Generalized Gilbert equation including inertial damping: Derivation from an extended breathing Fermi surface model

In a recent paper [M.-C. Ciornei et al., Phys. Rev. B 83, 020410(R) (2011)], it has been shown by mesoscopic nonequilibrium thermodynamics that, for relatively fast magnetization dynamics, Gilbert's equation of motion for the magnetization has to be supplemented by an inertial damping term that contains the second derivative of the magnetization with respect to time. In the present Brief Report, it is shown that such an additional damping term can be derived and evaluated very naturally also within a slightly extended breathing Fermi surface model for magnetization damping.