The fractional-dimensional space approach to impurity states in low-dimensional semiconductor heterostructures under magnetic fields

We have used the fractional-dimensional space approach to study the effects of applied magnetic fields on shallow-impurity states in GaAs-(Ga,Al)As quantum wells and superlattices. In this scheme, the actual anisotropic "impurity + heterostructure + magnetic field" system is treated as isotropic in an effective fractional-dimensional space, and the value of the fractional dimension is associated to the degree of anisotropy introduced both by the heterostructure barrier potential and applied magnetic field. No ansatz or fitting with experiment is involved and the fractional dimension is chosen using an analytical procedure. Theoretical fractional-dimensional calculations for shallow-impurity states in GaAs-(Ga,Al)As semiconductor quantum wells and superlattices, under magnetic-field applied along the growth direction, were shown to be in overall agreement with available experimental measurements.