Rashba-type spin accumulation near a void at a system edge
L. Y. Wang and C. S. Chu
Department of Electrophysics, National Chiao Tung University, Hsinchu 30010, Taiwan
(Received 1 February 2011; revised manuscript received 16 August 2011; published 30 September 2011) We consider the spatial distribution of spin accumulation Sznear a void (edge void) located at a system edge. The edge void is semicircular in shape with a radius R0≈ lso, the spin-relaxation length, and it is formed out of a Rashba-type two-dimensional electron system in the diffusive regime. The nonuniform driving field provides the essential condition, and diffusive contributions to the spin currents from spin polarizations provide the primary impetus for the spin accumulation. The edge void reveals an underlying asymmetry of the bulk-void counterpart, namely, a finite bulklike spin flow at the system edge. This bulklike spin flow gives rise to spin accumulation when open boundary occurs. The bulklike spin flow is found to exhibit a similar spatial profile as that of Szat the system edge. This bulklike spin flow is expected to be a primary impetus for local spin injection at the sample edge near the edge void when a local protrusion occurs. Spin orientation of the bulklike spin flow varies with the locations on the edge-void boundary.
DOI:10.1103/PhysRevB.84.125327 PACS number(s): 72.25.Dc, 71.70.Ej, 73.40.Lq
I. INTRODUCTION
All-electrical generations and manipulations of spin po-larization are the main goals of semiconductor spintronics. Rashba spin-orbit interaction1(RSOI) has been the key knob for achieving this goal due to its gate-tuning capability.2–4 However, in the diffusive regime (lso le), the RSOI’s
contribution to the edge-spin accumulation Sz is completely quenched due to the linear-k dependence of the SOI.5–9The edge-spin accumulation is an essential feature of spin Hall effect (SHE),10–27 and k, l
so, and le are, respectively, the
electron momentum, spin-relaxation length, and mean-free path. Although RSOI contributes to SHE in the mesoscopic ballistic regime28,29(l
so< leand the system dimension L < lφ, the phase coherent length), it remains important to seek for ways to restore the RSOI contribution to edge-spin accumulation (or SHE) in the diffusive regime. Furthermore, the edge-spin accumulation, if current induced, could lead to spin injection into a region where the driving field is absent,30 even though great care must be exercised when the RSOI is changed across the injection boundary.31
Effects of RSOI on SHE in the diffusive regime have been obtained at two corners of an electrode-sample interface7,32 and in its competing interplay with the cubic-k Dresselhaus SOI (DSOI).33However, the former has the spin accumulation restricted to within a lso region about the interface corners,
whereas the latter has the RSOI restricted to suppressing the spin accumulations due to the cubic-k DSOI.21,22Seeking for more flexible ways of RSOI’s contribution has prompted a recent study on a nonuniform field SHE.34 A void in the bulk of a Rashba-type two-dimensional electron system (2DES) and its surrounding nonuniform driving field were found to generate a spin accumulation Sz.34 The underlying physical process is different from the conventional one. While the conventional one is associated with a finite out-of-plane spin current (SC) Iz
ν, the key process in Ref.25is associated with an in-plane SC Iμ
ν, and with its vanishing at the edge-void boundary. Here, μ,ν∈ {x,y}, and the superscript (subscript) denotes spin (flow) direction.
Spin accumulation at a system edge, however, is even more interesting in its own right,26,27 and also in its possible
connection with spin injection.30,31,35 It is legitimate then to consider the spin accumulation near a void (edge void, see Fig.1) at a system edge to get a clear physical understanding of the physical processes that contribute to the phenomenon, and to make connection with spin injection. In this work, we find out that the edge void (EV) reveals an underlying asymmetry, namely, a bulklike spin flow (spin-current source in the next section) at the system edge. This asymmetry feature does not produce spin density Szin a bulk void along the longitudinal symmetry axis (y= 0),34 but it is the sole source for the Sz at the sample edge in the case of an EV. Furthermore, we show that the spatial profile of the bulklike spin flow is quite similar to that of Szat the system edge. As the bulklike spin flow provides a primary impetus for Szwhen open boundary occurs, it will provide a primary impetus for spin injection when an elongated protrusion occurs at the sample edge near the edge void. The edge-spin-accumulation profile will thus provide us a guide for the favorable sites for spin injection. Detailed investigation of the specific spin retrieval scheme will be carried out in future study.
A physical picture for the spin-accumulation processes is presented below, and it starts from the finding in Ref.25that the nonuniform driving field E(ρ) gives rise to an Edelstein-type spin polarization34
SEd = −N0ατ e/¯h ˆz× E(ρ), (1)
for which spin is in plane and spatial variation at positionρ is from the driving field. Here, N0, α, τ , and e are the energy
density per spin, RSOI coupling constant, mean-free time, and charge magnitude, respectively. Already, SEd satisfies the spin-diffusion equation, but the boundary condition for the SC has not. The SC Ii
n(S,J E
n) contains terms related to the spin polarization S and to the direct field-driving term JnE, and is given by22,34,36 Inν= −2D∇nSν− RνzνSz( ˆν· ˆn), (2) Inz= −2D∇nSz− ν=x,y RzννSν ( ˆν· ˆn) + JnE,
FIG. 1. An edge void of radius R0 is positioned at a system edge. Asymptotic driving field is E0and the origin of the coordinate coincides with the void center.
where the gradient terms correspond to diffusive contributions with D the diffusion constant, the Rνlmdenotes the precession of Slinto Sνwhen the flow is along ˆm, and JE
n corresponds to the direct effect of the driving field. Setting S= SEd leads to zero Iz
n but nonzero Inν. That the former SC is zero reflects the quenching of the conventional RSOI’s contribution to SHE, but the nonzero value for the latter SC shows that diffusive contributions from in-plane spin polarization provide the primary impetus for the SHE. A nonuniform driving field opens up this unconventional contribution of RSOI to SHE. The condition Iν,z
n = 0 at the EV boundary (ˆn normal to the boundary) generates an additional S including, most importantly, a nonzero spin accumulation Sz, as is evident from the Iν
n expression in Eq. (2). The conditions that the spin currents are zero at both the void boundary and the system edge, where the symmetries are incompatible, mandates a self-consistent procedure for the determination of the spin accumulation. To this end, we have devised a semianalytical approach, which provides us a transparent view of the physical processes involved.
The SC given by Eq. (2) has D= v2
Fτ/2, where vF is the Fermi velocity, and Rilm= 4τ
nilnh n kv
m k, where
iln is the Levi-Civita symbol, h
k the effective RSOI field,
and the overline denotes angular average over Fermi sur-face. The direct field-driving term in the SC is given by
JE n = ν 4τ2v n k(hk× ∂ hk
∂kl)zeN0∇lϕ(r), where the
nonuni-form driving field E= −∇ϕ(ρ) = σ0j has the electric current
density j satisfying the steady-state condition ∇ · j = 0 and the boundary condition jn= 0 for ˆn normal to the boundary. Here, σ0is the electric conductivity and E= E0xˆ− E0(R0/ρ)2(cos 2φ ˆx+ sin 2φ ˆy) outside the circular edge void,
with ρ and φ being the coordinates originated from the EV center.
In this paper, we calculate the spin polarization S in the vicinity of the EV. Our result is in the form S= SB+ SEV. The first term SB= SEd + SBis the spin polarization for a bulk circular void,34where Ii
n(S B,JE
n)= 0 at the EV boundary.
JnE’s sole contribution to SC is in i= z and is exactly canceled by that from SEd . Along the seemingly symmetry axis (φ= 0,π , or y= 0) of the bulk void, we find that the SCs Ix
n(S B, 0) and Iz n(S B,
0) are nonzero for ˆn= ˆy. This hidden asymmetry
in SC is revealed when the circular void is positioned at a system edge and is exhibited via its generation of an additional
SEV.
The calculation of SEV is carried out in a two-step procedure. The first step produces SE1, which, together with
SB, has Ii n(S
B+ SE1,JE
n)= 0 at the sample edge (y = 0). This step is solved analytically and SE1 is found to have already incorporated an essential part of the spin accumulation at the system edge, especially at the corners junctioning the edge and the void boundary. The second, and final, step is to find SE2such that, with SEV= SE1+ SE2, S satisfies the SC boundary conditions at both the void boundary and the system edge. All the S’s above for each step are solutions to the spin-diffusion equation [Eq. (3)], but each has to satisfy a designated SC boundary condition and each is driven by a designated SC source term. Nonzero SC at the boundary or edge in a step of our calculation will be treated as a SC source term for the determination of S in the next step.
We note, in passing, that the SC is used for the establishment of a boundary condition for the spin-diffusion equation. A con-ventional form of the spin-current operator Jli = (1/4)(Vlσi+
σiVl) is appropriate for hard wall boundary,22,37,38 where the kinetic velocity Vl= (1/i¯h)[ ˆx,H], and spin unit of ¯h is implied. As the boundary condition is applied to a region much shorter in distance than lsofrom the boundary, the effect
of spin torque39,40should be of secondary importance here. In the following, we present our model and theory for the nonuniform driving-field effects in the vicinity of an EV. Numerical results and discussions are presented in Sec.III. A conclusion is presented in Sec.IV.
II. MODEL AND THEORY
The spin-diffusion equation (SDE) for the case of nonuni-form driving field has been derived in Ref.25. For the case of RSOI, the Hamiltonian Hso= hk· σ has the effective SOI
field hk= −αˆz × k, where σ denotes the Pauli matrices. The SDE is given by D∇2Sν− νν ¯h2 Sν+ Rνzν ¯h ∇νSz− Mν0· ∇ 2¯h3 D 0 0 = 0, (3) D∇2Sz− zz ¯h2 Sz− Rzxx ¯h ∇xSx− Rzyy ¯h ∇ySy = 0, where S is in units of ¯h.
The spin-charge coupling term is −Mν0∇D0
0 where
Mν0= 4τ2h3
k ∂nν
k
∂ k, and ∇D00 becomes position dependent in
a nonuniform driving field. Here, D00= 2N0eϕ(ρ), ˆnk =
hk/ hk, and Rzνν = −Rνzν = −2hFvFτ for RSOI. Further-more, il= 4τh2
k(δil− n i kn
l
k) are the D’yakonov-Perel’ (DP) spin-relaxation rates for which xx= yy = zz/2= 2h2Fτ in the RSOI case.41 The boundary condition for the SDE, as mentioned above, is given by Ii
n= 0 for ˆn normal to either the system edge or the EV boundary.
Our main goal is to solve for SEV. Adding this term to
SB, the spin polarization for a bulk circular void, will give us the total spin polarization S. The expression for SB has been obtained analytically,34 but it is too lengthy and is not presented here. An essential part of SEV, namely, SE1, is the spin accumulation at the system edge, and it can be captured by first applying our SC boundary condition to the edge at
SE2, is obtained by imposing the SC boundary conditions at both the EV boundary and the sample edge. Thus, SEV=
j=1,2 S Ej
.
To address the SC boundary condition at the system edge, we start from the SDE for SEj(j = 1,2), which is obtained from Eq. (3), given by
∇2SEj x − 4S Ej x + 4∇xS Ej z = 0, ∇2SEj y − 4S Ej y + 4∇yS Ej z = 0, (4) ∇2SEj z − 8S Ej z − 4∇xS Ej x − 4∇yS Ej y = 0.
This equation has adopted a length unit lso =
√
Dτso, where τso= 2¯h2/(h2Fτ) and hFis the RSOI field at the Fermi surface. A Fourier transform solution to this SDE with respect to x is facilitated by writing SEjin the form
SiEj =
dk
q=1,2,3
ηq(j )(k) aiqeikxe−βqy, (5)
where index q denotes the qth mode of solution for the SDE and e−βqyindicates that SEjis localized near the y= 0 edge.
The eigenmodes, given by aiq, depend on k, and the amplitude for each such mode is attributed to ηq(j )(k).
By substituting SEjinto Eq. (4), we obtain
⎡ ⎢ ⎣ −k2+ β2 q− 4 0 4ik 0 −k2+ β2 q− 4 −4βq −4ik 4βq −k2+ β2 q − 8 ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ axq ayq azq ⎤ ⎥ ⎦ = 0, (6) where β1= √ k2+ 4, β 2= k2− 2 + 2i√7, and β 3= β2∗. The eigenmodes are (ai1)= (1,g1,0), (ai2)= (g2,g3,1),
and (ai3)= (−g2∗,g3∗,1) with g1= ik/
√
k2+ 4, g
2= ik(3 + i√7)/8, and g3= ig2
k2− 2 + 2i√7/k. The fact that SEj are real requires η1(j ) to be pure imaginary and η(j )2 (k)= [η(j )3 (−k)]∗. The amplitudes η(j )q (k) will be fixed by the SC boundary conditions.
The boundary condition Ini(S
B+ SE1,JE
n)= 0 at y = 0, and ˆn= ˆy is obtained from Eq. (2), given by
− ∂ ∂y SxE1+ SBx+ 2α ER 2 0 x3 y=0 = 0, −∂ ∂y SyE1+ SyB− 2 SzE1+ SzB y=0 = 0, (7) − ∂ ∂y SEz1+ SzB+ 2 SyE1+ SyB y=0 = 0, where E= eE0N0τ/¯h, and the term involving Eis due to SEd .
Note that, if the terms in Eq. (7) that involve SBand Ewere to add up to zero for their respective equations, then SE1 would be obviously zero. However, the contrary turns out to be the case here. Therefore, by moving these SBand Eterms to the right-hand side of Eq. (7), they become the SC sources
fi(1)for the SE1generation, as is given by −∂ ∂yS E1 x y=0 = f(1) x , − ∂ ∂yS E1 y − 2S E1 z y=0 = f(1) y , (8) − ∂ ∂yS E1 z + 2S E1 y y=0 = f(1) z , where fi(1)= −Ii y(S Ed + SB,JE)|y=0 for |x| R0 and fi(1)= 0 for |x| < R0. Analytical forms of fi(1) are obtained to be fx(1)(x)= 4π x {−X2+ 2 Im[gZ2]} − 2α E R02 x3, fy(1)(x)= 0, (9) fz(1)(x)= −4π |x|Im[Z1]+ 4π{X0+ 2 Im[gZ0]} + 4π{−X2+ 2 Im[gZ2]},
where g= γ2/(γ22+ 4), Xm= itxHm(1)(γ1|x|), and Zm = tzHm(1)(γ2|x|). Here, Hm(1)(z) denotes the Hankel function of the first kind, and the constants γ1 = 2i and γ2=
2+ 2i√7.34 Explicit expressions for tx and tzare not shown here, but they were defined for the expression of SBin a bulk circular void.34 Specifically, both tx and tz depend linearly on α, the RSOI coupling constant.
It is worth mentioning that fi(1)can also be interpreted as a bulklike spin flow at the system edge, which gives rise to the edge-spin accumulation, as is evident in Eq. (8). Furthermore,
fi(1) = 0 reflects an unexpected asymmetry. The symmetric
structure of a bulk circular void seems to suggest that all currents, including charge and spin, flowing normally to the symmetry axis (φ= 0,π) must be zero. This is indeed the case for the charge current, but is otherwise for the spin current. The reason is related to the fact that SBx and SBz are odd in y, while that for SBy is even.34As a consequence, according to Eq. (2), there are diffusive contributions to f(1)
x and fz(1) but not to
f(1)
y . Furthermore, there is an additional contribution to fz(1) through the spin precession of SBy. Thus, we have nonzero f
(1)
i except for i= y, as is shown in Fig.2(a).
FIG. 2. (Color online) Spin-current source terms fi(j )for j = 1,2 are plotted, respectively, in (a), (b), and with abscissas x and φ. In (a), empty symbols denote f(1)i , and solid lines denote SC Iyiat y= 0 when SE1is included. In (b), open symbols denote f(2)
i , and solid lines denote SC Ii
nfor ˆn= ˆρ and at ρ = R0when the total S is used.
The amplitudes η(1)q (k) for the q modes are determined by Fourier transforming Eq. (8), via the integral2π1 dx e−iκx, to obtain ⎡ ⎢ ⎣ β1 β2g2 −β3g2∗ β1g1 β2g3− 2 β3g∗3− 2 2g1 β2+ 2g3 β3+ 2g∗3 ⎤ ⎥ ⎦ ⎡ ⎢ ⎢ ⎣ η(1)1 η(1)2 η(1)3 ⎤ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎣ ˜ fx(1) 0 ˜ fz(1) ⎤ ⎥ ⎦ , (10)
and from it, the η(1)q (k). Here, ˜f
(1)
i (κ) is the Fourier transform of fi(1).
The amplitudes η(2)
q (k) for S
E2can be calculated similarly.
However, we need to address both the SC boundary conditions at the EV boundary and the system edge. The construction of
SE1 has removed any further need of introducing the SC source at the system edge. Yet, additional spin polarization is generated because the SC from SE1 does not satisfy the boundary condition at the EV boundary. In effect, this gives rise to a SC source at the EV boundary that, in turn, generates
SE2. In this work, we find out that it is convenient to replace the SC sources fi(2) at the EV boundary by an auxiliary SC source fiaux at y= 0, but located outside the system edge, for |x| < R0. This approach is in line with the concept of
introducing image charges for the electrostatic problem where the image charges that help satisfy the boundary condition for the electrostatic potential must locate outside the region of interest. Taking into account the parity of SB with respect to
x, the auxiliary SC sources fiaux(|x| < R0) are written in the
form fxaux(x)= fxaux0 + n=1,2,... Ax,nsin nπ x R0 , fyaux(x)= fyaux0 + n=1,2,... Ay,ncos (2n− 1)πx 2R0 , (11) fzaux(x)= fzaux0 + n=1,2,... Az,ncos (2n− 1)πx 2R0 ,
where fiaux0 = [fx(1)(R0) x/R0,0, fz(1)(R0)] is to make sure
that faux
i connects to f
(1)
i continuously to avoid the Gibbs phenomenon in the Fourier transformation. With the Fourier transformed auxiliary SC sources ˜fiaux substituted into the right-hand side of Eq. (10), the column vector on the left-hand side of the equation becomes η(2)i . Thus, ηi(2) and SE2 are expressed in terms of Aj,n. These Aj,n coefficients will then be fixed by the SC boundary condition at the EV boundary
Ii
n(ρ= R0)= 0, which is obtained from Eq. (2), given by
− ∂ ∂ρS EV x − 2 cos φSzEV ρ=R0 = 0, −∂ ∂ρS EV y − 2 sin φSzEV ρ=R0 = 0, (12) −∂ ∂ρS EV z + 2 cos φSxEV+ 2 sin φSyEV ρ=R0 = 0, where φ∈ (0,π). Similar to Eq. (10), the contributions from
SE1 in Eq. (12) will become the SC source fi(2)when it is moved to the right-hand side of the equation. Solving the equation by direct discretization leads us to the total spin polarization S= SB+ SEV.
III. NUMERICAL RESULTS
Numerical examples presented in this section are organized as follows. Figure 2 illustrates the effectiveness of our approach. Figures 3 and 4 present our main results that, respectively, spin accumulation Sz does occur near an EV due to RSOI and the Szexhibits similar spatial profile as the bulklike spin flow at the system edge. The spin accumulation
Szfor voids of large (R0 lso) and small (R0 lso) radii are
shown in Figs.5and6. Finally, Fig.7presents the distribution of the spin accumulation as probed by a scanning optical beam.
We have assumed material parameters that are consistent with GaAs. Specifically, the effective mass m∗= 0.067m0,
with m0the free-electron mass, electron density ne= 1 × 1012
cm−2, electron mean-free path le= 0.43 μm, the Rashba coupling constant α= 0.3 × 10−12 eV m,2,42 and the spin-relaxation length lso= 3.76 μm. Furthermore, the driving field E0= 40 mV/μm, and the EV structure radius R0= 0.5lsoin
Figs.3and4.
The effectiveness of our self-consistent procedure is illus-trated in Fig.2. We plot the SC Ii
n(solid curves) at the system edge (y= 0) and at the EV boundary (ρ = R0) in, respectively,
Figs.2(a) and2(b). For comparison, we plot the SC source terms fi(1) and f
(2)
FIG. 3. (Color online) Out-of-plane spin densities SBz, SE1z ,
SE2
z and spin accumulation Szare plotted in (a), (b), (c), and (d), respectively. Part (d) is the sum of (a), (b), and (c). The external electric field E is applied along ˆxand the EV structure has a radius
R0= 0.5 lso.
respectively. Our result that all the six solid curves, three each for Iyi and I
i
ρ, overlap on the zero abscissas shows that the SC boundary conditions are satisfied remarkably. Furthermore, the symmetries of the SC source are consistent with those derived from SB, as has been discussed in the previous section for the case of fi(1).
Our main results are presented in Figs. 3 and 4. The former is the out-of-plane spin polarization, while the latter is the connection between the edge-spin accumulation and the bulklike spin flow represented by f(1)
z . The spatial distribution of out-of-plane spin densities SBz, S
E1
z , and S E2
z are shown, respectively, in Figs. 3(a), 3(b), and3(c). The spin accumulation Sz, given by the sum of these three out-of-plane spin densities, is denoted by Fig.3(d). It is clearly shown that RSOI’s contribution to spin accumulation Sz can be turned on locally by an EV due to the nonuniform driving field. Basically, Figs.3(a)–3(c)provide a pictorial way of viewing the formation of the spin accumulation. Figure 3(a) shows the spin density SBz that equals that for a bulk void.34 The spin density is centered along the φ= π/2 direction and is
FIG. 4. (Color online) Spatial profiles of bulklike spin flow (SC source) f(1)
z (triangles), edge-spin accumulation Sz(solid line), and −∂Sz/∂y(dashed line) at the sample edge (y= 0) for the edge void in Fig.3. Numerical factors are introduced to facilitate comparison.
separated into two regions of opposite spin polarization: a core region and an outer region. The core region concentrates along the void boundary and has a radial thickness of about 0.3 lso∼ 1.1 μm. The outer region has a much wider spatial
extent, in the form of a curved spin cloud, and having its center located about a distance of one lsofrom the void boundary. This
spin density is driven by an in-plane SC of diffusive origin. The SC boundary condition is satisfied at the void boundary but not at the system edge. This results in a residual SC, or a SC source term fi(1)[shown in Fig.2(a)], at the system edge that drives the generation of SEz1in Fig.3(b). The spin density
SEz1concentrates mostly at the two corners of the EV with a range of about 0.5 lsoand with spin polarization opposite to
that of the core spin density in Fig.3(a). SEz1also contains a wide outer region of compensating spin cloud with much smaller spin-density magnitude. The SC boundary condition, however, is not satisfied at the void boundary. A residual SC, or a SC source term fi(2), then drives the generation of the spin density SEz2. Although the spin density S
E2
z , in general, has a smaller magnitude, it enhances the core region in Fig.3(a)
at the void boundary. By our design, SEz2and fi(2)together satisfy the SC boundary condition at both the system edge and the void boundary. This approach has the advantage that the self-consistently determined SEz2 is of relatively small magnitude, thus, the method is numerically more stable and efficient. The total spin accumulation Sz, as shown in Fig.3(d), is the sum of the three spin densities.
Figure 4 compares the spatial profiles of the bulklike spin flow f(1)
z [also shown in Fig. 2(a)] and the total spin
FIG. 5. (Color online) Total out-of-plane spin densities Sz for edge voids of radii R0= 0.2 lso and R0= 7 lso in (a) and (b), respectively.
accumulation Sz at the sample edge. Both characteristic behaviors of Sz, namely, Sz and −∂Sz/∂y, are included for the comparison, where −∂Sz/∂y denotes the diffusive contribution of Szto spin current. To facilitate the comparison, constant factors 2.5 and 0.5 are multiplied to Szand−∂Sz/∂y, respectively. The spatial variation of f(1)
z in Fig.4matches very well with −∂Sz/∂y, except for the region x≈ 2.5lso, where
their values are already quite small. Szalso exhibits a similar spatial variation as f(1)
z , except again at around x≈ 2.5lso,
when their respective values are small. The deviation comes from SE1
y , and the fact that it can contribute to I z
y is seen in Eq. (7). We stress here that this close spatial correspondence between the bulklike spin flow and the characteristic behaviors of Szin Fig.4collaborate the spin-current-driven nature of the edge-spin accumulation Sz. It also sheds light on spin injection, suggesting that the bulklike spin flow will be a driving agent for spin injection. We note that f(1)
z is chosen to represent the bulklike spin flow in the above consideration. The reason is that, in Fig.2(a), f(1)
y is zero and fx(1)does not couple with Sz, according to Eq. (7).
The magnitude of the out-of-plane Sz of EV is found to decrease in both large (R0 lso) and small (R0 lso) void
radii. These two regimes are shown in Fig. 5 as contour plots of Sz, and in Fig. 6 as the plots of the Sz’s spatial variations along two representing directions, namely, along
x and y, in Figs. 6(a) and 6(b), respectively. Comparing with the intermediate-radius regime (R0= 0.5 lso in Fig.3),
three spatial modification features are found in Sz in both the small-radius [R0 = 0.2 lso in Fig. 5(a)] and large-radius
[R0= 7 lso in Fig. 5(b)] regimes. That the magnitude of Sz
at the EV boundaries drops drastically is clearly shown in Figs.6(a)and6(b). The sizes of Szclouds at the EV corners, positive Sz in the small x− R0 region in Fig. 6(a), shrink
FIG. 6. (Color online) Spatial distribution of the total out-of-plane spin densities Sz for edge voids in Figs.3and 4along (a) the x axis and (b) the y axis. The abscissas are the distances from the void edge.
drastically from∼ lsoto∼ 0.2–0.3 lso. Finally, the thickness
of the Szat the void boundary, negative Szin the small y− R0
region in Fig.6(b), shrinks from∼ 0.3 lsoto less than 0.1 lso.
These modifications in Sz can be understood as diminishing the effects of the nonuniform driving field for the case of large void radius. In the small-radius regime, however, different physical origins are at work for the Sz modifications. Since the entire vicinity of the EV is within a lso, the SC source
terms fi(j ) can exert their effects to the region and, thus, no longer concentrated at the corner regions. This is supported, in Fig.5(a), by the shrinking of the positive Sz at the void corners and the appearance of a positive Szcloud around the
φ= π/2 direction. This positive Szcloud has on its two sides negative Szclouds, which are understood as the compensating spin cloud. Finally, this spin cloud configuration in Fig.5(a)
is expected to vanish as R0 approaches zero. The integration
of the SC source terms fi(j )over the system edge (not shown) decreases to zero as R0→ 0.
The spin accumulation Szcan be probed optically by Kerr rotation. We plot, in Figs.7(a) and7(b), the net number of out-of-plane spin within a circular probe as it is scanned, respectively, along the directions φ= π/2 and 0. The radius of the probe is the same as the void, and the distance between the probe and the void centers is d. For φ= π/2, the net spin number exhibits a negative dip at small-d region and a large positive peak in the larger-d region. The former dip, located at around d≈ 0.5 lso, picks up the core region, and the latter peak,
located at around d≈ R0+ lso, picks up the outer region of
the spin accumulation. The system edge has only a mild effect on this curve, as is evident from the close resemblance of this curve to that of a bulk void. In contrast, for the φ= 0 case, the
FIG. 7. (Color online) Net number of out-of-plane electron spins, from Sz, within a circular probe area of the same radius as the EV structure. The probe center is shifted by a distance d from the EV center (a) along ˆy (φ= π/2) and (b) along edge ( ˆx, or φ = 0).
positive peak in the small-d region and the negative dip in the larger-d region reflects the sole effect of the system edge. The former peak, located at around d≈ lso− 1.3 lso, picks up the
spin accumulation at the corners of the EV, while the latter dip, located at around d≈ 2R0+ 0.8 lso, corresponds to the
situation when the probe moves out of the core region.
IV. CONCLUSIONS
In conclusion, we have studied in detail the physical processes of the formation of spin accumulation near an edge void and have demonstrated their spin-flow-driven nature. The EV structure reveals an underlying asymmetry of the bulk void, a nonzero bulklike spin flow on the y= 0 axis. The spin accumulation consists of both bulklike (void-boundary) and edgelike (system-edge) characteristics. Even though the edgelike spin accumulation has comparable magnitude but
opposite signs compared with the bulklike spin accumulation, the two spin accumulations occur in different spatial locations so that the spin accumulation exhibits both characteristics simultaneously. Moreover, the spin accumulation and the bulklike spin flow correspond quite well in their spatial profiles at the system edge. This bulklike spin flow is expected to remain a driving agent for spin injection when a local protrusion occurs at the system edge. The EV structure could thus provide favorable sites for spin injection. These results should be of interest to further research in all-electric spintronics, both experimentally and theoretically.
ACKNOWLEDGMENTS
This work was supported by Taiwan NSC (Contract Nos. 96-2112-M-009-0038-MY3 and 100-2112-M-009-019), NCTS Taiwan, and a MOE-ATU grant.
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