Chiral zero-energy modes in the disordered ?-T3 lattice

The alpha-T3 lattice is an interpolate between the graphene (alpha = 0) and the Dice lattices (alpha = 1) and has a nondispersive flat band across the Dirac bands at the band center (E = 0). In this paper, we study the delocalization effect of the additional chiral zero-energy modes (CZEM) from the vacancy disorder in the alpha-T3 lattice and address the influence of the flat band on the CZEM. It is shown that the mere broadening of the flat band without the CZEM could produce a supermetallic phase around the band center E similar to 0 as well as an adjoined narrow localization regime. When the CZEM from the vacancy disorder is turned on, the transport property resembles the graphene case (alpha = 0) that electrons in the low-energy regime are fully localized except the Dirac point of E = 0, which is a critical delocalization point because the zero-energy conductivity (ZEC) is independent of the inelastic-scattering strength q. But the ZEC itself is shown to weakly rely on both the model parameter alpha and the vacancy density nc due to the broadening effect of the flat band. The extreme vacancy imbalance among each ABC site of a primitive cell is also studied, and the electron transport around E similar to 0 is changed to be either the pure inelastic disorder case without vacancy or the fully localized phase without the critical delocalization point of E = 0, which depends on the central site B atom replaced by vacancy or not.