# Investigation on the Effect of Femtosecond Laser Induced Spin Transfer Torque of GdFeCo Alloy
## 2 Method
included the uniaxial single-ion anisotropy energy and the exchange interaction energy:  
\[\mathcal{H}=\mathcal{H}_{ani}+\mathcal{H}_{exc}=-k_{u}\underset{i}{\sum}( \boldsymbol{S}_{i}\cdot\boldsymbol{e})^{2}-\underset{i<j}{\sum}J_{ij} \boldsymbol{S}_{i}\cdot\boldsymbol{S}_{j} \tag{1}\]  
where \(\mathcal{H}_{ani}\) is the spin Hamiltonian for uniaxial anisotropy energy, \(\mathcal{H}_{exc}\) is the Hamiltonian for exchange interaction energy, \(k_{u}\) is the anisotropy constant, \(\boldsymbol{S}_{i}\) is a unit vector representing the direction of atomic spin moment, \(\boldsymbol{S}_{j}\) represents the other atoms besides \(\boldsymbol{S}_{i}\) in the range of exchange interactions, \(\boldsymbol{e}\) is a unit vector describing the direction of the easy axis, \(J_{ij}\) is the exchange interaction between atomic sites \(i\) and \(j\). For all the calculations in this paper, the initial magnetization direction of GdFe alloy is along the \(Z\) axis in Cartesian coordinates. In order to balance the two conditions of lower Curie temperature and higher magnetic moment, the ratio of transition metal elements to rare earth elements is generally 3:1 in the practical application of magneto-optical recording. We built the atomic models according to this elements ratio. Figure 3 shows the atomic-scale model structures of different sizes, with side lengths of 1 nm and 10 nm respectively, which were established according to the Hamiltonian of GdFe amorphous alloy thin films system in Equation (1). The value and unit of critical physical parameters in atomistic models of different sizes are exactly the same, as shown in Table 1.  
Figure 2: Spin current conduction in a GdFe alloy. The magnetization dynamics are based on atomic-scale Landau-Lifshitz-Gilbert (LLG) equation. The spin variation of Fe sublattice is described by Equation (2), and the spin change of Gd sublattice is described by Equation (3):  
\[\frac{\partial\mathcal{S}_{i}}{\partial t}=-\frac{\gamma}{(1+\lambda^{2})} \left[\mathcal{S}_{i}\times\mathbf{H}_{eff}^{i}+\lambda\mathcal{S}_{i}\times \left(\mathcal{S}_{i}\times\mathbf{H}_{eff}^{i}\right)+\frac{Jc\mathbf{b}(\mathbf{p}\cdot \mathbf{m})}{\gamma eM_{s}}\mathcal{S}_{i}\times\left(\mathcal{S}_{i}\times \mathcal{S}_{j}\right)\right] \tag{2}\]  
\[\frac{\partial\mathcal{S}_{i}}{\partial t}=-\frac{\gamma}{(1+\lambda^{2})} \left[\mathcal{S}_{i}\times\mathbf{H}_{eff}^{i}+\lambda\mathcal{S}_{i}\times \left(\mathcal{S}_{i}\times\mathbf{H}_{eff}^{i}\right)^{\prime}\right] \tag{3}\]  
where \(\lambda\) is the microscopic damping parameter, \(\gamma\) is the gyromagnetic ratio, \(J_{e}\) is the current density, \(\mathbf{H}_{eff}^{i}=\zeta_{i}-\partial\mathcal{H}_{i}/\partial\mathcal{S}_{i}\) is the effective field at site \(i\) and includes stochastic thermal fluctuations, \(e\) is the absolute value of the electron charge, \(M_{s}\) is the saturation magnetization for Fe,\(|b(\mathbf{p}\cdot\mathbf{m})=\left[-4+(1+P)^{3}(3+\mathbf{p}\cdot\mathbf{m})/4P^{3/2} \right]^{-1}\) is a scalar function associated with magnetization unit \(\mathbf{p}\) and \(\mathbf{m}\), \(\mathbf{p}\) and \(\mathbf{m}\) represent the initial spin magnetization directions of Gd and Fe r2 espectively. For the third term to the right of Equation (2), we define it as the atomic spin transfer torque term \(\tau\) described as Equation (4). Equation (4) is corrected according to the macro current-induced magnetization dynamics proposed by [PERSON] et al. [23, 34, 36], the macro format is used to describe spin transfer torque of two different magnetic layers and we corrected it to the atomic format. \[\tau=\frac{J_{e}b(\mathbf{p}\cdot\mathbf{m})}{\gamma eM_{s}}\mathcal{S}_{i}\times \left(\mathcal{S}_{i}\times\mathcal{S}_{j}\right) \tag{4}\]  
We used the two-temperature model to simulate the effect of femtosecond laser heating on the electron-lattice temperature of the sample and the thermal coupling between the electrons and the lattices. The volume of the simulation for Gd\({}_{25}\)Fe\({}_{75}\) alloy is 10 \(\times\) 10 \(\times\) 10 nm\({}^{3}\)