Abstract
The intertwining between spin, charge, and lattice degrees of freedom can give rise to unusual macroscopic quantum states, including hightemperature superconductivity and quantum anomalous Hall effects. Recently, a charge density wave (CDW) has been observed in the kagome antiferromagnet FeGe, indicative of possible intertwining physics. An outstanding question is that whether magnetic correlation is fundamental for the spontaneous spatial symmetry breaking orders. Here, utilizing elastic and highresolution inelastic xray scattering, we observe a caxis superlattice vector that coexists with the 2\(\times\)2\(\times\)1 CDW vectors in the kagome plane. Most interestingly, between the magnetic and CDW transition temperatures, the phonon dynamical structure factor shows a giant phononenergy hardening and a substantial phonon linewidth broadening near the caxis wavevectors, both signaling the spinphonon coupling. By first principles and model calculations, we show that both the static spin polarization and dynamic spin excitations intertwine with the phonon to drive the spatial symmetry breaking in FeGe.
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Introduction
The combination of magnetism and characteristic electronic structures of the kagome lattice, including flatband^{1,2,3}, Diracfermion^{4,5,6,7}, and van Hove singularities^{8,9}, is a productive route to realize correlated and topological quantum states. Significant interests have been focused on a kagome superconductor AV_{3}Sb_{5} (A = K, Rb, Cs)^{10}, where van Hove singularities near the Fermi level trigger cascade time and spatialsymmetry breaking orders^{8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23}. Lately, a correlated version of AV_{3}Sb_{5} is realized in a kagome magnet FeGe^{24,25}. Like the AV_{3}Sb_{5} (A = K, Rb, Cs)^{10}, the electronic structure of FeGe features multiple van Hove singularities near the Fermi level, E_{F}^{24,25}. A charge density wave (CDW) establishes in the Atype antiferromagnetic (AAFM) phase and induces physical consequences, including anomalous Hall effect^{24} and robust edge modes^{25}, reminiscent to those observed in AV_{3}Sb_{5}^{16,18}. Below the CDW transition temperature, T_{CDW}, the static spin polarization is enhanced, indicating an intimate correlation between spin, charge, and lattice degrees of freedom^{24}. Despite these interesting observations, key questions yet to be answered. For instance, although CDW has been observed in correlated magnetic systems, such as the cuprate highT_{c} superconductors^{26,27} and spindensitywave systems^{28}, emergence of CDW well below the magnetic transition temperature is rare, suggesting a new correlation driven CDW mechanism in FeGe. Focusing on the kagome metals with van Hove singularities near E_{F}, it is also urged to determine the geometry of the CDW in FeGe and its possible connection with the loop current scenario^{11,12,13,14,15,16,17,24}. Here, we address these fundamental questions using advanced xray scattering and numerical calculations. We discover charge superlattice peaks at the AAFM wavevectors in FeGe, differentiating the CDW geometry in FeGe and AV_{3}Sb_{5} despite the same 2\(\times\)2\(\times\)2 superstructure^{20}. Most interestingly, the phonon dynamical structure factor shows giant phonon hardening and large phonon broadening effect near the caxis CDW wavevectors above the T_{CDW}. These phonon anomalies are in stark contrast with the phonon softening, known as Kohn anomaly, in the electronphonon coupled systems and the emergent amplitude mode that hardens below the T_{CDW}^{19}. Combining with densityfunctional theory (DFT) and model calculations, we show that the energetically favored 2\(\times\)2\(\times\)2 superstructure in FeGe involves mainly caxis lattice distortions in the Kagome plane, which is stabilized by the strong spinphonon interactions.
Results and Discussion
FeGe adopts a hexagonal structure with space group P6/mmm (No. 191). It is composed of a kagome lattice of Fe atoms with Ge1 centered in the hexagons. These kagome layers are stacked along the caxis and separated by honeycomb layers of Ge2. At T_{N} = 410 K, an AAFM kicks in with spin moment pointing along the caxis. Below T_{CDW} ~ 110 K, concomitant anomalous Hall effect and enhanced spin polarization are observed^{24}. Fig. 1b shows the density functional theory plus dynamical mean field theory (DFT + DMFT) calculated spectral function of FeGe. In agreement with angleresolved photoemission spectroscopy (ARPES) study^{24}, van Hove singularities at the M point (Fig. 1c) are pushed to the Fermilevel due to the local correlation effect (see also Supplementary Fig. 2). Fig. 1d, e show xray diffraction scans along highsymmetry directions at T = 10 K. Consistent with previous diffraction and scanning tunneling microscopy studies^{24,25}, CDW superlattice peaks are observed at \({Q}_{M}^{//}\) (H = 0.5, K, L=integer) and \({Q}_{L}\) (H = 0.5, K L=integer+0.5), where H, K, L are reciprocal lattice directions as shown in Fig. 1c (see Supplementary Fig. 3). While superlattice peak positions at \({Q}_{M}^{//}\) and \({Q}_{L}\) are the same as AV_{3}Sb_{5}^{19,20,22}, as shown in Fig. 1e, we observe a new charge superlattice peak at \({Q}_{A}^{\perp }\) = (0, 0, 2.5) that is absent in AV_{3}Sb_{5}^{19}. This new superlattice peak is narrow with a halfwidthathalfmaximum (HWHM) ~ 0.001 in reciprocal lattice units and doubles the unit cell along the crystal caxis.
Since \({Q}_{A}^{\perp }\) = (0, 0, L=halfinteger) overlaps with the Atype AFM peaks, it is necessary to prove that the observed peak at \({Q}_{A}^{\perp }\) is not due to the magnetic crosssection of xray scattering. For this purpose, we determine the temperature dependent superlattice peaks at \({Q}_{L}\) = (0, 0.5, 2.5), \({Q}_{M}^{//}\) = (0, 0.5, 3) and \({Q}_{A}^{\perp }\) = (0, 0, 2.5) and (0, 0, 4.5). Fig. 2a, c, e show θ−2θ scans below (90 K) and above (116 K) T_{CDW}. Fig. 2b, d, f show the full temperature dependent peak intensities and peak widths across the T_{CDW}. The same onset temperature for all four wavevectors proves that \({Q}_{A}^{\perp }\) peaks correspond to charge superlattice along the caxis. Since the xray scattering amplitude at Q = (0, 0, L) probes lattice distortions along the caxis, the superlattice peaks at \({Q}_{A}^{\perp }\) establish an outofphase lattice distortions along the caxis between adjacent FeGe layers at T_{CDW}. Insets of Fig. 2b, d show the hysteresisscans at \({Q}_{L}\) and \({Q}_{M}^{//}\) near T_{CDW}. The small hysteresis temperature, ΔT ~ 0.5 K, indicates that the transition at T_{CDW} is a weak firstorder transition^{20}. We note that we do not observe the temperature dependent hysteresis at \({Q}_{A}^{\perp }\), possibly due to its relatively weak peak intensity near T_{CDW} or an even smaller hysteresis temperature, ΔT. Given both \({Q}_{A}^{\perp }\) and \({Q}_{M}^{//}\) are present in FeGe, the 2\(\times\)2\(\times\)2 superstructure peaks at \({Q}_{L}\) should be considered as a superposition of \({Q}_{A}^{\perp }\) and the threeequivalent \({Q}_{M}^{//}\). As we continue to discuss below, an important consequence of the \({Q}_{A}^{\perp }\)peak is that it distinguishes the 2\(\times\)2\(\times\)2 charge modulations in FeGe and AV_{3}Sb_{5}, pointing to different electronic and structural origins of the CDWs in these two kagome metals.
The observation of charge superlattice peaks at \({Q}_{A}^{\perp }\) on top of the AAFM peaks naturally point to a spinphonon interaction in FeGe. We thus turn to determine the phonon dynamical structure factor, S(Q, ω), using meVresolution inelastic xray scattering (IXS). Fig. 3a, b show experimental and DFT calculated S(Q, ω) along the Γ(0, 0, 4)M(0.5, 0, 4)L(0.5, 0, 4.5)A(0, 0, 4.5)Γ(0, 0, 4) direction at 200 K. The overall agreements between IXS and DFT, including the phonon dispersion and intensity distribution, are good. However, we find that the phonon peakwidths near \({Q}_{A}^{\perp }\) are unusually broad, suggesting quasiparticle interactions^{29,30,31,32,33}. To reveal more details of this phonon anomaly, we show in Fig. 3c the phonon band dispersion at 200 and 420 K (open circles) and the extracted phonon peak width at 200 K (solid circles) along the MΓA direction. Interestingly, the phonon energy at the Apoint shows over 10% hardening from 420 to 200 K that accompanies with a phonon linewidth broadening. Fig. 3d, e show representative temperature dependent IXS spectra at the M and A point, respectively. The full temperature dependence of the extracted peak positions is shown in Fig. 3f. We find that the phonon peaks at the M point remain temperature independent within experimental error, whereas the phonon mode at the A point shows a giant 14% hardening from 420 to 110 K. Fig. 3g summarizes the temperature dependent phonon width at the A and M point. The fitted phonon peakwidth at the Apoint is broad even above the T_{N} with fullwidthathalfmaximum (FWHM) ~ 2 meV. The width continuously increases until temperature hits the T_{CDW}. At T = 150 K, the fitted FWHM ~ 3 meV (corresponding to ~2 meV intrinsic phonon width after deconvolution), yielding a Damping ratio~7%. In stark contrast, the phonon peak width at the M and L (see Supplementary Fig. 4) point is resolution limited in the entire temperature range, consistent with the absence of longitudinal and transverse acoustic phonon energy anomaly at the M and L point.
The observed phonon hardening and broadening above the T_{CDW} in FeGe are fundamentally different from the Kohn anomaly in electronphonon coupled CDW systems and the emergent amplitude mode below the T_{CDW}^{19, 29,30,31}. These phonon anomalies are, however, captured by the dynamical spinphonon coupling picture shown in Fig. 3h. The second order Feynman diagram depicts a phonon with energy, ω_{n}, and momentum, q, scatters into two magnons with (ξ_{m}, k) and (ω_{n}ξ_{m}, qk). As we show in more details in the Supplementary Discussion and Supplementary Fig. 11, this dynamical spinphonon interaction yields strong phonon selfenergy effect, including the phonon energy hardening and phonon linewidth broadening near the Apoint, in agreement with experimental observations. Interestingly, similar phonon anomalies were observed in Kondo insulator FeSi^{33} and spinPeierls compound CuGeO_{3}^{34,35}, supporting a ubiquitous phonon hardening and broadening in spinphonon coupled systems. The observation of superlattice peaks at \({Q}_{A}^{\perp }\) and the associated giant phonon anomalies constitute our main experimental observations.
Motivated by these experimental results, we perform DFT + U calculations for the AAFM phase of FeGe at the zero temperature to understand the interplay between static spinpolarizations and the lattice distortions in the CDW phase. Fig. 4a–c show the calculated phonon spectra of FeGe in the AAFM phase as increasing Hubbard U. We find that the experimentally observed phonon modes shown in Fig. 3a exhibit the most dramatic change as increasing U with an energy minimum at the L point for U < 2 eV. This observation indicates that stronger electronic correlations and spinpolarizations tend to induce a lattice instability in FeGe. Interestingly, this mode corresponds to atomic vibrations that are mainly composed of outofphase caxis lattice distortions between adjacent FeGe kagome layers, consistent with experimentally observed superlattice peaks at \({Q}_{A}^{\perp }\). To understand the nature of the 2\(\times\)2\(\times\)2 superstructure, we take the equal phase and amplitude superpositions of the experimentally observed phonon mode at the three equivalent Lpoints as shown in Fig. 4d. The arrows point the movement of Fe and Ge atoms. In kagome layers, Fe and Ge atoms move outofplane along the caxis. The Ge1 atoms are divided into outofphase Ge1a (blue) and Ge1b (light blue) groups, where Ge1a has a much larger atomic movement than that of Ge1b, forming Ge1 dimers along the caxis. The honeycomb layers of Ge−2 atoms (grey) show inplane Kekulétype distortions^{36,37}. Starting from this 2\(\times\)2\(\times\)2 supercell that preserves the P6/mmm space group, we relax the internal atomic positions. Fig. 4e shows the energy difference between the 2\(\times\)2\(\times\)2 superstructure and the ideal kagome phase, \(\Delta E={E}_{{Charge}D{imer}}{E}_{{Kagome}}\), which decreases as increasing U. Intriguingly, the 2\(\times\)2\(\times\)2 superstructure is already an energetically favored phase at U = 0 and becomes even more robust with increasing U accompanied by the increase in the static moment. These results suggest that the magnitude of the static spinpolarization is important to stabilize the 2\(\times\)2\(\times\)2 superstructure with large caxis lattice distortions in the Kagome plane. Furthermore, as we show in Fig. 4e, by forming this 2\(\times\)2\(\times\)2 superstructure, the ordered magnetic moment is further enhanced by 0.01 ~ 0.05 \({\mu }_{B}\)/Fe at U = 0 ~ 3 eV, consistent with the previous neutron scattering study^{24}. It is important to point out that the experimentally determined static spin moment is more consistent with DFT + U calculations at U = 0 (Fig. 4a), therefor the static spinmoment induced phonon softening effect at elevated temperature will be neglectable and the dominated phonon anomaly is expected to arise from the dynamical magnonphonon coupling as shown in Fig. 3.
Our experimental and numerical results support a spinphonon coupling picture for the emergence of CDW in FeGe. Near T_{CDW}, the energy gain by forming a 2\(\times\)2\(\times\)2 superstructure with enhanced static moment overcomes the energy cost of lattice distortions and gives rise to a weak first order phase transition^{38}. The presence of large itinerant electrons allows additional energy gain by removing the high densityofstates near E_{F}^{39}. We emphasize, however, that the AAFM induced van Hove singularity near E_{F} may only plays a minor role for the CDW in FeGe for the following reasons: first, the conventional electronphonon coupling tends to favor lattice distortions parallel to the nesting vectors different from the experimental and DFT observations. Second, the strong temperature dependent phonon anomaly near \({Q}_{A}^{\perp }\) and the absence of phonon anomaly at \({Q}_{M}^{//}\) are incompatible with a nesting driven CDW picture.
Methods
Sample preparation and characterizations
Single crystals of B35type FeGe were grown via chemical vapor transport method. Stoichiometric iron powders (99.99%) and germanium powders (99.999%) were mixed and sealed in an evacuated quartz tube with additional iodine as the transport agent. The quartz tube was then loaded into a twozone horizontal furnace with a temperature gradient from 600 °C (source) to 550 °C (sink). After 12 days growth, FeGe single crystals with typical size 1.5 × 1.5 × 3 mm^{3} can be obtained in the middle of the quartz tube.
Elastic Xray scattering
The single crystal elastic Xray diffraction was performed at the 4IDD beamline of the Advanced Photon Source (APS), Argonne National Laboratory (ANL). The incident photon energy was set to 11 keV, slightly below the Ge Kedge to reduce the fluorescence background. The Xrays higher harmonics were suppressed using a Si mirror and by detuning the Si (111) monochromator. Diffraction was measured using a vertical scattering plane geometry and horizontally polarized (σ) Xrays. The incident intensity was monitored by a He filled ion chamber, while diffraction was collected using a Sidrift energy dispersive detector with approximately 200 eV energy resolution. The sample temperature was controlled using a He closed cycle cryostat and oriented such that Xrays scattered from the (001) surface.
meVresolution inelastic Xray scattering
The experiments were conducted at beam line 30IDC (HERIX) at APS, ANL^{40}. The highly monochromatic xray beam of incident energy E_{i} = 23.7 keV (λ = 0.5226 Å) was focused on the sample with a beam cross section of ∼35 × 15 μm^{2} (horizontal × vertical). The overall energy resolution of the HERIX spectrometer was ΔE ∼ 1.5 meV (full width at half maximum). The measurements were performed in reflection geometry. Under this geometry, IXS is primarily sensitive the lattice distortions along the crystal caxis. This geometry selectively enhances the unstable phonon modes predicted in the DFT calculations. Typical counting times were in the range of 120 to 240 seconds per point in the energy scans at constant momentum transfer Q. H, K, L are defined in the hexagonal structure with a = b = 4.97 Å, c = 4.04 Å at the room temperature.
Curve Fitting
The total energy resolution ΔE = 1.5 meV is calibrated by fitting the elastic peak to a pseudovoigt function:
where the energy resolution is the FWHM.
IXS directly probes the phonon dynamical structure factor, \(S({{{{{\bf{Q}}}}}},\, \omega )\). The IXS crosssection for solid angle dΩ and bandwidth dω can be expressed as:
where k and \({{{{{\boldsymbol{\epsilon }}}}}}\) represent the scattering vector and xray polarization and \(i\) and \(f\) denote initial and final states. r_{0} is the classical radius of the electron. In a typical measurement, the energy transfer ω is much smaller than the incident photon energy (23.71 keV in our study). Therefore, the term \(\frac{{{{{{{\boldsymbol{k}}}}}}}_{f}}{{{{{{{\boldsymbol{k}}}}}}}_{i}}\) ~ 1, and \(\frac{{d}^{2}\sigma }{d\Omega {{{{{\rm{d}}}}}}{{{{{\rm{\omega }}}}}}}\propto S({{{{{\bf{Q}}}}}},\, \omega )\).
\(S({{{{{\bf{Q}}}}}},\, \omega )\) is related to the imaginary part of the dynamical susceptibility, \({\chi }^{{\prime\prime}}\left({{{{{\bf{Q}}}}}},\, \omega \right)\), through the fluctuationdissipation theorem:
Where \(\chi ^{\prime\prime}({{{{{\bf{Q}}}}}},\, \omega )\) can be described by the damped harmonic oscillator form, which has antisymmetric Lorentzian lineshape:
here i indexes the different phonon peaks.
The phonon peak can be extracted by fitting the IXS spectrum at constantmomentum transfer Q, using Eqs. (3) and (4). Due to the finite experimental resolution, the IXS intensity is a convolution of \(S\left({{{{{\bf{Q}}}}}},\, \omega \right)\) and the instrumental resolution function, R(ω):
Here R(ω) was determined by fitting of the elastic peak.
ARPES experiment
The ARPES experiments are performed on single crystals FeGe. The samples are cleaved in situ in a vaccum better than 5 × 10^{−11} torr. The experiment is performed at beam line 21ID1 at the NSLSII. The measurements are taken with synchrotron light source and a ScientaOmicron DA30 electron analyzer. The total energy resolution of the ARPES measurement is approximately 15 meV. The sample stage is maintained at T = 30 K throughout the experiment.
DFT + U calculations
DFT + U calculations are performed using Vienna ab initio simulation package (VASP)^{41}. The exchangecorrelation potential is treated within the generalized gradient approximation (GGA) of the PerdewBurkeErnzerhof variety^{42}. The simplified approach introduced by Dudarev et al. (LDAUTYPE = 2) is used^{43}. We used experimental lattice parameters of FeGe and FeSn^{24,44}. Phonon calculations are performed in the Atype AFM phase with a \(2\times 2\times 1\) supercell (with respect to the AFM cell), using both the densityfunctionalperturbation theory (DFPT)^{45} and frozen phonon approaches, combined with the Phonopy package^{46}. The two approaches yield identical results. The internal atomic positions of the chargedimerized \(2\times 2\times 2\) superstructure is relaxed with the initial atomic distortions shown in Fig. 4d, until the force is less than 0.001 eV/Å for each atom. Integration for the Brillouin zone is done using a Γcentered 8 × 8 × 10 kpoint grids for the \(2\times 2\times 2\) supercell and the cutoff energy for planewavebasis is set to be 500 eV. Besides the \(2\times 2\times 2\) lattice distortion ansatz, we have also employed other lattice distortion ansatz, including \(1\times 2\times 2\), \(\sqrt{3}\times \sqrt{3}\times 2\) and \(\sqrt{5}\times \sqrt{5}\times 2\). All these ansata yield ground state energies higher than the \(2\times 2\times 2\) superstructure and the original ideal Kagome structure.
DFT + DMFT calculations
The fully charge selfconsistent DFT + DMFT^{47} calculations are performed in the Atype AFM phase using an opensource code of DFT+embedded DMFT developed by Haule et al., based on Wien2k package^{48}. We choose a hybridization energy window from −10 eV to 10 eV with respect to the Fermi level. All the five \(3d\) orbitals on an Fe site are considered as correlated ones, and a local Coulomb interaction Hamiltonian of Ising form is applied with varied Hubbard U and Hund’s coupling \({J}_{H}\) as shown in the main text. We use the continuous time quantum Monte Carlo^{49} as the impurity solver and an “exact” double counting scheme by Haule^{50,51}. To compute the spectral function, the electron selfenergy on real frequency is obtained by the maximum entropy analytical continuation method. The SOC is not included in the DFT + DMFT calculations since the SOC strength of Fe\(3d\) orbitals is small and will rarely change the electronic correlations. All the calculations are performed at T = 80 K.
Electronphonon vs spinphonon driven CDW
From an energy point of view, the electronphonon coupling driven CDW emphasizes the competing energy scales of charge condensation energy and lattice deformation energy, whereas the spinphonon coupling highlights the magnetic energy gain by forming a CDW. To understand the spinphonon coupling driven CDW, one can consider a simplified 1D Heisenberg model:
Here \(J\) is the antiferromagnetic exchange energy, \({{{{{{\boldsymbol{S}}}}}}}_{i}\) is the local spin. \({\Delta }_{i}={(1)}^{i}\delta,\, \delta \ge 0\) is the lattice distortion at bond i, connecting sites i and i + 1, and \(k\) is the elastic constant. A CDW is energetically favored if the energy gain in the first magnetic term is greater than the energy cost of the second elastic term. This is rather static spinphonon coupling. When the system is magnetically ordered, the energetics of this system is described by quasiparticles, i.e. magnons and phonons, thus the dynamical spinphonon coupling becomes crucial. In the Supplementary Discussion (Section “Phonon lifetime by two magnon excitations”), we build a magnonphonon coupling model on a 1D AFM Heisenberg chain. One of the consequences of such dynamical spinphonon coupling appears as the phonon lifetime, which allows the direct comparison between experimental data and a theoretical prediction.
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
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Acknowledgements
We thank Matthew Brahlek, Pengcheng Dai, Jiangping Hu, H. C. Lei, Brain Sales, Jiaqiang Yan, Binghai Yan, Ming Yi, Zhida Song, and Jianzhou Zhao for stimulating discussions. This research was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division (xray and ARPES measurement and model analysis). This research used resources (beamline 4ID and 30ID) of the Advanced Photon Source, a U.S. DOE Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DEAC0206CH11357. ARPES measurements used resources at 21ID1 beamlines of the National Synchrotron Light Source II, a US Department of Energy Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under contract no. DESC0012704. T.T. Z. and S. M. acknowledge support from Tokodai Institute for Element Strategy (TIES) funded by MEXT Elements Strategy Initiative to Form Core Research Center Grants No. JPMXP0112101001, JP18J23289, JP18H03678, and JP22H00108. J.X.Y. acknowledges startup funding from the Southern University of Science and Technology. X.L.W. and A.F.W. acknowledge the support of the National Natural Science Foundation of China (Grant No. 12004056). T.T.Z. also acknowledges the support of the Japan Society for the Promotion of Science (JSPS), KAKENHI Grant No. JP21K13865. Y.L.W. acknowledges the support of the National Natural Science Foundation of China (No. 12174365). The DFT and DFT + DMFT calculations were performed on ThianHe1A, the National Supercomputer Center in Tianjin, China. R.T. acknowledges the support from The São Paulo Research Foundation, FAPESP (Grant No. 2021/111700).
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H.M., A.F.W., Y.L.W. and S.O. conceived the project. H.M., H.X.L, G.F., J.X.Y. and R.S. performed the elastic Xray scattering measurement. H.M., H.X.L., G.F. and A.S. performed the inelastic Xray scattering. H.M., H.X.L., H.N.L., T.Y., E.V. performed the ARPES measurement. X.L.W. and A.F.W. grew the single crystals of FeGe. T.T.Z. and S.M. calculated the phonon dynamical structure factor. Y.L.W. performed DFT+U and DFT+DMFT calculations for FeGe and FeSn. L.X.F., K.J. and S.O. performed the 1D model Analysis. H.M., T.T.Z., S.O. and Y.L.W. wrote the paper with inputs from all coauthors.
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Miao, H., Zhang, T.T., Li, H.X. et al. Signature of spinphonon coupling driven charge density wave in a kagome magnet. Nat Commun 14, 6183 (2023). https://doi.org/10.1038/s41467023419575
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DOI: https://doi.org/10.1038/s41467023419575
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