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Gan Liu, Xinran Ma, Kuanyu He, Qing Li, Hengxin Tan, Yizhou Liu, Jie Xu, Wenna Tang, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Libo Gao, Yaomin Dai, Hai-Hu Wen, Binghai Yan, Xiaoxiang Xi

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[Observation of anomalous amplitude modes in the kagome metal CsV3Sb5](https://mdr.nims.go.jp/datasets/3eb9bee9-a3aa-47e1-812b-7cad6fc0d61b)

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Observation of anomalous amplitude modes in the kagome metal CsV3Sb5ARTICLEObservation of anomalous amplitude modes in thekagome metal CsV3Sb5Gan Liu1, Xinran Ma1, Kuanyu He1, Qing Li 1, Hengxin Tan2, Yizhou Liu 2, Jie Xu1, Wenna Tang1,Kenji Watanabe 3, Takashi Taniguchi 4, Libo Gao 1,5, Yaomin Dai 1,5, Hai-Hu Wen 1,5,Binghai Yan 2✉ & Xiaoxiang Xi 1,5✉The kagome lattice provides a fertile platform to explore novel symmetry-breaking states.Charge-density wave (CDW) instabilities have been recently discovered in a new kagomemetal family, commonly considered to arise from Fermi-surface instabilities. Here we reportthe observation of Raman-active CDW amplitude modes in CsV3Sb5, which are collectiveexcitations typically thought to emerge out of frozen soft phonons, although phonon soft-ening is elusive experimentally. The amplitude modes strongly hybridize with other super-lattice modes, imparting them with clear temperature-dependent frequency shift andbroadening, rarely seen in other known CDW materials. Both the mode mixing and the largeamplitude mode frequencies suggest that the CDW exhibits the character of strong electron-phonon coupling, a regime in which phonon softening can cease to exist. Our work highlightsthe importance of the lattice degree of freedom in the CDW formation and points to thecomplex nature of the mechanism.https://doi.org/10.1038/s41467-022-31162-1 OPEN1 National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China. 2Department of Condensed MatterPhysics, Weizmann Institute of Science, Rehovot 7610001, Israel. 3 Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki,Tsukuba 305-0044, Japan. 4 International Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan.5Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China. ✉email: binghai.yan@weizmann.ac.il; xxi@nju.edu.cnNATURE COMMUNICATIONS |         (2022) 13:3461 | https://doi.org/10.1038/s41467-022-31162-1 | www.nature.com/naturecommunications 11234567890():,;http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-31162-1&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-31162-1&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-31162-1&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-31162-1&domain=pdfhttp://orcid.org/0000-0001-9612-4718http://orcid.org/0000-0001-9612-4718http://orcid.org/0000-0001-9612-4718http://orcid.org/0000-0001-9612-4718http://orcid.org/0000-0001-9612-4718http://orcid.org/0000-0003-3754-5170http://orcid.org/0000-0003-3754-5170http://orcid.org/0000-0003-3754-5170http://orcid.org/0000-0003-3754-5170http://orcid.org/0000-0003-3754-5170http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-7822-9812http://orcid.org/0000-0002-7822-9812http://orcid.org/0000-0002-7822-9812http://orcid.org/0000-0002-7822-9812http://orcid.org/0000-0002-7822-9812http://orcid.org/0000-0002-2464-3161http://orcid.org/0000-0002-2464-3161http://orcid.org/0000-0002-2464-3161http://orcid.org/0000-0002-2464-3161http://orcid.org/0000-0002-2464-3161http://orcid.org/0000-0003-0093-1625http://orcid.org/0000-0003-0093-1625http://orcid.org/0000-0003-0093-1625http://orcid.org/0000-0003-0093-1625http://orcid.org/0000-0003-0093-1625http://orcid.org/0000-0003-2164-5839http://orcid.org/0000-0003-2164-5839http://orcid.org/0000-0003-2164-5839http://orcid.org/0000-0003-2164-5839http://orcid.org/0000-0003-2164-5839http://orcid.org/0000-0002-8685-9267http://orcid.org/0000-0002-8685-9267http://orcid.org/0000-0002-8685-9267http://orcid.org/0000-0002-8685-9267http://orcid.org/0000-0002-8685-9267mailto:binghai.yan@weizmann.ac.ilmailto:xxi@nju.edu.cnwww.nature.com/naturecommunicationswww.nature.com/naturecommunicationsMaterials with a kagome lattice can host rich phenom-ena encompassing quantum magnetism1,2, Diracfermions3,4, nontrivial topology5–7, density waves,and superconductivity8–10. The recently discovered kagomemetals AV3Sb5 (A= K, Rb, or Cs)11,12 offer a new platform tostudy the interplay of these phenomena. These compoundshave Fermi levels close to Dirac points or van Hovesingularities12–14, leading to a plethora of possible intriguingground states. Indeed, charge-density waves and super-conductivity have been discovered12,15,16, with ample evidenceshowing that both types of orders are exotic. For example, theCDW transition is accompanied possibly by a large anomalousHall effect17,18, and the superconductivity features a pair-density wave state19.The nature of the CDW state and the mechanism for its for-mation have been under close scrutiny. In this work, we focus onCsV3Sb5, which has a CDW transition temperature TCDW= 94K12.Both hard-X-ray and neutron scattering showed the lack of softphonons (phonon modes that show frequency softening uponcooling toward a phase transition)20,21, although density functionaltheory (DFT) calculations found two phonon instabilities at the Mand L points of the Brillouin zone22–24. Considering that a 2 × 2modulation of the crystal lattice is well established12,19,25–27, theabsence of soft modes apparently breaks a pattern proven generic tomany known CDW systems—a soft phonon freezes to zero fre-quency and triggers the formation of a distorted lattice28. Currently,there is still no consensus on the form of the in-plane structure andthe c-axis periodicity27,29. The roles of Fermi-surface nesting andelectron–phonon coupling are also debated. Because the period ofthe 2 × 2 superlattice matches perfectly with the Fermiology of thevan Hove singularity, it is natural to ascribe the CDW transition toFermi surface nesting22,30–32, supported by the appreciable partialgapping of the Fermi surface observed experimentally33–37. How-ever, the calculated electronic susceptibility lacks the expecteddivergence38,39, and the effect of electron–phonon coupling maynot be dismissed21,39.Because the CDW features lattice distortions, studies of thelattice degree of freedom can offer insight into the mechanism.Raman scattering is a valuable tool in this respect. In well-studiedCDW systems, such as the transition metal dichalcogenides, as asoft phonon mode condenses to form a distorted lattice, newRaman-active collective excitations, known as amplitude modes,emerge, providing a direct probe of the CDW orderparameter40–42 (see Fig. 1a). Conversely, the observation ofamplitude modes is typically considered as evidence for the softmode. The temperature dependence of the amplitude modes aswell as the zone-folded modes, which become Raman-active dueto zone folding induced by the superlattice, can both reflect theCDW transition43–49. Combined with symmetry informationfrom polarization-resolved measurements, constraints can be seton the possible CDW ground state.Here, we report Raman scattering measurements on CsV3Sb5.We observe a multitude of CDW-induced modes, whose sym-metries and frequencies are in good agreement with DFT calcu-lations for a single-layer CsV3Sb5 under inverse Star of Daviddistortion. The observed temperature dependence of these modesand their calculated evolutions with varying lattice distortionallow us to identify two of them as amplitude modes, emergingfrom the predicted soft modes, although the soft modes are elu-sive experimentally. In contrast to mostly independent amplitudemodes and zone-folded modes in well-known CDWmaterials44–46, we show that they hybridize strongly in CsV3Sb5,causing spectral weight redistribution to the latter and renderingthem amplitude-mode-like. The anomalous hybridization and thelarge values of the amplitude mode frequencies provide evidenceof strong-coupling CDW, offering a possible explanation for thelack of soft modes. These results stress the importance of thelattice degree of freedom and electron–phonon coupling in theCDW formation in CsV3Sb5.ResultsRaman-active modes in CsV3Sb5. CsV3Sb5 crystallizes in ahexagonal lattice with the P6/mmm space group11. Figure 1bshows the unit cell of its crystal structure. The V atoms form akagome net interspersed by Sb atoms (labeled Sb1), all within theab-plane. The V atoms are further bonded by Sb atoms above andbelow the kagome plane (labeled Sb2). These V3Sb5 slabs areseparated by Cs layers, with weak coupling between them to forma quasi-two-dimensional (quasi-2D) structure. Factor groupanalysis yields three Raman-active phonon modes, ΓRaman=A1g+ E2g+ E1g. The former two can be detected when the pho-tons are polarized in the ab-plane, satisfied by the back-scatteringgeometry used in our experiment. These intense modes aremarked by dashed lines in Fig. 1c, d. They involve only the Sb2atoms, with their atomic vibrations along the c-axis and withinthe ab-plane for the A1g and E2g modes, respectively; see Fig. 1b.The E2g modes are a pair of degenerate vibrations with oppositecircling directions, i.e., opposite chiralities (see SupplementaryNote 1). These two types of symmetries can be distinguished bypolarization-resolved measurements. Specifically, the A1g modescan be detected in the XX and LL polarization configurations,whereas the E2g modes appear in the XX, XY, and LR config-urations. Here, XX and XY represent collinear and cross-linearpolarization for the incident and scattered photons, and LL andLR involve circularly polarized light with left (L) and right (R)helicity. A comparison of data in all four configurations isincluded in Supplementary Fig. 1.Below TCDW, multiple peaks emerge, highlighted by the dottedlines in Fig. 1c, d. Their origin will be discussed in the nextsections. These modes are rather weak compared to the mainlattice phonons. Their disappearance at 100 K suggests a closecorrelation with CDW formation. In contrast, many weak peak-like structures below 100 cm−1 lack temperature dependence,whose origin is unclear.Figure 1e compares the observed Raman mode frequencieswith those from DFT calculations for a single layer of CsV3Sb522,considering two possible forms of lattice distortion, the Star ofDavid (SD) and inverse Star of David (ISD, also referred to as tri-hexagonal) structures. Both of them show the same number ofA1g and E2g modes, but with different ordering. Overall, thecalculated ISD phonons agree much better with the experimentalresults, as shown in the figure and in Supplementary Tab. 1. Allthe five predicted A1g modes and five out of the eight predictedE2g modes are observed. The observed A1g mode below 50 cm−1is unaccounted for by our calculations. This mode was alsoobserved by pump-probe time-resolved spectroscopy, which,when compared with calculations taking into account interlayercoupling, was assigned as a Cs-mode due to CDW modulationalong the c-axis23. Except for this mode and the three missing E2gmodes due to their weak scattering cross section, the symmetryordering of all the other modes is in exact agreement between theexperiment and theory. These results suggest that the CDWground state consists of weakly coupled layers dominated by ISD-type distortion, but CDW modulation along the c-axis is alsoindispensable. Since the single-layer CsV3Sb5 holds the key tounraveling the CDW mechanism, we attempted creating atom-ically thin CsV3Sb5 by mechanical exfoliation. However, the lossof crystallinity impeded further investigation (SupplementaryFigs. 2 and 3). The almost non-detection of modes folded fromthe L-point may be attributed to weak interlayer interaction,because the M- and L-point instabilities differ only in theARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-31162-12 NATURE COMMUNICATIONS |         (2022) 13:3461 | https://doi.org/10.1038/s41467-022-31162-1 | www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsinterlayer ordering. Indeed, we have considered various forms ofc-axis modulation, and all of them predict a large number ofRaman modes far exceeding that observed experimentally(Supplementary Note 2). Polarization-angle dependent measure-ments (Supplementary Fig. 4) show that either the c-axismodulation is too weak to induce clear anisotropic Ramanresponse, or those candidate stacking orders with the D2h pointgroup can be ruled out.Temperature dependence of Raman modes. Figure 2a, b showsthe temperature-dependent Raman intensity color plot forCsV3Sb5, obtained in the LL and LR configurations, respectively.The intense A1g and E2g main lattice modes are the most con-spicuous features. Figure 2e–g shows the frequency (with thecorresponding value at 200 K subtracted), linewidth (full width athalf maximum), and normalized integrated area for both modes,extracted from Lorentzian fits of the peaks. The A1g frequencysharply increases below TCDW, whereas the E2g frequency exhibitsa subtle kink across the CDW transition. This is consistent withthe planar ISD lattice distortion mainly involving V atoms, for-cing the Sb2 atoms to displace along the c-axis, hence affectingthe out-of-plane vibration of the A1g mode more effectively. Thecalculated phonon vibration patterns and frequencies confirmthis picture (see Supplementary Fig. 5 and SupplementaryTable 1). The CDW transition also causes a faster decrease in thelinewidths below TCDW. This can be understood as being due tothe CDW-induced partial gapping of the Fermi surface33–37,which reduces the electron–phonon interaction. The integratedpeak intensity for both phonons increases upon warming, in linewith increased thermal phonon populations. The rate of increaseis faster when approaching TCDW from below, and interestingly,the value saturates below approximately 50 K. The renormaliza-tion of the phonon parameters across the CDW transition evi-dences sizable electron–phonon coupling.CDW-induced modes are labeled in Fig. 2a, b. Except for the A1mode, there appears to be two types of modes, represented by A2 andE3. A2 exhibits appreciable softening and broadening upon warmingtoward TCDW. It is overdamped before disappearing, visualized in thecolor plot in Fig. 2a as the streak of signal below 100 cm−1 between60 and 90 K (see also Supplementary Fig. 6). These are signatures of aCDW amplitude mode43–47, caused by the collapse of coherentCDW order near TCDW. E3 shows a smaller change of frequency andmuch less broadening, more consistent with the characteristics of azone-folded mode44, as this type of mode arises from folding a zone-boundary phonon to the zone center, making its temperaturedependence of the frequency as weak as that of normal phonons.Figure 2c, d compares the distinct temperature dependence of thesetwo types of modes. While A2 broadens significantly above 40 K, E3maintains its linewidth and suddenly vanishes above ~80 K. Thedramatic difference in the linewidth broadening is quantified inFig. 2i. Figure 2h shows the frequencies of all the observed Ramanmodes on the same scale. Upon warming, the CDW-induced modes(A1 excluded) soften more dramatically than the main lattice modes.While it is tempting to assign most of them as amplitude modesbecause of the apparent softening behavior, we show below that theyare in fact zone-folded modes, mixed with the amplitude modes topartially inherit their properties.Nature of CDW-induced modes. Although soft phonons were notdetected experimentally, our DFT results show that the formation ofCDW in CsV3Sb5 is similar to that in other well-known systems43–47,in the sense that a soft phonon mode at the CDW wavevectorcondenses and gives rise to a distorted lattice41. The imaginaryphonon modes of pristine CsV3Sb5 at three M points (see Supple-mentary Fig. 7) transform as irreducible representation Mþ1 (Ag) ofthe space group P6/mmm (little co-group D2h). Figure 3a shows thatthey form triply degenerate modes at Γ due to the artificial bandfolding without lattice distortion, in which these modes are notTωNormalCDWTCDWωAMωSM0Fig. 1 Raman-active phonon modes in CsV3Sb5. a Schematic illustration of the relation between the soft mode and amplitude mode in typical CDWmaterials, showing the latter emerges after the former freezes below TCDW. ωAM: amplitude mode frequency. ωSM: soft mode frequency. b Crystal structureof CsV3Sb5. Sb sites with different Wyckoff positions are labeled as Sb1 and Sb2. The arrows illustrate the vibration patterns of the main lattice A1g and E2gmodes. The E2g mode is doubly degenerate, and only one form is shown. c, d Raman spectra measured on the ab-plane at 100 K and 4 K in the LL and LRpolarization configurations. The dashed lines denote the main lattice phonons, and the dotted lines indicate the CDW-induced modes. e Comparison of themeasured (Expt.) and the calculated Raman mode frequencies for the inverse Star of David (ISD) and Star of David (SD) lattice distortions. The thick linesdenote the main lattice phonons. The dots indicate modes undetected in our experiment.NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-31162-1 ARTICLENATURE COMMUNICATIONS |         (2022) 13:3461 | https://doi.org/10.1038/s41467-022-31162-1 | www.nature.com/naturecommunications 3www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsmeasurable in Raman. Only when CDW appears, they come out asamplitude modes, characterizing the CDW transition. With CDWdistortion, they decompose to a singlet A1g mode and a doublet E2gmode under the point group D6h (see Supplementary Note 3):3Mþ1 ! A1g � E2g: ð1ÞDespite that intermediate structures in Fig. 3 are unstablestructures with finite atomic forces, calculated force constants arestill valid in the harmonic approximation (see “Methods”).Corresponding pseudo-phonon bands can provide useful insightsto understand the soft mode evolution with respect to CDWdistortion. As the lattice distorts from the pristine phase to theFig. 2 Evolution of the Raman modes in CsV3Sb5 across the CDW transition. a, b Temperature-dependent Raman intensity color plot for CsV3Sb5,measured in the LL and LR configurations. The normal phonon modes are labeled in black and the CDW-induced modes in white. The dashed lines markTCDW. c, d Temperature-dependent spectra for the A2 and E3 modes. e–g Frequency, linewidth, and amplitude for the E2g and A1g main lattice phonons. Thefrequency and amplitude are compared to the corresponding values at 200 K. h Temperature dependence of the Raman mode frequencies. i Temperaturedependence of the linewidth of the A2 and E3 modes. Error bars are standard deviations obtained from the least-squares fits to the phonon peaks.A2 A3A4A5A1g E2gE1 E2E3E4E2’ E3’10%40%60%90%100%A2 E1E1’Fig. 3 Phonon band structures and mode mixing in the process of CDW distortion. a Phonon band structures directly calculated by DFT. Here, 100%(0%) refers to the fully stable ISD (2 × 2 pristine) structure. 10% refers to the intermediate structure with 10% distortion from the pristine to ISD phases.After 2 × 2 × 1 band folding with no distortion, three imaginary modes (Mþ1 ) are folded to Γ. A weak ISD-type distortion lifts the degeneracy and leads to A1gand E2g modes. The ISD distortion gradually transforms imaginary modes to real. b Projections of the imaginary A1g (E2g) mode with 10% distortion to allthe other phonon modes at Γ, as evolving into the stable ISD phase (100%). We highlight all A1g and E2g modes by orange and blue dots, respectively, atthe Γ point. The dashed orange (blue) curve in (b) guides eyes to show the evolution of the imaginary A2 (E1) modes in the CDW distortion.ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-31162-14 NATURE COMMUNICATIONS |         (2022) 13:3461 | https://doi.org/10.1038/s41467-022-31162-1 | www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsstable ISD pattern, the imaginary A1g and E2g modes turn realwith positive frequencies, expected to be observable as twoRaman-active amplitude modes. The phonon displacementpatterns of the soft A1g and E2g modes are shown in Fig. 4a, b,dominated by vibrations of V atoms. The A1g mode is fullysymmetric, involving breathing-type motion for the V triangles, Vhexagons, and Sb2 atoms. The E2g mode involves circling motionfor the atoms forming the V hexagons, while the amplitude forthe Sb2 vibration is almost ten times smaller.DFT calculations further reveal that the amplitude modesstrongly hybridize with the other CDW-induced Raman modes(i.e., the zone-folded modes always at positive frequencies at the Γpoint in Fig. 3a), rendering them amplitude-mode-like, hence theirapparent temperature-dependent frequencies. Figure 4c, d showsthe real space displacement patterns of all the CDW-inducedmodes in the 2 × 2 × 1 ISD phase. A2−5 and E1−4 correspond tothose in Fig. 2, and E01�3 are undetected experimentally.Comparison with Fig. 4a, b shows that A2−5 (E1−4) all resemblethe A1g (E2g) soft mode, but the difference is also apparent. Thesimilarity results from hybridization of phonon modes.To quantify the mode mixing, we calculated the overlapbetween the soft modes and all the real modes of the stable ISDphase by projecting the phonon dynamical matrix eigenvectors,Pf ¼ jhuf j uSMij2, where juSMi refers to the eigenvector of thesoft modes shown in Fig. 4a, b and juf i refers to the eigenvectorof the mode at frequency f in the ISD phase. The results inFig. 3b show that as the soft A1g and E2g modes shift fromnegative to positive frequencies and turn into amplitude modes,they hybridize with most of the zone-folded modes belonging tothe same irreducible representation. The amount of calculatedprojection in the stable ISD phase correlates reasonably well withthe observed mode intensity in Fig. 1c, d. A2 and E1 are residualamplitude modes after mode mixing. E01�3 show minorprojection from the E2g soft mode because of negligibleeigenvector overlap, and accordingly their scattering crosssection is weak. E2−4 all involve V triangles (Fig. 4d), indicatingthat they have contributions unrelated with the E2g soft mode.Indeed, as discussed earlier, E3 shows clear experimentalsignatures of a zone-folded mode.The A2 mode was also observed by Wulferding et al. in theirRaman study50 and by time-resolved pump-probe spectroscopy23,51.However, in these works, it was suggested to emerge below~60 K23,50,51, hence ascribed to another phase transition associatedwith a unidirectional order19,25–27. According to our data (Fig. 2a),the A2 mode survives above 60 K, and there is no clear evidence fortwo distinct phase transitions. Its vibration pattern shown in Fig. 4cconfirms no relation with the unidirectional order. Ramanscattering, as a bulk probe, is probably not sensitive enough to theunidirectional order, due to its possible surface origin26,52 and itsexistence in nanoscale domains26. A second bulk transition wellbelow TCDW was recently revealed by multiple techniques53–55,which evaded detection by our Raman measurements, possibly alsodue to the lack of sufficient sensitivity. Another Raman study by Wuet al.56 reported a similar set of modes as ours, but with differentrelative intensities. They also observed extra modes that are possiblydue to stronger c-axis modulation in their sample.DiscussionThe anomalously large hybridization between the amplitudemodes and zone-folded modes is rare, because they are mostlydecoupled in the canonical CDW materials, with the amplitudemodes dominating the spectral intensity44–46. The hybridizationis highly unusual, because the A2 and E1 amplitude modes and thezone-folded Raman modes span a wide frequency range, and theydo not overlap in energy (except for E1 and E01) to exhibit thetypical anti-crossing57,58. Their strong coupling suggests that thehybridization occurs indirectly, through interaction with thecommon electronic system. As the Fermi surface instabilityassociated with the van Hove singularity is from the V bands22,modes mainly involving V (including A2−5 and E1−4) naturallymix with the amplitude modes, whereas those mainly involvingSb (including E01�2 and the A1g and E2g main lattice modes) donot. Similar mode mixing was also observed in the quasi-one-dimensional (quasi-1D) K0.3MoO3 using time-resolved pump-probe spectroscopy59, and a simple model based onGinzburg–Landau theory can well describe the entanglement ofthe electronic and lattice parts of the CDW order parameter. Thesimilar phenomena observed in two systems with differentSoft A1gSoft E2gA2 A3 A4 A5E1 E2 E3 E4E1’ E2’ E3’abbacVSb1Sb2cbadFig. 4 Real space displacement patterns of the imaginary soft modes and the stable CDW-induced Raman modes in the 2 × 2 × 1 ISD phase. a, b Softmodes with A1g and E2g symmetries, respectively. c, d CDW-induced A1g and E2g stable modes. The E2g modes are pairs of chiral phonons and only onechiral mode is shown. The radius of the circles represents the amplitude of the vibration, and the arrow on the circles stands for the initial phase of thevibration. Cs atoms are omitted in the crystal structure for clarity, because they do not contribute to lattice vibrations.NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-31162-1 ARTICLENATURE COMMUNICATIONS |         (2022) 13:3461 | https://doi.org/10.1038/s41467-022-31162-1 | www.nature.com/naturecommunications 5www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsdimensionality suggest the importance of electron–phonon cou-pling in both materials.However, a soft phonon is well established in K0.3MoO360,but shown to be absent in CsV3Sb520,21. In the mean-fieldweak-coupling theory28,41, the phonon softening, known asKohn anomaly61, is a direct consequence of the divergentelectronic susceptibility, which screens the phonon vibration atthe CDW wavevector. In reality, the singular electronic sus-ceptibility is smeared out, especially at dimensions higher thanone, and momentum dependent electron–phonon couplingdictates the phonon renormalization in certain systems such as2H-NbSe262,63. The lack of soft phonons in CsV3Sb5 seems torule out both mechanisms. Instead, CsV3Sb5 may fall into thestrong electron–phonon coupling regime64, in which the non-detection of Kohn anomaly in the quasi-1D (TaSe4)2I, NbSe3,and BaVS3 has also been reported65–67. In all these materials,strong electron–phonon coupling tends to localize electrons,violating the adiabatic Born-Oppenheimer approximation64used in DFT. Failure of the conduction electrons to screen thephonon vibration can naturally explain the absence of phononsoftening68. Possible phonon softening is also interrupted bythe first-order nature of the CDW transition53,69,70, precludingthe observation of complete softening to zero frequency. Thefirst-order transition may be understood as due to trilinearcoupling of the three components (3Q) of the CDW24 or theasymmetric double-well elastic potential for the ions22, bothcontributing a term to the Landau free energy which is odd inthe order parameter.The strong-coupling nature of the CDW in CsV3Sb5 isindeed supported by multiple facts, according to the qualitativecriteria discussed in ref. 28. The CDW-induced gap ΔCDW islarge, with 2ΔCDW/kBTCDW ≈ 22 according to infraredspectroscopy33, where kB is the Boltzmann constant. The latticedistortion is substantial (amounting to about 5% of the latticeconstant22), the distorted lattice exhibits clustering of V atomsto form trimers and hexamers, and the CDW locks with thepristine lattice to form a commensurate structure, all indicat-ing local chemical bonding22,38. Moreover, DFT shows that theelastic potential for the ions in the pristine structure featuresdouble minima deeper than the thermal energy kBTCDW at thetransition22, a defining feature of the strong-coupling theoryproposed by Gor’kov64. Such potential well traps the ions inone of its minima, precluding soft phonon condensation. Fromthe perspective of Raman scattering, the electron–phononcoupling constant λ can be estimated from the amplitude modefrequency ωAM and the unscreened soft mode frequency ω0SM asλ ¼ ðωAM=ω0SMÞ2, valid on the mean-field level41. The resultsfor CsV3Sb5 and a variety of other CDW materials are com-plied in Fig. 5. Notably, the four quasi-2D compounds 2H-NbSe2, 2H-TaSe2, CsV3Sb5, and 1T-TiSe2 are roughly locatedin the expected order according to their TCDW, ΔCDW, andcommensurability. The frequencies of the amplitude modes inCsV3Sb5 are large, only lower than that of the higher one in1T-TiSe2. Although the exact value of λ may not be meaningfulbeyond the weak-coupling limit, these results clearly indicatethe strong-coupling nature of the CDW in CsV3Sb5.Our Raman results offer informative insights into the CDWphase in CsV3Sb5, suggesting the dominance of the ISD-typedistortion revealed by the CDW-induced modes and eviden-cing strong electron–phonon coupling. Although these resultsfavor the local chemical bonding picture of the CDW transi-tion, a coherent understanding of the mechanism, whichshould reconcile with the evidence for the electronically-drivenscenario22,30–37, is apparently called for. CsV3Sb5 represents aunique case in which the amplitude modes emerge in theabsence of soft phonons. As important collective excitations ofthe CDW ground state, how they form without being driven byfolding of a soft phonon warrants further investigation. DFTaccurately predicts the CDW-induced Raman modes in thezero-temperature limit, meanwhile showing the typical corre-lation of the amplitude modes and soft modes illustrated inFig. 1a, suggesting that the experimentally elusive soft mode issomehow still relevant. Our work may stimulate further studiesof the interplay between CDW amplitude modes and possiblesuperconducting Higgs mode48,49,71 in the kagome metals andthe control of these symmetry-breaking states by ultrafastlight72–74.MethodsSample preparation. CsV3Sb5 single crystals were synthesized using the fluxmethod11. The freshly cleaved surface of the samples was used in the study of bulkcrystals. Raman scattering spectroscopy was performed using home-built confocalmicroscopy setups in the back-scattering geometry with 532 nm laser excitation.The normally incident light was focused on the sample to a micron-sized spot, andthe scattered light was directed through Bragg notch filters to access the low-wavenumber region. The Raman signal was collected using a grating spectrographand a liquid-nitrogen-cooled charge-coupled device. The samples were mounted ina vacuum chamber during data acquisition. Temperature control was achievedusing a Montana Instrument Cryostation.Calculations. The DFT calculation results by Tan et al.22 are used to compare withthe experiment. In addition, we calculated the force constants by Vienna ab-initioSimulation Package (VASP)75 and computed the phonon dispersion relation byPhonopy76. Perdew–Burke–Ernzerhof-type generalized gradient approximation(GGA) method has been used77 and the projected augmented wave (PAW)potentials with 9 valence electrons for the Cs atom, 5 valence electrons for V and Sbatoms are employed. The DFT-D3 correction78 is used to take interlayer van derWaals interactions into account. For the DFT calculation, a 5 × 5 × 5k mesh and anenergy cutoff of 400 eV were used. For the pseudo-phonon spectra of the inter-mediate structures in Fig. 3, we used the same frozen phonon method to calculateforce derivatives and obtained force constants, which is valid in the harmonicapproximation. Specifically, for each intermediate structure we calculated the forcedifferences between the slightly perturbed structure and the unperturbed one. TheFig. 5 Evidence of strong-coupling CDW in CsV3Sb5. Frequency of theamplitude mode ωAM in the zero-temperature limit and the unscreenedfrequency of the soft mode ω0SM far above TCDW for a collection of CDWmaterials. Some of the materials feature two amplitude modes, hence twodata points connected by a vertical line. Since no soft mode is observed inCsV3Sb5, ω0SM is taken to be its acoustic phonon frequency at 300 K21.Open (filled) symbols indicate the material is quasi-1D (quasi-2D). Thedashed lines mark electron–phonon coupling constant λ= 1 and 3 accordingto mean-field theory. Source of data: ZrTe380,81, TbTe357,82,K0.3MoO343,60, 1T-TiSe247,83, 2H-TaSe245,46,84, 2H-NbSe284–86.ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-31162-16 NATURE COMMUNICATIONS |         (2022) 13:3461 | https://doi.org/10.1038/s41467-022-31162-1 | www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsrepresentation decomposition relation in Eq. (1) is derived by directly calculatingthe characters of the folded modes (whose real space patterns are shown in Sup-plementary Fig. 7). See more details in Supplementary Note 3. The symmetryrepresentations of Raman-active modes are calculated using the methods in ref. 79as implemented in Bilbao Crystallographic Server.Data availabilityThe data in Figure 1e are provided in Supplementary Table 1. Other data are availablefrom the corresponding authors upon reasonable request.Received: 26 February 2022; Accepted: 3 June 2022;References1. Sachdev, S. 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B.Y. acknowledges the financial support by the European ResearchCouncil (ERC Consolidator Grant “NonlinearTopo”, No. 815869) and the ISF - Quan-tum Science and Technology (No. 1251/19).Author contributionsX.X. conceived the project. G.L., X.M., and K.H. performed the Raman experiments. Q.L.,Y.D., and H.-H.W. grew the CsV3Sb5 crystals. K.W. and T.T. grew the h-BN crystals. G.L.and X.X. analyzed the experimental data. H.T., Y.L., and B.Y performed the DFT cal-culations. J.X., W.T., and L.G. performed atomic force microscopy measurements. X.X.and B.Y. interpreted the results and co-wrote the paper, with comments from all authors.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version contains supplementary materialavailable at https://doi.org/10.1038/s41467-022-31162-1.Correspondence and requests for materials should be addressed to Binghai Yan orXiaoxiang Xi.Peer review information Nature Communications thanks the anonymous reviewers fortheir contribution to the peer review of this work. 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To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2022ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-31162-18 NATURE COMMUNICATIONS |         (2022) 13:3461 | https://doi.org/10.1038/s41467-022-31162-1 | www.nature.com/naturecommunicationshttps://doi.org/10.1038/s41467-022-31162-1http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/www.nature.com/naturecommunications Observation of anomalous amplitude modes in the kagome metal CsV3Sb5 Results Raman-active modes in CsV3Sb5 Temperature dependence of Raman modes Nature of CDW-induced modes Discussion Methods Sample preparation Calculations Data availability References References Acknowledgements Author contributions Competing interests Additional information