Effect of high pressure on the phonon spectra and superconductivity in ZrN and HfN

We report ab initio calculations of the phonon spectra, the electron-phonon interaction, and the superconducting transition temperature Tc for zirconium and hafnium nitrides under high pressure. The calculated phonon densities of states are qualitatively similar to Raman spectra measured at various pressures up to 32 GPa. The critical temperature Tc is determined by directly solving the Eliashberg equation using our calculated Eliashberg function. The pressure derivative of Tc for ZrN is in good agreement with the low pressure experimental result. In the case of HfN, for which there are no relevant experimental data, the available phenomenological estimate of Tc is significantly different from our first-principles prediction and we discuss the reasons for such a discrepancy. The calculated dependence Tc(p) becomes nonlinear at p>10 GPa. We show that in these compounds a decrease in Tc upon compression mainly occurs because of a decrease in the electron-phonon coupling constant λ, which is, in turn, due to an increase in the phonon frequencies.