Hot Electron Relaxation in Ferromagnetic Metals: Memory Function Approach

Abstract

This study leads to the investigation of the non-equilibrium electron relaxation in ferromagnetic metals. Here we consider the relaxation of electrons due to their coupling with magnons and phonons in a ferromagnet using the memory function approach. In the present model, electrons live at a higher temperature than that of the phonon and magnon baths, mimicking a non-equilibrium steady-state situation. Further we analyze theoretically the generalized Drude scattering rate within the framework of two temperature model and study the full frequency and temperature behavior for it. In zero frequency regime, the rate of electron-magnon scattering and electron-phonon scattering shows a linear temperature dependence at higher temperature values greater than Debye temperature. Whereas at lower temperature values, $T\ll \Theta _{D}$, corresponding scattering rates follow the temperature behavior as ($1/\tau _{e-p} \varpropto T^3$) and ($1/\tau _{e-m} \varpropto T^{3/2}$), respectively. In the AC regime, we compute that $1/\tau \propto \omega ^2$ for $\omega \ll \omega _{D}$ and for the values greater than the Debye frequency, it is $\omega$-independent. Also, in lower frequency and zero temperature limit, we have observed the different frequency scale of electron-magnon and electron-phonon scattering, i.e., ($1/\tau \propto \omega ^{3/2}$) and ($1/\tau \propto \omega ^{3}$). These results can be viewed with the pump-probe experimental setting for ferromagnetic metals.