We have used the variational and fractional-dimensional space approaches in a study of the virial theorem value and scaling of the shallow-donor binding energies versus donor Bohr radiusin GaAs-(Ga,Al)As semiconductor quantum wells and quantum-well wires. A comparison is made with previous results with respect to exciton states. In the case the donor ground-state wave function may be approximated by a D-dimensional hydrogenic wave function, the virial theorem value equals 2 and the scaling rule for the donor binding energy versus quantum-sized Bohr radius is hyperbolic, both for quantum wells and wires. In contrast, calculations within the variational scheme show that the scaling of the donor binding energies with quantum-sized Bohr radius is in general nonhyperbolic and that the virial theorem value is nonconstant.
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de Dios-Leyva, M., Oliveira, L.E. On the scaling of exciton and impurity binding energies and the virial theorem in semiconductor quantum wells and quantum-well wires. MRS Online Proceedings Library 692, 3641 (2001). https://doi.org/10.1557/PROC-692-H6.34.1