FIG. 1 : Schematic geometry of the spin Hall effect; a longitudinal current jx 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).
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.
FIG. 2 : Left - Static spin Hall conductivity as a function of εF , for disorder strengths ô-1 = 0, 0.2. (Inset) Integrand used to compute σsH(0); the error bars represent the amplitude of σo(ω)/ω fluctuations near the zero frequency. Right – Normalized histogram of the static spin conductivity for ô-1 = 0.2, showing a Gaussian distribution with mean value ⟨ σsH (ω)⟩ = 0.98 [−1/8π] and standard deviation ∆σsH (0) = 0.1 (solid line).
1) T. L. van den Berg, L. Raymond and A. Verga. “Dynamical spin Hall conductivity in a magnetic disordered system.” Phys. Rev. B, 84 : 245210, (2011).
2) T. L. van den Berg, L. Raymond, and A. Verga. “Enhanced spin Hall effect in strong magnetic disorder”, Phys. Rev. B, 86, 245420, (2012).