Effect of in-plane magnetic field on the spin Hall effect in a Rashba-Dresselhaus system

In a two-dimensional electron gas with Rashba and Dresselhaus spin-orbit\ncouplings, there are two spin-split energy surfaces connected with a degenerate\npoint. Both the energy surfaces and the topology of the Fermi surfaces can be\nvaried by an in-plane magnetic field. We find that, if the chemical potential\nfalls between the bottom of the upper band and the degenerate point, then\nsimply by changing the direction of the magnetic field, the magnitude of the\nspin Hall conductivity can be varied by about 100 percent. Once the chemical\npotential is above the degenerate point, the spin Hall conductivity becomes the\nconstant $e/8\\pi$, independent of the magnitude and direction of the magnetic\nfield. In addition, we find that the in-plane magnetic field exerts no\ninfluence on the charge Hall conductivity.\n