Thermodynamics of Tin Using AFLOW-APL*

Tin is one of the more interesting metallic elements, with a 13.2^oC phase transition from diamond-like "gray" (α-Sn) to "white" (the eponymous β-Sn). Density Functional Theory (DFT) finds a simple hexagonal phase (sh-Sn, hP1) nearly degenerate with the other phases, although it is never seen. sh-Sn is very close in energy to β-Sn at all volumes, as found by a variety of density functionals. While the simple hexagonal structure is a high pressure phase of Si, there it is not close to the ground-state structure at normal pressures.

To understand this behavior we used the high-throughput AFLOW-APL (AFLOW Harmonic Phonon Library) to find the energy-volume relationship, phonon spectra and free energy for tin in the $\alpha$-, $\beta$-, and $sh$-phases. Our results showed significant discrepancies between DFT and experiment. We find that DFT predicts the β-Sn phase as the ground state of tin, and a transition from β-Sn to sh-Sn at 200\,K. Both of these results contradict experiment. Our results are thus a cautionary tale about the use of DFT for understanding the behavior of systems with nearly degenerated crystal structures.