Extended Phonon Collapse and the Origin of the Charge-Density Wave in $2H\mathrm{\text{−}}{\mathrm{NbSe}}_2$

We report inelastic x-ray scattering measurements of the temperature dependence of phonon dispersion in the prototypical charge-density-wave (CDW) compound $2H\mathrm{\text{−}}{\mathrm{NbSe}}_2$. Surprisingly, acoustic phonons soften to zero frequency and become overdamped over an extended region around the CDW wave vector. This extended phonon collapse is dramatically different from the sharp cusp in the phonon dispersion expected from Fermi surface nesting. Instead, our experiments, combined with ab initio calculations, show that it is the wave vector dependence of the electron-phonon coupling that drives the CDW formation in $2H\mathrm{\text{−}}{\mathrm{NbSe}}_2$ and determines its periodicity. This mechanism explains the so far enigmatic behavior of CDW in $2H\mathrm{\text{−}}{\mathrm{NbSe}}_2$ and may provide a new approach to other strongly correlated systems where electron-phonon coupling is important.

Figure 1

Temperature dependence of the soft-phonon mode and the charge-density-wave superlattice peak near $q_{\mathrm{CDW}}=(0.329,0,0)$. (a)–(d) Energy scans at $\mathbf{q}=(3-h,0,0)$, $h=0.325$, for temperatures $8\mathrm{K}\le T\le 90\mathrm{K}$. Solid (red) lines are fits consisting of damped harmonic oscillators (inelastic) and a pseudo-Voigt function (elastic) (blue dashed lines). (e) Intensity of the charge-density-wave superlattice peak for $T\le 120\mathrm{K}$. The inset shows the phonon and superlattice peak intensities just above $T_c$. (f),(g) Phonon frequency ${\omega }_q$ and critical damping ratio $\Gamma /{\tilde{\omega }}_q$ of the soft-phonon mode, respectively, at $\mathbf{q}=(h,0,0)$ with $h=0.3$ (circle), 0.325 (triangle), and 0.35 (square). The solid line in (f) is a power law fit of the form $[(T-T_c)/T_c{]}^{\delta }$ yielding $\delta =0.48\pm 0.02$.

Figure 2

Wave vector dependence of the soft phonon at $T=33\mathrm{K}$. Energy scans at $\mathbf{Q}=(3-h,0,1)$, $h=0.275–0.375$. Solid (red) lines represent the total fit result consisting of a damped harmonic oscillator functions (inelastic) and a pseudo-Voigt function (elastic) (blue dashed lines).

Figure 3

Experimentally obtained dispersion and damping ratio of the soft-phonon branch in $2H\mathrm{\text{−}}{\mathrm{NbSe}}_2$ at four temperatures $8\mathrm{K}\le T\le 250\mathrm{K}$. Plotted are (a) the frequency of the damped harmonic oscillator ${\omega }_q=\sqrt{{\tilde{\omega }}_q^2-{\Gamma }^2}$ and (b) the damping ratio $\Gamma /{\tilde{\omega }}_q$. Lines are guides to the eye. Note that phonons at $h=0.325,0.35$ and $T=8\mathrm{K}$ were not detectable due to strong elastic intensities. The inset in (b) shows the experimentally observed damping $\Gamma$ of the damped harmonic oscillator (symbols) and scaled DFPT calculations (see Fig. 4) of $2\gamma$ (lines, offset 0.7 meV) with $\sigma =0.1\mathrm{eV}$ (black) and 1 eV (red).

Figure 4

Ab initio calculation for the soft-phonon mode along the crystallographic (100) direction in $2H\mathrm{\text{−}}{\mathrm{NbSe}}_2$. Shown are the calculated (a) dispersion, (b) $2\gamma$ [the contribution of the EPC to the phonon linewidth (FWHM)], and (c) the electronic JDOS. Calculations were done with $0.1\mathrm{eV}\le \sigma \le 1.0\mathrm{eV}$ (see the text). Note that calculated imaginary frequencies are shown as negative roots of the square phonon frequencies. Lines are guides to the eye.