Ab initio predictions of structure and physical properties of the Zr2GaC and Hf2GaC MAX phases under pressure

The electronic structure, structural stability, mechanical, phonon, and optical properties of Zr2GaC and Hf2GaC MAX phases have been investigated under high pressure using first-principles calculations. Formation enthalpy of competing phases, elastic constants, and phonon calculations revealed that both compounds are thermodynamically, mechanically, and dynamically stable under pressure. The compressibility of Zr2GaC is higher than that of Hf2GaC along the c-axis, and pressure enhanced the resistance to deformation. The electronic structure calculations reveal that M2GaC is metallic in nature, and the metallicity of Zr2GaC increased more than that of Hf2GaC at higher pressure. The mechanical properties, including elastic constants, elastic moduli, Vickers hardness, Poisson’s ratio anisotropy index, and Debye temperature, are reported with fundamental insights. The elastic constants C11 and C33 increase rapidly compared with other elastic constants with an increase in pressure, and the elastic anisotropy of Hf2GaC is higher than that of the Zr2GaC. The optical properties revealed that Zr2GaC and Hf2GaC MAX phases are suitable for optoelectronic devices in the visible and UV regions and can also be used as a coating material for reducing solar heating at higher pressure up to 50 GPa.

www.nature.com/scientificreports/ where n m is the thermally excited number of electrons and n e is the total number of valence electrons in the unit cell. k B and N(E F ) are the Boltzmann constant, and value of DOS at Fermi level in unit states/eV/unit cell, respectively. The Fermi energy of M 2 GaC MAX phases are used to estimate the velocity of electron ( v F ) near the Fermi level: Then use this value to estimate the conductivity (σ) as: where τ , m, n, e, and l are the time between two collisions, mass of electron, number of electrons, electron's charge, and mean free path of electron, respectively.
To calculate the bulk, shear, and Young's modulus, the following equations used within the Voigt (V) 57 , Russ (R) 58,59 , and Voigt-Russ and Hill (VRH) 60,61 approximation scheme: where Bv, Gv, and Br, Gr are the bulk and shear modulus in terms of Voigt and Russ approximation, respectively. The Values of Young's modulus and Poisson's ratio obtained by: The mechanical Anisotropy (A) calculated as follows: To calculate the hardness, the semi-empirical method based on Pugh's ratio, which is proposed by the Chen et al. 62 , was adopted as: where k is the Pugh's ratio (G/B), and G is the shear modulus. The Debye temperature determined by the Anderson method 63 : The transverse sound velocity (V t ), longitudinal sound velocity (V l ), and mean sound velocity (V m ) calculated by: (5) f m = n m n e = k B T × N(E F ) n e = 0.026 × N(E F ) n e (2(C 11 + C 12 ) + 4C 13 + C 33 ) (9) B R = (C 11 + C 12 )C 33 − 2C 2 12 (C 11 + C 12 + 2C 33 − 4C 13 ) (10) G V = 1 30 (C 11 + C 12 + 2C 33 − 4C 13 + 12C 44 + 12C 66 ) (11) G R = 5 2 (C 11 + C 12 )C 33 − 2C 2 12 2 C 55 C 66 3B V C 55 C 66 + (C 11 + C 12 )C 33 − 2C 2 12 2 (C 55 + C 66 ) www.nature.com/scientificreports/ where h is Planck's constant, k B is Boltzmann's constant, n represents the number of atoms per unit cell, V a is the atomic volume, respectively. The melting temperature (T m ) of MAX phase materials having hexagonal crystal structure was calculated from the elastic constants as follows 64 : The 3D Young's modulus surface was obtained using the following equation 65 : where S ij and l i represent the elastic compliance tensor of M 2 GaC MAX phases and direction cosine in the sphere coordination, respectively. The value of the imaginary part can be calculated from the moment matrix element between the occupied and unoccupied electronic states as: where e is the electronic charge, ω is the light frequency, u is the vector defining the polarization of the incident electric field, ψ c k and ψ v k are the conduction and valence band wave function at k, respectively.

Results and discussion
Structural properties and compressibility. The  where z M is known as the internal parameter. Figure 1 shows the optimized unit cell of the M 2 GaC MAX phase at 0 GPa pressure. The optimized parameters for each pressure up to 50 GPa with the pressure step of 10 GPa is tabulated in Table 1. With the increase in pressure, the lattice parameters, unit cell volume reduced while the internal parameter increases for both the MAX phases studied. The thermodynamic stability of M 2 GaC MAX phase materials is predicted in terms of formation enthalpy H cp by comparing the total energy M 2 GaC MAX phase to the energy of non-MAX competing phases (single elements, binary and ternary compounds). Table 2 shows the most competing phases considered for M 2 GaC MAX phases determined by using linear optimization procedure 66 . This linear optimization procedure has been successfully used for predicting many MAX phases 67,68 in which a phase is considered to be stable if �H cp < 0 . Based on identified competing phases, the Eq. (1) for Zr 2 GaC and Hf 2 GaC MAX phases can be rewritten as:  www.nature.com/scientificreports/ Both the Zr 2 GaC and Hf 2 GaC phases fulfill the criterion �H cp < 0 , indicating that the M 2 GaC MAX phases are thermodynamically stable and can be formed experimentally. Moreover, the calculated formation energy per atom for Zr 2 GaC and Hf 2 GaC at 0 GPa is − 7.59 eV/atom and − 7.45 eV/atom, respectively. There has been a increase in energy of formation for Zr 2 GaC and Hf 2 GaC as pressure is increased. Figure 2 shows the effect of pressure on the normalized lattice parameters a/a 0 , c/c 0 and volume V/V 0 ( a 0 , c 0 , and V 0 are the lattice parameters and volume at the 0 GPa pressure). The compressibility for both the M 2 GaC MAX phase along the c-axis is more than that of the a-axis as pressure increases from 0 to 50 GPa 32,33 . Similarly, the volume change ratio gradually (V/V 0 ) decreases with an increase in pressure, indicating that the compressibility of the M 2 GaC MAX phase system is strong. The compressibility results are in good agreement with other Zr and Hf based MAX phases 37,39,69 . Moreover, the V/V 0 of Zr 2 GaC and Hf 2 GaC was reduced to 21.1% and 19.1%, respectively. In other words, external pressure has a more significant effect on Zr 2 GaC than the Hf 2 GaC MAX phase. Zhao et al. 70 studied the properties of Ti 3 AC 2 (A = Al, and Si) MAX phases under pressure in which the ratio of V/V 0 of Ti 3 AlC 2 and Ti 3 SiC 2 was reduced to 18.7% and 16.9%, respectively. It is worth noticing that at high pressure, the volume ratio curve becomes steady, indicating that change in atomic distance is smaller, which results in stronger mutual repulsion as atoms come further closer; eventually, compression of the crystal becomes more difficult. Similarly, the reduction in lattice parameter ratios ( a/a 0 , c/c 0 ) are in the order of Zr 2 GaC > Hf 2 GaC. Figure 3 exhibits the normalized bond lengths l 1 l 10 and l 2 l 20 (where l 10 and l 20 are the bond lengths of M-Ga and M-C at 0 GPa, respectively) M-Ga and M-C atoms within the M 2 GaC (M = Zr, Hf) MAX phase unit cell versus pressure. It can be noted that the bond length M-Ga (M = Zr, Hf) becomes steeper than that of M-C, indicating that the direction along the M-Ga is easily compressed compared to the M-C bond. These results agreed with a weaker metallic bond between M-Ga atoms in the unit cell, which defines the lattice parameter c. The bond lengths of Zr-Ga and Zr-C reduced more than Hf-Ga and Hf-C, exhibiting that the Zr 2 GaC MAX phase is more compressible than Hf 2 GaC along the Zr-Ga direction.
Furthermore, the c/a and the internal parameter z were used to calculate the distortion within the structure. As mentioned earlier that the crystal structure of the MAX phase is hexagonal and constituted by [M 6 X] octahedron and [M 6 A] trigonal prism. For an ideal structure, the octahedra and trigonal parameters should be equal to one ( o r = p r = 1), the variation of o r and p r from 1 tells the distortion in these polyhedra. Moreover, Aydin et al. proposed that the smaller the distortion, the more stable the structure is 71 . In this work, the distortion parameters deduced from the optimized lattice parameters for M 2 GaC MAX are: o r = 1.071 and p r = 1.109 for Zr 2 GaC, and o r = 1.058 and p r = 1.128 for Hf 2 GaC, respectively. This indicates the distortion in the octahedra and trigonal prism in both proposed MAX phase structures and the distortion is small and similar because o r /p r = 0.93-0.97. This behaviour is explained as steric effect 72 . The reported o r and p r for Ti 2 GaC MAX phase are 1.088 and 1.081 with o r /p r = 1.00 53 . It can be seen that the octahedra and trigonal prisms' distortion in Ti 2 GaC is smaller than that of Zr 2 GaC and Hf 2 GaC MAX phases. Thus, it is concluded that the structure of Ti 2 GaC is more stable than the M 2 GaC (M = Zr, Hf) MAX phases. The distortion parameters for IV-B and V-B group transition metal MAX phases are deduced from their optimized lattice parameters in the literature for comparison 38,53 . We found that, with increasing the valance of M element (Ti → Zr → Hf , and V → Nb → Ta ), o r decreases towards value 1, but p r increases to values higher than 1 in M 2 GaC (See Table 3). In other words, increasing the valance of M   Electronic properties. The band structure for the M 2 GaC MAX phase at equilibrium lattice parameter within the GGA-PBE was calculated and discussed. Figure  In other words, the conductivity of MAX phase compounds is lower along the c-axis than to their basal planes. There is no apparent difference in band associated with Zr and Hf atoms in terms of energy level because electronegativity for Zr (1.33) and Hf (1.30) is almost the same. At different pressure, the increase in bandwidth was observed within the mentioned pressure limit. We found that with the increase in pressure from 0 to 50 GPa, the bands become looser, i.e., the bands www.nature.com/scientificreports/ above the Fermi level move upward while the bands below the Fermi level move downward. Moreover, the bands at the Fermi level increases with the increase in pressure.
To explain the bonding behavior of M 2 GaC (M = Zr, Hf) MAX phases, we examine the density of states (DOS). The partial and total density of states (TDOS) for M 2 GaC MAX phase compounds at pressure 0 GPa, 30 GPa, and 50 GPa are depicted in Fig. 5. In our previous work 42 , the hybridization of M (4d, 4p, 5s), Ga (3d, 4p, 4s), and C (2p, 2s) orbitals for M 2 GaC is explained at 0 GPa. The TDOS at Fermi level at 0 GPa for Zr 2 GaC and Hf 2 GaC are 3.00 states/eV/unit and 2.47 states/eV/unit, respectively. In the Zr 2 GaC MAX phase, the lowest valance bands of TDOS is formed by C-s with Zr-d, Zr-p in the energy ranges from − 10.85 to − 9.11 eV. The states range from − 8.01 to − 5.1 eV, and − 5 to − 1.90 eV are formed by Ga-s states and strong hybridization of Zr-d and C-s states, respectively. The highest valance bands in the created by Zr-d and Ga-p hybridization, which is relatively weaker     www.nature.com/scientificreports/ than of Zr-d and C-s states. These results are consistent with that of Cr 2 AlC MAX phase material, which has the maximum DOS at E F , i.e., 6.46 states/eV cell/unit 75 . Moreover, the density of states at the Fermi level increased with increased pressure, and obtained results are tabulated in Table 1. It is observed that the increase in TDOS for the Zr 2 GaC MAX phase is more significant than that of Hf 2 GaC. The TDOS values for M 2 GaC illustrate that these MAX phase materials are metallic, and their metallicity is in the order of Zr 2 GaC > Hf 2 GaC in the given range of pressure. However, there is an increase in bandwidth, and correspondingly, the intensity decreased with the increase in pressure. The density of states on the right side of the Fermi level moves rightwards, whereas the density of states on the left side of the Fermi level moves leftwards under pressure (See Fig. 6), which is in good agreement with the analysis of band structure. It is worth noticing that the main contribution at the Fermi level is from the M-4d electrons in both MAX phases, which is not affected by the pressure. This implies that Zr-d and Hf-d electrons mainly contribute to the conduction properties of MAX phase materials under pressure.
To understand the chemical bonding of Zr 2 GaC and Hf 2 GaC MAX phase, we investigate the Crystal Orbital Hamilton Population (COHP) for M-C, M-Ga, and Ga-C bonds, respectively. The COHP method has been widely applied to investigate the bonding and antibonding analysis of many MAX phases [76][77][78] . Figure 7 shows the COHP at the ground state of Zr 2 GaC and Hf 2 GaC MAX phases. It is observed that the COHP curves for both   Table 1. It is observed that the metallicity of both Zr 2 GaC and Hf 2 GaC phases increases with increasing pressure. According to Eq. (6), the conductivity mainly depends upon n v F ratio because e, l, and m are constants. In the given pressure range, the n v F ratio of M 2 GaC MAX phases is in order of n v F Hf 2 GaC > n v F Zr 2 GaC , hence we may conclude that the conductivity of Hf 2 GaC > Zr 2 GaC. To insight the chemical bonding of M 2 GaC MAX phases under pressure, the charge density distributions mapping along the (100) plane at 0 GPa, 30 GPa, and 50 GPa are plotted in Fig. 8. The bonding character of MAX phases is essential to understand the chemical bonding of their 2D derivatives (MXenes) 79, 80 . The preferential accumulation of charges (positive regions at the scale bar) between two atoms indicates the covalent bonds, while balancing the positive or negative (depleted regions) charges at atomic position exhibits the ionic bonding 81 . At 0 GPa, the strong charge accumulation regions were observed at C and M = Zr, Hf atoms, indicating the formation of a strong covalent bond between C-Zr and C-Hf atoms. The charge accumulation at these atomic positions increases with an increase in pressure due to the decrease in atomic distance and an increase in internal www.nature.com/scientificreports/ parameters (z) (See Fig. 3). Furthermore, there is a sign of charge balancing around the Zr and Hf atoms, with the C indicating the small degree of ionic bonding. It is also seen that another covalent bond is formed between the Ga-M = Zr, Hf atoms, which is comparatively weaker than that of C-M = Zr, Hf atoms. Therefore, the chemical bonding in the M 2 GaC MAX phase is predicted to be a mixture of covalent and ionic nature and degree of bonding increases with the increase in pressure.
Mechanical stability and dynamical properties. The mechanical properties of material help to predict the material's response under the application of load. The mechanical properties of MAX phase materials also contribute to predicting the usefulness in service and are critical in the fabrication process. The elastic constants (C ij ) for M 2 GaC MAX phase materials calculated in the pressure range from 0 to 50 GPa are shown in Fig. 9, and calculated mechanical properties are listed in Table 4. As we know that the M 2 GaC MAX phase has the hexagonal crystal structure, and there are six stiffness constants (C 11 , C 12 , C 13 , C 33 , C 44 = C 55, and C 66 ), but five of them are independent since C 66 = (C 11 − C 12 ) 2 82 . Our results are consistent with other MAX phases. The elastic moduli versus pressure for the M 2 GaC MAX phase are plotted in Fig. 10. It is found that the elastic constants and moduli increase monotonically with an increase in pressure, and the values of C 11 , C 33 , Young's modulus (E), and bulk modulus (B) increased significantly compared to other elastic constants. Contrary, the values of C 66 and shear modulus (G) vary slowly. It can also be noticed that C 11 and C 66 for the Zr 2 GaC MAX   55,83 . The mechanical stability of M 2 GaC MAX phases are predicted from the Born stability criteria 84 i.e., C 11 > 0, C 11 − C 12 > 0, C 44 > 0, C 66 > 0, (C 11 + C 12 ) C 33 − 2C 2 13 > 0. Both MAX phases satisfy the mechanical stability criteria in the mentioned range of pressure. The elastic constant C 33 for Zr 2 GaC and Hf 2 GaC increases by up to 462 GPa and 506 GPa, while the values for C 66 for both materials increased only by 112 GPa and 169 GPa, respectively. The rapid increase in C 33 and moderate C 66 infers the increasing insensitivity of the compression strain along the c axis, not the shear strain.
It is known that the moduli (B, G, and E) measure the resistance of the material to fracture, plastic deformation, and stiffness and are essential to understand the solid-state properties, i.e., structural stability, ductility, stiffness, and brittleness. In this work, the elastic moduli (B, G, and E) increased with an increase in pressure from 0 to 50 GPa, i.e., for Zr 2 GaC increase in moduli are B (122-302 GPa), G (94-126 GPa), and E (226-333 GPa) and that of Hf 2 GaC are B (142-347 GPa), G (112-168 GPa), and E (266-435 GPa), respectively. The elastic moduli for the M 2 GaC MAX phase are in the order of (B, G, E) Hf 2 GaC > (B, G, E) Zr 2 GaC in the given pressure range.
The brittle/ductile behavior of the M 2 GaC MAX phase is predicted from the Poisson's ratio (σ). It is the ratio between the transverse strain to longitudinal strain under tensile stress. It is an important tool to quantify the failure state in the solids. Frantsevich et al. 87 proposed a borderline value σ ~ 0.26, which separates the ductile and brittle materials. For the brittle materials, this value is small, whereas the material is considered to be ductile if Poisson's ratio is greater than 0.26. Our calculated values for σ for Zr 2 GaC and Hf 2 GaC are 0.192 and 0.187 at 0 GPa, respectively. These values increase linearly with an increase in pressure, as shown in Fig. 11a. There is a sharp increase in σ noticed for the Zr 2 GaC MAX phase when pressure increase from 40 to 50 GPa. In the pressure    69,85 . It is worth noticing that the grain size of the material has an essential effect on hardness, yield strength, tensile, and fatigue strength according to the Hall-Petch relation because grain boundaries hinder the movement of dislocations 91,92 . The effect grain size on the compressive strength of bulk Ti 2 AlC MAX phase followed the Hall-Petch relation under the dynamic and quasi-static loads 93 . Moreover, the oxidation resistance and mechanical properties of MAX phase thin films can be improved by increasing the grain boundaries 94,95 . Figure 11b illustrates the anisotropic index (A) of the M 2 GaC MAX phase in the function of pressure. Typically, a material is called to be isotropic if anisotropic index A = 1 and the deviation from 1 indicate the anisotropic nature of the material. Figure 11b shows that the values of A do not satisfy the isotropic criteria, and an increase in pressure results in a higher anisotropic index for both Zr 2 GaC and Hf 2 GaC MAX phases suggesting the anisotropic nature of M 2 GaC MAX phases in the given pressure range. In other words, the properties for the M 2 GaC MAX is not identical in all directions. This fact is in good agreement with the anisotropic properties of M 2 GaC MAX phases, i.e., higher compressibility along the c-axis compared to other basal planes.
According to Pugh's criteria, a material will behave ductile if the B/G > 1.75 and G/B < 0.57, otherwise it should be brittle 62,96 . For 0 GPa these ratios for M 2 GaC MAX phases are in the order (Hf 2 GaC) B/G = 1.26 < (Zr 2 GaC) B/G = 1.29 and (Hf 2 GaC) G/B = 0.78 > (Zr 2 GaC) G/B = 0.77, respectively. This indicates that the M 2 GaC phases behave in a brittle manner at 0 GPa: however, with an increase in pressure, M 2 GaC phases likely to be ductile (See Fig. 12). These results are consistent with studies available in the literature 28 . It is also worth noticing that the ductility of Zr 2 GaC phase increase abruptly when pressure is exceeded from 40 GPa and pressure has significant impact in term of brittle/ductile transition for Zr 2 GaC compared to that of Hf 2 GaC. Moreover, the ratio G/B can also be used to determine the chemical bond. For ionic materials, the value is G/B ≈ 0.6, and that of covalent materials G/B ≈ 1.1. In this work, G/B values change from 0.77 to 0.41 for Zr 2 GaC and 0.78-0.48 for Hf 2 GaC, suggesting that ionic bonding is crucial for M 2 GaC compounds.
The Debye temperature is characteristic of solids that can evaluate many physical properties of the material, including thermal conductivity, thermal expansion, specific heat, and melting temperature. It can be calculated by numerous methods, among which the Anderson method is simple and widely used 63 . In this method, the average sound velocity (V m ) is used to calculate the Debye temperature. The calculated Debye temperature and sound velocities for M 2 GaC MAX phases are listed in Table 5. It is noticed that the density and Debye temperature increases with increasing the pressure, and at a given pressure, the Debye temperature is always in the order of (Zr 2 GaC)θ D > (Hf 2 GaC)θ D . The calculated melting point for the M 2 GaC MAX phase is tabulated in Table 5. The melting temperature for Zr 2 GaC and Hf 2 GaC MAX phases understudied are 1481 K and 1648 K at 0 GPa and increases with an increase in pressure. The higher melting temperature values indicate that M 2 GaC MAX phases are suitable for high-temperature applications.
For the dynamical stability of M 2 GaC MAX phases, phonon calculation was performed along the highsymmetry directions in the Brillouin zone. The calculated phonon dispersion curves at 0 GPa, 30 GPa, and 50 GPa are shown in Fig. 13. There are eight atoms per unit cell in the 211 family of MAX phases. So, 24 phonon branches are produced; three are acoustic, and the rest are for optical modes. The optical branches are situated at the upper part of the dispersion curves, responsible for the optical behavior of MAX phase materials. These optical modes originate from the out-of-phase oscillations of atoms in lattice when one atom goes to the left and its neighbor to the right. In contrast, the acoustic branches are located at the lower part of phonon dispersion curves As mentioned earlier that the MAX phases compounds studied in this work have an elastic anisotropic nature, while the elastic anisotropy is not apparent. Figure 14 plots the 3D Young's modulus surfaces obtained by an open-source software package (AnisoVis) 97 of Zr 2 GaC (a, b, c) and Hf 2 GaC (d, e, f) at 0, 10, and 50 GPa, respectively. It can be observed that the shape begins to deviate from the sphere with an increase in pressure, and color varies in different regions, which indicates the elastic anisotropy of M 2 GaC MAX phases. The color variation (dark blue to yellow) exhibits that elastic modulus increases with an increase in pressure, and obtained values are mentioned at the top of each 3D plot, which agrees well with the previously calculated results (see Fig. 10). The Young's modulus at 0 GPa pressure for Zr 2 GaC and Hf 2 GaC are 226.57 GPa and 266.68 GPa and rises to 333.38 GPa and 435.43 GPa when pressure is increased to 50 GPa, respectively. The pressure effect on the elastic anisotropy of Hf 2 GaC is more significant than that of the Zr 2 GaC MAX phase. According to the deviation degree of spherical shape, the anisotropy of Hf 2 GaC is bigger than Zr 2 GaC (See Fig. 11b).
Optical properties. The optical properties for the M 2 GaC MAX phase (M = Zr, Hf) were determined for the first time by frequency-dependent dielectric functions within the photon energies up to 20 eV. The optical properties for The MAX phases are optically anisotropic 99,100 . Thus, two polarization directions, <100> and <001>, were chosen to investigate the optical properties. The MAX phase compounds under this study are metallic in nature (see band structure) so, the term Drude (plasma frequency 3 eV and damping 0.05 eV) has been used with Gaussian smearing of 0.5 eV for all calculations. The frequency-dependent dielectric function ε(ω) = ε 1 (ω) + iε 2 (ω)(where ε 1 (ω) is the real part and ε 2 (ω) is the imaginary part of the dielectric function) has a close relation to the band structure. Once the imaginary part is known, the real part can be derived using the Kramers-Kronig equation. Later, all the optical properties can be obtained using the ε 1 (ω) and ε 2 (ω) 101 , as shown in Figs. 15, 16, 17, 18 and 19.
Dielectric constant. The real part of the dielectric constant ε 1 (ω) is essential for optoelectronic devices because it corresponds to the primitivity component that measures the stored energy. The imaginary part of the dielectric constant ε 2 (ω) gives the information about the optical system's energy reduction in the function of frequency. Figure 15a,b shows the real ε 1 (ω) and imaginary part ε 2 (ω) of dielectric constants calculated for the < 100 > and < 001 > polarization directions for M 2 GaC. It is observed that the real part of dielectric constant ( ε 1 ) approaches to zero from below, while the imaginary part ( ε 2 ) gets to zero from above, which implies that the M 2 GaC MAX phases are metallic in nature 102 . In the real part of the dielectric constant, the spectra within the infrared region (I.R ≤ 1.7 eV) has the highest dielectric constant for <100> polarization compared to < 001 > polarization due to intra-band transition of electron. The sharp peaks were observed in ε 1 (ω) for Zr 2 GaC and Hf 2 GaC   www.nature.com/scientificreports/ phases along the <100> polarization at ∼1.01 eV and ∼1.04 eV, respectively. It is worth noticing that the spectra of ε 1 (ω) for different polarization directions exhibit different features in the photon energy range. There is no significant difference observed for both MAX phases in the ε 2 (ω) spectra for < 100 > and < 001 > polarization. Thus, we can deduce that the MAX phases studies here are optically anisotropic. Moreover, the value ε 1 (ω) approached zero from below for polarization < 100 > M 2 GaC at around 12.8-13.7 eV and for polarization < 001 > M 2 GaC , reaches zero at approximately 13.9-15.6 eV. While the value for ε 2 (ω) reaches zero from above for polarization <100> and <001> at photon energy ranges from 12.5 to 16.7 eV. www.nature.com/scientificreports/ Loss function. The loss function is peak corresponds to the bulk plasma frequency ( ω p ), which appears where ε 2 < 1 and ε 1 approaches to zero. It is the energy loss of the first electron traversing through a material, and bulk plasma frequency ( ω p ) is obtained from the loss function spectrum. The studied M 2 GaC MAX phases become transparent if the frequency of incident light is higher than that of plasma frequency. By analyzing, the energyloss function peaks of Zr 2 GaC and Hf 2 GaC phases was occurred at around 12.79 eV and 13.7 eV, respectively, for <100> polarization: and corresponding 14.00 eV and 15.61 eV for <001> polarization as shown in Fig. 16. It is noticed that the plasma frequent ( ω p ) of M 2 GaC for the <001> polarization is larger than that for <100> polarization 103 . Moreover, energy loss spectra for M 2 GaC MAX phases show no peaks in the photon energy range of 0-10 eV due to large ε 2 (ω) (see Fig. 15b). www.nature.com/scientificreports/ Absorption coefficient. The absorption coefficient gives the knowledge about the efficiency of the solar energy conversion, which is important for solar cell material. It corresponds to the amount of light of a specific wavelength into a solid before getting absorbed. Figure 17 depicts the energy-based absorption (α) spectra of M 2 GaC MAX phases. It is observed that the absorption spectra for both MAX phases are weak in the infrared region (I.R), increases monotonously in the visible region and dominant ultraviolet (UV) regions. The maximum value of α was observed for Zr 2 GaC and Hf 2 GaC at around 6.09 eV and 6.86 eV, respectively, for <100> polarization; and corresponding 6.34 eV and 6.97 eV for <001> polarization. Moreover, the light absorption of M 2 GaC in the <001> polarization direction is larger than that for <100> polarization. In other words, both Zr 2 GaC and Hf 2 GaC MAX phases absorb more light in the direction of <001> polarization compared to its counterpart <100> , indicating their optically anisotropic nature 98 . The rise in α was observed in the direction of the UV region, exhibits the high absorbent feature of the material. Based on the calculated absorption spectra of M 2 GaC MAX phases, it can be deduced that these materials are competing candidates for optoelectronic devices in both visible and UV regions.
Photoconductivity. Photoconductivity (σ) of material can be described as the increase in the electric conductivity due to absorbing photos. For M 2 GaC MAX phases σ is shown in Fig. 18. It is noticed that for both MAX phases under this study, photoconductivity increases exponentially when the photo energy goes to 0 eV as expected for metals because there is no band gap present in the M 2 GaC MAX phases. A sharp dip in photoconductivity of Zr 2 GaC and Hf 2 GaC for <100> polarization was observed at 0.37 eV and 0.39 eV, and that for polarization <001> was observed at 0.26 eV and 0.20 eV, respectively 88 . Peak heights for different polarization   Reflectivity. Finally, reflectivity spectra of M 2 GaC MAX phases for <100> and <001> polarization, as a function of incident light are demonstrated in Fig. 19. The reflectivity for M 2 GaC MAX phases shows the highest reflectivity in the I.R region and visible region ranges from 4.4 to 13.10 eV and then approaches zero for both phases in the incident photon energy ranges from 19 to 22 eV. However, it is worth noticing that the reflectivity is almost constant for <100> polarization of Zr 2 GaC and Hf 2 GaC MAX phases within the visible region, and values are above 45% and should appear as a metallic gray color. It is known that materials having constant reflectivity in the visible regions with an average value of about 44% are capable of reflecting the solar light, which results in a reduction in solar heating in the visible light region 104 . So, it may be concluded that Zr 2 GaC and Hf 2 GaC MAX phases can be used as the coating material for the purpose of solar heating reduction. However, the variable reflectivity within the visible region of different polarization indicated the optical anisotropy M 2 GaC MAX phase 105 . The dependence of reflectance on the pressure of M 2 GaC MAX phases was studied as well, and results for polarization <100> at a pressure range from 0 to 50 GPa is shown in Fig. 20. For the M 2 GaC MAX phase, the reflectance exhibits less change in the moderate range of the I.R region ranging from 0 to 0.48 eV at all pressures and show variable reflectivity in the rest I.R region. It is noticed that the reflectivity increases with an increase in pressure in the I.R region. However, the reflectivity of M 2 GaC decreased at higher pressure, but almost the same in the visible region then increases in the UV region more quickly and exhibits a higher value at 0 GPa. The  www.nature.com/scientificreports/ reflectance at pressure range 0-50 GPa remains above 40% in the visible zone. Thus, it is concluded that M 2 GaC MAX phase materials are ideal for coating materials under high-pressure conditions to avoid solar heating in the <100> polarization direction.

Conclusion
The effect of pressure on structural stability, mechanical, electronic, phonon, and optical properties of M 2 GaC MAX phases (M = Zr and Hf) in the pressure range from 0 to 50 GPa were calculated by using first-principles calculations. The formation of enthalpy with respect to its most competing phases showed that M 2 GaC MAX phases are thermodynamically stable. The band structure and total density of states exhibited that M 2 GaC MAX phases are metallic in nature, with an increase in bandgap at the Fermi level with an increase in pressure. The DOS at E F in the pressure range of 0-50 GPa are Zr 2 GaC = 2.96-4.27 states/eV/unit > Hf 2 GaC = 2.47-2.60 states/ eV/unit, which implies that the metallicity of Zr 2 GaC increased more than that of Hf 2 GaC with increasing the internal pressure. According to COHP analysis, M-C bonds are stronger than that M-Ga in both Zr 2 GaC and Hf 2 GaC MAX phases. The volume ratio and lattice parameters Zr 2 GaC and Hf 2 GaC decrease with increasing pressure, and the compressibility of Zr 2 GaC is better than that of Hf 2 GaC. Besides, the normalized bond lengths show that crystals compressed more easily along the M-Ga (Zr, Hf) direction under pressure. The effect of pressure on the mechanical properties of M 2 GaC MAX phases is pronounced. Both the Zr 2 GaC and Hf 2 GaC MAX phases revealed the brittleness behavior at 0 GPa pressure and tended to ductile when pressure increased from 10  www.nature.com/scientificreports/ to 50 GPa. Moreover, there is a linear increase in elastic constants, elastic moduli, Poisson's ratio, and a decrease in Vickers hardness was observed with the increase in pressure. The calculated Vickers hardness is found to be 18.23 GPa and 20.99 GPa for Zr 2 GaC and Hf 2 GaC, respectively. The phonon dispersion curves have confirmed the dynamical stability of compounds in the given pressure range. The optical properties of the MAX phase compound under this study reveals some interesting information. The absorption spectra of M 2 GaC increased to the maximum value in the visible region, and the UV region indicates its high absorbent capability and is suitable for optoelectronic devices in the visible and UV regions. Moreover, the reflectance curves show the constant values in the visible region with an average value above 44%. We conclude that these compounds can also be used as a coating material to avoid solar heating at even high pressure. To the author's best knowledge, no study had been made to predict the mechanical, electronic, thermal, phonon, and optical properties of M 2 GaC MAX phases under pressure. Hence, these results can serve as a reference for future theoretical and experimental research.