Stabilty of biskyrmions in centrosymmetric magnetic films

Motivated by the observation of biskyrmions in centrosymmetric magnetic films [X. Z. Yu et al., Nat. Commun. 5, 3198 (2014), W. Wang et al., Adv. Mater. 28, 6887 (2016)], 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 lambda, 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 lambda 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 N-z. The shape of biskyrmions has been obtained by the numerical minimization of the energy of interacting spins in a 1000 x 1000 x N-z 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.