  The temperature increment due to the Joule heating in a nanopillar spin
transfer torque system is investigated. We obtain a time dependent analytic
solution of the heat conduction equation in nanopillar geometry by using the
Green's function method after some simplifications of the problem. While Holm's
equation is applicable only to steady states in metallic systems, our solution
describes the time dependence and is also applicable to a nanopillar-shaped
magnetic tunneling junction with an insulator barrier layer. The validity of
the analytic solution is confirmed by numerical finite element method
simulations and by the comparison with Holm's equation.
