Off-Center Neutral and Negatively Charged Donor Impurities in Semiconductor Heterostructures: Fractal Dimension Method

Abstract

We reduce the problems of the on- and off-center D0 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/Ga0.7Al0.3As 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 Å. 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 Å.