An exactly solvable tight-binding billiard in graphene

Abstract

A triangular graphenic billiard is defined as a planar carbon polymer in the H\"uckeloid approximation of $\pi-$band electrons. It is shown that the equilateral triangle of arbitrary size and zig-zag edges allows for exact solutions of the associated spectral problem. This is done by a construction of wave superpositions similar to the Lam\'e solution of the Helmholtz equation in a triangular cavity, revisited by Pinsky. Exact wave functions, eigenvalues, degeneracies, and edge states are provided. The edge states are also obtained by a non-periodic construction of waves with vanishing energy. A comment on its connection with recent molecular models, such as triangulene, is given.