The densities of phonon states of thin Sn films on InSb substrates are determined during different stages of the $\alpha \text{−}\mathrm{Sn}$ to $\beta \text{−}\mathrm{Sn}$ phase transition using nuclear inelastic x-ray scattering. The vibrational entropy and internal energy per atom as a function of temperature are obtained by numerical integration of the phonon density of states. The free energy as a function of temperature for the nanoscale samples is compared to the free energy obtained from ab initio calculations of bulk tin in the $\alpha \text{−}\mathrm{Sn}$ and $\beta \text{−}\mathrm{Sn}$ phase. In thin films this phase transition is governed by the interplay between the vibrational behavior of the film (the phase transition is driven by the vibrational entropy) and the stabilizing influence of the substrate (which depends on the film thickness). This brings a deeper understanding of the role of lattice vibrations in the phase transition of nanoscale Sn.

• Received 28 March 2017
• Revised 16 July 2019

DOI:https://doi.org/10.1103/PhysRevB.100.075408