Spin Hall Effect on Edge Magnetization and Electric Conductance of a 2D Semiconductor Strip
A. G. Mal’shukov,1L. Y. Wang,2C. S. Chu,2and K. A. Chao31Institute of Spectroscopy, Russian Academy of Science, 142190, Troitsk, Moscow oblast, Russia 2Department of Electrophysics, National Chiao Tung University, Hsinchu 30010, Taiwan 3Solid State Theory Division, Department of Physics, Lund University, S-22362 Lund, Sweden
(Received 7 July 2005; published 27 September 2005)
The intrinsic spin Hall effect on spin accumulation and electric conductance in a diffusive regime of a 2D electron gas has been studied for a 2D strip of a finite width. It is shown that the spin polarization near the flanks of the strip, as well as the electric current in the longitudinal direction, exhibit damped oscillations as a function of the width and strength of the Dresselhaus spin-orbit interaction. Cubic terms of this interaction are crucial for spin accumulation near the edges. As expected, no effect on the spin accumulation and electric conductance have been found in case of Rashba spin-orbit interaction. DOI:10.1103/PhysRevLett.95.146601 PACS numbers: 72.25.Dc, 71.70.Ej, 73.40.Lq
Spintronics is a fast developing area to use electron spin degrees of freedom in electronic devices [1]. One of its most challenging goals is to find a method for manipulating electron spins by electric fields. The spin-orbit interaction (SOI), which couples the electron momentum and spin, can be a mediator between the charge and spin degrees of freedom. Such a coupling gives rise to the so-called spin Hall effect (SHE) which attracted much interest recently. Because of SOI the spin flow can be induced perpendicular to the dc electric field, as has been predicted for systems containing spin-orbit impurity scatterers [2]. Later, similar phenomenon was predicted for noncentrosymmetric semi-conductors with spin split electron and hole energy bands [3]. It was called the intrinsic spin Hall effect, in contrast to the extrinsic impurity induced effect, because in the former case it originates from the electronic band structure of a semiconductor sample. Since the spin current carries the spin polarization, one would expect a buildup of the spin density near the sample boundaries. In fact, this accumu-lated polarization is a first signature of SHE which has been detected experimentally, confirming thus the extrinsic SHE [4] in semiconductor films and intrinsic SHE in a 2D hole gas [5]. On the other hand, there was still no experi-mental evidence of intrinsic SHE in 2D electron gases. The possibility of such an effect in macroscopic samples with a finite elastic mean free path of electrons caused recently much debates. It has been shown analytically [6 –11] and numerically [12] that in such systems SHE vanishes at arbitrary weak disorder in dc limit for isotropic as well as anisotropic [10] impurity scattering when SOI is repre-sented by the so-called Rashba interaction [13]. As one can expect in this case, there is no spin accumulation at the sample boundaries, except for the pockets near the electric contacts [7]. At the same time, the Dresselhaus SOI [14], which dominates in symmetric quantum wells, gives a finite spin Hall conductivity [11]. The latter can be of the order of its universal value e=8@. The same has been shown for the cubic Rashba interaction in hole systems [12,15]. In this connection an important question is what
sort of the spin accumulation could Dresselhaus SOI in-duce near sample boundaries. Another problem which, as far as we know, was not discussed in literature, is how the
electric current along the applied electric field will change
under SHE. In the present work we will use the diffusion approximation for the electron transport to derive the drift-diffusion equations with corresponding boundary condi-tions for the spin and charge densities coupled to each other via SOI of general form. Then the spin density near the flanks of an infinite 2D strip and the correction to its longitudinal electric resistance will be calculated for Dresselhaus and Rashba SOI.
Let us consider two-dimensional electron gas (2DEG) confined in an infinite 2D strip. The boundaries of the strip are at y d=2. The electric field E drives the dc current in the x direction and induces the spin Hall current in the y direction. This current leads to spin polarization buildup near boundaries. Since d k1F , where kF is the Fermi wave vector, this problem can be treated within the semi-classical approximation. Moreover, we will assume that d is much larger than the electron elastic mean free path l, so that the drift-diffusion equation can be applied for descrip-tion of the spin and charge transport. Our goal is to derive this equation for SOI of general form
Hso hk ; (1)
where x; y; z is the Pauli matrix vector, and the effective magnetic field hk hk is a function of the two-dimensional wave vector k.
We start from determining linear responses to the mag-netic Br; t and electric Vr; t potentials. The magmag-netic potentials are introduced in order to derive the diffusion equation and play an auxiliary role. The corresponding one-particle interaction with the spin density is defined as
Hsp Br; t . These potentials induce the spin and charge densities, Sr; t and nr; t, respectively. Because of SOI the charge and spin degrees of freedom are coupled, so that the electric potential can induce the spin density [16] and vice versa. Therefore, it is convenient to introduce PRL 95, 146601 (2005) P H Y S I C A L R E V I E W L E T T E R S 30 SEPTEMBER 2005week ending
the four vector of densities Dir; t, such that D0r
nr; t and Dx;y;zr; t Sx;y;zr; t. The corresponding four vector of potentials will be denoted as ir; t. Accordingly, the linear response equations can be written in the form Dir; t Z d2r0dt0X j ijr; r0; t t0jr0; t0 D0 ir; t: (2)
The response functions ijr; r0
; t t0 can be expressed in a standard way [17] through the retarded and advanced Green functions Grr; r0 ; tand Gar; r0 ; t. In the Fourier representation we get ijr; r0; ! i! Z d!0 2 @nF!0 @!0 hTr G ar0 ; r; !0 iGrr; r0; !0 !j i; (3) where 0 1, i i at i x; y; z, and nF! is the Fermi distribution function. The time Fourier components of densities D0 ir; t at ! EFare defined as D0ir; ! iZ d2r0X j jr0; ! Z d!0 2nF! 0 hTr Grr; r0 ; !0iGrr0; r; !0j Gar; r0; !0 iGar0; k; !0j i: (4) The trace in Eqs. (3) and (4) runs through the spin varia-bles, and the angular brackets denote the average over the random distribution of impurities. Within the semiclassical approximation the average of the product of Green func-tions can be calculated perturbatively. Ignoring the weak localization effects, the perturbation expansion consists of the so-called ladder series [17,18]. At small ! and large jr r0j they describe the particle and spin diffusion pro-cesses. The building blocks for the perturbation expansion are the average Green functionsGrandGa, together with the pair correlator of the impurity scattering potential
Uscr. A simple model of the short-range isotropic poten-tial gives hUscrUscr0i r r0=N
0, where N0 is the electron density of states at the Fermi energy and 1=2. Within the semiclassical approach the explicit be-havior of the electron wave functions near the boundaries of the strip is not important. Therefore, the bulk expres-sions can be used for the average Green functions. Hence, in the plane wave representation
Grk; ! Gak; ! y ! E
k hk i1;
(5)
where Ek k2=2m E
F. Since the integral in (4) rap-idly converges at jr r0j & k1
F , D0ir; ! are given by the local values of potentials. From (4) and (5) one easily obtains the local equilibrium densities
D0
ir; ! 2N0ir; !: (6) In their turn, the nonequlibrium spin and charge densities are represented by the first term in Eq. (2). Within the diffusion approximation this term is given by the gradient expansion of (3) [18]. Such an expansion is valid as far as spatial variations of Dir; ! are relatively small within the length of the order of the mean free path l. The length scale for spin density variations near the boundaries of the strip is given by vF=hkF. Hence, the diffusion approximation can be employed only in the dirty limit hkF 1=. The diffusion equation is obtained after the ladder summation in the first term of Eq. (3) and multiplying this equation by the operator inverse to ijr; r0
; !, as it has been previ-ously done in [19,20]. After some algebraic manipulations one gets
X j
DijD
j D0j i!Di; (7) where the diffusion operatorDij can be written as
Dij ijDr2 ij Rijmr
m Mij: (8) The first term represents the usual diffusion of the spin and charge densities, while the second one describes the D’akonov-Perel’ [21] spin relaxation
ij 4h2
kij nikn
j
k ; (9)
where i; j 0, the overline denotes the average over the Fermi surface, and nk hk=hk. The third term gives rise
to precession of the inhomogeneous spin polarization in the effective field of SOI [19]
Rijm 4X l
"ijlhl
kvmF: (10)
The nondiagonal elements of the formDi0appear due to spin-orbit mixing of spin and charge degrees of freedom. They are collected in Mij. For Rashba SOI Mi0 have been calculated in [7,8]. In general case
Mi0h 3 k 2 @ni k @k r: (11)
When a time independent homogeneous electric field is applied to the system one has 0 eEx and D0
0 2N0eEx. At the same time, i 0 and, hence, D0i 0 at i x; y; z. Because of charge neutrality the induced charge density eD0 0. It should be noted that in the system under consideration the charge neutrality cannot be fulfilled precisely. The spin polarization accumulated at the strip boundaries gives rise to charge accumulation via the M0i terms in (7) and (8). The screening effect will, however, strongly reduce this additional charge, because the screening length of 2DEG is much less than the typical length scale of spin density variations. We will ignore such a small correction and set D0 0 in (7). In this way one PRL 95, 146601 (2005) P H Y S I C A L R E V I E W L E T T E R S 30 SEPTEMBER 2005week ending
arrives to the closed diffusion equation for three compo-nents of the spin density. This equation coincides with the usual equation describing diffusive propagation of the spin density [19], for exception of the additional term Mi0D0
0 2N0eEh3krxknik=2 due to the external electric
field. Its origin becomes more clear in an infinite system where the spin density is constant in space and only ijand Mij are retained in (7) and (8). Hence, the corresponding solution of (8) at ! 0 is Si Sbi, with Sbi D0 i=2 N0eE 2 X j 1ijh3 k @njk @kx ; (12)
where 1ijis the matrix inverse to (9). Such a phenome-non of spin orientation by the electric field was predicted in Ref. [16] and recently observed in [22]. In the special case of Rashba SOI hk kk z it is easily to get from
(12) the result of Ref. [16] Sb
y N0eE.
In addition to the diffusion equation one needs the boundary conditions. These conditions are that the three components of the spin flux Ixy; Iyy; Iyz flowing in the y direction turn to 0 at y d=2. The linear response theory, similar to (2), gives
Il ir; t Z d2r0dt0X j l ijr; r0; t t0jr0; t0; (13) where the response function is given by
l ijr; r0; ! i! Z d!0 2 @nF!0 @!0 hTr G ar0; r; !0 Jl iGrr; r0; !0 !j i; (14) with the one-particle spin-current operator defined by Jl
i iv
l vli=4 and the particle velocity vl kl m @ @kl hk : (15)
Taking into account (7) and (6), we obtain from (13) and (14) Iyir D@Si @y 1 2R ijyS j Sbj izIsH: (16) The first two terms represent the diffusion spin current and the current associated with the spin precession. The third term is the uniform spin Hall current polarized along the z axis. It is given by IsH 1 2R zjySb j eE N0 2v y F @h k @kx hk z : (17)
From (10) and (12) it is easy to see that for Rashba SOI both terms in (17) cancel each other making IsH 0, in accordance with [6 –12]. Therefore, in case of the strip the solution of the diffusion equation satisfying the boundary condition is Sj jySb
y. Hence, the spin density is uniform and does not accumulate near boundaries. It should be
noted that such accumulation can, however, take place in the ballistic regime of electron scattering [23]. At the same time, as shown in Ref. [11], even in the diffusive regime
IsH 0 for the Dresselhaus SOI. This inevitably leads to the spin accumulation. Taking Dresselhaus SOI in the form
hxk kxk2
y 2; h y
k kyk2x 2; (18) one can see that the bulk spin polarization (12) has a nonzero Sb
x component, Rzxy 0, while Rzyy 0. Hence, the solution of the diffusion Eq. (7) with the boundary condition Ixyd=2 Izyd=2 0 is Sx, Sz 0 Sy 0. Let us define Siy Siy Sb
i. The dependence of Sid=2 from the strip width, as well as an example of Sz coordinate dependence, are shown in Fig. 1. The damped oscillation in the d-dependence of the spin accu-mulation on the flanks of the strip can be seen for the Sz polarization. Similar oscillations take place also in the coordinate dependence. The length scale of these oscilla-tions is determined by the spin precession in the effective spin-orbit field.
The arbitrary units have been used in Fig. 1. For a numerical evaluation let us take E 104 V=m,
h2
kF q
=@ 0:1, and =kF 0:8 for a GaAs quantum
∆ κ κ κ ∆ ∆
FIG. 1 (color online). Spin densities Sid=2 Si for
i x; zon the boundaries of the strip, as functions of its width d, for =k 0:9, 1.0, and 1.3, respectively. The inset shows the dependence of Szy on the transverse coordinate y. Lengths
are measured in units of lso v2F@=2vFyhky.
PRL 95, 146601 (2005) P H Y S I C A L R E V I E W L E T T E R S 30 SEPTEMBER 2005week ending
well of the width w 100 A doped with 1:5 1015 m2 electrons. We thus obtain jSzd=2j ’ 5 1011 m2. The corresponding volume density Sz=w ’ 5
1019 m3, which is within the sensitivity range of the Faraday rotation method [4].
It should be noted that in the considered here ‘‘dirty’’ limit h2k
F q
=@ 1 the spin Hall current is suppressed by
the impurity scattering. As shown in [11,12] for Dresselhaus and cubic Rashba SOI, this current decreases as h2
kF
2=@2 down from its highest universal value. At the same time, an analysis of the diffusion equation shows that the accumulated at the flanks of the strip spin density decreases slower, as h2
kF q
=@. This explains why for the
considered above realistic numerical parameters, even in the dirty case, the noticeable spin polarization can be accumulated near the boundary.
Usually, the spin Hall effect is associated with the spin polarization flow, or the spin density accumulation on the sample edges, in response to the electric field. On the other hand, this effect can show up in the electric conductance as well. To see such an effect we take 0-projection of (13), which by definition is the electric current. The current flows along the x axis. The corresponding response func-tion x
0j is given by (14) with Jx0 vx. Using Eqs. (14), (15), and (7), and expressing i from (6) one gets the electric current density
Ix E A@Sz
@y ; (19)
where is the Drude conductivity and
A e 1 22 2vyF @h k @kx hk z vx F @h k @ky hk z : (20)
The total current is obtained by integrating (19) over y. Therefore, the spin Hall correction to the strip conductance
G A
ESzd=2 Szd=2
2A
E Szd=2: (21)
Hence, the dependence of G on the strip width coincides with that of the spin density shown in Fig. 1(a).
In conclusion, we employed the diffusion approximation to study the spin Hall effect in an infinite 2D strip. In case of the Dresselhaus spin-orbit interaction this effect leads to spin accumulation near the flanks of the strip, as well as to a correction to the longitudinal electric conductance. Both the spin accumulation and the conductance exhibit damped oscillations as a function of the strip width.
This work was supported by the Taiwan National Science Council NSC93-2112-M-009-036, NSC94-2811-M-009-010, and RFBR Grant No. 03-02-17452.
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