Anharmonic Gr\"uneisen theory based on self-consistent phonon theory: Impact of phonon-phonon interaction neglected in the quasiharmonic theory

TL;DR

This paper develops a self-consistent phonon theory for thermal expansion that nonperturbatively includes anharmonic effects, clarifying the limitations of the quasiharmonic approximation and explaining its success in certain materials.

Contribution

It introduces a nonperturbative SCP-based approach to thermal expansion, deriving explicit corrections to the quasiharmonic approximation and analyzing anharmonic effects on phonon frequencies.

Findings

The Gr"unseisen formula holds with SCP frequencies.

Phonon anharmonicity provides small corrections to thermal expansion.

Two correction terms to phonon frequency shifts tend to cancel in strongly anharmonic materials.

Abstract

We formulate a theory of thermal expansion based on the self-consistent phonon (SCP) theory, which nonperturbatively considers the anharmonic effect. We show that the Gr\"unseisen formula holds within the SCP theory by replacing the phonon frequency by the SCP frequency. By comparing it with the quasiharmonic approximation (QHA), we derive explicit formulas of the correction to the QHA result. We show that the phonon anharmonicity gives a small correction of $O(\langle{\hat{U}_4\rangle}/\langle{\hat{U}_2\rangle})$ to the thermal expansion coefficient $\alpha$, where $\hat{U}_2$ and $\hat{U}_4$ are the harmonic and the quartic terms of the potential energy surface. On the other hand, we show that the phonon anharmonicity gives two correction terms to the temperature ($T$)-dependent phonon frequency shift which are comparable to the original QHA term. In strongly anharmonic materials such as NaCl and MgO, these two correction terms tend to cancel out each other, which explains why the QHA sometimes gives reasonable values for the $T$-dependent phonon frequency shift while it fails for thermal expansion.