Instabilities of a spin-valve system with perpendicular polarizer and in-plane bias field

Under a perpendicular spin-transfer-torque (STT) polarizer and in-plane (IP) bias magnetic field, three-dimensional instability thresholds of the critical strength of STT and the field are derived from an effective one-dimensional free energy. The modified astroids derived from the STT and IP fields are quite different from the classical Stoner-Wohlfarth astroid. We find that the STT breaks the symmetry of the astroid seriously when the orientation of bias field is along the easy and hard axes. In particular, the modified astroid not only separates the region with two stable states from the region with only one stable state, but also delimits the region with a dynamical stable state (no stable equilibrium state) when the amplitude of the STT is larger than a critical value required to switch the magnetization at zero-bias field. Finally, the nucleation field, coercivity, and precessional critical field for uniaxial anisotropy are rigorously determined including the effect of the STT, and the hysteresis loops for various orientations of a bias field are computed and discussed.