Shallow donors in semiconductor heterostructures: Fractal dimension approach and the variational principle

We propose a simple method for calculating the energies of the shallow donor S states in semiconductor heterostructures. The trial function is taken as a product of the wave equation for an unbound electron in the heterostructure corresponding to the lowest level with a variational function that depends only on electron-ion separation. Starting from variational principle, we find that the latter is a solution of the radial wave equation for a hydrogen-like atom in an effective space with a non-power-law dependence of the Jacobian volume element on the ion-electron separation. By using the relation between the Jacobian volume element and the dimension of the space similar to the one proposed by Mandelbrot for fractal geometric objects, we find that the fractional dimension is related with the density of charge distribution for the electron ground state within the heterostructure. An excellent agreement between our results for donor binding energies in GaAs-(Ga,Al)As quantum wells, quantum well wires (QWW's), and quantum dots with a square-well potential and those previously obtained by means of the variational, Monte Carlo, and series-expansion methods is found when we use the fractal dimension defined locally. Additionally, donor binding energies curves versus GaAs-(Ga,Al)As QWW radius, for models with soft-edge-barrier and double-step potentials are presented.