Steady states of spin distributions in the presence of spin-orbit interactions

In the presence of spin-orbit interactions, the steady state established for spin distributions in an electric field is qualitatively different from the steady state for charge distributions. This is primarily because the steady state established for spin distributions involves spin precession due to spin-orbit coupling. We demonstrate in this work that the spin density matrix in an external electric field acquires two corrections with different dependencies on the characteristic momentum-scattering time. One part is associated with conserved spins, diverges in the clean limit, and is responsible for the establishment of a steady-state spin density in electric fields. Another part is associated with precessing spins, is finite in the clean limit, and is responsible for the establishment of spin currents in electric fields. Scattering between these distributions has important consequences for spin dynamics and spin-related effects, in general, and explains some recent puzzling observations, which are captured by our unified theory.

Figure 1

Effective field ${\mathbf{\Omega }}_{\mathit{k}}$ at the Fermi energy in the Rashba model (Ref. 57) (a) without (Ref. 63) $(E=0)$ and (b) with an external electric field $(E>0)$.