Dimensional crossover in excitons on a square lattice

Dimensional crossover between one-dimensional (1D) and two-dimensional (2D) excitons has been studied with a two-band tight-binding model on a square lattice by introducing anisotropy to both the transfer energies and the Coulomb interaction between an electron and a hole. It has been found that the binding energy of the lowest exciton (${\mathit{E}}_{\mathit{B}}$) takes a minimum value in the anisotropic regime rather than in the isotropic 2D case. In other words, a weak anisotropy reduces ${\mathit{E}}_{\mathit{B}}$ while a strong anisotropy increases ${\mathit{E}}_{\mathit{B}}$ as shown by comparing the isotropic 2D case with the 1D limit. On the other hand, the energy levels of higher exciton states have been found to cross each other as the degree of anisotropy increases, and a phase diagram is drawn with respect to the character of the lowest few excitons. These behaviors in the anisotropic region do not appear without the anisotropy of either the transfer energies or the Coulomb interaction which is present in real materials. The effect of dimensional crossover on optical properties is also discussed.