Lattice dynamical calculation of negative thermal expansion in ZrV₂O₇ and HfV₂O₇

We report lattice dynamics calculations of negative thermal-expansion (NTE) behavior of ZrV₂O₇ family, extending our previous work on the ZrW₂O₈ family. The two families of compounds differ in terms of the oxygen coordination around the V/W atoms leading to differences in the nature of soft phonons under compression that are responsible for the NTE. Our calculations quantitatively reproduce the negative expansion over a range of temperatures. We also discuss the relation of the soft phonons with the phase transitions observed in the ZrV₂O₇ family. Especially, the calculations show a soft-phonon mode at a wave vector of 0.31⟨1,1,0⟩, which is in excellent agreement with the known incommensurate modulation in ZrV₂O₇ below 375 K.