First-principles study of the effect of pressure on the physical properties of PbC

Silicon carbide has been used as a cutting material and as a semiconductor in lighting and power electronics. Results from some studies, carried out on IV-IV group carbides like GeC and SnC, allow to identify potential technological applications of these carbides in extreme environments, opening the possibility to find new carbides for similar applications. For this work, the PbC was studied under hydrostatic pressure in the framework of the Density Functional Theory, obtaining relevant information on its structural, electronic, mechanical, vibrational, thermodynamical, and optical properties. The optimized lattice parameter and volume, and electronic bands structures type agree with the available theoretical data at zero GPa. The calculated enthalpy values show a phase transition, from the B3 structure (CsCl-type) to the B1 structure (rocksalt or NaCl-type), at 23.5 GPa. The PbC is energetically, mechanically, and dynamically stable for all the pressure values in the studied range; it is a metallic, anisotropic, and brittle material with paramagnetic ionic-covalent bonds and good hardness (the highest mechanical resistance was found above T = 370 K). As the pressure increases, it was noted: (i) the increase of the electronic cloud around the C and Pb atoms, (ii) the DOS spread, (iii) the change to be a ductile material with a tendency to the metallic bonds and (iv) an increase of the hardness and the Young modulus, due to C 2p and Pb 6p-orbitals. Our results show that the PbC is a promising material for applications in the development of optical and optoelectronic devices, and to be used as a protective coating against the low frequencies in the UV and infrared and visible regions.


Introduction
The IV-group binary compounds with carbon (C) are called covalent or covalent-ionic carbides; their sp-valence electrons [1] form a heteropolar material. When C interacts with C their bonds are symmetric, but C with another IV-group element form non-symmetric bonds, due to the size and chemical potential differences, creating the ionic behavior [2]. These compounds have attracted great attention due to their potential technological applications [3][4][5]. Silicon carbide (SiC) is the most important among the IV-group binary carbides. This carbide is a semiconductor with a wide bandgap, that has been employed for technological applications as biosensors, photocatalytic processes, field emission transistors, nanoelectronics, membrane technology, photothermal or photodynamic activities, medical implants, etc [6]. On the other hand, theoretical studies about germanium carbide (GeC) and tin carbide (SnC) suggest that their nanostructures could be materials for protective coatings [7], microelectronics [8][9][10] and energy [11][12][13] applications and spintronic devices [14,15]. Then, the PbC is a promising material for technological applications since its properties could be similar to those of SiC, GeC and SnC.
In the IV-group, the natural chemical partner for C is Si [2], because the main difficulty to generate compounds with C, for the other elements, is the precipitation of one or more elements during the crystal growth. Another factor is the C's chemical potential, which competes more with the other atomic potentials (Ge, Sn, or Pb) but not with the Si's one; that makes the bonds asymmetric (table 1), even though each atom contributes four valence electrons. Benzair et al [2] published a theoretical study on GeC and SnC (zinc-blende type, B3 phase, figure 1(b)) where it was found that the valence charge is transferred, from the non-bonding region and the atomic sites to the bonding regions closer to the C atoms, developing more ionic bonds than those in SiC; this behavior resembles the III-V semiconductor compounds.
The natural bonding between C and Pb atoms also is very difficult to occur. The first report on the synthesis of lead carbide was made by Durand in 1923, who obtained PbC 2 by adding calcium carbide to an aqueous solution of lead acetate; but this compound was not successfully synthesized by reproducing this technique. Masson and Cadiot (1965) made a more tangible result, they reported the preparation of acetylene derivatives from lead without separating PbC crystals [16]. However, Sengupta et al (2006) accidentally found the formation of a PbC 2 layer at the graphite crucible after isothermal annealing of the eutectic lead-bismuth [17]. On the other hand, the theoretical studies made by Wang (2002) on the IV-group and III-V-groups compounds at the B3 phase [18], is the first work where some electronic properties of the crystalline PbC were described.
An experimental X-ray diffraction investigation on SiC [19] provided a direct probe on intermediate metastable states, at the high-pressure phase transition (P T ) from the B3 phase to the rock salt type (B1 phase, figure 1(a)), at 105 GPa. Theoretical studies predict this P T at 100 GPa [20], 66 GPa [21] and 65 GPa [5]. Also, the P T value was investigated theoretically for GeC and SnC [5], where these carbides' structure change from the B3 to the B1 phase at 89.4 GPa and 32.5 GPa, respectively. Another possible polytype for these compounds is the wurtzite (Wz) type, which was obtained from the theoretical studies carried out on SiC [22,23] and GeC [24].
For this manuscript, we are reporting the magnetic configuration of the different possible PbC polytypes (B1, B3, and Wz, figure 1(c)) to identify the most stable structure at zero GPa and to locate the P T from the values calculated for the enthalpy. The work is organized as follows: in section 2 we describe the computational calculations made, and the theory required for this work; the results and discussion on the structural, electronic, mechanical, vibrational, thermodynamical, and optical properties are available in section 3; and the summary of this work is given in section 4.  Wz

Computational details
The physical properties of the PbC polytypes (B1, B3, and Wz phases) were computed in the Density Functional Theory (DFT) framework [26], using the Cambridge Sequential Total Energy Package (CASTEP) code [27,28] which is based on a plane wave basis to build the wave functions and atomic orbitals. The base set expansion was truncated to a cut-off kinetic energy of 500 eV. The Vanderbilt-type ultrasoft pseudo-potentials [29] were employed to represent the core electrons of the Pb and C atoms [30]. The electron-electron exchangecorrelation interactions were considered using the Generalized Gradient Approximation (GGA) and the Perdew-Wang (PW91) functional [30]. Once the Kohn-Sham equations were solved, to calculate the total energy, the first Brillouin zone (BZ) integrations were performed using 9 × 9 × 9 k-grids on the Monkhorst-Pack scheme [31]. Likewise, the Broyden-Fletcher-Goldfarb-Shanno's (BFGS) algorithm was used for the geometry optimization [32]. The convergence tolerances were fixed as follows: the equilibrium forces on atoms were less than 0.01 eV Å, and the ionic displacement and the highest strain amplitude were reduced to 5 × 10 −3 Å and 0.02 GPa, respectively.
To determine the magnetic properties of the PbC, different magnetic configurations (MCs) were modeled. The MCs studied were arranged as follows: for the paramagnetic (PM) configuration, the formal spin was set equal to zero for each atom at each of the B1, B3, and Wz structural phases; for the B1 and B3 phases, the ferromagnetic (FM) and the I, II, III-type antiferromagnetic (AFM I, AFM II, and AFM III, respectively) behaviors were modeled as it was described in [33]; whereas for the Wz structural phase, the FM, as well as Cand G-type antiferromagnetic orders (AFM C and AFM G, respectively) were arranged as it was described in [34]. For the FM and AFM configurations, the formal spin of the atoms was considered. These MCs, not necessarily remain after the geometric relaxation of the structure models.
Once the most stable PbC structural phases were identified, their mechanical properties were obtained calculating the full tensor of second-order elastic constants, using the method of homogeneous deformations [35]. Besides, the phonon and optical properties were calculated using a cut-off energy of 820 eV and a BZ tested by 5 × 5 × 5 k-grids. The phonon dispersion curves were plotted using the data obtained from calculations under the Density Functional Perturbation Theory (DFPT), using the linear response method [36] with normconserving pseudo-potentials and a q-vector grid spacing of 0.05 1/Å. Finally, the optical properties were calculated with a plasma frequency of 10 eV for the polarization vector [1 0 0].

Theory
When a system reaches chemical equilibrium, the Gibbs' energy is minimized. This thermodynamic potential is defined in terms of the enthalpy (H), the entropy (S) and the temperature (T); but, when = T 0, the Gibbs energy is equal to H. The equation to obtain the enthalpy of the PbC system, and to determine the P T under the effect of the pressure, is given by: where U is the internal energy, P is the pressure and V is the volume.
The variation of the lattice parameter and the volume, as the pressure increases, are very important parameters for considering a possible phase transition. Therefore, the lattice parameter (a 0 ) and volume (V 0 ), at zero GPa, were used for this work, as are reported in reference [37].
An energetic stability criterion for a system involves the total energy, which must be smaller than the sum of the energy of the free atoms set, and the difference between these; this allows to define the cohesive energy (E coh ) [38] for the studied system by: There, E Pb and E C are the energies for the isolated Pb and C atoms; n and m are the number of Pb and C atoms in the system's cell, respectively; and E total is the total energy of the PbC systems. The phase stability and stiffness of a solid material can be obtained from the elastic constants (C ij ) [39,40]. Positive strain energies indicate mechanical stability, which implies that C ij must assure the Born-Huang stability criteria [38]. For a cubic crystal, three independent second-order elastic constants (SOEC) must be determined: C 11 , C 12 , and C 44 ; and their SOEC' values must satisfy the following mechanical stability criteria [35]: Macroscopic parameters can be obtained by means of the C ij constants. Such are the cases of the resistance to volumetric deformation (bulk modulus, B), the shear stress (shear modulus, G), as well as the relationship between stress and strain (Young's modulus, Y). Since a polycrystalline material can be considered a set of single crystals at random orientation, all these moduli can be obtained through the relationships used in [41]. The brittle or ductile character of a material can be evaluated using the empirical criteria of the Pugh's ratio (K ). A material is brittle if > K 0.5, but ductile if < K 0.5 [42]. The Poisson's ratio (ν) gives us signals of the type of structure. For covalent crystals n < 0.2, for ionic-covalent crystals n   0.2 0.3 and for metals n  0.33 [43]. Additionally, the hardness can be studied by means of the Vickers hardness (H V ) [42] . The K, ν, and H V values can be obtained utilizing: The mechanical stresses suffered by a material affect its vibrations (phonons). Besides, the upper limit of the phonons' frequency (ω) of a crystal lattice is known as the elastic Debye temperature (θ D ) and it depends on the averaged sound velocity (v m ). The relationships used to obtain these parameters are those described in [44]. The elastic anisotropy impacts a variety of physical processes, such as the microscale cracking in ceramics, the development of plastic deformation, the alignment or misalignment of quantum dots, the fluid transport in poroelastic materials, the mobility of positively charged defects, the phonon focusing in crystallites and the plastic relaxation of thin-film metals [45]. The anisotropy of the studied systems was obtained using the elastic anisotropy index (A U ) which is described in [46]. The phonon dispersion plot (PD, which is the frequency dependence on the wave vector), and the phonon dispersion relationship were obtained from calculations made applying the linear response theory [36,47]. The linear response calculations evaluate the dynamic matrix of a k vectors' set, based on the evaluation of the second-order change of the total energy, generated by atomic displacements [48]. The second order electronic energy is described in [49]. For crystals with n atoms, three branches at the PD are acoustic while n-3 branches are optical. Furthermore, a crystal is dynamically stable when all points at the PD are above zero [50].
The equilibrium structure of a crystal, without pressure exerted on it, can be found by minimizing the Helmholtz free energy (F), which is evaluated for all its geometric degrees of freedom a , If a harmonic crystal is considered, the internal energy (U) is the sum of the ground state total energy (E a i { }) and TS is the vibrational free energy (E vib ) [51]: The free energy dependence of the structure is completely contained in E a i { }if anharmonic effects are neglected, consequently the structure is independent of T. However, if the phonon frequencies depend on the structural parameters (quasi-harmonic approximation), the equilibrium structure could be obtained at any temperature T as described in [52]. The vibrational entropy (S), and the lattice contribution to the heat capacity (C v ) can be calculated as [53]; for the case of the Cv can be obtained as a function of the empirical Debye temperature (Θ D ) [54].
The electromagnetic radiation that hits the surface of a material can be transmitted, absorbed, or reflected; and the relative degree of these phenomena depends directly on the properties of the material. The optical properties can be obtained by calculating the frequency-dependent complex dielectric function (ε(ω)), which is formed by a real (ε 1 (ω)) and an imaginary part (ε 2 (ω)) as described in [55]. Moreover, the energy loss function, the adsorption (I), the reflectivity (R), the refractive index (N), and the conductivity (σ) were calculated for this work, as in [56].

Magnetic configurations
The electronic band structure (EBS) is useful to identify the electrical behavior (metal, semimetal, semiconductor, or insulator) of a material, as well as to identify non pairing electronic bands which tell us about its magnetic behavior. Then, from the corresponding EBSs is possible to see that all the B1 phase MCs are metallic compounds (figures 2(a)-(c)) where the bands of up/down spin are the same for each MC. On the other hand, the B3 phase shows two different behaviors: the NM (figure 2(d)), FM and AFM I cases (figure 2(e)), which are metallic in accordance with the results reported by S. Q. Wang and H. Q. Ye [18]. However, the AFM II and III cases are narrow gap semiconductors (figure 2(f)): their direct gap values are 0.007 (AFM II) and 0.004 eV (AFM III). Finally, the NM Wz phase (figure 2(g)) is a metallic compound while the FM (figure 2(h)) and C and G-type AFM (figure 2(i)) are semiconductors with narrow band gap values (direct gap of 0.006 eV, for both).
At the total and partial densities of states (DOS and PDOS, respectively) for the different MCs (figure 3) is shown that for the B1, the B2 and the Wz phases (FM and AFM cases), the up-and down-spin state contributions are the same. At the valence band (VB) there is a mixed states contribution from C and Pb atoms, while at the conduction band (CB) the Pb atoms contribute more states for energies displaced to higher values. The exception are the B3 FM/AFM I and Wz NM phases/MCs (figures 3(e) and (g), respectively), for which Pb 6p contributes the majority of states at the CB. From −18 to −15 eV there is a large contribution of Pb 5d-orbitals. Between energies −14 and −4 eV, most of the states come from the C 2 s and Pb 6s-orbitals, except for the NM Wz phase (figure 3(g)), since this phase has a greater contribution from Pb 6s-orbitals. At the energy intervals from −4 eV to 3 eV and from 3 to 9 eV the C 2p-orbitals and Pb 6p-orbitals contribute more states, respectively; except for the FM, the AFM I B3 phase and the NM Wz phase (figures 3(e) and (g), respectively), for which the Pb 6p-orbitals (from −4 to 9 eV) contribute the most states. At energies from 9 to 10 eV there is a large contribution of Pb 6s-orbitals for all cases.
Taking into account the criteria to determine the magnetic state through the calculated values of the 2 * Integrated Spin Density (2 * ISD) and the 2 * Integrated |Spin Density| (2 * I| SD |) [57,58], it is possible to indicate that the B1, B3 and Wz-type phases are paramagnetic compounds because the values of 2 * ISD and 2 * I| SD | are zero for all MCs. The fact that all the FM and AFM cases show the same contribution to PDOS from both spin  channels (up and down), and that the Pb 5d-orbitals are at lower energies in the valence band (closer to its nucleus), supports the results shown by the previous criteria.

Thermodynamic stability and structural parameters
To determine the equilibrium between the B1, B3, and Wz phases of the PbC, the H was calculated from zero to 50 GPa. From zero to 23.5 GPa, the B3 phase has associated the lower H value than those for the B1 and Wz phases (figure 4(a) and table 2); therefore, the H phases' difference ( ) was determined to identify the most stable phase transition (P T ). For pressures < < P 23. 5 50 GPa, the Wz phase has associated a higher enthalpy than those for the B3 and B1 phases. The enthalpy difference (figure 4(b)) indicates that the B3 to B1 transition is more stable (thermodynamically) than the B3 to Wz transition, since the P T for the former (P T(B3 to B1) ) is lower than the P T for the latter (P T(B3 to Wz) ) without crossing each other. This indicates that there is a P T(B3-B1) at 23.5 GPa and a possible P T(B3-Wz at 25 GPa. Since the B1 and B3 phases are cubic systems, the lattice parameters = = a b c; this value and the volume were evaluated (table 2) from zero to 50 GPa for both phases. The lattice parameter for the B3 phase diminished 7.8%, going from zero to 50 GPa; and this behavior was similar for the volume, which was reduced by 21.7%. Meanwhile, for the B1 phase, the decreases were of 9.0% and 24.5% for the lattice parameter and volume, respectively. These results indicate a greater effect of the pressure on the structural properties of the B1 phase. The results from the 8-atoms cell for the lattice parameter and volume (B3 phase), at zero GPa, are closer to those reported in [18]. Furthermore, the difference between the enthalpy for the primitive cell and the 8-atom cell is only 0.001 eV (0.0002%).
From the normalized lattice and volume parameters for the B3 and B1phases (figure 5(a)), it was found that |a| and |V| for the B3 phase decrease 5.9% and 16.8%, respectively, going from zero to P T ; then, at the P T , both parameters exhibit a discontinuity, with increments of 0.3% (|a|) and 1.0% (|V|). Meanwhile, |a| and |V| for the B1 phase, from P T to 50 GPa, decrease 3.5% (|a|) and 10.2% (|V|). The most significant changes at P T are associated to the bond lengths ( figure 5(b)): the Pb-C and Pb-Pb bonds diminished 5.9%, from zero to 23.5 GPa; however, at the P T , there is an abrupt change for the Pb-C and Pb-Pb bond lengths, increasing by 10.6% and 35.4%, respectively. From P T to 50 GPa, the trend of the bond lengths change is similar to that observed for the B3 phase: for the B1 phase diminished 3.5%. As an effect of the pressure, |a|, |V| and the bond lengths decrease slightly.
The pressure effect on the PbC crystal structure was analyzed using x-ray diffraction. For the B3 phase were found 12 main peaks, from 0 to 120°at zero GPa (figure 6(a)), and those with the higher intensities are at 29.1°, 33.7°, 48.5°, 57.7°and 60.3°. At 23.5 GPa, all the peaks move to the right and their intensities increase, except for the first peak ( figure 6(b)). The second peak was displaced 2.2°and its intensity increased 47.3%; while the eighth peak was displaced 6.3°and its intensity increased three times. At the transition from B3 to the B1 phase (P T ), there was only a change: the seventh peak shifts to the left and its intensity doubles (figure 6(c)), indicating the phase transition of the PbC. When the pressure on the B1 phase reached 50 GPa (figure 6(d)), the diffraction peaks moved to higher angles, but their intensities decreased by an average of 22%; additionally, an increase for the smaller peaks was observed, which might indicate that the C and Pb atoms are closer to each other and feel repulsion between them.

Electronic properties
The EBS, DOS, PDOS, and electron density differences were studied to identify the effect of pressure on the electron density of the PbC; also, the energy stability was analyzed by means of the calculated cohesion energy. The B3 phase at zero GPa (figure 7(a)) showed metallic behavior, being the Γ point the place where the CB and VB overlap slightly. The band gap, at the Γ point located around −15 eV (VB), is 2.9 eV. When the pressure reached 23.5 GPa ( figure 7(b)), the bands overlap was maintained at the Γ point, but the bands above the Fermi energy were shifted toward more positive energies, and bands below zero eV were shifted toward more negative energies. Another variation occurred for the band gap located around −15 eV; this was reduced to 1.6 eV. The transition to the B1 phase, at P T , sends the CBs at the Γ point toward more negative energies, and two bands pass the Fermi energy (figure 7(c)); on the contrary, the VBs close to zero eV moved toward more positive energies, and those bands which pass the L point surpassed the Fermi energy. Also, it was observed that the VBs between −5 and −15 eV were shifted to lower energies, and the band gap below these bands was reduced to 1 eV. At 50 GPa (figure 7(d)) the B1 phase's bands followed the same trend as at P T , but with the peculiarity that the band gap that was located around −15 eV disappeared. The VB moves toward lower energies by the effect of the pressure.  The EBS changes, observed for the B3 phase from zero to 23.5 GPa, are reflected on the 16% reduction of the states' concentration (around 6 eV) and, also, on the shift to more positive (negative) energies of the peaks of states' concentration at the CB (BV) (figures 8(a) and (b)). The Pb 6s-orbitals are the most affected by the pressure increase; the effect is reflected on the DOS increase, where the peaks located at 7.9 and −10 eV grow 86.1% and 66.9%, respectively. Furthermore, the DOS' peak at 2.2 eV, linked to the C 2s-orbital, increased by 32.8%. At the transition pressure, the C atom orbitals were the most affected for the B1 phase ( figure 8(c)). The peak associated to the C 2s-orbital, located at −9.5 eV, doubles their magnitude at the DOS; meanwhile, the peaks linked to the C 2p-orbital and located at 7.31 and −0.2 eV increase by 58.4% and 24.9%, respectively. However, the decrease of the C 2s-orbital's peak around −10 eV is linked to the states' distribution through the VB, when the pressure is increased to 50 GPa ( figure 8(d)). The peak at −9.8 eV of the C 2s-orbital decreases by 30.6%, producing a distribution of states from −18 to 10 eV. Furthermore, it is observed that the C 2s-and Pb 6s-orbitals are responsible of the inter-bands gap loss, around −15 eV, at 23.5 GPa.
The PDOS at the Fermi energy also changes. For the B3 phase at zero GPa ( figure 9(a)), the main contribution for the DOS comes from the C 2p-orbitals, followed by that from the Pb 6p-orbitals. When the pressure was increased to 23.5 GPa ( figure 9(b)), the DOS was reduced by 28.6%, and the C 2p-and Pb 6sorbitals were reduced 24.2% and 48.9%, respectively. When the transition to the B1 phase occurs ( figure 9(c)), the DOS for the VB at the L and X points close to the Fermi energy, are linked to the C 2p-orbital, followed by the  Pb 6p-orbital, and the incursion of the Pb 6s-and C 2s-orbitals. At 50 GPa (figure 9(d)), the DOS at the Fermi energy diminished by 21.3%, as well as the C 2p-and Pb 6p-orbitals by 27.9% and 34.8%, respectively; on the contrary, the DOS of the Pb 6 s and C 2s-orbitals were increased by 26.7% and 6.8%, leading the Pb 6s-orbital to provide a higher DOS than the Pb 6p-orbital.
The population analysis and the electron density difference density were carried out to obtain information about the interaction between the C and Pb atoms. The Mulliken charges from zero to 50 GPa ( figure 10(a)) indicate negative charges at the C atoms, whereas the positive charges are at the Pb atoms. This suggests that the electronic charge was transferred from the Pb to the C atoms. Also, the electronic cloud surrounding the carbon atoms (purple isosurface) extends towards the Pb atoms ( figure 10(b)); but there is an electronic cloud (green isosurface) around the Pb atoms which could indicate an ionic-covalent bond with a covalent tendency. When 10 GPa were applied, the charge transfer increased by 1.4%, which is maintained until 23.33 GPa. This small difference can be seen by means of the electron clouds around the C and Pb atoms (figures 10(b) and (c)), which become slightly denser at 23.5 GPa. When the compound changes from the B3 to the B1 phase, at 23.5 GPa, there is a charge transfer increase (28.8%) which indicates that the electronic clouds are located at each atom ( figure 10(d)). This behavior may be due to the abrupt increase of the bond lengths, perceived at the structural analysis, and agrees with the shift of the VBs to lower energies. At 25 GPa, the charge transfer is maintained at 0.94 |e| and it begins to increase considerably until it increases by 7.4% at 50 GPa. This behavior may explain the  distribution of the DOS throughout the CB and VB. Additionally, the electronic clouds are not only located at each atom, also are distributed throughout the crystal (figure 10(e)); even the appearance of a small purple isosurface could indicate a tendency to metallic bonds.
On the other hand, as more positive is the E coh value, more energetically stable is the structure; and it was found that, at zero GPa (figure 11), the B3 phase's energy is 2.6 eV; at 5 GPa decreases slightly (0.6%). From that point, up to 23.33 GPa, the energy decreases by 7.2%. When the compound turns into the B1 phase, there is an energy discontinuity, with a decrease of 12.3%, compared to that for the B3 phase at P T . Then the energy decreases almost linearly, until it is reduced to 1.8 eV. The PbC's energetic stability decreases as the applied pressure increases, being the B1 phase's the highest rate of decrease. This behavior agrees with the enthalpy increase for the PbC (figure 4(a)).

Mechanical properties
The Born stability criteria (equations (3)-(5)) must be fulfilled for a correct prediction of the mechanical properties. The values required to verify the stability criteria are greater than zero (table 3), from the B3 to the B1 phase, passing through P T and from zero to 50 GPa; then the PbC is a mechanically stable system. For the B3 phase at zero GPa, the constant C 11 is greater than C 44 and C 12 , which indicates that the system has greater resistance along [100] direction. Therefore, the lowest resistance occurs under transverse expansion and shear forces. This behavior could be due to the distribution of C and Pb atoms in the B3 phase. From zero to 23.5 GPa, the C 12 , C 11 and C 44 constants increase 3 times, 67.8% and 11.7%, respectively ( figure 12(a)). When the PbC changes from the B3 to the B1 phase (at P T ), C 11 and C 44 increase 25.2% and 44.8%, while C 12 reduces 14.2%.  The B1 phase behavior is like that for the B3 phase: C 12 and C 44 increase 67.9% and 17.5%; the C 11 case is different, because increases 45.7% from P T to 45 GPa, but at 50 GPa only increases 2.9%. The behavior of the system, under pressure, indicates a greater resistance along the longitudinal compression than the transverse expansion and shear stress.
On the other hand, for the B3 phase at zero GPa the Y is greater than B and G. It si in accordance with the elastic constants (C 11 , C 44 and C 12 ). From zero to P T , the B and G values ( figure 12(b)) increase 1.3 times and 2.6%, respectively, from zero to 23.5 GPa. Meanwhile, the Y value increases 15.15% from 0 to 10 GPa, and only grows 0.76% at 23.5 GPa. For the B1 phase, at the P T , the G, Y and B moduli increase 92.9%, 79.4% and 7.7%, respectively. When a pressure of 45 GPa are applied on the B1 phase, B and Y increase 48.2% and 29.9%; while G increases 26.4%. At 50 GPa, B and Y only increase 7.1% and 0.4%, but G does not change. For the PbC, B is relatively linear for both phases and Y is greater for the B1 than for the B3 phase; meanwhile, the module with the lowest values is G, behaving in a similar way to C 44 .
At zero GPa the system is brittle with ionic-covalent bonds ( figure 13(a)), it could be for the interaction between C 2p-orbitals and Pb 6p-orbitals and the higher cohesive energy in the system. However, when a pressure of 10 GPa is applied, the system change to a ductile material with ionic-covalent bonds and when it reaches a pressure of 15 GPa, the behavior is metallic. At 23.5 GPa, the K values decrease to 0.32 and ν values increase to 0.36. At the P T , K increases abruptly 53.4% and ν decreases 26.3%, making the B1 phase a brittle  Table 3. Defined mechanical stability criteria for a cubic crystal, these must be greater than zero. The values are in GPa units. 0  145  241  60  5  158  314  64  10  166  382  67  15  169  445  67  20  170  509  68  23.5  170  551  67  B1  23.5  263  590  97  25  270  609  99  30  293  668  103  35  326  728  107  40  358  799  111  45  376  874  114  50  370  919  114 material with ionic-covalent bonds; at 50 GPa it becomes a ductile material ( = K 0.45) with a tendency to metallic bonds ( = v 0.30) which is reflected in the electronic cloud distribution throughout the crystal. On the other hand, at zero GPa, the H V value is 9.8 ( figure 13(b)); as the pressure is increased up to 23.5 GPa, the hardness of the B3 phase decreases 56.2%. When the phase changes to the B1 the hardness value increases up to 13.2, and this value does not change much until the pressure reaches 30 GPa; from this point, the hardness increases to 13.6 at 40 GPa, but decreases 6% at 50 GPa.
These properties indicate that, at zero GPa, the bonds of B3 phase are ionic-covalent, describing a brittle material with good stress along the longitudinal compression and good hardness. When the pressure reaches 23.5 GPa the interaction between atoms changes, and the bonds tend to be metallic; this reflects a state's dispersion, the increase of the Pb 6s-orbitals around 7.9 and −10 eV at the DOS, and the change from a brittle to a ductile material with lower hardness. At the transition from B3 to B1, the bond lengths' change and the charge concentration at the atoms raise, leading to a bonds' change from metallic to ionic-covalent and turning the material to a brittle one, with greater hardness than that at zero GPa but with higher H and lower E coh . When the pressure reaches 50 GPa, the DOS and the charges distribution favor a tendency to metallic bonds, causing a ductile material with good hardness. The reason that Y is higher for the B1 phase could be explained by the fact that the material is at the border between a brittle and ductile material.
As the pressure increases, the V m decreases ( figure 14(a)). For the B1 phase, from zero to 5 GPa, V m increases 0.4% but then decrease slightly until it is reduced 5.1% at 23.5 GPa. The θ D follows a similar trend, from zero to 10 GPa there is an increase of 3.1% but decreases to 297.8 K at 23.5 GPa. At the P T , there are abrupt increases: 29.9% and 35.6% for the V m and θ D , respectively; since increasing the bond lengths also increases the density for  the B1 phase. From 23.5 to 45 GPa there are increases: 7.6% for V m and 10.9% for θ D ; but at 50 Gpa V m and θ D only decrease 0.5% and 0.3%, respectively. This behavior is like that for Y; the increase for V m is due to the capacity of the material to deform along the [100] direction. The PbC has associated the highest θ D at 45 GPa, which is the pressure where the material also has associated the highest microhardness.
The A U shows that the B3 phase is an anisotropic material; from zero to 23.5 GPa, its value increases 80.9% ( figure 14(b)). At the transition to B1, there is a 91.3% decrease, bringing PbC closer to an isotropic material. The anisotropy starts to increase slightly up to 45 GPa, showing that the arrangement of the atoms varies along the [100] direction, but at 50 GPa the anisotropy is reduced to 7.7%. The PD shown six branches ( figure 15): three acoustic and three optical; additionally, no negative frequencies were observed, indicating that PbC is a dynamically stable system. At zero GPa ( figure 15(a)) it was observed that, at the L and X points, there are located the maximum frequencies of the acoustic branches which are in the low frequency region (LFR). Meanwhile, the lowest points of the optical branches, in the high frequency region (HFR), are located at the Γ and L points. Additionally, the phonon bandgap is of 6.7 THz. As pressure is applied to the B3 phase, the bands shift to higher frequencies. At 23.5 GPa ( figure 15(b)), the lowest frequency of the optic branches is at the Γ point and the frequencies at LFR and HRF increase by 13.1% and 13.4%, while the  phonon bandgap increases to 8.6 THz. The transition from the B3 phase to the B1 phase (figure 15(c)) almost does not change the frequencies of the optical branches along the path; the LFR decreases 11.48% and the HFR and the phonon bandgap increased 0.43% and 21.3%, respectively. At 50 GPa ( figure 15(d)), the frequencies for the B1 phase's branches continue increasing; the LRF, HRF, and the phonon bandgap grow by 5.1%, 7.0% and 10.5%, respectively.

Vibrational properties
At zero GPa, the partial phonon density of states (PPDOS) shown that Pb atoms are associated to lower frequencies than those linked to C atoms ( figure 16(a)), as it was expected considering the mass difference; but the density peaks are higher for C atoms. The main peaks for Pb atoms, located at 2.7 and 4.3 THz, have associated values of 0.35 and 0.26; while those for C atoms, located at 14.5 and 15.9 THz, are 0.6 and 0.4. For the B3 phase, when the pressure reached 23.5 GPa ( figure 16(b)), the PPDOS moved to higher frequencies; the phonon density for Pb atoms (in LFR) disperses 11.7%, and that for the C atoms (HFR) increases up to 35.7%. Consequently, the intensity peaks linked to the Pb and C atoms decreased 12.5% and 26.7%. At P T ( figure 16(c)), the PPDOS for the B1 phase disperses 10.8% in the LFR for the Pb atoms, but contracts 3.2% for the C atoms in the HFR; this latter contraction could be due to the contribution increase of the C 2p-orbital located at 7.31 and −0.2 eV. Similarly, the LFR peaks decreased 12.2% for Pb atoms, and the HFR peaks decreased 3.8% for C atoms. At the PPDOS was observed a increasing spread trend of the distribution, up to 50 GPa (figure 16(d)): 16.5% for the Pb atoms and 26.6% for the C atoms. On the other hand, the peaks associated to the Pb and C peaks changed 14.5%. The frequencies' increase for the C and Pb atoms, and the PPDOS distribution, could explain why the X-ray predicted the PbC's crystalline phases separation at 50 GPa.

Thermodynamical properties
The thermodynamic properties for the PbC were calculated to identify the effects of the pressure (zero to 50 GPa), and high temperature (0 to 1000 K) on its behavior. The zero-point energy, also known as the ground state energy, was calculated for the compound at different pressures: the values for the B3 phase at zero and 23.5 GPa are 0.1119 and 0.1368 eV; and those for the B1 phase at 23.5 GPa and 50 GPa are 0.1368 and 0.1547 eV. The entropy (S) is a physical magnitude linked to the disorder; then, it is known that the greater the entropy, the greater the disorder in a crystalline structure. The S increases, for the B3 phase at zero GPa ( figure 17(a)), as the temperature increases. This behavior is not only due to the increase of the kinetic energy of the atoms, also because of a wider distribution of this energy due to the bond lengths decrease. When the pressure was increased to 23.5 GPa, the B3 and B1 phases have associated the same value of S, but it is less than the value associated when the pressure is zero GPa; this fact could be due to the bond lengths' decrease and the kinetic energy change as a pressure effect. At 50 GPa and for the B1 phase, S values are slightly greater than those at 23.5 GPa, up to around 400 K; however, as the temperature rises, their values are very similar. This behavior could be due to the degree of structural order and by the separation of the crystalline phase at 50 GPa, which is worse than that at lower pressures. At zero GPa, the H ( figure 17(b)) increases as temperature increases and this is because U is directly related to T. Also, when the pressure value increased, H began to decrease because of the V decrease caused by the pressure; particularly, the B1 and B3 phases at 23.5 GPa have associated the same H values. The Helmholtz free Figure 16. Partial, and total phonon density of states for the PbC compound (pink), and the Pb (green), and C (blue) atoms. energy (F) is a thermodynamic potential that measures the useful work that can be obtained from a thermodynamic system. The F corresponds to the difference between U and the transferred energy amount (TS product). For the B3 phase, at zero GPa, F decreases as the T increases ( figure 17(c)); this means that at higher temperatures, there is a more disordered final state (higher S) and less work is required to create it after the energy transfer. At 23.5 GPa, the B3 and B1 phases have associated the same F, greater than those at zero GPa: at higher pressure values S decreases and F increases. At 50 GPa, the B1 phase's Helmholtz free energy values are slightly lower than those at 23.5 GPa, as T increases, due to the U difference for the PbC caused by the pressure effect.
On the other hand, C v for the B3 phase at zero GPa ( figure 18(a)) increase rapidly: proportional to T 3 , at low temperatures and, above room temperature (300 K), grows exponentially approaching the classical Dulong-Petit value. The C v values for the B3 and B1 phases at P T are the same, being lower than those at zero GPa and indicating that the phases absorb less thermal energy. Once 50 GPa are applied to the B1 phase, the C v values from 0 to 115 K are greater than those at 23.5 GPa; this behavior may be linked to the increase of the dispersion of the PPDOS. For temperatures above 115 K, the C v values are lower than those at the P T .
Instead, the Θ D value is 360 K for the B3 phase, at zero GPa, when T = 0 K ( figure 18(b)). The changes for Θ D could be divided in two zones: first Θ D decrease but rapidly increase; then, at the second zone, it reaches values around 600 K; and a slight increase is observed at the third zone. The better mechanical resistance was found above = T 370 K. At P T , the B3 and B1 phases have associated the same Θ D value, greater than that at zero GPa.   mechanical resistance was found above = T 417 K. When the pressure reaches 50 GPa, the Θ D value reaches a minimum and its values are greater above = T 110 K. Also, it was observed that the Θ D has a value around 880 K, starting the second zone, and the greatest mechanical resistance was located above = T 600 K.

Optical properties
The ε(ω) has two components and provides information about the effects of an electric field that passes through a material; the real part (ε 1 (ω)) indicates the polarization degree, while the imaginary part (ε 2 (ω)) specifies the energy dissipation. At zero GPa ε 1 (ω) is negative at low frequencies ( figure 19(a)), but turns positive from 3. There are negative ranges in the infrared, visible, and low-frequency UV regions; meaning that the dielectric response of the material at these regions is such that the electric field or light at these frequencies cannot diffuse. For the B3 phase, at zero GPa in the infrared region, the ε 2 (ω) exhibits a maximum and decreases exponentially ( figure 19(b)), having an intermediate peak at 4.6 eV; above 37.9 eV, the value of the ε 2 (ω) is almost zero. As pressure increases, the intermediate peak decreases and shifts to higher frequencies. At the frequency 5.4 eV, the peak for the B3 and B1 phases was found at 23.5 GPa, while the peak for the B1 phase at 50 GPa was found at 4.4 eV. Considering that the electric behavior of the PbC is metallic, the higher values for ε 2 (ω) are mainly due to the electronic transition between the C 2p and the Pb 6p-orbitals. Furthermore, the fact that ε 2 (ω) decreases, with increasing the frequency and its value reaches zero, evidences that the PbC is transparent and optically anisotropic. The energy loss function (L) is the quantification associated to a fast electron which passes through a material. For the B3 phase ( figure 19(c)), at zero GPa and low frequencies, the L has low values and increases at the UV region, forming three main peaks located at 15.2, 23.1 and 36 eV; for frequencies above 37.7 eV, the L values are very close to 0. As the pressure increases, the peaks of L move towards higher frequencies. The first two main peaks decrease and are located at 16.3 and 24.9 eV, for the B3 and B1phases at P T ; but those peaks are located at 16.8 and 26.1 eV for the B1 phase at 50 GPa. The third main peak increases always, for the B3 and B1 phases at 23.5 GPa are located at 38.5 eV and for the B1 phase at 50 GPa is located at 39.0 eV. The L positive values region is known as the bulk plasma frequency (ω p ); for the infrared and visible regions PbC is a refractive material (frequencies below ω p ) while is transparent at the UV region (frequencies above ω p ).
The absorption coefficient (I) provides information on the penetration of a wavelength into a material before it dissipates. For the B3 phase at zero GPa, as the frequency value increases in the infrared region, the value of I increases, until it reaches a peak at 0.5 eV (figure 20(a)), and then decreases. The same trend was observed in the visible region, but the peak was found at 2.6 eV and is smaller than that in the infrared region. Beyond the visible region, the values for I at the UV region increase, with a maximum peak at 7.9 eV; then I decreases, getting two I 102.7 eV; for the frequencies between these regions there is a small zone of adsorption where its main peak is at 82.8 eV. The B3 and B1 phases, at 23.5 GPa, have associated similar values for I at the infrared region, but these values decrease for the visible region. At the UV region the main peaks increase 12.2%, 16.5%, and 46.3%; the regions without adsorption increase and move to 41.5-80.2 eV and > I 104.7 eV. At 50 GPa, I shows the same trend: its values decrease at the visible region, by 14.1% and the peaks increase again by 9.4%, 14.5% and 13.53%. Additionally, the regions without adsorption increased 4.9%, and are located at 41.8-80.7 eV and > I 104.0 eV. This indicates that in the UV region (low frequencies), followed by the infrared and visible regions, PbC has good wave adsorption and, as the pressure increases, the I increases; therefore, under these conditions, it is a promising material for applications in optical and optoelectronic devices.
The reflectivity (R) is the capacity of a material surface to reflect an incident wave. For the B3 phase ( figure 20(b)), at zero GPa and low frequencies, the R has values close to 1 and begins to decrease exponentially, with two intermediate main peaks of 0.68 and 0.17 at 9.2 and 35.9 eV, respectively. When a pressure of 23.5 GPa is applied, the B3 and B1 phases exhibit the same exponential trend; however, the peaks move to higher frequencies, located at 10.4 and 38.3 eV with values of 0.68 and 0.33. At 50 GPa, two peaks associated with the B1 phase are located at 11.0 and 38.9 eV, with values of 0.67 and 0.51. The results obtained for R reinforce those obtained for L: in the infrared, visible and low-frequency UV regions, the PbC is a refractory material and, as the pressure increases, the applications area becomes broader at higher frequencies.
The refractive index (N) quantifies the wavelength variation and has two components: the real part (n), that gives the phase velocity, and the imaginary part (extinction coefficient, k) that quantifies the absorption loss of the wave. For the B1 phase at 0 GPa, the refractive index at low frequencies in the infrared region has associated very high values and drops quickly (figure 20(c)); at the visible region the values of n rise and, at the UV region, there is a peak at 4.3 eV of 2.5, then drops finding a minimum point at 36.1 eV and rises rapidly to stay at values close to 0.94, starting at 66.0 eV. Applying pressure, the values of n shift slightly to higher frequencies, and only the peak at low frequencies in the UV region increases. At the P T (B3 and B1 phases) this peak value is 2.8 at 4.9 eV, while at 50 GPa (B1 phase) it is 2.9 at 5.4 eV. Then, at the visible region and at low frequencies in the infrared and UV regions the phase velocity is bigger and, consequently, a high refractive index. At 0 GPa, the k values (figure 20(d)) indicate that the B3 phase (at low frequencies) has a higher absorption loss and an exponential decrement; an intermediate peak was found at 36.4 eV with a value of 0.11. When the pressure increases to 23.5 GPa, the B3 and B1 phases' behavior is like that at 0 GPa, but the peak moves to 38.8 eV with a value of 0.12. For the B1 phase case, at 50 GPa, the peak changes to 39.1 eV with a value of 0.11. This adsorption loss, at low frequencies, supports the behavior shown through I.
The real part of the optical conductivity, or photoconductivity (σ), increases the electrical conductivity due to the adsorption of photons which promote electrons to the unoccupied states. The imaginary part represents a phase lag of the charge carriers to respond to rapid changes of the photons and can be considered as a type of capacitance. For the B3 phase (at zero GPa), the optical conductivity real part (figure 20(e)) values are high, at low frequencies in the infrared region, and decreases to 2.5 at 1.05 eV. At the visible region there is a peak of 3.8 at 2.2 eV. At low frequencies, the highest σ values are at the UV region (the highest peak is 7.8 at 4.7 eV); however, from 37.7 to 80.2 eV and above 82.8 eV there are not optical conductivity. As the pressure increases, the σ values move to higher frequencies: at 23.5 GPa (B3 and B1 phases) the peak increases to 8.9 at 6.7 eV; and the regions without optical conductivity move to 40.2-79.9 eV and above 83.8 eV. Also, the σ values diminish at the visible region. Once the pressure reaches 50 GPa, the mentioned peak changes to 11.1 at 7.4 eV and the regions without optical conductivity are located between 39.9 and 81.5 eV and above 84.5 eV. This behavior suggests less σ at the infrared and visible regions, as the pressure increases, but more at low frequencies in the UV region. The imaginary part of the σ (figure 20(f)), at low frequencies in the infrared region for the B3 phase at zero GPa, has associated high values from it decreases; at 4.0 eV it has a minimum (0.2) and rises to a maximum peak of 5.7 at 7.9 eV; later, its values decrease. As the pressure increases, the magnitudes increase and move to higher frequencies. At the P T (B3 and B1 phases), the value at the valley point is −1.6 at 4.7 eV and the peak is 6.3 at 9.4 eV. For the B3 phase at 50 GPa, the valley value is −3.1 at 5.2 eV and the peak value is 5.3 at 37.9 eV. This behavior indicates more phase lag at low frequencies in the infrared and UV regions; furthermore, the phase lag is small at high frequencies in the infrared region and also at the visible region.

Conclusions
For this study, we performed ab-initio calculations based on the DFT, to determine many properties of the PbC under pressure, considering three possible phases: B1, B3 and Wz. It was found that, at zero GPa, the most thermodynamically stable structure is the B3 phase. The PbC is a metallic and anisotropic material with paramagnetic behavior, which is energetically, mechanically, and dynamically stable. It was found a P T at 23.5 GPa (B3 to B1 phase transition). The calculated lattice parameter, the volume, and the bands structure are in good agreement with a previously reported theoretical result at zero GPa. By applying pressure, the volume and bond lengths decrease, causing an increase of their electronic clouds near the atoms; the DOS disperses, there are greater participation of C 2p and Pb 6p-orbitals and a possible separation of the crystalline phase; these changes reflect on the conversion from a brittle material with ionic-covalent bonds and good hardness (values around 13 for H V ) into a ductile material with metallic bonds' tendency and increased hardness and Y. Also, as the pressure value increases, the density and the anisotropy of the PbC were increased, reducing V m ; however, above P T , the V m increased and the anisotropy decreased due to the deformation along the [100] direction. Additionally, the microhardness is the highest at 45 GPa. Higher frequencies are associated to the C atoms whereas two frequencies are linked to the Pb atoms and, as the pressure increases, the frequencies disperse increasing the phonon bandgap. At high temperatures S and H increase, but F decreases; as pressure increases, S decreases but above P T it decreases slightly. H decreases due to the volume decrease, and F increases; but at 50 GPa there is a greater U change for the B1 phase than that at P T . The PbC's maximum thermal adsorption (12 cal/cell K) and the highest mechanical resistance were found above = T 130 K; as the pressure increases, the C v decreases slightly and the maximum resistance was found at higher temperatures. The higher phase velocity is linked to the infrared, visible and low frequency UV regions, where the PbC is a dielectric and refractory material, whereas at the high frequency UV regions the material is transparent. Furthermore, as the pressure increases, the optical applications area is wider, mainly at high frequencies. Also, the PbC is a promising material for applications, in developing optical and optoelectronic devices; at infrared, visible and UV (low frequencies) regions. The PbC has associated the peculiarity that, as the pressure increases, is less photoconductive in the visible and infrared regions; but the photoconductivity increases at low frequencies in the UV region. Additionally, there is more phase lag at low frequencies in the infrared and UV regions, and a small phase lag at high frequencies at the infrared and visible regions.