Thermal Expansion Coefficients of Bi$_2$Se$_3$ and Sb$_2$Te$_3$ Crystals from 10 K to 270 K

Lattice constant of Bi$_2$Se$_3$ and Sb$_2$Te$_3$ crystals is determined by X-ray powder diffraction measurement in a wide temperature range. Linear thermal expansion coefficients ($α$) of the crystals are extracted, and considerable anisotropy between $α_\parallel$ and $α_\perp$ is observed. The low temperature values of $α$ can be fit well by the Debye model, while an anomalous behavior at above 150 K is evidenced and explained. Grüneisen parameters of the materials are also estimated at room temperature.