Zero-point energies prevent a trigonal to simple cubic transition in high-pressure sulfur

Jack Whaley-Baldwin

doi:10.1088/2516-1075/abd487

Abstract

Recently published density functional theory results using the PBE functional (Whaley-Baldwin and Needs 2020 New J. Phys. 22 023020) suggest that elemental sulfur does not adopt the simple-cubic (SC) phase at high pressures, in disagreement with previous works (Rudin and Liu 1999 Phys. Rev. Lett. 83 3049--52; Gavryushkin et al 2017 Phys. Status Solidi B 254 1600857). We carry out an extensive set of calculations using a variety of different exchange–correlation functionals (both local and non-local), and show that even though under LDA and PW91 a high-pressure SC phase does indeed become favourable at the static lattice level, when zero-point energies (ZPEs) are included, the transition to the SC phase is suppressed in every case, owing to the larger ZPE of the SC phase; thus confirming the transition sequence as BCC, with no intervening SC phase. We reproduce these findings with pseudopotentials that explicitly include core electronic states, and show that even at these high pressures, only the n = 3 valence shell contributes to bonding in sulfur. We then compare our findings against the all-electron code ELK, which is in excellent agreement with our pseudopotential results, and examine the roles of the exchange and correlation contributions to the total energy. We further calculate anharmonic vibrational corrections to the ZPEs of the two phases, and find that such corrections are several orders of magnitude smaller than the ZPEs and are thus negligible. The effect of finite temperatures is also considered, and we show that the phase becomes even more unfavourable with an increase in temperature. Finally, the experimental consequences of our results on the equation of state of sulfur and its superconducting critical temperature are explicitly calculated.

Highlights

High-pressure first-principles investigations on sulfur [1,2,3] have identified single-atom simple-cubic P m3m sulfur as an energetically competitive phase in the range 300 − 500 GPa, where it competes with a single-atom trigonal R3m phase
Using the LDA [2], and PBE [3] exchange-correlation functionals, previous authors have concluded that a transition to the P m3m phase occurs at around 280 GPa, whereas [1] finds that the simple-cubic phase is not favourable at either the static-lattice or zero-point energies (ZPEs)-included level of theory
Having established that the trigonal → simple cubic transition does not occur for any XC-functional at the harmonic level, we considered whether anharmonic corrections to the zero-point energies of the P m3m and R3m structures would result in a transition with the LDA functional

Summary

INTRODUCTION

High-pressure first-principles investigations on sulfur [1,2,3] have identified single-atom simple-cubic P m3m sulfur as an energetically competitive phase in the range 300 − 500 GPa, where it competes with a single-atom trigonal R3m phase. Using the LDA [2], and PBE [3] exchange-correlation functionals, previous authors have concluded that a transition to the P m3m phase occurs at around 280 GPa, whereas [1] finds (using PBE) that the simple-cubic phase is not favourable at either the static-lattice or ZPE-included level of theory. All three authors are in agreement that sulfur eventually adopts a primitive BCC Im3m structure at a pressure of around 500 GPa. The existence of a simple-cubic phase in sulfur would be significant, as the P m3m space group is, even at high

COMPUTATIONAL DETAILS

ENERGIES OF THE TRIGONAL AND

ANHARMONIC CORRECTIONS TO ZERO-POINT ENERGIES

ELECTRONIC DENSITIES OF STATES

CONCLUSIONS