Upper Bound to the Energy of Cross-Tie Walls

Abstract

A simple mathematical model is suggested for the magnetization variation inside a cross-tie domain wall in ferromagnetic films. The exchange and anisotropy energies are calculated analytically for this model. The magnetostatic self-energy is overestimated by adding essentially the contribution of volume charge at an infinite film thickness to the contribution of the surface charge at zero film thickness, and this gives an upper bound of about 10 erg/cm2 for the total wall energy at any thickness. Neglecting the volume charge yields an energy of 0.27 erg/cm2 for zero thickness, while neglecting the surface charge yields 7.7 erg/cm2 for an infinite thickness. The latter should be an upper bound for the energy in any thickness, if the energy is a monotonic increasing function of the thickness. The radius of a Bloch line is found to be of the order of 100 Å, while the distance between crosses is found to be of the order of 10-4 cm in Permalloy, which is of the order of experimentally observed values.