In atomic GdFe$_2$ films capped by 4$d$ and 5$d$ transition metals, we show that skyrmions with diameters smaller than 12 nm can emerge. The Dzyaloshinskii--Moriya interaction (DMI), exchange energy, and the magnetocrystalline anisotropy (MCA) energy were investigated based on density functional theory. Since DMI and MCA are caused by spin--orbit coupling (SOC), they are increased with 5$d$ capping layers which exhibit strong SOC strength. We discover a skyrmion phase by using atomistic spin dynamic simulations at small magnetic fields of $\sim$1 T. In addition, a ground state that a spin spiral phase is remained even at zero magnetic field for both films with 4$d$ and 5$d$ capping layers.

As an outstanding information carrier, skyrmion is increasingly widely used in devices based on complex geometries. To achieve the skyrmion-based spintronic devices, a reasonable and feasible nanotrack is essential. In this paper, we conducted a study on the current-driven skyrmion movement in a circular-ring-shaped nanotrack. Our results suggest that the asymmetry of the inside and outside boundary of the circular ring changed the stable position of the skyrmion, causing it to move like the skyrmion Hall effect when driven by currents. Moreover, the asymmetric boundaries have advantages in enhancing or weakening the skyrmion Hall effect. Additionally, we also compared the skyrmion Hall effect from the asymmetric boundary of circular ring nanotracks with that from inhomogeneous DMI. It is found that the skyrmion Hall effect caused by the circular ring is significantly greater than that caused by the inhomogeneous DMI. These results contribute to our understanding of the skyrmion dynamics in confined geometries and offer an alternative method for controlling the skyrmion Hall effect of skyrmion-based devices.

In this review, we trace the evolution of the quantum spin-wave theory treating non-collinear spin configurations. Non-collinear spin configurations are consequences of the frustration created by competing interactions. They include simple chiral magnets due to competing nearest-neighbor (NN) and next-NN interactions and systems with geometry frustration such as the triangular antiferromagnet and the Kagomé lattice. We review here spin-wave results of such systems and also systems with the Dzyaloshinskii-Moriya interaction. Accent is put on these non-collinear ground states which have to be calculated before applying any spin-wave theory to determine the spectrum of the elementary excitations from the ground states. We mostly show results from a self-consistent Green’function theory to calculate the spin-wave spectrum and the layer magnetizations at finite T in two and three dimensions as well as in thin films with surface effects. Some new unpublished results are also included. Analytical details and the validity of the method are shown and discussed.