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Article

Theoretical Study of Pressure-Induced Phase Transitions in Sb2S3, Bi2S3, and Sb2Se3

by
Estelina Lora da Silva
1,*,†,
Mario C. Santos
2,3,†,
Plácida Rodríguez-Hernández
4,†,
Alfonso Muñoz
4,† and
Francisco Javier Manjón
3,†
1
IFIMUP, Institute of Physics for Advanced Materials, Nanotechnology and Photonics, Department of Physics and Astronomy, Faculty of Sciences, University of Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal
2
Instituto dos Pupilos do Exército, Estrada de Benfica, nº 374, 1549-016 Lisboa, Portugal
3
Instituto de Diseño para la Fabricación y Producción Automatizada, MALTA Consolider Team, Universitat Politècnica de València, 46022 València, Spain
4
Departamento de Física, Instituto de Materiales y Nanotecnología, MALTA Consolider Team, Universidad de La Laguna, 38200 Tenerife, Spain
*
Author to whom correspondence should be addressed.
†
These authors contributed equally to this work.
Crystals 2023, 13(3), 498; https://doi.org/10.3390/cryst13030498
Submission received: 23 February 2023 / Revised: 8 March 2023 / Accepted: 12 March 2023 / Published: 14 March 2023
(This article belongs to the Special Issue Pressure-Induced Phase Transformations (Volume II))
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Abstract

:
We report an ab initio study of Sb2S3, Sb2Se3, and Bi2S3 sesquichalcogenides at hydrostatic pressures of up to 60 GPa. We explore the possibility that the C2/m, C2/c, the disordered Im-3m, and the I4/mmm phases observed in sesquichalcogenides with heavier cations, viz. Bi2Se3, Bi2Te3, and Sb2Te3, could also be formed in Sb2S3, Sb2Se3, and Bi2S3, as suggested from recent experiments. Our calculations show that the C2/c phase is not energetically favorable in any of the three compounds, up to 60 GPa. The C2/m system is also unfavorable for Sb2S3 and Bi2S3; however, it is energetically favorable with respect to the Pnma phase of Sb2Se3 above 10 GPa. Finally, the I4/mmm and the disordered body-centered cubic-type Im-3m structures are competitive in energy and are energetically more stable than the C2/m phase at pressures beyond 30 GPa. The dynamical stabilities of the Pnma, Im-3m, C2/m, and I4/mmm structural phases at high pressures are discussed for the three compounds.

1. Introduction

A great deal of attention has been paid to the family of A 2 X 3 sesquichalcogenides in the last decade since the identification of the trigonal tetradymite-like R-3m phases of the group-15 sesquichalcogenides Sb 2 Te 3 , Bi 2 Se 3 , and Bi 2 Te 3 as 3D topological insulators [1,2]. Such topological insulators are a state of quantum matter where the bulk is a trivial insulator, whereas the surface states are topologically-protected conducting states due to time-reversal symmetry and strong spin-orbit coupling. Respective properties allow for potential applications in the fields of spintronics as well as for quantum computing [3].
These sesquichalcogenides have also been well studied in relation to their thermoelectric properties and as phase change materials [4,5].
Stibnite (Sb 2 S 3 ), bismuthinite (Bi 2 S 3 ) and antimonselite (Sb 2 Se 3 ) minerals belong to group-15 sesquichalcogenides. Such systems do not crystallize into the tetradymite-type structure at room conditions, but instead into the orthorhombic U 2 S 3 -type (Pnma) structure (Figure 1b). Several technological applications can be considered for such compounds. These include, applications for photovoltaic solar cells, X-ray computed tomography detectors, fuel cells, biomolecules and gas sensors, solid-state batteries, fiber lasers, and photoelectrochemical devices [6,7,8,9,10,11,12,13,14,15,16].
High-pressure (HP) studies of Sb 2 Te 3 , Bi 2 Se 3 , and Bi 2 Te 3 have shown that they undergo pressure-induced phase transitions (PTs) to monoclinic C2/m and C2/c structures, to the disordered solid solution Im-3m phase, and to the body-centered tetragonal I4/mmm structure [17,18,19,20,21,22,23,24,25,26]. These HP studies have shown a plethora of interesting properties, such as enhanced thermoelectric properties [27,28] and superconductivity, which could be of topological nature [29,30,31,32,33]. Therefore, there is a fundamental interest in identifying if the above mentioned HP phases can be associated to group-15 sesquichalcogenides with the Pnma structure crystallizing at room pressure. Since the Pnma structure has also been identified as a possible HP post-perovskite phase of the (Mg,Fe)SiO 3 and NaFeN 3 compounds [34,35], the study of Sb 2 S 3 , Bi 2 S 3 and Sb 2 Se 3 at HPs could also provide useful information about the structures and stability of ABO 3 orthorhombic minerals at pressure conditions close to those of the Earth’s mantle.
Initial experimental HP studies of Sb 2 S 3 , Bi 2 S 3 , and Sb 2 Se 3 have shown that the Pnma structure is stable under compression, with first-order PTs occurring at around 50 GPa [36,37,38,39,40,41]. A comparative study between experimental and theoretical analysis suggested that the Pnma structure of the three U 2 S 3 -type sesquichalcogenides should actually be stable up until 50 GPa [39]. Moreover, another work has reported that the crystalline Pnma phase for Sb 2 Se 3 is in fact stable even at 70 GPa [42]. Curiously enough, HP studies have found that Pnma-type Sb 2 Se 3 becomes a topological superconductor at around 10 GPa and at a low temperature value of 2.5 K [43], exhibiting highly conducting spin-polarized surface states, and similar to what occurs for Bi 2 Se 3 [44]. Moreover, HP superconductivity has been recently found in amorphous Sb 2 Se 3 and attributed to the crystallization of the material in a phase with possible “metavalent” bonding as that present in tetradymite-like Sb 2 Te 3 , Bi 2 Se 3 , Bi 2 Te 3 [45,46].
It must be noted, however, that some experimental HP studies have suggested that several first- and second-order PTs occur for Sb 2 S 3 up to 50 GPa [47,48,49]. Furthermore, it has also been suggested that the HP phases of Sb 2 S 3 could be similar to those observed for heavier sesquichalcogenides such as Bi 2 Se 3 , Bi 2 Te 3 , and Sb 2 Te 3 . Moreover, a theoretical study of Bi 2 S 3 at HP predicts the system to be unstable under compression, and decomposing into another stoichiometric system [50]. Therefore, there remains the question for whether the different structural phases (C2/m, C2/c, and disordered Im-3m and I4/mmm) observed for heavier cation sesquichalcogenides could be observed at HP on the three U 2 S 3 -type minerals, and at different pressure conditions.
In this work, we report theoretical simulations between 0 GPa and 60 GPa of the Pnma and hypothetical R-3m, C2/m, and C2/c; and the disordered Im-3m and I4/mmm phases for Sb 2 S 3 , Sb 2 Se 3 , and Bi 2 S 3 (Figure 1), with a view to assessing which, if any, are likely to be observed at HP. This work complements a previous study in which we examined the stability of the tetradymite-like (R-3m) phase with respect to the Pnma phase, up until 10 GPa, for the three U 2 S 3 -type minerals [51].

2. Theoretical Methodology

The six studied crystalline phases (Pnma, R-3m, C2/c, C2/m, disordered Im-3m, and I4/mmm) for Sb 2 S 3 , Bi 2 S 3 , and Sb 2 Se 3 were simulated within the framework of density-functional theory (DFT) [52]. The Vienna Ab initio Simulation Package (VASP) package [53] was used within the projector augmented-wave (PAW) scheme. The datasets included six valence electrons for S[3s 2 3p 4 ] and Se[4s 2 4p 4 ], and 15 valence electrons for Sb[4d 10 5s 2 5p 3 ] and Bi[5d 10 6s 2 6p 3 ]. Total energy convergence was achieved with a plane-wave kinetic-energy cut-off of 600 eV. The Perdew-Burke-Ernzerhof parameterization revised for solids (PBEsol), of the generalized-gradient approximation (GGA) exchange-correlation (xc) functional [54], was considered for all the calculations.
The sampling of the Brillouin-zone (BZ) was converged with Γ -centered Monkhorst-Pack [55] grids employing adequate meshes for the different structural phases of the three compounds: Pnma-6 × 10 × 6, R-3m-12 × 12 × 12, C2/m-6 × 12 × 6, C2/c-10 × 10 × 8, disordered Im-3m (using a C2/m conventional cell)-6 × 12 × 12, and I4/mmm-12 × 12 × 12.
The Im-3m phase is a body-centered cubic (bcc) disordered structure, considered as a disordered solid solution, and it has been theoretically predicted and experimentally found for Bi 2 Te 3 [17]. For sesquichacogenides with A 2 X 3 stoichiometry, the bcc lattice site (2a Wyckoff position) is randomly occupied by 40% of A cations and 60% of X anions. This means that such a structure is a disordered phase with a mixture of cations and anions randomly sharing the same bcc crystallographic position and forming an A-X substitutional alloy [17]. Due to the theoretical difficulty in simulating the disordered Im-3m structure, we have used a 9/10-fold C2/m structure (the formation of 9/10 chemical A-X bonds), as was previously employed for Bi 2 Te 3 [17] and Bi 2 Se 3 [22]. Moreover, it has been observed that the 9/10-fold C2/m structure presents a bcc-like structural order, in agreement with the observed XRD patterns [17,49], therefore giving support to employ the calculated intermediate bcc-like monoclinic C2/m phase to confirm the experimental presence of the disordered Im-3m system.
For the structural relaxations, the atomic positions and the unit-cell parameters were allowed to change during the ionic relaxation, for different volume values. From these relaxations, we obtained the respective external pressure for the specific isotropic volume compression. The pressure-volume (P-V) curves for all the compounds were fitted to a third-order Birch-Murnaghan equation of state [56,57] to obtain the equilibrium volume, bulk modulus, and respective pressure derivative. The enthalpy (H) curves were computed by considering the relation, H = E + p V , where E is the total electronic energy of the system, p is pressure, and V is the volume. The analysis and comparison of the H curves for the different polymorphs provides deep insight regarding the thermodynamic stability of each phase for increasing pressure values, up until the studied pressure range (60 GPa).
Lattice dynamics calculations, within the harmonic approximation, were performed at different pressure points, which were found to be energetically favorable. These were considered for the Pnma, the disordered Im-3m, and the tetragonal phases. We have also calculated the phonon band structure for the C2/m system of Sb 2 Se 3 to confirm the dynamical stability at the pressure point where the energetic stability is evidenced (20 GPa). The phonon properties were computed by using the supercell finite-displacement method implemented in the Phonopy package [58], with VASP being used as the second-order force calculator. Supercells were expanded up to 2 × 4 × 2 for the Pnma systems, and 2 × 2 × 2 for the disordered and tetragonal phases, enabling the exact calculation of frequencies at the zone center ( Γ ) and inequivalent zone-boundary wavevectors, which were then interpolated to obtain phonon-dispersion curves.
Since our calculations on the Pnma phases for the three compounds under study were in good agreement with the overall data found in the literature [51], we have proceeded in carrying out a theoretical study of the hypothetical R-3m, C2/m, and C2/c; and disordered Im-3m and I4/mmm phases of Sb 2 S 3 , Bi 2 S 3 , and Sb 2 Se 3 , to probe whether such polymorphs could be energetically competitive under hydrostatic pressure.

3. Results and Discussion

3.1. Energetic Stability

Figure 2a–c shows the pressure-dependence of the enthalpy differences relative to the stable phase at ambient pressure between the six above mentioned phases of Sb 2 S 3 , Bi 2 S 3 , and Sb 2 Se 3 , respectively. The values of the predicted transition pressures between the different phases are summarized in Table 1.
From the enthalpy plots (Figure 2), we observe that:
  • At 0 GPa, the orthorhombic Pnma structure is energetically stable for Bi 2 S 3 and Sb 2 S 3 ; however, for Sb 2 Se 3 , it is the trigonal R-3m phase that is the most energetically favorable phase at 0 GPa [51].
  • The two monoclinic C2/c and C2/m phases do not become energetically competitive with the ground-state system over the range of pressures examined. However, an exception occurs for the C2/m phase of the Sb 2 Se 3 structure, which competes energetically with the Pnma polymorph at 9.9 GPa.
  • The bcc-like disordered Im-3m structure is the most energetically stable phase at pressures above 35.1, 30.1, and 22.1 GPa for Sb 2 S 3 , Bi 2 S 3 , and Sb 2 Se 3 , respectively.
  • The body-centered tetragonal I4/mmm phase, although being quite close in energy to the bcc-like Im-3m structure (namely, for Bi 2 S 3 close to 30 GPa, and for Sb 2 Se3 between 30 and 50 GPa), is never energetically favorable within the studied pressure range, for either of the three studied compounds.
With respect to the first point, referring to Bi 2 S 3 and Sb 2 S 3 , our calculations predict that the Pnma structure is energetically the most stable phase, up to 30 GPa, and in good agreement with experimental evidences that show the observation of this phase, both at room and also at HP conditions [30,31,32,33,45,46]. Surprisingly, however, our simulations indicate that the Sb 2 Se 3  R-3m phase is the most stable polymorph at pressures below 4.8 GPa; being both the Pnma and R-3m phases, energetically competitive between 0 and 4.8 GPa [51].
Regarding the second point, our analysis further shows that the two monoclinic C2/c and C2/m phases are never energetically competitive in the three compounds throughout the studied pressure range up to 60 GPa; with the exception of the C2/m phase of the Sb 2 Se 3 close to 10 GPa. These results are consistent with experimental analysis obtained from Refs. [30,31,32,33,45,46], in which no PT had been observed for the Sb 2 S 3 and Bi 2 S 3 systems up until ∼50 GPa. However, the respective results are not consistent with three recent studies reporting evidences of PTs in Sb 2 S 3 [47,48,49]. A PT to an unknown phase was claimed to occur at around 15 GPa [47,48], and several transitions were also reported between 10 and 25 GPa, and tentatively proposed to be the R-3m, C2/c, and C2/m structural phases [49]. In this context, it must be stressed that our calculations are performed for pure hydrostatic conditions; therefore, it is not expected that a complete agreement occurs with experiments if non-hydrostatic conditions are considered for the experiments.
As for the third point, from a thermodynamic point of view, our results indicate that the bcc-like disordered Im-3m phase, initially identified for Bi 2 Se 3 , Bi 2 Te 3 , and Sb 2 Te 3 [17,19,22], seems to be energetically favorable at HP for our three materials of interest. These results are consistent with the observation of such a phase at around 50 GPa for Sb 2 Se 3 [36] and above 25 GPa for Sb 2 S 3 [47,48,49]. However, our results do not agree with those found for Bi 2 S 3 [37,40], for which a disorder, attributed mostly to a pressure-induced amorphization (PIA), has been observed above 50 GPa. Notably, PIA in Bi 2 S 3 is consistent with a recent theoretical work that claims that a respective system is unstable above 31.5 GPa, decomposing into a mixture of BiS 2 and BiS compounds [50]. In this context, it must be stressed that the existence of a bcc-like structure at HP is not only expected for A 2 X 3 sesquichalcogenides, but also for AX (A=Ge,Sn,Pb) chalcogenides [59,60]. All these compounds have in common the property of being cataloged as evidencing metavalent bonding at room pressure, such as SnTe, PbS, PbSe, PbTe, Sb 2 Te 3 , Bi 2 Se 3 , or Bi 2 Te 3 [46]; or as being compounds with p-type covalent bonds that develop metavalent bonding at HP when approaching a six-fold coordination, such as GeSe, SnSe, As 2 S 3 , and also U 2 S 3 -type sesquichalcogenides [61,62], and ultimately, they will reach eight-fold coordination typical of bcc-like metals at very HP.
With respect to the fourth point, the I4/mmm structure was firstly proposed for Bi 2 Se 3 by combining experimental and theoretical data [25]. It was found that the structural phase transition pathway for Bi 2 Se 3 followed the sequence: R-3m → C2/m → C2/c → I4/mmm, when the quasi-hydrostatic pressure was considered. The C2/c phase would, however, be suppressed when nonhydrostatic pressure conditions are taken into account [25]. These results are also compatible with Raman analysis and XRD data performed on Bi 2 Se 3 , where it was evidenced that the stability of the I4/mmm phase was up to 81.2 GPa [24]. Moreover, it has also been claimed [26], that the alloying of Bi 2 Se 3 with Bi 2 Te 3 enables transitions to occur from the monoclinic phases, up to the disordered Im-3m at ∼19 GPa, which then would be surpassed energetically by the tetragonal I4/mmm polymorph at ∼23 GPa. Curiously enough, the energetic competition between the C2/m, C2/c, and I4/mmm phases is observed at HP for all three U 2 S 3 -type sesquichalcogenides (if we discard the disordered Im-3m phase), and the I4/mmm structure would indeed be energetically the most favorable polymorph at very HP. However, the disordered Im-3m phase is always energetically more competitive than I4/mmm for the U 2 S 3 -type sesquichalcogenides, and so this latter phase is, a priori, not expected to be observed at HP.
Finally, we have to mention that we have also performed the enthalpy calculations by employing two further xc functionals, in order to confirm the present theoretical data (see Appendix A). The employed functionals were the PBE-D2 method of Grimme (which takes into account the dispersion correction term) [63], and LDA [64]. From the analysis of the plots (Figure A1 and Figure A2) we can infer similar energetic trends of the six polymorphs; however, mild differences are observed between the two competing phases of the disordered Im-3m and I4/mmm—detailed discussion is found in Appendix A.
In summary, the agreement of our results regarding the observation of the disordered Im-3m phase for Sb 2 Se 3 and Sb 2 S 3 , but not for Bi 2 S 3 , suggests that thermodynamic stability is not sufficient to explain the lack of the HP disordered phase for the latter compound. In the following section, we discuss the dynamical stability of the Pnma, C2/m, Im-3m, and I4/mmm phases as a function of pressure, in order to provide a deeper understanding regarding this question.

3.2. Dynamical Stability

Energetic stability is a necessary, but it is not a sufficient condition for a structural phase to be synthetically accessible. One should also probe the dynamical stability of the system, which requires the study of the phonon frequencies. If imaginary frequencies emerge (as represented by negative frequencies in the phonon dispersion curves), we can analyze the results by considering that the system is at a potential-energy maximum (transient state), undergoing a phase transition, and therefore cannot be kinetically stable at the given temperature and/or pressure conditions [65,66,67,68,69,70].
In this section, we consider the phonon properties of different phases for the three compounds, which were observed to be energetically the most favorable (Figure 2) at different pressure values, namely:
  • The C2/m phase of Sb 2 Se 3 at 20 GPa.
  • The disordered bcc-type Im-3m phase of the three compounds above 30 GPa.
  • The Pnma phase of the three compounds at 50 GPa.
  • The I4/mmm phase of the three compounds at HP.

3.2.1. The C2/m Phase of Sb 2 Se 3 at 20 GPa

By analyzing the enthalpies as a function of pressure (Figure 2), we observe that the C2/m phase in Sb 2 Se 3 is thermodynamically the most stable phase between 10 and 20 GPa. In order to confirm the dynamical stability, we have performed the phonon dispersion curves of respective system at 20 GPa (Figure 3). The dispersion curves do not evidence any imaginary modes; therefore, we may conclude that the C2/m phase in Sb 2 Se 3 is dynamically stable in this pressure range and could potentially be observed after the Pnma phase, although it has not yet been experimentally observed [36].

3.2.2. The Disordered bcc-Type Im-3m Phase of the Three Compounds above 30 GPa

To assess the possibility of dynamical stability for the disordered Im-3m phases of the three compounds at HP, we have evaluated the phonon dispersion curves at pressure values of 30 GPa, which is close to the transition pressures observed in Figure 2; and at higher pressures of 50 (Sb 2 Se 3 ) and 60 GPa (Sb 2 S 3 , Bi 2 S 3 ).
As illustrated in Figure 4, all three disordered structures at 30 GPa show imaginary modes along the dispersion curves, thus indicating that these structures are dynamically unstable at this pressure range.
The phonon dispersion curves of Sb 2 S 3 and Bi 2 S 3 still present imaginary modes at 60 GPa (Figure 4), thus indicating that neither compound is likely to adopt this phase up to this pressure range. We note, however, that the dynamical instabilities found for Sb 2 S 3 and Bi 2 S 3 both decrease (the imaginary, soft modes shift to higher frequency values, towards positive values) with increasing pressure, suggesting that this phase could in principle become stable at pressures above 60 GPa. In this context, we must note that Efthimiopoulos et al. [37] had observed a pressure-induced amorphization above 50 GPa for Bi 2 S 3 ; however, the authors were not able to identify the phase to be the disordered Im-3m structure, even at 65 GPa. On the other hand, the experimental data for Sb 2 S 3 , suggests that the disordered bcc-like phase exists between 28.2 and 50.2 GPa [49]. However, it must be noted that experimental measurements detailed in Ref. [49] were carried out under non-hydrostatic behavior due to the employed pressure-transmitting medium.
Finally, our calculations suggest that the Im-3m phase becomes dynamically stable in Sb 2 Se 3 already at 50 GPa; a result that is in agreement with the experimental observation of this phase at around 50 GPa [36].
In the light of the above results regarding the disordered Im-3m phase of Sb 2 S 3 , Bi 2 S 3 , and Sb 2 Se 3 , we can speculate that the stability of the disordered solid solution in group-15 sesquichalcogenides seems to be related to the sizes of cations and anions. Since the Im-3m phase is consistently being evidenced at HP for sesquichalcogenides with heavier cations and anions (Sb 2 Se 3 , Sb 2 Te 3 , Bi 2 Se 3 , and Bi 2 Te 3 ), the observation of such a HP phase could be related to the radius sizes [71] of Se, Te, Sb, and Bi (atomic radii: r Se = 117 , r Te = 137 , r Sb = 141 , and r Bi = 182 pm, respectively). Stemming on these values, we can infer that the disordered solid solutions are energetically favorable in sesquichalcogenides if the atomic radii of the cation and anion differ by less than ∼ 65 pm, or if the size ratio between them is smaller than ∼1.55 (case of Bi 2 Se 3 ). In this context, Sb 2 S 3 , which shows a radius difference of between r Sb and r S of 37 pm (141 − 104 = 37 pm) and a size ratio of 1.35, meets the criteria for the observation of the disordered Im-3m phase at HP. However, Bi 2 S 3 , which evidences a larger radius difference (78 pm) and ratio (1.75) between B and S, does not satisfy the above criteria. Such results would therefore be consistent with the amorphous phase observed for Bi 2 S 3 at HP [37,40].
Finally, it must be considered, and as suggested in Ref. [17], that the atomic radii between the anion and cation tend to become approximately equal at HP due to a higher probability of charge transfer from cation to anion. Therefore, HP inherently creates a favorable environment for the disordered solid solutions due to the decrease in the difference between the cation and anion atomic radii. This means that we cannot discard that the disordered solid solution in Bi 2 S 3 could occur at very HP values, namely, when the difference between the two radii decreases below 65 pm and when the ratio decreases below 1.55.

3.2.3. The Pnma Phase of the Three Compounds at 50 GPa

In order to study the dynamical stability of the well known low-pressure Pnma phase at HP, we present in Figure 5 the phonon dispersion curves of the respective phase for Sb 2 S 3 , Bi 2 S 3 , and Sb 2 Se 3 at 50 GPa.
Curiously enough, we find that the Pnma phase for Sb 2 Se 3 is already unstable at 50 GPa; therefore, a PT should occur at smaller pressures to the C2/m phase, and ultimately, to the disordered Im-3m phase, as previously commented. On the other hand, the Pnma phase for Sb 2 S 3 and Bi 2 S 3 is still dynamically stable at 50 GPa, although thermodynamically, it is not the most stable phase (Figure 2). These results, together with the dynamical instability observed for the disordered phase of Sb 2 S 3 and Bi 2 S 3 at 50 GPa (Figure 4) and the thermodynamic instability of the C2/m and C2/c phases, suggest that only the Pnma structure should be observed up to 50 GPa for both compounds, unless there is a decomposition, as theoretically predicted for Bi 2 S 3 [50].

3.2.4. The I4/mmm Phase of the Three Compounds at HP

By considering the enthalpy curves of the three compounds (Figure 2), we can infer that the I4/mmm phase is very close in energy with the disordered Im-3m system at pressures above 30 GPa. We therefore have carried out phonon calculations to probe the dynamical stability of this phase, and for all three compounds.
With regard to the Sb 2 S 3 system, we observe that only at 60 GPa is the I4/mmm structural phase dynamically stable (Figure 6). Below this pressure value, imaginary modes are observed in the vicinity of the Γ -point, which harden for increasing pressure points. This result, together with the dynamic instability of the disordered Im-3m phase still at 60 GPa, suggests that both the I4/mmm and Im-3m structures are not expected to be observed at pressures below 60 GPa; however, any of these systems could potentially be observed above this pressure range.
With respect to the Bi 2 S 3 compound (Figure 7), we observe that the I4/mmm phase is already dynamically stable at 50 GPa, maintaining the stability at 60 GPa. Therefore, the I4/mmm phase could in fact be observed at above 40 GPa, if there was no sample decomposition, as theoretically predicted [50].
Finally, we observed that the the tetragonal I4/mmm phase for Sb 2 Se 3 is only dynamically stable at 40 GPa (Figure 8), since there is a localized imaginary mode at the high-symmetry M-point, at any other pressure value. These results differ from the remaining two S-based compounds, and evidence that the disordered Im-3m phase will be the only (thermodynamically and dynamically) stable phase for the Sb 2 Se 3 compound at very HP (since the Pnma is also unstable at 50 GPa, Figure 5c), and as discussed in the previous subsection.

4. Conclusions

We have carried out a comprehensive set of total energy and lattice dynamics calculations in order to investigate the stabilities of six possible structural phases, viz. Pnma, R-3m, C2/m, and C2/c; and disordered Im-3m and I4/mmm for the Sb 2 S 3 , Bi 2 S 3 , and Sb 2 Se 3 sesquichalcogenides under hydrostatic pressures up to 60 GPa. Our theoretical results have been commented in the light of the available experimental data.
We find that the Pnma phase is energetically more stable at room pressure for the Sb 2 S 3 and Bi 2 S 3 compounds. Curiously, the trigonal R-3m phase is the most energetically favorable phase for Sb 2 Se 3 at 0 GPa, although from an experimental perspective, this compound is synthesized in the Pnma phase. In fact, we find that the Pnma phase is dynamically stable for Sb 2 S 3 and Bi 2 S 3 up to 50 GPa, but not for Sb 2 Se 3 at this pressure value.
From our calculations, we observe that the monoclinic C2/m and C2/c systems for the three compounds are energetically less favorable throughout the studied pressure range, and they are not expected to be observed at HP under hydrostatic conditions, except for Sb 2 Se 3 between 10 and 20 GPa. Moreover, the C2/m phase of Sb 2 Se 3 is dynamically stable at 20 GPa, so a pressure-induced phase transition from the Pnma to the C2/m system could be observed between 10 and 20 GPa under hydrostatic conditions.
The disordered bcc-like Im-3m phase is predicted to be the most energetically stable phase above 35, 30, and 22 GPa for Sb 2 S 3 , Bi 2 S 3 , and Sb 2 Se 3 , respectively; however, this structure is dynamically unstable for Sb 2 S 3 and Bi 2 S 3 up to 60 GPa, and for Sb 2 Se 3 up to 50 GPa. Therefore, this HP phase is expected to occur for Sb 2 Se 3 above 50 GPa, in good agreement with experiments, and perhaps above 60 GPa for the other two compounds.
With respect to the I4/mmm phase, we have learned that the structure is not energetically competitive with the Im-3m phase, throughout the studied pressure range, and for the three compounds; however, this phase is close in energy to the Im-3m phase for Bi 2 S 3 at 30 GPa, and for Sb 2 Se 3 , close to 38 GPa. On the other hand, the I4/mmm system is dynamically stable for Bi 2 S 3 above 50 GPa, and for Sb 2 Se 3 , quite close to 40 GPa. Therefore, this phase could potentially be observed for the former compound at HP; however, not for Sb 2 Se 3 .
Curiously enough, we must mention that through symmetry analysis, the I4/mmm →Fmmm→ C2/m structural transition pathway may occur as a second-order phase transition. We have, however, not considered the intermediate Fmmm system, since, and to the best of our knowledge, such a phase has not yet been experimentally observed for sesquichalcogenides at HP, and therefore, respective analysis would be out the scope of the present work.
We hope that this work will stimulate further investigation of the sesquichalcogenides at HP in order to clarify which pressure-induced phase transitions are observed under hydrostatic and non-hydrostatic conditions.

Author Contributions

E.L.d.S., M.C.S., P.R.-H., and A.M. performed the ab initio calculations. F.J.M. supervised the work and comparison with experimental data. E.L.d.S. and F.J.M. wrote the manuscript. All authors have read and agreed to the published version of the manuscript.

Funding

This publication is part of the Project MALTA Consolider Team network (RED2018-102612-T), financed by MINECO/AEI/10.13039/501100003329; by I+D+i project PID2019-106383GB-42/43, financed by MCIN/AEI/10.13039/501100011033; by projects PROMETEO CIPROM/2021/075 (GREENMAT) and MFA/2022/025 (ARCANGEL), financed by Generalitat Valenciana; and by the European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant, agreement No. 785789-COMEX.

Data Availability Statement

The data that support the work presented within this paper are available from the corresponding author upon reasonable request.

Acknowledgments

E.L.d.S acknowledges the High Performance Computing Chair—an R&D infrastructure (based at the University of Évora; PI: M. Avillez), endorsed by Hewlett Packard Enterprise, and involving a consortium of higher education institutions, research centers, enterprises, and public/private organizations; and the Portuguese Foundation of Science and Technology with the CEEC individual fellowship, 5th edition, with Reference 2022.00082.CEECIND.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A. Comparison of Enthalpy vs. Pressure Behavior for Different Exchange-Correlation Functionals

In order to confirm the enthalpy variation as the pressure increases for the six studied phases presented in Figure 2, and obtained by employing the PBEsol functional, we observe that the energetic trends are functional-independent. In Figure A1 and Figure A2, we present enthalpy results carried out by considering the PBE-D2 and the LDA functional, respectively. For PBE-D2 (Figure A1), the transition pressures are very close to those obtained with PBEsol. Moreover, for the Sb 2 Se 3 system, the energetic trends are also comparable to those of PBEsol, namely, that the R-3m phase is energetically the most stable at room pressure, and that the C2/m phase shows a small interval range of stability before the disordered Im-3m phase takes over the energetic configurational space. The I4/mmm system is energetically competitive, with the disordered phase at a slightly lower pressure range when compared to the results obtained with PBEsol. For Sb 2 S 3 and Bi 2 S 3 , the tetragonal I4/mmm phase becomes energetically the most stable at pressures close to 60 GPa. The results obtained with the LDA functional (Figure A2) also show similar trends; however, the transition pressures are slightly higher than for PBEsol. Moreover, a competition between the two monoclinic phases and I4/mmm seems to occur in Sb 2 Se 3 slightly, before the latter phase becomes the most stable phase. At around 60 GPa, the disordered phase stabilizes, becoming the most stable structure for Sb 2 Se 3 . In addition, for the Bi 2 S 3 compound, we observe a pressure range, at which the I4/mmm becomes energetically the most stable (∼30–45 GPa), which then is energetically taken over by the disordered Im-3m phase at above 45 GPa.
Crystals 13 00498 g0a1 550
Figure A1. PBE+vdW enthalpy vs. pressure curves, for the different possible phases (shown in Figure 1) of Sb 2 S 3 (a), Bi 2 S 3 (b), and Sb 2 Se 3 (c), relative to the lowest-energy phase at ambient pressure: the Pnma phase for Sb 2 S 3 and Bi 2 S 3 , and the R-3m phase for Sb 2 Se 3 .
Figure A1. PBE+vdW enthalpy vs. pressure curves, for the different possible phases (shown in Figure 1) of Sb 2 S 3 (a), Bi 2 S 3 (b), and Sb 2 Se 3 (c), relative to the lowest-energy phase at ambient pressure: the Pnma phase for Sb 2 S 3 and Bi 2 S 3 , and the R-3m phase for Sb 2 Se 3 .
Crystals 13 00498 g0a1
Crystals 13 00498 g0a2 550
Figure A2. LDA enthalpy vs. pressure curves, for the different possible phases (shown in Figure 1) of Sb 2 S 3 (a), Bi 2 S 3 (b), and Sb 2 Se 3 (c), relative to the lowest-energy phase at ambient pressure: the Pnma phase for Sb 2 S 3 and Bi 2 S 3 , and the R-3m phase for Sb 2 Se 3 .
Figure A2. LDA enthalpy vs. pressure curves, for the different possible phases (shown in Figure 1) of Sb 2 S 3 (a), Bi 2 S 3 (b), and Sb 2 Se 3 (c), relative to the lowest-energy phase at ambient pressure: the Pnma phase for Sb 2 S 3 and Bi 2 S 3 , and the R-3m phase for Sb 2 Se 3 .
Crystals 13 00498 g0a2

Appendix B. Evolution of the Lattice Parameters of Sb2S3, Bi2S3, and Sb2Se3 as a Function of Pressure

In Table A1, Table A2 and Table A3, we present the lattice parameters of the different studied compounds at pressure values where stability (thermodynamic and dynamical) is evidenced. Since the C2/c structure is never stable throughout the studied pressure range, we do not present the respective theoretical values for any of the compounds. As for Sb 2 S 3 and Bi 2 S 3 , we only present the lattice parameters related to the Pnma phase, at 0 and 30 GPa, since this is the pressure range for which the structure is energetically more favorable; and where none of the remaining phases satisfy both of the stability conditions that we mentioned previously.
Table
Table A1. Theoretical estimation (PBEsol) of the lattice parameters of the Sb 2 S 3 compound, for the Pnma structural phase, for different pressure values, at which the system is both energetically and dynamically stable (presented in units of Å).
Table A1. Theoretical estimation (PBEsol) of the lattice parameters of the Sb 2 S 3 compound, for the Pnma structural phase, for different pressure values, at which the system is both energetically and dynamically stable (presented in units of Å).
Sb 2 S 3 0 GPa10 GPa20 GPa30 GPa
a 0 11.2410.3010.029.82
b 0 3.833.723.633.58
c 0 10.9110.249.839.53
Table
Table A2. Theoretical estimation (PBEsol) of the lattice parameters of the Bi 2 S 3 compound, for the Pnma structural phase, for different pressure values, at which the system is both energetically and dynamically stable (presented in units of Å).
Table A2. Theoretical estimation (PBEsol) of the lattice parameters of the Bi 2 S 3 compound, for the Pnma structural phase, for different pressure values, at which the system is both energetically and dynamically stable (presented in units of Å).
Bi 2 S 3 0 GPa10 GPa20 GPa30 GPa
a 0 11.1910.5910.3810.22
b 0 3.963.823.743.67
c 0 10.9410.329.949.64
Table
Table A3. Theoretical estimation (PBEsol) of the lattice parameters of the Sb 2 Se 3 compound, for the different structural phases that are both energetically and dynamically stable within the range of studied pressures (presented in units of Å).
Table A3. Theoretical estimation (PBEsol) of the lattice parameters of the Sb 2 Se 3 compound, for the different structural phases that are both energetically and dynamically stable within the range of studied pressures (presented in units of Å).
Sb 2 Se 3
Pnma10 GPaa 0 = 10.77, b 0 = 3.83, c 0 = 10.69
15 GPaa 0 = 10.63, b 0 = 3.77, c 0 = 10.48
20 GPaa 0 = 10.50, b 0 = 3.72, c 0 = 10.35
R-3m0 GPaa 0 = 4.01, c 0 = 28.16
C/2m20 GPaa 0 = 13.15, b 0 = 3.55, c 0 = 8.06
disordered Im-3m50 GPaa 0 = 13.55, b 0 = 4.49, c 0 = 5.50
I4/mmm40 GPaa 0 = 3.26, c 0 = 15.84

Appendix C. Mode Mapping of the Disordered Im-3m Phase of Sb2Se3 at 30 GPa

By observing the phonon dispersion curves of the disordered Im-3m phase in Sb 2 Se 3 at 30 GPa (Figure 4c), we may notice that the instability mode is localized at a high-symmetry point, the M-point, which is defined as being a zone-boundary instability (or anti-ferroelectric instability, resulting in an anti-phase periodic distortion). Since the negative mode is localized, it is therefore possible to map out the anharmonic potential energy surfaces by following the eigenvectors associated with this instability/displacement, and thus, to be able to obtain a lower energy structure corresponding to the minima of the potential energy surface. For these calculations, we use the open-source MODEMAP package [72,73]. A sequence of displaced structures in a commensurate supercell (1 × 2 × 2) expansion is generated by displacing along the phonon eigenvectors over a range of amplitudes of the normal-mode coordinate Q. The total energies of the “frozen phonon” structures are evaluated from single-point DFT calculations. The E(Q) curves are then fitted to a polynomial function, with the number of terms depending on the form of the potential well.
The displacements associated with the disordered bcc high-pressure structure result from the condensation of these soft M-modes, leading to a lower energy system and an enlarged primitive unit-cell. The displacement involves two Se atoms at the center of the cell, namely at (0.50000, 0.75000, 0.50000) and (0.50000, 0.25000, 0.50000). Both atoms evidence mild distortions along the x- and z-directions, minimizing the energy to 3.63 meV when compared to the original disordered Im-3m phase at Q = 0 (Figure A3).
Crystals 13 00498 g0a3 550
Figure A3. Double −well potential−energy surfaces for the phonon instabilities associated with the high-symmetry M−point negative modes (Figure 4). The normal-mode coordinates Q have been normalized so that the minima are located at Q = ±1, and the energy differences are those calculated in a 1 × 2 × 2 supercell. The bottom figures represent the structure at Q = 0 (left) and at Q = 1 (right).
Figure A3. Double −well potential−energy surfaces for the phonon instabilities associated with the high-symmetry M−point negative modes (Figure 4). The normal-mode coordinates Q have been normalized so that the minima are located at Q = ±1, and the energy differences are those calculated in a 1 × 2 × 2 supercell. The bottom figures represent the structure at Q = 0 (left) and at Q = 1 (right).
Crystals 13 00498 g0a3

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Figure 1. Unit-cell representations of the crystal structures of the R-3m (Z = 1) (a), Pnma (Z = 4) (b), C2/m (Z = 2) (c), C2/c (Z = 2) (d), the C2/m 9-/10-fold structure used to model the disordered bcc-type Im-3m phase (Z = 2) (e), and the I4/mmm (Z = 1) (f), and A 2 X 3 sesquichalcogenide structures (A = Sb, Bi; X = S, Se). The A cations and X anions are shown as brown and yellow spheres, respectively.
Figure 1. Unit-cell representations of the crystal structures of the R-3m (Z = 1) (a), Pnma (Z = 4) (b), C2/m (Z = 2) (c), C2/c (Z = 2) (d), the C2/m 9-/10-fold structure used to model the disordered bcc-type Im-3m phase (Z = 2) (e), and the I4/mmm (Z = 1) (f), and A 2 X 3 sesquichalcogenide structures (A = Sb, Bi; X = S, Se). The A cations and X anions are shown as brown and yellow spheres, respectively.
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Figure 2. Calculated relative enthalpy vs. pressure curves, for the different possible phases (shown in Figure 1) of Sb 2 S 3 (a), Bi 2 S 3 (b), and Sb 2 Se 3 (c), relative to the lowest-energy phase at ambient pressure: the Pnma phase for Sb 2 S 3 and Bi 2 S 3 , and the R-3m phase for Sb 2 Se 3 .
Figure 2. Calculated relative enthalpy vs. pressure curves, for the different possible phases (shown in Figure 1) of Sb 2 S 3 (a), Bi 2 S 3 (b), and Sb 2 Se 3 (c), relative to the lowest-energy phase at ambient pressure: the Pnma phase for Sb 2 S 3 and Bi 2 S 3 , and the R-3m phase for Sb 2 Se 3 .
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Figure 3. Harmonic phonon dispersion curves of the C2/m phase of the Sb 2 Se 3 compound at 20 GPa.
Figure 3. Harmonic phonon dispersion curves of the C2/m phase of the Sb 2 Se 3 compound at 20 GPa.
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Figure 4. Harmonic phonon dispersion curves of the disordered bcc-like Im-3m phases of Sb 2 S 3 (a), Bi 2 S 3 (b), and Sb 2 Se 3 (c); and calculated at 30 GPa (left), and 50 (Sb 2 Se 3 ) or 60 GPa (Sb 2 S 3 and Bi 2 S 3 ; right). The BZ q−vector description represents the C2/m space−group, according to the symmetry of the employed cell.
Figure 4. Harmonic phonon dispersion curves of the disordered bcc-like Im-3m phases of Sb 2 S 3 (a), Bi 2 S 3 (b), and Sb 2 Se 3 (c); and calculated at 30 GPa (left), and 50 (Sb 2 Se 3 ) or 60 GPa (Sb 2 S 3 and Bi 2 S 3 ; right). The BZ q−vector description represents the C2/m space−group, according to the symmetry of the employed cell.
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Figure 5. Harmonic phonon dispersion curves of the Pnma phase of Sb 2 S 3 (a), Bi 2 S 3 (b), and Sb 2 Se 3 (c), calculated at 50 GPa.
Figure 5. Harmonic phonon dispersion curves of the Pnma phase of Sb 2 S 3 (a), Bi 2 S 3 (b), and Sb 2 Se 3 (c), calculated at 50 GPa.
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Figure 6. Harmonic phonon dispersion curves of the I4/mmm phase of the Sb 2 S 3 compound between 30 and 60 GPa.
Figure 6. Harmonic phonon dispersion curves of the I4/mmm phase of the Sb 2 S 3 compound between 30 and 60 GPa.
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Figure 7. Harmonic phonon dispersion curves of the I4/mmm phase of the Bi 2 S 3 compound between 30 and 60 GPa.
Figure 7. Harmonic phonon dispersion curves of the I4/mmm phase of the Bi 2 S 3 compound between 30 and 60 GPa.
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Figure 8. Harmonic phonon dispersion curves of the I4/mmm phase of the Sb 2 Se 3 compound between 30 and 60 GPa.
Figure 8. Harmonic phonon dispersion curves of the I4/mmm phase of the Sb 2 Se 3 compound between 30 and 60 GPa.
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Table
Table 1. Theoretical estimation of the pressure-induced phase transitions of R-3m →Pnma, Pnma→ disordered C2/m, C2/m→ disordered Im-3m, and Pnma→ disordered Im-3m for the Sb 2 Se 3 , Sb 2 S 3 , and Bi 2 S 3 compounds (presented in units of GPa).
Table 1. Theoretical estimation of the pressure-induced phase transitions of R-3m →Pnma, Pnma→ disordered C2/m, C2/m→ disordered Im-3m, and Pnma→ disordered Im-3m for the Sb 2 Se 3 , Sb 2 S 3 , and Bi 2 S 3 compounds (presented in units of GPa).
Sb 2 S 3 Bi 2 S 3 Sb 2 Se 3
R-3m →Pnma––4.8 *
Pnma→C2/m––9.9
C2/m→Im-3m––22.1
Pnma→Im-3m35.130.1–
* Ref. [51].
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da Silva, E.L.; Santos, M.C.; Rodríguez-Hernández, P.; Muñoz, A.; Manjón, F.J. Theoretical Study of Pressure-Induced Phase Transitions in Sb2S3, Bi2S3, and Sb2Se3. Crystals 2023, 13, 498. https://doi.org/10.3390/cryst13030498

AMA Style

da Silva EL, Santos MC, Rodríguez-Hernández P, Muñoz A, Manjón FJ. Theoretical Study of Pressure-Induced Phase Transitions in Sb2S3, Bi2S3, and Sb2Se3. Crystals. 2023; 13(3):498. https://doi.org/10.3390/cryst13030498

Chicago/Turabian Style

da Silva, Estelina Lora, Mario C. Santos, Plácida Rodríguez-Hernández, Alfonso Muñoz, and Francisco Javier Manjón. 2023. "Theoretical Study of Pressure-Induced Phase Transitions in Sb2S3, Bi2S3, and Sb2Se3" Crystals 13, no. 3: 498. https://doi.org/10.3390/cryst13030498

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