A theoretical study of the Pnma and R3̄m phases of Sb₂S₃, Bi₂S₃, and Sb₂Se₃

Abstract

We report a comparative theoretical study of the Pnma and R3̄m phases of Sb₂S₃, Bi₂S₃, and Sb₂Se₃ close to ambient pressure. Our enthalpy calculations at 0 K show that at ambient pressure the R3̄m (tetradymite-like) phase of Sb₂Se₃ is energetically more stable than the Pnma phase, contrary to what is observed for Sb₂S₃ and Bi₂S₃, and irrespective of the exchange–correlation functional employed in the calculations. The result for Sb₂Se₃ is in contradiction to experiments in which all three compounds are usually grown in the Pnma phase. This result is further confirmed by free-energy calculations taking into account the temperature dependence of unit-cell volumes and phonon frequencies. Lattice dynamics and elastic tensor calculations further show that both the Pnma and R3̄m phases of Sb₂Se₃ are dynamically and mechanically stable at zero applied pressure. Since these results suggest that the formation of the R3̄m phase of Sb₂Se₃ should be feasible under 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 Sb₂Se₃. Finally, the stability of the R3̄m phase across the three group-15 A₂X₃ sesquichalcogenides is analysed based on their van der Waals gap and X–X in-plane geometry.

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1 Introduction

Since the identification of the trigonal tetradymite-like R3̄m phases of group-15 sesquichalcogenides (i.e. Sb₂Te₃, Bi₂Se₃, and Bi₂Te₃; Fig. 1a) as 3D topological insulators,^1,2 the family of A₂X₃ sesquichalcogenides has attracted a great deal of attention from the scientific community. Three-dimensional topological insulators represent a new class of matter, with insulating bulk electronic states and topologically protected metallic surface states arising from time-reversal symmetry and strong spin–orbit coupling (SoC). These properties make them of potential interest for spintronics and quantum computing applications.³ Due to this fundamental interest and potential applications, the identification of new topological insulators and materials with superconducting properties is currently an important research area in condensed matter science.

The stibnite (Sb₂S₃), bismuthinite (Bi₂S₃), and antimonselite (Sb₂Se₃) minerals are also group-15 sesquichalcogenides but do not crystallize in the tetradymite-like R3̄m structure under ambient conditions and instead are reported to adopt an orthorhombic U₂S₃-type Pnma structure (Fig. 1b). Sb₂S₃, Bi₂S₃, and Sb₂Se₃ are semiconductors with band gap widths of 1.7, 1.3, and 1.2 eV, respectively.^4,5 These materials are used in a wide range of technological applications including photovoltaics (solar cells), X-ray computed tomography detectors, fuel cells, gas sensors, and for the detection of biomolecules.^6–12 Additionally, Sb₂Se₃ has recently found a number of other applications including in solid-state batteries, fiber lasers, and photoelectrochemical devices.^13–16

Fig. 1 Structures of the R3̄m (a) and Pnma (b) phases of the A₂X₃ sesquichalcogenides (A = Sb, Bi and X = S, Se). The A cations and X anions are shown as brown and yellow spheres, respectively.

Since several phases, including the Pnma phase, have been synthesized for Bi₂Se₃, which usually crystallizes in the tetradymite-like R3̄m structure,^17–20 it is natural to wonder whether the R3̄m structure could be adopted by other sesquichalcogenides that generally adopt the U₂S₃-type structure, viz. Sb₂S₃, Bi₂S₃, and in particular Sb₂Se₃. In fact, several theoretical studies have been performed over the years to investigate the properties of the hypothetical tetradymite-like Sb₂Se₃ structure. Some of these studies have suggested that this phase should undergo a topological quantum phase transition under compression,^21,22 while one found that tetradymite-like Sb₂Se₃ is dynamically stable and is a topological insulator at ambient pressure.²³ Interestingly enough, the R3̄m phases of group-15 sesquichalcogenides are characterized by a unique type of bonding termed “metavalent bonding” that not only underpins the topological properties but also makes these materials useful for phase-change memories, as highly efficient thermoelectrics, and for photonic devices.²⁴

In 2013 an experimental study claimed to have observed the R3̄m phase of Sb₂Se₃, with a topological transition occurring at 2 GPa,²⁵ but the tetradymite-like structure of Sb₂Se₃ was not confirmed beyond doubt. On the other hand, a comparative experimental and theoretical study of the three U₂S₃-type sesquichalcogenides suggested the Pnma structure to be stable up to 50 GPa.²⁶ This is supported by several experimental high-pressure studies on Sb₂Se₃ in which the Pnma structure was found to be stable up to 50 GPa and above.^27,28 However, one study observed a pressure-induced isostructural phase transition at 12 GPa and a further transition to a disordered Im3̄m structure above 50 GPa, followed by a pressure-induced amorphization on releasing the pressure.²⁹ Despite the apparent stability of the Pnma phase, experimental high-pressure studies have also found that Pnma Sb₂Se₃ becomes a topological superconductor at 2.5 K and around 10 GPa,³⁰ exhibiting highly conducting spin-polarized surface states similar to those observed for Bi₂Se₃.³¹ Furthermore, a recent study has claimed to have synthesized R3̄m of Sb₂Se₃ by atomic layer epitaxy on a buffer layer of Bi₂Se₃.³² We can therefore conclude that while the bulk of the experimental evidence suggests that the R3̄m phase of Sb₂Se₃ is not observed at high pressures, it is not conclusive as to whether this phase could potentially be formed under favourable synthesis conditions.

In light of the above studies, it is interesting to compare the stabilities of the Pnma and R3̄m structural phases of the three U₂S₃-type sesquichalcogenides under close to ambient conditions and to confirm whether or not the R3̄m phase could be synthetically accessible. In this work, we report a set of systematic density-functional theory (DFT) calculations on the Pnma and R3̄m phases of the three U₂S₃-type sesquichalcogenides under ambient conditions and at pressures up to 10 GPa. We show that the Pnma phases of Sb₂S₃ and Bi₂S₃ are energetically more stable than the R3̄m phases over this pressure range, but that, unexpectedly, the R3̄m phase of Sb₂Se₃ is predicted by several exchange–correlation (XC) functionals to be more stable than the Pnma phase close to ambient conditions. To aid in future experimental efforts to prepare the R3̄m phase of Sb₂Se₃, we also confirm its dynamical and mechanical stability and provide a theoretical structure and vibrational spectra to support its identification. Finally, we also discuss the only two studies that, to our knowledge, have claimed to have prepared the tetradymite-like phase of Sb₂Se₃ under close to ambient conditions.

2 Methods

The structural properties of the different crystalline phases of Sb₂S₃, Bi₂S₃, and Sb₂Se₃ were calculated within the framework of pseudopotential plane-wave density-functional theory³³ using the Vienna Ab initio Simulation Package (VASP) code.³⁴ The revised Perdew–Burke–Ernzerhof generalized-gradient approximation (GGA) functional for solids (PBEsol)^35,36 was used for all calculations. Additional calculations were also performed with the local-density approximation (LDA)³⁷ functional and the dispersion-corrected PBE-D2³⁸ functional to assess the impact of the XC treatment on the results. Projector augmented-wave (PAW) pseudopotentials including six valence electrons for S[3s² 3p⁴] and Se[4s² 4p⁴] and fifteen valence electrons for Sb[4d¹⁰ 5s² 5p³] and Bi[5d¹⁰ 6s² 6p³] were used to model the ion cores. The convergence of the total energy was achieved with a plane-wave kinetic-energy cut-off of 600 eV. The Brillouin zones (BZs) were sampled with Γ-centered Monkhorst-Pack³⁹ grids with 6 × 10 × 6 (Pnma) and 12 × 12 × 12 subdivisions (R3̄m).

Structural relaxations were performed by allowing the atomic positions and the unit-cell parameters to optimise at a series of fixed volumes in order to confirm the stability of both the Pnma and R3̄m phases over a pressure range from 0–10 GPa. At each volume, we obtain the (hydrostatic) external pressure for the applied compression and the corresponding structural parameters. The pressure–volume (p–V) curves for each of the compounds were fitted to a third-order Birch–Murnaghan equation of state^40,41 to obtain the equilibrium volume, the bulk modulus, and its pressure derivative. The enthalpy H as a function of volume was computed using the relationship H = E + pV, where E is the total electronic energy of the system, p is the pressure, and V is the volume. Comparison of the H curves of the different polymorphs can provide insight into the relative thermodynamic stabilities over the studied pressure range.

Lattice-dynamics calculations were performed on the Pnma and R3̄m phases of Sb₂Se₃ at a series of cell volumes corresponding to different applied pressures. The phonon frequencies were computed by using the supercell finite-displacement method implemented in the Phonopy package⁴² with VASP as the force calculator.⁴³ Supercell expansions of 2 × 4 × 2 for the Pnma phase and 2 × 2 × 2 for the R3̄m phases were used to enable the exact calculation of frequencies at the zone center (Γ) and unique zone-boundary wave vectors, which were interpolated to obtain phonon-dispersion curves together with density of states curves on uniform 50 × 50 × 50 Γ-centered q-point meshes.

Infrared (IR) and Raman spectra were calculated for the ground-state R3̄m phase of Sb₂Se₃ using the methods described in ref. 44 and implemented in the Phonopy-Spectroscopy package.⁴⁵ The spectral linewidths were obtained by computing the third-order force constants of a 2 × 2 × 2 expansion of the primitive cell, and following the many-body perturbative approach described in detail in ref. 46 and implemented in the Phono3py software.

Elastic tensors were computed to assess the mechanical stability of the Pnma and R3̄m phases of Sb₂Se₃ at zero pressure, by employing the central-difference method where the unique components of the elastic tensor are determined by performing six finite distortions of the lattice and deriving the tensor elements from the strain–stress relationship.⁴⁷ For these calculations, it was necessary to increase the plane-wave energy cutoff to 950 eV to converge the stress tensor. We then employed the ELATE software⁴⁸ to analyze the linear compressibility using the results.

3 Results and discussion

3.1 Structural properties of the Pnma phase

In order to verify the accuracy of our calculations as a prior step before attempting to study the potential R3̄m phases of Sb₂S₃, Bi₂S₃ and Sb₂Se₃, we first calculated the equilibrium lattice parameters, bulk moduli and pressure derivatives of the Pnma phases and compared them to other experimental and theoretical studies in the literature (Table 1).

Table 1 Calculated equilibrium lattice parameters (a₀, b₀ and c₀), bulk moduli (B₀) and pressure derivatives of the Pnma phases of Sb₂Se₃, Sb₂S₃ and Bi₂S₃ compared to experiments and other theoretical results from the literature

	Sb2Se3		Sb2S3		Bi2S3
	Theo.	Exp.	Theo.	Exp.	Theo.	Exp.
a This work. b Ref. 26. c Ref. 49. d Ref. 50. e Ref. 51. f Ref. 27. g Ref. 52. h Ref. 53. i Ref. 54. j Ref. 55. k Ref. 56. l Ref. 57. m Ref. 58. n Ref. 59. o Ref. 60. p Ref. 61. q Ref. 62. r Ref. 63. s Ref. 30. t Ref. 64. u Ref. 65.
a 0 (Å)	11.75^a		11.24^a		11.19^a
	11.80^b	11.80^f	11.27^b	11.30^b^j^k	11.41^b	11.27^p
	11.52^c	11.79^g	11.02^c	11.31^l^m	11.00^n	11.33^q
	11.91^d		11.30^h		11.58^o
	11.53^e		11.08^i
b 0 (Å)	3.98^a		3.83^a		3.96^a
	3.99^b	3.98^f	3.81^c	3.84^b^j^k	3.97^b	3.97^p
	3.96^c^e	3.99^g	3.84^h	3.84^l^m	3.94^n	3.98^q
	3.98^d		3.83^b^i		3.99^o
c 0 (Å)	11.30^a		10.91^a		10.94^a
	11.28^b	11.65^f^g	10.89^b	11.23^b^j^l^m	11.01^b	11.13^p
	11.22^c^e		10.79^c	11.24^k	10.83^n	11.18^q
	11.70^d		11.22^h		11.05^o
			10.81^i
V 0 (Å^3)	528.11^a		469.6^a		484.4^a
	531.1^b	547.1^f	470.4^b	486.0^b	498.3^b	498.4^p
	511.8^c	547.5^g	453.0^c	487.7^i^m^j	469.1^n	501.6^q
	598.1^r	552.5^s	529.9^r	488.2^k	510.1^o
					511.6^q
B 0 (GPa)	31.1^a		31.5^a		42.3^a
	70.5^c	30.0^f	32.2^b	37.6^b	83.6^n	36.6^p
		32.7^s	80.3^c	26.9^j	32.3^o	38.9^q
				27.2^k	36.5^q	37.5^u
				41.4^t
	6.6^a		6.6^a		6.8^a
		6.1^f	6.2^b	3.8^b	5.9^q	6.4^p
		5.6^s		7.9^j	6.4^o	5.5^q
				6.0^k		4.6^u
				7.8^t

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The Pnma phases of A₂X₃ sesquichalcogenides comprise layers stacked by weak interactions along the crystallographic a direction, the description of which is challenging for conventional DFT functionals.^26,51,63

The calculated lattice parameters of Sb₂Se₃ (a₀ = 11.75 Å, b₀ = 3.98 Å and c₀ = 11.30 Å) are in good agreement with the experimental measurements in ref. 27 and 52 (a₀ = 11.80 Å, b₀ = 3.97 Å, c₀ = 11.65 Å and a₀ = 11.79 Å, b₀ = 3.98 Å and c₀ = 11.65 Å, respectively), and also with other ab initio calculations.^26,49–51 The most notable deviation of our calculated values from experimental measurements is a ∼3% reduction of the c₀ parameter, which contributes to a ∼3–4% underestimation of V₀ compared to experiments. Our results are comparable to the theoretical results in ref. 26, where calculations were also carried out using PAW pseudopotentials and the PBEsol functional. The c₀ of 11.70 Å quoted in ref. 50 is considerably larger than the present results but closer to experiments, but the a₀ parameter has a larger error. We attribute this to the use of the PBE functional in this study, which has a tendency to overestimate volumes and has been shown to do so by ∼10% for antimony chalcogenides.⁶³ On the other hand, the LDA tends to underestimate volumes, as can be seen in the lattice parameters quoted in ref. 49, 51 and 59. Interestingly, the difference in the predicted and measured b-axis lengths is very small, which we attribute to the fact that this crystallographic direction corresponds to covalently bonded chains of atoms.

Our calculated lattice parameters for the Pnma phase of Sb₂S₃ show similar trends to those of Sb₂Se₃. As shown in Table 1, the calculated parameters agree well with experimental measurements^26,55 and other theoretical results.^26,49,53,54 We note that, however, for Sb₂Se₃, the lattice parameters obtained from LDA calculations tend to underestimate compared to experiments, resulting in discrepancies with the a₀ quoted in ref. 49 and 54, although the c₀ parameter is closer to our PBEsol results than to the PBE values quoted in ref. 53, which again agree better with experimental results.^26,55

Our results for Bi₂S₃ are also consistent with experimental measurements and other theoretical studies in the literature. We note that calculations performed on Bi₂S₃ using the Armiento and Mattsson 2005 parametrized GGA functional (AM05)^66–68 seem to show a slightly better reproduction of the c₀ parameter compared to experiments.²⁶

The calculated B₀ and values obtained by fitting the p–V curves of Sb₂Se₃ to a third-order Birch–Murnaghan equation are B₀ = 31.1 GPa ( = 6.6), which are close to the experimental values of B₀ = 30 GPa ( = 6.1) from ref. 27 and B₀ = 32.7 GPa ( = 5.6) from ref. 60. For Sb₂S₃, we obtained B₀ = 31.5 GPa ( = 6.6), which is within the range of experimental values^26,55,56,64 and consistent with the PAW/PBEsol calculations in ref. 26 These results are also close to those experimentally measured for the As-doped stibnite mineral.⁶⁹ Finally, our values of B₀ = 42.3 GPa ( = 6.8) for Bi₂S₃ are again consistent with other DFT calculations^60,62 and the experimental values reported in ref. 61, 62 and 65.

3.2 Energetic stability of the Pnma and R3̄m phases at 0 K and up to 10 GPa

Since our calculations on the Pnma phases were found to be in good agreement with experimental and theoretical studies, we proceeded to carry out a theoretical study of the potential R3̄m phases of Sb₂S₃, Bi₂S₃, and Sb₂Se₃ to probe whether this phase could be energetically competitive from ambient pressure up to 10 GPa.

Fig. 2a–c show the pressure dependence of the enthalpy differences between the R3̄m and Pnma phases of Sb₂S₃, Bi₂S₃, and Sb₂Se₃, respectively. We found that the orthorhombic Pnma phase is the most energetically stable phase of Bi₂S₃ and Sb₂S₃ at all the pressures examined, as expected from experiments that obtained this phase both under ambient conditions and at high pressures. Surprisingly, however, our simulations predict that the R3̄m phase of Sb₂Se₃ is more stable than the Pnma phase below 4.8 GPa, indicating that both the Pnma and R3̄m phases are energetically competitive over this range. This in principle contradicts existing experimental studies on Sb₂Se₃ that have so far consistently obtained the Pnma phase under ambient conditions. We note, however, that the energy difference between the two phases is only 22.71 meV per f.u., which is lower than k_BT ≈ 25 meV at 300 K and which should therefore make it accessible under ambient conditions.

Fig. 2 Calculated relative enthalpy vs. pressure up to 10 GPa for the Pnma and R3m phases (c.f.Fig. 1) of Sb₂S₃ (a), Bi₂S₃ (b), and Sb₂Se₃ (c) relative to the predicted lowest-energy phase at ambient pressure, viz. the Pnma phases of Sb₂S₃ and Bi₂S₃ and the R3̄m phase of Sb₂Se₃.

In order to rule out the possibility that these results could be due to inaccuracies with the PBEsol XC functional, we also computed the enthalpies of the two phases with two additional functionals, namely the LDA³⁷ and the PBE-D2 method.³⁸ The results of these calculations are presented as an Appendix (Section 5.1). The unit-cell volumes of the Pnma phases obtained with the two functionals are underestimated and overestimated with respect to the PBEsol results shown in Table 1, as expected. For PBEsol, the two additional functionals predict that at ambient pressure the R3̄m phase of Sb₂Se₃ is more energetically stable than the Pnma phase, and remains so up to ∼6 GPa (LDA) and 8 GPa (PBE-D2). On the other hand, both functionals predict that the Pnma phases of Sb₂S₃ and Bi₂S₃ are more stable than the R3̄m phase up to at least 10 GPa.

For the Sb₂Se₃ structure, we have also investigated the effect of SoC on the energy differences between the two phases. With SoC, the R3̄m phase is also predicted to be more stable than the Pnma phase with an energy difference of around 27.38 meV per f.u., a difference of around 4.67 meV per f.u. We are therefore confident that our results are not affected by the inclusion of relativistic effects.

3.3 Energetic stability of the low-pressure Pnma and R3̄m phases of Sb₂Se₃ at finite temperature

The calculations in the previous section were performed at 0 K without taking into consideration the contributions to the free energy from crystal vibrations (phonons). In order to probe whether these effects could alter the energy ordering between the Pnma and R3̄m phases of Sb₂Se₃, we performed lattice-dynamics calculations on the equilibrium and compressed structures to evaluate the constant-volume Helmholtz and constant-pressure Gibbs free energies at zero pressure (F/G; Fig. 3).

Fig. 3 Constant-volume Helmholtz (top) and Gibbs (bottom) free energies of the Pnma phase of Sb₂Se₃ (black, solid line) relative to the R3̄m phase (red, dashed line) as a function of temperature at zero applied pressure.

F is obtained by summing the lattice energy (here the DFT total energy) and the vibrational contributions to the internal energy and entropy from the zero-point atomic motion and thermal population of the harmonic phonon energy levels.⁷⁰ As shown in Fig. 3, F predicts that at zero pressure the R3̄m phase remains the most stable across the temperature range examined with no crossing of the free energy to suggest a transition to the Pnma phase. At 0 K, the difference in F between the two phases is 27.24 meV per f.u., which is ∼4.53 meV higher than the difference in H due to the addition of the zero-point energy (i.e. the differences in the phonon frequencies selectively stabilise the R3̄m phase). At 300 K, the difference between the phases shows a negligible increase from 0.11 meV per F.U. to 27.35 meV.

Another factor that can influence the ordering of two competing phases is thermal expansion. A variation of the lattice volume due to thermal expansion/contraction impacts both the lattice energy and the phonon contributions to the free energy. This can be accounted for through the quasi-harmonic approximation (QHA) where the thermal expansion of the lattice is predicted from the volume dependence of the lattice energy, phonon frequency spectrum and phonon free energy.^70,71 The free energy F is computed for a series of unit-cell volumes and the equilibrium volume and Gibbs free energy G at a finite temperature T are obtained by minimizing F for a given (constant) pressure. Fig. 3 shows the difference in G between the Pnma and R3̄m phases of Sb₂Se₃. When taking into account the thermal expansion, the R3̄m phase still remains stable phase with respect to the Pnma phase from 0–1000 K with a similar energy difference of 29.43 meV at 0 K to that predicted using the constant-volume F, and a slightly smaller difference of 26.96 meV at 300 K.

To investigate the effect of pressure on the free energy, we also computed the difference in the Gibbs energy between the two phases at applied pressures from 0–5 GPa (Fig. 4). For p = 3 and 4 GPa, these calculations predict the R3̄m phase to be the most energetically stable phase across the 0–1000 K temperature range examined, but it can be clearly seen that pressure reduces the energy differences between the two phases. At 4 GPa, the smallest energy difference between the two phases of ∼3.25 meV is predicted to occur between 650 and 700 K. Increasing the pressure slightly to 4.2 GPa results in the energies of the two phases becoming nearly equal at around 400 K, and at 4.3 GPa a phase transition from R3̄m to Pnma is predicted to occur around this temperature. At 4.5 GPa, the predicted transition temperature decreases to ∼200 K, and at 5 GPa, the Pnma phase becomes the most energetically favorable structure across the entire temperature range examined.

Fig. 4 Gibbs free energies of the Pnma phase of Sb₂Se₃ (black, solid line) relative to the R3̄m phase (red, dashed line) as a function of temperature at applied pressures from 3–5 GPa.

In summary, free-energy calculations including phonon contributions and thermal expansion at zero applied pressure also predict that the R3̄m phase of Sb₂Se₃ is more stable than the Pnma phase. Interestingly, at most temperatures both F and G predict an increase in the energy difference, from which we infer that phonon contributions to the free energy selectively stabilise the R3̄m phase, at least at zero applied pressure. At low finite pressures between 4.2 and 4.4 GPa, the Gibbs free energies predict a pressure-induced transition between the R3̄m and Pnma phases near room temperature, and predict the Pnma phase to be stable above 5 GPa, which is similar to the 4.8 GPa transition pressure predicted from the 0 K enthalpies without zero-point energy corrections (c.f. Fig. 2). The small decrease in the predicted transition pressure can be attributed to the different impacts of volume changes on the phonon spectra of the two phases.

3.4 Dynamical stability of the Pnma and R3̄m phases of Sb₂Se₃

Energetic stability is a necessary but not sufficient condition for a structural phase to be synthetically accessible. A second criterion is that one should also confirm the dynamical stability of the system, which can be performed by studying the phonon frequency spectrum. If imaginary frequencies are present in the phonon dispersion, this is an indication that the system is not a minimum on the structural potential-energy surface (and is instead e.g. a transition state or a hilltop in multidimensional space), and would spontaneously convert to a lower-energy structure and thus be kinetically unstable under a given set of conditions.^72–77

We therefore investigated the dynamical stability of the R3̄m phase of Sb₂Se₃ to confirm whether this structure could potentially be synthesized under or close to ambient conditions. To do so, we evaluated the phonon band dispersion and density of states (DoS) curves of the R3̄m phase at zero pressure and different temperatures using the QHA method described in the previous section (Fig. 5). For comparison, we also present the phonon band structure and DoS of the Pnma phase. Both structures show real frequencies across the whole of the Brillouin zone, indicating that both are dynamically stable under ambient conditions (i.e. 0 GPa and room temperature) and confirming that, as implied by the energetics calculations, both phases should be accessible under appropriate synthesis conditions.

Fig. 5 Quasi-harmonic phonon dispersion curves for Pnma (top) and R3̄m Sb₂Se₃ (bottom). The color gradient runs from blue (low T) to red (high T) for temperatures between ∼0 and 1000 K.

In this context, we note that these results are consistent with a recent theoretical study of the R3̄m phase of Sb₂Se₃, which reported phonon dispersion curves and confirmed the dynamical stability of this phase at 0 GPa and 0 K.²³ This study also reported formation energies as evidence for the stability of the R3̄m phase. In addition, ab initio molecular dynamics calculations also confirmed that the R3̄m phase remains structurally stable at 300 K.²³

3.5 Mechanical stability of the Pnma and R3̄m phases of Sb₂Se₃

Having established the energetic and dynamical stabilities of the phase R3̄m of Sb₂Se₃ at zero pressure, we also checked the mechanical (elastic) stability as a third condition that must be fulfilled for this system to be synthetically accessible. This is achieved by calculating the elastic constants and confirming that they obey the Born stability criteria when the solid is subjected to homogeneous deformations.^72,78,79Table 2 lists the calculated elastic constants of the Pnma and R3̄m phases of Sb₂Se₃ at zero pressure.

Table 2 Calculated elastic constants c_ij (GPa) of the Pnma and R3̄m phases of Sb₂Se₃ at zero pressure

	c 11	c 22	c 33	c 12	c 13	c 23	c 44	c 55	c 66
Pnma	30.92	81.65	55.20	17.32	15.10	26.89	17.83	25.21	7.69

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	c 11 = c22	c 33	c 12	c 13 = c23	c 15 = −c25 = c46	c 44 = c55	c 66
R3̄m	90.81	40.21	25.85	21.61	−12.00	25.10	32.48

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We also computed the linear compressibility of both phases using ELATE analysis tools.⁴⁸ For both phases, only directions corresponding to positive linear compressibilities were obtained, indicating that both phases are mechanically stable under ambient conditions. In the case of the R3̄m phase, we obtained linear compressibilities between β_min = 4.9 TPa⁻¹ (hexagonal a-axis) and β_max = 19.5 TPa⁻¹ (hexagonal c-axis) with an anisotropy value of 3.95. For the Pnma phase, the compressibilities fall between β_min = 3.7 TPa⁻¹ (b-axis) and β_max = 25.7 TPa⁻¹ (a-axis) with an anisotropy of 6.87. These values are of the same order as the experimental axial compressibilities of the Pnma phase of Sb₂Se₃ (β_a = 15.2 TPa⁻¹, β_b = 3.9 TPa⁻¹, and β_c = 8.3 TPa⁻¹).⁶²Table 3 summarises the elastic moduli calculated from the elastic constants, and we note that the bulk modulus of the Pnma phase of Sb₂Se₃ (31.8 GPa) is similar to that obtained from the Birch–Murnaghan fit (c.f.Table 1), which confirms that the elastic constants are adequately converged.

Table 3 Calculated elastic properties of the Pnma and R3̄m phases of Sb₂Se₃ at zero pressure obtained within the Voigt approximation using the ELATE analysis tool: bulk modulus, B₀, Young's modulus, E, shear modulus, G and Poisson's ratio, υ

	B 0 (GPa)	E (GPa)	G (GPa)	υ
Pnma	31.82	44.10	17.38	0.27
R3̄m	40.00	65.56	26.72	0.23

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The calculated elastic constants in Table 2 fulfill the necessary and sufficient Born criteria for the mechanical stability of orthorhombic (eqn (1)) and rhombohedral (eqn (2)) systems.⁷⁹ The calculated elastic constants therefore indicate that both the R3̄m and Pnma phases of Sb₂Se₃ are mechanically stable under ambient conditions.

c₁₁, c₄₄, c₅₅, c₆₆ > 0; c₁₁c₂₂ > c²₁₂; c₁₁c₂₂c₃₃ + 2c₁₂c₁₃c₂₃ − c₁₁c²₂₃ − c₂₂c²₁₃ − c₃₃c²₁₂ > 0 (1)

The elastic tensors of the Pnma phase of Sb₂Se₃ have previously been calculated.⁴⁹ However, in these calculations, the components were overestimated as clearly shown by the comparison of the bulk modulus with experiments. Part of the disagreement could be due to the low cut-off energy used in these calculations.

In summary, our calculations on Sb₂Se₃ demonstrate that the Pnma and R3̄m phases are energetically competitive under ambient conditions and that both phases are dynamically and mechanically stable. Given that it should in principle be possible to obtain the R3̄m phase, but it has yet to be reported experimentally, it may be that the R3̄m phase is difficult to form kinetically, i.e. that the Pnma phase is formed faster than the R3̄m phase under typical synthesis conditions. We note that the R3̄m phase did not appear on the pressure/temperature phase diagram prepared by Pfeiffer et al.,⁸⁰ although this study did not attempt to vary the synthesis conditions at close to ambient pressure, which the present calculations suggest would allow this phase to be formed. Since our calculations increase the possibility that the R3̄m phase could potentially be prepared under slightly non-equilibrium conditions, to help identify R3̄m Sb₂Se₃ in future experiments we provide in the following section a reference structure and vibrational spectra.

3.6 Crystal structure and vibrational spectra of R3̄m Sb₂Se₃

Table 4 lists the predicted equilibrium crystal structure (lattice parameters and atomic positions) of R3̄m Sb₂Se₃ obtained using the PBEsol XC functional. (The calculations with PBE + D2 and LDA yield slightly different lattice parameters, viz. PBE + D2 – a₀ = 4.02 Å, c₀ = 28.81 Å and V₀ = 403.89 ų; and LDA – a₀ = 3.99 Å, c₀ = 27.58 Å, and V₀ = 381.61 ų.) The predicted lattice parameters are in good agreement with those reported for other tetradymite-like sesquichalcogenides.⁸¹ The optimised a₀ and c₀ are slightly smaller than those of Bi₂Se₃ and much smaller than those of Sb₂Te₃.⁸¹

Table 4 Predicted equilibrium lattice parameters and atomic positions for the hexagonal unit cell of the R3̄m phase of Sb₂Se₃

	c 0 (Å)	V 0 (Å3)	B 0 (GPa)
4.01	28.16	392.16	50.56	4.16

---

	Site	Sym.	x	y	Z
Sb1	6c	3m	0.00000	0.00000	0.60082
Se1	3a	−3m	0.00000	0.00000	0.00000
Se2	6c	3m	0.00000	0.00000	0.78792

---

We have also computed the Raman and infrared (IR) spectra of the equilibrium structure to provide spectral signatures that could be used to identify the R3̄m phase in experiments using routine characterisation techniques (Fig. 6).

Fig. 6 Simulated Raman (top) and infrared (IR; bottom) spectra of equilibrium R3̄m Sb₂Se₃. Spectral lines have been broadened with the calculated intrinsic mode linewidths at 300 K.

The frequencies, irreducible representations and IR/Raman intensities associated with each of the zone-centre (Γ-point) vibrational modes are listed in Table 5.

Table 5 Calculated vibrational modes of the R3̄m phase of the equilibrium structure of Sb₂Se₃. The three acoustic modes span an irreducible representations of Γ_acoustic = A_2u + E_u, and the remaining 12 optic modes span a representation of Γ_optic = 2E_g (Raman) + 2A_1g (Raman) + 2E_u (IR) + 2A_2u (IR)

	Frequency (cm^−1)	Raman intensity (10^5 Å^4 amu^−1)	IR intensity (e^2 amu^−1)	Spectral linewidth (cm^−1)
E g	53.3	0.02	Inactive	1.02
A 1g	83.7	0.37	Inactive	2.11
E u	86.6	Inactive	3.47	4.32
E u	131.7	Inactive	0.03	3.13
E g	139.4	3.32	Inactive	1.89
A 2u	145.1	Inactive	0.18	2.94
A 2u	182.8	Inactive	0.29	5.38
A 1g	203.8	1.02	Inactive	2.16

---

The inversion symmetry in the R3̄m structure leads to a mutual exclusion between the IR and Raman activities of the modes, with each spectrum being characterized by four bands.⁸¹ The most intense Raman band occurs at around 139 cm⁻¹ (E_g), while a second prominent feature is predicted at ∼204 cm⁻¹ (A_1g). The frequency of this A_1g mode is higher than that in Bi₂Se₃ but lower than that in In₂Se₃, as expected from the difference in mass between In, Sb and Bi.^82,83 Lower-frequency E_g and A_1g modes with much lower intensities are also found around 53 and 84 cm⁻¹, respectively, which are again slightly blue-shifted when compared to the corresponding frequencies calculated for Bi₂Se₃.⁸² There are four IR-active modes, two with E_1u symmetry (87 and 132 cm⁻¹) and two with A_2u symmetry (145 and 183 cm⁻¹). Of these, the 87 cm⁻¹ mode is the most prominent in the spectrum, while the second E_u mode at 132 cm⁻¹ is comparatively weak. The two A_2u bands have moderate and comparable intensities and form a pair of smaller features at higher frequencies. As expected, given the mass difference, the IR-active modes in Sb₂Se₃ again have slightly higher frequencies than those calculated for Bi₂Se₃.⁸²

It is also worth comparing our predicted structural and vibrational properties of R3̄m Sb₂Se₃ to the published works pertaining to the possible synthesis of this phase. In 2013, Bera et al.²⁵ claimed to have observed the R3̄m phase of Sb₂Se₃ at room temperature. However, the experimental lattice parameters were not disclosed and the reported Raman spectrum is not consistent with our theoretical spectrum, although our spectrum is consistent with simulations performed in the same study. We also note that our calculations of the pressure coefficients for the Raman- and IR-active modes of tetradymite Sb₂Se₃ (not shown) indicate that all the modes should show positive pressure coefficients, which is again consistent with the theoretical results shown in ref. 25.

If we consider that the band gap of tetradymite-like Sb₂Se₃ is similar to or even larger than that of tetradymite-like Bi₂Se₃, we would expect the Raman spectrum of tetradymite-like Sb₂Se₃ to be similar. We may therefore conclude that the experimental Raman spectra reported in ref. 25 are not consistent with tetradymite-like Sb₂Se₃. In this vein, we note that the appearance of soft Raman modes with a negative pressure coefficient at low pressures as reported in ref. 25 is similar to those of Se and Te nano- or micro-clusters.^84–87 Such features are either formed during synthesis or induced by laser heating in Raman scattering measurements at high laser powers, as has recently been discussed.⁸⁷

Recently, Matetskiy et al.³² claimed to have observed the tetradymite-like phase of Sb₂Se₃ in MBE-deposited layers over a thick buffer layer of tetradymite-like Bi₂Se₃. This study reported a lattice parameter of a₀ = 4.048 Å for quintuple layers of tetradymite-like Sb₂Se₃. The quintuple layers were observed by scanning-tunnelling microscopy (STM) to be of ∼1 nm in thickness, meaning that the c₀ lattice parameter, corresponding to three quintuple layers, would be ∼30 Å. Both the reported a₀ and c₀ lattice parameters are consistent with our calculated values. Furthermore, the electron dispersion obtained from angle-resolved photoemission spectroscopy (ARPES) is compatible with the R3̄m phase. It is therefore highly probable that tetradymite-like Sb₂Se₃ was synthesized for the first time in this study. Unfortunately, this study did not report vibrational spectra that would allow for additional comparison to our predictions.

3.7 On the observation of the R3̄m phase of A₂X₃ sesquichalcogenides

An analysis of bonding in layered materials under ambient conditions, including tetradymite-like B^V₂X^VI₃ and A^VIB^V₂X^VI₄ chalcogenides and also transition metal dichalcogenides (TMDs), has recently been performed by plotting the van der Waals (vdW) gap spacing vs. the X–X plane spacing.⁸⁸ The vdW gap spacing is defined as the interplanar distance across the interlayer space, whereas the X–X plane distance is the interplanar width across the intralayer space.^88,89 In ref. 88, it was concluded from the strong correlation that TMDs show a pure interlayer vdW interaction, whereas the tetradymite-like chalcogenides do not because the vdW gap spacing is much smaller than expected for their X–X plane spacing, which is indicative of a stronger interlayer interaction than that expected for pure vdW interactions.

It has recently been suggested that the vdW gap spacings in tetradymite-like group-15 sesquichalcogenides are smaller than those found for pure vdW materials, e.g. GaSe, InSe and TMDs. This has been attributed to the presence of the extra delocalized electrons between the layers that contribute an electrostatic component to the bonding that is not present in pure vdW materials.^24,90 These delocalized electrons arise from a new type of bonding, termed “metavalent bonding”, between the cations and anions inside the quintuple layers of the tetradymite-like structure, which has been demonstrated by the observation of a net charge difference between the layers in group-15 sesquichalcogenides⁹¹ and through a topological study of the electronic charge density of SnSb₂Te₄.⁹² In essence, the metavalent bonding results in the cations and anions providing extra electrons to the space between the layers, which in turn supports a stronger interlayer interaction.

In Fig. 7, we compare experimental (where available) and theoretical vdW gap spacings against X–X plane spacings for the full set of tetradymite-like B^V₂X^VI₃ (B = As, Sb, Bi; X = S, Se, Te) sesquichalcogenides.^88,89 (The theoretical results are obtained with PBEsol and dispersion-corrected PBE.)

Fig. 7 Size of van der Waals gap vs. chalcogenide X–X plane spacing in the R3̄m phases of A₂X₃ sesquichalcogenides (A = As, Sb, Bi and X = Te, Se, S). The data obtained from our theoretical calculations using dispersion-corrected PBE are shown as circles with crosses. For comparison, we also show theoretical PBEsol calculations on α-Bi₂Se₃, α-Bi₂Te₃ and α-Sb₂Te₃ from ref. 89 (crosses) and experimental measurements on β-As₂Te₃, α-Bi₂Se₃, α-Bi₂Te₃ and α-Sb₂Te₃ from ref. 93–96 (dots).

Fig. 7 shows that the compounds usually synthesized in the tetradymite-like structure, viz. β-As₂Te₃, α-Sb₂Te₃, α-Bi₂Te₃ and α-Bi₂Se₃, have a very small vdW gap spacings (typically below 2.9 Å) and large X–X plane spacings (typically above 3.4 Å). It is particularly striking that the Te–Te plane spacing in β-As₂Te₃ is below 3.5 Å, which is very different to other Te sesquichalcogenides and close to the Se–Se plane spacings in Se sesquichalcogenides such as Bi₂Se₃. From this analysis, we can therefore inferr that all the compounds with a stable tetradymite-like structure under ambient conditions (see Fig. 5 of ref. 88) have relatively small vdW gap spacings and X–X plane spacings between 3.4 and 3.8 Å. Since As₂Se₃ and Sb₂Se₃ do not usually crystallize in the tetradymite-like structure, it can similarly be inferred that X–X plane spacings below 3.4 Å and vdW gap spacings above 2.9 Å are outside the limit of stability for tetradymite structures of the B^V₂X^VI₃ and A^VIB^V₂X^VI₄ chalcogenides. This is consistent with: (i) the locations of α-Bi₂Se₃ and β-As₂Te₃ close to the lower limit of the X–X plane spacing; (ii) the metastable tetradymite-like phase of β-As₂Te₃ being obtained under ambient conditions; and (iii) the metastable orthorhombic Pnma phase of Bi₂Se₃ (guanajuatite) being obtained under ambient conditions.^17,97–99 The latter is isostructural to Bi₂S₃, Sb₂S₃ and Sb₂Se₃ and has been observed in high-pressure studies on α-Bi₂Se₃ (paraguanajuatite) on decreasing the pressure.^54,69

Most importantly, the vdW gap spacings and X–X plane spacings calculated for Sb₂Se₃ with PBEsol and dispersion-corrected PBE are similar to those obtained for other group-15 tetradymite-like sesquichalcogenides. This lends further support to the idea that the R3̄m phase of Sb₂Se₃ should be synthetically accessible with a structure similar to that predicted in our calculations. Unfortunately, despite the fact that R3̄m Sb₂Se₃ does appear to have been synthesized recently,^32,88 a complete structural characterization has yet to be reported and therefore several of our calculated parameters cannot be compared to experimental measurements. We thus hope that the present study will stimulate further attempts to prepare and characterise the R3̄m phase of Sb₂Se₃.

Finally, the analysis in Fig. 7 shows that all the group-15 sesquisulphides for which the theoretical R3̄m structures have X–X plane spacings below 3.4 Å do not seem to be stable in this phase under ambient conditions according to our calculations. This is in good agreement with the high predicted enthalpies of these phases of Sb₂S₃ and Bi₂S₃. However, this does not necessarily mean that the R3̄m phase of these compounds cannot be synthesized, as R3̄m As₂Se₃ can be prepared at high pressure and high temperature, which indicates that the tetradymite-like structure of As₂Se₃ is metastable under ambient conditions.^80,100–102 It is therefore possible that tetradymite-like As₂S₃, Sb₂S₃ and Bi₂S₃ could also potentially be synthesized under the right conditions.

4 Conclusions

We have performed a comparative theoretical study of the Pnma and R3̄m phases of Sb₂S₃, Bi₂S₃, and Sb₂Se₃ at applied pressures of up to 10 GPa. Our calculations predict that at ambient pressure the R3̄m (tetradymite-like) phase of Sb₂Se₃ is energetically more stable than the Pnma phase, in contrast to Sb₂S₃ and Bi₂S₃. This result contradicts the fact that all three compounds are usually grown on the Pnma phase. Further energetic studies of both Sb₂Se₃ phases show that the higher energetic stability of the R3̄m phase with respect to the Pnma phase is predicted by three different XC functionals and is unaffected by the phonon contributions to the free energy and thermal expansion at finite temperature. Lattice dynamics and elastic tensor calculations further show that both the phases of Sb₂Se₃ are dynamically and mechanically stable at zero pressure. Our calculations therefore suggest that the formation of this phase should be feasible under close to ambient conditions.

To aid in its identification, we have provided a theoretical crystal structure and predicted IR and Raman spectra. We have also discussed our results against the only two published works, to the best of our knowledge, that have claimed to have synthesized tetradymite-like Sb₂Se₃, and concluded that there is a high probability that this phase has recently been synthesized by molecular beam epitaxy on a thick buffer layer of tetradymite-like Bi₂Se₃.

Finally, we have discussed the stability of the R3̄m structure for all group-15 sesquichalcogenides by comparing the vdW gap and X–X plane spacings, which again suggests that the R3̄m phase of Sb₂Se₃ should be synthetically accessible. We hope that this work will stimulate further investigation of tetradymite-like Sb₂Se₃ and the corresponding phases of As₂Se₃ and other sesquichalcogenides. These sesquichalcogenide phases could potentially show topological properties interesting for spintronics and quantum computation at moderate pressures, and also be of interest as phase change materials and highly-efficient thermoelectric materials, and for photonic devices.

5 Appendices

5.1 Enthalpy vs. pressure curves obtained using different exchange–correlation functionals

In order to verify that the variation of the enthalpy differences between the Pnma and R3̄m phases of the three sesquichalcogenides with pressure presented in Fig. 2 was not due to an issue with the PBEsol functional, we performed similar calculations using dispersion-corrected PBE (PBE-D2; Fig. 8) and the LDA (Fig. 9). For Sb₂Se₃ both functionals also predict the R3̄m phase to be the most energetically stable under ambient conditions. There is however some variation in the predicted transition pressure, with PBEsol and the LDA (Fig. 9) predicting similar pressures but PBE-D2 predicting a slightly higher transition point (Fig. 8).

Fig. 8 Calculated relative enthalpy vs. pressure up to 10 GPa for the Pnma and R3̄m phases of Sb₂S₃ (a), Bi₂S₃ (b) and Sb₂Se₃ (c) relative to the lowest-energy phases at ambient pressure, viz. the Pnma phases of Sb₂S₃ and Bi₂S₃ and the R3̄m phase of Sb₂Se₃, obtained using PBE-D2.

Fig. 9 Calculated enthalpy vs. pressure up to 10 GPa for the Pnma and R3̄m phases of Sb₂S₃ (a), Bi₂S₃ (b) and Sb₂Se₃ (c) relative to the lowest-energy phase at ambient pressure, viz. the Pnma phases of Sb₂S₃ and Bi₂S₃ and the R3̄m phase of Sb₂Se₃, obtained using the LDA.

5.2 Electronic properties of the R3̄m structural phase of Sb₂Se₃

Fig. 10 shows the electronic band structure and atom- and orbital-projected (partial) density of states of the R3̄m phase of Sb₂Se₃. We have considered spin–orbit coupling effects in this calculation due to their importance in determining the electronic properties of many potential topological insulators. The valence band maximum (VBM) is mostly composed of Se p states, whereas the conduction band minimum (CBM) is a hybridization of Se and Sb p states with a small contribution from Se s states. The band gap is direct and located along the Γ–Z segment of the dispersion in the vicinity of the Γ point. Both the VBM and CBM show a “Mexican-hat” feature - indentation of the band dispersion - which indicates that our DFT+SoC calculations predict the R3̄m phase of Sb₂Se₃ to be a topological insulator at ambient pressure. Sb₂Se₃ differs from similar topological insulators such as Bi₂Te₃¹⁰⁴ and Bi₂Se₃¹⁰⁵ for which at zero pressure this feature only occurs in the VBM but at moderate pressures the CBM also starts to develop a “Mexican-hat” feature. The calculated PBEsol band gap is around 0.23 eV. From a parabolic fitting of the band edges,¹⁰³ we obtain a hole effective mass m_h = −0.107 and an electron effective mass m_e = 0.098.

Fig. 10 Electronic band structure (left panel) and partial density of states (right panel) of the R3̄m phase of Sb₂Se₃ obtained using PBEsol and including spin–orbit coupling (SoC) effects. The figure was plotted using the Python toolkit SUMO.¹⁰³

Author contributions

E. L. d. S., M. C. S., J. M. S., P. R. H. and A. M. performed the ab initio calculations. E. L. d. S., M. C. S., J. M. S., P. R. H., A. M., D. M. G., R. V., and F. J. M. contributed to the interpretation and discussion of results. F. J. M. supervised the project. E. L. d. S., M. C. S., J. M. S., P. R. H., A. M., D. M. G., R. V., and F. J. M. contributed equally to the discussion and drafting of the paper. All authors have read and agreed to the published version of the manuscript.

Data availability

Raw data from this study can be obtained from the corresponding author on reasonable request.

License statement

For the purpose of open access, the author has applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This publication is part of the MALTA Consolider Team network (RED2018-102612-T) (MINECO/AEI/10.13039/501100003329), and is supported by I + D + i project PID2019-106383GB41/42/43 (MCIN/AEI/10.13039/501100011033), by the PROMETEO/2018/123(EFIMAT) and CIPROM/2021/075 (GREENMAT) projects (Generalitat Valenciana), and by the European Union Horizon 2020 research and innovation programme under a Marie Sklodowska-Curie grant agreement (785789-COMEX). E. L. d. S., A. M., and P. R.-H. acknowledge computing time provided on the MALTA-Cluster at the University of Oviedo and on the MareNostrum facility through Red Española de Supercomputación (RES) with technical support provided by the Barcelona Supercomputing Center (QCM-2018-3-0032). E. L. d. S. also acknowledges the Network of Extreme Conditions Laboratories (NECL), financed by FCT and co-financed by NORTE 2020 through the Portugal 2020 and FEDER programmes. J. M. S. is grateful to UK Research and Innovation for the support of a Future Leaders Fellowship (MR/T043121/1) and to the University of Manchester for the previous support of a Presidential Fellowship.

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