Physical Properties Investigations of Ternary-Layered Carbides M₂PbC (M = Ti, Zr and Hf): First-Principles Calculations

Abstract

We investigated structure optimization, mechanical stability, electronic and bonding properties of the nanolaminate compounds Ti₂PbC, Zr₂PbC, and Hf₂PbC using the first-principles calculations. These structures display nanolaminated edifices where MC layers are interleaved with Pb. The calculation of formation energies, elastic moduli and phonons reveal that all MAX phase systems are exothermic, and are intrinsically and dynamically stable at zero and under pressure. The mechanical and thermal properties are reported with fundamental insights. Results of bulk modulus and shear modulus show that the investigated compounds display a remarkable hardness. The elastic constants C₁₁ and C₃₃ rise more quickly with an increase in pressure than that of other elastic constants. Electronic and bonding properties are investigated through the calculation of electronic band structure, density of states, and charge densities.

1. Introduction

During the 1960s, Hans Nowotny and coworkers in Vienna achieved a momentous accomplishment [1], finding in excess of 100 new carbides and nitrides. Among these materials were the alleged H phases and their family members, the MAX phases. Despite this great achievement, these phases remained generally neglected until the 1990s, when several scientists started to take renewed interest. The breakthrough development that set off a renaissance came throughout the 1990s, when Barsoum and El-Raghy [2,3,4] synthesized nearly phase-pure samples of Ti₃SiC₂, and uncovered a material with an extraordinary mix of ceramic and metallic properties. Like metals, the material displayed large electrical and warm conductivity, and it was machinable. Furthermore, it was very impervious to oxidation and hot shock, similar to ceramics. At the point when they later found Ti₄AlN₃, these phases shared an obvious fundamental structure that gave them comparable properties. This accomplishment prompted the presentation of the nomenclature M_n+1AX_n (n = 1–3) or MAX phases, where M is an early transition metal, A is a group IIIA or IVA element, and X is either carbon (C) and/or nitrogen (N) [5,6,7,8]. The MAX phases are considered to have good resistance to warm shock and oxidation [9,10,11,12]. They show high strength at high temperatures, however, are relatively soft and can be machined, harnessing regular fast speed tools without grease. They likewise are great mild and electrical conductors [13,14,15,16,17,18]. Moreover, the two-dimensional MXene structures are derived from MAX phases, which possess practical utilization in Li-ion and sodium-ion batteries and supercapacitors [19,20].

The 211 MAX phases have been investigated by theoretical calculations [21,22,23,24,25,26,27,28,29,30,31,32], that have determined a large range of physical properties, including structural, mechanical, electronic, electrical, optical and bonding properties. Amongst them, more than 50 M₂AX phases have been found at present. M₂PbC, members of M₂AX phases with M = Ti, Zr, Hf, have recently attracted attention owing to their particular structural characteristics [24]. M₂PbC compounds were also proposed to have unusual physical properties, especially mechanical properties [18]. The crystal structure of M₂PbC was synthesized for the first time by Jeitschko et al. [33]. El-Raghy et al. [34] have synthesized M₂PbC using reactive HIPing of the stoichiometric combination of the equivalent elemental powders in the temperature range between 1200 °C to 1325 °C for 4 to 48 h. They reported the Vickers hardness and electrical resistivity of M₂PbC phases. On the theoretical research side, few works, mainly based on density functional theory (DFT) calculations, have explored the electronic, elastic, thermodynamic, and optical properties of M₂GaC MAX phases with M = Ti, Zr and Hf [24,35,36].

These previously cited experimental and theoretical investigations motivated us to investigate the structural and mechanical stability, elastic properties under pressure, as well as electronic, bonding and dynamical properties of Ti₂PbC, Zr₂PbC and Hf₂PbC MAX phases. In this contribution, we intend to report a theoretical study of the structural stability, electronic, mechanical, and phonon properties of the M₂PbC (M = Ti, Zr, Hf) MAX phase systems. The results revealed that the M₂PbC MAX phases are elastically and electronically anisotropic in nature, and appropriate for high-temperature application, optoelectronic devices and coating material.

2. Materials and Computational Methods

All calculations are implemented in the Quantum Atomistix ToolKit (quantumATK) [37] package using the density functional theory (DFT) within the local combination of the atomic orbitals (LCAO) approach. The generalized gradient approximation (GGA) of the Perdew, Burke, Ernzerhof (PBE) is employed for the exchange correlation functional [38]. The norm-conserving PseudoDojo [39] pseudopotential was chosen for characterizing the interaction between ion nuclei, and the valence electrons. The computation of the self-consistent field (SCF) was iterated until the difference in total energy less than 10⁻⁶ Ha was completed. The optimized geometry structures were obtained using the limited-memory Broyden-Fletcher-Goldfarb-Shanno minimization technique, with force on each atom site fewer than 0.05 eV/Å. For the geometry optimization, a 4 × 4 × 3 Monkhorst-Pack [40] k-grid is used, and for electronic property calculations, a 10 × 10 × 8 grid is used. Because of the strong relativistic effect owing to the existence of heavy Pb element, spin orbital coupling (SOC) contribution is taken account of the electronic structure calculations. Moreover, the phonon dispersion was computed using a dynamical matrix based on a FD method, implemented in quantum ATK [37].

3. Results and Discussion

3.1. Structure Optimization

M₂PbC materials crystallize in the hexagonal structure with space group P63/mmc. This latter can be described as an almost close-packed stacking of M₂PbC layers, interleaved with an unadulterated tin layer every fourth layer (Figure 1). The atoms forming the structure are situated at the accompanying Wyckoff positions, in which M atoms are at 4f (Z_M), Pb at 2d and C at 2a. The coordinate Z_M denotes the main internal free parameters in the unit cell structure. For the atoms in 4f positions, the ideal value of the internal dimensionless parameter Z is 1/12 for the metal atom M. Our quantified values for a, c and Z_M are listed in

Table 1. The equilibrium lattice parameters a and c, volume V, and relaxed atomic parameter ZM and the formation energy (∆Ef) for M₂PbC (M = Ti, Zr, Hf).

	a (Å)	c (Å)	c/a	V (Å3)	ZM	∆Ef (eV)
Ti2PbC	3.224 3.20 [34] 3.239 [24]	13.926 3.81 [34] 14.134 [24]	4.319	125.42	0.0776 0.0765 [24]	−2.477
Zr2PbC	3.413 3.38 [34] 3.384 [33] 3.41 [35]	14.868 14.66 [34] 14.67 [33] 14.96 [35]	4.356 4.333 [33] 4.39 [35]	150.02 150 [35]	0.0816 [35]	−2.940
Hf2PbC	3.386 3.55 [34] 3.358 [33] 3.416 [36]	14.584 14.46 [34] 14.47 [33] 14.712 [36]	4.307 4.308 [33] 4.307 [36]	144.80 148.663 [36]	0.0810 0.083017 [36]	−2.704

[33,34] for comparison. Contrasted to the experiment, the computed a and c lattice constants are overrated by about 0.96% and 1.2% for Ti₂PbC, respectively. In the case of Zr₂PbC, the variation from experimental values is 0.88% and 2.87%. The deviations are estimated to be about 6.02% and 1.40% for Hf₂PbC. Moreover, our results agree well with previous theoretical [24,31,32,35,36].

The stability of M₂PbC is then examined by calculating the formation energy and vibrational phonon spectrum. The formation energy is computed to examine the structural stability obtained from the difference between their total energies and the sum of the isolated atomic energies of the pure constituents. The numerical result listed in the table suggests that formation energy values are negative, evidencing exothermic and intrinsic stability. The large formation energy demonstrates the structural robustness of the M₂PbC. Next, the phonon band structures spectrum is calculated to check the dynamic stability of the investigated systems. Moreover, it is shown from Figure S1 from the Supplementary Materials that all branches in the phonon spectrum curves are positive, and the absence of imaginary phonon mode indicates that all investigated MAX structures are dynamically stable.

The impact of pressure on lattice parameters is investigated by varying pressure from 0 to 50 GPa with a step of 10 GPa, which is presented in Table S1 and reflected in Figure 2. In terms of the lattice parameters, the unit cell volume decreased with the rise in pressure, while the internal parameter increased for all studied systems. Figure 2 illustrates the effect of pressure on the ratio of equilibrium parameters a/a₀, c/c₀ and volume V/V₀, where a₀, c₀, and V₀ represent the lattice constants and volume at zero pressure. From Figure 2, a reduction in the lattice parameters ratio can be seen with growing pressure. Thus, deviations are observed to be linear interpolation for a/a₀ than that of c/c₀. It is also observed from Figure 2 that the V/V₀ ratio decreases with additional increase in pressure, indicating a strong compressibility. Moreover, the reduction in V/V₀ ratio is found to be 20.4%, 21.5% and 25.5% for Ti₂PbC, Zr₂PbC and Hf₂PbC, respectively, with a more significant effect on Zr₂PbC. It is worth noting that, at high pressure, the volume proportion bend turns out to be consistent, illustrating that change in atomic distance is more modest, which brings about more grounded shared repulsion as atoms come nearer; in the long run, compression of the structure becomes more difficult. A similar behavior is observed for lattice parameter ratios (a/a₀, c/c₀).

The bond length ratio of M–C and M–Pb atoms of the M₂PbC (M = Ti, Zr, Hf) MAX phase unit cell as function of pressure is shown in Figure 3. It is noticed that the bond length is shortened with the increase in pressure. Moreover, the bond length M-Pb becomes stiffer than that of M–C. The bond lengths in Zr₂PbC are more reduced compared with those of Ti²PbC and Hf₂PbC, indicating further compressibility along the Zr–Pb direction.

3.2. Elastic Properties and Mechanical Stability

The elastic constants are determined by the strain-energy strategy. A proper arrangement of strains is applied to the under-framed unit cell lattice with the relaxed system. Then, the elastic constants are calculated from the subsequent change in total energy on the distortion. For the hexagonal crystal structure, there are five independent stiffness constants Cij, C₁₁ in particular, C₁₂, C₁₃, C₃₃, C₄₄; while C₆₆ = (C₁₁ − C₁₂)/2). The elastic constants are computed by employing the Lagrangian strain and stress tensors. The obtained elastic constants are tabulated in

Table 2. The predicted elastic constants, Cij (in GPa), bulk moduli (B, in GPa), shear moduli (G, in GPa), Young’s moduli (Y, in GPa), and Poisson’s ratio (ν) for Ti₂PbC, Zr₂PbC and Hf₂PbC.

	C11	C33	C44	C12	C13	C66	B	G	Y	A	v
Ti2PbC	244.4	228.2	68.5	100.2	42.4	72.1	119.8	76.2	188.6	0.707	0.238
Zr2PbC	220.1 220.10 a	219.9 216.28 a	69.8 79.83 a	68.5 65.86 a	62.9 64.53 a	76.2 77.12 a	116.3 116.25 a	74.1 78.12 a	183.4 191.47 a	0.889	0.237 0.23 a
Hf2PbC	286.8 237.07 b	263.2 217.02b	88.1 79.50 b	65.4 71.49 b	77.4 70.77 b	110.7 82.79 b	141.6 124.03 b	98.1 80.18 b	239.1 197.91 b	0.892	0.219 0.23 b

^a ref [35], ^b ref [36].

with previous theoretical values for comparison. Small deviation is seen between the two theoretical results (zero pressure) because of the various methods for estimations. It is noticed that the elastic constants C₁₁ and C₃₃, which show stiffness against principal stains, decrease when M is varied from Ti, Hf to Zr. Therefore, the C₆₆, which depends on the resistance to shear in the {010} plane in the 〈110〉 direction, decreases, following the same trend as C₁₁ and C₃₃. The elastic stiffness C₄₄, which indicates the resistance to shear in the {010} or {100} plane in the 〈001〉 direction, increases when M is changed from Ti, Hf to Zr. More importantly, the mechanically stable phases or macroscopic stability are subject to the positive definiteness of the stiffness tensor [32]. For a stable hexagonal construction, its five stiffness constants ought to fulfill the criteria of Born stability [41], i.e., $C_{11}>0, C_{11}-C_{12}>0, \left(C_{11}+C_{12}\right)C_{33}-2C_{13}^2>0 and C_{44}>0$. It is found that the investigated systems satisfied the Born mechanical stability conditions since all their elastic constants are positive.

From the computed elastic constants, the macroscopic mechanical properties, namely their bulk B and shear G moduli, can be estimated using the Voigt–Reuss–Hill approximations [42,43,44,45,46]:

$${\mathrm{B}}_{\mathrm{V}}=\frac{1}{9}\left(2\left(C_{11}+C_{12}\right)+4C_{13}+C_{33}\right)$$

(1)

$${\mathrm{B}}_{\mathrm{R}}=\frac{\left(\left(C_{11}+C_{12}\right)C_{33}-2C_{12}^2\right)}{\left(C_{11}+C_{12}+2C_{33}-4C_{13}\right)}$$

(2)

$${\mathrm{G}}_{\mathrm{V}}=\frac{1}{30}\left(C_{11}+C_{12}+2C_{33}-4C_{13}+12C_{44}+12C_{66}\right)$$

(3)

$${\mathrm{B}}_{\mathrm{R}}=\frac{\frac{5}{2}\left[{\left(\left(C_{11}+C_{12}\right)C_{33}-2C_{12}^2\right)}^2\right]C_{44}{C}_{66}}{\left[3B_V{C}_{44}{C}_{66}+{\left(\left(C_{11}+C_{12}\right)C_{33}-2C_{12}^2\right)}^2\left(C_{44}+C_{66}\right)\right]}$$

(4)

$$\mathrm{B}=\frac{1}{2}\left({\mathrm{B}}_{\mathrm{V}}+{\mathrm{B}}_{\mathrm{R}}\right)$$

(5)

$$\mathrm{G}=\frac{1}{2}\left({\mathrm{G}}_{\mathrm{V}}+{\mathrm{G}}_{\mathrm{R}}\right)$$

(6)

where B_V, G_V, and B_R, G_R represent the bulk and shear modulus, which can be defined in the Voigt and Russ approximation, respectively. The Young’s modulus, Poisson’s ratio and elastic anisotropy are obtained by:

$$Y=\frac{9BG}{3B+G}$$

(7)

$$\nu =\frac{3B-2G}{2\left(3B+G\right)}$$

(8)

$$A=\frac{4C_{44}}{C_{11}+C_{33}-2C_{13}}$$

(9)

The computed bulk modulus, shear modulus, compressibility, Young’s modulus, and Poisson’s ratio of M₂PbC (M = Zr, Hf) are gathered in

Table 3. Calculated density, ρ, in g/cm³, longitudinal, transverse and average sound velocity, ${\vartheta }_t$, ${\vartheta }_l$, ${\vartheta }_m$, in km/s, and Debye temperatures, θ_D, in K of Ti₂PbC, Zr₂PbC and Hf2PbC.

	ρ	${{\mathit{\vartheta }}_{\mathit{t}}}_{}$	${{\mathit{\vartheta }}_{\mathit{l}}}_{}$	${{\mathit{\vartheta }}_{\mathit{m}}}_{}$	${\mathit{\theta }}_{\mathit{D}}$
Ti2PbC	8.509	3.089	5.198	3.318	393.5
Zr2PbC	9.037	2.919	4.920	3.137	350.0
Hf2PbC	13.540	2.450	4.180	2.638	298.2

according to the Voight–Reuss–Hill bounds [34,38,40]. It is observed that Hf₂PbC has a high bulk modulus, shear modulus, and Young’s modulus, than those of the Ti₂PbC and Zr₂PbC. The larger shear modulus is primarily owing to its bigger C₄₄, as this constant indicates the resistance to shear in the {010} or {100} plane in the 〈110〉 direction. Besides, Hf₂PbC shows a large value of C₁₁ and a small value of C₄₄, suggesting a low G and E, in spite of a generally massive B compared with other compounds. Moreover, our estimated Poisson’s ratio of the Hf₂PbC is significantly smaller than that of Ti₂PbC and Zr₂PbC. This value of Poisson’s ratio demonstrates that the Hf₂PbC is approximately steady against shear and a stronger level of covalent bonding, bringing about the greater hardness.

The elastic constants C_ij and elastic moduli are studied under pressure varying from 0 to 50 GPa, as shown in Figure 4 and Figure 5. It is observed from the plots that the elastic constants and moduli uniformly increase with pressure increases. Our findings also show that the computed C₁₁, C₃₃, Young’s modulus, and bulk modulus values increase remarkably when compared to other elastic constants, whereas the C₆₆ and shear modulus change more gradually. Note that C₄₄ for the Zr₂PbC increases uniformly up to 40 GPa, and further reduces when pressure reaches 50 GPa. It is also noticed that C₃₃ is larger than C₁₁ under pressure, stating that the chemical bonding strength grows significantly with an increase in pressure. It is also shown that an increase in pressure leads to a larger increase in bulk, shear and Young’s moduli for Ti₂PbC than that of Zr₂PbC and Hf₂PbC, indicating a good resistance to deformation for Ti₂PbC. This behavior is observed in M₂AC (A = Sn, Ga...) compounds [30,35,36]. Based on a substantial experimental dataset assessment, Pugh [47] suggested that G/B values less than 0.571 are correlated with ductility, whereas higher G/B values denote brittleness. As is reported in Table 3, the investigated systems are expected to behave essentially as brittle materials.

The 3D plots of Young’s modulus, shear modulus and Poisson’s ratio surfaces of Ti₂PbC, Ti₂PbC and Zr₂PbC obtained using the ELATE program [48] are displayed in Figure 6, Figure 7 and Figure 8. These 3D plots were obtained from the computed values of C_ij of the investigated systems at 0 and 50 GPa. Note that the plotting of 3D contours is an attractive method of introducing mechanical anisotropy where an ideal sphere assigns the ideal isotropy of solids and, starting from a spherical shape, estimates the degree of anisotropy. It is observed that, in general, the shape (Figure S1) begins to deviate from the sphere form with a rise in pressure, indicating the elastic anisotropy of M₂PbC MAX phases. Moreover, the pressure effect on the elastic anisotropy of Hf₂PbC is more significant than that of the Zr₂PbC and Ti₂PbC (see Table 2).

3.3. Thermodynamic Properties

One of the main criteria that decides the thermal qualities of materials is the Debye temperature (θ_D). At low temperatures, the vibrational excitations occur exclusively from acoustic vibrations. Along these lines, the Debye temperature, determined from elastic constants at low temperatures, is the same as that defined from specific heat estimations. Since the Debye temperature may be obtained from the averaged sound velocity, it can be calculated from the following expression [49]:

$${\theta }_D=\frac{h}{k_B}{\left[\frac{3n}{4\pi }\left(\frac{N_A\rho }{M_w}\right)\right]}^{1/3}{\vartheta }_m$$

(10)

where h, k_B, N_A, ρ, M_w and n denote the Planck’s constant, the Boltzmann’s constant, the Avogadro’s number, the density, the molecular weight, the number of atoms in the unit cell, respectively.

The longitudinal elastic wave velocity, ${\vartheta }_l$, transverse elastic wave velocity, ${\vartheta }_t$, and the mean sound velocity, ${\vartheta }_m$, are estimated by:

$${\vartheta }_l=\sqrt{\left(\frac{3G+4G}{3\rho }\right)}$$

(11)

$${\vartheta }_t=\sqrt{\frac{G}{\rho }}$$

(12)

$${\vartheta }_m={\left[\frac{1}{3}\left(\frac{2}{{\vartheta }_t^3}+\frac{1}{{\vartheta }_l^3}\right)\right]}^{-1/3}$$

(13)

The ${\vartheta }_l$ and ${\vartheta }_t$ can be computed from the polycrystalline shear modulus and bulk modulus [50]. The determined values of Debye temperatures are recorded in

Table 4. Octahedron, o_d, trigonal prism, p_d, distortion parameters, their ratio for M₂PbC (M = Ti, Zr, Hf). Atomic radius of M elements is also reported.

	od	pd	od/pd	RM (nm)	RPb (nm)
Ti2PbC	1.186618	1.061337	1.118041	1.40	1.80
Zr2PbC	1.132275	1.070055	1.058146	1.55	1.80
Hf2PbC	1.146912	1.07636	1.065546	1.55	1.80

. Our outcomes foresee that the considered MAX phases have a relatively large value of θ_D to show that they have a somewhat stiff lattice and, thus, greater thermal conductivity. The gradual diminishing of mean sound speeds in the Ti → Zr → Hf series further clarifies the inclination of bringing down Debye temperatures along a similar consecutive order.

3.4. Steric Effect

According to the 211 MAX phase structure shown in Figure 1, the unit cell crystal is formed of [M₆X] octahedra and [M₆A] trigonal triangular prisms. For a perfect cubic structure binary, the parameters of octahedra and trigonal should be equal to one (o_d = p_d = 1) with fourfold symmetry axes, whereas the octahedral block loses its fourfold axis in the ternary MAX, resulting in a relaxation due to this decreased symmetry. This distortion can be described by the distortion parameter, o_d,. The noncubic distortion of the octahedron can be quantified from the parameter o_d determined as [51]:

$$o_d=\frac{\sqrt{3}}{2\sqrt{4Z_M^2{\left(\frac{c}{a}\right)}^2+\frac{1}{12}}}$$

(14)

For the trigonal prism, the distortion parameter p_d is expressed by [29]:

$$p_d=\frac{1}{\sqrt{\frac{1}{3}+{\left(\frac{1}{4}-Z_M\right)}^2{\left(\frac{c}{a}\right)}^2}}$$

(15)

For 211 MAX phases like M₂PbC, an ideal packing of hard spheres of equivalent diameter leads to a ratio c/a ≈ 4.89. The calculated octahedra and trigonal prisms parameters, deduced from the optimized lattice constants, are listed in Table 4, along with the atomic radii R_M of the M elements [51]. It is found that both polyhedra are distorted for all systems. Moreover, the distortion of the two constituting polyhedra is minimal for elements M (Zr, Hf) with a large atomic radius. This distortion is similar. This behavior can be described as a steric effect, in which the M atoms constitute a large part of the lattice structure topology. This effect is also defined by the fact that lattice constants a and c are larger for Zr₂PbC and Hf₂PbC than that of Ti₂PbC (Table 1).

3.5. Electronic Properties

To get a further understanding about the electronic feature, we calculated the band electronic structure at optimized geometry at equilibrium volume by including SOC contribution, as illustrated in Figure 9. It can be evaluated from the energy band structure that no band gap is observed at the Fermi level, in which a large overlapping of occupied and unoccupied states appears. Therefore, the conductivity of studied materials is of metallic electronic nature. It is also observed that a strong anisotropic character is indicated with less c-axis energy dispersion, owing to the decreased dispersion along the short H–K and M–L directions. These findings are highly similar to those obtained for other MAX phase materials [29]. Note that SOC leads to an increased band dispersion and splitting along the investigated high symmetry directions.

Figure 10 displays the total and partial density of state (DOS) of Ti₂PbC, Zr₂PbC, and Hf₂SnC. The DOS plot around the Fermi level mainly lies in a dip. This area might part the occupied and unoccupied states, bringing rise to more robust cohesion, underlying strength [44] and higher bulk modulus. The electronic analysis reveals the contributions of C p situated within the −6 to −1.8 eV energy range, showing a similar form as the d states of M (M = Ti, Zr, Hf) atoms situated between C layers. This reveals a hybridization between M d and C p states, and consequently of a covalent interaction. In the range of −2.5 to −0.6 eV, the Pb 6p states interact primarily with the d state of M elements. However, these Pb p-M d mixtures show a higher energy than that of the C p-M d (for example, nearer to the Fermi level) implying that M-C bonds are more grounded than M-Pb bonds. The large mixing with C p states and M d lead to stabilize the structure of M₂PbC.

Indeed, for these ternaries which Pb electronic configuration is 5s²5p², the Fermi level is observed to be in a dip above M d-Pb p interaction and below the broad M d unoccupied band. Generally, the Pb p states are changed to lower energy as the filling of the p band is raised. At the Fermi level, the TDOS values for Ti₂PbC, Zr₂PbC, and Hf₂SnC, are 4.54, 3.10, 3.35 states per unit cell per eV, respectively. Moreover, the states close to the Fermi level are occupied by M d states as well as Pb p states.

To probe the light of the bonding nature, we calculated the charge densities (Figure 11), mapping in the (11–20) plane for Ti₂PbC, Zr₂PbC, and Hf₂SnC. The charge densities are computed using the full potential (linearized) augmented plane waves, plus the local orbitals (FP-(L) APW + lo) method [52] within the GGA-PBE approximation. The self-consistent procedure was performed with R_MT × K_max = 11. The plots show that metallic bonding is illustrated in M and Pb layers, while no free charge density distribution in C planes is noticed. In the plane, the charge density distribution reports detailed features on the interaction between different atoms. The interaction between M and C atoms in M₂PbC indicates covalent bonds, showing a very strong bonding. Simultaneously, the more electronegative character of C contrasted with Ti, Zr or Hf gives the presence of ionic bonding between Ti, Zr or Hf and C, whereas the more electropositive description of Pb affirms the ionic character between M and Pb. Thus, the bonding nature in M₂PbC is found to be a mixture of metallic–covalent–ionic behavior. It can be concluded that large melting points and modulus are anticipated from the strong existence of covalent and ionic bonding, whilst microscale plasticity and good electrical conductivity are expected from the presence of the metallic nature [53,54].

4. Conclusions

We have performed theoretical analysis of structural, elastic, electronic, thermodynamic and bonding properties of M₂PbC, with M = Ti, Zr and Hf using first principles calculation. It is revealed that the relaxed structural properties are found in excellent agreement with the available experimental data. The explored materials show an interesting bulk modulus compared with the MAX phases experimentally or theoretically investigated so far. Our analysis of stiffness constants reveals that all M₂PbC MAX phases show an intrinsic and mechanical stability. It is also shown that these materials are described by a small elastic anisotropy. Moreover, the shear modulus, Young’s modulus, and Poisson’s ratio for ideal polycrystalline M₂PbC MAX phases are calculated. Our findings exhibit that B > G, the bulk modulus is greater that the shear modulus for all compounds, which means that the shear modulus represents a parameter limiting the stability. The stability origin of the M₂PbC is associated with the interaction between M d-Pb p and M d-C p bonds. For the Pb element with the deepest filled p states, the M d-C p hybridization are lower in energy with stiffer bonding than the M d-Pb p. The electronic structure analysis shows that the M₂PbC MAX materials have a metallic character. Hence it is possible to deduce that Ti₂PbC is more conductive compared with the other MAX compounds. The charge density map analysis reveals that the bonding behavior in these compounds may occur from the blend between the covalent-ionic and metallic nature.