# Ultrafast Modulation of Magnetization Dynamics in Ferromagnetic (Ga, Mn)As Thin Films
## 3 Results
For the compressively-strained (Ga,Mn)As film studied in this work, the influence of magnetic anisotropy field modulation on magnetization precession frequency can be directly described by the expression for the FMR frequency. With the magnetization lying in the plane of the sample, the precession frequency can be expressed as [32; 34]:  
\[\left(\frac{\omega}{\gamma}\right)^{2}=H_{4\parallel}\bigg{(}4\pi M_{eff}+H_{4 \parallel}+\frac{H_{2\parallel}}{2}\bigg{)}, \tag{2}\]  
where \(\gamma\) is the gyromagnetic ratio (\(\gamma\) = 1.7588 Hz/Oe for g factor = 2.0023); \(H_{4\mid\,\mid}\) and \(H_{2\mid\,\mid}\) are the in-plane cubic and uniaxial anisotropy fields, respectively; and 4\(\pi\)_Meff_ is the effective perpendicular uniaxial anisotropy field, 4\(\pi\)_Meff_ = 4\(\pi\)_M_ - \(H_{2\perp}\), where \(H_{2\perp}\) is the out-of-plane perpendicular uniaxial anisotropy field. When the pump intensity increases, the in-plane magnetocrystalline anisotropy fields \(H_{4\mid\,\mid}\) and \(H_{2\mid\,\mid}\) are observed to decrease monotonically, presumably due to increasing temperature and increasing carrier density caused by increase of pump intensity. On the other hand, the dependence of 4\(\pi\)_Meff_ on temperature and carrier density shows variations in opposite directions (as argued above, 4\(\pi\)_Meff_ decreases with increasing temperature, but increases with increasing carrier density). This indicates that the magnetization precession frequency decreases linearly when the decreasing of in-plane magnetic anisotropy field dominates the behavior of magnetization precession either through thermal or nonthermal effects. This influence of the in-plane magnetic anisotropy field, however, cannot explain the nonlinear dependence of precession frequency on pump intensity shown in Figure 2. Thus, the strong increase of 4\(\pi\)_Meff_ due to increasing photoexcited carrier density offers the possibility of slowing down the rate of decline of magnetization precession frequency. To obtain a quantitative insight into the nonlinear influence of in-plane and out-of-plane magnetic anisotropy fields on magnetization precession, the magnetization precession was numerically simulated by employing LLG equation combined with two different MO effects, polar Kerr rotation  
Figure 3: Coordinate system used in this paper. The initial magnetization lies in the plane of the sample along [100] direction. The [100] direction, [010] direction and [001] direction are defined as the directions of \(x\) axis, \(y\) axis, and \(z\) axis, respectively. The magnetic anisotropy field \(H_{xy}\) is in the plane of the film, and the magnetic anisotropy field \(H_{z}\) is out of the plane of the film. The arrows in (Ga,Mn)As layer represent the direction of compressive strain in the plane. and magnetic birefringence (LLG-2 MO) [13; 23]. The micromagnetic modeling could be expressed as [14; 28; 35]  
\[\frac{\partial\overset{\rightarrow}{M}}{\partial t}=-\gamma\overset{ \rightarrow}{M}\times\overset{\rightarrow}{H_{eff}}+\frac{\alpha}{M}\overset{ \rightarrow}{M}\times\frac{\partial\overset{\rightarrow}{M}}{\partial t}, \tag{3}\]  
where \(M\) is the local magnetization contributed from Mn ions, \(H_{eff}\) is the effective field (which includes the in-plane and out-of-plane anisotropy fields), and \(\alpha\) is the Gilbert damping coefficient. For compressively-strained (Ga,Mn)As films, the initial magnetization before excitation lies along the easy axes direction in the plane of the sample. In the simulation program, the initial state of the magnetization is set to be along the x axis in the \(x\)-\(y\) plane (i.e., in the plane of the film), and z axis is normal to the film plane, as seen in Figure 3. Thus, \(H_{x\text{-}y}\) and \(H_{z}\) correspond to the effect of \(H_{4\,|\,|}\) and \(4\pi\)\(M_{eff}\), respectively. \(8H_{x\text{-}y}\) and \(8H_{z}\) correspond to the modulated value of \(H_{4\,|\,|}\) and \(4\pi\)\(M_{eff}\), respectively.