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

The conventional Landau-Lifshitz-Gilbert (LLG) equation is a widely used tool to describe the dynamics of local magnetic moments, viewed as classical vectors of fixed length, with their change assumed to take place simultaneously with the cause. Here we demonstrate that recently developed [M. D. Petrovic et al., Phys. Rev. Appl. 10, 054038 (2018)] self-consistent coupling of the LLG equation to a time-dependent quantum-mechanical description of electrons-where nonequilibrium spin density from time-dependent nonequilibrium Green function (TDNEGF) calculations is inserted within a torque term into the LLG equation while local magnetic moments evolved by the LLG equation introduce time-dependent potential in the quantum Hamiltonian of electrons-microscopically generates time-retarded damping in the LLG equation described by a memory kernel that is also spatially dependent. For sufficiently slow dynamics of local magnetic moments on the memory time scale, the kernel can be expanded into power series to extract the Gilbert damping (proportional to the first time derivative of magnetization) and magnetic inertia (proportional to the second time derivative of magnetization) terms whose parameters, however, are time-dependent in contrast to time-independent parameters used in the conventional LLG equation. We use examples of single or multiple local magnetic moments precessing in an external magnetic field, as well as field-driven motion of a magnetic domain wall (DW), to quantify the difference in their time evolution computed from the conventional LLG equation versus the TDNEGF+LLG quantum-classical hybrid approach. The faster DW motion predicted by the TDNEGF+LLG approach reveals that important quantum effects, stemming essentially from a finite amount of time that it takes for a conduction electron spin to react to the motion of classical local magnetic moments, are missing from conventional classical micromagnetics simulations. We also demonstrate a large discrepancy between the TDNEGF+LLG-computed numerically exact and, therefore, nonperturbative result for charge current pumped by a moving DW and the same quantity computed by a perturbative spin motive force formula combined with the conventional LLG equation.