# Interplay between phonons and anisotropic elasticity drives negative thermal expansion in PbTiO$_3$

## TL;DR

This study reveals that negative thermal expansion in PbTiO$_3$ results from a complex interaction between phonon behavior and anisotropic elasticity, challenging previous assumptions about the necessary conditions for NTE.

## Contribution

It demonstrates that phonon properties alone are not sufficient or necessary for NTE, highlighting the role of anisotropic elasticity in this phenomenon.

## Key findings

- NTE in PbTiO$_3$ involves interplay between phonon properties and elasticity.
- Rigid unit phonon modes are not essential for NTE.
- New insights into the thermal behavior of materials and NTE mechanisms.

## Abstract

We use first-principles theory to show that the ingredients assumed to be essential to the occurrence of negative thermal expansion (NTE) -- rigid unit phonon modes with negative Gr\"{u}neisen parameters -- are neither sufficient nor necessary for a material to undergo NTE. Instead, we find that NTE in PbTiO$_3$ involves a delicate interplay between the phonon properties of a material (Gr\"{u}neisen parameters) and its anisotropic elasticity. These unique insights open new avenues in our fundamental understanding of the thermal properties of materials, and in the search for NTE in new materials classes.