Trends of higher-order exchange interactions in transition metal trilayers

We present a systematic study of higher-order exchange interactions beyond the pairwise Heisenberg exchange in transition metal trilayers based on density functional theory calculations. We show that these terms can play an important role in magnetic trilayers composed of a single hexagonal Fe or Co atomic layer sandwiched between 4d and 5d transition metal layers. We study the dependence of the biquadratic and the three-site and four-site four spin interaction on the band filling of the 4d and 5d layers as well as the stacking sequence, i.e., fcc vs. hcp stacking. Our calculations reveal relatively small higher-order interactions for Co based trilayers. For Fe based trilayers with a Rh or Jr layer the higher-order terms can be on the same order of magnitude as pairwise Heisenberg exchange. The trends obtained for freestanding trilayers are used to understand the higher-order interactions in ultrathin film systems on surfaces that are experimentally accessible. It is shown that hcpRh/Fe/Ir(111) and hcp-Rh/Fe/Rh(111) exhibit the largest values for the biquadratic and the three-site four spin interaction of all systems under study. We further demonstrate that the three-site four spin interaction is responsible for the experimentally observed change of the magnetic ground state of Rh/Fe/Ir(111) from a spin spiral (single-Q) for fcc-Rh to a 2Q state for hcp-Rh. We find similar trends for Rh/Fe/Rh(111), i.e., replacing the Jr surface by the isoelectronic Rh surface. For Rh/Co/Ir(111), we obtain a negative value for the four-site four spin interaction which will lead to a reduced stability of magnetic skyrmions which are metastable in this film at zero magnetic field. In contrast, for Pd/Fe/Ir(111), the four-site four spin interaction is positive which leads to an enhanced stability of skyrmions.