Vacancy-induced brittle to ductile transition of W-M co-doped Al3Ti (M=Si, Ge, Sn and Pb)

We investigated the effect of vacancy formation on brittle (D022) to ductile (L12-like) transition in Al3Ti using DFT calculations. The well-known pseudogap on the density of states of Al3Ti migrates towards its Fermi level from far above, via a W − M co-doping strategy, where M is Si, Ge, Sn or Pb respectively. In particular, by a W − M co-doping the underline electronic structure of the pseudogap approaches an octahedral (L12: t2g, eg) from the tetragonal (D022: eg, b2g, a1g, b1g) crystal field. Our calculations demonstrated that (1) a W-doping is responsible for the close up of the energy gap between a1g and b1g so that they tend to merge into an eg symmetry, and (2) all M-doping lead to a narrower gap between eg and b2g (moving towards a t2g symmetry). Thus, a brittle to ductile transition in Al3Ti is possible by adopting this W − M co-doping strategy. We further recommend the use of W-Pb co-doped Al3Ti to replace the less anodic Al electrode in Al-battery, due to its improved ductility and high Al diffusivity. Finally this study opens a new field in physics to tailor mechanical properties by manipulating electron energy level(s) towards higher symmetry via vacancy optimization.

and (2) the d xz and d yz lower than the d xy level. An effective unit area as shown in Fig. 2a is defined as S = l x × l y , where l x and l y is respectively the shortest atomic distance along the x and y axis. By shrinking the unit cell (or d in Fig. 2a) along the z-axis or expanding in the xy-plane indicated by S, a tetragonal crystal structure may return to and approach an octahedral-like structure. Thus, by reducing the ratio r (Å −1 ) (r = d/S), a brittle-ductile transition may be facilitated. From first-principles calculations, we test this new strategy to achieve the designed reduction of r by adopting a W − M co-doping strategy. Key challenges in this approach are (1) to generate sufficient Al 1 School of Materials Science and Engineering, Tsinghua University, Beijing, 100084, P. R. China. 2 Graduate School at Shenzhen, Tsinghua University, Shenzhen, 518055, P. R. China. 3 Singapore University of Technology and Design, 487372, Singapore, Singapore. Correspondence and requests for materials should be addressed to P.W. (email: wuping@sutd.edu.sg) or Q.L. (email: liql@sz.tsinghua.edu.cn) vacancies in D0 22 -Al 3 Ti to make the brittle structure more deformable and (2) to manipulate specific electron energy levels to transfer the low symmetry tetragonal to a high symmetry octahedral-like crystal field.
Moreover, due to the decrease in Gibbs free energy, Al 3 Ti may also be used as the anode material to replace Al in Al-battery. Recently, electronic structure of pure D0 22 -Al 3 Ti 12,14 and Al diffusion mechanisms of D0 22 -Al 3 Sc 11 are reported. However, like many brittle intermetallics, short cycling life of a native D0 22 -Al 3 Ti electrode is expected due to the structure damages during the charge and discharge processes. In addition, high Al diffusivity is essential to Al-battery, which requires an easy formation of Al vacancy based on Shi's 11 findings that Al vacancies facilitate the Al diffusion in Al 3 Sc. Therefore, the current study on the formation of Al vacancies may provide practical solutions to enhance both the mechanical and electrochemical properties of Al 3 Ti for Al-battery applications.

Results
Crystal structure. Crystal structures of both the ductile (L1 2 ) and brittle (D0 22 ) phases are shown in the   To conduct a systematic study, the site preference of W in D0 22 -Al 3 Ti was investigated first by using a 2 × 2 × 1 supercell including 32 atoms. The first-principles calculations have been performed to calculated the total energies E tot for the pure D0 22 -Al 3 Ti supercell and E dope for [(Al 23 W)Ti 8 ] and [Al 24 (Ti 7 W)] structures. To determine the site preference of W, the substitution energy E sub is defined as: where μ i (i = Al, Ti and W) is the chemical potential of these atoms in their stable bulk phases. In this study, the stable phases are considered as Ti in hcp structure 17 where E M is the total energy of single M-atom occupying Ti site. While the substitution energies of W − M clusters can be written as: The results are showed in Table 1. ΔE are positive which means that the co-doping systems have much lower substitution energies than the single doping systems. It indicates that introducing W in pure D0 22 -Al 3 Ti structure will conduce to the substitution of Ti by M.

Vacancy formation energy. The crystal model with an Al vacancy were created by removing an individual
Al-atom from W − M co-doping supercell. In order to reduce the computation loads, we focus on the first-nearest neighbors, thus, the two possible Al vacancies are at V 1 and V 2 sites considering the system symmetry, shown as yellow spheres in Fig. 2a. The stability of the defected structures were studied by vacancy formation energy calculation after the atomic defects are relaxed completely. The formation energy of a neutral aluminum vacancy (hereafter simply referred to as an aluminum vacancy) (E V ) is estimated by the following equation (5): is the vacancy formation energy, E def is the total energy of D0 22 -Al 3 Ti/W supercell containing one M-atom and one Al vacancy simultaneously and E W−M is the total energy of W − M co-doping supercell. The last term represents the difference in the number of atoms from the W − M co-doping system, where n i denotes the number of atoms to be taken from or inserted into the supercell in order to take account of point defect generation. If a corresponding atom is inserted into the supercell, n i is negative and if such an atom is taken away from the supercell, n i is positive. i μ is the chemical potential of these atoms in their stable bulk phases. The calculated defect formation energies are tabulated in Table 2.
From Table 2, it can be easily observed that the vacancy formation energies of Al at V 1 site are higher than that at V 2 site. For a W − M co-doping system, when the Al vacancy occurs at V 2 site, the vacancy formation energies are negative under both Al-rich environment and Ti-rich environment, which indicates V 2 defects can be formed spontaneously during the fabrication of the alloy. In addition, both V 1 and V 2 defects are spontaneously formed by a W-Pb co-doping under either Al-rich or Ti-rich environment. The results attribute to the fact that the W − M co-doping cluster plays a vital role in the formation of Al vacancies.
Electronic states of defected structure. The calculated DOS are given in Fig. 3a for the pure D0 22 -Al 3 Ti, and in Fig. 3b-f for W-C, W-Si, W-Ge, W-Sn and W-Pb co-doping Al 3 Ti with an Al vacancy at the V 1 site, respectively. Δn is introduced to indicate the valley of psuedogap. The value of Δn is the energy of the lowest position on the calculated DOS curve. Thus, the positive value means the psuedogap is higher than the Fermi level. The result is shown in Table 3. A clear pseudogap is observed in the D0 22 -Al 3 Ti (circled part in Fig. 3a), which indicates the strong bonding-antibonding separation. The result shows a strong hybridization existing in the D0 22 structure as well as a strong directionality in bonding. Therefore, it is difficult to form the slip system in the tetragonal D0 22 structure and leads to brittleness. The partial DOS of D0 22 -Al 3 Ti around the pseudogap was investigated, shown in Fig. 3g. From the edges of the gap, the splitting 3d orbitals could be observed clearly, thus, b 1g (d 2 x−y ) energy is higher than a 1g (d 2 z ) on the right edge, while b 2g (d xy ) energy is higher than e g (d xy , d yz ) on the left edge, which appears a typical tetragonal crystal field.
From Fig. 3b-f, by adding different M elements and forming a W − M co-doping cluster with an Al vacancy at V 1 simultaneously, the pseudogap migrates from far above towards the Fermi level, indicated by the red arrows.   The results show that there are less bonding states which may favor a D0 22 to L1 2 -like transition. To carry out a more in-depth and detailed study, partial DOS crossing the pseudogap of the W-Pb co-doping system was calculated and shown in Fig. 3h. Contributions from W and Pb to bonding electrons were investigated separately. On the right edge of the pseudogap, it is observed that W-atom contributes a lot to form strong hybridization between the d 2 z and d 2 x−y levels. Similarly on the left edge of the pseudogap, the Pb-atom has a strong influence on rising the d xz energy and d yz energy towards the d xy level (or a strong hybridization among these 3d orbitals). Therefore, the vacancy-induced 3d-orbital-splitting tend to facilitate a ductile L1 2 -like structure, thus, e g (d 2 x−y , d 2 z ) and t 2g (d xy , d xz , d yz ).
To obtain the brittle to ductile transition, the tetragonal D0 22 structure is expected to transform into an octahedral-like structure, which could be realized by either a shrinking along z-axis or an expanding on the xy-plane or both. By representing the ratio r of z-axis d to the xy-plane S, the change in structures are quantified, as shown in Fig. 2b. Taking pure D0 22 -Al 3 Ti as the standard, it can be concluded that r decreased with the formation of Al vacancy at V 1 site, which indicates that the tetragonal crystal field tends to transform into an octahedral-like crystal field. As a result, the stable phase change from D0 22 to L1 2 -like ductile structures. When an Al vacancy forms at the V 2 site, S remains nearly a constant except for the W-C co-doping. The larger S in the W-C co-doping system is due to the small size of C. Among all M elements in Table 4, C is the only dopant whose size is smaller than that of Al (0.39 Å for Al 3+ ). More details will be outlined in Session 3 below.

Discussion
In order to enable a brittle to ductile transition, we proposed and validated a W − M co-doping mechanism to (1) generate sufficient Al vacancies in D0 22 -Al 3 Ti, and (2) simultaneously to manipulate specific electron energy levels to approach the high symmetry octahedral-like electronic structures. In particular, an equation for the lattice energy of W − M co-dopants is derived based on the E W−M (eV) given in Table 1 Similarly, an equation for the formation energy of V 2 -W − M co-dopants is derived based on the E V 2 (eV) data given in Table 2: Both equations (7) and (8) reasonably reproduce DFT calculations shown in Table 2. We derived V 2 -W − M equations only since they are stable (or having negative formation energy) for all the M elements. Like equation (6), both equations (7) and (8)  Finally, we have systematically investigated a series of W − M co-doping D0 22 -Al 3 Ti (M = C, Si, Ge, Sn and Pb) intermetallics using first-principles calculation method. The site preference of W in pure D0 22 -Al 3 Ti was first studied, it shows W (a d element) has a clear preference to substitute Al 1 (a sp element) site due to the strong crystal field. Then, we confirmed the [(Al 23 W)Ti 8 ] system is conductive to the subsequent doping of M-atom. Meanwhile, a M substitution of Ti reduces the stability of [(Al 23 W)Ti 8 ], which might benefit the intercalation and deintercalation of Al-ion during charge-discharge cycling in rechargeable Al-battery. The two possible Al vacancies were also investigated. In comparison to the vacancy formation energies of Li-ion in Li 3 N 19 (−0.14 ~ 0.52 eV), the Al vacancies in W − M co-doped Al 3 Ti have much lower formation energies, therefore, high Al diffusivity is expected.
The DOS of W − M co-doping Al 3 Ti with an Al vacancy at the V 1 site were investigated. The results show the pseudogap migrates towards the Fermi level from far above, indicating a tendency to transform into ductile L1 2 -like structure. By analyzing the partial DOS around the pseudogap, we found that W and Pb have almost independent contributions to the transition, thus, W mainly influents d 2 x−y and d 2 z while Pb have a strong effect on d xy , d xz and d yz . It shows the crystal splitting effect on the 3d orbitals plays a decisive role not only on the