We employ polarization-resolved electronic Raman spectroscopy and density functional theory to study the primary and secondary order parameters, as well as their interplay, in the charge density wave (CDW) state of the kagome metal $A{\mathrm{V}}_3{\mathrm{Sb}}_5$. Previous x-ray diffraction data at 15 K established that the CDW order in ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ comprises of a $2\times 2\times 4$ structure: one layer of inverse-star-of-David and three consecutive layers of star-of-David pattern. We analyze the lattice distortions based on the $2\times 2\times 4$ structure at 15 K, and find that the $U_1$ lattice distortion is the primarylike (leading) order parameter while $M_1^{+}$ and $L_2^{-}$ distortions are secondarylike order parameters for vanadium displacements. This conclusion is confirmed by the calculation of bare susceptibility ${\chi }_0^{\prime }\left(q\right)$ that shows a broad peak at around $q_z=0.25$ along the hexagonal Brillouin zone face central line ($U$ line). We also identify several phonon modes emerging in the CDW state, which are lattice vibration modes related to V and Sb atoms as well as alkali-metal atoms. The detailed temperature evolution of these modes' frequencies, half-width at half-maximums, and integrated intensities support a phase diagram with two successive structural phase transitions in ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$: the first one with a primarylike order parameter appearing at $T_S=94\mathrm{K}$ and the second isostructural one appearing at around $T^{*}=70\mathrm{K}$. Furthermore, the $T$ dependence of the integrated intensity for these modes shows two types of behavior below $T_S$: the low-energy modes show a plateaulike behavior below $T^{*}$ while the high-energy modes monotonically increase below $T_S$. These two behaviors are captured by the Landau free-energy model incorporating the interplay between the primarylike and the secondarylike order parameters via trilinear coupling. Especially, the sign of the trilinear term that couples order parameters with different wave vectors determines whether the primarylike and secondarylike order parameters cooperate or compete with each other, thus determining the shape of the $T$ dependence of the intensities of Bragg peak in x-ray data and the amplitude modes in Raman data below $T_S$. These results provide an accurate basis for studying the interplay between multiple CDW order parameters in kagome metal systems.

• Received 13 January 2022
• Accepted 14 March 2022

DOI:https://doi.org/10.1103/PhysRevB.105.155106