Anomalous spin Hall effects in Dresselhaus (110) quantum wells

Anomalous spin Hall effects that belong to the intrinsic type in Dresselhaus (110) quantum wells are discussed. For the out-of-plane spin component, antisymmetric current-induced spin polarization induces opposite spin Hall accumulation, even though there is no spin-orbit force due to Dresselhaus (110) coupling. A surprising feature of this spin Hall induction is that the spin accumulation sign does not change upon bias reversal. Contribution to the spin Hall accumulation from the spin Hall induction and the spin deviation due to intrinsic spin-orbit force as well as extrinsic spin scattering can be straightforwardly distinguished simply by reversing the bias. For the in-plane component, inclusion of a weak Rashba coupling leads to a new type of $S_y$ intrinsic spin Hall effect solely due to spin-orbit-force-driven spin separation.

Figure 1

(Color online) Spin deviation in $S_z$ due to spin-orbit force in (a) Rashba and (b) linear Dresselhaus (001) systems. (c) Spin Hall induction due to antisymmetric CISP along $\left[00\overline{1}\right]$ in Dresselhaus (110) systems may cause opposite $S_z$ accumulations as well, even though there is no spin-orbit force. (d) In the case of linear Dresselhaus (110) plus weak Rashba couplings, a coupled spin-orbit force given by Eq. (11) may lead to spin deviation in $S_y$.

Figure 2

(Color online) (a) Linear Dresselhaus (110) spin-orbit field in $k$ space. (b) Local charge density $⟨eN⟩$ (color shading) and local charge current density (arrows) in a $16a\times 8a$ Dresselhaus (110) sample subject to three terminals with a weak bias $e{V}_0={10}^{-3}{t}_0$. (c) Local spin density $⟨{\sigma }^z⟩$ (color shading) and its corresponding local spin current density $⟨{\mathbf{J}}^{S_z}⟩$ (arrows) in the same device as (b) with same conditions. Spin Hall induction in a $40a\times 40a$ Dresselhaus (110) sample with (d) upward bias and (e) downward bias; strong bias $e{V}_0=0.4t_0$ is applied. Note that in (b)–(e), the regions outside the dashed lines are simulating the leads (zero spin-orbit coupling and constant on-site energy set equal to the applied bias).

Figure 3

(Color online) Imaging of local spin densities $⟨{\sigma }^z⟩$ and $⟨{\sigma }^y⟩$ in a $60a\times 60a$ (110) sample with (a)–(b) top-to-bottom and (c)–(d) right-to-left orientations. A strong Dresselhaus (110) coupling $t_D=0.08t_0$ and a weak Rashba coupling $t_R=0.02t_0$ are used. Bias is set $e{V}_0=0.4t_0$.

Figure 4

(Color online) Local spin densities $⟨{\sigma }^z⟩$ and $⟨{\sigma }^y⟩$ in a $60a\times 60a$ sample as a function of the transverse position with various coupling parameters $t_D$ and $t_R$. The origin is set at the center of the sample. As indicated in panel (a), the coupling parameters from black to the lightest curves are $\left(t_R,t_D\right)=\left(0,0.1\right)t_0$, $\left(0.02,0.08\right)t_0$, $\left(0.04,0.06\right)t_0$, $\left(0.06,0.04\right)t_0$, $\left(0.08,0.02\right)t_0$, and $\left(0.1,0\right)t_0$. In each panel, the red (dark gray) thick curve correspond to Fig. 3.

Figure 5

(Color online) (Online color) Coupling ratios $t_D/t_0$ estimated by Eq. (12) for various QWs with widths $w=5$, 7.5, and 10 nm.