First-principles insights into thermo-mechanical, electronic, and optical properties of Hf₂AC (A = Cl, Br) MAX phases through A-site halogen tuning

Abstract

In this study, we conduct a comprehensive first-principles investigation of the mechanical, electronic, optical, and thermo-mechanical properties of Hf₂AC MAX phases, where the A-site halogen is varied between Cl and Br. We use density functional theory to look into how changing the A-site atom affects the structural stability and other functional properties of these layered ternary carbides. Structural analysis reveals that replacing Cl with the larger Br atom increases lattice constants and unit cell volume, aligning with the atomic size trend. Both phases satisfy key stability criteria, including thermodynamic, mechanical, and dynamic stability, confirmed by formation energies, elastic constants, and phonon dispersions. Mechanical property analysis shows that Hf₂BrC has slightly reduced stiffness compared to Hf₂ClC due to weaker Hf–Br bonding, though both maintain ductility and mechanical robustness. Thermo-mechanical parameters, such as Debye temperature, melting point, and thermal conductivity, are influenced by halogen mass and bonding character. Thermo-lattice properties, including heat capacity and the Grüneisen parameter, further support their thermal stability over a broad temperature range. The electronic band structures and density of states show that both compounds act like metals, with Hf d-orbitals at the Fermi level being the most important. Optical properties, derived from the dielectric function, indicate strong activity in the visible and ultraviolet regions. Br substitution causes a red shift in absorption and reflectivity spectra, enhancing plasmonic and photonic behavior. High reflectivity and photoconductivity beyond the plasma frequency highlight their potential in optoelectronic and UV shielding applications. The ability to fine-tune these materials through atomic-level modifications opens new pathways for designing MAX phases with tailored performance for use in coatings, electronics, and energy-related applications.

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1. Introduction

MAX phases, a unique class of layered ternary carbides and nitrides, have garnered significant scientific interest owing to their remarkable combination of metallic and ceramic characteristics. These materials are structurally resilient at high temperatures, exhibit excellent machinability, and demonstrate superior electrical and thermal conductivity along with exceptional resistance to thermal shock, corrosion, and oxidation.^1–3 Such a blend of properties makes MAX phases ideal for a wide array of high-performance industrial applications, including machinable refractories, heating elements, rotating electrical contacts, bearings, nozzles, die-pressing tools, and heat exchangers. Notably, their ability to withstand neutron irradiation has also led to investigations into their potential roles in nuclear reactor components.⁴ In recent years, several MAX phases have also been explored as potential candidates for thermal barrier coatings (TBCs), given their favorable thermal and mechanical behavior at elevated temperatures.^5,6

Structurally, MAX phases are characterized by a hexagonal layered lattice and are described by the general chemical formula M_n+1AX_n, where M is an early transition metal, A is an element from groups 13 to 17 of the periodic table, and X is either carbon or nitrogen.² The integer n indicates the number of M–X layers in the structure; for n = 1, the resulting formula becomes M₂AX, known as the 211-type MAX phase, which is the focus of this investigation.^2,5,7

Over 60 stable ternary MAX phases have been experimentally synthesized, and numerous solid solutions have been reported through substitution at the M and A sites. Common examples include (Nb, Zr)₂AlC, (Ti, V)₂SC, (Ti, Hf)₂SC, V₂(Al, Ga)C, Ti₂(Si, Ge)C, and Cr₂(Al, Ge)C.^8–15 However, A-site solid solutions are typically limited to elements from groups 13 to 16, such as Al, Si, or Ga. Recently, a new class of MAX phases has emerged where halogen elements (Cl, Br, I) replace the conventional A-site elements. These include compounds such as La₂Br(C, N), La₂Cl(C, N), Sc₂ClC, Sc₂BrC, Hf₂ClC, and Hf₂BrC, which indicate the structural feasibility of incorporating halogens at the A site in MAX-like frameworks.²

The unique characteristics of M₂AX phases, commonly referred to as “211” MAX phases, have captivated the scientific community due to their remarkable combination of metallic and ceramic properties. These materials exhibit a fascinating array of attributes: moderate hardness (Vickers hardness of 2.0–8.0 GPa), anisotropy, elastic stiffness, excellent machinability, damage tolerance, and high-temperature plasticity.¹⁶ They are lightweight, resistant to fatigue, creep, oxidation, and thermal shock, and demonstrate a transition from brittleness to plasticity at elevated temperatures. These extraordinary features make MAX phases highly desirable for diverse technological applications. Recent studies have further expanded the understanding of M₂AX phases, showcasing their multifunctionality. For instance, H. Kim et al.¹⁷ identified M₂AX phases (M = lanthanides, A = Al or Si, X = C or N) as exceptional candidates for bond coat applications. Similarly, O. S. Rijal et al.¹⁸ explored the properties of Ti₂CdC and Ti₂SC, revealing their potential in optoelectronics, advanced photonic devices, solar thermal collectors, and thermal insulation. The works of B. U. Haq et al.¹⁹ and M. N. Amin et al.²⁰ have highlighted the applications of W₂GaX and Cr₂AC phases, respectively, in high-temperature environments, optoelectronic devices, and other advanced technologies. Furthermore, W. Tan et al.²¹ demonstrated the potential of S-based M₂SX phases in thermoelectric applications, while M. A. Ali et al.²² and M. M. Uddin et al.²³ underscored the suitability of Hf₂SeC, Hf₂AlC, and Hf₂AlN for high-temperature and solar heating reduction applications.

Despite these advances, a comprehensive understanding of how halogen substitution at the A site influences the physical properties of MAX phases is still lacking.

The prior work by Ohmer et al.² focused on predicting the elastic constants and moduli of Hf₂ClC and Hf₂BrC within a broad high-throughput screening of 82 potential MAX phases, using basic thermodynamic descriptors to assess phase stability. While this study was foundational in identifying the existence of these compounds, it primarily addressed phase feasibility rather than functional behavior. In contrast, our present study provides a comprehensive, in-depth, and comparative investigation of the functional properties influenced by halogen substitution in Hf-based MAX phases, extending well beyond prior structural predictions and providing helpful information regarding their mechanical, electronic, thermal, and optical behavior. We systematically explore their thermo-mechanical responses by analyzing elastic anisotropy indices, Debye and melting temperatures, the Grüneisen parameter, and both lattice and minimum thermal conductivities, which are critical for understanding their thermal resilience and structural integrity. Additionally, we discuss their electronic structure through detailed assessments of band dispersion, directional charge carrier mobility, effective mass, and the nature of orbital hybridization—key insights necessary for electronic and energy-related applications. Furthermore, we present a complete evaluation of the optical behavior of these materials, including their dielectric functions, absorption characteristics, photoconductivity, reflectivity, and energy-loss spectra, which have not been previously reported. To the best of our knowledge, no earlier study has comprehensively examined the collective influence of halogen substitution (Cl versus Br) on such an extensive range of physical properties in Hf-based MAX phases. Our findings address this critical gap and present a valuable framework for the property-driven design of advanced MAX materials, highlighting their potential across multifunctional domains such as thermal barrier coatings, optoelectronics, and structural applications operating under extreme environments.

The arrangement of this paper is as follows: Section 2 describes the DFT computational setting. Section 3 presents and analyzes observations and their physical implications. Section 4 provides a brief summary of the present findings and concluding remarks.

2. DFT computational setting

In this work, we explore the impact of halogen substitution, specifically Cl and Br, at the A site of Hf₂AC (A = Cl, Br) MAX phases through comprehensive first-principles calculations based on density functional theory (DFT).²⁴ All calculations were performed using the CASTEP code,²⁵ a plane-wave-based density functional theory (DFT) software package. Ultrasoft pseudopotentials were employed for all atomic species (Hf, Cl, Br, and C), allowing accurate treatment of valence electron interactions while maintaining computational efficiency. To accurately capture the exchange–correlation interactions, we employed the generalized gradient approximation (GGA) using the Perdew–Burke–Ernzerhof (PBE) functional.²⁶ A plane-wave energy cutoff of 500 eV was selected to ensure convergence of total energy across all investigated systems. For Brillouin zone integration, a Monkhorst–Pack k-point mesh of 9 × 9 × 2 was used, offering an optimal balance between computational efficiency and numerical accuracy.²⁷ The interactions between valence electrons and ionic cores were described using ultrasoft pseudopotentials, which were carefully chosen to ensure reliable modeling of chemical bonding.²⁸ The valence states considered in our calculations were Hf (5d²5s²), Cl (3s²3p⁵), Br (4s²4p⁵), and C (2s²2p²), which play a critical role in defining the material properties of the Hf₂ClC and Hf₂BrC compounds. Geometry optimizations were carried out using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization algorithm.²⁹ The convergence thresholds were rigorously set to a total energy difference of 5 × 10⁻⁶ eV per atom, a maximum force of 0.01 eV Å⁻¹, a maximum stress of 0.02 GPa, and a maximum displacement of 5 × 10⁻⁴ Å. These strict criteria ensured that the optimized structures were robust and well-suited for subsequent property evaluations. Following structural optimization, we systematically computed various mechanical properties of Hf₂ClC and Hf₂BrC, including their elastic constants via the stress–strain method.³⁰ Phonon dynamic properties, including the phonon dispersion and phonon density of states (PDOS), were calculated using the finite displacement method.³¹ The crystallographic structures were visualized using the VESTA software package, providing insights into the atomic arrangements and symmetry.³²

3. Observations and physical implications

3.1 Crystallographic properties

The substitution of halogen (Cl, Br) in Hf₂AC (A = Cl, Br) MAX phases induce significant changes in both the structural characteristics and properties of these compounds due to differences in atomic size, electronegativity, and bonding behavior. As illustrated in Fig. 1, these compounds crystallize in the hexagonal crystal system with the space group P6₃/mmc (no. 194), with each unit cell containing eight atoms and two formula units. The atomic positions for Hf₂AC (A = Cl, Br) are defined by the Wyckoff coordinates: C atoms occupy the 2a site at (0, 0, 0), A-site atoms (Cl or Br) are located at 2c (1/3, 2/3, 1/4), and Hf atoms are positioned at 4f (2/3, 1/3, Z_M). The crystal structure is determined by two main lattice parameters, a and c, along with the internal atomic coordinate Z_M.

Fig. 1 Optimized crystal structures of Hf₂AC (A = Cl, Br) MAX phases: (a) standard orientation highlighting the layered hexagonal architecture, (b) projection along the a-axis emphasizing the stacking sequence, and (c) projection along the c-axis revealing the atomic arrangement within the basal plane.

The primary effect of substituting Cl and Br in the A-site on Hf₂AC MAX phases is observed in the c/a ratio and unit cell volume. Hf₂BrC exhibits a slightly higher c/a ratio and unit cell volume than Hf₂ClC, which is consistent with the larger atomic radius of Br compared to Cl. The trend of increasing c/a ratio and unit cell volume with larger atoms is further confirmed by comparing Hf₂ClC and Hf₂BrC with other Hf-based MAX phases in Table 1. These changes in the crystal structure reflect a typical trend observed in MAX phases, where the incorporation of larger atoms leads to expanded interlayer spacing and a corresponding increase in unit cell volume. The equilibrium crystal structures of Hf₂AC (A = Cl, Br) have been optimized by minimizing the total energy, with the resulting structural parameters provided in Table 1. Although direct experimental or theoretical benchmarks for Hf₂ClC and Hf₂BrC are currently unavailable, our optimized values show strong consistency with those reported for other well-known and experimentally verified MAX phases, thereby reinforcing the reliability of our calculations and their alignment with established trends in the field (see Table 1).

Table 1 Structural and energetic properties of Hf₂AC (A = Cl, Br) MAX phases, featuring optimized lattice parameters (a, c in Å), c/a ratio, internal lattice parameter (Z_M in Å), unit cell volume (V in ų), formation enthalpy (ΔH_f in eV), formation energy (ΔE_f in eV) and coordination sites (O_r, P_r) along with theoretical comparisons

Phases	a	c	c/a	ZM	V	ΔHf	ΔEf	Or	Pr	Ref.
Hf2ClC	3.432	12.947	3.77	0.0834	132.094	−7.13	−1.26	1.77	1.17	This
Hf2BrC	3.490	13.427	3.84	0.0833	141.636	−6.95	−1.10	1.75	1.16	This
Hf2AlC	3.275	14.362	4.39	0.0876						Expt.^36
Hf2SeB	3.522	12.478	3.54							Expt.^37
Hf2AlC	3.310	14.350	4.33	0.0890	136.345	—		—	—	Theo.^23
Hf2SC	3.423	12.196	3.56	0.1004	123.770	−8.16		—	—	Theo.^34
Hf2SnC	3.330	14.370	4.32	0.0879	138.006	—		—	—	Theo.^35
Hf2TlC	3.370	14.847	4.41	0.0825	146.025	—		—	—	Theo.^33

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3.2 Compound's stability

Since Hf₂AC (A = Cl, Br) are newly identified compounds with unexplored properties that have not yet been experimentally synthesized, our objective is to establish their theoretical foundation through density functional theory (DFT) calculations. This theoretical investigation aims to guide future experimental and computational studies. While we have predicted the stability of Hf₂ClC and Hf₂BrC using first-principles calculations, additional theoretical analyses may be necessary. Therefore, we conduct a comprehensive stability assessment of the Hf₂AC (A = Cl, Br) phases to provide insights for future experimental validation.

To evaluate the structural stability of Hf₂AC (A = Cl, Br), we examined the c/a ratio. For M₂AX-type phases, the ideal c/a value is approximately 4.89 (or 2√6).^7,38 As shown in Table 1, the computed c/a values for Hf₂ClC (3.77) and Hf₂BrC (3.84) are reasonably close to this ideal value, indicating favorable structural stability.

Furthermore, in MAX phases, the spacing between the M and X (X = C, N) layers is characterized by the internal structural parameter z_m, which ideally takes a value of approximately 0.083 (or 1/12).^7,38 Our calculations reveal that the z_m values for Hf₂ClC and Hf₂BrC are close to this ideal value, indicating minimal deviation and further supporting structural stability (see Table 1).

To further assess stability, we analyzed distortions in the unit cell using the octahedral site (O_r) and trigonal prism site (P_r) parameters, defined as:⁵

Ideally, both O_r and P_r should be equal to 1, with values greater or less than 1, indicating structural distortions. The O_r and P_r values for Hf₂ClC and Hf₂BrC are quite similar, with only slight differences. The slightly higher O_r and P_r values for Hf₂ClC compared to Hf₂BrC can be attributed to variations in atomic size, bonding characteristics, and structural distortions introduced by the halogen element (Cl vs. Br). Since Cl has a smaller atomic radius than Br, the interatomic spacing in the Hf₂ClC structure is slightly reduced compared to Hf₂BrC, leading to greater structural distortion. In contrast, the larger Br atoms in Hf₂BrC result in a more relaxed lattice structure with slightly reduced distortions, leading to lower O_r and P_r values. However, since these differences are small, both compounds remain structurally stable.

To evaluate their thermodynamic stability, we calculated the formation enthalpy (ΔH_f) and binding energy (ΔE_b), both of which serve as key indicators of stability. The formation enthalpy was computed using the following equations:³⁹

where, E_tot(Phase) represents the total energy of the Hf₂AC (A = Cl, Br) phase, while E_Hf, E_A, and E_C denote the atomic energies of Hf, A (Cl or Br), and C, respectively. The variables l, m, and n represent the number of atoms in the compound's empirical formula, while N indicates the total number of atoms in the formula unit. The calculated values of ΔH_f presented in Table 1, confirm the thermodynamic stability of both Hf₂ClC and Hf₂BrC. Moreover, similar negative energy values have been reported for other MAX phases.^{4,5,7,40–42} This consistency suggests that Hf₂AC (A = Cl, Br) phase in our study is not only energetically favorable and exothermic but also highly stable.

Although the formation enthalpy (ΔH_f) provide insights into thermodynamic stability, it does not fully capture the stability of a compound. It is also essential to examine phase stability with respect to competing phases. This approach is particularly important in materials science, chemistry, and engineering, where real-world applications depend on the compound's resistance to decomposition into other stable phases. To address this, we analyzed the phase stability by comparing the total energies of Hf₂ClC and Hf₂BrC with those of relevant competing phases, as obtained from the Open Quantum Materials Database (OQMD).⁴³ For both compounds, the same set of competing phases was considered, although their compositional contributions differ slightly as dictated by the overall stoichiometry. The decomposition reactions into competing stable phases are expressed as follows:

• Hf₂ClC → 0.5 × Hf₂CCl₂ + 0.1 × Hf₆C₅ + 0.4 × Hf

• Hf₂BrC → 0.5 × Hf₂CBr₂ + 0.1 × Hf₆C₅ + 0.4 × Hf

These competing phases were selected based on thermodynamic stability criteria and known structural feasibility. The calculated formation energies with respect to these phases are listed in Table 1. The negative values of the formation energies (ΔE_f in eV) confirm the relative stability of Hf₂ClC and Hf₂BrC, reinforcing the conclusions drawn from the formation enthalpy and binding energy analyses.

To further confirm the dynamical stability of Hf₂AC (A = Cl, Br) MAX phases, we conducted a phonon dispersion analysis at zero pressure and temperature. This evaluation is essential for determining the feasibility of synthesizing these compounds in a hexagonal structure. Using density functional perturbation theory (DFPT), we computed the phonon spectrum dispersion.³¹

The phonon dispersion curves (PDCs) with total density of states, shown in Fig. 2(a and b), offer vital information about stability. A fundamental criterion for dynamical stability is the absence of imaginary phonon frequencies in the dispersion curves. Since our results show no such imaginary frequencies, we confirm that Hf₂ClC and Hf₂BrC are dynamically stable and can potentially be synthesized experimentally.

Fig. 2 Calculated phonon dispersion spectra and the right panel shows the phonon density of states for (a) Hf₂ClC, and (b) Hf₂BrC.

Each graph displays three acoustic branches (red curves in the low-frequency region) and twenty-one optical branches (blue, green, and orange curves in the higher-frequency region). The longitudinal acoustic (LA) mode is seen progressing smoothly, while the transverse acoustic (TA) modes exhibit slight dispersion near the Γ-point. The transverse optical (TO) and longitudinal optical (LO) modes are located at higher frequencies for Hf₂ClC (∼20 THz, ∼15.35 THz) compared to Hf₂BrC (∼19.22 THz, ∼15.0 THz). This frequency reduction in Hf₂BrC is due to Br being heavier than Cl, which lowers the vibrational frequencies as per the inverse mass relation in lattice dynamics (i.e., ). As a result, Hf₂BrC has softer optical phonon modes, leading to slightly lower phonon energy compared to Hf₂ClC. The low-frequency acoustic phonons (red curves), responsible for lattice vibrations and heat conduction, are slightly more dispersed in Hf₂BrC compared to Hf₂ClC. Since Br is larger than Cl, it weakens the bonding strength in Hf₂BrC, making the structure slightly more flexible. This results in lower acoustic phonon velocities, which could impact thermal conductivity. In both cases, the optical and acoustic phonon branches remain well-separated, which indicates limited phonon scattering. However, Br induces softer phonon modes compared to Cl. This may enhance acoustic–optical phonon interactions, leading to increased phonon scattering and reduced thermal conductivity. This suggests that Hf₂BrC might exhibit lower thermal conductivity than Hf₂ClC.

Hf₂AC (A = Cl, Br) MAX phases satisfy all essential stability criteria, including formation energy, phonon dispersion curves, and mechanical stability conditions (Section 3.3). These results confirm their thermodynamic and dynamical stability, making them strong candidates for experimental synthesis. Several MAX phases, including carbides, nitrides, and borides, have already been successfully synthesized.⁴¹ Additionally, various Hf-based MAX phases, such as Hf₂AlC, Hf₃AlC₂, Hf₅Al₂C₃, Hf₂SnC, and Hf₃PbC₂ have been experimentally realized.^36,44–46 Given this trend, there is a strong potential for the successful synthesis of Hf₂ClC and Hf₂BrC, further expanding the family of Hf-based MAX phases.

3.3 Mechanical and elastic properties

Elastic constants are essential for evaluating the mechanical stability and stiffness of materials under external stress. The mechanical behavior of a crystal lattice is characterized by its second-order elastic stiffness constants, which play a crucial role in determining the mechanical stability of a material.⁴⁷ These constants are derived from the strain-energy relationship as:where E represents the energy of the crystal, V₀ is its equilibrium volume, and ε is the applied strain.

In hexagonal crystal structures, the stiffness matrix consists of 21 components, of which only six are independent: C₁₁, C₁₂, C₁₃, C₃₃, C₄₄, and C₆₆. Among these, C₆₆ is dependent and follows the relation C₆₆ = (C₁₁ − C₁₂)/2. The computed values of these elastic constants for Hf₂AC (A = Cl, Br) are presented in Table 2. To confirm mechanical stability, these values must satisfy the Born–Huang stability criteria for hexagonal structures:⁴⁷

C₁₁ > |C₁₂|, 2C₁₃² < C₃₃(C₁₁ + C₁₂), C₄₄ > 0 and C₆₆ > 0 (4)

Table 2 The calculated stiffness constants (C_ij in GPa), Cauchy pressure (C′′ in GPa), and Kleinman parameter (ζ) for Hf₂AC (A = Cl, Br) MAX phases along with some other theoretical results

Compounds	C11	C12	C13	C33	C44	C66	C′′	ζ	Ref.
Hf2ClC	165	56	75	140	27	54	29	0.48	This work
Hf2BrC	156	78	68	147	22	39	56	0.63
Hf2AlC	329	70	81	272	126	129	−56	—	Ref. 23
Hf2SC	320	95	121	330	149	113	−54	—	Ref. 34
Hf2SnC	293	87	101	278	110	103	−23	—	Ref. 35
Hf2TlC	278	69	62	230	71	104	−2	—	Ref. 33

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The computed values for Hf₂ClC and Hf₂BrC meet these criteria, indicating that they are mechanically stable. These results suggest that these MAX phases can be considered for further experimental exploration. Since no prior experimental or theoretical data exist for the elastic constants of Hf₂ClC and Hf₂BrC, a comparative analysis with other Hf-based MAX phases provides valuable insights into their mechanical behavior.

A comparison of the stiffness constants in Table 2 reveals that Hf₂ClC and Hf₂BrC exhibit lower mechanical stiffness than other Hf-based MAX phases. The values of C₁₁, which represent resistance to uniaxial deformation, are 165 GPa and 156 GPa for Hf₂ClC and Hf₂BrC, respectively. These values are significantly lower than those of other Hf-based MAX phases, indicating that halogen-based MAX phases are more flexible. Similarly, the values of C₃₃, which represent resistance to compression along the c-axis, are 140 GPa for Hf₂ClC and 147 GPa for Hf₂BrC, compared to much higher values in Hf₂AlC, Hf₂SC, Hf₂SnC, and Hf₂TlC. This reduction in stiffness along the c-axis suggests that halogen substitution weakens bonding in this direction, making these structures more compliant under mechanical stress. The shear modulus, represented by C₄₄ and C₆₆, is also significantly lower in Hf₂ClC (27 GPa, 54 GPa) and Hf₂BrC (22 GPa, 39 GPa) compared to other Hf-based MAX phases as mentioned in Table 2. This suggests that Hf₂ClC and Hf₂BrC have reduced resistance to shear deformation, with Hf₂BrC being even more flexible due to the larger atomic size and lower electronegativity of Br, which weakens the overall bonding strength.

The stiffness constants C_ij reveal the elastic anisotropy of Hf₂ClC and Hf₂BrC. The difference between C₁₁ and C₃₃ indicates stronger bonding along the a(b)-axis than the c-axis, making compression along the a-axis more difficult. The higher C₁₁ value confirms that bonding along [100] is stronger than along [001], similar to other Hf-based MAX phases.^23,33–35 The lower C₄₄ values suggest shear deformation is easier than linear compression, providing flexibility while maintaining structural integrity. The lower values of C₁₂ and C₁₃ compared to C₁₁ and C₃₃ indicate that stress along the a-axis significantly affects mechanical properties. The equation C₁₁ + C₁₂ > C₃₃ shows that the bonds are stronger bonding in the (001) plane than along the c-axis. The difference between two specific elastic constants, known as the Cauchy pressure, is expressed as C′′ = (C₁₂ − C₄₄) and serves as an indicator of a material's ductility or brittleness. A positive Cauchy pressure suggests ductility, while a negative value indicates brittleness. It also helps us understand how atoms bond in solids. A positive Cauchy pressure implies predominant ionic bonding, whereas a negative value suggests covalent bonding.⁴⁸ The positive C′′ values for Hf₂ClC (29 GPa) and Hf₂BrC (56 GPa) (see in Table 2) confirm their ductile nature, which enhances their resistance to mechanical failure under stress. The presence of ionic bonding improves their flexibility, making them suitable for applications requiring high toughness, such as wear-resistant and impact-resistant coatings. Their ductile behavior also ensures better machinability, enabling their use in cutting tools and structural components exposed to cyclic loading. Additionally, their combination of ductility and moderate stiffness makes them promising for thermal barrier coatings, where resistance to thermal shock and mechanical integrity are crucial.

The substitution of Cl and Br in A site of Hf₂AC (A = Cl, Br) significantly affects the Kleinman parameter , which describes the relative contributions of bond bending and bond stretching in response to mechanical deformation.⁴⁹ The Kleinman parameter is a dimensionless parameter that typically lies in the range 0 ≤ ζ ≤ 1. The calculated ζ values increase from 0.48 in Hf₂ClC to 0.63 in Hf₂BrC, indicating a shift toward greater bond bending dominance. This change is attributed to the larger atomic radius of Br, which weakens Hf–Br bonds compared to Hf–Cl, making it easier for bonds to bend rather than stretch under stress. A higher ζ value in Hf₂BrC suggests greater flexibility and the ability to absorb mechanical strain more effectively. This enhances its resistance to damage, making it suitable for applications requiring impact resistance and adaptability to thermal expansion. In contrast, the lower ζ in Hf₂ClC indicates a more balanced contribution of bond bending and stretching, resulting in higher stiffness and structural stability. This makes Hf₂ClC a better candidate for applications where mechanical strength and rigidity are prioritized over flexibility.

The polycrystalline elastic moduli, including the bulk modulus (B), shear modulus (G), Young's modulus (E), and Poisson's ratio (ν), were determined from the single-crystal elastic constants using the Voigt–Reuss–Hill (VRH) approximations.⁵⁰ The Voigt and Reuss equations provide the upper and lower bounds for these moduli, and Hill's approximation takes their average to define the polycrystalline properties. The bulk modulus is calculated as B_H = (B_V + B_R)/2, where B_V and B_R are the Voigt and Reuss bulk moduli, respectively. Similarly, the shear modulus is given by G_H = (G_V + G_R)/2.

Using these values, Young's modulus and Poisson's ratio are derived from the relationships E = 9B_HG_H/(3B_H + G_H) and v = 3B_H − 2 G_H/2(B_H + G_H). The computed values for Hf₂ClC and Hf₂BrC, along with other MAX phases, are presented in Table 3 and shown in Fig. 3(b).

Table 3 The calculated values of polycrystalline elastic moduli (B, G, Y in GPa), Pugh's ratio (B/G), Poisson's ratio (ν), hardness parameters (H_micro and H_macro in GPa) and machinability index (μ_M) of Hf₂AC (A = Cl, Br) phases along with other theoretical data

Phases	B	G	E	B/G	ν	Hmicro	Hmacro	μM	Ref.
Hf2ClC	97.90	37.51	99.78	2.61	0.33	4.24	2.43	3.62	This work
	110.86	45.24	119.48	2.45	0.32	5.42	—	—	Ref. 2
Hf2BrC	98.62	31.44	85.27	3.13	0.35	3.02	0.95	4.52	This work
	116.38	29.45	81.47	3.95	0.38	2.29	—	—	Ref. 2
Hf2AlC	154	122	297	0.79	0.19	—	—	—	Ref. 23
Hf2SC	183	123	300	1.49	0.23	—	—	—	Ref. 34
Hf2SnC	161	102	252	1.58	0.24	—	—	—	Ref. 35
Hf2TlC	129	87	214	1.48	0.22	—	—	—	Ref. 33

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Fig. 3 Comparison of (a) elastic constants and (b) elastic moduli of Hf₂ClC and Hf₂BrC with other Hf-based MAX compounds.

The bulk modulus (B) represents a material's resistance to uniform compression. From Fig. 3(b), it can be observed that Hf₂ClC (97.90 GPa) and Hf₂BrC (98.62 GPa) exhibit significantly lower bulk modulus (B) values compared to Hf₂AlC, Hf₂SC, Hf₂SnC and Hf₂TlC. This indicates that halogen-based MAX phases are more compressible and flexible. The lower B values suggest weaker interatomic bonding in Hf₂ClC and Hf₂BrC, making them more deformable under pressure. The bulk modulus (B) shows a slight increase of 0.73% upon Cl-to-Br substitution, indicating a minimal change in compressibility. This suggests that replacing Cl with the larger Br atom does not significantly alter the overall resistance to volumetric deformation. Despite this minor variation, both Hf₂ClC and Hf₂BrC remain highly compressible compared to other Hf-based MAX phases, reinforcing their potential for applications where flexibility and adaptability under mechanical stress are required.

The shear modulus (G), which measures resistance to shape deformation, is also lower in Hf₂ClC (37.51 GPa) and Hf₂BrC (31.44 GPa) compared to other Hf-based MAX phase.^23,33–35 This suggests that halogen-containing MAX phases have reduced resistance to shear forces, making them more ductile and less rigid than other Hf-based MAX phases. The value of G decreases significantly by 16.16% upon Cl-to-Br substitution, indicating that Hf₂BrC becomes more flexible and less resistant to shear deformation. This reduction in shear modulus can be attributed to the larger atomic radius and lower electronegativity of Br compared to Cl, which weakens interatomic bonding and reduces resistance to shear forces. The lower shear modulus enhances the material's ductility, making Hf₂BrC more adaptable under mechanical stress. The improved ductility makes Hf₂BrC suitable for environments subjected to mechanical shocks or vibrations, such as aerospace components and thermal barrier coatings.

Young's modulus (E), which defines the stiffness of a material, follows the same trend. This confirms that Hf₂ClC and Hf₂BrC are mechanically softer and more flexible compared to these Hf-based MAX phases.

The value of E decreases by 14.57% upon Cl-to-Br substitution, confirming reduced stiffness and structural rigidity in Hf₂BrC. This decline is primarily due to the weaker bonding strength introduced by the larger Br atoms, which reduce the overall bond strength and increase lattice flexibility. This reduction in stiffness enhances the material's toughness and damage tolerance, making Hf₂BrC a better candidate for applications requiring impact resistance and flexibility, such as protective coatings and shock-absorbing layers. The decreased E also suggests improved machinability, making it easier to process and shape for engineering applications.

In addition, Young's modulus (E) plays a crucial role in determining thermal shock resistance (R), as seen in the equation , where σ_b stands for strength and α for thermal expansion coefficient.⁵¹ A higher Young's modulus leads to a lower thermal shock resistance, meaning materials with greater stiffness are more susceptible to thermal stress fractures. Since Hf₂BrC has a lower Young's modulus (85.27 GPa) compared to Hf₂ClC (99.78 GPa), it is expected to have better thermal shock resistance. This is because lower stiffness allows the material to absorb thermal stress more effectively, reducing the likelihood of crack propagation under rapid temperature changes. The greater thermal shock resistance of Hf₂BrC makes it a better candidate for these applications, as it can withstand sudden temperature variations without failure. On the other hand, Hf₂ClC, with its higher Young's modulus, may be more suitable for applications where structural rigidity and mechanical strength are prioritized over thermal shock resistance, such as load-bearing components in high-performance machinery.

The B/G ratio (Pugh's ratio) and Poisson's ratio (ν) are key indicators of a material's mechanical properties, particularly its brittleness or ductility.^52,53 The B/G ratio and Poisson's ratio (ν) of Hf₂ClC and Hf₂BrC are greater than those of other Hf-based MAX phases, as mentioned in Table 3. Generally, if B/G > 1.75, the material tends to be ductile, and if B/G < 1.75, it is more likely to be brittle. Both Hf₂ClC and Hf₂BrC have B/G ratios above 1.75, indicating they are ductile materials. However, Hf₂BrC shows a higher B/G ratio (3.13) compared to Hf₂ClC (2.61), suggesting it is slightly more ductile. This change is attributed to the larger ionic radius of Br compared to Cl, which allows for more flexible bonding and less rigidity in the structure, enhancing ductility.

Poisson's ratio (ν) reveals the compressibility, bonding forces, and nature of the material. The value of v greater than 0.26 typically suggests ductility, while a value below 0.26 tends to indicate brittleness. Both values are above 0.26, indicating that both materials are ductile. The substitution of Cl with Br increases the Poisson's ratio slightly, which aligns with the trend seen in the B/G ratio and Cauchy pressure. This implies that the presence of Br in Hf₂BrC results in a slightly higher ductility compared to Hf₂ClC. This characteristic could make Hf₂BrC suitable for use in industries requiring durable, high-performance.

Moreover, the Poisson's ratio (ν) of Hf₂AC (A = Cl, Br) is critical in understanding the nature of the interatomic forces and the chemical bonding in these compounds. The values of ν for Hf₂ClC and Hf₂BrC are 0.33 and 0.35, respectively, both of which fall within the range of 0.25 to 0.50, indicating that the interatomic forces in these materials are central in nature.⁴⁹ This suggests that the material's deformation behavior is influenced by central forces between the atoms, rather than directional covalent forces.

The Poisson's ratio also gives an explanation for the bonding character. For purely covalent materials,⁵³ Poisson's ratio is typically around 0.10, while for purely ionic materials, it tends to be closer to 0.25. The calculated Poisson's ratios for Hf₂ClC and Hf₂BrC (both above 0.25) suggest that these compounds exhibit a significant ionic contribution to their chemical bonding. When considering the substitution of Cl with Br, the increase in Poisson's ratio from 0.33 to 0.35 (Hf₂ClC to Hf₂BrC) indicates a slight enhancement in the ionic bonding character. Br, being larger and less electronegative than Cl, may result in a more flexible atomic structure, which could lead to a slightly greater ionic interaction. This subtle increase in ionic bonding may contribute to Hf₂BrC's slightly greater ductility compared to Hf₂ClC, as reflected in its higher B/G ratio. Thus, the Cl-to-Br substitution slightly enhances the ionic character of the bonding in Hf₂BrC, which could improve its mechanical properties, making it more ductile.

The hardness of a material is a key factor in its ability to resist deformation, wear, and indentation under force. For Hf₂AC (A = Cl, Br) compounds, the hardness parameters (H_micro, H_macro) are calculated by following relation:^54,55

H_macro = 2(k²G)^0.585 − 3 (6)

where, k and G refers to Pugh's ratio (k = G/B) and shear modulus, respectively.

The calculated hardness values for Hf₂ClC (H_micro = 4.24 GPa, H_macro = 2.43 GPa) and Hf₂BrC (H_macro = 3.02 GPa, H_micro = 0.95 GPa) show that both compounds are relatively soft, with Hf₂BrC being softer than Hf₂ClC (see Table 3). The Cl-to-Br substitution in Hf₂AC (A = Cl, Br) leads to a reduction in hardness, with Hf₂BrC showing significantly lower hardness compared to Hf₂ClC. The lower hardness of Hf₂BrC can be attributed to the larger atomic size of Br compared to Cl, which likely results in weaker atomic bonds and a more easily deformable structure. The relatively soft nature of Hf₂ClC and Hf₂BrC makes these materials suitable for applications where high hardness is not a critical requirement, but good ductility and flexibility are desired. Additionally, Hf₂BrC, with its lower hardness, may find use in applications that require easier machining or shaping, such as in electronic components or catalysts where moderate mechanical strength suffices.

The machinability index (μ_M = B/C₄₄) is an important parameter for assessing the plasticity and lubricating properties of a material.⁵⁶ It is influenced by factors such as elastic properties, hardness, and the ability of a material to undergo plastic deformation. A higher μ_M value indicates improved lubrication, reduced friction, and enhanced machinability, making the material easier to shape and process. The calculated values of μ_M for Hf₂ClC and Hf₂BrC are presented in Table 3, showing that Hf₂BrC has better machinability compared to Hf₂ClC. This is due to Br's larger atomic size, which weakens atomic bonding, increasing plasticity and easing deformation. Hardness and machinability are inversely related, meaning materials with lower hardness are generally easier to machine. The decrease in hardness for Hf₂BrC is 28.77% (microhardness) and 60.91% (macro-hardness), while its machinability index is 24.86% higher than Hf₂ClC. This confirms that as hardness decreases, machinability improves due to weaker atomic bonding, allowing for easier deformation. The higher machinability of Hf₂BrC makes it suitable for high-speed machining, reducing tool wear and processing costs.

3.4 Elastic anisotropy

In this study, we explore the elastic anisotropy of Hf₂AC (A = Cl, Br), which is crucial for understanding its mechanical behavior under various conditions. Elastic anisotropy influences plastic deformation, crack propagation, and stress tolerance. The crystal structure (see Fig. 1) reveals distinct atomic arrangements along the a(b) and c directions, leading to differences in bond strength. The stiffness along a(b) is represented by C₁₁, while the stiffness along c is denoted by C₃₃. Since C₁₁ ≠ C₃₃, this confirms anisotropy in mechanical properties. The shear anisotropic factors (A₁, A₂, A₃) provide insight into the bonding strength variations across different atomic planes. The shear anisotropy factor for {100} shear planes between [011] and [010] directions is,⁵⁷for {010} shear planes between [101] and [001] directionsfor {001} shear planes between [110] and [010] directions

The calculated shear anisotropic factors of Hf₂AC (A = Cl, Br) are listed in Table 4. The calculated anisotropy values indicate that both Hf₂ClC and Hf₂BrC exhibit moderate anisotropy, with deviations from the ideal isotropic value (1). However, Cl-to-Br substitution significantly impacts the degree of anisotropy. For Hf₂BrC, A₁ and A₂ values decrease compared to Hf₂ClC, dropping from 0.69 to 0.52, indicating an increase in anisotropy in {100} and {010} shear planes. This suggests that Br's larger atomic size weakens directional bonding, leading to increased deformation along these planes. Despite these changes, A₃ remains 1.0 for both compounds, confirming isotropy in {001} plane. This comparison effectively highlights the practical implications of elastic anisotropy in Hf₂ClC and Hf₂BrC. Hf₂BrC's increased anisotropy suggests greater plastic deformation, making it more adaptable for flexible structural applications where some deformation can be tolerated. Conversely, Hf₂ClC's more uniform bonding structure enhances wear resistance and fracture toughness, making it more suitable for high-stress environments such as protective coatings or structural components.

Table 4 The calculated shear anisotropic factors (A₁, A₂ and A₃), percentage anisotropy in the compression and shear (A^B and A^G), universal anisotropic index (A^U), equivalent Zener anisotropic factor (A^eq), compressibility ratio (k_c/k_a), and universal log-Euclidean index (A^L) for studied compounds Hf₂AC (A = Cl, Br)

Compounds	A1	A2	A3	A^B	A^G	A^U	A^eq	kc/ka	A^L	Ref.
Hf2ClC	0.69	0.69	1.0	0.0	0.049	0.52	1.91	2.38	1.27	This work
Hf2BrC	0.52	0.52	1.0	0.0	0.045	0.48	1.86	2.12	1.08

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We have also calculated the percentage elastic anisotropy in shear (A^G) and compression (A^B), the universal anisotropy factor (A^U), the equivalent Zener anisotropy factor (A^eq) by the following common relation^58,59 and their estimated values are displayed in Table 4.

A^B = (B_V − B_R)/(B_V + B_R) (10)

A^G = (G_V − G_R) + (G_V + G_R) (11)

For an isotropic crystal, A^B = A^G = 0. However, the degree of anisotropy is represented by any variation larger than zero. For both materials, the percent elastic anisotropy in compression (A^B) is 0.0, meaning they are isotropic in compressibility, implying uniform bulk modulus distribution. However, the percentage elastic anisotropy in shear (A^G) is 0.049 for Hf₂ClC and 0.045 for Hf₂BrC, confirming slight anisotropy in shear deformation. The lower A^G value for Hf₂BrC suggests that Br substitution weakens the directional dependency of shear modulus, slightly reducing anisotropy in shear compared to Cl.

Ranganathan and Ostoja-Starzewski⁵⁹ have introduced the concept of the universal elastic anisotropic index (A^U) which is a widely used index for determining anisotropy due to its capacity to be applied to all potential crystal symmetries. The crystal is completely isotropic when the value of the universal anisotropy index (A^U) is exactly equal to zero (A^U = 0). The universal anisotropy factor (A^U) values of 0.52 (Hf₂ClC) and 0.48 (Hf₂BrC) further indicate that both compounds deviate from perfect isotropy. Hf₂BrC has a slightly lower A^U value, signifying that the introduction of Br reduces overall elastic anisotropy.

For an isotropic crystal, the equivalent Zener anisotropy factor is equal to one (A^eq = 1). The equivalent Zener anisotropy factor (A^eq) values of 1.91 (Hf₂ClC) and 1.86 (Hf₂BrC) support this trend, confirming moderate anisotropy. Br substitution weakens atomic bonding, which makes the material less stiff in one direction and a little more flexible in another. This makes Hf₂BrC more adaptable for applications requiring flexibility, while Hf₂ClC remains preferable for high-stress applications requiring superior mechanical stability.

The compressibility ratio provides insight into how a material deforms under stress along different crystallographic directions.⁶⁰ A higher ratio indicates greater compressibility along the c-axis compared to the a-axis, signifying directional anisotropy in response to external forces. For Hf₂ClC, k_c/k_a is 2.38, while for Hf₂BrC, it is 2.12 (see Table 4). This suggests that Hf₂ClC exhibits greater compressibility along the c-axis, making it more adaptable to deformation in this direction under applied stress. The substitution of Cl with Br in Hf₂AC (A = Cl, Br) leads to a decrease in the compressibility ratio (k_c/k_a) from 2.38 to 2.12, indicating an 11% reduction. This suggests that Hf₂BrC has lower anisotropy in compressibility compared to Hf₂ClC. Br is larger than Cl, so it doesn't weaken the bonding along the c-axis as much as Cl does. This makes the compressibility between the a- and c-axes more even. As a result, Hf₂BrC exhibits a more balanced mechanical response and improved structural stability under stress. On the other hand, Hf₂ClC, with its higher k_c/k_a ratio, is more compressible along the c-axis, making it more flexible and adaptable to deformation in this direction.

We have also calculated the universal log-Euclidean index (A^L) can be defined using the log-Euclidean formula.⁵⁸

where, the subscripts V and R represent the Voigt and Reuss approximations, respectively. For the universal log-Euclidean index calculation, the Voigt and Reuss values of C₄₄ are obtained from ref. 58 which are C^V₄₄ = [C^R₄₄ + (3/5)((C₁₁ − C₁₂ − 2C₄₄)²/(3(C₁₁ − C₁₂) + 4C₄₄))], and C^R₄₄ = [(5/3)((C₄₄(C₁₁ − C₁₂))/3(C₁₁ − C₁₂) + 4C₄₄)].

The universal log-Euclidean index (A^L) for Hf₂ClC and Hf₂BrC is 1.27 and 1.08, respectively, indicating a 15% reduction with Br substitution. Since most crystalline solids have A^L < 1.0, both compounds exhibit significant anisotropy, with Hf₂ClC being more anisotropic. The higher A^L and compressibility ratio (k_c/k_a) in Hf₂ClC (2.38) compared to Hf₂BrC (2.12) reflects its stronger layered character and pronounced anisotropic mechanical behavior. Br substitution weakens interlayer bonding due to its larger atomic radius, leading to a more uniform elastic response and reduced anisotropy in Hf₂BrC. This translates to enhanced structural flexibility and adaptability, making Hf₂BrC suitable for applications requiring moderate mechanical compliance. In contrast, Hf₂ClC, with its higher anisotropy and directional bonding, is suitable for high-stress environments where mechanical rigidity is crucial.

Thus, replacing Cl with Br reduces anisotropy, balancing structural properties and making Hf₂BrC a more isotropic material with improved mechanical versatility.

3.5 Thermo-mechanical properties

In the study of acoustics for crystalline solids, sound velocity is a critical parameter, especially in elastically anisotropic materials where sound velocities vary depending on the direction in which elastic waves propagate. These materials exhibit different velocities for longitudinal and transverse waves, which are key in understanding their mechanical, electrical, and thermal conductivities. The sound velocity is directly related to the material's elastic moduli and can be calculated using the longitudinal (v_l) and transverse (v_t) sound velocities. The average sound velocity, v_a, is derived from the following formula:⁶¹where the transverse sound velocity v_t is given by:And the longitudinal sound velocity v_l is:

Acoustic impedance (Z) of a medium is another important parameter that can be calculated from its density (ρ) and shear modulus (G):⁴⁹

Higher acoustic impedance values are observed in denser and stiffer materials. The unit of acoustic impedance is Rayl (1 Rayl = 1 kg m⁻² s⁻¹ = 1 Ns m³).

Finally, the radiation intensity (I) is also an important acoustical factor. It is proportional to the surface velocity of the material and scales with its density and modulus of rigidity. The radiation intensity can be expressed as:⁴⁹

The calculated values of sound velocities, acoustic impedance, and radiation factor for Hf₂AC (A = Cl, Br) are listed in Table 5 along with some other theoretical results. The substitution of Cl with Br in Hf₂AC (A = Cl, Br) significantly affects its acoustic properties due to the difference in atomic mass and bonding strength. The mass density increases from 10.17 g cm⁻³ in Hf₂ClC to 10.53 g cm⁻³ in Hf₂BrC, primarily due to the larger atomic weight of Br. This increase in density leads to a noticeable reduction in sound velocities. The transverse sound velocity decreases from 1921 m s⁻¹ in Hf₂ClC to 1729 m s⁻¹ in Hf₂BrC, reflecting a 10% reduction, while the longitudinal sound velocity drops from 3814 m s⁻¹ to 3654 m s⁻¹, indicating a 4.2% decrease. The average sound velocity also declines by approximately 9.7%, from 2154 m s⁻¹ to 1945 m s⁻¹, which suggests that the propagation of sound waves through the material becomes slower with Br substitution. Materials with lower acoustic radiation and slower phonon propagation tend to have lower thermal conductivity, making them suitable for thermal insulation or thermal barrier coatings. Additionally, because they minimize energy loss from lattice vibrations, they can enhance the efficiency and durability of devices operating at high temperatures or under thermal stress. Substituting Br for Cl increases density, slows sound propagation, and reduces acoustic radiation, making Hf₂BrC better at damping vibrations and insulating heat, which is advantageous for high-temperature and thermal barrier applications.

Table 5 The calculated mass density (ρ in gm cm⁻³), sound velocities (v_t, v_l and v_a in ms⁻¹), acoustic impedance (Z in Rayl), radiation factor (I in m⁴ kg⁻¹ s⁻¹) for Hf₂AC (A = Cl, Br)

Compounds	ρ	vt	vl	va	Z (10^6)	I	Ref.
Hf2ClC	10.17	1921	3814	2154	19.53	0.19	This work
Hf2BrC	10.53	1729	3654	1945	18.21	0.16
Hf2AlC	9.65	3560	5730	3920	—	—	Ref. 23
Hf2SC	11.02	3840	5300	3740	—	—	Ref. 34
Hf2TlC	13.04	2589	4345	2866	—	—	Ref. 33

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Furthermore, the acoustic impedance, which measures a material's ability to transmit sound, decreases from 19.53 × 10⁶ Rayl in Hf₂ClC to 18.21 × 10⁶ Rayl in Hf₂BrC, showing a 6.8% reduction. This suggests that Hf₂BrC exhibits weaker impedance matching, making it less effective in transmitting sound. Additionally, the radiation factor, which indicates a material's ability to radiate sound energy efficiently, decreases by 15.8%, from 0.19 to 0.16 m⁴ kg⁻¹ s⁻¹. The reduction in both acoustic impedance and radiation efficiency implies that Hf₂BrC has weaker bonding interactions and a lower capacity to propagate and radiate sound compared to Hf₂ClC. Overall, Cl-to-Br substitution results in a material with reduced sound transmission and increased structural flexibility, making Hf₂BrC more suitable for applications requiring lower acoustic impedance, while Hf₂ClC remains preferable for high-speed acoustic applications.

In conclusion, the incorporation of halogens like Cl and Br in Hf₂AC compounds leads to a reduction in sound velocities and mechanical stiffness compared to other hafnium-based materials, as shown in Table 5. This suggests that these halogen-substituted compounds may exhibit more flexibility or adaptability under certain conditions.

3.6 Thermo-lattice properties

The Debye temperature (Θ_D) is a crucial thermophysical parameter that provides significant insights into the thermal and mechanical behavior of materials. It is related to the material's lattice vibrations, thermal conductivity, interatomic bonding, and other properties such as melting temperature and specific heat capacity. The Debye temperature effectively serves as a cutoff for the phonon energy distribution and helps distinguish between the classical and quantum mechanical behavior of phonons. The values of Θ_D are obtained from the following equation:⁶²where h is Planck's constant, n is the number of atoms in the unit cell, k_B is the Boltzmann constant, and V₀ is the equilibrium volume. The Debye temperature (Θ_D) for Hf₂ClC is 250 K, while for Hf₂BrC, it is 221 K are listed in Table 6. This shows a reduction of approximately 11.6% in the Debye temperature when Cl is replaced with Br. The decrease in the Debye temperature suggests that the interatomic bonding in Hf₂BrC is weaker than in Hf₂ClC. This results in less efficient lattice vibrations and lower mechanical wave velocities in Hf₂BrC compared to Hf₂ClC. The reduced Debye temperature in Hf₂BrC indicates that the material has a lower thermal conductivity due to weaker atomic interactions. This behavior can be attributed to the larger atomic radius of Br compared to Cl, which introduces more spacing between the atoms, weakening the bonding and lowering the lattice thermal conductivity. These weaker atomic interactions in Hf₂BrC result in a lower Debye temperature and reduced thermal conductivity. In contrast, Hf₂ClC, with its stronger atomic interactions, is expected to exhibit a higher lattice thermal conductivity. The stronger bonds in Hf₂ClC allow for more efficient heat transfer through the crystal lattice. Therefore, the substitution of Cl with Br reduces the atomic bonding strength, leading to lower thermal conductivity in Hf₂BrC compared to Hf₂ClC.

Table 6 The calculated values of Debye temperature (Θ_D in K), melting temperature (T_m in K), thermal expansion coefficient (α in K⁻¹), heat capacity (ρC_P in J m⁻³ K⁻¹), Grüneisen parameter (γ), dominant phonon wavelength (λ_dom in m), minimum thermal conductivity (k_min in W m⁻¹ K⁻¹) and lattice thermal conductivity (k_ph in W m⁻¹ K⁻¹ at 300 K) for Hf₂AC (A = Cl, Br)

Compound	ΘD	Tm	γ	kph	kmin	α (×10^−5)	ρCP (×10^6)	λdom (×10^−12)
Hf2ClC	250	1059	1.98	2.61	0.722	4.27	2.51	89.51
Hf2BrC	221	1044	2.18	1.65	0.630	5.09	2.33	81.10

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The melting temperature⁶³ (T_m = 354 + 1.5 (2C₁₁ + C₃₃)), of a material is a key thermophysical property that reflects the strength of atomic interactions and the material's thermal stability. For Hf₂ClC, the calculated melting temperature is 1059 K, while for Hf₂BrC, it is 1044 K. This difference indicates a 1.4% decrease in the melting temperature when Cl is substituted with Br. The lower melting temperature in Hf₂BrC can be attributed to the relatively weaker atomic interactions compared to Hf₂ClC. As previously discussed, Br atoms are larger than Cl atoms, which increases the interatomic spacing and weakens the bonds between atoms. The correlation between melting temperature and Young's modulus (E) also supports this observation. A higher Young's modulus typically indicates stronger atomic bonding and greater material stiffness.

The Grüneisen parameter is an essential measure of anharmonic effects in crystalline solids, influencing thermal expansion, phonon interactions, and thermomechanical properties.⁶⁴ Grüneisen parameters for different polycrystalline materials are within the expected range [0.85–3.53] with the Poisson's ratio in the range of 0.05–0.46 which are good agreement with our calculated values.⁶⁴ The calculated values for Hf₂ClC and Hf₂BrC are 1.98 and 2.18, respectively, indicating that Hf₂BrC exhibits a higher degree of anharmonicity than Hf₂ClC. The 10.1% increase in the Grüneisen parameter when Cl is replaced with Br suggests that the lattice vibrations in Hf₂BrC are more sensitive to temperature changes and exhibit stronger anharmonic interactions. Additionally, the reduction in Debye temperature (by 11.6%) and melting temperature (by 1.4%) in Hf₂BrC further supports the trend of increased anharmonicity, as lower bond strength and structural rigidity contribute to greater phonon–phonon interactions.

Lattice thermal conductivity (k_ph) is one of the most fundamental and crucial properties of a material, especially for high-temperature applications, and describes how efficiently heat is conducted through phonon propagation inside a crystal. For Hf₂ClC and Hf₂BrC compounds, lattice thermal conductivity (κ_ph) is estimated at 300 K using Slack's formula:⁶⁵

Here, M_av is the average atomic mass (in kg mol⁻¹), δ is the cubic root of the average atomic volume (in units of m³), n is the number of atoms per unit cell, T is the absolute temperature, γ is the Grüneisen parameter, and A is a factor (in W-mol kg⁻¹ m⁻² K⁻³) dependent on γ. The coefficient A(γ) in Slack's model is determined by Julian's equation:⁶⁶

The calculated values of lattice thermal conductivities at room temperature under study are listed in Table 6. The k_ph value of Hf₂ClC and Hf₂BrC at 300 K is 2.61 W mK⁻¹ and 1.65 W mK⁻¹, respectively, indicating a significant reduction of approximately 36.8% when Cl is substituted with Br. This reduction in thermal conductivity aligns with the observed decrease in Debye temperature, as lower Debye temperatures correlate with weaker interatomic bonding and reduced phonon propagation efficiency. The larger Br atomic radius increases spacing, enhancing phonon scattering and reducing lattice thermal conductivity. As a result, Hf₂BrC exhibits greater thermal insulation properties compared to Hf₂ClC, making it a more suitable candidate for applications requiring thermal barrier coatings and thermoelectric materials. Conversely, Hf₂ClC, with its higher thermal conductivity, is better suited for heat dissipation applications in microelectronic and nanoelectronic devices.

The lattice thermal conductivity of a compound approaches a minimal value at high temperatures above the Debye temperature, which is known as minimum thermal conductivity (k_min), which is independent of further increasing temperature. The minimum thermal conductivity estimated by using the following equation:⁶⁷

where M is molar mass, and N_A is Avogadro's number.

The minimum thermal conductivity (k_min) of Hf₂ClC is 0.722 W mK⁻¹, while for Hf₂BrC, it is 0.630 W mK⁻¹, indicating a decrease of approximately 12.7% when Cl is replaced by Br. Since k_min is closely related to sound velocity and Debye temperature, the lower values in Hf₂BrC suggest that it has weaker atomic interactions and a less rigid lattice compared to Hf₂ClC. The minimum thermal conductivity of the studied MAX phase compounds, Hf₂ClC and Hf₂BrC, is significantly lower than the reference value of 1.25 W mK⁻¹, making them suitable candidates for thermal barrier coatings (TBC) applications. The lower k_min values indicate that these materials exhibit limited heat conduction at elevated temperatures, an essential property for TBCs used in aerospace and energy applications.

The substitution of Cl with Br in Hf₂AC (A = Cl, Br) significantly affects the thermal expansion coefficient, volumetric heat capacity, and dominant phonon wavelength, leading to variations in thermal performance. The calculated values of these parameters are listed in Table 6. The thermal expansion coefficient⁶⁰ of Hf₂BrC is higher than that of Hf₂ClC, indicating that Hf₂BrC undergoes greater expansion with increasing temperature. This increase is primarily attributed to the larger atomic radius of Br. Since the thermal expansion coefficient is inversely related to the melting temperature, the lower melting point of Hf₂BrC further supports this observation. In practical applications, a lower thermal expansion coefficient is desirable for thermal barrier coatings to prevent thermal stress and structural instability. Therefore, Hf₂ClC, with its lower expansion, is a more suitable candidate for coatings in high-temperature environments.

The volumetric heat capacity of Hf₂ClC is higher than that of Hf₂BrC, suggesting that it has a greater ability to store thermal energy and maintain thermal stability.⁴⁹ A higher heat capacity results in lower thermal diffusivity and contributes to better thermal conductivity. Since heat capacity plays a crucial role in determining how well a material responds to temperature fluctuations, the higher value in Hf₂ClC aligns with its greater lattice thermal conductivity. This characteristic makes Hf₂ClC more effective in heat dissipation applications, such as microelectronics and aerospace coatings, where efficient thermal management is essential.

The dominant phonon wavelength of Hf₂ClC is longer than that of Hf₂BrC, which indicates a higher average sound velocity and lower phonon scattering in the former.⁴⁹ A longer phonon wavelength facilitates heat transfer by reducing phonon–phonon interactions, leading to improved thermal conductivity. Conversely, the shorter phonon wavelength in Hf₂BrC implies increased phonon scattering, which hinders heat conduction and reduces thermal conductivity. This characteristic makes Hf₂BrC a potential candidate for thermal insulation and thermoelectric applications, where lower thermal conductivity is advantageous for energy conversion efficiency and thermal management.

Overall, the presence of Cl enhances thermal conductivity, stability, and lower expansion, making Hf₂ClC preferable for applications requiring efficient heat dissipation and structural reliability. On the other hand, Br substitution increases phonon scattering and thermal expansion while reducing heat capacity, making Hf₂BrC more suitable for thermal insulation and thermoelectric applications.

3.7 Electronic properties

This section provides a detailed comparison of the electronic band structures of Hf₂ClC and Hf₂BrC, highlighting the effect of substituting Cl with Br. The calculated band structures of Hf₂ClC and Hf₂BrC are shown in Fig. 4(a and b). The band structure calculations, performed using the GGA-PBE approximation, span an energy range from −4.0 to 4.0 eV, with the Fermi level set at 0 eV. The computed band structures along high-symmetry directions in the Brillouin zone indicate that both materials exhibit metallic behavior, as evidenced by multiple bands crossing the Fermi level. The dispersive nature of the band structure along different crystallographic directions provides crucial insights into the electronic transport properties of Hf₂ClC and Hf₂BrC. The dispersion of energy bands along the c-direction (Γ–A, H–K, and M–L) and the ab-plane (A–H, K–Γ, Γ–M, and L–H) differs significantly due to variations in bonding strength and orbital hybridization. Along the c-direction (Γ–A, H–K, and M–L), the band dispersion is relatively weaker in both compounds, indicating higher electronic localization and lower mobility along this axis. The Mulliken population analysis reveals distinct bonding characteristics in Hf₂AC (A = Cl, Br) MAX phases (Table 7). Hf atoms exhibit small positive charges (+0.56e for Hf₂ClC and +0.21e for Hf₂BrC), while C atoms carry strongly negative charges (−0.85e), confirming substantial charge transfer from Hf to C and the presence of strong polar covalent Hf–C bonds within the layers. In contrast, the halogen atoms display different behaviors: Cl is slightly negative (−0.26e), indicating an ionic Hf–Cl interaction, whereas Br is slightly positive (+0.44e), reflecting a reversal in charge transfer. This reversal arises from Br's lower electronegativity and larger size, which weaken the ionic interaction and introduce a more covalent, polarizable Hf–Br bond. Consequently, interlayer bonding in Hf₂BrC is weaker, leading to enhanced ductility, reduced stiffness, and stronger anisotropic electronic behavior compared to Hf₂ClC. However, the effect of Cl and Br substitution is evident: in Hf₂ClC, the bands in these directions are slightly more dispersive than in Hf₂BrC, suggesting stronger interlayer coupling and better charge transport along the c-axis. Br has a larger atomic radius, which increases the space between layers. This reduces orbital overlap and makes the bands flatter, which lowers the electrical conductivity along the c-direction.

Fig. 4 Calculated band structure of (a) Hf₂ClC and (b) Hf₂BrC along the high symmetry directions in the Brillouin zone at ambient conditions.

Table 7 The calculated orbital charges (electron), atomic Mulliken charges (electron), effective valence (electron), and Hirshfeld charge (electron) of the Hf₂AC (A = Cl, Br) MAX compounds

Compounds	Species	Mulliken atomic populations				Mulliken charge	Formal ionic charge	Effective valence	Hirshfeld charge	Effective valence
		s	p	d	Total
Hf2ClC	C	1.53	3.33	0.0	4.85	−0.85	−4	3.15	−0.36	3.64
	Cl	1.93	5.34	0.0	7.26	−0.26	−1	0.74	0.03	0.97
	Hf	0.36	0.17	2.91	3.44	0.56	+4	3.44	0.17	3.83
Hf2BrC	C	1.52	3.33	0.0	4.85	−0.85	−4	3.15	−0.36	3.64
	Br	1.31	5.25	0.0	6.56	0.44	−1	0.56	0.06	0.94
	Hf	0.43	0.47	2.90	3.79	0.21	+4	3.79	0.15	3.85

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On the other hand, the bands along the ab-plane (A–H, K–Γ, Γ–M, and L–H) show stronger dispersion in both compounds. This means that charge transport in the plane is more efficient. This is due to the presence of stronger covalent interactions within the layers, particularly between Hf and C atoms. However, a noticeable difference exists between Hf₂ClC and Hf₂BrC: the bands in Hf₂ClC are more dispersive, suggesting enhanced electronic delocalization and lower effective mass of charge carriers in the ab-plane. This higher dispersion contributes to better electronic conductivity, making Hf₂ClC more suitable for electronic applications requiring high carrier mobility. In Hf₂BrC, the bands in the ab-plane are slightly flatter, implying increased carrier scattering and reduced conductivity due to weaker orbital interactions. Overall, the comparison of band dispersion along the c-direction and ab-plane reveals that Hf₂ClC has greater electronic transport properties, particularly within the layers, due to stronger hybridization and better orbital overlap. On the other hand, Hf₂BrC exhibits more localized electronic states, leading to higher effective mass and reduced carrier mobility, especially along the c-direction. These findings highlight the anisotropic electronic nature of these materials and their potential applications, with Hf₂ClC being more suitable for conductive applications and Hf₂BrC potentially offering advantages in thermal barrier coatings or electronic devices where controlled conductivity is required. The anisotropic transport observed in Hf₂AC (A = Cl, Br) aligns well with trends seen in other MAX phases.^5,7,68,69

The density of states (DOS) plots in Fig. 5(a and b) provide deeper insight into the electronic structure of Hf₂ClC and Hf₂BrC, complementing the band structure analysis from Fig. 4. The variations in band dispersion along the c-direction (Γ–A, H–K, and M–L) and ab-plane (A–H, K–Γ, Γ–M, and L–H) can be directly linked to the differences in the projected DOS (PDOS) contributions of Hf, Cl, Br, and C atoms.

Fig. 5 Representation of the total and partial density of states for (a) Hf₂ClC and (b) Hf₂BrC.

From Fig. 5(a and b), it is evident that the Hf-5d orbitals play a dominant role in the electronic states near the Fermi level (E_F). These orbitals are mainly responsible for the dispersive nature of the bands and charge transport characteristics. In Hf₂ClC, the Hf-5d states are more delocalized, contributing to the stronger dispersion observed in the band structure, particularly along the ab-plane, where covalent interactions between Hf and C are more prominent. This increased dispersion corresponds to lower effective mass and higher electronic mobility.

In contrast, in Hf₂BrC, the DOS shows that the Br-4p states contribute differently compared to Cl-3p states in Hf₂ClC. The Br-4p orbitals are more localized, leading to reduced hybridization with Hf-5d states. This weaker interaction results in flatter bands near the Fermi level, especially along the c-direction, where interlayer interactions are already weak. As a result, electronic conductivity is lower in Hf₂BrC compared to Hf₂ClC. The localization of Br states also has an effect on the dispersion along the ab-plane, but not as much as it does along the c-direction. Furthermore, the contribution of C-2p orbitals remains relatively similar in both compounds, indicating that variations in electronic dispersion are primarily influenced by the substitution of Cl with Br. The key factor in the observed differences in band dispersion is the reduced hybridization between Hf-5d and Br-4p states. Therefore, the DOS analysis confirms that the Hf-5d and halogen (Cl-3p or Br-4p) orbitals govern the band structure variations.

The electronic properties of Hf₂AC (A = Cl, Br) MAX phases are strongly influenced by their total density of states (TDOS), which plays a crucial role in assessing electrical stability. The calculated N(E_F) values for Hf₂ClC and Hf₂BrC are 4.61 and 5.19 states per eV per unit cell, respectively. Lower N(E_F) values indicate greater electrical stability, suggesting that Hf₂ClC is slightly more stable than Hf₂BrC. These values align with the stability range (2–12 states per eV per unit cell) found in other MAX phases, confirming the robust electrical stability of both materials.⁷⁰ A key feature in both compounds is the presence of a pseudo-gap near E_F, which separates bonding and anti-bonding states. The pseudo-gap functions as a barrier, restricting electron transitions and enhancing stability. The Fermi level lies within the bonding region, further supporting the stability of these structures. The substitution of Cl with Br slightly reduces the pseudo-gap, which correlates with a higher N(E_F), leading to a minor decrease in stability for Hf₂BrC compared to Hf₂ClC. This modification occurs due to Br's larger atomic radius and lower electronegativity, which weaken the hybridization between Hf and A-site atoms, making electronic states more localized and reducing charge transport efficiency.

3.8 Charge density distribution map

The electronic charge density distribution maps for Hf₂ClC and Hf₂BrC reveal crucial insights into their chemical bonding and charge transfer characteristics. The color gradient in Fig. 6 illustrates the variation in electronic charge, with blue representing high charge density and red indicating low charge density. The substitution of Cl with Br influences the bonding nature within these MAX phases, affecting charge accumulation and hybridization.

Fig. 6 Electronic charge density distribution map of Hf₂ClC (left) and Hf₂BrC (right), respectively.

Significant charge accumulation between Hf and C atoms in both compounds demonstrates strong covalent bonding between these atoms. This strong covalent character arises from the hybridization of Hf-5d and C-2p orbitals, which is consistent with the band structure and density of states (DOS) analysis. The substitution of Cl with Br does not significantly alter this Hf–C bonding, as it remains the dominant interaction in both materials. The Hf–A (A = Cl, Br) bonds also exhibit covalent character but are relatively weaker than Hf–C bonds. The electronic charge density around Cl and Br atoms is more localized, indicating weaker hybridization with Hf compared to Hf–C. The larger atomic radius and lower electronegativity of Br compared to Cl reduce the extent of orbital overlap with Hf, weakening the Hf–Br bond relative to Hf–Cl. This reduced bonding strength is also reflected in the flatter bands in the electronic band structure of Hf₂BrC, which suggests lower charge carrier mobility. A noticeable feature in the charge density maps is the ionic nature of Cl–C and Br–C bonds. In both compounds, there is a clear separation of charge between A-site (Cl/Br) and C atoms, confirming the ionic bonding character. However, the charge distribution around Br appears more diffuse compared to Cl, indicating a more polarized bond due to its larger size and lower electronegativity. This increased ionic nature further reduces the overall hybridization in Hf₂BrC, leading to a less dispersive electronic structure compared to Hf₂ClC.

3.9 Bond population analysis

The Mulliken atomic population analysis provides valuable insights into the nature of chemical bonding and charge transfer in Hf₂AC (A = Cl, Br) MAX compounds. The deviation of Mulliken charges from their formal ionic states indicates significant charge redistribution among the atoms, suggesting the coexistence of covalent and ionic bonding. The substitution of Cl with Br affects the charge transfer and bonding characteristics due to differences in electronegativity and atomic size between the two halogen elements.

In Hf₂ClC, the Mulliken charge on Cl is −0.26e, while in Hf₂BrC, Br exhibits a positive charge of 0.44e. This indicates that Br has a lower tendency to attract electrons compared to Cl, leading to weaker charge transfer in Hf₂BrC. The charge distribution also affects the bonding nature; Hf–Cl bonding exhibits stronger hybridization than Hf–Br, as reflected in the calculated effective valence charge (EVC). The EVC values for all species in both compounds are greater than zero, confirming the partial covalency of the bonds.⁷¹ However, the effective valence of Hf in Hf₂BrC is slightly higher (3.79e) than in Hf₂ClC (3.44e), implying that the bonding in Hf₂BrC is less ionic and more covalent in nature. This phenomenon is primarily due to the larger atomic radius of Br, which reduces electrostatic attraction and weakens the bond strength between Hf and Br compared to Hf and Cl.

To further validate the bonding characteristics, the Hirshfeld population analysis (HPA) was performed, revealing variations from the Mulliken charge analysis.⁷² Despite the differences, both methods consistently indicate that the Hf–C bonds remain predominantly covalent, while the Hf–Cl and Hf–Br bonds exhibit partial covalency with some ionic character. The Cl–C and Br–C bonds, however, are primarily ionic in both cases. The substitution of Cl with Br also affects electronic properties, as weaker bonding in Hf₂BrC leads to lower charge transfer efficiency and increased carrier localization. This results in reduced electronic conductivity and higher anisotropy in transport properties. However, the relatively weaker Hf–Br bonding may lead to a slight reduction in structural robustness compared to Hf₂ClC. Overall, the replacement of Cl with Br alters the bonding nature by decreasing the degree of charge transfer and reducing hybridization effects. This leads to a shift in electronic and transport properties, making Hf₂BrC slightly less ionic and more covalent but with weaker bonding interactions compared to Hf₂ClC.

3.10 Optical properties

The optical properties of materials are important because they determine their electronic nature and potential applications in optoelectronic devices, solar cells, lasers, photodetectors, sensors, and reconfigurable photonics. The optical behavior of Hf₂ClC and Hf₂BrC MAX phases is analyzed through the frequency-dependent complex dielectric function, ε(ω) = ε₁(ω) + iε₂(ω), which directly correlates with the electronic structure across all photon energy levels.⁷³ The imaginary part ε₂(ω) can be calculated from the momentum matrix elements connecting to the occupied and unoccupied electronic states with the following relations:⁷⁴where, the integral is over the first Brillouin zone, ω refers the frequency of light, e indicates the electronic charge, u is the vector indicating the polarization of the incident electric field, ε₀ is the dielectric constant in free space and Ψ_k^c and Ψ_k^v are the conduction and valence band wave functions respectively. The real part ε₁(ω) was obtained from the imaginary part ε₂(ω) by using the Kramer–Kronig (K–K) relations:⁷³where P implies the principal value of the integral. In order to describe the material response to incident electromagnetic radiation, for studied MAX phase compounds Hf₂AC (A = Cl, Br) all the other energy-dependent optical parameters, like absorption coefficient α(ω), photoconductivity σ(ω), loss function L(ω), reflectivity R(ω) and refractive index, n(ω) were computed directly from ε₁(ω) and ε₂(ω) and displayed in Fig. 7, respectively. These properties were examined across a photon energy range of up to 30 eV. As in the case of many MAX phases, Hf₂AC (A = Cl, Br) exhibits metallic behavior, which significantly influences their optical properties. The conduction electrons in these materials affect the dielectric function by changing how they respond to light. Specifically, the intra-band contributions from free electrons play a dominant role, affecting the low-energy optical spectrum. In metallic or metal-like conductors, both intra-band and inter-band transitions shape the optical behavior, requiring a precise approach for accurate characterization. To achieve a reliable analysis of the optical properties of Hf₂ClC and Hf₂BrC, the conventional Drude model was employed.⁷⁵ This model effectively accounts for free-electron contributions to the dielectric function. For our calculations, we incorporated a Drude damping parameter of 0.05 eV and a plasma frequency of 3 eV for both compounds. These values are consistent with earlier reports for metallic MAX phases and related carbides/nitrides,^76–80 where such parameters accurately describe the free-electron contributions and Drude-like behavior at low photon energies. The Drude term primarily captures the intra-band transitions, which dominate at low photon energies. These transitions arise due to the free-electron nature of the material and contribute significantly to optical conductivity, reflectivity, and absorption spectra.

Fig. 7 The energy dependent (a) real part of dielectric function, (b) imaginary part of dielectric function, (c) refractive index, (d) extinction coefficient (e) absorption coefficient, (f) photoconductivity, (g) reflectivity, and (h) loss function of Hf₂ClC and Hf₂BrC, respectively.

3.10.1 Dielectric function. The optical behavior of Hf₂ClC and Hf₂BrC provides a compelling insight into their metallic nature, as depicted by the real ε₁(ω) and imaginary ε₂(ω) components of their dielectric function, which are depicted in Fig. 7(a and b). The presence of Cl and Br in these MAX phases leads to subtle yet significant differences in their optical responses. The real part ε₁(ω) exhibits a strong metallic character, showing negative values at low energy, which is consistent with the Drude-like behavior typically observed in conductive materials. This trend aligns well with the electronic band structure, reaffirming the metallic conductivity of these compounds. As shown in Fig. 7(a), ε₁(ω) attains large negative values at low photon energies and crosses zero at approximately 21.79 eV for Hf₂ClC and 20.77 eV for Hf₂BrC. This zero-crossing energy corresponds to the plasma frequency, where the absorption coefficient (Fig. 7(e)) vanishes, reflectivity (Fig. 7(g)) drops sharply, and the energy loss function (Fig. 7(h)) exhibits its first pronounced peak. Moreover, as depicted in Fig. 7(a) and (b), ε₁(ω) crosses zero from below and approaches zero from above, thereby reinforcing the metallic and highly reflective characteristics.

The imaginary part of the dielectric function, ε₂(ω), which represents optical absorption due to electron excitations, is shown in Fig. 7(b). The high ε₂(ω) values at low photon energies indicate strong absorption for both Hf₂ClC and Hf₂BrC. As seen in Fig. 7(b), ε₂(ω) initially drops to ε₂ = 15.59 at 0.57 eV for Hf₂ClC and to ε₂ = 12.69 at 0.56 eV for Hf₂BrC, before rising to their respective maxima of 16.84 at 0.97 eV and 15.65 at 1.13 eV. Beyond these peaks, ε₂(ω) gradually decreases almost exponentially across 1.38–6.49 eV and 8.24–15.77 eV for Hf₂ClC, and 3.35–7.93 eV for Hf₂BrC, reflecting their optical stability against excessive energy losses. The low-energy peaks originate from intra-band transitions of conduction electrons, while the high-energy peaks are attributed to inter-band transitions between occupied and unoccupied states, consistent with metallic characteristics. The high static dielectric constant of ε₂(ω) at low photon energies also signifies strong light–matter interactions, highlighting the potential of Hf₂ClC and Hf₂BrC for high-k dielectric, capacitor, and optical modulator applications.

3.10.2 Refractive index. The refractive index, n(ω), is a key optical parameter that governs how light propagates through a material, making it fundamental for photonic and optoelectronic applications. The variation of n(ω) with photon energy for Hf₂ClC and Hf₂BrC, shown in Fig. 7(c), exhibits a gradual decrease with increasing energy. Both compounds display remarkably high static refractive index values of n(0) = 84.33 for Hf₂ClC and 84.31 for Hf₂BrC, indicating strong light–matter interactions and high polarizability, which are advantageous for waveguides, photonic crystals, and energy-harvesting devices. The refractive index decreases sharply with photon energy, showing prominent peaks near 1.10 eV (n = 3.58) for both phases. Within the visible range (1.67–3.10 eV), n(ω) declines from 3.06 to 2.08 for Hf₂ClC and from 3.18 to 2.26 for Hf₂BrC. The slightly higher refractive index in Hf₂BrC can be attributed to Br's larger atomic radius and lower electronegativity, which modify the electronic structure and enhance polarization effects. This tunable refractive response, combined with metallic conductivity, demonstrates the advantages of these MAX phases for advanced optoelectronic and photonic devices requiring efficient light modulation and energy management.

3.10.3 Extinction coefficient. The extinction coefficient, k(ω), a key optical parameter that quantifies the attenuation of electromagnetic radiation as it propagates through a material, reflects intrinsic absorption losses. This property provides valuable insight into light–matter interactions and is particularly significant for photonics, optical coatings, and optoelectronic device applications. The calculated k(ω) spectra for Hf₂ClC and Hf₂BrC, presented in Fig. 7(d), exhibit distinct frequency-dependent trends. As shown in Fig. 7(d), both compounds display high static values, k(0) ≈ 102.64 for Hf₂ClC and 102.66 for Hf₂BrC, with strong absorption features in the infrared region. With increasing photon energy, k(ω) gradually decreases; however, Hf₂BrC reaches a slightly higher peak (k = 2.11 at 8.04 eV) compared to Hf₂ClC (k = 1.88 at 8.55 eV) in the visible range, indicating enhanced photon–electron coupling due to Br substitution. The pronounced extinction values signify efficient light absorption and strong photon-induced charge excitation, desirable for optical and photoactive devices. Additionally, k(ω) exhibits minor anisotropy within the 3.52–20.13 eV range, consistent with the layered structure of MAX phases. The observed local maxima in the k(ω) spectra coincide with the zero-crossing points of the real part of the dielectric function, confirming the correlation between energy loss and plasma resonance behavior.

3.10.4 Absorption coefficient. The absorption coefficient, α(ω), provides vital information about a material's ability to absorb incident photon energy, directly influencing its optoelectronic performance. The absorption spectra of Hf₂ClC and Hf₂BrC, illustrated in Fig. 7(e), reveal strong photon–matter interactions extending across the infrared (IR), visible, and ultraviolet (UV) regions. Due to their metallic character, absorption begins at 0 eV and exhibits primary peaks at 0.18 eV (α = 0.46 × 10⁵ cm⁻¹), followed by secondary peaks in the visible range near 2.69 eV, where α reaches approximately 0.84 × 10⁵ cm⁻¹ for both phases. Maximum absorption occurs in the UV region at 12.15 eV (α = 2.84 × 10⁵ cm⁻¹) for Hf₂ClC and 14.79 eV (α = 2.91 × 10⁵ cm⁻¹) for Hf₂BrC. Within the visible spectrum (1.67–3.10 eV), α(ω) increases steadily from 0.61 × 10⁵ cm⁻¹ to 0.83 × 10⁵ cm⁻¹ for Hf₂ClC and from 0.59 × 10⁵ cm⁻¹ to 0.81 × 10⁵ cm⁻¹ for Hf₂BrC, indicating excellent light-harvesting efficiency. This consistent absorption behavior highlights the strong electronic transition activity within the conduction band, driven primarily by Hf-d orbital contributions near the Fermi level. The broad and intense absorption response across a wide energy range underscores the potential of these MAX phases for high-performance photodetectors, UV-visible absorbers, and other optoelectronic and photonic devices.

3.10.5 Photoconductivity. The photoconductivity spectra of Hf₂ClC and Hf₂BrC, illustrated in Fig. 7(f), commence at zero photon energy, confirming the absence of a band gap and thereby validating their metallic nature. This behavior arises from the presence of free charge carriers that contribute to electrical conductivity even at low photon energies. The observed trends in photoconductivity are consistent with the absorption coefficient, electronic band structure, and density of states (DOS) analyses, all of which indicate substantial carrier mobility and metallic behavior. The frequency-dependent optical conductivity, σ(ω), of Hf₂AC (A = Cl, Br) further highlights strong photon–electron interactions essential for charge transport processes. The static optical conductivity of these compounds is about 20.93 S m⁻¹, which means that they can move charges around easily and excite electrons quickly when light hits them. Within the visible region, σ(ω) varies between 2.77–2.59 S m⁻¹ for Hf₂ClC and 2.82–2.75 S m⁻¹ for Hf₂BrC, with the latter exhibiting slightly enhanced conductivity. The maximum photoconductivity is observed at 8.0 eV for Hf₂ClC (σ = 6.31 S m⁻¹) and at 7.67 eV for Hf₂BrC (σ = 7.18 S m⁻¹), after which σ(ω) decreases gradually with energy. The combination of strong low-energy conductivity and photon-induced carrier excitation underscores the potential of these MAX phases for optoelectronic and photoactive applications, including photodetectors, solar energy converters, and optical sensors, where efficient light-to-charge conversion is vital.

3.10.6 Reflectivity. The reflectivity spectra of Hf₂ClC and Hf₂BrC, presented in Fig. 7(g), exhibit similar overall trends, characteristic of metallic materials. Both compounds have high static reflectivity values of R(0) ≈ 0.98, which shows that they are very good at reflecting low-energy electromagnetic radiation. As photon energy increases, the reflectivity decreases, reaching R = 0.46 at 0.82 eV for Hf₂ClC and R = 0.43 at 0.75 eV for Hf₂BrC. The spectra of both compounds follow a comparable decreasing trend up to approximately 3.44 eV, beyond which a slight anisotropy is observed, reflecting directional dependence in their optical response due to their layered crystal structure. Within the visible photon energy range (1.67–3.10 eV), Hf₂ClC and Hf₂BrC maintain moderate reflectivity levels between 43% and 32%, indicating controlled light reflection suitable for solar radiation management and optical coating applications. The high reflectivity in the low-energy region coupled with tunable optical anisotropy at higher energies suggests that these MAX phases could serve as efficient reflective coatings and thermal control materials for optoelectronic and high-temperature devices.

3.10.7 Loss function. The energy loss function, L(ω), represents the energy dissipated by fast-moving electrons traversing a material and provides critical insight into its plasmonic and dielectric behavior. As illustrated in Fig. 7(h), the L(ω) spectra for Hf₂ClC and Hf₂BrC exhibit no distinct peaks in the low-energy range (0–15 eV), which can be attributed to the relatively large values of the imaginary component of the dielectric function, ε₂(ω), that dampen collective oscillations. A prominent peak emerges between 15 and 28 eV for both compounds, corresponding to bulk plasma oscillations—collective excitations of conduction electrons within the material. The maximum peak in each spectrum indicates the bulk plasma frequency, where the real part of the dielectric function crosses zero, marking the transition from metallic to dielectric behavior. The calculated plasma frequencies are 22.21 eV for Hf₂ClC and 22.35 eV for Hf₂BrC, indicating a slightly stronger plasmonic response in the Br-substituted phase. These energy-loss peaks coincide with sharp decreases in reflectivity, signifying the point at which the material transitions from a reflective to a transparent regime. Beyond the plasma frequency, both compounds exhibit minimal energy loss and increased transparency, suggesting their potential for high-frequency optical and plasmonic applications where low optical damping and efficient charge oscillations are essential.

4. Conclusions

This first-principles study reveals that A-site halogen substitution is an effective approach for tailoring the multifunctional behavior of Hf₂AC (A = Cl, Br) MAX phases. Both Hf₂ClC and Hf₂BrC phases are confirmed to be structurally, mechanically, and dynamically stable. The substitution of Cl with Br reduces stiffness, hardness, and sound velocity, enhancing ductility, vibration damping, and thermal insulation performance, suitable for thermal barrier applications. In contrast, Hf₂ClC exhibits greater rigidity, higher conductivity, and superior thermal dissipation, making it better suited for structural and heat-management uses. Electronically, both compounds display metallic conductivity dominated by Hf–C bonding, while optically, Br substitution shifts the absorption toward lower energies, improving infrared and UV responsiveness. Overall, halogen engineering at the A-site emerges as a powerful strategy for tuning mechanical, thermal, and optical functionalities in Hf-based MAX phases for diverse high-temperature and optoelectronic applications.

Author contributions

Asif Mohammed Arfi: investigation, methodology, data curation, conceptualization, writing original manuscript; Mahamudun Nabi: writing – original draft, methodology, formal analysis, data calculations, validation. Md. Hasan Mia: writing – original draft, review & editing, conceptualization, formal analysis, validation, supervision. Omar Alsalmi: data curation, formal analysis, review – editing; Muhammad Athar Uddin data curation, formal analysis, review – editing; Md. Zahid Hasan: formal analysis, validation, supervision, review – editing.

Conflicts of interest

The authors declare that they have no known conflicting financial interests or personal ties that may have seemed to affect the work presented in this study.

Data availability

Relevant data from this study are available from the corresponding author upon a reasonable request.

Acknowledgements

The authors extend their appreciation to Umm Al-Qura University, Saudi Arabia for funding this research work through grant number: 25UQU4300099GSSR10. Funding statement: This research work was funded by Umm Al-Qura University, Saudi Arabia under grant number: 25UQU4300099GSSR10.

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