Linked skyrmions in shifted magnetic bilayer

TL;DR

This work proposes a shifted magnetic bilayer with mutually perpendicular DMI in two coupled layers to realize complex topological spin textures. By combining phase-space analysis, homotopy theory, and first-principles material guidance, it identifies a novel linked-skyrmion class stabilized by anti-aligned points, with topological charge $Q^ ext{D}=Q^ ext{T}-Q^ ext{B}$ that can be arbitrarily large. In addition to linked skyrmions, the system supports conventional skyrmion bags and $k\pi$-skyrmions, all tunable via interlayer coupling $J_C$ and external field $B_ ext{ext}$. A Ni/InAs(001) thin-film realization is proposed based on ab initio calculations, demonstrating feasible material parameters and characteristic scales (~130 nm × 45 nm) for observing these textures. The findings open avenues for high-density, topologically rich spin textures with potential for enhanced transport responses in skyrmion-based devices.

Abstract

We present a shifted magnetic bilayer that exhibits various magnetic phases and magnetic textures with arbitrarily large topological numbers. The proposed system is characterised by a mutually orthogonal Dzyaloshinskii-Moriya interaction (DMI) in two different layers which can be induced by suitably placing non-magnetic atom with spin-orbit coupling. At weak interlayer coupling, the ground state resembles a checker-board pattern containing regions with unfavourable magnetic alignment which we call anti-aligned points. At finite interlayer coupling and finite external magnetic field, the bilayer can demonstrate a new class of magnetic solitons where multiple magnetic solitons can be connected by topological point defects which we call linked skyrmion. In addition to that the model also demonstrates conventional skyrmion-bags and $kπ$-skyrmions. Finally, with rigorous first principle calculations, we propose a suitable material candidate where these magnetic configurations can be observed.

Figures (13)

• Figure 1: Schematic of the shifted magnetic bilayer. (a) Red, blue and yellow spheres show the top magnetic, bottom magnetic and non-magnetic sites. Resulting direction of the DMI vectors are shown with yellow arrows. Red and blue arrows show the favourable orientation of the spin due to the DMI. Green region shows the nonmagnetic spacer layer. (b) Top view of the bilayer. Dashed lines show the nearest neighbours from opposite layers.
• Figure 2: Phase diagram and different periodic ground states of the bilayer Hamiltonian (Eq.\ref{['H']}). (a) Phase diagram showing checker-board (magenta), spiral (yellow), skyrmion lattice (cyan) and saturated ferromagnetic (white) phase and the location of the representative configurations. (b) Schematic of the checker board in the decoupled limit ($J_C$=0) and in absence of external magnetic field ($B_\mathrm{ext}$=0). Orange ellipses show the anti-aligned points. (c) Colour code used in the figures demonstrated with an isotropic Néel skyrmion. $\odot(\otimes)$ shows the region with positive(negative) out of plane components denoted by black(white) colour. Layer resolved checker-board pattern with $B_\mathrm{ext}$=0.0 and (d)$J_C$=0.0$J$, (e)$J_C$=0.006 and (f)$J_C$=0.012$J$. $\bm{q}_\mathrm{T,B}$ shows the direction of the spiral $q$ vector in top,bottom layer. The numbers on the top show the optimal cell size. White lines show the trajectory of the zero out-of-plane magnetisation. Orange circles show the location of the anti-aligned points and red and blue arrows show the out of plane magnetic alignment from top and bottom layer in the corresponding region. (g) Skyrmion lattice phase obtained at ($J_C$=0.02$J$,$\mu B_\mathrm{ext}$=0.007$J$). The hexagonal region denotes the domain of the skyrmion (Eq.\ref{['Q']}) and the white arrows show the elongation of the skyrmion in individual layers. (h,i) Show two degenerate spin spiral configurations obtained at ($J_C$=0.03$J$,$\mu B_\mathrm{ext}$=0.002$J$).
• Figure 3: Linked skyrmion with different topological charges. (a) Enlarged view of the topological point defect. Red and blue arrows show the magnetic moment on top and bottom layer. The green arrow shows the average of the four magnetic moments of the top layer which constitutes the anti-aligned point confined within the grey region. Bottom panel shows the top view of the extended region containing the point defect where the grey box denotes the region of the anti-aligned point. (b-f) Different linked skyrmions with their topological charge from top ($Q^\mathrm{T}$) and bottom ($Q^\mathrm{B}$) layer and the corresponding charge of the topological point defect ($Q^\mathrm{D}$). All configurations are stabilised at ($J_C$=0.02$J$,$\mu B_\mathrm{ext}$=0.007$J$) in a 400$\times$400 mesh. The number on the top shows the dimesnsion of the plotting region. The small square region of the bottom panel of column (b) correspond to the configuration at the bottom panel of column (a).
• Figure 4: Different composite skyrmions without point defect. (a-c) $k\pi$-skyrmions and (d-l) skyrmion bags with different topological charges stabilised at ($J_C$=0.02$J$,$\mu B_\mathrm{ext}$=0.007$J$) in a 400$\times$400 mesh. The figures present the total magnetisation of both layers. The legends show the topological charge from top and bottom layer.
• Figure 5: Material realisation of the bilayer model. (a) The thin-film structure of Ni/InAs(001). Red spheres denote Ni (magnetic) layer and yellow spheres denote In (main source of SOC). As is shown in green. (b) Total energy of spin spiral where red and black dashed lines represent spiral along $y$ and $x$ direction on the top layer.