Observation of anomalous amplitude modes in the kagome metal CsV₃Sb₅

Kagome-lattice metals AV3Sb5 (A = K, Rb, Cs) host intertwined phenomena including charge-density waves (CDWs) and superconductivity. In CsV3Sb5 (T_CDW ≈ 94 K), scattering experiments report no soft phonon despite DFT-predicted phonon instabilities at M and L. The in-plane structure and c-axis periodicity of the CDW and the relative roles of Fermi-surface nesting versus electron-phonon coupling remain debated. Because CDWs involve lattice distortions, the authors use Raman spectroscopy—sensitive to collective amplitude modes and zone-folded phonons—to probe the lattice dynamics, symmetry, and temperature evolution across T_CDW, aiming to clarify the nature and mechanism of CDW formation in CsV3Sb5.

Prior work established CDW and superconductivity in AV3Sb5, with anomalous Hall effects and possible pair-density-wave superconductivity. While DFT finds phonon instabilities at M/L and a 2×2 superlattice is well established, hard X-ray and neutron scattering did not observe soft phonons. Fermi-surface nesting tied to van Hove singularities and partial Fermi-surface gapping has been proposed, but calculated electronic susceptibilities lack strong divergences, suggesting electron-phonon coupling may be significant. In canonical CDW materials, soft-phonon condensation yields Raman-active amplitude modes whose temperature dependences diagnose the order parameter; zone folding produces additional Raman modes. Reports on CsV3Sb5 show conflicting signatures about softening and secondary transitions, motivating a comprehensive Raman and DFT analysis.

- Samples: CsV3Sb5 single crystals grown by flux; freshly cleaved surfaces measured. Attempts at mechanical exfoliation led to loss of crystallinity. - Raman spectroscopy: Home-built confocal back-scattering setup with 532 nm excitation. Polarization-resolved measurements in XX, XY, LL, LR configurations; photons polarized in ab-plane. Notch filters enabled low-wavenumber access. Spectra collected with a grating spectrograph and LN2-cooled CCD. Temperature control via Montana Instrument Cryostation in vacuum; measurements from 4 K to above T_CDW. - Mode analysis: Lorentzian fits extracted frequencies, linewidths (FWHM), and integrated intensities versus temperature for main lattice phonons (A1g, E2g) and CDW-induced modes (labeled A1, A2, E1–E3). - First-principles calculations: Compared to DFT results for single-layer CsV3Sb5 under Star of David (SD) and inverse Star of David (ISD) distortions. Additional calculations performed with VASP (PBE-GGA, PAW; DFT-D3 for vdW), 5×5×5 k-mesh, 400 eV cutoff. Phonons via Phonopy using frozen-phonon force constants. Pseudo-phonon spectra computed for intermediate structures with 0–100% distortion between pristine 2×2 and ISD to track evolution of imaginary soft modes to real modes. Symmetry analysis and irreducible representations obtained using Bilbao Crystallographic Server. - Mode mixing quantification: Projected soft-mode eigenvectors u_s onto stable ISD eigenvectors u_f to compute P = (u_s·u_f)^2 across all Γ-point modes, correlating with observed Raman intensities and identifying amplitude-mode remnants after hybridization. - Symmetry/stacking considerations: Polarization-angle dependence assessed potential c-axis modulation; compared observed mode counts and symmetries to predictions for various stacking orders.

- Discovery of Raman-active CDW amplitude modes in CsV3Sb5 despite the experimental absence of soft phonon modes in prior scattering studies. - Temperature dependence: - A2 mode exhibits pronounced softening and strong broadening upon warming towards T_CDW ≈ 94 K; becomes overdamped between ~60–90 K before vanishing above T_CDW—hallmarks of an amplitude mode. - E3 mode shows modest frequency change and limited broadening; disappears near ~80 K, consistent with a zone-folded mode. - Main lattice phonons: A1g frequency increases sharply below T_CDW; E2g shows a subtle kink. Linewidths for both decrease faster below T_CDW, consistent with reduced electron-phonon scattering due to partial Fermi-surface gapping. Integrated intensities increase with temperature and saturate below ~50 K. - Symmetry and structure: - The set and ordering of observed Raman modes match DFT for ISD distortion in a single layer; five A1g and most E2g CDW-induced modes identified. One low-frequency A1g mode (<50 cm⁻1) likely associated with c-axis modulation (Cs mode). - Results support a CDW ground state dominated by ISD-type in-plane distortion with weak interlayer coupling; c-axis modulation present but weak. - Mode hybridization (anomalous): DFT shows soft modes at M (triply degenerate after folding) decompose into A1g and E2g amplitude modes under distortion, but these strongly hybridize with many zone-folded Raman modes of the same symmetry, spreading amplitude-mode character (frequency softening and broadening) across multiple modes. Projections P correlate with observed intensities; A2 and E1 are residual amplitude modes after mixing. - Evidence for strong electron-phonon coupling CDW: - Large amplitude-mode frequencies in CsV3Sb5 (among the highest in quasi-2D CDWs) and significant mode mixing are atypical of weak-coupling systems. - Absence of soft-mode phonon softening can be explained by strong coupling, violation of adiabatic screening, and/or first-order character of the transition preventing complete softening. - Additional indicators: substantial lattice distortion (~5% of lattice constant), commensurate 2×2 order with V atom clustering (trimers/hexamers), and large CDW gap ratio from optics. - Clarifications relative to prior reports: The A2 amplitude mode persists above ~60 K and is not tied to a separate unidirectional-order transition; Raman (a bulk probe) may miss secondary low-temperature transitions or nanoscale surface orders.

The observation of amplitude modes and their anomalous hybridization with zone-folded modes directly evidences a lattice-coupled CDW order parameter in CsV3Sb5. The strong temperature evolution and overdamping of A2 near T_CDW validate its amplitude-mode character. The broad redistribution of amplitude-mode character across many modes, especially those dominated by V-atom motion, implies strong coupling between lattice vibrations and the electronic system associated with van Hove singularities. This provides a natural route to reconcile the lack of an observed soft phonon (no clear Kohn anomaly) with a lattice-driven component of the CDW: in the strong-coupling regime, electron screening is ineffective and first-order transition behavior can preclude complete phonon softening. The excellent agreement between measured mode symmetries/frequencies and DFT for ISD distortion constrains the CDW ground state, indicating weak interlayer coupling with modest c-axis modulation. Overall, the results emphasize that electron-phonon coupling plays a central role in the CDW mechanism in CsV3Sb5, complementing and constraining purely electronically driven scenarios.

This work identifies Raman-active CDW amplitude modes in CsV3Sb5 and reveals unusually strong hybridization with zone-folded phonons, establishing a strong electron-phonon coupling CDW with ISD-type in-plane distortion. The findings explain the elusive soft-phonon behavior and highlight the lattice’s essential role in CDW formation. Future research should (i) clarify the microscopic origin of amplitude modes in the absence of observable softening, (ii) resolve the c-axis stacking/modulation and its impact on Raman spectra, (iii) explore the coupling between CDW amplitude modes and potential superconducting Higgs modes, and (iv) use ultrafast techniques to manipulate and disentangle electronic and lattice components of the order parameter.

- Direct soft-phonon softening was not observed experimentally in CsV3Sb5; conclusions rely on Raman signatures and DFT-based soft-mode evolution. - Bulk Raman lacks sensitivity to possible surface-limited unidirectional order and may miss a low-temperature secondary transition reported by other probes; nanoscale domain effects could be invisible. - Interlayer (c-axis) modulation remains weakly constrained; predicted mode counts for various stackings exceed observations, and some L-point folded modes are not detected. - Three predicted E2g modes were not observed, likely due to weak scattering cross-section. - Mechanical exfoliation attempts degraded crystallinity, limiting single-layer experimental verification.