Lattice dynamics and negative thermal expansion in the framework compound $\mathrm{ZnNi}{\left(\mathrm{CN}\right)}_4$ with two-dimensional and three-dimensional local environments

$\mathrm{ZnNi}{\left(\mathrm{CN}\right)}_4$ is a three-dimensional (3D) framework material consisting of two interpenetrating PtS-type networks in which tetrahedral $\left[\mathrm{Zn}{\mathrm{N}}_4\right]$ units are linked by square-planar $\left[\mathrm{Ni}{\mathrm{C}}_4\right]$ units. Both the parent compounds, cubic $\mathrm{Zn}{\left(\mathrm{CN}\right)}_2$ and layered $\mathrm{Ni}{\left(\mathrm{CN}\right)}_2$, 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 $\mathrm{ZnNi}{\left(\mathrm{CN}\right)}_4$ 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 ${\alpha }_a=-21.2\times {10}^{-6}{\mathrm{K}}^{-1}$ and ${\alpha }_c=+14.6\times {10}^{-6}{\mathrm{K}}^{-1}$, leading to an overall volume expansion coefficient, ${\alpha }_V$ of $-26.95\times {10}^{-6}{\mathrm{K}}^{-1}$, pointing towards pronounced NTE behavior. Analysis of the derived mode-Grüneisen 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 $\left[\mathrm{Ni}{\mathrm{C}}_4\right]$ and/or $\left[\mathrm{Zn}{\mathrm{N}}_4\right]$ rigid units, echoing what has previously been observed in $\mathrm{Zn}{\left(\mathrm{CN}\right)}_2$ and $\mathrm{Ni}{\left(\mathrm{CN}\right)}_2$. However, in $\mathrm{ZnNi}{\left(\mathrm{CN}\right)}_4$, modes below 10 meV have the most negative Grüneisen 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 $\mathrm{Zn}{\left(\mathrm{CN}\right)}_2$ results in modes with twisting motions, capable of inducing greater NTE.

Figure 1

Structural model of $\mathrm{ZnNi}{\left(\mathrm{CN}\right)}_4$ (showing the square-planar $\mathrm{Ni}{\left(\mathrm{CN}\right)}_4$ and tetrahedral $\mathrm{Zn}{\left(\mathrm{NC}\right)}_4$ units), which was used for ab initio calculations [14]. Key: Ni, blue, Zn, orange, C, brown, N, light blue.

Figure 2

Left: Comparison of the measured x-ray powder diffraction pattern (blue) with a theoretical pattern calculated from the tetragonal double interpenetrating structure [14]. Right: The total correlation function of $\mathrm{ZnNi}{\left(\mathrm{CN}\right)}_4$ at 15 and 295 K obtained from Fourier transformation of the $Q_{\mathrm{i}}\left(Q\right)$ measured by total neutron diffraction.

Figure 3

The generalized phonon density-of-states (GDOS) [23] of $\mathrm{ZnNi}{\left(\mathrm{CN}\right)}_4$ at 200, 300, and 400 K. Measurements were carried out with an incident wavelength of 4.14 Å on the IN6 spectrometer at the ILL. The spectra are vertically shifted with respect to each other to aid comparison.

Figure 4

Comparison between experimental (Exp), at 200 K, and calculated (Calc) phonon spectra of $\mathrm{ZnNi}{\left(\mathrm{CN}\right)}_4$. The calculated spectrum has been convoluted with a Gaussian of a full width at half maximum of 8% of the energy transfer to describe the effect of the experimental energy resolution.

Figure 5

Left: The calculated PDOS for each atom in $\mathrm{ZnNi}{\left(\mathrm{CN}\right)}_4$. N contributes the most below 30 meV, however C and Ni are dominant in the lowest-energy phonon band at 7 meV. Right: The calculated PDOS for each atom projected onto the $x$ ($y$) and $z$ axes. The CN stretching modes are omitted for clarity in the right panel.

Figure 6

Volume thermal expansion coefficient ${\alpha }_{\mathrm{V}}$ vs temperature in $\mathrm{ZnNi}{\left(\mathrm{CN}\right)}_4$, calculated using the QHA.

Figure 7

Contribution of modes of energy E to the thermal expansion in $\mathrm{ZnNi}{\left(\mathrm{CN}\right)}_4$.

Figure 8

Relaxed structure of $\mathrm{ZnNi}{\left(\mathrm{CN}\right)}_4$ with calculated thermal ellipsoids at 300 K representing a 75% probability of containing the atom. The Ni atom has a large oblate spheroid indicating it moves predominantly perpendicular to its square-planar environment. In contrast, the comparatively small and perfectly spherical thermal ellipsoid of Zn indicates it moves very little. The ellipsoid for N is isotropic perpendicular to the bond compared to that of C, which has an oblate shape more similar to that of Ni.

Figure 9

MSD with temperature for each atom type in $\mathrm{ZnNi}{\left(\mathrm{CN}\right)}_4$ (left). This represents the average of a single atom in the unit cell. The integrand of Eq. (3) representing the contribution to the MSD at 300 K of phonon modes with energy, E, for each atom type (right) and averaged over all atoms projected along the $x$ ($y$) and $z$ axes (inset, right). Low-energy modes contribute the most to the MSDs. N has the largest amplitude of motion as a function of temperature arising from modes between 10 and 20 meV, as a result of greater freedom of movement around the Zn. Ni has a larger MSD than Zn, dominated by modes around 7 meV. At this energy, Ni has a larger amplitude of motion than even C and N (see Fig. 14).

Figure 10

The calculated mode Grüneisen parameters, ${\gamma }_{\mathrm{i}}$, at $q$ points from a 40 × 40 × 20 $k$-point mesh over the whole Brillouin zone. A color-coded illustration for neighboring modes is used for contrast. The most negative Grüneisen parameters can be seen in the low-energy region, corresponding to the acoustic modes.

Figure 11

Schematic illustration of zone-boundary acoustic phonon mode with energy 1.12 meV, along Z (0, 0, ½), with the highest mode Grüneisen parameter of −79.40. The vectors represent the real displacement, however their amplitude has been exaggerated.

Figure 12

Left: The calculated phonon dispersion curves for $\mathrm{ZnNi}{\left(\mathrm{CN}\right)}_4$ along the indicated high-symmetry points (the CN stretching modes at ∼280 meV are not shown). Right: The dispersion curves with grayscale color coding according to values of the mode Grüneisen parameters. Black indicates the most negative value, whereas white represents a value ≥0. The more visible the dispersion curve the more it contributes to NTE in $\mathrm{ZnNi}{\left(\mathrm{CN}\right)}_4$.

Figure 13

Schematic illustrations of the zone center phonon modes viewed along the $c$ axis (top) and from the side (bottom), demonstrating how these motions shrink the $ab$ plane. The arrows represent the real displacement vectors of the atoms. The first and fourth modes are perfect rotational RUMs. The second and sixth are RUMs for the tetrahedral units and distortions of the square-planar units, whereas the third and fifth are RUMs for the square-planar units and distortions of the tetrahedra. Frequencies for each mode from left to right are (in meV): 12.20, 15.48, 19.05, 38.81, 41.55, 51.04. The corresponding mode Grüneisen parameters are: $-5.029,-5.163,-0.781,-4.367,-3.880,-2.066$, respectively.

Figure 14

The lowest energy optic mode at the zone center, which has a frequency of 6.9 meV and Grüneisen parameter of −10.67. This mode is the optic mode that contributes the most to the NTE, but it is not a RUM. The motion is possible due to the square-planar coordination of the Ni making it only constrained in two dimensions. (The lower Ni atom is moving into the plane page.)