Spintronics Meets Nonadiabatic Molecular Dynamics: Geometric Spin Torque and Damping on Dynamical Classical Magnetic...

We analyze a quantum-classical hybrid system of steadily precessing around the fixed axis slow classical localized magnetic moments (LMMs), forming a head-to-head domain wall, surrounded by fast electrons driven out of equilibrium by LMMs and residing within a metallic wire whose connection to macroscopic reservoirs makes electronic quantum system an open one. The model captures the essence of dynamical noncollinear magnetic textures encountered in spintronics, while making it possible to obtain the exact time-dependent nonequilibrium density matrix of electronic systems and split it into four contributions. The Fermi surface contribution generates dissipative (or dampinglike in spintronics terminology) spin torque on LMMs, as the counterpart of electronic friction in nonadiabatic molecular dynamics (MD). Among two Fermi sea contributions, one generates geometric torque dominating in the adiabatic regime, which remains as the only nonzero contribution in a closed system with disconnected reservoirs. Locally geometric torque can have nondissipative (or fieldlike in spintronics terminology) component, acting as the counterpart of geometric magnetism force in nonadiabatic MD, as well as a much smaller dampinglike component acting as "geometric friction." Such current-independent geometric torque is absent from widely used micromagnetics or atomistic spin dynamics modeling of magnetization dynamics based on the Landau-Lifshitz-Gilbert equation, while previous analyses of how to include our Fermi-surface dampinglike torque have severely underestimated its total magnitude.