---
title: 'EuAl4: CDW and Magnetic Phase Complexity'
url: https://www.emergentmind.com/topics/eual4
type: topic
---

# EuAl4: CDW and Magnetic Phase Complexity

EuAl\(_4\) is a BaAl\(_4\)-type rare-earth intermetallic whose low-temperature physics is governed by the coexistence of an incommensurate charge-density wave (CDW), multiple antiferromagnetic phases, and field-induced topological spin textures. At room temperature it crystallizes in the tetragonal space group \(I4/mmm\); below \(T_{\mathrm{CDW}} \approx 145\) K it develops a long-wavelength modulation with \(\mathbf{q}\parallel \mathbf{c}^{*}\), and below \(T_N=15.4\) K it enters a cascade of magnetically ordered states. A central result of recent diffraction work is that the CDW phase is non-centrosymmetric, with orthorhombic superspace symmetry \(F222(0\,0\,\sigma)00s\), which directly affects microscopic interpretations of the skyrmion phases and of the coupling between lattice, charge, and magnetism [2506.01633].

## 1. Crystal chemistry and baseline phase diagram

EuAl\(_4\) is a tetragonal intermetallic of the BaAl\(_4\) structure type, space group \(I4/mmm\) (No. 139), with three crystallographically independent atoms: Eu, Al1, and Al2. At 160 K the lattice parameters are \(a=b=4.3922(1)\,\text{\AA}\), \(c=11.1707(3)\,\text{\AA}\), and \(V=215.50(1)\,\text{\AA}^3\) [2506.01633]. Eu is divalent, \(\mathrm{Eu}^{2+}\), with localized \(4f\) moments and \(S=7/2\), while the itinerant states relevant to transport and the CDW are derived primarily from the Al network [2506.01633].

EuAl\(_4\) is metallic and has repeatedly been discussed as a topological semimetal or topological magnet. In the Eu(Ga\(_{1-x}\)Al\(_x\))\(_4\) series, EuAl\(_4\) is the Al-rich end member and one of the compositions that exhibits a clear CDW-like transport anomaly at ambient pressure; the zero-field resistivity is metallic and the residual resistivity ratio is approximately 70, consistent with high crystal quality [1804.02076].

| Regime | Characteristic temperature or vector | Established feature |
|---|---:|---|
| CDW onset | \(T_{\mathrm{CDW}} \approx 145\) K | Incommensurate modulation with \(\mathbf{q}\parallel \mathbf{c}^{*}\) |
| Magnetic ordering onset | \(T_{N1}=15.4\) K | First zero-field antiferromagnetic transition |
| Additional zero-field magnetic transitions | \(T_{N2}=13.2\) K, \(T_{N3}=12.2\) K, \(T_{N4}\approx 10.0\) K | Cascade of distinct magnetic phases |
| Field-induced textures for \(H\parallel[001]\) | \(\mathbf{Q}_1=(0.194,0,0)\), \(q\approx 0.085\) | Single-\(\mathbf{Q}\) spirals, rhombic and square skyrmion lattices, vortex and meron phases |

The magnetic nomenclature depends on the probe. In one convention, zero-field cooling passes through PM \(\rightarrow\) VII \(\rightarrow\) VI \(\rightarrow\) V \(\rightarrow\) I, while field along \([001]\) drives I \(\rightarrow\) II \(\rightarrow\) III \(\rightarrow\) IV \(\rightarrow\) forced FM; phases II and III are rhombic and square skyrmion lattices, respectively [2511.07079]. Resonant magnetic x-ray scattering resolves the four zero-field ordered states as AFM1–AFM4, each with single-\(\mathbf{k}\) incommensurate order [2403.10159].

## 2. Charge-density wave: modulation, transverse character, and phason disorder

Below \(T_{\mathrm{CDW}}\), satellite reflections appear in single-crystal diffraction with modulation vector \(\mathbf{q}\approx 0.17\,\mathbf{c}^{*}\). Representative refinements give \(\mathbf{q}=(0,0,0.1781(3))\) at 70 K, \(\mathbf{q}=(0,0,0.1741(2))\) at 20 K, and \(\mathbf{q}=0.1743(1)\,\mathbf{c}^{*}\) at 30 K [2202.10282;2506.01633]. No splitting or broadening of the fundamental Bragg peaks is observed in the CDW regime, so the lattice remains metrically tetragonal even though the full modulated structure has lower symmetry [2506.01633].

The modulation is transverse. For \(\mathbf{q}\parallel \mathbf{c}^{*}\), the dominant atomic displacements are within the \(ab\) plane, so that \(\mathbf{q}\cdot\mathbf{u}(\mathbf{r})\approx 0\). In superspace parameterization this is expressed through first-order harmonic displacement functions,
\[
u^{(j)}_\alpha(t)=A^{(j)}_{\alpha,1}\cos(2\pi t)+B^{(j)}_{\alpha,1}\sin(2\pi t),
\]
with the nonzero coefficients concentrated in in-plane components [2506.01633]. Neutron Laue diffraction had already shown that the CDW superlattice peaks are absent along the \((0\,0\,l)\) axis, which is consistent with a modulation mainly due to in-plane displacements of Al ions rather than longitudinal displacements along \(c\) [2104.07935].

A further refinement of the structural picture is the identification of phason disorder. In the 30 K single-crystal x-ray data, first-harmonic displacement modulation alone systematically overestimates the intensities of second-order satellites, producing the “\(\Delta F\) problem.” Introducing second-order harmonic modulation of the anisotropic displacement parameters resolves this discrepancy, whereas higher-harmonic displacement modulation does not. This is taken as the hallmark of phason dynamics in an incommensurate structure, so the CDW in EuAl\(_4\) is best described as a transverse CDW with significant phason disorder [2506.01633].

Real-space cryogenic 4D-STEM imaging is consistent with this description but emphasizes the internal structure of the modulation. It resolved two out-of-phase intra-unit-cell shear modulations with wavelength \(\lambda_{\mathrm{CDW}}\simeq 6.5\) nm, one associated with Al1–Al2 distortions and one with a \(yz\)-shear of the unit cell, showing directly that the long-wavelength CDW carries internal degrees of freedom beyond a single scalar amplitude [2311.17682].

## 3. Superspace symmetry and the inversion-symmetry problem

The symmetry of the CDW phase has been a central issue because earlier probes supported different superspace descriptions. Single-crystal x-ray diffraction first established an orthorhombic CDW on the tetragonal lattice and assigned the superspace group \(Fmmm(00\sigma)s00\), with the fourfold symmetry broken entirely by the modulation wave [2202.10282]. Subsequent inelastic x-ray scattering and lattice-dynamics work argued that the soft-mode eigenvector is most naturally described by \(Immm(00\gamma)s00\) [2402.15397]. Cryogenic 4D-STEM then showed that the modulation breaks inversion symmetry locally while preserving it on average, yielding local point groups compatible with non-centrosymmetric environments [2311.17682].

The decisive structural refinement uses second-order satellites. At 30 K the synchrotron dataset contains 207 unique main reflections, 380 unique first-order satellites, and 394 unique second-order satellites, of which 31 are observed above \(3\sigma\). Refinements over six candidate superspace groups show that the best agreement is obtained for the non-centrosymmetric orthorhombic superspace group
\[
F222(0\,0\,\sigma)00s,
\]
with \(R_F^{\mathrm{obs}(\mathrm{overall})}=2.51\%\) and \(R_F^{\mathrm{obs}(m=2)}=5.66\%\). The centrosymmetric alternatives \(Immm(0\,0\,\sigma)s00\) and \(Fmmm(0\,0\,\sigma)s00\) fit the second-order satellites significantly worse [2506.01633].

In this description the average lattice remains essentially tetragonal, but the modulation lowers the symmetry from \(4/mmm\) to \(222\). The transformation from the tetragonal \(I\)-cell to the orthorhombic \(F\)-cell is
\[
\mathbf{a}_F=\mathbf{a}_I+\mathbf{b}_I,\qquad
\mathbf{b}_F=-\mathbf{a}_I+\mathbf{b}_I,\qquad
\mathbf{c}_F=\mathbf{c}_I.
\]
The loss of inversion is accompanied by site splitting, notably Al1 \(\rightarrow\) Al1a + Al1b, and by symmetry-allowed differences in the modulation functions of those sites [2506.01633].

This resolves a long-running controversy. The CDW phase is not merely orthorhombic in a centrosymmetric sense; it is acentric in the full superspace description. A plausible implication is that local inversion breaking observed in microscopy and average acentricity established by diffraction are two descriptions of the same structural fact at different levels of resolution [2311.17682;2506.01633].

## 4. Magnetic order, skyrmion phases, and microscopic interpretations

Below \(T_{N1}=15.4\) K, EuAl\(_4\) develops multiple incommensurate antiferromagnetic phases. Time-of-flight neutron Laue diffraction found \(\mathbf{q}_2=(\delta_2,\delta_2,0)\) with \(\delta_2=0.085\) at 13.5 K, then an abrupt change below \(T_{N3}=12.2\) K to \(\mathbf{q}_1=(\delta_1,0,0)\) with \(\delta_1=0.17\) at 11.5 K and \(\delta_1=0.194\) at 4.3 K [2104.07935]. Resonant magnetic x-ray scattering later resolved AFM1 as an in-plane spin-density wave, AFM2 as coexistence of that SDW with a second SDW having moments along \(c\), AFM3 as a single-chirality magnetic helix, and AFM4 as a helix with reversed chirality; all four phases remain single-\(\mathbf{k}\) [2403.10159].

A distinctive low-temperature result is the spontaneous reversal of spin chirality. Below \(T_{N3}=12.2\) K the helix is stabilized with a single chirality across the sample, while below \(T_{N4}\simeq 10\) K the chirality reverses and the sample remains a single chiral domain. Concomitantly, the symmetry lowers to polar monoclinic, with uniaxial charge and spin strip domains. Group-theoretical analysis shows that the polar monoclinic symmetry is required to explain the asymmetry of the two chiral states and the chirality reversal [2403.10159].

Under field \(H\parallel[001]\), EuAl\(_4\) hosts a rhombic skyrmion lattice in phase II, a square skyrmion lattice in phase III, vortex–antivortex phases, meron–antimeron textures, and single-\(\mathbf{Q}\) spirals. The fundamental modulation vectors of phases III, VI, and VII are \(\mathbf{Q}_1=(q,q,0)\), \(\mathbf{Q}_2=(q,-q,0)\) with \(q\approx 0.085\), while phase I carries \(\mathbf{Q}_1=(0.194,0,0)\) and phase V \(\mathbf{Q}_1=(0.17,0,0)\) [2511.07079].

The microscopic origin of these textures is actively debated. The structural identification of the CDW phase as non-centrosymmetric \(F222(0\,0\,\sigma)00s\) means that ordinary Dzyaloshinskii–Moriya interactions are symmetry-allowed below \(T_{\mathrm{CDW}}\), so a more exotic mechanism is not required to account for skyrmions in the ordered state [2506.01633]. By contrast, soft-x-ray ARPES on Eu(Ga\(_{1-x}\)Al\(_x\))\(_4\) argues that multiple nesting vectors derived from a Z-centered Fermi-surface pocket match the periodicities and symmetries of the helical and skyrmion phases, suggesting a common origin in competing nesting-induced RKKY interactions [2604.12674]. This suggests that realistic models of EuAl\(_4\) must account simultaneously for symmetry-allowed DM terms and for strongly momentum-selective itinerant exchange.

## 5. Electronic structure, phonons, and transport renormalization

Band-structure calculations and ARPES consistently place EuAl\(_4\) in the class of three-dimensional topological semimetals. In the tetragonal basic structure a Dirac nodal crossing occurs above \(E_F\) along \(\Gamma\)–\(Z\), protected by \(C_{4z}\), while the partial density of states at \(E_F\) is dominated by Al-derived states and the Eu \(4f\) manifold lies well below \(E_F\) [2202.10282]. Soft-x-ray ARPES across the Eu(Ga\(_{1-x}\)Al\(_x\))\(_4\) series further identified a Lifshitz transition between EuGa\(_4\) and EuGa\(_{2.48}\)Al\(_{1.52}\), where a Z-centered electron pocket emerges; in EuAl\(_4\) this pocket supplies the nesting vectors that match the zero-field helix and the square skyrmion lattice [2604.12674].

The low-temperature electronic structure is strongly reconstructed by magnetism. Laser ARPES showed that EuAl\(_4\) undergoes band splitting, backfolding, the appearance of new Fermi sheets, and a large enhancement of quasiparticle lifetime across the AFM transitions, with the most dramatic changes at the AFM3 \(\rightarrow\) AFM4 transition rather than at \(T_{N1}\). This coincides with the largest drop in resistivity and indicates that the detailed magnetic structure, not merely the presence of order, controls carrier coherence [2409.16468].

The origin of the CDW is now tied to momentum-dependent electron–phonon coupling. Inelastic x-ray scattering revealed a broad softening of a transverse acoustic branch along \(\Gamma\)–\(Z\) that freezes out at \(T_{\mathrm{CDW}}\), and the eigenvector of that soft mode matches the displacement pattern of the modulated phase. The broad anomaly, together with the absence of a sharply peaked susceptibility, places EuAl\(_4\) in the “type II” category of EPC-driven CDWs rather than a simple Peierls nesting picture [2402.15397]. Comparative Wannier-based susceptibility calculations across BaAl\(_4\)-type compounds reached the same conclusion: the CDW in EuAl\(_4\) and SrAl\(_4\) requires strong EPC to a transverse acoustic mode at small \(q\) along \(\Gamma\)–\(Z\), in addition to a maximum in \(\mathrm{Re}\,\chi_0(\mathbf{q})\) [2306.15068].

Pressure and Raman spectroscopy expose the same hierarchy. High-pressure IXS under diamond-anvil conditions shows that the EPC responsible for the CDW is progressively suppressed by hydrostatic pressure, with \(dT_{\mathrm{CDW}}/dP\approx -54.7\) K/GPa and a critical pressure \(P_c\approx 2.5\) GPa for CDW suppression; the phonon self-energy analysis identifies a critical EPC amplitude \(A_{\max}=2.6\) meV at the transition [2602.14884]. Raman measurements, in turn, found that below \(T_c\sim 145\) K the Fano asymmetry \(1/|q|\) of the \(A_{1g}\) and \(B_{1g}\) phonons decreases with the free-carrier density, indicating weakened EPC in the CDW ground state, while the \(B_{1g}\) linewidth reveals enhanced phonon–phonon interactions and stronger lattice anharmonicity. The same Raman work identified shoulder-like anomalies around 50 K, suggesting a possible intermediate electronic state between the high-temperature metal and the fully developed CDW regime [2501.02171].

## 6. Tuning, family relationships, and unresolved directions

EuAl\(_4\) is unusually sensitive to weak symmetry-breaking perturbations. Compressive uniaxial stress along \([010]\) of only several tens of MPa enhances antiferromagnetic character, increases the helix wavevector in phase I from \(q=0.194\) at 0 MPa to \(q=0.201\) at 80 MPa, raises \(T_{N3}\) with slope \(dT_{N3}/d\sigma_{[010]}\approx 25\) K/GPa, suppresses phase V, and destabilizes the square skyrmion lattice in favor of other phases [2511.07079]. First-principles calculations in that study show that orthorhombic distortion reshapes the Fermi surface and changes the nesting vectors, supporting a direct route from lattice distortion to magnetic modulation.

Within the broader BaAl\(_4\) family, EuAl\(_4\) is closely related to SrAl\(_4\), which also hosts a non-centrosymmetric transverse modulation described by \(F222(0\,0\,\sigma)00s\), and to EuAl\(_2\)Ga\(_2\), whose CDW instead adopts the I-centered orthorhombic superspace group \(Immm(00\gamma)s00\). Despite that difference, both EuAl\(_4\) and EuAl\(_2\)Ga\(_2\) place the CDW primarily on the Al1-type layers [2309.08959;2408.13563]. In the Eu(Ga\(_{1-x}\)Al\(_x\))\(_4\) series, only \(x=1\) and \(x=0.50\) show CDW-like transport anomalies at ambient pressure, which was attributed to the combined effects of chemical order and chemical pressure [1804.02076].

Surface-sensitive probes add another layer. ARPES and STM on EuAl\(_4\) reveal a \(1\times 2\) surface reconstruction with ordered 50% Eu vacancies, quasi-one-dimensional modulations, and unidirectional replica bands orthogonal to the bulk CDW vector; these features disappear irreversibly on thermal cycling, indicating decoupled surface and bulk orders [2509.04742]. This establishes that the bulk incommensurate CDW does not exhaust the symmetry-lowering phenomena accessible in EuAl\(_4\).

Several questions remain open. A full symmetry-consistent refinement of all magnetic phases in the non-centrosymmetric CDW background is still required; the role of phason disorder in magnetic pinning and dynamics is unresolved; and the relative weights of CDW-enabled DM interactions and nesting-driven RKKY in stabilizing the multiple skyrmion phases remain under active discussion [2506.01633;2604.12674]. What is already clear is that EuAl\(_4\) is not adequately described as a simple tetragonal antiferromagnet with an incidental superstructure: its defining property is the mutual renormalization of a transverse incommensurate CDW, itinerant electronic structure, and unusually elaborate Eu-moment magnetism.

Source: https://www.emergentmind.com/topics/eual4