Analytical expressions for asymmetric double quantum wells and their application to semiconductor heterostructures

Abstract

Analytical solution of Schro¨dinger's equation for an asymmetric double quantum well structure is obtained for the first time in terms of a transcendental equation, roots of which give energy eigenstates and enable derivation of corresponding wave functions. Results obtained are in agreement with those reported in the literature by numerical methods and do away with the need of perturbation approximation used in such cases. As an example, GaAs/Al x Ga 1Àx As system has been discussed with respect to variations in barrier width, quantum well (QW) width, confining potential and material composition. The analysis shows that the coupled QWs can be matched for a given energy level to allow maximum transport between them by variation of one or more of these parameters so that the energy difference between it and the next nearest state is minimal. This condition shows a complete delocalization and the probability of finding the electron in the two QWs becomes equal, which amounts to an asymmetric system becoming symmetric with respect to probability density.