Skyrmion and Meron Crystals in Intermetallic Gd$_3$Ru$_4$Al$_{12}$: Microscopic Model Insights into Chiral Phases

TL;DR

The paper constructs a realistic microscopic model for the centrosymmetric skyrmion/meron host Gd$_3$Ru$_4$Al$_{12}$ by integrating neutron scattering, magnetization, and FMR data through a multi-target fit on an AB-stacked breathing kagome lattice. It identifies dipole-dipole interactions and easy-plane single-ion anisotropy as key stabilizers of a rich chiral phase diagram, including a hexagonal skyrmion lattice and commensurate meron–antimeron textures, the latter validated by polarized resonant x-ray diffraction. In the short-range regime, a codimension-two spiral spin-liquid emerges with chiral fluctuations driven by staggered DM interactions, linking to the anomalous Hall response. The work demonstrates the power of a lattice-resolved, data-constrained microscopic approach to unravel complex topological spin textures and offers pathways for engineered spintronic functionalities in frustrated magnets.

Abstract

Topological spin textures in frustrated intermetallics hold great promise for spintronics applications. However, understanding their origin and properties remains a significant challenge due to competing and often long-range interactions mediated by conduction electrons. Here, by combining neutron scattering experiments with theoretical modeling via unprecedented multi-target fits that further incorporate the ferromagnentic resonance data and magnetization curve, we construct a realistic microscopic model for the prototypical intermetallic skyrmion host \text{Gd}$_3$\text{Ru}$_4$\text{Al}$_{12}$. Beyond magnetic frustration, we identify the competition between dipolar interactions and easy-plane single-ion anisotropy as a key ingredient for stabilizing the rich chiral magnetic phases observed in this compound -- including a hexagonal skyrmion crystal and two distinct meron crystals. Remarkably, the meron crystal in lower field is revealed to be commensurate with the underlying lattice, and its unique three-meron-one-antimeron spin texture is verified by the polarized x-ray diffraction data. At elevated temperatures, the short-range spin correlations in \text{Gd}$_3$\text{Ru}$_4$\text{Al}$_{12}$ are well described by a codimension-two spiral spin-liquid. Perturbations from staggered Dzyaloshinskii-Moriya interactions give rise to chiral fluctuations that account for the temperature and field dependence of the anomalous Hall response. Our results highlight the unique power of neutron scattering, especially when combined with complementary experimental techniques, to unravel complex magnetic phase transitions and provide new insights into the rich variety of topological spin textures in frustrated systems.

Figures (24)

• Figure 1: Codimension-two spiral spin-liquid in Gd$_3$Ru$_4$Al$_{12}$. (a) The Gd$^{3+}$ ions with spin S = 7/2 in Gd$_3$Ru$_4$Al$_{12}$ form breathing kagome lattices in the $ab$ plane, which are AB-stacked along the $c$ axis. Curved solid lines in the figure indicate the exchange paths for the two shortest intralayer couplings ($J_1$ and $J_2$), the sixth-neighbor intralayer coupling ($J_6$), the three shortest interlayer couplings ($J_{\rm{c1}}$, $J_{\rm{c2}}$, and $J_{\rm{c3}}$), and the two shortest second-layer couplings ($J_{\rm{n1}}$ and $J_{\rm{n2}}$). (b) Under dominant ferromagnetic $J_1$ couplings, the three spins over the smaller triangles of the breathing kagome lattice can be viewed as an effective spin $S_{\rm{eff}}=3S$, leading to AB-stacked triangular lattices of effective spins. Curved solid lines indicate the exchange paths for the two shortest intralayer couplings ($J_1^{\rm{T}}$ and $J_2^{\rm{T}}$), the two shortest interlayer couplings ($J_{\rm{c1}}^{\rm{T}}$ and $J_{\rm{c2}}^{\rm{T}}$), and the shortest second-layer coupling ($J_{\rm{n1}}^{\rm{T}}$). (c) Equivalent honeycomb lattice formed by the effective spins. Curved solid lines indicate the exchange paths for the inter-sublattice couplings ($J_1^{\rm{H}}$ and $J_3^{\rm{H}}$) and the intra-sublattice couplings ($J_2^{\rm{H}}$ and $J_4^{\rm{H}}$). (d) Representative spiral rings in reciprocal space calculated for a $J_1^{\rm{H}}$-$J_2^{\rm{H}}$ honeycomb-lattice model with a frustration ratio of $|J_2^{\rm{H}}/J_1^{\rm{H}}|=0.2$ (green), 0.5 (blue), and 0.7 (red). (e) The panel at the bottom describes a real-space spin configuration surrounding a momentum vortex in the spiral spin-liquid state. Pseudocolor corresponds to the phase $\phi$ of the in-plane spin rotation. In the presence of antisymmetric DM interactions, spin fluctuations will become chiral as described in the panel on the top. (f, g) Diffuse neutron scattering patterns in the ($h$, $k$, 1) (f) and ($h$, 0, $l$) (g) planes measured on WAND$^2$ at $T = 20$ K. Gray lines mark the Brillouin zone boundaries in reciprocal space. (h, i) Simulated diffuse neutron scattering patterns in the ($h$, $k$, 1) (h) and ($h$, 0, $l$) (i) planes using the classical Monte Carlo simulations for the fitted $J_{126}$-$J_{\rm{c123}}$-$J_{\rm{n12}}$ model on the original breathing kagome lattice.
• Figure 2: Hierarchical spin excitations in Gd$_3$Ru$_4$Al$_{12}$. (a) INS spectra, $S$(Q, $\omega$), measured on SEQUOIA (SEQ) along the high symmetry lines at $T = 5$ K in the magnetically long-range ordered regime (left) and at $T=100$ K in the correlated paramagnetic regime (right). (b) Experimental constant-energy slices for the INS spectra measured at $T = 5$ K in the ($h$, 0, $l$) plane at $E$ = 1.0 (left) and 8.5 meV (right). (c) Landau-Lifshitz-Gilbert (LLG) dynamics simulations of the spectra using the fitted $J_{126}$-$J_{\rm{c123}}$-$J_{\rm{n12}}$ model. The simulated spectra have been convoluted by the instrumental energy resolution as described in the Supplemental Material supp. (d) Simulated constant-energy slices in the ($h$, 0, $l$) plane at $E$ = 1.0 (left) and 8.5 meV (right). Both the experimental and simulated spectra are integrated over an energy width of $\Delta E = \pm$0.15 meV.
• Figure 3: Topological phase transitions in Gd$_3$Ru$_4$Al$_{12}$. (a) Theoretical phase diagram for Gd$_3$Ru$_4$Al$_{12}$ calculated in a magnetic field along the $c$ axis. Calculations were performed for the equivalent triangular lattice model in Table \ref{['tab:exchange']} using classical Monte Carlo simulations. Pseudocolor corresponds to the negative skyrmion number, $-N_\mathrm{sk}$. The phase boundaries are determined from the calculated specific heat (red diamonds), magnetic susceptibility (green circles), d$M$/d$T$ (dark blue square) and d$N_\mathrm{sk}$/d$H$ (light blue triangles). The observed phases include the stripe, sinusoidal modulated spin density wave (SDW), spiral spin-liquid (SSL), meron-antimeron (M-AM), skyrmion, inverted meron (IM), and fan structures. The topological phase transition between the meron-antimeron phase and the skyrmion phase is delineated by a blue dashed line. (b) Temperature evolution of the simulated magnetic structure factor components along the $c$ axis (red) and in the $ab$ plane (blue). (c) Simulated (red line) and experimental (blue circles) field evolution of the magnetic long-range order wavevector $q/a^*$. Calculations were performed on a $44\times 44\times 2$ supercell of the effective triangular lattice. Experimental data are adapted from Ref. hirschbergerLatticecommensurate2024. (d) Prototype topological spin textures, including two types of merons ($N_\mathrm{sk} = -1/2$) and antimerons ($N_\mathrm{sk} = 1/2$), skyrmion ($N_\mathrm{sk} = -1$), and inverted meron ($N_\mathrm{sk} = -1/2$). (e-g) Simulated magnetic structures in the skyrmion phase (e), meron-antimeron phase (f), and inverted meron phase (g) on two successive layers of the equivalent AB-stacked triangular lattice viewed along the $c$ axis. Pseudocolor of the spins corresponds to the length of spin component along the $c$ axis. Pseudocolor in the background corresponds to the solid angle $\mathit{\Omega}$ as defined in the Supplemental Material supp. Unit cells of the magnetic structures are outlined by black lines. In panel (f), one magnetic unit cell is composed of three merons and one anti-meron, each components being outlined by rectangles of the same color as those of the prototype spin textures in panel (d).
• Figure 4: Identification of the commensurate M-AM lattice through comparison with the resonant x-ray diffraction data. (a)-(h) Calculated magnetic structure factor, $S(\mathbf{Q})$, and its decomposition for the SKL in (a)-(d) and the M-AM lattice in (e)-(h). Calculations were performed on the original breathing kagome lattice at zero temperature in a field of 1.0 and 1.7 T for the M-AM and SKL states, respectively. The components shown are the total structure factor $S^\mathrm{tot}(\mathbf{Q})$, the out-of-plane component $S^\mathrm{zz}(\mathbf{Q})$, the in-plane component perpendicular to the wave vector $S^\mathrm{\perp}(\mathbf{Q})$, and the in-plane component parallel to the wave vector $S^\mathrm{\parallel}(\mathbf{Q})$. (i, j) Comparison between the experimental (blue markers) and calculated values of $R\sin^2(2\theta)$ for the (i) SKL and (j) M-AM states. The experimental data, which were measured in the commensurate phase at $T = 7.8$ K in $H = 1.3$ T, are adapted from Ref. hirschbergerLatticecommensurate2024. Calculations were performed on the original breathing kagome lattice in a field of $H$ = 1.3 T. (k) Ellipsoid plot for the same polarized x-ray diffraction data (blue markers), for which the linear fit is shown by the solid blue line. Dashed yellow and red lines are calculated for the M-AM and SKL models at $H$ = 1.3 T, respectively. The shaded areas describe the variance of the calculated slope within the stable field range of 0.8 to 1.5 T and 1.2 to 2.0 T for the M-AM and SKL states, respectively, with the lower boundary corresponding to the lower field.
• Figure 5: Chiral spin fluctuations in the spiral spin-liquid regime. (a) Scalar spin chirality, $\chi_\Delta$ and $\chi_\nabla$, over the small and large triangles on the original breathing kagome lattice, respectively. DM interactions with strengths of $D_1$ and $D_2$ along the $c$ axis are indicated by blue and green circled dots, respectively. (b) Temperature and field-dependence of $\chi_\Delta$ calculated for the fitted $J_{126}$-$J_{\rm{c123}}$-$J_{\rm{n12}}$ model with $D_1 = 0.02$ meV and $D_2 = 0$. (c) Similar calculations for $\chi_\nabla$ with $D_1=0$ and $D_2 = 0.02$ meV. (d) Experimental chirality-driven Hall conductivity, $\sigma_{xy}^{\chi}$, as reproduced from the previous report kolincioKagome2023.