Current Flow in Uniform Ferromagnets: Bulk and Surface Heating Rates
  With spintronics applications in mind, we use irreversible thermodynamics to derive the rates of entropy production and heating near an interface when heat current, electric current, and spin current cross it. Associated with these currents are apparent discontinuities in temperature (\Delta T), electrochemical potential (\Delta \tilde {\mu}), and spin-dependent "magnetoelectrochemical potential" (\Delta \bar \mu_{\uparrow,\downarrow}). This work applies to magnetic semiconductors and insulators as well as metals, due to the inclusion of the chemical potential \mu, which usually is neglected in works on interfacial thermodynamic transport. We also discuss the (non-obvious) distinction between entropy production and heat production. Heat current and electric current are conserved, but spin current is not, so it necessitates a somewhat different treatment. At low temperatures or for large differences in material properties, the surface heating rate dominates the bulk heating rate near the surface. We also consider the case, noted by Rashba, where bulk spin currents occur in equilibrium. Although a surface spin current (in A/m^{2}) should yield about the same rate of heating as an equal surface electric current, production of such a spin current requires a relatively large "magnetization potential" difference across the interface. 
Matthew R. Sears, Wayne M. Saslow
cond-mat.mtrl-sci