Micromagnetics

General

Micromagnetism deals with the interactions between magnetic moments on sub-micrometre length scales. These are governed by several competing energy terms. Dipolar energy is the energy which causes magnets to align north to south pole. Exchange energy will attempt to make the magnetic moments in the immediately surrounding space lie parallel to one another (if the material is ferromagnetic) or antiparallel to one another (if antiferromagnetic). Anisotropy energy is low when the magnetic moments are aligned along a particular crystal direction. Zeeman energy is at its lowest when magnetic moments lie parallel to an external magnetic field.

Since the most efficient magnetic alignment (also known as a configuration) is the one in which the energy is lowest, these four energy terms will attempt to become as small as possible at the expense of the others, yielding complex physical interactions.

The competition of these interactions under different conditions is responsible for the overall behaviour of a magnetic system.

History

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

Landau-Lifshitz Gilbert equation

Generally, a form of the Landau-Lifshitz Gilbert equation:

${d{\vec {M}} \over dt}=-|\gamma |{\vec {M}}\times {\vec {H}}_{\mathrm {eff} }-{|\gamma |\alpha \over M_{s}}{\vec {M}}\times ({\vec {M}}\times {\vec {H}}_{\mathrm {eff} })\qquad (1)$

is used to solve time-dependent micromagnetic problems, where ${\vec {M}}$ is the magnetic moment per unit volume, ${\vec {H}}_{\mathrm {eff} }$ is the effective magnetic field, $\alpha$ is the Gilbert phenomenological damping parameter and $\gamma$ is the electron gyromagnetic ratio.

Landau-Lifshitz equation

If in (1) we put $\alpha =0$ , then we get the famous Landau-Lifshitz equation (LLE)

${d{\vec {M}} \over dt}=-|\gamma |{\vec {M}}\times {\vec {H}}_{\mathrm {eff} }.\qquad (2)$

References

Brown, William Fuller, Jr. (1963). Micromagnetics. New York: Wiley.{{cite book}}: CS1 maint: multiple names: authors list (link)

Gilbert, Thomas L. (2004). "A Phenomenological Theory of Damping in Ferromagnetic Materials". IEEE Transactions on Magnetics. 40 (6): 3443–3449. doi:10.1109/TMAG.2004.836740. ISSN 0018-9464.

External links

Magnetization dynamics applet
OOMMF - The Object-Oriented Micromagnetic Framework - a popular free micromagnetic simulation tool using finite-difference lattice discretisations of space and FFT.
Magpar - a parallelizable, finite element based, free micromagnetic simulation package.
Nmag - a parallelizable, finite element based, free micromagnetic simulator that is scriptable in Python.