Stabilty of biskyrmions in centrosymmetric magnetic films

Motivated by the observation of biskyrmions in centrosymmetric magnetic films [X. Z. Yu, Nat. Commun. 5, 3198 (2014)2041-172310.1038/ncomms4198, W. Wang, Adv. Mater. 28, 6887 (2016)ADVMEW0935-964810.1002/adma.201600889], we investigate analytically and numerically the stability of biskyrmions in films of finite thickness, taking into account the nearest-neighbor exchange interaction, perpendicular magnetic anisotropy (PMA), dipole-dipole interaction (DDI), and the discreteness of the atomic lattice. The biskyrmion is characterized by the topological charge Q=2, the spatial scale λ, and another independent length d that can be interpreted as a separation of two Q=1 skyrmions inside a Q=2 topological defect in the background of uniform magnetization. We find that biskyrmions with d of order λ can be stabilized by the magnetic field within a certain range of the ratio of PMA to DDI in a film having a sufficient number of atomic layers Nz. The shape of biskyrmions has been obtained by the numerical minimization of the energy of interacting spins in a 1000×1000×Nz atomic lattice. It is close to the exact solution of the Belavin-Polyakov model when d is below the width of the ferromagnetic domain wall. We compute the magnetic moment of a biskyrmion and discuss ways of creating biskyrmions in experiment.