Side-jumps in the spin-Hall effect: construction of the Boltzmann collision integral
Dimitrie Culcer, E. M. Hankiewicz, Giovanni Vignale, R. Winkler
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Side-jumps in the spin-Hall effect: construction of the Boltzmann collision integral
Dimitrie Culcer, E. M. Hankiewicz, Giovanni Vignale, and R. Winkler
4, 5 Condensed Matter Theory Center, Department of Physics,University of Maryland, College Park MD20742-4111 Institut f¨ur Theoretische Physik und Astrophysik,Universit¨at W¨urzburg, 97074 W¨urzburg, Germany Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211 Materials Science Division, Argonne National Laboratory, Argonne, IL 60439 Northern Illinois University, De Kalb, Illinois 60115 (Dated: November 3, 2018)We present a systematic derivation of the side-jump contribution to the spin-Hall current insystems without band structure spin-orbit interactions, focusing on the construction of the collisionintegral for the Boltzmann equation. Starting from the quantum Liouville equation for the densityoperator we derive an equation describing the dynamics of the density matrix in the first Bornapproximation and to first order in the driving electric field. Elastic scattering requires conservationof the total energy, including the spin-orbit interaction energy with the electric field: this resultsin a first correction to the customary collision integral found in the Born approximation. A secondcorrection is due to the change in the carrier position during collisions. It stems from the part of thedensity matrix off-diagonal in wave vector. The two corrections to the collision integral add up andare responsible for the total side-jump contribution to the spin-Hall current. The spin-orbit-inducedcorrection to the velocity operator also contains terms diagonal and off-diagonal in momentum space,which together involve the total force acting on the system. This force is explicitly shown to vanish(on the average) in the steady state: thus the total contribution to the spin-Hall current due to theadditional terms in the velocity operator is zero.
I. INTRODUCTION
Semiconductor spin electronics is an active area of re-search in which both theory and experiment have madesubstantial progress during the past decade.
The re-cent focus on electrically-induced phenomena in sys-tems with spin-orbit interactions has brought to lightmany unexplored and fascinating facets of semiconduc-tor physics. Considerable progress has been made in pastyears in the electrical manipulation of spins in semicon-ductors, where experimental and theoretical work haveyielded the prediction and discovery of the spin-Halleffect.
The spin-Hall effect, which is the main fo-cus of this paper, is the generation of a transverse spin-current at the edges of the sample as a response toan external electric field. Such a spin current leads to aspin-accumulation at the edges of the sample, and the re-lationship between spin currents and spin accumulationis nontrivial.
The first observations of the spin-Halleffect were followed by the measurement of the spin-Halleffect at room temperature by optical techniques andthe first successful measurement of the spin-Hall effect intransport in ballistic HgTe/HgCdTe quantum wells. Two main mechanisms have been shown to be re-sponsible for the spin-Hall effect. The presence of spin-orbit coupling in the impurity potentials gives rise tocontributions to the spin-Hall effect which are termed extrinsic . The spin-Hall effect observed in Refs.10,11 was shown by Engel et al. , Tse and Das Sarma and Hankiewicz and Vignale to be due to extrinsicmechanisms. Band structure spin-orbit interactions yieldcontributions termed intrinsic . The spin-Hall ef- fect observed in Ref. 12,18 is believed to be due to in-trinsic mechanisms. Extrinsic and intrinsic mechanismswere broadly discussed in the context of the anomalousHall effect (AHE), which is the generation of a transversecharge and spin-polarization current in response to anelectric field in a ferromagnetic medium.
In fact, ex-trinsic contributions to the anomalous Hall and spin-Halleffects are closely related.
The interplay of intrinsicand extrinsic contributions is a complicated problem. Itwas first addressed by Tse and Das Sarma using an ap-proach based on the diagrammatic Kubo formula. Thiswas followed by a series of publications and this topiccontinues to be an active area of research. However, ourfocus in this work is on the case of extrinsic spin-orbitinteractions alone and specifically the way they may beobtained from a kinetic equation approach.Extrinsic contributions to the spin current are of twokinds. The first and more intuitive contribution arisesfrom the asymmetric scattering of up and down spinsknown as skew scattering . This effect is found be-yond the first Born approximation, i.e., provided onegoes to at least third order in the electron-impuritypotential. The effect appears naturally in thestandard Boltzmann collision integral provided onegoes beyond the first-Born approximation. The associ-ated spin-Hall conductivity scales with disorder as theordinary Drude conductivity (i.e., proportional tothe electron-impurity scattering time), although it is ofcourse much smaller.The second extrinsic contribution has been known inthe literature as side-jump and has two main character-istics: (i) it appears already in the first Born approxi-mation for the electron-impurity potential, and (ii) theassociated spin Hall conductivity is independent of theelectron-impurity scattering time – a surprising univer-sality which will be fully explained below. In contrastto skew-scattering, the side-jump mechanism cannot bederived straightforwardly from the standard form of theBoltzmann equation. Very early on, Luttinger calcu-lated the side jump contribution to the charge conductiv-ity using a recursive density matrix approach, providinga thorough calculation yet leaving many questions unan-swered concerning the physical picture behind differentcontributions. Later, Berger used a wave-packet for-malism to build a semiclassical picture of side jump,identifying it with a shift of the center of mass of thewave packet during collisions. Nozi`eres and Lewiner used this picture and carefully studied the side jumpcontributions within a Boltzmann approach. These au-thors accounted for six contributions, some of which can-cel each other; yet they did not associate any physi-cal interpretation with the cancelations. In Ref. 15 theskew-scattering and side-jump terms were identified alsowithin the framework of a drift-diffusion approach. InRef. 23 it was claimed that the proper way to describethe side-jump effect is to replace the band energy in theusual Boltzmann collision integral by a modified bandenergy which includes corrections due to the spin-orbitinteraction with the electric field. However, the validityof this claim has not been formally shown to date. Skewscattering and side jump were rigorously studied by Tseand Das Sarma using a diagrammatic Kubo formalism,demonstrating that the side-jump term can be identifiedwith an anomalous current which gives rise to a renor-malization of the current vertex. A derivation based onthe Kubo formula was also presented in Ref. 49, findingthat the side-jump contribution for the conduction bandis independent of disorder and of the Coulomb potentialto all orders in the strength of these interactions.The purpose of this paper is to show that a rigorousalternative derivation of the side jump within the frame-work of the kinetic equation which does not require intu-itive approaches, and follows cleanly and clearly from theBoltzmann equation provided one constructs the collisionintegral with care. This construction, however, requiresthat we go beyond the Boltzmann equation formalismand resort to the full quantum Liouville equation for theone-particle density matrix, which we treat to second or-der in the electron-impurity potential and to first orderin the electric field. In order to focus on the essentialphysics, we assume that band structure spin-orbit inter-actions are negligible. Thus we only take into accountspin-orbit interactions with the impurities and with theexternal electric field that drives the current.Starting from the quantum Liouville equation we showrigorously that the conservation of energy in the Boltz-mann collision integral must be modified to include theeffective band energy, i.e., the bare band energy plus thespin-orbit interaction with the electric field. This modi-fication is reflected in the appearance of a correction to the scattering term usually found in the Born approxima-tion. This additional term, which is spin-dependent andlinear in the electric field, acts as a source for the spincurrent, yielding one half of the side jump contribution.The other half emerges when one takes into accountthe change in the position of the particle during colli-sions with impurities. The semiclassical picture of therenormalization of the trajectory of the electron duringcollisions will be derived rigorously from the scatteringof electrons off the impurity potential which involves theterms off-diagonal in the wave vector in the density ma-trix. It turns out that this process and the modifiedconservation of energy give equal contributions to thespin-Hall current, so that the full side jump contributioncan indeed be obtained by including twice the spin-orbitinteraction energy with the electric field in the δ -functionof conservation of energy in the ordinary Boltzmann col-lision integral, as suggested previously in Ref. 23. How-ever, the present derivation is rigorous, and constitutesformal validation of this heuristic approach. We notethat the factor of two emerges naturally from the dia-grammatic Kubo formula, as demonstrated in Ref. 20,when one takes into account the vertex renormalizationof the spin and charge currents.In addition to constructing the picture of side jumpoutlined above, the analysis expounded in this work sup-ports the argument that the full velocity operator con-tains the net force acting on the system, which must van-ish on physical grounds. To provide a formal derivationof this fact, we show that the total contribution of theadditional terms in the velocity operator to the spin-Hallcurrent is indeed zero. This is due to the fact that thevelocity operator contains a spin-dependent term linearin the electric field, which reflects the spin-orbit interac-tion with the electric field, as well as an additional spin-dependent term off-diagonal in wave vector, which re-flects the spin-orbit interaction with the impurities. Theimportance of this latter term was recognized already indiagrammatic linear response theory in Ref. 20. Withinthe kinetic equation formalism, the two terms in the ve-locity operator produce equal and opposite contributionsto the spin-Hall current which cancel each other out. Thephysical interpretation of this fact is that, in the steadystate, the average force acting on an electron must van-ish.In this paper, therefore, we strive to provide an under-standing of the side-jump mechanism in the absence ofintrinsic spin precession due to band-structure spin-orbitcoupling, which is rigorous and at the same time physi-cal. We believe that a rigorous physical understanding isa first step towards building a consistent picture, withinthe kinetic equation framework, of the interplay of spinprecession due to band structure spin-orbit coupling andspin-orbit coupling due to impurities. This interplay hasbeen studied by Tse and Das Sarma based on a dia-grammatic linear-response approach, and by Hu et al numerically. More recently Hankiewicz and Vignale constructed a phase diagram of this problem, while Rai-mondi and Schwab employed a Keldysh Green’s func-tion technique. The long-term aim of this work is to builda rigorous understanding, based on the kinetic equation,of intrinsic spin precession, skew scattering and side jumpon an equal footing.This article is organized as follows. In section II a ki-netic equation is derived starting from the quantum Li-ouville equation for the density operator. In section IIIwe explicitly construct the collision integral, includingall contributions in the first Born approximation arisingfrom the modification of the position operator. SectionIV discusses the contributions of the side-jump mecha-nism to the spin-Hall effect. It demonstrates that thecorrections to the velocity operator do not contribute tothe spin current. We end with conclusions. II. TIME EVOLUTION OF THE DENSITYMATRIX
We outline in this section the formalism that will beused to determine the collision integral and the way itgives rise to the side jump spin-Hall current. The systemis described by a density operator ˆ ρ , which obeys thequantum Liouville equation d ˆ ρdt + i ~ [ ˆ H + ˆ H E + ˆ H U , ˆ ρ ] = 0 . (1)In this equation ˆ H = ~ ˆ k m ∗ (2)is the Hamiltonian for a parabolic conduction band, with m ∗ the carrier effective mass, whileˆ H E = e E · ˆ r + λ e E · ˆ σ × ˆ k (3a) ≡ e E · ˆ r + ˆ σ · ˆ ∆ k (3b)represents the interaction with a constant uniform exter-nal electric field E . The term ˆ H E includes both the directinteraction and the interaction via the (material depen-dent) spin-orbit coupling of strength λ . We have usedthe notation ˆ ∆ k = 2 λ e ˆ k × E for the effective field char-acterizing the spin-dependent part of this interaction, aterm which, as we shall see later, gives rise to one half ofthe side jump contribution to the spin-Hall conductivity.Finally ˆ H U = U (ˆ r ) + λ ∇ U (ˆ r ) · ˆ σ × ˆ k (4)denotes the interaction with a set of randomly distributedimpurities, again both directly and via spin-orbit cou-pling.We project the Liouville equation onto a set of time-independent states {| k s i} of definite wave vector k andspin orientation s = ± z -axis. The matrixelements of ˆ ρ in this basis form the spin density matrix and are written as ρ kk ′ ≡ ρ ss ′ kk ′ = h k s | ˆ ρ | k ′ s ′ i , with cor-responding notations for the matrix elements of ˆ H , ˆ H E and ˆ H U . We assume that impurities are uncorrelated andthe normalization is such that the configurational averageof h k s | ˆ H U | k ′ s ′ ih k ′ s ′ | ˆ H U | k s i is ( n i /V ) | ¯ U kk ′ | δ ss ′ , where n i is the impurity density, V the crystal volume, and¯ U kk ′ = U kk ′ (cid:0) − iλ σ · k × k ′ (cid:1) . (5)Here U kk ′ are the matrix elements of the electron-impurity potential U (ˆ r ) between plane waves, while ¯ U kk ′ is reserved for the total potential of a single impurity in-cluding the spin-orbit contribution. In what follows spinindices will be suppressed and all quantities are assumedto be matrices in spin space. Note that we use the conven-tion that ¯ U kk ′ and U kk ′ have units of energy × volume.The density matrix ˆ ρ is divided into a part diago-nal in k and a part off-diagonal in k , given by ρ kk ′ = f k δ kk ′ + g kk ′ . These two parts of ˆ ρ satisfy a set of cou-pled equations df k dt + i ~ [ ˆ H, ˆ f ] kk = − i ~ [ ˆ H U , ˆ g ] kk (6a) dg kk ′ dt + i ~ [ ˆ H, ˆ g ] kk ′ = − i ~ [ ˆ H U , ˆ f ] kk ′ − i ~ [ ˆ H U , ˆ g ] kk ′ , (6b)where ˆ H ≡ ˆ H + ˆ H E . In the first Born approximationthe solution to Eq. (6b) for g kk ′ is g kk ′ = − i ~ lim η → Z ∞ dt ′ e − ηt ′ e − i ˆ Ht ′ / ~ [ ˆ H U , ˆ f ( t − t ′ )] e i ˆ Ht ′ / ~ (cid:12)(cid:12) kk ′ , (7)where η > f ( t − t ′ ) in the inte-gral by ˆ f ( t ), which is written simply as ˆ f , and satisfiesthe equation df k dt + i ~ [ ˆ H, ˆ f ] kk + ˆ J ( f k ) = 0 , (8)where the scattering term ˆ J ( f k ) isˆ J ( f k ) = ( i/ ~ ) [ ˆ H U , ˆ g ] kk (9a)= 1 ~ lim η → Z ∞ dt ′ e − ηt ′ [ ˆ H U , e − i ˆ Ht ′ / ~ [ ˆ H U , ˆ f ] e i ˆ Ht ′ / ~ ] (cid:12)(cid:12) kk . (9b)Equations (8) and (9) describe the dynamics of the den-sity matrix and constitute the complete set of tools werequire in order to derive the kinetic equation satisfiedby the density matrix in an electric field, including theside-jump contribution to the scattering term due to themodification of the position operator by the spin-orbitinteraction. III. DERIVATION OF THE COLLISIONINTEGRAL
We want to evaluate further the collision integral (9).For this purpose, we decompose the matrix f k into apart scalar in spin space and a spin-dependent part, thus f k = n k S k , with S k ex-pressible in terms of the Pauli matrices. We will show inthe following that, to first order in λ and in the electricfield, we can write the scattering term asˆ J ( f k ) = ˆ J ( n k ) + ˆ J a sj ( n k ) + ˆ J b sj ( n k ) + ˆ J ( S k ) . (10)The first of these terms comes from the band Hamilto-nian ˆ H , is a scalar in spin space and does not dependon λ or on the electric field. The second term reflectsthe fact that the total energy must be conserved duringscattering events, including the second term in Eq. (3).The third term comes from the direct interaction withthe electric field e E · ˆ r , and arises because r fails to com-mute with the spin-orbit interaction with the impurities.Physically this reflects the change in r during a collisionwith an impurity. The resulting variation of e E · r alsocontributes to the overall energy balance. Both ˆ J a sj ( n k )and ˆ J b sj ( n k ) are spin-dependent and are linear in λ and the electric field E . We will discuss each of the contri-butions in turn below. The last scattering term, ˆ J ( S k ),will be important in the kinetic equation below in de-termining the steady-state correction linear in E to thespin-dependent part of the density matrix, and thus tothe spin-Hall current. A. Scattering correction due to the conservation ofthe modified carrier energy
In this subsection we focus on the part of the scatteringterm which is linear in the electric field and arises fromthe addition of the spin-orbit interaction with the electricfield σ · ∆ k to the particle energy ε k ≡ ~ k / (2 m ∗ ).Since we are working to first order in λ and in the elec-tric field, in this subsection we only need to consider thescalar part of the impurity potential, U kk ′ . Moreoverthe time evolution operators in this subsection includethe side jump energy σ · ∆ k , but not the term e E · r ,which will be considered in the next subsection. The ma-trix elements of ˆ H in the exponents of the time evolutionoperators are thus diagonal in k .Writing out all the terms in the double commutator(9b) we find1 ~ [ ˆ H U , e − i ˆ Ht ′ / ~ [ ˆ H U , ˆ f ] e i ˆ Ht ′ / ~ ] kk = n i ~ Z d d k ′ (2 π ) d (cid:0) U kk ′ e − iH k ′ t ′ / ~ U k ′ k f k e iH k t ′ / ~ − U kk ′ e − iH k ′ t ′ / ~ f k ′ U k ′ k e iH k t ′ / ~ − e − iH k t ′ / ~ U kk ′ f k ′ e iH k ′ t ′ / ~ U k ′ k + e − iH k t ′ / ~ f k U kk ′ e iH k ′ t ′ / ~ U k ′ k (cid:1) , (11)where H k = ε k + σ · ∆ k and d is the dimensionality of the system. By expanding the exponentials of Pauli matrices,the product of time evolution operators can be written, to first order in λ , as e − iH k ′ t ′ / ~ e iH k t ′ / ~ = e i ( ε k − ε k ′ ) t ′ / ~ (cid:20) cos ∆ k t ′ ~ cos ∆ ′ k t ′ ~ − i σ · ˆ ∆ k ′ cos ∆ k t ′ ~ sin ∆ ′ k t ′ ~ + i σ · ˆ ∆ k sin ∆ k t ′ ~ cos ∆ ′ k t ′ ~ (cid:21) , (12)where ˆ ∆ k is a unit vector in ∆ k direction.The only task that remains is integration over the time variable t ′ , giving a series of δ -functions reflecting energyconservation. The overall result for this scattering term, to first order in λ , can be decomposed into a scalar partˆ J ( n k ) independent of λ , and spin-dependent parts ˆ J a sj ( n k ) + ˆ J ( S k ). These parts may be written as followsˆ J ( n k ) = πn i ~ Z d d k ′ (2 π ) d |U kk ′ | ( n k − n k ′ ) (cid:2) δ ( ǫ + − ǫ ′ + ) + δ ( ǫ − − ǫ ′− ) + δ ( ǫ + − ǫ ′− ) + δ ( ǫ − − ǫ ′ + ) (cid:3) (13a)ˆ J a sj ( n k ) = πn i ~ Z d d k ′ (2 π ) d |U kk ′ | ( n k − n k ′ ) { σ · ( ˆ ∆ k + ˆ ∆ k ′ ) [ δ ( ǫ + − ǫ ′ + ) − δ ( ǫ − − ǫ ′− )]+ σ · ( ˆ ∆ k − ˆ ∆ k ′ ) [ δ ( ǫ + − ǫ ′− ) − δ ( ǫ − − ǫ ′ + )] } (13b)ˆ J ( S k ) = 2 πn i ~ Z d d k ′ (2 π ) d |U kk ′ | ( S k − S k ′ ) δ ( ε k − ε k ′ ) . (13c)The full energies ǫ ± = ε k ± ∆ k / ǫ ′± = ε k ′ ± ∆ k ′ /
2, where ∆ k = | ∆ k | . The scalar term ˆ J ( n k ) reproducesthe ordinary Boltzmann-equation scattering term. The side-jump term ˆ J a sj ( n k ) constitutes a correction that reflectsthe presence of the spin-orbit interaction energy with the electric field in the condition for energy conservation. Weexpand the δ -functions in this scattering term in ∆ k as δ ( ǫ + − ǫ ′ + ) = δ ( ε k − ε k ′ ) + (cid:18) ∆ k − ∆ k ′ (cid:19) ∂∂ε k δ ( ε k − ε k ′ ) , (14)with corresponding expressions for the other combinations of δ -functions. Adding all contributions together thescattering term ˆ J a sj ( n k ) simplifies considerably and we obtain the final expressionˆ J a sj ( n k ) = 2 πn i ~ Z d d k ′ (2 π ) d |U kk ′ | ( n k − n k ′ ) 12 σ · ( ∆ k − ∆ k ′ ) ∂∂ε k δ ( ε k − ε k ′ ) . (15)The presence of this scattering term reflects the fact thatthe total energy including the spin-orbit interaction en-ergy with the electric field is conserved in elastic colli-sions. B. Scattering correction due to the change in r during collisions We have so far ignored the presence of the term e E · ˆ r in the time evolution operator. At this stage we wouldlike to determine the additional scattering term linearin E arising from it, which we denote by ˆ J b sj ( n k ). Toaccomplish this we use Eq. (7) to find the correction g b kk ′ to g kk ′ arising from the presence of e E · ˆ r in the timeevolution operator. Using Eq. (9a) we will then obtain ˆ J b sj ( n k ) as ( i/ ~ ) [ ˆ H U , ˆ g b ] kk .Starting from Eq. (7), we expand the time evolutionoperator to first order in the term e E · ˆ r . Using the ma-trix elements of the ordinary position operator ˆ r betweenBloch states h k | ˆ r | k ′ i = i ∂∂ k δ ( k − k ′ ) , (16)we obtain additional terms of the form t ′ e − iε k t ′ e E · ∂∂ k δ ( k − k ′ ) = (cid:18) i ∂∂ε k e − iε k t ′ (cid:19) e E · ∂∂ k δ ( k − k ′ ) . (17)We integrate over t ′ as before, and after some lengthybut straightforward algebra, we obtain g b kk ′ = − πe E · Z d d k ′ (2 π ) d (cid:18) ∂ ¯ U kk ′ ∂ k + ∂ ¯ U kk ′ ∂ k ′ (cid:19) ( n k − n k ′ ) ∂∂ε k δ ( ε k − ε k ′ ) . (18)We have not written out explicitly a contribution to g b kk ′ containing terms of the form ∂n k /∂ k . We find that suchterms drop out in the final evaluation of spin currents when the scattering potential is elastic, as a result of integratingover k and k ′ . In the final analysis these terms involve the square matrix element | ¯ U kk ′ | which does not have anycontributions linear in λ . We find that the leading contribution due to these terms is ∝ λ and may therefore beneglected. Evaluating the k -derivatives of the impurity potentials gives ∂ ¯ U kk ′ ∂ k + ∂ ¯ U kk ′ ∂ k ′ = − iλ U kk ′ σ × (cid:0) k − k ′ (cid:1) . (19)Substituting this into the Eq. (18) and subsequently evaluating ˆ J b sj ( n k ) = ( i/ ~ ) [ ˆ H U , ˆ g b ] kk we obtain the scatteringterm ˆ J b sj ( n k ) = 2 πn i eλ ~ Z d d k ′ (2 π ) d |U k ′ k | ( n k − n k ′ ) E · σ × (cid:0) k − k ′ (cid:1) ∂∂ε k δ ( ε k − ε k ′ ) , (20)which is easily seen to be exactly equal to ˆ J a sj ( n k ). Thesum of these terms constitutes the total side-jump scat-tering term ˆ J sj ( n k ) = ˆ J a sj ( n k )+ ˆ J b sj ( n k ) = 2 ˆ J a sj ( n k ), whichcontains the well-known factor of two associated with sidejump. We emphasize that we obtain this reinforcement of the side jump directly from the scattering term, andour work shows no evidence that it is related in any directway to the integral of the velocity operator [see Eqs. (29)and (35) below] over the time of a collision. IV. CONTRIBUTION OF THE SIDE-JUMPMECHANISM TO THE SPIN-HALL CURRENT
We have derived a contribution linear in the electricfield to the scattering term appearing in the kinetic equa-tion. This contribution is brought about by the spin-dependent interaction of the charge carriers with the elec-tric field due to the spin-orbit interaction. In this sectionwe will first evaluate the correction that this term yieldsin the spin-dependent part of the density matrix, and wewill show that this correction accounts fully for the side-jump spin-Hall current including the important factor oftwo.
This is done in subsection IV A. To show thatthis is the only side-jump contribution to the spin-Halleffect, subsection IV B will demonstrate that the modi-fications to the velocity operator do not contribute anyadditional terms to the spin current.
A. Contribution of the side-jump scattering term
We need to find the contribution to the spin-Hall cur-rent brought about by the additional scattering termˆ J sj ( n k ). In order to further evaluate Eq. (8) we let f k = f k + f E k , where f k ( ε k ) is the Fermi-Dirac func-tion, which is a scalar in spin space in the case un-der study, and f E k is the correction we will determinefrom the kinetic equation. Firstly, the kinetic energypart of the Hamiltonian, ε k , drops out of the com-mutator. Also, in the commutator [ σ · ∆ k , f k ] wenote that σ · ∆ k is first order in the electric field, sothe density matrix can be replaced with the equilibriumFermi-Dirac function, which is a scalar in spin space, so[ σ · ∆ k , f k ] = 0. Moreover, in the side-jump scatteringterm ˆ J sj ( n k ), which is also first-order in the electric field,we may replace n k by f k . Following some short andstraightforward algebra, the side-jump scattering termcan be written asˆ J sj ( f k ) = 1 τ p σ · ∆ k δ ( ε − ε F ) , (21)where τ p is the usual momentum relaxation time.Next, we decompose f E k into a part scalar in spinspace and a spin-dependent part, thus f E k = n E k S E k . The equation for n E k in the steady state is theordinary scalar Boltzmann equation. The term [ ˆ H E , ˆ f ] k becomes ( e/ ~ ) E · ( ∂f /∂ k ) which is a scalar and acts asthe source term for n E k . We write the scalar part of theBoltzmann equation asˆ J ( n E k ) = e E ~ · ∂f k ∂ k , (22)where the scattering term ˆ J ( n k ) has been defined in Eq.(13a). The solution for n E k is written as n E k = τ p e E ~ · ∂f k ∂ k = ~ τ p e E · k m ∗ ∂f k ∂ε k . (23) The equation for S E k in the steady state is ˆ J ( S E k ) = − ˆ J sj ( f k ), in which the RHS, ˆ J sj ( f k ), acts as a sourceterm for S E k . Substituting the explicit expressions forthe two scattering terms, this equation can be written ina simpler form as S E k τ p = − τ p σ · ∆ k δ ( ε − ε F ) , (24)and has the simple solution S E k = − σ · ∆ k δ ( ε − ε F ) . (25)We would like to draw attention to the fact that the ad-ditional side-jump collision integral is part of the sourcefor S E k . The source term itself contains a factor of 1 /τ p ,which cancels the 1 /τ p appearing on the LHS of the equa-tion for S E k . This explains why the end result for theside jump contribution to the spin-Hall current does notdepend on the strength and shape of the impurity poten-tial.Now that we have found the solution for S E k , in otherwords the spin-dependent part of the density matrix inan electric field, we can determine its contribution to thespin-Hall current. We denote the components of the spinvelocity as v ij which corresponds to a spin component i flowing along the direction j . In finding the contribu-tion to the spin-Hall current due to S E k we can restrictourselves to the term in the spin velocity to zeroth or-der in the electric field, which is v ij = ( ~ k j /m ∗ ) ~ σ i / E = ( E x , ,
0) the side-jump Hamiltonian σ · ∆ k becomes eλ E · σ × k = − eλE x k y σ z , (26)This gives a side-jump spin-Hall current as a result of themodification to the scattering term j zy (cid:12)(cid:12)(cid:12) sct = (cid:18) ~ (cid:19) Z d d k (2 π ) d ~ k y m ∗ tr( σ z S E k ) = neλ E x , (27)where the trace is taken over the spin components and n is the density. This term therefore gives a spin-Hallconductivity σ zyx | sct = neλ , which is the usual side-jumpterm in the spin-Hall current. This result isvalid in both two and three dimensions.
B. Vanishing contribution of the corrections to thevelocity operator
It will be shown in this subsection that the correctionto the velocity operator linear in the electric field doesnot yield any additional terms in the spin-Hall current.The velocity operator is defined as the time derivative ofthe physical position operator, which in turn is given byˆ r phys = ˆ r + λ ˆ σ × ˆ k . (28)Notice that all spin-orbit interactions can be most di-rectly derived by replacing r by r phys in the direct inter-actions and expanding to first order in λ .The velocity operator has a part which is diagonal in k and is given by v k = ( i/ ~ ) [ ˆ H, ˆ r phys ] k = ~ k m ∗ + i ~ [ eλ E · ˆ σ × ˆ k , ˆ r ] k + i ~ [ e E · ˆ r , λ ˆ σ × ˆ k ] k = ~ k m ∗ − eλ ~ σ × E . (29)The k -diagonal part of the spin velocity v ij , up to firstorder in the electric field, is thus v ij = ~ k j m ∗ ~ σ i − eλ { ( σ × E ) j , σ i } (30)where { A, B } ≡ AB + BA . For an electric field along ˆ x we obtain for the k -diagonal part of the spin velocity v zy the expression v zy = ~ ~ k y m ∗ σ z − eλ E x . (31)The E -dependent part of the spin velocity operator is ascalar in spin space, and its contribution to the spin-Hallcurrent is found by multiplying by the scalar part of theequilibrium density matrix f k . It gives us the term j zy (cid:12)(cid:12)(cid:12) vel,d = − eλ E x Z d d k (2 π ) d tr f k = − neλ E x (32)so its contribution to the spin-Hall conductivity is σ zyx | vel,d = − neλ .This is, however, not the full story. The velocity op-erator also has a term that is off-diagonal in the wavevector, which is referred to as v kk ′ and is given by v kk ′ = i ~ [ ˆ H U , ˆ r phys ] kk ′ . (33)The matrix elements of the impurity potential are givenby Eq. (5). The part of the matrix element v kk ′ origi-nating from ˆ r is easily seen to be i ~ [ ˆ U , ˆ r ] kk ′ = 1 ~ (cid:18) ∂U kk ′ ∂ k ′ + ∂U kk ′ ∂ k (cid:19) . (34) The disorder potential is the potential ˆ U due to the fullensemble of impurities present in the system. In the finalresult for the spin-Hall current, the k -off-diagonal partof the velocity operator will be traced with the k -off-diagonal part of the density matrix g kk ′ , which in thefirst Born approximation is also linear in ˆ U . Once thisis done, a configurational average will be performed overthe impurities. In the end we seek the result to first orderin λ . However, it proves more straightforward to work interms of the full potential ˆ U until the end. Only then wewill restrict ourselves to the terms which are first orderin λ .With these insights in mind we proceed, Eq. (33) yields v kk ′ = − iλ ~ σ × ( k − k ′ ) U kk ′ , (35)where we have written the matrix elements of the fullpotential ˆ U . Note that this expression has not been av-eraged over impurity configurations. This expression forthe k -off-diagonal part of the velocity operator holds be-cause, for elastic scattering, the scalar part of the scat-tering potential U kk ′ ≡ U ( k − k ′ ) depends only on thedifference k − k ′ . Its contribution to the velocity oper-ator is immediately seen to be zero. This can also beunderstood by noting that the scalar part of the impu-rity potential commutes with r and does not contributeto the velocity.The k -off-diagonal part of the velocity operator con-tributes to the side-jump spin-Hall current. To find itscontribution we return to Eq. (7) and integrate over timeto find g kk ′ = iπ δ ( ε k − ε k ′ ) U kk ′ ( f k − f k ′ ) . (36)This expression also contains the matrix elements of thefull impurity potential and has not been averaged overimpurity configurations.The contribution of the k -off-diagonal part of the ve-locity operator to the spin current is found by takingthe trace of the spin velocity arising from Eq. (35) withthe k -off-diagonal part of the density matrix given in Eq.(36). This yields for the spin-Hall current j zy (cid:12)(cid:12)(cid:12) vel,od = ~ σ z Z d d k (2 π ) d Z d d k ′ (2 π ) d v y kk ′ g k ′ k (37a)= − ( λ ~ ) 2 π ~ Z d d k (2 π ) d ( k x − k ′ x ) h U kk ′ U k ′ k i δ ( ε k ′ − ε k )( f k − f k ′ ) , (37b)where the bracket denotes the average over impurity configurations. At this stage we introduce the simplification thatwe require only terms to first order in λ . We note that, since the entire term in Eq. (37) already contains λ , the otherterms in this equation are needed only to zeroth order in λ . Consider first the term proportional to k x , which, to firstorder in λ , can be written as − ( λ ~ ) Z d d k (2 π ) d k x (cid:20) π ~ Z d d k ′ (2 π ) d h U kk ′ U k ′ k i δ ( ε k ′ − ε k )( n k − n k ′ ) (cid:21) = − ( λ ~ ) Z d d k (2 π ) d k x ˆ J ( n k ) (38a)= − ( λ ~ ) Z d d k (2 π ) d k x (cid:18) eE x ~ ∂f k ∂k x (cid:19) . (38b)where the last replacement follows from the scalar Boltz-mann equation, as written in Eq. (22), assuming, as be-fore, that E k ˆ x . Further, Eq. (37) also contains a termproportional to k ′ x , which is easily seen to give exactly thesame contribution if one swaps k and k ′ in the summa-tion. The contribution of the k -off-diagonal part of thevelocity operator to the spin Hall current to first orderin λ is therefore j zy (cid:12)(cid:12)(cid:12) vel,od = − λeE x Z d d k (2 π ) d k x ∂f k ∂k x = neλ E x . (39)The spin-Hall conductivity originating from this termis σ zyx | vel,od = neλ and it exactly cancels the contribu-tion σ zyx | vel,d from the k -diagonal E -dependent part ofthe velocity operator. The physical explanation of thiscancellation is the following. Notice that the full spin-dependent part of the velocity operator, from Eqs. (29)and (35) is − eλ ~ σ × E − iλ ~ σ × ( k − k ′ ) U kk ′ (40)which contains the total force acting on the system. Ac-cording to Ehrenfest’s theorem, the expectation values ofposition and momentum obey time evolution equationsanalogous to those of classical mechanics. Consequentlythe expectation value of the force should be zero in thesteady state, consistent with the earlier suggestion thatthe total force acting on the system does not contributeto the spin current. We note also that the presence ofthe velocity terms off-diagonal in wave vector is crucialin obtaining the correct side-jump contribution in the di-agrammatic Kubo-formula approach, as demonstrated inRef. 20.
V. SUMMARY AND DISCUSSION
We have completed the formal derivation of the side-jump spin-Hall current, where we have considered (in theabsence of intrinsic spin precession) all contributions tothe kinetic equation in the first Born approximation. Wewill now discuss our findings and their implications. It isevident from our analysis that, within the kinetic equa-tion framework, the side-jump spin-Hall current origi-nates solely from the modification of the Boltzmann col-lision integral due to the spin-dependent interaction en-ergy of an electron with the external electric field and the impurity field. The density-matrix formulation ofthe problem shows that the spin-orbit interaction withthe electric field alters the condition for energy conser-vation, since the total energy conserved during collisionsmust include the spin-dependent part. In addition, thespin-orbit coupling with the impurities causes a changein the position of the electron during scattering process,which again affects the energy balance via the interac-tion energy e E · r . This effect doubles the size of theside-jump current.The understanding of the side-jump effect emergingfrom this derivation differs from the conventional expla-nation, according to which this phenomenon is attributedto the linear-in- E modification of the velocity operator.It is clearly seen in the previous section that the full ve-locity operator in the presence of spin-orbit interactionscontains an extra term due to the impurity potential,which is off-diagonal in wave vector. This term, and the k -diagonal velocity operator result in two corrections tothe spin current that are equal in magnitude but op-posite sign so they cancel out. This could be justifiedinformally using the fact that the velocity operator con-tains the total force acting on the system (or rather theterm to leading order in λ of this force), and thereforeshould vanish in the steady state . Therefore, contraryto conventional assumptions, the side-jump contributionto the spin-Hall current is traced to the qualitatively dif-ferent carrier spin dynamics during collisions. It is nottraced to any linear-in- E correction to the velocity oper-ator. The linear-in- E correction to the velocity operatoris canceled by the contribution of the off-diagonal in k correction to the velocity operator. The importance ofthis off-diagonal-in- k correction in the velocity operatorin obtaining the correct side-jump current was recognizedin the diagrammatic Kubo-formula approach as shown inRef. 20.We emphasize that in obtaining our results we haveused a rigorous quantum mechanical formulation, start-ing with the quantum Liouville equation and making,in the course of the derivation, the same assumptionsthat are characteristically made in linear response theory.Therefore our formalism could in some sense be regardedas being built from the ground up. The approach we haveused demonstrates that the derivation of the side jumpdoes not need to rely on intuitive semiclassical ideas aslong as the collision integral is derived rigorously fromthe fundamental starting point of all transport theories.In contrast, Ref. 37 using semiclassical Boltzmann argu-ments counted six terms contributing to side jump, butdid not clearly indicate which terms should cancel. Sincefor the conduction band the side-jump contributions havethe same magnitude but opposite signs, it led to freedomin choosing which terms cancel and still obtaining thecorrect amplitude of the side-jump. For example Ref. 19counted the terms from the anomalous velocity and theshift of the position operator. Our analysis shows exactlywhich terms are non-zero and which contributions can-cel and is in agreement with the Kubo derivation of sidejump contributions presented in Refs. 20,49. Our resultsare also in agreement with recent extensive studies of theanomalous Hall effect in magnetic semiconductors. 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