Fig. 4

Energy dispersion of spin spirals for ultrathin Co films. a Energy dispersion of planar homogeneous spin spirals for Co/Ir(111), Rh_fcc/Co/Ir(111) and Rh_hcp/Co/Ir(111) along the high symmetry directions \({\bar{M}} - \bar{\mathit{\Gamma}} - {\bar{K}}\) of the two-dimensional Brillouin zone without spin–orbit coupling (SOC). Energies are given with respect to the FM state. The symbols (open gray circles for Co/Ir(111), red filled circles for Rh_fcc/Co/Ir(111) and green triangles for Rh_hcp/Co/Ir(111)) represent the DFT calculations while the solid lines are the fits to the Heisenberg exchange interaction beyond nearest neighbors (see Supplementary Note 2 for details). b Zoom around the FM state (\({\bar{\it{\Gamma }}}\)-point) of a, where dashed lines represent a q² fit in the case of Co/Ir(111) and a fit including an additional q⁴ term in the case of Rh/Co/Ir(111). c Zoom around the FM state (\({\bar{\it{\Gamma }}}\)-point) of the energy dispersion for cycloidal spin spirals including the effect of spin–orbit coupling. The DMI leads to the local energy minima for clockwise rotating spin spirals along both high symmetry directions and the magnetocrystalline anisotropy energy is responsible for the constant energy shift of the spin spirals with respect to the FM state