Lattice dynamical calculation of negative thermal expansion in ${\text{ZrV}}_2{\text{O}}_7$ and ${\text{HfV}}_2{\text{O}}_7$

We report lattice dynamics calculations of negative thermal-expansion (NTE) behavior of ${\text{ZrV}}_2{\text{O}}_7$ family, extending our previous work on the ${\text{ZrW}}_2{\text{O}}_8$ 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 ${\text{ZrV}}_2{\text{O}}_7$ 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 ${\text{ZrV}}_2{\text{O}}_7$ below 375 K.

Figure 1

(Color online) The calculated phonon-dispersion relation up to 10 meV for cubic ${\text{ZrV}}_2{\text{O}}_7$ and ${\text{HfV}}_2{\text{O}}_7$ along the (100), (110), and (111) directions. The solid and dashed lines correspond to ambient pressure and 3 kbar, respectively.

Figure 2

Calculated partial density of states of various atoms in ${\text{ZrV}}_2{\text{O}}_7$ and ${\text{HfV}}_2{\text{O}}_7$.

Figure 3

(a) Calculated pressure variation of phonon density of states up to 10 meV in ${\text{ZrV}}_2{\text{O}}_7$ and ${\text{HfV}}_2{\text{O}}_7$. (b) Calculated Grüneisen parameter $\Gamma \left(E\right)$ averaged over phonons of energy $E$. For comparison, the calculated values (Ref. 9) of $\Gamma \left(E\right)$ for ${\text{ZrW}}_2{\text{O}}_8$ are also shown.

Figure 4

The calculated volume thermal expansion (solid line) in the cubic ${\text{ZrV}}_2{\text{O}}_7$ along with separate contributions from the two lowest phonon branches (dotted line) and all the phonons below 9 meV (dashed line).

Figure 5

Comparison between the calculated and experimental thermal expansion behaviors of ${\text{ZrV}}_2{\text{O}}_7$ (Ref. 3) and ${\text{HfV}}_2{\text{O}}_7$ (Ref. 15). The low-temperature phase of $M{\text{V}}_2{\text{O}}_7$ $\left(M=\text{Zr},\text{Hf}\right)$ below about 400 K has positive thermal-expansion coefficient. The calculations have been carried out in high-temperature phase.

Figure 6

Contribution of phonons of energy $E$ to the volume thermal expansion as a function of $E$ at 500 K.

Figure 7

The calculated contribution to the mean squared amplitude of various atoms arising from phonons of energy $E$ at $T=500\text{K}$ in ${\text{ZrV}}_2{\text{O}}_7$. The atoms are labeled as indicated in Table .