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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.

Using the effective mass approximation in a parabolic two-band model, we studied the effects of the geometrical parameters, on the electron and hole states, in two truncated conical quantum dots: (i) GaAs-(Ga,Al)As in the presence of a shallow donor impurity and under an applied magnetic field and (ii) CdSe–CdTe core–shell type-II quantum dot. For the first system, the impurity position and the applied magnetic field direction were chosen to preserve the system’s azimuthal symmetry. The finite element method obtains the solution of the Schrödinger equations for electron or hole with or without impurity with an adaptive discretization of a triangular mesh. The interaction of the electron and hole states is calculated in a first-order perturbative approximation. This study shows that the magnetic field and donor impurities are relevant factors in the optoelectronic properties of conical quantum dots. Additionally, for the CdSe–CdTe quantum dot, where, again, the axial symmetry is preserved, a switch between direct and indirect exciton is possible to be controlled through geometry.

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