Origin of the Incommensurate Modulation in Te-III and Fermi-Surface Nesting in a Simple Metal

Inelastic x-ray scattering experiments have been performed on incommensurately modulated Te-III at high pressure and reveal a pronounced phonon anomaly. The anomaly is reproduced in first-principles lattice dynamics calculations of unmodulated, body-centered monoclinic (bcm) Te, which is shown to be dynamically unstable. The calculated Fermi surface of bcm Te exhibits surprisingly effective nesting for a simple, electronically three-dimensional metal. The combined experimental and theoretical results corroborate recent proposals that the modulated crystal structure of Te-III and other chalcogens is the manifestation of a pressure-induced charge-density wave state.

Figure 1

(a) Crystal structure of Te-III. The dashed lines indicate four bcm unit cells, light and dark atoms are near the unit cell corners and centers, respectively, cylinders connect nearest neighbors, and solid lines highlight the modulation. (b) IXS spectra of Te-III at 8.5 GPa ($T=295\mathrm{K}$) for three different momentum transfers $q$. The circles indicate the measured intensities, the lines denote the fitted spectra and the contributing components.

Figure 2

(a) Measured phonon dispersion relation of Te-III at 8.5 GPa for momentum transfers $\mathit{q}=(0\xi 0)$ parallel to the modulation vector. The solid symbols represent the results for the LA phonon dispersion branch, and small open symbols mark additional weak features observed in the IXS spectra. The solid line is a guide to the eye; the dashed line shows a sine-type dispersion relation and the dash-dotted line $\Delta$ shows the difference between the sine-type dispersion and the observed LA dispersion. (b) Calculated phonon dispersion curves of bcm Te (solid symbols). Phonons with imaginary frequencies are displayed with negative energies. The experimental results for the LA phonon branch in Te-III at 8.5 GPa are shown by small open symbols.

Figure 3

Calculated Fermi surface of bcm Te, which comprises three sheets shown individually in (a)–(c) and combined in (d). Thick light-gray lines show the edges of the Brillouin zone, while the thin dark lines mark the Wigner-Seitz cell derived from the conventional bcm crystallographic unit cell. The reciprocal lattice vectors refer to the conventional bcm unit cell.

Figure 4

Calculated Fermi surface of bcm Te (a) in the ${\mathit{b}}^{*}{\mathit{c}}^{*}$ plane through $\Gamma$ and (b) in the (001) plane through $\frac{1}{2}{\mathit{c}}^{*}$. The gray-shaded rectangles are the same as shown in Fig. 3a. Dotted, dashed, and solid lines correspond to the FS sheets shown in Figs. 3a, 3b, 3c, respectively. The inset in (a) illustrates the idealized parallel sections of the Fermi surface and the “double nesting” of two sets of FS sheets with the same nesting vector. The gray-shaded circle shows the size of the free-electron Fermi sphere.