Geometry of the charge density wave in the kagome metal $A{\mathrm{V}}_3{\mathrm{Sb}}_5$

Kagome lattice is a fertile platform for topological and intertwined electronic excitations. Recently, experimental evidence of an unconventional charge density wave (CDW) is observed in a $Z_2$ kagome metal ${A\mathrm{V}}_3{\mathrm{Sb}}_5$ $(A=\text{K}, \mathrm{Cs}, \mathrm{Rb})$. This observation triggers wide interest in the interplay between frustrated crystal structure and Fermi surface instabilities. Here, we analyze the lattice effect and its impact on CDW in ${A\mathrm{V}}_3{\mathrm{Sb}}_5$. Based on published experimental data, we show that the $2\times 2\times 2$ CDW breaks the sixfold rotational symmetry of the crystal due to the phase shift between kagome layers and can explain the twofold symmetric CDW peak intensity observed by scanning tunneling spectroscopy. The coupling between the lattice and electronic degrees of freedom yields a weak first-order structural transition without continuous change of lattice dynamics. Our result emphasizes the fundamental role of lattice geometry in proper understanding of unconventional electronic orders in ${A\mathrm{V}}_3{\mathrm{Sb}}_5$.

Figure 1

Crystal structure and CDW-induced lattice distortion of ${A\mathrm{V}}_3{\mathrm{Sb}}_5$. (a) and (b) side view and top view of the ${A\mathrm{V}}_3{\mathrm{Sb}}_5$ structure. DFT calculations find that the star-of-David (SD) is a local energy minimum at zero temperature, while ISD is the global minimum. (c) shows the SD and ISD lattice distortion of the V kagome lattice. Purple and black arrows show atomic distortions of ${\mathrm{V}}^{1–6}$ and ${\mathrm{V}}^{7–12}$ [see (b)], respectively. Distortions of ${\mathrm{V}}^{1–6}$ and ${\mathrm{V}}^{7–12}$ are out of phase. The ISD pattern corresponds to ${\mathrm{V}}^{1–6}\text{and}{\mathrm{V}}^{7–12}$ moving toward or outward from the center of the V hexagon. (d) reproduces the STS determined CDW peak intensity near $Q=0$ [10]. The inset shows the Bragg (green) and CDW (purple) peaks in the momentum space. The coordinates are shown in reciprocal lattice units. (e) shows the longitudinal acoustic phonon dispersion at 300 and 50 K [18]. The dashed curves show acoustic phonon anomalies that are expected for strong electron-phonon driven CDW [23, 24, 25, 26, 27].

Figure 2

CDW-induced lattice distortions in real space and their corresponding diffraction patterns in momentum space. (a) and (b) ISD distortion on V-kagome sublattice. (c) and (d) V-stretching distortion on V-kagome sublattice, both of which preserve the $D_{6h}$ symmetry in the kagome plane. (e) and (f) Sb1/Cs distortion that has $C_{3h}$ symmetry. In the diffraction calculations, the lattice distortions were set to be 1% deviating from the original positions. The units and directions of the momentum space are shown in Fig. 1. The size of the dots is proportional to the intensity of the diffraction pattern.

Figure 3

$C_2$ CDW peak intensity. (a) and (b) show calculated CDW superlattice peak intensity at $L=1.5$ and 2, respectively. (c) shows the fundamental Bragg peak intensity. The calculation is based on the theoretically refined ISD structure [12]. The size of the dots is proportional to the intensity of the diffraction pattern.

Figure 4

CDW lattice coupling induced a weak first order phase transition. At high temperature, the ideal kagome lattice is stable and corresponding to the free-energy minimum. At zero temperature, ISD is the energy minimum while SD is a local minimum. Near $T_{\text{CDW}}$, the asymmetric lattice free energy adds a cubic term in the CDW free energy through electron-phonon coupling, which, consequently, drives the CDW transition to a weak first-order transition.

Figure 5

Simulated diffraction intensity. (a) V-SD distortion and (b) V-ISD distortion. In the diffraction calculations, the lattice distortions were set to be 1% deviating from the original positions. The size of the dots are proportional to the intensity of the diffraction pattern.