A first-principles study of the electronic, mechanical, vibrational, and optical properties of the zirconium carbide under high pressure

Zirconium carbide is a compound widely used in cutting tools, nuclear reactors, field emitter arrays and solar energy receivers; additionally, combined with other materials, it can be used in rocket technology and the aerospace industry. For this work was studied the effect of the high hydrostatic pressure on the electronic, mechanical, vibrational, and optical properties of the ZrC, from first principles calculations based on the Density Functional Theory. The calculated enthalpy and cohesive energy data show a B1 (NaCl) to B2 (CsCl) phase transition at 297 GPa. For the B1 phase, results for the calculated equilibrium lattice parameters, bands structure, electron and phonon densities of states, elastic moduli constants, entropy, enthalpy, Gibbs free energy, heat capacity, reflectivity, loss function, conductivity, and dielectric function are consistent with the available experimental and theoretical data. Our results for phonons show that the B1 phase is dynamically stable; in contrast, the B2 phase is not stable. Furthermore, when pressure is applied, the calculated density of electronic states shows that the C 2p-orbitals around the Fermi energy contribute significantly to the conduction band, turning the compound into a ductile the material, with a mixture of metallic and ionic-covalent bonds. On the other hand, the study of the mechanical properties of the B1 phase shows a highest mechanical resistance and maximum thermal absorption, above 356 K and 638 K, respectively; but these switch to higher temperatures as pressure is applied. Finally, the B1 phase of the ZrC is a good coating material and a photon detector at low frequencies in the UV region, but also at the visible and infrared regions; although, increasing the pressure, the values of the optical properties increase. The increase of the parameters’ values of the studied properties, as the pressure increases, indicates that the ZrC could be more efficient in a wider range of applications.


Introduction
The transition metal carbides (TMC) have attracted attention because these are materials with peculiar thermal, electronic, and mechanical properties [1]; among these properties, the high melting point (around 3670 K [2]) is too notorious and allows to tag the TMC as refractory carbides [3]. Another properties linked to the TMC are the extreme hardness, the superconductivity, the magnetic susceptibility, and the electrical and chemical resistance; all these properties make those compounds very attractive for many technological applications, as: high-speed steel cutting tools, information storage, spintronic devices and protective coatings for optical and mechanical components. Also, those carbides are used in the petroleum and aerospace industries [3][4][5][6][7]. The more stable phase crystallization, for most of the TMC, is the rock salt type (B1), which exhibits a contribution mixture from

Theory
To analyze the structural properties of the ZrC, under pressure, the volume and lattice parameters were normalized, using as reference those for the systems at zero GPa, as it was described in [22,23]. The Gibbs' free energy relationship was used to determine the P T value associated with the ZrC's structural phase change; nevertheless, when T = 0 K, the Gibbs' free energy is equal to the enthalpy (H): where U is the internal energy, P is the pressure and V is the volume. The cohesive energy (E coh ) [24] per atom was chosen as the thermodynamic stability criterion, and this is defined by: being E Zr and E C the total energies for isolated Zr and C atoms; n and m the number of Zr and C atoms, respectively, in the system's cells; and E total is the ground-state total energy for the ZrC. The elastic constants (C ij ) were used to define the materials' mechanical resistance, as well as the stability of the elastic materials' phase. Therefore, three independent second-order elastic constants (SOEC) were determined: C 11 , C 12 , and C 44 ; and their SOEC'values must satisfy the following mechanical stability criteria [5]: ) () C C 0 3 11 12 + > ( ) () C C 2 0 4 11 12 > ( ) C 0 5 11 > ( ) C 0 6 44 On the other hand, the bulk (B 0 ) and shear (G) Hill's moduli provide information on the hardness of the material, concerning different types of deformation. In contrast, Young's modulus (Y) is a measure of the material's elastic behavior, subjected to a tensile or compressive force. The formulas to obtain these values can be found in many references [9,25]. The Pugh's and Poisson's ratios indicate the ductility and brittleness of a material [26]: Pugh's ratio values greater than 0.5 are linked to brittle materials, and lower values are linked to ductile materials [27][28][29]; the relevant values for the Poisson's ratio (ν) are those 0.33 for metals, while those from 0.16 to 0.30 are related to ionic-covalent compounds [27][28][29]. Vickers hardness (H V ) is a measure used to determine the hardness of a material [30], and the K, ν, and H V values can be obtained using: Thermal energy or external forces produce vibrations at the material's lattice. These mechanical waves called phonons are responsible for transporting heat and sound through the crystal [31]. The frequency (ω) dependence on the wave vector is known as the phonon dispersion (PD), and the phonon dispersion relations were achieved within the linear response theory to calculate the response to periodic perturbations [32,33]. The PD was obtained by finite displacements of atomic planes, using a combination of supercells, maximized by a cut-off radius of the force constants [34]. The phonon dispersion was calculated assuming that the mean equilibrium position of each ion in a BZ site, and the amplitude of the atomic displacements, are small compared to the interatomic distances. The phonons' properties are described using the harmonic approximation (equation (10)) based on the dynamical matrix related to the interatomic force constants [35]: The dynamical matrix (D αi,βj ) depends on the mass of the ith and jth atoms (M i M j ), the force constants (C αi,βj ) and the summation over the lattice vectors within the cut-off radius (R).
The finite displacement method allows to calculate the constant force matrix using the difference equation described in [35]. Usually, the phonons are described calculating their values at points in the first BZ [36]. There will be 3 N phonons per k-point. The lowest three modes represent the acoustic branch; these modes correspond to the translation of the crystal lattice and tend to a value of zero at the Γ point. This behavior is characteristic of sound waves. The remaining vibrational modes (3 N−3) are optical modes because they can interact with electromagnetic radiation in ionic crystals. Unlike the acoustic modes, the frequencies of optical modes are finite at the point q = 0. The crystal structure is dynamically unstable when the PD curves have imaginary (negative) frequencies or soft phonon modes [37].
The minimized Helmholtz free energy, concerning all its geometric freedom degrees {ai} without external pressure, can be found for any temperature (T) as F({ai},T) = U−TS. For the perfectly harmonic crystal, its U was replaced by the sum of the ground-state total energy (E tot ), and the second term was replaced by the vibrational free energy (E vib ) [38]: If anharmonic effects are not considered, the phonon frequencies do not depend on the lattice parameters; consequently, the free energy dependence on the structure is enclosed only in E tot . On the other hand, the zeropoint vibrational energy (E zp ), the phonon density of states (F(ω)), the Boltzmann constant (k B ), and the Planck constant (ħ) are part of the E vib (T): where E zp can be evaluated as The contribution from vibrational entropy (S) can be calculated using the following formula [35]: Furthermore, the lattice contribution to the heat capacity (C v ) can be obtained using: The empirical Debye temperature (Θ D ), which is obtained by experimental data, indicates that at higher temperatures the modes begin to be excited, and below this value, the modes start to freeze. Considering Θ D , we can obtain the following equation: where ω D is the maximum phonon frequency. Then, if x = ħω/k B T , we can write the C v as a function of the Debye temperature, as [39]: When electromagnetic radiation strikes a material, it can be absorbed, reflected or transmitted; and the relative extent of these effects is directly dependent on the material. The frequency-dependent complex dielectric function was employed for the calculation of optical properties, and this is defined by: e w e w e w = + ( ) ( ) ( ) ( ) i 18 1 2 were the second term (equation (18)) was obtained from the occupied and unoccupied electronic states (data taken from the moment matrix) [35], which is represented by the following equation: Where Ω represents the until cell volume, u is the polarization of the incident electric field and ψ k c and ψ k v are the wave functions for the conduction and valence bands, respectively. The first part of the equation (18) (ε 1 (ω)) can be obtained from the value of ε 2 (ω), using the Kramers-Kroning transformation. In addition, the adsorption spectrum, the loss function, the reflectivity, and the refractive index were evaluated as in [35].

Magnetic configurations
The electronic bands structure (EBS) allows to identify the electronic material's behavior (metal, semiconductor or insulator). The calculated EBS (this work) for the B1 phase shows a metallic compound. This behavior agrees with that found for the paths L-Γ and W-K-Γ, reported by Yang et al [2] (figure 1(a)) and calculated with the Vienna Ab initio Simulation Package (VASP) code; however, at point 'X' the two bands close to the E F , calculated with the VASP code, differ. This may be due to the approximation and functionals taken into consideration since no significant changes in the partial and total densities of states (PDOS and DOS, respectively) are observed near the E F . The EBS for the B2 phase ( figure 1(d)), also defines a metallic compound.
The PDOS and DOS are shown for the different MCs (figures 1(b), (c), (e) and (f)). To model the B1 and B2 phases, the formal charges and spin channels were considered to generate the FM and the A, C, and G AFM initial magnetic types. Once the ground state of each system was reached, the DOS for the FE and the AFM cases shows the same states' contributions from the up and down spin channels. For the B1 and B2 phases (FE and the AFM cases), states' contributions from the up and down spin channels are the same. The B1 phase has a higher From the criterion proposed in [40,41], to determine the magnetic state through the calculated values for the 2 * Integrated Spin Density (2 * ISD) and the 2 * Integrated |Spin Density| (2 * I |SD|), it is possible to indicate that all the MCs (for both phases) are paramagnetic, because the values of 2 * ISD and 2 * I |SD| are zero for all cases. That criterion is supported by the fact that all MCs have associated the same total energy value, and contributions to PDOS (FM and AFM cases) from both spin channels are the same.

Thermodynamic stability and structural parameters
From the obtained enthalpy values (figure 2(a) and table 1) and the enthalpy difference between the B1 and B2 phases (figure 2(b)), as pressure is applied, it is possible to identify that the B1 phase is more stable than the B2 phase, at zero GPa; that agree with many other reported studies [4,11,[42][43][44][45][46], and the behavior remains until the exerted pressure is more significant than 297 GPa. Then, the B2 phase becomes more stable; therefore, P T was identified as the pressure where the transition from B1 to B2 occurs (figure 2(a)). That value is almost the same than those reported in references [11,42,43] but lower than those pressures reported in [4] (525 GPa) and [47] (358 GPa). These differences can be attributed to the different pseudo-potential used.
Among the calculated lattice parameter, volume, and enthalpy values, obtained for both phases as a function of the applied pressure (table 1), the a lattice parameter values agree with values reported in other works [4,11,[42][43][44][45][46]: 4.075 Å for the B1 phase (at zero GPa) and 2.919 Å for the B2 phase.
The normalized lattice parameter and volume for ZrC exhibit discontinuities at P T (figure 3(a)), which indicates a phase transition. For the B1 phase, from zero GPa to P T , the a lattice parameter and the volume decrease 15% and 40%, respectively; then, from P T to 500 GPa (B2 phase), those decrease 5% and 14%, respectively. Furthermore, from zero to 297 GPa, the Zr-C and Zr-Zr bond lengths decrease 15% (figure 3(b)) for the B1 phase. The Zr-C bond length increases abruptly by 6%, but the Zr-Zr bond length decreases by 13%. From P T to greater pressures (B2 phase), the Zr-C and Zr-Zr bond lengths drop by 5%; this behavior indicates the stabilization of the structure.

Electronic properties
The bands structure provides information that allows to identify a system as conductor, semiconductor or insulator. It has been observed that the ZrC ceramic material has a metallic nature [5]. It does not change its metallic behavior due to the applied high pressure (figure 4). At zero GPa (B1 phase), the CB closer (CB-1) to the Fermi energy (E F ) has positive energy values only at the Γ point, while the VB closer to E F (VB + 1) has a value of 0.4 eV at the Γ point ( figure 4(a)). When the structure is subjected to 297 GPa (figure 4(b)), the CB valley moves to the top of the VB crest and, at P T , the VB crest values vary from 1.0 (B1 phase) to 2.4 eV (B2 phase). On the other hand, the CB valley splits into two: one at 2.1 eV (between the Γ and X points) and the second at 1.1 eV, closer to the E F , between the R and Γ points (figure 4(c)). The same trend is observed for the B2 phase at 500 GPa (figure 4(d)): the VB crest value increases until it reaches 2.9 eV, and the CB valley energy decreases to 0.8 eV. At the partial (per orbital), and total densities of states (PDOS and DOS, respectively; figure 5) for ZrC, it is possible to observe a states' bunching at −2.3 eV (VB, at zero GPa). This contribution is mostly provided by the C 2p-orbitals (52%) and Zr 3d-orbitals (38%). Three areas at the CB are notable: the highest states' bunching, between 4 and 5 eV, followed by a peak around 2.6 eV and another at 1.1 eV. These electronic states come from the Zr 4d-orbitals, which contribute 80%, and the C 2p-orbitals which contribute 17% ( figure 5(a)). Furthermore, the B1 phase's electronic states at the DOS (297 GPa) are localized mostly at four zones around −2.7, 4.0, 5.0, and 5.3 eV ( figure 5(b)). The VB peak located at −2.7 eV is reduced by around 51%, with respect to that of the DOS at zero GPa (figures 6(a) and (b)). Likewise, the states' density at the CB, between 4 and 5 eV, was reduced by 50% and located at 5.3 eV. This redistribution is observed on the PDOS: contributions from C (41% from C 2p-orbitals) and Zr (47% from the Zr 4d− and 10% from the Zr 4p-orbitals) to the VB; being the C 2p-orbitals and the Zr 4d-orbitals closer to the E F . Moving away from E F (around 5.6 eV), the contribution of the Zr 3d-orbitals increases to 67%, and the C 2p-orbitals' contribution decreases to 22%. At P T the structure changes to the B2 phase (figure 5(c)); then, the DOS accumulation at −2.7 eV (VB) moves to −0.5 eV (figures 5(b) and (c)). Closer to E F , the states' concentration is provided by the C 2p-orbitals (55%), Zr 4d, and Zr 5s-orbitals by 35 and 10%, respectively. For the CB, at 5 eV, the Zr 4d (56%) and 5 s (14%) orbitals contribute the most. The accumulation of states at the VB decreases by 18%: 46% from C 2p-orbitals and 39% from Zr 4dorbitals ( figure 5(d)). On the other hand, the highest accumulation at the CB decreases by 54%; in this zone, the largest contribution comes from the Zr 4d and 5s-orbitals by 71% and 15%, respectively. The decrease of the  [44]. c [45]. d [11]. e [42]. f [46]. g [43]. h [4].
concentration of states at the DOS indicates a greater distribution of states along the CB and the VB; in addition, the displacement of the concentration of states from the VB to values closer to the E F from the C 2p-orbital is observed. The Mulliken population analysis and the electron density difference could help to understand the interaction between the atoms in the system; both were computed ( figure 6(a)). It is observed that, at zero GPa ( figure 6(b)), the C atoms gain a charge of −0.79 |e| while the Zr atoms exhibit a charge of 0.79 |e|, indicating a strong coulombic interaction (ionic character). According to other works [2,44], for which this behavior manifests, a covalent-ionic bond has been reported. C atom's charge increases to −0.8 |e|, at 100 and 200 GPa, indicating that there is a greater interatomic force at these pressures. At 297 GPa, the C atoms charge decreases to 0.79 |e|; there is also a charge gain at the Zr-C bond (figure 6(c)) compared to charges at zero GPa; this is supported by the decrease of the bond lengths and the greater states distribution around E F described for the PDOS. An abrupt 7.6% decrease of the C atoms' charge is observed when the phase changes to B2 at P T . Compared to the phase B1, the B2 phase exhibits a charge accumulation between Zr atoms and a charge loss between the C atom and their Zr neighboring atoms (figure 6(d)); these gain/loss of charge explain the abrupt change of the Zr-Zr (decrease) and Zr-C (increment) bond lengths. The 0.73 |e| is maintained from P T to 500 GPa; however, there is a charge increase around the C atom ( figure 6(e)), indicating a greater contribution of  the 2p-orbitals around E F and, consequently, the decrease of the Zr-C bond length. For the case of charge increase and bond decrease between Zr atoms, they cause an increase in the distribution of their 4d-orbitals.
The E coh is a measure of the energetic stability of a system, and is defined as the electrostatic energy that holds all atoms together to build up a structure. The ZrC stability diminishes as the applied pressure increases (figure 7), and this behavior follows its enthalpy increase. The cohesive energy decreases 35% for the B1 phase, when the pressure goes from zero to 297 GPa. There is an abrupt energy change (19%) at P T , reaffirming the transition from the B1 phase to the B2 phase. From 297 to 500 GPa, a similar cohesive energy diminishing trend is observed; the energy is reduced by 50%.

Mechanical properties
The ZrC fulfillment of the mechanical stability criteria (equations (3)- (6)) is essential for the correct prediction of the structures' mechanical properties under pressure. The calculated elastic constants' values (table 2) for ZrC, at zero GPa, are in good agreement with results from theoretical [5] and experimental [48] studies on the ZrC. The C 11 and C 12 values of the elastic constants for ZrC gradually increase, up to 297 GPa; for the C 44 case, a slight decrease is observed. It is evident a transition from the B1 phase to the B2 phase, since the C 11 value decreases up to 75%; at the same time, the C 12 and C 44 values increase 2.5 and 3 times, respectively ( figure 8(a)). At 400 GPa, C 12 decreases 24%, C 11 rises 4 times up the initial value, while C 44 only changes 9%. The reported experimental  B 0 at zero GPa has a value of 223 GPa [49], which is consistent with our results (table 2). B 0 holds a linear increase up to 297 GPa; moreover, G and Y exhibit similar behavior since these show a maximum value at 200 GPa and decrease slightly at P T ( figure 8(b)). When the ZrC structure changes from the B1 to the B2 phase, an increase (of 1%) is observed for B 0 , while G and Y increase more than twice their values. B 0 shows the same behavior for both phases (B1 and B2); in contrast, other modules have a more irregular behavior. G and Y increase 6% and 5%, respectively, from P T to 300 GPa; but decrease to 39% and 34% at 400 GPa. However, both moduli increase 35% their values when the applied pressure reach 500 GPa.
The Pugh's ratio (K ) values exhibit the brittleness of the B1 phase, at zero GPa, as well as their turning to ductile with increasing pressure (figure 9(a)). The B2 phase, is brittle at pressures of 297 and 300 GPa, but ductile at 400 and 500 GPa. The B1 phase is ionic-covalent, at zero and 100 GPa, while it is metallic at 200 and 297 GPa ( figure 9(a)). Additionally, the calculated Poisson's ratio (ν) agrees with the electronic properties described. On the other hand, the B2 phase is ionic-covalent at 297 and 300 GPa, but metallic for the remaining pressures. Then, the material is brittle at low pressures, due to the high interatomic forces involved to bond the atoms. Furthermore, as the pressure increases, the bonds weaken, the charges are delocalized; however, accumulating states near the E F make the material more ductile. On the other hand, H V values gradually decrease until 59% at 297 GPa (B1 phase); however, when the phase changes to B2, H V increases 5 times its value ( figure 9(b)). As could be seen at the rest of the mechanical properties, from P T , the material stabilizes.

Vibrational properties
The PD, associated to the ZrC under pressure, was investigated at high symmetry points and along the lines that connect these points    figure 10(a)). No imaginary phonon frequencies or soft phonon modes were observed at the whole BZ, indicating dynamical stability, and agreeing with another reported studies [36,50]. The optical mode frequencies at the Γ point are 6.78, 7.88, 12.17, 13.32, and 15.66 THz. The dynamical stability of the B1 phase is backed by the PD at 297 GPa ( figure 10(b)). The branches occupy a wider interval at the low-frequency region (LFR) but are localized at a compressed one at the high-frequency region (HFR), as the applied pressure rises. Frequencies increase at the Γ point when the pressure reaches P T . Frequencies of 6.79 and 10.67 THz are in the LFR; while 25.43, 26.30, and 27.95 THz are in the HFR. At 297 GPa, when occurs the transition to the B2 phase, the system is dynamically unstable due to negative frequencies in the acoustic modes. On the other hand, the frequencies of the optical mode approach each other to values of 22.7, 22.9, and 23.1 THz at the Γ point. ZrC is also unstable at 500 GPa; however, negative frequencies only occur along the Γ-M-X-Γ path ( figure 10(d)). The calculated partial phonon density of states (PPDOS), for ZrC at zero GPa, is divided into two frequency regions ( figure 11(a)): the LFR (from 1 to 9.9 THz) and the HFR (from 12 to 20.1 THz). The Zr atoms' vibrations contribute to the phonon branches in the LFR, with a maximum peak at 6.4 THz; meanwhile, the C atoms'   figure 11(b)), the LFR is 67% wider, with a maximum at 12 THz; however, the HFR decreases to 69%, with a maximum located at 26.5 THz. Also, the forbidden phonon band increases almost four times. At P T , from the B1 to the B2 phase ( figure 11(c) figure 11(d)), unlike the other pressures, Zr and C atoms contribute to the vibrations in the LFR; these vibrations intersect at 20 THz. On the other hand, the C atom's vibrations contribute to the HFR. Furthermore, the maximum peak for the LFR is at 8 THz and the corresponding for the HFR is at 34.4 THz, contributed by Zr and C vibrations, respectively. The bond lengths decrease, as the pressure increases, generating a vibration unbalance on each atom. C atoms increase their frequency more than the Zr atoms, because the atomic C mass is smaller than the Zr atomic mass. Then, vibrations are an indicator that neighboring chains between C atoms cause the negative frequencies in the G-X, X-M and M-G lines, along the   Figure 11. Partial, and total phonon density of states for the ZrC compound (green), and the Zr (blue), and C (gray) atoms.

Thermodynamical properties
The PPDOS is necessary to evaluate the dependence of the entropy (S), enthalpy (H), Helmholtz free energy (F), heat capacity (C v ), and Debye temperature (Θ D ) of a crystal in the quasi-harmonic Debye model [39], on temperature. Then, the phonon contribution to the F, H, S, and C v at temperatures from 0 to 1000 K is shown at figure 12. The calculated zero-point energies (B1 phase) are 0.5383 and 0.9330 eV, at 0 and 297 GPa, respectively; these values agree with those reported in [51]. For the B2 phase, the obtained zero-point energies are 0.1929 and 0.2386 eV, at 297 and 500 GPa, respectively.
The calculated values for S and H increase continually (figures 12(a) and (b)), as temperature increases; this trend also was reported in [36]. However, those values drop at the transition (from B1 to B2) driven by the applied pressure. The entropy associated to B2 is smaller than that associated to B1 ( figure 12(a)), notoriously at high temperatures; this fact was reported, also, in [52]. In other words, the B2 phase lattice order is better than that for the B1 phase under pressure at high temperatures. On the contrary, the F gradually decreases with increasing temperature, but increases under pressure ( figure 12(c)), after the transition. This result indicates that the B1 phase is more stable than the B2 phase at high temperatures.
The C v increases with the temperature but decreases abruptly at the transition (from B1 to B2, figure 13(a)). First, when the temperature is lower than 500 K, C v increases rapidly obeying the well-known Debye model; later, for temperatures higher than 500 K, C v increases slowly with the temperature and tend to the asymptotic limit (so-called the Dulong-Petit limit) of 46.4 cal cell −1 K, which agrees with many other reported studies [52][53][54]. On the other hand, for the B1 phase at room temperature, C v depends on the pressure; when pressure varies from 0 to 297 GPa, C v changes from 24.6 cal cell −1 K to 5.9 cal cell −1 K. At zero GPa (B1 phase), the zero Θ D is 329.5 K ( figure 13(b)). As the temperature varies from 0 to 1000 K, the Θ D increases quickly and then reach a constant value at about 286 K. At 297 GPa, the zero Θ D is 0 K, it increases dramatically at 15 K; however, it decreases and stabilizes as a constant at 407 K. For the B2 phase, the zero Θ D are 198.5 and 282 K for 297 and 500 GPa, respectively; Θ D increases rapidly up to room temperature and increases steadily from this point.

Optical properties
The absorption coefficient provides information to know how far the light, with a specific wavelength, penetrates a material before it is dissipated; therefore, this coefficient provides helpful information on the ideal efficiency for solar energy conversion [55]. The ZrC at zero GPa [56] exhibits two absorption peaks (at 10.3 and 31.2 eV, figure 14(a)) in the UV region [53]; besides, the peaks in the visible region are very weak. Once the pressure reaches the value 297 GPa, the absorption in the UV region increases: four peaks are found at 11, 16.1, 32.3, and 34.8 eV. Above P T , the absorption by the B2 phase slightly decreases, in a region between 11.5 and 40.6 eV. The UV absorption region grows at 500 GPa, ranging from 7.6 to 42.2 eV, and the absorption values also increase. Therefore, at high pressures, ZrC increases the photon adsorption range, indicating that it could be used for optical and optoelectronic devices in the UV region.
On the other hand, being the reflectivity the capability of a surface to reflect the incident radiation [57], it was found that the highest reflectivity peak is in the infrared zone (at 0.01 eV, value of 0.99 for zero GPa,    figure 14(b)); but the reflectivity decreases considerably, reaching a minimum at 25.1 eV, in the UV zone. Only two great peaks were found for higher frequencies: at 31.8 eV (value of 0.4) and 35.3 eV of (value of 0.45). From 0 to 8 eV, the calculated reflectivity agrees with that reported by J F Alward et al [58]. There is a reflectivity decrease, at low frequencies, for the B1 phase at 297 GPa; however, the reflectivity increases at the UV zone, with a valley at 22.6 (0.1) between two peaks at 18.9 eV (0.61) and 36.4 eV (0.57). For the B2 phase, the reflectance increases, starting from the infrared zone to frequencies of 42.1 eV (value of 0.62) and 45.1 eV (value of 0.18) for pressures of 297 GPa and 500 GPa, respectively. The high reflectivity values at low frequencies reveal a refractory material; and, as the pressure increases,the possibility of use this as a coating material increases.
The energy loss function is a dielectric parameter that describes the collective excitations produced by the swift charges. The frequency at the maximum energy loss is known as the bulk plasma frequency (ω p ) and occurs at ε 1 = 0 and ε 2 < 1 [59]. For frequencies below ω p , the materials are reflective; whereas, above this frequency, the material is transparent [60]. The maximum bulk plasma frequencies ( figure 14(c)) were found at 36.  [62]. The shift of ω p to higher frequencies, and the energy loss function values, depend on the applied pressure rise.
An incident wave on a material could be refracted or bent, and the wavelength variation amount is quantified by means of the refraction index; their real and imaginary (extinction coefficient) parts indicate the phase velocity and the attenuation amount, respectively [57]. At zero GPa, there is a higher phase velocity in the infrared spectrum (0.01 eV), decreasing considerably at higher frequencies (figure 14(d)); however, there are two peaks within the visible (2.9 eV) and UV (4.5 eV) regions. At 297 GPa, the B1 phase maintains the highest phase velocity in the infrared region, but there is an 65% increase of the phase velocity and two peaks shift to frequencies of 1.2 and 3.3 eV. The B2 phase's peaks show another change at 3.8 and 5.3 eV frequencies, at the phase transition. For 500 GPa, there is another 24% increase for these two peaks and a change at 2.1 and 5.2 eV frequencies.
On the other hand, the greatest wave attenuation (figure 14(g)) made by the ZrC at zero GPa was found in the infrared region, followed by another in the visible region. At the UV region, waves are less attenuated, except those at 31.2 eV; those of 42 eV are almost nil. At the pressure of 297 GPa, attenuation increases around 40% in the UV region; and a plateau was formed between 25.5 and 34.5 eV. Additionally, there is almost no attenuation at 50.2 eV. When phase changes to B2 at P T , the attenuation of waves decreases as it was observed for B1, but the decreasing trend occurs without fluctuations. At 500 GPa, attenuation increases 20% if it is compared to that at 297 GPa.
The optical conductivity (or photoconductivity) is the optoelectronic phenomena which increases the electrical conductivity, as a result of the photons' absorption by the electrons, at the occupied states and the excitement of those into unoccupied states [55]. From the real and imaginary parts of the studied ZrC's photoconductivity (figures 14(e) and (h), respectively) was found that photoconductivity is higher in the infrared region, indicating the metallic nature of the ZrC [63]. The real part of the conductivity (at zero GPa) decreases after the infrared region, except for two peaks in the UV region (7 and 30.6 eV). The obtained peak at 3.1 eV has been reported in other works [58,64], while the peak at 7 eV is close to that reported by A Delin et al (6.5 eV) [65]. At 297 GPa, the conductivity after the infrared region increases by approximately 60%; however, our results are larger by 26% and 50%, respectively, and shifted to frequencies of 9.9 and 31.8 eV. Once the compound changes to the B2 phase, the first peak shifts to 5.9 eV, and the conductivity steadily decreases having four intermediate peaks between 25 and 40 eV. At 500 GPa, only the first peak was found, at 5.9 eV and increased 45%. Regarding the imaginary part (at zero GPa), two peaks are observed in the UV region (10.4 and 31.5 eV) and a valley (28.1 eV) between these. Three peaks (11.2, 16.3 and 35.6 eV) are formed when pressure is 297 GPa, and the valley moves to the visible region (3.6 eV). Above P T , for the B2 phase at 297 GPa, there are a conductivity reduction (20%), a slight displacement of the three peaks (11.7, 15.6, and 35.7 eV) and the formation of two valleys (3.6 and 5.1 eV) with negative values. At 500 GPa, a stable conductivity is noted from 8.5 to 43 eV and a valley at 5 eV.
The dielectric function provides information about the effects of the energy loss and the polarizability of a material while an electric field passes through it: the real part ε 1 (ω), indicates the polarization extent and the imaginary part ε 2 (ω) its absorption (equation (19)). The material is transparent if ε 2 (ω) = 0 and absorption begin when ε 2 (ω)≠0 [66]. At low frequencies, ε 1 (ω) exhibits negative values ( figure 14(f)) and, passing through the visible and UV zones, exponentially tends to zero. This behavior is the same for all the studied pressure range and the two phases. In addition, the obtained results show a zero upward crossing at 20.2 eV, which is similar to that reported by F Espinosa et al (19.4 eV) [62]. This crossover indicates a well-defined plasmon at the dominant peak observed by means of the loss function.
Regarding the imaginary part of the dielectric function ( figure 14(i)), the behavior is the opposite of that for the real part; the former tends to zero as the frequency increases. This behavior implies that the ZrC, at high frequencies, becomes almost transparent with very little absorption; and this agree with the loss function. Other works, also, verify the well-defined shape of the ε 2 (ω) curve [58,64].

Conclusions
We have performed ab-initio calculations, based on the DFT, to study the structural, electronic, mechanical, vibrational, thermodynamical, and optical properties of the ZrC, under high pressure, for the rock salt (B1) and CsCl (B2) structures. The ZrC system shows a paramagnetic behavior at zero GPa. This B1 system is structural, thermodynamical, energetical, and dynamically stable; however, it is less stable as the pressure increases. A phase transition (from B1 to B2) was identified: at the pressure 297 GPa (P T ); above this pressure, the system is dynamically unstable.
The lattice parameters and volume decrease as pressure increases. The calculated lattice parameters, electronic bands structure, phonon dispersion curves, elastic, thermodynamic, and optical values are in good agreement with experimental and theoretical results published elsewhere for the B1 phase. The band structure, DOS, and electron density difference show that the B1 and B2 phases are metallic in the studied pressure interval.
The Poisson's ratio indicates that the ZrC has ionic-covalent bonds, and, as the pressure increases, the bond lengths decrease, favoring an increase of the C2p-orbitals near the EF. Pugh's ratio and Vickers hardness indicate that ZrC is a ceramic material (hard and brittle) and tend to be ductile under pressure. C atoms' vibrations contribute to HFR while Zr atoms' vibrations impact the LFR. As pressure increases, the phonons' band gap decreases; additionally, at 500 GPa, the C and Zr atom's vibrations contribute to the LFR. S, H, and θ_D values increase as the pressure increases while those for F and C_v decrease. This compound presents its highest mechanical resistance T > 356°K and the maximum thermal absorption (46.4 cal cell −1 ) T > 638°K but when pressure is applied, these are found at T > 1000°K and the máximum termal absorption decrease to 46.4 cal cell −1 . ZrC is a refractive material below 36.3 eV and is transparent above this frequency but increases to 45.1 eV as it increases to 500 GPa; it has associated a high phase velocity, wave attenuation and photoconductivity in the infrared interval and it has a defined plasmon in the UV region. In addition, ZrC is a good coating material and a photon detector in the UV region. Also, it was observed a shift of the frequencies towards higher values and increments of the values for the optical properties, as the pressure increases.