Exciton fractional dimension in semiconductor heterostructures arising from variational principle

We present a simple method for calculating the ground-state energy of an exciton in quantum confined structures. We express the exciton wave function as a product of the electron and hole one-particle wave functions with a variationally determined envelope function which describes the exciton intrinsic properties. Starting from the variational principle, we derive an one-dimensional wave equation for this envelope function and show that it describes a hydrogen-like atom in an effective isotropic space with the non-integer running dimension. We establish that this dimension runs from three, as the electron-hole separation is small for all heterostructures, to two in quantum well one in quantum well-wire and to zero in quantum dot, as the separation is large. The exciton ground-state energies are calculated for different confining potential shapes. Our results for GaAs-(Ga, Al)As heterostructures with square-well potential are in an excellent agreement with those obtained previously by means of other methods.