By combining the time-dependent nonequilibrium Green function formalism (NEGF) for the evolution of the electronic density matrix, implemented [1] using an efficient algorithm which scales linearly in the number of time steps, with the classical Landau-Lifshitz-Gilbert equation (LLG) describing the motion of magnetic moments in ferromagnetic materials, we study the domain wall (DW) dynamics initiated by injection of ultrashort dc current pulses. Unlike the usually employed steady-state NEGF+LLG formalism for problems where dc charge current is injected, our time-dependent NEGF+LLG makes it possible to study injection of current pulses, as well as pumping of spin current into the electrodes due to the DW motion. We also demonstrate the limits of steady-state NEGF+LLG approach to study DW motion when dc current is injected, which naively assumes that electronic Hamiltonian commutes at different times. Thus, our approach opens new avenues for theoretical and computational optimization of the memory technologies [2] based on DW dynamics.

[1] B. S. Popescu and A. Croy, New J. Phys. 18, 093044 (2016).

[2] S. Parkin and S.-H. Yang, Nat. Nanotech. 10, 195 (2015).