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Nonvanishing spin Hall currents in the presence of magnetic impurities Pei Wang, You-Quan Li, and Xuean Zhao Zhejiang Institute of Modern Physics and Department of Physics, Zhejiang University, Hangzhou 310027, P. R. China 7 (Dated: February 6, 2008) 0 0 The intrinsic spin Hall conductivity in a two dimensional electron gas with Rashba spin-orbit 2 coupling is evaluated by taking account of anisotropic coupling between magnetic impurities and itinerantelectrons. Inourcalculation Kubo’slinearresponse formalism isemployedandthevertex n correction is considered. In the semiclassical limit µ ≫ 1/τ, a non-vanishing spin Hall conductiv- a ity σ is found to depend on the momentum relaxation time τ, spin-orbit splitting ∆ and the J sH anisotropiccoefficientofinteractionbetweenitinerantelectronsandmagneticimpurities. Theclean 4 limit of σ is in theregion of e/8π∼e/6π, dependingon the anisotropic coefficient. 1 sH PACSnumbers: 72.25.Dc,72.25.-b,72.25.Rb ] l l a h I. INTRODUCTION the external electric field modifies the SO-induced pseu- - domagnetic field such that the spins are tilted out of the s e RecentlythespinHalleffect(SHE)[1,2,3,4,5,6,7,8] 2DEG plane in directions that are opposite for positive m hasattractedmuchattentionduetoitspotentialapplica- andnegativelateralmomentumstates. Thiscorresponds to a flow of σ = 1/2 and σ = 1/2 spins in oppo- t. tioninspintronics. In the spinHalleffect, a longitudinal site directionswz ithout acorreszpond−ingnetchargetrans- a electricfieldcreatesatransversemotionofspinswiththe m spin-upandspin-downcarriersmovinginoppositedirec- port. Inthepresenceofisotropicimpurityscattering,the spinHallcurrentisproportionaltothetimederivativeof - tions, which leads to a transverse spin current perpen- d the spin polarization [13] which vanishes in a stationary dicular to the external electric field. To understand the n state. Whereas, this relation does not fulfil for magnetic spinHalleffect,oneneedstostudytheintrinsicspinHall o impurities, leaving over an opportunity for nonvanishing effect (ISHE) that has been discussed intensively. Theo- c spin Hall conductivity. As we are aware, the case con- [ retically ISHE may existin the p-type semiconductor [1] sidered by Inoue has not been developed to anisotropic and two-dimensional electron gas (2DEG) [2]. Sinova 2 model in the semiclassical limit µ 1/τ. And Inoue et al. [2] predicted a universal spin Hall conductivity in v ≫ et al. adopted a simplified approximation in calculating clean 2DEG as 6 the integral of Green’s functions. It is therefore obliged 7 e 7 σsH = . to study suchkindmodels andinvestigatespin Hallcon- 0 8π ductivity carefully. 1 In this paper we calculate the spin Hall conductivity Fromthenon,manyresearchershavedevotedtotheissue 6 for a two-dimensional magnetically disordered Rashba- of whether this result can be modified in the presence of 0 electrongaswherethemagneticinteractionisanisotropic / impurity. Some authors found that an arbitrarily small t in the limit of large Fermi circle µ 1/τ,∆. The XXZ- a concentrationof impurities would suppress the spin Hall ≫ type interaction between the magnetic impurity and the m conductivity to zero due to the vertex corrections [9, 10, electronspinisadopted. Ourcalculationiscarriedoutby 11,12,13]. Whileothers[14,15,16]arguedthatthespin - consideringthevertexcorrectionswiththehelpofKubo’s d Hall conductivity was robustin the presence of disorder, n fallingtozeroonlywhenthelifetimebroadeningislarger linearresponseformalismwithintheself-consistentBorn o approximation. The paper is organized as follows: In than the spin orbit splitting of the bands, i.e., 1/τ >∆. c Sec. II, we introduce the model by taking into account Grimaldi [17] considered the situation of sufficiently low v: electron density, and found that the vertex corrections of magnetic impurities. In Sec. III, the calculation pro- i cedure of the spin Hall conductivity is presented. In X nolongersuppressedthespinHallconductivitywhenthe Sec.IV,wegiveacomparisonbetweenourresultandthe Fermi energy is comparable to or smaller than the spin- r resultinotherdisorderedsystemanddiscussthepossible a orbit energy. Other authors revealed that the vanishing reasons that cause the vanishing spin Hall conductivity. ofσ wasapeculiarfeatureofthelinearRashbamodel. sH Takingintoaccountofnonlinearmomentumdependence of the spin-orbit interaction α(p) [18] or a nonquadratic bandspectrumε(p)[19],σ wasrobustagainstimpurity II. THE MODEL WITH MAGNETIC IMPURITY sH scattering. Very recently, an important step was made by Inoue We consider the 2DEG with Rashba spin-orbit cou- etal. [20]who foundthe spinHallconductivity wasnot pling in the presence of impurities, whose Hamiltonian zerointheexistenceofmagneticimpuritiesinthelimitof consists of two parts H = H +V with H the sum 0 dis 0 ∆τ 1. Physically, the acceleration of the electrons by of the kinetic and Rashba terms, and V the potential dis ≫ 2 caused by impurities. The main contribution of a short- ranged magnetic impurity can be described by an inter- actionbetweenanitinerantelectronandalocalmagnetic moment[21]whoseorientationisdefinedbypolarandaz- imuthal angles (θ,φ) (see Fig. 1). A quite general form of such kind interaction is of XXZ type whose second quantization form is given by the following Hamiltonian N V = dr2uδ(r R ) (1) dis i Z − Xi=1 FIG.1: Blochsphererepresentationofanmagneticimpurity. γcosθ sinθ e−iφi ψˆ†(r) i i ψˆ(r), (cid:18)sinθieiφi γcosθi (cid:19) − spin current is defined as where (R ,θ ,φ ) denote the position and orientation of i i i theithimpurity,N thetotalnumberofimpurities;uand 1 Jz(p)= (v σ +σ v ) γ refer to, respectively, the strength and the anisotropy y 4 y z z y of the coupling between the itinerant electrons and the 1dε(p) = σ . (6) impurities. 2 dp z y ψˆ(r)= ψˆ↑(r) . (2) On the basis of Kubo’s formalism, the spin Hall conduc- (cid:18)ψˆ (r) (cid:19) tivity can be expressed as [13] ↓ The interaction becomes isotropic for the special value e dω σ (ω)= 1Tr n (ω+ω )Jz γ =1 which corresponds to the model considered by In- sH ωV Z 2π n F 1 y oue [20]. We assume that the distribution of impurities Gr(ω+ω ) Ga(ω+ω ) j Ga(ω ) is homogeneous and their orientations are isotropic, ac- 1 − 1 x 1 (cid:2) (cid:3) cordingly, + n (ω )JzGr(ω+ω )j Gr(ω ) Ga(ω ) , (7) F 1 y 1 x(cid:2) 1 − 1 (cid:3)o P(R θ φ ,R θ φ , ,R θ φ ) 1 1 1 2 2 2 ··· N N N where n (ω) = 1/(eβ(ω−µ) +1) is the Fermi function, 1 F =( )NdR dΩ dR dΩ dR dΩ , (3) d 4πV 1 1 2 2··· N N jx(p)= [ε(p)+α(p)(σxpy σypx)]istheone-particle dp − x where dΩ = sinθ dθ dφ and V denotes the area of chargecurrentoperator,Gr andGa aretheretardedand i i i i the 2DEG. Note that the model given by Eq. (1) for advancedGreen’s functions. The trace is takenovermo- magnetic impurity is not reducible to that for nonmag- mentum and spin indices. The diagrammatic method is netic impurity since the matrix between ψˆ†(r) and ψˆ(r) employedto calculatethe averagespinHallconductivity for the former is a traceless matrix while that for the over the distribution of impurities. The trace in Eq.(7) latter is an unit matrix. For a general discussion, the is expanded into the sum of diagrams (see Fig. 2). strength of Rashba coupling is momentum-dependent i.e., H = ε(p)+α(p)(σ p σ p ). This Hamiltonian 0 x y y x H can be diagonalized to be− A. Average Green’s functions 0 ε (p)=ε(p) pα(p), (4) ± ∓ Now we calculate the averageGreen’s functions in the by the unitary matrix self-consistent Born approximation. For simplicity, the Fermienergyµissettobe the zeropointofenergysince 1 1 1 it can be combined in the dispersion relation. Then the U(p)= , (5) √2(cid:18)ieiϕp ieiϕp (cid:19) bottom of conduction band goes to negative infinite in − the large Fermi circle limit. The free Green’s function in where ϕ denotes the angle between the momentum p p chiral bases can be expressed as and the x axis. The momentum-dependent Rashba coupling hasimportanteffects onthe spin-Hallconductivity 1 0 innotthreelperveasnetncteootfhneofinnmaalgrneseutilctsicnatttheereprrse[s1e8n]t,wcahsielewititihs Gr0(ch)(p,ω)= ω−ε+(p)+iη 1 , 0 magnetic scatterers.  ω ε (p)+iη   − −  (8) III. SPIN HALL CONDUCTIVITY where ε (p) = ε(p) pα(p). The Feynman diagram ± ∓ for the magnetic impurities is shown in Fig. 3. The As is well known that spin angular momentum is not self-consistent Born equation for the averaged retarded conservedin the presence of spin-orbitcoupling,and the Green’s function in chiral bases is given by 3 G¯a G¯a G¯a Tr[JyzGrjxGa]= Jyz t djx + Jyz t pppppppppppppppppppppp djx + Jyz t pppppppppppppppppppp pppppppppppppppppppp djx +··· G¯r G¯r G¯r FIG. 2: The spin Hall conductivity is calculated in terms of diagrams. The first diagram contributes to σ0 , while the others sH contributetotheladdercorrection forspinHallconductivity. G¯r andG¯a denotetheaverageGreen’sfunctionstakingoverthe distribution of impurities. pppppp!appMpppppppppppppp- = Z 41πdθdφsinθ„sγincθoesiθφ s−inγθceo−siθφ «× ppppp!apppppppppppppppp-p ×„sγincθoesiθφ s−inγθceo−siθφ « FIG. 3: The Feynman diagram for the magnetic impurity (left hand side of the diagram equation) is different from that for the nonmagnetic impurity (right hand side). Each dot line connected to the same impurity gives an extra matrix and the integration dθdφsinθ/4π for θ and φ is performed. R sinθNu2 γcosθ sinθe−iφ G¯r (p,ω)=Gr (p,ω) + dθdφ Gr (p,ω)U†(p) U(q) (ch) 0(ch) Z 4π V2 0(ch) (cid:18)sinθeiφ γcosθ (cid:19) Xq − γcosθ sinθe−iφ G¯r (q,ω)U†(q) U(p)G¯r (p,ω), (9) (ch) (cid:18) sinθeiφ γcosθ (cid:19) (ch) − where N denotes the number of impurities. Eq. (9) has The Green’s functions in σ bases are z a solution 1 0 G¯r(ch)(p,ω)= ω−ε+(p)+ 2iτ 1 ,  0   ω−ε−(p)+ 2iτ  (10) G¯r(p,ω)=U(p)G¯r (p,ω)U†(p), ch where τ denotes the momentum-relaxation time and G¯a(p,ω)=U(p)G¯a (p,ω)U†(p). (12) 1/τ = n u22πN (γ2 + 2)/3, n = N/V the impurity ch i F i concentrationandN the density ofstates atFermisur- F face. Similarly, the averaged advanced Green’s function is 1 0 G¯a(ch)(p,ω)= ω−ε+(p)− 2iτ 1 .  0 ω ε (p) i  Then we calculate σs0H, which corresponds to the one-  − − − 2τ  loop diagram in Fig. 2, at zero temperature and zero (11) frequency 4 e dω σ0 = lim 1 Tr θ( ω ω )Jz(p)[G¯r(p,ω+ω ) G¯a(p,ω+ω )]j (p)G¯a(p,ω ) sH ω→0Vω Z 2π Xp n − − 1 y 1 − 1 x 1 + θ( ω )Jz(p)G¯r(p,ω+ω )j (p)[G¯r(p,ω ) G¯a(p,ω )] − 1 y 1 x 1 − 1 o e 1 = (1 ), (13) 8π − 1+(∆τ)2 where ∆ = 2p α(p ) is the spin-orbit splitting at the correction with one advanced and one retarded Green’s F F Fermi surface, and p is the Fermi momentum. functions is considered. The sum of ladder diagrams at F |ε(pF)=µ In deriving the last line of Eq. (13), we took the integral ω =0 gives with respect to ω and the summation with respect to p 1 in the limit of largeFermicircleµ 1/τ,∆by means of e thesamemethodinRef.[13]. Clear≫ly,σ0 isthesameas σsLH = 2−πV Tr[JyzG¯r(p,0)jx(p)G¯a(p,0)], (14) sH Xp that derived by Dimitrova [13] for nonmagnetic impuri- e ties. But it differs from e/8π derived by Inoue [20], who where the sum of the series of vertex corrections to the takes the product of Green functions as a δ-function for Jz(p) is denoted by the matrix Jz (see Fig. 4). simplicity. Suchanapproximationisvalidwhen∆τ 1, y y ≫ but it is inappropriate to take this approximationbefore e taking account of the vertex corrections to the spin Hall conductivity. B. Vertex correction Jyz = t!a!appppppp + at!a!appppppp!a!ppppppppppppp +··· e Furthermore, we calculate the vertex correction σL sH to the spin Hall conductivity, which corresponds to the FIG. 4: The vertex of spin current with vertex corrections. ladderdiagraminFig.2. Vertexcorrectionstotheterms in Eq.(7) with two advanced or two retarded Green’s functions vanish as µ 1/τ,∆ [13]. Only the vertex The self-consistent equation of Jz is ≫ y e Jz = niu2 dθdφ 1 sinθ γcosθ sinθe−iφ G¯a(p,0)(Jz(p)+Jz)G¯r(p,0) γcosθ sinθe−iφ . (15) y V Z 4π (cid:18)sinθeiφ γcosθ (cid:19) y y (cid:18)sinθeiφ γcosθ (cid:19) Xp − − e e In the limit of large Fermi circle µ ∆,1/τ, the sum- and advanced Green functions. ≫ mation over momentum in Eq.(15) can be evaluated by taking integral. As a result, we have A= −3ivF∆τ , 4(γ2+2)[1+(∆τ)2] γ2 2 2 γ2 3 1 1 (Jz) =( B+ C)(Jz) +( B+ C)(Jz) , B = ( + ), y ↑↑ 3 3 y ↑↑ 3 3 y ↓↓ γ2+2 2 2[1+(∆τ)2] e 2 γ2 e γ2 2 e 3 1 1 (Jz) =( B+ C)(Jz) +( B+ C)(Jz) , C = ( ). (17) y ↓↓ 3 3 y ↑↑ 3 3 y ↓↓ γ2+2 2 − 2[1+(∆τ)2] e γ2 γ2 e e (Jz) = A B(Jz) , y ↑↓ − 3 − 3 y ↑↓ From these equations, we find (Jyz)↑↑ =(Jyz)↓↓ and e γ2 γ2 e (Jz) = A B(Jz) , (16) e e y ↓↑ 3 − 3 y ↓↑ (Jz) = (Jz) = ivF∆τ . (18) e e y ↑↓ − y ↓↑ 2+(8/γ2+6)[1+(∆τ)2] where v =dε(p)/dp is the Fermi velocity. A, B, e e F |ε(p)=µ andCaremomentumintegralsoverproductsofretarded The vertex correction to the spin Hall conductivity is 5 evaluated as a Rashba-split2DEG. Itwas assumedthat the magnetic impuritieswereshort-rangedandtheorientationsoftheir σL = −ie∆τ(Jyz)↑↓[1 1 ] local moments were distributed uniformly. Unlike the sH 4πv − 1+(∆τ)2 caseofnonmagneticimpurities,thevertexcorrectionσL Fe sH (∆τ)2 doesnotcancelσs0H,leadingtoanon-vanishingspinHall = σ0 . conductivity(see Eq.(20))whichdepends uponthe spin- 1+(4/γ2+3)[1+(∆τ)2] sH orbitsplitting∆andthemomentumrelaxationtimeτ in (19) thelimitoflargeFermicircleµ ∆,1/τ. Thescattering ≫ changes both the momentum and spin directions from a Summing Eq.(13) and Eq.(19), we obtain the spin Hall magnetic impurity and lead to some correlationbetween conductivity them. Therefore, the average spin tilting cannot cancel the spin current completely. In the dirty limit τ 1/∆, e (∆τ)2 ≪ σsH =σs0H +σsLH = 8π 3γ2+4 . (20) σsH goes to zero. While in the clean limit τ → ∞, the 1+ (∆τ)2 spin Hall conductivity is 4γ2+4 e 4γ2+4 This is our main conclusion. lim σ = , (22) Now we discuss the relation between our work and a τ→∞ sH 8π3γ2+4 recent work on such topics. Inoue et al. [20] considered the vertex correctionsdue to magnetic impurities, which which depends on the coefficient γ. When γ changes correspond to the isotropic case γ = 1 of our model. In from 0 to , the clean limit of σsH changes from e/8π ∞ this special case, our result becomes to e/6π, and is e/7π for the isotropic interaction. This result differs from the universal value e/8π for the ideal (∆τ)2 clean 2DEG and also differs from the result σ = 0 in σ =(1+ )σ0 sH sH 8+7(∆τ)2 sH nonmagnetically disordered system. e (∆τ)2 1 Concerning to the vertex corrections, the spin Hall = (1+ )(1 ). (21) conductivity in nonmagnetically disordered or magnet- 8π 8+7(∆τ)2 − 1+(∆τ)2 ically disordered systems does not go back to the universal value e/8π even if the impurity concentration is This differs from the result obtained by Inoue et al. [20]. arbitrarilysmall. Thisresultcomesfromthe infinite size Intheirpaper,theproductofGreenfunctionsinEq.(13) was taken to be a δ-function so that their σ0 and σL ofsystemandinfinite decoherencelengthLϕ assumedin sH sH 1 calculation. As τ , the correctedvertex (Jz) goes lack the factor (1 ). Such an approxima- →∞ y ↑↓ − 1+(∆τ)2 tozero,butthe summationofGreen’sfunctions diverges e tion does not affect the total spin Hall conductivity in because of lack of a cut-off in momentum space. This the existenceofnonmagneticimpurities becausethe σ0 divergenceleadstoanon-vanishingσL andchangesthe sH sH and σL cancel each other. However, the σ0 and σL total spin Hall conductivity. In real systems, the quan- sH sH sH do not cancel each other in the magnetically disordered tum interference contributing to the vertex corrections 1 only happens inside the decoherence length. So the spin system, and hence the factor (1 ) enters the − 1+(∆τ)2 Hall conductivity depends on the decoherence time τϕ final result. Even in the situation of τ∆ 1, the ap- when the momentum-relaxation time is large enough to ≫ proximation condition of Ref. [20], our σ does be comparable with it τ τ . The σ derived in this sH|τ∆≫1 ∼ ϕ sH not have the same lower order approximation as that workisonlyvalidwhenτ isnottoolarge. Furtherstudies e (∆τ)2 are in progress. (1+ ) in Ref. [20]. 8π 8+7(∆τ)2 IV. 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