First-principles study of M₄AlC₃ (M = Ti, Zr) MAX phases under hydrostatic pressure: material design for industrial applications

Abstract

MAX phase compounds, combining metallic and ceramic properties, are ideal for high-pressure environments due to their excellent electrical and thermal conductivity, corrosion and oxidation resistance, and damage tolerance. This study investigates the structural, mechanical, electronic, thermal, and optical properties of M₄AlC₃ (M = Ti, Zr) under hydrostatic pressure. Negative formation energies and positive phonon dispersion confirm thermodynamic and dynamic stability, while mechanical stability aligns with Born's criteria. Increasing stiffness constants and moduli (bulk, shear, Young's), along with Poisson's and Pugh's ratios, suggest enhanced mechanical performance. Ti₄AlC₃ and Zr₄AlC₃ are brittle below 60 GPa and 40 GPa, respectively, but become ductile above these pressures. A rising machinability index with pressure supports industrial applicability. Anisotropy is confirmed via 3D plots, and DOS analysis reveals metallic nature. Strong UV absorption and conductivity highlight their potential in UV-optical devices. Reflectivity above 60% and high IR reflectance suggest use in thermal coatings and solar heat management. Increasing Debye and melting temperatures under pressure further indicate their suitability for high-temperature applications. These findings support their use in extreme conditions such as aerospace, deep-sea exploration, and ultra-hard ceramic development.

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1. Introduction

MAX phases have recently attracted significant interest from materials scientists due to their unique combination of chemical, physical, electrical, and mechanical properties. Composed mainly of carbides and nitrides, these compounds follow the general formula M_n+1AX_n (n = 1, 2, 3…), where M is a transition metal, A is an element from groups IIA or IVA, and X is carbon or nitrogen. They crystallize in a layered hexagonal structure with P6₃/mmc symmetry.¹ MAX phases are uniquely blend metallic and ceramic characteristics. While metals typically offer high thermal and electrical conductivity, plastic deformability, and thermal shock resistance, ceramics provide high stiffness, excellent high-temperature performance, and superior resistance to corrosion and oxidation. MAX phases combine these features, earning the term “metallic ceramics” in literature.^2–8 Because of these exceptional properties, they are suitable for demanding applications such as gas burner nozzles, high-temperature foil bearings, and heating elements. In high-temperature conditions, they serve as alternatives to graphite and are used in tools for dry concrete drilling and various industrial roles.⁹ Their rising prominence stems from potential uses in advanced technologies including sensors, high-temperature ceramics, electrical contacts, catalytic systems, and protective coatings. Consequently, MAX phases have become a major focus in materials research.^9,10 Some MAX phase compounds are applicable to nuclear technology, for example, Ti₃AlC₂ and Ti₃SiC₂, are found to be resistant to radiation damage.¹¹ First documented by Nowotny et al. in the 1960s, around 100 ternary nitrides and carbides have been identified.¹² MAX phases are generally grouped based on the integer n into M₂AX (211), M₃AX₂ (312), and M₄AX₃ (413) types,^13–15 with examples including Ti₂AlC and Ti₂AlN (211), Ti₃AlC₂ and Ti₂GeC₂ (312), and Ti₄AlN₃ and Nb₄AlC₃ (413). Other variants like Ti₅SiC₄ (514), Ta₆AlC₅ (615), and Ti₇SnC₆ (716) also exist. Differences among these subgroups arise from variations in M-layer stacking, resulting in distinct structural features.^16–19

The MAX phase materials have been first synthesized in the 1960s belonged to the 211 family.²⁰ However, significant advancements began in the mid-1990s when Barsoum and collaborators highlighted their high-temperature stability combined with excellent machinability. Since then, extensive studies have focused on their synthesis and properties, including the impact of pressure on their structural, elastic, electrical, optical, and thermodynamic behaviors.^21–25 However, compared to the huge studies on 211 and 312 MAX phases, relatively limited work has been published on the 413 MAX phases. In 2000, the first 413 MAX phase, Ti₄AlN₃, was discovered which play significant role in the exploration of these type materials.⁶ Though there is a relatively small amount of research on, the unique properties and potential applications of 413 MAX phases still attract interest. Ti₄SiC₃ and Ti₄GeC₃ are highlighted by their excellent resistance to oxidation and corrosion which make them highly effective as protective coatings in thin film technology. These materials are particularly useful in environments where resistance to oxidized chemicals and high temperatures is essential.^25–27 On the other hand, materials like Ta₄AlC₃, Nb₄AlC₃, Ti₄AlN₃, and V₄AlC₃ are usually used in bulk form due to their excellent damage tolerance and mechanical properties. These MAX phases also exhibit high resistance to fracture and wear which makes them suitable for applications in such an environment where durability is critical. For instance, they are good in high-temperature applications, structural components in high-stress systems and other applications where materials should survive under repeated mechanical stresses and thermal shocks.^27–31 Li, Z., Zhang et al. studied the physical properties of MAX phase's materials Ti₂AlC and Cr₂AlC used for highly dense passivation enhanced corrosion resistance and coatings for ATF.^32,33 Hu et al. contribute to the family of 413 MAX phases by identifying Nb₄AlC₃. Studies revealed that Nb₄AlC₃ has same hexagonal layered crystal structure as Ti₄AlN₃. Nb₄AlC₃ exhibit high-temperature stability, damage tolerance and high resistance to oxidation, which make it suitable for various industrial and technological applications.^30,31 Recently, ab initio method has been employed to find the elastic anisotropy, hardness, thermal conductivity anisotropy and electronic structure of Ti₄AlC₃, Zr₄AlC₃, and Hf₄AlC₃. Moreover, a comparative analysis was conducted to understand their potential characteristics. These three compounds have not been synthesized experimentally but their structural and chemical features are studied in details via theoretic modeling. The hardness of Zr₄AlC₃ is found lower than Ti₄AlC₃. The thermal conductivity decreases from Ti₄AlC₃ to Zr₄AlC₃ at high temperature. Same trend is also followed by the Debye temperature.³⁴ In recent years, the study of solid materials under high-pressure conditions has gained significant attention. Researchers are increasingly exploring how extreme pressure influences the configurational, bulk, electronic, and thermal features of various materials. Studies^21–24 have investigated the impact of pressure on the structural, elastic, optical, electrical, and thermodynamic properties of M₂AX phases (M = Ti, V, Nb, Ta, A = In, Sn, Ge, Ga, and Tl). Li and Wang studied the structural, electronic, and mechanical properties of nanolaminate phase Ti₄GeC₃ with pressure.³⁵ Recently, pressure effect on different properties of 312 type MAX phases, Zr₃InC₂ and Hf₃InC₂ have been studied which ensured their application as TBC material.³⁶ However, the research about pressure effect on different properties is limited on 413 MAX phases. Among them, the impact of pressure on the physical properties of Nb₄AlC₃ has been studied recently.³⁷ This research has explored how external pressure influences structural stability, mechanical behavior, electronic properties, and thermodynamic characteristics. MAX phases are considered as highly promising structural materials for industries operating in extreme environments. Also changing physical properties under pressure is related to industrial applications. Many previous studies^22,23,37 show that MAX phase materials transit from brittle to more ductile behavior under pressure. This is important because enhanced ductility improves their suitability for structural components in critical environments. As a result, studying their mechanical behavior under pressure conditions and assessing their structural stability is significant. This scientific understanding helps enhancing their practical application in demanding fields such as aerospace, nuclear energy, and high-temperature engineering. These studies have played a significant role in inspiring the investigation of pressure effects on the physical properties of M₄AlC₃ (M = Ti, Zr). As far as we know, no research has yet been conducted on how the physical properties of M₄AlC₃ (M = Ti, Zr) compounds are affected by changes in pressure. Therefore, studying the various characteristics under pressure of M₄AlC₃ (M = Ti, Zr) is essential. Understanding how the physical features of the selected compounds behave under pressure, will provide critical insights into their potential applications in high-performance environments. This research investigate the dynamical and thermal stability, changes in structural features, mechanical steadiness and changes in bulk properties as well as optical and thermal characteristics of the M₄AlC₃ (M = Ti, Zr) compound under high-pressure conditions, reaching up to 70 GPa. The goal of the existing work is to search and characterize the behavior of M₄AlC₃ (M = Ti, Zr) under such pressures so that this investigation can contribute to a deeper understanding of their performance in extreme environments.

2. Computational details

To analyze various properties of the studied compounds, calculations were conducted using the plane-wave pseudopotential approach within the Generalized Gradient Approximation (GGA) framework developed by Perdew, Burke, and Ernzerhof (PBE),³⁸ based on Density Functional Theory (DFT).³⁹ All simulations were performed using the CASTEP code.⁴⁰ Ultrasoft pseudopotentials were applied to describe interactions between ionic cores and valence electrons,⁴¹ considering the valence configurations of Ti (3p⁶ 3d² 4s²), Zr (5s² 4p⁶ 4d²), Al (3s² 3p¹), and C (2s² 2p²) in the pseudopotential setup. For Brillouin zone sampling and convergence accuracy, a plane-wave cut-off energy of 400 eV with a 19 × 19 × 2 k-point mesh was used for Ti₄AlC₃, and 420 eV with an 18 × 18 × 2 mesh for Zr₄AlC₃. Structural optimizations were carried out using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization method, employing convergence thresholds of 5 × 10⁻⁶ eV per atom for energy, 0.01 eV Å⁻¹ for maximum force, 0.02 GPa for stress, and 5 × 10⁻⁴ Å for maximum displacement.⁴² Elastic constants (C_ij) were computed using the stress–strain method, while polycrystalline elastic moduli were evaluated using the Voigt–Reuss–Hill approximation.⁴³ The convergence parameters for these calculations were set to an energy tolerance of 4 × 10⁻⁶ eV, force tolerance of 0.01 eV Å⁻¹, and displacement tolerance of 4 × 10⁻⁴ Å.⁴⁴ Additionally, Mulliken bond populations, electronic density of states (DOS), and optical properties were derived directly from the CASTEP output. Debye temperatures were estimated based on the calculated elastic constants. Phonon dispersion and phonon density of states were determined using Density Functional Perturbation Theory (DFPT) combined with the finite displacement method, as implemented in CASTEP.

3. Results and discussion

3.1 Structural properties

Both Ti₄AlC₃ and Zr₄AlC₃ crystallize in a hexagonal structure, one of the seven Bravais lattice types. They are categorized under the space group P6₃/mmc (No. 194), which features a primitive lattice along with mirror and glide symmetry planes.³⁴ The specific Wyckoff positions for each element are provided in Table 1, with Ti/Zr and C atoms occupying two distinct lattice sites. The unit cell of both compounds contain 16 atoms having formula unit, , where 8 atoms are present per primitive cell of M₄AlC₃ (M = Ti, Zr). The optimized structures of both Ti₄AlC₃ and Zr₄AlC₃ compound are drawn by using lattice parameters. Fig. 1(a) and (b) show the conventional structures of Ti₄AlC₃ and Zr₄AlC₃ respectively.

Table 1 Wyckoff positions of Ti, Al, C in Ti₄AlC₃ and Zr, Al, C in Zr₄AlC₃

System	Compounds	Atomic positions					Ref.
		Element	Wyckoff	x	y	z
Hexagonal	Ti4AlC3	Ti	4f	0.33	0.667	0.055	34
			4e	0	0	0.16
		Al	2c	0.33	0.667	0.25
		C	2a	0	0	0
			4f	0.667	0.33	0.11
	Zr4AlC3	Zr	4f	0.33	0.667	0.055
			4e	0	0	0.16
		Al	2c	0.33	0.667	0.25
		C	2a	0	0	0
			4f	0.667	0.33	0.11

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Fig. 1 Conventional crystal structures of (a) Ti₄AlC₃ and (b) Zr₄AlC₃ compounds.

The lattice constants, unit cell volume, and normalized volume under different pressures are presented in Table 2. The calculated lattice parameters for Ti₄AlC₃ and Zr₄AlC₃ closely matched with the experimental values at zero pressure, with only minor deviations attributed to differences in computational methods. A slight overestimation by the GGA–PBE approach is noted, which is a known characteristic of this method and supports the reliability of the current results. Fig. 2 (a–d) illustrates how the lattice parameters ‘a’ and ‘c’, the c/a ratio, and the normalized volume change with increasing pressure. Here we observed that as pressure increased, both lattice constants and cell volume are decreased. This is due to enhanced interactions among Ti, Al, and C atoms in Ti₄AlC₃ and among Zr, Al, and C atoms in Zr₄AlC₃, leading to stronger bonding, shorter bond lengths, and thus a contraction of the lattice structure. As shown in Table 2, the lattice parameter ‘c’ decreases more rapidly than ‘a’, indicating that the materials undergo greater compression along the c-axis than along the a-axis.

Table 2 Lattice constants a (Å), c (Å), volume V (ų), normalized volume V/V₀ and formation energy ΔE_f (eV per atom) at high pressure

Phases	Pressure (GPa)	a (Å)	c (Å)	c/a	V (Å^3)	V/V0	Formation energy, ΔEf (eV per atom)	Ref.
Ti4AlC3	0	3.060	23.45	7.66	198.80	1.00	−8.68	This study
		3.038	23.234	7.66	—	—	—	34
	10	3.01	22.98	7.63	180.45	0.91	−8.67	This study
	20	2.97	22.62	7.62	173.08	0.87	−8.68
	30	2.93	22.32	7.61	167.04	0.84	−8.57
	40	2.91	22.08	7.59	161.88	0.81	−8.50
	50	2.88	21.87	7.59	157.45	0.79	−8.42
	60	2.86	21.67	7.57	153.41	0.77	−8.33
	70	2.84	21.48	7.56	149.77	0.75	−8.24
Zr4AlC3	0	3.310	25.18	7.60	239.69	1.00	−8.92	This study
		3.298	24.972	7.57	—	—	—	34
	10	3.26	24.65	7.56	226.94	0.95	−8.90	This study
	20	3.21	24.23	7.55	216.89	0.90	−8.84
	30	3.18	23.87	7.51	209.01	0.87	−8.76
	40	3.15	23.55	7.48	201.74	0.84	−8.67
	50	3.12	23.22	7.44	195.26	0.81	−8.55
	60	3.09	22.94	7.42	189.57	0.79	−8.43

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Fig. 2 Structural parameters of the Ti₄AlC₃ and Zr₄AlC₃; (a) lattice constant a (Å) (b) lattice constant c (Å) and (c) c/a ratio under pressure (d) normalized cell volume ratio (V/V₀) under pressure.

Determining the formation energy is essential for assessing the crystal stability of a solid. To evaluate the pressure-dependent stability of the investigated compounds, the formation energy was calculated using the following equation:

where, M = Ti, Zr.

Here, E_s(M), E_s(Al), E_s(C) and E_total (M₄AlC₃) represent the energy of M, (M = Ti, Zr), Al, C and M₄AlC₃, where (M = Ti, Zr) and N denotes the number of atoms in the unit cell. For being thermodynamically stable, the titled phases need a negative formation energy.⁴⁵ The calculated negative values of the formation energy (Table 2) for both compounds throughout all the pressure confirms the chemical and thermodynamic stability.

Every materials have chance to going dynamically unstable as they are normally experienced time-relevant mechanical stress.⁴⁶ Since the check in dynamical stability is very imperative for device applications. The dynamical stability of materials is understand by phonon dispersion curve (PDC) analysis where the absence of imaginary frequency ensures the dynamically stable nature and the presence of imaginary frequency inform the dynamical instability of material. Fig. 3(a and b) indicate the phonon dispersion curves with phonon density of states of M₄AlC₃ (M = Ti, Zr) along the high symmetry direction Γ–A–H–K–Γ–M–L–H. After analyzing the Fig. 3(a and b) we have noticed that the vibrational frequencies of these materials are positive confirming the dynamical stable nature of M₄AlC₃ (M = Ti, Zr).

Fig. 3 Phonon dispersion curves of (a) Ti₄AlC₃ and (b) Zr₄AlC₃.

Since the unit cell of M₄AlC₃ (M = Ti, Zr) contains 16 atoms (8 atoms per primitive cell) therefore there exist 48 modes according to 3n formula. Among 48 modes there're 45 optical modes and 3 acoustic modes whereas the upper energy's optical modes are created by lighter atoms (C, Si) and lower energy's optical modes and acoustic modes are created by heavier atoms (Ti, Zr) with little bit contribution of lighter atoms (C, Si). The dispersive manner of low energies' optical phonon modes have been observed which are overlapped with the acoustic branches. The phononic band gap between the lower optical modes and high frequency's optical modes is observed in materials Ti₄AlC₃ and Zr₄AlC₃ which is mainly raised from mass difference. Here three different regions are perceived in the phonon dispersion curves recommended the phononic characteristics of Ti₄AlC₃ and (b) Zr₄AlC₃ materials.

3.2 Mechanical constancy and bulk features

Elastic properties describe a material's ability to return to its original shape after deformation. Analyzing these properties helps determine a material's suitability for engineering, construction, and manufacturing, ensuring better performance, durability, and reliability. The elastic rigidity coefficients are very important parameters as they associated with brittleness, ductility, mechanical stability and stiffness of materials.^47,48 These properties provide valuable information about how atoms within a substance are bonded, how they respond to external forces and how the material maintains its stability under various conditions.⁴⁹ Hence, in order to understand the mechanical properties of M₄AlC₃ (M = Ti, Zr) and response of these materials under pressure, it is very important to investigate the elastic constants.

The mechanical response of the MAX-phase compounds M₄AlC₃ (M = Ti, Zr) has been analyzed using the strain–stress methodology.⁵⁰ This method is incorporated within the CASTEP computational framework. The elastic constants can be determined using the mathematical relationship:

where σ_ij represents the stress tensor corresponding to a specific set of applied strains δ_ij. Since both Ti₄AlC₃ and Zr₄AlC₃ belongs to hexagonal structure, they possess only five independent elastic constants: C₁₁, C₁₂, C₁₃, C₃₃ and C₄₄. Additionally, it has one dependent elastic constant C₆₆, defined by the relation C₆₆ = (C₁₁ − C₁₂)/2. Before further analysis, it is essential to verify the mechanical stability of both hexagonal compounds Ti₄AlC₃ and Zr₄AlC₃. This can be done by applying the well-established stability criteria formulated by Born⁵¹ which is grounded on the stiffness constants (C_ij):

However, to demonstrate mechanical stability under pressure, a hexagonal crystal system must meet the following conditions:²⁴

The elastic constants of M₄AlC₃ (M = Ti, Zr) are calculated up to 70 GPa and 60 GPa for Ti₄AlC₃ and Zr₄AlC₃ have been listed in Table 3 with previous calculated data available in literature.³⁴ The calculated elastic constants show good agreement with the previous study at zero pressure. A slight discrepancy is observed in elastic parameters between the present calculated and previous theoretical values at zero pressure. This deviation occurred due to the usage of different methodology. For ensuring the accuracy of our investigated result, we have listed some similar types of MAX compounds (Nb₄AlC₃, Hf₄AlC₃) for comparison. Almost analogous values have been observed of the studied phases M₄AlC₃ (M = Ti, Zr) compared to other similar types of compounds. Both the compounds Ti₄AlC₃ and Zr₄AlC₃ satisfied all the necessary conditions for mechanical stability under pressure ensures the mechanical stability of the titled phases. However, this stability is based solely on theoretical analysis as there is no experimental data available in the literature regarding the material's mechanical properties under pressure.

Table 3 Elastic stiffness constants, C_ij (in GPa) of M₄AlC₃ (M = Ti, Zr) compounds with the variation of pressure

Phases	Pressure (GPa)	C11	C12	C13	C33	C44	C66	Cp (001) = C12 − C66	Cp (100) = C13 − C44	Ref.
Ti4AlC3	0	400.47	84.69	80.74	339.81	142.39	157.89	−73.20	−61.65	This study
		400.30	76.30	70.40	312.0	155.4	—	—	—	34
Hf4AlC3	0	411.80	91.70	96.90	330.8	158.1	—	—	—
Nb4AlC3	0	427.05	116.81	128.60	354.03	169.15	155.12	—	—	37
Ti4AlC3	10	467.93	113.57	122.03	413.14	187.32	177.18	−63.61	−55.15	This study
	20	514.86	137.78	149.73	452.76	194.82	188.54	−50.76	−45.09
	30	563.30	166.63	186.74	511.67	215.97	198.34	−31.71	−29.23
	40	614.30	189.07	215.04	547.96	235.90	212.62	−23.55	−20.86
	50	651.57	218.43	244.23	589.75	245.52	216.57	1.86	−1.29
	60	693.46	249.78	277.69	628.89	256.62	221.84	27.94	21.07
	70	740.05	274.57	305.57	634.0	278.66	232.74	41.83	26.91
Zr4AlC3	0	348.19	79.26	79.90	283.31	113.89	134.47	−52.21	−33.99	This study
		355.30	75.40	78.90	287.60	133.00	—	—	—	34
Hf4AlC3	0	411.80	91.70	96.90	330.8	158.1	—	—	—
Nb4AlC3	0	427.05	116.81	128.60	354.03	169.15	155.12	—	—	37
Zr4AlC3	10	402.28	108.32	115.39	341.58	139.66	146.98	−38.66	−24.27	This study
	20	454.78	135.77	153.08	393.52	166.76	159.51	−23.74	−13.68
	30	494.11	164.70	183.89	420.93	180.61	164.71	−0.01	3.28
	40	518.44	186.44	212.64	419.36	195.35	166.00	20.44	17.29
	50	542.76	215.60	242.99	433.62	207.26	163.58	52.02	35.73
	60	558.18	246.19	267.34	483.81	207.78	155.95	90.24	59.56

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The variation of elastic constants C_ij of M₄AlC₃ (M = Ti, Zr) are shown in Fig. 4(a and b). As shown in Fig. 4(a and b), the elastic constants of M₄AlC₃ (M = Ti, Zr) show increased manner with pressure. For pressure up to 60 GPa (for Ti₄AlC₃) and up to 30 GPa (for Zr₄AlC₃) the variations in the values of C_ij follow a linear trend, maintaining the relationship described by Hooke's law, (where stress is proportional to strain). This linear increase suggests a typical elastic response under pressure. However, after 60 GPa (for Ti₄AlC₃) and 30 GPa (for Zr₄AlC₃) a slight deviation from the linear response is observed which indicate the system is no longer follows Hooke's law. This could be indicative of a phase transition occurring at around 60 GPa and 30 GPa in Ti₄AlC₃ and Zr₄AlC₃ respectively. The higher limit of pressure was selected based on the observed pressure induced phase transition. Ti₄AlC₃ exhibit phase transition near 60 GPa, so properties calculation was extended up to 70 GPa. On the other hand, Zr₄AlC₃ shows phase transition after 30 GPa, so calculations were extended up to 60 GPa. This helps to study the behavior after phase transition. Since the transition pressure was different for different compounds, the higher limit was chosen accordingly.

Fig. 4 Elastic stiffness parameters of (a) Ti₄AlC₃ and (b) Zr₄AlC₃ under pressure.

The elastic constant C₁₁ indicates the material's resistance to uniaxial compression or tension along the a-axis (⁵² direction), while C₃₃ represents resistance along the c-axis ([001] direction).^53–55 As shown in Fig. 4(a and b), both compounds exhibit an increase in these stiffness constants with rising pressure. A higher C₁₁ implies stronger resistance to deformation along the a-axis, while an increasing C₃₃ indicates enhanced resistance along the c-axis, suggesting stronger interlayer bonding.^55,56 Throughout the pressure range of 0–70 GPa, C₁₁ remains consistently greater than C₃₃, implying that more pressure is needed to induce plastic deformation along the a-axis than the c-axis. This trend aligns with typical MAX phase behavior, where in-plane bonding is stronger than interlayer interactions.^21,55

C₁₂ and C₁₃ are referred to as the in-plane and cross-plane coupling moduli, respectively. They describe the material's response to shear stress along the a-axis when uniaxial strain is applied along the b- and c-axes. As shown in Fig. 4(a and b), both C₁₂ and C₁₃ increase with applied pressure, indicating that the materials become more resistant to shear deformation in the b- and c-directions under stress along the a-axis.⁵⁶ C₄₄ measures the resistance to shear deformation within the basal plane and is closely associated with material hardness.⁵⁷ The calculated values of C₄₄ of both the materials M₄AlC₃ (M = Ti, Zr) are greater than 100 GPa confirming high strength against shear deformation. Here the phase Ti₄AlC₃ exhibits higher C₄₄ therefore it is harder than the other phase Zr₄AlC₃.

These elastic stiffness constants are essential for calculating polycrystalline mechanical properties such as bulk modulus (B), shear modulus (G), Young's modulus (E), Poisson's ratio (ν), and Pugh's ratio (B/G). Understanding how these properties vary with pressure is vital for assessing mechanical performance. The bulk and shear moduli are commonly derived using the Voigt (V),⁵⁰ Reuss (R),⁵⁸ and Voigt–Reuss–Hill (VRH)⁵⁹ averaging methods, based on the following equations:

B = ½(B_V + B_R) (8)

G = ½(G_V + G_R) (9)

In this context, B_V and G_V represent the bulk and shear moduli from the Voigt model, while B_R and G_R are derived from the Reuss model. The averaged values of bulk (B) and shear (G) moduli are then used to compute Young's modulus (E) and Poisson's ratio (ν) using standard formulas:⁵¹

The machinability index can be obtained by following equation:⁶⁰

The hardness of both materials is pressure-dependent and estimated using standard formulas.^61,62

Table 4 presents the calculated values of B, G, E, B/G, μ_m, ν, and H_v under varying pressures. The zero-pressure results align well with previous studies, supporting the accuracy of the present calculations. As shown in Table 4 and Fig. 5(a and b), the values of bulk (B), shear (G), and Young's (E) moduli are increased with increasing pressure, indicating enhanced bonding strength. On the basis of Bulk, shear and Young's modulus at zero pressure, the studied and referenced compounds can be arranged as Nb₄AlC₃ > Hf₄AlC₃ > Ti₄AlC₃ > Zr₄AlC₃. Here, among all the materials Nb₄AlC₃ has highest bulk, shear and Young's moduli indicating its strongest resistance towards volume deformation, shear deformation and more stiffness respectively.

Table 4 Computed values of bulk modulus (B, in GPa), shear modulus (G, in GPa), Young's modulus (E, in GPa), Pugh's ratio (B/G), machinability index (μ_m = B/C₄₄), Poisson's ratio (ν), and hardness (H_v, in GPa) for M₄AlC₃ (M = Ti, Zr) compounds under different pressure conditions

Phase	Pressure	B	G	E	B/G	μm	ν	Hv	Ref.
Ti4AlC3	0	180.90	147.87	348.81	1.22	1.27	0.18	25.15	This study
		170.60	153.70	354.60	1.11	—	0.15	—	34
Hf4AlC3	0	190.90	152.50	361.40	1.25	—	0.19	—
Nb4AlC3	0	216.78	153.15	371.87	1.42	1.28	0.21	29.71	37
Ti4AlC3	10	229.10	175.93	420.24	1.30	1.22	0.19	26.49	This study
	20	261.81	184.77	426.91	1.41	1.34	0.21	24.94
	30	301.94	198.40	488.26	1.52	1.39	0.23	24.16
	40	334.81	212.67	526.52	1.57	1.42	0.25	24.42
	50	367.27	219.09	548.25	1.67	1.50	0.26	22.42
	60	402.77	225.89	570.93	1.78	1.57	0.27	22.12
	70	431.14	236.53	599.89	1.82	1.55	0.27	22.29
Zr4AlC3	0	161.29	121.29	290.94	1.33	1.42	0.20	19.88	This study
		162.10	131.70	311.00	1.23	—	0.18	—	34
Hf4AlC3	0	190.90	152.50	361.40	1.25	—	0.19	—
Nb4AlC3	0	216.78	153.15	371.87	1.42	1.28	0.21	29.71	37
Zr4AlC3	10	202.27	138.67	338.63	1.46	1.45	0.22	19.68	This study
	20	242.71	155.19	383.78	1.56	1.46	0.24	19.67
	30	274.47	162.25	406.62	1.69	1.52	0.25	18.58
	40	296.89	164.48	416.53	1.81	1.52	0.27	17.43
	50	323.57	165.39	423.95	1.96	1.56	0.28	15.96
	60	350.87	165.09	428.12	2.13	1.69	0.30	14.51

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Fig. 5 Elastic moduli B, G and E of (a) Ti₄AlC₃ and (b) Zr₄AlC₃ under pressure.

To measure the resistance of solids to volume changes, the bulk moduli, B is a good indicator can be used to predict the fracture and compressing resistance of solids.⁶³ The shear modulus, G is used to measure the resistance of solids to shape changes and measure the plastic deformation depending on the value of C₄₄.⁶⁴ On the other hand, the Young's modulus, E explains the ratio of stress and strain, and measures the rigidity of solids. The superior Young's modulus indicates the more solid's ability to deform.⁶⁵ Fig. 5 (a and b) indicates that the values of B, G, and E are increased with pressure, indicating that the resistance to volume, shape and longitudinal deformation increased linearly with pressure. The value of B exceeded the critical value 100 GPa for both compounds, confirming that they are hard materials. Ti₄AlC₃ consistently shows higher B, G and E values than Zr₄AlC₃, suggesting greater resistance to both volumetric and shear deformation, and improved structural stability under stress.

Young's modulus (E) is a key bulk property that measures a material's stiffness and its resistance to linear deformation under stress. It also correlates with thermal shock resistance (R), as defined by the following equation:⁶⁶

In the thermal shock resistance equation, σ_f is the bending strength, ν is Poisson's ratio, and α is the thermal expansion coefficient. Since thermal shock resistance (R) is inversely related to Young's modulus (E), materials with lower E are better suited for thermal barrier coatings (TBCs). Given their high E values, Ti₄AlC₃ and Zr₄AlC₃ are not ideal for TBC applications.

Pugh's ratio (B/G), which compares bulk to shear modulus, indicates ductility-materials with B/G > 1.75 are ductile, while lower values suggest brittleness.⁶⁴ As shown in Table 4 and Fig. 6(a), both compounds initially exhibit brittle behavior. Ti₄AlC₃ transitions to ductile above 60 GPa, and Zr₄AlC₃ above 40 GPa. This pressure-induced ductility enhances their suitability for industrial applications under high-pressure conditions.

Fig. 6 (a) Pugh's ratio (B/G) and (b) Poisson's ratio (ν) of M₄AlC₃ (M = Ti, Zr) compounds at different pressure.

Poisson's ratio (ν) is another indicator of a material's ductility, with 0.26 as the critical threshold—values above indicate ductility, below indicate brittleness.⁶⁷ In this study, Ti₄AlC₃ transitions from brittle to ductile at 60 GPa, and Zr₄AlC₃ at 40 GPa, showing improved deformability under pressure. Poisson's ratio also reflects bonding type: metallic bonds typically have ν ≈ 0.10, covalent ≈ 0.33.⁶⁸ Values between 0.18–0.30 for both compounds suggest mixed metallic-covalent bonding. Moreover, Poisson's ratio can distinguish central force solids (ν = 0.25–0.50) from non-central ones.⁶⁹ Ti₄AlC₃ and Zr₄AlC₃ are non-central force solids below 40 GPa and 30 GPa respectively, becoming central force solids beyond those pressures.

Cauchy Pressure (CP), calculated as C₁₃–C₄₄ (for the (100) plane) or C₁₂–C₆₆ (for the (001) plane), further indicates bonding and ductility.⁷⁰ Positive CP suggests metallic and ductile behavior; negative CP indicates covalent and brittle nature.⁷¹ CP becomes positive at 60 GPa for Ti₄AlC₃ and 40 GPa for Zr₄AlC₃, consistent with B/G and ν trends. However, at 50 GPa (Ti₄AlC₃) and 30 GPa (Zr₄AlC₃), CP shows mixed values across planes—suggesting a transitional bonding state requiring deeper analysis. The machinability index (MI = B/C₄₄) assesses how easily a material can be machined—higher MI indicates better machinability and lower tool wear. MI increases with pressure for both compounds, with Ti₄AlC₃ showing better machinability than Zr₄AlC₃. From Fig. 7(a and b), ductility and machinability improve under pressure.

Fig. 7 (a) Machinability index, μ_m and (b) hardness, H_v of M₄AlC₃ (M = Ti, Zr) compounds at different pressure.

Hardness (H_v), which measures resistance to plastic deformation, inversely affects machinability. It is an essential macroscopic factor used for technological and industrial applications lower hardness supports easier shaping and cutting whereas a hard material has ability to resist plastic deformation from the irreversible motion of atom. The calculated hardness from zero to high pressure of the studied phases are listed in Table 4. As pressure increases, the hardness of both materials decreases, which aligns with the inverse relationship between Pugh's ratio (B/G) and hardness. A higher B/G indicates greater ductility and lower hardness, improving machinability. Ti₄AlC₃ has higher hardness than Zr₄AlC₃ due to stronger atomic bonding from its smaller atomic radius. This is significant for industrial applications, as materials with higher machinability improve production efficiency, reduce tool wear, and lower energy consumption. The consistency between hardness, Pugh's ratio, and machinability index under pressure confirms the reliability of these parameters for high-pressure machining.

3.3 Insights of anisotropy

Material properties vary due to crystal anisotropy, which affects phenomena such as reaction kinetics, phonon behavior, and fracture mechanics. Understanding anisotropy helps explain material responses under different conditions.⁷² Table 3 shows that C₁₁ > C₃₃, confirming anisotropic elasticity, which also impacts polycrystalline moduli.⁷³ MAX-phase carbides display anisotropy due to their atomic arrangement, leading to directional variations in mechanical properties.⁷⁴ To quantify this, we use indices like universal anisotropy (A^U), shear anisotropic factors (A₁, A₂, A₃), percentage anisotropy (A_B, A_G), and 3D plots of elastic stiffness parameters.⁷⁵A₁, A₂ and A₃ represent shear anisotropy in the {100}, {010}, and {001} planes, linked to specific crystallographic directions. For isotropic crystals, A_i = 1 (i = 1, 2, 3); deviations indicate anisotropy. As shown in Table 5, both Ti₄AlC₃ and Zr₄AlC₃ exhibit elastic anisotropy, with shear anisotropy factors varying with pressure depending on the crystal plane. Both compounds show a similar trend, suggesting a parallel pressure response. To further quantify anisotropy, percentage anisotropy for bulk (A_B) and shear modulus (A_G) is calculated using Ranganathan and Ostoja-Starzewsky's method.⁷⁶

Table 5 Various anisotropy indices—shear factors (A₁, A₂, A₃), percentage anisotropy (A_B and A_G), universal anisotropy (A^U), Zener's index (A_Z), and compressibility ratio (k_c/k_a) for Ti₄AlC₃ and Zr₄AlC₃

Elements	Pressure	A1	A2	A3	AB	AG	A^U	AZ	kc/ka	Ref.
Ti4AlC3	0	0.985	0.984	0.889	0.304	0.21	0.02	0.902	1.25	This study
	10	0.818	1.176	0.865	0.113	0.30	0.03	1.057	1.16
	20	0.821	1.170	0.848	0.080	0.12	0.03	1.033	1.17
	30	0.777	1.230	0.846	0.030	0.43	0.04	1.240	1.10
	40	0.734	1.289	0.815	0.135	0.15	0.06	1.109	1.12
	50	0.728	1.304	0.825	0.040	0.68	0.07	1.134	1.10
	60	0.708	1.338	0.819	0.030	0.81	0.08	1.157	1.10
	70	0.634	1.460	0.759	0.140	1.42	0.15	1.197	1.23
Zr4AlC3	0	0.987	0.966	0.836	0.420	0.44	0.05	0.847	1.32
	10	0.874	1.089	0.830	0.220	0.28	0.03	0.950	1.24
	20	0.765	1.230	0.800	0.100	0.24	0.03	1.045	1.18
	30	0.706	1.320	0.774	0.005	0.69	0.07	1.097	1.23
	40	0.591	1.525	0.696	0.290	1.92	0.20	1.177	1.35
	50	0.526	1.691	0.666	0.350	2.86	0.30	1.267	1.43
	60	0.564	1.640	0.750	0.130	2.33	0.24	1.332	1.25

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Ranganathan and Ostoja-Starzewsky proposed the universal anisotropy index (A^U) as a standard metric to quantify elastic anisotropy in crystals, calculated using the following equation:⁷⁷

B_V, B_R, G_V, and G_R are derived from Voigt and Reuss approximations. Zero values of A_B, A_G, and A^U indicate isotropy, while non-zero values confirm anisotropy. As shown in Table 5, both Ti₄AlC₃ and Zr₄AlC₃ exhibit anisotropic behavior under pressure. Positive A_B and A_G, reflect directional dependence of bulk and shear moduli, with shear anisotropy showing a stronger pressure response. Additionally, the increasing A^U with pressure indicates growing elastic anisotropy due to unequal changes in bulk and shear moduli across crystallographic directions. The Zener anisotropy factor, A_Z is a crucial parameter in determining the mechanical behavior of a solid as it indicates whether the material exhibits direction-dependent or direction-independent elastic properties. It is calculated by following equation:⁷⁸

A Zener factor of 1 indicates isotropy, while values differing from 1 reflect anisotropy. As shown in Table 5, the Zener anisotropy factor (A_Z) for both compounds exceeds 1 and rises with pressure, indicating increasing anisotropy under compression. In hexagonal structures, compressibility anisotropy is also described by the ratio k_c/k_a, comparing linear compressibility along the c- and a-axes.⁷⁹

A k_c/k_a value of 1 indicates equal compressibility along the c- and a-axes, reflecting isotropy. However, Table 5 shows values consistently above 1 for both compounds under pressure, suggesting greater compressibility along the c-axis and confirming anisotropic behavior.

To visualize the anisotropic behavior of Ti₄AlC₃ and Zr₄AlC₃, 3D plots of Young's modulus, shear modulus, and Poisson's ratio were generated using ELATE,^80,81 highlighting directional mechanical variations. Fig. 8 and 9 depict this under pressure, with corresponding maxima and minima detailed in Table 6. In isotropic materials, elastic properties remain uniform in all directions, resulting in spherical 3D plots of elastic moduli.⁸² In contrast, anisotropic materials show deviations from this shape, reflecting directional dependence. As shown in Fig. 8 and 9, increasing pressure distorts the spherical shape, indicating enhanced anisotropy. The green surfaces highlight maximum Young's modulus, while blue regions represent maximum shear and Poisson's ratio, with inner green areas denoting their minimum values. From Table 6, for Ti₄AlC₃, Young's modulus increases from 370.03 GPa (max) and 312.93 GPa (min) at 0 GPa to 653.31 GPa and 449.94 GPa at 70 GPa. Shear modulus shows a similar rise. For Zr₄AlC₃, values increase from 315.77 GPa/253.43 GPa (Young's) and 134.47 GPa/113.88 GPa (shear) at 0 GPa to 479.13 GPa/306.11 GPa and 207.78 GPa/124.63 GPa at 60 GPa. These trends align with the universal anisotropy index, confirming that pressure enhances anisotropy and validating the reliability of the study.

Fig. 8 3D distributions of Young's modulus (E), shear modulus (G), and Poisson's ratio (ν) for Ti₄AlC₃ under varying pressures.

Fig. 9 3D distributions of Young's modulus (E), shear modulus (G), and Poisson's ratio (ν) for Zr₄AlC₃ under varying pressures.

Table 6 The anisotropy of Young's modulus (E), shear modulus (G), and Poisson's ratio (ν) for Ti₄AlC₃ and Zr₄AlC₃ under pressure is calculated

Elements	Pressure	Young's modulus (GPa)		AY	Shear modulus (GPa)		AG	Poisson's ratio		Aν
		Emin	Emax		Gmin	Gmax		νmin	νmax
Ti4AlC3	0	312.93	370.03	1.182	142.39	157.89	1.109	0.166	0.197	1.183
	10	361.92	435.09	1.202	158.35	187.33	1.183	0.143	0.242	1.690
	20	386.91	436.94	1.129	164.73	181.04	1.099	0.207	0.261	1.266
	30	416.12	511.18	1.228	174.35	215.97	1.239	0.171	0.292	1.711
	40	504.56	553.74	1.097	203.95	225.95	1.108	0.221	0.279	1.258
	50	452.63	582.08	1.286	186.81	245.52	1.314	0.173	0.326	1.889
	60	465.38	610.22	1.311	190.21	256.62	1.349	0.176	0.343	1.947
	70	449.94	653.31	1.452	187.63	278.66	1.485	0.150	0.378	2.531
Zr4AlC3	0	253.43	315.77	1.246	113.88	134.47	1.181	0.174	0.233	1.337
	10	289.42	350.63	1.211	127.01	146.98	1.157	0.191	0.273	1.432
	20	315.50	369.45	1.171	133.40	152.36	1.142	0.212	0.299	1.409
	30	384.40	441.30	1.267	140.28	184.48	1.315	0.188	0.336	1.791
	40	291.06	458.32	1.575	124.43	195.35	1.570	0.136	0.410	3.020
	50	277.90	477.54	1.718	118.18	207.26	1.754	0.112	0.451	4.027
	60	306.11	479.13	1.565	124.63	207.78	1.667	0.133	0.405	3.046

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3.4 Chemical bonding and electronic characteristics

The electronic structure reveals key insights into conductivity, bonding, and stability, influenced by valence and conduction electrons. Band structures, showing energy dispersion E(k) across the Brillouin zone, classify materials as metals, semimetals, or insulators. For M₄AlC₃ (M = Ti, Zr) MAX phases, band structures were calculated under pressure. Fig. 10 and 11 display bands for Ti₄AlC₃ (0–70 GPa) and Zr₄AlC₃ (0–60 GPa), with green lines marking electronic states and the Fermi level at 0 eV. Multiple bands cross the Fermi level at all pressures, confirming metallic behavior without a band gap. Initially, bands near the Fermi level are moderately dispersed but widen with pressure, indicating stronger orbital hybridization and increased metallicity, especially in Zr₄AlC₃. The band structure also shows anisotropic conductivity, with lower dispersion along the c-axis (Γ–A, H–K, M–L) and higher dispersion in basal plane directions (A–H, K–Γ, Γ–M, L–H), reflecting pressure-dependent directional electronic behavior.⁸³

Fig. 10 Electronic band structures of Ti₄AlC₃ at (a) 0, (b) 30, (c) 50, and (d) 70 GPa.

Fig. 11 Electronic band structures of Zr₄AlC₃ at (a) 0, (b) 20, (c) 40, and (d) 60 GPa.

This study analyzes the electronic density of states (DOS), including total (TDOS) and partial (PDOS), to understand bonding and hybridization in Ti₄AlC₃ and Zr₄AlC₃.^84,85 Fig. 12 shows TDOS under various pressures, considering outer electron configurations: Ti (3p⁶ 3d² 4s²), Zr (5s² 4p⁶ 4d²), Al (3s² 3p¹), and C (2s² 2p²). TDOS is evaluated from −5 to 5 eV at pressures of 0, 30, 50, and 70 GPa for Ti₄AlC₃, and 0, 20, 40, and 60 GPa for Zr₄AlC₃. Both compounds show metallic behavior with TDOS at the Fermi level. For Ti₄AlC₃, TDOS at the Fermi level decreases with pressure: 3.86, 3.60, 3.30, and 3.26 states per eV per unit cell (Fig. 12(a)). For Zr₄AlC₃, TDOS increases: 3.85, 4.40, 5.30, and 5.58 states per eV per unit cell (Fig. 12(b)). PDOS for Ti, Zr, Al, and C (Fig. 13 and 14) further detail atomic contributions from −10 to 10 eV, with the Fermi level (0 eV) marked by black dotted lines.

Fig. 12 Total density of states for (a) Ti₄AlC₃ and (b) Zr₄AlC₃ under varying pressures.

Fig. 13 Partial density of states for Ti₄AlC₃ at (a) 0, (b) 30, (c) 50, and (d) 70 GPa.

Fig. 14 Partial density of states for Zr₄AlC₃ at (a) 0, (b) 20, (c) 40, and (d) 60 GPa.

In Ti₄AlC₃ (Fig. 13), the main TDOS peak in the valence band (−5 to −1.8 eV) arises from Ti-3d and C-2p orbitals, with minor Al-3s involvement, reflecting strong Ti–C hybridization. Near the Fermi level, Ti-3d and Al-3p orbitals dominate, while Ti-4s, Al-3s, and C-2p contribute less. At the Fermi level, Ti-3d is the primary contributor; in the conduction band, Ti-3d remains dominant with minor Al-3p and C-2p inputs. Increasing pressure reduces Ti-3d contributions, lowering TDOS. This matches prior MAX phase studies.⁸⁶ The stronger Ti–C covalent bond contrasts with the weaker, more metallic Ti–Al bond. For Zr₄AlC₃ (Fig. 14), similar trends appear: Zr-4d dominates valence, Fermi level, and conduction bands; Al-3p gains influence with pressure, increasing TDOS, while C-2p and Zr-4p remain minimal contributors.

The bonding characteristics of M₄AlC₃ (M = Ti, Zr) were analyzed using Mulliken population analysis, which distinguishes between covalent and ionic bonds.⁸⁷ Table 7 presents bond lengths, populations, and charges for Ti₄AlC₃ (0–70 GPa) and Zr₄AlC₃ (0–60 GPa). Positive bond populations confirm covalent bonding, while the negative charge on C and Al and positive charge on Ti/Zr indicate charge transfer from metal to non-metal atoms.⁸⁸ Ti₄AlC₃ maintains only C–Ti bonds under pressure, showing no structural change. In contrast, Zr₄AlC₃ forms a new Al–Zr bond at 60 GPa, indicating pressure-induced bonding adaptation. Both compounds show decreasing bond population and bond length with pressure, reflecting weaker covalent interactions due to atomic compression.

Table 7 Mulliken atomic populations of M₄AlC₃ (M = Ti, Zr) under various pressure

Compounds	Pressure, P (GPa)	Species	s	p	d	Total	Charge	Bond	Population	Length (Å)
Ti4AlC3	0	C	1.47	3.20	0	4.67	−0.67	C–Ti	1.22	2.06
		Al	1.07	1.95	0	3.03	−0.03	—	—	—
		Ti	2.13	6.61	2.63	11.36	0.64	—	—	—
	20	C	1.46	3.21	0	4.67	−0.67	C–Ti	1.16	2.01
		Al	1.00	2.01	0	3.01	−0.01	—	—	—
		Ti	2.09	6.60	2.67	11.37	0.63	—	—	—
	40	C	1.45	3.22	0	4.67	−0.67	C–Ti	1.11	1.98
		Al	0.97	2.04	0	3.01	−0.01	—	—	—
		Ti	2.07	6.59	2.71	11.37	0.63	—	—	—
		C	1.44	3.22	0	4.67	−0.67	C–Ti	1.03	1.95
		Al	0.94	2.08	0	3.02	−0.02	—	—	—
		Ti	2.03	6.57	2.77	11.37	0.63	—	—	—
Zr4AlC3	0	C	1.50	3.26	0	4.76	−0.76	C–Zr	1.26	2.24
		Al	1.14	1.92	0	3.05	−0.05	—	—	—
		Zr	2.19	6.47	2.62	11.27	0.73	—	—	—
	20	C	1.49	3.27	0	4.76	−0.76	C–Zr	1.18	2.19
		Al	1.07	1.97	0	3.04	−0.04	—	—	—
		Zr	2.15	6.43	2.68	11.27	0.73	—	—	—
	40	C	1.49	3.28	0	4.77	−0.77	C–Zr	1.11	2.15
		Al	1.04	2.00	0	3.03	−0.03	—	—	—
		Zr	2.13	6.40	2.74	11.26	0.74	—	—	—
	60	C	1.49	3.28	0	4.71	−0.77	C–Zr	1.03	2.13
		Al	1.02	1.98	0	3.00	−0.0	Al–Zr	0.31	2.67
		Zr	2.10	6.36	2.79	11.26	0.74	—	—	—

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To investigate atomic charge distribution and bonding in M₄AlC₃ (M = Ti, Zr), charge density maps were generated. Fig. 15 displays these maps for Ti₄AlC₃ at 0 and 70 GPa, and Zr₄AlC₃ at 0 and 60 GPa, with red indicating high and blue low electron density.

Fig. 15 Electron charge density of Ti₄AlC₃ at (a) 0 GPa, (b) 70 GPa, and Zr₄AlC₃ at (c) 0 GPa, (d) 60 GPa.

From Fig. 15(a) and (b), at 0 GPa high electron density (red regions) is primarily concentrated around the Ti–C bonds indicating strong covalent interactions. As pressure increases to 70 GPa, the electron density around Ti decreases slightly suggesting a reduction in covalent character and bonding becoming more ionic in nature. From Fig. 15(c) and (d), at 0 GPa high electron density is observed around Zr–C bonds similar to Ti₄AlC₃. However, at 60 GPa a noticeable charge density is observed around Al–Zr which signify the formation of an additional bonding interaction at high pressure. These charge density mappings show good agreement with Table 7 as discussed earlier. The observed trends in charge distribution have consistency with the computed bond population and Mulliken charge values.

3.5 Optical properties

Studying the optical properties of MAX phases M₄AlC₃ (M = Ti, Zr) reveals their electronic structure and interaction with light. Key parameters—photon absorption, reflectivity, dielectric function, refractive index, optical conductivity, and loss function—are crucial for optoelectronic devices like solar cells and LEDs. This work examines Ti₄AlC₃ and Zr₄AlC₃'s optical response from 0 to 30 eV under pressure, providing the first theoretical insight into their pressure-dependent behavior for industrial and energy applications. Optical behavior is described by the frequency-dependent dielectric function:

ε(ω) = ε₁(ω) + iε₂(ω) (23)

Here, ε₁(ω) is the real part of the dielectric function, indicating the material's ability to store electromagnetic energy, while ε₂(ω) is the imaginary part, representing energy loss from electronic transitions and absorption. The imaginary part is calculated using momentum matrix elements considering all transitions between occupied and unoccupied states, as expressed by the following equation.^89,90In this equation, the integral spans the first Brillouin zone. Here, u is the polarization vector of the incident electric field, ω the light frequency, e the electronic charge, and ψ_k^c and ψ_k^v the conduction and valence band wave functions at k-point k. The real part, ε₁(ω), is derived using the Kramers–Kronig relation.^91,92The integral is denoted as M. From the complex dielectric function ε₂(ω), key optical parameters—refractive index n₁(ω), absorption coefficient α(ω), reflectivity R(ω), and energy loss spectrum L(ω)—are derived using the following relations:^92,93

A Gaussian smearing of 0.5 eV was applied in all optical calculations for accuracy and smooth spectra. For metallic Ti₄AlC₃ and Zr₄AlC₃, both inter- and intra-band transitions impact the dielectric function. A Drude term with a 4 eV plasma frequency and 0.05 eV damping was used to model free-electron effects, improving low-energy optical response.⁹⁴ The dielectric function, a key optical property, refer to in what way a material reacts to an exterior electromagnetic (EM) field. It has two components: the real part, specifying the material's capability to store electric energy, whereas imaginary part, related to energy absorption and dissipation. For M₄AlC₃ (M = Ti, Zr) compounds, both components were analyzed under varying pressure and plotted in Fig. 16(a, b) as functions of photon energy. At low energies, the negative real part confirms their metallic nature. As energy increases, ε₁(ω) becomes positive, peaks, then decreases and turns negative again—demonstrating a Drude-like response typical of metals. This behavior is consistent across all pressures, indicating that both compounds retain their metallicity, which aligns with earlier band structure results. Under pressure, distinct peaks appear in the dielectric function: for Zr₄AlC₃ between 2.21–2.35 eV (0–60 GPa) and for Ti₄AlC₃ between 1.6–2.52 eV (0–70 GPa), all within the visible range. This confirms their sustained optical activity under pressure, highlighting potential for applications in optical coatings, sensors, and energy-efficient window technologies. For both compounds, the imaginary part of the dielectric function, ε₂(ω), shows a large positive value at low photon energies, indicating strong absorption due to inter-band transitions and free-electron behavior. As photon energy increases, ε₂(ω) gradually decreases and approaches zero around 24–26 eV—marking the plasma frequency, where the material transitions from absorption to transparency. At this point, both the absorption coefficient and reflectivity drop sharply. This plasma frequency is crucial for understanding optical transitions, especially in plasmonic applications, optical coatings, and electromagnetic shielding.

Fig. 16 Pressure-dependent dielectric function of (i) Zr₄AlC₃ and (ii) Ti₄AlC₃: (a) real part, (b) imaginary part, and (c) refractive index.

The refractive index, a dimensionless quantity, is the fraction of the speed of light in a vacuum to its speed in a material, indicating how much light slows down and bends upon entering. It is key to understanding optical behaviors like reflection, refraction, and transmission. A higher refractive index indicates that light travels more slowly in the medium.⁹⁵ As shown in Fig. 16(c), the refractive indices of M₄AlC₃ (M = Ti, Zr) under different pressures are notably high in the infrared region, ranging from 3.8 to 5.02. These values suggest strong interaction with infrared light. With increasing pressure, variations in peak values indicate enhanced ability to guide and manipulate infrared light, making these materials promising for optical communication, infrared sensors, fiber optics, and laser systems.

The absorption coefficient α(ω) measures a material's light absorption efficiency across wavelengths, key for solar and optoelectronic uses. For M₄AlC₃ (M = Ti, Zr), pressure-dependent absorption spectra appear in Fig. 17(a) for Ti₄AlC₃ and Fig. 18(a) for Zr₄AlC₃. Absorption starts at zero in low photon energies, confirming metallicity, rises through the infrared and visible regions, and peaks sharply in the UV range. For Ti₄AlC₃, peaks appeared between 7.78–8.51 eV (at 0–70 GPa), while for Zr₄AlC₃, peaks occurred between 10.93–12.74 eV (0–60 GPa). Notably, maximum absorption coincides with minima in reflectivity and energy loss, indicating high efficiency in light absorption. This makes both compounds strong candidates for UV optoelectronic applications such as UV sensors, LEDs, photodetectors, and optical communication systems where effective UV light control is essential.

Fig. 17 Pressure-dependent (a) absorption, (b) loss function, (c) conductivity, and (d) reflectivity of Ti₄AlC₃.

Fig. 18 Pressure dependent (a) absorption, (b) loss function, (c) conductivity and (d) reflectivity of Zr₄AlC₃ compound.

Optical conductivity reflects how photon excitation increases a material's electrical conductivity.^96,97 The real part, σ(ω), versus photon energy under pressure is shown in Fig. 17(b) for Ti₄AlC₃ and Fig. 18(c) for Zr₄AlC₃. In both, σ(ω) starts at zero energy, indicating overlapping valence and conduction bands at the Fermi level, confirming their metallic nature and lack of band gap, consistent with band structure results. This allows continuous electron excitation, enabling photocurrent generation across a wide energy range—ideal for photodetectors and photovoltaic devices. For Ti₄AlC₃, major conductivity peaks appear between 5.92–6.41 eV (0–70 GPa), while for Zr₄AlC₃, peaks lie between 7.25–9.02 eV (0–60 GPa). With increasing pressure, both peak intensity and position shift upward. Notably, conductivity minima align with the lowest absorption points. Since the peaks fall in the UV range, both materials demonstrate strong potential for UV optoelectronic applications like sensors, LEDs, and photodetectors.

Reflectivity is a crucial optical property for evaluating the practical applications of MAX phase materials. The reflectivity spectra, R(ω), under pressure are shown in Fig. 17(d) for Ti₄AlC₃ and Fig. 18(d) for Zr₄AlC₃, illustrating how reflectivity varies with photon energy. This analysis helps determine the suitability of M₄AlC₃ (M = Ti, Zr) as protective coatings for minimizing solar heat absorption. According to Li et al.,⁹⁸ a reflectivity of at least 44% is required for effective thermal barrier coatings (TBCs). Both compounds show reflectivity around 60%, indicating strong potential for such applications. The highest reflectivity is observed in the infrared region, confirming their effectiveness in reflecting low-energy radiation and reducing heat absorption. In contrast, reflectivity decreases in the visible range, while small peaks appear in the UV region—between 7–10 eV for Ti₄AlC₃ and 9–13 eV for Zr₄AlC₃. Overall, their high infrared reflectivity makes them promising candidates for use in aerospace coatings, energy-efficient building materials, and high-temperature industrial protection.

The electron energy loss function, L(ω), describes the energy lost by fast-moving electrons as they pass through a material and provides insight into electron-material interactions.⁹⁹ Fig. 17(c) and 18(b) show the loss spectra of Ti₄AlC₃ and Zr₄AlC₃ under pressure. In the 0–15 eV range, no distinct peaks are observed, indicating the absence of significant plasmonic excitations and minimal energy dissipation—suggesting a stable electronic response, which is advantageous for devices requiring efficient charge transport and low electron scattering. The most prominent energy loss occurs at the plasma frequency, where ε₁(ω) = 0 and ε₂(ω) < 1, as seen in the dielectric function analysis.¹⁰⁰ In both materials, peaks in the loss function appear between 19–25 eV, corresponding to sharp decreases in reflectivity, absorption, and conductivity. Beyond this energy, the materials become transparent. This transparency above the plasma frequency highlights their potential for UV optics, transparent conductive coatings, and high-frequency optoelectronic devices.

3.6 Thermodynamic properties

MAX phase compounds, valued for their strength, toughness, and thermal stability, are promising for high-temperature applications. Studying their thermal properties is essential to assess performance and limits. The Debye temperature (θ_D) reflects the highest atomic vibrational mode and relates to specific heat, melting point, and thermal expansion.⁵² Anderson's method, a reliable and simple approach, calculates θ_D using the average elastic (sound) wave velocity as follows:¹⁰¹In this equation, h, k_B, N_A are Planck's constant, Boltzmann constant, and Avogadro's number, respectively; ρ and M denote the material's density and molecular weight; n is the number of atoms per unit cell; and v_m is the average sound velocity. The average sound velocity in crystals is calculated as follows:¹⁰²

Longitudinal (v_l) and transverse (v_t) sound velocities in crystals are calculated from shear (G) and bulk (B) moduli using these formulas:¹⁰¹

And

Table 8 lists density (ρ), transverse (v_t), longitudinal (v_l), average sound velocity (v_m), and Debye temperature (θ_D) for Ti₄AlC₃ and Zr₄AlC₃ under various pressures, with trends shown in Fig. 19 and 20. Both materials' densities rise with pressure due to volume reduction, and v_m follows suit, closely tied to θ_D. Ti₄AlC₃'s Debye temperature steadily increases with pressure, while Zr₄AlC₃'s stabilizes near 30 GPa, indicating limited further effect on atomic vibrations. Higher θ_D reflects stronger bonding, hardness, and thermal conductivity. Ti₄AlC₃ has significantly higher θ_D highlights its superior bonding and thermal properties, making it ideal for high-performance uses like coatings, thermal management. Also the value of Debye temperature of Ti₄AlC₃ increases with pressure making it suitable for application in aerospace engineering technology. According to the value of Debye temperature listed in Table 8, these compounds can be arranged as Ti₄AlC₃ > Nb₄AlC₃ > Zr₄AlC₃ > Hf₄AlC₃.

Table 8 Evaluated properties of M₄AlC₃ (M = Ti, Zr) compounds include density (ρ), transverse (v_t) and longitudinal (v_l) sound velocities, average sound velocity (v_m), Debye temperature (θ_D), melting temperature (T_m), and minimum thermal conductivity (k_min)

Elements	Pressure (GPa)	ρ (gm/cm^3)	vt (m s^−1)	vl (m s^−1)	vm (m s^−1)	θD (K)	kmin (W m^−1 K^−1)	Tm (K)	Ref.
Ti4AlC3	0	4.453	5762.54	9214.12	5851.94	827.86	1.941	2065.13	This study
Hf4AlC3	0	6.926	3684.90	5924.02	—	496.81	1.154	—	34
Nb4AlC3	0	6.930	4701.66	7795.24	5382.06	681.78	1.340	2166.18	37
Ti4AlC3	0	8.618	5792.27	9054.56	—	837.32	2.070	—	34
	10	4.684	6128.60	9949.42	6190.97	897.02	2.124	2377.50	This study
	20	4.883	6151.38	10 201.43	6227.09	914.91	2.197	2577.72
	30	5.059	6262.37	10 581.75	6349.64	943.97	2.294	2811.41
	40	5.221	6382.29	10 882.97	6475.41	972.74	2.389	3018.84
	50	5.368	6388.60	11 083.19	6491.65	984.21	2.440	3193.34
	60	5.509	6403.42	11 304.12	6516.15	996.35	2.493	3377.72
	70	5.643	6474.23	11 501.75	6590.90	1015.03	2.562	3525.15
Zr4AlC3	0	5.929	4522.95	7381.03	4585.33	602.20	1.299	1823.54
		6.802	4656.61	7455.78	—	623.76	1.434	—	34
Hf4AlC3	0	6.926	3684.90	5924.02	—	496.81	1.154	—
Nb4AlC3	0	6.930	4701.66	7795.24	5382.06	681.78	1.340	2166.18	37
Zr4AlC3	10	6.262	4705.81	7863.04	4779.98	640.00	1.405	2073.21	This study
	20	6.552	4866.82	8284.01	4950.43	643.52	1.500	2308.62
	30	6.799	4885.06	8496.33	4978.29	691.71	1.546	2467.73
	40	7.044	4832.22	8560.48	4932.57	685.52	1.569	2538.36
	50	7.278	4774.81	8646.28	4882.77	684.11	1.589	2632.71
	60	7.497	4692.63	8727.11	4809.32	681.86	1.595	2754.26

---

Fig. 19 Evaluated transverse (v_t), longitudinal (v_l), and average (v_m) sound velocities of (a) Ti₄AlC₃ and (b) Zr₄AlC₃ under varying pressure.

Fig. 20 Calculated (a) density, ρ and (b) Debye temperature, θ_D for Ti₄AlC₃ and Zr₄AlC₃ different pressure.

Minimum thermal conductivity (k_min) defines the lowest limit of heat conduction in MAX phases, relevant at extreme temperatures where phonon transport is minimal. It reflects atomic response to thermal variations and is directly linked to average sound velocity (v_m). Higher v_m implies greater k_min. It is calculated using the following equation:¹⁰³

The symbols in this equation retain the same meanings as defined in eqn (32).

The computed minimum thermal conductivity (k_min) values for M₄AlC₃ (M = Ti, Zr) are listed in Table 8, with their pressure-dependent trends shown in Fig. 21(a). At 0 GPa, Ti₄AlC₃ exhibits a k_min of 1.94 W m⁻¹ K⁻¹, notably higher than Zr₄AlC₃'s 1.29 W m⁻¹ K⁻¹. With increasing pressure, k_min rises significantly—reaching 2.562 W m⁻¹ K⁻¹ for Ti₄AlC₃ at 70 GPa and 1.595 W m⁻¹ K⁻¹ for Zr₄AlC₃ at 60 GPa. Across all pressure levels, Ti₄AlC₃ maintains a higher k_min than Zr₄AlC₃. Notably, even at 60 GPa, Zr₄AlC₃'s k_min remains below the ambient value of Ti₄AlC₃. Fig. 21(a) also highlights a steeper increase in Ti₄AlC₃'s k_min with pressure, suggesting a stronger pressure sensitivity. For thermal barrier coating (TBC) applications, materials with k_min ≈ 1.25 W m⁻¹ K⁻¹ are ideal.¹⁰⁴ While Zr₄AlC₃ exhibits lower thermal conductivity than Ti₄AlC₃ and Nb₄AlC₃ (see Table 8), making it promising for insulation, its increasing k_min under pressure limits its suitability for high-pressure environments.

Fig. 21 Calculated (a) minimum thermal conductivity (k_min) and (b) melting temperature (T_m) of Ti₄AlC₃ and Zr₄AlC₃ under varying pressure.

Melting temperature (T_m)—the point where a solid becomes liquid at standard pressure—is another key thermal metric. Fine et al.¹⁰⁵ proposed an empirical method using elastic constants to estimate the melting point of hexagonal crystals, including MAX phases.

T_m = 354 + 1.5 (2C₁₁ + C₃₃) (37)

The calculated melting temperatures of M₄AlC₃ (M = Ti, Zr) at various pressures, presented in Table 8 and Fig. 21(b), show a consistent increase with pressure. As E is closely related with C₁₁ and C₃₃, this trend aligns with the rise in Young's modulus (E) discussed in Section 3.2, indicating enhanced structural rigidity under compression. Ti₄AlC₃ exhibits higher melting temperatures than Zr₄AlC₃ across all pressure levels, highlighting its superior thermal resilience. This makes Ti₄AlC₃ more suitable for extreme heat environments such as aerospace and energy systems. While Zr₄AlC₃ remains viable for high-temperature applications, its relatively lower melting point limits its use in more thermally demanding conditions.

4. Conclusions

The structural, mechanical, electrical, thermal, and optical properties of M₄AlC₃ (M = Ti, Zr) MAX phase compounds were investigated under 0–70 GPa pressure using first-principles DFT calculations via the CASTEP code. At ambient conditions, computed structural parameters aligned well with previous theoretical and experimental data. With increasing pressure, both lattice parameters and unit cell volumes decreased, while elastic constants (C_ij) satisfied Born's criteria, confirming mechanical stability. Pressure enhanced bulk, shear, and Young's moduli, indicating stronger bonding, greater resistance to deformation, and increased stiffness. Pugh's and Poisson's ratios revealed a brittle-to-ductile transition at ∼60 GPa for Ti₄AlC₃ and ∼40 GPa for Zr₄AlC₃. The machinability index (μ_m) also improved with pressure, especially for Ti₄AlC₃, suggesting better industrial workability, although hardness declined, indicating reduced plastic deformation resistance. Anisotropic indices confirmed high elastic anisotropy under pressure. Electronic band structures and DOS confirmed metallic behavior across the pressure range. Optical properties—including dielectric function, absorption, and conductivity—further supported their metallic nature. High UV absorption and photoconductivity suggest potential for coatings to reduce solar heating and for use in optoelectronic devices, transparent conductive films, and smart windows. Thermal properties such as Debye temperature (Θ_D), minimum thermal conductivity (k_min), and melting temperature (T_m) all increased with pressure, indicating stronger bonding and higher thermal resilience. Ti₄AlC₃ consistently outperformed Zr₄AlC₃ in Θ_D and T_m, making it more suitable for extreme thermal environments. Overall, this study offers valuable insights into pressure-tuned behavior of M₄AlC₃ compounds, supporting their potential in high-temperature applications, optoelectronics, UV detection, thermal barrier coatings, and other advanced technologies.

Data availability

Data will be sent on request to the corresponding author Md. Atikur Rahman (atik E-mail: 0707phy@gmail.com)

Conflicts of interest

The authors ensure that there is no conflict to declare about the publication of this article.

Acknowledgements

The authors extend their appreciation to University Higher Education Fund for funding this research work under Research Support Program for Central Labs at King Khalid University through the project number CL/PRI/A/6.

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