Thermodynamic $\mu T$ model of ultrafast magnetization dynamics

Exciting a ferromagnetic sample with an ultrashort laser pulse leads to a quenching of the magnetization on a subpicosecond time scale. On the basis of the equilibration of intensive thermodynamic variables, we establish a powerful model to describe the demagnetization process. We demonstrate that the magnetization dynamics is mainly driven by the equilibration of chemical potentials. The minimum of magnetization is revealed as a transient electron equilibrium state. Our method identifies the slowing down of ultrafast magnetization dynamics by a critical region within a magnetic phase diagram.

Figure 1

Typical results of the $\mu T$ model, transient magnetization (upper panel), chemical potentials (central panel), and temperatures (lower panel). The blue curves correspond to a low fluence $F_0=2.5\mathrm{mJ}/{\mathrm{cm}}^2$, whereas the red curves are calculated after excitation with twice of that fluence, $2\times F_0$. In the upper panel, experimental results [21] are shown for comparison. The vertical lines indicate the respective time where the magnetization dynamics suffer a minimum. Characteristic points (i) to (v), as marked for the blue solid demagnetization curve, are analyzed in the text.

Figure 2

Magnetization dynamics for a rectangular laser pulse with the duration of $50\mathrm{fs}$. Two fluences $F=1\times F_0$ and $F=2\times F_0$ are applied. For a direct comparison, the approximation $m\left(t\right)\approx m_0+\frac{1}{2}\frac{d^{2}m}{d{t}^2}{t}^2$ is depicted in the diagram.

Figure 3

Phase diagram of ultrafast magnetization dynamics. The black curve is the equilibrium magnetization cuve $m\left(T\right)$. The gray curves result from the $\mu T$ model for the fluences $F/F_0=1.0,1.2,...,2.4,2.8,3.2,3.6$ with $F_0=2.5\mathrm{mJ}/{\mathrm{cm}}^2$. The symbols mark several times at $t=0,0.3,0.5,1,2,5,10,25\mathrm{ps}$. The background color labels the relaxation time towards the equilibrium magnetization.