Analytic modeling of switching time dynamics of monodomain ferromagnets with biaxial energy landscape

Assuming the macrospin model, we develop analytic models to describe the magnetization dynamics of an in-plane-anisotropy ferromagnet driven by spin-transfer-torque with spin polarization collinear to the easy axis orientation. Thus far, the physics of in-plane magnets has been analyzed using numerical solution of the Landau Lifshitz Gilbert (LLG) equation, while analytic expressions of switching time probability, which are needed for memory design and optimization, are lacking. In the limit of small torque, low damping and zero temperature, we construct an average energy flow equation to describe the dynamics of the in-plane magnet. We approximate the elliptic integrals in the flow equation with rational functions and obtain the switching time of the magnetization as a function of energy landscape, material parameters, and input spin current. We also evaluate analytical expressions for switching time probability and cumulative distribution functions assuming a Boltzmann equilibrium distribution of magnetization in the initial energy basin. Good agreement between the model and numerically evaluated results based on Monte Carlo simulations of the LLG equation is demonstrated.