Oscillations of the bandgap with size in armchair and zigzag graphene quantum dots

We determine here the evolution of the bandgap energy with size in graphene quantum dots (GQDs). We find oscillatory behaviour of the bandgap and explain its origin in terms of armchair and zigzag edges. The electronic energy spectra of GQDs are computed using both the tight binding model and ab initio density functional methods. The results of the tight binding model are analyzed by dividing zigzag graphene quantum dots into concentric rings. For each ring, the energy spectra, the wave functions and the bandgap are obtained analytically. The effect of inter-ring tunneling on the energy gap is determined. The growth of zigzag terminated GQD into armchair GQD is shown to be associated with the addition of a one-dimensional Lieb lattice of carbon atoms with a shell of energy levels in the middle of the energy gap of the inner zigzag terminated GQD. This introduces a different structure of the energy levels at the bottom of the conduction and top of the valence band in zigzag and armchair GQD which manifests itself in the oscillation of the energy gap with increasing size. The evolution of the bandgap with the number of carbon atoms is compared with the notion of confined Dirac Fermions and tested against ab initio calculations of Kohn-Sham and TD-DFT energy gaps.