Hole spin relaxation and coefficients in Landau-Lifshitz-Gilbert equation in ferromagnetic (Ga,Mn)As

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 of 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 beta is around 0.1-0.3 at low temperature, which is consistent with the experiment [J.-P. Adam et al., Phys. Rev. B 80, 193204 (2009)]. As the temperature increases, beta monotonically increases. We show that the Gilbert damping coefficient a increases with temperature below the Curie temperature, showing good agreement with the experiments [J. Sinova et al., Phys. Rev. B 69, 085209 (2004);Kh. Khazen et al., ibid. 78, 195210 (2008)]. Moreover, we also calculate the temperature dependences of the spin stiffness and vertical spin stiffness.