| 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 unconditionallyunique 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. |
