Skyrmions with Attractive Interactions in an Ultrathin Magnetic Film

We determined the parameters of a classical spin Hamiltonian describing an Fe monolayer on Pd(111) surface with a ${\mathrm{Pt}}_{1-x}{\mathrm{Ir}}_x$ alloy overlayer from ab initio calculations. While the ground state of the system is ferromagnetic for $x=0.00$, it becomes a spin spiral state as Ir is intermixed into the overlayer. Although the Dzyaloshinsky-Moriya interaction is present in the system, we will demonstrate that the frustrated isotropic exchange interactions play a prominent role in creating the spin spiral state, and these frustrated couplings lead to an attractive interaction between Skyrmions at short distances. Using spin dynamics simulations, we show that under these conditions the individual Skyrmions form clusters, and that these clusters remain stable at finite temperature.

Figure 1

(a) Calculated isotropic exchange interactions $J_{ij}$ in a (${\mathrm{Pt}}_{1-x}{\mathrm{Ir}}_x$)Fe bilayer on Pd(111) as a function of the distance $d$ between the Fe atoms, given in lattice constants of the triangular lattice $a$. (b) Energy per spin in the spin spiral state $E_{\mathrm{SS}}(\mathit{q})/N$ as a function of the wave vector along the $[1\overline{1}0]$ direction, relative to the ferromagnetic state $E_{\mathrm{FM}}/N$. Inset shows a slight anisotropy favoring the $[1\overline{1}0]$ axis over the $[2\overline{1}\overline{1}]$ direction for $x=0.10$.

Figure 2

Attractive interaction between Skyrmions for $x=0.10$ and $B=4.22\mathrm{T}$. The field-polarized state represents the ground state for $B>4.18\mathrm{T}$ [44]. (a) Interaction energy between two Skyrmions along the $[2\overline{1}\overline{1}]$ and $[1\overline{1}0]$ directions. Squares and circles denote data points; the interpolated lines are guides to the eye. Inset shows a close-up of the first minimum. (b) Real-space spin configuration of two isolated Skyrmions (squares, circles) and two interacting Skyrmions (line), with the centers of the Skyrmions at the same lattice points in the two cases. $S_i$ is the projection of the vector ${\mathit{S}}_i$ on the $[2\overline{1}\overline{1}]$ line connecting the centers, shown by arrows in the legend. The color bar describes the orientation of the spins in the legend and in Fig. 3.

Figure 3

(a) Ordered initial configuration of Skyrmions at $T=0\mathrm{K}$, for $x=0.10$ and $B=4.22\mathrm{T}$. (b),(c) Final configuration after a thermalization of $t=605\mathrm{ps}$ at $T=4.7\mathrm{K}$, for different coupling parameters: (b) attractive Skyrmions at $x=0.10$ and $B=4.22\mathrm{T}$, (c) repulsive Skyrmions at $x=0.00$ and $B=0.00\mathrm{T}$. The lattice size is $N=128\times 128$ atoms.

Figure 4

Pair correlation function $f(r)$ of Skyrmions after a thermalization of $t=484\mathrm{ps}$. (a) Difference between repulsive ($x=0.00$, $B=0.00\mathrm{T}$) and attractive ($x=0.10$, $B=4.22\mathrm{T}$) Skyrmions at $T=4.7\mathrm{K}$. (b) Temperature dependence of $f(r)$ for $x=0.10$ and $B=4.22\mathrm{T}$. The system contained 31 Skyrmions on an $N=128\times 128$ atomic lattice.