Introduction

The combination of magnetism and characteristic electronic structures of the kagome lattice, including flat-band1,2,3, Dirac-fermion4,5,6,7, and van Hove singularities4) 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.

Fig. 3: Giant phonon anomalies near the charge dimerization wavevectors.
figure 3

a and b experimental and DFT 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. As described in Methods, the IXS intensity is dominated by the lattice distortions along the crystal c-axis. The IXS data shown in a were collected at 200 K. c Extracted phonon dispersion along the M-Γ-A direction at 200 (cyan) and 420 K (orange). The dashed rectangle highlights the temperature dependent phonon energy renormalization near the A point. d and e temperature dependent IXS spectra at the M and A point. Dashed curves are fittings of the experimental data (see Methods). Blue, green and orange circles represent 110, 300 and 420 K, respectively. f Temperature dependence of the fitted phonon peak positions at the A (open purple circles) and M (open green squares) point. g Temperature dependence of the fitted phonon peak width at the A (open purple circles) and M (open green squares) point. Dashed line represents the instrumental energy resolution. h Dynamical spin-phonon coupling. Top panel shows the second-order Feynman diagram for the phonon self-energy. Dashed and solid lines represent the phonon and magnon Green’s functions, respectively. iωn and \(i{\xi }_{{{{{{\rm{m}}}}}}}\) are bosonic Matsubara frequencies. Bottom panel shows a schematic of the dynamical spin-phonon coupling in an effective 1-dimensional spin chain with A-AFM. The magnon-phonon scattering induces strong the phonon self-energy effects and yields a phonon-energy hardening and phonon-linewidth broadening near the charge dimerization A-point (see Supplementary Fig. 11). The vertical error bars shown in cf represent 1-standard deviation from either Poissonian statistics or least-squares fitting. The vertical error bars shown in g represent the experimental step size that is about 3 times larger than the fitting error bars.

The observed phonon hardening and broadening above the TCDW in FeGe are fundamentally different from the Kohn anomaly in electron-phonon coupled CDW systems and the emergent amplitude mode below the TCDW19, 29,30,31. These phonon anomalies are, however, captured by the dynamical spin-phonon coupling picture shown in Fig. 3h. The second order Feynman diagram depicts a phonon with energy, ωn, and momentum, q, scatters into two magnons with (ξm, k) and (ωnm, q-k). As we show in more details in the Supplementary Discussion and Supplementary Fig. 11, this dynamical spin-phonon interaction yields strong phonon self-energy effect, including the phonon energy hardening and phonon linewidth broadening near the A-point, in agreement with experimental observations. Interestingly, similar phonon anomalies were observed in Kondo insulator FeSi33 and spin-Peierls compound CuGeO334,35, supporting a ubiquitous phonon hardening and broadening in spin-phonon 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 A-AFM phase of FeGe at the zero temperature to understand the interplay between static spin-polarizations and the lattice distortions in the CDW phase. Fig. 4a–c show the calculated phonon spectra of FeGe in the A-AFM 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 spin-polarizations tend to induce a lattice instability in FeGe. Interestingly, this mode corresponds to atomic vibrations that are mainly composed of out-of-phase c-axis lattice distortions between adjacent Fe-Ge 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 L-points as shown in Fig. 4d. The arrows point the movement of Fe and Ge atoms. In kagome layers, Fe and Ge atoms move out-of-plane along the c-axis. The Ge-1 atoms are divided into out-of-phase Ge-1a (blue) and Ge-1b (light blue) groups, where Ge-1a has a much larger atomic movement than that of Ge-1b, forming Ge-1 dimers along the c-axis. The honeycomb layers of Ge−2 atoms (grey) show in-plane Kekulé-type distortions36,37. 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 spin-polarization is important to stabilize the 2\(\times\)2\(\times\)2 superstructure with large c-axis 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 study24. 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 spin-moment induced phonon softening effect at elevated temperature will be neglectable and the dominated phonon anomaly is expected to arise from the dynamical magnon-phonon coupling as shown in Fig. 3.

Fig. 4: Static spin-polarization-assisted 2 × 2 × 2 superstructure in FeGe.
figure 4

ac, The DFT + U calculated phonon spectra of FeGe in the AFM phase as increasing Hubbard U. The phonon spectra are plotted with respect to the non-magnetic BZ of FeGe. The calculated ordered magnetic moments per Fe atom, M, are 1.50, 2.35 and 2.57 μB for U = 0, 2 and 3 eV, respectively. The calculated M at U = 0 eV is closer to the experimental value. The red curve corresponds to the experimentally observed phonon modes which show the most dramatic change as the spin-polarization is enhanced. The red circles highlight the \({B}_{1u}\) phonon mode at the L-point, which has the lowest energy along the A-L-H-A direction. The energy of the \({B}_{1u}\) mode and nearby phonon dispersion at U = 0 matches the IXS determined dynamical structure factor shown in Fig. 3a. d The equal phase and amplitude superposition of the \({B}_{1u}\) modes at the three equivalent L-points yields a charge-dimerized 2\(\times\)2\(\times\)2 superstructure. The arrows indicate the movements of Fe and Ge atoms. In kagome layers, Fe and Ge atoms move out-of-plane to form dimers along c-axis. Ge-1a (blue) and Ge-1b (light blue) have out-of-phase vibrations. Ge-1a has much larger movement than Ge-1b. The honeycomb layers of Ge−2 (grey) atoms show in-plane Kekulé-type distortions. e Left y-axis shows the DFT + U calculated energy difference between the charge-dimerized 2\(\times\)2\(\times\)2 superstructure and the ideal Kagome phase, \(\Delta E={E}_{{Charge}-{Dimer}}-{E}_{{Kagome}}\). Right y-axis shows the calculated ordered magnetic moment of the 2\(\times\)2\(\times\)2 superstructure (blue solid) and ideal Kagome (red dashed) phases, respectively. The magnetic moments are enhanced by 0.01 ~ 0.05 \({\mu }_{B}\)/Fe by forming the 2\(\times\)2\(\times\)2 superstructure.

Our experimental and numerical results support a spin-phonon coupling picture for the emergence of CDW in FeGe. Near TCDW, 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 transition38. The presence of large itinerant electrons allows additional energy gain by removing the high density-of-states near EF39. We emphasize, however, that the A-AFM induced van Hove singularity near EF may only plays a minor role for the CDW in FeGe for the following reasons: first, the conventional electron-phonon 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 B35-type 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 two-zone 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 mm3 can be obtained in the middle of the quartz tube.

Elastic X-ray scattering

The single crystal elastic X-ray diffraction was performed at the 4-ID-D beamline of the Advanced Photon Source (APS), Argonne National Laboratory (ANL). The incident photon energy was set to 11 keV, slightly below the Ge K-edge to reduce the fluorescence background. The X-rays 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) X-rays. The incident intensity was monitored by a He filled ion chamber, while diffraction was collected using a Si-drift energy dispersive detector with approximately 200 eV energy resolution. The sample temperature was controlled using a He closed cycle cryostat and oriented such that X-rays scattered from the (001) surface.

meV-resolution inelastic X-ray scattering

The experiments were conducted at beam line 30-ID-C (HERIX) at APS, ANL40. The highly monochromatic x-ray beam of incident energy Ei = 23.7 keV (λ = 0.5226 Å) was focused on the sample with a beam cross section of ∼35 × 15 μm2 (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 c-axis. 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 pseudo-voigt function:

$$R\left(\omega \right)=\left(1-\alpha \right)\frac{I}{\sqrt{2\pi }\sigma }{e}^{-\frac{\omega }{2{\sigma }^{2}}}+\alpha \frac{I}{\pi }\frac{\Gamma }{{\omega }^{2}+{\Gamma }^{2}}$$
(1)

where the energy resolution is the FWHM.

IXS directly probes the phonon dynamical structure factor, \(S({{{{{\bf{Q}}}}}},\, \omega )\). The IXS cross-section for solid angle dΩ and bandwidth can be expressed as:

$$\frac{{d}^{2}\sigma }{d\Omega {{{{{\rm{d}}}}}}{{{{{\rm{\omega }}}}}}}=\frac{{k}_{f}}{{k}_{i}}{r}_{0}^{2}{{{{{\rm{|}}}}}}\vec{{\epsilon }_{i}}\cdot \vec{{\epsilon }_{f}}{{{{{{\rm{|}}}}}}}^{2}S({{{{{\bf{Q}}}}}},\, \omega )$$
(2)

where k and \({{{{{\boldsymbol{\epsilon }}}}}}\) represent the scattering vector and x-ray polarization and \(i\) and \(f\) denote initial and final states. r0 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 fluctuation-dissipation theorem:

$$S\left({{{{{\bf{Q}}}}}},\, \omega \right)=\frac{1}{\pi }\frac{1}{(1-{e}^{\omega /{k}_{B}T})}\chi {{{{{\rm{{{\hbox{'}}}}}}}{{\hbox{'}}}}}({{{{{\bf{Q}}}}}},\, \omega )$$
(3)

Where \(\chi ^{\prime\prime}({{{{{\bf{Q}}}}}},\, \omega )\) can be described by the damped harmonic oscillator form, which has antisymmetric Lorentzian lineshape:

$${\chi }^{\prime\prime}\left({{{{{\bf{Q}}}}}},\, \omega \right)=\mathop{\sum}\limits_{i}{I}_{i}\left[\frac{{\Gamma }_{i}}{{(\omega -{\omega }_{Q,i})}^{2}+{\Gamma }_{i}^{2}}-\frac{{\Gamma }_{i}}{{(\omega+{\omega }_{Q,i})}^{2}+{\Gamma }_{i}^{2}}\right]$$
(4)

here i indexes the different phonon peaks.

The phonon peak can be extracted by fitting the IXS spectrum at constant-momentum 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):

$$I\left({{{{{\bf{Q}}}}}},\, \omega \right)=S\left({{{{{\bf{Q}}}}}},\, \omega \right)\otimes R(\omega )$$
(5)

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 21-ID-1 at the NSLS-II. The measurements are taken with synchrotron light source and a Scienta-Omicron 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 exchange-correlation potential is treated within the generalized gradient approximation (GGA) of the Perdew-Burke-Ernzerhof variety42. The simplified approach introduced by Dudarev et al. (LDAUTYPE = 2) is used43. We used experimental lattice parameters of FeGe and FeSn24,44. Phonon calculations are performed in the A-type AFM phase with a \(2\times 2\times 1\) supercell (with respect to the AFM cell), using both the density-functional-perturbation theory (DFPT)45 and frozen phonon approaches, combined with the Phonopy package46. The two approaches yield identical results. The internal atomic positions of the charge-dimerized \(2\times 2\times 2\) 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 Γ-centered 8 × 8 × 10 k-point grids for the \(2\times 2\times 2\) supercell and the cutoff energy for plane-wave-basis 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 self-consistent DFT + DMFT47 calculations are performed in the A-type AFM phase using an open-source code of DFT+embedded DMFT developed by Haule et al., based on Wien2k package48. 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 Carlo49 as the impurity solver and an “exact” double counting scheme by Haule50,51. To compute the spectral function, the electron self-energy 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.

Electron-phonon vs spin-phonon driven CDW

From an energy point of view, the electron-phonon coupling driven CDW emphasizes the competing energy scales of charge condensation energy and lattice deformation energy, whereas the spin-phonon coupling highlights the magnetic energy gain by forming a CDW. To understand the spin-phonon coupling driven CDW, one can consider a simplified 1D Heisenberg model:

$$H=J\mathop{\sum }\limits_{i=1}^{N}\left(1+{\Delta }_{i}\right){{{{{{\boldsymbol{S}}}}}}}_{i}\cdot {{{{{{\boldsymbol{S}}}}}}}_{i+1}+\frac{k}{2}\mathop{\sum }\limits_{i}^{N}{\Delta }_{i}^{2}$$

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 spin-phonon coupling. When the system is magnetically ordered, the energetics of this system is described by quasiparticles, i.e. magnons and phonons, thus the dynamical spin-phonon coupling becomes crucial. In the Supplementary Discussion (Section “Phonon lifetime by two magnon excitations”), we build a magnon-phonon coupling model on a 1D AFM Heisenberg chain. One of the consequences of such dynamical spin-phonon coupling appears as the phonon lifetime, which allows the direct comparison between experimental data and a theoretical prediction.