Cross-Tie Domain Wall Ground State in Thin Films

Cross-tie domain wall is a complex two dimensional magnetic domain wall structure, often found experimentally in thin soft ferromagnetic films with thicknesses of about several exchange lengths. The structure of such a wall can roughly be imagined as a sequence of magnetic vortices and anti-vortices, arranged along a straight line. In this work, equilibrium energies of different one-dimensional magnetic domain wall configurations, existing in thin soft ferromagnetic films, are calculated (fully taking into account the exchange and long range dipolar interaction), based on the various classical Ritz models. They are compared to the energy of two-dimensional cross-tie domain wall, thus, estimating the region of the film thicknesses, where the cross-tie structure is the lowest energy domain wall configuration. The upper boundary of this region is estimated here for the first time. Convenient approximate analytical energy expressions in terms of elementary functions are given here both for classical and for cross-tie domain walls.