Modeling Second Order Anisotropy of Monodomain Magnetic Body

The switching of a ferromagnet’s magnetization direction is a fundamental aspect in modeling spintronic devices. The switching can be achieved by applying an external magnetic field or spin current, and one of the key determinants of the switching process is the magnetocrystalline anisotropy. To model the magnetization dynamics, considering only the first order magnetocrystalline perpendicular anisotropy contributions had been sufficient thus far as it’s contributions are predominant. However, as the dimensions of the magnetic body are scaled down, or subject to voltage control of anisotropy (VCMA) effect, the magnetocrystalline perpendicular anisotropy and the out-of-plane demagnetizing field both end up having equal order, and nowadays low barrier magnets are purposefully designed to have magnetic anisotropy close to zero for special applications. In these cases, the contribution of the second order anisotropy of the ferromagnet need to be considered while studying the magnetization dynamics. In this paper, we developed a physics-based SPICE-compatible compact model of a monodomain magnetic body considering the second-order magnetic anisotropy contributions. The magnetization dynamics are described by the stochastic Landau-Lifshitz-Gilbert (sLLG) equation and can be used to study both perpendicular magnetic anisotropy (PMA) magnets and in-plane magnetic anisotropy (IMA) magnets. Here, we studied the effect of second order anisotropy in the critical switching field and current for PMA magnet and found that the second-order contributions to the switching field and current become significant when it is in the same order as the first order anisotropy.