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 $1000\times 1000\times N_z$ 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.

Figure 1

Computer-generated spin field in a Bloch-type byskirmion in a ferromagnetic film.

Figure 2

The plot of the function $f\left(p\right)$ given by Eq. (24).

Figure 3

DDI and effective DDI between columns of spins in a simple cubic lattice for the film thickness $N_z=100$. Upper panel: Logarithmic scale. Lower panel: Linear scale at short distances.

Figure 4

The energy landscape of a Belavin-Polyakov biskyrmion (upper panel) and biantiskyrmion (lower panel). The biskyrmion has a local energy minimum at a finite separation, $d>0$, while the biantiskyrmion for the same parameters has no energy minimum at all.

Figure 5

Upper panel: Spin field in a $Q=-2$ BP biantiskyrmion ($d>0$). Lower panel: Spin field in a $Q=-2$ BP antiskyrmion ($d=0$). In the presence of DDI, PMA, and the applied magnetic field, the initial state with $d>0$ in the course of relaxation evolves to the minimal-energy state with $d=0$.

Figure 6

Magnetic-field dependence of the effective radius of a stable skyrmion bubble for the “original” and rescaled models.

Figure 7

Magnetic field dependencies of the parameters $\lambda$ and $d$ of stable biskyrmions in a film of thickness $N_z=100$ with $D/J=0.001$ and $\beta =1$. Exchange energy $E_{\text{ex}}$ and the total energy $\mathrm{\Delta }E$ (with respect to the uniformly magnetized spins with spins down) are shown on the right $y$ axis. For stronger fields, the biskyrmion is smaller and close to the BP biskyrmion, as shown by the energies.

Figure 8

Computer-generated images of the spin field in a stable BP biskyrmion (upper panel) and a biskyrmion bubble (lower panel) numerical solutions of the centrosymmetric Heisenberg lattice model with the ferromagnetic exchange, DDI, PMA, and external field. The values of parameters are written above the figures. Orange/green color indicates positive/negative $z$ components of the spin field. The in-plane spin components are shown as white arrows.

Figure 9

Magnetic field dependencies of the parameters $\lambda$ and $d$ of stable biskyrmions in a film of thickness $N_z=100$ with $D/J=0.001$ and $\beta =0.5$.

Figure 10

Magnetic-field dependencies of the parameters $\lambda$ and $d$ of stable biskyrmions (left axis) and the exchange energy (right axis) in films of thickness $N_z=20$ and 10 with $D/J=0.001$ and $\beta =1.$