Stability Analysis of Spin-Torque Nano-Oscillator in the Rotating Frame

Spin-torque nano-oscillators (STNOs) have become one of the emerging and novel microwave devices with the high performance and tunability of GHz range frequency. The nanopillar structure with an out-of-plane (OP) spin polarizer and an in-plane (IP) magnetized free layer (FL) has been considered as a good candidate for the STNOs. Using the local rotational coordinate transformation, a nonstationary process describing magnetization dynamics in the laboratory frame is therefore transformed into a stationary one in the rotating frame. In this way, the state phase diagram of this type of STNOs is well established as a function of an applied current and external field, which is also evidenced by the macrospin simulations. Also, we show that the frequency tunability of the STNOs through electrical current can be well elevated by applying a static magnetic field anti-parallel to the spin-polarizer vector.