Maxwell Equations and Landau–Lifshitz Equations

Abstract

This chapter will present the theoretical, mathematical and computational fundamentals for micromagnetics. The target of micromagnetics is to clarify the motion of magnetic moments in ferromagnetic materials and devices, which is described by the nonlinear Landau–Lifshitz equations, or the equivalent Landau–Lifshitz–Gilbert (LLG) equations. In the LLG equations, the time derivative of the magnet moment in a micromagnetic cell is controlled by the local effective magnetic field. The effective magnetic field contains the terms determined by the fundamental and applied magnetism in a magnetic material, including the external field, the crystalline anisotropy field, the exchange field, the demagnetizing field, and the magneto-elastic field. Among these field terms, the most time-consuming one in computation is the demagnetizing field, which will be calculated by the Green’s function method following the Maxwell’s equations.