In this paper, we report a recent theoretical study of the calculation of the binding energy and photoionization cross section of a single dopant in a spherical hollow or core/shell quantum dot taking into account the interaction of the electron with longitudinal optical phonons. Using Frolich approach and Lee-low Pines transformation, we determine the impact of different parameters such as shell thickness and dopant position on the energy and optical response of a bound polaron for two types of ionic II–VI semiconductors CdTe and ZnSe with different phonon coupling constants. Regardless of the material used, the electron–phonon interaction visibly reduces binding energy. For photoionization cross section, a redshift of resonance peaks was found when the effect of phonons is taken into consideration or when the donor is moved away from the shell center. These calculations provide us insights when choosing between materials for optoelectronic applications.

The Lusternik–Schnirelmann category has been described in different ways. Two major ones, the first by Ganea, the second by Whitehead, are presented here with a number of variants. The equivalence of these variants relies on the axioms of Quillen's model category, but also sometimes on an additional axiom, the so-called “cube axiom”.