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 $(\mathrm{M}\text{--}\mathrm{CN}\text{--}{\mathrm{M}}^{\ensuremath{'}})$ 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 ${\mathrm{Zn}}_{x}{\mathrm{Cd}}_{1\ensuremath{-}x}{(\mathrm{CN})}_{2}$, which we show to exhibit a NTE effect over the temperature range $25--375\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ more than double that of materials such as $\mathrm{Zr}{\mathrm{W}}_{2}{\mathrm{O}}_{8}$.