The spin current operator are defined by
Jli ≡ (1/2) (Vlσi+ σiVl) (3.42)
and each spin unit ~/2 is not included here. The velocity operator is given by
Vl≡ pl
m∗ + ∂hp· σ
∂pl , (3.43)
where m∗ denotes the effective mass of electron. The first term in right-hand side of Eq. (4.3) is classical kinetic term and the second term is spin-dependent velocity due to SOI. The spin current Jli stands for the electron moving with the velocity vl = (pl/m∗) and spin state σi. After some algebra, one can obtain the expression for spin current densities zero temperature (ω0 = EF), the spin current densities can be simplified in the form of
Ili = 1
2πN0
For the SHE, it is most important to study spin currents flowing along y direction when a static electric field is applied along x axis. To obtain the spin current densities Iiy has to calculate Xyij and Yyij
The detailed calculation is shown in Appendix E.It is found that the last term of Xyij is exactly cancelled out the contribution of Yyij. Eqs. (22), (25) and (26) are allowed to write down the correct spin current density expressions
Iyi(r) = −2D∂Si
∂y − Rijy(Sj − Sjb) + 2eIsHδiz, (3.50)
which are associated with spin densities Si. The first term of Iyi describes the normal
additional term IsHB is totally contributed from the external magnetic field. Naturally, these two terms are proportional to the linear electric field E because the origin of SHE is coming from spin-charge coupling by SOI. Their expressions are given by
The explicit boundary conditions of the spin current for the case of Rashba SOI are expressed as
Another boundary conditions of the spin current for the case of Dresselhaus SOI are expressed as
For Rashba SOI case, it is easily to check that IsH vanishes without an external magnetic field eBk. Furthermore, the bulk spin density Syb is equal to Syb(0) such that the total spin-Hall current eIsH is still zero even in the presence of external magnetic field. For Dresselhau SOI case, IsH is finite even without eBk. However, IsHB is dependent on eBk and can modulate eIsH by tuning either the strength or the direction of eBk.
In the cases of a 2D strip, the hard-wall boundary conditions Iyi(y = ±d/2) = 0 are imposed. The boundary conditions indicate that both of the spin and charge current cannot penetrate the edges. The solutions of spin densities can be obtained by solving
Si =P
jAijeiλjy for indices j = 1 ∼ 6, i = x, y, and z. One can solve Aij and λj by using the Eq. (3.40) and boundary conditions.
Furthermore, the SHE is associated with the spin polarization flow, or the spin density accumulation on the strip edges, in response to the electric field. In the other word, the SHE can show up in the electric conductance as well. In Eq. (3.44), i = 0 and l = x indicate the charge flowing along x axis with the velocity operator Vx = px/m∗ + ∂hp · σ/∂px. One can obtain the electric current density
Ix = σDE + A∂Sz
∂y (3.54)
where σD is the Drude conductivity and
A = e 2Γ2
"
2vFy µ∂hp
∂px × hp
¶
z
+ vFx µ∂hp
∂py × hp
¶
z
#
. (3.55)
The detailed calculation is shown in Appendix F. The total current is obtained by inte-grating Eq. (3.54) over y. Therefore, the spin-Hall correlation to the strip conductance is given by
∆G = A
E [Sz(d/2) − Sz(−d/2)] = 2A
ESz(d/2) . (3.56)
It is the evidence that the spin accumulations feedback to modify the traditional electric current in x direction due to the intrinsic SHE.
3.5 Summary
In summary, we have derived the diffusion equations for spin densities Si with or without an in-plane magnetic field in the case of either Rashba or Dresselhaus SOI. It is emphasized that the electron spin relaxation length lso is much larger than the electron mean free path lmean in the diffusive regime. In the weak magnetic field limit, the diffusion equation is proportional to linear magnetic field. 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.
The conventional electric current is also modified by the spin-charge coupling due to the intrinsic SHE.
The case of intrinsic SHE without the external magnetic field will be studied in Chapter4. Another case of intrinsic SHE with the in-plane magnetic field will be studied in Chapter5. Both cases are described by the diffusion equations which are obtained in this chapter.
the magnetic field on a two-dimensional strip
In this chapter, the intrinsic spin Hall effect (SHE) on spin accumulation and electric conductance in a diffusive regime has been studied for a 2D strip with a finite width d, shown in Fig. 4.1. It is found that the spin polarization near the edges of the strip exhibits damped oscillations as a function of the width and strength of the Dresselhaus spin-orbit interaction (SOI) while an electric current is applied in the longitudinal direction. Cubic terms of Dresselhaus SOI are crucial for spin accumulation near the edges. As expected, no effect on the spin accumulation and electric conductance have been found in the case of Rashba SOI. At the same time, the conventional electric current can be correlated by the SHE. This correlation is associated with the magnitude of the spin accumulations on the edges.
4.1 Introduction
Starting from 1990, Datta and Das first proposed a quantum device to manipulate the electron spins through the spin-orbit interaction (SOI) produced by a tunable-biased gates atop the semiconductor [5]. The field of spintronics becomes attractive and emerging in the solid state physics. The SOI plays an important role of coupling the electron orbital motion and the spin degree of freedom in the semiconductor through a driving electric field. It is because the strength of SOI is much larger in the semiconductor than in the vacuum [13].
The spin densities can accumulate near the transverse boundaries y = ±d/2 in a semi-conductor with SOI by applying a longitudinal electric field due to SHE. The SHE can be understood that an electron spin encounters a transverse force which is induced by a longitudinal driving electric field [77]. It is different from the extrinsic SHE induced by impurities scattering, however, the intrinsic SHE is owe to either Rashba [12] or Dressel-haus SOI [11] coupling the electric field and the electron spin. For linear Rashba SOI, the spin accumulation near the sample boundaries due to the intrinsic SHE can produce a universal spin Hall conductivity e/(8π~) in the ballistic regime [33]. However, the in-trinsic SHE vanishes [37–40] at the arbitrary weak disorder in dc limit for isotropic as well as anisotropic [78] impurity scattering while the sample is in presence of the linear Rashba SOI in the asymmetric quantum well. However, the spin accumulation can occur for cubic Rashba SOI in the hole system [41]. At the same time, the cubic Dresselhaus SOI gives rise a finite spin Hall conductivity in the symmetric quantum well [40].
In our study, we consider the diffusion equation for spin densities Si (for i = x, y, and z). Instead of Boltzmann equation, the Green’s functions are used in the diffusion approach, in which the spin relaxation length lso is larger than the mean free path lmean. We treat this disorder system by taking averaging over all impurity positions. The spin densities and spin currents are computed in linear response of the electric field E. The bulk spin densities Sxb(0) and Syb(0) are finite in Dresselhaus and Rashba SOI cases, respectively.
y= d/2
x
y
y= − d/2 y= d/2
x
y
x
y
y= − d/2
Figure 4.1: The 2D strip of the width d is applied a electric field along x axis. The transverse boundaries at y = ±d/2.
Furthermore, the spin Hall current vanishes leading to zero spin accumulation in the case of Rashba SOI. On the other hand, the spin Hall current is finite resulting in spin accumulation at edges y = ±d/2. The spatial distributions of Sx and Sz are shown the symmetric and anti-symmetric properties, respectively, in Dresselhaus SOI case. However, the spatial distribution of Sy is zero in this case. It is remarkable that the spin polarization of Sz can be changed sign at the same time, by changing either the electron density n or the quantum well thickness w. Several boundary effects are considered for SHE with interfaces [42, 79–81]. In a 2D strip, spin currents have to be zero for hard-wall boundaries [42]. Based on the boundary conditions, the spin accumulation near edges can be obtained in a 2D strip.
It is also addressed that the conventional electric current is correlated by the intrinsic SHE. Because the spin-charge mixing induces the transverse spin Hall current resulting