Quantized static micromagnetic theory and its applications to two-dimensional topological multiferroic materials

The original micromagnetism is quasi-classical since it is based on the assumption that the magnetization magnitude is unchanged in the whole magnetic system, so the theory is only strictly valid at zero temperature. To generalize its applicability to elevated temperatures, Garanin proposed the Landau–Lifshitz–Bloch equation by further considering the random fields arising from the heat effects and allowing the length variations of magnetic moments. However, the expressions for the fields and changes are usually complicated, or even impossible to be derived for materials with complex spin structures. In the present work, a quantized static micromagnetic theory is proposed and integrated with the quantum computational methods we developed before, and afterwards used to simulate the topological ferromagnetic (FM) and ferroelectric (FE) textures in the CoFeB-like multiferroic thin films consisting of spins with S = 1/2, 1, 3/2, 2, and 5/2 respectively. Consequently, both FM and FE skyrmionic lattices (SkLs) of the Néel- and Bloch-types are observed simultaneously at low temperatures, except for the case of S = 1/2 where only FM SkLs can be induced; the topological charge per FM skyrmion (SkN) is calculated to be -1.0, whereas those of FE SkNs are quantized fractional numbers; and the formation temperatures of the FM and FE SkLs are proportional to J S ( S + 1 ) . With this quantized micromagnetic theory, we are able to simulate and display FM and FE textures in mesoscopic scale at nonzero temperatures without the need to derive extra mathematic expressions, however with greatly reduced computer memory and expedited computational speed. This work possibly paves a new way for computational magnetism.