Analytical study of spin current density and spin-transfer torque in semi-Dirac heterostructures

Abstract

Using a low energy effective Hamiltonian for a ferromagnetic semi-Dirac material the energy eigenvalues and eigenvectors for a heterostructure including ferromagnet/normal metal/ferromagnet (FM/NM/FM) made of semi-Dirac materials have been obtained. Then analytic expressions for spin current density components and spin transfer torques for the heterostructure as a function of the ferromagnetic magnetization, chemical potential as well as length of the spacer layer have been obtained. It is shown that the spin current density and spin transfer torques are very sensitive functions of the length and chemical potential of the spacer layer. More interestingly,a beating pattern of oscillations, in terms of the thicknessof the NM spacer, occurs for the y-component of both the STT and spin current density. The variation of spin transfer torque components versus the angular and azimuthal angles for the second magnetization direction is also noteworthy, where we get symmetric behaviour but different magnitudes. The obtained calculations make possible to study the spin current density and spin transfer torques regarding the arrangement of the magnetizations in the two ferromagnetic layers.