Fractional-dimensional approach for excitons in ${\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}-\mathrm{G}\mathrm{a}}_{1-x}{\mathrm{Al}}_x\mathrm{As}$ quantum wells

The fractional-dimensional approach, in which the real semiconductor heterostructure system is substituted by an effective isotropic environment with a fractional dimension, was used in the study of ground and excited excitonic states in GaAs-(Ga,Al)As quantum wells. The fractional-dimensional formalism was extended to include the possibility of dealing with excited states and varying effective masses across the heterostructure interfaces, with the fractional dimension chosen in a systematic way. Theoretical fractional-dimensional results for ground-state $1s$-like exciton states in GaAs-(Ga,Al)As quantum wells were shown to be in good agreement with previous detailed calculations and recent experimental measurements. Moreover, theoretical results within the fractional-dimensional scheme were found in excellent agreement with the recent experimental high-resolution spectroscopic studies on excited-exciton states of shallow ${\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}-\mathrm{G}\mathrm{a}}_{1-x}{\mathrm{Al}}_x\mathrm{As}$ quantum wells with Al concentration in the range of 1–4.5 %.