Abstract
Owing to its unique geometry, the kagome lattice hosts various manybody quantum states including frustrated magnetism, superconductivity, and chargedensity waves (CDWs). In this work, using inelastic Xray scattering, we discover a dynamic shortrange \(\sqrt{3}\times \sqrt{3}\times 2\) CDW that is dominant in the kagome metal ScV_{6}Sn_{6} above T_{CDW} ≈ 91 K, competing with the \(\sqrt{3}\times \sqrt{3}\times 3\) CDW that orders below T_{CDW}. The competing CDW instabilities lead to an unusual CDW formation process, with the most pronounced phonon softening and the static CDW occurring at different wavevectors. Firstprinciples calculations indicate that the \(\sqrt{3}\times \sqrt{3}\times 2\) CDW is energetically favored, while a wavevectordependent electronphonon coupling (EPC) promotes the \(\sqrt{3}\times \sqrt{3}\times 3\) CDW as the ground state, and leads to enhanced electron scattering above T_{CDW}. These findings underscore EPCdriven correlated manybody physics in ScV_{6}Sn_{6} and motivate studies of emergent quantum phases in the strong EPC regime.
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Introduction
Quantum materials are typically strongly correlated or topologically nontrivial, giving rise to unconventional superconductivity^{1,2,3,4,5}, electronic nematicity^{6,7}, topological phases of matter^{8,9}, and quantum criticality^{10,11}. A common hallmark of quantum materials is the presence of competing electronic instabilities, such as the competition between a ferromagnetic metal and a paramagnetic insulator in the manganites that lead to colossal magnetoresistance^{12,13}, and the competition between chargedensity wave (CDW) and superconductivity in the cuprates^{14,15,16,17}.
Whereas the physics in many quantum materials are derived from strong electronic correlations, the unique geometry of the kagome lattice leads to geometric frustration, Dirac cones, magnon/electronic topological flat bands, and van Hove singularities^{18,19,20,21,22,23,24,25,26,27}, the combination of which gives rise to nontrivial electronic topology and correlated manybody states. As exemplified by AV_{3}Sb_{5} (A = K, Rb, Cs)^{28} and FeGe^{29,30}, kagome metals could exhibit an unconventional CDW breaking both timereversal and rotational symmetries^{31,32,33,34,35} coexistent with a superconducting ground state^{31,36,37}, and a CDW that coexists with antiferromagnetism that enhances the ordered moment^{29,30}, demonstrating the kagome lattice to be amenable to unconventional CDWs. Furthermore, the CDWs in both AV_{3}Sb_{5} and FeGe are associated with a 2 × 2 inplane ordering^{31,38,39,40,41}, indicating a prominent role of nesting between neighboring van Hove singularities^{42,43,44}.
Recently, CDW was discovered in the bilayer kagome metal ScV_{6}Sn_{6}^{45}, a member of the HfFe_{6}Ge_{6}type compounds. Similar to AV_{3}Sb_{5}, V atoms in ScV_{6}Sn_{6} form kagome layers with VV distances in the range 2.73–2.75 Å, the V dorbital bands cross the Fermi level, and there are no local moments^{45}. In contrast to AV_{3}Sb_{5} and FeGe, the V atoms in ScV_{6}Sn_{6} form kagome bilayers [Fig. 1a], and the CDW is associated with a \(\sqrt{3}\times \sqrt{3}\) inplane ordering [Fig. 1b], and a tripling of the unit cell along the caxis. Furthermore, whereas the CDW in AV_{3}Sb_{5} is dominated by inplane displacements of the V atoms^{40} and hosts a superconducting ground state, the CDW in ScV_{6}Sn_{6} is mostly driven by displacements of the Sc and Sn atoms along the caxis^{45}, and no superconductivity is observed up to pressures of 11 GPa^{46}. Optical reflectivity measurements and electronic structure calculations indicate that the CDW in ScV_{6}Sn_{6} is unlikely to result from Fermisurface nesting, and the CDW does not exhibit a prominent charge gap formation^{47}, distinct from AV_{3}Sb_{5}^{48,49}. This view is reinforced by electronic structure measurements, which in addition identify the lattice or a Lifshitz transition as instrumental for the CDW in ScV_{6}Sn_{6}^{50,51}.
Phonons play crucial roles in the CDWs of the kagome metals AV_{3}Sb_{5} and FeGe, and understanding their behaviors offered critical insights regarding the mechanism underlying CDW formation^{38,52,53,54}. Whereas CDWs in both the weak and strongcoupling limits are expected to exhibit soft phonons above the CDW ordering temperature, inelastic Xray scattering (IXS) measurements of AV_{3}Sb_{5} reveal an absence of such phonon softening, suggesting an unconventional CDW near the van Hove filling^{38,53}. Inelastic neutron scattering unveils the hardening of a longitudinal optical phonon inside the CDW state of CsV_{3}Sb_{5}, implicating a key role of electronphonon coupling (EPC) in the CDW formation^{52}. IXS measurements of FeGe uncover a charge dimerization and significant spinphonon coupling, which intertwine with magnetism to drive the CDW formation^{54}. In the case of ScV_{6}Sn_{6}, theoretical calculations find competing lattice instabilities^{55} and the softening of a flat phonon mode is observed via a combination of experimental and theoretical techniques^{56}.
Here, we use IXS to study the lattice dynamics related to CDW formation in ScV_{6}Sn_{6}, revealing a clear phonon softening above the firstorder CDW ordering temperature T_{CDW} ≈ 91 K [Fig. 1c]. Whereas longrange static CDW order occurs at \({{{{{{{{\bf{q}}}}}}}}}_{{{{{{{{\rm{s}}}}}}}}}=(\frac{1}{3},\frac{1}{3},\frac{1}{3})\), corresponding to a \(\sqrt{3}\times \sqrt{3}\times 3\) CDW (q_{s}CDW), the phonon softening is most prominent at \({{{{{{{{\bf{q}}}}}}}}}^{*}=(\frac{1}{3},\frac{1}{3},\frac{1}{2})\), corresponding to a shortrange \(\sqrt{3}\times \sqrt{3}\times 2\) CDW (q^{*}CDW). q^{*}CDW gains in intensity upon cooling, but becomes suppressed below T_{CDW}, replaced by q_{s}CDW via a firstorder transition [Fig. 1d]. These observations depict a CDW formation process in ScV_{6}Sn_{6} distinct from known nestingdriven or typical EPCdriven CDWs^{57}, with the softest phonon occurring at q^{*}, while the static CDW occurs at a distinct wavevector q_{s} [Fig. 1e]. Firstprinciples calculations reveal that although q^{*}CDW is energetically more favorable at the density functional theory level, a qdependent EPC promotes q_{s}CDW as the ground state, and also leads to strong electron scattering above T_{CDW}, accounting for the large resistivity drop upon cooling below T_{CDW}. These findings underscore the importance of EPCdriven manybody physics in ScV_{6}Sn_{6} and provide a further example of unconventional CDW on the kagome lattice.
Results
Competition between two distinct CDWs
Elastic scattering in ScV_{6}Sn_{6} was measured by setting the energy transfer in IXS to zero, with results presented in Fig. 2. For T ≳ 100 K, clear diffuse scattering centered around (0, 0, 6) + q^{*} are observed in lscans [Fig. 2a]. Scans along \((\frac{1}{3}+h,\frac{1}{3}+h,6\frac{1}{2})\) and \((\frac{1}{3}+h,\frac{1}{3}h,6\frac{1}{2})\) confirm the shortrange nature of these peaks along two orthogonal inplane directions [Fig. 2b, c]. The q^{*}CDW peak is significantly broader in the hkplane than along l, and the peak asymmetry in Fig. 2b likely results from the variation in structure factors of the associated soft phonons in different Brillouin zones, since q^{*} (H) is a highsymmetry point and the energies and damping rates of phonon modes should be symmetric around it (Supplementary Note 1 and Supplementary Fig. 1). These diffuse scattering centered around (0, 0, 6) + q^{*} evidence an unreported q^{*}CDW in ScV_{6}Sn_{6}, distinct from q_{s}CDW in its ground state^{45}. As the temperature is lowered, a weak peak around (0, 0, 6) + q_{s} is first observed at 104 K and quickly gains in intensity upon further cooling. In contrast, the q^{*}CDW peak at q^{*} is no longer discernible at 90 K [Fig. 2d]. The temperature evolution of the integrated intensities are compared for q_{s}CDW and q^{*}CDW in Fig. 1d, clearly revealing their competition. At T = 88 K (below T_{CDW}), the peak intensity of q_{s}CDW is at least 3 orders of magnitude larger than the maximum peak intensity of q^{*}CDW (occurring at ≈94 K), accounting for why only q_{s}CDW was detected in lab source Xray diffraction measurements^{45}.
The fullwidths at halfmaximum (FWHM) of the measured CDW peaks along l are compared in Fig. 2e, revealing that q^{*}CDW remains shortrange down to 94 K. By fitting the Lorentzian function to lscans of q^{*}CDW, we find the extracted correlation lengths is around 20 Å for T ≲ 110 K. In the case of q_{s}CDW, the associated peaks are also broad for T ≳ 100 K, but sharpen for T ≲ 90 K, with a correlation length exceeding 100 Å. We note the peaks associated with q_{s}CDW in Fig. 2a appear slightly away from q_{s} in some measurements, which may result from a distribution of shortrange q_{s}CDW clusters, domain formation due to the lowering of lattice symmetry below T_{CDW}, or a small sample misalignment.
Lattice dynamics associated with the formation of CDWs
To probe the lattice dynamics associated with the CDW formation in ScV_{6}Sn_{6}, IXS measurements were carried out at (0, 0, 6) + q_{s} and (0, 0, 6) + q^{*} [Fig. 3a, b], clearly revealing soft phonons at both positions. Whereas the soft phonons form two peaks centered around the elastic line at T = 200 K, they further soften upon cooling and form a single quasielastic peak. To quantitatively analyze the phonon spectra, the phonon contributions in Fig. 3a, b are fit using the general damped harmonic oscillator (DHO)^{58,59}:
shown as solid lines. In the DHO model, A is an intensity scale factor, E_{0} is the undamped phonon energy, and γ is the damping rate (peak FWHM when γ ≪ E_{0}). The fit values of E_{0} decrease markedly with cooling for both q_{s} and q^{*}, with the phonons at q^{*} softer than those at q_{s} [Fig. 3c]. In contrast, the damping rate γ changes relatively little with temperature, with the phonons at q_{s} slightly more strongly damped than those at q^{*}. The observation of phonon softening in tandem with the growth of q^{*}CDW suggests it is dynamic in nature, and the diffuse character of q^{*}CDW is a result of softening over an extended region in momentum space. On the other hand, while q_{s}CDW develops at T = 104 K in the elastic channel [Fig. 2a], the corresponding q_{s} phonon mode retains welldefined energy at a similar temperature (105 K), indicating q_{s}CDW develops via the growth of an elastic central peak, rather than phonons softening to zero energy. The shortrange q_{s}CDW precursors detected at T ≳ 100 K [Fig. 2a, e] suggest the firstorder transition at T_{CDW} is likely orderdisorder type, as suggested for AV_{3}Sb_{5}^{53}.
IXS measurements at q_{s} and q^{*} were also carried out in the (220) Brillouin zone [Fig. 3e, f], which is dominated by phonons polarized in the abplane. In contrast, measurements in the (006) Brillouin zone are dominated by caxis polarized phonons. For both q_{s} and q^{*}, soft phonons are hardly detectable in the (220) Brillouin zone, although the presence of q^{*}CDW is evidenced by the more intense elastic peak at 100 K relative to 85 K. For comparison, the elastic peak at q_{s} gains in intensity upon cooling from 100 K to 85 K, due to the appearance of q_{s}CDW. The opposing temperature evolution of elastic peaks in Fig. 3e, f are consistent with the competition between q_{s}CDW and q^{*}CDW revealed in Fig. 2. The much weaker soft phonons in the (220) Brillouin zone suggest q^{*}CDW is associated with dominantly caxis polarized lattice vibrations, similar to q_{s}CDW which is mostly due to Sc and Sn displacements along the caxis^{45}. Two additional phonon branches are also detected in Fig. 3e, f, with phonon energies at q_{s} slightly higher than those at q^{*}. The fact these phonons hardly change across T_{CDW} suggests they are likely associated with inplane vibrations of the lattice. Additional phonon modes that do not change significantly across T_{CDW} are also detected in several Brillouin zones (see Supplementary Note 2 and Supplementary Fig. 2), the energies of these phonon modes are shown in Fig. 4d and Supplementary Fig. 3.
Firstprinciples calculations
Firstprinciples calculations were employed to understand the experimentally observed CDWs in ScV_{6}Sn_{6}, with the calculated electronic structure shown in Fig. 4a. We find the electronic structure close to the Fermi level is dominated by V3d orbitals, which can also be seen in the projected density of states (DOS) [Fig. 4b], in agreement with the previous study^{55}. Characteristic features of the kagome lattice are identified in the electronic structure, including Dirac cones at K (~−0.1 eV and −0.04 eV) and H (~−0.5 eV), and topological flat bands around −0.5 eV at M and L, which manifest as van Hovelike features around −0.5 eV in the electronic DOS [Fig. 4b].
To probe the origins of the competing CDWs in ScV_{6}Sn_{6}, the nesting function \(J({{{{{{{\bf{q}}}}}}}})=\frac{1}{{N}_{{{{{{{{\bf{k}}}}}}}}}}{\sum }_{\nu,\mu,{{{{{{{\bf{k}}}}}}}}}\delta ({\epsilon }_{\mu {{{{{{{\bf{k}}}}}}}}})\delta ({\epsilon }_{\nu {{{{{{{\bf{k}}}}}}}}+{{{{{{{\bf{q}}}}}}}}})\) is computed, where ϵ_{μk} is the energy (with respect to the Fermi energy) of band μ at k. As can be seen in Fig. 4c, the most prominent feature of J(q) is at the Mpoint, which does not correspond to a CDW instability [Fig. 4d], and multiple marginal features are observed in the \({q}_{z}=\frac{1}{3}\) and \(\frac{1}{2}\) planes [Fig. 4c]. Most importantly, in the \({q}_{z}=\frac{1}{3}\) plane, no peak is present at \({{{{{{{{\bf{q}}}}}}}}}_{{{{{{{{\rm{s}}}}}}}}}=(\frac{1}{3},\frac{1}{3},\frac{1}{3})\), suggesting that Fermisurface nesting is completely irrelevant in the formation of q_{s}CDW, consistent with previous findings^{47,55}. In the \({q}_{z}=\frac{1}{2}\) plane, hot spots are found around \((\frac{1}{6},\frac{1}{6},\frac{1}{2})\) and \({{{{{{{{\bf{q}}}}}}}}}^{*}=(\frac{1}{3},\frac{1}{3},\frac{1}{2})\), indicating a possible contribution of nesting towards q^{*}CDW. The results in Fig. 4a–c are obtained without spinorbit coupling (SOC), and adding SOC leads to only marginal changes (Supplementary Note 3 and Supplementary Figs. 4 and 5).
In addition to Fermisurface nesting, EPC can also drive a CDW transition. To elucidate the role of phonons in the competing CDWs of ScV_{6}Sn_{6}, we calculated its phonon spectrum using DFPT^{60}, shown in Fig. 4d. The calculations reproduce several experimentally measured phonons modes at A and H [circles in Fig. 4d], demonstrating consistency between theory and experiment. In particular, several calculated phonon modes are nearly degenerate around 12.8 meV and 16.8 meV at H (q^{*}), as well as around 13.0 meV and 17.0 meV at q_{s}. These phonons match the experimental observations in Fig. 3e, f, and are dominated by the inplane motion of Sn atoms. Similar to previous calculations^{55}, imaginary phonon modes are present along A − L − H, with the imaginary A_{1} mode at H lowest in energy. In addition to the soft phonons experimentally observed at q_{s} and q^{*}, a low energy ~ 4.0 meV phonon mode without softening is detected experimentally at \(A=(0,\, 0,\, \frac{1}{2})\), occurring at a higher energy than the calculated ~2.5 meV mode [Fig. 4d]. Furthermore, two new phonon modes are identified experimentally at A upon entering the q_{s}CDW state (Supplementary Note 2 and Supplementary Figs. 2 and 3).
In most cases, the imaginary phonon mode with the lowest energy would drive a CDW transition, which is clearly not the case in ScV_{6}Sn_{6}, since static CDW occurs at q_{s}, rather than at q^{*} (H) which has the lowest phonon mode. More surprisingly, our calculations indicate an absence of imaginary phonons at q_{s} [Fig. 4d and Supplementary Fig. 7], suggesting that at the level of density functional theory, q_{s}CDW is also less competitive than the undistorted P6/mmm structure. This is reflected in the recovery of the undistorted structure when relaxing the supercell modulated by the lowest energy phonon mode at q_{s} (see “Methods”).
To address this problem, we calculated the phonon selfenergy \({\Pi }_{{{{{{{{\bf{q}}}}}}}}\nu }^{{\prime\prime} }\) (proportional to the phonon peak width in energy) and qdependent EPC strength λ_{qν} for the lowest phonon mode (ν = 1) along KH at different electron temperatures (see “Methods” and Supplementary Note 4), which are related to the EPC matrices \({g}_{mn}^{\nu }({{{{{{{\bf{k}}}}}}}},{{{{{{{\bf{q}}}}}}}})\) via^{61,62}:
and
We find that at a high electron temperature (~0.1 Ry), the P6/mmm structure of ScV_{6}Sn_{6} is stable without imaginary phonons [top of Fig. 4e and Supplementary Fig. 6], while both \({\Pi }_{{{{{{{{\bf{q}}}}}}}}}^{{\prime\prime} }\) and λ_{q} for the lowest phonon mode exhibit humps around q_{s} [middle and bottom of Fig. 4e], evidencing a qdependent EPC. This is further corroborated by calculations at a low electron temperature (~0.01 Ry), where the hump in \({\Pi }_{{{{{{{{\bf{q}}}}}}}}}^{{\prime\prime} }\) becomes further enhanced [Supplementary Note 4 and Supplementary Fig. 7]. In combination with the absence of features in the nesting function [Fig. 4c], these results suggest that the qdependent EPC plays a key role in selecting q_{s}CDW as the ground state in ScV_{6}Sn_{6}.
Discussion
CDWs usually occur via phonon softening, corresponding to coherent lattice oscillations that gradually become more competitive in energy, or the growth of a central peak that reflects the ordering of local CDW patches. The development of CDWs in one dimension as modeled by Peierls^{57}, and in twodimensional systems such as 2HNbSe_{2}^{63} and BaNi_{2}As_{2}^{64,65}, are accompanied by prominent phonon softening. While such phonon softening is limited to a small range in momentum in Peierls’ model, it occurs over an extended range in 2HNbSe_{2} and BaNi_{2}As_{2}, similar to the observed behavior of q^{*}CDW in ScV_{6}Sn_{6}. On the other hand, orderdisorder CDW transitions have been reported in systems such as (Ca_{1−x}Sr_{x})_{3}Rh_{4}Sn_{13}^{66} and AV_{3}Sb_{5}^{53}, and likely characterize the formation of q_{s}CDW in ScV_{6}Sn_{6}. Thus, the CDW formation process in ScV_{6}Sn_{6} is unique in that both prominent phonon softening and the growth of a central peak are observed, with the two effects associated with different wavevectors, a result of competing CDW instabilities.
There are three implications that directly result from our experiments and firstprinciples calculations. First, while q^{*}CDW is energetically favored in DFT calculations, q_{s}CDW is the ground state of ScV_{6}Sn_{6}. This apparent inconsistency could result from a qdependent EPC selecting q_{s}CDW as the ground state. This is because the calculated electronic states and phonon energies are “bare” particles, without full consideration of EPC, which leads to considerable electron/phonon selfenergies in the strongcoupling limit. We argue that if the phononinduced electronic selfenergy is properly taken into consideration in manybody theories beyond DFT, q_{s}CDW should become energetically more competitive than q^{*}CDW. This view of a qdependent EPC favoring q_{s}CDW as the ground state is supported by an enhancement of the phonon selfenergy \({\Pi }_{{{{{{{{\bf{q}}}}}}}}\nu }^{{\prime\prime} }\) around q_{s} upon decreasing the electron temperature (Supplementary Note 4 and Supplementary Fig. 7). Second, both q_{s}CDW and q^{*}CDW are associated with the A_{1} phonon mode, for which the V kagome lattice is mostly unaffected. Since the electronic states near the Fermi level are dominated by the V3d orbitals, gapopening associated with either q_{s}CDW or q^{*}CDW is unlikely to be prominent in ScV_{6}Sn_{6}. Third, our findings explain the substantial drop in resistivity below T_{CDW} [Fig. 1c]: the qdependent EPC and extended phonon softening revealed in this work both enhance electron scattering above T_{CDW}, and the removal of these effects below T_{CDW} strongly reduces electron scattering, consistent with optical conductivity measurements^{47}.
Furthermore, it is interesting to consider whether the competition between CDW instabilities in ScV_{6}Sn_{6} could be tilted in favor of q^{*}CDW via external tuning. In this regard, electrical transport measurements in pressurized ScV_{6}Sn_{6} reveal that the sharp drop in resistivity associated with q_{s}CDW persists up to ~2.0 GPa, beyond which it is suddenly replaced by a much weaker kink in resistivity, before becoming fully suppressed at ~2.4 GPa^{46}. The sudden qualitative change in resistivity anomaly above ~2.0 GPa is suggestive of a change in the ground state, and the much less pronounced resistivity anomaly between ~2.0 GPa and ~2.4 GPa suggests the associated transition being secondorder. In such a scenario, a distinct possibility is that q^{*}CDW becomes the ground state between ~2.0 GPa and ~2.4 GPa, and since q^{*}CDW develops through phonon softening [Fig. 3] as in 2HNbSe_{2}^{63} and BaNi_{2}As_{2}^{64,65}, the corresponding resistivity anomaly would be likewise rather subtle.
In conclusion, we uncovered competing CDW instabilities in the kagome metal ScV_{6}Sn_{6}, which lead to a unique CDW formation process with the dominant soft phonons and the ground state CDW occurring at different wavevectors, distinct from typical phonondriven CDWs. The two CDWs develop in highly different manners, suggestive of distinct mechanisms, and differentiate ScV_{6}Sn_{6} from CDWs in other kagome metals. As the q_{s}CDW ground state is not captured in firstprinciples DFT calculations, it is likely a correlated manybody effect driven by a qdependent EPC. Our findings demonstrate a strong EPC on the kagome lattice could lead to nearly degenerate ground states, a setup primed for the emergence of unusual phases of matter.
Methods
Experimental details
Single crystals of ScV_{6}Sn_{6} were grown using the selfflux method with Sc:V:Sn = 1:6:40^{45}. Distilled dendritic scandium pieces (99.9%), vanadium pieces (99.7%), and tin shot (99.99+%) were placed in an alumina crucible and sealed under vacuum in a quartz ampoule. The ampoule was placed in a furnace and heated to 1150 °C for 12 h, then held at 1150 °C for 20 h. The sample was then cooled to 750 °C at a rate of 1 °C/h, at which point the excess Sn flux was removed with the aid of a centrifuge. The resulting crystals are platelike with the caxis normal to the plates, and typical sample dimensions are around 1 × 1 × 0.5 mm^{3}. Electrical resistivity was measured with the standard fourpoint method.
Inelastic Xray scattering measurements were carried out in the transmission geometry using the BL35XU beamline^{67} at SPring8, Japan. The incident photon energy is 21.7476 keV. A ~70μmthick sample [Supplementary Fig. 8a], comparable to the attenuation length of ~20 keV Xrays in ScV_{6}Sn_{6}, was prepared and mounted on a Cu post using silver epoxy. The instrumental energy resolution was measured using a piece of polymethyl methacrylate (PMMA) and parametrized using a pseudoVoigt function. The instrumental resolution function R(E) is then obtained by normalizing the pseudoVoigt function to unit area [Supplementary Fig. 8b]. The fullwidth at halfmaximum (FWHM) of R(E) is ~1.38 meV. For temperatures around T_{CDW}, measurements were consistently carried out upon warming. Momentum transfer is referenced in reciprocal lattice units, using the hightemperature hexagonal P6/mmm cell of ScV_{6}Sn_{6}, with a = b ≈ 5.47 Å and c ≈ 9.16 Å^{45}. All measured scattering intensities are normalized by a monitor right before the sample. For momentum scans of elastic scattering, an attenuator was used to avoid saturating the detector when needed, which can be corrected for via the calibrated attenuation of the attenuator. These corrections have been applied to the data in Fig. 2.
Analysis of the experimental data
The elastic scans in momentum around q^{*} are fit to a Lorentzian function:
where b is a small constant, A is the integrated area, Q^{*} = (0, 0, 6) + q^{*} is the center of peak, and Γ is the fullwidth at halfmaximum (FWHM). For temperatures with detectable q_{s}CDW intensity, regions around \((\frac{1}{3},\frac{1}{3},6\frac{1}{3})\) and \((\frac{1}{3},\frac{1}{3},6\frac{2}{3})\) are masked in the fit. Integrated intensities and FWHMs for q_{s}CDW are then determined numerically from the data around \((\frac{1}{3},\frac{1}{3},6\frac{1}{3})\), after subtracting the fit to the Lorentzian function. For temperatures without detectable q^{*}CDW, the integrated intensities and FWHMs for q_{s}CDW are likewise obtained numerically, after subtracting a small constant term determined from the mean of data points away from the q^{*}CDW peaks. The obtained integrated intensities and FWHMs for q_{s}CDW and q^{*}CDW are shown in Figs. 1d and 2e.
Using the least squares method, all measured experimental phonon intensities are fit to the expression:
where n is the minimum number of phonon modes that capture the experimentally measured data, and the integrals correspond to convolutions with the instrumental resolution R(E). In practice, since R(E) has a FWHM of ~1.38 meV, the integrals are numerically carried out in the energy range [−20, 20] meV. The above equation contains a small constant term b, a resolutionlimited elastic peak cR(E) and n general damped harmonic oscillators (DHOs)^{58,59}, with each phonon mode represented by the DHO S_{i}(E):
where A_{i} is the intensity scale factor, E_{0i} is the phonon energy in the absence of damping, and γ_{i} is the damping rate, all for phonon mode i. In the limit of γ_{i} → 0 (or the phonon mode is resolutionlimited), the above equation for S_{i}(E) can be replaced by:
In addition, the difference between the actual zero energy and the nominal zero energy is contained in our model as a free parameter δE to account for shifts in energy between different scans. Possible shifts of energy within each scan are considered to be negligible and ignored in our analysis. The data in Fig. 3 and Supplementary Fig. 2 have been shifted by δE obtained in the fits.
As the temperature is cooled and q_{s}CDW develops, phonons at q_{s} become difficult to measure due to the elastic tail of the q_{s}CDW peak. A result of this is that the 95 K and 100 K data become almost resolutionlimited around the elastic line, and no phonons are contained in the corresponding fits [Fig. 3a and Supplementary Fig. 9a]. On the other hand, while the soft phonon mode at q^{*} becomes a single peak centered around the elastic peak at 95 K, 100 K, and 105 K, they are broader in energy than the instrumental resolution [Fig. 3b and Supplementary Fig. 9b]. Although these soft phonons can be described by the DHO model, it is not possible to reliably extract E_{0i} and γ_{i}.
Firstprinciples calculations
Electronic structure calculations were carried out using density functional theory (DFT) implemented in Quantum Espresso^{68}. The exchangecorrelations function was taken within the generalized gradient approximation (GGA) in the parameterization of Perdew, Burke and Ernzerhof^{69}. The energy cutoff of planewave basis was up to 64 Ry (720 Ry for the augmentation charge). The 3s, 3p, 3d, and 4s electrons for Sc and V atoms and 4d, 5s, and 5p electrons for Sn are considered valence electrons in the employed pseudopotentials. For the undistorted structures, the charge density was calculated selfconsistently with a Γcentered 12 × 12 × 6kpoint mesh with a Gaussian broadening of 0.01 Ry (low electron temperature). The lattice constants and atomic coordinates were fully relaxed until the force on each atom was less than 1 meV/Å and the internal stress less than 0.1 kbar.
The bare electronic susceptibility was calculated with the Lindhard formula:
where μ, ν are band indexes, ε_{μk} is the energy eigenvalue of band μ at k, f(ε_{μk}) is the FermiDirac distribution.
The imaginary part of the bare electron susceptibility (\({\chi }_{0}^{{\prime\prime} }(\omega,\, {{{{{{{\bf{q}}}}}}}})\)) is related to the nesting function J(q) through:
The phonon spectrum is calculated using density functional perturbation theory (DFPT)^{60} on a 4 × 4 × 3qgrid. The electronphonon coupling strength λ_{qν} and phonon selfenergy \({\Pi }_{{{{{{{{\bf{q}}}}}}}}\nu }^{{\prime\prime} }\) were calculated on a 48 × 48 × 24 Wannierinterpolated kgrid using the EPW package^{62}. Bands derived from Sc3d, V3d and Sb5p orbitals from DFT calculations were fit to a tightbinding Hamiltonian using the Maximally Projected Wannier Functions method [Supplementary Fig. 10], which were then used in the EPW calculations^{61}.
We have also performed calculations for the distorted structures associated with q^{*} and q_{s}CDWs. For q_{s}CDW, the initial structure was obtained by imposing the lowest energy phonon mode modulation on a rhombohedral supercell with lattice vectors \({{{{{{{{\bf{A}}}}}}}}}_{1}^{{\prime} }={{{{{{{{\bf{A}}}}}}}}}_{1}+{{{{{{{{\bf{A}}}}}}}}}_{3}\), \({{{{{{{{\bf{A}}}}}}}}}_{2}^{{\prime} }={{{{{{{{\bf{A}}}}}}}}}_{2}+{{{{{{{{\bf{A}}}}}}}}}_{3}\) and \({{{{{{{{\bf{A}}}}}}}}}_{3}^{{\prime} }=({{{{{{{{\bf{A}}}}}}}}}_{1}+{{{{{{{{\bf{A}}}}}}}}}_{2})+{{{{{{{{\bf{A}}}}}}}}}_{3}\), where A_{i} are the lattice vectors for the undistorted P6/mmm structure (A_{3}⊥A_{1,2} and ∠(A_{1}, A_{2}) = 120°). For q^{*}CDW, the initial structure was obtained by imposing the lowest energy phonon mode (A_{1}) modulation on a 3 × 3 × 2 hexagonal supercell with lattice vectors \({{{{{{{{\bf{A}}}}}}}}}_{1}^{{\prime\prime} }=2{{{{{{{{\bf{A}}}}}}}}}_{1}+{{{{{{{{\bf{A}}}}}}}}}_{2}\), \({{{{{{{{\bf{A}}}}}}}}}_{2}^{{\prime\prime} }={{{{{{{{\bf{A}}}}}}}}}_{1}+2{{{{{{{{\bf{A}}}}}}}}}_{2}\) and \({{{{{{{{\bf{A}}}}}}}}}_{3}^{{\prime\prime} }=2{{{{{{{{\bf{A}}}}}}}}}_{3}\). See Supplementary Fig. 11 for a comparison between primitive unit cells for the undistorted P6/mmm structure and the distorted structures. The initial structures were then fully relaxed so that the force on each atom was less than 1 meV/Å and the internal stress less than 0.1 kbar. The fully relaxed q^{*}structure is ~7.5 meV/f.u. lower in energy than the undistorted structure, whereas the q_{s}structure relaxes back to the undistorted P6/mmm structure, consistent with our calculations that show an absence of imaginary phonon at q_{s}.
The high electron temperature phonon calculations were simulated with a larger Gaussian smearing of 0.1 Ry (Fig. 4e and Supplementary Figs. 6 and 7)^{65}.
Data availability
The IXS data generated in this study are provided in the Source Data file. Source data are provided with this paper.
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Acknowledgements
Y.S. and C.C. acknowledge support from the National Key R&D Program of China (No. 2022YFA1402200), the Pioneer and Leading Goose R&D Program of Zhejiang (2022SDXHDX0005), the Key R&D Program of Zhejiang Province, China (2021C01002), and the National Natural Science Foundation of China (No. 12274363, 12274364). Y.L. acknowledges support from the Fundamental Research Funds for the Central Universities (Grant No. 2021FZZX00103). Measurements at the BL35XU of SPring8 were performed with the approval of JASRI (Proposal No. 2022B1283). The calculations were performed on the HPC facility at the Center for Correlated Matter, and partially on the HPCC at Hangzhou Normal University.
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Y.S. and C.C. conceived and led the project. S.C. prepared the samples. S.C., H.F. and T.M. carried out the experiments. S.C. and Y.S. carried out the data analysis. C.X., Y.D. and C.C. carried out the firstprinciples calculations. All authors discussed and interpreted the results. Y.S., C.C., Y.L. and M.S. wrote the paper with input from all authors.
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Cao, S., Xu, C., Fukui, H. et al. Competing chargedensity wave instabilities in the kagome metal ScV_{6}Sn_{6}. Nat Commun 14, 7671 (2023). https://doi.org/10.1038/s41467023434541
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DOI: https://doi.org/10.1038/s41467023434541
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