Edge-dependent selection rules in magic triangular graphene flakes

The electronic shell and supershell structure of triangular graphene quantum dots has been studied using density functional and tight-binding methods. The density functional calculations demonstrate that the electronic structure close to the Fermi energy is correctly described with a simple tight-binding model where only the p_z orbitals perpendicular to the graphene layer are included. The results show that (i) both at the bottom and at the top of the p_z band a supershell structure similar to that of free electrons confined in a triangular cavity is seen, (ii) close to the Fermi level the shell structure is that of free massless particles, (iii) triangles with armchair edges show an additional sequence of levels ('ghost states') absent for triangles with zigzag edges while the latter exhibit edge states, and (iv) the observed shell structure is rather insensitive to the edge roughness

Electronic structure of triangular, hexagonal and round graphene flakes near the Fermi level

The electronic shell structure of triangular, hexagonal and round graphene quantum dots (flakes) near the Fermi level has been studied using a tight-binding method. The results show that close to the Fermi level the shell structure of a triangular flake is that of free massless particles, and that triangles with an armchair edge show an additional sequence of levels ("ghost states"). These levels result from the graphene band structure and the plane wave solution of the wave equation, and they are absent for triangles with an zigzag edge. All zigzag triangles exhibit a prominent edge state at the Fermi level, and few low-energy conduction electron states occur both in triangular and hexagonal flakes due to symmetry reasons. Armchair triangles can be used as building blocks for other types of flakes that support the ghost states. Edge roughness has only a small effect on the level structure of the triangular flakes, but the effect is considerably enhanced in the other types of flakes. In round flakes, the states near the Fermi level depend strongly on the flake radius, and they are always localized on the zigzag parts of the edge