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 S_z accumulation about the current axis in the absence of external magnetic field. Here we employ the real space Landauer-Keldysh formalism [B. K. Nikolić et al., 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 t_SO^R, the breaking process is clearly seen. The influence of the Rashba interaction on the magnetization of the 2DEG is studied herein. For coexisted Rashba t_SO^R and Dresselhaus t_SO^D spin-orbit couplings in the absence of B, interchanging t_SO^R and t_SO^D reverses the entire accumulation pattern.