Competing interplay between Rashba and cubic-k Dresselhaus spin-orbit interactions
in spin-Hall effect
R. S. Chang,1C. S. Chu,1,2 and A. G. Mal’shukov1,2,3
1Department of Electrophysics, National Chiao Tung University, Hsinchu 30010, Taiwan 2National Center for Theoretical Sciences, Physics Division, Hsinchu 30043, Taiwan 3Institute of Spectroscopy, Russian Academy of Science, 142190 Troitsk, Moscow oblast, Russia
共Received 5 March 2009; revised manuscript received 20 April 2009; published 14 May 2009兲 Focusing on the interplay between the Rashba and cubic-k Dresselhaus spin-orbit interactions共SOI兲, we calculate the spin accumulation Szand the spin polarizations Si
B
at, respectively, the lateral edges and in the bulk of the two-dimensional electron gas. Their dependences on both the ratio between the Rashba and the Dresselhaus SOI coupling constants and the electron densities are studied systematically. Strong competition features in Szare found. In the Dresselhaus-dominated regime Szchanges sign when the electron density is large enough. In the Rashba-dominated regime Szis essentially suppressed. Most surprising is our finding that the Rashba-dominated regime occurs when␣⬇2˜, where ␣ and ˜ are the Rashba and the effective linear-k Dresselhaus SOI coupling constants, respectively. For the spin polarizations SiB, the Rashba-dominated regime occurs when␣ⱖ˜. Our results point out that decreasing 兩␣兩 leads to the restoration of the spin accumulation
Sz.
DOI:10.1103/PhysRevB.79.195314 PACS number共s兲: 72.25.Dc, 71.70.Ej, 75.40.Gb, 85.75.⫺d
I. INTRODUCTION
Spin-orbit interaction共SOI兲 provides the key leverage for the recent strive for all electrical generations and manipula-tions of spin densities in semiconductors.1–9 Intrinsic SOIs,
such as the Rashba SOI共RSOI兲 共Refs. 3,7, and9–12兲 and
the Dresselhaus SOIs共DSOIs兲,6,13,14,23are of particular
inter-est. It is due to their tunability, gate tuning for the RSOI and either sample thickness or electron-density tuning for the DSOI, and to their physical origins, being independent of disorder that requires the presence of SOI impurities. Yet the ever present background scatterers do play a subtle role in the intrinsic spin-Hall effect.15 In spin-Hall effect共SHE兲, an
external electric field induces a transverse spin current and, in turn, an out-of-plane spin accumulation Sz at lateral edges.1,5–9For intrinsic SOIs, the background scatterers lead
to a complete quenching of the edge spin accumulation Sz when the SOI depends only linearly on the electron momen-tum k,15 but S
z maintains finite and dependent on the mo-mentum relaxation time when the SOI has a cubic-k dependence.14–17Thus, separately considered, the RSOI does
not contribute to edge spin accumulation Szwhile the cubic-k DSOI does. For a more realistic situation, when the two SOIs coexist in a sample, RSOI could exert its effect on the edge spin accumulation Sz, but that would have to be mediated through the cubic-k DSOI. It is of great interest to see whether this effect would be reinforcing or competing for Sz. Thus, in this work, we focus upon the interplay between the RSOI and the cubic-k DSOIs in their combined, or com-peting, effects on both the edge spin accumulation Szand the bulk spin density Si
B
. Bulk spin density Si B
, formed in an external electric field, is another important physical quantity of interest that is closely related to the intrinsic SOIs. The subscript i denotes the vector component of spin. The effect of the background scatterers on Si
B
is less subtle than that on Sz: Si
B
, remains finite for all intrinsic SOIs and depends on also.3 Intuitively, up to leading order in the SOI coupling
constant one might expect this Si B
feature to arise from a SOI-effective magnetic field.4 It turns out to be the case
when there is only one dominated SOI and the SOI depends on k linearly. Take, for instance, a Rashba-type two-dimensional electron gas共2DEG兲 in the diffusive regime, the k-dependent effective magnetic field becomes 具hk典=−␣z ⫻具k典 when 具k典 is averaged over the electron distribution given by a shifted Fermi sphere f共⑀, k兲= f0共⑀, k兲
−eបk·Emⴱ ␦共⑀F−⑀兲, where ␣, f0, and mⴱ are, respectively, the
Rashba coupling constant, Fermi-Dirac distribution, and electron mass and for e⬎0. With 具hk典=␣e/បzˆ⫻E, the bulk spin density, in units of ប, is given by SB= −N
0␣e/បzˆ⫻E,
which was first obtained by Edelstein.3In the above
expres-sion the density of states per spin is denoted by N0. Beyond
leading order or linear k dependence in the SOIs, or for the coexistence of different types of SOIs, the derivation of SiB becomes more involved. In this work, we calculate the Si
B within a spin-diffusion equation approach and perform a sys-tematic study on the competing interplay between the RSOI and the cubic-k SOIs.
Interplay between the RSOI and the linear-k DSOI in a sample has attracted much attention lately.18–29Earlier work
studied the effect of ␣=˜ , where˜ is the effective linear-k DSOI coupling constant, on the magnetoconductivity.18
More recent work on the same␣=˜ regime pointed out that the spin becomes a good quantum number, independent of k, and has a long relaxation time.19The D’yakonov-Perel’共DP兲
mechanism1 for spin relaxation is suppressed. This finding
led to proposals for spintronic transistor that would manipu-late polarized spin transport in the diffusive regime.19,20 It
was later shown, within the same ␣/˜ =1 regime, that the Fermi circles of opposite spins are connected by a wave vec-tor Q that depends only on the SOI constant and the effective mass.24 This leads to the persistent spin-helix state.24 Since
the ratio ␣/˜ is important for the development of spintron-ics, and the transport is anisotropic when both ␣ and˜ are 1098-0121/2009/79共19兲/195314共6兲 195314-1 ©2009 The American Physical Society
finite,22,23,26–29 a number of experiments were designed to
extract this ratio by the monitoring of the spin photocurrent.22,26,27 Most of the studies dealt with the
linear-k SOIs. One of the exceptions is a weak localization experiment that has extracted ␣, ˜ , and also the cubic-k DSOI coupling constant from comparing the magnetocon-ductance data with a weak localization theory.21The delicate
interplay between the RSOI and the cubic-k DSOI is yet to be explored and is much needed in either the spin transport, as is briefed above, or in the spin accumulations, as is related to SHE.
To study the interplay between the RSOI and the cubic-k DSOI in the diffusive regime, we extend our previous studies on the spin diffusive in a 2DEG strip to include both types of the SOI.17The diffusive regime has l
e⬍Lso, where Lsoand le are, respectively, the typical spin-relaxation length due to either the RSOI or the DSOI and the momentum-relaxation length. We study in detail the variations in Sz and Si
B with respect to␣/˜ and to the electron density. Our result shows Dresselhaus-dominated and Rashba-dominated regimes are determined primarily by the ratio ␣/˜ . In the intermediate regime, intricate interplay between the RSOI and cubic-k DSOI is clearly shown as the electron density is varied. The edge spin accumulation Sz is essentially suppressed in the Rashba-dominated regime. Most surprisingly is our finding that the Rashba-dominated regime occurs when ␣⬇2˜ for the edge spin accumulation Sz and when ␣=˜ for the bulk spin polarization Si
B
. Our result points to a possible way to restore the DSOI’s contribution to the SHE, namely, to lower 兩␣/˜ 兩 to values well below unity. In Sec. II we present the spin-diffusion equation and the analytical solutions. In Sec.
IIIwe present our numerical results and discussions. Finally, in Sec.IV, we will present our conclusion.
II. THEORY
The system we consider is a 2DEG confined in an infinite strip with transverse boundaries at y =⫾d/2. The thickness of the strip wⰆd. An electric field E in the x direction in-duces the SHE. The phenomenon is described by a spin-diffusion equation14,17 which has been derived from the
Keldysch nonequilibrium Green’s function method.30 It has
also been extended to the case of an in-plane magnetic field.31Detail of the derivation is not repeated here, but we
will describe the physical meaning of the terms in the spin-diffusion equation. For our purpose here, the SOI magnetic field hkincludes both the RSOI and the cubic-k DSOI and is separated into linear-k and cubic-k terms hk= hk,1+ hk,3. In a 2DEG, hk lies on the two-dimensional plane after we aver-age it with the lowest subband wave function over its thick-ness. Explicitly, we have
hk,1=␣共ky,− kx兲 +2共− kx,ky兲,
hk,3=共kxk2y,− kykx2兲. 共1兲
Here,is the DSOI coupling constant,2=具kz 2典, and k
xand kyare along, respectively, the关100兴 and 关010兴 directions for
a zinc-blende crystal.32It is convenient to define the effective
linear-k DSOI coupling constant ˜ =2. The SOI hamil-tonian Hso= hk·, whereis the Pauli-matrix vector.
Equation共1兲 provides us a simple way to get at the
direc-tion of the effective magnetic field hk for a given electron distribution in the k space. This is important for an intuitive understanding of the spin-diffusion equation. From hk= −h−k, the effective magnetic field is zero when the k-space
occupation is symmetric, as it is for the equilibrium case. If the deviation from equilibrium is a shifted distribution char-acterized by a wave vector Q, then the effective magnetic field hR due to RSOI will be along the direction of Qˆ ⫻zˆ. The DSOI case is less straight forward, but when Q is along either kxor ky, then hDwill be along or opposite to Qˆ . Spe-cifically, in the low electron-density 共kFⰆ兲 regime hDwill be opposite共along兲 to Q when Q is along kx共ky兲. The direc-tion of hD will be reversed in the high-density 共kFⰇ兲 re-gime. Here kF is the Fermi wave vector.
The stationary spin-diffusion equations are given by
D 2 y2Sz+ Rzxy ប ySx+ Rzyy ប ySy− ⌫zz ប2Sz= 0, D 2 y2Sy+ Ryzy ប ySz− ⌫yy ប2Sy− ⌫yx ប2Sx− Cy ប2= 0, D 2 y2Sx+ Rxzy ប ySz− ⌫xx ប2Sx− ⌫xy ប2Sy− Cx ប2= 0, 共2兲
where diffusion constant D =vF2/2, and Siis the spin density in units of ប. Since kFleⰇ1, charge neutrality is maintained by the condition of zero net charge density throughout.
The DP spin-relaxation rates ⌫il= 4h k 2共␦il− n k i nk l兲 for i, l 苸1, 2, and 3, for unit vector nk= hk/hk. The overline denotes the angular average over the Fermi surface. Specifically, we have ⌫xx=⌫yy=⌫zz/2=2kF 2共␣2+˜2−1 2˜2˜k2+ 1 8˜2˜k4兲 and ⌫ xy =⌫yx= −␣kF 2˜ 共4−k˜2兲 for k˜=k
F/. The diagonal components of the DP spin-relaxation rate receive independent contribu-tions from the individual SOI. The off-diagonal DP compo-nents, however, involve both SOIs together, as they are pro-portional to ␣˜ . Furthermore, k˜ serves as an agent that carries the cubic-k effects of the DSOI. For example, the term that has ˜2˜k2 is resulted from mixing the linear-k and
the cubic-k effects of the DSOI, whereas the term that has

˜2˜k4is due solely to the cubic-k effect of the DSOI, and in its
second order.
Spin precession arising from spatial nonuniformity in spin densities is characterized by coefficients Rilm= 4兺
nilnhk n
vF m , whereilnis the Levi-Civita symbol. Specifically, the RSOI’s contributions are Ryzy= −Rzyy=2kF2
mⴱ␣, and the DSOI’s contri-butions are Rzxy= −Rxzy=2kF2
mⴱ 共˜ − 1
4˜ k˜2兲. As we will explore
the interplay between the two SOIs by varying␣while keep-ing ˜ fixed, it is more convenient to define the length scale lso= 2D/Rzxyaccording to the strength of˜ only. The
coeffi-cient Rilm, if not zero, causes the precession of S
spatial variation along mˆ , to rotate into Si. That Rzyy, for instance, receives sole contribution from RSOI can be under-stood from our aforementioned shifted electron-distribution picture. Taking that Sy共qy兲 is represented by a shifted distri-bution with Qˆ =−yˆ, the effective magnetic field hR due to RSOI will be along Qˆ ⫻zˆ=−xˆ, leading to the precession of Syabout xˆ clockwise. On the other hand, the effective mag-netic field hDdue to DSOI for this case will be along −yˆ, assuming low electron-density regime, and cannot lead to the precession of Sy. Similar argument can be applied to explain
why Rzxy, for instance, receives contribution from DSOI only.
The effect of the driving electric field on the above spin diffusion enters through the coefficients Ci, for i苸1,2. It is given by Ci= 1 2Mxi0D00/x, where Mxi0= 42hk3 n k i kx incorporates
the spin-charge coupling and D00= −2N0eEx is a local
equi-librium density.17 The bulk spin densities can be solved
di-rectly from Eq. 共2兲. We obtain Sy B =−Cx⌫yx+Cy⌫xx ⌫xy⌫yx−⌫xx⌫yy, Sx B = Cx⌫yy−Cy⌫xy ⌫xy⌫yx−⌫xx⌫yy, and Sz B
= 0. The full expressions are given by
Sx B =
冉
N0eE ˜ 2ប冊
⫻ 共8␣4− 16␣2˜2+ 8˜4兲 + 共2␣4+ 8␣2˜2− 10˜4兲k˜2+共␣2˜2+ 3˜4兲k˜4+共
−3 8␣2˜2+ 1 8˜4兲
˜k6− 5 16˜4˜k8+ 3 64˜4˜k10 共− 4␣4+ 8␣2˜2− 4˜4兲 + 共− 4␣2˜2+ 4˜4兲k˜2− 2˜4˜k4+1 2˜ 4˜k6− 1 16˜ 4˜k8 , Sy B =冉
N0eE␣ 2ប冊
⫻ 共8␣4− 16␣2˜2+ 8˜4兲 + 共12␣2˜2− 12˜4兲k˜2+共␣2˜2+ 3˜4兲k˜4−1 4˜ 4˜k6+ 3 16˜ 4˜k8 共− 4␣4+ 8␣2˜2− 4˜4兲 + 共− 4␣2˜2+ 4˜4兲k˜2− 2˜4˜k4+1 2˜ 4˜k6− 1 16˜ 4˜k8 . 共3兲Equation 共3兲 reduces to the pure RSOI result3 when ˜ =0,
and to the pure cubic-k DSOI result17 when ␣= 0. If the
cubic-k term in the DSOI is dropped 共k˜=0兲, Eq. 共3兲 gives
SLRDB = −Nប0eE共˜ ,␣兲 so that it points to the third quadrant in the x-y plane. For␣=˜ , SLRDB forms 45° with the −x axis.
We solve the spin-diffusion equation for the spin density Si across the semiconductor strip. The boundary condition we use is derived from requiring the local spin-current den-sity Iy
i
, which is expressed in terms of both Siand its spatial derivative Si/y, to be zero in its transverse flow Iy
i at the lateral edges.17 This is appropriate for a hard-wall boundary.33,34 Extended to include both SOIs, the
spin-current density is given by
Iy z = − 2D ySz− R zyy共S y− Sy B兲 − Rzxy共S x− Sx B兲 + I sH␦iz, Iy y = − 2D ySy− R yzyS z, Iy x = − 2D ySx− R xzy Sz. 共4兲
The spin-current density in Eq.共4兲 has contribution from
spin diffusion, via the spatial gradients in Si, spin precession, via the Rilm coefficients, and the electric field, via the bulk spin-current density IsH. It is given by
IsH= − RzyySy b − RzxySx b + 42eEN0vF y
冉
hk kx ⫻ hk冊
z . 共5兲III. NUMERICAL RESULTS AND DISCUSSIONS
In this section, we present the edge spin accumulation Sz and the bulk spin polarizations SBfor a 2DEG semiconductor strip that consists of both the RSOI and the cubic-k DSOI. For definiteness, material parameters are chosen to be con-sistent with GaAs: effective mass mⴱ= 0.067m0, with m0the
electron mass; Dresselhaus SOI = 27.5 eV Å3,32 and ˜ 共
⬅2兲=2.22 eV m for quantum well thickness w=300 Å.
The width of the strip is d = 30 m, and the mean free path le= 1 m. Typical value of lso for n = 1⫻1015 m−2, or k˜
= 0.76, is lso= 2.20 m. The electrons occupy only the
low-est subband in the quantum well. An electric field E = 25 mV/m is applied along x to set up the spin-Hall phe-nomenon.
Figure1presents the spatial profile of Szacross the semi-conductor strip. Besides the well-known odd-parity feature of Szin the transverse coordinate y, Figs.1共a兲–1共d兲show, for the given physical parameter ranges, that the spin accumula-tion Szis sensitive to the ratio␣/˜ . Szhas the largest mag-nitude in Fig.1共a兲, where␣/˜ =0; it exhibits sign changes in Figs.1共b兲and1共c兲, where␣/˜ =0.5 and 1, respectively; and it is essentially suppressed in Fig.1共d兲, when␣/˜ =2.0. The fact that the RSOI dominates as early as␣/˜ =2.0 is surpris-ing.
Dependence of Sz on the electron density, or k˜, is also shown in Fig. 1. For the case of pure Dresselhaus SOI, the edge spin accumulation Szincreases with k˜. This feature cor-roborates with the fact that the cubic-k SOI is the major contributor to Sz. On the other hand, Szcan change sign by either increasing the electron density, as is shown by the k˜ = 1.20 curve in Fig.1共b兲, or by increasing the␣/˜ ratio, as is
shown by the curves in Fig.1共c兲. This sign change in Szis a manifestation of the competition between the RSOI and the cubic-k DSOI. That the RSOI joins forces with the cubic-k DSOI to compete with the pure cubic-k DSOI feature is an intriguing result we have found here. This is supported by Fig. 1共b兲, when the RSOI is of intermediate strength. The sign change in Szoccurs for the k˜ =1.20 curve of which the cubic-k effect is the strongest. For larger RSOI, as in Fig.
1共c兲, the sign change occurs for all curves shown, including those of smaller k˜ values. All these characteristics, and the results in the following, have prompted us to categorize the spin accumulation into three regimes: the Dresselhaus-dominated regime, Fig.1共a兲; the Rashba-dominated regime, Fig. 1共d兲; and the intermediate regime, Figs. 1共b兲 and 1共c兲. The ratio ␣/˜ is the key parameter that helps define these regimes. We note in passing that we have treated␣and k˜ as though they were independent parameters whereas in prac-tice they may be connected. It has been demonstrated experi-mentally, however, that the two parameters can be decoupled by a two gates technique.35
The dependence of the edge spin accumulation Sz −⬅S
z共y = −d/2兲 at the sample edge on the regime parameter␣/˜ is presented in Fig.2. The first feature that we want to address about these curves is their even parity in␣/˜ . This confirms our expectation nicely because the symmetry of the system seems to demand a parity symmetry in Sz− with respect to ␣/˜ . However, the fact that Sz−⫽0 at ␣/˜ =0 rules out the possibility for a Sz
−
of odd parity, leaving us the even parity as the only choice. There are other features, which are equally important, in the general dependence of Sz− on␣/˜ . Starting from a maximum at ␣/˜ =0, Sz− passes through a minimum of negative value and then diminishes to small values at ␣/˜ =2. The minima occur within a region 0.5
⬍␣/˜ ⬍1. These minima of Sz− are resulted from the com-petition between the two SOIs. In contrast, the maximum of Sz
−
is a pure DSOI feature. The maximum value of Sz −
in-creases with k˜ up to around k˜=1.20. Beyond this, the maxi-mum value of Sz− takes on a different course and decreases with increasing k˜ to negative values, as is indicated by the k
˜ =1.51 curve. Suppression on Sz−
is already quite significant when␣/˜ ⬎1.5. On the other hand, taking both directions of Sz−into account,兩␣/˜ 兩⬍1 is the optimal region for the exhi-bition of spin accumulation.
Presented in Fig.3is the dependence of Sz−on both k˜ and
␣/˜ . It is meant to be a comprehensive presentation, with Sz−
versus k˜ curves plotted together for different values of␣/˜ 共0ⱕ␣/˜ ⱕ2兲. The pure DSOI case is denoted by the circular dots and the␣/˜ =1 case is denoted by the open circles. The
␣/˜ =2 case forms the boundary of the group of curves of increasing M, where the variation in Sz
−
with k˜ is weak and the magnitudes small. It is convenient to describe the general features separately in two regions: the 0ⱕ␣/˜ ⱕ1 and the 1ⱕ␣/˜ ⱕ2 regions. In the former region the Sz− increases with k˜ initially, following quite closely with the ␣/˜ =0 curve, before it reaches its maximum. Beyond this point, Sz− deviates from the ␣/˜ =0 curve and decreases to pass through zero and into the negative value region. The increas-ing of ␣/˜ results in the negative shifting in the k˜ values of the zero of Sz
−
. There is a tendency, as k˜ further increases, for the Sz−to conform to the pure DSOI behavior, namely, that Sz− increases its magnitude with k˜. This tendency, however, gradually fades out, for ␣/˜ ⬎0.4, and results in a much weaker dependence on k˜ when␣/˜ ⬇2.
The domination of the pure DSOI in the small k˜ region is consistent with the understanding that the DSOI is the sole contributor to Sz
−
there and the effect of RSOI needs to be −15 −10 −5 0 5 10 15 −3 −2 −1 0 1 2 3 (b) α/ ˜β = 0.5 −15 −10 −5 0 5 10 15 −2 −1 0 1 2 3 (a) α/ ˜β = 0.0 S − z(µm − 2) −15 −10 −5 0 5 10 15 −3 −2 −1 0 1 2 3 (c) α/ ˜β = 1.0 y (µm) S − z(µm − 2) −15 −10 −5 0 5 10 15 −3 −2 −1 0 1 2 3 (d) α/ ˜β = 2.0 y (µm) ˜k = 0.76 ˜k = 0.93 ˜k = 1.20
FIG. 1. 共Color online兲 Spatial profile of spin density: Szversus
y. Szin units ofm−2, y in units ofm, and the width of the strip
d = 30 m. k˜=0.76 共black solid line兲, 0.93 共blue diamond兲, and
1.20共red triangle兲 correspond to electron density n=1.0, 1.5, and 2.5⫻1015 m−2. The ratio ␣/˜=0.0, 0.5, 1.0, and 2.0 in 共a兲–共d兲, respectively. −2 −1 0 1 2 −4 −3 −2 −1 0 1 2 3 α/ ˜β S − z(µm − 2) ˜k = 1.51 ˜k = 1.20 ˜k = 0.93 ˜k = 0.76 ˜k = 0.68
FIG. 2. 共Color online兲 Edge spin accumulation Sz− versus␣/˜.
k
˜ =1.51 共black cross兲, 1.20 共red triangle兲, 0.93 共blue diamond兲, 0.76
共black solid line兲, and 0.68 共black triangle兲 correspond to electron density n = 3.5, 2.5, 1.5, 1.0, and 0.8⫻1015 m−2, respectively. The curves show the symmetry Sz−共␣兲=Sz−共−␣兲.
mediated by the cubic-k DSOI. The effects of RSOI emerge in larger k˜ values, where the location of the zero of Sz
−
is negatively shifted with the increasing of␣/˜ . The zero of Sz −
can be understood with the vanishing of the effective mag-netic field h = 0. Taking the ␣/˜ =0 case as an example, 具hD典=具关k
x共−˜ +ky 2兲,k
y共˜ −kx
2兲兴典. In a driving E field, the
averages 具kx典=−e/បE and 具ky典=0 so that 具hD典 becomes zero if 具˜ −ky2典=0. This condition is satisfied if k˜=
冑
2, which is quite close to the value k˜ =1.47 for the pure DSOI curve. We point out that the averages on h that enter into the contribution to the various processes considered in this work are much complicated than the one that we have just shown. But this nice correspondence in the k˜ values for the␣/˜ =0 case convinces us that the effective magnetic field concept is at work. This effect of ␣ on the zero of Sz−
shows that the effect of RSOI is a competing one. Similar competing nature causes the suppression of the pure DSOI feature in the k˜ ⬎1.2 region. As is demonstrated by the ␣/˜ =1 curve, the pure DSOI trend in this region no longer prevails but is largely suppressed. It is also of interest to see that Sz
−
is negative in the entire shown values of k˜.
For the 1ⱕ␣/˜ ⱕ2 region, Sz−decreases monotonically in its magnitude with the increasing of␣/˜ , while exhibiting a converging behavior as␣/˜ increases. Except for a residual positive value for Sz
−
in the large k˜ region when␣/˜ =2, the Sz−is essentially suppressed.
In Figs.4and5we present both the bulk spin polarization
SBand its linear-k counterpart S LRD B
and their dependence on
␣/˜ and k˜. Relative magnitude between each vector pairs is shown, with the magnitude of SLRDB chosen as unity. For a
fixed k˜, the two vector pairs become more aligned in both their directions and magnitudes when ␣/˜ increases. For a fixed␣/˜ the general feature is that the vector pairs deviate more from each other as k˜ increases. This is consistent with the understanding that the deviation comes from the cubic k in the DSOI. There is an interesting intermediate region where the SB⬇0. Starting from ␣/˜ =0, the zero SB is around k˜ ⬇1.1. With the increasing of␣/˜ , the k˜ value for zero SBdecreases. This zero SBfeature ties up nicely with the zero Sz
−
feature in Fig. 3in that both of the k˜ values for the corresponding zeros decrease with the increasing of ␣/˜ . Beyond the region ␣/˜ ⱖ1, the zero SB feature subsides while the vector pairs align nicely with each other, both in
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 −5 −4 −3 −2 −1 0 1 2 3 S − z(µm − 2)
˜k
increasing N • : α/ ˜β=0 ◦ : α/ ˜β=1.0 increasing MFIG. 3. 共Color online兲 Spin densities Sz− versus k˜. The case of pure Dresselhaus SOI共␣/˜=0兲 is denoted by black circular dots. The case of␣/˜=1.0 is indicated by blue open circles. Intermediate between them we have curves for␣/˜=N⫻⌬, where ⌬=321. From ␣/˜=1 to ␣/˜=2 we have curves for ␣/˜=1+M ⫻⌬. Sz
−
is essen-tially suppressed when␣⬇2˜.
0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 α/ ˜β ˜ k
FIG. 4.共Color online兲 Bulk spin polarization SBas a function of ␣/˜ 共abscissa兲 and k˜ 共ordinate兲, and for 0ⱕ␣/˜ⱕ1.2. Directions of SBare shown with x and y axes along the abscissa axis共pointing
right兲 and the ordinate axis 共pointing upward兲, respectively. SLRDB 共black arrows兲 denotes the linear-k SOIs, and SB共red diffused ar-rows兲 denotes the full SOIs. The magnitudes of the vector pairs are normalized between themselves such that the magnitude of the bulk spin polarization is兩SB兩/兩S LRD B 兩, and that of S LRD B is unity. 1 1.2 1.4 1.6 1.8 2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
α/ ˜β
˜ k
FIG. 5.共Color online兲 Bulk spin polarization SBas a function of ␣/˜ and k˜, and for 0.9ⱕ␣/˜ⱕ2.0.
direction and magnitude, unless for very large k˜. Thus the RSOI helps to make the linear-k SOI dominates in the for-mation of SB for␣/˜ ⬎1.
IV. CONCLUSION
In conclusion, we have studied systematically the compet-ing interplay between the RSOI and the cubic-k DSOI in their contribution to the edge spin accumulation and the bulk spin polarization. There are three regimes, namely, the Dresselhaus-dominated regime 共兩␣/˜ 兩⬍0.5兲; the
Rashba-dominated regime 共兩␣/˜ 兩⬇2兲; and the intermediate regime 共0.5⬍兩␣/˜ 兩⬇1兲. The optimal restoration of the spin accu-mulation occurs in the region 兩␣/˜ 兩⬍1.0. While the RSOI alone cannot give rise to spin accumulation, it can still exert its effect via the cubic-k DSOI, and thus provide the needed tunability for spin accumulations.
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.
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