Nonvanishing spin Hall currents in the presence of magnetic impurities

0.16 MB·English

by  Pei Wang

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 cou-
pling 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 uni-
versal 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. SUMMARY AND DISCUSSION Acknowledgments
Using Kubo’s linear-response theory, we have evalu- This work was supported by the NSFC No. 10225419,
ated,inthe ladderapproximation,thevertexcorrections 10674116,10674117,60471052;andthe ZhejiangProvin-
of magnetic impurities on the spin Hall conductivity in cial Natural Foundation M603193.
[1] S. Murakami, N. Nagaosa, S. C. Zhang, Science, 301, wirth, A. H. MacDonald, Phys. Rev. Lett., 92, 126603
1348 (2003). (2004).
[2] J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jung- [3] Y. K. Kato, R. C. Myers, A. C. Gossard, D. D.
6
Awschalom, Science, 306, 1910 (2004). [14] J.Schliemann,D.Loss,Phys.Rev.B,69,165315(2004).
[4] J. Wunderlich, B. Kaestner, J. Sinova, T. Jungwirth, [15] A.A.Burkov,A.S.Nunez,A.H.MacDonald,Phys.Rev.
Phys.Rev.Lett., 94, 047204 (2005). B, 70, 155308, (2004).
[5] H. A. Engel, B. I. Halperin, E. I. Rashba, Phys. Rev. [16] K. Nomura, J. Sinova, T. Jungwirth, Q. Niu, and A. H.
Lett., 95, 166605 (2005). MacDonald, Phys. Rev.B, 71, 041304(R) (2005).
[6] O.Chalaev, D. Loss, Phys.Rev.B, 71, 245318 (2005). [17] C. Grimaldi, E. Cappelluti, F. Marsiglio, Phys. Rev. B,
[7] W. K. Tse, S. DasSarma, Phys. Rev. Lett., 96, 056601 73, 081303(R) (2006).
(2006). [18] K.Nomura,J.Sinova,N.A.Sinitsyn,A.H.MacDonald,
[8] S.Zhang, Phys. Rev.Lett., 85, 393 (2000). Phys. Rev.B, 72, 165316 (2005).
[9] E. G. Mishchenko, A. V. Shytov, B. I. Halperin, Phys. [19] P. L. Krotkov, S. DasSarma, Phys. Rev. B, 73, 195307
Rev.Lett., 93, 226602 (2004). (2006).
[10] J. I. Inoue, G. E. W. Bauer, L. W. Molenkamp, Phys. [20] J. I. Inoue, T. Kato, Y. Ishikawa, H. Itoh, G. E. W.
Rev.B, 67, 033104 (2003). Bauer, L. W. Molenkamp, Phys. Rev. Lett, 97, 046604
[11] J. I. Inoue, G. E. W. Bauer, L. W. Molenkamp, Phys. (2006).
Rev.B, 70, 041303(R) (2004). [21] P. W. Anderson, J. Phys. C 3, 2436 (1970). K. Yosida,
[12] R. Raimondi, P. Schwab, Phys. Rev. B, 71, 033311 Theory of Magnetism (Springer, New York1996).
(2005).
[13] O.V. Dimitrova, Phys.Rev. B, 71, 245327 (2005).