Negative thermal expansion and low-frequency modes in cyanide-bridged framework materials

We analyze the intrinsic geometric flexibility of framework structures incorporating linear metal–cyanide–metal $\left(\mathrm{M}\text{–}\mathrm{CN}\text{–}{\mathrm{M}}^{\prime }\right)$ linkages using a reciprocal-space dynamical matrix approach. We find that this structural motif is capable of imparting a significant negative thermal expansion (NTE) effect upon such materials. In particular, we show that the topologies of a number of simple cyanide-containing framework materials support a very large number of low-energy rigid-unit phonon modes, all of which give rise to NTE behavior. We support our analysis by presenting experimental verification of this behavior in the family of compounds ${\mathrm{Zn}}_x{\mathrm{Cd}}_{1-x}{\left(\mathrm{CN}\right)}_2$, which we show to exhibit a NTE effect over the temperature range $25–375\mathrm{K}$ more than double that of materials such as $\mathrm{Zr}{\mathrm{W}}_2{\mathrm{O}}_8$.

Figure 1

A representation of local vibrational modes responsible for NTE behavior in structures containing (a) single-atom and (b) diatomic linkages. The filled circles represent heavy (usually metal) atoms, and the open circles represent light bridging atoms, such as oxygen (O) or cyanide (CN) species. In both cases, population of these vibrational modes leads to a decrease in the overall metal⋯metal distance.

Figure 2

Framework flexibility in the 2D perovskite topology containing (a) single-atom and (b) diatomic linkages. The framework in (a) is such that the rotation of any chosmen rigid unit induces a rotation of equal magnitude in all units, such that adjacent units rotate in opposite directions. This motion corresponds to the single RUM possessed by the framework topology (which occurs for the single wave vector $\mathbf{k}={⟨\frac{1}{2}\frac{1}{2}⟩}^{*}$). On the other hand, incorporation of diatomic linkages within the framework allows each rigid unit to rotate independently. This is a property of the large concomitant increase in the number of RUMs: The structure in (b) possesses a RUM at all wave vectors.

Figure 3

Temperature-dependent structural behavior of the ${\mathrm{Zn}}_x{\mathrm{Cd}}_{1-x}{\left(\mathrm{CN}\right)}_2$ family as determined by single crystal x-ray diffraction. (a) The unit cell volume change (relative to $100\mathrm{K}$) for $\mathrm{Zn}{\left(\mathrm{CN}\right)}_2$ (open circles), ${\mathrm{Zn}}_{0.80}{\mathrm{Cd}}_{0.20}{\left(\mathrm{CN}\right)}_2$ (solid line), and $\mathrm{Cd}{\left(\mathrm{CN}\right)}_2$ (filled circles). The thermal expansion behavior of $\mathrm{Zr}{\mathrm{W}}_2{\mathrm{O}}_8$ (broken line) is included for comparison (data taken from Ref. 9). (b) Atomic displacement (thermal) parameters for $\mathrm{Zn}{\left(\mathrm{CN}\right)}_2$ (relative to $100\mathrm{K}$): the transverse (open circles) and longitudinal (filled circles) displacement of the C and N atoms (open circles); and the isotropic (by symmetry) displacement of the Zn atoms (broken line). The more rapid increase in the transverse displacement parameters of the C and N atoms indicates that transverse vibrational modes of the $\mathrm{Zn}\text{–}\mathrm{CN}\text{–}{\mathrm{Zn}}^{\prime }$ bridge are populated more rapidly with increasing temperature than longitudinal modes. The inset illustrates the $\mathrm{Zn}\text{–}\mathrm{CN}\text{–}{\mathrm{Zn}}^{\prime }$ moiety: the two Zn atoms have tetrahedral coordination. The arrows give the directionality of each of the three displacement parameters.