium state of photo-excited electrons: theinterband recombination. In principle, there also existother mechanisms limiting electron lifetime in the band.Among them is intervalley electron scattering via impu-rities or phonons, where such transitions require largevalues of electron momentum transfer resulting in longelectron lifetimes (comparable with τr). Another mecha-nism is electron capture by impurities. These phenomenamay play an important role in the photoinduced VHEand require a separate consideration.

Our results suggest an explanation of the operationof an optical transistor based on the VHE reported inan experimental work [3]. There, the authors show aquasi-linear dependence of photoconductivity σH on thedensity of the photoinduced electrons ne. It is demon-strated that the slope of the curve σH(ne) is controlledby the gate voltage, and the change of the inclinationangle cannot be explained by the equilibrium formulafor the intrinsic photoconductivity (see Fig. 3 in Ref. [3]and the corresponding discussion). Neither such (exper-imental) behavior of σH(ne) can be interpreted by theside-jump contribution, in either equilibrium or nonequi-librium cases. However, our formulas Eqs. (18) and (19)provide a possible interpretation of the experimental be-havior as we show that the main contribution to thephotoconductivity originates from skew scattering. In-deed, the slope of σH(ne) is proportional to the meanenergy of the photo-excited electrons 〈ε〉 = (~ω −∆)/2.Since with the increase of the gate voltage the c-bandbecomes more populated, the chemical potential µ (or,more precisely, the electron quasi-Fermi level, which de-

pends on the gate voltage) enters the band. Then, theMoss-Burstein effect [29, 30] results in a shift ∆ → 2µ(where µ is measured from the middle of the gap) as somelow-energy states in the c-band become occupied. Hence,the ratio σH/ne ∝ 〈ε〉 becomes gate-voltage dependent.Moreover, the increase of the electron density results inan enhancement of the scattering processes. These ar-guments elucidate the role that skew scattering plays inthe VHE in nonequilibrium situations and support ourtheoretical findings.

CONCLUSIONS

We have developed a microscopic theory of the pho-toinduced VHE in two-dimensional Dirac materials byemploying the Keldysh nonequilibrium diagrammatictechnique and analyzing the impurity scattering mecha-nisms under nonequilibrium conditions. We have demon-strated that while in ballistic samples the intrinsic Berryphase-related term is dominant, in disordered samplesthe main contribution to the Hall photoconductivitystems from asymmetric skew scattering, with the otherprincipal contributions canceling each other out. In thisway, one is able to explain the operation of optical tran-sistors based on the VHE.

We thank M. Glazov, L. Golub, and A. Parafilo for use-ful discussions and important advice, J. Rasmussen (RE-CON) for a critical reading of our paper, and E. Savenkofor help with the figures. We have been supported by theInstitute for Basic Science in Korea (Project No. IBS-R024-D1) and the Russian Science Foundation (ProjectNo. 17-12-01039).

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Nonequilibrium theory of the photoinduced valley Hall effectAbstract Introduction General theory Intrinsic contribution Side-jump contribution Coherent skew scattering Asymmetric skew scattering Discussion Conclusions References