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The thermal properties of materials are critically important to varioustechnologies and are increasingly the target of materials design efforts.However, it is only relatively recent advances in first-principlescomputational techniques that have enabled researchers to explore themicroscopic mechanisms of thermal properties, such as thermal expansion. We usethe Gr\"{u}neisen theory of thermal expansion in combination with densityfunctional calculations and the quasiharmonic approximation to uncovermechanisms of thermal expansion in PbTiO$_3$ thin-films in terms of elastic andvibrational contributions to the free energy. Surprisingly, we find thatalthough the structural parameters of PbTiO$_3$ thin-films evolve withtemperature as if they are dominated by linear elasticity, PbTiO$_3$ thin-filmsare strongly anharmonic, with large changes in the elastic constants andGr\"{u}neisen parameters with both misfit strain and temperature. We show thata fortuitous near-cancellation between different types of anharmonicity givesrise to this behavior. Our results illustrate the importance of high-orderphonon-strain anharmonicity in determining the temperature-dependent structuralparameters of PbTiO$_3$ thin-films, and highlight the complex manner in whichthermal expansion, misfit strain and elastic and vibrational properties areintertwined.

Understanding mechanisms of thermal expansion in PbTiO$_3$ thin-films from first principles: role of high-order phonon-strain anharmonicity