Single-$q$ Cycloid and Double-$q$ Vortex Lattices in Layered Magnetic Semimetal EuAg$_4$Sb$_2$

TL;DR

The paper resolves three zero-field magnetic textures in EuAg$_4$Sb$_2$: a single-$q$ cycloid (ICM1) and two inequivalent double-$q$ vortex lattices (ICM2, ICM3), using strain-controlled domain selection and polarized neutron techniques to determine spin orientation and propagation vectors. A momentum-space phenomenological model, incorporating an $xy$-type interaction and a four-spin term, explains the near-degeneracy and field-tunable transitions among ICM1–ICM3, linking easy-plane anisotropy and spin moiré formation to the observed textures. The work demonstrates a tunable spin-moiré platform in a quasi-2D magnetic semimetal, with strain and field offering routes to domain control and potential skyrmion-like states, advancing design principles for topological spin textures in layered materials.

Abstract

Recently, a host of exotic magnetic textures such as topologically protected skyrmion lattices has been discovered in several bulk metallic lanthanide compounds. In addition to hosting skyrmion phases, a hallmark of this class of materials is the appearance of numerous spin textures characterized by a superposition of multi-$q$ magnetic modulations: spin moiré superlattices. The nuanced energy landscape thus motivates detailed studies to understand the underlying interactions. Here, we comprehensively characterize and model the three zero-field magnetic textures present in one such material, EuAg$_4$Sb$_2$. Systematic symmetry breaking experiments using magnetic field and strain determine that the ground state incommensurate magnetic phase (ICM1) is single-$q$. In contrast, ICM2 and ICM3 are both double-$q$, \textit{i.e.}, spin moiré superlattices. Further, through application of polarized small angle neutron scattering and spherical neutron polarimetry, we demonstrate that ICM1 is a single-$q$ cycloid and ICM2 and ICM3 are double-$q$ vortex lattices, with Eu moments lying in the $ab$-plane in zero field and with a ferromagnetic component at finite field. Despite the quasi-2D nature of EuAg$_4$Sb$_2$, the modulations propagate out of the \textit{ab}-plane, leading to a shift of the spin texture between triangle lattice planes. Further, the ICM3 to ICM2 transition includes an unusual 45$^\circ$ rotation of the magnetic vortex lattice. Motivated by the coexistence of such drastically different phases in this compound, we conclude by developing a phenomenological model to understand the stability of these states. Our experimental probes and theoretical modeling definitively characterize three different and tunable phases in one material, and provide insight for the design of new topological spin-texture materials.

Figures (6)

• Figure 1: a Rhombohedral ABC stacked europium triangle lattice layers viewed along the $c$-axis. The unit cell is indicated with a black outline. b$H||c$ phase diagram for EuAg4Sb2 containing the three incommensurate phases ICM1-3, from kurumaji2025electronic. Legend indicates the measurement from which each indicated point was derived. c-e 3D SANS isosurface plots of ICM1-3 highlighting one magnetic domain each in red. The data are taken in zero field at 2.1 K, 8 K, and 9.5 K, respectively. The data are inversion symmetrized and has a high temperature background subtracted. f Schematic of the polarization-analysis setup used in the SANS experiments. g A schematic of Cryopad used to perform spherical neutron polarimetry (SNP) measurements. Note the conventional coordinate system used in polarization measurements: $\bm{x}$ is along the scattering vector $\bm{Q}$, $\bm{z}$ is the vertical axis in the instrument, and $\bm{y}$ completes the orthogonal coordinate system. h Polarization-dependent intensity of peaks in 50 mT of field along the $c$-axis as a function of temperature in different phases. i Polarization-dependent intensity of peaks at 2 K for various fields along the $c$-axis in different phases. In h,i, the data are background subtracted and corrected for imperfect polarization and analysis efficiency. SF is indicated with circles while NSF is indicated with triangles. Error bars are the 95% confidence interval from a fit of the peak intensity. ICM3 (yellow) corresponds to the $H0L$-type peak while ICM3' (purple) corresponds to the $HHL$-type peak (see e).
• Figure 2: a-b single-$q$ and double-$q$ order for a transverse modulated spin structure. The color-scale is the curl (vorticity) of the texture. c Various possible spin textures associated with one spin modulation. From left to right: longitudinal, proper screw spiral, cycloid, conical, and transverse spin structures. d,g,h SANS pattern measured under tension along the vertical direction (see green arrows in d) at 2 K in ICM1, 8 K in ICM2, and 9.5 K in ICM3. e,h,k SANS pattern measured with no applied strain (after the crystal broke) under the same conditions. f,i,l SANS pattern measured under uniaxial compression along the vertical axis (see green arrows in f). A cartoon (not to scale) of the strained crystal is shown with gray hexagons. The domain structure is indicated with red, white, and black lines. Crossed lines indicate a domain of double-$q$ states. SANS data are inversion symmetrized, smoothed with a half pixel gaussian kernel, and has a 15 K background measurement subtracted.
• Figure 3: a-b Geometry of the crystal and neutron spin polarization analysis axes for the (000) and (006) satellite peaks. The coordinate axes of the polarization are depicted with red, green, and blue arrows, and the scattered neutron path is indicated in yellow. The yellow arrow corresponding to the scattering vector $\bm{Q}$. c 3D SANS intensity of ICM1 for reference, with one domain highlighted in red. d The polarization matrix elements for different magnetic satellite peaks and the corresponding model intensities from the model depicted in e. The incident and outgoing neutron polarizations are indicated with the inner and outer marker color, respectively. Red, green, and blue correspond to $x$, $y$, and $z$, respectively. The upwards and downwards triangles indicate a positive and negative incident neutron polarization, respectively. Measured matrix elements are indicated with triangles, while the elements calculated from the model are indicated with squares. All errorbars include both statistical and estimated systematic uncertainties. e Schematic spin structure of ICM1. A single Eu-layer is shown for simplicity. Propagation direction is also depicted.
• Figure 4: a Geometry of the crystal and polarization axes for the (000) satellite peak. The coordinate axes of the polarization are depicted with red, green, and blue arrows, and the scattered neutron path is indicated in yellow. The yellow arrow corresponding to the scattering vector $\bm{Q}$. b 3D SANS intensity of ICM2 for reference, with one domain highlighted in red. c The polarization matrix elements for different magnetic satellite peaks and the corresponding model intensities from the model depicted in d. The incident and outgoing neutron polarizations are indicated with the inner and outer marker color, respectively. Red, green, and blue correspond to $x$, $y$, and $z$, respectively. The upwards and downwards triangles indicate a positive and negative incident neutron polarization, respectively. Measured matrix elements are indicated with triangles, while the elements calculated from the model are indicated with squares. All errorbars include both statistical and estimated systematic uncertainties. e The sum of the pattern depicted in d superimposed with the corresponding pattern for the second ICM2 propagation vector.
• Figure 5: a-b The polarization matrix elements for different magnetic satellite peaks of the a$(H0L)$ and b$(HH0)$ type, and the corresponding model intensities from the models depicted in c and d, respectively. The incident and outgoing neutron polarizations are indicated with the inner and outer marker color, respectively. Red, green, and blue correspond to $x$, $y$, and $z$, respectively. The upwards and downwards triangles indicate a positive and negative incident neutron polarization, respectively. Measured matrix elements are indicated with triangles, while the elements calculated from the model are indicated with squares. All error-bars include both statistical and estimated systematic uncertainties. Inset of c,d depict the geometry of the crystal and polarization axes for the (000) satellite peak. The coordinate axes of the polarization are depicted with red, green, and blue arrows, and the scattered neutron path is indicated in yellow. The yellow arrow corresponding to the scattering vector $\bm{Q}$. e The superposition of the patterns depicted in c-d, representing the complete ICM3 texture. f 3D SANS intensity of ICM3 for reference, with one domain highlighted in red.