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 $\mathrm{Zr}{\mathrm{W}}_2{\mathrm{O}}_8$.