Electrical Spin Orientation, Spin-Galvanic, and Spin-Hall Effects in Disordered Two-Dimensional Systems

In disordered systems, the hopping conductivity regime is usually realized at low temperatures where spin-related phenomena differ strongly from the cases of delocalized carriers. We develop the unified microscopic theory of current-induced spin orientation, spin-galvanic, and spin-Hall effects for the two-dimensional hopping regime. We show that the corresponding susceptibilities are proportional to each other and determined by the interplay between the drift and the diffusion spin currents. Estimations are made for realistic semiconductor heterostructures using the percolation theory. We show that the electrical spin polarization in the hopping regime increases exponentially with the increase of the concentration of localization sites and may reach a few percent at the crossover from the hopping to the diffusion conductivity regime.

Figure 1

Illustration of CISP in the 2D hopping regime: electrons rarely hop between localization sites (green areas). In the presence of electric current $j_x$, the quantum interference between the direct and indirect hopping paths, shown respectively by magenta and blue arrows, leads to spin polarization $S_y$.

Figure 2

Microscopic mechanism of CISP. (a) Electric current flow leads to the drift spin current ${\mathcal{J}}_{\mathrm{dr}}$ due to the SHE. In the strongly inhomogeneous system under study, it is partially compensated by the diffusion spin current ${\mathcal{J}}_{\mathrm{diff}}$. (b) Spin current $\mathcal{J}$, accompanied by the spin precession in spin-orbit field ${\mathbf{\Omega }}_{SO}$, results in electron spin polarization $S_y$ due to the KKM effect.

Figure 3

Mechanism of SGE. (a) Spin polarization leads to the drift spin current ${\mathcal{J}}_{\mathrm{dr}}$ due to the inverse KKM effect. It is partially compensated by the diffusion spin current, ${\mathcal{J}}_{\mathrm{diff}}$. (b) Spin current $\mathcal{J}$ results in the electric current $j_x$ due to ISHE.

Figure 4

Drift, diffusion, and total spin currents (in arbitrary units) as functions of hyperfine-induced spin relaxation time for $n_s{a}_b^2=0.01$. The black curve shows the function $f(n_s,{\tau }_s)$ in absolute units.