Stackings and effective models of bilayer dice lattices

We introduce and classify nonequivalent commensurate stackings for bilayer dice or ${\mathcal{T}}₃$ lattices. For each of the four stackings with vertical alignment of sites in two layers, a tight-binding model and an effective model describing the properties in the vicinity of the threefold band-crossing points are derived. Focusing on these band-crossing points, we found that although the energy spectrum remains always gapless, depending on the stacking, different types of quasiparticle spectra arise. They include those with flat, tilted, anisotropic semi-Dirac, and ${C}₃$-corrugated energy bands. We use the derived tight-binding models to calculate the density of states and the spectral function. The corresponding results reveal drastic redistribution of the spectral weight due to the interlayer coupling that is unique for each of the stackings.