Electronic band evolution between Lieb and kagome nanoribbons

We investigate the electronic properties of nanoribbons made out of monolayer Lieb, transition, and kagome lattices using the tight-binding model with a generic Hamiltonian. It allows us to map the evolutionary stages of the interconvertibility process between Lieb and kagome nanoribbons by means of only one control parameter. Results for the energy spectra, the density of states, and spatial probability density distributions are discussed for nanoribbons with three types of edges: straight, bearded, and asymmetric. We explore for different nanoribbon terminations: (i) the semiconductor-metallic transition due to the interconvertibility of the Lieb and kagome lattices, (ii) the effect of both nanoribbon width and inclusion of the next-nearest-neighbor hopping term on the degeneracy of the quasi-flat states, (iii) the behavior of the energy gap versus the nanoribbon width, (iv) the existence and evolution of edge states, and (v) the nodal spatial distributions of the total probability densities of the non-dispersive states.