Electric field dependence of quantum-well eigenstates

Two very accurate methods are developed, one based on the shooting method and the other on the relaxation method, for calculating the eigenenergies and eigenfunctions of states in a quantum well with an applied electric field. These methods, which give accuracies greater than 0.001 meV, are well controlled, give the quantum-well eigenfunctions, and are easily applied to situations of varying potential and effective mass. Comparisons with the variational approach of Bastard and others are made. These techniques allow one to follow the development of the quantum-well eigenstate outside the well and to determine the validity of the quasi-bound state approximation. Recent results in the literature showing that the ground-state hole eigenfunction becomes unbound at moderate electric fields are shown to be erroneous. Detailed calculations are presented for the electron (ground and first excited) and hole (ground) eigenstates of a quantum well with width 85 Å, and barrier heights of 240 (conduction band) and 160 meV (valence band) for applied electric fields varying from 0 to 150 kV/cm. Also, we have calculated the overlap integrals and the dipole matrix elements appropriate to quantum confined Stark effect modulators and infrared quantum-well detectors, respectively.