Confined electron and hydrogenic donor states in a spherical quantum dot of GaAs-${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$${\mathrm{Al}}_{\mathit{x}}$As

According to hydrogenic-effective-mass theory, exact solutions and quantum-level structures are presented for confined electron and hydrogenic donor states in a spherical quantum dot (SQD) of GaAs-${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$ ${\mathrm{Al}}_{\mathit{x}}$As. Calculated results reveal that the values of the quantum levels of a confined electron in a SQD can be quite different for cases with finite and infinite barrier heights. The quantum-level sequence and degeneracy for an electron in a SQD are similar to those of a super- atom of GaAs-${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$ ${\mathrm{Al}}_{\mathit{x}}$As but different from those in a Coulomb field. There is stronger confinement and larger binding energy for a hydrogenic donor in a SQD of GaAs-${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$ ${\mathrm{Al}}_{\mathit{x}}$As than in the corresponding quantum-well wires and two-dimensional quantum-well structures. The binding energy and its maximum of the ground state of a donor at the center of a quantum well are found to be strongly dependent on the well dimensionality and barrier height.