Anatomy of the magnetic anisotropy energy mediated by tight-binding Rashba electrons

The magnetic anisotropy is a fundamental quantity, defining the orientational stability of the magnetic state. Due to its importance, many different approaches have been put forth to explain its properties and behavior, ranging from simple models to fully first-principles calculations based on the electronic structure of a chosen material. In this work, we focus on a simple tight-binding model of spin-orbit-coupled electrons exchange-coupled to a background ferromagnetic order, meant to abstract the essential physics at the interface between a ferromagnetic layer and a heavy-metal layer. We propose a new method to calculate the magnetic anisotropy energy, based on the spin susceptibility of the electrons. This is compared to other approaches, such as energy differences between different orientations of the ferromagnet, or the anisotropy of the spin-orbit energy. We investigate not only their compatibility but also what physical insights can be gained from each of them. In particular, we establish under what conditions perpendicular magnetic anisotropy is favored, and how it can be maximized in the present model. The computation of the magnetic anisotropy energy from the spin susceptibility is general, and can be readily extended to realistic electronic structure calculations.