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 E_tot 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_demcan be calculated as a convolution of the magnetization distribution inside a ferromagnet with the dipolar interaction kernel

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

DV_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)