Bilayer triple-Q state driven by interlayer higher-order exchange interactions

Using first-principles calculations and an atomistic spin model, we predict the stabilization of a bilayer triple-Q state in an atomic Mn bilayer on Ir(111) due to interlayer higher-order exchange interactions. Based on density functional theory (DFT) we study the magnetic interactions and ground state in a Mn monolayer and bilayer on the Ir(111) surface. We calculate the energy dispersion of spin spirals (single-Q states) to scan a large part of the magnetic phase space and to obtain constants of pairwise exchange interactions. By including spin-orbit coupling, we determine the strength of the Dzyaloshinskii-Moriya interaction. To reveal the role of higher-order exchange interactions in these films, we consider multi-Q states obtained by a superposition of spin spirals. For the Mn monolayer in fcc stacking on Ir(111), the triple-Q state exhibits the lowest total energy in DFT, while the Néel state is most favorable for hcp stacking. For the Mn bilayer on Ir(111), two types of the triple-Q state are possible. In both magnetic configurations, a triple-Q state occurs within each of the Mn layers. However, only in one of them, the spin alignment between the layers is such that nearest neighbor spins of different layers also exhibit the tetrahedron angles that characterize the triple-Q state. We denote this state—which has the lowest total energy in our DFT calculations—as the ideal bilayer triple-Q state. This state exhibits significant topological orbital moments within each of the two Mn layers which are aligned in parallel resulting in a large topological orbital magnetization. We interpret the DFT results within an atomistic spin model, which includes pairwise Heisenberg exchange, the Dzyaloshinskii-Moriya interaction, as well as higher-order exchange interactions. We classify the different types of higher-order interactions into intralayer terms, i.e., acting only between spins within one of the layers, and interlayer interactions, in which spins in both layers are involved. We point out the role of an odd or even distribution of spins in the multispin interactions between the layers. Finally, we demonstrate that the stabilization of the ideal bilayer triple-Q state in a Mn bilayer on Ir(111) can only be explained upon taking the effect of interlayer higher-order exchange interactions into account.