Photogalvanic Effect in 2D Dichalcogenides Under Double Illumination
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Photogalvanic Effect in 2D Dichalcogenides Under Double Illumination
M.V. Entin ∗ and L.I. Magarill † Rzhanov Institute of Semiconductor Physics, Siberian Branch,Russian Academy of Sciences, Novosibirsk, 630090 Russia andNovosibirsk State University, Novosibirsk, 630090 Russia
V.M. Kovalev ‡ Rzhanov Institute of Semiconductor Physics, Siberian Branch,Russian Academy of Sciences, Novosibirsk, 630090 Russia andNovosibirsk State Technical University, Novosibirsk, 630072 Russia (Dated: November 6, 2018)We study the photogalvanic effect caused by a simultaneous action of circular-polarized interbandand linearly-polarized intraband illuminations. It is found that, in such conditions, the steadyphotocurrent appears. The effect originates from the valley-selective pumping by the circular light,the trigonal asymmetry of the valleys together with the even asymmetry of the linearly-polarizedlight, that produces a polar in-plane asymmetry of the electron and hole distribution functions,leading to the photocurrent. The approach is based on the solution of the classical kinetic equationfor carriers with accounting for the quantum interband excitation.
Introduction.
The transition metal dichalcogenides(TMDs), such as
M oSe , M oS etc. attracted muchattention as real 2D materials consisting of single molec-ular layers [1]. These materials demonstrate unique op-tical and transport properties. Having the indirect-bandstructure in a 3D phase they become the direct band ma-terials in a 2D phase. The band structure of these mate-rials consists of two independent valleys coupled by thetime-reversal symmetry [2]. The large distance betweenvalleys in the reciprocal space suppresses the intervalleyscattering processes resulting in the conservation of val-ley quantum number. This property opens the way to usethe valley quantum number as an additional to the spindegree of freedom in modern applications for electronicdevices based upon quantum laws. The new research di-rection, valleytronics , attracts much attention in moderncondensed matter physics [3, 4].Lately, the main attention has been paid to study thetransport effects in these materials [5–8]. One of the in-triguing transport phenomena is the so-called valley Halleffect [5–7]. If the TMD monolayer is placed in the in-plane static electric field, the Hall current appears in eachvalley directed across the field. This current depends onthe valley quantum number and has its opposite directionin valleys having the opposite value of valley quantumnumber. The time-reversal symmetry coupling differentvalleys results in the vanishing of the net Hall currentin equilibrium. To destroy this symmetry, the circular-polarized external electromagnetic (EM) field, having thefrequency exceeding the bandgap of TMD layer, shouldbe applied to the sample. Due to the valley selectivityof optical transitions, such EM field creates the imbal-ance of electron populations in different valleys resulting ∗ [email protected] † [email protected] ‡ [email protected] in the nonzero valley Hall current in the system. Usually,the circularly-polarized EM field is assumed to be weakenough and it is studied by means of the perturbationtheory. Contrary, the valley Hall effect under the reso-nant interband excitation due to the strong EM field hasbeen analyzed in detail [9].Recently [10] we have developed the quantum field the-ory of the coherent photogalvanic valley Hall effect inTMD materials. As for the valley Hall effect in [9], theexternal EM field pumping the valleys has been consid-ered in the nonperturbative manner. It was shown thatif a sample is illuminated by two light sources havingcircular (with basic frequency) and linear polarizations(with double frequency), the stationary valley Hall cur-rent arises. This effect needs the intercoherence betweenthe light sources.The other mechanism of photocurrent which does notneed two coherent light sources is considered in thepresent paper. The main idea is as follows. The to-tal electron spectrum E k has its time-reversal symmetry,while the spectrum of an individual valley ε p = E K + p ( K is the position of the valley center) does not have it. Atthe same time, the symmetry of the electron dispersion ε p towards p ↔ − p , where momentum p is counted fromthe valley minimum, exists only approximately, near theband extrema. The lack of this symmetry means alsothe absence of symmetry towards the spatial reflectionwhich is responsible for the photogalvanic effect (PGE).The valley selection by the circular-polarized light leadsto a loss of the spatial symmetry and, hence, the PGEbecomes permitted.Here we will assume that the TMD layer is illuminatedby two light sources. The first circularly-polarized pump-ing field populates the only valley, whereas the secondlinearly polarized probe field, creates the stationary cur-rent in the system due to intraband electron processes.Both fields are supposed to be weak, and the currentarises as a second order response to the linear-polarizedradiation. The frequency of circular polarized light Ωcorresponds to the interband transitions, the frequencyof linear-polarized light ω gets into the domain of free-electron absorption and will be assumed to be less thanthe characteristic energy of photoexcited carriers.The current density in the microwave probe electricfield E ( t ) = E cos( ωt ) is described by a phenomenolog-ical expression j i = β ijk E j E k . (1)Depending on the initial state of the dichalcogenidemonolayer, one should distinguish two cases. The firstone occurs if the monolayer is in the n − or p − dopedregime. In that case the circularly-polarized EM fieldcreates the nonequilibrium electron and hole densitiesadditional to the equilibrium ones. Formally, under theaction of linearly-polarized field the PGE current in agiven valley consists of the two contributions. The firstone is due to equilibrium electron ( n − doped) or hole( p − doped) densities, whereas the second one is relatedto the photogenerated particles. The total current beingsummarized over the valleys does not contain the termassociated with the equilibrium carriers because the lat-ter is canceled out. Thus, in the rest of the paper weanalyze the current part due to the noneqiulibrium car-riers generated by the circularly-polarized EM field.The photogalvanic tensor β ijk in this case is propor-tional to ηαJ/ ~ Ω, where the light, causing the interbandpumping, is characterized by the frequency Ω, intensity J , absorption coefficient α , and the degree of circular po-larization η = η + − η − ( η ± are the fractions of a photonwith right or left polarizations).The symmetry of point K in a dichalcogenide is C v .Taking into account the fact that the linear tensor β ijk originates from the spectrum warping, we find the non-zero components of β ijk : β yxy = β yyx = β xyy = − β xxx = − β . Thus, it is enough to calculate the component β only.For simplicity, below we neglect the bands spin split-ting. The spin is a robust quantum number for interbandtransitions. The theory will be developed for a single spinprojection. The generalization to both spin projectionsis reduced to collecting the corresponding contributionsto the current.The other simplification is the neglection of weak non-selectivity of interband transitions caused by the inter-band matrix element dependence on momentum p . Thisassumption is valid if Ω only slightly exceeds the badgap∆, Ω − ∆ ≪ ∆ (we set ~ = 1). The generalization to thefinite Ω − ∆ can be done by means of introducing γ , theselectivity of excitation into valley K . In the two-bandDirac Hamiltonian [2, 5, 9] disregarding the possible spinsplitting of the valence band, γ = 2∆Ω / (∆ + Ω ). Atthe absorption threshold Ω = ∆, γ = 1, and at Ω ≫ ∆, γ → K is reflected in the electron spectrum ε p = ǫ p + w p , where ǫ p = p / m is the electron en-ergy, w p = C ( p x − p x p y ) = C p cos(3 φ p ) is the trigo-nal warping correction, p = p (cos φ p , sin φ p ) (this choicecorresponds to the reflection symmetry relative to theaxis x ). It is known that the asymmetry of the parti-cle dispersion leads to the second harmonic generation[11], purely valley currents [12] and alignment [13] of thephotoexcited carriers in gapless materials (graphene). Inthe present paper just the quantity w p is responsible forPGE.The field E ( t ) interaction with electrons is describedin the framework of classical Boltzmann equation. Thescattering by neutral and charged impurities will be con-sidered. Valley pumping.
The possibility of PGE is deter-mined by the disbalance between the valley populations n K − n − K . The valley concentrations n ± K are controlledby the circular polarized interband radiation. In the equi-librium n K = n − K ; the same is true if two circular com-ponents are mixed at an equal proportion. The station-ary values of n K are determined by the balance betweengeneration and recombination or intervalley scatteringprocesses. We will assume that the probability of theseprocesses is much less than the intravalley scattering. Inthis case, the intravalley equilibrium is established, whilethe disbalance between valleys remains. The balance isdescribed by the equation δn − K − δn K τ v − δn K τ r + g K = 0 , (2)where δn K is a concentration of photoexcited carriers, τ v , τ r are intervalley and recombination times, g K is aphotogeneration rate, accordingly. The quantity g K isdetermined by g ± K = αJη ± γ/ ~ Ω . (3)From Eq.(2) we find δn K − δn − K = τ αJηγ/ ~ Ω , (4)where 1 /τ = 1 /τ r + 2 /τ v . If there are finite valley pop-ulations at equilibrium ( n, p − doped regime), the relax-ation times do not depend on the external illumination,whereas, in the absence of carriers at equilibrium, therelaxation time depends on the δn ± K and, as a result onpumping field intensity J . Note, that the values of τ forelectrons and holes (and, consequently, their concentra-tions) can differ. Calculation of PGE coefficient.
Our analysis of PGEcoefficient β is based on the classical kinetic equationapproximation. For simplicity, we consider one type ofcarriers, namely electrons, keeping in mind that the total (cid:1)(cid:2)(cid:3)(cid:0) (cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9) (cid:10)(cid:11) (cid:12) (cid:13) (cid:14) (cid:15)(cid:16) (cid:17)(cid:18)(cid:19)(cid:20) (cid:21) (cid:22) (cid:23) (cid:24) (cid:25) (cid:26) FIG. 1. (Color online) a). Circular-polarized light with the right polarization selectively pumps the electrons between thevalence and conduction valleys in point K . b). The Brillouin band of dichalcogenides. The K electron valley is occupiedby photoexcitation, while − K remains empty. The probe linear-polarized microwave radiation together with the trigonalasymmetry of the spectrum results in the in-plane photogalvanic current. c). Sketch of predicted effect. current is determined by the summation of contributionsfrom different types of carriers and valleys. Let us havethe valley K to be populated, while − K is empty. Notehere that the case when the pumping field has its arbi-trary polarization is described by the same final equationswith the additional factor η reflecting the circular polar-ization degree. In specific cases of positive or negativecircular polarization η = ± F ( p , t ) reads ∂ t F + e E ∇ p F = ˆ I F . (5)The quantity ˆ I represents the collision operator with im-purities. In the Born approximationˆ I F ( p ) = 2 πn i Z d p ′ π | V ( p − p ′ ) | δ ( ε p − ε p ′ )( F ( p ) −F ( p ′ )) . (6)Here V ( p ) is the Fourier transform of the impurity po-tential, n i is the impurity concentration. Independentlyfrom the spectrum asymmetry, the collision operator van-ishes if F ( p ) depends only on the electron energy (inparticular, if it is the equilibrium distribution function).The collision operator bears the valley asymmetry via theelectron spectrum and can be separated into two parts,ˆ I = ˆ I + + ˆ I − . The first one, ˆ I + contains the isotropicpart of electron spectrum, ǫ p , whereas ˆ I − is determinedby the spectrum asymmetry. Assuming ˆ I − ≪ ˆ I + andexpanding Eq.(6) with respect to w p , we findˆ I − F ( p ) = 2 πn i Z d p ′ π | V ( p − p ′ ) | × δ ′ ( ǫ p − ǫ p ′ )( w p − w p ′ )( F ( p ) − F ( p ′ )) . (7)The solution of Eq.(5) is executed by the expansion inelectric field powers. The stationary correction appearsin the second-order with respect to the electric field. Itis given by F (2) = e (cid:16) ˆ I − E ∇ p h ( iω + ˆ I ) − E ∇ p f ( p ) i(cid:17) . (8) Here overline stands for the time averaging, f ( p ) is thestationary distribution function of photogenerated car-riers depending on the particle spectrum warping. Thisfunction is produced by the pumping. It is assumed that,due to long intervalley and recombination times, f ( p ) isquasiequilibrium function within the pumped valley. Atthe same time, this function is anisotropic in the momen-tum space due to the spectrum warping.The contribution to the stationary current density ofelectrons is expressed via F (2) ( p ) as j = e Z ( ∇ p ε p ) F (2) ( p ) d p π . (9)The photogalvanic current Eq.(9) arises due to theasymmetry of the spectrum directly and via the collisionoperator. Assuming the smallness of w p we expand thecurrent with respect to C . The expansion touches on thevelocity operator ∇ p ε p , stationary distribution function f ( p ), and the collision operator ˆ I . Thus, we have ∇ p ε p = p m + ∇ p w p , f ( p ) ≈ f ( p ) + w p ∂f ( p ) ∂ε p , (10)where f ( p ) is the stationary particle distribution func-tion taken at w p = 0. The collision integral can be alsoexpanded with respect to the asymmetric partˆ I − = ˆ I − − ˆ I − ˆ I − ˆ I − + ..., (11)1 iω + ˆ I = 1 iω + ˆ I + − iω + ˆ I + ˆ I − iω + ˆ I + ... Thus, we have three contributions to the current whichcomes from the velocity operator, stationary distributionfunction and the collision operator, respectively.At C = 0 the system is isotropic in the ( x, y ) plane.In this case, the action of ˆ I + onto the M -th angular har-monics of distribution function F ( p ) ∝ e iMφ p reduces tothe multiplication by the corresponding relaxation rate:ˆ I + F ( p ) = −F ( p ) /τ M , (12)where1 τ M = 2 πmn i Z | V ( p − p ′ ) | (1 − cos( M θ )) dθ. (13)Here p = p ′ , θ is an angle between p and p ′ . For thefurther calculation we use the identity1 α + ˆ I + [ g ( p ) cos( M ϕ p )] = g ( p ) cos( M ϕ p ) α − /τ M (14)valid for any function g ( p ).In the following section we calculate these contribu-tions assuming the simplified τ − constant approximationof the isotropic collision integral ˆ I + = − /τ which cor-responds to the electron scattering by short-range im-purities. This approximation is also valid for Coulombscattering when the screening length becomes less thanthe electron wavelength. Although this condition can behardly fulfilled if impurities are screened by the same car-riers, it is valid if there is an additional screening mecha-nism, for example due to the presence of gate electrode.The generalization to the case of unscreened Coulombimpurities will be given in a further section. Short-range impurities.
The contribution to the cur-rent which corresponds to the correction ∇ p w p is givenby j = − e τ ω τ ) Z ( ∇ p w p ) d p (2 π ) ( E · ∇ p ) f ( p ) (15)It can be readily shown that the other contributions tothe PGE current vanish.For the scattering on short-range impurities, 1 /τ = mn i U does not depend on the electron energy. Substi-tuting w p in Eq.(15), we find j x = e τ E x C ω τ ) Z ( p x − p y ) d p (2 π ) ∂ f ( p ) ∂p x . (16)The last integral can be directly expressed via the totaldensity of photo-generated electrons in a given valley n K ≡ n = Z d p (2 π ) f ( p ) . Finally, the total PGE coefficient including the summa-tion over carrier kinds is β = 3 e ηγ (cid:20) n e τ e C e ω τ e − n h τ h C h ω τ h (cid:21) , (17)where the subscripts e and h indicate carrier types.It is interesting to note that this result is analogues tothe photon drag effect [14] with the change of 3 C ηγ → q/ ( ωm ), where q, ω are the in-plane component of pho-ton wavevector and its frequency. Coulomb impurities.
Here the scattering caused by thenon-screened charge-impurities with concentration n i isstudied. The Fourier transform of the impurity potential is V ( q ) = 2 πe / ( κq ) ( κ is the dielectric constant). Thequantity τ M is1 τ M = πme n i κ p Z dφ − cos( M φ )1 − cos φ = π e n i κ ǫ p | M | . (18)The dependence of the Coulomb scattering times onthe electron energy (see Eq.(18)) brings additional con-tributions to the current. Eq.(7) converts toˆ I − F ( p ) = m n i e C κ p π Z dφ ddp ′ " ( F ( p ) − F ( p ′ )) × p cos(3 φ ) − p ′ cos(3 φ ′ ) p ′ + p − pp ′ cos( φ − φ ′ ) p ′ = p . (19)Further calculations are performed with the use of an-gular harmonics and Eqs. (14, 18, 19). We found thecurrent for the Boltzmann distribution function of pho-toexcited carriers. The resulting PGE coefficient reads β = 16 e ηγτ ( n h C h − n e C e ) F ( ωτ ) , (20) F ( y ) = 132 ∞ Z x e − x dx x y × (cid:20) x + 2 x + 10 − x y + x y − x y (1 + x y ) (cid:21) , (21)where τ = R ∞ τ e − ǫ/T dǫ R ∞ e − ǫ/T dǫ = T κ π e n i . (22)The function F ( y ) having the asymptotic behaviour F ( y ) ≈ − y if y ≪ , (23) F ( y ) ≈ y if y ≫ , is presented in Fig.2. For unscreened Coulomb impuri-ties scattering the relaxation times are identical for elec-trons and holes, Eq.( ?? ). In that case the total PGEof electrons and holes is determined by the same func-tion F ( y ) in Eq.(20) with the renormalized coefficient τ nC → τ e n e C e − τ h n h C h .In a simple situation n e = n h for interband transitions.In a two-band model C e = C h . The Coulomb scatteringalso does not depend on the charge sign, and τ e = τ h . Insuch case the PGE current should vanish. Nevertheless, C e = C h due to the influence of other bands on theelectron spectrum (see, e.g. [2]). Besides, n e = n h ifthese quantities are controlled by different mechanismsof carrier capture.As seen in Fig.2 (unlike the case of short-range im-purity scattering) the PGE experiences a non-uniformscaled drop with the frequency. This is the consequenceof concurrence between different contributions to the F ( y ) FIG. 2. Function F ( y ) according to Eq.(21). PGE. One of them appears as a result of asymmetricscattering of the stationary even corrections to the dis-tribution function, producing the odd stationary distri-bution function responsible for the current. The otherresults from the high-frequency odd correction to the dis-tribution function, which, after the asymmetric scatter-ing, gives rise to the even high-frequency function and,after an additional action of microwave field, converts tothe stationary odd function. The corrections to the cur-rent are controlled by dimensionless parameters ωτ or ωτ , and that results in their different relative strengthin different frequency domains.Let us estimate the effect in M oS by means of Eq.(20). The parameters C e and C h have values C h = − . eV ˚ A, C e = − . eV ˚ A [2]. Choosing the valuesof other necessary parameters τ e = τ h = 10 − s , n e = n h = 10 cm − and taking η = γ = 1, we find β =2 . µA · cm/V at ω = 0. Conclusions and discussions.
The effect studied herediffers from the known photogalvanic effect (see [15] andreferences therein) by its origin from the spectrum asym-metry rather than the asymmetry of interaction potential or the crystal-induced Bloch wave function asymmetry.The circular polarized light causes the selective popula-tion of valleys; the linear-polarized microwave illumina-tion converts the trigonal asymmetry of the spectrum in asingle valley to the polar asymmetry of the photocurrent.The valley population lives during a relatively long inter-valley time, as compared with the momentum relaxationtime (responsible for usual PGE). This circumstance am-plifies the effect.The photocurrent is determined by both electrons andholes. As seen in Eq.(17), the e-h asymmetry (in par-ticular, difference between τ e and τ h ), together with adifferent direction of partial currents, results in the com-plex frequency dependence of the photocurrent, up to itsalternating sign.It is desirable to compare the effect predicted here tothe other PGE. The usual PGE appears in the secondorder in the external alternating electric field. It requiresthe medium asymmetry. The coherent PGE [10] appears,at least, in the third order of electric field, does not re-quire the asymmetry of the media, but needs the inter-coherence of two light sources with the first and secondharmonics.As compared to the coherent PGE, the effect con-sidered here has the forth order in the external field,which can be produced by two independent light sources.Hence, the inter-coherence of two light sources is not as-sumed.The considered effect depends on different parametersdescribing the system, namely, the asymmetry of valleyspectrum, intervalley scattering time, momentum relax-ation time and valley pumping selectivity. All these fac-tors contribute to the effect and, hence, can be studiedby the effect measurements. Acknowledgments.
This research was supported by theRSF grant No 17-12-01039. [1] A.V. Kolobov and J. Tominaga,
Two-DimensionalTransition-Metal Dichalcogenides , Springer Series in Ma-terial Science (2016).[2] A. Kormanyos, G. Burkard, M. Gmitra, J. Fabian, V.Zolyomi, N. Drummond and V.Fal’ko, 2D Mater. ,049501 (2015).[3] A. Rycerz, J. Tworzydlo, and C. W. J. Beenakker, Nat.Phys. , 172 (2007).[4] J. Karch, S. A. Tarasenko, E. L. Ivchenko, J. Kamann, P.Olbrich, M. Utz, Z. D. Kvon, and S. D. Ganichev, Phys.Rev. B , 121312(R) (2011).[5] D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Phys.Rev. Lett., , 196802 (2012).[6] K. F. Mak, K. L. McGill, J. Park, and P. L. McEuen,Science 344(6191), 14891492 (2014).[7] J. Lee, K. F. Mak, and J. Shan, Nat Nano advance onlinepublication (2016). [8] S. Konabe and T. Yamamoto, Phys. Rev. B , 075430(2014).[9] V.M. Kovalev, W.-K. Tse, M.V. Fistul and I.G. Savenko,New J. Phys. , 565 (2017).[11] L. E. Golub and S. A. Tarasenko, Phys. Rev. B ,201402(R) (2014).[12] L.E. Golub, S.A. Tarasenko, M.V. Entin and L.I. Maga-rill, Phys. Rev. B , 195408 (2011).[13] R.R. Hartmann and M.E. Portnoi, Optoelectronic Prop-erties of Carbon-based Nanostructures: Steering electronsin graphene by electromagnetic fields , LAP LAMBERTAcademic Publishing, Saarbrucken, (2011).[14] E.L.Ivchenko, Phys. Status Solidi B , 2538 (2012).[15] M.M. Glazov, S.D. Ganichev, Physics Reports535