micromagnetic

Micromagnetics is the continuum description of magnetism in ferromagnetic materials at sub-micrometer length scales. In this framework, the magnetization M(r,t) is treated as a continuous vector field with fixed magnitude M_s, the saturation magnetization. The equilibrium configuration minimizes the micromagnetic energy, while the dynamics are described by the Landau-Lifshitz-Gilbert equation for the time evolution of the magnetization.

The effective magnetic field H_eff driving the dynamics includes contributions from exchange interactions, magnetocrystalline anisotropy, demagnetizing (magnetostatic) fields, and external applied fields. The exchange energy, governed by the exchange stiffness A, favors alignment of neighboring spins; anisotropy energy, characterized by constants such as K1 and easy-axis directions, sets preferred orientations; demagnetizing energy arises from magnetic charges and is a long-range interaction; Zeeman energy comes from external fields.

The Landau-Lifshitz-Gilbert equation can be written, in suitable units, as dm/dt = -gamma m x H_eff + alpha m x dm/dt, where m is the normalized magnetization, gamma is the gyromagnetic ratio, and alpha is the Gilbert damping parameter. This equation describes precession of m about H_eff and its damping toward equilibrium.

Numerical micromagnetics discretizes space into small cells or finite elements. The cell size is chosen to resolve the exchange length and is typically a few nanometers. The demagnetizing field is computed using FFT-based methods for periodic systems or boundary-element methods for other geometries.

Applications include the study of domain structures, domain walls, vortices, and spin textures in thin films, nanowires, and magnetic memory devices. It informs the design of magnetic storage, MRAM, sensors, and spintronic components. Popular software tools include OOMMF and MuMax3, among others.