Evolution of bandgap with size in armchair and zigzag graphene quantum dots*

We present here the evolution of the bandgap energy with size in armchair and zigzag graphene quantum dots (GQDs). The results of the tight binding model are analyzed by dividing zigzag graphene quantum dots into concentric rings. For each ring, the solutions are obtained analytically and then 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 GQD. This causes a difference in the nature of the wave functions between 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.