Ultra-fast magnetisation rates within the Landau-Lifshitz-Bloch model

The ultra-fast magnetisation relaxation rates during the laser-induced magnetisation process are analyzed in terms of the Landau-Lifshitz-Bloch (LLB) equation for different values of spin $S$. The LLB equation is equivalent in the limit $S \rightarrow \infty$ to the atomistic Landau-Lifshitz-Gilbert (LLG) Langevin dynamics and for $S=1/2$ to the M3TM model [B. Koopmans, {\em et al.} Nature Mat. \textbf{9} (2010) 259]. Within the LLB model the ultra-fast demagnetisation time ($\tau_{M}$) and the transverse damping ($\alpha_{\perp}$) are parameterized by the intrinsic coupling-to-the-bath parameter $\lambda$, defined by microscopic spin-flip rate. We show that for the phonon-mediated Elliott-Yafet mechanism, $\lambda$ is proportional to the ratio between the non-equilibrium phonon and electron temperatures. We investigate the influence of the finite spin number and the scattering rate parameter $\lambda$ on the magnetisation relaxation rates. The relation between the fs demagnetisation rate and the LLG damping, provided by the LLB theory, is checked basing on the available experimental data. A good agreement is obtained for Ni, Co and Gd favoring the idea that the same intrinsic scattering process is acting on the femtosecond and nanosecond timescale.