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Theoretical study of the stability of the tetradymite-like phases of Sb$_2$S$_3$, Bi$_2$S$_3$, and Sb$_2$Se$_3$

We report a comparative theoretical study of the \textit{Pnma} and \textit{R-3m} phases of Sb$_2$S$_3$, Bi$_2$S$_3$, and Sb$_2$Se$_3$ close to ambient pressure. Our enthalpy calculations at 0 K show that at ambient pressure the \textit{R-3m} (tetradymite-like) phase of Sb$_2$Se$_3$ is energetically more stable than the \textit{Pnma} phase, contrary to what is observed for Sb$_2$S$_3$ and Bi$_2$S$_3$, and irrespective of the exchange-correlation functional employed in the calculations. The result for Sb$_2$Se$_3$ is in contradiction to experiments where all three compounds are usually grown in the \textit{Pnma} phase. This result is further confirmed by free-energy calculations taking into account the temperature dependence of the unit-cell volumes and phonon frequencies. Lattice dynamics and elastic tensor calculations further show that both \textit{Pnma} and \textit{R-3m} phases of Sb$_2$Se$_3$ are dynamically and mechanically stable at zero applied pressure. Since these results suggest that the formation of the \textit{R-3m} phase for Sb$_2$Se$_3$ should be feasible at close to ambient conditions, we provide a theoretical crystal structure and simulated Raman and infrared spectra to help in its identification. We also discuss the results of the two published works that have claimed to have synthesized tetradymite-like Sb$_2$Se$_3$. Finally, the stability of the \textit{R-3m} phase across the three group-15 \textit{A$_2$X$_3$} sesquichalcogenides is analysed based on their van der Waals gap and X-X in-plane geometry.