J Kailasvuori- Boltzmann approach to the spin Hall effect revisited and electric field modified collision integrals

8/3/2019 J Kailasvuori- Boltzmann approach to the spin Hall effect revisited and electric field modified collision integrals

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ournal of Statistical Mechanics:An IOP and SISSA journal

Theory and Experiment

Boltzmann approach to the spin Halleffect revisited and electric eldmodied collision integrals

J KailasvuoriDepartment of Physics, Freie Universit at Berlin, Arnimallee 14, 14195 Berlin,GermanyE-mail: [email protected]

Received 18 June 2009Accepted 21 July 2009Published 5 August 2009

Online at stacks.iop.org/JSTAT/2009/P08004doi:10.1088/1742-5468/2009/08/P08004

Abstract. The intrinsic contribution to the spin Hall effect in a two-dimensionaldegenerate electron gas (2DEG) with non-magnetic impurities is studied in aquantum Boltzmann approach. It is shown that if the steady state response isperturbative in the spinorbit coupling parameter , then the precession termvital for DyakonovPerel relaxation and the key to the spin Hall effect in previoussimilar Boltzmann studiesmust be left out to rst order in spinorbit coupling.In such a case one would have that to lowest order in the parameters electriceld, spinorbit coupling, impurity strength and impurity concentration there isno intrinsic contribution to the spin Hall effect, not only for a Rashba couplingbut also for a general spinorbit coupling. To cover all possible lowest order terms

we also consider electric eld induced corrections to the collision integral in theKeldysh formalism. However, these corrections turn out to be of second order in. For comparison we derive some familiar results in the case when the responseis not assumed to be perturbative in . We also include a detailed discussion of why a relaxation time approximation of the collision integral fails. Finally wemake a comment on pseudospin currents in bilayer graphene.

Keywords: graphene (theory), Boltzmann equation, quantum transport

ArXiv ePrint: 0812.4800

c 2009 IOP Publishing Ltd and SISSA 1742-5468/09/P08004+ 22$30.00
mailto:[email protected]://stacks.iop.org/JSTAT/2009/P08004http://dx.doi.org/10.1088/1742-5468/2009/08/P08004http://dx.doi.org/10.1088/1742-5468/2009/08/P08004http://arxiv.org/abs/0812.4800http://arxiv.org/abs/0812.4800http://dx.doi.org/10.1088/1742-5468/2009/08/P08004http://stacks.iop.org/JSTAT/2009/P08004mailto:[email protected]


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Boltzmann approach to the spin Hall effect

Contents

1. Introduction 2

2. The modelintrinsic versus extrinsic 5

3. Semiclassical description of a spinorbit coupled system 5

4. Derivation of the collision integral in the Keldysh formalism 7

5. The collision integral to linear order in spinorbit coupling 9

6. Solving the Boltzmann equation 10

7. General arguments for a vanishing SHE 148. Assumption of a linear response in the spinorbit coupling 14

9. Boltzmann equation without the precession term 15

10. Electric eld induced corrections to the collision integral 17

11. Conclusions 18

Acknowledgments 19

Appendix A. Left-hand side of equation (29) 19

Appendix B. Failure of a relaxation time approximation 19

References 21

1. Introduction

The idea in spintronics is to manipulate the electron spin for information storage andtransfer, as done with the electron charge in electronics. Possible advantages could forexample be smaller resistive losses or the additional richness that comes from the non-scalar nature of spin. In the study of spin currents particular attention has been given tothe spin Hall effect (SHE) in a two-dimensional degenerate electron gas (2DEG) with spin

orbit coupling. Perpendicular to an applied electric eld, opposite spins travel in oppositedirections, thus creating a spin current without a net charge current. Such a spin currentcan lead to an accumulation of spin at the edges of a sample, although in contrast to theelectrical charge analogy this is not given. An individual spin can change its direction, forexample due to spin precession, and therefore the accumulated spin polarization is not aconserved quantity. The SHE opens up one possibility of manipulating spin with electricelds.

The experiments on the SHE are few and recent [ 1][6]. In contrast, much workhave been devoted to the theoretical aspects (see e.g. the reviews [ 7][9]) and even inrecent years there has been an intense discussion about the different mechanisms and howto compare results reached by different formalisms. This discussion is closely relatedto the one on the anomalous Hall effect (AHE) [10], for example in the distinctionbetween extrinsic (impurity related, e.g. skew scattering, side-jump etc) and intrinsic

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(band structure related) mechanisms. The latter could be due to structure inversionasymmetry of the conning potential (leading, for example, to Rashba coupling) or dueto bulk inversion asymmetry (allowing for Dresselhaus coupling).

The earliest theoretical works on the SHE were done by Dyakonov and Perel in1971 [11], who showed that spinorbit coupling leads to a spin current perpendicularto the electric eld. The name spin Hall effect was coined in 1999 [12]. A universal spinHall conductivity in a 2DEG was proposed [13][15]. However, impurities were neglectedin these studies. Many studies have later shown that universality is lost when impuritiesare taken into account [ 16][28]. On the other hand, there can still be topological edgemodes, responsible for the quantum SHE (see e.g. [ 29]).

The effect of non-magnetic impurities on the intrinsic contribution to the SHEhas been studied using a Boltzmann approach [ 16][21] as well as diagrammaticmethods [22][28] like the Kubo formalism. The diagrammatic methods are moresystematic and have a more general range. The Boltzmann approach offers more intuition,but has typically been implemented by identifying distinct processes or hand-pickingdifferent contributions rather than systematically covering all possible contributionsthrough a formal approach. Within the Boltzmann language one nds several differentapproaches. We are going to use the quantum Boltzmann approach, which treats spincoherently.

This paper is going to deal with the intrinsic SHE in a steady state calculation to rstorder in spinorbit interaction. We attempt in a Keldysh derivation of the Boltzmannequation to account for all contributions present to lowest order in the electric eld, inspinorbit splitting and in the strength and concentration of non-magnetic impurities(sections 35 and 10). The spinorbit interaction H SO = b is chosen to be of the isotropic form b = b(k)b(), where the unit vector b has a winding number N ,i.e. bx + i by = e i0 +i N with 0 being a constant.

Many recent theoretical studies conclude that the SHE vanishes for N = 1 (e.g. forthe Rashba and linear Dresselhaus spinorbit couplings) for point-like impurities [ 16][23],[25][28] as well as nite range impurities [17][19], [21]. General arguments have beenproposed to explain this vanishing [27, 28] (section 7). A nonzero result can be foundin [23, 24]. With an alternative denition of spin currents, a nonzero result is also foundin [18].

For other odd N the SHE is nonzero [18, 19], assumes a universal value in a speciclimit [19], but depends otherwise on the range of the impurity potential and is thereforein general not universal, though it is independent of the spinorbit splitting b, the overallstrength of the impurity potential and of the impurity concentration [19]. We reproducethese results in section 6. Additionally, we give explicit results on the polarization andspin currents for the components of the spin parallel to the plane and calculate equilibriumspin currents.

We will also show that the above results on the SHE are based on the response notbeing perturbative in the spinorbit coupling. If perturbativeness is assumed (section 8),the spin precession termseemingly the prerequisite for a spin Hall currentmust be leftout to linear order in spinorbit coupling, suggesting that to lowest order the intrinsic SHEis zero for arbitrary winding N (section 9). Possible alternative contributions to the SHE

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Table 1. Summary of spin densities and spin currents in the nonperturbativeand the perturbative cases for the cases N = 1 and for other odd N . Theentries present the (phase space)/(real space) quantities, respectively. NZ standsfor a (typically) nonzero result whereas uncommented zeros stand for triviallyvanishing components (e.g. due to angular considerations). Nontrivial cases arecommented.

Nonperturbativesx,y s z j x,y j z

|N | = 1 f eq NZ/ 0 0/ 0 NZ/ NZa 0/ 0f (E ) NZ/ NZb 0c/ 0 NZ/ 0 0c/ 0d

|N | = 1 f eq NZ/ 0 0/ 0 NZ/ 0 0/ 0f (E ) NZ/ 0 NZ/ 0 NZ/ 0e NZ/ NZ

Perturbativesx,y s z j x,y j z

|N | = 1 f eq NZ/ 0 0/ 0 NZ/ NZf 0/ 0f (E ) NZ/ 0f 0/ 0 NZ/ 0 0/ 0

|N | = 1 f eq NZ/ 0 0/ 0 NZ/ 0 0/ 0f (E ) NZ/ 0 0/ 0 NZ/ 0e 0/ 0

a j yx , j xy = O(3 ), see under ( 26).b See (39).c Since G = 0.d General arguments in section 7.e Vanishing for N odd. Could, however, be nonzero for |N | = 2.f

See end of section 9. Relies on the introduced spin relaxation term. Here jyx , j

xy = O().

from electric eld induced corrections to the collision integral are discussed in section 10.A contribution that we believe has not been discussed before in the Boltzmann approachturns out to be the only candidate when the precession term is absent. However, thiscontribution turns out to be of second order in spinorbit splitting.

Leaving out the precession term leads to some formal difficulties in the case N = 1.The Boltzmann equation becomes unsolvable. However, this can be remedied by includinga small spin relaxation term (section 9). This could suggest that for N = 1, non-

magnetic impurities are not enough for a consistent steady state solution in the spinpolarization in the case that the response is perturbative in the spinorbit coupling. Acomparison of spin densities and spin currents in both phase space and real space is givenin table 1. In this context we also make a comment on pseudospin currents in bilayergraphene.

The spinorbit corrections to the collision integral make the analytical treatmentconsiderably more complicated. For the case |N | = 1 but not for the case |N | = 1 thiscomplication is necessary even for a qualitatively correct understanding of the vanishingSHE. In appendix B we discuss the failure of the relaxation time approximation, that inmany other problems is useful for a simple qualitative understanding and as a startingpoint for deriving real space equations, for example describing thermoelectric effects.In particular, we contrast with the derivation of the DyakonovPerel spin relaxationmechanism.

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2. The modelintrinsic versus extrinsic

For a semiclassical Boltzmann description (see e.g. [ 30][34]) one needs the Wignertransformed one-particle Hamiltonian. For the spinorbit (SO) coupled electrons we aregoing to study, it is

H (x , p , t ) = a(k) + b(k ) + e(x , t ) (1)

with e < 0 and k (x , p , t) = p eA (x , t ). We want to describe a 2D system with x and p chosen to lie in the x, y-plane. Throughout the paper = 1. In the absence of spinorbit coupling the dispersion is given by a k , typically a = k2/ 2m. For the Rashbainteraction b := |b| = k and b := b/b = , with parametrizing the strength of thecoupling. However, we want to consider an arbitrary odd-integer winding number N inb = b(k)b() (with bx + i by = e i0 +i N ). The energy bands are sk = a + sb with s = giving the sign of the spin along the spin quantization axis b, i.e. b | bs = s | bs .

The total Hamiltonian H tot = H + H imp also includes an impurity potential V imp (x ) =n U (x x n ) of charged, non-magnetic impurities at positions x n . Including the spin

orbit coupling experienced at impurities one has

H imp = V imp + ext k V imp . (2)

H and H imp are treated very differently in the Boltzmann approach. H enters to linearorder in the kinetic equation, whereas H imp is in the Keldysh machinery turned into animpurity averaged self-energy to appear to quadratic order in the collision integral.

A spinorbit coupling enters both through the intrinsic (i.e. band related) term band in the extrinsic (i.e. impurity related) term ext k V imp , and the consequencesof the two are usually studied separately in the literature. This paper deals only with theintrinsic contribution to the SHE.

3. Semiclassical description of a spinorbit coupled system

In a Boltzmann description of an electron system with spin, the spatial degrees of freedom are treated semiclassically, whereas the treatment of the spin remains quantummechanical. The state of the system is given by the 2 2-matrix-valued distribution

function f (x , p , t ), here with the spin index = z , z . It is related to the equal timedensity matrix (x1, x2)| t2 = t1 = (x 2, t1)(x 1, t1) by a Wigner transformation (see

e.g. [30][33]). In the absence of scattering one can derive the Boltzmann equation for f by applying Heisenbergs equation of motion on (x1, x2), then identifying t2 = t1, Wignertransforming the result and gradient expanding it to rst order. The approximation tostop at rst order in gradient expansion is the semiclassical approximation, which relieson the external perturbations, such as electromagnetic potentials, changing negligibly onlength and timescales of the de Broglie wavelength B and time B = B /v F .

From the matrix elements of the distribution function f one extracts the densitiesand current densities of charge and spin. The matrix elements are most convenientlyexpressed in the decomposition f = 1f 0 + f = f 0 + f in Pauli matrices (with = x,y,z ). (Throughout this paper we use the convention of summation over repeatedindices.) Furthermore, we nd it convenient to decompose the vector f = f b b + f c c + f z z

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into its components along the basis vectors b(), z and c () = z b(), analogous to thecylindrical basis vectors k () := k /k , z and () := z k ().

The charge density en and current density e j in phase space are derived fromen = Tr( fH/ ) and e j = Tr( fH/ A ), which yields

n(x , k , t) := Tr f = 2 f 0 = n+ + n

j i(x , k , t) := Tr( v i f ) = 2 f 0 ki a + 2 f ki b = n+ v+i + n vi +

2Nbk

f c i(3)

with i = x, y. Here we introduced the velocity matrices v i := ki H = ki a + ki b.The spin-independent part of the velocity is k a =: v0. The band velocities arev s := k s = bs |v | bs = vs k . The intra-band elements n := b | f | b = f 0 f b givethe density of each spin band s = . The inter-band elements b | f | b = f z if care important for the coherent treatment of spin and are, for example, present in the lastterm of (3)1, containing the zitterbewegung of the spinorbit coupled electrons.

The real space densities are obtained by integrating the phase space densities overmomentum, e.g.

j (x , t) = d2k(2)2 j (x , k , t ). (4)When not otherwise stated, densities in this paper are always assumed to be phase spacedensities.

The spin density, i.e. the polarization, is given by s = ( / 2) Tr( f ) = f (with

= 1). There is not a unique way to dene the spin current because spin polarization isnot a conserved quantity. (For a proposal for a conserved spin current, see [35]. For itsimplications for the SHE, see [18].) When band velocities coincide, i.e. v s = v0, then it isclearly j = f v0. For the general case we choose the common denition

j i =14 Tr {v i , f } = f ki a + f 0 ki b (5)

(with {A, B } = AB + BA ). The SHE is a real space current of z-component spins

j z = d2k(2)2 f z v0 = 1e SH z E (6)perpendicular to an applied electric eld E along the plane. SH is the spin Hallconductivity.

The Boltzmann equation in matrix form is given by

i[H, f ] + T f + 12 {v i , x i f } + eE i ki f zij eBz12 {v i , kj f } = J [f ] (7)

where the matrix-valued functional J is the collision integral. In components it reads(from now on the charge e in eE and eB is absorbed into the elds)

t f 0 + x i f 0 ki a + x i f ki b + E i ki f 0 + zij Bz ( ki f 0 kj a + ki f kj b) = J 02f b + t f + x i f ki a + x i f 0 ki b + E i ki f + zij Bz( ki f kj a + ki f 0 kj b) = J .

(8)

1 Note also that this term is equally shared between the two bands; bs | 12 {v , f }| bs = ns v s + k 1 bf c .

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However, by virtue of denition ( 5) the equations ( 8) can be compactly written as

t n + x j + k (nE + j B ) = 2J 02(s b) + t s + x j + k (s E + j B ) = J .

(9)

The left-hand side of the rst equation is the same as for charged, spinless particles inan electromagnetic eld. Apart from the spin precession term, the second equation isof similar form. This is what one would expect, since the electromagnetic eld does notinteract with the spin in the considered model but only with the charge that the spin sitson. The spin enters in a nontrivial way only through the precession term and through thecollision integral.

4. Derivation of the collision integral in the Keldysh formalism

The presence of, for example, two-body interactions or disorder averaged impurityinteraction is described in the Boltzmann approach by the collision integral J . It isassumed that one is in the kinetic regime , where the de Broglie wavelength B = 1 /k Fis much shorter than the scattering length . Like [19] we use the Keldysh formalism(see e.g. [30][33]), but among other general methods we can mention the non-equilibriumstatistical operator formalism [ 30]. For disorder averaged impurities, see also the compactderivation in [21].

The Keldysh derivation of the semiclassical equations starts with relating f to theWigner transformed Keldysh Greens function

f (x , p , t ) = 12

+ d4i GK(x , p , t, ). (10)The equation of motion for GK is given by the Dyson equation. Integrating the equationover the frequency results in the semiclassical Boltzmann equation with a collision term.This should be contrasted with the quasiclassical Boltzmann approach (see e.g. [ 33]) wherethe integration is instead performed over |k | to obtain a distribution function f (x , p , t, ).This is the approach for example in [19] and [20].

The Keldysh equation derived from the Dyson equation can be written in twoequivalent forms

1 = (i t H ) G or 1 = G (i t H ) . (11)The product involved here is the convolution product, the identity stands for 1 :=1 (x 1 x 2 )(t 1 t 2 ) and quantities are written in Keldysh matrix space with

G = GR GK

0 GA and i t H =i t H R K

0 i t H A. (12)

Each element in these matrices is an innite-dimensional matrix in real space indices, andin our case also a 2 2 matrix in spin indices.

The two equations in ( 11) contain the same information. To derive kinetic equationsone takes the difference of them, which for the Keldysh component yields

(i t H )GK GK (i t H ) = R GK GK A GR K + K GA . (13)

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The right-hand side is going to give the collision integral, which we only treat to lowestorder in impurity concentration and impurity strength (the rst Born approximation),

p = n imp d2 p(2)2 |U (| p p |)|2G0p , (14)with G0 = G( = 0). ( is diagonal in momentum space due to the disorder averaging of the impurity interaction.) According to the self-consistent Born approximation we replaceG0K by GK . A crucial approximation comes with choosing the generalized KadanoffBaymansatz [36]

GK = i( GR h hG A) + (15)where h := GK | t2 = t1 , which generalizes the quasiparticle approximation GK =

h(x , p , t )( ) for spinless electrons. The approximation can be considered to be anexpansion in relaxation times of the system and corresponds to a Markov approximation.The convolution product takes after Wigner transformation A(x 1, t1, x 2, t2)

A(x , p , t, ) the form AB = Ae(i / 2)D B with the Poisson-bracket-like gradient D. In agauge invariant treatment valid when the electromagnetic elds are weak and vary slowly(see e.g. p 344, vol 1 in [30] or ch 7 in [34]), one introduces k ( p , x , t ) = p A and(, x , t ) = and lets {x , k , t ,} become the new set of independent variables(i.e. x i k = 0). This changes the gradient into

D = x i ki

ki x i +

t

t + E i(

ki

ki ) + ijl B i

kj kl (16)

with X Y := ( X )Y and X

Y := X (Y ). Gradient expanding the left-hand sideof (13) to rst order and integrating over the frequency yields the left-hand side of ( 7). The

right-hand side, the collision part, is usually taken to zeroth order in gradient expansion.Inserting ( 15) into (13), and using that combinations such as d GR [. . .]GR vanish, onearrives at the collision integral (with W kk := 2n imp |U (|k k |)|2)

J = k W kk2 d2 (GRk fG Ak + GRk fG Ak )= k W kk2 d2 (G0Rk fG 0Ak + G0Rk fG 0Ak ) + (17)

where in the last row only terms of second order in the interaction strength were kept.

The shorthand notations f = f (k , x , t) f (k , x , t ) and (d2k / (2)2) =: k wereintroduced.In the expression ( 17) we also need the retarded and advanced components. For this

one should take the sum of the two equations ( 11), which for GR ( = 0) after Wignertransformation leads to

2 = (+ H 0)e(i / 2)D G0R + G0R e(i/ 2)D (+ H 0), (18)with + := + i (to take care of the boundary conditions provided by the imaginarypart of R when = 0) and with H 0 := H . To zeroth order in gradient expansion itis solved by

G0R =s=

S bs+ s

S bs :=12 (1 + s b) (19)

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where S is the spin projection operator. With this zeroth order G0R , the last line in ( 17)can be reformulated to be identical to the result derived in [ 21] and essentially also to theone derived in [19].

To the knowledge of the author, one always nds in the literature the zeroth ordersolution for G0R , maybe because in the common cases like spinless electrons or Zeemancoupled electrons (with a constant magnetic eld) the rst order contribution to ( 18) (andthus to G0R ) vanishes. However, for a spinorbit coupled system it does not. We nd therst order contribution

G0R = zNbE b Bz( a) kb

k(+ + )2(+ )2= [Bz = 0]

= zN E 4k

s( sb

1

)1

+ s , (20)

with E := E in the polar decomposition E = E k k + E . In section 10 we showhow the contribution ( 20) modies the collision integral. We will also investigate whetherthe corresponding correction could be an alternative source to the SHE not relying on theprecession term 2.

5. The collision integral to linear order in spinorbit coupling

In this section the Boltzmann equation is expanded to rst order in spinorbit coupling,as done in [19] and [21], hence assuming b(kF ) F . The subscript F indicates the valueof the corresponding quantity at the Fermi surface determined by a = F , where F = at low temperatures kBT F .

The collision integral is taken to the habitual zeroth order in gradient expansion,meaning (17) with (19). To slim down the often lengthy expression for collision integrals,some more shorthand notation is introduced. x means that the quantity x depends onprimed variables such as k , s etc, whereas x correspondingly depends on k , s. Forexample, S = 12 (1 + s bk ). Also, x := x x ; for example =

sk sk or

( sb) = sb s b .

Inserting ( 19) into (17) gives

J 0 = k W kk 12 ss ( ) 1 + ss b b2 f 0 + s b + s b2 f + X 0J = k W kk 12 ss ( ) 1 + ss Bkk2 f + sb + s b2 f 0 + X

(21)

2

Note that if in contrast to equation (18) one takes the difference of the equations ( 11), the equation for G0R

isidentical to ( 13) with = 0. This equation does not determine G0R , but given ( 18), the precession term (in theequation for G0R ) implies the presence of ( 20) to rst order in electromagnetic elds.

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with the matrix B kk := b(b )T + b (b)T b b .3 Not written out in equation ( 21) are theprincipal part terms

X = k W kk 12 ss P 1 ss b b2 ( f f 0) ( s b)2 f (22)which are not considered to be part of the elastic collision integral but to be related torenormalization corrections. Such terms can usually left out if one is interested in theinteraction only to lowest order (however, see [ 38]) and it is beyond the scope of thepresent paper to discuss them. (See [ 39] for a lucid treatment on how to interpret andhandle them in the case of spinless electrons interacting through a two-body interaction.See also the derivation of Bloch spin relaxation equations in [40].)

The delta functions ( sk

sk

) connecting Fermi surfaces at different |k | make itdifficult to nd an analytical solution. However, in the considered limit b(kF ) F onecan use the expansion

( ) = ( a) + ( sb) ( a) + O(2) (23)

after which one is left with the spin-independent delta function (ak ak ) implying|k | = |k |. With this expansion the collision integral reads

J 0 = k W kk

( a) f 0 +

O ( 2 )

( a) b f

J = k W kk [( a) f + ( a) b f 0] .(24)

The terms with ( a) = a ( a) are made sense of by integration by parts.If f = O() (i.e. if the polarization vanishes as 0), the term indicated as of order

O(2) can be neglected to linear order in . This resulting collision integral is essentiallythe one found in [21] and is similar to the one in [19]4. For a physical interpretation of spinorbit coupling dependent contributions, see [ 19].

In section 10 we discuss corrections to the collision integral when one goes beyondzeroth order in gradient expansion.

6. Solving the Boltzmann equation

We now set out to solve the uniform, steady state Boltzmann equation

2 f b + E k f = k W kk [( a) f + ( a) b f 0] (25)3 The structure of B is relevant for graphene (one Dirac cone). Not surprisingly, an equivalent structure is foundin [37] We note that one gets a different matrix B if one instead uses the non-equilibrium statistical operatormethod, used for example in [38]. This difference, to be discussed elsewhere [47], has no inuence on the presentpaper since the term involving B does not contribute to rst order in spinorbit coupling.4 The quasiclassical approach of [19] makes a straightforward comparison not obvious, but it seems like there isno analogue of the term containing a f 0 that we obtain after integration by parts.

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without implementing the assumption of perturbativeness in , to be discussed insection 8. The distribution function f = f eq + f (E ) is linearized only in the electriceld and not in the spinorbit coupling.

The distribution n = f 0 f b of the respective spin band is in equilibrium given bythe FermiDirac (FD) distribution. The equilibrium distribution is therefore given by

f eq0 f eqb

= f FD i.e. f eq =

S b f FD , (26)

with vanishing inter-band elements f eqc = f eqz = 0. Time reversal symmetry requires thereal space equilibrium polarization f eq = k f eq(k ) to be zero, which also follows triviallyfrom the vanishing angular part of the integral. The real space spin current, on the otherhand, need not vanish since it is even under time reversal symmetry. For |N | = 1 the realspace spin current is trivially zero, but for N = 1 we nd j xy = j yx = m23/ 2 + O(5)for a quadratic dispersion a = k2/ 2m.

The equilibrium distribution is from now on taken to linear order in , i.e.

f eq0 = f FD ( )

2 = f FD (a ) + O(2)

f eqb

= f FD ( )

2 = b a f FD (a ) + O(2)

= f eq0 = f FD

f eq = b a f FD(27)

where from now on f FD f FD (a ). From ( 27) one sees that a small spinorbit couplingdoes not change the charge density but induces a small polarization at the Fermi surface.The distribution f eq = f FD + b a f FD satises (25) for E = 0.

The charge part of ( 25) does not depend on the polarization. When a uniform, staticelectric eld is applied, one nds the usual solution f (E )0 = tr E k k f FD where tr is thetransport relaxation time. Gathering the known terms on the left-hand side one can writethe polarization part of the equation as

E k f eq + k W kk ( a) b f (E )0 = 2 b f (E ) k W kk ( a) f (E ) . (28)It turns out (see the appendix) that the equation can be written in the form

F (E k b E c ) + G(E k b + E c ) = 2 b f (E )

K f (E ) |k = k , (29)

with the shorthand := d2 . The functions F and G depend on k and are proportionalto . They do not depend on , on E , on the impurity concentration n imp or on the overallstrength of the impurity potential U . However, they depend on the range of the potentialthrough dimensionless fractions of the Fourier components of the functions 5.

K (k, ) := D(a)W kk |k = k =:m

eim K m

K (k, ) := D(a)kv0[ a W kk ]k = k =:m

eim K m .(30)

5 For comparison with [ 19], note that a W kk |k = k = ( v0 k) 1 tan 2 ( / 2) W kk |k = k .

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In particular, for N = 1 we nd for a general dispersion a k that

F = a f FD2

( 2) 1tr + 112 2 tr 112 101 101 + 112 tr

+k k a f FD

21 + tr 112

G = 0

(31)

with 1tr = K 0 K 1 and introducing shorthands 112 := K 1 K 2, 101 := K 0 K 1 and

112 = K 1 K 2. We also nd that G = 0 when b is not proportional to k. The vanishingof G is going to imply the vanishing of the SHE for Rashba coupling.

The combination E k bE c has a winding N 1. In particular, for the Rashba case

b = (i.e. N = 1) one has

E k b E c = b(k )T c( )T E = sin 2 cos2cos 2 sin2 E

E k b + E c =0 11 0 E , (32)

i.e. the left-hand side of ( 29) has an angle independent term.The solution can be found by Fourier decomposition in the basis {b, c , z }

f (E ) =n

ein (bf bn

+ c f c n + z f zn ) =n

ein b c z

f bnf c nf zn

(33)

where the Fourier coefficients {f bn , f c n , f zn } of course only depend on k and not on .With E := E x + i E y one has 2(E k b E c) = e iE (b ic ) + e iE (b ic ) and thereforethe left-hand side of ( 29) can be written as

12 e

iE b c z

F + GiG iF

0

+ c .c., (34)

which contains only the n = 1 Fourier component and the complex conjugate n = 1

component. This is going to imply that f bn , f c n , f zn = 0 for |n | = 1. (Choosing a Cartesianbasis in (33), in contrast, couples the equation for component n with the componentsn N .) This fact has some direct implications for the electric eld induced contributionsto the real space densities. For example, the real space density of z-spins f (E )z is triviallyzero. For |N | = 1 the in-plane components f (E )x and f (E )y also vanish trivially in realspace, whereas for |N | = 1 they can be nonzero. The real space spin Hall current j z canbe nonzero for all N , whereas the contribution to j x and j y vanishes trivially for all N .

On the right-hand side of ( 28) we nd

K f | (E )k = k = n ein b f bn K (1 cos N cos n )+ i f c n K sin N sin n

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+ c f c n

K (1 cos N cos n ) if bn

K sin N sin n

+ z f zn K (1 cos n ) .Here, b = b cos N c sin N was used and terms odd in were left out, using thatK is even in . Including the precession term 2 b(z f c c f z) the equation for the n = 1Fourier components of f (E ) becomes

E

2

F + GiG iF

0

=

1cos i 1sin 0

i 1sin 1cos 2b0 2b 1tr

f b1f c 1f z1

(35)

where

1cos := K (1 cos N cos ) = K 0 (K N 1 + K N +1 )/ 2 1sin := K sin N sin = ( K N 1 K N +1 )/ 2

(36)

(possibly negative) were introduced.With the -dependent b, the matrix and hence the solution will be inhomogeneous

in . For |N | = 1 the solution nonetheless goes to zero when 0. For point-likeimpurities ( 1cos =

1tr = K 0 and 1sin = 0) we nd

f z1 = i N E 2k a f FD

4k22 + K 20= j z = k v0f z = Nk

2F 22(4k2F 2 + K 20 )

( E y , E x ) (37)

for the real space spin current. As in [19] one recovers in the clean limit 1tr = K 0 kF a universal spin Hall conductivity SH = ( N/ 8), whereas SH decreases to zero in theopposite (dirty ) limit.

For the case N = 1 one has for arbitrary impurity range that cos = sin =: 2 02 .The determinant 4 b2 1cos +

1tr ( 2cos

2sin ) of the matrix in ( 35) becomes singular at b = 0.

For N = 1 we nd the solution

f b1

= E 02(F + G) +G

4 tr b2

f c 1 = iE G

4 tr b2

f z1 = iE G2b

.

(38)

The z-component of the real space spin density k f z must vanish trivially as notedafter equation ( 34), the in-plane spin components resulting in a electric-eld-inducedpolarization in real space

s =

k

(bf b + cf c ) = z E

da D 02(F + G) +

G

2 tr b2 . (39)

Remember that the contribution from the equilibrium polarization is zero in real space.

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According to (31), G = 0 for N = 1 for arbitrary dispersion and an arbitrary rangeof impurities. Thus, f z = 0 for Rashba coupling, implying a zero spin Hall current. Asimilar analysis applies to N = 1, e.g. for linear Dresselhaus coupling, since a model of winding N is related to one of winding N through reection (say along the x-axis).

If for Rashba coupling there was a nonzero SHE, i.e. G = 0, then f z and consequentlythe spin Hall current would be independent of (since in general, G ). Furthermore,the in-plane components f x and f y of the polarization would diverge as 1 as 0,i.e. one would not recover an unpolarized distribution when sending the spinorbitcoupling to zero. However, in the derivation of the Boltzmann equation there were onlyassumptions of and f being small enough, and no assumptions of them not beingtoo small. Therefore, to the extent that a diverging polarization in such a case is anunthinkable result, the vanishing of the SHE for the Rashba case is a natural implication.

7. General arguments for a vanishing SHE

The vanishing of the SHE for the Rashba case has been found by numerous previousstudies (see section 1). To the authors knowledge it has not been related to the nitenessof the in-plane polarization as done in the argument above. In linear response studiesgeneral arguments have been given in the case of quadratic dispersion a k2 [27, 28],where it is noted from [y , ( p )z] = 2ipyz that for Heisenberg operators

ddt

y = i[y , H + V imp ] = 2 pyz = 2 m j zy (40)

for non-magnetic impurities. The steady state condition (d/ dt)y = 0 forces the spinHall current j zy to be zero. A similar argument applies to j zx .

The analogue to this argument in the Boltzmann approach can be established for thereal space densities. In phase space, ( s b)y = bx sz = j zy bx /v 0y . Only for N = 1 andfor b/v0 independent of k is it possible for bx /v 0y to be a constant, here m . Accordingto (9) one has t sy = 2(s b)y E k s y + J in the uniform case. In the integrationover momentum the last two terms vanish, resulting in t sy = 2mj zy in real space forN = 1. Likewise, t sx = 2mj zx . In a steady state, t s = 0 implies j z = 0.

In phase space, on the other hand, the steady state condition 0 = t sy = 2(s b)y E k s y . . . does not imply a vanishing j zy . We could have had G = 0 as long as

da Dv 0G/b = 0 guaranteed the vanishing in real space. However, in section 6 we foundthat j z = 0 also in phase space.8. Assumption of a linear response in the spinorbit coupling

We are now going to study the implications of a basic assumption, namely that the steadystate response of a spinorbit coupled system in an electric eld is perturbative in thesmall parameter (vF is kept constant), analogously to the usual assumption of a linearresponse in the electric eld strength E . The latter assumption means that one canexpand the solution f eq + f (E ) + f (E

2 ) + in non-negative powers in the electric eldand solve the equation iteratively by solving equations that are homogeneous in orders of E (see e.g. equation (28)). With the assumption now to be studied, the same is expected

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to apply also for the spinorbit coupling parameter (vF is kept constant), in which casethe solution can be expanded in powers of E and

f =m,n 0

f (m ,E n ) (41)

assuming that the solution is analytic both in and E .The equations are going to be solved order by order in both parameters, as illustrated

in (45). In the expansion of the Boltzmann equation in a small spinorbit coupling thereis in this case no ambiguity about which terms to include in a given equation. This shouldbe contrasted with the previous section, where we chose to include the precession term2b f a term of order O(2)though we discarded other terms of the same order, forexample in (24). The reproduced results seem to be based on hand-picking terms with

physical insight.

9. Boltzmann equation without the precession term

Due to the assumption in section 8, not only the equation but also the solution is linearizedin the spinorbit coupling. Together with the habitual linearization in the electric eldthis means that we consider

f = f (0) + f ( ) + f (E ) + f (,E ) (42)

where the superscripts denote the order in and E , respectively (e.g. f (E ) 0E 1). The

equilibrium distribution is f eq

= f (0)

+ f ( )

, where f (0)

= 1f FD .The left-hand side of the Boltzmann equation ( 8) is in the static, uniform case

df dt

= E k f + 2 O ( 2 )

f b (43)where the precession term must be neglected to order . However, leaving out theprecession term leads to some formal trouble in the case when N = 1. Thederivative E k = e i(E x iE y)( k + i k 1 ) + c .c. comes with a winding number 1. For a spinorbit coupled system the equilibrium polarization f eq = f (E = 0) alsohas a winding number, here N . Thus, the combination E k f comes with terms of

winding N 1. For N = 1 (e.g. Rashba) or N = 1 (e.g. linear Dresselhaus) the left-hand side therefore contains terms without angular dependence, as seen in ( 32). Suchterms cannot be matched by the collision integral, essentially because an equation like1 = d (f () f ( )) has no solution. (A collision integral cannot be a source/drain.)This lack of a solution might be related to the discarding of the principal part termsX in (21), but we have not been able to investigate this. As a simple remedy, we introduceinstead a small spin relaxation term, which could come from spin relaxation processes notrelated to the spinorbit coupling. The Boltzmann equation then reads

1S f + E k f = k W kk [( a) f + ( a) b f 0] (44)where it is assumed that 1S is much smaller than 1tr , is independent of and is anumber, though in general it could be a matrix.

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We now solve the Boltzmann equation ( 44) order by order in and E as shown inequation ( 45)

0E 0: 0 = k W kk ( a) f (0)1E 0: 1S f

( ) = k W kk ( a) f ( ) + ( a) b f (0)00E 1: E k f (0) = k W kk ( a) f (E )1E 1: 1S f

(,E ) + E k f ( )

=

k

W kk ( a) f (,E ) + ( a) b f (E )0 .

(45)

Note that terms known to be zero were left out 6. We are out to nd f (,E ) , which is of order 1E 1 and therefore the rst contribution that incorporates the combined effects of an electric eld and spinorbit coupling 7. However, to solve the 1E 1 equation one needsto solve the previous equations to nd out f ( ) and f (E ) .

The 0E 0 equation is consistent with f (0) = 1 f FD from (27). The spin relaxationterm in the 1E 0 equation decreases the equilibrium polarization in ( 27) into f ( ) = b a f FD with := (1 + 1S tr ) 1 1. The 0E 1 equation is solved by f (E ) = 1 tr E k kf FD . So nally at the 1E 1 equation one already knows the terms E k f ( )

and

k W kk ( a) b f

(E )0 . Collecting these contributions on the left-hand side we

get an equation of the same form as ( 29), but with the spin precession term replaced bythe spin relaxation term, and with F and G modied due to the factor in E k f ( ) .The equation for the n = 1 Fourier coefficients turns into

E

2

F + GiG iF

0

=

1S + 1cos i 1sin 0

i 1sin 1S + 1cos 0

0 0 1S + 1tr

f b1f c 1f z1

. (46)

For N = 1 one has cos = sin for a general impurity potential, leading the determinantof the matrix to be zero unless 1S = 0. (With

1S = 0 there is either no solution when

G = 0 or multiple solutions when G = 0.) For |N | = 1 one has cos = sin for realisticimpurity potentials, in which case 1S = 0 is not needed.

It is important to remember that the matrix does not depend on . The polarizationf (,E ) is therefore proportional to . It is also clear that f z = 0 and the spin Hall current j z are zerofor an arbitrary spinorbit coupling.

For N = 1 we can adopt the derivation in section 7 to show that t sy = mj zy

1S sy in real space. For our steady state case we therefore nd sy j zy = 0

in real space for N = 1. Likewise, sx = 0. For |N | = 1 the real space polarizationis trivially zero (see under equation ( 34)). Summarizing we have that for no N in theperturbative case does the electric eld alter any real space densities to lowest order in

6 For example f ( E ) = 0 since the electric eld alone leads to no polarization, therefore 1S f could be left out inthe equations of order 0 E x .7 Note that f (

2 ) or f ( E2 ) cannot contribute to the SHE. Therefore we do not need consider them, though one of

them need not be subleading to f ( ,E ) .

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. The only nonzero real space densities are the equilibrium spin currents j x and j y for|N | = 1. From f eq = f FD + b a f FD for N = 1 we nd j xy = j yx = (1 / 4)( 1)bF kF .

Table 1 gives a summary and offers a comparison of the different cases considered inthis paper. In this paper we do not consider the case N even, but we note that for |N | = 2an electric-eld-induced contribution to the real space spin currents j x and j y is not forcedto be zero by simple angular considerations. (This is analogous to s x(E ) and s y(E ) beingallowed to be nonzero only for |N | = 1.) The case N = 2 is relevant for pseudospincurrents for one valley in bilayer graphene. Note that this is not the SHE pseudospincurrent j z discussed in [41]. Actually, with the denition ( 5) a nonzero (pseudo)spincurrent j z is not possible when a = 0, the latter being the case in single and double layergraphene.

10. Electric eld induced corrections to the collision integralThe precession term has so far seemed like the only term that could involve the f zcomponent and lead to nonzero spin current j z . In this section we are going to see thatthe electric eld E modies the collision integral in a way that involves the f z component.However, the correction turns out to be of order 2.

In trying to incorporate in our Boltzmann equation all terms present to rst orderin our parameters we have so far left out terms by gradient expanding the right-handside of the Dyson equation ( 13) for GK only to zeroth order and not to rst order in theelectric eld. First order corrections have been accounted for, for example, in the case of electronphonon renormalization of the ac conductivity [42] (see also [33]) and have alsobeen discussed in the SHE and AHE literature (see e.g. [ 18] or [43]). However, a derivativesuch as 2E k S bs = sk

1E b , which occurs in the derivation of these corrections,gives a vector that remains in the plane. The collision integral becomes more complicatedbut does not involve the f z component.

The contribution discussed in this paper comes about in a slightly subtler way, andhas to the authors knowledge not been discussed previously. It comes from gradientexpanding the equation of motion also for the retarded Greens function G0R , which resultsin the correction G0R given by (20). With Bz = 0 one obtains

J =

k

W kk2

d2

(G0Rk fG 0Ak + G0Rk fG 0Ak + G0Rk fG 0Ak + G0Rk fG 0Ak )

= ss k W kk8 [(z S + S f z)E ( a sb 1)( )

+ ( z S + S f z )E ( a s b 1)( )]

= 12 k ( a) (W kk [zM f + M f z ]) (47)where for the last line the expansion ( 23) was used and where

M (k , k ) := k 1E b + k 1E b. (48)

Note that J 0 = 0, i.e. there is no contribution to the charge part of the equation onlyto the polarization part. Note particularly that J z = 0, which would give a nontrivialequation for f z . However, since M E and f this correction contributes to order

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2E as announced, in which case the result f z = 0 in section 9 is not changed to lowestorder.

One can ask if this correction could change the result f z = 0 for the Rashba case inthe nonperturbative scenario of section 6 where homogeneity in was not an issue. Thecorrection would then enter as an imaginary element of order 2E replacing the zero inthe bottom of the vector with F and G in (35). However, since 2cos

2sin = 0 for N = 1,

the inverse matrix has a zero zz-element. Therefore, the correction cannot contribute tof z1 in the Rashba case. In this respect the contributions discussed in the beginning of this section, on the other hand, could contribute, but we have not investigated them ina systematic way. The real space spin current j z would in any case remain zero, at leastfor a quadratic dispersion, due to the arguments in section 7.

11. Conclusions

This paper studied the intrinsic contribution to the spin Hall current in a spinorbitcoupled 2DEG by deriving a Boltzmann equation in the Keldysh formalism and solvingit in the uniform steady state case. The vector b determining the spinorbit couplingwas assumed to be of the form b = b(k)b() with b = k and bx + i by ei0 +i N . Wereproduced the common result that SHE vanishes for N = 1 (e.g. for Rashba coupling)but not for other N . We were able to give a new perspective to this vanishing by pointingout that a nonzero result leads the in-plane components of the polarization to divergewhen 0.

The mentioned treatment does not assume the response to be perturbative in . Wetherefore found it interesting to study the implications of assuming the response to beperturbative not only in the electric eld but also in . The precession termpreviouslythe prerequisite for the spin Hall currentmust then be left out to rst order in spinorbit splitting. The out-of-plane polarization f z becomes trivially zero and there seemsto be zero SHE for any winding. We also saw that all other real space densities have zeroelectric eld induced contributions.

Leaving out the spin precession term gives a Boltzmann equation for in-planepolarization ( f x , f y) that is unsolvable for N = 1. The unsolvability might be relatedto the left out principal parts, the inclusion of which would have been beyond the scope

of the present study. As an ad hoc remedy the precession term was replaced by a smallspin relaxation term. This could suggest that if the response is perturbative in , thennon-magnetic impurities are not enough for the existence of a steady state solution forthe polarization.

To cover all contributions to rst order in electric eld, in spinorbit splitting and inimpurity strength and concentration, we considered corrections to the collision integralthat come from going to rst order in electric eld in the gradient expansion of the self-energy side of the Dyson equation. One of the corrections, to our knowledge not discussedbefore, actually involves the f z component. However, this contribution is of order 2.Thus, the vanishing of the spin Hall current to lowest order in in the perturbative caseis not changed by these corrections.

Finally, the paper includes a detailed discussion of why a relaxation timeapproximation fails and a comment on pseudospin currents in bilayer graphene.

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Acknowledgments

The author wishes to acknowledge discussions with M L uffe, D Culcer, F Gethmann,A G Malshukov, K Morawetz, T Nunner, P Schwab, G Vignale, F von Oppen andR Winkler. This work was supported by the Swedish Research Council. The authoralso wishes to acknowledge a visitor grant of the Max Planck Institute for the Physics of Complex Systems.

Appendix A. Left-hand side of equation ( 29)

For b = k and bx + i by = e i0 +i N and f = b a f FD one has

E k f = E k k +1

kE k b a f FD = (E k b + NE c) a f FD + kE k b k a f FD .

(A.1)

The term k W kk ( a) b f (E )0 is here for brevity only evaluated for a point-likeimpurity potential, i.e. W kk = W constant, and constant density of states D(a) = m/ 2.Hence

1tr = k ( a)W kk (1 cos( )) = DW. (A.2)With ( a) = a ( a) = a ( a) a partial integration gives

k W ( a) b f (E )0 = da

d2 ( a) a (DW b f

(E )0 )

= da ( a) k a kf FD d2 E k b + a (k ) kf FD d2 b E k= |N | ,1

2

(E k b + NE c )(k a kf FD + a f FD )

= |N | ,12

(E k b + NE c )(k k a f FD + a f FD ) (A.3)

where it was used that b = b cos N c sin N and k = k cos sin . In thelast line it was used that a k = a ((d a/ dk) a ) = ( a v0) a + v0 2a = (( 1)/k ) a + k a

for v0 k 1

. For a non-constant D (i.e. for = 2) the result in ( A.3) is modied.Note also that it is only for point-like impurities that the contribution ( A.3) vanishes for|N | = 1.

Adding up (A.1) and (A.3) one obtains for N = 1 that F = 12 ( 2) a f FD +12 k k a f FD and G = ( 1)( a f FD +

12 k k a f FD ). For |N | = 1 the contribution ( A.3)

vanishes and one gets for example G F = N a f FD , needed for the result ( 37).

Appendix B. Failure of a relaxation time approximation

A useful approximation for the collision integral found in standard applications of theBoltzmann equation is the relaxation time approximation (RTA)

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where the relaxation time depends only on the absolute value k of the momentum. Itexpresses that the role of the collision integral is to relax the deviation from equilibriumf k := f k f eqk . For angle independent (i.e. momentum independent) W kk it can bederived from the collision integral to the left in ( B.1). Thus RTA should be a goodapproximation for weakly momentum dependent impurity potentials.

A prerequisite for the RTA is that it is consistent with the conserved quantities. Inparticular, the momentum integral of the Boltzmann equation should give the continuityequation t n + x j = 0 expressing the conservation of particle number. The right-hand side (given W kk = W k k ) indeed vanishes identically k J [f k ] = 0, expressingthat the collision term cannot in real space act as a source (or drain) of particles. Forthe RTA particle conservation is not automatic, since k f k / would be nonzero if f contained an angle independent component. However, in typical applications the deviationf = n f n exp(in) contains to lowest order only angle dependent terms.The RTA can be implemented also in the case of spin in the same way as in ( B.1), ormore generally by letting 1 be a matrix acting on f to allow for different relaxationtimes for different spin components of f . The RTA has been useful for example for thederivation of the DyakonovPerel spin relaxation (DPSR) mechanism [ 44] (see also [45]and [46]), found by deriving for initially polarized electrons the Bloch equations from theuniform, time-dependent Boltzmann equation for a Rashba-type spinorbit interaction.In that problem the electric eld is absent.

The general collision integral ( 17) still satises k J = 0 for all components, expressingthat not only particle number but also spin is conserved in collisions with non-magneticimpurities. Thus, a sensible RTA must still satisfy

k

f k / = 0. That is to say f cannot contain an angularly constant term. In the derivation of the DPSR this is satisedalthough n = 0 components are present and essential. The RTA is only needed in thepart of the Boltzmann equation that is of rst power in , and f ( ) is by simple inspectionseen to only contain the angular components n = 1.

We note that in the SHE setup studied in this paper the situation is very differentcompared to the DPSR problem. It turns out that a RTA always implies a nonzero SHE.For N = 1 we therefore do not even qualitatively reproduce the correct spin current.For |N | = 1 the SHE is on the other hand nonzero and can be captured by a RTA. In theN = 1 case the nonzero SHE is intimately related to the RTA failing to conserve spin.With the simple RTA J [f ] f / the real space equations for the Rashba/Dresselhaus

case in section 7 get modied to

t sy = 2mj zy sy/. (B.2)

A steady state no longer implies j zy = 0 but instead j zy = sy / 2m . Thus, if apolarization in real space is induced by the electric eld, then there is a nonzero SHE.The result in ( 39) shows that such a polarization is actually induced. Thus, the collisionintegral transformed into RTA form would enter as a nonzero spin source, simultaneouslyallowing for a nonzero SHE. Therefore, in contrast to the DPSR problem with Rashbacoupling, the RTA fails when applied to the SHE problem with Rashba coupling.

For N = 1 in the perturbative case, a RTA and equation ( B.2) do not lead to thesame contradiction since sx = sy = 0 in real space (see table 1). In the rest of thisappendix we give some further details on the nonperturbative case.

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In the SHE problem we look at a steady state situation and the expansion is donearound the equilibrium distribution instead of an initial polarization as in the DPSRproblem. The Boltzmann equation is expanded in powers of E rather than in powers of .The lowest order deviation f (E ) (from the equilibrium polarization f eq) will now containangle independent terms, causing the non-vanishing of k f (E ) in (39). Contrast this withordinary steady problems with an electric eld, where constant terms are not present inf (E ) E v f eq since f eq is independent of and E v comes with angular componentsn = 1. In the SHE problem on the other hand f eq b comes with angular components N , and in the Rashba/Dresselhaus case in particular N = 1. In the Rashba case f (E )can therefore contain angular components n = 1 1 = 2, 0, 2 in the Cartesian basis .(In the rotating basis given by {b, c , z } used in this paper this translates into the Fouriercomponents n = 1.) In particular, the presence of the n = 0 term means that a RTA

would be inconsistent with the impurity scattering having to conserve spin.The other side of the coinas displayed in ( B.2)is that the RTA allows for f z1 = 0and consequently a nonzero SHE. The analogue of equation ( 35) is with a general matrix 1 with constant elements is given by

E

2

k f eqbi 1k f

eqb

0

=

1bb 1bc

1bz

1cb 1cc 2b + 1cz 1zb 2b + 1zc 1zz

f b1f c 1f z1

. (B.3)

Note that now the left-hand side derives only from the term E k f eq , whereas the left-hand side of (35) had an additional contribution from the term

k W kk ( a) b f

(E )0

as seen in (28). This latter term cannot be captured by a RTA . However, this term wasessential for giving the left-hand side in ( 35) the crucial structure that all the componentsof the vector are proportional to each other, i.e. linearly dependent, as followed with G = 0.This structure makes it possible to nd a solution f z1 = 0 for suitable matrix elements in .(In (35) we had cos = sin .) In the vector of the left-hand side of ( B.3), on the other hand,the components k f eqb and k

1f eqb

are linearly independent. There is no natural choiceof matrix elements of 1 that with such a left-hand side can result in the cancellationsneeded to make f z1 vanish. (The elements of 1 are assumed to be at the most weaklydependent on k and should certainly not contain factors of FermiDirac distributions.)For the same reasons also a simple collision integral like J =

k W kk ( a) f fails

to reproduce the vanishing SHE because it gives a Boltzmann equation with the same

left-hand side as in (B.3).For the case |N | = 1 the term k W kk ( a) b f (E )0 vanishes for W constant(corresponding to constant), as seen in appendix A. The collision integral enters the

Boltzmann equation only with the simple contribution k W kk ( a) f , which for W constant can be put in the RTA form. Therefore, the RTA can qualitatively reproducethe correct result. Neither can there be any angularly constant terms in f (E ) for |N | = 1.Here the RTA works well, in contrast to the |N | = 1 case.

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