DFT-based study of the structural, optoelectronic, mechanical and magnetic properties of Ti3AC2 (A = P, As, Cd) for coating applications

The first-principles approach has been used while employing the Perdew–Burke–Ernzerhof exchange-correlation functional of generalized gradient approximation (PBE-GGA) along with the Hubbard parameter to study the structural, optoelectronic, mechanical and magnetic properties of titanium-based MAX materials Ti3AC2 (A = P, As, Cd) for the first time. As there is no band gap found between the valence and conduction bands in the considered materials, these compounds belong to the conductor family of materials. A mechanical analysis carried out at pressures of 0 GPa to 20 GPa and the calculated elastic constants Cij reveal the stability of these materials. Elastic parameters, i.e., Young's, shear and bulk moduli, anisotropy factor and Poisson's ratio, have been investigated in the framework of the Voigt–Reuss–Hill approximation. The calculated values of relative stiffness are found to be greater than ½ for Ti3PC2 and Ti3AsC2, which indicates that these compounds are closer to typical ceramics, which possess low damage tolerance and fracture toughness. Optical parameters, i.e., dielectric complex function, refractive index, extinction coefficient, absorption coefficient, loss function, conductivity and reflectivity, have also been investigated. These dynamically stable antiferromagnetic materials might have potential applications in advanced electronic and magnetic devices. Their high strength and significant hardness make these materials potential candidates as hard coatings.


Introduction
Recently, MAX phases have captured the attention of a lot of researchers due to the discovery of MXenes, 1-5 which are a novel 2D material made from transition metal carbides and nitrides. MXenes offer higher shielding effectiveness due to their moderate electrical conductivity and can be obtained by the selective etching of the A-layer from a MAX material. 6,7 MXenes have the capability to provide perfect guidance to design shielding materials for electromagnetic interference, 8 and can be used as an electrode material to enhance the capacitance of super-capacitors. 9 The family of MAX materials exhibiting properties of metals and ceramics simultaneously was rst reported in the early 1960s by Nowtony and co-workers. Later, in 1996, Barsoum et al. discovered a fascinating MAX material, titanium silicon carbide T 3 SiC 2 . [10][11][12][13] Titanium-based MAX materials like Ti 3 SiC 2 , Ti 3 AlC 2 and Ti 2 AlC have demonstrated unique properties such as high temperature strength, low density and good oxidation resistance. 14,15 Moreover, lithiumbased MAX materials like Li 2 Ti 3 O 7 , LiNiO 2 , Li x CoO 2 , and Li x -TiO 2 have extraordinary technological applications. 16,17 MAX materials are denoted as M n+1 AX n , where M represents a transition metal, A represents an element from groups XIII-XVI, X is either carbon or nitrogen and n may vary from 1-3. 18,19 MAX compounds are categorized in distinct phases, namely, 211, 312 and 413, with respect to the value of n. 20 The foremost distinction among MAX alloys depends upon the number of inserting A layers per M layers. 21 Mechanically, MAX compounds are different from MX carbides and nitrides. 22 The incredible characteristics of MAX phases solely depend on M-X bonds with covalent-metallic nature, and are extremely strong when compared with M-A bonds. 23 Ti 3 AlC 2 from the Ti-Al-C family is a very interesting MAX material due to its greatly tailorable properties. 24 Experimental and computational investigations to explore the properties of MAX phase materials have been reported. Various MAX materials like Ti 3 SiC 2 , Zr 2 AlC 2 , V 2 AlC, V 4 AlC 3Àx , V 12 Al 3 C 8 , Mo 2 TiAlC 2 , Mo 2 Ti 2 AlC 3 , Ti 3 AlC 2 , Ti 2 InC, Zr 2 InC and Hf 2 InC have demonstrated unique and promising structural, mechanical, electrical and optical properties. [25][26][27][28][29][30][31] MAX phases have great importance in shielding and coating applications, such as the in situ growth of MAX phase coatings on carbonised wood and their terahertz shielding properties, 32 exfoliation and defect control of the two-dimensional few-layer MXene Ti 3 C 2 T x for electromagnetic interference shielding coatings, 33 highly conductive and scalable Ti 3 C 2 T x -coated fabrics for efficient electromagnetic interference shielding, 34 "beyond Ti 3 C 2 T x : MXene for electromagnetic interference shielding" 35 and MAX phase-based electroconductive coating for high-temperature oxidizing environment. 36 As per the literature, there is no experimental or computational study reported till date on the novel Ti 3 AC 2 (A ¼ P, As, Cd) combination of titanium-based MAX materials. Since MAX phases have great importance in shielding and coating applications, as discussed above, this challenge has motivated us to computationally inspect the structural, optoelectronic, mechanical and magnetic properties of these compounds for the rst time using an ab initio approach, where calculations have been performed using the CASTEP simulation code.

Computational methodology
The rst-principles simulation was performed opting for the plane wave pseudo-potential based on DFT, 37 as employed in the CASTEP code. 38 The structural parameters were investigated by considering electron-ion interactions with a cut-off energy of 600 eV using norm conserving pseudo-potentials. 39,40 A Monkhorst-Pack grid of 10 Â 10 Â 2 was chosen to model the Brillouin zone (BZ). 41 For settling down ionic positions, the conjugate gradient method was used. 42 The structural parameters of the system in anticipation of Hellmann-Feynman forces were found to be signicantly less than 0.02 eVÅ À1 in an energy convergence criterion of 1 Â 10 À5 eV. 43 The PBE-GGA functional in addition to the Hubbard parameter U was utilized, particularly to evaluate the electronic and magnetic properties of the system. 44,45 The values of U for Ti and Cd atoms were opted as 2.5 eV and 2.0 eV, respectively. The U-values opted for these calculations are taken from the standard parameterization, as mentioned in the CASTEP code. Optical and mechanical properties were calculated under the umbrella of Kramer-Kronig relations 46,47 and the Voigt-Reuss-Hill approximation. 48,49 To determine the thermal stability of Ti 3 AC 2 (A ¼ P, As, Cd), phonon frequencies have been illustrated while utilizing density functional perturbation theory. 50

Structural and electronic properties
The structural investigations revealed that Ti 3 AC 2 (A ¼ P, As, Cd) exhibited a hexagonal crystal structure with the point group (D 6h , 6/mmm, 6/m 2/m 2/m) and space group (P6 3 /mmc, P 6c2c). Interestingly, the structure of Ti 3 AC 2 (A ¼ P, As, Cd) is found to be similar to many MAX materials, such as Ti 3 SiC 2 and Ti 3 GeC 2 , 24,51 with its lattice parameters given in Table 1. The Wyckoff positions of Ti, A and C atoms are located at 2a, 2b and 4f, respectively, as shown in Fig. 1.
The search for geometric optimization and structural stability is the initial step in any rst-principles simulation. For this reason, energy versus volume graphs for each Ti 3 AC 2 (A ¼ P, As, Cd) material are plotted in Fig. 2, and the data is tted rendering to the rule of energy of state (EOS) owing to Birch-Murnaghan. 52 The calculated formation energy of the considered Ti 3 AC 2 (A ¼ P, As, Cd) materials is negative, which indicates the structural stability of these MAX materials. 53 The negative ground state energy values for Ti 3 AC 2 (A ¼ P, As, Cd) at static equilibrium are found to be À11 919.74 Ry, À19 595.75 Ry and À32 935.30 Ry, respectively.
The band structures of Ti 3 AC 2 (A ¼ P, As, Cd) have been anticipated within the 1st Brillouin zone (BZ) along with high symmetry lines from the calculated structural parameters. Fig. 3 depicts the electronic band structures of Ti 3 PC 2 , Ti 3 AsC 2 and Ti 3 CdC 2 , and reveals the metallic behaviour of these compounds. As a matter of fact, no band gap has appeared across the Fermi level (E F ), and the electronic states of valence and conduction bands are overlapping. Such metallic behaviour of the Ti 3 AC 2 compounds is quite analogous to a few already reported materials with MAX phases. 54,55 Ti 3 AC 2 (A ¼ P, As, Cd) can offer excellent electrical, thermal and metallic conductivity. The total density of states (TDOS) of Ti 3 PC 2 , Ti 3 AsC 2 and Ti 3 CdC 2 determined at E F revealed that these materials exhibit 5.87, 6.22 and 2.95 states per eV, respectively, as depicted in Fig. 4 along with the PDOS. Moreover, it has been observed that the value of TDOS for Ti 3 AsC 2 is slightly greater than those of Ti 3 PC 2 and Ti 3 CdC 2 , indicating the more conductive nature of this compound. The occupied valence states of Ti 3 PC 2 , Ti 3 AsC 2 and Ti 3 CdC 2 are observed to be at À6.13, À6.00 and À8.35 eV, respectively, with respect to the E F , as shown in Fig. 3. Fig. 4(a) demonstrates the partial density of states (PDOS) for Ti 3 PC 2 . Hybridization in the valence band from À11.6 to À9.7 eV occurs due to the 3s states of P and 2s states of C. The hybridization between the 3d states of Ti, 3p states of P and C in the valence band from À4.2 to À2.0 eV resulted in raising the top of the valence band towards the E F . As for the conduction band, the contribution of the 3d states of Ti near the E F appears as a higher density of states in the region from 1.4 to 2.8 eV. Fig. 4(b) demonstrates the PDOS for Ti 3 AsC 2 . Hybridization of the s states of As and C appeared from À10.6 to 9.6 eV. The top of the valence band rises towards the E F due to the mix p states of As, C and the d states of Ti in the valence band from À3.6 to À1.6 eV. The maximum density of states in the conduction band appears due to the 3d states of Ti from 1.9 to 2.7 eV. Fig. 4(c) shows that the hybridization in Ti 3 CdC 2 resulted from the 2s states of C with the 3p and 3d states of Ti from À11.5 to À9.5 eV. The maximum density of states within the valence band appears due to the 3d states of Cd from À12.6 to À10.7 eV. The interaction between the s and p states of Ti and C from À7.9 to À5.8 eV raises the valence band towards the E F . The strong hybridization between the 3d states of Ti and 2p states of C from À4.2 to À2.4 eV is evidence for Ti-C covalent bonds in Ti 3 CdC 2 . The maximum density of states in the conduction band from 1.5 to 2.4 eV appeared due to the 3d states of Ti. 56 3.2. Optical and dynamical properties 3.2.1 Optical properties. It is well known that optical properties of materials have strong correlations with their electronic band structure. 56 The optical properties of the Ti 3 AC 2 (A ¼ P, As, Cd) compounds were calculated with a Gaussian smearing of 0.5 eV and variation in energy from 0 to 20 eV. The interaction of electromagnetic (EM) radiation is supposed to be plane polarized along [1 0 0] with the material and the optical parameters, i.e., refractive index, extinction coefficient, absorption, loss function, optical conductivity and reectivity, have thus been investigated. 57 3.2.1.1 Complex dielectric function. The complex dielectric constant is an energy dependent function that has major contributions towards the optical properties of materials. It has real (3 1 (u)) and imaginary parts (3 2 (u)) that provide information regarding the polarization/dispersion and absorption of light, respectively, in relation to the electronic band structure of materials. In Fig. 5  at 0 eV, respectively. The imaginary part also demonstrated the metallic behaviour of the MAX compounds.
3.2.1.2 Refractive index and extinction coefficient. The refractive index represents the phase velocity and the extinction coefficient describes the absorption losses when EM radiation passes through the material. Fig. 6(a) shows the variation in refractive index for Ti 3 PC 2 , Ti 3 AsC 2 and Ti 3 CdC 2 as a function of photon energies. It has been observed that Ti 3 AsC 2 demonstrates the maximum value of refractive index among the studied compounds due to considerable interaction between the valence electrons and incident photons. This interaction ultimately resulted in polarization within the material. 58 Fig. 6(b) represents the static values of the extinction coef-cient calculated for Ti 3 CdC 2 , Ti 3 PC 2 and Ti 3 AsC 2 , which are found to be 0.7, 2 and 2.4, respectively. Ti 3 PC 2 shows two strong peaks at 0.5 and 2.2 eV, Ti 3 AsC 2 only shows a single peak at 0.7 eV, and Ti 3 CdC 2 shows two peaks at 1.2 and 2.9 eV. The extinction coefficients for the three considered materials follow a decreasing trend for energies ranging from 5 to 20 eV. Among these three materials, the values of the extinction coefficients allude to the fact that Ti 3 AsC 2 absorbs more radiation as compared to Ti 3 PC 2 and Ti 3 CdC 2 .
3.2.1.3 Absorption coefficient and energy loss function. It is well known that the absorption coefficient of any material dictates the amount of photon energy absorbed. Instead of absorption, the extinction coefficient helps to estimate the conversion efficiency of many optical materials for applications in solar devices. 59 The absorption spectra of Ti 3 PC 2 , Ti 3 AsC 2 and Ti 3 CdC 2 , as shown in Fig. 7(a), reveal the metallic behaviour of the compounds due to the absorption of incident photons of all energies. A sharp increasing trend in the absorption coefficient has been observed from almost 5 eV, and at energies about 14 eV, it decreases drastically. Ti 3 AsC 2 exhibits maximum absorption as a result of transitions from the p state of As and the 3d state of Ti 60 when compared with Ti 3 PC 2 and Ti 3 CdC 2 . The sudden drop of absorption above 13 eV might lead to plasma resonance.
The loss function describes the plasma resonance frequencies that appear due to effects like dispersion, heating and scattering. At plasma frequencies, the loss function demonstrates its maximum value. Fig. 7(b) shows the loss functions and bulk plasma frequencies for Ti 3 PC 2 , Ti 3 AsC 2 and Ti 3 CdC 2 to be 7.8 (13.3 eV), 29 (15.7 eV) and 7.3 (14.3 eV), respectively. At higher frequencies, the loss function for Ti 3 CdC 2 attains its minimum value; hence, this compound could be a suitable dielectric material. If the plasma frequencies are slightly lower than that of the incident photons, the compounds are considered transparent. 59 3.2.1.4 Optical conductivity and reectivity. Optical conductivity describes the conductivity of EM radiation with a threshold frequency through the material's surface by interand intra-band transitions. It approximately follows a similar trend as that for the absorption spectra of MAX materials. Fig. 8(a) depicts the optical conductivity of the Ti 3 AC 2 (A ¼ P, As, Cd). The optical conductivity uctuates between 0.2 to 7.0 eV, and attains its maximum value, i.e., 2.1, 2.4 and 2.7 eV for   Ti 3 PC 2 , Ti 3 AsC 2 and Ti 3 CdC 2 , respectively, followed by a sharp decrease and a dip between 9 and 10 eV.
Reectivity helps to explain the surface behavior of MAX materials. Reectivity is a ratio of the energy possessed by incident photons to that of reected photons. Maximum reectivity is observed specically in the UV and in the moderate IR regions. In the visible region, the considered MAX materials offer 44% reectivity and are potential candidates to reduce solar heating. 54 The static values of reectivity for Ti 3 PC 2 , Ti 3 AsC 2 and Ti 3 CdC 2 are found to be 0.75, 0.78 and 0.63, respectively, at an incident energy of 0 eV, as shown in Fig. 8(b). A sharp increasing trend in reectivity towards its corresponding maximum values at 8.9 eV (0.99), 10 eV (0.99), and 9.2 eV (0.94) has been observed. Aerwards, sharp and spiky dips appeared around 12 eV followed by further enhancement in the reectivity values.
3.2.2 Dynamical properties. It is worth mentioning that neither theoretical nor experimental efforts have been made so far to explore the dynamical properties of these materials. Therefore, at present, a comparative study of the observed results with the literature is not possible. However, the outcomes pertaining to calculated phonon frequencies could be benecial for determining the dynamical and thermodynamical properties of these MAX phases experimentally.
As there are 12 atoms in Ti 3 AC 2 (A ¼ P, As, Cd), resulting in 36 phonon branches or modes of vibration, three modes at zero frequency are recognized as acoustic modes and the remaining 33 modes are called optical modes of vibration. From these 33 optical modes of vibration, 12 modes are found to be Raman active, 9 modes are IR active and 12 calculated modes are found to be inactive modes. Our calculated total modes for the Ti 3 AC 2 (A ¼ P, As, Cd) phases are consistent with a previous theoretical study of the lattice dynamics of Al-containing carbides M 3 AlC 2 (M ¼ Ti, V, Ta). 61 However, a select 7 (out of 12) Raman and 6 (out of 9) IR active modes for each phase are given in Table 2, and the remaining Raman and IR active modes can be called degenerate modes, as mentioned above. These compounds are found to be dynamically stable because no imaginary frequency or so mode 51,57 appears at the gamma (G) point. The calculated symmetries and phonon frequencies for the considered  compounds, i.e., Ti 3 AC 2 (A ¼ P, As, Cd), are listed in Table 2. For these compounds, the modes of vibration are categorized by irreducible representations 62 of the point group symmetry D 6h 6/ mmm and the space group symmetry P6 3 /mmc in the hexagonal phase of the crystal structure. The highest frequencies of the Raman modes are observed at 621, 656 and 644 cm À1 , while the highest IR modes are observed at 542, 569 and 603 cm À1 for Ti 3 PC 2 , Ti 3 AsC 2 and Ti 3 CdC 2 , respectively. The modes in the range of frequency between 175 cm À1 to 546 cm À1 are due to the similar motion of Ti and C atoms in all the structures. The highest frequency modes are due to the motion of carbon atoms. The low frequency Raman modes around 98 cm À1 are due to the motion of the heavy atoms Ti and As in the Ti 3 PC 2 and Ti 3 AsC 2 phases, respectively. However, the low frequency Raman modes at 55 cm À1 are due to the motion of the Cd atom in the Ti 3 CdC 2 system.

Mechanical properties
Aer experiencing external stress, a material's response may correspond to its elastic properties, i.e., bond strength and mechanical stability. The mechanical properties of Ti 3 AC 2 (A ¼ P, As, Cd) at various pressure values, i.e., 0 GPa, 5 GPa and 10 GPa, are listed in Tables 3 and 4. Shear deformation can illustrate the ductility of a material. 63 Hooke's law, tted to shapeless crystals, has been used in the framework of the robust CASTEP simulation code along with the stress-strain behavior in order to calculate the elastic parameters. As reported, the strain pattern of the materials can be reduced signicantly for crystals with a low symmetry having mutually independent elastic constants. 64 The elastic constants of the considered MAX materials satisfy Born's criterion 57 with positive values, leading to the assessment that these materials are mechanically stable. The anisotropy factor for the hexagonal phase of MAX materials is dened as K c /K a ¼ (C 11 + C 12 À 2C 13 )/(C 33 À C 13 ), where K c and K a are the compressibility coefficients along the c-and a-axes, respectively. Bulk and shear moduli are used to determine the hardness, 65 whereas Young's modulus is used to estimate the stiffness of    0  318  237  58  382  197  5  335  105  63  365  118  10  297  95  111  348  184  Ti 3 AsC 2  0  340  62  79  368  146  5  343  67  84  367  148  10  347  71  80  376  147  Ti 3 CdC 2  0  354  135  211  265  140  5  355  135  202  263  137  10  392  125  240  256  176 solid materials. 56 The elastic constants of the considered MAX materials reveal their anisotropic nature. Table 4 summarizes the mechanical properties of the MAX materials using the Voigt-Reuss-Hill approximation. [66][67][68] The Young's (E), bulk (B) and shear (G) moduli can be expressed as follows: and The values of Young's modulus follow the trend of Ti 3 AsC 2 > Ti 3 PC 2 > Ti 3 CdC 2 , revealing that Ti 3 AsC 2 is a bit stiffer than the Ti 3 PC 2 and Ti 3 CdC 2 compounds. The values of the bulk modulus follow the trend of Ti 3 CdC 2 > Ti 3 AsC 2 > Ti 3 PC 2 . The calculated value for shear modulus follows the order of Ti 3 AsC 2 > Ti 3 PC 2 > Ti 3 CdC 2 . In solids, Pugh's ratios B/G and G/B decide the brittle or ductile nature of materials. If B/G > 1. 75 and B/G < 0.5, then the material is considered ductile; otherwise, it is brittle. 52,58 According to Pugh's criteria, Ti 3 PC 2 and Ti 3 AsC 2 possess a brittle nature, while Ti 3 CdC 2 is found to be a ductile material. The value of the machinability index 69,70 follows the order of Ti 3 CdC 2 > Ti 3 AsC 2 > Ti 3 PC 2 .
Poisson's ratio y, the shear anisotropy factor A and the linear compressibility coefficient can be calculated using the expressions given below: Poisson's ratio determines the degree of covalent bonding in materials. 51 It has a value of 0.25 for ionic materials and 0.1 for covalent materials. Table 4 shows that Poisson's ratios for the studied MAX materials are around 0.25; therefore, the materials exhibit ionic character. The anisotropy factor helps to understand the isotropic or anisotropic nature of materials. If its value is 1, then the material is considered isotropic; otherwise, it is an anisotropic material. 51 In the current study, the MAX materials are observed to be anisotropic, and might have potential applications in crystal physics and engineering sciences. 71 Table 4 reveals that the considered MAX materials exhibit large linear compressibility (a) along the a-axis instead of along the c-axis. 51 3.3.1 Bond stiffness. Generally, Pugh's ratio is not considered a good indicator for MAX phase materials when it comes to exhibiting their high damage tolerance and fracture toughness experimentally. Thus, the computational mode of bond stiffness can well predict the damage tolerance of these MAX phases using the ratio of bond stiffness of the weakest and strongest bonds. This criterion is also applied nowadays to ternarylayered borides such as MoAlB. 72,73 In MAX phases, chemical bonding involves the incorporation of ionic, covalent and metallic bonding, wherein bonding between M-X slabs is deemed stronger than bonding among the M-A slabs. 22,72 Thus, bond stiffness of the M-A slabs decreases with the increase of that of the M-X slabs in MAX phases. 73 Furthermore, according to mechanics, for a classical spring, a relation between load and deformation is dened by Hooke's law. 73 It is assumed here that a similar relation exists for chemical bonds in a solid, and so bond stiffness can be utilized to characterize and quantify bond strength. Specically, the bond length d as a function of pressure P can be estimated using lattice parameters and internal coordinates. As variation in P causes changes in the bond strength, the relative bond lengths (d/d 0 ), where d 0 denotes the bond length at 0 GPa, should be linked to P by a quadratic curve 74 obeying eqn (7), shown below. The slope of such a curve is dened as 1/k, where k represents the bond stiffness. 73 Young's, bulk and shear moduli, Pugh's ratio, machinability index, Poisson's ratio, shear anisotropic factor, and linear compressibility coefficients of Ti 3 AC 2 (A ¼ P, As, Cd) where C i (i ¼ 0, 1, 2) is the quadratic tting coefficient. Fig. 9(a)-(c) displays the behavior of normalized bond lengths calculated for Ti 3 AC 2 (A ¼ P, As, Cd) as a function of pressure from 0 GPa to 20 GPa. The declining patterns of all bonds on increasing pressure fortify our general viewpoint that bond strength increases if pressure grows systematically. Table  5 presents the various bond lengths of the considered materials, i.e., Ti 3 AC 2 (A ¼ P, As, Cd), calculated at a diverse range of pressures, ranging from 0 GPa to 20 GPa. As expected, the variation of bond strength in the Ti 3 AC 2 (A ¼ P, As, Cd) compounds follows the universal change of electronic conguration due to the increase of the atomic radii of the A elements, i.e., P / Cd. The results reveal that the bond length decreases with increasing pressure. In addition to this, C-Ti bonds are  found to be stiffer than P-Ti bonds, while P-Ti bonds are stronger than As-Ti and Cd-Ti bonds in the Ti 3 AC 2 (A ¼ P, As, Cd) family. Since shorter bond lengths lead to strong bonding, the M-C slabs possess stronger bonds as compared to bonds in the M-A slabs. It should be noted that at 0 GPa, the C1-Ti4 bond (2.10523Å) is 3.2% shorter than its counter bond C2-Ti1 (2.18472Å) in Ti 3 PC 2 . This implies that the C1-Ti4 bond is stronger. Similar analyses on the A elements (A ¼ P, As, Cd) reect that the P1-Ti6 bond (2.48509Å) is 3.0% and 19.9% shorter than the As1-Ti3 (2.56021Å) and Cd1-Ti6 bonds (2.97993Å), respectively, indicating that the P1-Ti6 bond is comparatively stronger than its counterparts, i.e., As1-Ti3 and Cd1-Ti6. Table 6 discloses that the P2-Ti5 bond possesses a larger magnitude of bond stiffness, i.e., 654 GPa, compared to that of the As2-Ti5 bond (552 GPa) but less than the bond stiffness calculated for Cd2-Ti5 (1193 GPa). Moreover, Table 6 lists the coefficients of the second order polynomial t of relative bond lengths as a function of pressure for Ti 3 PC 2 , Ti 3 AsC 2 and Ti 3 CdC 2 . Here it is noteworthy that the negative magnitudes of coefficient C 1 and positive magnitudes for C 2 reveal an increase in the deformation resistance to compression with increasing pressure. This result is obvious from Fig. 9, which illustrates the decreasing trend of bond length. However, some negative values of coefficient C 2 demonstrate that an increase in Table 6 Coefficients (C 1 , C 2 ) of the second order polynomial fit of relative bond length as a function of pressure and bond stiffness (k) for Ti 3 AC 2 (A ¼ P, As, Cd) the deformation resistance to compression with increasing pressure is relatively slow, as noticed particularly for the Ti2-Ti6 bond. In addition, it can be noticed that the relative stiffness (i.e., the ratio of the bond stiffness of the weakest bond to that of the strongest bond) is greater than 1 2 for Ti 3 PC 2 and Ti 3 AsC 2 but lower than 1 2 for the Ti 3 CdC 2 compound (Table 6). This means that Ti 3 PC 2 and Ti 3 AsC 2 are closer to typical ceramics, which possess low damage tolerance and fracture toughness. 72,75 However, in the case of the Ti 3 CdC 2 compound, a few unusual properties might be expected, such as its unusual stiffness of 4526 GPa for the Ti2-Ti6 bond.

Magnetic properties
The interaction of the constituent elements of a solid and the crystal elds produces signicant effects, which are duly recognized as magnetism. 76 To account for these effects in the considered MAX compounds, the DFT + U functional was implemented. Regarding this, the spin-polarized DOS of Ti 3 PC 2 , Ti 3 AsC 2 and Ti 3 CdC 2 were estimated and are shown in Fig. 10(a)-(c). The Fermi level was adjusted at 0 eV. It is observed that the spin up [ states are exact replicas of the spin down Y states in each studied compound, assuring the antiferromagnetic behavior of these materials. Moreover, due to such a symmetric behavior of the spin up [ and spin down Y states, the net magnetic moment for all compounds is observed to be zero. The existence of few states at the E F further endorses the metallic nature of the compounds. It is important to mention here that prior to this study, neither theoretical nor experimental observations regarding the magnetic properties of these materials have been reported.
In Fig. 11(a)-(c), the charge difference calculation (isosurface charge density) plots clearly demonstrate the charge accumulation (yellow color) and charge depletion (cyan color). It has also been noticed from the charge density plots that most of the charge is either accumulated or depleted in between the interlayers and on the Ti atoms in the case of the Ti 3 PC 2 and Ti 3 AsC 2 composites, while in the case of Ti 3 CdC 2 , an inadequate amount of charge is accumulated and depleted on the C atoms. These results illustrate the antiferromagnetic behavior of the studied compounds.

Conclusions
In the present article, a DFT-based rst-principles study has been carried out to investigate the structural, optoelectronic, mechanical and magnetic properties of Ti 3 AC 2 (A ¼ P, As, Cd). As no signicant band gap exists between the valence and conduction bands, this fact depicts that the considered materials behave like conductors. The presence of DOS around the E F describes the contribution of Ti 3d states to the conduction mechanism. The results pertaining to the elastic behavior indicated that these materials are not only thermally stable but mechanically stable as well. As per the data from Pugh's ratio, Ti 3 PC 2 and Ti 3 AsC 2 demonstrate a brittle nature, while Ti 3 CdC 2 is ductile in nature. The values of relative stiffness greater than 1 2 reveal that Ti 3 PC 2 and Ti 3 AsC 2 are closer to typical ceramics, which possess low damage tolerance and fracture toughness. The magnitude of reectivity reveals that these materials can be potential candidates for coating various devices to prevent solar heating. Moreover, the studied materials are dynamically stable and exhibit antiferromagnetic behavior. Henceforth, these MAX materials with hexagonal phases would be very appropriate potential candidates for coating applications.

Conflicts of interest
There are no conicts to declare.