Boltzmann approach to the spin Hall effect revisited and electric field modified collisionintegrals

Abstract

The intrinsic contribution to the spin Hall effect in a two-dimensional degenerate electron gas(2DEG) with non-magnetic impurities is studied in a quantum Boltzmann approach. It isshown that if the steady state response is perturbative in the spin–orbit coupling parameterλ, then the precession term—vital for Dyakonov–Perel relaxation and the key to the spinHall effect in previous similar Boltzmann studies—must be left out to first order inspin–orbit coupling. In such a case one would have that to lowest order in the parameterselectric field, spin–orbit coupling, impurity strength and impurity concentration there is nointrinsic contribution to the spin Hall effect, not only for a Rashba coupling butalso for a general spin–orbit coupling. To cover all possible lowest order termswe also consider electric field induced corrections to the collision integral in theKeldysh formalism. However, these corrections turn out to be of second order inλ. Forcomparison we derive some familiar results in the case when the response is not assumed to be perturbativein λ. We also include a detailed discussion of why a relaxation time approximation of thecollision integral fails. Finally we make a comment on pseudospin currents in bilayergraphene.