Ab initio study of the structure, elastic, and electronic properties of Ti3(Al1−nSin)C2 layered ternary compounds

The MAX phase materials such as layered ternary carbides that simultaneously exhibit characteristics of metallic and ceramic materials have received substantial interest in recent years. Here, we present a systematic investigation of the electronic, structural stabilities, and elastic properties of Ti3(Al1−nSin)C2 (n = 0,1) MAX phase materials using the ab initio method via a plane-wave pseudopotential approach within generalized-gradient-approximations. The computed electronic band structures and projected density of states show that both Ti3SiC2 and Ti3AlC2 are metallic materials with a high density of states at the Fermi level emanating mainly from Ti-3d. Using the calculated elastic constants, the mechanical stability of the compounds was confirmed following the Born stability criteria for hexagonal structures. The Cauchy pressure and the Pugh’s ratio values establish the brittle nature of the Ti3SiC2 and Ti3AlC2 MAX phase materials. Due to their intriguing physical properties, these materials are expected to be suitable for applications such as thermal shock refractories and electrical contact coatings.

www.nature.com/scientificreports/ distribution of charge density on the (1120) plane of Ti 3 AlC 2 , where robust directional Ti-C-Ti-C-Ti covalent bond chains were observed that linked to fairly weaker Ti-Al covalent bindings. In a similar study of electronic structure and bonding properties of Ti 3 AlC 2 , Wang and Zhou 29 reported that the electrical conductivity of Ti 3 AlC 2 decreases with increasing pressure, and over the whole pressure range, the material was found to exhibit elastic anisotropy. Son et al. 30 have used density functional theory (DFT) to analyze the structural, elastic, and thermodynamic properties of Ti 3 SiC 2 and Ti 3 AlC 2 crystals. In order to discover the finite-temperature properties of these crystals, the vibrational, mechanical, quasi-harmonic contributions, and anharmonic adjustment to the total free energy of the systems were determined and extrapolated and the functions of electron localization, charge densities, electronic and vibrational densities have been studied. Zhou and Zhimei investigated the electronic structure and chemical bonding in layered machinable Ti 3 SiC 2 31 . According to them, bonding within Ti 3 SiC 2 is facilitated by metallic, covalent, and ionic bonding due to the strong Ti-C-Ti-C-Ti covalent bond strings in the structure 31 . In recent years, several studies have been carried out on the mechanical properties, and structural stabilities of Ti 3 SiC 2 and Ti 3 AlC 2 32-34 that reported their excellent structural properties that are suitable for many practical applications. Synchrotron x-ray diffraction measurements showed that Ti 3 SiC 2 and Ti 3 AlC 2 are stable materials under pressure from 0 to 61 GPa at room temperature 35 . Thermal stability of bulk Ti 3 AlC 2 has been investigated 36 within 1100-1400 °C, and hydrogen has been found to alter the properties and stability of the MAX phase. Analogous facts have also been noticed in the temperature range 1473-1673 K in bulk Ti 3 SiC 2 in the hydrogen atmosphere and it was found that the dissociation of Ti 3 SiC 2 was accelerated by hydrogen 37 .
Herein, we have investigated Ti 3 SiC 2 and Ti 3 AlC 2 using plane-wave pseudopotentials (PW-PP) approach in the framework of DFT. Since hardness varies from one material to another as commonly acknowledged by materials scientists, materials with Vickers hardness greater than 40 GPa are categorized as superhard 38,39 . We have achieved a result which by far characterizes Ti 3 SiC 2 and Ti 3 AlC 2 as superhard materials which we feel none of the studies conducted so far could address.

Result and discussion
Structural properties. The layered ternary Ti 3 (Al 1−n Si n )C 2 (n = 0, 1) compounds are based on the layers of hexagonally close-packed Si/Al and Ti layers with C occupying octahedral centers between the Ti layers as depicted in Fig. 1. The end phases could also be characterized as alternating stacking of two layers of a planar close-packed Si/Al and Ti 6 C octahedral layers. The Ti atom is found to be located at 4f. (0.33, 0.67, z), Al/Si atoms are positioned at 2b (0, 0, 0.25) whereas the atom of C is at 4f. (0.33, 0.67, z) Wyckoff positions. Figure 1 illustrates the crystal symmetries of the studied compounds and their computed structural parameters as well as the experimental results from available literature(s) are summarized in Table 1. The results of the equilibrium lattice constants, bulk modulus, and its pressure derivative are computed by fitting the obtained data of the equilibrium energy as well as volume to the second-order Birch-Murnaghan's equation of state (EOS) 40 . The obtained results showed the reasonability of our calculations.
(1) www.nature.com/scientificreports/ One can easily note that the difference between our obtained results and experimental data of equilibrium lattice parameters is less than 1%, showing that our results obtained at the level of the Perdew-Burke-Ernzerhof (PBE) type of generalized gradient approximations functional are sufficiently reliable. In Table 1, the bulk modulus of Ti 3 SiC 2 is higher than that of Ti 3 AlC 2 , showing that Ti 3 SiC 2 is harder than Ti 3 AlC 2 . Figure 2 demonstrates the band structures and total density of states (TDOS) computed along the high symmetry points in the brilluoin zone (BZ) using the equilibrium lattice parameters. It is seen that both valence bands and conduction bands overlap significantly resulting in no energy gap at the Fermi level. Thus, the studied compounds demonstrate metallic character which is a common feature of the MAX phase materials. However, there are more valence electrons in the Ti 3 SiC 2 unit cell than in Ti 3 AlC 2 . This gives rise to the further occupation of the bonding states near the Fermi level. The substitution of Si by Al in Ti 3 AlC 2 presents additional valence electrons per atom, and consequently, the Fermi level is moved to a higher energy level. This suggests that the increased extra valence electrons fill in the Si/Al-Ti p-d hybridized bonding states as well as the metal to metal d-d consequential bonding.

Electronic properties.
Accordingly, the filling of the bonding orbitals rises the strength of the bond and thereby increasing the bulk moduli. The energy band also exhibits a highly anisotropic character along with lesser c-axis energy dispersion. The anisotropy of the band structure near and below the Fermi level implies that, for single crystals, both Ti 3 SiC 2 and Ti 3 AlC 2 are conductors and anisotropic, and electrical conductivity is lowered along c direction than the ab-plane similar to the observed trend in the literature 28 .
The investigated total densities of states (TDOS) plot for Ti 3 SiC 2 and Ti 3 AlC 2 presented in Fig. 2 points out that the peak structures and corresponding heights of the peaks are equivalent, signifying resemblance in chemical bonding. The TDOS per unit cell at the Fermi level for Ti 3 SiC 2 and Ti 3 AlC 2 are 4.029 and 6.855 states/eV, respectively. Therefore, there is an increasing trend in the DOS at the Fermi level with an increasing number of  Elastic properties. Investigations of elastic constants are vital for applications related to the mechanical properties of solids. They provide information on stability, bonding, ductility, brittleness, anisotropy, compressibility, Vicker's hardness, and stiffness of solids 44,45 . For hexagonal crystals structures, five independent elastic constants ( C 11 , C 12 , C 13 ,C 33 , C 44 ) are required. Table 2 summarizes our computed results of the five independent elastic constants of Ti 3 SiC 2 and Ti 3 AlC 2 alongside available experimental and theoretical data. A stable hexagonal crystal must satisfy the following Born-Huang stability criteria 46 ; (2) C 11 > 0; C 11 − C 12 > 0; C 44 > 0; (C 11 + C 12 )C 33 − 2C 2 13 > 0  www.nature.com/scientificreports/ Table 2 demonstrates that the computed results of the independent elastic constants for Ti 3 SiC 2 and Ti 3 AlC 2 MAX phase compounds satisfy the mechanical stability criteria which signify that all the compounds are mechanically stable. It is also well known that elastic constants C 11 , and C 33 shows linear compression resistances along a and c directions, respectively, whereas C 12 , C 13, and C 44 are related to the shape elasticity. Consistent with Table 2, the value of C 11 is higher than C 33 for both Ti 3 SiC 2 and Ti 3 AlC 2 compounds which agrees well with literature results.
From the computed elastic constants, several polycrystalline elastic moduli comprising, Bulk, Shear, Young moduli, and Poisson's ratio were evaluated using Voigt 48 , Reuss 49 , and Hill 50 approximations. It is assumed that, in the Voigt scheme, the strain is uniform all along the polycrystalline materials aggregating to external strain. By following this approach, for the hexagonal lattices, the Voigt shear modulus (G V ) and Reuss shear modulus (G R ) are expressed as:

And Voigt bulk modulus (B V ), Reuss bulk modulus (B R ) by:
Hill showed that Voigt/Reuss averages give upper and lower bounds, and therefore, proposed that real effective moduli can be approximated by the arithmetic mean of the two bounds 51 . Thus, using Hill's approximations We have also computed Y , and η , which are commonly evaluated for polycrystalline materials to study their hardness. Both Y and η are defined by the following expressions as; The computed Bulk modulus, Young's modulus, Shear moduli, and Poisson's ratio of both Ti 3 SiC 2 and Ti 3 AlC 2 as defined in Eqs. (3)-(8) are listed in Table 3. The calculated values for the bulk modulus of Ti 3 SiC 2 and Ti 3 AlC 2 are 139 GPa and 182 GPa respectively. These values agree well with the reported value by Gray et al. 47 , with less than 13% and 7% deviation respectively for Ti 3 AlC 2 and Ti 3 SiC 2 . Moreover, our results for Shear modulus of 87 GPa for Ti 3 AlC 2 although are lower than the reported experimental value in Table 3, the results of Ti 3 SiC 2 of 121 GPa are in good agreement with the reported value. From comparing Tables 1 and 3, it can be seen that the calculated value of B obtained from the single crystal elastic constants summarized in Table 3 has approximately the same value as the one obtained from the data fitting in the Murnaghan's equation of state (Table 1). This indicates the accuracy and reliability of our computed elastic constants for both Ti 3 SiC 2 and Ti 3 AlC 2 MAX phase compounds.
Following the Pugh ratio, B/G shows the brittle or ductile character of materials. Pugh's critical value is 1.75. The calculated ratio B/G for Ti 3 AlC 2 and Ti 3 SiC 2 are 1.60 and 1.56, respectively, which are less than Pugh's critical value. As such, these compounds have a brittle feature which agreed well with the result given in Table 3 52 . Cauchy relation defined as: C c = C 13 − C 44 , is another parameter signifying ductility or brittleness of a material.
Positive values of C c shows ductility otherwise the material is brittle 53 . The evaluated C c of the ternaries are -44 and -28 GPa respectively. From these values, one can conclude that the studied materials are brittle in nature C 11 + C 12 + 2C 33 + 4C 13 www.nature.com/scientificreports/ which confirmed the Pugh's result. Consequently, the brittle nature of Ti 3 AlC 2 and Ti 3 SiC 2 can be related to their ceramic character. Young's modulus (Y) measures the stiffness of a material. The higher the Y, the stiffer a material is. Our result presented in Table 3 shows that there is good agreement with the reported values of 215 GPa and 297 GPa for Ti 3 AlC 2 and Ti 3 SiC 2, respectively. Information about the bonding forces can be obtained via Poisson's ratio (η). The(η) for Ti 3 AlC 2 and Ti 3 SiC 2 are 0.24 and 0.23 respectively, which shows the interatomic forces within studied materials are central since upper and lower limits of the Poisson's ratio is 0.5 and 0.25 respectively, and the calculated values fall within the two limits. Our results are closer to the experimental value of 0.178 for Ti 3 AlC 2 and 0.248 for Ti 3 SiC 2 52 . We have further calculated the Vickers' hardness H v 54 of studied compounds. Vickers's hardness is another key mechanical property of solids that explains stability, which is predicted using Eq. (9). It is reported that solids with Vickers hardness H V > 40 GPa 38 are graded as super hard solids. The calculated H v of Ti 3 AlC 2 and Ti 3 SiC 2 MAX phase compounds are 40.28 GPa, and 46.75 GPa respectively (Table 3). Therefore, these crystals, have an excellent ability to withstand dents or scratches.

Method
Ab initio calculations were used to investigate the elastic, and electronic properties of Ti 3 SiC 2 and Ti 3 AlC 2 using PW-PP as implemented in Quantum Espresso 55 . Generalized gradient approximation (GGA) parametrized by Perdew-Burke-Ernzerhof (PBE) is used to treat exchange and correlation (XC) energy 56 . The core ion and valence electrons interactions were described using ultrasoft-pseudopotentials (UPP). A 600 Ry kinetic energy cut-off of the plane wave is used in the calculations. The electronic configurations: 3s 2 , 4s 2 , 3p 6 , 3d 2 for Ti, 3s 2 , 3p 2 for Si, 3p 1 , 3s 2 for Al and 2s 2 , 2p 2 for C were considered for the valence electrons. For the Brillouin zone (BZ) integration, 12 × 12 × 12 k-points mesh was generated using the Monkhorst-Pack scheme 57 . These parameters were found to be adequate to converge total energies up to 10 -8 eV. Both studied materials were fully relaxed in terms of cell parameters and atomic positions. Analysis of independent elastic constant (C ij ) were performed using thermo_pw 45 . C ij delineates response of materials to macroscopic stress. In computing elastic constants, a small strain, e is applied to a material and the variation of total energy per volume, U of the material is obtained 58 : where V 0 and E represent the equilibrium volume and the difference between the initial and deformed total energy of the system respectively. The hexagonal Ti 3 SiC 2 and Ti 3 AlC 2 MAX phase compounds are characterized by five independent elastic constants which include C 11 , C 12 , C 13 ,C 33 and C 44 . Therefore, the elastic matrix of the hexagonal system is written as 59,60 ;

Conclusion
In this work, the structural stability, electronic, and mechanical properties were investigated using ab-initio calculations. The complete set of independent elastic constants C ij, shear modulus, bulk modulus, Poisson's ratio, and Young's modulus were calculated. Our results showed that the studied ternaries are mechanically stable and are super hard materials with Vicker's hardness as large as 46.75 GPa and 40.28 GPa for Ti 3 SiC 2 and Ti 3 AlC 2 respectively. The investigated electronic band structures, TDOS, and PDOS showed the metallic behavior of these compounds. In Ti 3 SiC 2 , the top of the VB and bottom of the CB were found to be dominated by the Si-3p, C-2p, and Ti-3d energy states while for the Ti 3 AlC 2 the top and bottom of VB and CB were respectively found to be shaped by Al-3p, C-2p, and Ti-3d orbitals. We expect that our findings will provide suitable guidance for experimental and theoretical studies on these interesting MAX phases. . . . C 11 C 13 . . . . .