Newly synthesized MAX phase Zr₂SeC: DFT insights into physical properties towards possible applications

Abstract

A DFT study of the synthesized MAX phase Zr₂SeC has been carried out for the first time to explore its physical properties for possible applications in many sectors. The studied properties are compared with prior known MAX phase Zr₂SC. The structural parameters (lattice constants, volume, and atomic positions) are observed to be consistent with earlier results. The band structure and density of states (DOS) are used to explore the metallic conductivity, anisotropic electrical conductivity, and the dominant role of Zr-d states to the electrical conductivity at the Fermi level. Analysis of the peaks in the DOS and charge density mapping (CDM) of Zr₂SeC and Zr₂SC revealed the possible variation of the mechanical properties and hardness among them. The mechanical stability has been checked using elastic constants. The values of the elastic constants, elastic moduli and hardness parameters of Zr₂SeC are found to be lowered than those of Zr₂SC. The anisotropic behavior of the mechanical properties has been studied and analyzed. Technologically important thermodynamic properties such as the thermal expansion coefficient (TEC), Debye temperature (Θ_D), entropy (S), heat capacity at constant volume (C_v), Grüneisen parameter (γ) along with volume (V) and Gibbs free energy (G) are investigated as a function of both temperature (from 0 to 1600 K) and pressure (from 0 to 50 GPa). Besides, the Θ_D, minimum thermal conductivity (K_min), melting point (T_m), and γ have also been calculated at room temperature and found to be lowered for Zr₂SeC compared to Zr₂SC owing to their close relationship with the mechanical parameters. The value of the Θ_D, K_min, T_m, and TEC suggest Zr₂SeC as a thermal barrier coating material. The optical properties such as dielectric constant (real and imaginary part), refractive index, extinction coefficient, absorption coefficient, photoconductivity, reflectivity, and loss function of Zr₂SeC are computed and analyzed to reveal its possible applications.

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1. Introduction

Since the 1990s, one of the most widely known classes of transition metal carbides or nitrides is the so-called MAX phase materials with general formula M_n+1AX_n (M – early transition metal, A – A-group element, X – C or N; n = 1–4).^1–4 MAX phase materials can attract great attention owing to their exceptional performance combining both ceramics (elastically rigid, lightweight, creep and fatigue resistant as ceramic materials) and metals (machinable, electrically and thermally conductive, not susceptible to thermal shock, plastic at high temperature and exceptionally damage-tolerant);^1,2 consequently, they are promising candidates such as in high-temperature technology as components, sliding electrical contacts, and contacts for 2D electronic circuits, Li-ion batteries, wear and corrosion-resistant coatings, superconducting materials, spintronics, and nuclear industry.^2,5–9 The hybrid properties of MAX phases are due to the existence of strong covalent M–X bonds and relatively weak metallic M–A bonds within their structure.^2,10,11 These challenging properties are always motivating scientists; consequently, more than 150 MAX phases have been discovered.¹ Moreover, researchers are also trying to manipulate the composition and structure to achieve better combination of the properties such as different alloys/solid solutions,^12–20 M₂A₂X^21–23 and M₃A₂X,²⁴ rare-earth i-MAX phases,^25,26 212 MAX phases,^27–29 314 MAX phases,^27,30 MAX phase borides^6,31–37 and two-dimensional (2D) MAX phase derivatives termed MXenes.^38,39

As an A element, chalcogen S containing MAX phases have drawn attention owing to their comparatively strong M–A(p–d) bonding which is usually weak for MAX phases. So far known, the S-containing ternary MAX phases are M₂SC (M = Ti, Zr, Hf, and Nb) and M₂SB (M = Zr, Hf, and Nb).^1,13–15 The existence of strong M–A bonding for the S-containing MAX phases is reflected from the higher values of Young's modulus, bulk modulus, and shear modulus of M₂SC (M = Ti, Zr, Hf) compared to that of M₂AlC (M = Ti, Zr, Hf).^34,40–42 All the S-containing phases have been studied completely by the different research groups. Amini et al.⁴⁰ have synthesized and studied the mechanical properties of Ti₂SC. Bouhemadou et al.⁴¹ have performed a first-principles investigation of M₂SC (M = Ti, Zr, Hf) compounds. A comprehensive study of elastic properties of 211 MAX phases has been carried out by Cover et al.,⁴² where the Ti₂SC and Zr₂SC were included. M₂SB (M = Zr, Hf, Nb) borides have been synthesized by Rackl et al.,^32,33 which has been further subjected for the comprehensive study of the physical properties of M₂SX (X = C and B).³⁴ For each case, the S-containing MAX phases have enhanced mechanical properties compared to corresponding Al-containing MAX phases.

Recently, a new chalcogen (Se) containing MAX phase (Zr₂SeC) has been synthesized by Chen et al.,⁴³ where they have investigated the only electronic density of states, charge density, electrical resistivity, and thermal conductivity of Zr₂SeC. These limited results are not enough to explore the Zr₂SeC thoroughly for practical application in many sectors where many other MAX phases have already been used. For example, the MAX phases have the potential to be used as structural components at high-temperature^1,44,45 where the knowledge of mechanical properties is essential. MAX phases are also using in high-temperature technology [e.g., as thermal barrier coating (TBC) material] where some prior knowledge of Debye temperature, minimum thermal conductivity as well as melting temperature is required.^46,47 Moreover, the MAX phases are potential candidates for use as cover materials to diminish solar heating⁴⁸ where the knowledge of optical properties is fundamental. Thus, to predict its possible relevance in other sectors, a detailed study of Zr₂SeC is of scientific importance. Therefore, a detailed study of Zr₂SeC needs to be performed to take the full advantages for possible use in many sectors.

Therefore, the structural, electronic, mechanical, thermal, and optical properties of Zr₂SeC have been presented in this paper, and the properties of Zr₂SeC have been compared with those of the Zr₂SC MAX phase. It is found that the Zr₂SeC is soft with low Vickers hardness like other phase materials. Moreover, it is suitable to be used as a TBC material. Furthermore, it can also be used as a cover material for spacecraft to reduce solar heating. The rest of the article is organized as follows: the detailed Computational methodology is given in Section 2, Results and discussion are presented in Section 3, and important Conclusions are drawn in Section 4.

2. Computational methodology

A density functional theory based on the plane-wave pseudopotential method was implemented in the Cambridge Serial Total Energy Package (CASTEP) code^49,50 to calculate the physical properties of Zr₂SeC. The exchange and correlation functions were treated by the generalized gradient approximation (GGA) of the Perdew–Burke–Ernzerhof (PBE).⁵¹ The pseudo-atomic calculations were performed for C – -2s²2p², Se – 4s²4p⁴, and Zr – 4d²5s² electronic orbitals. A k-point⁵² mesh of size 10 × 10 × 3 was selected to integrate the Brillouin zone, and the cutoff energy was set as 500 eV. The Broyden Fletcher Goldfarb Shanno (BFGS) technique⁵³ was used for structure relaxation, and density mixing was used for electronic structure calculation. The structure was relaxed with the following parameters: the self-consistent convergence of the total energy is 5 × 10⁻⁶ eV per atom, the maximum force on the atom is 0.01 eV Å⁻¹, the maximum ionic displacement is set to 5 × 10⁻⁴ Å, and maximum stress of 0.02 GPa. The optical properties of the Zr₂SeC compound were calculated using the complex dielectric function ε(ω) = ε₁(ω) + iε₂(ω). The time-dependent perturbation theory (first-order) is used to obtain the imaginary part of the dielectric function. The equation used to calculate the imaginary part ε₂(ω) is given by:where Ω and ε₀ correspond to the volume of the unit cell and dielectric constant of the free space, u and r stand for the incident electric field vector (polarized) and position vector. ω, e and ψ_k^c and ψ_kv stand for the light frequency, electronic charge, and conduction and valence band wave functions at k, respectively. Here, the sum k was used to present the Brillouin zone’ sampling in the k space. The sums v and c were used to represent the contribution from the unoccupied conduction band (CB) and occupied valence band (VB). The Kramers–Kronig relations were used to estimate the real part ε₁(ω) from the imaginary part ε₂(ω). The real and imaginary parts of the dielectric function were further used to obtain the other optical constants: refractive index, extinction coefficient, absorption spectrum, reflectivity, and energy-loss spectrum based on the relations found elsewhere.⁵⁴

3. Results and discussion

3.1 Structural properties

The unit of Zr₂SeC is shown in Fig. 1, which is crystallized in the hexagonal system with space group P6₃/mmc (194).² The unit cell contains two formula units, and there are eight atoms in the unit cell. The Zr₆C octahedron interleaved between two atomic layers of the Se atom is shown in Fig. 1. For 211 MAX phases, the atomic positions of Zr, Se, and C atoms in the unit cell are (1/3, 2/3, 0.0965), (1/3, 2/3, 3/4), and (0, 0, 0). The unit of Zr₂SeC is optimized geometrically to calculate the physical properties further. The lattice constants of the optimized cell are presented together with previous experimental and theoretical results⁴³ that assure the parameters used for calculations. For example, our calculated values of a, c, and V are only 0.1%, 0.18%, and 1.08%, respectively, higher than those of experimental values (Table 1).

Fig. 1 Crystal structure (unit cell) of the Zr₂SeC compound.

Table 1 Calculated lattice parameters (a and c), c/a ratio, volume (V), and atomic positions of Zr₂SeC MAX phase

a (Å)	c (Å)	c/a	V (Å^3)	Ref.	Positions	Zr	Se	C
3.4655	12.5406	3.618	130.429	This study	x	1/3	1/3	0
3.462	12.518	3.615	129.029	Expt.^43	y	2/3	2/3	0
3.487	12.631	3.622	132.080	Theo.^43	z	0.0965, 0.0963 (ref. 43)	3/4	0

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3.2 Electronic properties and bonding nature

The electronic band structure of Zr₂SeC of optimized structure within the GGA-PBE is estimated and analyzed as shown in Fig. 2(a) along the high symmetry lines of the Brillouin zone (Γ–A–H–K–Γ–M–L–H). None existence of a bandgap close to the Fermi level owing to the overlapping of the conduction and valence band reveals the metallic nature of the titled compound like other MAX phases.⁵⁵ The band structure also reveals the anisotropic nature of electronic conductivity. It is seen that the energy dispersion along Γ–A, K–H, and L–M directions, which are along the c-direction, are small. The conductivity in the basal plane is exhibited by H–K, Γ–M, and L–H that are larger than those of c-direction. Thus, the conductivity is higher in the basal plane than that of c-direction for Zr₂SeC, which is typical for MAX phases.^20,29,56

Fig. 2 The electronic band structure of (a) Zr₂SeC, (b) total and partial DOS of Zr₂SeC, (c) the total DOS of Zr₂AC (A = S, Se) and (d) charge density mapping of Zr₂AC (A = S, Se).

After band structure, the next step for insight into the electronic properties of Zr₂SeC is to investigate its density of states (DOS). Fig. 2(b) shows the DOS of Zr₂SeC with a finite value of DOS (1.84 states per unit cell per eV) at the Fermi level that is slightly higher than the value (1.48 states per unit cell per eV) obtained by Chen et al.,⁴³ might due to the difference in the functionals used. The DOS of Zr₂SeC is slightly higher than that of Zr₂SC (1.79 states per unit cell per eV); a similar result is also reported by Chen et al.⁴³ The value of DOS at the Fermi level (E_F) can provide information regarding electrical conductivity as it is directly related to the DOS and electron mobility.² The partial DOS is used to explore the contribution from different states to the total DOS. For instance, the DOS at the E_F is contributed from Zr-d states and involved in the conduction of Zr₂SeC. The C and Se do not contribute to the DOS at E_F and are not involved in the conduction properties. The contribution of different atoms to the DOS at different energy states is also observed from the partial DOS (PDOS). The DOS profile is almost similar to that of the reported DOS.⁴³ The hybridization among the different Zr, Se, and C electronic states is observed from PDOS. The lowest energy band (−9 to −12 eV) comes from the hybridization between Zr-d and C-s orbital's electrons with a dominating role of C-s states. The peak in the energy range −4.5 to −7.5 eV is accredited by the strong hybridization of Se-p and Zr-d states with a minor contribution from Zr-s and C-p states. The strongest peak in the range – 1.7 to 4.5 eV results from the hybridization between Zr-d, Se-p, and C-p states with dominant contribution from C-p states.

The highest valence band in the range 0 to 1.7 eV is dominated by Zr-d states with a small contribution from Se-p and C-p states. The PDOS of Zr₂SC is not presented here due to similar nature. The TDOS of Zr₂AC (A = S, Se) is shown in Fig. 2(c) to disclose the effect of the A element. As seen in Fig. 2(b), the S/Se contributed to the TDOS mostly in the energy range of −4.5 to −7.5 eV. The peak found in this region is shifted towards lower energy (indicated by the green line and black arrow) for Zr₂SC compared to Zr₂SeC, indicating a higher bonding M–A strength for Zr₂SC compared to that in Zr₂SeC. The peaks position due to Zr and C remain the same for both compounds. Thus, higher values of the mechanical properties characterizing parameters are expected owing to the presence of stronger M–A bonding in Zr₂SC than in Zr₂SeC. To strengthen this statement, we have calculated the charge density mapping (CDM) for both phases, as shown in Fig. 2(d). Fig. 2(d) shows the CDM for both Zr₂AC (A = S, Se), where the red and blue colors indicate the highest and the lowest value, respectively. The directional covalent bonding occurs between Zr and C atoms where the charges are geometrically localized, and this boding is comparatively stronger, required more energy to break this bonding. As seen in the figure, the charge density at Zr and C position is unchanged for both Zr₂SC and Zr₂SeC. The charges are accumulated to a greater extent at S positions of Zr₂SC than that of Se positions of Zr₂SeC. The greater charges at the S position the greater bonding strength between Zr–S than Zr–Se which is in good agreement with DOS results. Now it's time to prove the statement that the Zr– S bonding stronger than Zr–Se which will be proven in the following section. The electronic band structure and DOS of Zr₂SeC and Zr₂SC are similar to that of M₂SC (M = Hf, Nb).^32,41

3.3 Mechanical properties

Some of the physical properties of solids, for example, the mechanical stability, bonding strength, deformation, failure mode, stiffness, anisotropic nature in bonding strength, etc., can be brought out by studying mechanical properties characterizing parameters that are subjected in this section. At first, the stiffness constants (C_ij) have been calculated using the well-known strain–stress method.^{29,30,57–59} The stiffness constants, which are five in number as independent due to the hexagonal nature of Zr₂SeC and presented in Table 2 together with those of other S-containing MAX phases Zr₂SX (X = C, B). One of the prime importance of stiffness constants is the use of checking the mechanical stability of solids. Max Born⁶⁰ proposed some conditions on the stiffness constants of solids for being mechanically stable, having a limitation that they are not enough to predict the stability for all crystal systems correctly. The limitation is overcome by Mouhat et al.⁶¹ and the stability conditions for a hexagonal systems becomes:

C₁₁ > 0, C₁₁ > C₁₂, C₄₄ > 0, (C₁₁ + C₁₂)C₃₃ − 2(C₁₃)² > 0. (2)

Table 2 The elastic constants, C_ij (GPa), bulk modulus, B (GPa), shear modulus, G (GPa), Young's modulus, Y (GPa), macro, H_macro (GPa), micro hardness, H_micro (GPa), Pugh ratio, G/B, Poisson ratio, ν and Cauchy pressure, CP (GPa) of Zr₂SeC, together with those of M₂SC (M = Zr, Hf, Nb)

Phase	C11	C12	C13	C33	C44	B	G	Y	Hmacro	Hmicro	G/B	ν	CP	Ref.
Zr2SeC	260	97	96	293	128	154	100	247	14.85	17.79	0.65	0.23	−30	This study
Zr2SC	295	89	102	315	138	166	115	280	17.89	21.57	0.69	0.22	−49	34
Hf2SC	311	97	121	327	149	181	120	295	17.35	21.72	0.66	0.23	−52	34
Nb2SC	316	108	151	325	124	197	105	267	11.58	15.84	0.53	0.27	−16	34

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The C_ij presented in Table 2 satisfy the above relations, and hence Zr₂SeC is expected to be mechanically stable like M₂SC (M = Zr, Hf, Nb). It should be noted here that Zr₂SeC is already experimentally realized; thus, the question regarding stability is not expected at all. But, checking mechanical stability is necessary for practical application under load. The importance of C_ij is not only limited to mechanical stability checking but also provides significant information. For example, the stiffness of solids along [100] and [001] directions are defined by C₁₁ and C₃₃, respectively, thus, comparatively low pressure is required to deform Zr₂SeC along crystallographic a-axis than c-axis because of C₁₁ < C₃₃. The resistance to shear deformation is measured by the value of C₄₄. It is evident from Table 2 that C₄₄ < C₃₃ and C₁₁, indicating a low pressure required to shear deformation than axial deformation. Moreover, C₄₄ is also a hardness predictor, in fact, directly related to the hardness compared to other elastic moduli.⁶² C₄₄ of Zr₂SeC is smaller than M₂SC (M = Zr, Hf) but larger than Nb₂SC, thus, the hardness of Zr₂SeC is expected to be lower than M₂SC (M = Zr, Hf) but higher than Nb₂SC. Furthermore, the unequal values of C₁₁ and C₃₃ reveal the anisotropy in the bonding strength in the a and c-axis owing to the difference in the atomic arrangement in a and c-axes. The difference in the atomic arrangement is mainly responsible for the mechanical anisotropy that will be presented in the letter.

Bulk modulus (B) defines the property related to the incompressibility of the materials. The material's resistance to change its volume keeping shape unchanged under hydrostatic pressure is measured by its B. The resistance of solids to plastic deformation at constant volume is provided by its shear modulus (G). The stiffness of solids is measured by its Young's modulus (Y); the higher value of Y indicates stiffer solids and vice versa. These moduli are also important to realize the hardness, stiffness, brittleness/ductileness of solids. From these points of view, we have calculated the B and G using Hill's approximation⁶³ based on the Voigt⁶⁴ and Reuss⁶⁵ models as follows:

B = (B_V + B_R)/2, (3)

where B_V = [2(C₁₁ + C₁₂) + C₃₃ + 4C₁₃]/9 and B_R = C²/M; C² = (C₁₁ + C₁₂)C₃₃ − 2C₁₃²; M = C₁₁ + C₁₂ + 2C₃₃ − 4C₁₃.

B_V represents the upper limit of B (Voigt bulk modulus) and B_R represents the lower limit of B (Reuss bulk modulus). Like B, average values of Voigt (G_V) and Reuss (G_R) were used to calculate G using the following equations:

G = (G_V + G_R)/2, (4)

where G_V = [M + 12C₄₄ + 12C₆₆]/30 and C₆₆ = (C₁₁ − C₁₂)/2.

The Young's modulus (Y) and Poisson's ratio (υ) were also calculated using their relationships with B and G:³⁴

Y = 9BG/(3B + G) (5)

and

υ = (3B − Y)/(6B) (6)

The lower values of B and G of Zr₂SeC than those of Zr₂SC revealed that low pressure is required to volume and plastic deformation for Zr₂SeC than Zr₂SC and lower value of Y for Zr₂SeC than Zr₂SC, indicating that Zr₂SeC is less stiff than Zr₂SC. The elastic moduli were further used to study the hardness of Zr₂SeC using the equations:^66,67

H_macro = 2(k²G)^0.585 − 3 (7)

(where Pugh's ratio, k = G/B) and

The study of the hardness parameter is beneficial for those materials which are used as structural components like MAX phases. The replacement of Se by S is enough to increase the M–A bonding strength that must improve the hardness of Zr₂SC than Zr₂SeC. The obtained values of H_macro and H_micro = 14.85 (17.89) and 17.79 (21.57) GPa for Zr₂SeC (Zr₂SC). Thus, the enhancement of hardness for Zr₂SC is observed. Though the elastic moduli (B, G, Y) of Zr₂SeC are smaller than that of Nb₂SC but the hardness parameters of Zr₂SeC are larger than those of Nb₂SC owing to the larger value of C₄₄ (ref. 62) (C₄₄ of Zr₂SeC and Nb₂SC are 128 and 124 GPa, respectively). The elastic moduli and hardness parameters of Hf₂SC are larger than those of Zr₂SeC. The lowering of the bonds strength (Zr–C and Zr–Se bond) for Zr₂SeC compared to Zr₂SC (Zr–C and Zr–S bond) is responsible for the lowering of elastic moduli and hardness for the same. The bond length of Zr–C (2.339 Å) in Zr₂SeC is higher than in Zr₂SC (2.326 Å), and Zr–Se (2.776 Å) in Zr₂SeC is higher than Zr–S in Zr₂SC (2.687 Å) and hence the bonds (Zr–C, Zr–S) of Zr₂SC are stronger than those of Zr₂SeC (Zr–C, Zr–Se). Thus, lower hardness parameters for Zr₂SeC compared to Zr₂SC is expected as reflected from the hardness parameters presented in Table 2. To be more confirmed, we have calculated the Vickers hardness (H_v) using the Gou et al.⁶⁸ formula that is based on the Mulliken bond population and geometrical averages of the bonds present within the crystal. The relevant formula can be found elsewhere.⁶⁹ The calculated values of H_v for Zr₂SeC and Zr₂SC are 2.52 and 4.33 GPa, respectively. Again, the Vickers hardness of Zr₂SeC is lower than that of Zr₂SC as expected. The obtained elastic moduli and hardness are in good agreement with the DOS and CDM results.

3.4 Mechanical anisotropy

For hexagonal crystal, the elastic anisotropy is related to microcracks and anisotropic plastic deformation, which plays a vital role in understanding the mechanical stability of the material under service. The mechanical anisotropy in MAX phases is normally due to unequal elastic constants, i.e..C₁₁ ≠ C₃₃, which indicates that the mechanical properties in all crystallographic planes are not identical.⁷⁰ Moreover, the knowledge of anisotropy offers the information necessary to enhance further the stability of solids for many applications.⁷¹ Therefore, it is necessary to study the anisotropic behavior of mechanical properties in Zr₂SeC. For this purpose, the 2D and 3D visualization of Young's modulus, compressibility, shear modulus, and Poisson's ratio of Zr₂SeC MAX phase is presented ESI [Fig. S1(a–d)†] in comparison with those of Zr₂SC [Fig. S2(a–d)†] by using the open-source software packages ELATE and AnisoVis.^72,73 The extent of anisotropy is estimated by the variable value of elastic properties in all directions. The 2D projections and 3D plots are perfectly circular and spherical for isotropic materials, and variation from circular/spherical shape demonstrates the degree of anisotropy of the corresponding elastic property. The obtained results for Zr₂SeC and Zr₂SC are tabulated in Table 3. The unit value of the anisotropic index indicates the identical mechanical properties in all directions where the value higher or lower than 1 (one) measures the degree of anisotropy. Table 3 confirms the anisotropic nature of the mechanical properties and Zr₂SeC is more anisotropic compared to Zr₂SC.

Table 3 The minimum and maximum values of Young's modulus, Y (GPa), linear compressibility, K (TPa⁻¹), shear modulus, G (GPa), and Poisson's ratio, υ and their anisotropic indices, A of Zr₂AC (A = Se, S)

Phases	Ymin	Ymax	AY	Kmin	Kmax	AK	Gmin	Gmax	AG	υmin	υmax	Aυ	Ref.
Zr2SeC	208.93	277.50	1.32	1.913	2.288	1.19	81.071	127.59	1.57	0.078	0.343	4.394	This work
Zr2SC	250.08	306.39	1.22	1.78	2.12	1.19	101.19	137.27	1.35	0.110	0.290	2.53	34

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We have also investigated the different anisotropic factors for the {100}, {010} and {001} planes that are computed using the eqn (9)–(11), respectively and presented in Table 4:⁷⁴

Table 4 The anisotropic factors, A₁, A₂, A₃, B_a, B_c, k_c/k_a, and universal anisotropic index A^U of Zr₂AC (A = Se, S)

Phase	A1	A2	A3	kc/ka	Ba	Bc	A^U
Zr2SeC	0.73	1.57	1.14	0.84	383	904	0.218
Zr2SC	0.73	1.34	0.98	0.85	412	955	0.113

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Since the values of A_i's are not equal to 1 (one), thus Zr₂SeC and Zr₂SC are anisotropic because A_i = 1 implies the isotropic nature. The bulk modulus for a and c-direction are computed using the eqn (12) and (13), respectively:⁷⁵

where Λ = 2(C₁₁ + C₁₂) + 4C₁₃α + C₃₃α² and . The unequal values of B_a and B_c [Table 4] indicates the anisotropic nature of Zr₂AC (A = Se, S). The linear compressibility (k) along the a and c-axis are calculated by the equation:⁷⁶

The obtained values of k_c/k_a are not equal to 1 (k_c/k_a = 1 for isotropic materials) and hence Zr₂AC (A = Se, S) are anisotropic. Another important anisotropic factor, the universal anisotropic index A^U is estimated by the following equation:⁷⁷

where B and G are obtained by Voigt and Reuss models. Since, the values of A^U are greater than zero, indicating the anisotropic nature of Zr₂AC (A = Se, S).

3.5 Thermal properties

The thermodynamic properties of the material at high temperatures and pressures are of scientific and technical significance, which help to predict the material's applications at elevated temperatures and pressure. Here we have investigated the thermodynamic properties of the newly synthesized Zr₂SeC MAX phase in comparison with Zr₂SC over the wide temperature (from 0 to 1600 K) and pressure (from 0 to 50 GPa) by using the quasi-harmonic Debye approximation.^78,79 To disclose the important thermodynamic properties, the thermal expansion coefficient (TEC), heat capacity at constant volume (C_v), entropy (S), and Grüneisen parameter (γ) have been investigated in this temperature and pressure range. The quasi-harmonic Debye model remains valid in this given range of temperature and pressure and has been successfully used to calculate the thermodynamic properties of MAX phases.^80,81

The temperature dependence of the thermal expansion coefficient (TEC) is displayed in Fig. 3. It is seen that the TEC increases rapidly with the increase in temperature up to 300 K. The increase in TEC of Zr₂SeC and Zr₂SC becomes less sensitive of temperature at the T ≥ 300 K. However, at the constant temperature, TEC decreases with the increase in pressure. The calculated value of TEC for Zr₂SeC (3.88 × 10⁻⁵ K⁻¹) < Zr₂SC (4.03 × 10⁻⁵ K⁻¹) at zero pressure and T = 300 K.

Fig. 3 The temperature effect on the thermal expansion coefficient of (a) Zr₂SeC and (b) Zr₂SC at different pressure.

The specific heat is significant to predict the density of states, band structure, and lattice vibrations. The effect of temperature on the specific heat at constant volume (C_v) for Zr₂SeC and Zr₂SC is given in Fig. 4. The heat capacity for both MAX phases satisfies the Debye T³ power-law at the low temperatures, which is mainly due to an exponential increase in the number of exciting phonon modes. At high temperatures, the specific heat curves become converged to get the classical Dulong-petit (DP) limit (C_v = 3nN_Ak_B). The DP limit for Zr₂SeC and Zr₂SC is found to be at ∼197.12 J mol⁻¹ K⁻¹ and ∼196.80 J mol⁻¹ K⁻¹. The heat capacity decreases linearly with the increase in pressure. The calculated value of C_v for Zr₂SeC (156.46 J mol⁻¹ K⁻¹) is greater than that of Zr₂SC (151.32 J mol⁻¹ K⁻¹) at zero pressure, and T = 300 K. It is also observed in Fig. 4 that the heat capacity of Zr₂SeC reached the DP limit at a lower temperature compared to Zr₂SC. The values of Θ_D (Table 5) and H_v of Zr₂SeC are smaller than those of Zr₂SC; thus, it is expected to reach the DP limit at a lower temperature for Zr₂SeC than that of Zr₂SC. A similar result is also found for other solids.⁸²

Fig. 4 The temperature on the heat capacity at constant volume, C_v of (a) Zr₂SeC and (b) Zr₂SC at different pressure.

Table 5 The calculated volume (V), Debye temperature (Θ_D), melting temperature (T_m), TEC (α), Grüneisen parameter (γ), minimum thermal conductivity (K_min), specific heat (C_v) and entropy (S) of Zr₂SeC together with those of M₂SC (M = Zr, Hf, Nb) MAX phases at temperature of 300 K and pressure of 0 Pa

Phases	V (Å^3)	ΘD (K)	Tm (K)	α (×10^−5 K^−1)	γ	Kmin (W m^−1 K^−1)	Cv (J mol^−1 K^−1)	S (J mol^−1 K^−1)
a Ref. 34.
Zr2SeC	129.7	679	1571	3.88	3.11	1.3	156.5	126.5
Zr2SC	126.3	727	1712^a	4.03	3.12	1.12^a	151.3	115.9
Hf2SC	130.1	598	1778^a	3.59	3.11	0.85^a	164.8	146.7
Nb2SC	113.7	821	1790^a	3.12	3.12	1.09^a	141.0	98.1

---

The Grüneisen parameter (γ) for Zr₂SeC and Zr₂SC as a function of temperature is depicted in Fig. 5. It is an important parameter that is used to calculate the thermal states and can also quantify the anharmonic effect. The Grüneisen parameter for Zr₂SeC and Zr₂SC increases with the increase in temperature and decreases with the increase in pressure. The effect of temperature on γ for Zr₂SeC and Zr₂SC at higher pressure is less significant. The trend of γ at 50 GPa is nearly constant with the increase in temperature. The calculated value of γ for Zr₂SeC (3.11) is smaller than that of Zr₂SC (3.12) at zero pressure and T = 300 K.

Fig. 5 The effect of temperature and pressure on the Grüneisen parameter of (a and c) Zr₂SeC and (b and d) Zr₂SC, respectively.

We have further calculated pressure and temperature dependence of volume (V), Gibbs free energy (G) Debye temperature (Θ_D) and entropy (S) of Zr₂SeC and the results are presented in ESI (see ESI Fig. S3–S6†). Moreover, we have also calculated the parameters discussed above for M₂SC (M = Zr, Hf, Nb) and the results are presented in ESI† (to save the journal space) together with those of Zr₂SeC (see Fig. S3–S9†).

Furthermore, we have also calculated three important parameters regarding the application at high temperatures, such as the Debye temperature (Θ_D), minimum thermal conductivity (K_min) and melting temperature (T_m) at room temperature. The Θ_D is calculated using the quasi-harmonic Debye model. It is found that the Θ_D for Zr₂SeC (679 K) is smaller than that of Zr₂SC (727 K) which is good agreement with statement that Debye temperature is higher for harder solids and vice versa. The thermal conductivity of the materials which are used at high temperatures (e.g., ceramics) reduces to the smallest value (K_min) at high temperatures. The K_min of Zr₂SeC is computed by the equation:⁸³

where k_B, v_m, N_A and ρ are Boltzmann constant, average phonon velocity, Avogadro's number and density of crystal, respectively. The obtained value of K_min is 1.3 (W m⁻¹ K⁻¹) for Zr₂SeC which is comparable with those of M₂SC (M = Zr, Hf, Nb)³⁴ as well as other MAX phase materials.^7,29,30 The thermal conductivity of Zr₂SeC (18.30 W m⁻¹ K⁻¹) is also lower than that of Zr₂SC (21.10 W m⁻¹ K⁻¹) as measured by Chen et al.⁴³ They have also measured the electronic contribution to the electrical conductivity, which is important for predicting high-temperature application. Accordingly, the electronic contribution to the thermal conductivity is 25% for Zr₂SeC and 19.3% for Zr₂SC; thermal conductivity is dominated by phonon contribution. Thus, K_min of Zr₂SeC and Zr₂SC is also dominated by phonon contribution and the low values of K_min could be useful to predict its suitability as TBC material. Elastic constants can be used to calculate the melting temperature (T_m) of Zr₂SeC and Zr₂SC by using the expression:⁸⁴

T_m = 3C₁₁ + 1.5C₃₃ + 354 (17)

It is found that the melting temperature of Zr₂SeC (1571 K) is lower than that of Zr₂SC (1712 K), Hf₂SC (1778 K) and Nb₂SC (1790 K),³⁴ and is comparable with the melting point of Zr₂GaC (1481 K), Hf₂GaC (1648 K), and Hf₃SnC₂ (1773 K).^59,70 Recently, the research has been focused to prove the MAX phases as TBC materials.^47,85 The calculated values of the parameters related to the TBC applications for Zr₂SeC are: Θ_D ∼ 679 K, TEC ∼ 2.81 × 10⁻⁶ K⁻¹, K_min – 1.3 W m⁻¹ K⁻¹, and T_m ∼ 1571 K. The values for a promising TBC material, Y₄Al₂O₉ are: Θ_D ∼ 564 K, TEC ∼ 7.51 × 10⁻⁶ K⁻¹, K_min – 1.13 W m⁻¹ K⁻¹, and T_m ∼ 2020 K.^86,87 A comparison of these values indicates that the Zr₂SeC MAX phase can be used as a TBC material for high-temperature applications. However, all the thermal properties characterizing parameters of Zr₂SeC calculated at ambient condition are presented in Table 5 together with those of M₂SC (M = Zr, Hf, Nb) for comparison.

3.6 Optical properties

The optical properties of the Zr₂SeC compound are calculated and presented in Fig. 6 for [100] and [001] polarization directions up to 25 eV incident phonon energy. These two polarization directions represent the related electric field which is in perpendicular and parallel directions to the c-axis of unit cell structure. As confirmed that the titled compound is metallic; thus, a correction (intra-band) is needed to the low energy region of the spectrum. This correction is done by setting up a plasma frequency of 3 eV and damping of 0.5 eV. Besides, Gaussian smearing of 0.5 eV was used with the intention that the k-points be more effective on the Fermi level. The optical constants of Zr₂SeC are compared with those of M₂SC (M = Zr, Hf, Nb) for [100] direction only.

Fig. 6 (a) Real part ε₁ and (b) imaginary part ε₂ of dielectric function ε (ω), (c) refractive index n, (d) extinction coefficient k, (e) absorption coefficient α, (f) photoconductivity σ, (g) reflectivity R, and (h) loss function LF of Zr₂SeC together with those of M₂SC (M = Zr, Hf, Nb) MAX phases as a function of photon energy.

Fig. 6 shows the (a) real and (b) imaginary part of the complex dielectric function of Zr₂SeC, which is very important and used to explain the material's response to the electric field.⁴⁷ As seen, the high negative static value of the real part and high positive static value of the imaginary part of the dielectric function indicates the metallic nature of Zr₂SeC; is consistent with the band structure result. The ε₁(ω) and ε₂(ω) of Zr₂SeC are too much similar to those of M₂SC (M = Zr, Hf, Nb). Strong anisotropy is observed in the low energy part, which tends to be isotropic in the high-energy region. The peaks in the ε₂(ω) are the results of electrons excitation, such as intra-band transitions (within M-d states) and inter-band transitions due to absorption of photon incident on it. The first two peaks for [001] direction of Zr₂SeC are due to electron transition within Zr-d states. The third peak for [001] and the only peak for [100] direction assumed to be due to the inter-band transition of electrons. Approaching of zero by ε₁(ω) from below [at 14.91 and 15. 25 eV for [100] and [001] directions, respectively] and ε₂(ω) from above [at 19 eV for both directions] is another indication for the metallic nature of Zr₂SeC. The energy at which the ε₁(ω) touches zero from below are 15.42, 16.5, 16.44 eV for M₂SC (M = Zr, Hf, Nb), respectively. Fig. 6(c) shows the refractive index n(ω) of Zr₂SeC. The value of static n(0) is 19 and 6 for [100] and [001] directions, respectively. As the value of n(ω) is an indication for the light velocity propagating the materials, thus, lower value of n(ω) is good for optical devices such as waveguides. The value of n(ω) is 6 for [001] direction implies the use of Zr₂SeC in parallel to the field direction (parallel to the c-axis) is better than that of [100] direction (perpendicular to the c-axis). However, the velocity is reduced by the interaction with electrons during propagation through the Zr₂SeC, as confirmed from values greater than the unit value. Fig. 6(d) shows the imaginary part of the refractive index, known as extinction coefficient, k(ω), also used to measure the absorption capability of solids. The spectra of k(ω) follow the trend of ε₂(ω) for each of the MAX phases presented here. The static value of k(ω) is 2.8 and 0.23 for [100] and [001] directions, respectively. The k(ω) decreases to zero at ∼16 eV, indicating that the intrinsic oscillation frequency of Zr₂SeC is 16.0 eV. The higher value of k(ω) than n(ω) indicates the energy range wherein light cannot propagate, and Zr₂SeC exhibit the behavior belongs to the reflective metal.^88,89

Fig. 6(e) shows the absorption coefficient (α) of Zr₂SeC [100] and [001] for two polarization directions along with that of M₂SC (M = Zr, Hf, Nb) for [100] direction only. The metallic behavior of Zr₂SeC and M₂SC (M = Zr, Hf, Nb) is reflected from the starting of the spectra, started at zero photon energy owing to no existence of the bandgap. For [100] directions, α increases with the rise in photon energy, exhibits a peak in the range of 1.5–1.9 eV, and then decreases slightly 3.3 to 3.6 eV, finally reaches maximum values at 7 to 9 eV. The energy range wherein α attains a value of greater than 10⁵ cm⁻¹ is the zone of strong absorption. The strong absorption zone of Zr₂SeC is 3.5 to 14 eV and 2.8 to 14.4 eV for [100] and [001], respectively. The absorption coefficient of Zr₂SeC is similar to that of M₂SC (M = Zr, Hf, Nb) for [100] direction. The photoconductivity (σ) of Zr₂SeC is shown in Fig. 6(f), in which the curve is noted to be started in association with the photon incident, which implies the zero bandgap of the titled compound. The σ is found to be similar to that of k(ω) in peaks position as σ is the results of photon absorption that is exhibited by k(ω). Strong anisotropy is also observed in the low energy region that tends to be lowered in the high region.

Fig. 6(g) demonstrates the reflectivity (R) Zr₂SeC that is similar to that of M₂SC (M = Zr, Hf, Nb) for [100] direction. The static values of R are 0.81 and 0.5 for [100] and [001] directions, respectively. The inset of Fig. 6(g) shows the R up to 6 eV in which a horizontal redline is drawn at 0.44 (44%) of R to demonstrate the suitability for use as a cover material to diminish solar heating. It was suggested by Li et al.^48,90 that the MAX phase with reflectivity greater than 44% is suitable for use cover materials for the same mentioned before. As seen, R decreases to lower than 0.44 at 1.7 and 2.8 eV for [001] and [100] directions, respectively. However, R of Zr₂SeC decreases to lower than 0.44 at slightly lower energy than that of M₂SC (M = Zr, Hf), but the lowest value (34%) is higher than that of M₂SC (M = Zr, Hf, Nb). R increases to a maximum of 0.91 at 10 eV, followed by another peak of 0.7 at 14.1 eV and finally declined sharply towards zero, and the titled compound becomes transparent for the incident light. The Zr₂SeC shows reflective behavior in the energy range of 7.5 to 14 eV in which the refractive index is very low, and the reflectivity is greater than 70%. A similar reflectivity is also shown by M₂SC (M = Zr, Hf, Nb) MAX phases. Fig. 6(h) shows the loss function of Zr₂SeC that demonstrates how fast electrons losing their energy when traversing the Zr₂SeC compound. A peak corresponding to the plasma frequency (ω_p) is observed at the energy where ε₁(ω) and ε₂(ω) approaches zero from below and above respectively, and reflectivity shows trailing edges. The ω_p of Zr₂SeC are 14.7 and 15.1 eV for [001] and [100] directions, respectively. The ω_p of M₂SC (M = Zr, Hf, Nb) MAX phases are slightly higher than that of Zr₂SeC for [100] direction. This plasma frequency defined a critical value at which the material changes its behavior from metallic to transparent dielectric.

4. Conclusions

A comparative study of the structural, electronic, mechanical, thermal, and optical properties of synthesized MAX phase Zr₂SeC with prior known MAX phase Zr₂SC has been performed by density functional theory for the first time. The obtained lattice constants, volume, and atomic positions are noticed to be agreed with experimental and theoretical results. The metallic behavior has been confirmed from the band structure and density of states (DOS). The less energy dispersion along the c-direction confirmed the anisotropic behavior of electrical conductivity, and partial DOS revealed the dominant role of Zr-d states to the electrical conductivity. Shifting of the peak in the DOS owing to the replacement of Se by S and difference in charge density mapping (CDM) revealed that the strength of bonds present in Zr₂SeC is lower than the bonds in Zr₂SC. The mechanical stability of the synthesized phase is further confirmed by the stiffness constants. The stiffness constants, elastic moduli, and hardness parameters of Zr₂SeC are found lower than those of Zr₂SC. The lowering of the parameters is explained based on the bond populations, and bond lengths present within them in an association with the DOS and CDM results. Both the directional projections of the elastic moduli and anisotropy indices confirm the anisotropic behavior of Zr₂SeC and Zr₂SC; Zr₂SeC is more anisotropic than Zr₂SC. The temperature and pressure-dependent properties exhibit the expected variation for the same. The value of Θ_D, K_min, T_m, and TEC of Zr₂SeC are lower than those of Zr₂SC, C_v of Zr₂SeC reach classical DP limit at a lower temperature than that of Zr₂SC; is consistent with the parameters used to characterize mechanical properties. A comparison of the values of Θ_D, K_min, T_m, and TEC for Zr₂SeC with those of a promising TBC material (Y₄Al₂O₉)⁸⁷ reveals its possibility to be used as TBC material. The real and imaginary parts of the dielectric constant and the absorption and photoconductivity curves agree with the band structure results and confirm the metallic nature of Zr₂SeC. The possible relevance of Zr₂SeC for use as shielding material to diminish solar heating has been confirmed from the analysis of the studied optical properties. Like electrical conductivity and mechanical properties, the optical properties of Zr₂SeC also exhibit anisotropic behavior.

Data availability

The data sets generated and/or analyzed in this study are available from the corresponding author on reasonable request.

Author contributions

M. A. A. designed and supervised the project, performed the calculation and analysis of a part of the results, and wrote a part of the draft manuscript. M. W. Q. performed the calculation and analysis of a part of the results and wrote a part of the draft manuscript. Both authors finalized the draft of the manuscript.

Conflicts of interest

There are no conflicts to declare.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d1ra02345d

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