Ab initio lattice dynamics and electron-phonon coupling in the refractory compounds ZrN and HfN

We present a theoretical calculation of the phonon dispersions and electron-phonon coupling in the refractory compounds ZrN and HfN using modern ab initio techniques. We find very good agreement with high-precision tunneling measurements of the Eliashberg function for ZrN, thus resolving a longstanding contradiction with previous theoretical results. To clarify some qualitative differences in the dispersion of the optical branches between our results and available experimental data for ZrN, we have also performed a neutron-scattering measurement of the generalized phonon-density of states, which corroborated our theoretical findings.

Figure 1

Phonon dispersion curves of ZrN and HfN along the high-symmetry directions $\Delta \left(100\right)$, $\Sigma \left(110\right)$, and $\Lambda \left(111\right)$. Theoretical results are shown as lines, while open symbols denote experimental data of inelastic neutron scattering experiments taken from Refs. 3, 12 for ZrN and HfN, respectively. Longitudinal modes are indicated by full lines or by triangles, and transverse modes by dashed lines or by diamonds. Note that in both cases there is a break in the frequency scale.

Figure 2

Generalized phonon density of states of ZrN. Shown are results of our neutron-scattering experiment and the theoretical spectrum, which has been artificially broadened using a Gaussian width of $2\mathrm{meV}$ for a better comparison. The spectra below $40\mathrm{meV}$ are multiplied by a factor of 4 for clarity.

Figure 3

Calculated Eliashberg functions of ZrN and HfN.