Conserved spin Hall conductance in two dimensional electron gas in a perpendicular magnetic field

Using the microscopic theory of the conserved spin current [Phys. Rev. Lett. $\mathbf{\text{96}}$, 076604 (2006)], we investigate the spin Hall effect in the two dimensional electron gas system with a perpendicular magnetic field. The spin Hall conductance ${\sigma }_{\mu \nu }^s$ as a response to the electric field consists of two parts, i.e., the conventional part ${\sigma }_{\mu \nu }^{s0}$ and the spin torque dipole correction ${\sigma }_{\mu \nu }^{s\tau }$. It is shown that the spin-orbit coupling competes with Zeeman splitting by introducing additional degeneracies between different Landau levels at certain values of magnetic field. These degeneracies, if occurring at the Fermi level, turn to give rise to resonances in both ${\sigma }_{\mu \nu }^{s0}$ and $\sigma_{\mu\nu }^{s\tau}$ in spin Hall conductance. Remarkably, both of these two components have the same sign in the wide range of variation in the magnetic field, which result in an overall enhancement of the total spin Hall current. In particular, the magnitude of ${\sigma }_{\mu \nu }^{s\tau }$ is much larger than that of ${\sigma }_{\mu \nu }^{s0}$ around the resonance.