  We propose a theoretical model of magnetization switching in a ferromagnetic
multilayer by both electric current and microwaves. The electric current gives
a spin transfer torque on the magnetization, while the microwaves induce a
precession of the magnetization around the initial state. Based on numerical
simulation of the Landau-Lifshitz-Gilbert (LLG) equation, it is found that the
switching current is significantly reduced compared with the switching caused
solely by the spin transfer torque when the microwave frequency is in a certain
range. We develop a theory of switching from the LLG equation averaged over a
constant energy curve. It was found that the switching current should be
classified into four regions, depending on the values of the microwave
frequency. Based on the analysis, we derive an analytical formula of the
optimized frequency minimizing the switching current, which is smaller than the
ferromagnetic resonance frequency. We also derive an analytical formula of the
minimized switching current. Both the optimized frequency and the minimized
switching current decrease with increasing the amplitude of the microwave
field. The results will be useful to achieve high thermal stability and low
switching current in spin torque systems simultaneously.
