Joule heating in spin Hall geometry
  The theoretical formula for the entropy production rate in the presence of spin current is derived using the spin-dependent transport equation and thermodynamics. This theory is applicable regardless of the source of the spin current, for example, an electric field, a temperature gradient, or the Hall effect. It reproduces the result in a previous work on the dissipation formula when the relaxation time approximation is applied to the spin relaxation rate. By using the developed theory, it is found that the dissipation in spin Hall geometry has a contribution proportional to the square of the spin Hall angle. 
Tomohiro Taniguchi
cond-mat.mes-hall cond-mat.mtrl-sci cond-mat.stat-mech