Ultrafast Demagnetization Dynamics Due to Electron-Electron Scattering and Its Relation to Momentum Relaxation in Ferromagnets

We analyze theoretically the demagnetization dynamics in a ferromagnetic model system due to the interplay of spin-orbit coupling and electron-electron Coulomb scattering. We compute the $k$-resolved electronic reduced spin-density matrix including precessional dynamics around internal spin-orbit and exchange fields as well as the electron-electron Coulomb scattering for densities and spin coherences. Based on a comparison with numerical solutions of the full Boltzmann scattering integrals, we establish that the $k$-resolved reduced spin-density matrix dynamics are well described using a simpler generalized relaxation-time ansatz for the reduced spin-density matrix. This ansatz allows one to relate the complicated scattering dynamics underlying the demagnetization dynamics to a physically meaningful momentum relaxation time $\tau$. Our approach reproduces the behaviors of the demagnetization time ${\tau }_m\propto 1\mathrm{/}\tau$ and ${\tau }_m\propto \tau$ for the limits of short and long $\tau$, respectively, and is also valid for the intermediate regime. The ansatz thus provides a tool to include the correct demagnetization behavior in approaches that treat other contributions to the magnetization dynamics such as transport or magnon/phonon dynamics.