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

D. Schmeltzer
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Abstract

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 $\frac{e}{4π}$ which disagree with the result $\frac{e}{8π}$ 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 $ΔH_{2}$ in the $i=2$ direction and to measure a Charge- Hall current $\frac{e}{h}(\frac{g}{2})μ_{B}ΔH_{2}$ in the $i=1$ direction.

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D. Schmeltzer. 2005-04-01. The Non-Dissipative Spin-Hall Conductivity and The Identification of the Conserved Current. https://arxiv.org/abs/cond-mat/0504035

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