Design of 2D Skyrmionic Metamaterial Through Controlled Assembly

TL;DR

This work tackles the challenge of creating nontrivial, high-order skyrmionic textures with tailor-made topologies by introducing a simulated controlled assembly framework for 2D skyrmionic metamaterials in a Pd/Fe/Ir(111) monolayer. It combines high-throughput ASD simulations with AI-assisted texture design to assemble building blocks—skyrmions, antiskyrmions, and skyrmioniums—into lattice-like, flake-like, and cell-like metamaterials, and analyzes their stability under varying $B^{ext}$ and $T$. The study demonstrates long-lived textures with $Q=-2,-5,-12$ in lattices, and $Q=-22$/$-23$ honeycomb flakes, including nesting-based texture nesting and defect considerations, while showing robustness to parameter perturbations. The results offer a pathway to engineer complex, metastable magnetic textures with potential spintronic applications, and suggest experimental routes for controlled assembly using gradient fields, STM, or temperature gradients, supported by publicly available data and code.

Abstract

Despite extensive research on magnetic skyrmions and antiskyrmions, a significant challenge remains in crafting nontrivial high-order skyrmionic textures with varying, or even tailor-made, topologies. We address this challenge, by focusing on a construction pathway of skyrmionic metamaterials within a monolayer thin film and suggest several skyrmionic metamaterials that are surprisingly stable, i.e., long-lived, due to a self-stabilization mechanism. This makes these new textures promising for applications. Central to our approach is the concept of 'simulated controlled assembly', in short, a protocol inspired by 'click chemistry' that allows for positioning topological magnetic structures where one likes, and then allowing for energy minimization to elucidate the stability. Utilizing high-throughput atomistic-spin-dynamic simulations alongside state-of-the-art AI-driven tools, we have isolated skyrmions (topological charge Q=1), antiskyrmions (Q=-1), and skyrmionium (Q=0). These entities serve as foundational 'skyrmionic building blocks' to form the here reported intricate textures. In this work, two key contributions are introduced to the field of skyrmionic systems. First, we present a a novel combination of atomistic spin dynamics simulations and controlled assembly protocols for the stabilization and investigation of new topological magnets. Second, using the aforementioned methods we report on the discovery of skyrmionic metamaterials.

Figures (6)

• Figure 1: Pair-wise interaction between topological building blocks. (a) Pictorial representations of pair interaction dynamics as a function of applied magnetic field, for all pair-combinations of skyrmions, antiskyrmions, and skyrmionium. For comprehensive visualizations, we refer to the supplementary movies S1-S4. The red and blue stars indicate the calculated equilibrium (stationary) distances for each pair as influenced by the external magnetic field, with distances expressed in units of the Fe-Fe equilibrium distance $d_{\rm Fe-Fe} = 0.272$ nm. Blue stars indicate that for shorter distances the pair interaction energy is repulsive (equilibrium distance). Red stars, conversely, denote that for shorter distance, the spin textures either merge or annihilate (critical distance). (b) Temporal evolution of the total energy for each pair assembly process, as derived from ASD simulations. The $y$-axis quantifies the total energy. Accompanying subplots provide snapshots of the spin configurations during the assembly process. Color scheme: The adjoining colorbar clarifies the spin orientation in these configurations: dark blue color (-1) signifies spins pointing downwards, away from the viewer, while red color (1) represents spins pointing upwards, towards the viewer. Whit color indicates that the spin is parallel to the plane.
• Figure 2: Construction and stability of lattice-like skyrmionic metamaterials. Construction of an individual high-order antiskyrmion (the prefabricated building block) and subsequent construction of extended skyrmionic periodic textures (also called "lattice-like skyrmionic metamaterials" in the following) by combining many individual high-order antiskyrmions. The ASD simulation system size is $60 \times 60$ spins in all subfigures. The color scheme is the same as in Fig. 1. (a) Illustration of the step-by-step formation of a triangular high-order antiskyrmion via controlled assembly. The left diagram presents the initial- and post-relaxation energies at each stage, with relaxation times increasing from 50 ps (step 1) to 200 ps (step 5). Top and bottom subfigures display the corresponding initial and postrelaxation spin configurations. Additional details regarding the step-by-step formation of a high-order triangular antiskyrmion are available in Supplementary Movie S5. (b-d) Three distinct examples of lattice-like skyrmionic metamaterials. Here, the rotation of each prefabricated building block is hindered due to their position within an extended texture and implied interaction with adjacent prefabricated building blocks. In the top panels of figures (b-d), the lattice-like skyrmionic metamaterials in real space are shown. The bottom panels each display a stability map analysis of the corresponding lattice-like skyrmionic metamaterial as a function of applied magnetic field and temperature. A value of 1 indicates stability after 200 ps ASD simulation, while 0 denotes lattice disintegration occuring at some point before the 200 ps mark. (b) lattice-like skyrmionic metamaterial formed from $Q=-2$ high-order antiskyrmions. (c) A lattice-like skyrmionic metamaterial formed from $Q=-5$ high-order antiskyrmions. (d) An lattice-like skyrmionic metamaterial composed of $Q=-12$ high-order antiskyrmions.
• Figure 3: Building process and stability of flake-like skyrmionic metamaterials. (a) Two distinct methodologies for assembling a high-order ring-shaped antiskyrmion with $Q=-6$. The top pathway involves combining six antiskyrmions with one skyrmionium, whereas the bottom pathway utilizes one high-order antiskyrmion ($Q=-5$) paired with an additional antiskyrmion. Additional details regarding the formation of a high-order ring-shaped antiskyrmion are available in Supplementary Movie S6. (b) Temporal progression of a skyrmionic honeycomb lattice at 1 K. The left image depicts the initial spin configuration, while the right image illustrates the state after 200 ps ASD time evolution. (c) and (d) The left-most panel presents a flawless honeycomb flake-like skyrmionic metamaterial alongside a variant with a 5-7 ring defect (indicated by red dots). The central subplots in (c) and (d) display the real-space spin configurations of a skyrmionic honeycomb flake after 200 ps ASD simulations at four distinct temperatures. The far-right panels in (c) and (d) showcase the respective stability phase diagrams. Here, a value of 1 indicates stability after 200 ps of ASD simulations, while 0 denotes that the flake vanished. The color scheme in the spin configurations is the same as in Fig. 1.
• Figure 4: Cell-like skyrmionic metamaterials with distinct topological charges. (a-d) skyrmion bags with topological charges ranging from $-1$ to $-4$; (e) skyrmion bags with topological charges ranging from $-11$ to $-24$. Additional details regarding the formation of those textures are available in Supplementary Movie S7. The color scheme in the spin configurations is the same as in Fig. 1.
• Figure 5: Tests of the robustness and general applicability of the controlled assembly process. (a) Subplots 1 to 4 display the final states after 200 ps of an initial state consisting of two skyrmionium and one antiskyrmion (the same for all four subplots). The color scheme is the same as in Fig. 1.(b) Results under random perturbations of varying maximum amplitudes on each nearest shell applied to $\mathbf{D}_{ij}$, $J_{ij}$, and $K^U$. (c) Final states after scaling of the exchange interaction parameters $J_{ij}$ and damping $\alpha$ (see Supplementary Note 1), following 200 ps ASD simulations. The color from white to dark blue and the inside number from 1 to 4 indicates the final state displayed in (a). (d-f) Final states with random perturbations ($\pm8\%$) applied to combinations of $\mathbf{D}_{ij}$-$K^{\mathrm{U}}$, $J_{ij}$-$K^{\mathrm{U}}$, and $J_{ij}$-$\mathbf{D}_{ij}$, respectively.