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Solid State Communications

Total delay time for a particle's tunneling through a smooth potential barrier

L. V. Chebotarev

Published May 1, 1996• 2 citations
Publisher
DOI
10.1016/0038-1098(96)00131-7

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The total delay time for a particle tunnelling through a non-rectangular potential barrier is calculated as an explicit function of the potential energy U(x), on a rigorous mathematical basis. The resulting expression is valid for sufficiently smooth potentials. It represents a physically well defined quantity that is measured in the experiment directly. In particular, an explicit expression for the tunnelling time related to the potential is obtained. It is valid if the particle's wavelength, taken far away from the barrier, is small compared with the barrier's effective length l, irrespective of the value of the barrier penetration integral. Exact sufficient conditions for the validity of Connor's parabolic connection formulae are established.
The energy dependence of the delay time associated with tunnelling through smooth (symmetric) double and triple barriers is investigated. Sharp resonances in the delay are found to mirror resonances in the transmission rate. In the strong-localization limit, particles with off-resonant energies do not perceive the inner potential relief and tunnel through the given double, or triple, barrier in much the same way as they would tunnel through a simple, unstructured barrier. Verification of the derived relations through a high-precision numerical solution of the Schrödinger equation confirms the quality of the obtained expressions in wide energy ranges up to the barrier top.