Spin orientation and spin-Hall effect induced by tunneling electrons

Spin orientation and spin-Hall effect induced by tunneling electrons

A. G. Mal’shukov1,2,3and C. S. Chu2

1_{Institute of Spectroscopy, Russian Academy of Science, 142190 Troitsk, Moscow Oblast, Russia} 2_{Department of Electrophysics, National Chiao Tung University, Hsinchu 30010, Taiwan}

3_{National Center for Theoretical Sciences, Physics Division, Hsinchu 30043, Taiwan}

共Received 26 July 2007; published 27 December 2007兲

It is shown that a flux of unpolarized electrons across a symmetric double barrier quantum well induces a spin polarization inside the well. Besides, the transmitted current acquires a spin polarized component and the spin-Hall current flows in the planar direction. These phenomena are due to a combined effect of Dresselhaus interaction and the spin-orbit interaction induced by gradients of heterostructure material parameters. In con-trast to previous studies of the spin filtering effect, we predict that it can be observed in the case of an isotropic distribution of incident electrons.

DOI:10.1103/PhysRevB.76.245326 PACS number共s兲: 73.40.Lq, 71.70.Ej, 72.25.Dc

I. INTRODUCTION

The spin-orbit interaction共SOI兲 is a fundamental quantum relativistic phenomenon which recently attracted much inter-est in connection with spin transport of electrons in semicon-ductors and metals. Due to SOI, an electric field can influ-ence the spin degree of freedom, thereby giving rise to a number of transport phenomena which have potential for ap-plication in spintronics. One of them is the spin-Hall effect 共SHE兲, which recently has been intensively studied both theoretically共for a review see Ref.1兲 and experimentally.2,3

A standard system to study this effect is a two-dimensional 共2D兲 electron gas confined within a quantum well. Due to SHE the electric current in the quantum well gives rise to a perpendicular flux of the spin polarization, as well as to the out-of-plane spin density near sample edges. Closely related to SHE is the electric spin orientation, which is a bulk in-plane spin polarization induced by the dc electric current.4_In

both cases SOI is realized either via impurity scattering or due to an intrinsic spin-orbit coupling mechanism. The latter consists of two parts. The first one is the Dresselhaus interaction5_{which is inherent to all zinc-blende}

semiconduc-tors. The second contribution is determined by gradients of material parameters and the electric potential across a heterointerface.6,7_{In quantum wells共QWs兲 these interactions}

are averaged with wave functions of confinement coordinate in the heterostructure growth direction. After such an aver-aging procedure the second term transforms into the Rashba SOI 共Ref. 8兲 which is not zero only in asymmetric in z-direction 共growth direction兲 heterostructures. The Rashba and averaged Dresselhaus interactions are basic SOI widely used in works on SHE, as well as in works on spin-dependent transport in general.

In this work we will consider spin-orbit effects on elec-tron transport from a different point of view. Namely, we will consider the electric current parallel to the growth direction, rather than parallel to the quantum well. An appropriate model for studying such a situation is a double barrier quan-tum well. A principal difference from the conventional 2D system is that one cannot use neither Rashba nor Dresselhaus interactions averaged over QW confinement. For example, a part of the latter interaction, which is proportional to the z

component of the electron momentum operator, turns to 0 when averaged with a wave function of a confined state in the well. At the same time it is finite for tunneling states. Considering these states it is also easy to see that SOI asso-ciated with electric fields at heterointerfaces does not reduce to the single parameter Rashba interaction. The explicit de-pendence on z of the initial spin-orbit interaction becomes important. To make this point more clear we deliberately considered a symmetric heterostructure, where Rashba SOI is zero. For such a model of a symmetric double barrier quantum well we found out that the electric current in the z direction induces a parallel to the z-axis spin density, as well as a current of spins polarized in a planar direction. The spin current flows parallel to heterointerfaces. The former effect is an analog of the electric spin orientation, while the latter is the spin-Hall effect. Besides, we found the transmittance of the double barrier structure to be dependent on the spin ori-entation. A similar spin filtering effect has been considered before within various models.9,10_{It was shown there that this}

effect can be observed only in the case of an anisotropic distribution of incident electrons. It was suggested to create such a distribution applying a planar electric field. In contrast to these works, we predict the spin filtering effect for an isotropic distribution of electrons tunneling through a sym-metric double barrier structure. Moreover, our spin filter makes electrons polarized in the z direction, instead of the planar polarization in Refs.9and10. Such fundamental dis-tinctions arise from the⬃kzterm of the Dresselhaus

interac-tion which has been neglected in Ref.10.

The general Hamiltonian of the problem will be derived in Sec. II. In Sec. III we will present our results related to the spin-Hall effect, spin orientation, and the spin filtering effect. A brief conclusion is presented in Sec. IV.

II. SPIN-ORBIT HAMILTONIAN

Let us consider a quantum well 共QW兲 of the width 2d separated from the left共z⬍−d−b兲 and right 共z⬎d+b兲 parts of a doped semiconductor system by two equal barriers of the thickness b. These parts are assumed to be thermal res-ervoirs with respective chemical potentials ␮l and ␮r.

consider an electron gas confined in the y direction. For sim-plicity, the adiabatic case will be considered when the con-finement width w slowly increases from the QW towards reservoirs. This situation is realized when the confinement is achieved by depleting the electron gas with the help of elec-trodes on top of the QW.

The spin-orbit interaction in such a system is represented by two Hamiltonians H_so共1兲 and H_so共2兲. The former is the Dresselhaus SOI, while the latter is SOI due to change of the band-gap width and other material parameters across hetero-interfaces. Usually, in narrow gap semiconductors H_so共2兲 is much stronger. Therefore we will consider the Dresselhaus interaction within the first order perturbation theory. Only the part of H_so共1兲which is proportional to the z component of the electron momentum operator will be taken into account. Hence for关001兴 growth direction

Hso共1兲= − i␥␴z共kx

2 − ky

2_兲⳵

⳵z, 共1兲

where␴zis the Pauli matrix and␥is the coupling parameter,

which is assumed to be z independent. An important property of Hamiltonian共1兲 is that its expectation values taken with tunneling states incident from the left and from the right reservoirs have opposite signs, while it does not change sign when kx, ky→−kx, −ky. Alternatively, other parts of the

Dresselhaus SOI, which have been omitted in Eq. 共1兲, are even functions of kˆz and odd functions of kx, ky. Since the

interaction represented by H_so共2兲is of the same symmetry, the omitted terms of H_so共1兲do not add particularly new qualitative features to spin dependent electron tunneling, as well as to other effects considered below. At the same time, the sym-metry difference of Eq. 共1兲 and H_so共2兲 has important conse-quences for these effects. This is also the main reason why the results of Ref. 10, where interaction 共1兲 has been ne-glected, are qualitatively different from those presented be-low.

Following Ref.6 the Hamiltonian H_so共2兲can be written as

Hso共2兲= 1 k储 共␴xky−␴ykx兲h共z兲, 共2兲 where h共z兲 = k储 ⳵␤ ⳵z 共3兲 and k储=

冑

kx 2 + ky

2_{. The parameter h共z兲 denotes SOI strength} which varies across the heterostructure depending on semi-conductor material parameters and the electric potential. Ig-noring the electron energy, which is much less than the gap value,␤共z兲 can be written as6

␤共z兲 = 1 2m共z兲

⌬共z兲 3Eg共z兲 + ⌬共z兲

, 共4兲

where Eg共z兲 and ⌬共z兲 are respective values of the band gaps

and split off energies.

The major effect of SOI 共1兲 is that it gives rise to spin precession around the z axis. This precession takes place during particle transmission through the double barrier

struc-ture. Since the width of this structure is small in comparison with the spin precession length, the effect of the spin preces-sion is expected to be small. In order to get explicitly the corresponding small parameter, the Hamiltonian can be transformed using an appropriate unitary transformation. Taking into account that the kinetic energy operator in the z direction is 1 2 ⳵ ⳵z 1 m共z兲 ⳵ ⳵z 共5兲

one can apply the unitary transformation

H→ U−1_{HU, 共6兲} with U = ei␴z␽共z兲_, 共7兲 where ␽共z兲 = −␥共kx2− ky2兲

冕

0 z m共z兲dz. 共8兲

This transformation removes Eq.共1兲 from the Hamiltonian. At the same time, applying it to Eq.共2兲 one obtains, up to the linear in␽共z兲 terms, the spin orbit interaction

Hso= Hso共2兲+

2␽共z兲 k储

共␴xkx+␴yky兲h共z兲. 共9兲

For the model under consideration, with rectangular symmet-ric barriers and a rectangular QW, h共z兲 becomes

h共z兲 = k储兵共␤r−␤b兲关␦共z − b − d兲 −␦共z + b + d兲兴 + 共␤b−␤w兲

⫻关␦共z − d兲 −␦共z + d兲兴其. 共10兲

The parameters␤r,␤b, and␤wdenote SOI strengths for

res-ervoirs, barriers, and QWs, respectively.

The transmission wave functions are represented by two sets of functions incident from the left共␺l_{兲 and from the right}

共␺r_{兲 of the double barrier structure. In the zeroth order, when}

the second term in Eq.共9兲 is ignored, these functions can be conveniently written using the chiral basis. In the case of the quantum wire confinement this basis corresponds to the spin quantization axis directed along the y axis. Outside the double barrier structure the scattering eigenstates are repre-sented by incident, transmitted, and reflected plane waves, with transmission and reflection amplitudes t_␴l/r and r_␴l/r, re-spectively, where␴= 1 , 2 denotes the spin projection in the corresponding chiral basis. This projection is conserved upon the scattering, as far as the second term in Eq. 共9兲 is ne-glected. The wave vector of the scattering states is denoted as k =

冑

2mr共E−E储兲, where mris the electron effective mass in

the left and right reservoirs, E is the total energy, and E储 is

the energy of motion in x, y directions.

III. SPIN CURRENT AND SPIN POLARIZATION A. Spin current

The nonzero spin current Jn s

, where s and n denote the spin polarization and current direction, respectively, can be

calculated already in the zeroth order with respect to␽共z兲, while the latter will be important below in the calculation of the spin density and spin dependent transmission. Hence in this section we entirely neglect the presumably weak Dresselhaus SOI. Let us take the chiral component of Jn

s

. That means that we look for the flux of spins polarized per-pendicular to the flux direction that can be expressed as Jn

s_{= J␧}snz_{, where ␧}snz _{is the antisymmetric tensor. Using}

the conventional definition of the spin current operator Jˆ_ns=兵vn,␴s/4其, where the spin dependent part of

vn= kn m共z兲+␧ niz_␴ i h共z兲 k储 共11兲 is obtained from Eqs.共2兲 and 共3兲, the spin current density can be written as Jn s_{共z兲 =}␧ snz 2 k

兺

储,␥

冕

0 ⬁ _dk 2␲

冋

k储 2m*共z兲共兩␺1 ␥_兩2_−兩␺ 2 ␥_兩2_兲 +h共z兲 k储 共兩␺1␥兩2+兩␺␥2兩2兲

册

nF␥共E兲, 共12兲

where␥= l, r and n_F␥共E兲 is the Fermi distribution function for the left and right reservoirs. The first term in square brackets represents the bulk spin current density distributed in the QW, barriers and outside, while the second term is the “sur-face” term which, according to Eq. 共10兲, is finite only on heterostructure interfaces. From Eq.共12兲 it becomes imme-diately evident that the spin current is not zero in the equi-librium state when nF

l_共E兲=n F

r_{共E兲. For example, the surface}

current at each interface is given by␳共z兲⌬␤共z兲, where␳共z兲 is the local equilibrium electron density and⌬␤共z兲 is the dif-ference of the spin-orbit coupling parameters␤on both sides of the interface, as follows from Eq. 共10兲. In a symmetric QW the surface currents on opposite interfaces flow in op-posite directions, so that they cancel each other. It is easy to see that the total equilibrium current, obtained by integration of Eq.共12兲 over z, is identically zero in case of a symmetric heterostructure. That follows from the symmetry relation

␺1 l/r_共z兲=␺ 2 r/l_{共−z兲 which makes J} n s_{共z兲 an odd function of z. At}

the same time, one cannot expect the total current to be zero in an asymmetric heterostructure, as has been shown by Rashba11for confined states. It should be noted that Rashba found that the total equilibrium current is cubic with respect to the spin-orbit coupling constant, while the current density given by Eq.共12兲 is linear. The latter becomes evident from the above expression for the surface current. That means that, at least, linear and quadratic terms vanish after integra-tion of Eq.共12兲 over z.

By convention, the “nonequilibrium” current could be de-fined as a part of Eq.共12兲 which is proportional to nF

l

− nF r

. In a symmetric QW this current density is an even function of z and, hence, the corresponding total nonequilibrium current is finite. However, one cannot define unambiguously the dissi-pative part of the spin current using only its definition共12兲. Calculation of the spin accumulation at the sample boundary would be helpful to clarify the physical meaning of Eq.共12兲.

B. Spin orientation

In order to calculate the spin density induced by the tun-neling current, the second term in Eq.共9兲 must be taken into account. It causes spin flip processes upon transmission and reflection of particles incident onto the double barrier struc-ture. Therefore we label spin variables of wave functions by two indices, as␺_␣␤, where␣denotes the spin polarization of the incident wave. Treating such functions as matrices, the spin density can be expressed as

S共z兲 =1 2

兺

k_储,␥

冕

0 ⬁ _dk 2␲Tr关␺ ␥+_␴␺␥_兴n F ␥_{共E兲, 共13兲}

where␴=共␴x,␴y,␴z兲 is the vector of Pauli matrices. We

cal-culate Eq. 共13兲 in the first order perturbation theory with respect to␽共z兲. Since a commutator of two terms in Eq. 共9兲 is proportional to␴z, one should expect the z component of

S共z兲 to be finite. Further, the time inversion symmetry

dic-tates

兺

␥=l,rTr关␺k储 ␥+_␴␺ k_储 ␥_{兴 = −}

兺

␥=l,rTr关␺−k储 ␥+_␴␺ −k储 ␥ _{兴. 共14兲}

Applying this relation to Eq.共13兲 the latter is transformed to Sz共z兲 =

兺

k储

冕

0 ⬁ _dk 4␲Tr关␺ l+_␴ z␺l兴关nF l_{共E兲 − n} F r_{共E兲兴. 共15兲}

It immediately follows from this expression that the spin density is zero in the equilibrium state.

The first order correction to the wave function can be written in terms of the retarded Green function, so that

␺␣␤l 共z兲 =␺␣l共z兲␦␣␤+

冕

dz

⬘

G␤共z,z

⬘

兲V␤␣共z

⬘

兲␺␣l共z

⬘

兲, 共16兲 where V_␤␣共z

⬘

兲 is a matrix element of the second term in Eq. 共9兲 and ␺_␣l共z兲 is the unperturbed wave function. In its turn, the Green function is given by

G_␤共z,z

⬘

兲 = − imr kt_␤关␺␤ l_共z兲␺ ␤ r_共z

_⬘

_{兲␪共z − z}

_⬘

_兲 +␺_␤r共z兲␺_␤l共z

⬘

兲␪共z

⬘

− z兲兴. 共17兲 In the considered case of a symmetric heterostructure, it is easy to see that the transmission coefficient t1= t2⬅t. At the same time, the reflection is spin dependent through its phase. We note that, according to Eq.共8兲, the matrix elements V_␤␣ are proportional to kx

2 − ky

2

. After integration in Eq.共13兲 over angles of the vector k储this expression turns to 0. It does not

necessarily happen for other than关001兴 crystal orientations. We will not consider such an opportunity here. Also, we will not discuss here other than Dresselhaus’s SOI effects, for example, due to strain, or due to potential gradients along x, y axes. Instead, let us consider a situation when electrons are confined, say, in the y direction. In this case k_x2− k_y2 becomes kx

2 − ky

2

n, where the overline and the label n denote averaging

of the momentum operator over the nth quantum eigenstate in the y direction. Thus the symmetry between x and y di-rections is broken and Sz does not turn to zero. Below we

the y direction the spin-orbit interaction共9兲 depends only on kx, with k储=兩kx兩.

Substituting Eq.共17兲 into Eq. 共16兲 and then into Eq. 共15兲 we express Sz共z兲 through unperturbed eigenstates. The latter

are calculated for a square barrier structure described above. For an order of magnitude evaluation of the spin density, the result can be written in an analytical form. Calculations are strongly simplified if only resonance terms in Eq.共15兲 are taken into account. One may also make use of the small parameter kw2/2mwU, where kw is the wave vector in the z

direction within a QW and U is the barrier height. At the lowest transmission resonance kw= k0⬇␲/2d. By this way, in the leading approximation the spin density in the center of the well共z=0兲 and the center of the wire 共y=0兲 can be writ-ten as Sz= − 4 ␲

兺

k_x,n 兩␾n共0兲兩2

冕

0 ⬁ dk ⌫ 4 关共kw− k0兲2+⌫2兴2 mrmb ␬k ⫻A␽n共d兲h2cosh2␬b sinh 2␬b关nF

l_{共E兲 − n} F r_共E兲兴,

共18兲 where h = kx共␤w−␤b兲, ␬⬇

冑

2mbU. The phase ␽n共z兲 is

ob-tained from Eq. 共8兲 by averaging ky

2

with oscillatory wave functions␾n共y兲. The width of the transmission resonance in

k space is given by⌫=共m_b2/mrmw兲共kkw/␬2d兲sinh−2␬b, where

mr, mb, mw are effective masses in reservoirs, barriers, and

QWs, respectively.A is a dimensionless function of k. The value of this function is close to 1 in the range of parameters under consideration.

For a numerical evaluation the following parameters have been taken: d = 100 Å, b = 40 Å, electron density in reservoirs n = 1023m−3, the quantization energy of the parabolic con-finement ប␻= 4 meV, and ␥= 27 eV Å3_.7 _{Other parameters}

correspond to the InAs quantum well, In0.53Ga0.47As barrier, and In_0.9Ga_0.1As reservoirs. With such parameters we calcu-lated ⌫=104cm−1 and ␽共d兲⬇10−3. From Eq. 共18兲, in the linear regime ⌬␮=␮l−␮rⰆEF, the spin density can be

evaluated as Sz⬇0.1⌬␮meV−1␮m−3. Although the spin

density is not high, nevertheless, in the range of 1 mV ter-minal voltages it can be detected by the Kerr rotation method.3 _{The parameters of the heterostructure can also be}

optimized to reach the higher density. Experimentally, the total spin polarization can be enhanced in superlattices.

C. Spin dependent transmittance

The second term in Eq.共9兲 causes spin flips of a particle transmitting through the double barrier structure. The spin dependent transmission is obtained from Eqs.共17兲 and 共16兲 where ␺l is calculated at z⬎d+b. By this way, near the resonance, the spin flip transmittance becomes

⌬t1,2= − i16t2␽共d兲Bmb 3_m r mw2 h2kw 2 ␬3_k cosh3␬b sinh␬b , ⌬t2,1= −⌬t1,2, 共19兲

whereB⬃1 is a dimensionless factor. It is easy to see that, according to Eq.共19兲, ⌬t1,2transforms an unpolarized flux of

electrons, say, from the left reservoir into a flux of spin po-larized electrons, with the polarization in the z direction. In-deed, according to our choice of the spin basis, the z polar-ization is obtained by taking an average of the ␴y spin

operator. Equation共19兲 gives rise to two spinors correspond-ing to two possible initial spin polarizations: ␺1=共t,⌬t1,2兲 and␺₂=共⌬t2,1, t兲. For an unpolarized source, after summing up averages of ␴y with these spinors, one obtains a

finite result, taking into account that near the resonance t = i⌫关kw− k0+ i⌫兴−1⬇1. It should be noted that besides Eq. 共19兲 ⌬t1,2contains one more term which, however, does not result in a resonant polarized transmission. This term has been neglected. The transmission coefficient obtained from Eq. 共19兲 is symmetric with respect to x→−x, y→−y and, hence, remains finite after the angular averaging, in contrast to a spin filtering effect considered in Refs.9 and10. Close to the Fermi energy and with the same numerical parameters used for evaluation of Eq.共18兲 we obtain ⌬t1,2⬇3⫻10−3_{. In} spite of its small value, this coefficient can lead to a notice-able accumulation of the spin polarization in a reservoir with small spin relaxation rate. For example, let us take a reser-voir of 1␮m3 _{volume. The spin polarization flux carried} from the left reservoir through the wire of the length L is given by Is= L ␲

兺

k_x,n

冕

0 ⬁ dk k mr Im关t*_{⌬t1,2兴关n} F l_{共E兲 − n} F r_{共E兲兴. 共20兲}

Taking␮l−␮r= 1 meV, L = 1␮m, the typical spin relaxation

time in bulk semiconductors 1 ns,12 _{and all the rest of the}

parameters as in the above evaluations, we get the spin den-sity in the reservoir around 0.5␮m−3_{, which is within the} optical detection range.3

IV. CONCLUSION

We considered the spin orbit effects associated with reso-nant tunneling of electrons through a symmetric double bar-rier structure. Two contributions to SOI have been taken into account, namely, the Dresselhaus SOI and the sporbit in-teraction induced by gradients of heterostructure material pa-rameters. We found out that the vertical transport of electrons gives rise to the spin current flowing parallel to heterointer-faces, as well as to the spin polarization within the QW. These effects are analogous to the spin-Hall effect and the electric spin orientation, intensively studied recently for 2D electron gas. A distinction with these traditional studies is that instead of electron wave functions confined in a QW we employ eigenstates corresponding to resonant electron trans-mission. Moreover, instead of an electric field parallel to the QW, the vertical bias has been considered. In compliance with such a 3D model, we calculated a distribution in the z direction of the spin-Hall current density and spin polariza-tion. This dependence reveals an interesting structure, such as “surface” currents flowing along the heterointerfaces. The spin-Hall current density does not turn to 0 with the external bias, thus signaling existence of the equilibrium spin current density, which was found to be linear with respect to the spin-orbit interaction. At the same time, in a symmetric

double well structure the net equilibrium current, obtained by integration of the current density over z, turns to 0. Although the spin-Hall and the equilibrium currents take place in the absence of the Dresselhaus SOI, the latter is necessary to obtain the finite out-of-plane spin density within the QW. It was found out that the most important is a part of the Dresselhaus SOI which is proportional to the z-component of the electron momentum operator. This SOI also gives rise to spin dependent transmission. Due to interplay of the Dressel-haus SOI and the spin-orbit interaction induced by gradients of heterostructure material parameters, an unpolarized and isotropic in the x, y directions beam of electrons becomes polarized in the z direction after tunneling a through double barrier QW.

The effects discussed in this paper can be interesting in the application to metal surfaces with strong spin-orbit ef-fects associated with surface states.13The STM set up with a magnetic tip could be employed for studying spin related effects. The above model, however, must be modified to take into account spin-orbit interactions typical for a particular metal surface.

ACKNOWLEDGMENTS

This work was supported by RFBR Grant No. 060216699, the National Science Council of ROC under Grants No. NSC95-2112-M-009-004, and No. NSC93-2119-M-007-002 共NCTS兲, and the MOE-ATU Grant.

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