Biskyrmion lattices in centrosymmetric magnetic films

We develop a theoretical framework to construct biskyrmion lattices which were observed in nonchiral magnetic films. Skyrmions and antiskyrmions in a pure two-dimensional (2D) exchange model can be described in terms of a complex function that is analytic except for a simple pole. In the same 2D pure exchange model, biskyrmion lattices are naturally described by the Weierstrass ℘ and ζ elliptic functions. We study films of finite thickness containing up to 1000×1000×100 spins starting with the biskyrmion lattice as the initial state. We investigate how the biskyrmion lattice evolves toward a minimum-energy state in zero magnetic field in the presence of perpendicular magnetic anisotropy (PMA) and dipole-dipole interaction (DDI). Metastable biskyrmion lattices exist at lower PMA strengths. At higher PMA strengths, we observe stable triangular lattices of biskyrmion bubbles containing Bloch lines, whose energies are lower than the energy of the uniformly magnetized state.