Magneto-spin Hall conductivity of a two-dimensional electron gas M. Milletar`ı, R. Raimondi CNISM and Dipartimento di Fisica ”E. Amaldi”, Universita` Roma Tre, 00146 Roma, Italy P. Schwab Institut fu¨r Physik, Universita¨t Augsburg, 86135 Augsburg, Germany (Dated: February 3, 2008) It is shown that the interplay of long-range disorder and in-plane magnetic field gives rise to an out-of-planespinpolarizationandafinitespinHallconductivityofthetwo-dimensionalelectrongas inthepresenceofRashbaspin-orbitcoupling. Akeyaspectisprovidedbytheelectric-fieldinduced 8 in-plane spin polarization. Our results are obtained first in the clean limit where the spin-orbit 0 splitting is much larger than the disorder broadening of the energy levels via the diagrammatic 0 evaluationoftheKubo-formula. Thentheresultsareshowntoholdinthefullrangeofthedisorder 2 parameter αp τ by means of the quasiclassical Green function technique. F n a J It is well established that the peculiar linear-in- tually yields out-of-plane spin polarization and spin Hall 1 momentum dependence of the Rashba (and of Dres- effect. In contrast to Ref.[9] we do not have to assume 1 selhaus) spin-orbit coupling leads in a two-dimensional a non-parabolicity of the energy bands or an energy de- electron gas (2DEG) to the vanishing of the spin Hall pendence of the scattering probability. ] l conductivity[1, 2, 3, 4]. This can be directly recognized Our analysis is carried out in two steps. In the first l a by considering the continuity-like equation for the in- step,wecalculatetheout-of-planespinpolarizationusing h plane spin polarization, where the spin-nonconserving the diagrammatic approach of Ref.[3], valid in the clean - s terms can be written as the spin current associated to limit when the spin-orbit splitting is much larger than e the out-of-plane spin polarization and to the spin Hall the disorder-induced broadening, 2αp (cid:29)τ−1. In order m F effect[5, 6, 7]. In this paper, we show that the inter- tomakecontactwiththeanalysisofRef.[9]performedin at. play of an in-plane magnetic field, B, taken parallel to the opposite dirty limit (αpF (cid:28)τ−1), we present, in the the electric field, E, (say along the eˆ axis) and long- second step, a derivation based on the Eilenberger equa- m x range disorder changes this behavior providing then a tion for the quasiclassical Green function in the presence - d potential handle on the spin Hall effect. In particular, of spin-orbit coupling[11]. The advantage of so doing is n we show that while the out-of-plane spin polarization is thattheanalysisisvalidforanarbitraryvalueofthepa- o linear in the magnetic field, the spin Hall conductivity rameter2αp τ andalsoallowstodeterminetheeffective F c is quadratic. Our analysis is valid in the standard good Bloch equations for the spin density. [ metallic regime (cid:15) τ/(cid:126) (cid:29) 1 with spin-orbit effects taken The Hamiltonian of a 2DEG perpendicular to the eˆ - F z 1 into account to first order in α/v . Here α and τ are axis reads F v the spin-orbit coupling and the elastic quasiparticle life- 6 p2 time due to impurity scattering, respectively, while v , H = +b(p)·σ+V(x) (2) 8 F 2m 7 pF and (cid:15)F =vFpF/2 are the parameters of the 2DEG in 1 the absence of spin-orbit coupling. where b(p) = αp×eˆ −ω eˆ is the effective magnetic z s x 1. Our proposal of a magnetic field-induced spin Hall ef- field including both the Rashba spin-orbit coupling and 0 fect differs from related previous suggestions both for theexternalmagneticfield. InEq.(2),V(x)describesthe 8 the analytical treatment of it[8] and for the microscopic potentialscatteringfromtheimpuritiesand,accordingto 0 mechanism responsible of the effect[9]. The difference the established procedure in the literature, will be taken : v of our proposal with respect to Ref.[9] is closely related as a random variable. i to the electric field-induced in-plane spin polarization, Thepossibilityofanon-vanishingspinHallconductiv- X which for short-range disorder scattering is given by [10] ityinthepresenceofanin-planefieldmaybeappreciated r a byconsideringtheequationofmotionfortheeˆ -axis(in- y s =−N α|e|E τ. (1) y 0 x plane) spin polarization, which yields InEq.(1)N =m/(2π)isthefreedensityofstatesofthe 0 ∂s ∂ 2DEG in the absence of spin-orbit interaction. Contrary y + ·jy =−2mαjz +2ω s , (3) ∂t ∂x s s,y s z towhatonecouldexpect,thegeneralizationofEq.(1)for long-range disorder, as we will show later, is not the re- where ji is the eˆ -axis component of the spin current s,γ γ placement of the elastic quasiparticle lifetime τ with the thatispolarizedalongtheeˆ -axis. Understationaryand i transporttime,τ ,asithasbeenassumedinRef.[9]. We uniform conditions, the above equation implies, in the tr find then that long-range disorder leads to a non-trivial absence of the magnetic field, a vanishing spin current modification of the effective Bloch equations and even- and hence a vanishing spin Hall conductivity, with the 2 corresponding to the eigenvalues E =p2/2m±αp with ± tan(ϕ) = p /p . In terms of the transformation matrix, y x U, defined by Eq.(8), the Pauli matrices transform as Uσ U† = pˆ σ +pˆ σ x y z x y pˆ ≡ cos(ϕ) Uσ U† = −pˆ σ +pˆ σ , x . (9) y x z y y pˆ ≡ sin(ϕ) Uσ U† = −σ y z x FIG. 1: Diagrams to be evaluated in zeroth and first order As a consequence the spin density vertex, the magnetic in the magnetic field. The dashed line ending with a cross indicatesthemagneticfieldinsertion. Underimpurityaverage field insertion and the charge current vertex become the (spin density and charge current) vertices are dressed by 1 the standard ladder resummation. Us U† = − σ , (10) z 2 x U(−ω σ )U† = −ω (pˆ σ +pˆ σ ), (11) s x s y z x y (cid:16) p (cid:17) latter defined by jsz,y = σs,HEx. In the presence of an UjcxU† = mσ0+ασz pˆx−σypˆy. (12) in-plane magnetic field one may have a bulk spin Hall current determined by Uponimpurityaveragingthespinandchargeverticesget ω renormalized. In terms of the renormalized quantities, jz = s s . (4) s,y αm z Eq.(5) becomes to first order in the Zeeman field Innettihcefifeoldllowwhinicghwime pevliaelsuathteatszoftojzfirsttoosredceorndinotrhdeerm.ag- s =−iω |e|Ex µ(cid:88)=±pˆ µ(Γ GR−Γ GA)GRGAJ , s,y z s 4π x µµ¯ µ¯ µ¯µ µ¯ µ µ c,µµ As anticipated, we begin by sketching the calculation p performed with the diagrammatic approach. To linear (13) order in the electric field, the Kubo formula for the zero- where the first (second) term in the brakets refers to the temperatureexpressionoftheout-of-planespinpolariza- magnetic field insertion in the top (bottom) Green func- tion reads tionlineandµ=±labelstheeigenstates. Thequantities Γ and J are the dressed vertices corresponding to 1 (cid:88) (cid:104) (cid:105) µµ(cid:48) c,µµ(cid:48) sz =−2π Trσ szGR(p)jcxGA(p) |e|Ex, (5) sz andjcx,respectively,andtheGreenfunctionsareeval- p uated via the self-energy given in Eq.(6). Apart from the spin vertex Γ , all the other quantities have been where the Green functions can be obtained from Eq.(2), µµ(cid:48) evaluated in Ref.[3], where it has been shown that the the vertices are s =(1/2)σ , jx =(p /m)σ −ασ , and z z c x 0 y off-diagonal matrix element J vanishes and the self- the trace is over the associated spin indices. In Eq.(5), c,µµ¯ energy is diagonal in the eigenstate basis with thebarindicatestheaverageovertheimpuritypotential, which,attheleveloftheself-consistentBornapproxima- (cid:18) (cid:19) −(+)i V α 1 tion, yields the self-energy ΣR±(A) = 2τ , τ± =τ 1± V1v ,τ =2πN0V0. ± 0 F ΣR,A(p)=(cid:88)|V(p−p(cid:48))|2GR,A(p(cid:48)), (6) (14) The spin vertex obeys the equation p(cid:48) |V(q)|2 being the Fourier transform of V(x)V(x(cid:48)). In ν(cid:88)=±µ 1+µνcos(ϕ−ϕ(cid:48)) Γ =1+ Γ GRGA|V|2 , (15) order to consider the effect of long-range disorder, we µµ¯ νν¯ ν¯ ν 2 p(cid:48) expand the above scattering probability as which yields |V|2 =V +2V cos(ϕ−ϕ(cid:48))+2V cos(2ϕ−2ϕ(cid:48))+··· , (7) 0 1 2 iµ V whereϕ−ϕ(cid:48)istheanglebetweenthetwomomentapand Γ =1+ 1. (16) p(cid:48). The harmonics, V , V , V are functions of |p| and µµ¯ 2αpFτ V0 0 1 2 |p(cid:48)|. In the following we will ignore this dependence[12] By using GRGA = iτ (GR −GA), integrating over the and take both momenta at p . To first order in the µ µ µ µ µ F energy, ξ = p2/2m−µ, and keeping terms up to order magneticfieldthediagramstobeevaluatedareshownin α/v , Eq.(13) becomes Fig.1. Notice that all the vertices and propagators ap- F pneetaircinfigelidn.tIhteisdtiahgernamcosnvmeunsietnbte, feovllaolwuaintegdRaetf.z[3e]r,otomuasge- sz =−|e|Ex8αω2sτ (cid:18)1− VV1(cid:19) (cid:88) µNµpJ2µτµ, (17) the disorder-free Hamiltonian eigenstates 0 µ=± µ 1 where N = N (1 ∓ α/v ), p = p (1 ∓ α/v ) are |p±(cid:105)= √ (±iexp(−iϕ)|p↑(cid:105)+|p↓(cid:105)) (8) ± 0 F ± F F 2 the density of states and the Fermi momentum of the 3 two spin subbands and J (p ) = J pˆ are the Fermi- is made by integrating over the energy (cid:15), which is the ±± ± ± x surface expressions of the charge vertices. Finally, bor- Fourierconjugatedvariableofthetimedifferencet −t . 1 2 rowing from Ref.[3] the expression for the vertices Forinstance,theout-of-planespindensityisgivenbythe angular average of the Keldysh component[13] (cid:18) (cid:19) V α V +V J± =vF V0−0V1 ∓ vF V00−V22 , (18) s =seq− N0 (cid:90) d(cid:15)(cid:104)Tr(σ g)(cid:105), (cid:104)...(cid:105)≡(cid:90) 2π dφ... . (24) z z 8 z 2π one gets the out-of-plane spin polarization 0 Inordertosolve,tolinearorderintheelectricfield,the 1 ω N V −V s =− |e|E s 0 1 2 (19) Keldysh component of the Eilenberger equation (21), we z 2 xαp p V −V F F 0 2 usetheminimalsubstitution∂ g →−|e|E eˆ ∂ g where x x x (cid:15) eq g =tanh((cid:15)/2T)(gR−gA)withgR =−gA =1−∂ b ·σ and, via Eq.(4), the spin Hall conductivity eq eq eq eq eq ξ 0 is the equilibrium quasiclassical Green function. As in |e| (cid:18) ω (cid:19)2 V −V the diagrammatic treatment previously developed, we σ =− s 1 2. (20) sH 4π αp V −V finditconvenienttotransformtheequationstotheeigen- F 0 2 statebasisviaEq.(9). Afterexpressingthequasiclassical Remarkably the above central result shows that the sign Green function as a four-dimensional column vector of the spin Hall conductivity depends on the relative strength of the harmonics of the scattering probability. g˜=UgU† =g˜0σ0+g˜·σ →(cid:0)g˜0 g˜3 g˜1 g˜2 (cid:1)t, (25) Thismayexplainthesignchangeinthenumericalevalua- the Eilenberger equation (21) can be then written as a tionofRef.[8]. Ontheotherhand,Eq.(19)isinconsistent linear system of four equations for the components of g˜ with Ref.[9], where for the disorder model of Eq.(7) one would expect a vanishing out-of-plane polarization. 1 1 ∂ g˜=− (M +M )g˜+ (1+N)(cid:104)Kg˜(cid:105)+S +S , (26) Recall that we derived Eqs.(19) and (20) in the clean t τ 0 1 τ 0 1 limit. We now rederive the above results by means of where, the kinetic equations approach of Ref.[11] and find that   Eqs.(19) and (20) are valid for all values of the disorder 0 1 0 0 parameterαpFτ. Furthermorewewillderivetheeffective N = − α 1 0 0 0 (27) Bloch equations in the dirty limit. We start with the vF 0 0 0 0 Eilenberger equation 0 0 0 0   ∂tgˇ = −21µ(cid:88)=±(cid:110)pmµ + ∂∂p(bµ·σ),∂∂xgˇµ(cid:27) M0 = 1+ VV01N +2αpFτ000 000 000 001 (28) − i (cid:88)[b ·σ,gˇ ]−i(cid:2)Σˇ,gˇ(cid:3) (21) 0 0 −1 0 µ µ   0 0 0 0 µ=± for the quasiclassical Green function (gˇ≡gˇt1t2(pˆ;x)) M1 = 2ωsτ vαF0pˆx pˆ0x −0pˆx −0pˆy . (29) i (cid:90) (cid:18)GR G (cid:19) 0 0 pˆy 0 gˇ= dξGˇ (p,x), Gˇ = (22) π t1t2 0 GA In Eq.(26), (cid:104)...(cid:105) denotes angle integration over ϕ(cid:48) with thescatteringkernelthatcanbeexpandendintoangular where Gˇ (p,x) is the Wigner representation of the t1t2 harmonics as Green function, which has both matrix structure in the Keldysh (denoted by the check symbol) and spin spaces. K(ϕ−ϕ(cid:48))=K(0)+cos(ϕ−ϕ(cid:48))K(a)+sin(ϕ−ϕ(cid:48))K(b)+··· [,] and {,} indicate commutator and anticommutator. (30) As for the diagrammatic approach, the index µ = ± la- each coefficient being itself a matrix bels the two spin subbands (b = b(p )). In integra- tionslikeinEq.(22)thecorrespo±ndingpo±lesintheGreen 1 0 0 0  0 0 0 0 functions yield the two-component decomposition of the K(0) =0 VV01 0 0 ,K(b) = V0−V2 0 0 0 1 quasiclassical Green function 0 0 1 0  V0 0 0 0 0 0 0 0 V1 0 −1 0 0 gˇ = 1(cid:110)1(1±bˆ ·σ),gˇ(cid:111). (23) V0 ± 2 2 0 and   The”0”subscriptdenotesevaluationattheFermisurface 2V 0 0 0 1 intheabsenceofspin-orbitcoupling. Inthefollowingwe K(a) = 1  0 V0+V2 0 0 . are going to use Eq.(21) to first order in the parame- V0  0 0 2V1 0  ter |b |/(cid:15) . The connection to the physical observables 0 0 0 V +V 0 F 0 2 4 Finally the source electric-field dependent terms are At last we study the combined effect of magnetic and electric field in the diffusive regime, ω τ,αp τ (cid:28) 1.     s F pˆ 0 x The effective Bloch equations for the spin density are S0 =E−−vvαα0FFppˆˆxy ,S1 = vFωpsFEpˆpxˆ02xpˆy , ∂∂ttssxy == −−ττss−−11((cid:2)ssxy−+ssex0qV)0/(V0−V2)−1(cid:3)+2ωssz((3376)) with E = |e|E v ∂ (2tanh((cid:15)/2T)). Notice that, consis- ∂ s = −2τ−1s −2ω [s +s τ /τ], (38) x F (cid:15) t z s z s y 0 tr tently with the accuracy we are working, one may use for the charge density component the solution obtained where τ−1 = 2(αp )2τ and seq = N ω . Eqs.(36) and s F tr x 0 s in the absence of both spin-orbit coupling and magnetic (38) agree with what was found in Ref.[9], the only dif- field ference is in the term proportional to the electric field (i.e. s ) in Eq.(37). The stationary spin polarization as 1 V 0 jc,x ∼(cid:104)pˆxg˜0(0)(cid:105)= 2V −0V τE, (31) a function of magnetic field is now determined as 0 1 where the characteristic transport time renormalization s = − V0s0 1+2ωs2τs2(V0−V2)(V0−V1)−1(39) τ = τV /(V −V ) appears. We seek now a stationary y V −V 1+2ω2τ2 tr 0 0 1 0 2 s s solutionofEq.(26)whichisevaluatedinfirstorderinthe V s V −V ω τ magnetic field g˜=g˜(0)+g˜(1)+.... We then get sz = −V 0−0V V1−V21+2sωs2τ2 (40) 0 2 0 1 s s (cid:104)g˜(1)(cid:105)=(M −K(0))−1(τ(cid:104)S (cid:105)−(cid:104)M g˜(0)(cid:105)). (32) 0 1 1 showing an out-out-plane contribution as observed ex- perimentally in Ref.[14]. According to the transformation of Eq.(10), the out-of- plane spin polarization is related to In conclusion, we have shown that the combined effect of an in-plane magnetic field, long-range disorder and (cid:18) (cid:19) s ∼(cid:104)g˜(1)(cid:105)=− ωs 1 E+ 1 V0−V1(cid:104)pˆ g˜(0)(cid:105) , spin-orbit coupling gives rise to an out-of-plane spin po- z 1 2αp v p αp τ V x 3 larization and finite spin Hall conductivity, whose value F F F F 0 (33) does not depend on the concentration of defects as long which is expressed in terms of (cid:104)pˆxg˜3(cid:105) evaluated at zero as the 2DEG is in the metallic regime. To obtain the magneticfield. Thislatterquantityisnothingbutthein- correct value of the electric-field induced in-plane spin planespinpolarization(cf. thesecondlineofEq.(9)). By polarizationitisessentialtotakeintoaccountthediffer- multiplyingthesecondcomponentofthesystemEq.(26) ent transport times in the two spin-orbit splitted bands. by pˆx and performing the angle average, one obtains the We acknowledge financial support by the Deutsche generalization, for long-range disorder, of the Edelstein ForschungsgemeinschaftthroughSFB484andSPP1285 result[10] andbyCNISMunderProgettiInnesco2006. R.R.thanks thekindhospitalityoftheICTS,JacobsUniversity, Bre- α V sy ∼(cid:104)pˆxg˜3(0)(cid:105)=−v V −0V τE. (34) men, where this work was initiated. F 0 2 Finally, by using Eqs.(33,34) into Eq.(24) one recovers the result (19) of the diagrammatic approach, which is nowmanifestlyvalidforanystrengthofthedisorder. To [1] J.I.Inoue,G.E.W.Bauer,andL.W.Molenkamp,Phys. understand the meaning of Eq.(34), it is useful to recall Rev. B 70, 041303(R) (2004). the origin of the in-plane polarization[10]: In the pres- [2] E. G. Mishchenko, A. V. Shytov, and B. I. Halperin, ence of an electric field the Fermi surface is shifted by Phys. Rev. Lett. 93, 226602 (2004). δpx ∼ |e|Exτ. As a result the total spin of the electrons [3] R. Raimondi and P. Schwab, Phys. Rev. B 71, 033311 neitherintheplusnorintheminusbandaddsuptozero. (2005). Althoughthecontributionsofbothbandstendtocancel, [4] A. Khaetskii, Phys. Rev. Lett. 96, 056602 (2006). a finite spin polarization remains due to the α/v cor- [5] E. I. Rashba, Phys. Rev. B 70, 201309(R) (2004). F [6] O. V. Dimitrova, Phys. Rev. B 71, 245327 (2005). rectionsinthedensityofstates. Forlong-rangedisorder, [7] O.ChalaevandD.Loss,Phys.Rev.B71,245318(2005). the Fermi surface shift is proportional to the transport [8] Q. Lin, S. Y. Liu, and X. L. Lei, Appl. Phys. Lett. 88, time τ , so one might expect the transport time also tr 122105 (2006). in the in-plane spin polarization. However due to the [9] H. A. Engel, E. I. Rashba, and B. Halperin, Phys. Rev. α/v corrections each band has its own effective trans- Lett. 98, 036602 (2007). F port time, τ ≡J τ /v and the explicit result reads [10] V. M. Edelstein, Solid State Commun. 73, 233 (1990). tr,± ± ± F (s =αN |e|E τ) [11] R. Raimondi, C. Gorini, P. Schwab, and M. Dzierzawa, 0 0 x Phys. Rev. B 74, 035340 (2006). v V [12] Hence,weneglecttheeffectsdescribed,inRef.[9],bythe s = F(N τ −N τ )|e|E =− 0 s . (35) y 4 + tr,+ − tr,− x V −V 0 scattering kernel Ke. 0 2 5 [13] Within the quasiclassical formalism, g describes the dy- namic part only to which add the equilibrium part. [14] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Phys. Rev. Lett. 93, 176601 (2004).