Virial theorem, scaling properties and magnetic-field effects on Coulomb-bound states in semiconductor quantum wells
We have used the variational and fractional-dimensional space approaches, in the effective-mass approximation, in order to investigate the effects of applied magnetic fields on Coulomb-bound states, i.e. donor and exciton states, confined in GaAs-(Ga, Al)As quantum wells. In the variational procedure, we have used a simple hydrogenic-like envelope wavefunction whereas the anisotropic Coulomb-bound state+quantum well+magnetic-field system is modelled through an effective isotropic medium in the fractional-dimensional scheme. The magnetic fields are applied along the heterostructure growth direction, and calculations were performed for the binding energies, virial-theorem values and scaling properties. A virial-theorem value equal to 2 and a hyperbolic scaling for binding energies of Coulomb-bound states versus quantum-confined Bohr radii are obtained if one assumes a ground-state envelope wavefunction as a D-dimensional hydrogenic wavefunction, in contrast with results using the variational approach. Moreover, theoretical results within the variational approach lead to exciton-energy diamagnetic shifts in good agreement with available experimental measurements.