Local density functional calculations of the electronic structures of Ti2AlC and Ti3AlC Samir F. Matar, Yann Le Petitcorps and Jean Etourneau Institut de Chimie de laMatie`re Condense�e de Bordeaux–CNRS, Cha�teau Brivazac, Avenue du Docteur Schweitzer, F33600 Pessac, France Local density functional calculations are used to address the electronic structures and the properties of chemical bonding of two definite phases formed within the ternary system Ti, Al and C: Ti2AlC and Ti3AlC.From the analyses of the density of states and of the crystal orbital overlap populations of the respective phases within the ASW method the role of C is assessed. Moreover, the bonding within TiC is discussed concomitantly. These calculations are of interest in the composite field to understand the mechanisms of formation of new compounds at the matrix/reinforcement interface.Carbon fibre-reinforced titanium–aluminium intermetallic ably large atomic sphere radii leading to a large overlap in the ASA. composite materials are of interest for space and aeronautics applications. In recent years several works have been devoted In k-space, the Brillouin zone integration is achieved on a uniform mesh of points in the irreducible wedge of the relevant to their investigation, both experimentally1–3 and theoretically. 4,5,6 From the latter point of view, we modelled recently Bravais lattice. The matrix elements are constructed involving solutions of the Schro�dinger equation up to the secondary l the influence of substituted and inserted carbon within the alloy lattice of TiAl on its electronic structure.6 Based on quantum number, lmax+1, where lmax=2 for Ti and Al and 1 for C and the ES.The contributions associated with the lmax+1 quantitatively resolved chemical bonding criteria we proposed that carbon should substitute preferentially for aluminium higher angular momenta are relative to non-explicitly calculated terms in the limited ASW basis set7 but should always when it enters the TiAl lattice.This is supported by the actual occurrence of Ti-rich carbide compounds such as Ti2AlC in be lower than 0.1 electron in order to ensure a convergence of the immediate neighbourhood of the intermetallic matrix of TiAl. Moreover, the growth of such a carbide phase from this alloy leads to an enrichment of Al atoms at the intermetallic/ ternary compound interface.For the titanium-rich alloy Ti3Al, carbon is inserted to give Ti3AlC. In all cases TiC is formed in the vicinity of the carbon fibre (Fig. 1). Thus Ti–C interactions form TiC, whereas Ti2AlC and Ti3AlC are formed respectively with the TiAl–C and Ti3Al–C couples, according to a diffusion path determined by Clochefert.3 In this, the second part of our investigation of carboncontaining TiAl, we address the electronic properties of the titanium-rich carbides Ti2AlC and Ti3AlC actually forming in the {Ti–Al–C} phase diagram, with the objective of examining the influence of carbon on the chemical bonding within the alloy lattice.Method of calculation As in our earlier investigation,6 the electronic properties of all carbon-containing alloy systems were calculated using the ab initio self-consistent augmented spherical wave (ASW) method.7 The ASW method allows one to describe the electronic properties of a material starting from those of its atomic constituents.The calculations are based on the density functional theory in which the effects of exchange and correlation are treated in the local density approximation within the scheme of von Barth and Hedin, and Janak.8 The ASW method uses the atomic sphere approximation (ASA) where each atom is surrounded by a sphere.Within the atomic spheres the potential is assumed to be spherically symmetric. The ASA imposes a unit-cell volume equal to the total volume of the spheres, leading to their overlap. This is unproblematic for close-packed crystal structures, but for loosely packed ones the empty space must be represented by use of ‘empty spheres’ (ES), i.e., pseudo-atoms with Z=0 atomic number and no core Fig. 1 Electron micrographs of the chemical interaction between states. ES are introduced in order to account for the interstitial carbon and Ti–Al intermetallics. Upper: C–TiAl; lower: C–Ti3Al.(Reproduced with permission from ref. 3.) space in the lattice and to avoid the use of otherwise unreason- J. Mater. Chem., 1997, 7(1), 99–103 99the charges. The self-consistent cycle is carried out until the is an anion at face centres of the cube; space group Pm3m). Therefore Ti3AlC [Fig. 1(b)] can be described as an antiper- following convergence criteria are reached: DE=10-8 Ry [1 Ry (rydberg)=13.6 eV] for the total energy and DQ=10-8 ovskite because Ti atoms occupy the face centres, Al and C being at corner and cube-centre positions respectively.From for the charge difference between two successive cycles. In this work all calculations were carried out at the experimental the point of view of coordination polyhedra, the structure can be regarded as a three-dimensional array of Ti6C octahedra lattice constants obtained from ref. 3. Furthermore, in this work the chemical bonding features are discussed based sharing corners. It is hence a poorly packed structure because the midpoints of the edges, D 0 0, 0 D 0 and 0 0 D, are vacant on the so-called COOP (read CO-OP: crystal orbital overlap populations), of which a comprehensive account was given by sites where ES had to be introduced in the ASA. Before examining the electronic structures of these two Hoffmann from the quantum chemistry standpoint (extended Hu�ckel calculations).9 This allows for the DOS features to be carbides, it is relevant to consider the coordination polyhedra in TiC.We stress that this binary carbide is modelled here in discussed on bases of chemical bonding criteria by weighting them with the sign and magnitude of the overlap integral a 151 composition although it is known to be sub-stoichiometric in carbon, i.e.TiCx with 0.56<x<0.98. between the relevant orbitals. We recently implemented the COOP in the ASW method10 with the objective of obtaining TiC crystallizes in the NaCl-type structure (space group Fm3m) with four formula units per unit cell [Fig. 2(c)]. Ti and more precise information on the chemical bonding from first principles. C are at the origin and D D D positions, respectively. Ti and C are octahedrally coordinated with each other; consequently, Ti6C octahedra share edges. Crystal structures and setup of the unit cells for By comparing the three structures an interesting observation ASW calculations appears: dimensionality increases from Ti2AlC to Ti3AlC and TiC.In contrast to several Ti2AlM (M=Nb, V, Cr, Mn) compounds Ad hoc and non-unique choices of the atomic spheres radii which crystallize in a tetragonal structure,5 Ti2AlC is hexagonal in the ASA were such that: rTi/rES=1.26, rAl/rES=1.30 and with a large c axis and two formula units per cell.It crystallizes rC /rES=1.10. Such values were tested as one choice which in the Cr2AlC-type structure11 with the P63/mmc space group simultaneously minimizes the overlap between the spheres and and Ti at (4f ), Al at (2c) and C at (2a) Wyckoff positions. The yields in converged lmax+1 residual charges. structure is shown in Fig. 2(a). It can be regarded as an alternating stacking of triangular prisms and octahedral Ti polyhedra containing Al and C atoms, respectively, along the Calculations and Results c-axis.Ti6C octahedra share edges and form two-dimensional Partial charges layers perpendicular to the c axis. From this low dimensionality, the structure is poorly packed and in the ASA, ES had to Table 1 gives the partial charges for Ti2AlC and Ti3AlC (DQ be introduced between the layers at sites related to those of Ti designates the deviation from neutrality).The overall features general positions. of charge transfer are similar, in that it occurs from the two With one formula unit per unit cell, Ti3AlC has a structure metallic atoms towards the non-metal and the empty spheres, derived from the cubic perovskite ABX3 (where A and B are i.e.Ti,Al�C,ES. The averure from neutrality per large and small cations at corner and centre positions, and X metal is then ca. 0.73 in Ti2AlC and ca. 0.79 in Ti3AlC. In as far as carbon receives ca. 0.6 electrons in both carbides, this charge excess leads to the larger occupancy of ES in the latter. However, the differences which characterize the DQ values of Ti and Al in each compound should be addressed.They arise from the fact that Al exhibits a larger d character in Ti2AlC (represented by the higher d occupancy) than in Ti3AlC. This should indicate a larger hybridization between Ti and Al in Ti2AlC with respect to Ti3AlC. By virtue of this mixing there is an enhancement of the sp character of Ti which could be due to its interaction with Al and/or with C.This establishes a covalent character of the bonding in these materials, to be further illustrated in next section. Density of states (DOS) The upper panels (a) of Fig. 3 and 4 show the site-projected DOS of Ti2AlC and Ti3AlC. Energy reference along the x axis Table 1 Site and l-projected partial charges for Ti2AlC and Ti3AlC s p d f DQ Ti2AlC(ES)2:a Al 1.08 0.82 0.23 (0.02) -0.84 Ti 0.37 0.56 2.33 (0.08) -0.67 C 1.40 3.05 0.15 (0.02) 0.62 ES 0.54 0.20 (0.04) — 0.78 Ti3AlC(ES)3:b Al 0.88 1.25 0.15 (0.01) -0.69 Ti 0.32 0.51 2.28 (0.07) -0.82 C 1.29 3.07 0.21 (0.03) 0.60 ES 0.62 0.19 (0.05) — 0.85 Fig. 2 (a) Hexagonal structure of Ti2AlC. (b) Perovskite-derived structure of Ti3AlC. (c) NaCl-type structure of TiC. (Reproduced with aQ=2(-0.67)-0.84+0.62+2(0.78)=0 (neutrality).bQ=3(-0.82)- 0.69+0.60+3(0.85)=0 (neutrality). permission from ref. 3.) 100 J. Mater. Chem., 1997, 7(1), 99–103is taken with respect to the Fermi level (EF) within a reduced energy range (-8 to +8 eV), i.e. excluding the low lying C 2s states, to make the presentation clear. The y axis gives the DOS per atom and unit energy (atom-1 eV-1).In both carbides, the Fermi level crosses the lower part of the Ti 3d states centred above EF because of the nearly empty d band. These states show much larger structures towards the lower energies than in TiAl,6 where they interact solely with Al s,p states because of the extra interaction with carbon. This is shown by the peaks between -6 and -4 eV in Ti2AlC and -7 and -3 eV in Ti3AlC.From a preliminary crystal-field analysis, the peaks in Ti DOS at -2, 1 and 2 eV arise mainly from in-plane xy and x2-y2 d orbitals. However, it is difficult to separate totally the contributions of the five different d orbitals because of their hybridization and of the collective character of the electrons. The DOS at EF , n(EF), are dominated by Ti 3d and show a sharp peak in Ti2AlC, probably due to Ti–C interactions (see next section) whereas such a feature is absent in Ti3AlC where n(EF ) are three times lower. There is a larger contribution from Al states at and above EF in Ti2AlC which agrees with our discussion of the charges, leading to a mixing between Ti 3d and Al 3p.In Ti3AlC the DOS are dominated by carbon and Ti on one hand and Ti d on the other hand below and above EF, respectively.Al plays a less important role at EF in this carbide and its ‘sp’ DOS are seen in the energy windows -8 to -6 eV and -4 to -2 eV. A relevant feature is the broadness of the band over the energy range -8 to-2 eV as opposed to the sharp peaks in the same range in Ti2AlC. In both compounds the DOS of the ES closely follow those of the other species, which is consistent with charge transfer into them from the other sites.Fig. 3 Ti2AlC: (a) site projected densities of states in atom-1 eV-1 (solid line Ti; dashed line Al; dotted line C; dash-dotted line ES); (b) At this point the discussion of the mixing features solely partial COOP for pair interactions: Ti–C (solid line), Ti–Al (dashed from the partial DOS cannot give more information about the line), Al–C (dotted line) chemical bonding in the two compounds.A further step must be undertaken, by examining the COOP. Crystal orbital overlap populations (COOP) The features of chemical bonding can be assessed further by using the COOP. In the lower panels (b) of Fig. 3 and 4, the COOP are shown for the interactions between the different atoms in the two compounds plotted in the same energy range, i.e.for Ti–C, Ti–Al and Al–C. Along the y(COOP) axis, positive, negative and zero values point to bonding, antibonding and non-bonding states, respectively. In Ti2AlC, below EF, Ti–C interactions predominate, whereas in TiAl Ti–Al interactions are the driving interaction for the bonding.6 They exhibit largely bonding character in the range -6 to-2 eV, and follow exactly the Ti DOS in the same energy range in which C 2p states dominate.Thus the sp character introduced into Ti (cf. Table 1) mainly arises from its interaction with carbon. The antibonding counterpart can be seen in the conduction band (2 to 6 eV). The large separation between the bonding and antibonding peaks is indicative of a strong interaction assimilated with a s-like interaction.This somehow opposes the Ti–C interaction in Ti3AlC, where less localized bonding states are seen to extend over a wide band in a larger energy window (s- and p-like bonding). Interestingly, carbon is engaged not only in Ti–C interactions but also in Al–C ones, Ti–C and Al–C interactions having bonds in the same energy range.This is in contrast to Ti2AlC, where only Ti–C interactions are present in the valence band. This is supported experimentally, whereby the solubility of carbon is much more important in Ti3Al than in TiAl.11 However, the Al–C bond seems weaker because it is largely antibonding from -4 to -2 eV whereas Ti–C is bonding over a wider energy range. For the Ti–Al interaction, the distance between these two Fig. 4 Ti3AlC: (a) site projected densities of states in atom-1 eV-1 sites is 23% shorter in Ti2AlC than in Ti3AlC. This should (solid line Ti; dashed line Al; dotted line C; dash-dotted line ES); (b) explain the differences appearing between the two panels for partial COOP for pair interactions: Ti–C (solid line), Ti–Al (dashed line), Al–C (dotted line) the Ti–Al interaction and should assess the d character brought J.Mater. Chem., 1997, 7(1), 99–103 101into Al by its bonding with Ti (cf. Table 1). As a matter of fact, Ti–Al becomes important only around EF, i.e. in the DOS region where Ti d states predominate. In contrast, Ti–Al interactions in Ti3AlC are mainly seen in the energy range of Al 3p states, around -2 eV, in the same energy range as Ti–C and Al–C bonds with the largest bonding contribution at the top of the valence band, whereas they are nearly absent in the Ti–C energy range in Ti2AlC owing to the nearly two-dimensional array of Ti6C octahedra.Comparison with TiC At this point a comparison with the electronic structure of TiC is in order. Fig. 5(a) gives the site-projected DOS of TiC. They are in good agreement with those of Blaha and Schwarz (ref. 12 and refs. cited therein) who gave a full account of the electronic structures of TiX (X=C, N, O) compounds by use of a linearized APW (augmented plane waves) method. The feature of the n(EF) minima at the Fermi level is related to the refractory nature of TiX and their stability.13 From -6 eV to EF, C 2p states predominate, whereas from EF to 8 eV, Ti 3d states with their t2g (4 eV) and eg (>4 eV) components show the major contribution to the DOS.In the valence band and from 4 eV to higher energies, C and Ti states have similar shapes, which is indicative of a covalent interaction between them. The DOS of ES follow the same evolution as the Ti and C ones, indicating that charge transfer into them is from both species.In the purely Oh point symmetry, Ti 3d orbitals split into two types of manifold: t2g and eg. The projection of the DOS along them is shown in Fig. 6. In the valence band, eg orbitals Fig. 6 Oh crystal-field decomposition of the Ti d-orbital DOS: (a) Ti d(t2g); (b) Ti d(eg) have a larger contribution with respect to t2g ones; they are involved with pds-type bonding with carbon whereas pdp bonding should be less involved.Since metal Ti–Ti interactions are of the dds type, one expects little bonding of this type in the valence band. This is explained more quantitatively by examining the COOP shown in the same energy window as the DOS in Fig. 5(b). They are resolved for three kinds of interactions in the lattice, namely Ti–C, Ti–Ti and C–C. The latter two types of interactions show fewer bonding features than the former; Ti–Ti interactions are clearly less predominant (for the reasons argued above) than C–C interactions, which exhibit bonding and antibonding states in the valence band whereas Ti–Ti bonding features can only be seen in the conduction band.Thus, in TiC 3d–2p bonding is the driving bonding force. Two types of 3d–2p bonding are found with increasing energy, i.e.pds predominates over pdp in the valence band, whence the directional character of the bonding in this compound. Since the former are stronger the antibonding counterparts are reversed, following energetical order: pdp* resembled by the antibonding peak at 4 eV and pds* at higher energies. In our two carbide systems, the directionality of the bond is reduced by the presence of Al, which acts through its p and d states in its bonding to Ti.From this there is an increase in the amount of d character in the valence band. This is indicated by the larger occupation of Ti 3d with 0.52 and 0.47 electrons in Ti2AlC and Ti3AlC, respectively, in comparison to the Ti d-band occupation in TiC. As a matter of fact, it explains the larger n(EF) of the former and its vanishing value in TiC.Discussion and Conclusion Experimentally, it was found that Ti–C bonding was the driving force which controlled the C–TiAl and C–Ti3Al interactions. 3 These experimental features agree rather well with Fig. 5 TiC: (a) site projected densities of states (solid line Ti; dashed the results of our calculations, indicating strong Ti–C inter- line C; dash-dotted line ES); (b) partial COOP for pair interactions: Ti–C (solid line), Ti–Ti (dashed line), C–C (dotted line) actions which are reminiscent of the formation of TiC in the 102 J.Mater. Chem., 1997, 7(1), 99–103immediate vicinity of the carbon fibre. This should destabilize lations was performed within the MNI pole of intensive computations.the actual alloy lattice, leading to the formation of precipitates. Our investigation has shown that carbon plays different roles in Ti2AlC and in Ti3AlC. While it bonds mainly to Ti in the former, both Ti–C and Al–C bonds are present in the References valence band for the latter. This is supported experimentally 1 D. Vujic, Z. Li and S. H. Wang, Metall.T rans. A, 1988, 19, 2445. because the solubility of carbon is higher in Ti3AlC when it 2 M. Morinaga, J. Saito, N. Yukawa and H. Adach, Acta Metall. enters the alloy lattice. Valence and conduction bands are Mater., 1990, 38, 25. largely separated by non-bonding states at EF in Ti3AlC, 3 L. Clochefert, PhD Thesis, Universite� Bordeaux 1, 1995. whereas bonding metallic Ti–Al interactions are present at EF 4 S.R. Chubb, D. A. Papaconstantopoulos and B. M. Klein, Phys. in Ti2AlC. This could point to a higher ‘ionic’ character in Rev. B, 1988, 38, 12120. 5 H. Erschbaumer, R. Podloucky, P. Ro�gl, G. Temnitschka and Ti3AlC and a higher ‘metallic’ character in Ti2AlC. R. Wagner, Intermetallics, 1993, 1, 99. Our comparison of the bonding within Ti2AlC and Ti3AlC 6 S. F. Matar and J. Etourneau, J. Alloys Compd., 1996, 233, 112. to its characteristics in TiC shows that Al reduces the direc- 7 A. R. Williams, J. Ku�bler and C. D. Gelatt Jr., Phys. Rev. B, 1979, tionality of the bonding mainly through its p states, as well as 19, 6094. its d ones which interact with Ti d states. From this there is 8 U. von Barth and L. Hedin, J. Phys. C, 1972, 5, 1629; J. F. Janak, an increase in the amount of d character in the valence band. Solid State Commun., 1978, 25, 53. 9 R. Hoffmann, Angew. Chem., Int. Ed. Engl., 1987, 26, 846. To conclude, the present study has brought a new insight 10 V. Eyert and S. F. Matar, 1994, unpublished results. into the bonding features in the C–Ti–Al ternary system, 11 W. B. Pearson, Acta Crystallogr., Sect. A, 1980, 36, 724. allowing for a more quantitative description on the chemical 12 P. Blaha and K. Schwarz, Int. J. Quantum Chem., 1983, 23, 1535. bonds. 13 J. Ha�glund, G. Grimvall, T. Jarlborg and A. Fernandez Guillermet, Phys. Rev. B, 1991, 43, 14400. Facilities provided by the computer centre of the Universite� Bordeaux 1 (CRIBx1) are acknowledged. Part of the calcu- Paper 6/05113H; Received 23rd July 1996 J. Mater. Chem., 1997, 7(1), 99&nda