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We reduce the problems of the on-and off-center D 0 and D --S-states in semiconductor heterostructures to the similar ones in an isotropic effective space with variable fractional dimension starting from the variational principle. The dimension of this space is defined as a scaling parameter that relates the radii of a set of spherical boxes to the charge densities within induced by the free electron ground state in the heterostructure. Explicit expressions for the effective space dimensionality in a quantum well (QW), quantum-well wire (QWW) and a quantum dot (QD) are found by using this definition. To solve the wave equations for the free electron ground state in the heterostructure and for the hydrogen-like atom S-states in the fractional-dimensional space, we use the numerical trigonometric sweep method. The three-parameter Hylleraas trial function is used to solve the similar problem for a negative-hydrogen-like ion in the effective space. Ground state binding energies for off-center neutral and negatively charged donors in QWs and spherical QDs are calculated. Our results are in a good agreement with those of the variational and Monte Carlo methods. In addition, novel results for the D --binding energy as a function of the cylindrical GaAs/Ga 0.7 Al 0.3 As QWW radius and the magnetic field intensity are presented. It is found that the D --binding energy in the wire increases from 0.055 Ry * up to about 1.230 Ry * as the radius decreases to 30 A. It is also shown that the magnetic field produces a considerable enhancement of negative-donor binding energy in QWW only for radii greater than 100 A.

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Neutral and charged shallow donor states confined by a parabolic potential in the plane of a GaAs quantum well are studied in the presence of a perpendicular magnetic field. The influence of the position of the donor on the energy of the states is studied, i.e., we consider both the cases when the donor impurity is on and off the center of the dot. We investigate the dependence of the ground and excited states on the confinement potential and on the presence of an external magnetic field. Two different theoretical approaches are used: the finite difference technique and the fractional-dimension method. We found that if the donor is displaced from the center of the quantum well, the presence of the lateral confinement shifts the magnetic field-induced angular momentum transitions and the unbinding of the charged donor state to a lower magnetic field.