Dynamic behavior of electron tunneling and dark current in quantum well systems under an electric field

A time evolution method is used to study the time dependence of electron tunneling in a quantum well under an electric field. A comparison of phase-shift analysis and the full time-dependent calculation is presented. Good agreement between the two on a large time scale is found, but discrepancy exists on the small time scale especially at high fields. Our studies confirm that the direct tunneling rate (Formula presented) evaluated by the phase-shift analysis is reliable at low fields. The phonon-assisted tunneling rate (Formula presented) is also calculated. We found that (Formula presented) dominates at low fields, while (Formula presented) increases exponentially with increasing field and becomes dominant at high fields. Contributions due to the thermionic emission, direct tunneling as well as phonon-assisted tunneling to the dark current are also evaluated.