Theoretical background

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 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_dem can be calculated as a convolution of the magnetization distribution inside a ferromagnet with the dipolar interaction kernel.

 Notations:
    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 - exchange weakening, a_ij - angle between M_i and M_j)

4. Stray field (W_ij - interaction coeff. between cells i and j)