Temperature rise due to Joule heating in a spin transfer torque nano-pillar structure

Considering that the spin-transfer-torque-induced magnetization dynamics in a nano-pillar structure usually requires a large current density of $10^{11}$ A/m$^2$, it is desired to have an accurate estimation of the temperature rise caused by the current-induced Joule heating. We investigate the current- induced heating effect in the nano-pillar by analytical and numerical methods. We employ the Green's function method to obtain analytic solution of the heat conduction equation. With proper approximations, we derive a simple analytic relation that expresses the temperature in term of the current density, the geometry of the nano-pillar, and material properties. The validity of the analytic expression is confirmed by the comparis on with commercial finite element method software.