Dynamical spin Hall conductivity in a magnetic disordered system

We investigate the intrinsic spin Hall effect in a quantum well semiconductor doped with magnetic impurities, as a means to manipulate the carriers’ spin. Using a simple Hamiltonian with Rashba spin-orbit coupling and exchange interactions, we analytically compute the spin Hall conductivity. It is demonstrated that using the appropriate order of limits, one recovers the intrinsic universal value. Numerical computations on a tight-binding model, in the weak disorder regime, confirm that the spin Hall effect is preserved in the presence of magnetic impurities. The optical spin conductivity shows large sample to sample fluctuations in the low-frequency region. As a consequence, for weak disorder, the static spin conductivity is found to follow a wide Gaussian distribution with its mean value near the intrinsic clean value.

Figure 1

Schematic geometry of the spin Hall effect; a longitudinal current $j_x$ driven by an external electric field induces a transversal spin current that results in a polarization of the carrier spins following the sign of their momentum (spheres with arrows).

Figure 2

The conductivity is calculated in the noncrossing approximation, as a zero-order loop with disorder averaged Green's functions, plus the ladder diagrams. The ladder diagrams are calculated by renormalization of the spin current vertex.

Figure 3

Real and imaginary parts of the intrinsic spin Hall conductivity for a pure system (13) and (14). The four branches $\pm {\omega }_{\pm }$ are indicated by vertical dashed lines, their separation is $|{\omega }_{+}+{\omega }_{-}|=4{\lambda }^2$ (arrows).

Figure 4

Dynamical conductivity ${\sigma }_{sH}$ as a function of $\omega$, for $\tau$ varying between 40 and 190 (light to dark lines). (Inset) As $\omega \to 0$ the real part tends to $1/8\pi$ for $\omega >1/\tau$, but for $\omega <1/\tau$ it increases up to $8/7\approx 1.14$.

Figure 5

Static spin Hall conductivity as a function of the disorder strength from Eq. (22), for two values of $\Delta$. The dot at ${\tau }^{-1}=0$ corresponds to the clean static limit.

Figure 6

Convergence of the conductivity for values of $\omega$ near zero, ranging from ${10}^{-3}$ to ${10}^{-9}$. The expression of $8\pi \text{Re}\left[\sigma \left(\omega \right)\right]$ for $\omega ={10}^{-n}$ converges to zero only for $\tau >{10}^{n+1}$. The horizontal line in the graph corresponds to the numerical factor $8/7$.

Figure 7

Density of states for $c{J}_s^2=0$ (clean system), and $c{J}_s^2=0.01,0.06$; $\lambda =0.4$. The spin-orbit interaction is responsible for the splitting of the chiral states (central peaks).

Figure 8

Optical spin conductivity as a function of the frequency, for different disorder strengths. For each ${\tau }^{-1}=c{J}_s^2=0.002,0.01,0.06$, three values of the concentration, $c=0.04,0.06,0.08$, were used; ${\epsilon }_F=2$, $\lambda =0.2$.

Figure 9

Optical spin conductivity as a function of frequency and ${\epsilon }_F$, for two different disorder strengths ${\tau }^{-1}=0.002,0.2$. As the disorder increases, the optical conductivity decreases and spreads out over $\omega$.

Figure 10

Static spin Hall conductivity as a function of ${\epsilon }_F$, for disorder strengths ${\tau }^{-1}=0,0.2$. (Inset) Integrand of Eq. (33), used to compute ${\sigma }_{sH}\left(0\right)$; the error bars represent the amplitude of ${\sigma }_o\left(\omega \right)/\omega$ fluctuations near the zero frequency.

Figure 11

Normalized histogram of the static spin conductivity for ${\tau }^{-1}=0.2$, showing a Gaussian distribution with mean value $\langle {\sigma }_{sH}\left(\omega \right)\rangle =0.98[-1/8\pi ]$ and standard deviation $\Delta {\sigma }_{sH}\left(0\right)=0.1$ (solid line).