Soft Phonon Charge-Density Wave Formation in the Kagome Metal KV$_3$Sb$_5$

Abstract

A range of of unusual emergent behaviors have been reported in the charge-density wave (CDW) state of the $A$V$_3$Sb$_5$ ($A=~$K, Rb, Cs) kagome metals, including a CDW formation process without soft phonons, which points to an unconventional CDW mechanism. Here, we use inelastic x-ray scattering to show that the CDW in KV$_3$Sb$_5$ forms via phonons that soften to zero energy at the CDW ordering vector ($L$-point) around $T_{\rm CDW}=78$~K. These soft phonons exhibit a remarkable in-plane anisotropy, extending over a much larger momentum range along $L$-$A$ relative to $L$-$H$, which leads to diffuse scattering common among $A$V$_3$Sb$_5$. Using first-principles calculations, we find that the momentum-dependent electron-phonon coupling (EPC) is peaked at $L$ and exhibits the same in-plane anisotropy as the phonon softening. Conversely, the electronic susceptibility is not peaked at $L$ and shows the opposite in-plane anisotropy. Our findings favor momentum-dependent EPC as the driving mechanism of the CDW in KV$_3$Sb$_5$, with a CDW formation process similar to that of transition metal dichalcogenides.

Figures (4)

• Figure 1: (a) The crystal structure of KV$_3$Sb$_5$. (b) The Brillouin zone of KV$_3$Sb$_5$. (c) Calculated electronic structure of KV$_3$Sb$_5$, the arrows mark van Hove points near the Fermi level. (d) The Fermi surface of KV$_3$Sb$_5$ for $k_z=\frac{1}{2}$. The arrows represent the nesting between van Hove points at $M$ ($L$). (e) Normalized resistivity $\rho(T)/\rho(\rm{300~K})$ of KV$_3$Sb$_5$ single crystals. The inset shows $d\rho/d T$, with a clear anomaly at $T_{\rm CDW}=78$ K. (f) Temperature dependence of elastic x-ray scattering in KV$_3$Sb$_5$ at ${\bf Q}=(3.5,0,0.5)$.
• Figure 2: (a) Color map of scattering intensities for ${\bf Q}=(1.5,1.5,4.5)$ at various temperatures and energies. (b) IXS spectra at various temperatures (vertically shifted), with the corresponding fits shown as solid lines. The resolution-limited elastic peak and phonon modes are shown as dashed and solid lines, respectively. The soft phonon mode is modeled using a DHO, and is represented by the shaded area. The scans at 80 K, 85 K and 90 K are scaled by the factors indicated in the figure. In fitting the 80 K and 85 K data, some parameters were constrained SI. (c) Temperature dependence of the fitted phonon energies $E_{\rm ph}$. The solid blue line is a power law fit. (d) Damping ratio $\gamma/(2E_0)$ of the soft mode in KV$_3$Sb$_5$, compared with the results for $2H$-NbSe$_2$Weber2011.
• Figure 3: (a) Schematic of the reciprocal space probed in this work, with the dashed area zoomed in on the right. Dots indicate reciprocal space positions where IXS scans were performed. Color maps of IXS scans along $A-L$ at (b) 85 K and (c) 110 K. (d) Color map of IXS scans along $H-L-H$ at 85 K. See Ref. SI for detailed IXS scans and fits corresponding to (b)-(d). (e) Dispersion of phonon energies $E_{\rm ph}$ along $A-L$ at the two temperatures. (f) Integral of inelastic scattering intensities for points along $A-L-A$ and $H-L-H$, obtained from data in (b)-(d) after a resolution-limited elastic peak is subtracted. The data for $L-A$ is mirrored from that of $A-L$.
• Figure 4: (a) Calculated phonon dispersions in KV$_3$Sb$_5$, for both low and high electronic temperatures. (b) The bare susceptibility $\chi_0(q)$ in the $q_z=\frac{1}{2}$ plane. (c) ${\bf q}$-dependent EPC $\lambda_{\bf q}$ for the soft phonon mode in the $q_z=\frac{1}{2}$ plane, computed at the high electronic temperature.