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Hole spin relaxation and coefficients in Landau-Lifshitz-Gilbert equation in ferromagnetic GaMnAs

We investigate the temperature dependence of the coefficients in the Landau-Lifshitz-Gilbert equation in ferromagnetic GaMnAs by employing the Zener model. We first calculate the hole spin relaxation time based on the microscopic kinetic equation. We find that the hole spin relaxation time is typically several tens femtoseconds and can present a nonmonotonic temperature dependence due to the variation of the interband spin mixing, influenced by the temperature related Zeeman splitting. With the hole spin relaxation time, we are able to calculate the coefficients in the Landau-Lifshitz-Gilbert equation, such as the Gilbert damping, nonadiabatic spin torque, spin stiffness and vertical spin stiffness coefficients. We find that the nonadiabatic spin torque coefficient $β$ is around $0.1\sim 0.3$ at low temperature, which is consistent with the experiment [Adam {\em et al.}, Phys. Rev. B {\bf 80}, 193204 (2009)]. As the temperature increases, $β$ monotonically increases and can exceed one in the vicinity of the Curie temperature. In the low temperature regime with $β<1$, the Gilbert damping coefficient $α$ increases with temperature, showing good agreement with the experiments [Sinova {\em et al.}, Phys. Rev. B {\bf 69}, 085209 (2004); Khazen {\em et al.}, {\em ibid.} {\bf 78}, 195210 (2008)]. Furthermore, we predict that $α$ decreases with increasing temperature once $β>1$ near the Curie temperature. We also find that the spin stiffness decreases with increasing temperature, especially near the Curie temperature due to the modification of the finite $β$. Similar to the Gilbert damping, the vertical spin stiffness coefficient is also found to be nonmonotonically dependent on the temperature.