3.2 Gilbert Equation of Motion

The minimization of Eq. (3.1) can find an equilibrium magnetization distribution. However, the energy landscape of micromagnetic systems is usually very complicated and contains many local maxima, minima, and saddle points. Therefore, the choice of the initial magnetization distribution has a strong influence on the result. A more physical approach to the problem and a more realistic approach of the system to its equilibrium in a local minimum is provided by a dynamic description of the path through the energy landscape.

The motion of a magnetic moment in a magnetic field is mainly governed by its Larmor precession around the local magnetic field. The damping of the precession causes the relaxation to equilibrium. There are many processes which contribute to the damping in a magnetic solid like magnon-magnon and magnon-phonon interactions, interactions between localized and itinerant electrons and eddy currents, for example [35,36,37].

The Gilbert equation [38,39] describes the precession and combines all damping effects in a phenomenological damping term with a single damping constant $\alpha$

$$\frac{d \boldsymbol{J}}{dt}=-\gamma \boldsymbol{J} \times \... ...hrm{s}}\boldsymbol{J} \times\frac{d \boldsymbol{J}}{dt} \quad,$$ (3.7)

where $\gamma=2.210173 \times 10^{5} \frac{\mathrm{m}}{\mathrm{As}}$ is the gyromagnetic ratio.

This formulation is equivalent to the Landau-Lifshitz-Gilbert (LLG) equation

$$\frac{d \boldsymbol{J}}{dt}=-\gamma' \boldsymbol{J} \times ... ...dsymbol{J} \times(\boldsymbol{J} \times \boldsymbol{H}) \quad,$$ (3.8)

with

$$\gamma'=\frac{\gamma}{1+\alpha^2} \quad.$$ (3.9)

The intrinsic timescale is determined by the Larmor frequency $\omega=\gamma \boldsymbol{H}_\mathrm{eff}$, which is usually in the order of GHz (cf. Sec. 9.4). Thus, the precession time is smaller than a nanosecond, which requires time steps in the order of picoseconds or even less. This limits the maximum simulated time to about 100 ns.