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 $(1/{\tau }_t)$ evaluated by the phase-shift analysis is reliable at low fields. The phonon-assisted tunneling rate $(1/{\tau }_p)$ is also calculated. We found that $1/{\tau }_p$ dominates at low fields, while $1/{\tau }_t$ 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.