Relativistic theory of magnetic inertia in ultrafast spin dynamics

The influence of possible magnetic inertia effects has recently drawn attention in ultrafast magnetization dynamics and switching. Here we derive rigorously a description of inertia in the Landau-Lifshitz-Gilbert equation on the basis of the Dirac-Kohn-Sham framework. Using the Foldy-Wouthuysen transformation up to the order of $1\mathrm{/}{c}^4$ gives the intrinsic inertia of a pure system through the 2${}^{\mathrm{n}\mathrm{d}}$ order time-derivative of magnetization in the dynamical equation of motion. Thus, the inertial damping $\mathcal{I}$ is a higher order spin-orbit coupling effect, $\sim 1\mathrm{/}{c}^4$, as compared to the Gilbert damping $\mathrm{\Gamma }$ that is of order $1\mathrm{/}{c}^2$. Inertia is therefore expected to play a role only on ultrashort timescales (sub-picoseconds). We also show that the Gilbert damping and inertial damping are related to one another through the imaginary and real parts of the magnetic susceptibility tensor respectively.