Three-dimensional micromagnetic calculations for magnetite using FFT

An unconstrained 3-D micromagnetic model for magnetite grains with resolutions up to 23 x 23 x 63 is presented. The model has been used to investigate the magnetic domain states of submicron parallelepipeds with various elongations. The method of fast Fourier transformation (FFT) has been implemented in three dimensions to accelerate the computation of the magnetostatic energy and its gradient, assuming constant magnetization for each subcube of the model. A 3-D implementation of the exchange energy and its gradient, using a five-point formula to approximate the Laplace operator, was chosen. Special attention has been paid to single-domain (SD) or flower states and pseudo-single-domain (PSD) configurations. A circular configuration, called vortex state, has been found to have the lowest energy of various PSD states. As a local energy minimum (LEM), the free energy of the vortex state is compared to that of a single-domain state. A comparison of these energies is used to determine a lower and an upper threshold size for the SD to PSD transition. In the interval between these two threshold sizes both configurations, SD and PSD, coexist. Above the upper threshold size we found metastable double-vortex configurations which seem to represent three-domain structures with closure domains.