Details

The Landau–Lifshitz equation describes the magnetization dynamics of ferromagnetic materials within the framework of micromagnetism. The magnetization of a ferromagnetic material is modeled by a continuous vector field.

The results of micromagnetism operate best on nanometer length scales under the assumptions that the norm of the magnetization vector is equal to the saturation magnetization everywhere and that the energy functional varies smoothly on atomic length scales.

The dynamics of the system are closely tied to an effective field , which is defined as the functional derivative of the energy over the magnetization field:

.

From this definition of the effective field, the Landau–Lifshitz equation can be written as:

.

Again, is the gyromagnetic ratio, is the damping parameter, and is the saturation magnetization. Some examples of common terms that contribute to include the exchange field, the demagnetizing field and the external field. We are only considering the dynamics of a single magnetization vector in the presence of a uniform external field in this Demonstration. For simplicity, the saturation magnetization is fixed at 1, and the components can be changed by varying the polar angle and the azimuthal angle . The values of , and each of the three Cartesian components of the field can also be varied.

Snapshot 1: Using a small, nonzero (), loses energy slowly as it relaxes toward the low-energy configuration. As a result, precesses around many times, and the trajectory resembles a tightly wound spiral.

Snapshot 2: Here a larger value of () is used. loses energy more quickly as it relaxes toward the low-energy configuration. As a result, does not precess around as many times, and the trajectory resembles a loosely wound spiral.

Snapshot 3: Setting the initial magnetization direction opposite of the external field direction, the energy of the system is maximized. However, , so the configuration is stable. Likewise, as when the initial magnetization is in its lowest energy configuration, the lowest energy configuration is also stable.

Snapshot 4: The system is placed in a configuration with and . The system does not lose energy, and the magnetization precesses around the external field.

References

[1] L. Landau and E. Lifshitz, "On the Theory of the Dispersion of Magnetic Permeability in Ferromagnetic Bodies," Physikalische Zeitschrift der Sowjetunion, 8, Part 2, 1935 pp. 153–164. doi:10.1016/B978-0-08-036364-6.50008-9.

[2] J. M. D. Coey, Magnetism and Magnetic Materials, Cambridge: Cambridge University Press, 2010.

[3] C. Abert. "Micromagnetism." (Jun 13, 2023) micromagnetics.org/micromagnetism.