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

Abstract

<jats:title>Abstract</jats:title> <jats:p>Recently published density functional theory results using the PBE functional (Whaley-Baldwin and Needs 2020 <jats:italic>New J. Phys.</jats:italic> 22 023020) suggest that elemental sulfur does not adopt the simple-cubic (SC) <jats:inline-formula> <jats:tex-math><?CDATA $Pm\bar{3}m$?></jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>P</mml:mi> <mml:mi>m</mml:mi> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> <mml:mo>̄</mml:mo> </mml:mover> </mml:mrow> <mml:mi>m</mml:mi> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="estabd487ieqn1.gif" xlink:type="simple" /> </jats:inline-formula> phase at high pressures, in disagreement with previous works (Rudin and Liu 1999 <jats:italic>Phys. Rev. Lett.</jats:italic> <jats:bold>83</jats:bold> 3049--52; Gavryushkin <jats:italic>et al</jats:italic> 2017 <jats:italic>Phys. Status Solidi</jats:italic> B <jats:bold>254</jats:bold> 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 <jats:inline-formula> <jats:tex-math><?CDATA $R\bar{3}m\to $?></jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>R</mml:mi> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> <mml:mo>̄</mml:mo> </mml:mover> </mml:mrow> <mml:mi>m</mml:mi> <mml:mo>→</mml:mo> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="estabd487ieqn2.gif" xlink:type="simple" /> </jats:inline-formula> 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 <jats:italic>n</jats:italic> = 3 valence shell contributes to bonding in sulfur. We then compare our findings against the all-electron code <jats:monospace>ELK</jats:monospace>, 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 <jats:inline-formula> <jats:tex-math><?CDATA $Pm\bar{3}m$?></jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>P</mml:mi> <mml:mi>m</mml:mi> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> <mml:mo>̄</mml:mo> </mml:mover> </mml:mrow> <mml:mi>m</mml:mi> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="estabd487ieqn3.gif" xlink:type="simple" /> </jats:inline-formula> 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.</jats:p>