Impact of higher-order exchange on the lifetime of skyrmions and antiskyrmions

Abstract

Reliable control of skyrmion lifetime is essential for realizing spintronic devices, yet the role of higher-order exchange - which can lead to skyrmion stabilization - remains largely unexplored. Here we calculate lifetimes of isolated skyrmions and antiskyrmions at transition-metal interfaces based on an atomistic spin model that includes all fourth-order exchange terms. Within harmonic transition-state theory, we evaluate both energetic and entropic contributions and find substantially enhanced lifetimes when higher-order exchange is included. The four-spin four-site interaction raises the energy barrier and lowers the curvature of the energy landscape at the collapse saddle point, increasing the pre-exponential factor. We show that skyrmions and antiskyrmions can remain thermally stable even without Dzyaloshinskii-Moriya interaction (DMI), and that tuning the four-spin term by a small amount modulates the prefactor over orders of magnitude. Our results identify higher-order exchange as a promising route to stabilize topological magnetic textures - in particular in systems lacking DMI - and to engineer their thermally activated decay.

Figures (8)

• Figure 1: Higher-order exchange interactions and harmonic transition state theory. a Meta-stable skyrmion at $B_\perp-B_C=3.95$ 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 $\mathbf{m}_i\cdot\mathbf{m}_j$) and curved lines (combinations of products) (see Eq. (\ref{['eq:hamiltonian']})). c,d Total energy including all terms in Eq. (\ref{['eq:hamiltonian']}) (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 $B_\perp-B_C=3.95$ 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 ($\mathbf{m}_i$, $\mathbf{m}_j$, $\mathbf{m}_k$, $\mathbf{m}_l$) 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.
• Figure 2: Effect of higher-order exchange interactions on skyrmion and antiskyrmion lifetime.a,b,c Saddle point configuration for the collapse of the isolated skyrmion at $B_\perp-B_C=1.25$ T (a) and $B_\perp-B_C=3.25$ T (b) and antiskyrmion at $B_\perp-B_C=3.25$ T (c). d Radius of skyrmions (blue) and antiskyrmions (red) in fcc-Pd/Fe/Ir(111) with (solid lines and circles) and without (dashed line) explicitly considering HOI as a function of the magnetic field. The magnetic field is given relative to the field $B_C$ for the onset of the field polarized phase for the respective energy model. e Energy barriers associated with the skyrmion and antiskyrmion collapse as a function of the magnetic field with and without (see Ref. malottki2019paul2020) explicitly considering HOI. The field regime where the Chimera collapse of the skyrmion is favoured is colored green. f Pre-exponential factor $\nu_0$ in units of temperature $T$ plotted over $B_\perp-B_C$ for skyrmions (blue) in the regime of the radial collapse and antiskyrmions (red) for the energy model considering HOI (circles) and not explicitly taking HOI into account (dashed lines) g Mean lifetime $\tau$ (see colorbar) for different magnetic fields and different temperatures for the regime of radial skyrmion collapse. As a reference, the contour line for a lifetime of $1$ h for skyrmions without considering HOI explicitly is also given (see Ref. malottki2019). h Mean lifetime of antiskyrmions taking HOI into account.
• Figure 3: HOI and curvature of the energy surface for the skyrmion.a,b Local part of the eigenvalue spectrum of the Hessian matrix of the skyrmion (a, blue) and the skyrmion saddle point (SP) (b, green and blue) in fcc-Pd/Fe/Ir(111) including HOI as a function of the magnetic field $B_\perp-B_C$. Eigenvalues above the magnon gap are indicated by the orange painted area. For reference the corresponding local eigenvalue spectra in the model without considering HOI explicitly are shown as gray dashed lines. Selected eigenmodes are labeled while the number in brakets gives their multiplicity (see surrounding text). c,e Curvature associated with the local eigenvalue spectrum (see Eq. \ref{['eq:curvature']})) for the skyrmion (blue, c) and the corresponding SP (green and blue, e) for the energy model including HOI (solid line). The local curvatures of the SP for the model neglecting HOI are drawn with dashed gray lines. In e only the regime of the radial collapse is shown. d Energy along the seesaw eigenmodes of the SP including HOI (blue, SP in f) and of the SP excluding HOI (gray, SP in h). g Configuration obtained after displacing the SP in f along the indicated direction (black arrow) using the corresponding eigenvector of the seesaw mode. i Total curvature difference of the minimum and the SP for the regime of the radial collapse mechanism.
• Figure 4: Lifetimes of skyrmions and antiskyrmions as a function of the 4-site HOI.a,b Energy barrier (black) and pre-exponential factor $\nu_0/T$ (purple) for the skyrmion (a) and the antiskyrmion (b) annihilation in fcc-Pd/Fe/Ir(111) for a magnetic field of $B_\perp=6$ T (skyrmions) and $B_\perp=4$ T (antiskyrmions) as a function of the 4-site HOI $K_1$. The value $K_1^{\text{DFT}}$ corresponds to the one determined in DFT calculations paul2020. b,d Mean lifetime $\tau$ for isolated skyrmions (b) and antiskyrmions (d) for various temperatures $T$ and values of $K_1$ calculated within HTST. The one-week- and one-hour-isoline of $\tau$ are given as black lines.
• Figure 5: Skyrmion and antiskyrmion lifetime including HOI and neglecting DMI.a Energy barriers (black) and pre-exponential factors $\nu_0$ (purple) of skyrmions (filled circles) and antiskyrmions (open circles) in fcc-Pd/Fe/Ir(111) considering HOI and vanishing DMI for varying the magnetic field relative to the critical field $B_\perp-B_C$. b Radius of skyrmions and antiskyrmions as a function of the magnetic field. c: Mean lifetime $\tau$ (see colorbar) of the above mentioned isolated magnetic textures as a function of the temperature $T$ and the external magnetic field $B_{\perp}-B_C$.