Magnetization dynamics in the inertial regime: Nutation predicted at short time scales

The dynamical equation for magnetization has been reconsidered by enlarging the phase space of the ferromagnetic degrees of freedom to the angular momentum. The generalized Landau-Lifshitz-Gilbert equation that includes inertial terms, and the corresponding Fokker-Planck equation, are then derived in the framework of mesoscopic nonequilibrium thermodynamics theory. A typical relaxation time $\tau$ is introduced describing the relaxation of the magnetization acceleration from the inertial regime toward the precession regime defined by a constant Larmor frequency. For time scales larger than $\tau$, the usual Gilbert equation is recovered. For time scales below $\tau$, nutation and related inertial effects are predicted. The inertial regime offers new opportunities for the implementation of ultrafast magnetization switching in magnetic devices.

Figure 1

Numerical solution of Eq. (12) with ${\tau }_1=2$ ps. (a) Trajectories at two different fields with $\left|\alpha \right|=0.05$ (dashed line) and curves deduced from the LLG equation (lower line). (b) Trajectory of the magnetization with changing suddenly the effective fields from $H=1$ MA/m to $H=3$ MA/m and $H=0$, with damping (continuous lines) and without (dotted lines). (c) Time derivative of the azimuth angle $\varphi$ plotted as a function of time for the trajectory of (b).