Strong anharmonicity induces quantum melting of charge density wave in $2H-{\mathrm{NbSe}}_2$ under pressure

The pressure and temperature dependence of the phonon dispersion of $2H-{\mathrm{NbSe}}_2$ is measured by inelastic x-ray scattering. A strong temperature dependent soft phonon mode, reminiscent of the charge density wave (CDW), is found to persist up to a pressure as high as 16 GPa, far above the critical pressure at which the CDW disappears at 0 K. By using ab initio calculations beyond the harmonic approximation, we obtain an accurate, quantitative description of the $(P,T)$ dependence of the phonon spectrum. Our results show that the rapid destruction of the CDW under pressure is related to the zero mode vibrations—or quantum fluctuations—of the lattice renormalized by the anharmonic part of the lattice potential. The calculations also show that the low-energy longitudinal acoustic mode that drives the CDW transition barely contributes to superconductivity, explaining the insensitivity of the superconducting critical temperature to the CDW transition.

Figure 1

Phase diagram of ${\mathrm{NbSe}}_2$. Squares indicate the limit of the CDW phase measured in Ref. [19] with the addition of the point at 4.6 GPa from Ref. [18], triangles indicate the superconducting $T_c$ measured in [20], and circles indicate the pressure and temperature of our IXS measurements. Inset: crystallographic structure of $2H-{\mathrm{NbSe}}_2$ [3].

Figure 2

Phonon dispersion at 3.5 GPa (a) and 16 GPa (b). Closed (respectively, open) dots represent the experimentally determined longitudinal acoustic (longitudinal optic) soft phonon dispersions, and lines represent the results of phonon calculations. Close to the $\mathrm{\Gamma }$ point, the phonon dispersion clearly shows an optical branch. Closer to the $M$ point, the energy of the phonon observed corresponds to the acousticlike branch calculated. Colored thick and thin lines represent the SSCHA results for the longitudinal acoustic and optical anharmonic phonons, respectively, at several temperatures. The thin black lines represent other phonon modes present in this energy range calculated within the SSCHA at 20 K, which are very harmonic and barely depend on temperature (see SM).

Figure 3

Pressure dependence of the square of the measured energy of the longitudinal acoustic soft phonon at the CDW ordering wave vector and at 20 K (red dots) compared to the square of the energy calculated in the harmonic approximation (black diamonds). Lines are linear fits. The red region indicates the range of pressure where a CDW is observed experimentally at 20 K. The orange region indicates the range of pressure where harmonic calculations predict a CDW ground state.

Figure 4

(a) Calculated phonon dispersion at 7 GPa and 0 K with the SSCHA. The size of the bars is proportional to the electron-phonon contribution to the phonon linewidth (the bar is twice the FWHM). (b) The Eliashberg function ${\alpha }^{2}F\left(\omega \right)$ and the integrated electron-phonon coupling $\lambda \left(\omega \right)$. (c) The phonon density of states decomposed into the different atoms. In all figures the gray background denotes the integration area used to infer that the soft acoustic longitudinal mode contributes 17% of the total electron-phonon coupling.