Spin susceptibilities, spin densities, and their connection to spin currents

Spin susceptibilities, spin densities, and their connection to spin currents

Sigurdur I. Erlingsson, John Schliemann, and Daniel Loss

Department of Physics and Astronomy, University of Basel, Klingelbergstrasse 82, CH-4056, Switzerland 共Received 22 June 2004; published 14 January 2005兲

We calculate the frequency-dependent spin susceptibilities for a two-dimensional electron gas with both Rashba and Dresselhaus spin-orbit interactions. The resonances of the susceptibilities depend on the relative values of the Rashba and Dresselhaus spin-orbit constants, which could be manipulated by gate voltages. We derive exact continuity equations, with source terms, for the spin density and use those to connect the spin current to the spin density. In the free electron model the susceptibilities play a central role in the spin dynamics since both the spin density and the spin current are proportional to them.

DOI: 10.1103/PhysRevB.71.035319 PACS number共s兲: 73.63.⫺b, 85.75.⫺d

I. INTRODUCTION

The ability to manipulate spin states in semiconducting and metallic nanostructures is now the focus of much experi- mental and theoretical attention. Some spintronics applica- tion are already in use^1,2and more have been proposed, rang- ing from spin field-effect transistors^3–5 to spin qubits.^6,7 In many cases the spin-orbit interaction can be used to manipu- late the electron spin via electronic means.⁸This sort of elec- tronic manipulation is important since technologically, elec- tric field control of spins is preferred over magnetic field control. Also, electronic control of spins has revealed many interesting physics in experiments involving semiconductor heterostructures.^9,10

Some time ago it was proposed that electric fields could lead to magnetization in antiferromagnetic materials fulfill- ing certain symmetries.¹¹This is the so-called magnetoelec- tric effect. The magnetoelectric effect was considered for conductors with special symmetry properties¹²and for pyro- electric superconductors.¹³Electric-field-induced spin orien- tation in semiconductors due to linear in momentum spin- orbit interaction was also discussed in Refs. 14–16. There has been renewed interest in this subject since this induced spin polarization might serve as spin injectors in certain semiconductor heterostructures.^17,18Only recently there have been experiments which seem to demonstrate such current induced polarization of spins via spin-orbit interaction.^19,20

A somewhat related effect is the so-called spin-Hall ef- fect. In the normal Hall effect the electrons are deflected by the Lorentz force, but in the spin-Hall effect spins are scat- tered by impurities preferentially to the left 共right兲 if their spins point “up”共“down”兲due to spin-orbit interaction.^21,22 This will result in a spin current, but no net charge current, flowing perpendicular to the applied charge current. Such an extrinsic transverse spin current was already investigated by Dyakononov and Perel some time ago in Ref. 23 and more recently by Hirsch.²⁴Presently there is much interest in spin- orbit mediated spin-Hall effect in semiconducting hetero- structures. Here the effect can appear in hole-^25–31 or electron-doped^32–42 semiconductors due to band structure properties and impurities are not necessary. Thus, the term intrinsic spin Hall effect is used to distinguish this from the previously discussed mechanism since it occurs already in the absence of impurities.^26,32Due to differences in the band

structure of the holes and electron there is an important dis- tinction between the two since in the electron system the spin Hall conductance共the ratio of the transverse spin current and the applied electric field兲 reaches a universal value of e / 8␲ in clean systems.³² However, impurities are believed to modify this universal value^33,40 and in addition, the exact influence of vertex corrections on the spin-Hall conductivity is currently under investigation.^38,42,43

In this paper we consider the spin susceptibilities of a 2DEG with Rashba and Dresselhaus spin-orbit coupling.

Due to the strong 2DEG confinement the Dresselhaus cou- pling reduces to terms linear in momentum. For such linear momentum spin-orbit coupling the spin susceptibility can be used to characterize other transport properties. We calculate the susceptibilities using a free electron model and relate them to the electric-field-induced spin density. Also, we de- rive continuity equations 共with source terms兲 for the spin density and spin current, similar to the equations already derived for only Rashba coupling.³⁶These equations are ex- act operator identities and via them we can relate the spin current to the spin density. Via these relations the spin cur- rent共which is nontrivial to measure兲can be connected to the spin density, or magnetization, which is easier to detect. The susceptibilites play a central role in the free electron model, since the Fourier transform of the spin density and the spin current are proportional to the susceptibilities.

II. THE MODEL

We consider a two-dimensional electron gas共2DEG兲with Rashba and Dresselhaus spin-orbit coupling. In the absence of external fields the Hamiltonian may be written as

H =p_x²+ p_y² 2m +␣

ប共p_y␴x− p_x␴y兲+ ␤

ប共p_y␴y− p_x␴x兲, 共1兲 where␣^and␤ are the Rashba and Dresselhaus共linear兲 co- efficient in a 2DEG. It is easy to see that 关p_x, H兴=关p_y, H兴

= 0, and thus we seek eigenstates of the form

␺k,s共r兲 ⬅ 具r兩ks典=e^ik·r

冑

Au_s共k兲 共2兲 where u_s共k兲is spinor to be determined and A is the system area. Since the momenta are conserved it is possible to in- troduce an effective magnetic field

⌫=

冢

^{−␤␤kkyx−0+␣␣kkxy}

冣

^{. 共3兲}

The eigenspectrum can be written in terms of this effective magnetic field, the eigenenergies of the Hamiltonian being 共s = ± 1兲

E_s共k兲=ប²k²

2m + s兩⌫⁺兩, 共4兲 and the corresponding spinor

u_s共k兲= 1

冑

2

冢

^{s兩⌫⌫1++兩}

冣

^{, 共5兲}

where⌫⁺=⌫x+ i⌫y. Note that Eqs.共4兲and共5兲are also valid for an in-plane magnetic field B using the substitution ⌫

→⌫+¹₂g␮BB, where g and␮Bare the effective g-factor and Bohr magneton, respectively.

Writing the quasimomentum in polar coordinates k

= k共cos␪^{, sin}␪兲, the eigenspectrum of the Hamiltonian in Eq.

共1兲becomes

E_s共k兲=ប²k²

2m + sk

冑

共␣²⁺␤²兲g共␪兲, 共6兲 where g共␪兲= 1 − sin共2␾兲sin共2␪兲 determines the anisotropy of the Fermi surfaces and the corresponding eigenspinors are

u_s共k兲= 1

冑

2

冢

^{− scos共␾兲e−i1␪g共+ i sin␪兲 共␾兲ei␪}

冣

^{. 共7兲}

Here we have introduced the following parametrization of the spin-orbit coupling strength:

sin共␾兲= ␣

冑

␣²⁺␤^2, ␾苸关−␲^/2,␲^/2兴. 共8兲 The eigenfunctions in Eq.共7兲 have the interesting property that they depend on the spin-orbit coupling parameters␣^and

␤ only via the angle ␾. This allows one to take the limit

␣^,␤→0 such that the angle␾remains fixed and the result- ing eigenvectors in Eq.共9兲are also共degenerate兲eigenvectors of the free electron Hamiltonian.⁴⁴ As was pointed out in Ref. 37, the Kramers conjugate state of ␺k,s共r兲 is ␺−k,s共r兲, i.e., they belong to the same branch.

III. SPIN SUSCEPTIBILITIES AND CURRENT-INDUCED MAGNETIZATION

For a weak driving field the response of the system is obtained by the Kubo formalism. Due to the spin-orbit cou- pling, a pure electric field driving results in a nonzero mag- netic response. Since the spin-orbit term in Eq.共1兲 is linear in momenta, both the response functions due to magnetic and electric perturbation can be expressed with spin susceptibili- ties. The␩= x , y , z component of the spin density operator is defined as

␳_␩共r兲=

兺

n

␴n,␩共r兲=

兺

n

␴n,␩␦共r − r_n兲, 共9兲 where r_nand␴n,␩are the position operator and Pauli matrix, respectively, of the nth electron. For a translationally invari- ant system, the wave vector and frequency共q ,␻兲dependent susceptibilities are

␹_␩␩⬘共q,␻兲=

冕

0

⬁

dte^i␻t i

បA具关␳_␩共q,t兲,␳_␩⬘共− q兲兴典, 共10兲

=

冕

0

⬁

dte^i␻t1

A

兺

_k,s ^if_ប^{ks具关␴␩共q,t兲,␴␩}⬘共− q兲兴典ks, 共11兲 where f_ks= f关E_s共k兲兴with f being the Fermi distribution func- tion and we used the notation 具¯典ks=具ks兩¯兩ks典. The fre- quency should be viewed as i␻→i共␻^{+ i}ប⁻¹⑀兲 to regularize the integral. The operators in Eq.共11兲refer to single particle operators. This susceptibility is a spin density response func- tion and to get the magnetization response function, each spin density operator in Eq.共11兲 should be multiplied with the electron effective magnetic moment g␮B/ 2. Using Eq.

共11兲 and the eigenspectrum represented by Eqs.共6兲and共7兲 the susceptibilities for a spatially homogeneous perturbation 共q = 0兲become

␹xx共␻兲= 1 共2␲兲²ប

冕

0

2␲

d␪^共␣^sin␪⁻␤^cos␪兲² 共␣²⁺␤²兲^1/2

冑

g共␪兲

冕

k₊共␪兲

k₋共␪兲

dk

⫻ k²

4共␣²⁺␤²兲g共␪兲k²−共ប␻^{+ i}⑀兲², 共12兲

␹xy共␻兲= 1 共2␲兲²ប

冕

0

2␲

d␪^共␣^sin␪⁻␤^cos␪兲共␤^sin␪⁻␣^cos␪兲 共␣²⁺␤²兲^1/2

冑

g共␪兲

⫻

冕

k₊共␪兲 k₋共␪兲

dk k²

4共␣²⁺␤²兲g共␪兲k²−共ប␻^{+ i}⑀兲². 共13兲 Note that all q = 0 dependence has been dropped for clarity.

Here we assumed zero temperature and the Fermi distribu- tion function was replaced by a step function. The resulting Fermi contours k_±共␪兲are the solutions of

k_F²= k_±共␪兲²± k_±共␪兲k_SO

冑

g共␪兲, 共14兲 where k_F²= 2m⑀F/ប² is the squared Fermi momentum and k_SO= m

冑

_␣²+␤^2/ប². The k integral results in a linear term and a term involving an inverse tangent in k_±共␪兲 and the subse- quent angular integrals cannot be solved analytically. The inverse tangent can be expanded in powers of k_SO/ k_FⰆ1, resulting in the following lowest order result:

␹xx共␻兲=␹yy共␻兲= m

2␲ប²

冉

^{1 +}

兿

^s

^冑

^{共ប␧s2␻−+ i共ប⑀␻兲2+ i⑀兲2}

冊

^,

共15兲 where the resonance energies are ⑀±

2= 8⑀SO⑀F关1 ± sin共2␾兲兴, with⑀SO= m共␣²⁺␤²兲/ប². Using the same procedure we can calculate the off-diagonal susceptibilities in a similar manner

␹xy共␻兲=␹yx共␻兲=␦␹共␻兲−␹xx共␻兲

sin共2␾兲 − sin共2␾兲␦␹^共␻兲, 共16兲 where we have defined

␦␹共␻兲= m 2␲ប²

8␧SO␧F

兿

^s

^冑

^{␧s2−共ប␻+ i⑀兲2. 共17兲}

The magnetization is related to the spin density in Eq.共9兲 through m共r兲=₂¹g␮B␳共r兲 which leads to the standard linear response relation

m_␩共q,␻兲=

冉

^12g␮B

冊

^{2␹␩␩⬘共q,␻兲B␩⬘共q,␻兲. 共18兲}

To obtain the Pauli paramagnetic susceptibility one should take the following order of limits:⁴⁵

lim

q→0

␻→lim0␹_␩␩⬘共q,␻兲= m

␲ប²␦_␩,␩⬘. 共19兲 The diamagnetic contribution can be disregarded since we assume an in-plane magnetic field. The susceptibilities in Eqs. 共12兲 and 共13兲 are calculated for the reverse order of limits done in Eq.共19兲. These␻⫽0 susceptibilities are the spin-orbit contribution coming from the region in k space where only one共s = −1兲 of the two branches is occupied.⁵⁵ Also, the spin current 共see discussion below兲 which results from the spin-orbit interaction is nonzero due to contribu- tions from the k space area between the two Fermi contours.^32,33,41 Thus we only focus on this contribution when we relate the susceptibilities to the spin-orbit mediated spin densities and the spin currents.

The value of the resonance frequency is determined by

冑

8␧SO␧F⬇0.16 meV= 40 GHz for typical GaAs para- meters:⁴⁶ ␣^{= 0.5}⫻10⁻⁹meV m and electron density n_e= 4

⫻10¹⁵m⁻². For lower frequencies the susceptibilities remain nominally constant. In the limit␣Ⰶ␤the lowest order con- tribution to Eqs.共15兲and共16兲become

␹xx共␻兲= m 2␲ប²

1 1 −共ប␻^{+ i}⑀兲²

8␧SO␧F

, 共20兲

␹xy共␻兲= − m 2␲ប²

␣

␤

1 −共ប␻^{+ i}⑀兲² 4␧SO␧F

冉

^{1 −共ប8␻}␧SO^{+ i}␧^⑀F^兲2

冊

^{2. 共21兲}

Here we have not included impurities and thus the regular- ization parameter⑀can strictly only be attributed to an adia- batic turning on of the external electric or magnetic field.

In the absence of electric and magnetic fields, the spin- orbit interaction does not give rise to a net magnetization.

Even though the spin-orbit interaction has the form of a momentum-dependent magnetic field, the total contribution averages to zero.^47,48 However, although the spin-orbit in- duced splitting does not give rise to an equilibrium magne-

tization, there is an asymmetry in the local magnetic field in momentum space, i.e., the local magnetic field is odd under wave vector reversal, and any translation of the Fermi sphere away from the⌫point will induce a magnetization.¹⁵Apply- ing a homogeneous electric field E共r , t兲= E⁰e^−i␻tto the sys- tem will give rise to the following time-dependent perturba- tion:

V共t兲= − e i␻^E

0e^−i␻t· j共q = 0兲, 共22兲 where j共q兲 is the Fourier transform of the current density operator

j共r兲=

兺

n

j_n共r兲=

兺

n

1

2兵␦共r − r_n兲,vn其, 共23兲 and the velocity operatorv_nfor the Hamiltonian in Eq.共1兲is given by

v_n=

冢

^{mm11ppn,xn,y−+␣ប␣ប␴␴n,yn,x−+␤␤បប␴␴n,xn,y}

冣

^{. 共24兲}

From linear response theory the Fourier transform of the electric field induced spin density may be written as

具␳x共␻兲典=eE_x共␻兲

i␻

冉

^␣ប␹xy共␻兲+␤ ប␹xx共␻兲

冊

−eE_y共␻兲

i␻

冉

^␣ប␹xx共␻兲+␤

ប␹xy共␻兲

冊

^{, 共25兲}

具␳y共␻兲典=eE_x共␻兲

i␻

冉

^␣ប␹xx共␻兲+␤ ប␹xy共␻兲

冊

− eE_y共␻兲

i␻

冉

^␣ប␹xy共␻兲+␤

ប␹xx共␻兲

冊

^{. 共26兲}

This result is reminiscent of the pure magnetic field induced spin density, except here eE_y共␻兲/ i␻ plays the role of mag- netic field, via the spin-orbit coupling. As was pointed out in Refs. 15 and 41 the dc limit corresponds to replacing the frequency with momentum scattering i␻→−1 /␶^.

Multiplying Eqs. 共25兲 and 共26兲 by the sample area will give the total number of induced magnetic moments, mea- sured in units of g␮B/ 2. Applying an electric field E

⬇100 V / cm to a GaAs 2DEG’s with high mobility 共ប/␶

⬇10⁻²meV兲 and a sample area of A =共500␮^m兲², the num- ber of magnetic moments 共Bohr magnetons兲 would be around 2.5⫻10⁷. For a 2DEG thickness of a few nm these magnetic moments produce a magnetic field of the order 10⁻⁶T. Probing 2DEG properties using ESR techniques has been succesfully used to determine the spin-orbit splitting⁴⁹ and other 2DEG spin properties,^50,51Using similar ESR tech- niques, the spin-orbit coefficients␣^,␤ could in principle be determined by measuring different spin density component for different direction of driving current, as a function of␣

which could be tuned by gate voltages.^46,52,53 Such an in- duced spin density could be detected by Faraday rotation measurements.¹⁹

IV. CONNECTION TO SPIN CURRENT

The usual way of deriving the operator version of the particle continuity equation is to start from the definition of the density operator

␳共r,t兲=

兺

n

␦关r − r_n共t兲兴, 共27兲 and from there one can derive the standard continuity equa- tion,

⳵

⳵^t␳共r,t兲+ ⵜ · j共r,t兲= 0 共28兲 from the Heisenberg equation of motion for␳共r , t兲. The form of the current density in Eq.共28兲is uniquely determined by the Heisenberg equation of motion. For the Hamiltonian in Eq. 共1兲 the current density is given by Eq. 共23兲, using the velocity operator in Eq.共24兲. In the case of the spin density operator

␳_␩共r,t兲=

兺

_n ^{␴n,␩共t兲␦关r − rn共t兲兴, 共29兲}

the same procedure of evaluating the Heisenberg equation motion will not result in a unique definition of the associated spin current density. The reason is that the precession due to the momentum dependent spin-orbit magnetic field intro- duces additional terms into the equation of motion. In order to proceed one has to postulate a form for the spin current density. The most widely used definition, and physically rea- sonable, is the following:

j^␩共r兲=

兺

n

1

2兵␴n,␩,j_n共r兲其. 共30兲 This form of the spin current is Hermitian and reduces to the standard spin current form when the velocity operator is spin independent.⁵⁴ Having determined the form of the spin cur- rent the resulting continuity equations become

⳵

⳵^t␳x共r,t兲+ ⵜ · j^x共r,t兲= −2m␣

ប² j_x^z共r,t兲+2m␤ ប² j_y^z共r,t兲,

共31兲

⳵

⳵^t␳y共r,t兲+ ⵜ · j^y共r,t兲= −2m␣

ប² j_y^z共r,t兲+2m␤ ប² j_x^z共r,t兲,

共32兲

⳵

⳵^t␳z共r,t兲+ ⵜ · j^z共r,t兲= +2m␣

ប² 关j_y^x共r,t兲+ j_x^y共r,t兲兴

−2m␤

ប² 关j_x^x共r,t兲+ j_y^y共r,t兲兴. 共33兲 Similar equations have already been derived for pure Rashba

coupling.³⁶The above Eqs.共31兲–共33兲are exact relation for a systems with spin-orbit coupling linear in momentum and including impurities would not change their form.^56,57

Taking the thermal average of Eqs.共31兲–共33兲gives partial differential equation connecting the spin densities and spin currents. Based on these equations one can make a few ob- servation on the nature of the spin current, without explicitly solving them. First of all, for a homogeneous system in the stationary limit the right-hand side of Eq. 共33兲must vanish for all values of ␣^, ␤. This is trivially satisfied for all 具j^␩典

= 0, but more interestingly also when具j_y^x典= −具j_x^y典 and 具j_x^x典=

−具j_y^y典. It is easy to show that the latter case is true in equi- librium

具j_x^x典= −具j_y^y典= 1 3␲

␤

ប

冉

ប^m2

冊

^{2共␣2−␤2兲, 共34兲}

具j_x^y典= −具j_y^x典= 1 3␲

␣

ប

冉

ប^m2

冊

^{2共␣2−␤2兲, 共35兲}

which covers the results of Ref. 37 as special cases. In par- ticular, the current expectation values vanish at␣^{= ±}␤^{due to} the additional concerved quantity arising at these points.⁴ Furthermore, Eq. 共33兲 shows these equilibrium currents do not act as source terms for the spin density, since the rhs always vanishes.

Let us now consider a homogeneous system such that the divergence terms vanish. By using Eqs. 共31兲 and 共32兲 one can derive the following identity for the Fourier transform of the x and y component of the average z-polarized spin cur- rent:

具j_x^z共␻兲典=ប²i␻关␣具␳x共␻兲典+␤具␳y共␻兲典兴

2m共␤²⁻␣²兲 , 共36兲 具j_y^z共␻兲典=ប²i␻关␤具␳x共␻兲典+␣具␳y共␻兲典兴

2m共␤²⁻␣²兲 . 共37兲 These relations establish a connection between the spin cur- rent components 具j_x,y^z 典 and the x , y components of the spin density, in the frequency domain. This is quite useful since the spin current, which is hard to detect, is related to a quan- tity which is easier to measure. Also, Eqs.共36兲and共37兲is a good starting point for connecting spin current and spin den- sity response functions using standard Kubo formalism.

Let us now assume a homogeneous electric field applied in the y direction. The spin conductivities are defined as the ratio of the spin current and applied electric field

␴_␩y z 共␻兲=

ប 2具j_␩^z共␻兲典

E_y共␻兲 , 共38兲 where the factorប/ 2 in the definition of the spin conductiv- ity is due to our definition of the spin current in terms Pauli matrices and not the spin operators, i.e., S =ប␴^{/ 2. Using} Eqs.共25兲and共26兲to relate the spin density to the suscepti- bilities we obtain the following result for the ac spin conduc- tivities:

␴xy

z 共␻兲=eប² 4m

␣²⁻␤²

␤²⁺␣²␦␹共w兲, 共39兲

␴yy

z 共␻兲= −␣²⁻␤² 2␤␣

eប²

4m关␦␹共w兲−␹xx共w兲兴. 共40兲 These equations for the spin conductivities, along with Eqs.

共15兲and共17兲, in the␻→0 limit reproduce the result in Ref.

34 up to a sign convention for␤. For either pure Rashba or Dresselhaus we have the following result:

␴xy

z 共␻兲= ± e

8␲

冢

^{1 −共ប8␻1⑀SO+ i␧⑀F兲2}

冣

^{, 共41兲}

␴yy

z 共␻兲= 0 共42兲

the upper共lower兲sign refers to a pure Rashba共Dresselhaus兲. Taking the limit ⑀→0 recovers the universal limit of spin Hall conductance␴xy

z 共␻→0兲= e / 8␲^.32

V. CONCLUSION

We calculated the frequency-dependent spin susceptibili- ties for a 2DEG with both Rashba and Dresselhaus spin-orbit coupling. The suspectibilities have resonance peaks whose position depends on the relative magnitude of the Rashba

and Dresselhaus coefficients. The position of the resonance could be tuned via the gate dependence of␣. We derive a connection between spin density and the spin current starting from the Heisenberg equation of motion for the spin density.

Unlike the particle density, the resulting continuity equations have spin current source terms due to the spin-orbit interac- tion. Using these relation we connect the spin current to the spin density. In the free electron model the spin current and the spin-Hall coefficient are proportional to the susceptibility.

Indeed, using the continuity equations and the susceptibili- ties calculated here we recover the universal共ballistic兲value of the spin-Hall conductance. Such relations关Eqs. 共36兲and 共37兲兴 might help elucidate the nature of spin currents in a similar way to Ref. 41 which discussed the relation between the spin current and the dielectric function. By calculating the spin density with the correct impurity contribution would automatically give the spin current. We are confident that these and similar considerations will contribute to a deeper understanding of the role of impurities in the spin-Hall ef- fect.

ACKNOWLEDGMENTS

The authors acknowledge financial support from the NCCR Nanoscience, the Swiss NSF, DARPA, ARO, ONR, and the Spintronics RTN. We would like to thank J. Carlos Egues, Daniel Saraga, and Oleg Chalaev for enlightening discussions.

1S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnár, M. L. Roukes, A. Y. Chtchelkanova, and D. M.

Treger, Science 294, 1488共2001兲.

2G. A. Prinz, Science 282, 1660共1998兲.

3S. Datta and B. Das, Appl. Phys. Lett. 56, 665共1990兲.

4J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett. 90, 146801共2003兲.

5J. C. Egues, G. Burkard, and D. Loss, Appl. Phys. Lett. 82, 2658 共2003兲.

6D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120共1998兲.

7Semiconductor Spintronics and Quantum Computation, edited by D. D. Awschalom, D. Loss, and N. Samarth共Springer-Verlag, Berlin, 2002兲.

8Y. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Nature共London兲 427, 50共2004兲.

9J. M. Kikkawa and D. D. Awschalom, Nature共London兲 397, 139 共1999兲.

10G. Salis, Y. Kato, K. Ensslin, D. C. Driscoll, A. C. Gossard, and D. D. Awschalom, Nature共London兲 414, 619共2001兲.

11I. E. Dzyaloshinkii, Sov. Phys. JETP 10, 628共1959兲.

12L. S. Levitov, Y. V. Nazarov, and G. M. Éliasberg, Sov. Phys.

JETP 61, 133共1985兲.

13V. M. Edelstein, Phys. Rev. Lett. 75, 2004共1995兲.

14A. G. Aronov and Y. B. Lyanda-Geller, JETP Lett. 50, 431 共1989兲.

15V. M. Edelstein, Solid State Commun. 73, 233共1990兲.

16L. I. Magarill and M. V. Entin, JETP Lett. 72, 195共2000兲.

17A. Voskoboynikov, S. S. Lin, C. P. Lee, and O. Tretyak, J. Appl.

Phys. 87, 387共2000兲.

18X. Cartoixà, D. Z.-Y. Ting, E. S. Daniel, and T. C. McGill, Su- perlattices Microstruct. 30, 309共2001兲.

19Y. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, cond-mat/0403407共unpublished兲.

20S. D. Ganichev, S. N. Danilov, P. Scheider, V. V. Bel’kov, L. E.

Golub, W. Wegscheider, D. Weiss, and W. Prettl, cond-mat/

0403641共unpublished兲.

21C. M. Hurd, The Hall Effect in Metals and Alloys共Plenum, New York, 1973兲.

22The Hall effect and its Applications, edited by C. L. Chien and C.

R. Westgate共Plenum, New York, 1980兲.

23M. I. Dyakonov and V. I. Perel, JETP Lett. 13, 467共1971兲.

24J. E. Hirsch, Phys. Rev. Lett. 83, 1834共1999兲.

25S. Murakami, N. Nagaosa, and S.-C. Zhang, Science 301, 1348 共2003兲.

26D. Culcer, J. Sinova, N. A. Sinitsyn, T. Jungwirth, A. H. Mac- Donald, and Q. Niu, cond-mat/0309475共unpublished兲.

27B. A. Bernevig, J. Hu, E. Mukamel, and S.-C. Zhang, cond-mat/

0311024共unpublished兲.

28L. Hu, J. Gao, and S.-Q. Shen, cond-mat/0403231共unpublished兲.

29J. Schliemann and D. Loss, cond-mat/0405436, Phys. Rev. B共to be published兲.

30J. Hu, B. A. Bernevig, and C. Wu, Int. J. Mod. Phys. B 17, 5991 共2003兲.

31S. Murakami, Phys. Rev. B 69, 241202共R兲 共2004兲.

32J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A.

H. MacDonald, Phys. Rev. Lett. 92, 126603共2004兲.

33J. Schliemann and D. Loss, Phys. Rev. B 69, 165315共2004兲.

34N. A. Sinitsyn, E. M. Hankiewicz, W. Teizer, and J. Sinova, cond- mat/0310368共unpublished兲.

35S.-Q. Shen, cond-mat/0310368共unpublished兲.

36A. A. Burkov, A. S. Núñes, and A. H. MacDonald, cond-mat/

0311328共unpublished兲.

37E. I. Rashba, Phys. Rev. B 68, 241315共R兲 共2003兲.

38J. Inoue, G. E. W. Bauer, and L. W. Molenkamp, Phys. Rev. B 70, 041303共R兲 共2004兲.

39S.-Q. Shen, M. Ma, X. C. Xie, and F. C. Zhang, cond-mat/

0403005共unpublished兲.

40Y. Xiong and X. C. Xie, cond-mat/0403083共unpublished兲.

41E. I. Rashba, Phys. Rev. B 70, 161201共R兲 共2004兲.

42O. Dimitrova, cond-mat/0405339共unpublished兲.

43S. Murakami, cond-mat/0405003共unpublished兲.

44L. W. Molenkamp, G. Schmidt, and G. E. W. Bauer, Phys. Rev. B 64, 121202共R兲 共2001兲.

45G. D. Mahan, Many-Particle Physics, 3rd ed.共Kluwer Academic, New York, 2000兲.

46J. B. Miller, D. M. Zumbühl, C. M. Marcus, Y. B. Lyanda-Geller, D. Goldhaber-Gordon, K. Campman, and A. C. Gossard, Phys.

Rev. Lett. 90, 076807共2003兲.

47R. Winkler, Phys. Rev. B 69, 045317共2004兲.

48R. Winkler, cond-mat/0401106共unpublished兲.

49D. Stein, K. von Klitzing, and G. Weinmann, Phys. Rev. Lett. 51, 130共1983兲.

50M. Dobers, K. v. Klitzing, and G. Weimann, Phys. Rev. B 38, 5453共1988兲.

51N. Nestle, G. Denninger, M. Vidal, C. Weinzierl, K. Brunner, K.

Eberl, and K. von Klitzing, Phys. Rev. B 56, R4359共1997兲.

52D. Grundler, Phys. Rev. Lett. 84, 6074共2000兲.

53T. Koga, J. Nitta, T. Akazaki, and H. Takayanagi, Phys. Rev. Lett.

89, 046801共2002兲.

54A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Eur. Phys. J. B 22, 99共2001兲.

55Assuming that v_FqⰆk_BTⰆ冑_␧SO␧F, the dominant contribution comes from the region between the two Fermi contours 共the spin-orbit contribution兲ifប␻Ⰷ␧Fq / k_F.

56When Hamiltonians representing nonmagnetic impurites H_imp

= H_imp共r兲are added to Eq.共1兲, the commutator in the Heisenberg equation of motion for an operator O共r ,␴兲 becomes 关O共r ,␴兲, H + H_imp兴=关O共r ,␴兲, H兴, where O is any operator that does not explicitly depend on p. Both the particle and spin den- sity are such operator and thus the equations of motion remain the same in the presence of impurities. The complications due to the impurities appear in the calculation of the average values of the operators.

57Homogeneous共q = 0兲 spin systems driven by a constant field E₀ will exhibit damping of magnetization, i.e.,⳵t具␳_␩共t→⬁兲典= 0 for any finite amount of dissipation. Then, from Eq.共31兲 and the relation lim_t→⬁⳵t具␳x共t兲典= lim_s→0s²具␳˜_x共s兲典 for the Laplace trans- form, we get 0 = lim_s→0s²␳˜_x共s兲=共2m␤E₀/ប²兲␴˜_xy^z共s→0兲, where we used 具˜j

x

z共s兲典=共ប/ 2兲␴˜_xy^z共s兲E₀s and ␣= 0. Thus, the dc spin conductivity vanishes for any dissipation mechanism. The same follows directly from the other magnetization components and in the case of␣⫽0.