Role of electronic structure on observed phonon anomalies of transition-metal carbides

The possible role of electronic structure on observed phonon anomalies in high-temperature superconducting transition-metal carbides is studied by means of accurate ab initio calculations of the conduction electron response function. From augmented-plane-wave determinations of the electronic band structure, density of states, and Fermi surface of NbC and TaC, the wave-vector-dependent generalized susceptibility, $\chi \left(\overrightarrow{\mathrm{q}}\right)$, is calculated in the constant-matrix-element approximation. For both NbC and TaC, $\chi \left(\overrightarrow{\mathrm{q}}\right)$ has strong maxima at precisely those $\overrightarrow{\mathrm{q}}$ values at which soft modes were observed by Smith and Gläser. Maxima in $\chi \left(\overrightarrow{\mathrm{q}}\right)$ are predicted for other directions. The locus of these ${\overrightarrow{\mathrm{q}}}_{max}$ values can be represented by a warped cube of dimension $\sim 1.2\left(\frac{2\pi }{a}\right)$ in momentum space—in striking agreement with the soft-mode surface proposed phenomenologically by Weber. In sharp contrast, the $\chi \left(\overrightarrow{\mathrm{q}}\right)$ calculated for both ZrC and HfC—for which no phonon anomalies have been observed—fall off in all symmetry directions away from the zone center. In agreement with Phillips, we thus interpret the phonon anomalies in the transition-metal carbides as due to an "overscreening" effect resulting from an anomalous increase of the response function of the conduction electrons.