Broken spin-Hall accumulation symmetry by magnetic field and coexisted Rashba and Dresselhaus interactions

The spin-Hall effect in the two-dimensional electron gas (2DEG) generates symmetric out-of-plane spin Sz accumulation about the current axis in the absence of external magnetic field. Here we employ the real space Landauer-Keldysh formalism [B. K. Nikolić, Phys. Rev. Lett. 95, 046601 (2005); Phys. Rev. B 73, 075303 (2006)] by considering a four-terminal setup to investigate the circumstances in which this symmetry is broken. For the absence of Dresselhaus interaction, starting from the applied out-of-plane B corresponding to Zeeman splitting energy Δ=0-0.5 times the Rashba hopping energy tSOR, the breaking process is clearly seen. The influence of the Rashba interaction on the magnetization of the 2DEG is studied herein. For coexisted Rashba tSOR and Dresselhaus tSOD spin-orbit couplings in the absence of B, interchanging tSOR and tSOD reverses the entire accumulation pattern.