🤖 AI Summary
This work addresses the limitations of conventional one-dimensional Fokker–Planck models in accurately predicting the write error rate (WER) of MRAM devices incorporating in-plane magnetic fields, field-like torques, or asymmetric energy barriers. The authors present the first two-dimensional Fokker–Planck solver formulated on the unit sphere using a finite volume method, supporting four discretization schemes: central differencing, Scharfetter–Gummel, upwind, and a novel hybrid adaptive blending strategy. Validation against million-trajectory stochastic Landau–Lifshitz–Gilbert (sLLG) simulations demonstrates that central differencing faithfully reproduces realistic WER under strong two-dimensional effects. Furthermore, the proposed hybrid adaptive scheme achieves both high accuracy and numerical stability across a wide range of Péclet numbers, significantly enhancing the fidelity and unbiasedness of switching dynamics predictions for next-generation magnetic memory devices.
📝 Abstract
The Fokker--Planck (FP) equation is essential for predicting write error rates (WER) in STT and SOT-MRAM devices, but traditional 1D projections fail when symmetry is broken by in-plane fields, field-like torques, or anisotropic barriers. We develop a 2D finite-volume (FVM) solver on the unit sphere and validate it against $10^6$-trajectory stochastic Landau--Lifshitz--Gilbert (sLLG) simulations. The solver supports four discretization schemes---central, Scharfetter--Gummel (SG), upwind, and hybrid adaptive blending---each with different Péclet-dependent accuracy and monotonicity properties. We demonstrate that central differencing recovers ground-truth WER for STT and SOT geometries where 2D effects dominate, and show that the choice of discretization scheme directly affects predicted WER. For magnetic simulations, we recommend hybrid adaptive blending as the optimal balance of accuracy and stability across variable Péclet regimes. These results establish that customizable discretization is critical for accurate, unbiased predictions of switching dynamics in next-generation magnetic memory.