UNIFIED FORMULATION OF EXCITONIC ABSORPTION-SPECTRA OF SEMICONDUCTOR QUANTUM-WELLS, SUPERLATTICES, AND QUANTUM WIRES

The problem of hydrogenic systems placed into strongly anisotropic med ia is solved exactly by using a metric space with a noninteger dimensi on alpha (1 < alpha). This appraoch is an elegant and convenient way t o treat the case of Wannier-Mott excitons confined in semiconductor su perlattices, quantum wells, and quantum-well wires. Indeed, the relati ve motion of the electron-hole pair which constitutes such excitons ca n never be considered strictly one dimensional (1D), 2D, or 3D. In thi s paper, we propose a quantitative analysis of the shape of the optica l-absorption edge near an excitonic energy gap, for any arbitrary valu e of alpha. We present an exact generalization of the calculations per formed in the effective-mass approximation for allowed transitions by Elliot [Phys. Rev. 108, 1384 (1957)] in the three-dimensional case, an d by Shinada and Sugano [J. Phys. Soc. Jpn. 21, 1936 (1966)] for two-d imensional media: this model includes contributions of bound states an d of the so-called unbound states, which are responsible for an enhanc ed absorption continuum above the interband energy gap. At high energi es, this continuum tends to behave like the alpha-dimensional valence- to-conduction joint density of states. The versatility of this approac h 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 modul ated.