Introduction

The intricate relationship between lattice geometry and topological electronic behavior determines the ground state properties of materials. The non-trivial band topology of the kagome lattice is being extensively explored as candidates to engineer Dirac fermions1, topological flat bands2, magnetic Weyl semimetals3 or quantum spin liquid behavior4,5. Besides, the combination between the non-trivial topology and strong electronic correlations, originated from the interplay of orbital, charge and spin degrees of freedom, introduces topological correlated physics6 with more exotic phenomena, like topological Majorana modes.

Recently, a new family of non magnetic kagome metals AV3Sb5 (A=K, Rb, Cs) has emerged as a fertile playground to investigate correlation-driven topological phases7,8,9. Derived from the kagome structure, this family of materials features a quasi 2D electronic band structure with van Hove singularities (vHs) and Dirac crossings close to the Fermi level10,11. Superconductivity sets in at low temperature and is intertwined with an exotic charge density wave phase (CDW)12. Although the electronic structure obtained from density functional theory (DFT) calculations is well established, controversy surrounds the origin and stabilization of the CDW, and the mechanism of superconductivity and its gap symmetry13,14,15.

The rich phase diagram of the kagome lattice was investigated with respect to Fermi surface instabilities within the Hubbard model, revealing spin and CDW orders and unconventional superconductivity emerging upon tuning the band filling to the vHs16,17. Diffraction experiments and scanning tunneling microscopy (STM) unraveled multiple-q CDW modulations with an inverse Star of David structure (SoD) and a trihexagonal (TrH) distortions18,19, following the phonon instabilities at the M and L points predicted by DFT14,20. The q vector at M connects neighbouring vHs in the vicinity of the Fermi surface of AV3Sb5, suggesting that the CDW is triggered by the Peierls mechanism10,11. On the other hand, the CDW of CsV3Sb5 has been shown to break the time reversal symmetry21, unveiling an entanglement of superconductivity and charge order chirality19. Nevertheless, the chiral CDW has been questioned by recent STM experiments22, pointing to a sample/surface dependent fermiology or the presence of dynamical disorder. This is particularly appealing, since the sample quality and A-site occupancy has a strong impact on the Tc9 and might also influence the possible origin and stabilization of the charge modulations. Further, recent experiments reported a new unidirectional modulation charge stripe order with 4a0 wavelength that breaks the rotational symmetry22, nematicity23 and pair density wave order24.

Central to the interplay between CDW and \({{\mathbb{Z}}}_{2}\) band topology is the nature of the CDW phase. From the thermodynamic point of view, specific heat measurements8,9 show a small release of entropy through the CDW transition, suggesting a first order phase transition, and tight binding models describe the CDW as electronically mediated25,26. In contrast, DFT predicts soft modes at M and L points of the Brillouin zone14, for which, however, inelastic x-ray scattering (IXS) does not observe any anomaly of the low energy acoustic branches27. However, ARPES data have reported deviations from the linear electronic dispersion28, providing that the electron-phonon coupling gives sufficient strength for the CDW formation. Furthermore, optical spectroscopy ascribed the opening of the CDW gap to the suppression of density of states at the M point29 with strong phonon renormalizations30 without electronic anomalies at L. Finally, in addition to the enigmatic driving force of the CDW, the multiple qCDW and the different energy scales31 hinder a detailed study of the ground state of AV3Sb5, and the origin and stabilization of the CDW phase transition remains unclear.

Here, we use a combination of diffuse scattering (DS) and inelastic x-ray scattering (IXS) to identify frozen phonon fluctuations of the 2 × 2 × 2 (L point) 3D-CDW in AV3Sb5 at T > TCDW that condense into a triple-QL CDW through a first order phase transition, preserving the six-fold symmetry of the kagome lattice. The precursor of the charge order is absent for the 2 × 2 (M point) and 2 × 2 × 4 superstructures, indicating that the 2 × 2 × 2 charge modulation is the leading Fermi surface instability. Furthermore, the intensity of the quasi-elastic central peak (CP) of IXS follows the diffuse scattering (DS) intensity, showing that the CDW is of an order-disorder transformation type. This is corroborated by the absence of a clear softening, but small broadening of the optical branch at L. Our results solve the dispute about the primary order parameter and highlight the critical role of the order-disorder phase transition to understand the controversial experimental reports.

Results

x-ray diffraction

Figure 1a and b sketches the structure of CsV3Sb5 (P6/mmm space group). The unit cell is described as individual sublattices of a kagome network of V coordinated by Sb1 atoms and a hexagonal net of Sb2 above and below each kagome layer. The lattice parameters obtained from the DS at room temperature (a = b = 5.49 Å, c = 9.34 Å) are consistent with the reports in the literature7,8. The temperature dependence of the CDW instability at the L point is displayed in the Fig. 1e, showing a sharp onset at ~95 K, indicative of a first order-like phase transition. Bragg reflections with the same temperature dependence (not shown) were also measured with propagation vectors (0.5 0 0) and (0.5 0 0.25) r.l.u.27.

Fig. 1: Charge density wave in CsV3Sb5.
figure 1

a Chemical structure of AV3Sb5. Orange, red and cyan balls stand for A (Cs, Rb), V and Sb atoms. b Top view of the kagome structure, showing the V-kagome net interlaced with the Sb atoms. c Calculated phonon dispersions of fully-relaxed CsV3Sb5. Soft modes are observed at M and L. The red ellipse marks the region where phonons are measured in Fig. 4d. d Sketch of the Brillouin zone along the main symmetry directions. e Temperature dependence of the (2.5 0.5 0.5) CDW reflection in CsV3Sb5, showing a sharp drop of intensity at the TCDW = 95 K. (Inset, rocking scans).

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Diffuse Scattering, DS

Besides the sharp and intense CDW diffraction spots in x-ray diffraction, diffuse intensity is present at T > TCDW due to lattice vibrations and correlated disorder. Figure 2 summarizes the processed DS maps for the CsV3Sb5 sample at 90 and 100 K for the (h k 0.5), (h k 0) and (h 0 l) planes. The image reconstruction of the (h k 0.5) plane at 100 K (Fig. 2a, bottom) reveals the presence of DS corresponding to freezing of a transverse component of phonons (streaks highlighted in the yellow square), with most pronounced intensity around the (3 0 0.5) reciprocal lattice vector. Other satellites are equivalent according to the symmetry operations of the P6/mmm space group. The DS is visible below 150 K and condense into a 3QL order at TCDW (Fig. 2a, top). The fluctuating precursor observed in Fig. 2a along the < 1 0 1 > direction is in agreement with the phonon calculations that predict a lattice instability at the L point, Fig. 1c14,20. The correlation length of the charge precursor (ξ) perpendicular to the streak of diffuse intensity remains nearly constant as we cool down to TCDW and extends to 4-5 unit cells, while along the streak reduces to less than one unit cell, Fig. 3b. Remarkably, DS is absent for T > TCDW in the (h k 0) plane (Fig. 2b, bottom) and (h 0 l) plane (Fig. 2c, bottom) that maps the longitudinal component of the (0.5 0 0.25) and (0.5 0 0.5) CDWs19,22,27. However, CDW reflections are again present at T < TCDW, matching the hard x-ray diffraction data of Fig. 1e. The presence of diffuse intensity as satellites at only the (0.5 0 0.5) reciprocal lattice vector indicates that the Fermi surface instability at the L position is the leading order parameter and drives the CDW formation in the kagome metal AV3Sb5 (see Supplementary Fig. 1 for the DS maps of RbV3Sb5).

Fig. 2: Diffuse scattering maps of CsV3Sb5.
figure 2

a (h k 0.5) plane at 90 K (top), showing the CDW reflections (weak spots highlighted within the yellow circle in the zoomed in area) and the precursor of the 3D CDW at 100 K (bottom), marked with arrows. b, c (h k 0) and (h 0 l) planes, respectively (no precursor). The DS maps of RbV3Sb5 are shown in the Supplementary Fig. 1.

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The reconstruction of the (h k 0.5) plane for CsV3Sb5 at different temperatures around the (3 0 0.5) cut is shown in Fig. 3a-c. Figure 3d shows the profile of the diffuse precursor along the diffuse streak of the Fig. 3b. The critical scattering was modelled assuming a sum of Bragg and a Lorentzian profile that accounts for the peak sharpening as T approaches TCDW from the high temperature. The temperature dependence of the DS intensity is extracted by integrating the diffuse precursor and plotted in Fig. 3f. It starts at temperatures below 150 K and increases down to 100-105 K, below which it sharply diverges at the critical temperature, following the temperature dependence of the energy integrated experiment of Fig. 1e.

Fig. 3: Temperature dependence of the diffuse scattering and IXS central peak.
figure 3

a 90 K, b 100 K and c 150 K around the (3 0 0.5) cut. The sketch in c shows the reciprocal lattice vectors and the IXS scan direction through the CDW, as indicated in the red line. d Profile of the diffuse scattering at 120 K and its fitting to a sum of Bragg and Lorentzian profiles. e Temperature dependence of the IXS spectra at the (2.5 0.5 0.5) reciprocal lattice vector, highlighting the enhancement of the elastic line and drop of intensity of the low-energy phonon at 96 K. f Temperature dependence of the intensity of the DS signal, elastic central peak and optical phonon. Inset, temperature dependence of the generalized susceptibility. T* denotes the change of slope of the electronic susceptibility, ~105 K, presumably related to percolation of CDW domains. The error bars are the fit to uncertainty.

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Inelastic x-ray scattering, IXS

To disentangle static and dynamic contributions and to get further insights about the origin of the observed diffuse features, we have carried out IXS scanning the (2.5 0.5 0.5) satellite. As shown in Fig. 3e, the IXS spectra consists on a (quasi)-elastic central peak (zero energy loss) evolving in intensity below 120 K and a low energy non-dispersive phonon at 6 meV. The intensity of the CP sharply drops upon warming from 95 K and follows the intensity of the pretranslational fluctuations detected by DS above 105 K. The CP of CsV3Sb5 shows parallels to the critical scattering observed in SrTiO332,33 and transition metal dichalcogenides34 exhibiting a critical divergence on approaching TCDW. On the other hand, directly visualized in the raw data of Fig. 3e, the intensity of the low-energy mode increases smoothly upon cooling and drops below 100 K due to the modification of the dynamic structure factor on approaching the CDW phase, making the quasi-elastic contribution more important than the phonon intensity to the DS. Following the applicability of the Ornstein-Zernike correlation function35, the generalized susceptibility, χ(q), proportional to the q Fourier component of the displacement-displacement correlation function 〈∣uq∣2〉 = kBTχ(q), displays a linear Curie-Weiss behavior for the quasi-elastic component of the CP (inset of Fig. 3f) and vanishes at the critical temperature, indicating a temperature dependence of the density of states. Nevertheless, χ(q) of the DS changes slope at T* ~ 105 K, coinciding with the drop of intensity of the low-energy phonon and the deviation of the DS and CP intensities (we tentatively associate T* as the temperature where the CDW domains start to percolate.).

Focusing on the inelastic part of the IXS spectra, Fig. 4a shows the momentum dependence of the phonon dispersion along the (h h 0.5) direction (ΔE=3 meV). The spectra consist on 3 weakly dispersive branches with 2.5 (γ), 6 (ω) and 11 meV (δ) at h = 0. The lowest energy mode merges with the 6 meV branch at 0.1 r.l.u. and, at the L point, only two optical-like modes, ω and δ, are observed. These two modes are better visualized in Fig. 4b and its energy dispersion, obtained from the fitting to damped harmonic oscillators (DHO) convoluted with the experimental resolution, is displayed in the Fig. 4c. They exhibit a weak dispersion in reciprocal space and can be described as a system of independent quantum oscillators or localized atomic vibrations (Einstein-type phonon modes). To isolate the contribution of each mode, we increased the energy resolution down to 1.5 meV and work at the (3.5 2.5 0.5) reciprocal lattice vector. Figure 4d shows the temperature dependence of the ω branch. Indeed, the ω phonon in Fig. 4b consists on 2 modes, which we label here as ω1 and ω2, in good agreement the DFPT calculations in Fig. 1c (Supplementary Fig. 2). From these data, the energy of ω1 and ω2 remains constant as the temperature approaches the CDW transition. This clearly demonstrates the absence of a Kohn anomaly and discards a soft phonon nature of the CDW phase transition; i.e, it is not phonon driven but of order-disorder type. This does not preclude the presence of a soft mode below TCDW, which is, in fact, a fingerprint of both order-disorder and displacive phase transitions. We need to stress here that it is not the crystal quality, but the inherent dynamical disorder that stabilizes the CDW in CsV3Sb5. On the other hand, a detailed analysis of the phonon linewidth reveals an anomalous broadening at TCDW. At 150 K, ω1 is not resolution limited and develops a small intrinsic broadening of ~0.5 meV (a DHO with 0 meV intrinsic width corresponds to the instrumental resolution of 1.5 meV). Upon cooling, the width of the phonon decreases and becomes resolution limited at 105 K. However, its width increases again (not resolution limited) below 100 K, coinciding with T*, a signature of a proximity to a Fermi surface instability.

Fig. 4: Einstein modes and phonon broadening.
figure 4

a Color map of the low-energy phonons of CsV3Sb5 along the (3-h h 0.5) direction at 120 K. The three modes identified at 2.5 (γ, asterisks), 6 (ω, triangles) and 11 meV (δ, squares) are superimposed. b IXS spectra and c dispersion along the (2+h 1-h 0.5) direction (ΔE = 3 meV) of the δ and ω branches. d Temperature dependent high resolution IXS spectra (ΔE= 1.5 meV) at the 3.5 2.5 0.5 reciprocal lattice vector. Inset, fitting details of the scan at 150 K. Note that, ω in b is now resolved as 2 modes, ω1 and ω2. e Temperature dependence of the linewidth of ω1. The error bars represent the fit uncertainty. See Supplementary for the fitting details and Supplementary Fig. 3.

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Discussion

The results presented here indicate that weak coupling limit theories, \({\omega }_{2{k}_{F}}{\tau }_{eh} < \)1 (\({\omega }_{2{k}_{F}}\) and τeh are the bare phonon frequency and electron-hole pair lifetime, respectively), are not adequate to interpret the absence of the Kohn anomaly and the major contribution of the CP to diffuse scattering data. This locates the CDW instability of AV3Sb5 in the strong coupling non-adiabatic or in the strong electron-phonon interaction (EPI) regimes. Indeed, recent reflectivity measurements have reported a much larger CDW gap than the weak-coupling BCS value29,36. The effect of the dynamical disorder was theoretically modelled by Yu and Anderson37 for the strong interaction of electrons and single dispersionless (Einstein-type) optical modes and further extended by Gor’kov38 for any arbitrary ionic displacements. Within this scenario, the CDW in AV3Sb5 realizes in two stages. First, an ionic displacement in the kagome plane introduces a strong EPI that traps the electronic cloud near an ion, changing the local electronic environment and breaking the adiabatic approximation. This gives rise to a quasi-elastic response above the CDW transition, an associated order-disorder dynamics and the absence of a Kohn anomaly. In an order-disorder transformation, the harmonic potential well deforms and the phase transition is achieved by the critical growth and percolation of quasi-static domains on approaching the critical temperature39. In each domain, the potential within the atom moves is quasi-harmonic and phonon anomalies are expected. This naturally explains the change of slope in the diffuse susceptibility of Fig. 3f and the small broadening of the low-energy mode in Fig. 4d, e. Strong EPI mechanisms have been considered to explain the CDW formation in the one dimensional Peierls instability of BaVS340,41, transition metal dichalcogenides42, H-bond ferroelectrics43 and the controversial charge order in high-Tc cuprates44. On the other hand, in the non-adiabatic dynamics, \({\omega }_{2{k}_{F}}\) fluctuates so rapidly that cannot couple with the electron-hole condensate during the electron-hole pair lifetime. The rapidly fluctuating phonon reduces the screening between atoms and prevents the development of the Kohn anomaly at TCDW. Following this argument, the suppression of the Peierls transition of BaVS3 under pressure was interpreted as a reduction of the τeh45. This is a particularly appealing scenario since the TCDW of CsV3Sb5 is completely suppressed at low pressures and evolves towards a double superconducting dome46,47,48,49.

We would like to finish our discussion by placing our results within the physics of the kagome metals AV3Sb5. The identification of the charge order fluctuations at T > TCDW indicates that the condensation of the CDW order parameter at the L point in (Cs,Rb)V3Sb5 is the leading instability, favoring the alternating Star of David and trihexagonal (LLL) pattern50. The absence of DS at the M point suggests that the strength of the coupling parameter of the L and M modes in the Landau free energy expansion, γML, is rather small or the phase transition is more strongly first-order (the condition for the susceptibility at the M point to diverge is ξ2 ~ γML, where ξ is the correlation length of the L order. If ξ never becomes too large, one would not expect an enhancement of the M susceptibility above TCDW. R. M. Fernandes, Private Communication). Our DS results also indicate that the 6-fold symmetry is preserved, discarding single QM/double QL orders (staggered SoD and staggered TrH). Rather, our data seem to agree with the experiments that support a combination of triple QM/triple QL order (LLL+MMM pattern)51,52 at low temperature.

On the other hand, the importance of the EPI has been highlighted recently by neutron scattering53 and, perhaps, acts concomitantly with nesting to enhance the charge correlations. Coherent phonon spectroscopy reported a condensation of M and L-point phonons, suggesting an order-disorder crossover at lower temperature54 to a different ordering pattern as observed here. An order-disorder transition is expected to be of first order type, thus the small release of entropy at TCDW8,9 fits within this scenario. Moreover, the data presented here does not conflict with the possible chiral charge density wave, but put constrains on the nematic order22,23. Broken rotational symmetries can be affected by local dynamic disorder, fluctuating domains and defects that introduce pinning potentials. These pinning centers and local distortions introduce variations of the amplitudes of the local atomic displacements, which can be detected by pair distribution function (PDF) analysis55.

In conclusion, we have presented diffuse scattering and inelastic x-ray scattering data to demonstrate that the CDW in the kagome metal AV3Sb5 is of order-disorder type without phonon softening. A precursor of the CDW is observed in diffuse scattering with propagation vector (0.5 0 0.5), indicating that the CDW instability at the L point is the primary order parameter, discarding a broken six-fold rotational symmetry at high temperature and pointing to an LLL pattern. Furthermore, we demonstrate that the diffuse signal contains intensity of the quasi-elastic central peak, likely due to the growth of fluctuating domains, thus favouring strong coupling theories to describe the CDW ground state of AV3Sb5. Our results present a step forward and provide crucial information to understand the enigmatic charge order of (Cs,Rb)V3Sb5.

Methods

Single crystals of AV3Sb5 (A=Cs, Rb) with TCDW=95 and 104 K, were synthesized by the self-flux growth method7. Diffuse scattering measurements were performed with a Dectris PILATUS3 1M X area detector at the ID28 beamline at the European Synchrotron Research Facility (ESRF) with an incoming energy Ei=17.8 keV. Orientation matrix refinement and reciprocal space reconstructions were done using the CrysAlis software package. Final DS images were processed with an in-house code. Low energy phonons were measured by inelastic x-ray scattering (IXS) at the ID28 IXS station at ESRF (Ei=17.8 keV, ΔE=3 meV) and at the HERIX beamline at the Argonne Photon Source (APS) (Ei=23.72 keV, ΔE=1.5 meV). The components (hkl) of the scattering vector are expressed in reciprocal lattice units (r.l.u.), (hkl)= ha* + kb* + lc*, where a*, b*, and c* are the reciprocal lattice vectors.

Lattice dynamic calculations were performed by Density Functional Perturbation Theory (DFPT) as implemented in QUANTUM ESPRESSO package. The forces were calculated in 8 × 8 × 4 supercells using the ultrasoft parametrization of the PBE exchange-correlation functional. We set the energy cutoffs of 60 and 600 Ry for the basis-set and treated the Van der Waals corrections within the Grimme’s semiempirical approach. An 8 × 8 × 4 k-point grid and a Marzari-Vanderbilt smearing of 0.01 Ry was used for the Brillouin zone integration. The convergence threshold was set to 10−19.