Year: 2023
Author: Jin Lan, Panchi Li, Rui Du, Lei Yang, Jingrun Chen, Jin Lan, Rui Du, Jingrun Chen
Numerical Mathematics: Theory, Methods and Applications, Vol. 16 (2023), Iss. 1 : pp. 182–203
Abstract
Electron spins in magnetic materials have preferred orientations collectively and generate the macroscopic magnetization. Its dynamics spans over a wide range of timescales from femtosecond to picosecond, and then to nanosecond. The Landau-Lifshitz-Gilbert (LLG) equation has been widely used in micromagnetics simulations over decades. Recent theoretical and experimental advances have shown that the inertia of magnetization emerges at sub-picosecond timescales and contributes significantly to the ultrafast magnetization dynamics, which cannot be captured intrinsically by the LLG equation. Therefore, as a generalization, the inertial LLG (iLLG) equation is proposed to model the ultrafast magnetization dynamics. Mathematically, the LLG equation is a nonlinear system of parabolic type with (possible) degeneracy. However, the iLLG equation is a nonlinear system of mixed hyperbolic-parabolic type with degeneracy, and exhibits more complicated structures. It behaves as a hyperbolic system at sub-picosecond timescales, while behaves as a parabolic system at larger timescales spanning from picosecond to nanosecond. Such hybrid behaviors impose additional difficulties on designing efficient numerical methods for the iLLG equation. In this work, we propose a second-order semi-implicit scheme to solve the iLLG equation. The second-order temporal derivative of magnetization is approximated by the standard centered difference scheme, and the first-order temporal derivative is approximated by the midpoint scheme involving three time steps. The nonlinear terms are treated semi-implicitly using one-sided interpolation with second-order accuracy. At each time step, the unconditionally unique solvability of the unsymmetric linear system is proved with detailed discussions on the condition number. Numerically, the second-order accuracy of the proposed method in both time and space is verified. At sub-picosecond timescales, the inertial effect of ferromagnetics is observed in micromagnetics simulations, in consistency with the hyperbolic property of the iLLG model; at nanosecond timescales, the results of the iLLG model are in nice agreements with those of the LLG model, in consistency with the parabolic feature of the iLLG model.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.OA-2022-0080
Numerical Mathematics: Theory, Methods and Applications, Vol. 16 (2023), Iss. 1 : pp. 182–203
Published online: 2023-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 22
Keywords: Inertial Landau-Lifshitz-Gilbert equation semi-implicit scheme second-order accuracy micromagnetics simulations.
Author Details
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