Vanishing spin-Hall current in a diffusive Rashba two-dimensional electron system: A quantum Boltzmann equation approach

0.21 MB·English

by  S. Y. Liu

Vanishing spin-Hall current in a diffusive Rashba two-dimensional electron system: A
quantum Boltzmann equation approach
S. Y. Liu,1,2,∗ X. L. Lei,1 and Norman J. M. Horing2
1Department of Physics, Shanghai Jiaotong University, 1954 Huashan Road, Shanghai 200030, China
6 2Department of Physics and Engineering Physics,
0 Stevens Institute of Technology, Hoboken, New Jersey 07030, USA
0 (Dated: February 2, 2008)
2
We present a quantum Boltzmann equation analysis of the spin-Hall effect in a diffusive Rashba

n two-dimensional electron system. Within theframework of theself-consistent Born approximation,
a we consider the roles of disorder-induced quasiclassical relaxation, collisional broadening of the
J
quasiparticles, and the intracollisional field effect in regard to spin-Hall dynamics. We present an

6 analytical proof that the spin-Hall current vanishes, independently of the coupling strength, of the
2 quasiparticlebroadening,oftemperatureandofthespecificformoftheisotropicscatteringpotential.
A sum relation of thecollision terms in a helicity basis is also examined.
]
l
l PACSnumbers: 72.10.-d,72.25.Dc,73.50.Bk
a
h
-
s I. INTRODUCTION
e
m
The spin-Hall effect refers to the appearance of a net polarized spin flow along the direction perpendicular to an
.
t external applied dc electric field. This phenomenon suggests a possible mechanism to control spin dynamics using
a
m an electric field and may be employed to resolve the challenge of spin injection in the emerging field of spintronics.
In early studies, the spin-Hall effect was associated with the disorder-induced spin-orbit (SO) interaction, of an
-
d extrinsic nature.1,2 More recently, another disorder-free spin-Hall effect was predicted, respectively, in bulk p-doped
n semiconductors3 and in Rashba two-dimensional (2D) systems.4 This intrinsic effect essentially arises from disorder-
o independent SO interactions, such as the Rashba SO coupling, the SO interaction involved in the Luttinger model,
c etc. Experimentally, the spin-Hall effect has been observed in a bulk n-type semiconductor5 and in a 2D heavy-hole
[
system.6

2 In Rashba 2D electron systems, the spin-Hall current can be strongly affected by a spin-conserving interaction
v between electrons and impurities. This issue has been investigated in the diffusive regime extensively by various
5 methods, including the Kubo formula,7,8,9,10,11,12 the spin-density matrix method,13 and a nonequilibrium Green’s
6
function approach.14,15,16 Physically, to examine the effect of disorder on spin-Hall current, we must simultaneously
3
take into account collisional broadening and quasiclassical relaxation, as well as the intracollisionalfield effect17,18,19
9
0 produced by the action of the electric field during the course of electron-impurity scattering. The most important
5 disorder effect on spin-Hall current is associated with quasiclassical relaxation. This scattering process yields an
0 additionaltermto the spin-Hallcurrent,causingitto vanish.9,10,11,12,13,14,15 Inaddition,the collision-inducedspread
/ of the quasiparticle density of states (DOS)-namely, collisional broadening-also influences the spin-Hall effect. It
t
a reduces the value of the intrinsic spin-Hall conductivity with increasing broadening of the DOS.7,8 For short-range
m
disorder, the total spin-Hall current still vanishes when collisional broadening in both the intrinsic and quasiclassical
- terms of the spin-Hall current is considered.9,10,11,12 There is yet another disorder-induced quantum effect-namely,
d
the intracollisional field effect-which can not be neglected.17,18,19 During the scattering process, the DOS of the
n
o quasiparticles can be modified by the external electric field, whereupon relaxation becomes dependent on this field
c even in linear response. To date, only Sugimoto et al. have tried to investigate the spin-Hall effect including these
: three processessimultaneously.16 Performinga numericalcalculation of the Keldyshnonequilibrium Green’s function
v
i in a spin basis, they found a generally nonvanishing spin-Hall current: the value of the spin-Hall conductivity they
X
obtained depends on the spin-orbit coupling constant, the transport lifetime, and the Fermi momentum.
r Inthispaper,weanalyticallyinvestigatethespin-HalleffectinRashba2DsystemsusingthenonequilibriumGreen’s
a function method, essentially the same as that used by Sugimoto et al.,16 but in a helicity basis. We carefully analyze
thenonequilibriumretardedandlesserGreen’sfunctionsbysimultaneouslytakingaccountofquasiclassicalrelaxation,
collisional broadening, and the intracollisional field effect. We prove analytically that the spin-Hall current vanishes,
irrespective of the spin-orbit coupling strength, of the form of the isotropic impurity potential, of temperature, and
of the collisional broadening of the DOS. We also discuss a sum relation of the relaxation terms in the helicity basis.
This paper is organized as follows. In Sec. II, we present the kinetic equation for the ”lesser” Green’s function.
The vanishing of spin-Hall current is provenin Sec. III. Finally, we review our results in Sec. IV. Severalappendixes
provide requisite details which could be read before Eq.(3) by readers desiring a ”proof first” exposition.
2
II. KINETIC EQUATION FOR A 2D RASHBA SYSTEM
We consider a 2D Rashba electron system with a single-particle noninteracting Hamiltonian given by20
p2
hˇ (p)= +αp (n σ). (1)
0
2m · ×
Here, p (p ,p ) (pcosφ ,psinφ ) is the 2D electron momentum, m is the effective mass, σ (σ ,σ ,σ ) are
x y p p x y z
≡ ≡ ≡
the Paulimatrices,α is a spin-orbit coupling constant,and n is a unit vectorperpendicular to the 2Delectronplane.
By a local unitary spinor transformation U ,
p
1 1 1
U = , (2)
p √2 ieiφp ieiφp
(cid:18) − (cid:19)
the Hamiltonian (1) can be diagonalized as hˆ (p) = U+hˇ (p)U = diag(ε (p),ε (p)) with ε (p) = p2 +( 1)µαp
0 p 0 p 1 2 µ 2m −
(µ=1,2) as dispersion relations of two spin-orbit-coupledbands. In this paper, we investigate the spin-Hall effect in
a 2D Rashba electron system driven by a constant, uniform electric field E along the x axis. In the Coulomb gauge,
the electric field can be described by a scalar potential, V eE r, with r as the electron coordinate.
≡− ·
In a realistic 2D system, electrons experience scattering by impurities. We assume that the interaction between
electrons and impurities can be characterizedby an isotropic potential, V(p k), corresponding to scattering of an
| − |
electron from state p to state k.
The nonequilibrium Green’s functions Gˇr,< in spin basis for 2D electron systems with Rashba SO coupling are
p
defined as usual. They are2 2 matrices andobey Dysonequationspresentedin Appendix A. For brevity,hereafter,
×
we employ a subscript p to denote the arguments of the Green’s functions and self-energies, (p,ω). It is most
convenientto study the lesserGreen’s function inthe helicity basis,Gˆ< =U+Gˇ<U , inwhich the unperturbed lesser
p p p p
Green’s function is diagonal. Using the transformation from spin basis to helicity basis, the Dyson equation for Gˆ<
p
can be written as
∂ ∂ iα ∂
ieE +p Gˆ<+ieE [Gˆ<, U+U ]+αp[σ ,Gˆ<]+ eE Bˆ< =Iˆ, (3)
· ∂p ∂ω p · p ∇p p p z p 2 · ∂ω p l
(cid:18) (cid:19)
with Aˆ< = [N ,Gˆ<], Bˆ< = N ,Gˆ< , N = U+(n σ)U , and Iˆ as a scattering term. In general, Iˆ has a
p p p p { p p} p p × p l l
complicatedformbecausethemomentumandtimevariablesoftheGreen’sfunctionsandself-energiesareintertwined
due to the presence of the electric field.22 Physically, this feature involves the intracollisional field effect17,18,19 and
results in an additional electric-field-dependent scattering. In this paper, we restrict our considerations to the linear
response regime. Based on this, the scattering term can be expressed as the sum of three terms: Iˆ =Iˆ +Iˆ +Iˆ .
l l1 l2 l3
Thefirstterm,Iˆ doesnotinvolvetheexplicitappearanceoftheelectricfield(althoughitisimplicitinGˆr,< andself-
l1 p
energiesΣˆr,<). ThetermsIˆ andIˆ arisefromthefirst-ordergradientexpansioninvolvingexplicitlineardependence
p l2 l3
on the electric field, and they are a manifestation of the intracollisional field effect. The explicit forms of quantities
Iˆ and Iˆ are similar to Iˇ and Iˇ , respectively, but with helicity-basis Green’s functions and self-energies in place
l1 l2 l1 l2
of the corresponding spin-basis ones (the forms of Iˇ and Iˇ are provided in Appendix A). The term Iˆ arises from
l1 l2 l3
the local nature of the transformation and its form is obtained from Iˆ by using i φ [ ,σ ] instead of operator
l2 2∇p p ··· x
.
p
∇
Fromthe explicit formof the scatteringterm, it may be seenthat Iˆ relatesto the retardedGreen’s function Gˆr in
l p
which the collisional broadening is considered. Detailed properties of Gˆr are analyzed in Appendix B. We find that
p
the linear electric field correction to the retarded Green’s function Gˆr in a 2D Rashba system does not vanish in
1p
general (although it has vanishing diagonal elements). Also, we make clear that disorder scattering is not essential
for the nonvanishing of Gˆr : Gˆr exists even when the electron-impurity collisions are ignored.
1p 1p
It is well known that the linear electric field correction to the retarded Green’s function vanishes for a one-band
electrongas.21However,fortwo-bandsystems,ifatransitionbetweenthetwobandsispresent,Gˆr nolongervanishes
1p
(see, for example, Ref.22 p. 215). In the 2D Rashba system that we study, there are two spin-orbit-coupled bands
arising from the structure asymmetry and a polarization process can be induced between these two bands by a dc
field.23 Accordingly, the linear dc electric field correction to Gˆr in a 2D Rashba electron system is nonvanishing.
p
InEq.(3), the electron-impurityscatteringis embedded inthe self-energiesΣˆr,<. Inthe presentpaper, weconsider
p
the disorder collisions only in the self-consistent Born approximation. It is widely accepted that this treatment is
3
sufficiently accurateto analyze transportproperties in the diffusive regime. On this basis, the self-energiesin helicity
basis take the forms
Σˆr,< =n Tˆ(p,k)Gˆr,<Tˆ+(p,k), (4)
p i k
k
X
where Tˆ(p,k)=U+(k)V(p k)U(p) and n is the impurity density.
i
−
Further,weassumethattheapplieddcfieldisweakenoughthatonlythelinearresponseisrequiredtobeconsidered.
After linearizing Eq.(3), we can write the kinetic equation for the linear electric field correctionto the lesser Green’s
function, Gˆ< =Gˆ< Gˆ< , as
1p p − 0p
1
αpCˆ +ieE Gˆ< eE φ Dˆ +R=Iˆ(1), (5)
− 1 ·∇p 0p− 2 ·∇p p 0 l
where the matrices Cˆ and Dˆ are given by
1 0
0 2(Gˆ< )
Cˆ = − 1p 12 , (6)
1 2(Gˆ<1p)21 0 !
0 (Gˆ< ) (Gˆ< )
Dˆ = 0p 11− 0p 22 . (7)
0 (Gˆ<0p)22−(Gˆ<0p)11 0 !
R is a remainder term which can be expressed as a total derivative with respect to ω. Gˆ< is the unperturbed
0p
equilibrium lesser Green’s function. It is a diagonal matrix and simply relates to the retarded Gˆr by the Kubo-
0p
Martin-Schwinger relation24
Gˆ< = 2in (ω)ImGˆr , (8)
0p − F 0p
with n (ω) as the Fermi function.
F
The scattering term Iˆ(1) is the sum of Iˆ(1), Iˆ(1), and Iˆ(1), the linear electric field parts of Iˆ , Iˆ and Iˆ . The
l l1 l2 l3 l1 l2 l3
explicit forms of these quantities are presented in Appendix C. It should be noted that Iˆ(1) and I(1) are diagonal
l2 l3
and off-diagonal matrices, respectively. In deriving Eq.(5), the properties of the retarded Green’s function in the
presence of collisionalbroadening are employed: its linear electric field correctionis an off-diagonalmatrix, while the
unperturbed one is diagonal.
From Eq.(5) we can formally express the off-diagonalelement of Gˆ< as
1p
1 1
(Gˆ< ) = eE φ [(Gˆ< ) (Gˆ< ) ]
1p 12 2αp 2 ·∇p p 0p 11− 0p 22
(cid:26)
(1)
+(I ) R . (9)
l 12− 12
o
(1)
At the same time, the diagonal elements of I take the form,
l
(I(1)) =ieE (Gˆ< ) +R , (10)
l µµ ·∇p 0p µµ µµ
with µ=1,2.
III. VANISHING SPIN-HALL CURRENT
We are interested in a spin current polarized along the z-direction and flowing along the y axis when a dc electric
field E is applied along the x axis-i.e. Jz. In spin basis, it relates to the single-particle spin current operator ˇjz by
y y
dω
Jz = Tr ˇjzIm(Gˇ< ) . (11)
y 2π y 1p
p Z
X (cid:2) (cid:3)
4
Therehasbeensomedebateaboutthedefinitionofthespincurrentˇjz inrecentyears.25,26,27,28 Itseemsreasonableto
y
definethespincurrentfromacontinuityequationrelatingtothespin,asshowninRef.28. Conventionally,thesingle-
particlespincurrentinspinbasisisdefinedasˇjz = 1 v ,σ withthevelocityoperatorv . Zhangetal. suggestedthat
y 4{ y z} y
the spin current operator has an additional term arising from the spin torque density, (ˇjz) = (y/2i)[σ ,hˇ (p)].28
y add z 0
Actually, the contribution fromthis term to spin-Hall currentvanishes in a homogeneous2D Rashba electronsystem
tolinearorderinthe dcfield. This canbeseenfromthe factthatthe additionalspin-Hallcurrentoperator,explicitly
given by (ˇjz) = αyp σ = iα ∂ p σ, is invariant under inversion of momentum p, i.e. p p. However, p
y add · − ∂py · → −
reversal leads to a sign change of the linear electric field correction to the lesser Green’s function: Gˆ< = Gˆ<
1,−p − 1p
[this result can be seen from Eq.(5) by inverting the momentum]. Based on this, it follows from the p integration of
Eq.(11) that the contribution to spin-Hall current arising from the additional spin-current term vanishes. Hence, we
only need to consider the conventional component of the spin-Hall current operator.
Consequently, we write the spin current operator Jz in the simple helicity-basis description as15
y
dωp
Jz = yIm(Gˆ<) . (12)
y 2π m p 12
p Z
X
Substituting Eq.(9) into this equation, we obtain
dω 1 dω 1
Jz = sinφ eE φ Im (Gˆ< ) (Gˆ< ) + sinφ Im(I(1)) . (13)
y 2π 4mα p ·∇p p 0p 11− 0p 22 2π2mα p l 12
Xp Z h i Xp Z
In this, the ω integration over the term R vanishes.
To further simplify the expression for Jz, we analyze the second term on the right-hand side of Eq.(13). In
y
connection with this, we find that there is a sum relation among the elements of I(1) in the self-consistent Born
l
approximation:
dω dωcosφ
sinφ Im(I(1)) = p Re(I(1)) Re(I(1)) . (14)
2π p l 12 2π 2 l 11− l 22
Xp Z Xp Z h i
Furthermore, separate relations also hold for each component of I(1):
l
dω dωcosφ
(1) p (1) (1)
sinφ Im(I ) = Re(I ) Re(I ) , (15)
2π p l1 12 2π 2 l1 11− l1 22
Xp Z Xp Z h i
dω dωcosφ
sinφ Im(I(1)) = p Re(I(1)) Re(I(1)) . (16)
2π p l3 12 2π 2 l2 11− l2 22
Xp Z Xp Z h i
The detailed proof of these equations is presented in Appendix D.
Based on the sum relation Eqs.(14) and (10) for diagonal components of I(1), we obtain
l
dωsinφ
Jz = peE φ Im (Gˆ< ) (Gˆ< )
y 2π 4mα ·∇p p 0p 11− 0p 22
Xp Z h i
dωcosφ
peE Im (Gˆ< ) (Gˆ< )
− 2π 4mα · ∇p 0p 11− 0p 22
Xp Z (cid:16) h i(cid:17)
dωeE cosφ
= ·∇p pIm (Gˆ< ) (Gˆ< )
− 2π 4mα 0p 11− 0p 22
Xp Z h i
dωcosφ
peE Im (Gˆ< ) (Gˆ< ) . (17)
− 2π 4mα · ∇p 0p 11− 0p 22
Xp Z (cid:16) h i(cid:17)
It is apparent from Eq.(17) that the integrand on its right-hand side becomes a total derivative with respect to p.
Consequently, the vanishing of the spin-Hall current is obtained immediately upon the momentum integration. We
notethatthis analyticalresultissignificantlydifferentfromthatofthe numericalanalysisofSugimatoet al.,16 which
5
purportstoexhibitanonvanishingspin-Hallcurrent. Ouranalysisshowsthatthisisnotthecaseevenwiththe”new”
spin current term included.
It is obvious from Eq.(17) that the spin-Hall current vanishes independently of the strength of SO coupling, of
the specific form of the isotropic scattering potential, and of the lattice temperature. Moreover, it is valid even
for arbitrary broadening of the DOS. However, this result holds only in the diffusive regimes-namely, in the regime
l k > 1 (l is diffusion length and k is the Fermi wave vector), because we restricted treatment of the electron-
D F D F
impurity scattering to the self-consistent Born approximation (otherwise, contributions from the maximally crossed
diagrams of electron-impurity scattering would have to be taken into account).
IV. CONCLUSIONS
The spin-Hall effect in 2D diffusive Rashba systems has been investigated by means of a quantum Boltzmann
approachin this paper. In the self-consistentBornapproximation,we haveconsideredalleffects inducedby electron-
impurity scattering-namely, quasiclassical relaxation, collisional broadening, and the intracollisional field effect. We
have proven analytically that the spin-Hall current vanishes, irrespective of the strength of SO coupling, of the
broadening of DOS, and of the form of the isotropic scattering potential. The sum relation of the scattering terms in
the helicity basis has been analyzed.
Our treatment is applicable at nonvanishing temperatures (not restricted to T = 0), and it is not restricted in
anyway to short-range scattering potentials. We have also made it clear that the spin-Hall effect vanishes even when
correctly considering the influence of the electric field in the scattering process (intracollisionalfield effect).
It shouldbe noted that this proofofvanishing of spin-Hallcurrentalsoapplies for a Dresselhauslinear-momentum
SO coupling, since it relates simply to the Rashba SO interaction by a global transformation. Furthermore, we have
also examined the spin-Hall effect when both the Rashba and Dresselhaus SO interactions are active, again finding
zero spin-Hall current.
Nevertheless, we note that our proof of vanishing of the spin-Hall effect has its validity limited to 2D systems with
linear-momentumSOcoupling. Also,this proofrequiresˇh tobe parabolicintheabsenceofSOcoupling. Otherwise,
0
the spin-Hall effect should be examined anew. This is to say that our result does not conflict with the nonzero
experimental spin-Hall observations5,6: the Hamiltonians of bulk and 2D heavy-hole systems in which nonvanishing
spin-Hall currents were observed differ significantly from the Hamiltonian hˇ (p) studied in this paper.
0
Note added. After this work was completed and submitted, an unpublished paper of Krotkov and Das Sarma
reported a nonvanishing spin-Hall effect in 2D Rashba electron systems by considering a nonparabolic effect.29
Acknowledgments
One of the authors (SYL) would like to gratefully acknowledgethe invaluable discussions with Dr. M. W. Wu, Dr.
W. Xu, Dr. W. S. Liu, and Dr. Y. Chen. This work was supported by projects of the National Science Foundation
of China and the Shanghai Municipal Commission of Science and Technology and by the Youth Scientific Research
Startup Founds of SJTU. N.J.M.H. is supported by the Department of Defense through the DURINT program
administered by the U.S. Army Research Office, DAAD Grant No. 19-01-1-0592.
APPENDIX A: NONEQUILIBRIUM GREEN’S FUNCTIONS IN SPIN BASIS
In spin space, the 2 2 nonequilibrium Green’s functions Gˇ(r ,τ ;r ,τ ) obey Dyson equations given by22
1 1 2 2
×
∂ 1
i + ~2 +iα (n σ)+eE r Gˇ(r ,τ ;r ,τ )=
∂τ 2m∇r1 ∇r1 · × · 1 1 1 2 2
(cid:26) 1 (cid:27)
δ(1 2)+ dτ′ dr′Σˇ(r ,τ ;r′,τ′)Gˇ(r′,τ′;r ,τ ), (A1)
1 1 2 2
−
ZC Z
∂ 1
Gˇ(r ,τ ;r ,τ ) i + ~2 iα (n σ)+eE r =
1 1 2 2 − ∂τ 2m∇r2 − ∇r2 · × · 2
(cid:26) 2 (cid:27)
6
δ(1 2)+ dτ′ dr′Gˇ(r ,τ ;r′,τ′)Σˇ(r′,τ′;r ,τ ). (A2)
1 1 2 2
−
ZC Z
In these equations, the electron-impurity interaction is embedded in the self-energies, Σˇ(r′,τ′;r ,τ ).
2 2
To further simplify Eqs.(A1) and(A2), we introduce relative and center-of-massvariables r=r r , τ =τ τ ,
1 2 1 2
− −
R = (r +r )/2, and T = (τ +τ )/2. Following this, we construct gauge-invariant retarded and lesser Green’s
1 2 1 2
functions Gˇr,< in momentum-frequency space following established procedures.22 Under homogeneous and steady-
p
state conditions, these Green’s functions satisfy the following equations:
p2 1 ∂
2 ω Gˇr +iα ip Bˇr+ eE Aˇr =2+Iˇ +Iˇ , (A3)
− 2m p · 2 · ∂ω p r1 r2
(cid:20) (cid:21) (cid:20) (cid:21)
∂ 1 ∂
i eE +p Gˇ<+iα ip Aˇ<+ eE Bˇ< =Iˇ +Iˇ , (A4)
· ∇p ∂ω p · 2 · ∂ω p l1 l2
(cid:20) (cid:18) (cid:19)(cid:21) (cid:20) (cid:21)
where Aˇr,< [n σ,Gˇr,<] and Bˇr,< n σ,Gˇr,< arise from the Rashba spin-orbit interaction term in the
p ≡ × p p ≡ { × p }
Hamiltonian. The first terms Iˇ and Iˇ on the right-hand sides of Eqs.(A3) and (A4), respectively, do not involve
r1 l1
the explicit appearance of the electric field
Iˇ =ΣˇrGˇr +GˇrΣˇr, (A5)
r1 p p p p
Iˇ =ΣˇrGˇ< Gˇ<Σˇa GˇrΣˇ<+Σˇ<Gˇa. (A6)
l1 p p − p p− p p p p
ThetermsIˇ andIˇ arisefromthe first-ordergradientexpansioninvolvingexplicitlineardependenceonthe electric
r2 l2
field,
i ∂ ∂
Iˇ = eE Σˇr Gˇr Σˇr Gˇr
r2 2 · ∇p p∂ω p− ∂ω p∇p p
(cid:18)
∂ ∂
+ Gˇr Σˇr Gˇr Σˇr , (A7)
∇p p∂ω p− ∂ω p∇p p
(cid:19)
i ∂ ∂
Iˇ = eE Σˇr Gˇ< Σˇr Gˇ<
l2 2 · ∇p p∂ω p − ∂ω p∇p p
(cid:18)
∂ ∂ ∂
Gˇ< Σˇa + Gˇ< Σˇa Gˇr Σˇ<
−∇p p∂ω p ∂ω p∇p p−∇p p∂ω p
∂ ∂ ∂
+ Gˇr Σˇ<+ Σˇ< Gˇa Σˇ< Gˇa . (A8)
∂ω p∇p p ∇p p∂ω p− ∂ω p∇p p
(cid:19)
Incidentally, ignoring the collisional broadening, i.e., the terms involving ∂/∂ω, and employing the generalized
Kadanoff-Baym ansatz, Eq. (A3) reduces to the kinetic equations of the distribution functions presented in the
previous studies.30
APPENDIX B: RETARDED GREEN’S FUNCTION
To analyzethe propertiesof the retardedGreen’s function, we begin from the Dysonequation in the helicity basis:
p2 iα ∂
2 ω Gˆr +αp σ ,Gˆr + eE Aˆr =2+Iˆ +Iˆ +Iˆ . (B1)
− 2m p { z p} 2 · ∂ω p r1 r2 r3
(cid:20) (cid:21)
This equation is obtained from Eq.(A3) by the transformation from spin basis to the helicity basis. The quantities
Iˆ and Iˆ are the helicity-basis analogues of the quantities Iˇ and Iˇ , respectively. Iˆ has a form similar to Iˆ ,
r1 r2 r1 r2 r3 r2
but with i φ [ ,σ ] in place of the operator .
2∇p p ··· x ∇p
7
Obviously, in the helicity basis, the noninteracting retarded Green’s function gˆr is diagonal
p
gˆr =diag (ω ε (p)+iδ)−1,(ω ε (p)+iδ)−1 . (B2)
p − 1 − 2
Including collisional broadening in the defi(cid:0)nition of the unperturbed retarded G(cid:1)reen’s function Gˆr , it obeys the
0p
Dyson equation in the absence of the electric field, as
p2 αp 1
ω Gˆr + σ ,Gˆr =1+ (Σˆr Gˆr +Gˆr Σˆr ). (B3)
− 2m 0p 2 { z 0p} 2 0p 0p 0p 0p
(cid:20) (cid:21)
The solution of this equation is also a diagonal matrix, (Gˆr ) = (Gˆr ) = 0, with elements independent of the
0p 12 0p 21
direction of momentum. To verify this, we examine the form of the self-energy Σˆr below.
0p
Let us first discuss the forms of self-energies Σˆr,< in general. Installing the actual form of matrix U into Eq.(4),
p p
we find
1
Σˆr,< = n V(p k)2 a Gˆr,<
p 2 i | − | 1 k
Xk n
+a σˆ Gˆr,<σˆ +ia [σˆ ,Gˆr,<] . (B4)
2 x k x 3 x k
o
Explicitly, the self-energies can also be rewritten as
n
(Σˆr,<) = i V(p k)2 a (Gˆr,<) +
p µν 2 | − | 1 k µν
Xk n
a (Gˆr,<) +ia [(Gˆr,<) (Gˆr,<) ] . (B5)
2 k µ¯ν¯ 3 k µ¯ν − k µν¯
o
In these expressions, µ¯ = 3 µ and a (i = 1,2,3) are factors associated with the directions of momenta, a =
i 1
−
1+cos(φ φ ), a =1 cos(φ φ ), and a =sin(φ φ ).
p k 2 p k 3 p k
− − − −
From Eq.(B5), we see that the diagonal elements of the Green’s functions appearing in nondiagonal elements of
Σˆr,< are always combined with the factor a . If the solution of Eq.(B3), Gˆr , is diagonal and independent of
p 3 0p
the direction of momentum, the self-energy term involving a vanishes under momentum integration. Hence, the
3
self-energy, Σˆr , also becomes diagonal and independent of momentum direction. As a result, Eq.(B3) reduces to a
0p
momentum-direction-independent equation for the diagonal elements of Gˆr . This implies that the assumed form of
0p
Gˆr isconsistentwithEq.(B3). FromuniquenessofthesolutionoftheDysonequationweimmediatelyconcludethat
0p
such a Gˆr is just the needed solution. It should be noted that, physically, the momentum-direction independence of
0p
theunperturbedretardedGreen’sfunctionisassociatedwiththerotationalsymmetryoftheunperturbedHamiltonian
in helicity basis.
Considering the angular independence of Gˆr , an important relation can be derived:
0p
V(p k)2sin(φ φ )[(Gˆr ) (Gˆr ) ]=0. (B6)
| − | p− k 0k 22− 0k 11
k
X
This equation is needed to prove the sum relation of the scattering term (Appendix D).
To the first order in the dc field, the Dyson equation for the linear electric field correction to the retarded Green’s
function, Gˆr =Gˆr Gˆr , can be written as
1p p− 0p
p2 αeE ∂
2 ω +( 1)µδ αp (Gˆr) +(1 δ ) sinφ (Gˆr) (Gˆr)
− 2m − µν 1 µν − µν 2 p∂ω 0 11− 0 22
(cid:20) (cid:21) h i
=(Iˆ(1)) +(Iˆ(1)) +(Iˆ(1)) . (B7)
r1 µν r2 µν r3 µν
Iˆ(1), Iˆ(1), and Iˆ(1) are the linear electric field parts of Iˆ , Iˆ and Iˆ . Iˆ(1) has the form
r1 r2 r3 r1 r2 r3 r1
(Gˆr ) (Σˆr ) [(Gˆr ) +(Gˆr ) ](Σˆr )
Iˆ(1) =2 0p 11 1p 11 0p 11 0p 22 1p 12 +(Gˆr Σˆr ,Σˆr Gˆr ), (B8)
r1 [(Gˆr ) +(Gˆr ) ](Σˆr ) (Gˆr ) (Σˆr ) 0p → 0p 1p → 1p
(cid:18) 0p 11 0p 22 1p 21 0p 22 1p 22 (cid:19)
8
where Σˆr = n Tˆ(p,k)Gˆr Tˆ+(p,k). Iˆ(1) vanishes due to the diagonal nature of the unperturbed retarded
1p i k 1k r2
Green’s function and self-energy. Iˆ(1) is off-diagonal and symmetric, (Iˆ(1)) = (Iˆ(1)) . From this Dyson equation
P r3 r3 12 r3 21
andtheexplicitformsofself-energies(B5),wefindthatGˆr becomesanoff-diagonalmatrixwithsymmetricelements,
1p
(Gˆr ) = (Gˆr ) . Also, Σˆr and hence Iˆ(1) have vanishing diagonal elements. At the same time, Eq.(B7) reduces
1p 12 1p 21 1p r1
to an equation for nondiagonal elements of Gˆr .
1p
Actually, Gˆr depends on the direction of momentum through a sine function. The expression for Gˆr can be
1p 1p
formally written as
eEsinφ
(Gˆr ) = p
1p µµ¯ ω p2/2m τ−1
p
− −
α ∂
[(Gˆr ) (Gˆr ) ]+[(Gˆr ) +(Gˆr ) ]Λ , (B9)
× −2 ∂ω 0p 11− 0p 22 0p 11 0p 22 p
(cid:26) (cid:27)
with Λ = 2n V(p k)2cos(φ φ )(ˆr ) , τ−1 = 2n V(p k)2[(Gˆr ) + (Gˆr ) ], and ˆr =
p i k| − | p − k G1p 12 p i k| − | 0p 11 0p 22 G1p
Gˆr /[eEsinφ ] as a quantity independent of the direction of momentum. In the case of short-range disorders,
1p p P P
the Λ and τ become momentum-independent constants. From Eq.(B9) it is obvious that Gˆr does not vanish.
p p 1p
We note that the electron-impurity scattering is not essential for the nonvanishing of Gˆr . In this, the second
1p
term on the left-hand side of Eq.(B7) plays a key role. In the absence of the electron-impurity collisions, Gˆr can be
1p
obtained analytically as
αeE ∂
(Gˆr ) =(Gˆr ) = sinφ (Gˆr ) (Gˆr ) . (B10)
1p 12 1p 21 −4(ω p2/2m) p∂ω 0p 11− 0p 22
− h i
Incidentally, from Eq.(B10) we can derive the retardedGreen’s function presented in our previous work.15 Using the
equality (Gˆr ) (gˆr ) in the absence of electron-impurity collisions, we rewrite Eq.(B10) as (Gˆr ) = (Gˆr ) =
0p ≡ 0p 1p 12 1p 21
+ with
1 2
A A
eE
= sinφ (Gˆr ) (Gˆr ) (B11)
A1 −4αp2 p 0p 11− 0p 22
h i
and
eE ∂
= sinφ (Gˆr ) +(Gˆr ) . (B12)
A2 −4p p∂ω 0p 11 0p 22
h i
is just the term that we employed in Ref.15, and is a total derivative with respect to ω. In the absence of
1 2
A A
electron-impurity scattering, the term has no effect on the spin dynamics of the considered system, because the
2
quantity directly associated with spin-HAall current, Gˆ< , takes the form
1p
Gˆ< = 2in (ω)Im + , (B13)
1p − F A1 R
with as a remainder term which can be expressed as a total derivative with respect to ω [this result is obtained
R
from Eq.(9)].
APPENDIX C: EXPLICIT EXPRESSION FOR Iˆ(1)
l
The scattering term Iˆ(1) can be expressed as the sums of Iˆ(1), Iˆ(1), and Iˆ(1), the corresponding linear electric
l l1 l2 l3
field parts of Iˆ , Iˆ , and Iˆ . Further, Iˆ(1) can be rewritten as Iˆ(1) = Fˆ +Fˆ , with an off-diagonal matrix Fˆ
l1 l2 l3 l1 l1 1 2 1 ≡
Σˆr Gˆ< Gˆ< Σˆa Gˆr Σˆ< +Σˆ< Gˆa , or explicitly,
1p 0p− 0p 1p− 1p 0p 0p 1p
(Fˆ ) =(Σˆr ) (Gˆ< ) (Σˆa ) (Gˆ< ) (Gˆr ) (Σˆ< ) +(Gˆa ) (Σˆ< ) , (C1)
1 µµ¯ 1p µµ¯ 0p µ¯µ¯− 1p µµ¯ 0p µµ− 1p µµ¯ 0p µ¯µ¯ 1p µµ¯ 0p µµ
and a general matrix Fˆ with elements
2
(Fˆ ) =2i[Im(Σˆr ) (Gˆ< ) Im(Gˆr ) (Σˆ< ) ], (C2)
2 µµ 0p µµ 1p µµ− 0p µµ 1p µµ
9
(Fˆ ) =[(Σˆr ) (Σˆa ) ](Gˆ< ) [(Gˆr ) (Gˆa ) ](Σˆ< ) , (C3)
2 µµ¯ 0p µµ− 0p µ¯µ¯ 1p µµ¯− 0p µµ− 0p µ¯µ¯ 1p µµ¯
where linear electric field correction to the lesser self-energy, Σˆ< , takes the form: Σˆ< =n Tˆ(p,k)Gˆ< Tˆ+(p,k).
1p 1p i k 1k
Furthermore, the diagonal Iˆ(1) and off-diagonal I(1) matrices have the forms
l2 l3 P
∂ ∂
(Iˆ(1)) = ieE Re(Σˆr ) (Gˆ< ) Re(Σˆr ) (Gˆ< )
l2 µµ · ∇p 0p µµ∂ω 0p µµ− ∂ω 0p µµ∇p 0p µµ
(cid:18)
∂ ∂
Re(Gˆr ) (Σˆ< ) + Re(Gˆr ) (Σˆ< ) , (C4)
−∇p 0p µµ∂ω 0p µµ ∂ω 0p µµ∇p 0p µµ
(cid:19)
e ∂ ∂
(Iˆ(1)) = ( 1)µ E φ ( 1)ν (Gˆ< ) Re(Σˆr ) (Gˆ< ) Re(Σˆr )
l3 µµ¯ − 4 ·∇p p − 0p νν∂ω 0p νν − ∂ω 0p νν 0p νν
ν (cid:20)
X
∂ ∂
Re(Gˆr ) (Σˆ< ) +Re(Gˆr ) (Σˆ< ) +(Rˆ ) , (C5)
−∂ω 0p νν 0p νν 0p νν∂ω 0p νν 2 µµ¯
(cid:21)
From the definition of the lesser Green’s function, we know that (Gˆ< ) = (Gˆ< )∗ . This implies that its
1p µν − 1p νµ
diagonal elements are pure imaginary-i.e., Re(Gˆ< ) = 0. Hence, we see from Eq.(B5) that the diagonal elements
1p µµ
of the linear electric field part of the self-energy,Σ< , are also pure imaginary. Consequently, the diagonalmatrix Fˆ
1p 2
is pure imaginary, as well as the matrix Iˆ(1).
l2
APPENDIX D: PROOF OF Iˆ(1) SUM RELATION
l
First we prove Eq.(15). Since Im(Iˆ(2)) is the sum of (Fˆ ) and Im(Fˆ ) , we first consider the integral
l1 12 1 12 2 12 I1 ≡
sinφ (Fˆ ) . Explicitly, it takes the form
p p 1 12
P = sinφ Re(Σˆr ) [Im(Gˆ< ) Im(Gˆ< ) ] Re(Gˆr ) [Im(Σˆ< ) Im(Σˆ< ) ] . (D1)
I1 p 1p 12 0p 22− 0p 11 − 1p 12 0p 22− 0p 11
Xp n o
Substituting the self-energies into Eq.(D1), we have
= sinφ V(p k)2 Re(Gˆr ) [Im(Gˆ< ) Im(Gˆ< ) ]
I1 p| − | 1k 12 0p 22− 0p 11
Xpk n
+cos(φ φ )Re(Gˆr ) [Im(Gˆ< ) Im(Gˆ< ) ] . (D2)
p− k 1p 12 0k 11− 0k 22
o
Interchanging the dummy integration variables p and k, can be rewritten as
1
I
= V(p k)2(sinφ sinφ cos(φ φ ))Re(Gˆr ) [Im(Gˆ< ) Im(Gˆ< ) ]
I1 | | − | | p− k p− k 1k 12 0p 22− 0p 11
pk
X
= V(p k)2cosφ sin(φ φ )Re(Gˆr ) [Im(Gˆ< ) Im(Gˆ< ) ]. (D3)
| | − | | k p− k 1k 12 0p 22− 0p 11
pk
X
From relation (B6) it can be seen that =0.
1
I
Next, we analyze the remaining component of the integral, sinφ Im(Fˆ ) ,
I2 ≡ p p 2 12
= sinφ Re(Gˆ< ) [Im(Σr )P+Im(Σr ) ]
I2 p 1p 12 0p 11 0p 22
Xp n
+Im(Gˆ< ) [Re(Σr ) +Re(Σr ) ] (Σˆ Gˆ) . (D4)
1p 12 0p 11 0p 22 − ↔
o
Again substituting the self-energies into Eq.(D4), this integral can be simplified as
= V(p k)2cosφ sin(φ φ )Re(Gˆ< ) [Im(Gˆr ) +Im(Gˆr ) ] (D5)
I2 | − | k p− k 1p 12 0k 11 0k 22
p,k
X
1
[cosφ cosφ cos(φ φ )][Im(Gˆr ) +Im(Gˆr ) ][Im(Gˆ< ) Im(Gˆ< ) ].
−2 k− p p− k 0p 11 0p 22 1k 11− 1k 22
p,k
X
10
Here, we have used Eq.(B6). In this, the vanishing of the real parts of the diagonal elements of the distribution
function-i.e., Re(Gˆ< ) = 0-has also been considered. Combining the terms proportional to cosφ cos(φ φ ) in
1k µµ p p − k
the second line of Eq.(D5) with a similar term in the first line, we obtain
1
= cosφ [Im(Σˆ< ) Im(Σˆ< ) ][Im(Gˆr ) +Im(Gˆr ) ]
I2 2 p 1p 11− 1p 22 0p 11 0p 22
Xp n
[Im(Σˆ< ) +Im(Σˆ< ) ][Im(Gˆr ) Im(Gˆr ) ] . (D6)
− 0p 11 0p 22 1p 11− 1p 22
o
Further, we note that in the self-consistent approximation,there is a vanishing quantity given by
K
cosφ [Im(Gˆr ) Im(Gˆr ) ][Im(Σˆ< ) +Im(Σˆ< ) ] (Σ G) . (D7)
K≡ p 0p 11− 0p 22 1p 11 1p 22 − ↔
Xp n o
The vanishing of = 0 can be shown by inserting the explicit forms of the self-energies, Eq.(B5), into and
K K
employing Eq.(B6). Finally, adding to the right hand side of Eq.(D6), we find
K
1
= cosφ ( 1)µ+1 Im(Σˆ< ) Im(Gˆr ) Im(Σˆ< ) Im(Gˆr )
I2 2 p − 1p µµ 0p µµ− 0p µµ 1p µµ
Xpµ n o
cosφ
= p[Re(Fˆ ) Re(Fˆ ) ], (D8)
2 11 2 22
2 −
p
X
which is Eq.(15).
In the following, we prove Eq.(16). From Eq.(C5), the involved integral, denoted as , can be written as
3
I
i dω ∂ ∂
= ( 1)νcosφ eE (Gˆ<) Re(Σˆr) (Gˆ<) Re(Σˆr)
I3 4 2π − p ·∇p 0 νν∂ω 0 νν − ∂ω 0 νν 0 νν
pµνZ (cid:26) (cid:20) (cid:21)
X
∂ ∂
Re(Gˆr) (Σˆ<) +Re(Gˆr) (Σˆ<) . (D9)
− ∂ω 0 νν 0 νν 0 νν∂ω 0 νν
(cid:20) (cid:21) (cid:21)
Here, the momentum integral of Eq.(C5) has been performed by parts integration,and the contribution from (Rˆ )
2 12
vanishes due to its total derivative nature with respect to ω. Each part of the integrand involves two functions Gˆr,<
0
and Σˆr,<. Further integration by parts with respect to ω in terms of the form ∂ A B ([with A and B being
0 ∂ω∇p
(Gˆr,<) and (Σˆr,<) , respectively, or vice versa] results in the operators and ∂/∂ω acting individually on A or
0 µµ 0 νν ∇p (cid:0) (cid:1)
B-i.e., ∂ A B. Thus, we obtain
∂ω ∇p
(cid:0) (cid:1) i dω ∂ ∂
= ( 1)νcosφ eE (Gˆ<) Re(Σˆr) (Gˆ<) Re(Σˆr)
I3 2 2π − p · ∇p 0 νν∂ω 0 νν − ∂ω 0 νν∇p 0 νν
pµνZ (cid:20)
X
∂ ∂
Re(Gˆr) (Σˆ<) + Re(Gˆr) (Σˆ<) . (D10)
−∂ω 0 νν∇p 0 νν ∇p 0 νν∂ω 0 νν
(cid:21)
Using Eq.(C4), Eq.(16) can be derived immediately.
∗ Electronic address: [email protected]
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