Two dimensional landscape of ferromagnetic domains and the resulting magnetization curves

Recently, Kittel's theory of ferromagnetic domains in thin films in one dimension, i. e. with the domains extended infinitely over one in-plane direction and with the anisotropy axis oriented perpendicular to the film was revisited by considering the film thickness and unbalanced domains with up and down magnetization, yielding to the computation of magnetic hysteresis [C. M. Teodorescu, Res. Phys. 46 (2023) 106287]. In this work, the above study is extended to samples featuring two-dimensional domain landscapes, for materials with strong magnetic anisotropy, typically characterized by a superunitary ratio between the anisotropy energy and the stray field energy densities. This allows one to compute the most stable structures for vanishing average magnetization < M > together with hysteresis curves for thin films with perpendicular magnetic anisotropy and two-dimensional rectangular domains and also for thin films with in-plane magnetic anisotropy. For two-dimensional films with perpendicular magnetic anisotropy, the most stable structure for < M > = 0 is found to be that with domains infinitely elongated along one in-plane direction, i. e. the one-dimensional case treated in the preceding work. For thin stripes with in-plane magnetization, the domain size l is approximately linear with the stripe lateral size d for low film thickness, while for large film thicknesses it follows a Kittel-like law, but as function of the stripe size l similar to d(1/2). For in-plane magnetized thin films of infinite lateral extent, the most stable structure is the single domain. As for hysteresis curves, the two-dimensional case with perpendicular magnetic anisotropy is shown to evolve from a 2D landscape derived from the checkerboard structure, but with unbalanced domains for magnetization near saturation, towards one-dimensional domain structures for lower magnetization. In some cases and depending also on the demagnetization factor, the one-dimensional case is not reached, and the film exhibit 2D structures on the whole range of the magnetization curve. The hysteresis obtained for thin magnetic stripes with in-plane magnetization also can exhibit a rich structure, with minor cycles on the wings of the magnetization curves evolving towards ,,normal" hysteresis (again, depending on the film thickness, stripe lateral size and demagnetization factor). An infinite thin film with in-plane anisotropy features a steplike magnetization dependence on the applied field.