Paradoxical Topological Soliton Lattice in Anisotropic Frustrated Chiral Magnets

Abstract

Two-dimensional chiral magnets are known to host a variety of skyrmions, characterized by an integer topological charge. However, these systems typically favor uniform lattices as a thermodynamically stable phase composed of either skyrmions (Q = -1) or antiskyrmions (Q = 1). In isotropic chiral magnets, skyrmion-antiskyrmion coexistence is typically transient due to mutual annihilation, making the observation of a stable, long-range ordered lattice a significant challenge. Here, we address this challenge by demonstrating a skyrmion-antiskyrmion lattice as a magnetic field-induced topological ground state in chiral magnets with competing anisotropic interactions, specifically Dzyaloshinskii-Moriya and frustrated exchange interactions. This unique lattice exhibits a net-zero global topological charge due to the balanced populations of skyrmions and antiskyrmions. Furthermore, density functional theory and spin-lattice simulations identify 2Fe/InSb(110) as an ideal candidate material for realizing this phase. This finding reveals new possibilities for manipulating magnetic solitons and establishes anisotropic frustrated chiral magnets as a promising material class for future spintronic applications.

Figures (20)

• Figure 1: Diversity of systems with frustrated exchange and chiral interactions.a, The diagram illustrates the diversity of systems with frustrated Heisenberg exchange interactions and DMI, including both isotropic and anisotropic cases. The case studied in the present work corresponds to the system with anisotropic interaction parameters, $i.e.$, the anisotropic frustrated chiral magnet. b, The schematic representation of the minimal model for the 2D magnet with isotropic frustrated exchange interactions on a square lattice where nearest and the next after-nearest neighbor exchange constants have opposite sign, $J_1>0$ (ferromagnetic) and $J_2<0$ (antiferromagnetic), see the spin-lattice Hamiltonian \ref{['spin_hamiltonian']} in Methods. c, The schematic representation of a rectangular lattice with anisotropic frustrated exchange interactions, characterized by unequal exchange coupling constants along the $x$- and $y$-directions. d, The DMI vectors between the nearest neighbors on a square lattice for the interfacial type DMI. For an isotropic system, the absolute values of the DMI vectors in both $x$ and $y$ directions are identical. e, The system with anisotropic DMI on a rectangular lattice, where the magnitude of the DMI differs along the $x$- and $y$-directions. f, A cycloidal spin spiral, often resulting from interfacial DMI in chiral magnets, characterized by a varying polar angle $\Theta$ with respect to the $z$-axis along the propagation direction. g, A cone spin spiral, characterized by a fixed polar angle $\Theta$ and a varying azimuthal angle along the propagation direction.
• Figure 2: Achieving S-AL ground states within model \ref{['Micro_Ham']} using anisotropic parameters: Through micromagnetic simulations utilizing direct energy minimization of our model, we obtain optimized unit cells (white rectangular boxes) for both SL and S-AL phases, as presented in a and b, respectively. The optimization scheme is detailed in fig. S9 supplementary. The magnetization vector field is visualized by standard color code. Here, $Q_\textrm{UC}=0$ signifies the rectangular unit cell which represents the net-zero topological charge lattice. c, Left and right spin textures depict individual skyrmion and antiskyrmion, respectively, obtained from the optimized S-AL configuration in b. d, Energy profiles for the SL and S-AL phases as a function of $\beta$. The S-AL phase is energetically favored over the SL phase for $\beta < \beta_\textrm{c}~(\approx 0.55$). The energy curves of the SL and S-AL phases intersect at the critical value $\beta_\textrm{c}$. The significant energy gain observed in the S-AL phase through shape optimization ($\theta$ optimization) is the primary reason for the high value of $\beta_\textrm{c}$. All calculations are performed under a constant external magnetic field of $h=0.35$. e, Energies for all competing non-collinear states--cycloidal-SS, cone-SS, SL, and S-AL--as well as saturated FM state are plotted as a function of $h$ for anisotropic frustrated chiral magnets. Here, we fix both $\alpha$ and $\beta$ to 0.1. The S-AL phase is identified as the ground state within the red-shaded regions, corresponding to the lowest energy states for specific ranges of $h$. For a comparison of the system's behavior at higher anisotropy, $\beta > \beta_\textrm{c}$, see fig. S10 supplementary.
• Figure 3: The phase diagrams in the $L_\mathrm{H}/L_\mathrm{D}$-$h$ plane.a, The phase diagram for the isotropic magnets. The parameter space is clearly divided into two regions: one dominated by exchange frustration and the other by DMI. Decreasing the ratio $L_\mathrm{H}/L_\mathrm{D}$ (enhancing exchange frustration) stabilizes the cycloidal-SS phase in the absence of external magnetic field $h$. Further, under an external field $h$, a cone-SS phase appears, with SL and S-AL as metastable states. In this region, the system exhibits three energetically favored phases: cycloidal-SS, cone-SS, and FM. The dashed line between cone-SS and FM denotes the second-order phase transition between them. Dominant DMI interactions are evidenced by the expansion of the cycloidal-SS ground state region at higher $L_\mathrm{H}/L_\mathrm{D}$ ratios. Typically, the cone-SS phase disappears under an external field in the strong DMI limit, while the cycloidal-SS phase undergoes a first-order phase transition to the SL phase. b, The phase diagram for the anisotropic frustrated chiral magnet. At $L_\mathrm{H}/L_\mathrm{D}\approx0.56$, the SL and S-AL phases exhibit equal energy across a range of $h$, defining a boundary between these two distinct lattice phases. The SL phase becomes energetically favorable compared to the S-AL phase upon increasing further $L_\mathrm{H}/L_\mathrm{D}$ ratios. This can be attributed to the fact that the critical value of $\beta_c$ remains below 0.1 in this regime.
• Figure 4: 2Fe/InSb(110), a prototypical 2D magnetic heterostructure:a, The slab geometry: a two-atomic-layer-thick Fe film grown on an InSb(110) substrate, forming a magnet/semiconductor heterostructure. b, The thin magnetic layer on the surface of a semiconductor adopts a lattice structure with 2D crystallographic axes, [$1^\prime00$] and [$01^\prime0$]. c, Within atomistic lattice model \ref{['spin_hamiltonian']}, energy lines representing all competing phases are plotted against the external magnetic field $B_\textrm{ext}$, applied perpendicular to the heterostructure. The S-AL phase, highlighted by the red region, is the ground state within a specific range of $B_\textrm{ext}$. The first-order phase transitions occur at the boundaries of this range: from the cycloidal-SS to the S-AL at low fields, and from the S-AL to the cone-SS at higher fields. The rightmost panel further confirms the second-order phase transition between the cone-SS and FM phases, as evidenced by the merging of energy lines beyond the vertical line. d, Spin configuration of the zero-field cycloidal-SS ground state, obtained within spin-lattice simulations. e, The S-AL phase at $B_\textrm{ext}$ = 0.5 Tesla. In this materials, the S-AL is also a charge (topological) neutral state due to equal number $Q=\pm 1$ topological charges. Similar to the micromagnetic model, the representing unit cell carries net-zero topological charge $i.e.,$$Q_\textrm{UC}=0$. f, Cone-SS phase at $B_\textrm{ext}$ = 0.9 Tesla. To improve visibility, the identical magnetic states in both Fe layers are spatially separated.
• Figure S5: Conical SS with unit vector corresponding to each spin, $\mathbf{n}(\mathbf{r})=\left[ \cos(\mathbf{q}\!\cdot\!\mathbf{r})\sin(\Theta),\sin(\mathbf{q}\!\cdot\!\mathbf{r})\sin(\Theta),\cos(\Theta) \right]$. The polar angle $\Theta$, measured with respect to the $z$-axis, remains fixed at each lattice site. The planar view in the bottom panel depicts a spin spiral with a period of $L=2\pi/q$, where the wavevector $\mathbf{q}$ is parallel to the $x$-axis. This cone-SS phase is subjected to an external magnetic field, $\mathbf{B}_\mathrm{ext}$, applied along the $z$-axis. The in-plane component makes an angle $\Phi$ ($\equiv \mathbf{q\cdot r}$) with the propagation direction.