Micromagnetism

Micromagnetics is a field of physics dealing with the prediction of magnetic behaviors at sub-micrometer length scales. The length scales considered are large enough for the atomic structure of the material to be ignored (the continuum approximation), yet small enough to resolve magnetic structures such as domain walls or vortices.

Micromagnetics can deal with static equilibria, by minimizing the magnetic energy, and with dynamic behavior, by solving the time-dependent dynamical equation.

Micromagnetics as a field (i.e., that deals specifically with the behaviour of (ferro)magnetic materials at sub-micrometer length scales) was introduced in 1963 when William Fuller Brown, Jr. published a paper on antiparallel domain wall structures. Until comparatively recently computational micromagnetics has been prohibitively expensive in terms of computational power, but smaller problems are now solvable on a modern desktop PC.

The purpose of static micromagnetics is to solve for the spatial distribution of the magnetization M at equilibrium. In most cases, as the temperature is much lower than the Curie temperature of the material considered, the modulus |M| of the magnetization is assumed to be everywhere equal to the saturation magnetization M_s. The problem then consists in finding the spatial orientation of the magnetization, which is given by the magnetization direction vector m = M/M_s, also called reduced magnetization.

The static equilibria are found by minimizing the magnetic energy,

subject to the constraint |M|=M_s or |m|=1.

The contributions to this energy are the following:

The exchange energy is a phenomenological continuum description of the quantum-mechanical exchange interaction. It is written as:

where A is the exchange constant; m_x, m_y and m_z are the components of m; and the integral is performed over the volume of the sample.

The exchange energy tends to favor configurations where the magnetization varies only slowly across the sample. This energy is minimized when the magnetization is perfectly uniform.

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