We discuss the quasiclassical Green function method for a two-dimensional
electron gas in the presence of spin-orbit coupling, with emphasis on the
meaning of the $\xi$-integration procedure. As an application of our approach,
we demonstrate how the spin-Hall conductivity, in the presence of spin-flip
scattering, can be easily obtained from the spin-density continuity equation.Comment: 3 pages, Submitted to Physica 

Cosimo Gorini

Edelstein

Inoue

Michael Dzierzawa

Peter Schwab

Raimondi

Rammer

Roberto Raimondi

Shelankov

Wang

English

arXiv

Gorini, Cosimo

Schwab, Peter

Dzierzawa, Michael

Raimondi, Roberto

OPUS - Augsburg University Publication Server

Quasiclassical approach and spin–orbit coupling

OPUS Augsburg

We discuss the quasiclassical Green function method for a two-dimensional electron gas in the presence of spin–orbit coupling, with emphasis on the meaning of the ξξ-integration procedure. As an application of our approach, we demonstrate how the spin-Hall conductivity, in the presence of spin-flip scattering, can be easily obtained from the spin-density continuity equation

University of Regensburg Publication Server

ARTICLE IN PRESS1386-9477/$ - sedoi:10.1016/j.phCorrespondE-mail addrPhysica E 40 (2008) 1078–1080www.elsevier.com/locate/physeQuasiclassical approach and spin–orbit couplingCosimo Gorinia, Peter Schwaba, Michael Dzierzawaa, Roberto Raimondib,aInstitut fu¨r Physik, Universita¨t Augsburg, 86135 Augsburg, GermanybDipartimento di Fisica ‘‘E. Amaldi’’, Universita` di Roma Tre, Via della Vasca Navale 84, 00146 Roma, ItalyAvailable online 24 August 2007AbstractWe discuss the quasiclassical Green function method for a two-dimensional electron gas in the presence of spin–orbit coupling, withemphasis on the meaning of the x-integration procedure. As an application of our approach, we demonstrate how the spin-Hallconductivity, in the presence of spin-ﬂip scattering, can be easily obtained from the spin-density continuity equation.r 2007 Elsevier B.V. All rights reserved.PACS: 72.25.Ba; 72.25.DcKeywords: EP2DS-17; Manuscript; LaTeX-2e; Style ﬁles1. IntroductionThe quasiclassical technique is one of the most powerfulmethods to tackle transport problems. Its main virtue reliesin the fact that starting from a microscopic quantumformulation of the problem at hand it aims at deriving asimpler kinetic equation resembling the semiclassicalBoltzmann one. In deriving such an equation some of theinformation at microscopic level is suitably incorporated ina set of parameters characterizing the physical system atmacroscopic level. Since the ﬁrst application to super-conductivity, this equation is known as the Eilenbergerequation (for a review see for instance Ref. [1]). We haverecently derived [2] such an equation for a two-dimensionalelectron gas in the presence of spin orbit coupling withHamiltonianH ¼ p22mþ b  r, (1)where bðpÞ is a momentum dependent internal magneticﬁeld. In the case of Rashba spin–orbit coupling b ¼ap ^ e^z. In Ref. [2] we adopted the standard x-integrationprocedure to arrive at the Eilenberger equation, and,though this leads to correct results, we feel the need for ae front matter r 2007 Elsevier B.V. All rights reserved.yse.2007.08.006ing author. Tel.: +3906 55177032; fax: +39 06 5579303.ess: raimondi@ﬁs.uniroma3.it (R. Raimondi).deeper understanding, which we provide in the presentpaper. In so doing we follow an analysis carried out byShelankov [3]. Finally, we use the Eilenberger equation tostudy the response to an external electric ﬁeld in thepresence of magnetic impurities.2. The quasiclassical approachIn deriving the Eilenberger equation a key observation isthat, by subtracting from the Dyson equation its hermitianconjugate, one eliminates the singularity for equal space–time arguments and gets a simpler equation for thex-integrated Green functiongðp^; xÞ ¼ ipZdx Gðp; xÞ; x ¼ ðpÞ  m. (2)Here Gðp; xÞ is the Green function in Wigner space, i.e. theFourier transform of Gðx1;x2Þ with respect to the relativecoordinate r ¼ x1  x2. The ‘‘check’’ indicates that theGreen function is a 2 2 matrix in the Keldysh space [1].To shed some light on the meaning of the x-integration, letus consider ﬁrst the space dependence of the two-pointretarded Green function for free electrons in the absence ofspin–orbit couplingGRðx1; x2Þ ¼Xpeipro xþ i0þ ; r ¼ x1  x2. (3)ARTICLE IN PRESSC. Gorini et al. / Physica E 40 (2008) 1078–1080 1079At large distances, the integral is dominated by the extremaof the exponential under the condition of constant energy.This forces the velocity to be parallel or antiparallel to theline connecting the two space arguments, qpðpÞ / r. It isthen useful to consider the momentum components parallel(pk) and perpendicular (p?) to r. Given the presence of thepole, one can expand the energy in powers of the twomomentum components xðpk; p?Þ ¼ vFðpk  pFÞ þ p2?=2m.In the case of the retarded Green function, the importantregion is that with velocity parallel to r. We then getGRðx1;x2Þ ¼Zdp?dpkð2pÞ2eipkro vFðpk  pFÞ  p2?2mþ i0þ¼  i eiðpFþo=vFÞrvFZdp?2peip2?r=2pF¼ ﬃﬃﬃﬃﬃﬃﬃ2pipFrsN0eiðpFþo=vFÞr; N0 ¼m2p. ð4ÞOne sees how the Green function is factorized in a rapidlyvarying term eipFr= ﬃﬃﬃﬃﬃﬃﬃpFrp , and a slow one, eiðo=vFÞr. Thissuggests to write quite generallyGRðx1;x2Þ ¼ ﬃﬃﬃﬃﬃﬃﬃ2pipFrsN0eipFrgRðx1;x2Þ¼ GR0 ðr;o ¼ 0ÞgRðx1;x2Þ, ð5Þwhere gRðx1; x2Þ is slowly varying and GR0 indicates the freeGreen function. Explicitly, in the present equilibrium casegRðx1;x2Þ ¼ i2pZdxeixr=vFo xþ i0þ ¼ eior=vF . (6)For the advanced Green function one can go through thesame steps with the difference that the integral isdominated by the extremum corresponding to a velocityantiparallel to r, so that one has the ingoing wave replacingthe outgoing one. In the non-equilibrium case Shelankovhas shown thatgRðx1;x2Þ ¼ i2pZdxeixr=vFGRðp;xÞ; p ¼ pr^ (7)and furthermore that the quasiclassical Green functioncorresponds to the symmetrized expressiongRðp^; xÞ ¼ limr!0ipZdx cosxrvF GRðp;xÞ, (8)when sending to zero the relative coordinate r.When the spin–orbit coupling is present the Greenfunction becomes a matrix in spin space and the Fermisurface splits into two branches ðpÞ ¼ ðp2=2mÞ  jbj. Wealways assume this splitting to be small compared to theFermi energy, i.e. jbj=F51. In the case of the Rashbainteraction we writeGRðx1;x2Þ ¼ Xﬃﬃﬃﬃﬃﬃﬃ2piprsNeipr12fP; ~gRðx1;x2Þg, (9)where P ¼ jihj is the projector relative to the  energybranch and the curly brackets denote the anticommutator.This ansatz allows us to proceed in Wigner space as before,while retaining the information on the coupling andcoherence of the two bands. Eq. (9) is the equivalent inreal space of the ansatz for the Green function Gðp;xÞ usedin Ref. [2]. With such an ansatz, Eq. (9), we obtain fromEq. (7)gRðx1;x2Þ ¼X12N0fNP; ~gRðx1;x2Þg. (10)What we have explicitly shown for the retarded componentof the Green function can be extended to the advanced andKeldysh components too. Notice that gR and ~gR coincidein the absence of spin–orbit coupling, since in that caseN ¼ N0. The derivation of the Eilenberger equation cannow be done following the steps detailed in Ref. [2]. We donot repeat them here and give just the ﬁnal resultXn¼qt gn þ12pnmþ qpðbn  rÞ; qx gnn oþ i½bn  r; gn ¼ i½ S; g, ð11Þwhere gn ¼ ð12ÞfPn; gg, g ¼ gþ þ g and both the momen-tum pn and the internal ﬁeld bn are evaluated at then-branch of the Fermi surface. Finally, S is the self-energy.It is often convenient to expand g in terms of Paulimatrices, g ¼ g0 þ g  r, to explicitly separate charge andspin components. Physical quantities like charge and spindensities and currents are related to the Keldysh compo-nent of g. For example the spin current for sl , l ¼ x; y; z isjlsðx; tÞ ¼ 12pN0Zd2pJK ls ð; x; tÞ, (12)whereJls ¼Xn¼12pnmþ qpðbn  rÞ; gnn o l, (13)and h  i is the angle average over the directions of p.3. Magnetic impurities and spin currentsFocusing on the Rashba interaction, we study the effectsof magnetic impurities on spin currents. In Refs. [4,5] theproblem has been recently tackled via diagrammatictechniques. We show how analogous results can beobtained in a simple and rather elegant way relying onEq. (11). As it is well known, spin currents arising from thespin Hall effect in such a system are completely suppressedby the presence of non-magnetic scatterers. By taking theangular average of Eq. (11), one obtains a set of continuityequations for the various spin components which let oneeasily understand the origin of this cancellation. Explicitly,by assuming s-wave and non-magnetic impurities randomlyARTICLE IN PRESSC. Gorini et al. / Physica E 40 (2008) 1078–10801080distributed in the systemV1ðxÞ ¼XiUdðx RiÞ, (14)the self-energy in the Born approximation turns out to beS1 ¼ ih gi=2t, 1=t being the momentum scattering rate.The continuity equations for the l ¼ x; y; z spin compo-nents then readqth gli þ qx  Jls ¼ 2hb0 ^ gil . (15)A rather important peculiarity of the Rashba Hamiltonianis that it lets one write the vector product appearing abovein terms of the various spin currents, so that, by choosingfor example l ¼ y, we are left withqth gyi þ qx  Jys ¼ 2ma Jzs;y. (16)Under stationary and homogeneous conditions this impliesthe vanishing of the Jzs;y spin current. As soon as magneticimpurities are introduced in the system, their presencechanges the self-energy and leads to the appearance ofadditional terms in Eq. (16). We assume the magneticscatterers to be also isotropic and randomly distributedV2ðxÞ ¼XiB  rdðx RiÞ, (17)and, proceeding again in the Born approximation, weobtain the self-energyS ¼ S1 þ S2 ¼ i2th gi  i6tsfX3l¼1slh gisl . (18)Here 1=tsf is the spin-ﬂip rate. With this, and byconsidering again stationary and homogeneous conditions,Eq. (16) becomes2maJKzs;y þ43tsfhgKy i ¼ 0, (19)which in terms of the real spin current and polarizationmeansjszy ¼ 23matsfsy. (20)By assuming a low concentration of magnetic impurities,we can use in Eq. (20) the value of the y-spin polarizationvalid in their absence, sy ¼ jejEatN0 [6], E being theexternal, homogeneous electric ﬁeld. We then get the spinHall conductivity to ﬁrst order in t=tsfssH ¼jej3pttsf, (21)a results that differs from those on Refs. [4,5]. This is notsurprising for Ref. [5], which neglects normal impurityscattering and then considers the opposite limit. The reasonwhy our result does not agree with the low magneticimpurity-concentration limit of Eq. (20) of Ref. [4] is notclear to us and deserves further investigation.AcknowledgementsThis work was supported by the Deutsche Forschungs-gemeinschaft through SFB 484 and SPP 1285 and byCNISM under Progetto d’Innesco 2006.References[1] J. Rammer, H. Smith, Rev. Mod. Phys. 58 (1986) 323.[2] R. Raimondi, C. Gorini, P. Schwab, M. Dzierzawa, Phys. Rev. B 74(2006) 035340.[3] A.I. Shelankov, J. Low Temp. Phys. 60 (1985) 29.[4] J. Inoue, T. Kato, Y. Ishikawa, H. Itoh, G.E.W. Bauer,L.W. Molenkamp, Phys. Rev. Lett. 97 (2006) 046604.[5] P. Wang, Y. Li, X. Zhao, Phys. Rev. B 75 (2007) 075326.[6] V.M. Edelstein, Solid State Commun. 73 (1990) 233;V.M. Edelstein, J. Phys. Condens. Matter 5 (1993) 2603.

Quasiclassical approach and spin-orbit coupling

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We discuss the quasiclassical Green function method for a two-dimensional electron gas in the presence of spin–orbit coupling, with emphasis on the meaning of the ξ-integration procedure. As an application of our approach, we demonstrate how the spin-Hall conductivity, in the presence of spin-flip scattering, can be easily obtained from the spin-density continuity equation

Quasiclassical approach and spin-orbit coupling

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