Soft Mode Origin of Charge Ordering in Superconducting Kagome CsV$_3$Sb$_5$

TL;DR

This study resolves the origin of the charge-density-wave order in CsV$_3$Sb$_5$ by combining high-resolution inelastic X-ray scattering and thermal diffuse scattering with non-perturbative anharmonic first-principles calculations. Structure-factor guided measurements reveal a soft phonon branch along the M–L direction with the strongest softening at the L point, where elastic intensity also rises upon cooling, signaling a lattice-driven CDW. The phonon softening is reproduced by SSCHA-based calculations that incorporate lattice anharmonicity and electron-phonon coupling, establishing a soft-mode instability at the $L$ point as the driving mechanism. These findings highlight the central role of lattice dynamics in kagome metals and provide a framework for understanding the intertwined CDW and superconductivity in CsV$_3$Sb$_5$ and related materials, with implications for topology and correlated electron phenomena.

Abstract

Charge-density-wave (CDW) order and superconductivity coexist in the kagome metals AV$_3$Sb$_5$ (A=K, Cs, Rb), raising fundamental questions about the mechanisms driving their intertwined phases. Here we combine high-resolution inelastic X-ray scattering with first-principles calculations to uncover the origin of CDW formation in CsV$_3$Sb$_5$. Guided by structure factor analysis, we identify a soft phonon mode along the reciprocal M-L direction, with the strongest effect at the L point, where the elastic scattering intensity also grows most rapidly upon cooling. First-principles calculations incorporating lattice anharmonicity and electron-phonon coupling reproduce these observations and establish a soft-mode instability at the L point as the driving mechanism of CDW formation. Despite the weakly first-order character of the transition, our results unambiguously demonstrate that the CDW in CsV$_3$Sb$_5$ originates from a softened phonon, clarifying its microscopic origin and highlighting the central role of lattice dynamics in kagome metals.

Figures (5)

• Figure 1: Calculated phonon intensities in $\Gamma_{420}$ and $\Gamma_{103}$ for both harmonic and anharmonic calculations. (a,c) The calculated dynamic structure factor from harmonic DFPT calculations along the line $\Gamma$-M-L in a) $\Gamma_{420}$ c) $\Gamma_{103}$. The harmonic phonon frequencies are marked with a thin white line. (b,d) Spectral weight along the line M-L as calculated within the SSCHA formalism at 100 K (see Supplementary Information for details), which includes both the structure factor as well as a realistic broadening of the phonon peaks due to anharmonic and electron-phonon interactions, for b) $\Gamma_{420}$ d) $\Gamma_{103}$. The position of the expected peak of the spectral function is marked with a thin white line as a reference.
• Figure 2: Thermal diffuse scattering just above $T_{\text{CDW}}$. (a-c) The thermal diffuse scattering at $95K$ in (a) $\Gamma_{420}$ (b) $\Gamma_{103}$, where all of the facets of the hexagon are crystallographically equivalent, (c) $\Gamma_{300}$. Some A and L points are indicated in black and red text, respectively.
• Figure 3: Raw IXS spectra. (a-c) Non-normalized IXS spectra, vertically offset for clarity, measured at temperatures between $97K$ and $300K$ at (a) a low-symmetry point $\mathbf{Q} = (0.35,\ 0.56,\ 2.59)$ (b) the M point $\mathbf{Q} = (0.5,\ 0.5,\ 3)$ (c) the L point $\mathbf{Q} = (0.5,\ 0.5,\ 2.5)$. The y-axis scale is not common across the subfigures. (d) A magnified, non-offset view of the low-energy range of (c) with overlays of the measured instrumental resolution scaled separately to the $302K$ and $97K$ data. Arrows indicate areas of enhanced low-energy spectral weight at 97 K which are not consistent with only a resolution-limited elastic peak.
• Figure 4: Fitted IXS spectra and phonon energies at the M and L points as a function of temperature. The y-axis scale is not common among the panels. (a-h) Fitted IXS spectra at the (a-d) M and (e-h) L points at (a,e) $302K$, (b,f) $197K$, (c,g) $127K$ and (d,h) $102K$ showing individual phonon modes, the elastic line and the overall sum of all of these contributions (full fit). (i,j) The fitted phonon energies as a function of temperature for (i) the M point (j) the L point alongside calculated anharmonic values for the modes which show softening with temperature (stars). The lines represent guides to the eye.
• Figure 5: Order of the transition, scale of the softening effect in reciprocal space and growth of the elastic peak. (a) A log-log plot of the energy of Phonon B at the L point as a function of the reduced temperature $T-$$T_{\text{CDW}}$, along with calculated values for the softening mode before and after the avoided crossing (stars). (b) The energy of Phonon B along the high symmetry line $\mathbf{Q} = (0.5,\ 0.5,\ L)$ as a function of temperature. The lines are guides to the eye. Some points are omitted at $97K$ due to the phonon becoming overdamped, with data from 102 K being shown instead for the L point (triangle). The shaded region indicates approximate boundaries for the energy at 97 K for these points. (c) The intensity of the fitted elastic peak as a function of temperature along the line $\mathbf{Q}~=~(0.5,\ 0.5,\ L)$.