Dynamic Landau-Lifshitz-Bloch-Slonczewski equations for spintronics

The atomistic Landau-Lifshitz-Gilbert equation is challenged when modeling spintronic devices where Joule heating is significant, due to its core assumption of a constant magnetization magnitude. Based on a statistical framework that treats the magnetization magnitude as a dynamic variable coupled to a thermal bath, we derive a dynamic Landau-Lifshitz-Bloch-Slonczewski set of equations for torques, that captures the transient, heating-induced demagnetization that occurs during high-current operation. Integrating these dynamic equations and comparing them to their stochastic equivalents reveals that both the energy landscape and switching dynamics in high-anisotropy systems are similarly modified. This approach yields accurate and accelerated predictions of critical currents and switching times.