Cascade of Spin Moiré Superlattices with In-Plane Field in Triangle Lattice Semimetal EuAg$_4$Sb$_2$

EuAg$4$Sb$2$ is a rhombohedral europium triangle lattice material that exhibits a rich phase diagram of spin moiré superlattices (SMS) and single-$q$ magnetic phases. In this paper, we characterize the incommensurate phases accessible with field applied in the plane with small angle neutron scattering (SANS). A variety of phases with unusual SANS patterns are accessible with magnetic field applied along the $a$ and $a^*$ directions. Many of these phases can be understood to be multi-$q$ phases. One phase in particular, ICM2b (ICM=incommensurate magnetic phase), is rather unconventional in that it is an anisotropic multi-$q$ phase that can rotate freely within the $ab$-plane, dependent on magnetic field direction and history. The stabilization of tunable multi-$q$ incommensurate spin textures \textit{via} in-plane field sets this class of materials apart from conventional skyrmion materials. We further identify that the propagation vectors of the in-plane phases have a significant commensuration with the diameter of the smallest pocket of the Fermi surface ($2k{\text{F}}$). The multi/single-$q$ nature is also correlated with the enhancement of resistivity, suggesting that a gap opens in the electron bands at $q=2k{\text{F}}$. We also compare with a phenomenological model of the phase diagram. The richness of phases revealed in this study hint at the frustrated nature of the incommensurate magnetism present in EuAg$_4$Sb$2$ and motivate further probes of these phases and the origin of the stability of spin moiré superlattices. Finally, the coupling of the multi-$q$ nature and $q=2k{\text{F}}$ commensuration condition reveals the key requirements for a strong SMS transport response.

💡 Research Summary

EuAg₄Sb₂ is a rhombohedral quasi‑two‑dimensional semimetal composed of a triangular lattice of Eu ions. The material hosts a rich set of incommensurate magnetic phases (ICMs) that are intimately linked to its electronic structure, in particular to the smallest cylindrical hole pocket of the Fermi surface. In zero field or with magnetic field applied along the crystallographic c‑axis (H‖c), three well‑characterized phases appear: ICM1, a single‑q cycloidal state; ICM2 and ICM3, both double‑q vortex‑lattice states. All three possess propagation vectors that satisfy the condition q≈2kF, where 2kF is the diameter of the smallest Fermi‑surface pocket. This commensuration opens a superzone gap at the Fermi surface, leading to a marked increase in electrical resistivity.

The present work explores how the phase diagram evolves when the magnetic field is rotated into the basal plane (H‖a or H‖a*). Using small‑angle neutron scattering (SANS) on high‑quality single crystals, the authors map out the propagation vectors, symmetry, and multi‑q nature of the newly accessed phases. The in‑plane field splits the original ICM2 into three distinct sub‑phases—ICM2a, ICM2b, and ICM2c—while ICM3 transforms into a single‑q state denoted ICM3a. The field‑temperature phase diagrams for H‖a and H‖a* are remarkably similar, indicating an almost isotropic energy landscape within the basal plane.

ICM3a is a single‑q phase with q=(0.100,0,0.03) (reciprocal‑lattice units) aligned parallel to the applied field. It can be reached directly from ICM1, ICM2, or ICM3 when a sufficiently strong in‑plane field is applied. The transition is interpreted as the suppression of the longitudinal component of the cycloid, leaving a transverse spin modulation that co‑exists with a uniform ferromagnetic component along the field direction. This state preserves the q≈2kF condition, and its resistivity is comparable to that of the zero‑field phases.

ICM2b exhibits a highly unconventional low‑symmetry SANS pattern. Two strong Bragg peaks appear at q₁≈(0.079,0.018,‑0.02) and q₂≈(‑0.063,0.099,0), which are approximately orthogonal, together with weaker half‑order satellites (e.g., q₁+½q₂). The pattern is consistent with a double‑q texture in which the two primary propagation vectors form a right angle. The presence of sub‑harmonic peaks suggests that higher‑order harmonics, likely mediated by conduction‑electron exchange, play a significant role. Importantly, the orientation of the two domains can be rotated simply by changing the direction of the in‑plane field, and the domains can be aligned or re‑oriented by field‑cooling protocols. This remarkable tunability points to an extremely flat energy landscape for the propagation vectors, a hallmark of strong magnetic frustration.

ICM2c appears at higher in‑plane fields and displays a dominant peak at (0.093,0,‑0.025) together with eight weaker peaks arranged in a hexagonal motif around the c‑axis. The authors propose that this phase is a triple‑q (or possibly a 5‑q quasicrystalline) state, where three propagation vectors of comparable magnitude coexist. The hexagonal arrangement of the weaker peaks suggests that the three vectors are separated by ~120°, forming a more isotropic spin‑density‑wave network. Notably, ICM2c is only observed for H‖a*, indicating that the trigonal magnetocrystalline anisotropy of EuAg₄Sb₂ selectively stabilizes this configuration.

The transport measurements reveal that the resistivity sharply rises in the ICM2b and ICM2c regimes, consistent with the opening of a superzone gap at q≈2kF. The correlation between the multi‑q nature and the resistivity enhancement provides compelling evidence that the magnetic modulation directly reconstructs the electronic bands.

To rationalize the experimental findings, the authors construct a phenomenological Landau‑type free‑energy model that includes four‑spin interactions and anisotropic exchange terms. By imposing the commensuration condition q≈2kF, the model reproduces the observed sequence of phase boundaries and the stability of the multi‑q states under in‑plane fields. The agreement between theory and experiment underscores the pivotal role of Fermi‑surface geometry, indirect RKKY‑type exchange, and higher‑order magnetic interactions in stabilizing spin‑moiré superlattices.

In summary, EuAg₄Sb₂ demonstrates that an in‑plane magnetic field can be used to continuously rotate and switch multi‑q spin‑moiré textures, a capability that is absent in conventional skyrmion‑hosting materials. The intimate coupling between the magnetic propagation vectors and the Fermi‑surface diameter not only governs the magnetic phase diagram but also drives a pronounced magnetotransport response. This work establishes EuAg₄Sb₂ as a versatile platform for exploring tunable spin‑moiré physics, offering new avenues for engineering electronic band structures via magnetic superlattices and for developing functional devices that exploit the interplay of spin, charge, and lattice degrees of freedom.