Spin-Hall effects: From the two-channel model to Dyakonov-Perel equations

The Dyakonov - Perel equations that describe spin-Hall and related effects are derived in the framework of Meso-scopic Non-Equilibrium Thermodynamic (MNET) theory, with the use of the Onsager reciprocity relations. It is shown that the two channel model is able to describe the problem without loss of generality, providing that only the density of electric charges is spin-dependent (no conductivity asymmetry and no anisotropic conductivity). Furthermore, it is shown that the spin-dependent Hall voltage measured corresponds to zero pure spin-current. In the framework of the two spin-channel model, an ensemble of electric carriers of charge e is composed of two sub-populations: the charge carriers of spin up ↑ and the charge carriers of spin down ↓.^1-4 The index ⇕ that differentiates the two populations defines the internal degrees of freedom, with respect to a quantification axis fixed along the unit vector m虠. We study how this system can be put out-of-equilibrium in a stationary regime by applying an electric potential on a branch of the device, and measuring the Hall-voltage created on a transverse electrode in the Hall configuration. A first non-equilibrium regime of moving electric charges is due to the gradient of the electrochemical potential μvarr; imposed throughout the device. The corresponding force ∇虠μ⇕ generates an electric current J虠⇕^e for each spin population ⇕ (i.e. each spin-channel) at each point {x, y, z} of the material (under the assumption of local equilibrium^5-7). The conductivity tensor σ⇕ is then introduced, that verifies the Ohm's law: J虠^e⇕ = -σμμ⇕/e.