Micromagnetics
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 originated from a 1935 paper by Lev Landau and Evgeny Lifshitz on antidomain walls.
[1]: 133 [2][3][4][5][6]: 440 Micromagnetics was then expanded upon by William Fuller Brown Jr. in several works in 1940-1941[1]: 133 [3][non-primary source needed][7][8][6]: 440 using energy expressions taken from a 1938 paper by William Cronk Elmore.
[10]: 41 [11] The field prior to 1960 was summarised in Brown's book Micromagnetics.
[10]: 44 The purpose of static micromagnetics is to solve for the spatial distribution of the magnetization
The static equilibria are found by minimizing the magnetic energy,[13]: 138 subject to the constraint
The exchange energy tends to favor configurations where the magnetization varies slowly across the sample.
[1]: 135 The exchange term is isotropic, so any direction is equally acceptable.
[1]: 83 Magnetic anisotropy arises due to a combination of crystal structure and spin-orbit interaction.
, the anisotropy energy density, is a function of the orientation of the magnetization.
In this approximation, called uniaxial anisotropy, the easy axis is the
The Zeeman energy favors alignment of the magnetization parallel to the applied field.
In particular, on the edges of the sample, the magnetization tends to run parallel to the surface.
This interaction arises when a crystal lacks inversion symmetry, encouraging the magnetization to be perpendicular to its neighbours.
plane interfacial DMI takes the form and for materials with symmetry class
the energy contribution is This term is important for the formation of magnetic skyrmions.
There exists a preferred local distortion of the crystalline solid associated with the magnetization director
For a simple small-strain model, one can assume this strain to be isochoric and fully isotropic in the lateral direction, yielding the deviatoric ansatz[13]: 128 [16]: 250–251
Here the elastic response is assumed to be isotropic (based on the two Lamé constants
The purpose of dynamic micromagnetics is to predict the time evolution of the magnetic configuration.
In variational terms, a change dm of the magnetization and the associated change dE of the magnetic energy are related by: Since m is a unit vector, dm is always perpendicular to m. Then the above definition leaves unspecified the component of Heff that is parallel to m.[12] This is usually not a problem, as this component has no effect on the magnetization dynamics.
From the expression of the different contributions to the magnetic energy, the effective field can be found to be (excluding the DMI and magnetoelastic contributions):[1]: 178 This is the equation of motion of the magnetization.
It describes a Larmor precession of the magnetization around the effective field, with an additional damping term arising from the coupling of the magnetic system to the environment.
It can be shown that this is mathematically equivalent to the following Landau-Lifshitz (or explicit) form:[17][1]: 181–182 where
, as[1]: 181 The interaction of micromagnetics with mechanics is also of interest in the context of industrial applications that deal with magneto-acoustic resonance such as in hypersound speakers, high frequency magnetostrictive transducers etc.
FEM simulations taking into account the effect of magnetostriction into micromagnetics are of importance.
Such simulations use models described above within a finite element framework.
[18] Apart from conventional magnetic domains and domain-walls, the theory also treats the statics and dynamics of topological line and point configurations, e.g. magnetic vortex and antivortex states;[19] or even 3d-Bloch points,[20][21] where, for example, the magnetization leads radially into all directions from the origin, or into topologically equivalent configurations.
In this discipline, numerical methods such as finite-element analysis are used to analyze the electric/magnetic fields generated by the stimulation apparatus; then the results are validated or explored further using in-vivo or in-vitro neuronal stimulation.
