Non-Abelian gauge fields in the gradient expansion: Generalized Boltzmann and Eilenberger equations

Non-Abelian gauge fields in the gradient expansion: Generalized Boltzmann and Eilenberger equations

C. Gorini,¹P. Schwab,^1,2R. Raimondi,³and A. L. Shelankov^4,5

1Institut für Physik, Universität Augsburg, 86135 Augsburg, Germany

2Institut für Mathematische Physik, Technische Universität, Braunschweig, Germany

3CNISM and Dipartimento di Fisica “E. Amaldi,” Università Roma Tre, 00146 Roma, Italy

4Department of Physics, Umeå University, Umeå, Sweden

5Ioffe Physicotechnical Institute, RAS, St. Petersburg, Russia 共Received 7 April 2010; published 15 November 2010兲

We present a microscopic derivation of the generalized Boltzmann and Eilenberger equations in the presence of non-Abelian gauges for the case of a nonrelativistic disordered Fermi gas. A unified and symmetric treat- ment of the charge关U共1兲兴and spin关SU共2兲兴degrees of freedom is achieved. Within this framework, just as the U共1兲Lorentz force generates the Hall effect, so does itsSU共2兲counterpart gives rise to the spin Hall effect.

Considering elastic and spin-independent disorder we obtain diffusion equations for charge and spin densities and show how the interplay between an in-plane magnetic field and a time-dependent Rashba term generates in-plane charge currents.

DOI:10.1103/PhysRevB.82.195316 PACS number共s兲: 72.25.⫺b

I. INTRODUCTION

Spin-charge-coupled dynamics in two-dimensional elec- tron共hole兲gases has been the focus of much theoretical and experimental work over the last two decades.^1,2Its rich phys- ics belongs to the field of spintronics and shows much po- tential for applications. Thanks to spin-orbit coupling all- electrical control of the spin degrees of freedom of carriers, as well as magnetic control of the charge one, is, in principle, possible.³Particularly interesting from this point of view are phenomena such as the spin Hall effect and the anomalous Hall effect. For a review of both, see Refs.4–6, respectively.

In general terms the theoretical problem at hand is that of describing spin-charge-coupled transport in a disordered sys- tem. In the semiclassical regime, defined by the condition

␭FⰆl, a Boltzmann-type treatment is sensible and expected to provide physical transparency. Here␭Fis the Fermi wave- length and l a typical length scale characterizing the system—say, the mean free path or that defining spatial in- homogeneities due to an applied field. The Boltzmann equa- tion is a versatile and powerful tool for the description of transport phenomena,⁷ and various generalizations to the case in which spin-orbit coupling appears have been proposed.^8–10 More general Boltzmann-type equations have also been obtained.^11–13 In both cases though, much of the physical transparency is lost due to a complicated structure of the velocity operator and of the collision integral. A semi- classical approach based on wave-packet equations^14,15 can partially circumvent these complications, though it is limited to the regime ⌬so␶/បⰇ1 with ⌬so the spin-orbit energy and ␶ the quasiparticle lifetime. On the other hand it was pointed out in different works^16–22 that Hamiltonians with a linear-in-momentum spin-orbit coupling term can be treated in a unified way by introducing SU共2兲 gauge potentials in the model. Taking as an example the Rashba Hamiltonian

H_R= p²

2m+␣共p_y␴^x−px␴^y兲, 共1兲 where␣ is the spin-orbit coupling constant, one can identi- cally transform it to

HR=关p+␥AR a␴^a/2兴²

2m + const. 共2兲

Here summation over a=x,y,z is implied, ␥ is the SU共2兲 coupling constant, and the components of the SU共2兲 vector potential are ␥共AR兲y

x= −␥共AR兲x

y= 2m␣^, 共AR兲x x=共AR兲y

y= 0.

From this point of view a different Hamiltonian, say, the Dresselhaus one, simply corresponds to a different choice of the vector potential. An additional advantage of this ap- proach is that it ensures the proper definition of physical quantities such as spin currents and polarizations.^21,23 More generally, the use of the non-Abelian language shows flex- ibility and potential and has already proven useful in differ- ent contexts. For example, in Ref. 19it was used to predict the existence of a “persistent spin helix” in systems with equal strength Rashba and Dresselhaus couplings. Such a helix was later observed²⁴ and soon after exploited.²⁵ The authors of Ref. 20 on the other hand employed it in their proposal of a perfect spin filter based on mesoscopic inter- ference circuits. Finally, since non-Abelian potentials can also be created optically,^26–29the range of applications of the approach goes beyond systems described by Hamiltonians like Eq. 共2兲. Indeed, even in solid-state systems higher- dimensional models such as the one considered in Ref. 30 would fit into the picture.³¹

Our goal in the present paper is therefore to put the non- Abelian approach in the semiclassical regime on firm ground, in order to obtain kinetic equations with a clear physical structure and as broad a field of application as pos- sible. More precisely, we derive an SU共2兲⫻U共1兲 covariant Boltzmann equation in the framework of the Keldysh³² mi- croscopic formalism. In the covariant approach a completely

symmetric treatment of the charge and spin degrees of free- dom is achieved.³³Also, we discuss the more general Eilen- berger equation^34–36 derived with the help of the so-called

␰-integrated Green’s function technique. The latter allows one to justify the Boltzmann equation in the case when the momentum is not a good quantum number due to impurity or other scattering, and the notion of particles with a given mo- mentum is ill defined. The results obtained hold in the me- tallic regime⑀FⰇប/␶^with⑀Fthe Fermi energy and ប/␶^the level broadening due to disorder, and as long the spin split- ting due to the关SU共2兲兴gauge fields is small compared to the Fermi energy, ⌬soⰆ⑀F, but for arbitrary values of ⌬so␶/ប.

We emphasize that, within this approach, not only the ap- plied electric and magnetic fields but also the internal spin- orbit-induced ones can be position and time dependent.

Our guideline for the present work is the familiar U共1兲 gauge-invariant Boltzmann equation. This reads⁷

冉

^⳵T+pm^ⴱ·ⵜR+F·ⵜpⴱ

冊

^{f共T,R,pⴱ兲=I关f兴, 共3兲}

where the electron-distribution function f at timeT and po- sition R is a function of the gauge-invariantkinematic mo- mentump^ⴱ=p+eA共T,R兲 共rather than the canonical momen- tump兲, and the Lorentz forceF= −e关E+共p^ⴱ/m兲∧B兴appears.

The right-hand side 共rhs兲 of Eq. 共3兲 contains the collision integral.

The paper is organized as follows. In Sec.II we start by recalling the quantum derivation of the Boltzmann equation, which allows us to introduce the general formalism in a pur- poseful way. In Sec. III the generalized Boltzmann and Eilenberger equations are obtained. In Sec. IVthe diffusive regime is discussed and spin-charge-coupled diffusion equa- tions are derived. Finally, Sec.Vshows two example calcu- lations. The first involves a study of the Bloch equations in the static limit in the presence of in-plane electric and mag- netic fields whereas the second is concerned with a novel effect, in which an in-plane charge current is generated by the interplay of an in-plane magnetic field and a time- dependent Rashba term.

We use a system of units where the Planck constant ប= 1 and e=兩e兩.

II. GRADIENT EXPANSION

The original quantum-mechanical derivation of the classi- cal Boltzmann equation by Keldysh³²has been exploited and extended by many authors, in particular, by Langreth³⁷ and Altshuler.³⁸Since it is very instructive, we outline the proce- dure following Ref.34and consider for simplicity’s sake the case of free electrons in a perfect lattice. Our aim will be to generalize it to the non-Abelian case and to later introduce disorder. The main character is the Green’s function in Keldysh spaceGˇ

Gˇ =

冉

^G0^{R G}G^KA

冊

^{. 共4兲}

G^R,A are the standard retarded and advanced Green’s func- tions whereasG^Kis the Keldysh Green’s function which car-

ries the statistical information about the occupation of the energy spectrum. One starts from the left-right-subtracted Dyson共quantum kinetic兲equation

−i关G₀⁻¹共1,1⬘^兲丢, Gˇ共1⬘^{,2兲兴= 0, 共5兲} where 1 , 1⬘, 2 are generalized coordinates containing space and time coordinates as well as spin and Keldysh space共and possibly additional兲 indices. The square brackets denote the commutator, the symbol “_丢” indicates convolution/matrix multiplication over the internal variables/indices and

G₀⁻¹共1,1⬘兲=

再

^{i⳵t1− 关−iⵜx12m+eA共1兲兴2+e⌽共1兲}

冎

^{␦共1 − 1⬘兲}

共6兲 describes free electrons coupled to an external electromag- netic field. In order to introduce the gradient expansion we write the Green’s function in the mixed representation in terms of Wigner coordinates

Gˇ共X,p兲=

冕

^{dxe−ipxGˇ共X,x兲, 共7兲}

whereX=共关t1+t₂兴/2 ,关x1+x₂兴/2兲is the center-of-mass coor- dinate and x=共t₁−t₂,x₁−x₂兲the relative one,

X=共T,R兲, x=共t,r兲, p=共⑀,p兲, px= −⑀^t+p·r.

Notice that in the presence of both translational symmetry with respect to time and space, the dependence on X drops out and convolution products as those in Eq. 共5兲 would re- duce to simple products in Fourier space. In the presence of external fields or in nonequilibrium conditions, Fourier trans- forms, as defined in Eq.共7兲, of convolution products can be systematically expanded in powers of derivatives with re- spect to the center-of-mass coordinates. To leading order, the gradient expansion applied to Eq.共5兲yields

−i关G₀⁻¹_丢, Gˇ兴 ⬇⳵_⑀^G0−1⳵TGˇ −⳵TG₀⁻¹⳵_⑀^{Gˇ −}ⵜpG₀⁻¹·ⵜRGˇ +ⵜRG₀⁻¹·ⵜpGˇ. 共8兲 Notice that such an expansion is in our case justified by the assumption that pF共⑀F兲 is the biggest momentum 共energy兲 scale of the problem. On the rhs in the above bothGandG₀⁻¹ are functions of 共X,p兲 with

G₀⁻¹共X,p兲=⑀− 关p+eA共X兲兴²

2m +e⌽共X兲. 共9兲 Integrating the Keldysh component of Eq. 共8兲 over the en- ergy ⑀ leads to the lhs of the Boltzmann equation for the distribution function

f共X,p兲 ⬅1

2

冋

^{1 +}

^冕

^{2d␲⑀iGK共X,p兲}

册

^{. 共10兲}

Note that the above defined quantity is not gauge invariant.

For this reason if one is to obtain a result in the form of Eq.

共3兲 a shift of the whole equation must formally be performed—i.e., one must send p→p^ⴱ in Eq. 共8兲. This is done in Ref. 37, though such a shift could also have been

performed before the gradient expansion. The latter is the way followed in Ref.38, where the mixed representation is right from the beginning defined in terms of the kinematic momentum

Gˇ共X,p兲→

冕

^{dxe−i关p+eA共X兲兴xGˇ共X,x兲, A=共⌽,A兲. 共11兲}

Unfortunately the simple and convenient concept of a “shift”

does not work when non-Abelian gauges are considered.

This is because the nature of the transformation关Eq.共11兲兴is actually geometric, a fact that manifests itself only when dealing with noncommuting fields. At its core lies the Wilson lineU_⌫共2 , 1兲, which is the exponential of the line integral of the gauge potential along the curve ⌫ going from 1 to 2 共see Ref. 39兲

U_⌫共2,1兲 ⬅Pe^{−i␩兰dyA共y兲}. 共12兲 Here the symbolPstands for path ordering along⌫whereas

␩is a general coupling constant—in theU共1兲case it reduces toe. In the spirit of the gradient expansion the integral in Eq.

共12兲is evaluated for small values of the relative coordinatex, and it is thus reasonable to pick ⌫ as the straight line con- necting the two points. For theU共1兲gauge it is seen that

U_⌫共2,1兲 ⬇e^{−ieA共X兲x} 共13兲 which is precisely the phase factor appearing in Eq.共11兲. In other words, the “shifting” of Eq.共8兲should properly be seen as the transformation

关G₀⁻¹共1,1⬘^兲丢, Gˇ共1⬘^,2兲兴

→U_⌫共X,1兲关G₀⁻¹共1,1⬘兲丢, Gˇ共1⬘^,2兲兴U_⌫⬘共2,X兲, 共14兲 where ⌫共⌫⬘兲 is a straight line from 1共X兲 toX共2兲 共Ref. 40兲 and matrix multiplication over the internal indices between U_⌫,U_⌫⬘and the commutator is implied.

With this hindsight about the nature of the shift required to obtain Eq.共3兲, it is possible to generalize the construction to the non-Abelian case. A general gauge transformation is defined as a local rotation of the second-quantized annihila- tion fermionic field␺

␺⬘^{共1兲=V共1兲}␺共1兲, V共1兲V^†共1兲= 1. 共15兲 For the gauge potential one has

␩^A⬘共1兲=V共1兲关␩A共1兲+i⳵1兴V^†共1兲. 共16兲 Notice that this is now a tensor with both real space and gauge indices

␩A共1兲=共e⌽+␥⌿^at^a/2, eA+␥A^at^a/2兲, 共17兲 where we found convenient to separate the Abelian coupling constant e from its non-Abelian counterpart ␥^{. The non-} Abelian scalar potential␥⌿^at^a/2 describes a Zeeman term—

i.e., of the kindb·s. Heresis the spin of the carriers whereas bcould be an applied magnetic field or, in a ferromagnet, the exchange field due its magnetization. Thet^a/2’s are the gen- erators of the given symmetry group, which in the SU共2兲 case become the Pauli matrices,t^a=␴^a, a=x,y,z. In the fol- lowing boldfaced quantities will indicate vectors in real

space whereas the presence of italics will denote a gauge structure—which, as in the above, will sometimes be written down explicitly. A sum over repeated indices is always im- plied. Since a Wilson line transforms covariantly, i.e., U_⌫⬘共2 , 1兲=V共2兲U_⌫共2 , 1兲V^†共1兲, it is possible to define a Green’s function

G˜ˇ共1,2兲 ⬅U_⌫共X,1兲Gˇ共1,2兲U_⌫共2,X兲 共18兲 which islocallycovariant, i.e.,

G˜ˇ⬘共1,2兲=V共X兲G˜ˇ共1,2兲V^†共X兲. 共19兲 In terms of G˜^Kwe can define a distribution function

f共X,p兲 ⬅1

2

冋

^{1 +}

^冕

^{2d␲⑀iG˜K共X,p兲}

册

^共20兲

which will be the natural generalization of f共X,p^ⴱ兲 from Eq.共3兲. The procedure is then clear:共1兲transform the kinetic equation according to Eq. 共14兲;共2兲expand the Wilson lines 共see below兲; 共3兲 perform a gradient expansion and write everything in terms of G˜^K共X,p兲; and共4兲integrate over the energy ⑀ to obtain the Boltzmann equation or over

␰⬅p²/2m−␮to end up with the Eilenberger equation.

Postponing the discussion of the last point to the next section, we now consider the general expression

G₀⁻¹共X,p兲=⑀^−H共X,p兲

=⑀^{−关p+eA共X兲+}␥A^a共X兲t^a/2兴²

2m +e⌽共X兲

+␥⌿^a共X兲t^a/2. 共21兲

In the Rashba model, Eq. 共2兲, one, for example, identifies t^a=␴^a, A^a=AR

a. 共22兲

The procedure outlined above共points 1–3兲leads to a locally covariant equation for G˜ˇ accurate to order 关共⳵X⳵p兲共A⳵p兲,共A⳵p兲²兴 共see Ref.41兲: in the mixed representa- tion language we have formally two expansion parameters,

⳵X⳵pⰆ1—the standard gradient expansion one—and A⳵pⰆ1—coming from the gauge fields. In the SU共2兲 case the latter corresponds to the physical assumption that the spin-orbit energy be small compared to the Fermi one,

⌬so/⑀FⰆ1. Even though our treatment is valid for any non- Abelian gauge, we now pick the SU共2兲 gauge for definite- ness’ sake. In this case steps 1–3 lead to

冉

^˜⳵T+m^p ·ⵜ˜

R− p

2m·兵关eE+␥E兴⳵⑀, ·其+1

2兵F·ⵜp, ·其

冊

^{G˜ˇ = 0,}

共23兲 where the symbol兵· , ·其 denotes the anticommutator. The co- variant共wavy兲derivatives are

⳵

˜_T=⳵T−i␥关⌿, ·兴, ⵜ˜_R=ⵜR+i␥关A, ·兴 共24兲 whereas the generalized Lorentz force reads

F= −e

冋

^E+p∧mB

册

U共1兲 SU共2兲

−␥

冋

^E+p∧mB

册

^.

共25兲 The fields are given as usual in terms of the field tensor F, but this has now anSU共2兲⫻U共1兲structure

Ei=F_0i⁰ = −⳵TAi−ⵜR_i⌽, 共26兲 Ei

a=F_0i^a = −⳵TAi

a−ⵜR_i⌿^a+i␥关⌿,Ai兴^a, 共27兲

Bi=1

2⑀ijkF0jk, Bi a=1

2⑀ijkFjk

a, 共28兲

Fjk0 =ⵜR_jAk−ⵜR_kAj, 共29兲 Fjk

a =ⵜR_jAk

a−ⵜR_kAj

a+i␥关Aj,Ak兴^a. 共30兲 Note that in order to obtain Eq.共23兲it is sufficient to expand the Wilson lines to first oder inx, e.g.,

U_⌫共X,1兲 ⬇1 +␩^Aix

2. 共31兲

This is not true for a general convolution of the kind 关F共1 , 1⬘^兲丢, G共1⬘^{, 3兲兴 with F共1 , 1}⬘^兲 a function with a more complicated structure than that of G₀⁻¹共1 , 1⬘兲. Such a case would require a second order expansion, e.g.,

U_⌫共X,1兲 ⬇1 +␩^Aix

2 +␩⳵XAix²

8 −␩^2A2x2

8 共32兲

and would lead to a rather more complicated equation.

To complete our preparatory work for the derivation of the kinetic Boltzmann or Eilenberger equations, we need to introduce the effect of disorder. Within the Keldysh formal- ism this is done by the addition of a self-energy contribution on the rhs of Eq. 共5兲

−i关⌺ˇ共1,2兲丢, Gˇ共2,3兲兴, 共33兲 which can be manipulated just as the “free” 共G₀⁻¹兲 term. In spin-orbit-coupled systems the presence of disorder can have a number of interesting effects. Indeed, phenomena such as the spin Hall effect, anomalous Hall effect, or related ones can have both an intrinsic and an extrinsic origin.^3–5 This depends on whether they arise from fields due to the band or device structure or from those generated by impurities. In the latter case skew-scattering and side-jump contributions to the dynamics appear.^42,43 For a discussion of these issues see Refs.5and44–47. In the following we limit ourselves to the treatment of intrinsic effects in the presence of spin- independent disorder. We consider elastic scattering with probability W=W共p−p⬘兲 and quasiparticle lifetime

␶⁻¹= 2␲兺p⬘␦共⑀p−⑀p⬘兲W共p−p⬘兲. In the Born approximation, the disorder self-energy in the mixed representation reads

⌺ˇ共X,p,⑀兲=

兺

p⬘

W共p−p⬘^兲Gˇ共X,p⬘^,⑀兲. 共34兲

From Eq. 共34兲 one obtains that the locally covariant self- energy ⌺˜ˇ is

⌺˜ˇ共X,p,⑀兲=

兺

p⬘

W共p−p⬘兲G˜ˇ共X,p⬘^,⑀兲, 共35兲 Which, in turn, implies

˜I关G˜兴= −i关⌺˜ˇ,G˜ˇ兴. 共36兲 Note that for a leading order description of the coupling be- tween spin关SU共2兲兴and charge关U共1兲兴, correctionsO共A⳵p兲in the collision integral are enough. Notice also that, whereasGˇ is peaked at the different folds of the spin-split Fermi sur- face, the peaks of G˜ˇ are “shifted” and thus located on the Fermi surface in the absence of spin-orbit coupling.

III. BOLTZMANN AND EILENBERGER EQUATIONS The question of whether to integrate the locally covariant kinetic equation with respect to⑀or to␰^=p2/2m−␮depends on the physical situation. If the spectral densityi共G^R−G^A兲is not “␦-like” as a function of⑀the energy integration is for- mally impracticable. The ␰integration on the other hand is capable of justifying a Boltzmann-type approach even when the first approach fails or looks severely limited.^34,35For the case considered of a degenerate gas of free electrons collid- ing elastically with impurities both procedures are viable, provided the condition ⑀F␶Ⰷ1 holds—since, as mentioned before, all quantities appearing in Eq.共36兲are peaked at the

⌬so= 0 Fermi surface.

A. Boltzmann (⑀integration)

Energy integration of the Keldysh component of Eqs.共23兲 and 共36兲 yields a Boltzmann-type kinetic equation for the 2⫻2 matrix distribution functionf共X,p兲

冉

^{˜⳵T+mp ·ⵜ˜R+21兵F·ⵜp, ·其}

冊

^{f共X,p兲=˜I关f兴 共37兲}

with the covariant derivatives and the generalized Lorentz forceF defined, respectively, as in Eqs.共24兲and共25兲, and where the collision integral reads

˜I关f兴= − 2␲

兺

p⬘

W共p−p⬘兲␦共⑀p−⑀p⬘兲关f共X,p兲−f共X,p⬘兲兴. 共38兲 Notice that Eqs.共37兲and共38兲are formally valid both in two and three dimensions. However, since the physical system we have in mind is a two-dimensional electron gas, from now on we restrict ourselves to two dimensions. Observable properties are conveniently expressed via the matrix density,

␳, and current, J,

␳共X兲=

冕

共2^d␲^2p兲²f共X,p兲, 共39兲

J共X兲=

冕

共2^d␲^2p兲² p

mf共X,p兲, 共40兲 which obey the following generalized continuity equation:

⳵

˜_T␳共X兲+ⵜ˜

R·J共X兲= 0, 共41兲

derived by integrating Eq.共37兲over the momentum. Observ- ables such as the particle and spin densities,nands^a, and the particle and spin currents, j⁰andj^a, can be evaluated as

n共X兲= Tr关␳共X兲兴, 共42兲

s^a共X兲=1

2Tr关␴^a␳共X兲兴, 共43兲

j⁰共X兲= Tr关J兴, 共44兲

j^a共X兲=1

2Tr关␴^aJ兴. 共45兲 One can check that these expressions agree with their micro- scopic definitions.^18,21

Equation共37兲is the first main result of the paper. Though the idea of rewriting spin-orbit interaction in terms of non- Abelian gauge fields is no novelty, we are not aware of a Boltzmann formulation in the above form. Whereas in Refs.

8 and9the collision integral and the velocity are nondiago- nal in the charge-spin indices, here their structure is simpler.

The gauge fields appear only in the covariant derivatives, describing precession of the spins around the external mag- netic field and the internal spin-orbit one, and in the gener- alized Lorentz force, which couples the spin and charge channels.

B. Eilenberger (␰integration)

The integration over␰^{of Eqs.}共23兲and共36兲yields in two dimensions the Eilenberger equation

冉

^{˜⳵T+vFpˆ·ⵜ˜R−12⳵⑀}

再 ^冋

^{pm共⑀兲·共eE+␥E兲}

^册

^{, ·}

冎

+ 1 2pF

兵F共pF,␸兲·关−pˆ+␸^ˆ⳵␸兴, ·其

冊

^˜gK

= − 2␲^N0

冕

^d2␸␲^⬘W共␸⁻␸⬘兲关g˜^K共␸兲−˜g^K共␸⬘兲兴, 共46兲 wherepˆ=共cos␸^{, sin}␸兲,␸^ˆ=共−sin␸^{, cos}␸兲,W共␸⁻␸⬘兲is the scattering amplitude at the Fermi surface and ˜g^K is the Keldysh component of the covariant quasiclassical Green’s function

g

˜ˇ共X,␸^,⑀兲 ⬅ i

␲

冕

^{d␰G˜ˇ共X,␸,⑀,␰兲. 共47兲}

Notice that the energy derivative ⳵⑀acts on the whole anti- commutator, i.e., on˜g^Ktoo. Just as in the Boltzmann case, and as opposed to what happens in the literature,^12,13 the velocity and the collision integral have here a simple diago-

nal structure, whereas the gauge fields appear only in the covariant derivatives and force terms. The collision integral will be extensively discussed in the Appendix to make an explicit comparison with Ref.12possible. Integration of Eq.

共46兲 over the energy and the angle leads again to the conti- nuity Eq. 共41兲, this time with densities and currents ex- pressed in terms of˜g

␳共X兲= −N₀

2

冕

^{d⑀具g˜K典, 共48兲}

J共X兲= −N₀

2

冕

^{d⑀vF具pˆ g˜K典, 共49兲}

where 具¯典 denotes the angular average. Recall that when expressing physical quantities in terms of the standard qua- siclassical Green’s function gˇ=i/␲兰d␰^Gˇ equilibrium high- energy contributions are missed.^34,36 For instance, Eq. 共48兲 for the particle density when only U共1兲 fields are present would be written, in terms ofg^K, as

␳共X兲= −N₀

2

冕

^{d⑀具gK典+N0e⌽共X兲 共50兲}

with the second term due to the scalar potential originating from the high-energy part. A virtue of the present formula- tion is that such contributions are by construction included in the covariant˜g. Moreover notice that, whereas in the pres- ence of spin-orbit coupling the usual normalization condition gˇ²= 1 is modified and becomes momentum dependent,¹³ in the covariant formulation ˜ˇg²= 1 holds—see the Appendix.

The normalization condition is established by direct calcula- tion “at infinity,” i.e., where, far from the perturbed region, the Green’s function reduces to its equilibrium form. It plays the role of a boundary condition imposed on Eq. 共46兲, and thus defines its solution uniquely.^48,49 In the presence of in- terfaces between different regions wave functions have to be matched, and this can be translated into a condition to be fulfilled by the quasiclassical Green’s function on either side of the interfaces.^50,51 Recently some very general such boundary conditions for multiband systems were obtained,⁵² though valid only as long as the spin and charge channels are decoupled. When this is not anymore the case, things are complicated by the momentum dependence of the normaliza- tion and, as far as we are aware of, beyond the present treat- ment of boundaries. The covariant formulation in terms of˜g suggests, however, the possibility for a nontrivial extension of the known boundary conditions to the case in which spin and charge channels are coupled, precisely because of the simple normalization of˜g.

IV. DIFFUSIVE REGIME

Our goal in this section is the discussion of spin-charge- coupled dynamics in the diffusive regime. Formally, the

“non-Abelian” Boltzmann Eq. 共37兲can be solved just as in the U共1兲 case. We expand the angular dependence of the distribution fin harmonics, f=具f典+ 2pˆ·f+¯and use the ex- pansion in Eq.共37兲to obtain an explicit expression forf,

f⬇−␶trp 2mⵜ˜

R具f典−␶tr

2 具pˆ兵F·ⵜp,具f典其典

−␶tr

2具pˆ兵F·ⵜp,共2pˆ·f兲其典 ⬅f_diff+f_drift+f_Hall. 共51兲 In the above ␶tris the usual transport time

1

␶tr

= 2␲^N0

冕

^d2␸␲^⬘W共␸⁻␸⬘兲关1 − cos共␸⁻␸⬘兲兴, 共52兲 which depends on the energy ␰⁼⑀p−⑀F through the scatter- ing probability W共␸⁻␸⬘兲=W共␰^,␰⬘^;␸⁻␸⬘兲兩_␰=␰⬘. The diffu- sion term f_diff is related to the 共covariant兲 derivative of the angular average of the distribution function, i.e., to the de- rivative of the charge and spin densities. The drift termf_drift arises from the second term on the rhs of Eq.共51兲, in which only the “electric” part of the Lorentz force, i.e.,

−关eE+␥E兴, contributes. The Hall component f_Hallcomes in- stead from the third term on the rhs of Eq.共51兲and is due to the “magnetic” part of the Lorentz force, −p∧关eB+␥B兴/m.

Using Eq.共51兲into Eq.共40兲one finally has J=

冕

共2^d␲^2p兲²

p

m关具f典+ 2pˆ·f+¯兴

⬇

冕

共2^d␲^2p兲² p

m2pˆ·关fdiff+f_drift+f_Hall兴

⬅Jdiffusion+Jdrift+JHall. 共53兲

The drift current is straightforwardly computed Jdrift=1

2兵␴共␮兲,eE+␥E其, ␴共␮兲= −N₀D共␮兲, 共54兲 whereN₀is the density of states,D共␮兲the energy-dependent diffusion constant, and␮^the共spin-dependent兲electrochemi- cal potential. Since we assume fields that are small compared to the Fermi energy, it is often sufficient to replaceD共␮兲by its value at the Fermi energy and in the absence of theU共1兲 andSU共2兲 fields. In the examples we discuss it will be im- portant to go one step beyond this simple approximation, for which we obtain

D共␮兲 ⬇D共⑀F兲+⳵_␰^D共␳^−N0⑀F兲/N₀ 共55兲 with

D共⑀F兲=v_F²␶tr

2 , ⳵_␰^D=␶tr

m共1 +␥0/2兲, 共56兲

␥0/2 =mv_F²

␶tr

⳵␰␶tr. 共57兲 The factor ␥0 is defined to make direct contact to Ref. 53.

Notice that due to the expansion in Eq. 共55兲 the diffusion constantD共␮兲becomes a spin-dependent object,

D共␮兲=D⁰+D^a␴^a 共58兲 withD⁰⬇D共⑀F兲andD^a⬇⳵_␰^Dsa/N₀.

The calculation of the diffusion current is slightly more involved: the momentum integration is delicate, since the

integrand has a nontrivial matrix structure and is out of equi- librium. In order to “extract” such a structure we first write

Jdiffusion⬅−1

2兵D,ⵜ˜␳其, 共59兲 thus defining a diffusion constant Dwhich is now a matrix.

Extending the Einstein relation to the present non-Abelian case will giveDan explicit form. At equilibrium one has

␳eq=N₀关e⌽+␥⌿兴+N₀⑀F, 共60兲 and as—again, at equilibrium—the diffusion current bal- ances out the drift one

Jdrift= −Jdiffusion=1

2兵D,ⵜ˜␳eq其= −1

2兵D,N0关eE+␥E兴其 共61兲 there follows:

D=D共␮兲 共62兲 as to be expected.

The Hall term f_Hall can be obtained from the equation implicit in Eq.共51兲

f_Hall= ␶tr

2m兵eB+␥B,∧f其 共63兲 from which we find

JHall= 1

4m兵eB+␥B,∧兵␶tr共␮兲,J其其 共64兲 with

␶tr共␮兲=␶tr+⳵␰␶tr共␳^−N0⑀F兲/N0 共65兲

=␶tr+␥0

2

␶tr

mv_F²

␳^−N0⑀F

N₀ . 共66兲

To be more explicit we give the expressions for the particle current j⁰and spin currentj^a

j⁰= −D共ⵜn+ 2eN₀E兲− 2D^a

冉

^{关ⵜ˜ s兴a+ ␥N20Ea}

冊

−e␶tr

m j⁰∧B−␥␶tr

m j^a∧B^a, 共67兲

j^a= −1

2D^a共ⵜn+ 2eN₀E兲−D

冉

^{关ⵜ˜ s兴a+␥N}2⁰E^a

冊

−e␶tr

m j^a∧B−␥␶tr

4mj⁰∧B^a. 共68兲

Here we included the Hall current only in the leading ap- proximation, i.e.,␶tr共␮兲⬇␶tr共⑀F兲. The diffusion equations for charge and spin are obtained by inserting Eqs.共67兲and共68兲 into the continuity Eq.共41兲.

V. TWO EXAMPLES A. Effect of an in-plane magnetic field

As a first simple example that shows how the formalism works, we obtain and solve the Bloch equations for a Rashba two-dimensional electron gas 共2DEG兲 driven by an electric field alongxand in the presence of an in-plane Zeeman field along x. This is the same geometry considered in Refs. 47, 53, and 54.

TheU共1兲fields read

E=共E,0,0兲, B= 0 共69兲 while, since the Zeeman field enters the Hamiltonian through the scalar potential␥⌿^x␴^x/2⬅b^x␴^x/2, theSU共2兲 ones are

␥E= 2m␣b^x共␴^z/2,0,0兲, ␥B= −共2m␣兲²共0,0,␴^z/2兲. 共70兲 From the expressions for the currents derived in the previous section, one obtains in the homogeneous limit a set of Bloch equations which generalizes those appearing in Ref.47to the case of angle-dependent scattering—but in the absence of extrinsic effects—namely,

s˙= −⌫ˆ关s−bN₀/2 +e␣␶trN₀zˆ∧E兴

−关b− 2e␣␶tr共1 +␥0/2兲zˆ∧E兴∧s. 共71兲 Here ⌫ˆ= 1/␶DPdiag共1 , 1 , 2兲 is the relaxation matrix with 1/␶DP=共2m␣兲²Dthe Dyakonov-Perel relaxation rate. Notice that the electric field in the first and in the second term on the rhs of Eq.共71兲has a different origin. While the first term is traced back to the 共spin兲 Hall current and therefore to the SU共2兲magnetic field, the second term can be traced back to the drift current. The factor␥0that appears due to the energy dependence of the scattering time has an important impact on the static solution of the Bloch equations. When ␥0= 0 we find

s=bN₀/2 −e␣␶trN₀zˆ∧E, 共72兲 i.e., the effects of the Zeeman and the electric field on the spin polarization are simply additive. This is not anymore the case if ␥0⫽0, in which case we find in the limit of weak electric and magnetic fields共in our geometry both inxdirec- tion兲

s^x=s_eq^x , s^y= −e␣␶trN₀E, s^z= −␥0

e␣␶trN₀Eb^x 4共2m␣兲²D,

共73兲 that is, in-plane fields generate an out-of-plane spin polarization.^53,55In the aboves_eq^x =b^xN₀/2.

B. Charge current from time-dependent spin-orbit coupling The Rashba spin-orbit coupling constant␣arises from the potential confining the 2DEG and is thus tunable by a gate voltage: if the latter is time dependent so is the former. Let us then consider the Rashba Hamiltonian for a time-dependent Rashba parameter, ␣→␣共T兲. In the non-Abelian language

this means that theSU共2兲vector potential becomes time de- pendent, and therefore a spin-dependent electric field is gen- erated. Explicitly we have

␥E= 2m␣^˙共␴^y/2,−␴^x/2,0兲, 共74兲

␥B= −共2m␣兲²共0,0,␴^z/2兲 共75兲 with␣^˙=⳵T␣^{. TheSU}共2兲electric field leads to the appearance of in-plane spin currents, as discussed in Ref.56. However, it does not generate a charge current, since it acts with opposite sign on particles with different spin: the net field obtained after averaging over all particles is zero. This is not anymore the case if a magnetic field is also present.

Say the latter points in x direction, then a nonzero average electric field in the y direction appears, given by

␥具Ey典= −m␣^˙具␴x典= −2m␣^{˙ sx}/n 共here具¯典 denotes the average over all particles兲. We then expect a particle current in ydirection of the order jy= −2DN₀␥具Ey典. We now make the argument quantitative. Let us apply an in-plane Zeeman field along x, as shown in Fig. 1. Then the SU共2兲 electric and magnetic fields are

␥E= 2m共␣^˙␴^y/2 +␣^bx␴^z/2,−␣^˙␴^x/2,0兲, 共76兲

␥B= −共2m␣兲²共0,0,␴^z/2兲. 共77兲 Note that the structure of the Bloch Eq.共71兲isnotmodified so that the stationary spin density is

s^x=s_eq^x , s^y= 0, s^z= 0 共78兲 withs_eq^x =N₀b^x/2 as before. As expected, theSU共2兲 electric field generates a particle current flowing alongy,

y z

y x

V (t)in V_out(t)

V_out(t) 0 0

b

^x

j

_y

b

^x

j

_y

FIG. 1. 共Color online兲The Rashba spin-orbit coupling constant

␣ is made time dependent by applying a time-dependent gate po- tential V_in共t兲. The light 共blue兲 area represents a two-dimensional electron gas inside a heterostructure. When an in-plane magnetic field along x, b^x, is also switched on, a charge current j_y⁰flowing alongyand proportional to␣˙ is generated, its actual sign depending on the sign of␣˙. The induced voltage drop in the transverse direc- tionV_out共t兲can then be used to measure the strength of the Rashba interaction.

j_y⁰= − 2D^xN₀ 2 ␥Ey

x=␶tr␣^{˙ N}0b^x共1 +␥0/2兲, 共79兲 having used D^x=␶tr/m共1 +␥0/2兲s^x. Finally, for a general di- rection of the in-plane magnetic fieldbthe charge current is given by

j⁰=␶trN₀␣^˙共1 +␥0/2兲zˆ∧b. 共80兲 Such an effect could provide an alternative way of estimating the strength of the Rashba interaction, since other spin-orbit mechanisms would not gain any time dependency from a modulated confining potential.

VI. CONCLUSIONS

We showed how to microscopically derive the generalized Boltzmann and Eilenberger equations in the presence of non- Abelian gauge fields. In the SU共2兲 case such equations can be used to describe spin-charge-coupled dynamics in two- dimensional systems whose Hamiltonians include linear-in- momentum spin-orbit coupling terms. All degrees of free- dom are treated symmetrically and the proper identification of the physical quantities follows naturally from the form of the continuity equation. Considering elastic disorder, we ob- tained results which hold as long as ⑀Ⰷ1/␶^,⌬soand for ar- bitrary values of⌬so␶. In particular, we showed that by using the covariant quasiclassical Green’s function, the collision integral in the kinetic equation is not affected by the gauge fields, which only appear to modify the hydrodynamic de- rivative. We expect that this nice disentanglement of gauge fields and disorder effects in the Boltzmann and Eilenberger equations may prove very useful when considering quantum corrections.^57,58We also expect the approach to allow for a generalization of the boundary conditions for the Eilenberger equation to the case in which spin and charge channels are coupled. When discussing the diffusive regime, we first ob- tained Bloch-type equations for the spin and charge, and then exploited them to predict a novel effect. Finally, we note that by making the non-Abelian coupling constant momentum dependent,␥→␥共p兲, it may be possible to extend the present formalism to include Hamiltonians with more general forms of spin-orbit interaction—i.e., not limited to being linear in momentum.

ACKNOWLEDGMENTS

We thank U. Eckern, J. Rammer, and M. Dzierzawa for discussions. This work was supported by the Deutsche Forschungsgemeinschaft through SFB 484 and SPP 1285 and partially supported by EU through Grant No. PITN-GA- 2009-234970.

APPENDIX: QUASICLASSICAL NORMALIZATION CONDITION AND THE COLLISION INTEGRAL Consider a quantity F共1 , 2兲 which is nonlocally covariant, i.e., which under the gauge-transformation Eq.共15兲transforms according toF⬘^{共1 , 2兲=}V共1兲F共1 , 2兲V^†共2兲.

Its locally covariant counterpart reads F˜共1 , 2兲

⬅U_⌫共X, 1兲F共1 , 2兲U_⌫共2 ,X兲, where X=共关t₁+t₂兴/2 ,关x₁ +x₂兴/2兲. In Wigner coordinates, up toO共A⳵p兲accuracy, one has

F˜ =F−1

2兵A⳵p,F其. 共A1兲

We define the ␰-integrated functions f共⑀,␸,X兲= i

␲

冕

^{d␰F共p,⑀,X兲, 共A2兲}

f˜共⑀^,␸^,X兲= i

␲

冕

^{d␰F˜共p,⑀,X兲. 共A3兲}

Let us start by assuming for simplicity A=共0 ,A^a␴^a/2兲—that is, we have neither electric nor mag- netic fields, only spin-orbit coupling—and setting theSU共2兲 coupling constant to one,␥= 1. Moreover, the functionsF,F˜ are assumed to be peaked at the Fermi surface␰= 0 or in its vicinity. Thus, by␰-integrating Eq.共A1兲by parts one has

f˜=f+1

2

再

^{ApF·pˆ −Ap·F␸ˆ⳵␸,f}

冎

^共A4兲

with pF the Fermi momentum in the absence of spin-orbit coupling. The presence of a U共1兲 vector potential can be handled just the same way whereas the inclusion of the scalar potentialse⌽+⌿^a␴^a/2 is trivial and amounts to a shift of the energy argument of f

f˜=f−1

2兵共e⌽+⌿兲⳵⑀,f其. 共A5兲 Therefore, in the presence of a general four-potential A=共e⌽+⌿,eA+A^a␴^a/2兲, one has

f˜=f−1

2兵共e⌽+⌿兲⳵⑀,f其 +1

2

再

^{共eA+pA兲F ·pˆ −共eA+pA兲F ·␸ˆ⳵␸,f}

冎

^{. 共A6兲}

We now use Eqs. 共A1兲–共A6兲 to show 共a兲 how the

␰-integrated Green’s functionsg and˜g are related, and what this implies for the latter’s normalization 共b兲 that the colli- sion integral Eq. 共36兲is equivalent to the one appearing in Ref. 12.

1. Aboutgandg˜ TakeF=Gˇ. From Eq.共A4兲one obtains

˜ˇg=gˇ+ 1

2

再

^{Ap·Fpˆ −Ap·F␸ˆ⳵␸,gˇ}

冎

^{, 共A7兲}

where we did not write down explicitly all the dependencies, since no confusion should arise. Direct calculations show¹³

g^R,A= ⫾

冉

^{1 −Ap}F^·pˆ

冊

^{, 共A8兲}

g_eq^K = 2 tanh共⑀/2T兲

冉

^{1 −A}pF^·pˆ

冊

^{, 共A9兲}

whereTbeing the temperature and the result for the Keldysh component being valid at equilibrium. It follows that to order 兩A兩/pF—i.e., ⌬so/⑀F—the locally covariant ␰-integrated Green’s function has no SU共2兲 共spin兲structure

˜g^R,A= ⫾1, 共A10兲

g

˜_eq^K = 2 tanh共⑀/2T兲 共A11兲 and satisfies the standard normalization condition˜ˇg²= 1ˇ. Re- call this has the meaning of a boundary condition satisfied by

˜ˇg and so is not affected by the introduction of driving elec- tromagnetic关U共1兲兴fields⁴⁹or of a Zeeman term. Indeed, we saw that including the scalar potentials e⌽ and ⌿ simply

“shifts” the energy argument ofg˜_eq^K

˜g_eq^K =

冋

^{1 −12兵共e⌽+⌿兲⳵⑀, ·其}

册

^{2 tanh共⑀/2T兲. 共A12兲}

2. Collision integral

Take F= −i关⌺ˇ,gˇ兴^K⬅C and so F˜= −i关⌺˜ˇ,G˜ˇ兴^K⬅C˜. The ␰ integration delivers

c

˜= −1

␶^{关具K典˜gK−具Kg˜K典兴 共A13兲} with the kernelK共␸⁻␸⬘^兲⬅2␲^N0␶W共␸⁻␸⬘兲, and where具¯典 is shorthand for angular average. One then uses the inverse of Eq.共A6兲to calculate the corresponding expression in the standard 共“nontilde”兲 language. We consider separately the effects of spin-orbit 共A兲 and of a Zeeman field 共⌿兲, since they add linearly.

First, spin-orbit. Starting from c=˜c− 1

2

再

^{Ap·Fpˆ −Ap·F␸ˆ⳵␸,c˜}

冎

=−1

␶^{关˜g−具Kg˜典兴−} 1

2␶

再

^{ApF·pˆ −Ap·F␸ˆ⳵␸,g˜−具Kg˜典}

冎

^,

共A14兲 the translation fromg˜togis done by means of Eq.共A4兲. The calculation is easy but some care is needed, so this is done step by step. First recall that

˜g−1

2

再

^{ApF·pˆ −Ap·F␸ˆ⳵␸,g˜}

冎

^{=g 共A15兲}

so that Eq.共A14兲becomes c= −1

␶^{关g−具Kg˜典兴+} 1

2␶

再

^{ApF·pˆ −Ap·F␸ˆ⳵␸,具Kg典}

冎

^.

共A16兲 Then consider the具Kg˜典 term, whereK=K共␸⁻␸⬘^兲

具Kg˜典=

冕

^d2␸␲^⬘g共␸⬘兲 + 1

2

冕

^d2␸␲^⬘

再

^{Ap·Fpˆ⬘K+Ap·F␸ˆ⬘K,⳵␸⬘g共␸⬘兲}

冎

=具Kg典+1

2

冕

^d2␸␲^⬘

再 ^冋

^{Ap·Fpˆ⬘+A·共p⳵F␸⬘␸ˆ⬘兲}

^册

^K

+ A·␸^ˆ⬘ pF

关⳵␸⬘K兴,g共␸⬘兲

冎

=具Kg典+1

2

冕

^d2␸␲^⬘

再

^{Ap·F␸ˆ⬘,关⳵␸⬘K兴g共␸⬘兲}

冎

^,

having performed a partial integration and used that g共␸

= 0兲=g共␸^{= 2}␲兲, ⳵_␸⬘␸^ˆ⬘^{= −pˆ}⬘. This way Eq. 共A16兲reads c= −1

␶^{关g−具Kg典兴+} 1

2␶

再

^{ApF·pˆ,具Kg典}

冎

− 1

2␶

冕

^d2␸␲^⬘

再

^{Ap·F␸ˆ⬘,共⳵␸⬘K兲g共␸⬘兲}

冎

− 1

2␶

再

^{Ap·F␸ˆ⳵␸,具Kg典}

冎

^{. 共A17兲}

Now work on the last term. Recall the assumption that the scattering amplitude depend only on the momentum transfer, i.e.,K共p,p⬘^{兲=K共p−p}⬘兲. This implies

␸^ˆ pF

⳵␸K= −pF

mpˆ⳵␰K−

冋

^{pmFpˆ⬘⳵␰+␸pˆF⬘⳵␸⬘}

册

^{K. 共A18兲}

From the last term of Eq.共A17兲one therefore has

− 1

2␶

再

^{Ap·F␸ˆ⳵␸,具Kg典}

冎

^=21␶

再

^{Am·p0,具⳵␰Kg典}

冎

+ 1

2␶

冓再

^{Am·p0,⳵␰Kg}

冎冔

+ 1

2␶

冕

^d2␸␲^⬘

再

^{Ap·F␸ˆ⬘,关⳵␸⬘K兴g共␸⬘兲}

冎

^.

共A19兲 Substitution back into Eq. 共A17兲gives

c= −1

␶^{关g−具Kg典兴+} 1

2␶

再

^{ApF·pˆ,具Kg典}

冎

+ 1

2␶

再

^{Am·p0,具⳵␰Kg典}

冎

^+21␶

冓再

^{Am·p0,⳵␰Kg}

冎冔

^.

共A20兲 This expression can also be obtained by a direct␰integration of the collision integral in the standard共nontilde兲 language.

It agrees with the one appearing in Ref. 12 for the case of parabolic bands when one identifies b·␴/2 =A·p/m,bbe- ing the internal spin-orbit field in the language of Ref. 12.

Besides the first two terms, in which no spin-orbit contribu-