Fig. 1: Higher-order exchange interactions and harmonic transition state theory.
From: Impact of higher-order exchange on the lifetime of skyrmions and antiskyrmions
a Meta-stable skyrmion at B⊥ − BC = 3.95 T in fcc-Pd/Fe/Ir(111) including higher-order exchange interactions (HOI). The color-code for the orientation of magnetic moments used in this work is shown in the lower-left corner. b The interactions of magnetic moments due to the biquadratic (2-site, gray), the 3-site (cyan) and the 4-site (pink) HOI are schematically depicted by straight lines (products mi ⋅ mj) and curved lines (combinations of products) (see Eq. (3)). c, d Total energy including all terms in Eq. (3) (black) and HOI energy contributions (gray, cyan, pink) to the total energy as a function of the geodesic distance R along the minimum energy path (MEP) of a skyrmion (c) and antiskyrmion annihilation (d) at B⊥ − BC = 3.95 T. The minima (saddle points (SPs)) are visualized by red (green) circles and the energy barrier is depicted by a vertical black line. e Schematic two-dimensional energy landscapes with low curvature at the minimum (red) and large curvature at the SP (green) in the upper left corner and vice versa in the lower right corner. f Schematic adjacency pattern for a sub-matrix of the matrix of second-order derivatives for four magnetic moments (mi, mj, mk, ml) in nearest neighbor distance of each other. For the mid row of panel b chosen as an example, each block is colored according to the specific HOI term contributing to its values.
