Voltage dependence of Landau-Lifshitz-Gilbert damping of a spin in a   current driven tunnel junction
  We present a theory of Landau-Lifshitz-Gilbert damping $\alpha$ for a localized spin ${\vec S}$ in the junction coupled to the conduction electrons in both leads under an applied volatege $V$. We find the voltage dependence of the damping term reflecting the energy dependence of the density of states. We find the effect is linear in the voltage and cotrolled by particle-hole asymmetry of the leads. 
Hosho Katsura, Alexander V. Balatsky, Zohar Nussinov, Naoto Nagaosa