Abstract
We use first-principles density functional theory to conduct an extensive structure search using the AIRSS package for elemental sulfur in the range 50–550 GPa. We then obtain the low-temperature phase diagram of sulfur in the same pressure range, including vibrational effects through the harmonic approximation. We do not find any structures lower in energy than those already reported in experiment, although the phase diagram below 100 GPa is found to be crowded with structures separated by only a few meV. We report the transition sequence ICM and obtain accurate pressures for each transition, although we find the second-order transition particularly difficult to define. Contrary to previous first-principles works (Pavel et al; Rudin and Liu 1999 Phys. Rev. Lett. 83 3049–52), we do not reproduce a trigonal simple cubic transition at either the static lattice or harmonic level. We also undertake a detailed analysis of the incommensurately modulated (ICM) phase of sulfur phase using a commensurate approximant found in the structure search. We find that the modulation amplitude is zero above 96 GPa; some 40 GPa below the experimentally reported transition to the unmodulated phase. We find that the body-centred atoms in the relaxed ICM approximant are, in addition to the dominant transverse modulation (which is a frozen-in optical phonon mode), slightly displaced longitudinally in the b-direction. We subsequently discover that this (small) longitudinal modulation is coupled to the transverse mode, and hence report previously unnoticed weak-mode coupling between transverse and longitudinal optical phonons in the ICM phase.
Highlights
The low-temperature phase diagram of elemental sulfur from ambient pressure up to around 200 GPa is reasonably well understood from experiment
We do not reproduce a transition to the simple-cubic P m3m phase at any pressure, the relatively close approach (≈ 10 meV apart) of the two curves at 380 GPa invites us to consider whether zero-point energy contributions could induce a phase transition
It can be seen that even after the inclusion of zero-point energies, a transition still does not occur, and the effect of this is to increase the stability of the R3m phase relative to the simple cubic (SC) phase, owing to the latter’s higher ZPE
Summary
The low-temperature phase diagram of elemental sulfur from ambient pressure up to around 200 GPa is reasonably well understood from experiment. At around 36 GPa, the trigonal P 3221 structure undergoes a phase transition to a body-centered tetragonal structure with space group I41/acd, consisting of helical square chains that run along the tetragonal c-axis. Sulfur is experimentally observed to undergo a transition to an incommensurately-modulated (ICM) superconducting phase at 83 GPa, which can be viewed as a distorted body-centered monoclinic structure (space group C2/m). This phase is reported to exist, with a modulation amplitude that decreases monotonically with pressure, up to 135 GPa, at which point sulfur adopts the trigonal β-Po structure with space group R3m. This transition is not, observed in the neighbouring Chalcogens Selenium and Tellurium, which are experimentally observed [3, 4, 8] to instead transform to a body-centered cubic phase
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