P. Lefebvre et al., GENERAL FORMALISM FOR EXCITONIC ABSORPTION EDGES IN CONFINED SYSTEMS WITH ARBITRARY DIMENSIONALITY, Journal de physique. IV, 3(C5), 1993, pp. 377-380
A metric space with a noninteger dimension alpha (1 < alpha) is used t
o describe bound and unbound states of strongly anisotropic Wannier-Mo
tt excitons, such as those confined in semiconductor superlattices, qu
antum wells and quantum-well wires. Indeed, the relative motion of the
electron-hole pair which constitutes such excitons can never be consi
dered strictly 1D, 2D or 3D. We calculate the optical absorption spect
rum, near a Van Hove singularity, for any arbitrary value of the dimen
sionality alpha. The whole absorption spectrum is obtained from a sing
le compact equation, much simpler than the existing models. This model
is an exact generalisation of the calculations performed, in the effe
ctive-mass approximation, for allowed transitions, by Elliott [Phys. R
ev. 108, 1384 (1957)], in the 3-dimensional case, and by Shinada and S
ugano [J. Phys. Soc. Japan 21, 1936 (1966)], for 2-dimensional media.