Formation of magnetic skyrmions with tunable properties in PdFe bilayer deposited on Ir(111)

We perform an extensive study of the spin-configurations in a PdFe bilayer on Ir(111) in terms of ab initio and spin-model calculations. We use the spin-cluster expansion technique to obtain spin model parameters, and solve the Landau-Lifshitz-Gilbert equations at zero temperature. In particular, we focus on effects of layer relaxations and the evolution of the magnetic ground state in an external magnetic field. In the absence of a magnetic field, we find a spin-spiral ground state; while applying an external magnetic field, skyrmions are generated in the system. Based on energy calculations of frozen spin configurations with varying magnetic field we obtain excellent agreement for the phase boundaries with available experiments. We find that the wavelength of spin-spirals and the diameter of skyrmions decrease with increasing inward Fe layer relaxation, which is correlated with the increasing ratio of the nearest-neighbor Dzyaloshinskii-Moriya interaction and the isotropic exchange coupling, $D/J$. Our results also indicate that the applied field needed to stabilize the skyrmion lattice increases when the diameter of individual skyrmions decreases. Based on our observations, we suggest that the formation of the skyrmion lattice can be tuned by small structural modification of the thin film.

Figure 1

Calculated Fe-Fe isotropic exchange interactions for PdFe/Ir(111) as a function of the interatomic distance measured in units of the in-plane lattice constant $\left(a_{2D}\right)$ for different Fe layer relaxations.

Figure 2

Magnitudes of the Fe-Fe DM vectors for PdFe/Ir(111) as a function of the interatomic distance measured in units of the in-plane lattice constant $\left(a_{2D}\right)$ for different Fe layer relaxations. The inset shows a sketch of the in-plane components of the calculated DM vectors between a central Fe atom $\left(C\right)$ and its nearest and next nearest Fe neighbors at $-5\%$ relaxation.

Figure 3

Energies of frozen spin-spiral (SS), selected skyrmion phase (SkX), and ferromagnetic (FM) spin structures in external magnetic field $\left(B_{\mathrm{ext}}\right)$ perpendicular to the surface for $-5\%$ relaxation of the Fe layer in PdFe/Ir(111). All the energies correspond to the $128\times 128$ lattice used for the spin-dynamics simulations. The vertical lines denote the phase boundaries between the three different states.

Figure 4

Calculated skyrmion numbers $\left(N_{\mathrm{sk}}\right)$ on a $128\times 128$ lattice at zero temperature as a function of external magnetic field $\left(B_{\mathrm{ext}}\right)$ in cases of $-5\%$ and $-10\%$ relaxations of the Fe layer in PdFe/Ir(111). The points represent skyrmion numbers averaged over 15 independent spin-dynamic simulations and the lines denote an interpolated curve between the points. The magnetic phases as described in the text are indicated.

Figure 5

(a) Zero-field ground state spin-spiral configuration of PdFe/Ir(111) system in case of $-5\%$ relaxation and (c) in case of $-10\%$ relaxation as obtained from spin-dynamics simulation. An external magnetic field leads to skyrmion lattice formation; (b) represents the skyrmion lattice for $-5\%$ Fe layer relaxation and (d) for $-10\%$ relaxation. Small red and blue arrows indicate magnetic moments with dominating out-of-plane and in-plane components, respectively.