Theoretical study of phase transitions in Sb2S3, Bi2S3 and Sb2Se3 under compression
E. Lora da Silva, J. M. Skelton, P. Rodríguez-Hernández, A. Muñoz, D. Martínez-García, F. J. Manjón
TTheoretical study of phase transitions in Sb S , Bi S and Sb Se under compression E. Lora da Silva, ∗ J. M. Skelton, P. Rodríguez-Hernández, A. Muñoz, D. Martínez-García, and F. J. Manjón Instituto de Diseño para la Fabricación y Producción Automatizada,MALTA Consolider Team, Universitat Politècnica de València, València, Spain School of Chemistry, University of Manchester,Oxford Road, Manchester M13 9PL, United Kingdom Departamento de Física, Instituto de Materiales y Nanotecnología,MALTA Consolider Team, Universidad de La Laguna, Tenerife, Spain Departamento de Física Aplicada - ICMUV,MALTA Consolider Team, Universitat de València (Dated: November 19, 2019)
Abstract
We report a theoretical study of Sb S , Sb Se and Bi S sesquichalcogenides at hydrostatic pressures up to 60 GPa. Weexplore the possibility that the R-3m , C2/m , C2/c and the disordered
Im-3m phases observed in sesquichalcogenides withheavier cations, viz. Bi Se , Bi Te and Sb Te , could also be formed in Sb S , Sb Se and Bi S , as suggested by recentexperiments. Our calculations show that the C2/m and
C2/c phases are energetically unstable for any of the three compoundsover the entire range of pressures examined. In contrast, the disordered bcc-like
Im-3m phase is energetically stable at highpressures; however, it is only for Sb Se that the disordered phase presents dynamical stability below 60 GPa. Our calculationsfurther show that at ambient pressure the Pnma phase is the most energetically favourable for Sb S and Bi S whereas,and surprisingly, for Sb Se it is the R-3m phase which presents the lowest enthalpy energy at 0 GPa, in contradiction toexperimental evidence. From lattice dynamics and elastic tensor calculations we observe that both
Pnma and
R-3m phases aredynamically and mechanically stable at 0 GPa. These results suggest that the formation of the
R-3m phase for Sb Se couldbe feasible at close to ambient conditions. Furthermore, and to aid the identification of this phase, we provide the theoreticalcrystal structure (lattice and atomic parameters) and complete infrared and Raman spectra. PACS numbers:
I. INTRODUCTION
Since the identification of the trigonal tetradymite-like
R-3m phases of group-15 sesquichalcogenides (i.e.Sb Te , Bi Se , Bi Te ) as 3D topological insulators, the family of A X sesquichalcogenides has attracted agreat deal of attention from the scientific community.Three-dimensional topological insulators represent a newclass of matter, with insulating bulk electronic statesand topologically-protected metallic surface states due totime-reversal symmetry and strong spin-orbit coupling,and present potential interest for spintronics and quan-tum computing applications. Due to this fundamen-tal interest and potential applications, identifying newtopological insulators and materials with superconduct-ing properties is currently among the most widely-studiedtopics in condensed matter science.Stibnite (Sb S ), bismuthinite (Bi S ) and antimon-selite (Sb Se ) minerals are also group-15 sesquichalco-genides; however, they do not crystallize at room con-ditions in the tetradymite-like structure, but in the or-thorhombic U S -type ( Pnma ) structure (Fig. 1.a).Sb S , Bi S and Sb Se are semiconductors with band-gap widths of 1.7, 1.3, and 1.2 eV, respectively. Thesematerials are used in a wide range of technological ap-plications including photovoltaic solar cells, X-ray com-puted tomography detectors, fuel cells, gas sensors andfor detection of biomolecules.
The
Pnma structure has been identified as a possiblepost-post-perovskite phase of (Mg,Fe)SiO minerals and of NaFeN at high pressure (HP). Thus, the study ofSb S , Bi S and Sb Se at HP could also provide usefulinformation about the HP behaviour of the ABO min-erals, which are found in the mantle of the Earth. In thiscontext, initial experimental HP studies of Sb S , Bi S and Sb Se have shown that the Pnma structure is sta-ble under compression, with first-order phase transitions(PTs) occurring around 50 GPa.
Curiously enough,recent HP studies have found that Sb Se becomes atopological superconductor at around 10 GPa and 2.5K, exhibiting highly conducting spin-polarized surfacestates similar to Bi Se . Moreover, three further studieshave suggested that several first- and second-order PTsoccur for Sb S up to 50 GPa. Furthermore, it hasalso been suggested that the HP phases of Sb S could besimilar to those observed for heavier sesquichalcogenidessuch as Bi Se , Bi Te and Sb Te . Therefore, it is ofinterest to examine and compare the stability of differ-ent structural phases (
R-3m , C2/m , C2/c and disordered
Im-3m ) observed for heavier cation sesquichalcogenides(i.e. Bi Se , Bi Te and Sb Te ) on our three miner-als of interest, Sb S , Bi S and Sb Se , and at differentpressure conditions.On the other hand, several theoretical studies per-formed on the R-3m structure of Sb Se have suggestedthat this phase should undergo a topological quantumphase transition under compression. In fact, it hasbeen reported that such a topological transition was ob-served experimentally at 2 GPa. Furthermore, recentcalculations suggest that the tetradymite-like Sb Se a r X i v : . [ c ond - m a t . m t r l - s c i ] N ov IG. 1: Images of the
Pnma (a),
R-3m (b),
C2/m (c),
C2/c (d), disordered bcc-type
Im-3m (e) A X sesquichalcogenidestructures (A = Sb, Bi; X = S, Se). The C2/m nine/ten-fold structure used to model the disordered
Im-3m phase is also shownfor comparison (f). The A cations and X anions are shown as brown and yellow spheres, respectively. structure becomes a topological insulator at ambientpressure. Consequently, HP studies performed on thesegroup-15 sesquichalcogenides are highly relevant to theresearch on topological states, therefore possible HPphases of these compounds should be thoroughly eval-uated. It is worthy of mentioning, that a recent workperformed on Bi S predicts the system to be unstableunder compression, and decomposing into another stoi-chiometric system In this work, we report theoretical simulations at 0 K ofthe
Pnma and hypothetical
R-3m , C2/m , C2/c and
Im-3m phases for Sb S , Sb Se and Bi S (Figure 1), with a view to assessing which, if any, are likely to fulfil eitherof the stability conditions under hydrostatic pressure. II. THEORETICAL METHODOLOGY
The structural properties of the different crystallinephases of Sb S , Bi S and Sb Se were calculated withinthe framework of density-functional theory (DFT). The Vienna
Ab-initio
Simulation Package (VASP) code was employed to perform simulations with the projec-tor augmented-wave (PAW) scheme including six valence2lectrons for S[3s ] and Se[4s ] and fifteen va-lence electrons for Sb[4d ] and Bi[5d ].Convergence of the total energy was achieved witha plane-wave kinetic-energy cut-off of 600 eV. Thegeneralized-gradient approximation (GGA) functionalwith the Perdew-Burke-Ernzerhof parameterization re-vised for solids (PBEsol), was used for all the calcu-lations.The Brillouin-zone (BZ) was sampled with Γ -centeredMonkhorst-Pack grids employing adequate meshes forthe different structural phases of the three compounds: Pnma - 6 × × R-3m - 12 × × C2/m - 6 × × C2/c - 10 × ×
8, and
Im-3m (using a
C2/m conventional cell) - 6 × × Im-3m phase is a body-centered cu-bic (bcc) disordered structure, and has been theoreti-cally predicted and experimentally found for Bi Te in2011. For sesquichacogenides with A X stoichiometry,the bcc lattice site (2a Wyckoff position) is randomly oc-cupied by 40% of A cations and 60% of X anions. Thismeans that such a structure is a disordered phase witha mixture of cations and anions randomly sharing thesame bcc crystallographic position and forming a A-Xsubstitutional alloy. Due to the theoretical difficulty insimulating the disordered
Im-3m structure, we have useda 9/10-fold
C2/m structure (formation of 9/10 chemicalA-X bonds), as was previously employed for Bi Te and Bi Se . Moreover it has been observed that the9/10-fold
C2/m structure presents a bcc-like structuralorder, in agreement with the observed XRD patterns; therefore giving support to employ the calculated inter-mediate bcc-like monoclinic
C2/m phase to confirm theexperimental presence of the disordered
Im-3m phase.Structural relaxations were performed by allowing theatomic positions and the unit-cell parameters to changeduring the ionic relaxation, at different volume values(compressions). From these we obtain the respectiveexternal pressure for the specific applied compression(isotropic volume compression) and hence respective setof crystal structures. The pressure-volume (P-V) curvesfor all the compounds were fitted to a third-order Birch-Murnaghan equation of state to obtain the equi-librium volume, bulk modulus and, respective pressurederivative. The enthalpy, H , curves were computed byconsidering the relation, H = E + pV , where E is thetotal 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 can provide insightsregarding the thermodynamic stability of each phase forincreasing pressure values, up until the studied pressurerange (60 GPa).Lattice-dynamics calculations were performed for theenergetically favourable polymorphs at different pressurevalues, namely the Pnma and
R-3m phases of Sb Se at room pressure, and for the disordered Im-3m phasesof all three compounds, as explained in detail in Secs.III B and III C. The phonon properties were computedby using the supercell finite-displacement method imple- mented in the Phonopy package with VASP used as theforce calculator. Supercells were expanded up to 2 × × Pnma phase, and 2 × × R-3m and disordered phases; to allow the exact calculation offrequencies at the zone center ( Γ ) and inequivalent zone-boundary wavevectors, which were then interpolated toobtain phonon-dispersion curves and density of states ona uniform 50 × × Γ -centered q -point mesh.To correct for the long-range Coulomb interaction(LO-TO splitting), a non-analytical correction, basedon the Born effective-charge tensors and the electronic-polarization component of the macroscopic static dielec-tric tensor, was applied. These quantities were ob-tained using the density-functional perturbation theory(DFPT) method implemented in VASP. Infrared (IR) and Raman spectra were calculated forthe ground-state
R-3m phase of the Sb Se structure byemploying the methods described in Ref. 45 and im-plemented in the Phonopy-Spectroscopy package. Thelinewidths were obtained by computing the third-orderforce constants and following the many-body perturba-tive approach described in detail in Refs. 45 and 47 andimplemented in the Phono3py software. Elastic tensors were computed to assess the mechani-cal stability of the two energetically favourable phases ofSb Se at 0 GPa, namely the Pnma and the
R-3m poly-morphs. Respective calculations were carried out by em-ploying the central-difference method, where the uniquecomponents of the elastic tensor are determined by per-forming six finite distortions of the lattice and derivingthe tensor elements from the strain-stress relationship. For these calculations, it was necessary to increase theplane-wave energy cutoff to converge the stress tensor ad-equatly, which was achieved by systematically increasingthe plane-wave cutoff up until 950 eV. We then furtheremployed the ELATE software to analyse the linearcompressibility using the computed stress tensors. III. RESULTS AND DISCUSSIONA. Structural properties of the
Pnma phase
The
Pnma phase of the A X structures are com-posed by weak stacking interactions which hold the lay-ers along the a -axis together, which description becomeschallenging for conventional DFT functionals. Wehave compared the equilibrium lattice parameters, bulkmoduli and pressure derivatives calculated for the
Pnma phases of Sb Se , Sb S and Bi S , with existing exper-imental and theoretical results found in literature (Tab.I), to verify the accuracy of our theoretical calculationsas a prior step before attempting the study of the HPphases.By observing Tab. I we may find that our calculatedlattice paramters of Sb Se , calculated at room pressure(a = 11.75 Å, b = 3.98 Å and c = 11.30 Å), arefound to be in good agreement with experimental values3 ABLE I: Calculated equilibrium lattice parameters (a , b and c ), equilibrium bulk moduli (B ) and pressure deriva-tives (B (cid:48) ), at 0 GPa, of the Pnma phase of Sb Se , Sb S and Bi S . Values are compared to experimental and othertheoretical results found in literature. Sb Se Sb S Bi S a (Å) a a a Theo. Exp. Theo. Exp. Theo. Exp. b f b b,j,k b p c g c l,m n q d h o e i b (Å) a a a Theo. Exp. Theo. Exp. Theo. Exp. b f c b,j,k b p c,e g h l,m n q d b,i o c (Å) a a a Theo. Exp. Theo. Exp. Theo. Exp. b f,g b b,j,l,m b p c,e c k n q d h o i V (Å ) a a a Theo. Exp. Theo. Exp. Theo. Exp. b f b b b p c g c i,m,j n q r s r k o q B (GPa) a a a Theo. Exp. Theo. Exp. Theo. Exp. c f b b n p s c j o q k q u t B (cid:48) a a a Theo. Exp. Theo. Exp. Theo. Exp. f b b q p s j o q k u t a This work, b Ref. 18, c Ref. 52, d Ref. 53, e Ref. 51, f Ref. 15, g Ref. 54, h Ref. 55, i Ref. 56, j Ref. 57, k Ref. 22, l Ref. 58, m Ref. 59, n Ref. 60, o Ref. 61, p Ref. 62, q Ref. 16, r Ref. 50, s Ref. 20, t Ref. 24, u Ref. 19 detailed in Refs. 15 and 54 (a = 11.80 Å, b = 3.97 Å,c = 11.65 Å and a =11.79 Å, b = 33.98 Å and c = 11.65 Å, respectively), and also with values obtainedfrom ab initio calculations found in literature. Wemust mention that the most notable deviation of our cal-culated values from experimental measurements is a ∼ parameter (not surprisingly leadingto an underestimation of the V of roughly ∼ obtainedfrom Ref. 53 of 11.70 Å is quite high when comparedto the present results, however closer to experimental re-sults; whereas the a parameter has a larger error whencompared to experimental values. This fact has to dowith the use of the GGA-PBE functional, which is wellknown to overestimate volumes (therefore overestimat-ing the volumes up to ∼ b -axis, the difference between theo-retical and experimental values are very small, since thisis the crystallographic direction where covalently bondedchains prevail.Similar results were obtained for our calculated latticeparameters of Sb S . As shown on Tab. I, the calcu-lated parameters for this compound agrees with exper-imental measurements as well as with other theo-retical results. We must note however, and asalready described for Sb Se , that the lattice parametersobtained from LDA calculations tend to underestimaterespective values, and therefore the discrepancy foundfor a presented from Refs. 52 and 56. The c parameteris however closer to our PBEsol results than the GGA-PBE values described in Ref. 55, which actually providesa better agreement to experimental results. Results obtained for the Bi S system is also consistentwith several data found in literature, both experimentalas well as theoretical. We note that calculations per-formed on Bi S by employing the Armiento and Matts-son 2005 parametrized GGA functional (AM05) seems to show a slightly better reproduction of c withrespect to experimental measurements. The last two rows of Tab. I show the calculated B and B (cid:48) results of the Pnma phase of the three com-pounds and respective comparison to experimental mea-surements and other calculations. The calculated valuesobtained by fitting the P-V curves of Sb Se to a third-order Birch-Murnaghan equation are B = 31.1 GPa (B (cid:48) = 6.6), which is close to the experimental values of B =30 GPa (B (cid:48) = 6.1) from Ref. 15 and B = 32.7 GPa (B (cid:48) = 5.6) from Ref. 61.For Sb S , we have obtained B = 31.5 GPa (B (cid:48) = 6.6),which is within the range of experimental values, and also with PAW-PBEsol calculations of Ref. 18.Moreover, these results are also close to those experi-mentally measured for the As-doped stibnite mineral. Finally, our values for Bi S result in B = 42.3 GPa(B (cid:48) = 6.8), which is consistent with DFT data andexperimental values reported in Refs. 62, 16 and 19. B. Energetic Stability
Since our calculations on the
Pnma phases were foundto be in good agreement with the overall data found in lit-erature, we have proceeded in carrying out a theoreticalstudy of the hypothetical
R-3m , C2/m , C2/c and disor-dered
Im-3m phases of Sb S , Bi S and Sb Se to probewhether such phases could be energetically competitiveunder hydrostatic pressure. Figs. 2a, 2b and 2c show thepressure-dependence of the enthalpy differences, relativeto the stable phase at ambient pressure, between the fiveabove mentioned phases of Sb S , Bi S and Sb Se , re-spectively. Values of the predicted transition pressuresbetween the different phases are summarized in Tab. II.From the enthalpy plots (Fig. 2) we may observe that:4 IG. 2: Calculated enthalpy vs pressure curves, for the dif-ferent possible phases (shown in Fig 1) of Sb S (a), Bi S (b)and Sb Se (c), relative to the lowest-energy phase at ambi-ent pressure: the Pnma phase for Sb S and Bi S and the R-3m phase for Sb Se . TABLE II: Theoretical estimation of the pressure-inducedphase transitions of R-3m → Pnma and
Pnma → disordered Im-3m for the Sb Se , Sb S and Bi S compounds (pre-sented in units of GPa). Sb Se Sb S Bi S R-3m → Pnma
Pnma → Im-3m
1. At 0 GPa the orthorhombic
Pnma phase is ener-getically stable for Bi S and Sb S ; however forSb Se it is the trigonal R-3m phase the mostfavourable phase at 0 GPa.2. The two monoclinic
C2/c and
C2/m phases do notbecome energetically competitive with the ground-state phase, over the range of pressures examined,in any of the three compounds.3. The bcc-like disordered
Im-3m structure, whichcan be understood as a disordered solid solutionof atoms, is the most energetically stable phase atpressures above 35, 30 and 21 GPa for Sb S , Bi S and Sb Se , respectively.With respect to the first point, referring to Bi S andSb S , our calculations predict that the Pnma struc-ture is energetically the most stable phase throughoutthe whole range of studied pressures, as expected fromexperimental evidences that show the observation of re-spective phase, both at ambient and at high pressure.Surprisingly, however, our simulations indicate that the
R-3m phase of Sb Se is the most stable phase at pres-sures below 4.8 GPa, being both Pnma and
R-3m phasesenergetically competitive between 0 and 4.8 GPa. Thisfeature contradicts the experimental findings of Sb Se consistently crystallizing to the Pnma phase at ambientconditions. We must note however that at 0 K and 0GPa, the energy difference between the two phases isonly 22.71 meV (per f.u.), which is lower than the ther-mal barrier (k B T ∼
25 meV at 300 K) required for thephase transition to occur under ambient conditions. Inorder to probe whether the vibrational contributions tothe free energy could alter the energy ordering betweenthe two phases, we have further plotted the free energies,where the entropy terms are obtained from our lattice dy-namics calculations, and which will be discussed in moredetail in Sec. III C.Regarding the second point, our analysis further showsthat for the three studied compunds, the two monoclinic
C2/c and
C2/m phases are never energetically compet-itive up until 60 GPa. These results are compatiblewith the fact that no phase transition had previouslybeen observed in Sb S , Bi S or Sb Se under compres-sion up to ∼
50 GPa.
However, three recent stud-ies have reported low-pressure phase transitions occuringfor Sb S . A transition to an unknown phase wasobserved around 15 GPa, and several other transi-5ions were also evidenced between 10 and 25 GPa, andtentatively proposed to be the
R-3m , C2/c and
C2/m structural phases. It must however be clarified that thislatter study applied an ethanol-methanol mixture as thepressure-transmitting medium, therefore there is a pos-sibility that the observed transitions could have been in-duced by non-hydrostatic pressure effects.Finally, as for the third point, from a thermodynamicpoint-of-view our results indicate that the bcc-like disor-dered
Im-3m phase, initially identified for Bi Se , Bi Te and Sb Te , seem to be energetically favourable atHP for our three materials of interest. These results areconsistent with the observation of such a phase at around50 GPa for Sb Se and above 25 GPa for Sb S . However, for the Bi S structure, our results do notagree with those found in Refs. 16 and 19, where disor-der has been observed above 50 GPa although attributedmostly to a pressure-induced amorphization. Moreover,a more recent work claims that Bi S is unstable above31.5 GPa, decomposing into a mixture of BiS and BiScompounds. In summary, the agreement of our results regarding theobservation of the disordered
Im-3m phase for Sb Se and Sb S , but not for Bi S , suggests that thermody-namic stability is not sufficient to explain the lack of theHP disordered phase for the latter compound. In the fol-lowing section we discuss the dynamical stability of the Im-3m phase as a function of pressure in order to providea deeper understanding regarding this question.
C. Dynamical Stability
Energetic stability is a necessary, but not sufficient con-dition for a structural phase to be synthetically accessi-ble. One should also probe the dynamical stability ofthe system, which requires the study of the phonon fre-quencies. If imaginary frequencies emerge (usually rep-resented by negative frequencies in the phonon disper-sion curves), this would indicate that the system is at apotential-energy maximum (transient state), undergoinga phase transition and thus cannot be kinetically stable atthe given temperature and/or pressure conditions.
In this section, we consider the phonon properties ofdifferent phases for the three compounds, which were ob-served to be energetically the most favourable (Fig. 2)at different pressure values, namely:1. The disordered
Im-3m phases of the three materi-ales at different pressure ranges.2. The
Pnma phase of the three systems at 50 GPa.3. The
Pnma and
R-3m phases of Sb Se at 0 GPa.
1. The Disordered BCC-Type Im-3m Phase
To assess the possibility of dynamical stability for thedisordered
Im-3m phases of the three compounds at HP, we have evaluated the phonon dispersion curves at pres-sure values of 30 GPa, which is close to the transitionpressures observed in Fig. 2; and at higher pressures of50 (Sb Se ) and 60 GPa (Sb S , Bi S ).As illustrated in Fig. 3, at 30 GPa all three disor-dered structures show negative modes along the disper-sion curves, thus indicating that these structures are dy-namically unstable at this pressure range.At 60 GPa the phonon dispersion curves of Sb S andBi S present imaginary modes (Fig. 3), indicating thatneither compound is likely to adopt this phase for pres-sures, at least until 60 GPa. We note however that thedynamical instabilities found for Sb S and Bi S bothdecrease (the negative modes shift to higher frequencyvalues, towards positive values) with increasing pressure,suggesting that these phases could in principle becomestable at pressures above 60 GPa. In this context, wemust note that Efthimiopoulos et al ., had observed apressure-induced amorphization above 50 GPa for Bi S ,however the authors were not able to identify the phaseto be the disordered Im-3m structure, even at 65 GPa.On the other hand, experimental data for Sb S , sug-gests that the disordered bcc-like phase exists between28.2 and 50.2 GPa. However, it must be noted, thatexperimental measurements detailed in Ref. 24 were car-ried out under non-hydrostatic behaviour, due to the em-ployed pressure-transmitting medium.Finally, our calculations suggest that Sb Se becomesdynamically stable already at 50 GPa; a result that is inagreement with the Im-3m phase being observed experi-mentally around 50 GPa. To close this point, we can speculate that the stabil-ity of the disordered solid solution of sesquichalcogenidesseems to be related to the size of cations and anions sincethe
Im-3m phase is consistently being observed at HPfor sesquichalcogenides with heavier cations and anions(Sb Se , Sb Te , Bi Se , and Bi Te ). It seems that thepossibility of occuring such a HP phase could be relatedto the radii size 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 inferthat the solid solutions are energetically favourable insesquichalcogenides if the atomic radii of the cation andanion differ by less than ∼ pm, or if the size ratiobetween them is smaller than 1.55 (case of Bi Se ). Itis thus likely that the disordered Im-3m phase of Sb S could indeed stabilize, because the radii difference be-tween r Sb and r S is 37 pm (141-104=37 pm) and the sizeratio is 1.35, and therefore within the above mentionedthresholds. However, the larger radius of Bi results in alarger radii difference (78 pm) and ratio (1.75) with re-spect to S, which could therefore evidence the instabilityof such a disordered phase for Bi S at HP.Moreover and as suggested in Ref. 36, the atomic radiibetween the anion and cation tends to become approxi-mately equal under pressure due to a higher probabilityof charge transfer from cation to anion. Therefore, HPinherently creates a favourable enviroment for the disor-6 IG. 3: Simulated phonon dispersion curves of the disordered bcc-like
Im-3m phases of Sb S (a), Bi S (b) and Sb Se (c),and calculated at 30 GPa (left) and 50 (Sb Se ) or 60 GPa (Sb S and Bi S ; right). The BZ q -vector description representsthe C2/m space-group, according to the symmetry of the employed cell. dered phase due to the decrease of the difference betweencation and anion atomic radii. Consequently, the transi-tion to the Bi S disordered solid solution could probablybe induced for very high pressure values, namely whenthe difference between the two radii decreases below 65pm and the ratio decreases below 1.55.
2. The Low-Pressure Pnma Phase at High-Pressure
In order to study the dynamical stability of the wellknown low-pressure
Pnma phase at HP, we present inFig. 4 the phonon dispersion curves of the respectivephase for Sb S , Bi S and Sb Se at 50 GPa.Curiously enough, we find that for Sb S and Bi S thesystem is still dynamically stable at 50 GPa, althoughthermodynamically it is not the most stable phase (Fig. 2). These results, together with the dynamical instabil-ity observed for the disordered phase of Sb S and Bi S at 50 GPa (Fig. 3), and the thermodymical instability ofthe C2/m and
C2/c phases, suggest that only the
Pnma structure should be observed at 50 GPa for both com-pounds; upto a plausible phase-transition should occurat higher pressures values.For Sb Se however, we note that at 50 GPa the Pnma structure starts presenting negative frequencies, localisedaround the zone-centre, therefore evidencing dynamicalunstability at the same pressure range where the disor-dered
Im-3m phase is already dynamically stable (Fig.3). Therefore, our dynamical and thermodynamical re-sults of Sb Se , clearly suggests that at HP a transitionfrom the Pnma phase to the disordered
Im-3m phase islikely to occur, in good agreement with experiment.7
IG. 4: Simulated phonon dispersion curves of the
Pnma phases of Sb S (a), Bi S (b) and Sb Se (c), calculated at50 GPa.
3. The Low-Pressure Phases of Sb Se By considering the enthalpy energies of the
Pnma vs R-3m phases of Sb Se (Fig. 2) we have shown that the R-3m phase is energetically the most favourable phaseup to ∼ Pnma phase of only 22.71 meV (per f.u.) at0 GPa.In order to verify if the entropy contributions to the
FIG. 5: Constant-volume Helmoltz (top) and Gibbs (bot-tom) free energies of the
R-3m phase (green) relative to the
Pnma phase (red) of Sb Se as a function of temperature. DFT total energies could affect the energetic stabilityfound for
R-3m with respect to
Pnma (the experimen-tally observed phase), we have evaluated the constant-volume (Helmholtz) Free energy at 0 GPa (Fig. 5, top).The Helmholtz Free energy ( F ) is obtained by summingthe lattice energy (DFT total energy) and the vibra-tional contributions from the population of the harmonicphonon energy levels. From Fig. 5 (top), we may observe that at 0 GPa the
R-3m phase is the most stable phase at any tempera-ture range, and no transitioning is observed to the
Pnma structure. At 0 K, the Free energy difference betweenthe two phases is 27.24 meV, which is ∼ G ) at a finite temperature is thus8btained by minimising the Helmoltz Free energy for agiven (constant) pressure. The theoretical background ofthe QHA is detailed in Refs. 75 and 76 and therefore willnot be extended in the present work.Fig. 5 (bottom) shows the difference of G between thetwo phases of interest. One may observe that by tak-ing into account the thermal expansion, the R-3m phasestill remains the most stable phase with respect to the
Pnma phase, with an energy difference of 29.43 meV at0 K (very similar behaviour to that obtained from F ).At room temperature (300 K) the energy difference be-tween the phases decreases slightly down to 26.96 meV.In summary, at room pressure, our Free energy resultsshow that R-3m is always more stable than
Pnma atany temperature. Neither for F nor for G do the dif-ferences decrease below the k B T limit, and therefore theenergy barrier is higher than that required for the phasetransition to be spontaneous given the available thermalenergy (which does not occur when the zero-point energyis not considered).We also present the G differences of Sb Se for pres-sure values between 0 and 5 GPa (Fig. 6), in order toanalyse the energetic ordering between the two phasesand probe if a pressure-induced phase transition couldbe observed as a function of temperature. We observethat the R-3m phase persists in the energetically stabil-ity at any temperature range up to 1000 K, for 3 and4 GPa. However we must note that the energy differ-ences between the two phases decreases considerably forincreasing pressures and at high temperature values. At4 GPa the lowest energy difference ( ∼ Pnma nearly reaching the energy of
R-3m at around400 K. In fact, at 4.3 GPa the phase transition from
R-3m to Pnma is observed around 400 K; at 4.5 GPa thetransition temperature decreases to ∼
200 K. Finally at5 GPa, the
Pnma phase becomes the most energeticallyfavourable structure for the Sb Se system.Based on the analysis from G , we conclude that thepressure-induced transition between the R-3m to Pnma is favoured at low pressures (between 4.2-4.4 GPa) nearroom temperature conditions. This conclusion is verysimilar to that evidenced from the enthalpy plots at 0K (Fig. 2) where the phase transition is predicted to bearound 4.8 GPa.After confirming that in fact the
R-3m phase is ther-modynamically more stable than the
Pnma phase at 0GPa, at any temperature range, we probe the dynam-ical stability of the
R-3m compound in order to assesswhether this structure could potentially be synthesizedfor Sb Se at/or close to ambient conditions.For this purpose, we have evaluated the phonon banddispersion and density of states (DoS) of the R-3m phasephase at 0 GPa, for different temperature values (Fig.7). We also present the phonon band structure and DoSfor the
Pnma phase for sake of comparison. Our resultsshow that there are no imaginary frequencies through- out the whole of the Brillouin-zone, thus indicating thatboth phases to be dynamically stable under ambientconditions (0 GPa and room temperature) and confirm-ing that, as implied by the energetics comparison, bothphases could potentially coexist.In this context, we must note that our results confirma recent theoretical work performed on the
R-3m Sb Se phase reporting the phonon dispersion curves, and con-firming dynamical stability of this phase at 0 GPA and 0K. In this work, the formation energies of Sb Se werealso computed evidencing the R-3m phase to be energet-ically stable. Moreover, ab initio molecular dynamics confirm that the R-3m phase remains unchanged at finitetemperature (300 K), once again favouring our presentedresults.
D. Mechanical Stability
Mechanical or elastic stability is the third conditionthat should be satisfied for a system to be potentiallysynthesized. Such a study is carried out by probing if theelastic constants obey the Born stability criteria when thesolid is submitted to homogeneous deformations.
We therefore evaluate the mechanical stability of the
Pnma and
R-3m phases of Sb Se by calculating andcomparing the elastic tensors between the two low-pressure phases (Tab. III).To confirm the accuracy of our calculated elastic con-stants, we have computed the linear compressibility ofboth phases at 0 GPa using the ELATE analysis tools.For both phases, only directions corresponding to pos-itive linear compressibilities were obtained, indicatingboth phases to be mechanically stable under ambientconditions. In the case of the R-3m phase, we have ob-tained linear compressibilities between β min = 4.9 TPa − (hexagonal a-axis) and β max = 19.5 TPa − (hexago-nal c-axis) with an anisotropy value of 3.95. For the Pnma phase, the compressibilities fall between β min =3.7 TPa − (b-axis) and β max = 25.7 TPa − (a-axis) withan anisotropy of 6.87. These values are of the same orderas the experimental axial compressibilities of the Pnma phase of Sb Se ( β a = 15.2 TPa − , β b = 3.9 TPa − , β c = 8.3 TPa − ). We must stress that the bulk mod-ulus of the
Pnma phase of Sb Se calculated from theelastic-constant tensor (31.8 GPa), and summarized inTab. IV together with other elastic moduli, is similar tothat obtained from the Birch-Murnaghan fit (c.f. Tab.I), as expected, thus demonstrating the adequate conver-gence criteria employed throughout the present calcula-tions. The calculated elastic constants in Tab. III fulfillthe necessary and sufficient Born criteria for the mechan-ical stability of orthorhombic (Eq. 1) and rhombohedral(Eq. 2) systems, respectively. Therefore, our calculatedelastic constants indicate that both the
R-3m and
Pnma phases of Sb Se are mechanically stable under ambientconditions.9 IG. 6: Gibbs free energies of the
R-3m phase (green) relative to the
Pnma phase (red) of Sb Se as a function of temperature,and for different pressure values.TABLE III: Calculated elastic constants c ij (GPa) of the Pnma and
R-3m phases of Sb Se at 0 GPa.c c c c c c c c c Pnma = c c c c = c c = -c = c c = c c R-3m
Pnma and
R-3m phases in Sb Se at 0 GPa as obtained within the Voigtapproximation using the ELATE analysis tool: bulk modulus,B , Young modulus, E , shear modulus, G and Poisson’s ratio, υ . B (GPa) E (GPa) G (GPa) υ Pnma
R-3m c , c , c , c > c c > c ; c c c + 2 c c c − c c − c c − c c > (1) c > | c | ; c > c < c ( c + c ); c < c ( c − c ) = c c (2)We note that elastic tensors of Sb Se have been pre-10 IG. 7: Quasi-harmonic phonon dispersion curves for
Pnma -Sb Se (top) and R-3m -Sb Se (bottom). The color gradientruns from blue (low T) to red (high T) across the temper-atures associated with the volume expansions considered inour calculations. viously calculated, although as for the bulk modulus,respective components were overestimated as well. Webelieve a reason for the disagreement could be due to thelow cut-off energy used in those calculations. E. Lattice Parameters, Infrared and Raman Spec-tra of the
R-3m phase of Sb Se Our calculations show that the
R-3m phase of Sb Se is energetically competitive with the Pnma phase and isboth mechanically and dynamically stable, all of whichsuggest this phase should be a ground-state structure un-der ambient conditions.The inconsistency found between the theoretical andexperimental data regarding Sb Se , can be based onthe possibility that the Pnma phase forms faster thanthe
R-3m phase, under the usual synthesis conditions.In fact, the
R-3m phase has not been proposed on thepressure/temperature phase diagram prepared by Pfeif-
TABLE V: Predicted lattice constants and atomic positionsfor the hexagonal unit cell of the
R-3m phase in Sb Se at 0GPa. a (Å) c (Å) V (Å ) B (GPa) B (cid:48) Site Sym. x y zSb
6c 3m 0.00000 0.00000 0.60082 Se
3a -3m 0.00000 0.00000 0.00000 Se
6c 3m 0.00000 0.00000 0.78792TABLE VI: Calculated vibrational modes of the
R-3m phaseof Sb Se at 0 GPa. The three acoustic IR-active modes areformed by the irreducible representations of Γ acoustic = A + E u and the remaining 12 optical modes are Γ optical = 2E g (Raman) + 2A (Raman) + 2E u (IR) + 2A (IR). Irr.eps. Frequency Raman Intensity Intensity(cm − ) (10 Å amu − ) (e amu − )E g A E u E u E g A A A fer et al. , although this study did not attempt to varythe synthesis conditions at close to ambient pressure, asthe present calculations suggest. In any case, it is note-worthy of mentioning that our calculations indicate the R-3m phase of Sb Se is energetically competitive withthe Pnma phase at close to ambient conditions. Thisraises the possibility that the
R-3m phase could poten-tially be prepared under slightly non-equilibrium condi-tions. We must note that Bera et al ., have claimed ofhaving observed such a phase at room temperature, al-though such an observation has not been confirmed byany other experimental group up untill now.In order to assist with the possible experimental syn-thesis of this phase, we provide the calculated latticeparameters and atomic positions of our optimised zero-pressure R-3m structure in Table V.We have also computed the IR and Raman spectra toaid in the identification of the spectral signatures thatshould distinguish the
R-3m from the
Pnma phase (Fig.8). The frequencies, irreducible representations andIR/Raman intensities associated with each of the zone-centre ( Γ -point) vibrational modes are listed in Table VI.The inversion symmetry in the R-3m structure leadsto mutual exclusion between the IR and Raman activ-ity of the modes, with each spectrum being characterisedby four bands. The most intense Raman band occursaround 139 cm − (E g ), while a second prominent fea-ture is predicted at ∼
204 cm − (A ). The frequency of11 IG. 8: Simulated infrared (IR; top) and Raman (bottom)spectra of the
R-3m phase of Sb Se at 0 GPa. The spectrallines have been broadened with the calculated intrinsic modelinewidths at 300 K. this A mode is higher than in Bi Se but lower thanin In Se , as expected from the difference in mass be-tween In, Sb and Bi. Lower-frequency E g and A modes with much lower intensities are also found around53 and 84 cm − , respectively, which are again slightlyhigher than the corresponding frequencies calculated forBi Se . There are four IR-active modes, two with E symmetry (87 and 132 cm − ) and two with A bands(145 and 183 cm − ). Of these, the 87 cm − mode isthe most prominent in the spectrum, while the secondE u mode at 132 cm − is very weak. The two A bandshave comparable, moderate intensities and form a pair ofsmaller features at higher frequencies. As expected giventhe mass difference, the IR-active modes in Sb Se againhave slightly higher frequencies than those calculated forBi Se . IV. CONCLUSIONS
In summary, we have carried out a comprehensive setof calculations to investigate the stability of five possiblephases, viz . Pnma , R-3m , C2/m , C2/c and disordered
Im-3m of the Sb S , Bi S and Sb Se sesquichalco-genides under hydrostatic pressures up to 60 GPa.For the three coumpounds we find that the monoclinic C2/m and
C2/c phases are energetically less favourablethroughout the studied pressure range and are not ex-pected to be observed at HP under hydrostatic condi-tions. On the other hand, the disordered bcc-like
Im-3m phase is predicted to be the most energetically stablephase of the three compounds at HP. However, calculatedphonon dispersion curves indicate that such a structuralphase remains dynamically unstable up to at least 60GPa for Sb S and Bi S . Moreover, the Pnma phaseis stable for the two compounds at HP, even up to 50GPa; therefore, we do not expect to observe the disor-dered
Im-3m phase in either of these two compoundsbelow 50 GPa. This conclusion agrees with results fromRef. 16, where the disordered phase has not been ob-served for Bi S up to 65 GPa. However, our resultsdisagree for Sb S with what was reported in Ref. 22, re-garding the observation of multiple high-pressure phases.This effect could have been caused by the use of thespecific pressure-transmitting medium resulting in non-hydrostatic behaviour during the experimental measure-ments.For Sb Se our calculations predict a transition to oc-cur from the Pnma to the disordered
Im-3m phase above21 GPa. Unlike the former two compounds, at 50 GPa,the
Pnma phase begins to evidence negative phonons atthe Γ -point, thus indicating dynamical instability at HP;whereas the disordered Im-3m phase stabilizes at thispressure range. Our calculations therefore support theconclusion that a phase transition occurs for Sb Se fromthe Pnma to the
Im-3m phase at HP, in good agreementwith experimental findings. By probing the low-pressure regions, we find that the
Pnma phase is the most stable phase for Bi S and Sb S ,in good agreement with experiments. However, and un-expectedly, for Sb Se , it is the R-3m phase that possesslower energy at 0 GPa, being surpassed by the
Pnma at a moderate pressure range, slightly below 5 GPa andaround room-temperature conditions. Since the
Pnma phase is experimentally obtained for Sb Se , we sug-gest that the orthorhombic phase is stabilized by ther-mal energy at room temperature, which suggests thepossibility of synthesizing the R-3m phase under opti-mized conditions. We would expect the trigonal phaseto show topological insulating properties under ambientconditions, which would likely make such an undertakinghighly worthwhile. We therefore provide theoretical lat-tice parameters and atomic positions along with referenceIR and Raman spectra to aid future experiments to iden-tify and characterize this phase. We hope that this workwill stimulate further investigation of the sesquichalco-genides at high pressure, especially to the lesser-knownAs analogues of the compounds examined in this work.
V. ACKNOWLEDGEMENTS
This research was supported by the SpanishMinisterio de Ciencia, Innovación y Universidades12nder the projects MAT2016-75586-C4-1-P/2-P/3-Pand RED2018-102612-T (MALTA-Consolider-Team net-work), by the Generalitat Valenciana under projectPROMETEO/2018/123 (EFIMAT); and by the Euro-pean Union Horizon 2020 research and innovation pro-gramme under Marie Sklodowska-Curie grant agreementNo. 785789-COMEX. JMS is grateful to the Universityof Manchester for the support of a Presidential Fellow-ship. The authors acknowledge the use of the MALTAComputing Cluster at the University of Oviedo and thecomputer resources at MareNostrum with technical sup-port provided by the Barcelona Supercomputing Center(QCM-2018-3-0032). 13
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