An exactly solvable tight-binding billiard in graphene

TL;DR

This paper presents an exact solution for the spectral problem of a triangular graphene billiard with zig-zag edges, providing explicit wave functions, eigenvalues, and edge states, and connecting these findings to molecular models like triangulene.

Contribution

It introduces a novel exactly solvable model of a graphene billiard with specific boundary conditions, enabling precise analysis of its spectral properties.

Findings

Exact solutions for eigenvalues and wave functions are derived.

Edge states with zero energy are explicitly constructed.

The model connects to molecular structures like triangulene.

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.