Skyrmionium meta-matter: a topologically heterogeneous magnetic crystal with emergent hybrid dynamics

TL;DR

This work reveals that pure skyrmionium lattices are intrinsically unstable and tend to spiral states, but mixtures of skyrmioniums ($Q=0$) and skyrmions ($Q=-1$) form metastable Skm$_n$Sk$_m$ crystals with diverse plane-group symmetries. By formulating a micromagnetic energy functional and mapping the phase diagram, the authors show that lattice-spacing tuning can drive transitions between polymorphs, while mutual interactions stabilize mixed lattices that would be unstable in isolation. The collective spin dynamics of these mixed lattices under out-of-plane and in-plane AC fields exhibits a rich spectrum of hybrid modes, including deformation-assisted rotations and orbital motions, spanning from sub-GHz to tens of GHz, enabling tunable magnonic responses. Overall, skyrmionium meta-matter provides a versatile platform for topologically heterogeneous magnetic lattices with reconfigurable structure and dynamics, offering potential for advanced spintronic and magnonic applications.

Abstract

We introduce and systematically investigate a new class of topological magnetic textures, skyrmionium meta-matter, composed of skyrmioniums (Skm, $Q=0$) and skyrmions (Sk, $Q=-1$) arranged in periodic lattices mimicking the richness of atomic materials. Pure skyrmionium lattices are unstable against elongation distortions and relax into the spiral phase, but even a small fraction of skyrmions acts as topological "pins" that stabilize diverse mixed Skm--Sk crystals. We classify these states by topological stoichiometry (Skm$_n$Sk$_m$) and show that each composition hosts multiple metastable polymorphs with distinct plane-group symmetries. Structural transformations between polymorphs are achieved by varying the lattice spacing, suggesting experimental control via pressure or strain. The collective spin dynamics is explored for both in-plane and out-of-plane AC magnetic fields. The resulting absorption spectra show resonant modes beyond the two rotational and one breathing mode of conventional skyrmion lattices. We identify hybrid excitations unique to Skm--Sk crystals, including (i) deformation-assisted rotations, where skyrmions acquire polygonal shapes and rotate, and (ii) orbital modes, where breathing skyrmioniums induce circular motion of confined skyrmions without changing their size. Mode frequencies span sub-GHz to above 10 GHz, consistent with exchange and DMI energy scales. Our results establish skyrmionium-based meta-matter as a versatile platform for tunable, topologically heterogeneous magnetic lattices with rich structural and dynamical properties, paving the way for reconfigurable magnonic and spintronic applications.

Figures (10)

• Figure 1: (a) Schematic of isolated Néel skyrmioniums in polar magnets with C$_{nv}$ symmetry or in multilayers with induced DMI. An isolated skyrmionium is a complex object composed of two nested skyrmions with opposite topological charges, $Q=+1$ and $Q=-1$, resulting in a total $Q=0$. (b) Structural evolution of isolated Skms within the field range $c-a$ of the phase diagram shown in (c), exhibiting Skm collapse at point $a$ and expansion at point $c$. (c) Phase diagram of states, highlighting the region of isolated Skms alongside the regions of stable SkL and the spiral phase (see text for details). The red line $0$–$B$ at $h=0$ delineates the region of elliptical instability of the skyrmion lattice. (d) Field-driven evolution of characteristic radii at the levels of the magnetization with $\theta=\pi/2, \pi, 3\pi/2$ for both SkmL and isolated skyrmioniums ($k_u=0$). In general, all radii diverge at the critical line $c - B$. The width of the circular domain wall $W$ [bottom spin configuration in (b)], however, passes without any interruption over the critical field value. The table summarizes the coordinates of various critical points introduced in the phase diagram and discussed in the text.
• Figure 2: Instability of the pure skyrmionium lattice. $h=0.2$, $k_u=0$. (a) Energy densities of the square and hexagonal SkmLs, shown as black curves with red dots #1 and #2 marking the energy minima. Snapshots #1 and #2 illustrate the corresponding magnetization distributions at the energy minima for fixed unit-cell geometries. Blue curves show the energy variation as a function of the unit-cell size, revealing the instability of the hexagonal SkmL under lattice deformation: the SkmL elongates toward configuration #4 and eventually transforms into a spiral state. The elongation path toward configuration #3 is separated by a small energy barrier. (b) Evolution of the magnetization patterns along the path connecting the two energy minima corresponding to the hexagonal and square SkmLs (see text for details).
• Figure 3: (a) Expansion of isolated skyrmions and skyrmion lattices as the system approaches the critical anisotropy value $k_u(B)$ from the right and left, respectively. The blue dashed curves show the core size of isolated skyrmions determined according to Lilley’s definition, the red solid lines represent the skyrmion core size within a skyrmion lattice, and the green solid lines indicate the lattice period. For a fixed magnetic field, increasing $k_u$ leads to a critical point where the lattice releases isolated skyrmions. This transition corresponds to the intersection of the red and blue curves, where the lattice period diverges and the skyrmion core size exhibits a discontinuous jump. For $h=0$, the skyrmion lattice and isolated skyrmions remain disconnected. (b) Angular profiles $\theta(r)$ of isolated skyrmions in zero magnetic field for different values of the easy-axis uniaxial anisotropy $k_u$, illustrating the progressive expansion of the skyrmion cores as $k_u$ approaches the critical value $k_u(B)=0.61685$ from the right. (c), (d) Snapshots showing the expansion of the SkL and isolated skyrmions for different values of the uniaxial anisotropy at $h=0$. The depicted area is the same for all structures and measures $30 \times 30$.
• Figure 4: (a) Energy-density “fingerprint” calculated as a function of the unit-cell dimensions. The black line traces the energy density of the hexagonal SkL for a fixed ratio of lattice parameters. The absence of a distinct local energy minimum corresponding to the hexagonal SkL—which becomes discernible only on the two-dimensional energy surface—indicates its instability and tendency to elongate into a spiral state (highlighted by the dotted orange arrow). (b) Snapshots illustrating the transition from a perfectly hexagonal skyrmion lattice to the elliptically unstable configuration.
• Figure 5: Examples of skyrmionium-based meta-matter with varying ratios of skyrmioniums and skyrmions. (a) A staggered square lattice with an equal number of Skms and Sks (SkmSk). (b) A square lattice Skm$_2$Sk with a reduced fraction of Sks. (c) A compound Skm$_2$Sk$_3$ with nonuniform Skm sizes and slight elongation. (d), (e) Polymorphs of the SkmSk$_2$ meta-matter exhibiting different planar symmetries. The corresponding symmetry elements are indicated within one unit cell for each mixed lattice. These structures exemplify the diversity of topological stoichiometries and plane-group symmetries possible in the skyrmionium–skyrmion compounds.