Spin-transfer and spin-orbit torques in the Landau-Lifshitz-Gilbert equation

Abstract

Dynamic simulations of spin-transfer and spin-orbit torques are increasingly important for a wide range of spintronic devices including magnetic random access memory, spin-torque nano-oscillators and electrical switching of antiferromagnets. Here we present a computationally efficient method for the implementation of spin-transfer and spin-orbit torques within the Landau-Lifshitz-Gilbert equation used in micromagnetic and atomistic simulations. We consolidate and simplify the varying terminology of different kinds of torques into a physical action and physical origin that clearly shows the common action of spin torques while separating their different physical origins. Our formalism introduces the spin torque as an effective magnetic field, greatly simplifying the numerical implementation and aiding the interpretation of results. The strength of the effective spin torque field unifies the action of the spin torque and subsumes the details of experimental effects such as interface resistance and spin Hall angle into a simple transferable number between numerical simulations. We present a series of numerical tests demonstrating the mechanics of generalised spin torques in a range of spintronic devices. This revised approach to modelling spin-torque effects in numerical simulations enables faster simulations and a more direct way of interpreting the results, and thus it is also suitable to be used in direct comparisons with experimental measurements or in a modelling tool that takes experimental values as input.

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• Spinor formulation of the Landau-Lifshitz-Gilbert equation with geometric algebra
• Dynamic Landau-Lifshitz-Bloch-Slonczewski equations for spintronics
• Investigating spin and orbital effects via spin-torque ferromagnetic resonance