Anharmonic phonon effects on linear thermal expansion of trigonal bismuth selenide and antimony telluride crystals

We adopted and extended an efficient Gr\"uneisen formalism to study the phonon anharmonicity and linear thermal expansion coefficients (TECs) of trigonal bismuth selenide (Bi$_2$Se$_3$) and antimony telluride (Sb$_2$Te$_3$). Anharmonicity of the systems is studied via extensive calculation of Gr\"uneisen parameters that exploit symmetry-preserving deformations. Consistent with experimental findings, a large anisotropy between the TECs in the $a$ and $c$ directions is found. The larger anharmonicity inherent in Sb$_2$Te$_3$, as compared to Bi$_2$Se$_3$ is offset by the volumetric effect, resulting in comparable temperature dependence of their linear TECs. The Debye temperatures deduced from our first-principles data also agree very well with the existing tabulated values. The highly efficient methodology developed in this work, applied for the first time to study the linear TECs of two trigonal thermoelectric systems, opens up exciting opportunities to address the anharmonic effects in other thermoelectrics and other low-symmetry materials.