On the spontaneous magnetization of two-dimensional ferromagnets

Ferromagnetism is typically discussed in terms of the exchange interaction and magnetic anisotropies. Yet real samples are inevitably affected by the magnetostatic dipole-dipole interaction. Because of this interaction, a theorem [R.B. Griffiths, Free Energy of interacting magnetic dipoles, Phys. Rev. 176, 655 (1968)] forbids a spontaneous magnetization in, nota bene, three-dimensional bodies. Here we discuss perpendicularly and in-plane magnetized ferromagnetic bodies in the shape of a slab of finite thickness. In perpendicularly magnetized slabs, magnetic domains are energetically favored when the lateral size is sufficiently large, i.e., there is no spontaneous magnetization. For in-plane magnetization, instead, spontaneous magnetization is possible below a critical thickness which, in very thin films, could be as small as few monolayers. At this critical thickness, we predict a genuine phase transition to a multi-domain state. These results have implications for two-dimensional ferromagnetism.