One-dimensional Lieb superlattices: from the discrete to the continuum limit†

The Lieb lattice is one of the simplest lattices that exhibits both linear Dirac-like and flat topological electronic bands. We propose to further tailor its electronic properties through periodic 1D electrostatic superlattices (SLs), which, in the long wavelength limit, were predicted to give rise to novel transport signatures, such as the omnidirectional super-Klein tunnelling (SKT). By numerically modelling the electronic structure at tight-binding level, we uncover the evolution of the Lieb SL band structure from the discrete all the way to the continuum regime and build a comprehensive picture of the Lieb lattice under 1D potentials. This approach allows us to also take into consideration the discrete lattice symmetry-breaking that occurs at the well/barrier interfaces created by the 1D SL, whose consequences cannot be explored using the previous low energy and long wavelength approaches. We find novel features in the band structure, among which are intersections of quadratic and flat bands, tilted Dirac cones, or series of additional anisotropic Dirac cones at energies where the SKT is predicted. Our results show that the universal SKT is absent when the lattice details are considered. Such features are relevant to experimental realizations of electronic transport in Lieb 1D SL realized in artificial lattices or in real material systems like 2D covalent organic/metal–organic frameworks and inorganic 2D solids.