Broken inversion symmetry in the charge density wave phase in EuAl$_4$

TL;DR

This study determines the symmetry of the CDW phase in EuAl4 using temperature-dependent single-crystal X-ray diffraction with access to second-order satellite reflections. The CDW is found to be non-centrosymmetric and transverse, best described by the orthorhombic superspace group $F222(0\,0\,\sigma)00s$, and phason disorder is evidenced by second-order harmonic modulation of the anisotropic displacement parameters. The breaking of inversion symmetry implies that Dzyaloshinskii-Moriya interactions can stabilize skyrmions in EuAl4, aligning with observations of skyrmion lattices under applied fields. These results clarify the CDW’s structural nature, its location on the Al network, and its interplay with magnetic order, influencing the microscopic mechanisms behind the material’s complex phase diagram.

Abstract

EuAl$_4$ exhibits a complex phase diagram, including the development of a charge density wave (CDW) below $T_{CDW} = 145$ K. Below $T_{N}=15.4$ K, a series of antiferromagnetically (AFM) ordered phases appear, while non-trivial topological phases, like skyrmion lattices, are stabilized under an applied magnetic field. The symmetries of the variously ordered phases are a major issue concerning the understanding of the stabilization of the ordered phases as well as concerning the interplay between the various types of order. EuAl$_4$ at room temperature has tetragonal symmetry with space group $I4/mmm$. The CDW phase has an incommensurately modulated crystal structure described by the modulation wave vector $\mathbf{q} \approx 0.17\mathbf{c}^{*}$. On the basis of various experiments, including elastic and inelastic x-ray scattering, and second-harmonic generation, it has been proposed that the symmetry of the CDW phase of EuAl$_4$ could be centrosymmetric orthorhombic, non-centrosymmetric orthorhombic or non-centrosymmetric tetragonal. Here, we report temperature-dependent, single-crystal x-ray diffraction experiments that show that the CDW is a transverse CDW with phason disorder, and with non-centrosymmetric symmetry according to the orthorhombic superspace group $F222(0\,0\,σ)00s$.Essential for this finding is the availability of a sufficient number of second-order ($2\mathbf{q}$) satellite reflections in the x-ray diffraction data set. The broken inversion symmetry implies that skyrmions might form due to Dzyaloshinskii-Moriya (DM) interactions, instead of a more exotic mechanism as it is required for centrosymmetric structures.

Figures (7)

• Figure 1: (a) Crystal structure of EuAl$_4$ with space group $I4/mmm$ in the periodic phase at 160 K. Depicted is an $I$-centered unit cell with basis vectors $\mathbf{a}_I$, $\mathbf{b}_I$ and $\mathbf{c}_I$. (b) Basic structure of EuAl$_4$ in the CDW phase at 30 K, showing an $F$ centered unit cell with the with basis vectors $\mathbf{a}_F$, $\mathbf{b}_F$ and $\mathbf{c}_F$ and space group $F222$. The relation between the $I$-centered and $F$-centered unit cells is: $\mathbf{a}_F = (\mathbf{a}_I + \mathbf{b}_I)$, $\mathbf{b}_F = (-\mathbf{a}_I + \mathbf{b}_I)$ and $\mathbf{c}_F = \mathbf{c}_I$. Orange spheres correspond to the Eu atoms; blue spheres represent Al1/Al1a atoms; green spheres represent Al1b atoms; and pink spheres stand for Al2 atoms.
• Figure 2: Single crystal of EuAl$_4$ as separated from the Al flux. Yellow lines form a mesh of $1\times 1$ mm$^{2}$.
• Figure 3: Undistorted view of the $(1\,k\,l)$ reciprocal lattice plane of (a) the SXRD-160 data, and (b) the SXRD-30 data. (c) Enlarged view of panel (b) about reflection $(1\, {-}3\, -1)$, with clearly visible first-order and second-order satellite reflections. Images have been generated from the measured data by the software CrysAlisPro crysalis. Dark bands are due to gaps (insensitive area) between the active modules of the Lambda 7.5M area detector.
• Figure 4: Flow chart showing the relations between tetragonal and orthorhombic space groups and superspace groups, where the basic-structure space group is a subgroup of $I4/mmm$. Furthermore, the acentric orthorhombic superspace groups (bottom row) are subgroups of the superspace groups in the middle row. "t2" indicates that the point group is a subgroup of index two. Centrosymmetric superspace groups are in light blue and acentric superspace groups are in light brown color.
• Figure 5: Perspective view of layers of aluminum atoms in EuAl$_4$. (a) Layer of Al atoms in the periodic phase, where the shortest distances are within the Al2-Al2 dumbbells with $d$[Al2--Al2] = 2.562(3) Å at 160 K. Next shortest distances are $d$[Al2--Al1] = 2.666(2) Å and $d$[Al1--Al1] = 3.1058(1) Å. (b) The CDW phase featuring Al1a and Al1b independent atoms in the basic structure of symmetry $F222$. Basic-structure distances in the modulated CDW phase at 30K are $d$[Al2--Al2] = 2.566(2) Å, $d$[Al2--Al1a] = $d$[Al2--Al1b] = 2.662(1) Å and $d$[Al1a--Al1b] = 3.1027 Å. The modulation of these distances is given as t-plots in Fig. \ref{['fig:eual4_t_plot_al']}.