Low dimensionality semiconductors: modelling of excitons via a fractional-dimensional space

An interaction space with a fractionnal dimension is used to calculate in a simple way the binding energies of excitons confined in quantum wells, superlattices and quantum well wires. A very simple formulation provides this energy versus the non-integer dimensionality of the physical environment of the electron-hole pair. The problem then comes to determining the dimensionality alpha. We show that the latter can be expressed from the characteristics of the microstructure. alpha continuously varies from 3 (bulk material) to 2 for quantum wells and superlattices, and from 3 to 1 for quantum well wires. Quite a fair agreement is obtained with other theoretical calculations and experimental data, and this model coherently describes both three-dimensional limiting cases for quantum wells (L(w) --> 0 and L(w) --> infinity) and the whole range of periods of the superlattice. Such a simple model presents a great interest for spectroscopists though it does not aim to compete with accurate but often tedious variational calculations.