Momentum-dependent electron-phonon coupling in charge density wave systems

Abstract

Many charge density wave (CDW) systems exhibit $q(T)$ electron-hole modulations continuously varying with $T$ and saturating upon cooling at an incommensurate value even if the maximum occurring in the electron-hole Lindhard response does not exhibit such a thermal shift. Using a simple RPA argument we show that the experimental $q(T)$ can be understood if the electron-phonon coupling (EPC) $g(q)$, necessary to set coupled electronic and structural modulations, is momentum dependent. In this analysis, the sense of variation of $q(T)$ depends upon the sign of $\frac{\ensuremath{\partial}g(q)}{\ensuremath{\partial}q}$ and its amplitude of thermal variation is controlled by the electron-hole coherence length (or CDW rigidity) in the modulation direction. This model quantitatively accounts for the thermal dependence of $q(T)$ in the one-dimensional (1D) CDW system ${\mathrm{K}}_{0.3}\mathrm{Mo}{\mathrm{O}}_{3}$ (blue bronze) both in its CDW ground state and in its pretransitional CDW fluctuation regime. We suggest that such a general analysis can be extended to account for the $q(T)$ dependence observed in other 1D and 2D CDW systems such as the transition metal di- and trichalcogenides as well as the lanthanide and rare-earth tritellurides. Using a detailed analysis of the low frequency phonon spectrum of the blue bronze, we then propose a new scenario for the $q$ dependent EPC, where $g(q)$ is due to a momentum-dependent hybridization between the critical phonon branch bearing the Kohn anomaly and other low-lying phonon branches. This allows obtaining a sign of $\frac{\ensuremath{\partial}g(q)}{\ensuremath{\partial}q}$ in agreement with that deduced from the analysis of $q(T)$. Finally, we propose that similar hybridization effects could also be relevant for other 1D and 2D CDW systems exhibiting a thermally dependent modulation.