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

TL;DR

EuAg4Sb2 hosts tunable spin moiré superlattices (SMS) under in-plane field, revealed by SANS to include multiple in-plane phases (ICM2a/b/c, ICM3a) with ICM2b displaying a freely rotating anisotropic multi-$q$ texture. A momentum-space spin Hamiltonian and simulated annealing reproduce the observed phases and predict additional intermediate states, highlighting a highly frustrated energy landscape. The study finds a strong link between SMS propagation vectors and electronic structure, with $|q_{xy}| ightarrow$ $2k_F$ gaps correlating with enhanced resistivity, especially in multi-$q$ SMS, underscoring a route to SMS-driven transport control. These results position EuAg4Sb2 as a platform for designing tunable SMS and SMS-based electronic properties with potential spintronic applications, distinct from conventional skyrmion systems due to in-plane field stabilization and rich multi-$q$ textures.

Abstract

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.

Figures (5)

• Figure 1: a Schematic of the real and reciprocal lattice vectors of EuAg4Sb2 in relation to the rhombohedral morphology of the synthesized single crystals. The orientation of the Eu triangle lattice layers is indicated in pink. $a$, $b$, and $c$ are the lattice vectors for a hexagonal unit cell and $a^*$ and $b^*$ are the reciprocal lattice vectors. b Schematic of the SANS geometry. The sample is rocked about the horizontal and vertical axes to collect a 3D diffraction pattern. The magnetic field (orange) is transverse to the neutron beam and is rocked with the crystal. Two crystals in different orientations were used to measure the $H||a$ and $H||a^*$ phases. c-f Schematic 2D bandstructure of a parabolic electron band subject to a c,e single-$q$ or d,f double-$q$ (SMS) modulation with propagation vector c,d equal to ($q=2k_F$) or e,f slightly larger than ($q>2k_F$) the Fermi momentum. The color maps to the band velocity $|v|=\sqrt{v_x^2+v_y^2}=1/\hbar \nabla_{\bm{k}} E$. g The temperature-field phase diagram for $H||c$ (reproduced from kurumaji2025electronic). PM: paramagnetic; FP: field-polarized; and ICM1-3: incommensurate magnetic modulation states. h-i The temperature-field phase diagram for $H||a^*$ and $H||a$, respectively. ICM2a-c and ICM3a are field-induced phases. Boundaries observed from upwards (downwards) field-sweeps are indicated with upwards (downwards) pointing triangles. Boundaries from field cooling are indicated with leftward pointing triangles. See supplementary section 1 for more information on the mapping of the in-plane phase diagrams.
• Figure 2: Overview of representative in-plane-field SANS patterns and their corresponding real-space textures. a SANS diffraction pattern in sample S1 for ICM3a at 2.15 K with 0.8 T of field applied along the $a^*$ direction (horizontal axis). A set of two diffraction peaks corresponding to the single-$q$ propagation vector (see green arrow) are observed along the $a^*$ direction. b The corresponding simulated real space spin texture for ICM3a, with the colorscale showing the vorticity (see Eq. \ref{['eq:vorticity']}). Cartesian coordinates $x$ and $y$ are defined, where the $x$ axis is along the $a^*$ direction. c The in-plane phase diagram for magnetic $H||a^*$. The temperature and field conditions of a,d,e,g are indicated with red crosses. d The SANS diffraction pattern for ICM2b at 6 K and 0.25 T with field sweeping up after zero field cooling and e after ramping field to 1.1 T, and then ramping field back down to 0.225 T. In both patterns, each double-$q$ domain is formed of two primary peaks at an approximate right angle with weaker half-order peaks. The propagation vectors are depicted with green arrows. f The corresponding real space spin texture for ICM2b corresponding to the diffraction pattern shown in e, with the colorscale showing the vorticity. g The SANS diffraction pattern for ICM2c at 0.3 T with field sweeping up after zero field cooling. The strongest diffraction peaks are along the $a^*$ direction, while eight additional diffraction peaks are present split about positions forming a hexagon with the primary peaks. h The corresponding real space spin texture for ICM2c corresponding to the diffraction pattern shown in g, with the colorscale showing the vorticity. All SANS data is inversion symmetrized, smoothed, $q_z$ integrated, and has a high temperature background subtracted.
• Figure 3: a-f Magnitude of the in-plane component of the magnetic propagation vector as a function of in-plane field for magnetic field applied along the a-c$a$ and d-f$a^*$ directions measured in sample S2 and sample S1, respectively. The temperature of each field dependence is indicated within each panel. The magnetic phase is labeled and indicated with colored shading. The nature of the propagation vector is indicated with the color of the mark (see supplemental information for more information on the different propagation vectors). The electronic commensuration condition ($q=2k_F$) is highlighted with a purple haze. Multi-$q$ states (the SMS condition) are highlighted with a green haze. g-i The longitudinal resistivity along the $a$ axis for field applied along $a$ (red) and $b^*$ (blue, perpendicular to the $a$ axis and crystallographically equivalent to $a^*$) at 1.8 K, 6 K, and 8 K, respectively. The resistivity enhancement is emphasized with a purple/green haze.
• Figure 4: Model phase diagram for field applied along the $a$ direction. a Propagation vectors considered in model. Black arrows correspond to the ICM1 (and ICM3) vectors, green arrows correspond to the ICM2 vectors, and blue arrows correspond to the ICM3 vectors. $q_x$ and $q_y$ are Cartesian coordinates in the reciprocal space, where $q_y$ is along the $a^*$ direction. b-d The model peak intensity for each of the peaks as a function of applied magnetic field. e-g Real-space model spin structure in several phases for increasing field. The in-plane spin is depicted with an arrow, and the $z$ component is indicated with red.
• Figure 5: Model phase diagram for field applied along the $a^*$ direction. a Propagation vectors considered in model. Black arrows correspond to the ICM1 (and ICM3) vectors, green arrows correspond to the ICM2 vectors, and blue arrows correspond to the ICM3 vectors. b-d The model peak intensity for each of the peaks as a function of applied magnetic field. e-h Real-space model spin structure in several phases for increasing field. The in-plane spin is depicted with an arrow, and the $z$ component is indicated with red. $x$ and $y$ are Cartesian coordinates in the $ab$ plane, where $x$ is along the $a$ axis.