I. Basic micromagnetic concept and main energy contributions
Theoretical micromagnetics as founded by W.F.Brown is a phenomenology for the evaluation of the total magnetic free energy Etot of a ferromagnetic body if its geometry, material parameters and the magnetization configuration are known. In its 'minimal' version micromagnetics takes into account four energy contributions:
Energy in the
external field E
_{ext}:
Magnetocrystalline anisotropy energy E
_{an} (e.g., uniaxial case):
Exchange stiffness energy E
_{exch} (isotropic exchange assumed):
Magnetodipolar interaction energy (stray or demagnetizing) field energy E
_{dem}:
where the demagnetizing field H
_{dem} can be calculated as a convolution of the magnetization distribution inside a ferromagnet with the dipolar interaction kernel.
II. Finite-difference approximations of the energy contributions used in MicroMagus
Notation:
M_{i} - magnetization in the cell
# i,
m_{i} = M_{i}/M_{i}
H_{i}^{ext} - external field averaged over the cell
# i
D
V_{i} - cell volume
1.
External field
2.
Anisotropy (
K_{i} – anisotropy constants)
- uniaxial (n_{i} – unit vector of the anis. axis)
- cubic (p_{x,y,z} – components of the unit magn. vector in a local coord. system)
3.
Exchange (
J_{ij} – exchange coeff.,
k_{ij} – exch. weakening,
a_{ij} –
angle between
M_{i} and
M_{j})
4.
Stray field (
W_{ij} – interaction coeff. between cells
i and
j)