Asymmetries in intrinsic spin-Hall effect in low in-plane magnetic field

Asymmetries in intrinsic spin-Hall effect in low in-plane magnetic field

L. Y. Wang,1 _{C. S. Chu,}1,2_{and A. G. Mal’shukov}1,2,3

1_{Department of Electrophysics, National Chiao Tung University, Hsinchu 30010, Taiwan} 2_{Physics Division, National Center for Theoretical Sciences, Hsinchu 30043, Taiwan} 3_{Institute of Spectroscopy, Russian Academy of Science, 142190 Troitsk, Moscow oblast, Russia} 共Received 22 June 2008; revised manuscript received 2 September 2008; published 2 October 2008兲 Effects of low in-plane magnetic field on bulk spin densities and edge spin accumulations of a diffusive two-dimensional semiconductor stripe are studied. Focusing upon the Dresselhaus-type intrinsic spin-orbit interaction共SOI兲, we look for the symmetry, or asymmetry, characteristics in two magnetic-field orientations: along and transverse to the stripe. For longitudinal field, the out-of-plane spin density S_zexhibits odd parity across the stripe and even parity in the magnetic field and is an edge accumulation. For transverse field, the out-of-plane Szbecomes asymmetric in both spatial and field dependencies and has finite bulk values for finite magnetic fields. Our results support utilizing low in-plane magnetic fields for the probing of the underlying SOI.

DOI:10.1103/PhysRevB.78.155302 PACS number共s兲: 72.25.Dc, 71.70.Ej, 73.40.Lq

I. INTRODUCTION

Generation and manipulation of spin densities by electri-cal means are major goals of semiconductor spintronics that are made possible by spin-orbit interactions 共SOI兲.1–8 _SOIs being considered are either intrinsiclike: the Rashba2,5,7,9–11 and the Dresselhaus SOIs4,8,12,13 _{or extrinsiclike: the} impurity-induced SOI.1,3,6,10,14 _{These SOIs contribute, in an} external electric field, to either spin densities in the bulk or spin accumulations at lateral edges, or both.

Out-of-plane spin polarization is of particular interest be-cause it permits efficient optical probe by Kerr rotation. The edge spin accumulation, according to the spin-Hall effect 共SHE兲, has an out-of-plane component and is resulted from a transverse spin current induced by the electric field.1,4–6 However, for the case of intrinsic SOI, it is understood that the SHE is quenched by background scatterers, be they iso-tropic or anisoiso-tropic,15 as long as the intrinsic SOIs consist of only linear-momentum dependence term. Meanwhile, no out-of-plane bulk spin densities are expected in an electric field.2,10,16 _{When applying an in-plane magnetic field to a} two-dimensional 共2D兲 system, one might be led by the in-plane nature of the effective spin-orbit magnetic field heff

=具hk典⫽0, obtained by averaging the spin-orbit effective

field over the distribution of the electron momentumបk, to expect that there were no out-of-plane spin densities. This is shown not to be the case by Engel et al.11_{for a Rashba-type} 2D system, where out-of-plane spin densities are found when the external in-plane magnetic field is longitudinal: a con-figuration studied by recent experiments.17,18_{However, either} the scatterer has to be anisotropic or the electron dispersion has to be nonparabolic for the effect to hold.11Here, longi-tudinal denotes the direction parallel to the electric field and its orthogonal counterpart in the 2D plane is denoted trans-verse.

In this paper, we have shown that out-of-plane bulk spin density can be generated in another system configuration with less restrictive assumptions. The configuration is a Dresselhaus-type 2D system and the external in-plane mag-netic field is in a transverse orientation. More importantly,

the effect holds for isotropic background scatterers and for parabolic dispersion for electrons. Our calculation has in-cluded the cubic Dresselhaus SOI.

This paper also addresses the symmetrical properties of the spin densities and spin accumulations in a weak in-plane magnetic field. We believe that this is important for distin-guishing the dominant type of SOI in a particular sample. Out-of-plane spin accumulations at the lateral edges of an extrinsic SOI two-dimensional electron gas共2DEG兲 are sym-metric with respect to a transverse magnetic field.6_The sup-pression that it exerts on the spin accumulations is exhibited in the position-dependent Hanle profiles.19 _{Study on the} same field configuration, but in an extrinsic SOI normal metal, has found similar field suppression in another physical quantity: the out-of-plane spin-Hall potential.20 _{For the} in-trinsic SOI, studies on the in-plane magnetic-field effects have focused on the spin-Hall conductivity21,22_{and the bulk} spin densities11,23_{but not on the symmetry properties. Thus it} is legitimate to perform a thorough and systematic investiga-tion on both the spatial as well as the field-dependent sym-metry characteristics of the spin distributions for the case of intrinsic SOI.

Our results show, for the case of a Dresselhaus-type 2DEG stripe, strong anisotropy in the symmetry characteris-tics with respect to the field orientations. For longitudinal field, the out-of-plane spin density Sz exhibits odd parity across the stripe and even parity in the magnetic field and is an edge accumulation. For transverse field, Sz becomes asymmetric in both spatial and field dependencies and has finite bulk values for finite magnetic fields. As the Rashba and the extrinsic SOIs do not depend on the crystal orienta-tions, while the Dresselhaus SOI does, the strong anisotropy in the symmetry characteristics obtained in this work is dis-tinct for the Dresselhaus SOI. Our work thus serves to com-mence the notion of utilizing low in-plane magnetic fields as a characterizing tool for the probing of the underlying SOI in a particular sample.

In this paper, we consider a diffusive Dresselhaus-type 2DEG stripe in a weak in-plane magnetic field as shown in Fig.1. The diffusive regime has lsoⰇle, where lsoand leare, respectively, the spin-relaxation length due to the SOI and

momentum relaxation length. Spin distributions across the entire width of the stripe are investigated for all three spin directions including out-of-plane and in-plane components. In Sec. II we present the spin-diffusion equation and the associated boundary conditions. In Sec. III, we present our numerical results. Finally, in Sec. IV, we will present a sum-mary and discussion of our results.

II. SPIN-DIFFUSION EQUATION

Following on the procedure of deriving the spin-diffusion equation for B = 0 from the Keldysh nonequilibrium Green’s function technique,16_{we extend the derivation to include an} in-plane magnetic field. The spin-dependent term in the Hamiltonian is given by HB·␴=共hk+ B˜ 兲·␴, where␴ is the

Pauli-matrix vector, hk= −h−kis the SOI effective field and a

function of the 2D wave vector k, and B˜ =gⴱ␮BB/2 is the Zeeman term. Here gⴱis the effective g factor and␮Bis the Bohr magneton. We consider the weak magnetic-field regime where E_FⰇhk⬎B˜.

A brief outline of the derivation is presented below. Start-ing with the perturbation from a four potential given by H

⬘

=兺i⌽i共r,t兲␶i, where the 2⫻2 matrices ␶0= 1 and ␶x,y,z =␴x,y,zand the four densities Di共r,t兲=−i Tr关␶iG−+共r,r,t,t兲兴 are expressed in terms of the full Green’s function. Within the linear-response regime, and for␻ⰆEF, it becomes

Di共r,␻兲 =

冕

d2r

⬘

兺

j

⌸ij共r,r

⬘

,␻兲⌽j共r

⬘

,␻兲 + Di

0_{共r,␻兲. 共1兲}

With 2N0, the electron density of states Di

0_共q,␻兲

= −2N0⌽i共q,␻兲 are easily understood as the local equilib-rium densities.16 _{This term turns out to be the driving term} for the spin-Hall effect; and within the linear response, it suffices to neglect the correction due to HB·␴in Di0.

The response function in the k representation is

⌸ij共q,␻兲 = i␻

兺

p₁k₁

冕

d␻

⬘

2␲ ⳵fFD共␻

⬘

兲 ⳵␻

⬘

具Tr关G a_共k 1,p1− q,␻兲 ⫻␶i Gr共p1,k1+ q,␻+␻

⬘

兲␶j兴典, 共2兲

where fFD共␻

⬘

兲 is the Fermi-Dirac distribution function at

en-ergy ␻

⬘

and the angular brackets denote averaging over the random impurity configurations. The averaged Green’s func-tion is given by Gr共0兲_{共p,␻兲=1/共␻− E}

p− HB·␴+ i⌫兲, where

EF is the Fermi energy and ⌫=1/2␶. In the following, we

consider the regime E_FⰇ⌫⬎hk. Evaluating Eq. 共2兲 within the ladder series24 leads to the summation, up to all orders, of a basic diagram ⌿_␮␭␣␥共␻,␻

⬘

, q兲=ci

V兩Vsc兩2兺pG_␥␣r 共p,␻

+␻

⬘

兲G_␮␭a 共p−q,␻

⬘

兲. The response function becomes ⌸ij共q,␻兲 = i␻ 2␲

兺

j

冕

d␻

⬘

⳵fFD ⳵␻

⬘

冉

␲N₀ ⌫

冊

␶␮␣ i _␶ ␤␯j ⌿␮␭␣␥共␻,␻

⬘

,q兲 ⫻兵关1 − ⌿共␻,␻

⬘

,q兲兴−1其_␭␯␥␤, 共3兲 where 1_␭␯␥␤⬅␦_␥␤␦_␭␯and⌫/共␲N0兲=ci兩Vsc兩2/V. Here, V is total

area of the sample and ciand Vscare, respectively, the

rity density and the Fourier transform of a short-range impu-rity potential at q = 0. Making use of the transformation ⌿_␭␭␥␥_⬘⬘=共1/2兲兺ij␶_␥␭i ⌿ij␶_␭⬘␥⬘

j

, Eqs.共1兲 and 共3兲 together gives

共1 − ⌿兲il_共D l− Dl 0_{兲 = i␻␶⌿}il Dl 0 , 共4兲 where ⌿il = ⌫ 2␲N0

兺

p⬘ Tr关␶i Gr共0兲共p

⬘

,␻+␻

⬘

兲␶lGa共0兲共p

⬘

− q,␻

⬘

兲兴. 共5兲 The charge neutrality is maintained by the condition D₀= 0, since E_F␶Ⰷ1 and␻= 0.

The spin-diffusion equation can be obtained by expanding ⌿il _{in lower orders of q and then by obtaining the Fourier} transform of Eq.共4兲 to the position representation.

Expand-ing⌿il_{up to lower orders in h}

kand to first order in B˜ results in a total of five terms given by

⌿il =

兺

␯=1 5 ⌿_␯il , 共6兲

where q and␻dependences are not shown. The term⌿₁il=共1+i␻␶− D␶q2_兲␦

ilwith D =vF

2_{␶/2 produces}

the regular diffusion equation. The second term ⌿₂i⬘l = iqm␶Ri⬘lm, where Rilm= 4␶兺n␧ilnhk

n

v_Fm and ␧iln _{as the} Levi-Civita symbol, causes the spin densities to precess about its local variations. The overline denotes the angular average over the Fermi surface and m , n are the component indices. The third term ⌿₃il= −␶⌫il_{, where ⌫}il_{= 4␶h}

k 2_共␦il_{− n} k i nk l_{兲 for}

i , l⫽0, and for unit vector nk= hk/hk, describes the Dyakonov-Perel’ spin relaxation.1The last two terms contain new contributions from the in-plane magnetic fields. In the fourth term, we have ⌿₄il=␶RB

ilm

, where RB ilm_{= −兺}

m2␧ilmB˜m and m is the field component index. This gives rise to pre-cession of the spin densities about the magnetic field. In the fifth term we have the spin-charge coupling ⌿5 i⬘0 =共−iq兲共MB i⬘0 − Mi⬘0兲. Here Mi0= 4␶3hk 3 ⳵nki ⳵k and MB i0 = 2␶2_共B˜ x ⳵hky ⳵k− B˜y ⳵hkx

⳵k兲␦iz. The latter contains the effect of the in-plane magnetic field. Up to this point we have kept a FIG. 1. Top-view schematic illustration of the 2D stripe. The 2D

stripe has a width d. In the system, electric field E and in-plane magnetic field B are applied. The direction along the stripe, or xˆ, is denoted longitudinal and that along yˆ, transverse.

generic form of the SOI. Applying to the case of Rashba SOI, when hk=␣k⫻zˆ, we find no out-of-plane bulk spin densities and no edge spin accumulations, which is consis-tent with previous findings.11_{A more interesting case is the}

Dresselhaus SOI, where hk=␤关kx共ky

2

−␬2兲,ky共␬2− kx

2_兲兴.25_Here ␬2_=具k

z

2_{典 is the average over the thickness of the 2DEG and}

具hk

z_{典=0. From Eqs. 共}

4兲 and 共6兲 and the form of hk, we obtain the static spin-diffusion equations

冦

D⳵ 2 ⳵y2Sx+ Rxzy ប ⳵ ⳵ySz− ⌫xx ប2Sx+ 2 បB˜ySz− C₁ ប2 = 0, D ⳵ 2 ⳵y2Sy− ⌫yy ប2Sy− 2 បB˜xSz= 0, D ⳵ 2 ⳵y2Sz+ Rzxy ប ⳵ ⳵ySx− ⌫zz ប2Sz− 2 បB˜ySx+ 2 បB˜xSy− B ˜ y បC2= 0,

冧

共7兲

where Si= Di/2 and the homogeneity along x is assumed. Effects of the in-plane magnetic field enter Eqs.共7兲 in two

places. The first is the precession effect given exactly by

d␴/dt=共2/ប兲B˜⫻␴. The second is through the coefficient

C2, which is originated from the spin-charge coupling term MB

i0

. Its expression is given by C2=␶共⳵hk x_/

⳵kx兲共⳵D0 0_/⳵

x兲,

where D₀0= −2N0eEx for e⬎0 represents the effects of the

driving electric field.

Expressions for other coefficients in Eqs. 共7兲 are the

Dyakonov-Perel’ spin relaxation rates ⌫xx=⌫yy=⌫zz/2 =␤2_␶k

F

6_共1/4−C2_{+ 2C}4_{兲, where C=␬/k}

F. Precessions about

local variation in spin densities are given by the coefficients

Rzxy_{= −R}xzy₌␤␶kF4

mⴱ共2C

2_{− 1/2兲=2D/l}

so. Finally, the

spin-charge coupling that originates from Mi0 _{is the coefficient}

C1= Mx0D00/2.

The bulk spin densities obtained from Eqs.共7兲 are

冦

Sz b =⌳y

冉

− 1 2C2+ C₁ ⌫xx

冊

冒

共1 + 2⌳x2+ 2⌳y2兲, Sy b = − 2⌳xSz b , Sx b_{= 2⌳} ySz b − C1 ⌫xx,

冧

共8兲

where⌳, the dimensionless B˜, is given by ⌳i= B˜i/⌫xx. Sbis checked to reproduce the correct B = 0 limit.16_{Except for the} ⌳yC2 term in Sz

b

, which is originated from the spin-charge coupling, all other terms in Sb_{that are proportional to⌳}

iare related to spin precessions about B˜ .

The boundary condition for the spin-diffusion equation is established in the following by connecting the spin current to the spin densities and their spatial gradients and then requires the transverse flow of the spin current to be zero at the lateral edges.16 _{This is appropriate for hard wall boundary.}26,27 _We start from the conventional form of the spin current operator

J_li⬅共1/2兲共Vl␴i+␴iVl兲, where spin unit of ប/2 is implied. The velocity operator is given by

Vl⬅ kl mⴱ+ ⳵hk·␴ ⳵kl , 共9兲

wherevl=共kl/mⴱ兲. The expression for the spin current is16

Il i 共q,␻兲 = i␻

冕

d␻

⬘

2␲ dNF d␻

⬘

兺

k,k⬘

冓

冉

vl␴i+ ⳵hk i ⳵kl

冊

⫻Gr

冉

k +q 2,k

⬘

+ q 2,␻+␻

⬘

冊

␶ j ⫻Ga

冉

k

⬘

− q 2,k − q 2,␻

⬘

冊

冔

⌽j共q,␻兲, 共10兲 where the summation convention for repeated indices is adopted. In the dc limit 共␻= 0兲 and at zero temperature 共␻

⬘

= EF兲, the spin current is related to the four densities in

the form Il i = 1 mⴱ关Xl ij⬘ Dj⬘− Xl i0_D 0 0_{+ Y} l ij⬘ Dj⬘− Yl i0_D 0 0_{兴, 共11兲}

where j

⬘

denotes the spin indices. The operators to the spin densities are Xl ij_⬅

冉

⌫ 2␲N0

冊

兺

k kl ⫻Tr

冋

␶i Gr共0兲

冉

k +q 2,␻+ EF

冊

␶ j Ga共0兲

冉

k −q 2,EF

冊

册

, 共12兲 and

Yl ij_⬅

冉

⌫ 2␲N₀

冊

兺

k ⳵hk i ⳵kl ⫻Tr

冋

Gr共0兲

冉

_{k +}q 2,␻+ EF

冊

␶ j_Ga共0兲

冉

_{k −}q 2,EF

冊

册

. 共13兲 Specifying to the flow of spin along y, we calculate Xy

ij and Yy ij to give Xy ij = − mⴱ

冉

iqyD␦ij+ 1 2R ijy_共␦ iz+␦jz兲

冊

− 2iqxmⴱ␶2vF y

冉

hk⫻ ⳵hk ⳵kx

冊

z ␦iz␦j0− ⳵hk i ⳵ky ␦j0, 共14兲 and Yy ij = ⳵hk i ⳵ky ␦j0, 共15兲

with the latter being exactly canceled by a term in Eq.共14兲.

Finally, substituting Eqs. 共13兲 and 共14兲 into Eq. 共10兲, we

arrive at the spin current expression that provides us the boundary condition I_yi= 0 at the lateral edges for the spin-diffusion equation with

Iy i_{共r兲 = − 2D}⳵Si ⳵y − Rijy ប 共Sj− Sj b_{兲 +}IsH ប ␦iz. 共16兲 The first term of Iy

i

describes the spin diffusion due to spatial variation in Si, the second term is the spin precession prompted by the SOI, and IsH␦izis the bulk spin current with

IsH= − RzjySj b_{+ 4␶}2_eEN 0vF y

冉

⳵hk ⳵kx ⫻ hk

冊

z . 共17兲

Equations共16兲 and 共17兲 appear to be the same as their

coun-terparts for the B = 0 case;16 but the magnetic field contrib-utes, in its lowest order, via the spin density Sj

b

in Eq.共8兲. It

is worth pointing out here that the primary purpose of deriv-ing Eq.共17兲 is to apply it to a region within a distance much

less than lsofrom the sample boundary. As such, the effect of

spin torque28,29 _{on the boundary condition should be of} sec-ondary importance, and the results in this work should also remain intact. An eventual exploration on this issue, how-ever, is left for future study.

III. NUMERICAL RESULTS: IN-PLANE B FIELD IN A DRESSELHAUS 2D STRIPE

In this section, we present the electric-field-induced bulk spin densities and edge spin accumulations in a Dresselhaus-type 2DEG stripe acted upon by an in-plane magnetic field. Symmetries, or asymmetries, of the spin distributions with respect to spatial coordinates and the magnetic field are pre-sented in two field orientations: longitudinal and transverse. For definiteness, we use material parameters consistent with GaAs: effective mass mⴱ= 0.067m0, with m0 the

elec-tron mass; effective g factor gⴱ= 0.44 共Ref. 30兲; and the

Dresselhaus SOI␤= 27.5 eV Å3_.25_{Other typical parameters} are electron density n = 2.4⫻1015 _m−2_{, quantum well}

thick-ness w = 300 Å, le= 1 ␮m, and lso= 2.9 ␮m. The electrons

occupy only the lowest subband in the quantum well. An electric field E = 25 mV/␮m is applied along x to set up the spin-Hall phenomenon.

Longitudinal field orientation case is presented in Figs.

2共a兲–2共c兲. Shown here are the spatial variations in all spin components of Si across the stripe. Sx has both finite bulk spin density and edge spin accumulation. It exhibits even parity in its spatial variation and remains so for finite field

Bx. The magnetic field causes only a minor change to the Sx profile while it has an even parity in its Bxdependence. Syis zero at Bx= 0 and has an edge spin accumulation in finite Bx. It is of odd parity in both its spatial and field dependencies.

Sz has an edge spin accumulation. It is of odd and even parities in its spatial and field dependencies, respectively. Overall, except for Sy, the effects of Bxfor the chosen range of field strengths is weak. That the spatial profile of Sy for finite fields mirrors that of Szcorroborates a spin precession picture as suggested by Eqs. 共7兲. Following the d␴/dt

=共2/ប兲B˜⫻␴time evolution, the precession of Szcontributes to Sy.

Transverse field orientation case is presented in Figs.3共a兲 and 3共b兲. The field effects on the spin distributions and on the parity of the Siprofiles are much more dramatic. In short, the Sxand Szprofiles become asymmetric in both their spatial and field dependencies. Sy, however, remains zero in all these cases. Qualitative understanding of these changes can be ob-tained again from the spin precession picture. We take, for instance, the Bx= −300 mT curve for Sz in Fig. 3共b兲. The FIG. 2. 共Color online兲 Spin densities Siversus y, in units of lso, for the case of a longitudinal in-plane magnetic field. Spin densities

Sx, Sy, and Sz in units of ␮m−2 are shown in 共a兲, 共b兲, and 共c兲, respectively, for magnetic fields B_x= −300 mT 共black/ triangles兲,

Bx= 0 mT共blue/solid curve兲 and Bx= 300 mT共red/dashed curve兲. The edges of the stripe are at y =⫾5lso.

out-of-plane spin density Sz⯝1.5 ␮m−2 in the bulk is re-sulted from the precession of the zero-field Sxand also from a spin-charge coupling term in Eq. 共8兲. On the other hand,

the Szedge spin accumulation is resulted from two spin pre-cession processes, if we treat Sias individual entities. First, the magnitude of Szedge accumulation is reduced due to its own precession. However, it may be increased due to the precession of Sx. As the zero-field Sx is even and the zero-field Szis odd in their spatial parity, it is inevitable that the magnitude of Sz will receive enhancement at one edge and suffer suppression at another. This leads to the breaking of the spatial parity of the Sz profile as is confirmed in Fig.3. The zero-field Sithus play a pivotal role in the shaping of the low in-plane magnetic-field Siprofile.

Figure4 presents the edge spin accumulations of Si⫾and their parity in their field dependencies. S_i⫾denote edge spin densities at y =⫿d/2. For the longitudinal field orientation depicted in Figs.4共a兲and4共b兲, Sx⫾and Sz⫾are of even parity in Bx, whereas Sy⫾is of odd parity in Bx. The magnitude of the variation is comparable for Sy⫾and Sz⫾, a feature consis-tent with our spin precession picture. More detailed symme-tries can be read off from Eq.共7兲 and is given in the

follow-ing: Sy共z兲 + = −Sy共z兲 − , Sx + = Sx − , Sx⫾共z兲共Bx兲=Sx⫾共z兲共−Bx兲, and

Sy⫾共Bx兲=−Sy⫾共−Bx兲. For the transverse field orientation de-picted in Figs.4共c兲and4共d兲, Sx⫾and Sz⫾become asymmetric in their field dependencies, whereas Sy⫾= 0. The extremum points in S_z⫾at By= 50 and −50 mT in Figs. 4共c兲and 4共d兲, respectively, demonstrate the competition between the two spin precession processes: decreasing in magnitude due to its own precession and increasing in magnitude due to spin pre-cession in S_x⫾. Finally, if we include both the spatial and the

field reversals, we obtain symmetries Sz+共By兲=−Sz−共−By兲 and

Sx

+_共B

y兲=Sx

−_共−B

y兲.

The entire spatial and field symmetries of the out-of-plane spin densities are presented in the contour plots in Fig.5. In Fig. 5共a兲, the longitudinal field case exhibits even parity in

Bxand odd parity in y. In contrast, the transverse field case, depicted in Fig. 5共b兲, exhibits much richer features. Even though the asymmetry of Sz with respect to Byand y, indi-vidually, is evident, the symmetry Sz共By, y兲=−Sz共−By, −y兲 is also clearly shown. At the lateral edges, the highest spin densities are shifted from By= 0. It is resulted from the two competing spin precession processes. Near the center of the sample, Szis odd in Byand its magnitude increases with the field as indicated in Eq.共8兲 already.

FIG. 3. 共Color online兲 Spin densities S_iversus y for the case of a transverse in-plane magnetic field. Spin densities Sx and Sz in units of␮m−2_{are shown in共a兲 and 共b兲, respectively, for magnetic} fields By= −300 mT共black/triangles兲, By= 0 mT共blue/solid curve兲, and By= 300 mT共red/dashed curve兲. Sy remains zero in all these cases.

FIG. 4. 共Color online兲 Edge spin densities S_i⫾versus magnetic field for both field orientations: longitudinal B_xcases in共a兲 and 共b兲 and transverse Bycases in共c兲 and 共d兲. Si⫾denotes spin densities at the edges y =⫿d/2. S_x is labeled by the dashed curve, S_y by the dashed-dotted curve, and Szby the solid curve.

−5 −4 −3 −2 −1 0 1 2 3 4 5 −400 −200 0 200 400 B x (mT) y( _so) −5 −4 −3 −2 −1 0 1 2 3 4 5 −400 −200 0 200 400 B y (mT) l −3 −2 −1 0 1 2 3 l y( _so) S_z(µm−2) (b) (a)

FIG. 5. 共Color online兲 Contour plot of Szon the Bi-y plane for 共a兲 the longitudinal and 共b兲 the transverse field orientations.

The strong in-plane magnetic-field anisotropy in the sym-metry characteristics of the Siprofiles shown here is distinct for the Dresselhaus SOI. For the edge spin accumulation Sz in a transverse magnetic field, the Dresselhaus SOI leads to an asymmetric field dependence, whereas extrinsic SOI leads to a symmetric field dependence.6_{This symmetry} character-istic for the extrinsic SOI is clearly seen in the experiment of Kato et al.6_{关Fig. 1共c兲 in Ref.6兴, and also in their} demonstra-tion the Sz profile fits well to a Lorentzian function

A0/关共␻L␶s兲2+ 1兴 which depends on even power of B through

the square of the electron Larmor precession frequency ␻L.

The factor A0 is a proportionality constant and ␶s is the

electron-spin lifetime. As for the Rashba SOI, symmetry governs that we turn to longitudinal magnetic field. We find no Szboth in the bulk and at edges, which is consistent with previous finding.11 _{In contrast, we find that the Dresselhaus} SOI leads to an even-parity field dependence. Nonvanishing bulk spin density Szdue to Rashba SOI, but for the case of either anisotropic scatterers or nonparabolic electron disper-sion, has been obtained by Engel et al.11_{and the field} depen-dence is of odd parity.11,17_{Thus we commence the notion of} utilizing low in-plane magnetic field for the determination of the underlying SOI in a particular sample, without the need

to prepare controlling samples of different crystal orienta-tions.

IV. CONCLUSION

In conclusion, we have performed a systematic and com-prehensive study on the effects of a weak in-plane magnetic field on the bulk spin densities and edge spin accumulations in a diffusive Dresselhaus-type 2D stripe. Our results show that out-of-plane spin density can be generated in the case of transverse field orientation without assuming anisotropic scatterers or nonparabolic electron dispersion relations. The breaking of the parity of the spin distributions with respect to their spatial and field dependencies provide a unique signa-ture for the Dresselhaus SOI. This work thus points to the possibility of invoking weak in-plane magnetic fields for the determination of the SOI in a particular sample.

ACKNOWLEDGMENTS

This work was supported by Taiwan NSC共Contract No. 96-2112-M-009-0038-MY3兲, NCTS Taiwan, Russian RFBR 共Contract No. 060216699兲, and a MOE-ATU grant. We are grateful to the Centre for Advanced Study in Oslo for hospitality.

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