Excitons and shallow impurities inGaAs−Ga1−xAlxAssemiconductor heterostructures within a fractional-dimensional space approach: Magnetic-field effects

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The fractional-dimensional space approach is extended to study exciton and shallow-donor states in symmetric-coupled ${\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}\ensuremath{-}\mathrm{G}\mathrm{a}}_{1\ensuremath{-}x}{\mathrm{Al}}_{x}\mathrm{As}$ multiple quantum wells. In this scheme, the real anisotropic ``exciton (or shallow donor) plus multiple quantum well'' semiconductor system is mapped, for each exciton (or donor) state, into an effective fractional-dimensional isotropic environment, and the fractional dimension is essentially related to the anisotropy of the actual semiconductor system. Moreover, the fractional-dimensional space approach was extended to include magnetic-field effects in the study of shallow-impurity states in ${\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}\ensuremath{-}\mathrm{G}\mathrm{a}}_{1\ensuremath{-}x}{\mathrm{Al}}_{x}\mathrm{As}$ quantum wells and superlattices. In our study, the magnetic field was applied along the growth direction of the semiconductor heterostructure, and introduces an additional degree of confinement and anisotropy besides the one imposed by the heterostructure barrier potential. The fractional dimension is then related to the anisotropy introduced both by the heterostructure barrier potential and magnetic field. Calculations within the fractional-dimensional space scheme were performed for the binding energies of $1s$-like heavy-hole direct exciton and shallow-donor states in symmetric-coupled semiconductor quantum wells, and for shallow-impurity states in semiconductor quantum wells and superlattices under growth-direction applied magnetic fields. Fractional-dimensional theoretical results are shown to be in good agreement with previous variational theoretical calculations and available experimental measurements.