Energy levels in quantum wells with capping barrier layer of finite size: Bound states and oscillatory behavior of the continuum states

The problem of a quantum-well structure in the vicinity of a high potential was studied theoretically. This problem corresponds to real devices with a finite-size capping layer. Results are obtained for the bound and the continuum states of a single quantum well, the continuum states of a multiple quantum well (MQW), and the continuum states with an electric field applied to the structure. For the bound states with no electric field, an equation including the cap-layer thickness (i.e., the distance of the well from the high potential) as one of its parameters is derived, and solved numerically for various well widths and cap-layer thicknesses, showing that significant deviations from the ‘‘regular’’ quantum-well levels are only expected for very thin cap layers (of the order of about 10 nm). The wave functions of the continuum states are found analytically, and used to calculate the probability of finding the carriers in the various regions of the device [cap layer, well(s), or buffer]. The theory predicts large energy-dependent oscillations in these probabilities for energies ranging up to more than 100 meV in the continuum; their periods depend on the cap-layer thickness and the carrier effective mass. Carrier segregation is then expected among the various regions for energies above the confining barriers. The contrast in probability between the regions increases for thin wells, deep potentials, and small effective masses, and is also enhanced by the presence of additional wells in the case of MQW’s. For MQW’s, the theory also predicts that the well(s) nearest to the surface will be more energy selective in capturing carriers in the continuum. This may result in different capture efficiencies for the various wells of a MQW structure. If an applied electric field lowers the potential away from the surface of the structure, the probability peaks shift linearly in energy with the magnitude of the field, and for small energies above the continuum edge, the probability of finding carriers in the well region increases.