Unified formulation of excitonic absorption spectra of semiconductor quantum wells, superlattices, and quantum wires

The problem of hydrogenic systems placed into strongly anisotropic media is solved exactly by using a metric space with a noninteger dimension α (1<α). This appraoch is an elegant and convenient way to treat the case of Wannier-Mott excitons 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 one dimensional (1D), 2D, or 3D. In this paper, we propose a quantitative analysis of the shape of the optical-absorption edge near an excitonic energy gap, for any arbitrary value of α. We present an exact generalization of the calculations performed in the effective-mass approximation for allowed transitions by Elliot [Phys. Rev. 108, 1384 (1957)] in the three-dimensional case, and by Shinada and Sugano [J. Phys. Soc. Jpn. 21, 1936 (1966)] for two-dimensional media: this model includes contributions of bound states and of the so-called unbound states, which are responsible for an enhanced absorption continuum above the interband energy gap. At high energies, this continuum tends to behave like the α-dimensional valence-to-conduction joint density of states. The versatility of this approach should be particularly useful for modeling and improving the dynamic properties of optical modulators, for which not only the energy gap, but also the dimensionality of the excitonic absorption onset is modulated.