The Non-Dissipative Spin-Hall Conductivity and The Identification of the Conserved Current

Original · EN

The two dimensional Rashba Hamiltonian is investigated using the momentum representation.One finds that the SU(2) transformation which diagonalizes the Hamiltonian gives rise to non commuting Cartesian coordinates for K=0 and zero otherwise.This result corresponds to the Aharonov -Bohm phase in the momentum space. The Spin -Hall conductivity is given by e/4π which disagree with the result e/8π given in the literature.Using Stokes theorem we find that the Spin -Hall current is carried equally by the up and down electrons on the Fermi surface. We identify the Magnetic current and find that for an electric field with a finite Fourier component in the momentum space a non zero Spin -Hall current is obtained.For the electric field which is constant in space, the orbital magnetic current cancels the spin current. In order to measure the Spin-Hall conductance we propose to apply a magnetic field gradient ΔH₂ in the i=2 direction and to measure a Charge- Hall current e/h(g/2)μBΔH₂ in the i=1 direction.