Octahedral tilt distortion in negative thermal expansion in the fluorides CaZrF6 and ScF3

Understanding the mechanism of negative thermal expansion (NTE) and controlling the thermal expansion properties of materials at the atomistic level have long captured the attention of scholars. In this regard, the NTE mechanism of fluorides such as monometallic $\mathrm{Sc}{\mathrm{F}}_{3}$ and bimetallic fluorides like $\mathrm{CaZr}{\mathrm{F}}_{6}$ is particularly interesting; despite sharing similar crystal structure, $\mathrm{CaZr}{\mathrm{F}}_{6}$ exhibits a thermal expansion coefficient twice that of $\mathrm{Sc}{\mathrm{F}}_{3}$. Why is this so? Here, we investigate the structural vibrations of these two substances in real and reciprocal space at atmospheric and zero pressure, respectively, using machine-learning based deep potentials in combination with density-functional theory. Our study reveals that the structural flexibility of $\mathrm{CaZr}{\mathrm{F}}_{6}$ results in a greater number of vibration modes exhibiting negative Gr\"uneisen parameters. Moreover, the combined impact of octahedral distortion and tilt vibration leads to a substantially greater NTE for $\mathrm{CaZr}{\mathrm{F}}_{6}$ compared to $\mathrm{Sc}{\mathrm{F}}_{3}$. This work offers insights into understanding and tailoring NTE by considering polyhedral distortion in other open-framework materials.