Determining the spin Hall conductance via charge transport

We propose a scheme where measurements of charge current and its noise can be used to determine spin-dependent transport properties. The scheme does not require the detection of individual spin components of the current and no magnetic or optical elements are needed. To illustrate the scheme we use it to determine the spin Hall conductance in a four-terminal setup, and also the spin polarization of a current injected from a ferromagnet into a semiconductor. Starting from the scattering formalism we express the spin current, and the relevant spin-dependent transport quantities, in terms of spin-dependent transmission coefficients. We derive expressions relating these coefficients to the charge current and noise. Measurements of current and noise thus allows the determination of the spin dependent transport quantities via these expressions.

Figure 1

Spin Hall effect in a four-terminal setup where a charge current flows from leads 1 to 3, resulting in spin currents in leads 2 and 4. No charge current flows in lead 4, see text for details. Only the hatched central region has spin-orbit (SO) coupling and is connected to leads 2 and 4 by single mode QPCs. The chemical potentials are ${\mu }_i={\mu }_0+e{V}_i$.

Figure 2

A schematic setup of spin injection from a magnetic (FM) lead $L$ into a semiconducting central region lead $C$ which is connected to lead $R$ via the QPC. The interface between $L$ and $C$ can be clean or disordered.