Analytical magnetic distributions and Stoner-Wohlfarth theory for near-uniformly magnetised rectangular nanomagnets

The micromagnetic equations which govern the magnetisation distribution have been studied analytically for nano-sized magnets. By exploiting the fact that in such magnets the magnetisation is near-uniform these equations can be linearised. The problem then reduces to the solution of two uncoupled Poisson equations with given source terms subject to certain surface boundary conditions. These source terms are particularly simple and have analytical forms when one restricts the analysis to rectangular nanomagnets. Solutions to the Poisson equations have been found analytically by expansion in Fourier series which automatically satisfies the boundary conditions. The total energies obtained using these Fourier solutions are compared with the energies obtained from numerical solutions to the full non-linear micromagnetic equations in the case of square cross-section nanomagnets (the only case that has been considered numerically) and shown to be in agreement to the order of a few percent. These solutions may now be used to calculate the total energy in the presence of an external magnetic field and in this way extend the Stoner-Wohlfarth theory to near-uniformly magnetised rectangular bodies.