Quantum mechanical tunneling time and its relation to the Tsu-Esaki formula

Document Type

Conference Proceeding


Various approaches have been used to calculate the quantum-mechanical tunneling time through potential barriers including the phase-delay method first introduced by Bohm and Wigner, Buttiker's analysis of the Larmor clock and its generalization, the dwell time of Smith, and numerical studies of wavepacket propagation through potential barriers among others. Most of those previous estimates have only dealt with the tunneling time through simple obstacles, including delta-potential scatterers and simple rectangular barriers under zero bias condition. Even for these simple cases, the agreement among the various estimates is far from being satisfactory. In this manuscript, the transmission line technique is employed to solve the time-dependent Schrodinger equation and an expression of the quantum-mechanical tunneling time is derived for an arbitrary potential profile under non zero bias condition. An exact analytical expression of the tunneling time through a rectangular barrier is derived and shown to be identical to the one obtained recently by Spiller et. al. using the Bohm's quantum potential approach. The tunneling time through resonant tunneling structures under zero bias is also calculated and is shown to be minimum at the quasi-boundstate energy. Finally, the quantum-mechanical tunneling time through the emitter-base junction of a typical heterojunction bipolar transistor is shown to be larger than its semiclassical counterpart.


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