GENERAL FORMALISM FOR EXCITONIC ABSORPTION EDGES IN CONFINED SYSTEMS WITH ARBITRARY DIMENSIONALITY

A metric space with a noninteger dimension alpha (1 < alpha) is used to describe bound and unbound states of strongly anisotropic Wannier-Mott excitons, such as those confined in semiconductor superlattices, quantum wells and quantum-well wires. Indeed, the relative motion of the electron-hole 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 alpha. 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.