In this section, the intrinsic SHE give rise the spatial distribution of the spin density Si
(i = x, y, z) exhibits significant symmetry properties on the 2D semiconductor strip in the absence of an in-plane magnetic field. The spin accumulation strongly depends on the specific SOI form. For the case of Rashba SOI, there is no spin density Sz accumulating near the edges because the spin-Hall current is exactly cancelled by arbitrary weak disorder in the diffusive regime. Only the bulk spin density Syb = −2τ αN0eE is induced by Rashba SOI through a driving electric field E ˆx. However, the spin density Sz(x) shows the anti-symmetric (symmetric) accumulation behavior on the 2D strip for the case of
electron mass. We choose the electric field E = 25mV /µm. Now, it is convenient to define the electron density n0 = 1015(1/m2) such that the units of the Fermi wave vector and the Fermi velocity are kF 0 =√
2πn0 = 7.92 × 107 (1/m) and vF 0 =1.36 × 105 (m/s), respectively. The typical mean free path is lmean = 1 µm so that the unit scattering time τ0 = 7.3 × 10−12(s) is given by lmean = vF 0τ0. The Dresselhaus SOI constant is β = 27.5eV ˚A3[13] and the DP relaxation energy is given by Γxx = 0.0042(C4−C2/2+1/8) (meV). The unit of quantum well thickness is w0 = 1 × 10−8 m such that the nth subband energy are εnz = ~2(nπ/w0)2/2m∗. By above definition, we can study the variation of spin densities in various parameters.
The electron density is n = n∗n0 and quantum well thickness is w = w∗w0, where n∗, w∗ are dimensionless numbers. The total electron energy is restricted to be lower than the second subband energy of quantum well leading to ~2k2F/2m∗+ ε1z > ε2z in a 2D system.
The Fermi wave vector is kF = kF 0√
n∗ and the parameter denotes C = C0/√
X, where X = nw2 and C0 = κ0/kF 0. This restriction of energy gives us X < 3C02. Secondly, the spin relaxation length lso in Eq. (4.6) is larger than the electron mean free path lmeanin the diffusive regime. Therefore we have the another restriction for X > [4C02−13.16w2, 0]max. The Fig. 4.2 shows that spin accumulation Sz± respecting to y = ±d/2 are plotted as a function of X in various quantum well thickness (a) w = 2 × 10−8m, (b) 2.5 × 10−8m, and (c) 3 × 10−8m with an electric field E = 25mV /µm. This criterion in Fig. 4.2 shows that Sz± can be changed the polarization direction by increasing X cross a critical value Xc. Because the spin-Hall current depends on parameter X and Xc = 34.15 is fixed for various thickness w. The spin-Hall current in Eq. (4.12) vanishes at the critical point
-10 -5 0 5 10
-6 -3 0 3 6
0 10 20 30 40
-5.0 -2.5 0.0 2.5 5.0
XC
XC
Sz +
Sz− ISH
(a) w=2x10
-8m
10.4
XC
S
za n d I
SH(D /l
SO) ( 1 / µ m
2)
(b)w=2.5x10
-8m
X=nw
2(c)w=3x10
-8m
Figure 4.2: Spin densities Sz± (1/µm2) and spin-Hall current ISH in unit of τ βkF2N0eED/(~lso) are plotted as a function of X = nw2 for various quantum well thickness: (a) w = 2 × 10−8m, (b) 2.5 × 10−8m, and (c) 3 × 10−8m. Sz± is the spin accumulations for y = d/2 and y = −d/2, respectively. These bold(red) arrows indicate the allowed ranges of parameter X: (a) 10.4 < X < 47.28, (b) 0 < X < 47.28, and (c) 0 < X < 47.28. The corresponding electron density is given by n = X/w2.
spin densities Syb and Szb are equal to zero due the characteristics of a cubic Dresselhaus SOI. The dependence of spin densities ∆Sx(y = d/2) at left-hand side edge y = d/2, is represented as a function of the strip width d in the Fig. 4.3 for a fixed quantum well thickness w = 3w0, where w0 = 1 × 10−8m. The spin density at y = d/2 is shown in the unit of 1/µm2. All the length scales of the width d and the transverse coordinate y normal to the electric field E ˆx are in the unit of a spin relaxation length lSO. The blue (solid), red (dashed), and green (dotted) curves are plotted for different parameters X =22, 30, and 40, respectively. The relation κ/kF = C0/√
X gives us the ratio of κ/kF = 0.84, 0.72, and 0.63 corresponding to X =22, 30, and 40, respectively. These curves are effectively corresponded to the variation of electron densities (a)n = 2.4n0, 3.3n0, and 4.4n0 in Fig. 4.3 with n0 = 1 × 1015(1/m2). The spin density ∆Sx is symmetric in transverse coordinate y to indicate ∆Sx(−d/2) = ∆Sx(−d/2). It is shown that magnitude of ∆Sx(d/2) saturate for a fixed X when the strip width d is beyond several lSO.
The dependence of spin densities ∆Sz(y = d/2) at left-hand side edge y = d/2, is represented as a function of the strip width d in the Fig. 4.4. The spin density is shown in the unit of 1/µm2. The blue (solid), red (dashed), and green (dotted) curves are plotted for different parameters X =22, 30, and 40, respectively. It is different from ∆Sx because the spin density ∆Sz is anti-symmetric in transverse coordinate y to indicate
∆Sz(y = d/2) = −∆Sz(y = −d/2). In cases of X = 22 and 30, the spin accumulations
∆Sz(y = d/2) show the same polarization direction due to X < Xc. However, in case of X = 40, ∆Sz(y = d/2) show the opposite polarization direction to cases of X = 22 and 30 due to X > Xc. The general features also show that magnitude of ∆Sz(y = d/2) saturate
0 2 4 6 8 10 -2
0 2 4 6 8 10
∆ S
x(y = d /2 ) (1 / µ m
2)
d (l
so)
X=40 ( κ /k
F
=0.63) X=30 ( κ /k
F=0.72) X=22 ( κ /k
F
=0.84)
Figure 4.3: Spin densities ∆Sx(y = d/2) are plotted as a function of the strip width d in unit of lSO for various values of X(κ/kF) in a fixed w = 3w0, where the unit of thickness denotes w0 = 1 × 10−8m. The blue (solid), red (dashed), and green (dotted) curves are represented for X =22, 30, and 40, respectively. The spin densities ∆Sx(y = −d/2) have the same values with respect to ∆Sx(y = d/2) due to even parity property of ∆Sx(y).
0 2 4 6 8 10 -3
-2 -1 0 1 2 3 4 5
∆ S
z( y = d /2 ) (1 / µ m
2)
d ( l
so )
X=40 ( κ /k
F
=0.63) X=30 ( κ /k
F
=0.72) X=22 ( κ /k
F
=0.84)
Figure 4.4: Spin densities ∆Sz(y = d/2) are plotted as a function of the strip width d in unit of lSO for various values of X(κ/kF) in a fixed w = 3w0, where the unit of thickness denotes w0 = 1 × 10−8m. The blue (solid), red (dashed), and green (dotted) curves are represented for X =22, 30, and 40, respectively. The spin densities ∆Sz(y = −d/2) have the same values but opposite sign with respect to ∆S (y = d/2) due to odd parity
the spin coming far from the edge is completely relaxed. Thus the accumulation of spin densities ∆Si(y = d/2) is dominated by the electron spin in the region within the several lSO.
The Fig. 4.5 and Fig. 4.6 present that the total spin densities Si (i = x, z) which include the bulk spin densities Sib(0). According to the Eq. (4.7) and Eq. (4.11), the total spin density Sx(y) exhibits the symmetric property to the transverse coordinate y. On the other hand, the spin density Sz(y) exhibits the anti-symmetric property to the transverse coordinate y. The blue (solid), red (dashed), and green (dotted) curves are plotted in a fixed w = 3w0 for various parameters X = 22, 30, and 40, respectively. The Sx(y) are shift by the bulk values Sxb(0) which are related to different values of X. It is easily found that the polarization direction of Sz(y) near two edges is reversed for X = 40 respecting to cases of X = 22, and X = 30.
4.6 Summary
In summary, we have studied the spatial distribution of the spin density Si without an in-plane magnetic field for the case of either Rashba or Dresselhaus SOI. In the case of zero magnetic field, the spin there is no spin accumulation occurring near a 2D strip edges for Rashba SOI. However, the spin densities Sz and Sx accumulate near a 2D strip edges for cubic Dresselhaus SOI. We also find that spatial distribution of Sx demonstrates the symmetric property in y axis. On the other hand, spatial distribution of Sz demonstrates the anti-symmetric property in y axis, corresponding to the intrinsic SHE.
-5 -4 -3 -2 -1 0 1 2 3 4 5 0
1 2 3 4 5 6 7 8 9 10
S
x( 1 / µ m
2)
y ( l
SO )
X=40 ( κ /k
F
=0.63) X=30 ( κ /k
F
=0.72) X=22 ( κ /k
F
=0.84)
Figure 4.5: Total spin densities Sx(y) are plotted as a function of transverse coordinate y in unit of lso. The blue (solid), red (dashed), and green (dotted) curves are represented for X = 22, 30, and 40, respectively. The total spin densities Sx(y) exhibit the symmetric behavior. The bulk values of Sb(0) depend on the values of X.
-4 -2 0 2 4 -3
-2 -1 0 1 2 3
S
z(y ) (1 / µ m
2)
y ( l
SO )
X=40 ( κ /k
F
=0.63) X=30 ( κ /k
F=0.72) X=22 ( κ /k
F
=0.84)
Figure 4.6: Total spin densities Sz(y) are plotted as a function of transverse coordinate y in unit of lso. The blue (solid), red (dashed), and green (dotted) curves are represented for X = 22, 30, and 40, respectively. The total spin densities Sz(ξ) exhibit the anti-symmetric behavior. The bulk values of Szb(0) = 0 in the absence of an in-plane magnetic field.
in-plane magnetic field on a two-dimensional strip
In this chapter, we studied the intrinsic spin-Hall effect (SHE) induced by a driving electric field E ˆx, in the presence of an in-plane magnetic field Bk = Bxx + Bˆ yy on a 2D strip. Inˆ the diffusive regime, the spatial distribution of the spin density Si(i = x, y, z) is calculated from a spin diffusion equation derived from the nonequilibrium Green’s function. In the presence of the in-plane magnetic field, the z-component spin density Sz normal to the 2D strip remains zero with or without Bk field for the case of Rashba spin-orbit interaction (SOI). For the case of Dresselhaus SOI, the spatial distribution of spin density show either symmetric or asymmetric features which depend on the direction of the in-plane magnetic field. By applying the longitudinal magnetic field Bx, the spatial distributions of spin densities Sx and Sz show the even parity in Bx but Sy shows the odd parity in Bx. The asymmetric property of Sz versus By is demonstrated for the intrinsic SHE in case
applying an in-plane magnetic field.
5.1 Introduction
More recently, the most important issue is to generate and control the spin-polarized electrons in the achievement of spin-based semiconductor devices [1, 44]. Among the different methods, spin-orbit coupling, which couples the electron spin to its momentum is attracted a lot of remarkable interest. Because the energy gap E0 in a semiconductor is much larger than the effective energy gap m0c2 in the vacuum (m0 is the free electron mass and c is the light speed) such that the ratio of the SOI is proportional to E0/m0c2 ∼ 106. In conclusion, the strength of SOI is much larger in a semiconductor than in the vacuum [13].
In the spin-orbit coupling system, a nonzero spin current is predicted in the direction perpendicular to the applied electric field due to the intrinsic SOI or extrinsic impurities scattering, referring to the intrinsic and the extrinsic SHE, respectively. The intrinsic SHE is involved with either Rashba SOI [12] or Dresselhaus SOI [11], or both, and the behavior of spin accumulations sensitively depends on the different type of SOI. In contrast to intrinsic SHE, the extrinsic SHE is contributed by skew-scattering processes, which induce the spin-dependent transport perpendicular to the electric field [31, 58]. Recently, the several experiments succeed to measure the SHE by either electronic [82] or optical detections. So far, the intrinsic SHE was demonstrated for the p-doped 2D electron gas [35]. Most experiments demonstrated the extrinsic SHE [36, 59].
The 2D strip with two edges at y = ±d/2 is sketched in Fig. 5.1. The in-plane magnetic field Bk with a angle θ respecting to the electric field E. The intrinsic SHE vanishes [37–40] for the disorder approach in the dc limit with Rashba SOI. At the same time, the Dresselhaus SOI gives a finite spin Hall conductivity due to the crystalline inversion asymmetry [40]. Instead of detecting spin current, one realistic way to detect the SHE is measure the spin accumulations in a semiconductor [36]. It is important to
We have studied that the spin accumulations are induced by the intrinsic SHE with an applied in-plane magnetic field in this chapter. The spin transport and relaxation of the intrinsic SHE with a perpendicular magnetic field have been studied in the diffusion ap-proximation for Rashba SOI [83, 84]. Recently, Rashba et al. studied the time-dependent electric field with a static in-plane magnetic field to produce a z-component spin accumu-lation via either non-parabolic band or the anisotropic scatterer [85]. Lin et al. studied the spin current and spin-Hall conductivity for short-range and remote impurities in the case of the intrinsic SHE with an in-plane magnetic field [86]. As known, the spin cur-rent is not conserved and its definition still remains an issue [54]. However, the spin accumulations can be realistically measured in the recent experiment[36]. Therefore, it is interesting to study the the behavior of the spin accumulation versus an in-plane mag-netic field near the boundaries. The symmetric property of spin accumulations have been observed experimentally when an in-plane magnetic field normal to the electric field is applied with the same magnitude but in the opposite direction [36, 59, 76]. This sym-metric spin accumulation is explained as the extrinsic SHE in the presence of an in-plane magnetic field [76]. As know, the extrinsic SHE produces the zero bulk spin density Sz which is perpendicular to 2DEG due to the spin-dependent distribution being propor-tional to linear electron momentum [87]. Therefore, the lowest-order spin accumulation Sz is expected up to the second order of the in-plane magnetic field resulting in symmetric Sz(y) to the in-plane magnetic field [36, 59]. In Sec. 5.2, the diffusion equations of spin densities are studied for the intrinsic SHE in the presence of an in-plane magnetic field. In Sec. 5.3, the spin currents are calculated to satisfy the boundary conditions. The solution
y= d/2
E B
xB
yB
||x
y y= − d/2
θ
y= d/2
E B
xB
yB
||x
y
x
y y= − d/2
θ
Figure 5.1: Top-view schematic illustration of the 2D strip with a width d. The longitu-dinal driving electric field is applied in the x-axis. The tunable in-plane magnetic field Bk can be applied in this 2D strip. The angle θ is between the in-plane magnetic field and the electric field.