Magnetic Domain - Landau-Lifshitz Energy Equation

Landau-Lifshitz Energy Equation

The contributions of the different internal energy factors described above is expressed by the free energy equation proposed by Lev Landau and Evgeny Lifshitz in 1935, which forms the basis of the modern theory of magnetic domains. The domain structure of a material is the one which minimizes the Gibbs free energy of the material. For a crystal of magnetic material, this is the Landau-Lifshitz free energy, E, which is the sum of these energy terms:

where

• E_ex is exchange energy: This is the energy due to the exchange interaction between magnetic dipole molecules in ferromagnetic, ferrimagnetic and antiferromagnetic materials. It is lowest when the dipoles are all pointed in the same direction, so it is responsible for magnetization of magnetic materials. When two domains with different directions of magnetization are next to each other, at the domain wall between them magnetic dipoles pointed in different directions lie next to each other, increasing this energy. This additional exchange energy is proportional to the total area of the domain walls.

• E_D is magnetostatic energy: This is a self-energy, due to the interaction of the magnetic field created by the magnetization in some part of the sample on other parts of the same sample. It is dependent on the volume occupied by the magnetic field extending outside the domain. This energy is reduced by minimizing the length of the loops of magnetic field lines outside the domain. For example, this tends to encourage the magnetization to be parallel to the surfaces of the sample, so the field lines won't pass outside the sample. Reducing this energy is the main reason for the creation of magnetic domains.

• E_λ is magnetoelastic anisotropy energy: This energy is due to the effect of magnetostriction, a slight change in the dimensions of the crystal when magnetized. This causes elastic strains in the lattice, and the direction of magnetization that minimizes these strain energies will be favored. This energy tends to be minimized when the axis of magnetization of the domains in a crystal are all parallel.

• E_k is magnetocrystalline anisotropy energy: Due to its magnetic anisotropy, the crystal lattice is "easy" to magnetize in one direction, and "hard" to magnetize in others. This energy is minimized when the magnetization is along the "easy" crystal axis, so the magnetization of most of the domains in a crystal grain tend to be in either direction along the "easy" axis. Since the crystal lattice in separate grains of the material is usually oriented in different random directions, this causes the dominant domain magnetization in different grains to be pointed in different directions.

• E_H is Zeeman energy: This is energy which is added to or subtracted from the magnetostatic energy, due to the interaction between the magnetic material and an externally applied magnetic field. It is proportional to the negative of the cosine of the angle between the field and magnetization vectors. Domains with their magnetic field oriented parallel to the applied field reduce this energy, while domains with their magnetic field oriented opposite to the applied field increase this energy. So applying a magnetic field to a ferromagnetic material generally causes the domain walls to move so as to increase the size of domains lying mostly parallel to the field, at the cost of decreasing the size of domains opposing the field. This is what happens when ferromagnetic materials are "magnetized". With a strong enough external field, the domains opposing the field will be swallowed up and disappear; this is called saturation.

Some sources define a wall energy E_W equal to the sum of the exchange energy and the magnetocrystalline anisotropy energy, which replaces E_ex and E_k in the above equation.

A stable domain structure is a magnetization function M(X), considered as a continuous vector field, which minimizes the total energy E throughout the material. To find the minimums a variational method is used, resulting in a set of nonlinear differential equations, called Brown's equations after William Fuller Brown Jr. Although in principle these equations can be solved for the stable domain configurations M(X), in practice only the simplest examples can be solved. Analytic solutions do not exist, and numerical solutions calculated by the finite element method are computationally intractable because of the large difference in scale between the domain size and the wall size. Therefore micromagnetics has evolved approximate methods which assume that the magnetization of dipoles in the bulk of the domain, away from the wall, all point in the same direction, and numerical solutions are only used near the domain wall, where the magnetization is changing rapidly.