Modulated Dirac bands and integer hopping ratios in a honeycomb lattice of phenalenyl-tessellation molecules

A family of nanographene molecules called phenalenyl-tessellation molecules (PTMs) exhibits two types of zero modes: a $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ type that spreads over the entire molecule and a vacancy-localized type. A periodic system of PTMs is expected to have low-energy bands that strongly reflect the properties of the zero modes of PTMs as effective atoms. In this study, we show that the low-energy Dirac bands in a class of honeycomb PTMs (H-PTM) can be represented by an effective honeycomb model which is determined only by the connections between neighboring effective atoms. The hopping parameters of H-PTM in each direction take positive integer ratios according to the connection order between two PTMs. By structurally designing each PTM, we can change the connection order of the PTMs and hence modulate the energy gap and the Fermi velocity of the Dirac band of the H-PTM. Moreover, we confirm that Dirac bands coexist with vacancy-localized zero modes in the H-PTM with vacancies. The result indicates that the nanographene structure arranging PTMs as effective atoms extends material design freedom that effectively generates a modulated Dirac electron system with coexisting localized electron spins for graphene-based electronic and quantum devices.