Structural, mechanical and optoelectronic properties of B6X (X = Se, S) chalcogenides under hydrostatic pressure

Structural, mechanical and optoelectronic properties of B6Se and B6S chalcogenides under hydrostatic pressure were studied by the HSE06 hybrid functional in the DFT approach. The calculation results show that both materials preserve a high hardness behavior with a slight decrease of around 11% in the Vickers hardness, from zero to 100 GPa of external hydrostatic pressure. The electronic band structures show that B6Se and B6S chalcogenides have a semiconductor behavior with an electronic bandgap of 3.89 eV and 3.77 eV at zero GPa, respectively; decreasing 30% when 100 GPa of external pressure is applied. The optoelectronic properties evidence the possible modulation of the electronic band gap by the appliance of pressure as well as the refractive index in the visible region from nω=2.8 to nω=4.4 while the absorption coefficient suggest the potential applicability of both chalcogenides under extreme pressure conditions as UV sensors in an interval from 7.5 eV to 30 eV.


Introduction
Boron-based materials have received particular attention because of their interesting physical properties as high hardness materials with potential applications such as scratching, polishing and cutting tools [1,2]. In transition metal-boron materials such as ReB 2 , MoB 3 , WB 3 , TaB 3 , OsB 2 or MnB 4 ; their hardness behavior is understood by the sigma bonding interaction between boron atoms (s and p orbitals) and the electronic density distribution of the transition metal (d orbital population) where a typical threshold in hardness occurs due to the presence of ionic and metallic contributions between boron and metal transition atoms [1][2][3][4][5][6][7][8][9]. When boron is bonded with nitrogen, it forms the cubic boron nitride phase (c-BN), one of the hardest materials. The boron-nitrogen interaction promotes the hybridization between s and p electron density orbitals to achieve a diamond-like crystal structure [10][11][12]. Furthermore, the presence of boron icosahedra clusters inside the crystal structure promotes higher mechanical stability and hardness properties, as evidenced in B 6 O, B 4 C and boron allotropes [13][14][15][16][17][18][19].
Most of the studies focused on transition metals or rare-Earth chalcogenides have received attention because of their potential applications in renewable energy technology, solar energy conversion, nonlinear optics, spintronics and semiconductor detectors [20,21]. By contrast, there are fewer reports related to boron chalcogenides; these materials with the general form B x M x (with x = 1, 2, 3 and M = S, Se) are suggested to have promising physical properties like superconductivity, thermoelectric efficiency, high storage capacity, anode functionality in lithium-ion batteries or photocatalyst [22,23].
Recently, new boron-rich B 6 Se and B 6 S chalcogenides were synthesized at high pressure (6.1 GPa) and high temperature (2700 K) by Cherednichenko et al [24]. According to their experimental x-ray diffraction data, both compounds have an orthorhombic symmetry with a space group Pmna with 24 boron atoms in four independent Wyckoff sites, figure 1. The Raman spectra, at atmospheric pressure, show bands from Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 150 to 1110 cm −1 . Cherednichenko also presented theoretical calculations on the crystal structure of the B 6 Se chalcogenide in the interval from zero to 35 GPa; estimating a bulk modulus (B 0 ) of 144 GPa at zero GPa [25] and suggesting the possibility of being a high hardness material. Subsequently, Hossain and co-workers [26] reported a comprehensive theoretical study of the mechanical and physical properties of both B 6 Se and B 6 S chalcogenides, at zero GPa. According to the calculated results, both chalcogenides have a Vickers hardness (H v ) in the interval from 31 to 35 GPa. Additionally, a semiconductor nature was evidenced with an energy band gap (E g ) of 3.32 eV for the B 6 Se and E g = 3.22 eV for the B 6 S.
In this work, theoretical calculations under hydrostatic pressure are presented for the newly synthesized B 6 Se and B 6 S chalcogenides to describe their mechanical and optoelectronic properties, evidencing their structural stability and the modulation of optical properties when external compression is applied, to provide useful information for future research in potential applications.

Computational details
Computational calculations were performed within the density functional theory (DFT) framework [27,28] as it is implemented in the Cambridge Serial Total Energy Package code [29]. Geometry relaxation of the crystal structure, vibrational and optical properties were performed under the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof for solids (PBESOL) exchange-correlation functional together with the linear response density functional perturbation theory (DFPT) [30], considering for boron, sulfur, and selenium atoms the 2s 2 2p 1 , 3s 2 3p 4 and 4s 2 3d 10 4p 4 orbitals as valence electrons, respectively.
The tightly bound core electrons were represented by a norm-conserving pseudopotential of the Vanderbilt type [31]. Electronic properties were computed within the Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional as it was proposed by Krakau et al [32][33][34]. The wave functions were expanded in a plane-wave basis set with a cut-off energy of 720 eV with k-point sampling inside the first Brillouin zone constructed using the  Monkhorst-Pack scheme with 2 ×3 × 2 grids [35]. The equilibrium properties were obtained via the geometry optimization process in the Broyden-Fletcher-Goldfarb-Shanno (BFGS) minimization scheme. The tolerance parameters to achieve convergence were 5 × 10 −6 eV atom −1 for the energy difference, 0.01 eV Å −1 for the forces between atom pairs, 5 × 10 −4 Å for the ionic displacement and 0.02 GPa for stress. The B 6 Se and B 6 S bulk materials were modeled with orthorhombic symmetry, Pmna space group (No. 53), with 24 boron atoms in four independent Wyckoff sites (4h and 8i) forming B 12 clusters. In contrast, the four selenium (sulfur) atoms are located in one independent Wyckoff position (4h). For the B 6 Se the initial lattice parameters were: a = 5.9463 Å, b = 5.3579 Å and c = 8.3824 Å, while for the B 6 S were; a = 5.8170 Å, b = 5.3025 Å and c = 8.2135 Å [25].

Results and discussion
The normalized lattice parameters as a function of pressure, figure 2, show a significant reduction in a and b directions compared against c, in both B 6 S and B 6 Se compounds. Additionally, both chalcogenides have a 32% volume diminution at 100 GPa. For comparison purposes, the experimentally recorded variation of the lattice parameters under pressure for the B 6 Se [25], was plotted with the theoretical data obtained in the present work, figure 3.
As it can be seen, the difference between experimental and calculated data is less than 1.5% denoting good consistency in the computational methodology.  Nine independent elastic constants were obtained through the linear finite strain-stress approximation ) The mechanical stability is determined from the Born criteria for an orthorhombic system, which establish that all the elastic constants must comply [36][37][38]: 0 . 11 The obtained C ij values satisfy these stability criteria (table 1), indicating that B 6 Se and B 6 S chalcogenides are mechanically stable from zero to 100 GPa. Based on the Voigt-Reuss-Hill approximation, the polycrystalline bulk (B) and shear (G) moduli for an orthorhombic crystal symmetry were calculated from the elastic constants, according to the equations [37][38][39]: 1 55 Where the following parameters were defined:  Poisson ratio (n) and Cauchy pressure terms [6,9] were calculated from the elastic constants.
According to the calculated values of the elastic constants, it could be observed for both chalcogenides that C , 11 C 22 and C 33 are the larger ones ( figure 4 and table 1), this indicates that B 6 Se and B 6 S have a major deformation resistance along axial directions. Moreover, the relation C C C 33 11 22 > > is followed in the interval of pressure from zero to 30 GPa, indicating that the bonding strength along the [001] direction is stronger than those along [100] and [010]; however, for pressure values above 50 GPa, it is observed that C C , 11 22 < suggesting that the bonding strength changes in both related directions. The C , 44 C 55 and C 66 elastic constants keep almost unaltered in all the pressure interval ( figure 4) compared against the C , 11 C 22 and C ; 33 being an insight of the resistance to shear deformation in the (100), (010) and (001) planes.
Finally, table 1 shows the elastic constant at zero, 50 and 100 GPa for both chalcogenides, exhibiting a good correlation with previous calculated elastic constant values at zero pressure [24,26].
The difference between experimental and theoretical bulk moduli for the B 6 Se chalcogenide is less than 5% while for the B 6 Se, it is around 2% at zero GPa. Both results indicate the reliability of the present calculations, similar to that previously observed for the structural parameters.
Despite the random behavior of the Vickers hardness from zero to 100 GPa, figures 5(a) and (b), a small change in magnitude could be seen between the Chen and Tian models. At zero pressure, both chalcogenides have similar Vickers hardness values. However, as a function of pressure, in Chen's model the hardness decreases around 5% in B 6 Se and 12% in B 6 S; while in the Tian model, the hardness increases by 5% in the B 6 Se and decreases by 0.8% in B 6 S.
A lack of hardness is observed in the Tian description, perhaps due to the approach implemented in the construction of the model, which considers the ionic contribution of the materials [41]. It is observed that there is a similar rate response in their E, B and G moduli between B 6 Se and B 6 S chalcogenides, figures 5(c) and (d).
The Young´s modulus increases from almost 300 GPa (zero pressure) to 700 GPa (at 100 GPa), giving an insight of the increment in the stiffening response of both materials.
The B and G moduli increase at 300 GPa and 150 GPa, respectively; evidencing that both chalcogenides (B 6 Se and B 6 S) have a good response under shear deformations. These results correlate well with the hardness stability under pressure.
Poisson ratio, Pugh criteria and Cauchy pressure terms (P , 100 ( ) P 010 ( ) and P 001 ( ) ) are parameters that describe the brittleness or ductility of materials in addition to an indirect description of the chemical bond character. It is commonly stated that 0.3 n < is indicative of a brittle material; otherwise, the material tends to be ductile; similar results are ruled out for G B values greater than 0.5. The Pettifor criteria [42] establish that a positive value for the Cauchy pressure is indicative of a metallic character, and thus, ductile behavior; in contrast, a negative value is associated with directional bonding, and consequently, the materials tend to be brittle. According to the criteria previously described, the B 6 Se and B 6 S chalcogenides are far from ductility in the pressure interval from zero to 100 GPa, figures 6(a)-(d). Consequently, their brittle nature is evidenced, and a high covalence character is suggested. Furthermore, it is seen that even at the high-pressure value of 100 GPa the three Cauchy pressure terms are in the negative range. At the same time, hardness does not change significantly (table 2), indicating that both materials preserve their covalent character and suggesting a highly stable hardness behavior. However, an abrupt behavior in the pressure range between 30 and 60 GPa is identified by the Cauchy pressure terms (P 100 ( ) ) and (P 010 ( ) ); maybe related to an internal structural response because of the atomic interaction in the a and b directions, suggesting the existence of a slight discrepancy in the physical properties between both chalcogenides even when the mechanical response is almost similar.
Additionally, the cohesive energy (E coh ), defined as the difference between isolated atom energy and crystal energy, was computed according to the equation [43][44][45]: are the energies corresponding to the isolated atoms and n , B n x are the number of atoms in B 6 Se and B 6 S unit cell. The cohesive energy values suggest the presence of a strong interaction between their constituents in the interval of pressure from zero to 100 GPa, because all the E Coh values are positive; and thus, shedding light on structural stability by considering that a crystal can only be stable if its total energy is lower than the total energy of the isolated atoms. Additionally, it is observed that the applied pressure has a greater influence on B 6 S than on B 6 Se, in concordance with the mechanical stable parameters such as Cauchy pressure, Pugh criteria and Poisson ratio.
The electronic band structure for the B 6 Se and B 6   pressures a direct bandgap at G is observed. However, at 100 GPa, the material has a transition mediated by an indirect band gap in the T -G symmetry points. The alteration in the electronic bandgap as a function of pressure from different symmetry points for the B 6 S chalcogenide related to different electronic configurations  is correlated with the mechanical behavior between 30 to 60 GPa observed in hardness and Cauchy pressure terms. The results presented suggest that the electronic band gap of both chalcogenides could be modulated by pressure, preserving their structural properties and improving mechanical properties. This offers a wide spread of possible technological applications for both chalcogenides [46].   region. For the B 6 S, the main contribution to the valence band comes from the p states; signifying that there is a major s-p interaction in B 6 Se compared against the B 6 S. On the other hand, the boron PDOS for both chalcogenides, figures 8(c) and (d), shows a strong relationship between the s and p states in the valence and conduction bands. The pressure effect on the B 6 Se and B 6 S chalcogenides DOS promotes a shift in energy that is noticeable at −12 eV and 2.5 eV regions for the s and p states, respectively; figures 9(a)-(d), show their preponderance contribution to the bandgap reduction. These results prove the stability of both chalcogenides in their electronic structure under compression, with a good correlation to the structural and mechanical behavior.
Simulated R a m a n g g g g 1 2 3 G = + + + Figures 10(a)-(f) shows the most representative Raman intensities for both chalcogenides; at zero pressure, Raman spectra are in good concordance with the experimental measurements done by Cherednichenko [24].
It could be seen that the pressure effect on B 6 S and B 6 Se causes the intensity reduction of A g modes at 426 cm −1 , 592 cm −1 and 908 cm −1 for B 6 Se and 452 cm −1 , 615 cm −1 and 948 cm −1 in B 6 S, figures 10(a) and (d).
Due to the symmetry nature of the D 2h point group, the assigned Raman and IR modes are predominantly related to the B 12 icosahedron coordination, figure 11 [47]. Since a vibrational active Raman mode is proportional to the polarizability and a populated electronic distribution is harder to polarize than a diffuse one, the reduction of Raman intensities comes from the s states of the Se and S because these are the most affected by the application of pressure, according to the DOS results, figure 9. Additionally, from figures 11(a)-(f), analogous behavior is obtained for the simulated vibrational IR modes at which the B 2u vibrational modes are the most sensible to the applied pressure; the remaining modes hold almost unaltered. Figure 12(b) shows that B 2u is related to the Se and S atoms while the B 1u and B 3u IR modes are constrained to the B atoms. It is remarkable to notice that even at 100 GPa the vibrational IR and Raman modes are globally unaltered.
The optical properties of B 6 Se and B 6 S chalcogenides under pressure were calculated using the dielectric function Re Im e w ew ew where Im e w [ ( )] represents the absorption energy by the medium whereas Re e w [ ( )] is related with the electronic polarizability [48]. Im e w [ ( )] can be determined from the electronic structure (equation (13)) and Re e w [ ( )] calculated from Im e w [ ( )] by using the Kramers-Kronig relation (equation (14)) [49][50][51][52], where M is the dipole matrix, i and j denote the initial and final states, E i is the energy of the electron in the ith state related with the f i Fermi distribution, n is the refractive index, a is the absorption coefficient and l is the free space wavelength of light:  and Im 1, e w < [ ( )] at which a negative value mean that light is reflected and the electric field of light is screened by the electrons close to the surface. Thus, it could be seen the major effect of pressure is in the dielectric properties of both materials below p w while the Im e w [ ( )] is associated with the energy absorption due to the transitions between valence optical band to conduction optical band.
The refractive index n w ( ) value under pressure calculated for B 6    B 6 S chalcogenides in the interval from 16 eV to 24 eV (far UV-region) is around to n 0.5 = without considerable deviation. Moreover, the most representative variation of n w ( ) is obtained with the application of pressure for both chalcogenides from 1.6 eV to 4.0 eV, inset figure 14; which is indicative of a potential adaptive material with a modulated refractive index inside the visible region, that does not yield into deformation by the application of pressure in an interval from zero to 100 GPa, according to the calculated mechanical properties; and thus, with potential applications in the visible region for both B 6 Se and B 6 S chalcogenides.
The diminishing response of the refractive index for values eV 8 w > could be understood because light in the material starts to be absorbed, as shown by the absorption coefficient , a w ( ) figures 15(a) and (c). It could be inferred that both materials are transparent to infrared -visible light because of the low a w ( ) values obtained below 4 eV, insets of figures 15(a) and (c), however, in the UV region the material absorption coefficient increases considerably from approximately 7.5 eV to 30 eV in the UV spectra, and because the refractive index holds almost unaltered, both materials could be used as UV sensors in extreme conditions of pressure. Finally, the energy loss computed, figure 15 puts into evidence the resonance between light and the characteristic plasma frequency of both B 6 Se and B 6 S materials.

Conclusions
This work presents a theoretical study of the B 6 Se and B 6 S chalcogenides in terms of their structural, mechanical and optoelectronic properties. According to all the mechanical parameters calculated, the B 6 Se and B 6 S chalcogenides are stable, brittle, have a high covalence character, and have hardness values that are practically unaltered in the pressure interval studied. The optoelectronic properties evidence a semiconductor nature with an energy band gap of 3.89 eV for B 6 Se and 3.77 eV for B 6 S, at zero GPa, and a 30% decrement when 100 GPa of pressure is applied. Finally, the refractive index in the visible region could be modulated from 2.8 to 4.4 with a UV absorption behavior, suggesting their applicability as UV sensor materials under high-pressure conditions, in the interval from 7.5 eV to 30 eV.