Structural Stability, Electronic, Mechanical, Phonon, and Thermodynamic Properties of the M2GaC (M = Zr, Hf) MAX Phase: An ab Initio Calculation

The novel ternary carbides and nitrides, known as MAX phase materials with remarkable combined metallic and ceramic properties, offer various engineering and technological applications. Using ab initio calculations based on generalized gradient approximation (GGA), local density approximation (LDA), and the quasiharmonic Debye model; the electronic, structural, elastic, mechanical, and thermodynamic properties of the M2GaC (M = Zr, Hf) MAX phase were investigated. The optimized lattice parameters give the first reference to the upcoming theocratical and experimental studies, while the calculated elastic constants are in excellent agreement with the available data. Moreover, obtained elastic constants revealed that both the Zr2GaC and Hf2GaC MAX phases are brittle. The band structure and density of states analysis showed that these MAX phases are electrical conductors, having strong directional bonding between M-C (M = Zr, Hf) atoms due to M-d and C-p hybridization. Formation and cohesive energies, and phonon calculations showed that Zr2GaC and Hf2GaC MAX phases’ compounds are thermodynamically and dynamically stable and can be synthesized experimentally. Finally, the effect of temperature and pressure on volume, heat capacity, Debye temperature, Grüneisen parameter, and thermal expansion coefficient of M2GaC (M = Zr, Hf) are evaluated using the quasiharmonic Debye model from the nonequilibrium Gibbs function in the temperature and pressure range 0–1600 K and 0–50 GPa respectively.


Introduction
A new class of nanolayered transition metal carbides and/or nitrides was first discovered by H. Nowotny and coworkers, which were called H-phases in 1960, now known as MAX phases [1]. The general formula M n+1 AX n typically represents these ternary compounds (where M is an early transition metal, A is an IIIA-or IVA-group element, and X is either C or N, 1 ≤ n ≤ 3). MAX phase crystallizes in the hexagonal crystal structure (space group P6 3 /mmc) in which MX 6 octahedral crafted with a pure A-group atom layer grown in c-direction, which bring forth a mixture of a strong covalent M-X bond and a relatively weak metallic M-A bond [2,3]. Due to a unique layered structure, MAX phases possess hybrid properties of both metals and ceramics, such as low density, machinability, thermal and electrical conductivity, damage and irradiation tolerance, oxidation, and corrosion resistance [4][5][6][7]. These remarkable properties make MAX phases suitable for versatile applications from microelectronics to aerospace [8][9][10][11]. Moreover, these ternary compounds can be exfoliated into Table 1. Calculated lattice parameters (a) and (c) in Å, c/a, bulk modulus (GPa), no. of density of states (DOS)at E F (states/eV/unit cell), and formation energy E for (eV/atom) for M 2 GaC MAX phase (M = Zr, Hf) obtained by generalized gradient approximation (GGA) and local density approximation (LDA).
Materials 2020, 13   The computed equilibrium volume for Zr2GaC and Hf2GaC is 136.92 Å 3 and 134.20 Å 3 , and lattice parameters (a, c) are 3.33 Å, 14.25 Å for Zr2GaC, and 3.32 Å, 14.02 Å for Hf2GaC, respectively. The study made by Sun Zhimei et al. [34] on Ga-containing MAX Phases M2GaC (where M = Ti, Zr, Hf, V, Nb, Ta, Cr, Mo, W) did not contain useful information related to optimized lattice parameters or volume, and there is no experimental data available in the literature so far. Hence, we predict the physical properties of the M2GaC MAX phase for the very first time.
The effect of pressure on the equilibrium volume of M2GaC, ranging from 0-50 GPa with a step of 10 GPa, is shown in Figure 3, where V0 is the volume at the zero-pressure equilibrium structural parameter. It is noticed that the volume ratio (V/V0) was reduced in the order of with an increase in pressure. Therefore, the compressibility of the M2GaC MAX phase system is strong, and the change of external pressure has a more significant impact on Zr2GaC.   The computed equilibrium volume for Zr2GaC and Hf2GaC is 136.92 Å 3 and 134.20 Å 3 , and lattice parameters (a, c) are 3.33 Å, 14.25 Å for Zr2GaC, and 3.32 Å, 14.02 Å for Hf2GaC, respectively. The study made by Sun Zhimei et al. [34] on Ga-containing MAX Phases M2GaC (where M = Ti, Zr, Hf, V, Nb, Ta, Cr, Mo, W) did not contain useful information related to optimized lattice parameters or volume, and there is no experimental data available in the literature so far. Hence, we predict the physical properties of the M2GaC MAX phase for the very first time.
The effect of pressure on the equilibrium volume of M2GaC, ranging from 0-50 GPa with a step of 10 GPa, is shown in Figure 3, where V0 is the volume at the zero-pressure equilibrium structural parameter. It is noticed that the volume ratio (V/V0) was reduced in the order of with an increase in pressure. Therefore, the compressibility of the M2GaC MAX phase system is strong, and the change of external pressure has a more significant impact on Zr2GaC.  The effect of pressure on the equilibrium volume of M 2 GaC, ranging from 0-50 GPa with a step of 10 GPa, is shown in Figure 3, where V 0 is the volume at the zero-pressure equilibrium structural parameter. It is noticed that the volume ratio (V/V 0 ) was reduced in the order of (V/V 0 ) Zr 2 GaC = 9% > (V/V 0 ) H f 2 GaC = 7% with an increase in pressure. Therefore, the compressibility of the M 2 GaC MAX phase system is strong, and the change of external pressure has a more significant impact on Zr 2 GaC.  is −6.328 eV/atom and −6.162 eV/atom, respectively (see Table 1). It is worth noticing that the formation and cohesive energies for M2GaC are negative, indicating that these MAX phases are energetically favorable from the thermodynamic point of view. Moreover, these structures can be experimentally formed by different synthesis methods.

Band Structure
Based on optimized lattice parameters, the band structure for M2GaC (M = Zr, Hf) was calculated and obtained. Figure 4 shows the band structure from −15 eV to 6 eV energy range along the high symmetry lines of the Brillouin zone. The appearance of the obtained band structure for both Zr2GaC and Hf2GaC MAX phases is similar to the other metallic MAX phases, such as Cr2AlC [47] and Ti2AlC [48] because of considerable overlapping of bands without having bandgap in the vicinity of the The energy of formation per atom (E M 2 GaC f or ) was calculated to investigate the phase stability of the researched M 2 GaC MAX phase, which is defined as follows [46]: where: x, y, z is the number of atoms for M, Ga, and C element in the unit cell, i.e., x = 4, y = 2, and z = 2 and E M 2 GaC total , E M solid , E Ga solid , and E C solid are the total energy of M 2 GaC MAX phase, M, Ga, and C atoms in the solid form respectively in their stable structures. The calculated formation energy for Zr 2 GaC and Hf 2 GaC is −7.59 eV/atom and −7.45 eV/atom, respectively, given in Table 1.
Cohesive energy (E Coh ) is defined as the energy required to break the crystal into an isolated atom and used to identify the structural stability of the M 2 GaC MAX phases. The E Coh is computed using Equation (2).
where: E M 2 GaC Coh , E M Iso , E Ga Iso , and E C Iso are the total energy of the M 2 GaC MAX phase and the energies of isolated M, Ga, and C atoms, respectively. The calculated E Coh for Zr 2 GaC and Hf 2 GaC is −6.328 eV/atom and −6.162 eV/atom, respectively (see Table 1). It is worth noticing that the formation and cohesive energies for M 2 GaC are negative, indicating that these MAX phases are energetically favorable from the thermodynamic point of view. Moreover, these structures can be experimentally formed by different synthesis methods.

Band Structure
Based on optimized lattice parameters, the band structure for M 2 GaC (M = Zr, Hf) was calculated and obtained. Figure 4 shows the band structure from −15 eV to 6 eV energy range along the high symmetry lines of the Brillouin zone. The appearance of the obtained band structure for both Zr 2 GaC and Hf 2 GaC MAX phases is similar to the other metallic MAX phases, such as Cr 2 AlC [47] and Ti 2 AlC [48] because of considerable overlapping of bands without having bandgap in the vicinity of the Fermi level. It can also be assessed that there is a strong anisotropic behavior with less c-axis energy dispersion. In other words, less energy dispersion was observed along the short H-K and M-L directions for both MAX phase compounds indicating the electronic anisotropic nature of M 2 GaC MAX phases (see Figure 4), which means conductivity will be lower along the c-axis as compared to their basal planes. The electronic anisotropic behavior of M 2 GaC MAX phases is in good agreement with the available data of other MAX phases [21,49]. This is a consequence of the nanolaminated structure of M 2 GaC MAX phases. Vincent et al. [50] investigated the electronic anisotropy of Ti 2 AlC MAX phase and found that resistivity (ρ = 1/σ) along the c-axis is much higher than polycrystalline bulk material or thin-film along the (0001) orientation. Fermi level. It can also be assessed that there is a strong anisotropic behavior with less c-axis energy dispersion. In other words, less energy dispersion was observed along the short H-K and M-L directions for both MAX phase compounds indicating the electronic anisotropic nature of M2GaC MAX phases (see Figure 4), which means conductivity will be lower along the c-axis as compared to their basal planes. The electronic anisotropic behavior of M2GaC MAX phases is in good agreement with the available data of other MAX phases [21,49]. This is a consequence of the nanolaminated structure of M2GaC MAX phases. Vincent et al. [50] investigated the electronic anisotropy of Ti2AlC MAX phase and found that resistivity (ρ = 1/σ) along the c-axis is much higher than polycrystalline bulk material or thin-film along the (0001) orientation.

Density of States (DOS)
To investigate the nature of chemical bonding in the M2GaC (M = Zr, Hf) MAX phases, the density of states (DOS) was calculated and studied. The total and partial density of states is shown in Figures 5 and 6 and the results obtained are tabulated in Table 1. It is observed that the total DOS at EF for Zr2GaC and Hf2GaC are 3.002 states/eV/unit cell and 2.475 states/eV/unit cell, respectively, suggesting that the both Zr2GaC and Hf2GaC MAX phases are metallic and the metallicity of M2GaC MAX phase compounds can be measured by DOS at EF. The metallicity of Zr2GaC and Hf2GaC phases at ambient temperature is investigated by: where nm and ne are the thermally excited number of electrons and the total number of valence electron in the unit cell while kB, T and N(EF) are the Boltzmann constant, temperature, and value of DOS at EF in unit states/eV/unit cell, respectively. The calculated values of fm are listed in Table 1. Moreover, the conductivity of the M2GaC MAX phase is investigated from the DOS at EF as follows:

Density of States (DOS)
To investigate the nature of chemical bonding in the M 2 GaC (M = Zr, Hf) MAX phases, the density of states (DOS) was calculated and studied. The total and partial density of states is shown in Figures 5 and 6 and the results obtained are tabulated in Table 1. It is observed that the total DOS at E F for Zr 2 GaC and Hf 2 GaC are 3.002 states/eV/unit cell and 2.475 states/eV/unit cell, respectively, suggesting that the both Zr 2 GaC and Hf 2 GaC MAX phases are metallic and the metallicity of M 2 GaC MAX phase compounds can be measured by DOS at E F. The metallicity of Zr 2 GaC and Hf 2 GaC phases at ambient temperature is investigated by: where n m and n e are the thermally excited number of electrons and the total number of valence electron in the unit cell while k B , T and N(E F ) are the Boltzmann constant, temperature, and value of DOS at E F in unit states/eV/unit cell, respectively. The calculated values of f m are listed in Table 1. Moreover, the conductivity of the M 2 GaC MAX phase is investigated from the DOS at E F as follows: where v F and m are the velocity of electrons near the Fermi level and mass of electrons, respectively.
From v F , the conductivity (σ) of material is estimated as: where: n, e, and l are the number of electrons, electrons' charge, and mean free path of electrons, respectively. τ is the time between two collisions. In Equation (5), e, l, and m are constants; thus, conductivity mainly depends upon the n/v F ratio. It is observed that the (n/v F ) Zr 2 GaC > (n/v F ) H f 2 GaC ; we may conclude that the conductivity of the M 2 GaC MAX phase is in the order of Zr 2 GaC > Hf 2 GaC. However, the values for total density of states (TDOS) obtained for M 2 GaC (M = Zr, Hf) is much smaller than that of Cr 2 AlC (6.46 states/eV/unit cell) [43], the maximum TDOS measured among the MAX phases so far. It is noted that the main contribution in the DOS at E F is from the M-4d electrons in both cases, indicating that d bands of transition metal mainly contribute to the conduction properties during the electrical transport. These results are consistent with the previous studies on MAX phases [46].

From
F v , the conductivity (σ) of material is estimated as: where: n, e, and l are the number of electrons, electrons' charge, and mean free path of electrons, respectively.  is the time between two collisions. In Equation (5) ; we may conclude that the conductivity of the M2GaC MAX phase is in the order of Zr2GaC > Hf2GaC. However, the values for total density of states (TDOS) obtained for M2GaC (M = Zr, Hf) is much smaller than that of Cr2AlC (6.46 states/eV/unit cell) [43], the maximum TDOS measured among the MAX phases so far. It is noted that the main contribution in the DOS at EF is from the M-4d electrons in both cases, indicating that d bands of transition metal mainly contribute to the conduction properties during the electrical transport. These results are consistent with the previous studies on MAX phases [46]. For Zr2GaC, the total and partial density of states (TDOS and PDOS) from Figure 5a shows that the lowest states range from −10.85 eV to −9.11 eV of the TDOS is formed by C-s with Zr-d, Zr-p along with a small portion of Zr-s. The higher states from −8.01 eV to −5.1 eV are entirely formed by Ga-s states. The valance band in the range −5 eV to −1.90 eV is formed by the strong hybridization of C-s and Zr-d states. The highest valance band is related to the relatively weak hybridization of Zr-d, Zrp, and Ga-p, which shows the covalent interaction between the Zr-d and Ga-p. The TDOS and PDOS for Hf2GaC shown in Figure 5b is similar to that of Zr2GaC with two common sharing features: Firstly, as M from Hf to Zr, the hybridization peaks of C-s and C-p with M-d states shifts from right to left towards the lower energy level and C-s, C-p peaks become narrow (See Figure 6). This expresses that the weakened M-d and C-s, C-p covalent interaction, results in decreased bulk modulus for the M2GaC MAX phase, as given in Table 1. The indicated peak shift is seen in the M2GaC MAX phase TDOS in Figure 6. Secondly, the M-d (M = Zr, Hf) and C-p hybridization peak lies between −5 eV and Fermi level is set to 0 eV.

Mechanical Properties
A brief knowledge about the elastic constants of the crystalline materials helps to predict its behavior under the application of external stress. It contributes to a critical understanding of the many solid-state properties, i.e., ductility, brittleness, stiffness, structural stability, and anisotropy. The elastic constants (Cij) of M2GaC (M = Zr, Hf) were calculated from the PBE, PW91, and LDA, and For Zr 2 GaC, the total and partial density of states (TDOS and PDOS) from Figure 5a shows that the lowest states range from −10.85 eV to −9.11 eV of the TDOS is formed by C-s with Zr-d, Zr-p along with a small portion of Zr-s. The higher states from −8.01 eV to −5.1 eV are entirely formed by Ga-s states. The valance band in the range −5 eV to −1.90 eV is formed by the strong hybridization of C-s and Zr-d states. The highest valance band is related to the relatively weak hybridization of Zr-d, Zr-p, and Ga-p, which shows the covalent interaction between the Zr-d and Ga-p. The TDOS and PDOS for Hf 2 GaC shown in Figure 5b is similar to that of Zr 2 GaC with two common sharing features: Firstly, as M from Hf to Zr, the hybridization peaks of C-s and C-p with M-d states shifts from right to left towards the lower energy level and C-s, C-p peaks become narrow (See Figure 6). This expresses that the weakened M-d and C-s, C-p covalent interaction, results in decreased bulk modulus for the M 2 GaC MAX phase, as given in Table 1. The indicated peak shift is seen in the M 2 GaC MAX phase TDOS in Figure 6. Secondly, the M-d (M = Zr, Hf) and C-p hybridization peak lies between −5 eV and −1.90 eV, while M-d and Ga-p is located between −1.90 eV and 0 eV. This indicates that bonding between the M-d and C-p states is stronger than that of the M-d and Ga-p states.

Mechanical Properties
A brief knowledge about the elastic constants of the crystalline materials helps to predict its behavior under the application of external stress. It contributes to a critical understanding of the many solid-state properties, i.e., ductility, brittleness, stiffness, structural stability, and anisotropy. The elastic constants (C ij ) of M 2 GaC (M = Zr, Hf) were calculated from the PBE, PW91, and LDA, and the obtained results are tabulated in Table 2. The hexagonal structure of MAX phases has six independent elastic constants (C 11 , C 12 , C 13 , C 33 , C 44 =C 55, and C 66 ), but five of them are listed since C 66 = (C11− C12) 2 [51]. Moreover, it is observed that the obtained elastic constants (C ij ) are positive and satisfy the mechanical stability criteria known as Born stability (C 11 > 0, C 11 −C 12 > 0, C 44 > 0, C 66 > 0, (C 11 + C 12 ) C 33 −2C 2 13 > 0) [52] showing that the M 2 GaC MAX phase is mechanically stable. In order to comprehend the mechanical properties further, the bulk modulus (B), shear modulus (G), Young's modulus (E), G/B or B/G (Pugh ratio), elastic anisotropy (A), and Poisson's ratio (ν) are calculated from the obtained elastic constants and results are given in Table 3. The Voigt (V) [53], Russ (R) [54,55], and Voigt-Russ and Hill (VRH) [56,57] approximation scheme were used to determine the parameters concerning properties. The following equations are used to calculate these quantities: and where Bv, Gv and Br, Gr are the B and G in terms of the Voigt and Russ approximation respectively and calculated by given Equations (8)-(11) and B R = (C 11 + C 12 )C 33 − 2C 2 12 (C 11 + C 12 + 2C 33 − 4C 13 ) (10) To calculate the anisotropy index (A) following expression is used The Poisson's ratio can be calculated by: Generally, the calculated parameters B, G, and E measure the material's resistance to fracture, resistance to plastic deformation, and stiffness of the material, respectively. The bulk modulus (B) calculated in terms of elastic constants is in good agreement with the bulk modulus (B*) obtained from the Birch-Murnghan equation of state (EOS), indicating that our estimated elastic constants for Zr 2 GaC and Hf 2 GaC are accurate and precise. Moreover, the calculated bulk modulus obtained from LDA is in good agreement with the available study [34]. Furthermore, B, G, and E for Hf 2 GaC > Zr 2 GaC, which means the effect to resist the deformation of Hf 2 GaC is better than that of Zr 2 GaC. Therefore, the reduction of volume ratio (V/V 0 ) for Zr 2 GaC is higher than Hf 2 GaC (See Figure 3).
Another parameter, the Pugh's ratio (B/G and G/B) [58], separates the ductile to brittle nature of the material. It is known that if B/G > 1.75 and G/B < 0.5, the material will be ductile; otherwise, it will be brittle [59]. In our cases (Zr 2 GaC and Hf 2 GaC), the B/G < 1.75 and G/B > 0.5; consequently, the compounds under this study are predicted to be brittle like Ta 2 GaC [20]. The anisotropy index (A) gives the knowledge about the anisotropic nature of the materials. If the value of A is equal to 1, then the material is said to be isotropic; otherwise, material will be anisotropic if the value of A is higher or lower than 1. Our calculated results from Table 3 showed that the M 2 GaC MAX phases are anisotropic.
One of the most essential elastic parameters is Poisson's ratio (σ), defined as the ratio between the transverse strain to longitudinal strain under the applied tensile stress. It gives knowledge about the material's chemical bonding and is linked to its stability against the shear stress. Whether the material is brittle or ductile can be predicted from the Poisson's ratio (σ) value, and the 0.33 [60] value is set for ductile material; otherwise, the material is called brittle if σ < 0.33. In the results of both investigated Zr 2 GaC and Hf 2 GaC MAX phases, the Poisson's ratio (σ) value is smaller; consequently, both compounds are predicted to be brittle.
For the phase stability of M 2 GaC MAX phases, the calculated phonon dispersion curves along the high-symmetry directions in the Brillouin zone are shown in Figure 7. The M 2 GaC MAX phases have eight atoms in the unit cell; therefore, the phonon dispersion curve shows twenty-four branches (three acoustic and twenty-one optical). The optical frequencies (longitudinal optical (LO) and transverse optical (TO)) at Γ are 14.58 THz and 16.00 THz for Zr 2 GaC and 16.56 THz and 18.32 THz for Hf 2 GaC. We have not observed any negative or imaginary frequency in the phonon dispersion curves of both MAX phases indicating that M 2 GaC MAX phases are dynamically stable. To the best of the authors' knowledge, phonon dispersion for Zr 2 GaC and Hf 2 GaC have not yet been investigated theoretically and experimentally.
Materials 2020, 13, x FOR PEER REVIEW 11 of 19 or lower than 1. Our calculated results from Table 3 showed that the M2GaC MAX phases are anisotropic.
One of the most essential elastic parameters is Poisson's ratio (σ), defined as the ratio between the transverse strain to longitudinal strain under the applied tensile stress. It gives knowledge about the material's chemical bonding and is linked to its stability against the shear stress. Whether the material is brittle or ductile can be predicted from the Poisson's ratio (σ) value, and the 0.33 [60] value is set for ductile material; otherwise, the material is called brittle if σ < 0.33. In the results of both investigated Zr2GaC and Hf2GaC MAX phases, the Poisson's ratio (σ) value is smaller; consequently, both compounds are predicted to be brittle.
For the phase stability of M2GaC MAX phases, the calculated phonon dispersion curves along the high-symmetry directions in the Brillouin zone are shown in Figure 7. The M2GaC MAX phases have eight atoms in the unit cell; therefore, the phonon dispersion curve shows twenty-four branches (three acoustic and twenty-one optical). The optical frequencies (longitudinal optical (LO) and transverse optical (TO)) at Γ are 14.58 THz and 16.00 THz for Zr2GaC and 16.56 THz and 18.32 THz for Hf2GaC. We have not observed any negative or imaginary frequency in the phonon dispersion curves of both MAX phases indicating that M2GaC MAX phases are dynamically stable. To the best of the authors' knowledge, phonon dispersion for Zr2GaC and Hf2GaC have not yet been investigated theoretically and experimentally.

Thermodynamic Properties
The thermal properties of Zr 2 GaC and Hf 2 GaC are computed in the temperature ranges 0-1600 K and the pressure in the range 0-50 GPa with the step of 10 GPa, whereas the quasiharmonic Debye model remains valid [41,42] in this temperature range. This model has been successfully applied to calculate the thermodynamic properties of other MAX phases as well. The Debye temperature (θ D ) is calculated from elastic constants and it depends on the mean propagation sound velocity (V m ). A couple of equations are used to calculate the θ D which are given below: [61][62][63] where and 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 while v m , v t , and v l are mean sound velocity, transverse and longitudinal sound velocities respectively. The calculated relationship between volume and temperature, heat capacity and temperature, Debye temperature and temperature, Grüneisen parameter (γ) and temperature, and thermal expansion coefficient and temperature for Zr 2 GaC and Hf 2 GaC is given in Figures 8-12. The calculated v m , v t , v l , Debye temperature (θ D ), thermal expansion coefficient CTE (α), and heat capacity at constant volume (C V ) and constant pressure (C P ) for Zr 2 GaC and Hf 2 GaC at 300 K are presented in Tables 1-4.
The Grüneisen parameter (γ) is used to calculate the thermal state and is a dimensionless quantity. The trend for γ is almost similar to that of volume. As shown in Figure 11 that γ decreases with an increase in pressure while increasing linearly with temperature. However, γ remains constant in the temperature range from 0-100 K. The obtained γ0 values for Zr2GaC and Hf2GaC are 1.76 and 1.74 at 0 GPa, respectively. Finally, the dependence of the thermal expansion coefficient (α) on temperature is depicted in Figure 12. At a given temperature, α decreases with the increase in pressure; this may be due to reduced unit cell volume. The results of α at ambient temperature and 0 GPa for Zr2GaC and Hf2GaC are 1.14 × 10 −5 K −1 and 0.84 × 10 −5 K −1, respectively. However, in the range 0-300 K, α increases rapidly and reaches a plateau at higher temperature, indicating that the temperature effect is more important at lower temperature. Table 4. The calculated transverse elastic wave velocity (vt), longitudinal elastic wave velocity (vl), the average wave velocity (vm), Debye temperature (θD), thermal expansion coefficient (α), and heat capacities at constant volume and constant pressure (Cv and Cp) at 300 K for M2GaC MAX phase (M = Zr, Hf).

Conclusions
In this study, electronic, structural, elastic, phonon, thermodynamical, and mechanical properties of the M2GaC MAX phase (M = Zr, Hf) were comprehensively investigated by employing first-principle calculations based on GGA and LDA exchange-correlation functional. The thermodynamic properties were computed using the quasiharmonic Debye model in the pressure ranges 0-50 GPa, and temperature in the range 0-1600 K. The results showed that the formation and cohesive energies for both Zr2GaC and Hf2GaC are found to be negative, i.e., −7.59 eV/atom and −6.32 eV/atom for Zr2GaC and −7.45 eV/atom and −6.16 eV/atom for Hf2GaC. The PDOS revealed that the main contribution in electric transport at EF was by transition metal M-4d electrons and hybridizations between M-d and C-p states is stronger than M-d and Ga-p states. The DOS at EF for Zr2GaC and Hf2GaC was 2.96 states/eV/unit and 2.47 states/eV/unit, respectively, indicating that these MAX phases are predicted to be electrical conductors while band structures pointed out their anisotropic nature. Moreover, the formation and cohesive energies, elastic constants, and phonon calculations showed that both compounds are thermodynamically, mechanically, and dynamically    From Figure 8, it is noticed that a small change in volume takes place from the temperature 0 to 100 K at a given pressure, and volume linearly increases with the rise in temperature while a decrease in volume is observed at high pressure at a given temperature. The relationship between the Debye temperature and temperature is shown in Figure 9. The linear decline in Debye temperature could be seen with the increase in temperature and increases with pressure. For both Zr 2 GaC and Hf 2 GaC MAX phase, the value of θ D at a given temperature and pressure is in the order of (θ D ) Zr 2 GaC > (θ D ) H f 2 GaC . Many physical properties of solidity such as hardness, thermal expansion [60], and heat capacity depend on Debye temperature. The hardness of the Zr 2 GaC and Hf 2 GaC MAX phase in terms of Debye temperature is in the order of Zr 2 GaC > Hf 2 GaC because the θ D value for Zr 2 GaC and Hf 2 GaC is 787.91 K and 720.78 K, respectively, at 0 GPa and ambient temperature.
The effect of temperature on heat capacity (Cv, Cp) of Zr 2 GaC and Hf 2 GaC MAX phase is depicted in Figure 10. The heat capacity is the ability of the material to absorb heat from the surroundings, and one can get useful information about the density of states, lattice vibration, energy band structure, etc., from heat capacity. It is observed that both Cv and Cp increase abruptly with temperature when temperature limit T ≤ 300 K, beyond the 300 K, Cv and Cp increases slowly and finally get converged at higher temperature to obey the Dulong-Petit limit. The effect of temperature and pressure on heat capacity is the opposite; however, temperature change has a more significant impact than pressure. The obtained heat capacity values of Zr 2 GaC and Hf 2 GaC MAX phase at constant volume corresponding to ambient temperature and 0 GPa are 72.33 J/mol·K and 76.00 J/mol·K, respectively.
The Grüneisen parameter (γ) is used to calculate the thermal state and is a dimensionless quantity. The trend for γ is almost similar to that of volume. As shown in Figure 11 that γ decreases with an increase in pressure while increasing linearly with temperature. However, γ remains constant in the temperature range from 0-100 K. The obtained γ 0 values for Zr 2 GaC and Hf 2 GaC are 1.76 and 1.74 at 0 GPa, respectively. Finally, the dependence of the thermal expansion coefficient (α) on temperature is depicted in Figure 12. At a given temperature, α decreases with the increase in pressure; this may be due to reduced unit cell volume. The results of α at ambient temperature and 0 GPa for Zr 2 GaC and Hf 2 GaC are 1.14 × 10 −5 K −1 and 0.84 × 10 −5 K −1, respectively. However, in the range 0-300 K, α increases rapidly and reaches a plateau at higher temperature, indicating that the temperature effect is more important at lower temperature.

Conclusions
In this study, electronic, structural, elastic, phonon, thermodynamical, and mechanical properties of the M 2 GaC MAX phase (M = Zr, Hf) were comprehensively investigated by employing first-principle calculations based on GGA and LDA exchange-correlation functional. The thermodynamic properties were computed using the quasiharmonic Debye model in the pressure ranges 0-50 GPa, and temperature in the range 0-1600 K. The results showed that the formation and cohesive energies for both Zr 2 GaC and Hf 2 GaC are found to be negative, i.e., −7.59 eV/atom and −6.32 eV/atom for Zr 2 GaC and −7.45 eV/atom and −6.16 eV/atom for Hf 2 GaC. The PDOS revealed that the main contribution in electric transport at E F was by transition metal M-4d electrons and hybridizations between M-d and C-p states is stronger than M-d and Ga-p states. The DOS at E F for Zr 2 GaC and Hf 2 GaC was 2.96 states/eV/unit and 2.47 states/eV/unit, respectively, indicating that these MAX phases are predicted to be electrical conductors while band structures pointed out their anisotropic nature. Moreover, the formation and cohesive energies, elastic constants, and phonon calculations showed that both compounds are thermodynamically, mechanically, and dynamically stable. Furthermore, the values of E, B, and G in terms of elastic constants are in the order of Hf 2 GaC > Zr 2 GaC. From the B/G, Poisson's ratio, and anisotropy index, it is concluded that the MAX phases investigated in this study are brittle and anisotropic. As far as thermodynamic properties of the M 2 GaC MAX phase are concerned, the volume and Grüneisen parameter increases linearly with an increase in temperature and decreases with an increase in pressure while the Debye temperature has an inverse trend compared to them. The heat capacities (Cv and Cp) and the thermal expansion coefficient increases rapidly up to the temperature ranges 0-300 K and then reaches a plateau at higher temperature. To the best of the authors' knowledge, no data related to the electronic, structural, phonon, and thermodynamic properties in the literature exist so far. Hence, the results about the M 2 GaC (M = Zr, Hf) MAX phase can serve as a reference for future theoretical and experimental research.