Topological and spin-orbit effects on orbital moments in ultra-thin magnetic films

Topological orbital moments (TOMs) are a direct hallmark of a magnetic texture with a non-trivial spin topology. In addition to giving insight into the topology of the magnetic texture, TOMs could also be used to manipulate magnetic structures with a compensated total spin moment. Experimental evidence of TOMs has been provided via transport measurements of the Hall effect in intercalated van-der-Waals materials. However, a direct observation of TOMs is still missing. Another complication arises for the unambiguous proof of the topological origin since orbital moments can also occur due to spin-orbit coupling. Here, we use first-principles electronic structure theory to investigate the origin of orbital moments in different compensated spin structures with a non-trivial topology. We focus on ultrathin magnetic films at surfaces such as Pd/Mn bilayers on Re(0001) which represent ideal model systems for the detection of TOMs since it is possible to apply experimental techniques with local resolution of magnetic properties such as spin-polarized scanning tunneling microscopy. Due to its trivial topology we use the row-wise antiferromagnetic (RW-AFM) state in a hexagonal monolayer to analyze the spin-orbit induced contributions. The triple-Q (3Q) state is a superposition state of three RW-AFM (1Q) states and thus electronically similar, however, due to its non-trivial spin topology it exhibits TOMs. The comparison between these two spin states allows us to disentangle the topological and spin-orbit contributions to the orbital moments. We find that TOMs have an important contribution in spin-compensated systems, since the spin-orbit coupling induced orbital moments are nearly compensated. First-principles calculations for atomic-scale skyrmion lattices in Fe monolayers on different surfaces exhibit the same general trend found for the 3Q state.