Enhanced spin Hall effect in strong magnetic disorder

We consider a two-dimensional electron gas in an inversion asymmetric layer and in the presence of spatially distributed magnetic impurities. We investigate the relationship between the geometrical properties of the wave function and the system's spin-dependent transport properties. A localization transition, arising when disorder is increased, is exhibited by the appearance of a fractal state with finite inverse participation ratio. Below the transition, interference effects modify the carrier's diffusion, as revealed by the dependence on the scattering time of the power law exponents characterizing the spreading of a wave packet. Above the transition, in the strong disorder regime, we find that the states are spin polarized and localized around the impurities. A significant enhancement of the spin current develops in this regime.

Figure 1

Time evolution of the wave function; we use a logarithmic scale $\log |{\psi }_{\uparrow }(\mathit{x},t)|$ to visualize the small amplitude region. The lattice size is ${2048}^2$. The exchange constant is $J_s=0.1$ (top row) and $J_s=1.0$ (bottom row); times are, from left to right, 160, 288, 416, 544, 672, 800. The concentration of magnetic impurities is $c=0.1$, and the Rashba constant $\lambda =0.2$. The color scale is fixed, and spans the interval $({10}^{-4},5\times {10}^{-2})$.

Figure 2

Width of the wave function showing a time power law with exponents depending on the disorder strength from quasi-free-dispersion ($\beta =0.95$, for $J_s=0.1$) to diffusion ($\beta =0.51$ for $J_s=1.0$). Exchange constant $J_s=0.1$, 0.2, 0.4, 1.0, and 2.0 (from top to bottom).

Figure 3

Measure of $D_2$ as a function of time for different disorder strengths ($J_s$ in the legend).

Figure 4

Inverse participation ratio $P_2$ as a function of time for different disorder strengths ($J_s$ in the legend). When $J_s\gtrsim 1$, $P_2\left(t\right)$ remains finite for long times, indicating the existence of a localization state.

Figure 5

Spin up and down local density of states for $J_s=0.1$ (top row) and $J_s=1$ (bottom row). The contours (blue and red) are at the location of the impurities, where the $z$ component of their magnetic moment is $n_z=-0.5$ (blue) and $-0.5$ (red). In the neighborhood of the impurities, at weak disorder strength, ${\rho }_{\uparrow }\left(i\right)$ and ${\rho }_{\downarrow }\left(i\right)$ compensate to give a value close to the one of the total density of states $\rho$; at variance, for a stronger disorder, the spin of the quantum states is strongly correlated with the orientation of the impurity magnetic moment. The Fermi energy is around 2, and the bottom color bar gives the scale of $\rho$; only a fraction $32\times 32$, of the total lattice is shown.

Figure 6

The three Hikami boxes for the weal localization corrections to the spin conductivity. The spin and charge current vertex are renormalized.

Figure 7

Diagrammatic representation of the Cooperon equation.

Figure 8

Quantum corrections to the spin Hall conductivity for the first, second, and third (identical) Hikami boxes, and the total corrections. In units of $(-e/8\pi )(\Delta /{\epsilon }_F)$ the quantum corrections are a function of $x=\Delta \tau$ only.

Figure 9

Static spin Hall conductivity as a function of the Fermi energy for different values of the disorder strength. The inset show the variation of ${\sigma }_{sH}$ at fixed ${\epsilon }_F=2$; in the strong disorder regime it increases with the disorder strength.

Figure 10

Local and total density of states for (left) $J_s=1$, at the localization transition, and (right) $J_s=4$, in the presence of impurity bands. The (unnormalized) local density of states is represented for two arbitrary sites in the lattice, occupied by impurities.