Magnon edge states of skyrmion crystal in non-uniform magnetic field

TL;DR

The paper demonstrates that a skyrmion crystal in a thin ferromagnetic film supports chiral magnon edge states at interfaces between regions of different magnetic field strengths. Using a combination of stereographic projection, semiclassical spin-wave theory, and a reduced extended Dirac model, it shows that a topological transition in the SkX magnon spectrum governs the emergence and localization of these edge modes. Full numerical calculations in a stripe geometry under nonuniform fields confirm two intra-gap edge states with opposite group velocities, whose localization persists over a range of fields and can extend over multiple skyrmions. The findings suggest controllable, field-tunable magnon waveguides in SkX systems, with potential applications in magnonics and spin-based information transport.

Abstract

A regular lattice of magnetic skyrmions is the ground state of thin ferromagnetic films with Dzyaloshinskii-Moriya interaction in a relatively wide range of external magnetic fields. It was previously theoretically shown that upon the increase of magnetic field a topological transition in the magnon spectrum of such skyrmion crystal (SkX) may occur. Non-uniform magnetic field may lead to localized magnon states emerging at the interface between two half-planes of SkX. Using semiclassical quantization and the stereographic projection approach, we study such appearing edge states both in a full band structure calculation and in simplified effective model. The latter effective model described by extended Dirac equation is applicable to two relevant magnon bands near $Γ$ point. We show that both the chirality of emerging edge states and the degree of its localization at the interface is controlled by magnetic field profile. We demonstrate that the localization length may be as small as a few inter-skyrmion distances.

Figures (5)

• Figure 1: Sketch (a) of a phases shows order of a different magnetic phases in a model \ref{['eq:classicalenergy2']}: helix, SkX and uniform configuration. Panels (b), (c) and (d) show evolution of Br and CCW branches with increasing of external magnetic field $b$. (e) dependencies parameters of Hamiltonian \ref{['eq:redham']} of magnetic field $b$.
• Figure 2: (a)-(c) are dispersion of excitation for reduced model \ref{['eq:disp']} for different bottom and top magnetic fields, gray lines show interface mode dispersion tha appears on the interface, gradient from white to black corresponds to IPR value of wave functions, see Eq.\ref{['eq:ipr1']}.
• Figure 3: (a) Skyrmion configuration for $b_{+}=0.76$ and $b_{-}=0.26$, gray arrows shown in-plane component of local magnetization, shades of gray demonstrate value $n_z$, where white corresponds to $n_z=1$, and black to $n_z=-1$. Red and blue circles demonstrate skyrmion radii. (b) Red circles and blue dots correspond to position of small and large skyrmions, gray line demonstate boundaries of primitive stripe-cell.
• Figure 4: Gray dots on panels (a)-(c) are dispersions of elementary excitations in the stripes under different values of external magnetic fields. Red and blue areas show dispersion of normal modes of uniform SkX under $b_{+}$ and $b_{-}$ respectively.
• Figure 5: The inverse participation ratio and the shape of the wave function for the edge states at the interface. Panels (a) and (b) correspond to the reduced model, Eq.\ref{['eq:redham']}, showing (a) the dependence of IPR Eq.\ref{['eq:ipr1']} on fields $b_-$ and $b_+$, (b) the modulus square of wave function for three combinations of the fields, $b_-$ and $b_+$, here shades of gray for curves correspond to shades of gray in the panel (a). Panels (c) and (d) show the same quantities as in panels (a) and (b), but calculated in the full model Eq.\ref{['eq:BdGeq']}. The panel (c) shows IPR value, Eq.\ref{['eq:ipr2']}, for different values of $b_-$ and $b_+$, and the panel (d) depicts the profile of densities $\int dx(|u(\mathbf{r})|^2-|v(\mathbf{r})|^2)$. Here dashed lines correspond to spurious states localized away from the interface, due to imposed cyclic boundary conditions, these states would disappear in the infinite sample geometry. Thin vertical lines correspond to positions of skyrmions' centers, see Fig. \ref{['fig:statconfig']}(b).