Excitons in anisotropic solids: The model of fractional-dimensional space

Wannier-Mott excitons in anisotropic or confined systems are studied using the model of fractional-dimensional space. The excitons in an anisotropic solid are treated as ones in an isotropic fractional-dimensional space, where the dimension is determined by the degree of anisotropy. By solving the simple hydrogenic Schr\"odinger equation in the fractional-dimensional space, exciton wave functions, bound energies, and associated optical spectra are obtained as a function of spatial dimensionality. Dimensional behavior in binding energy, radial density, and angular momentum is discussed. The model provides a quantitative measure of anisotropy by a fractional dimension, as viewed from exciton dynamics, which can be determined experimentally from interband optical spectra. The results obtained here are also applicable to hydrogenic impurities in anisotropic solids.