Relationship between phonons and thermal expansion in Zn(CN)2and Ni(CN)2from inelastic neutron scattering andab initiocalculations

Abstract

Zn (CN)${}_{2}$ and Ni(CN)${}_{2}$ are known for exhibiting anomalous thermal expansion over a wide temperature range. The volume thermal expansion coefficient for the cubic, three-dimensionally connected material, Zn(CN)${}_{2}$, is negative (${\ensuremath{\alpha}}_{V}=\ensuremath{-}51\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6} {\mathrm{K}}^{\ensuremath{-}1}$) while for Ni(CN)${}_{2}$, a tetragonal material, the thermal expansion coefficient is negative in the two-dimensionally connected sheets (${\ensuremath{\alpha}}_{a}=\ensuremath{-}7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6} {\mathrm{K}}^{\ensuremath{-}1}$), but the overall thermal expansion coefficient is positive (${\ensuremath{\alpha}}_{V}=48\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6} {\mathrm{K}}^{\ensuremath{-}1}$). We have measured the temperature dependence of phonon spectra in these compounds and analyzed them using ab initio calculations. The spectra of the two compounds show large differences that cannot be explained by simple mass renormalization of the modes involving Zn (65.38 amu) and Ni (58.69 amu) atoms. This reflects the fact that the structure and bonding are quite different in the two compounds. The calculated pressure dependence of the phonon modes and of the thermal expansion coefficient, $\ensuremath{\alpha}$${}_{V}$, are used to understand the anomalous behavior in these compounds. Our ab initio calculations indicate that phonon modes of energy \ensuremath{\sim} meV are major contributors to negative thermal expansion (NTE) in both compounds. The low-energy modes of \ensuremath{\sim}8 and 13 meV also contribute significantly to the NTE in Zn(CN)${}_{2}$ and Ni(CN)${}_{2}$, respectively. The measured temperature dependence of the phonon spectra has been used to estimate the total anharmonicity of both compounds. For Zn(CN)${}_{2}$, the temperature-dependent measurements (total anharmonicity), along with our previously reported pressure dependence of the phonon spectra (quasiharmonic), is used to separate the explicit temperature effect at constant volume (intrinsic anharmonicity).