A single equation describes excitonic absorption spectra in all quantum-sized semiconductors

The relative motion of the electron-hole pairs which constitute Wannier-Mott excitons in semiconductor quantum wells, superlattices, and quantum wires can never be considered strictly 1-D, 2-D, or 3-D. We propose an exact generalization of the well-known calculations of Elliott in the 3-dimensional case, and of Shinada and Sugano for 2-dimensional media-we calculate the absorption spectrum by bound and unbound excitonic states, by using a metric space with a noninteger dimension α>1. Whatever the dimensionality, i.e., for any quantum-sized structure, the whole optical density spectrum is obtained from a single compact equation, in excellent agreement with experimental data and with the most accurate available theories. We present examples of calculated spectra for quantum wells under applied perpendicular electric fields, and for quantum wires