Lattice dynamics and negative thermal expansion in the framework compound ZnNi(CN)4 with two-dimensional and three-dimensional local environments

$\mathrm{ZnNi}{(\mathrm{CN})}_{4}$ is a three-dimensional (3D) framework material consisting of two interpenetrating PtS-type networks in which tetrahedral $[\mathrm{Zn}{\mathrm{N}}_{4}]$ units are linked by square-planar $[\mathrm{Ni}{\mathrm{C}}_{4}]$ units. Both the parent compounds, cubic $\mathrm{Zn}{(\mathrm{CN})}_{2}$ and layered $\mathrm{Ni}{(\mathrm{CN})}_{2}$, are known to exhibit 3D and 2D negative thermal expansion (NTE), respectively. Temperature-dependent inelastic neutron scattering measurements were performed on a powdered sample of $\mathrm{ZnNi}{(\mathrm{CN})}_{4}$ to probe phonon dynamics. The measurements were underpinned by ab initio lattice dynamical calculations. Good agreement was found between the measured and calculated generalized phonon density-of-states, validating our theoretical model and indicating that it is a good representation of the dynamics of the structural units. The calculated linear thermal expansion coefficients are ${\ensuremath{\alpha}}_{a}=\ensuremath{-}21.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}\phantom{\rule{0.16em}{0ex}}{\mathrm{K}}^{\ensuremath{-}1}$ and ${\ensuremath{\alpha}}_{c}=+14.6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}\phantom{\rule{0.16em}{0ex}}{\mathrm{K}}^{\ensuremath{-}1}$, leading to an overall volume expansion coefficient, ${\ensuremath{\alpha}}_{V}$ of $\ensuremath{-}26.95\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}\phantom{\rule{0.16em}{0ex}}{\mathrm{K}}^{\ensuremath{-}1}$, pointing towards pronounced NTE behavior. Analysis of the derived mode-Gr\"uneisen parameters shows that the optic modes around 12 and 40 meV make a significant contribution to the NTE. These modes involve localized rotational motions of the $[\mathrm{Ni}{\mathrm{C}}_{4}]$ and/or $[\mathrm{Zn}{\mathrm{N}}_{4}]$ rigid units, echoing what has previously been observed in $\mathrm{Zn}{(\mathrm{CN})}_{2}$ and $\mathrm{Ni}{(\mathrm{CN})}_{2}$. However, in $\mathrm{ZnNi}{(\mathrm{CN})}_{4}$, modes below 10 meV have the most negative Gr\"uneisen parameters. Analysis of their eigenvectors reveals that a large transverse motion of the Ni atom in the direction perpendicular to its square-planar environment induces a distortion of the units. This mode is a consequence of the Ni atom being constrained only in two dimensions within a 3D framework. Hence, although rigid-unit modes account for some of the NTE-driving phonons, the added degree of freedom compared with $\mathrm{Zn}{(\mathrm{CN})}_{2}$ results in modes with twisting motions, capable of inducing greater NTE.