Control of the spin Hall current in two dimensional electronic gas
T. O. Cheche and E. Barna
Citation: Applied Physics Letters 89, 042116 (2006); doi: 10.1063/1.2234742
View online: http://dx.doi.org/10.1063/1.2234742
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Control of the spin Hall current in two dimensional electronic gas
T. O. Chechea兲
University of Bucharest, Faculty of Physics, Bucharest 077125, Romania and National Chiao Tung University, Department of Electrophysics, Hsinchu 300, Taiwan
E. Barna
University of Bucharest, Faculty of Physics, Bucharest 077125, Romania
共Received 16 April 2006; accepted 2 June 2006; published online 28 July 2006兲
The intrinsic spin Hall conductivity is obtained for a two-dimensional electronic gas共2DEG兲 in the presence of strain, Rashba coupling, and an external in-plane applied magnetic field. The conduction electrons of 关001兴 oriented quantum well are used to model the 2DEG. The spin current value is dependent on the stress applied in direction关111兴. © 2006 American Institute of Physics.
关DOI:10.1063/1.2234742兴
The spin current in the spin Hall effect means a flow of spin angular momentum perpendicular to an applied electri-cal field with a spin accumulation of opposite magnetization at each edge of the sample and no net charge current. The spin Hall conductivity共SHC兲 has both intrinsic 共absence of scattering兲 and extrinsic 共induced by scattering兲 contribu-tions. The intrinsic spin Hall effect was recently predicted as being generated by the splitting of conduction or valence bands, which is induced by intrinsic spin-orbit interactions.1–3The spin Hall effect has been detected in ex-periments dealing with strain effect,4 and there are reports which place the spin Hall current in the intrinsic regime.5–8It seems that a general agreement exists regarding the vanish-ing of the dc SHC of two-dimensional electronic gas共2DEG兲 with k-linear Rashba coupling and parabolic dispersion in the absence of magnetic field and the presence of scattering, even in the limit of weak disorder.9,10 On the other hand, in clean samples共where the transport scattering rate−1is small compared to the spin-orbit splitting ⌬ expressed in time units兲 modeled by a Rashba 共k-linear兲 spin-orbit coupling, one finds an intrinsic dc SHC value of e / 8, independent of details of the impurity scattering, for ac electric field of fre-quency in the range−1⬍⬍⌬, in the usual case where both spin-orbit split bands are occupied.7,10,11 Avoiding speculation that the above intrinsic SHC value would gener-ally be obtained for all k-linear coupling systems, we calcu-late in this work the intrinsic SHC of a specific k-linear coupling system, namely, a 2DEG in the presence of strain, Rashba spin coupling, and in-plane magnetic field. Though a more realistic model would involve the presence of impuri-ties and consideration of finite-size effect and of electron-electron interaction, clean samples are considered as experi-encing prominently an intrinsic spin Hall effect7,8 and thus, this theoretical work is motivated from experimental point of view. The 2DEG is obtained using a simple quantum-well 共QW兲 model.12
We consider for discussion the case of 关001兴-oriented QW and stress applied in关111兴 direction.
Firstly, the model of 2DEG is described. Secondly, the intrinsic SHC is calculated with numerical data matching InAs semiconductor. Thirdly, conclusions regarding the spin current are drawn.
For the electron Hamiltonian in the conduction band, we consider the following Hamiltonian:
H = P 2
2mc
+ V共z兲 + HR+ HP+ HB, 共1兲
where the first two terms describe the orbital motion of the electron confined in the z direction共perpendicular on the xy QW plane兲 by the V共z兲 potential. In Eq. 共1兲 e is the elemen-tary charge, mcis the electron effective mass, P = p + eA, with
p = −iប共/x ,/y ,/z ,兲 the canonical electron momentum,
A = zB共sin, −cos, 0兲 is the vector potential, B is the am-plitude of magnetic field B, is the angle between B and x, and x is the direction of the electric field. HR=ប·⍀R/ 2, with⍀R=␣R共P⫻n兲/ប, is the Rashba spin-orbit coupling,␣R is the Rashba coupling factor, n is the unit vector of the z direction, andiwith i = x , y , z are the Pauli matrices in the z representation. HP=ប·⍀P/ 2, with ⍀Px=a共xyPx−xzPz兲 + bPx共yy−zz兲/ប, is the bulk Pikus interaction 共responsible for the strain effect兲 written for 关001兴-oriented QW, a and b are constants which determine the magnitude of the splitting,13 and ij is the ij component of the strain tensor. Strain applied in directions关001兴 and 关110兴 does not yield a
z component of the strain Hamiltonian 共an effective mag-netic field in the z direction兲. On the other hand, strain ap-plied in关011兴, 关101兴, or 关111兴 direction induces such a com-ponent. The strain Hamiltonian for关011兴 or 关101兴 direction, even for the simpler case B = 0, introduces an additional pa-rameter, e.g., exxzz= b共xx−zz兲/2 for 关011兴 direction of the applied stress 共for this case the Hamiltonian reads HP= −exxzzpyy− eyzpyz兲. In the case of 关111兴 direction of the applied stress the only one necessary strain-dependent pa-rameter exy facilitates discussion and interpretation of pos-sible experimental results too. This case of关111兴 direction of applied stress, where xx=yy=zz and xy=yz=xz 共Ref. 14兲, is considered by this work. HB=xx+yy with x = 2−1BgB cos,y= 2−1BgB sinis the Zeeman term, g is the effective g factor, andB= eប/共2m0兲 is the Bohr
magne-ton. For a less computational effort we consider a semipara-bolic confinement potential shape in the z direction, namely,
V共z兲=mcE
2
z2/ 2 for z艌0 and V共z兲=⬁ for z⬍0; the Schrödinger equation for the orbital motion in the z direction yields wave functions of harmonic oscillator, confined by
V共z兲, having a displaced origin of the z axis.15Noticing that momenta pxand pyare constants of the motion for H given
a兲Author to whom correspondence should be addressed; Division of
Me-chanics, Research Center for Applied Sciences, Academia Sinica, Taiwan; electronic mail: cheche@gate.sinica.edu.tw
APPLIED PHYSICS LETTERS 89, 042116共2006兲
0003-6951/2006/89共4兲/042116/3/$23.00 89, 042116-1 © 2006 American Institute of Physics
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by Eq. 共1兲, the effective Hamiltonian H¯ is obtained by a quantum average over the wave function of the ground state of the orbital motion in the z direction1共z兲 and of the free
motion in the xy plane. The effective Hamiltonian reads
H¯ = ប2共kx 2 + ky 2兲/2m c+ 3ប0/2 +xx+yy+zz, where x=␥x+␣Rប共1+rf兲ky, y=␥y−␣Rប共1+rf兲kx, z =␣Rrf关ប共kx− ky兲+共␣s−␣c兲z¯兴 with ␥x=x+␣R共1+rf兲␣c¯,z ␥y =y−␣R共1+rf兲␣s¯,z ␣c= eB cos, ␣s= eB sin, rf ⬅␣xy/ 2␣R, and 0 2 =E 2 +c 2
, withc= eB / mc. The magni-tude of rfmay be modified by adjusting the values of stress. The electron spin precession around the momentum-dependent effective magnetic field is a useful tool for a men-tal visualization of the spin Hall effect. In our discussion the effective magnetic field in the z direction changes the explicit form of Bloch equations共as proposed by Ref. 2兲 used to find the SHC. A supplementary tilt of the precession axes共absent in the case of strain applied in关001兴 or 关110兴兲 has an impact on the value of spin Hall current.
The present analysis of the spin Hall effect of electrons in QW structures is based on the Kubo formalism for a spa-tially homogenous electric field.16 Next, we provide an ex-pression of the spin Hall conductivity for the Hamiltonian H¯ in the narrow QW limit, z¯→0. The narrow QW limit case
can capture the physics of the problem17 and the amount of algebra necessary to solve this case is moderate. The corre-sponding Hamiltonian, which models the present problem, reads H¯ =ប 2共k x 2+ k y 2兲 2mc +3ប0 2 +关x+␣Rប共1 + rf兲ky兴x+关y −␣Rប共1 + rf兲kx兴y+␣Rrfប共kx− ky兲z. 共2兲 For zero temperature and noninteracting conduction-band electrons the spin Hall conductivity is given by
xy Sz共 兲 =eប A
兺
k,⫽⬘ 共f,k− f⬘,k兲 ⫻ Im关具k,兩jx S,z共t兲兩⬘
,k典具k,⬘
兩v y兩⬘
,k典兴 关E共k兲 − E⬘共k兲兴关E⬘共k兲 − E共k兲 − ប− i兴. 共3兲 where f,kis the T = 0 K Fermi distribution function for en-ergy E共k兲 at wave vector k in a dispersion surface labeled by= ±, and A the xy area. The velocity operators are given by v = i关具H典,r兴/ប, where r is the position operator and 具¯典 means a quantum average over1共z兲. The spin current op-erator for the spin moment polarized along the z direction and flowing in the y direction when an electric field is ap-plied in the x direction is given by the generally accepted expression jxyS,z= 4−1共zvx+ vxz兲. The eigenvalues E±共k兲 and
eigenvectors兩k, ±典 of Hamiltonian H are as follows:
E±共k兲 = ប2k2 2mc +3ប0 2 ±
冑
⌳1+⌳2, 共4a兲 where ⌳1=关␣Rប共1+rf兲兴2共kx− ky兲2, ⌳2=关␣Rប共1+rf兲兴2k2 + 4−1共BgB兲2+␣R共1+rf兲BgB共kycos− kxsin兲艌0, and兩k, ± 典 =
冑
1 1 +±2冋
±exp共i␦±兲
1
册
, 共4b兲where ±=兩␣Rប共1+rf兲共kx− ky兲⫿
冑
⌳1+⌳2/冑
⌳2兩 and sin␦±=−␣Rប共1+rf兲共kx− ky兲⫿
冑
⌳1+⌳2关␣Rប共1+rf兲kx−x兴/冑
⌳2±.A general analytical expression of the spin Hall conduc-tivity may be obtained in the framework of the above 2DEG model. The presence of stress in 关111兴 direction makes the two surfaces E±共k兲 crossing only for particular orientations
of the magnetic field, namely, = 3/ 4 or = −/ 4 at k0
=共k0x, k0y兲. These magnetic field orientations are obtained by imposing⌳1共k0兲=⌳2共k0兲=0 in Eq. 共4a兲.18For the following discussion, we will consider this crossing surface case. With the translation of the origin defined by k = K + k0, the
cross-ing of the surfaces E±共K兲 in the K frame by the Fermi energy EF yields contours which are found with
KF±± =− AF±±
冑
AF±2 − 4B
F
2 , 共5兲
where AF±=
冑
2k0共cos␣+ sin␣兲±2ប−1mc␣R共1 + rf兲
冑
rf2共1+r
f兲−2共1−sin 2␣兲+1, BF= −2ប−2me共EF− 3ប0/ 2兲
+ k02, and ␣ is the polar angle in the K frame and k0=兩k0兩
=BgB /关2ប␣R共1+rf兲兴. As KF±
±
must be positive real, condi-tion AF±2 − 4BF艌0 must be fulfilled. For the simplest case,
BF⬍0ÛEF⬎ប2k02/ 2mc, with Eq. 共5兲 one finds the integral contour defined by K苸关min KF±+ , max KF±+ 兴. The intrinsic dc SHC obtained with Eq.共3兲, within the limits,→0, and then
→0 is xy Sz =eប 3␣ R共1 + rf兲 Amc ⫻
兺
k 共f+k − f−k兲 kx关+共1 +−2兲sin␦+−−共1 ++2兲sin␦−兴 关E+共k兲 − E−共k兲兴2共1 ++2兲共1 +−2兲 . 共6兲 The integral form of Eq.共6兲 in the K variable for the cross-ing surfaces case readsxy Sz = eប 162mc␣R共1 + rf兲 ⫻
冕
0 2 d␣冕
min KF±+ max KF±+ dK K cos␣共k0x+ K cos␣兲 rg2共1 − sin 2␣兲 + 1 ⫻冋
rg共sin␣− cos␣兲 −冑
rg 2共1 − sin 2␣兲 + 1 1 +−2 −rg共sin␣− cos␣兲 +冑
rg 2共1 − sin 2␣兲 + 1 1 ++2册
, 共7兲where rg⬅rf共1+rf兲−1. In the limits B→0 and rf→0, the model Hamiltonian from Eq. 共2兲 describes the 2DEG with
k-linear Rashba coupling and parabolic dispersion, and the
expression of intrinsic dc SHC from Ref. 15 is recovered. The plane of Fermi energy, which determines the inte-gration contour in Eq.共7兲, is chosen as being situated above the crossing point of the branches共in this case integration is independent of EF, see Ref. 19兲. An energy description of the problem can be found in Ref. 15. Figure 1 shows the intrinsic dc SHC obtained by integration of Eq.共7兲. We find that the intrinsic dc SHC is practically independent of the in-plane magnetic field for usual values of the magnetic field, B 苸关0,10兴 T. As shown in Fig. 1 the intrinsic dc SHC may be
042116-2 T. O. Cheche Appl. Phys. Lett. 89, 042116共2006兲
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changed by applying stress in direction关111兴. Thus, the spin current ranges between 0, forxySz= 0, and maximum value, for xy
Sz
= e / 8. From experimental point of view, an impor-tant conclusion is that by variation of rf 共induced by varia-tion of stress兲 in the interval 关−1,0兴 the spin current is modi-fied from zero to the maximum value in clean QW samples. On the other hand, for values of rf out of the interval 关−1,0兴 the variation of the intrinsic dc SHC decreases with the applied stress. Consequently, the accuracy of controlling the spin Hall current also decreases with the applied stress. The model described by the Hamiltonian of Eq.共2兲 predicts 关by an analytical integration of Eq. 共7兲兴 that xy
Sz共r f→ ±⬁兲 = e /共
冑
38兲.Interesting is the fact that the anisotropy itself of the dispersion branches is not sufficient to induce a strain-dependent spin Hall current. In Ref. 20, for a similar model excepting stress presence, but considering scattering effect, increasing electron density 共corresponding to increasing Fermi energy兲 yields a less pronounced variation of the dc SHC 共intrinsic plus extrinsic component兲 with the in-plane magnetic field. This is not in contradiction with the indepen-dency of the intrinsic dc SHC of k-linear Hamiltonian with-out effective magnetic field in the z direction, when the Fermi energy level is situated above the crossing point.15,19 The z component of the effective magnetic is necessary to obtain a strain-dependent value of intrinsic SHC. On the other hand, a remarkable analogy between the effect of strain on SHC of 2DEG and strain-induced spin relaxation of the electrons of conduction band for the bulk case may be ob-served: a stress applied in 关001兴 direction has an effect on neither intrinsic dc SHC nor spin relaxation time共as
calcu-lated in Ref. 13兲, but the stress applied in direction 关111兴 affects both quantities. As the two phenomena, the spin Hall effect and spin relaxation are considered for QW and bulk, respectively, this analogy does not hold for the directions 关011兴, 关101兴, and 关110兴. The three directions are equivalent for the bulk treatment of spin relaxation, but not for the QW case involved by our discussion on the spin Hall effect. Only directions关011兴 and 关101兴 induce an effective magnetic field in the z direction and consequently can generate a strain-dependent intrinsic dc SHC.
In conclusion, for a 2DEG, in the presence of Rashba coupling and the absence of Dresselhaus coupling, we pre-dict that the intrinsic dc SHC is dependent on both the in-plane magnetic field and applied stress. For 关001兴 direction of QW generating the 2DEG, the intrinsic dc SHC changes between 0 and the universal constant,xy
S,z
= e / 8as a func-tion of the magnitude of stress applied in 关111兴 direction, when the in-plane magnetic field is oriented at angles = 3/ 4 and −/ 4 from the dc electric field direction. Con-sequently, the magnitude of the intrinsic spin Hall current may be controlled by applying stress in direction关111兴.
The authors are grateful to M. C. Chang and A. G. Mal’shukov for numerous helpful discussions. This work is supported by the National Science Council of Taiwan under Contract No. NSC 0940033709.
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FIG. 1. Variation of intrinsic dc SHC with the parameter rf. The Fermi
energy EF= 2.7ប2k02/共2mc兲+3h02/ 2 is situated above the crossing point of
the two branches E±共K兲. The following numerical values are used: B=5 T,
g = 15, mc= 0.024m0共m0is the electron mass兲,␣R= 91 156 m / s, rf= 0.5, B
= 10 T, and= 3/ 4.
042116-3 T. O. Cheche Appl. Phys. Lett. 89, 042116共2006兲
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