General formalism for excitonic absorption edges in confined systems with arbitrary dimensionality

Abstract
A metric space with a noninteger dimension α (1 < α) is used to describe bound and unbound states of strongly anisotropic Wannier-Mott excitons, such as those confïned in semiconductor superlattices, quantum wells and quantum-well wires. Indeed, the relative motion of the electronhole pair which constitutes such excitons can never be considered strictly 1D, 2D or 3D. We calculate the optical absorption spectrum, near a Van Hove singularity, for any arbitrary value of the dimensionality [MATH] . The whole absorption spectrum is obtained from a single compact equation, much simpler than the existing models. This model is an exact generalisation of the calculations performed, in the effective-mass approximation, for allowed transitions, by Elliott [Phys. Rev. 108, 1384 (1957)], in the 3-dimensional case, and by Shinada and Sugano [J. Phys. Soc. Japan 21, 1936 (1966)], for 2-dimensional media.