Graphene flakes with defective edge terminations: Universal and topological aspects, and one-dimensional quantum behavior

Abstract

Systematic tight-binding investigations of the electronic spectra (as a function of the magnetic field) are presented for trigonal graphene nanoflakes with reconstructed zigzag edges, where a succession of pentagons and heptagons, that is 5-7 defects, replaces the hexagons at the zigzag edge. For nanoflakes with such reczag defective edges, emphasis is placed on topological aspects and connections underlying the patterns dominating these spectra. The electronic spectra of trigonal graphene nanoflakes with reczag edge terminations exhibit certain unique features, in addition to those that are well known to appear for graphene dots with zigzag edge termination. These unique features include breaking of the particle-hole symmetry, and they are associated with nonlinear dispersion of the energy as a function of momentum, which may be interpreted as nonrelativistic behavior. The general topological features shared with the zigzag flakes include the appearance of energy gaps at zero and low magnetic fields due to finite size, the formation of relativistic Landau levels at high magnetic fields, and the presence between the Landau levels of edge states (the socalled Halperin states) associated with the integer quantum Hall effect. Topological regimes, unique to the reczag nanoflakes, appear within a stripe of negative energies E_b < E < 0, and along a separate feature forming a constant-energy line outside this stripe. The lower bound (E_b) specifying the energy stripe is independent of size. A main finding concerns the limited applicability of the continuous Dirac-Weyl equation, since the latter does not reproduce the special reczag features. (See also the extended abstract in the paper.)