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Energy levels of triangular and hexagonal graphene quantum dots: a comparative study between the tight-binding and the Dirac approach

The Dirac equation is solved for triangular and hexagonal graphene quantum dots for different boundary conditions in the presence of a perpendicular magnetic field. We analyze the influence of the dot size and its geometry on their energy spectrum. A comparison between the results obtained for graphene dots with zigzag and armchair edges, as well as for infinite-mass boundary condition, is presented and our results show that the type of graphene dot edge and the choice of the appropriate boundary conditions have a very important influence on the energy spectrum. The single particle energy levels are calculated as function of an external perpendicular magnetic field which lifts degeneracies. Comparing the energy spectra obtained from the tight-binding approximation to those obtained from the continuum Dirac equation approach, we verify that the behavior of the energies as function of the dot size or the applied magnetic field are qualitatively similar, but in some cases quantitative differences can exist.