Unified cluster-based description of valence bands in AlIr, RuAl2, RuGa3 and Al–TM quasicrystalline approximants

Formation of the valence bands in AlIr, RuAl2 and RuGa3 was analysed on the basis of Wannier functions. It was sufficient to consider nine (s-, p- and d-like) Wannier functions per cluster centred on each transition metal for describing the valence bands. As in the case of Al-Cu-Ir quasicrystalline approximant, which we reported previously [Kitahara K, Takagiwa Y and Kimura K 2015 J. Phys. Soc. Jpn. 84 014703-1-8], covalent bonds between clusters form where the distance between neighbouring clusters is about 0.3 nm, making conduction states as antibonding functions. sp3d2-, sp3- and py-like Wannier functions are used for the covalent bonds in AlIr, RuAl2 and RuGa3, respectively.


Introduction
Formation of pseudogap or narrow bandgap in some of group 13 element-transition metal (TM) intermetallic compounds including Al-TM quasicrystalline approximants had extensively been studied, and importance of hybridisation between sp-functions of group 13 elements (or free electrons in jellium) and d-functions of TM atoms were realised [1]. However the concept of sp-d hybridisation cannot completely explain the formation of bandgap, especially the number of valence bands. For example, in the case of RuAl 2 , there are seven valence bands per ruthenium atom without considering spin degeneracy. While five out of the seven bands may be associated with sp-d bonding functions, the origin of the other two bands is unclear.
We studied the electronic band structures in AlIr (CsCl-type), RuAl 2 (TiSi 2 -type), RuGa 3 (CoGa 3 -type) and Al-Cu-Ir quasicrystalline approximant as typical intermetallic compounds with pseudogap or narrow bandgap. To analyse the band structures, we used maximally localised Wannier functions [2], which are representative functions corresponding to the bands in a specified energy range. We have already reported analysis on the Al-Cu-Ir approximant [3], where we described the valence and a part of the conduction bands on the basis of Wannier functions centred on clusters of 0.2 nm radius, and the number of valence bands (173) was decomposed in terms of the numbers of TMs (t = 23), clusters (c = 16) and covalent bonds between the clusters (b = 6) per primitive unit cell as 5t + 4c − b without considering spin degeneracy. As we shall show, fundamentally the same description can be applicable to the other three intermetallic compounds, AlIr, RuAl 2 and RuGa 3 . [001] Ru Ga Figure 1. Crystalline structures of (a) AlIr, (b) RuAl 2 and (c) RuGa 3 .

Results
We constructed nine Wannier functions per iridium atom for AlIr. As shown in Figure 2

Discussion
To summarise our results, it is convenient to consider clusters of which centres are the same as those of the constructed Wannier functions. In the cases of AlIr, RuAl 2 and RuGa 3 , the centres of such clusters are the sites of TMs; therefore, the numbers of TMs (t) and clusters (c) per primitive unit cell are t = c = 1 for AlIr, t = c = 2 for RuAl 2 and t = c = 4 for RuGa 3 . For each cluster, we constructed one s-, three p-and five d-like Wannier functions as candidates of valence states. It corresponds to a part of our counting rule 5t + 4c, i.e. five d-like functions per TM and one s-and three p-like functions per cluster. Formation of pairs of bonding and antibonding functions between Wannier functions at neighbouring sites can be considered as covalent bonds between the clusters. Such covalent bonds form where the distance between two clusters is about 0.3 nm as discussed in [3]. We should note that, Yannello and Fredrickson proposed 18−n rule for the number of valence electrons per TM in TM-main group intermetallic phases [9], where n is the number of electrons in bonding functions between TMs. This rule is a special case of our 5t + 4c − b rule, which reduces to 18 − n rule when t = c; in such case, 2b = tn, and therefore 2(5t + 4t − b) = t(18 − n). In the present cases, we do not need to consider TMs and clusters separately, and therefore 18 − n rule is sufficient. However, in some cases (at least in the case of Al-Cu-Ir quasicrystalline approximant [3]), the numbers of TMs and clusters can differ, and 5t + 4c − b rule is required.
To associate all the valence states with TMs implies that formal charges of iridium in AlIr, ruthenium in RuAl 2 and ruthenium in RuGa 3 are −3, −6 and −9, respectively.Öǧüt and Rabe [10] calculated the Born effective charge tensor of ruthenium in RuAl 2 , and the directional average of the tensor was about −6.2. This fact supports the formal charge of ruthenium in RuAl 2 . To examine the role of aluminium or gallium in formation of the electronic structures, we also calculated the electronic band structures in TMs-in-jellium models, where each aluminium or gallium in AlIr, RuAl 2 and RuGa 3 are replaced with homogeneous charges of 3+ and three free electrons. In Figures 5(a)-(c), the densities of states in these approximated models are compared with those of the original models. In the cases of AlIr and RuAl 2 , the densities of states of both the original and the approximated models are almost the same. Even in the case of RuGa 3 , where 75% of atoms are replaced with jellium in the approximated model, overall structure of the density of states is not drastically changed, and gap-like structure remains around 1 eV above the Fermi energy. It indicates that the role of aluminium or gallium is, as a first approximation, just to offer free electrons to the system. Finally, as our description is based on a short-range mechanism, it is very interesting whether it can be applicable to quasicrystals or higher order approximants, too; if not, there is a room to consider a long-range mechanism.