Symmetry and optical selection rules in graphene quantum dots

Graphene quantum dots (GQD's) have optical properties which are very different from those of an extended graphene sheet. In this Article we explore how the size, shape and edge--structure of a GQD affect its optical conductivity. Using representation theory, we derive optical selection rules for regular-shaped dots, starting from the symmetry properties of the current operator. We find that, where the x- and y-components of the current operator transform with the same irreducible representation (irrep) of the point group - for example in triangular or hexagonal GQD's - the optical conductivity is independent of the polarisation of the light. On the other hand, where these components transform with different irreps - for example in rectangular GQD's - the optical conductivity depends on the polarisation of light. We find that GQD's with non-commuting point-group operations - for example dots of rectangular shape - can be distinguished from GQD's with commuting point-group operations - for example dots of triangular or hexagonal shape - by using polarized light. We carry out explicit calculations of the optical conductivity of GQD's described by a simple tight--binding model and, for dots of intermediate size, \textcolor{blue}{($10 \lesssim L \lesssim 50\ \text{nm}$)} find an absorption peak in the low--frequency range of the spectrum which allows us to distinguish between dots with zigzag and armchair edges. We also clarify the one-dimensional nature of states at the van Hove singularity in graphene, providing a possible explanation for very high exciton-binding energies. Finally we discuss the role of atomic vacancies and shape asymmetry.