Spin Hall effect in a symmetric quantum well by a random Rashba field

Dopant ion concentrations in the sides of a symmetric quantum well are known to create a random Rashba-type spin-orbit coupling. Here, we demonstrate that, as a consequence, a finite size spin Hall effect is also present. Our numerical algorithm estimates the result of the Kubo formula for the spin Hall conductivity by using a tight-binding approximation of the Hamiltonian in the framework of a time-dependent Green’s function formalism well suited for very large systems.

Figure 1

Spatial dependence of the Rashba spin-orbit interaction for different values of $n_u$ and $n_d$, which are the concentration of the carriers on the up and down sides of the well, respectively. Left: $n_u={10}^{12}{\text{cm}}^{-2}$ and $n_d=5\times {10}^{11}{\text{cm}}^{-2}$. Right: $n_{d,u}=5\times {10}^{12}{\text{cm}}^{-2}$. In both figures, the system size is $400\times 400\text{nm}$ and $z_0$ is fixed to 4 nm. The energetic units are relative to $t_0$, which is the hopping integral.

Figure 2

(Color online) The density of states of a clean system with constant Rashba coupling $V_R$ is calculated for a square lattice with a size of $200\times 200a_0$. The energy is expressed in hopping-integral units, $t_0$.

Figure 3

(Color online) Left: The spin Hall conductivity for a clean system with a size of $200\times 200a_0$, with a constant $V_R$. The results are obtained by averaging over 200 initial wave functions. Right: $\eta$ dependence of the spin Hall conductance for a fixed energy $E=0.5t$ and a Rashba coupling $V_R=0.2t$. The error bars are smaller than the size of the symbols.

Figure 4

(Color online) The spin Hall conductivity when both up-down doping layers are present. For each Rashba field distribution, the spin-Hall conductivity was averaged over 200 initial wave functions at each energy. The figure presents averages over 100 random impurity configurations.