# Relativistic theory of magnetic inertia in ultrafast spin dynamics

## TL;DR

This paper derives a rigorous relativistic theory of magnetic inertia in ultrafast spin dynamics, showing inertia as a higher-order spin-orbit effect relevant on sub-picosecond timescales, and relating it to magnetic susceptibility.

## Contribution

It provides a first-principles derivation of magnetic inertia within the Dirac-Kohn-Sham framework, highlighting its order of magnitude and relation to Gilbert damping.

## Key findings

- Inertia arises from 1/c^4 order spin-orbit coupling effects.
- Inertia affects ultrafast magnetization dynamics on sub-picosecond timescales.
- Inertia is related to the real part of magnetic susceptibility.

## Abstract

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 $1/c^4$ gives the intrinsic inertia of a pure system through the 2$^{\rm nd}$ 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, $\sim 1/c^4$, as compared to the Gilbert damping $\Gamma$ that is of order $1/c^2$. 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.