Kinetic theory of spin-polarized systems in electric and magnetic fields with spin-orbit coupling: I. Kinetic equation and anomalous Hall and spin-Hall effects

Abstract: The coupled kinetic equation for density and spin Wigner functions are derived including spin-orbit coupling, electric and magnetic field as well as selfconsistent Hartree meanfields suited for SU(2) transport. The interactions are assumed to be with scalar and magnetic impurities as well as scalar and spin-flip potentials among the particles. The spin-orbit interaction is used in a form suitable to solid state physics with Rashba or Dresselhaus coupling, graphene, extrinsic spin-orbit coupling as well as effective nuclear matter coupling. The deficiencies of the two-fluid model are worked out consisting in the appearance of an effective in-medium spin-precession. The stationary solution of all these systems shows a band splitting controlled by an effective medium-dependent Zeeman field. The selfconsistent precession direction is discussed and a cancellation of linear spin-orbit coupling at zero temperature is reported. The precession of spin around this effective direction caused by spin-orbit coupling leads to anomalous charge and spin currents in an electric field. Anomalous Hall conductivity is shown to consists of the known results obtained from Kubo formula or Berry phases and a new symmetric part interpreted as inverse Hall effect. Analogously the spin-Hall and inverse spin-Hall effect of spin currents are discussed which are present even without magnetic fields showing a spin accumulation triggered by currents. The analytical dynamical expressions for zero temperature are derived and discussed in dependence on the magnetic field and effective magnetizations. The anomalous Hall and spin-Hall effect changes sign at higher than a critical frequency dependent on the relaxation time.