Electronic structure, bonding characteristics, and mechanical properties in (W2/3Sc1/3)2AlC and (W2/3Y1/3)2AlC i-MAX phases from first-principles calculations

With the recent discovery of in-plane chemically ordered MAX phases (i-MAX) of the general formula ()2AC comes addition of non-traditional MAX phase elements. In the present study, we use density functional theory calculations to investigate the electronic structure, bonding nature, and mechanical properties of the novel (W2/3Sc1/3)2AlC and (W2/3Y1/3)2AlC i-MAX phases. From analysis of the electronic structure and projected crystal orbital Hamilton populations, we show that the metallic i-MAX phases have significant hybridization between W and C, as well as Sc(Y) and C states, indicative of strong covalent bonding. Substitution of Sc for Y (M2) leads to reduced bonding strength for W–C and Al–Al interactions while M2–C and M2–Al interactions are strengthened. We also compare the Voigt–Reuss–Hill bulk, shear, and Young’s moduli along the series of M1  =  Cr, Mo, and W, and relate these trends to the bonding interactions. Furthermore, we find overall larger moduli for Sc-based i-MAX phases.

In this work, we have used first-principles calculations to investigate the electronic, vibrational, and mechanical properties of (W 2/3 Sc 1/3 ) 2 AlC and (W 2/3 Y 1/3 ) 2 AlC i-MAX phases, motivated by their recent discovery, and being the first W-based MAX phase materials. Both (W 2/3 Sc 1/3 ) 2 AlC and (W 2/3 Y 1/3 ) 2 AlC are stable with a calculated formation enthalpy of −27 and −22 meV/atom, respectively [30]. However, neither of the ternary MAX phases W 2 AlC, Sc 2 AlC or Y 2 AlC have been synthesized, explained by Meshkian et al showing that these are far from being theoretically stable, with a calculated formation enthalpy of +148, +118, and +185 meV/atom, respectively [30]. The finding of i-MAX phases, allowing introduction of non-traditional MAX phase elements like Sc, Y and W, may alter or introduce new properties as compared to previously known MAX phases. This motivates their exploration, for fundamental understanding, and potential future property tailoring.

Computational methods
All calculations were performed within the framework of density functional theory as implemented in the Vienna ab initio simulation package [32][33][34], combined with the Perdew-Burke-Ernzerhof generalized gradient approximation (PBE-GGA) [35] and the projector augmented wave (PAW) method [36,37]. The unit cells of both compounds were converged to an accuracy of 0.1 meV/atom, using a Monkhorst-Pack [38] 13 × 7 × 5 k-point grid and a plane wave cutoff energy of 400 eV. To calculate the density of states (DOS) and the projected crystal orbital Hamilton population (pCOHP) for each alloy, we used the LOBSTER code [39]. Dynamic stability in terms of phonon dispersion for both i-MAX phases was calculated from 1 × 2 × 1 supercells using the finite displacement method. PHONOPY [40] was used both to create the displacements and for the postprocessing analysis.
Using the energy-strain method [41], we derived and calculated the single crystal elastic constants, where a number of different strains are applied to the crystal lattice, followed by a calculation of the energy associated with each strain. For the monoclinic C2/c structure of (W 2/3 Sc 1/3 ) 2 AlC and (W 2/3 Y 1/3 ) 2 AlC, 14 different strains are needed to obtain the 13 independent elastic constants C ij ,. Here we used strain parameters of 0, ±0.01, and ±0.02. From C ij we calculated the bulk modulus (B), shear modulus (S), and Young's modulus (E) within the Voigt (V) and Reuss (R) model as expressed by the equations where the C ij 's are elastic constants and the S ij 's are compliance constants given by an inversion of the elastic-constant matrix. The upper boundary (Voigt) is found assuming that the strain is everywhere uniform while the lower boundary (Reuss) is found assuming that the stress is everywhere uniform. Arithmetic averages of Voigt and Reuss moduli are interpreted as the ratio of average stress and average strain within the composite. The stress and strain are generally unknown in the material and are expected to be nonuniform. Schematics were produced with VESTA [42].  [29]. The close relation between the C2/c and Cmcm structures have also been shown to be almost degenerate in energy [28,29,43]. The monoclinic unit cell of (W 2/3 Sc 1/3 ) 2 AlC and (W 2/3 Y 1/3 ) 2 AlC contains 48 atoms, 12 formula units (Z = 12). Table 1 show the calculated lattice parameters together with atomic positions. Our calculated results are in good agreement with the experimental values with deviations for (W 2/3 Sc 1/3 ) 2 AlC lattice parameters a, b, c, and β as −0.45%, −0.13%, +0.06%, and +0.36%, respectively. From table 1 we observe that the exchange of Sc for Y results in increased lattice parameters, as expected with larger metallic radius for Y (1.80 Å) as compared to Sc (1.62 Å).

Electronic structure and bonding analysis
The electronic structure and nature of bonding is essential to explain and understand many physical properties of materials. In addition to calculations of the electronic band structure, see figure 1 and an evident metallic character of both compounds, the bonding characteristics of monoclinic (W 2/3 Sc 1/3 ) 2 AlC and (W 2/3 Y 1/3 ) 2 AlC have been analyzed in terms of the DOS and the pCOHP, as shown in figure 2. In order to facilitate interpretation and to preserve the analogy for crystal orbital overlap population (COOP) analysis, the results are here presented as -pCOHP, rather than pCOHP. Except for the W-W (2) interaction across the Al-layer, see notation in figure 2, only nearest-neighbor interactions are considered in the pCOHP analysis, depicted in figures 2(g) and (h), since these are by far the strongest. Both DOS and pCOHP display five main regions; (i) from −13 to −11 eV, showing localized C-s states interacting mainly with W and Sc or Y, (ii) between −9 and −2 eV, from localized Al-s states in Al-Al interactions, (iii) between −7 and −3.4 eV, with Al-p states in Al-Al interactions as well as the bonding (interacting) states of M-d (W, Sc, Y) and C-p, (iv) from −3.4 to −1 eV, from M-d and Al-p states in bonding Al-Al, W-Al, Sc-Al, Y-Al interactions, (v) and states from −1 eV up to the Fermi level (E f ), mainly dominated by W-d with some contribution from Sc-d or Y-d. The last region shows a significant number of electrons, (N[E f ] = 2.25 states/fu for M 2 = Sc and 2.08 states/fu for M 2 = Y) despite the low contrib ution from pCOHP, which mainly consists of W-Sc and W-Y interaction. This is indicative of non-bonding electrons, mainly from the transition metals W and Sc or Y. The non-zero contribution at E f is also a strong indication of metallic character of these i-MAX phases.
From pCOHP we also find anti-bonding interactions below E f ; around −5 to −4 eV by C-C and from −2.4 eV up to E f by W-C. This is an indication of a non-optimized electronic structure which could be related to an unbalanced number of electrons. Here we have assumed complete occupation of the carbon sites, however, theoretical as well as experimental investigation of C occupancy is the scope of future work. Since the contribution from anti-bonding interactions is rather small, a small amount of, e.g. carbon vacancies could possibly counteract this interaction. In related MAX phase materials, i.e. V 4 AlC 3 [44,45] and Nb 4 AlC 3 [46], carbon vacancies have been shown theoretically and experimentally to be possible, and to stabilize the materials.  By integrating pCOHP up to E f , it is possible to get a rough estimate of the relative bond strengths within each i-MAX phase. IpCOHOP in figures 2(c) and (f) suggests the following order, in terms of bond strength for interactions defined in figures 2(g) and (h), for (W 2/3 Sc 1/3 ) 2 AlC: C-W > Al- , C-C. The two phases thus differ primarily in that the Al-Sc bonds are significantly weaker relative to most other bonds in (W 2/3 Sc 1/3 ) 2 AlC, in contrast to the corresponding Al-Y bonds in (W 2/3 Y 1/3 ) 2 AlC. Moreover, similar to regular MAX phases [47], the M-X bonds are stronger than the M-A bonds. On the other hand, the in-plane nearest neighbor bonds are comparatively weak, with the exception for the Al-Al nearest neighbor bonds.

Phonon dispersion and density of states
The phonon dispersion between high symmetry points in (W 2/3 Sc 1/3 ) 2 AlC and (W 2/3 Y 1/3 ) 2 AlC are shown in figures 3(a) and (c), with only positive frequencies implying their dynamic stability, i.e. stability with respect to lattice vibrations. Their monoclinic crystal structure contains 48 atoms, giving rise to totally 144 phonon branches out of which three are acoustic modes and 141 optical modes. Both i-MAX phases exhibit similar phonon dispersions, which can be related to the similar bonding characteristics discussed in a previous section, with a band gap around 13 THz separating high-frequency contribution, that mainly comes from the light carbon atoms (figures 3(b) and (d)), from the heavier transition metals and Al below the band gap. Figures 3(b) and (d) shows the phonon partial density of states (PHDOS) of (W 2/3 Sc 1/3 ) 2 AlC and (W 2/3 Y 1/3 ) 2 AlC where lowest frequency peaks are mainly attributed to W followed by Y(Sc) and Al. The contribution from W is similar for both i-MAX phases. However, upon substitution of Sc for Y the phonon DOS contribution from Al and Y is shifted to lower frequencies. The observed shift for Y, compared to Sc, can be attributed to the fact that it is heavier than Sc. The shift for Al can be attributed to the stronger Al-Y bonds as compared to Al-Sc, seen in figure 2.

Mechanical and elastic properties
The mechanical and elastic properties are dependent on the crystal structure and bonding between atoms. In table 2 we present calculated single crystal elastic constants C ij at 0 GPa for (W 2/3 Sc 1/3 ) 2 AlC and (W 2/3 Y 1/3 ) 2 AlC along with polycrystalline Voigt moduli; bulk modulus B, shear modulus G, and Young's modulus E as derived from C ij using equations (1), (3) and (5). In addition, we also present the Reuss moduli, calculated from the compliance constants obtained by inverting the 6 × 6 elastic-constant matrix, derived using equations (2), (4) and (5), together with the arithmetic means of the Voigt and Reuss moduli, i.e. the Voigt-Reuss-Hill (VRH) averages. Both i-MAX phases investigated herein fulfill the mechanical stability criterion for monoclinic structures [48].
First to note is that substitution of Sc for Y in general leads to decreased moduli. This decrease is greater for the Reuss moduli than for the Voigt moduli. We also note that the Reuss moduli are significantly lower than the Voigt moduli, in particular when it comes to the bulk modulus: B R is 37% lower than B V for (W 2/3 Sc 1/3 ) 2 AlC, and 45% lower than B V for (W 2/3 Y 1/3 ) 2 AlC. A similarly large difference is seen for the two i-MAX phases (Mo 2/3 Sc 1/3 ) 2 AlC and (Mo 2/3 Y 1/3 ) 2 AlC, which suggests that it is important to calculate both Voigt and Reuss moduli for i-MAX phases [27].
Comparison of calculated moduli of (W 2/3 Sc 1/3 ) 2 AlC with the hypothetical W 2 AlC (B V = 214 GPa, G V = 110 GPa, E V = 280 GPa) and Sc 2 AlC (B V = 88 GPa, G V = 57 GPa, E V = 140 GPa) in [49] shows that all moduli are in-between the two M 2 AX phases. Both G V and E V are strengthened by ~10% as compared to the arithmetic mean for the two M 2 AX phases. Generally, a large shear modulus G of a material is an indication of well-defined directional bonding between atoms. Here, G V is 100 and 99 GPa for (W 2/3 Sc 1/3 ) 2 AlC and (W 2/3 Y 1/3 ) 2 AlC, respectively. This indicates rather similar bonding, or more likely, a comparable total bond strength in the two i-MAX phases, which is confirmed by the bonding analysis in figure 2 where W-C and Al-Al is found to be slightly stronger for (W 2/3 Sc 1/3 ) 2 AlC compared to (W 2/3 Y 1/3 ) 2 AlC. This is, however, compensated by the latter having stronger interactions between C-Y and Al-Y as compared to C-Sc and Al-Sc.  [27,29] while IpCOHP values for Mo-based i-MAX phases are taken from [27]. All interactions, except for Al-M 2 and C-M 2 , are stronger when M 2 = Sc. For both Sc and Y, we find that both C-M and Al-M interactions are strengthened when going from Cr to Mo to W. For the C-M 2 , Al-Al, and Al-M 2 there is a decrease when going from Cr to Mo followed by no change or a slight increase in their interaction when going from Mo to W. These trends in interaction strengths are reflected in the trends found for the moduli in figure 4(b). The moduli for Cr-based i-MAX are in most cases higher than for Mo-based ones, which can be related to the comparatively stronger Al-Al and C-M 2 interactions. The increased moduli when going from Mo to W can be explained by the steady decrease in Al-M interaction strength and the dominating C-M interaction while the other interactions are unaffected. These non-linear trends found for moduli and for some of the bonding interactions for Cr, Mo, and W may be a result from size differences for M 1 and M 2 as well as different electronegativities. The size of the M 1 atom (1.28 Å for Cr and 1.39 Å for Mo and W) may explain why selected bonding interactions are changed from Cr to Mo but not from Mo to W. This is most notable for the Al-Al interaction. Moreover, the electronic properties will also influence the bonding interactions where both Mo (2.16) and W (2.36) are significantly more electronegative than Cr (1.66). Finally, comparing the moduli with a wellknown traditional MAX phase, Ti 2 AlC with B = 138 GPa, G = 113 GPa, E = 267 GPa [50], we find similar B but lower G and E for the i-MAX phases.

Conclusions
Two new atomically laminated W-based compounds, (W 2/3 Sc 1/3 ) 2 AlC and (W 2/3 Y 1/3 ) 2 AlC, belonging the family of i-MAX phases, have recently been synthesized, and are herein theoretically explored in terms of electronic structure, bonding characteristics, and mechanical properties. Both phases are described by a monoclinic structure of space group C2/c (#15) and are dynamically stable. They also display a clear metallic character. Substitution of Sc for Y (M 2 ), however, leads to reduced bond strength for W-C and Al-Al interactions, while M 2 -C and M 2 -Al interactions are strengthened, resulting in slightly decreased moduli; from B = 143 GPa, G = 94 GPa, and E = 231 GPa to B = 130 GPa, G = 91 GPa, and E = 219 GPa, respectively. Extending the comparison to other Y-based i-MAX phases, the moduli increases along the series where M 1 = Cr, Mo, and W.