Γ -Limit for Two-Dimensional Charged Magnetic Zigzag Domain Walls

Charged domain walls are a type of domain wall in thin ferromagnetic films which appear due to global topological constraints. The non-dimensionalized micromagnetic energy for a uniaxial thin ferromagnetic film with in-plane magnetization m∈ S¹ is given by Eε[m]=ε‖∇m‖L22+1ε‖m·e2‖L22+πλ2|lnε|‖∇·(m-M)‖H˙-122,where M is an arbitrary fixed background field to ensure global neutrality of magnetic charges. We consider a material in the form a thin strip and enforce a charged domain wall by suitable boundary conditions on m. In the limit ε→ 0 and for fixed λ> 0 , corresponding to the macroscopic limit, we show that the energy Γ -converges to a limit energy where jump discontinuities of the magnetization are penalized anisotropically. In particular, in the subcritical regime λ≦ 1 , one-dimensional charged domain walls are favorable, in the supercritical regime λ> 1 , the limit model allows for zigzaging two-dimensional domain walls.