Mo₂TiAlC₂ crystallizes in a hexagonal structure with a space group of P6₃/mmc and is isostructural with Ti₃SiC₂. Figure 1 gives an idea about the crystal structure of Mo₂TiAlC₂. In Mo₂TiAlC₂ there are 12 atoms in one unit cell (Z = 2). The calculated lattice constants a and c as well as equilibrium unit cell volume V along with atomic positions for Mo₂TiAlC₂ are listed in Table 1. The present theoretical results show the closeness to the experimental and theoretical data. The deviations from the data obtained via the Rietveld analysis of the powder x-ray pattern in the experiment^[17] for the lattice constants and unit cell volume are within 0.48% and 0.55%, respectively. On the other hand, the deviations of theoretical data obtained in previous VASP calculation^[19] are more than 0.50% and 1.25%, respectively. Therefore, the present first-principles calculations are highly reliable.

Fig. 1.

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	Fig. 1. Crystal structure of layered ordered new MAX compound Mo2TiAlC2.

Table 1.

Table 1.

Table 1. Structural properties obtained by using the DFT-based first-principles calculations together with corresponding experimental data. . Structural properties Experimental data Theoretical data SAED[17] Rietveld[17] CASTEP Calc.this VASP Calc.[19] a/Å 2.933 2.997 2.998 3.0082 c/Å 18.970 18.661 18.751 18.757 V/Å3 141.326 145.157 145.955 147.00 Space group P63/mmc Atomic position (Rietveld) Mo 4f (2/3, 1/3, 0.13336) (2/3, 1/3, 0.13316) Ti 2a (0, 0, 0) (0, 0, 0) Al 2b (0, 0, 1/4) (0, 0, 1/4) C 4f (1/3, 2/3, 0.06875) (1/3, 2/3, 0.06857)

Table 1. Structural properties obtained by using the DFT-based first-principles calculations together with corresponding experimental data. .

The elastic properties are directly related to the crystal structure and the nature of bonding between atoms within the system. These in turn largely determine the phonon spectrum and the Debye temperature of the compound. The predicted single crystal elastic constants C_ij under 0 GPa are presented in Table 2.

Table 2.

Table 2.

Table 2. Calculated values of single crystal elastic constants Cij (in unit GPa), polycrystalline bulk modulus B (in unit GPa), shear modulus G (in unit GPa), Young modulus Y (in unit GPa), Pugh’s ratio G/B, Poisson’s ratio v, and shear anisotropy factor A of Mo2TiAlC2. . Single crystal elastic properties Polycrystalline elastic properties CASTEP Calc.this VASP Calc.[19] CASTEP Calc.this VASP Calc.[19] C11 393 386 B 217 220 C12 132 143 G 140 131 C13 133 140 Y 346 329 C33 371 367 G/B 0.65 0.60 C44 160 150 v 0.23 0.25 C66 131 131 A 1.29 1.27

Table 2. Calculated values of single crystal elastic constants Cij (in unit GPa), polycrystalline bulk modulus B (in unit GPa), shear modulus G (in unit GPa), Young modulus Y (in unit GPa), Pugh’s ratio G/B, Poisson’s ratio v, and shear anisotropy factor A of Mo2TiAlC2. .

The polycrystalline elastic properties such as bulk modulus B, shear modulus G, Young’s modulus Y, Pugh’s ratio G/B, Poisson’s ratio v, and shear elastic anisotropy factor A derived from C_ij are also listed in Table 2 along with available results converted into aggregate values using Hill approximation.^[28] The positive values of C₁₁, C₄₄, (C₁₁ − C₁₂) and {(C₁₁ + C₁₂)C₃₃ − 2(C₁₂)²} justify the mechanical stability^[32] of newly discovered nanolaminate Mo₂TiAlC₂. The comparatively large Young’s modulus of Mo₂TiAlC₂ suggests that it should have a high resistance to pressure and tension in the range of elastic deformation. The directional bonding between atoms in this new compound should also be stronger due to having the large value of G. Generally, the large shear modulus of a material is an indication of its well-defined directional bonding between atoms. Pugh^[33] suggested G/B ratio as an index of the plastic behavior of a material and proposed the value ∼ 0.57 as the critical threshold value for separating the brittle and ductile nature of a material. A polycrystalline material would bear its mechanical property dominated by brittleness if G/B > 0.57, otherwise it would behave as a ductile material. In addition, the Frantsevich’s rule^[34] related to Poisson’s ratio recommends the value ∼ 0.33 as the border line between the ductile and brittle nature of a material. If the Poisson’s ratio is over 0.33, the material would show the nature of ductility. Otherwise, it would exhibit the characteristics of a brittle material. The present calculated results are evident that the new compound Mo₂TiAlC₂ like most of the MAX phases^[35–37] is brittle in nature in accordance with both the Pugh’s criterion and Frantsevich’s rule. Indeed, the brittleness is the inherent trend of MAX materials. It is seen that the new ordered MAX phase Mo₂TiAlC₂ possesses relatively low value of Poisson’s ratio and it is indicative of its high level of directional covalent bonding.

Elastic anisotropy of a crystal reflects a characteristic feature of bonding in different directions. Inherently, most of the known crystals are elastically anisotropic, and a precise depiction of such an anisotropic character has, therefore, an essential implication in crystal physics and science of engineering. It shows a relationship with the possibility of existence of microcracks in the crystals. For hexagonal crystals, a shear anisotropy factor associated with the (100) shear plane between the 〈011〉 and 〈010〉 directions is defined as A = 4C₄₄/(C₁₁ + C₃₃ − 2C₁₃). For an isotropic crystal, A is found to be unity. The deviation of A from unity assesses the elastic anisotropy possessed by the crystal and the amount of deviation measures the level of elastic anisotropy. The shear anisotropic factor for Mo₂TiAlC₂ is given in Table 2, which deviates from unity by ∼ 28%. It means that both the in-plane and out-of-plane inter-atomic interactions differ from each other significantly. Another anisotropic factor for hexagonal crystal is defined as the ratio between the linear compressibility coefficients along the c and a axes and expressed as k_c/k_a = (C₁₁ + C₁₂ − 2C₁₃)/(C₃₃ − C₁₃). This factor is also evaluated. The present (previous) value of 1.09 (1.10) discloses that the compressibility along the c axis is slightly greater than that along the a-axis for this new MAX compound.

The Debye temperature θ_D is an influential parameter directly related to various physical properties including melting temperature and specific heat, and is used to make a distinction between high- and low-temperature regions for a solid material. The Debye temperature well determines a demarcation between quantum and classical behavior of phonons. When the temperature T of a solid is raised over θ_D, all modes of vibrations are expected to have energy equal to k_BT. At T < θ_D, the high-frequency modes are found to be stationary. The vibrational excitations at low temperature appear only from acoustic vibrations. Consequently, the Debye temperature calculated from elastic constants is seen to be the same as that estimated from measured specific heat.^[38] The Debye temperature along with the related quantities including longitudinal, transverse and average sound velocities calculated within the present formalism is listed in Table 3. The high Debye temperature is an indication of strong covalent bonding in Mo₂TiAlC₂.

Table 3.

Table 3.

Table 3. Calculated density (ρ in units gm/cm3), longitudinal, transverse and average sound velocities (vl, vt, and vm in units km/s) and Debye temperature (θD in unit K) of Mo2TiAlC2. . ρ vl vt vm θD 6.297 8.006 4.715 3.196 413.6

Table 3. Calculated density (ρ in units gm/cm3), longitudinal, transverse and average sound velocities (vl, vt, and vm in units km/s) and Debye temperature (θD in unit K) of Mo2TiAlC2. .

A proper depiction of electronic structure is essential to explain many physical phenomena such as optical spectra of materials. The calculated electronic energy band structure of Mo₂TiAlC₂ at equilibrium lattice parameters along the high-symmetry lines in the first Brillouin zone is shown in Fig. 2(a). The new MAX carbide keeps its Fermi surface just below the valence band maximum around the Γ point. The occupied valence bands extend broadly from −8.7 eV to the Fermi level E_F. A lot of valence bands cross the Fermi level and partly cover the conduction bands. Accordingly, no band gap appears at the Fermi level and Mo₂TiAlC₂ should behave as a metallic compound.

Fig. 2.

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	Fig. 2. Electronic structure of Mo2TiAlC2. (a) Band structure at optimized cell parameters along with the high symmetry lines. (b) Total and partial DOSs. The Fermi level is set to be 0 eV.

To give a clear view of the nature of the electronic band structure, the total and partial energy density of states (DOS) are also calculated and explained as indicated in Fig. 2(b). A large value of total DOS is observed at the Fermi level and estimated as 5.6 states per eV per unit cell, which originates mainly from Mo 4d and Ti 3d states, indicating that the metallic bond is existent in Mo₂TiAlC₂ and it is electrically conductive. It is seen that several distinct peaks form the wide valence band in total DOS. The lowest lying valence band from −13.1 eV to −10.2 eV originates from C 2s electron with hybridization of Mo 4d and Ti 3d electrons. The next lowest peak structure situated between −8.8 eV and −7.1 eV is mainly composed of Al 3s states and shows the hybridization with Al 3p electrons. The peak on the left side of the highest peak ranges from −7.1 eV to −5.4 eV and arises due to C 2p states with hybridization of Mo 4d states. The highest peak located between −5.4 and −3.3 eV is derived from the strong hybridization of C 2p with Mo 4d and Ti 3d states. The intense peak on the right side of the highest peak consists of the dominant contributions from Mo 4d and Al 3p states with a small contribution from Mo 5p states. In the lowest valence band, the peaks due to Ti 3d and C 2s states are weaker than those of Mo 4d and C 2s states. Hence, the Mo–C bond is stronger than the Ti–C bond. It is seen that the hybridization of Al 3p states with Mo 4d/5p states appears in the highest energy range. Therefore, Mo-Al bond is weaker than both Mo–C and Ti-C bonds. The overall picture of electronic structure suggests that the new MAX compound Mo₂TiAlC₂ should appear to have strong covalent character and heavy metallic conductivity. Also some ionic natures would be expected due to the difference in electronegativity between the constituent atoms as predicted in most of the MAX phases.^[36,37,39] Strong covalent bonding in Mo₂TiAlC₂ as predicted from high Debye temperature is again verified now. The general features of the electronic structures obtained in the present and previous studies are almost similar.

Table 4.

Table 4.

Table 4. Population analysis of Mo2TiAlC2. . Species Mulliken atomic populations Effective valence Charge (e) s p d Total Charge (e) C 1.45 3.19 0.00 4.64 −0.64 – Al 0.91 1.79 0.00 2.70 0.30 2.70 Ti 2.04 6.45 2.64 11.12 0.88 2.12 Mo 2.23 6.70 5.02 13.95 0.05 3.95

Table 4. Population analysis of Mo2TiAlC2. .

The charge transfer and Mulliken atomic populations are also calculated in order to obtain the bonding characters in Mo₂TiAlC₂. Listed in Table 4 are the atomic populations along with effective valence. The effective valence is a measure of the difference between the formal ionic charge and the Mulliken charge on the anion species in the crystal and identifies a bond as being either covalent or ionic. The zero value of effective valence indicates an ideal ionic bond while a value greater than zero refers to an increasing level of covalency. The calculated effective valence listed in the last column in Table 4 signifies the remarkable covalency in bonding within Mo₂TiAlC₂. The results also show that C carries the negative charge and the positive charge is carried by the other three metal atoms. These suggest two possible paths for electron transfer. The one guides to s-d hybridized covalent bonding between Mo and C as well as Ti and C atoms, and the other one makes the metallic or weak covalent bonding between Mo and Al atoms. The calculated bond overlap population shown in Table 5 also predicts that a high overlap population implies a high level of covalency, whereas, a low value signifies the potency of ionicity in the chemical bonds. A value of zero population suggests that there is no significant interaction between the electronic populations of the two atoms. Positive and negative values refer to bonding and antibonding states, respectively.

Therefore, it can be seen that the Mo–C bond is more covalent than the Ti–C bond, and Ti–C bond holds stronger covalent bonding than the Mo–Al bond. The values of degree of metallicity^[31,40] defined as f_m = P^μ′/P^μ for Mo–C, Ti–C, and Mo–Al bonds are 0.021, 0.033, and 0.063, respectively. Hence, the metallicity ranking of the bonds is Mo–Al > Ti–C > Mo–C; the Mo–Al bond has the highest metallicity, yielding strong metallic bonding. Thus, based on the above discussion, one can determine that the bonding nature of the new compound is dominated by the covalent and metallic bonds. The calculated theoretical hardness is also listed in Table 5. The hardness of a material is closely related to its crystal structure and chemical bonding. The higher hardness value of 9.01 GPa ensures the strong covalent bonding within Mo₂TiAlC₂. Moreover, the bond-length of Mo–C bond is shorter than those of Ti–C and Mo–Al bonds. Thus considering the general relationship between bond-length and hardness, it is expected that Mo–C bond is harder than the other two bonds Ti–C and Mo–Al. It is observed that the lowest bond volume causes the highest hardness of the bond.

Table 5.

Table 5.

Table 5. Calculated values of Mulliken bond number nμ, bond length dμ, bond overlap population Pμ, bond volume , and bond hardness of μ-type bond and metallic population Pμ′ and Vickers hardness Hv of Mo2TiAlC2. . Bond nμ dμ/Å Pμ Pμ′ Hv/GPa Mo–C 4 2.1124 1.21 0.025 8.3452 25.54 9.01 Ti–C 4 2.1560 0.76 0.025 8.8727 14.30 Mo–Al 4 2.7921 0.40 0.025 19.2709 2.00

Table 5. Calculated values of Mulliken bond number nμ, bond length dμ, bond overlap population Pμ, bond volume , and bond hardness of μ-type bond and metallic population Pμ′ and Vickers hardness Hv of Mo2TiAlC2. .

To further clarify the nature of chemical bonding in Mo₂TiAlC₂, the electron charge density distribution is investigated and the contour of electron charge density in plane is given in Fig. 3. In the adjacent scale the blue and red color indicate the low and high electron density, respectively. An atom with large electronegativity (electronic charge) attracts electron density towards itself.^[41] Due to the large differences in electronegativity and radius between atoms, the electronic charges around C (2.55) and Mo (2.16) are greater than those around Ti (1.54) and Al (1.61). The higher electronegativities of C and Mo show strong accumulation of electronic charges whereas the relatively low charge density indicates weak charge accumulation of the other elements.

Fig. 3.

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	Fig. 3. Electronic charge density in plane of Mo2TiAlC2.

The contour of electron charge density reveals a strong directional Mo–C–Mo covalent bond chain with each pair of the chains linked by a relatively weak Ti–C bond. Because of a large difference in electronegativity, the electronic charge around Mo atoms is attracted towards C atoms and a strong covalent–ionic bonding along Mo and C direction is induced. The hybridized Mo 4d–C 2p states are, in fact, responsible for the forming of these bonds. Additionally, the electron charge density of Mo just overlaps with that of Al, which is an indication of a relatively weak bonding between Mo and Al. The present results are in good agreement with the findings that MAX phases characteristically have the remarkably strong M–X bonds and rather weak M–A bonds.^[1]

Figure 4 shows the Fermi surface of Mo₂TiAlC₂ in the equilibrium structure at P = 0. The Fermi surface consists of both electron and hole-like sheets centered along the Γ–A direction of the Brillouin zone. The inner sheets are completely cylindrical with circular cross section and give rise to three-dimensional Fermi pockets. The outer sheets expand along Γ–K direction and shrink along Γ–M directions. No additional sheets appear at the corners of the Brillouin zone The Fermi surface is formed mainly by the low-dispersive Mo 4d-like bands, which is responsible for the conductivity in the compound.

Fig. 4.

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	Fig. 4. Fermi surface of new ternary MAX phase carbide Mo2TiAlC2.