Magnetization nutation in magnetic semiconductors: Effective spin model with anisotropic RKKY exchange interaction

Abstract: We demonstrate that the magnetization in magnetic semiconductors exhibits nutational motion when subjected to an external magnetic field. This behavior originates from the splitting of the conduction-electron band which induces anisotropic, distance-dependent exchange coupling between localized spins. To investigate this phenomenon, we examine a general system that includes both charge and spin degrees of freedom, characteristic of a magnetic semiconductor. This system is composed of two subsystems: (1) a gas of noninteracting conduction electrons and (2) a ferromagnetic array of localized spins, coupled through the Vonsovskii (\textit{sd}) local interaction. The entire system is subject to external electrical and magnetic disturbances. Through the Feynman-Schwinger formalism, we integrate out the faster (Grassmann) charge degrees of freedom associated with the conduction electrons to obtain the effective Hamiltonian for the localized spins in the form of an XXZ spin model. We then provide general analytical formulas for the corresponding anisotropic exchange couplings, expressed in Fourier and direct spaces, as functions of the effective field that induces conduction-band splitting. .... We hope this study will motivate further research into nutational phenomena in magnetic semiconductors, possibly resulting in improvements in the accurate control of magnetization dynamics within spin-based electronic devices.