Role of alkali ions in the near-zero thermal expansion of NaSICON-type AZr2(PO4)3

Zero thermal expansion (ZTE) is a rare phenomenon of great importance in the field of materials design. $A{\mathrm{Zr}}_{2}{(\mathrm{P}{\mathrm{O}}_{4})}_{3}$ $(A=\mathrm{Na},\mathrm{K},\mathrm{Rb},\mathrm{Cs})$ exhibit near-zero thermal expansion. In this work, we perform first principles calculations in $A{\mathrm{Zr}}_{2}{(\mathrm{P}{\mathrm{O}}_{4})}_{3}$ $(A=\mathrm{Na},\mathrm{K},\mathrm{Rb},\mathrm{Cs})$ and ${\mathrm{Zr}}_{2}{(\mathrm{P}{\mathrm{O}}_{4})}_{3}$ compounds to elucidate the effects of $A$ cations on the thermal expansion behavior. Structural and vibrational analysis shows that the near-zero thermal expansion is strongly related to the dynamics of the ``lantern'' units of ${[\mathrm{Zr}}_{2}{(\mathrm{P}{\mathrm{O}}_{4})}_{3}$], where the ${\mathrm{ZrO}}_{6}$ and ${\mathrm{PO}}_{4}$ polyhedra are rigid. The calculated Gr\"uneisen parameters reveal that different $A$-site alkali metal atoms heavily affect the phonon modes of $A$ atoms and the rotation of ${\mathrm{ZrO}}_{6}$ and ${\mathrm{PO}}_{4}$ polyhedra, as well as the acoustic phonon modes, thus resulting in a different thermal expansion behavior of $A{\mathrm{Zr}}_{2}{(\mathrm{P}{\mathrm{O}}_{4})}_{3}$ $(A=\mathrm{Na},\mathrm{K},\mathrm{Rb},\mathrm{Cs})$. The absence of $A$ atoms in ${\mathrm{Zr}}_{2}{(\mathrm{P}{\mathrm{O}}_{4})}_{3}$ facilitates the structural flexibility and therefore the occurrence of a stronger negative thermal expansion. This work provides insights into the modulation of thermal expansion of these ${\mathrm{NaZr}}_{2}{(\mathrm{P}{\mathrm{O}}_{4})}_{3}$-type compounds.