Coherent phonon spectroscopy and interlayer modulation of charge density wave order in the kagome metal ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$

The recent discovery of the $A{\mathrm{V}}_3{\mathrm{Sb}}_5$ ($A=\mathrm{K}$, Rb, Cs) material family offers an exciting opportunity to investigate the interplay of correlations, topology, and superconductivity in kagome metals. The low-energy physics of these materials is dominated by an unusual charge density wave phase, but little is understood about the true nature of the order parameter. In this work, we use a combination of ultrafast coherent phonon spectroscopy and first-principles density functional theory calculations to investigate the charge density wave order in ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$. We find that the charge density wave is the result of a simultaneous condensation of three optical phonon modes at one $M$ and two $L$ points. This distortion can be described as tri-hexagonal ordering with an interlayer modulation along the $c$ axis. It breaks the $C_6$ rotational symmetry of the crystal and may offer a natural explanation for reports of uniaxial order at lower temperatures in this material family.

Figure 1

Coherent phonon spectroscopy data for ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$. (a) Transient reflectivity curves for a series of temperatures above and below the CDW phase transition. Temperatures are approximately evenly spaced and scans are offset for clarity. Above $T_{\mathrm{CDW}}$, the reflectivity increases after the pump pulse and a single oscillation frequency is apparent. Below $T_{\mathrm{CDW}}$, in contrast, the reflectivity decreases after the pump pulse and the presence of multiple oscillation frequencies results in a complex beat pattern. (b) Magnitude of the Fourier transform of the reflectivity oscillations after subtraction of a double exponential background. Curves are offset for clarity. Three resonances are identified and labeled $\alpha$ (1.3 THz), $\beta$ (3.1 THz), and $\gamma$ (4.1 THz). (c) Two-dimensional temperature-frequency map of the Fourier magnitude of the coherent phonon oscillations. While $\gamma$ is present at all temperatures, $\alpha$ only becomes active below $T_{\mathrm{CDW}}$. Below $T^{*}\approx 60\mathrm{K}$, the broad $\beta$ resonance appears and gradually grows in amplitude. A weak softening of all three frequencies is apparent with increasing temperature.

Figure 2

Behavior near the CDW phase transition. (a) Transient reflectivity curves for a finely spaced series of temperatures across the CDW phase transition. Temperatures are approximately evenly spaced. Curves are not offset and finite $\mathrm{\Delta }R/R$ values for $t<0$ are real. Inset shows pronounced changes in the transient optical response above (red), at (green), and below (blue) $T_{\mathrm{CDW}}$. (b) Average $\mathrm{\Delta }R/R$ value for time delays between $-1$ ps and $-0.2$ ps versus temperature using a 2 $\mu \mathrm{s}$ pulse period. A metastable-like divergence in the lifetime of the transient optical response is observed at $T_{\mathrm{CDW}}$. Insets are cartoons of the free energy and the gray curve is a guide to the eye. (c) Average $|\mathrm{\Delta }R/R|$ value for negative time delays versus pump-probe pulse period at the CDW phase transition. The fit is described in the main text.

Figure 3

First-principles phonon calculations. (a) Unit cell for ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$. The vanadium atoms (red) form a perfect kagome net and are coordinated with in-plane and out-of-plane antimony atoms. (b) First Brillouin zone of the hexagonal lattice, with high-symmetry points labeled. (c) Calculated phonon dispersion relations. There are two unstable modes with imaginary frequencies: one at the $M$ point with irreducible representation $M_1^{+}$, and the other at the $L$ point with irreducible representation $L_2^{-}$. Right panel shows the phonon density of states. (d) Illustration of the fully symmetric ${\mathrm{\Gamma }}_1^{+}$ phonon at 4.10 THz experimentally detected at all temperatures. (e) Illustration of the $L_2^{-}$ phonon at 1.27 THz experimentally detected below $T_{\mathrm{CDW}}$.

Figure 4

Energy lowering by $3Q$ distortions. For MMM and MLL configurations, positive displacements correspond to tri-hexagonal distortions and negative displacements correspond to “Star of David” distortions. The lowest energy occurs for the MLL configuration consisting of in-plane tri-hexagonal distortions that are laterally shifted in neighboring planes. Curves are least-squares fits to sixth-degree polynomials.