We examine the Dyakonov and Perel theory of the Spin Hall effect (SHE) from the viewpoint of irreversible thermodynamics, which is more constraining than the symmetry arguments of pure phenomenology. As thermodynamic driving forces we include the thermal gradient, the gradient of the electrochemical potential (rather than the potential gradient and density gradient separately), and the “internal” magnetic field that is thermodynamically conjugate to the magnetization. In turn, we obtain the form of bulk transport coefficients relating the fluxes to the thermodynamic forces. Relative to Dyakonov and Perel, in addition to the new terms due to thermal gradients, the Onsager relations require three new (nonlinear) terms in the current density, and minor revisions in the current density and spin current density. For a longitudinal current along a strip, the center-to-edge transverse voltage difference, due both to the $-\beta \vec{P}\times \vec{E}$ term of of the number current density $\vec{q}$ and to one of the new current density terms, is calculated. An ac capacitative probe likely would not significantly disturb this effect. An appendix explicitly relates the anomalous Hall Effect to the term in the (vector) number flux that is Onsager related to the SHE term in the (tensor) spin flux.

• Received 29 May 2014
• Revised 28 August 2014

DOI:https://doi.org/10.1103/PhysRevB.91.014401