Time-retarded damping and magnetic inertia in the Landau-Lifshitz-Gilbert equation self-consistently coupled to electronic time-dependent nonequilibrium Green functions

The conventional LLG equation is a widely used tool to describe dynamics of local magnetic moments, viewed as classical vectors. Here we demonstrate that recently developed [M. D. Petrovic et al., arXiv:1802.05682] self-consistent coupling of the LLG equation to time-dependent quantum electrons using time-dependent nonequilibrium Green function (TDNEGF) microscopically generates time-retarded damping in the LLG equation described by a memory kernel. For sufficiently slow dynamics, the memory kernel can be expanded to extract a time dependent Gilbert damping and magnetic inertia terms. Using examples of precessing single or multiple magnetic moments, as well as field-driven motion of a magnetic domain wall, we quantify the difference in their time evolution computed from conventional LLG equation vs. our TDNEGF+LLG approach. The faster DW motion predicted by TDNEGF+LLG approach reveals that important quantum effects, are missing from conventional classical micromagnetics simulations. We also demonstrate large discrepancy between TDNEGF+LLG-computed nonperturbative result for charge current pumped by a moving DW and the same quantity computed by perturbative spin motive force formula combined with the conventional LLG equation.