Bridging atomistic spin dynamics methods and phenomenological models of single-pulse ultrafast switching in ferrimagnets

We bridge an essential knowledge gap on the understanding of all-optical ultrafast switching in ferrimagnets, namely, the connection between atomistic spin dynamics methods and macroscopic phenomenological models. All-optical switching of the magnetization occurs after the application of a single femtosecond laser pulse to specific ferrimagnetic compounds. This strong excitation puts the involved degrees of freedom, electrons, lattice, and spins out-of-equilibrium between each other. Atomistic spin models have quantitatively described all-optical switching in a wide range of experimental conditions, while having failed to provide a simple picture of the switching process. Phenomenological models are able to qualitatively describe the dynamics of the switching process. However, a unified theoretical framework is missing that describes the element-specific spin dynamics as atomistic spin models with the simplicity of phenomenology. Here, we bridge this gap and present an element-specific macrospin dynamical model which fully agrees with atomistic spin dynamics simulations and symmetry considerations of the phenomenological models.

Figure 1

Equilibrium magnetization of a GdFeCo alloy for Gd concentration, $x_{\mathrm{Gd}}=25\%$. Element-specific normalized equilibrium magnetization and net equilibrium magnetization, $M\left(T\right)=x_{\mathrm{Gd}}{\mu }_{\mathrm{Gd}}{m}_{\mathrm{Gd}}-x_{\mathrm{Fe}}{\mu }_{\mathrm{Fe}}{m}_{\mathrm{Fe}}$, where ${\mu }_{\mathrm{Gd}\left(\mathrm{Fe}\right)}$ is the atomic magnetic moment of Gd(Fe). Lines correspond to the mean-field approximation with renormalized exchange parameters. Symbols correspond to atomistic spin dynamics simulations.

Figure 2

(a) Electron and lattice temperature dynamics for two laser pulse power values, $P_0$ and $2P_0$. Both electron and lattice temperature are kept constant, $T=300\mathrm{K}$, for $t<0$. At $t=0$, a laser pulse is applied and the dynamics of the electron and lattice temperature heat up. The dynamics of those temperatures are theoretically described by the two-temperature model. (b) Element-specific magnetization dynamics induced by the heat profile at (a). The dynamics are calculated using atomistic spin dynamics methods. For lower laser powers $P_0$, the magnetization of both sublattices demagnetizes rapidly and remagnetizes towards the new equilibrium. For laser power $2P_0$, the magnetization of both sublattices demagnetizes and switches. After switching, they relax towards the thermal equilibrium state. GdFeCo alloys with $x_{\mathrm{Gd}}=25\%$ are calculated.

Figure 3

Element-specific magnetization dynamics of GdFeCo calculated using atomistic spin dynamics (symbols) and macroscopic Baryakhtar-like equation (solid lines) for two laser pulse power values (a) $P_0$ and (b) $2P_0$. Both electron and lattice temperature are kept constant, $T=300$ K, for $t<0$. At $t=0$, a laser pulse is applied. In the Baryakhtar-like model, the relativistic relaxation parameters ${\alpha }_a^{\mathrm{B}}$ have a value different from the Gilbert damping in ASD simulations, $(\gamma /{\mu }_{\mathrm{Fe}}){\alpha }_{\mathrm{Fe}}^{\mathrm{B}}=0.005$ and $(\gamma /{\mu }_{\mathrm{Gd}}){\alpha }_{\mathrm{Gd}}^{\mathrm{B}}=0.01$. The exchange-relaxation parameter is varied, ${\alpha }_e^{\mathrm{B}}/{\alpha }_{\mathrm{Fe}}^{\mathrm{B}}=0,0.3$, and 3. The relaxation to the thermal state ($t<0$) is only well described for the Fe sublattice. (a) For $P_0$, the laser-induced dynamics is well described by ${\alpha }_e^{\mathrm{B}}/{\alpha }_{\mathrm{Fe}}^{\mathrm{B}}=0.1$. (b) For $2P_0$, the demagnetization phase of both sublattices is relatively well described in comparison to ASD simulations. Switching is also possible; here, for instance, for a value ${\alpha }_e^{\mathrm{B}}/{\alpha }_{\mathrm{Fe}}^{\mathrm{B}}=3$.

Figure 4

Element-specific magnetization dynamics of GdFeCo calculated using atomistic spin dynamics (symbols) and macroscopic LLB equation (solid lines) for two laser pulse power values (a) $P_0$ and (b) $2P_0$. For $t<0$, the electron and lattice temperatures are $T=300\mathrm{K}$, and at $t=0$, a laser pulse is applied. The exchange-relaxation parameter is varied, ${\alpha }_e/{\alpha }_a=0,0.1$, and 1, where ${\alpha }_a=0.01$ and $a=\mathrm{FeCo}$ or Gd. The initial relaxation dynamics is well described by ${\alpha }_e/{\alpha }_a=0$. (a) For laser power $P_0$, the element-specific dynamics is well described for ${\alpha }_e/{\alpha }_a=0.1$. (a) For ${\alpha }_e/{\alpha }_a=1$, exchange relaxation dominates and the element-specific dynamics are similar. (b) For laser power $2P_0$, the switching dynamics is not described by the LLB model.

Figure 5

Thermal equilibrium value of the relativistic exchange parameters ${\alpha }_{\mathrm{Fe}}$ and ${\alpha }_{\mathrm{Gd}}$ and the exchange-relaxation parameter ${\alpha }_{\mathrm{ex}}$ as a function of temperature.

Figure 6

Element-specific magnetization dynamics of GdFeCo calculated using atomistic spin dynamics (symbols) and the unified phenomenological model derived here, following Eq. (17) (solid lines), for two laser pulse power values (a) $P_0$ and (b) $2P_0$. Both electron and lattice temperatures are kept constant, $T=300$ K, for $t<0$. At $t=0$, a laser pulse is applied, which is the same as in Fig. 2. GdFeCo alloys with $x_{\mathrm{Gd}}=25\%$ are calculated.