Resonant Delay in Tunnelling through Smooth Double and Triple Barriers

Abstract

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 Schro È dinger equation confirms the quality of the obtained expressions in wide energy ranges up to the barrier top.Die Energieabha È ngigkeit der Verzo È gerungszeit, die im Tunneln durch glatte (symmetrische) Doppel-und Dreierbarrieren zum Vorschein kommt, ist untersucht worden. Scharfe Resonanzen in der Verzo È gerungszeit, die sich nach den Resonanzen in der Transmissionswahrscheinlichkeit richten, werden gefunden. Im Grenzfall einer starken Lokalisierung nehmen Teilchen mit au%er Resonanz liegenden Energien das innere Potentialrelief nicht wahr und gehen durch die vorhandene Doppeloder Dreierbarriere in ungefa È hr derselben Weise, wie sie eine einfache, nicht strukturierte Barriere durchdringen wu È rden. Die Ûberpru È fung der abgeleiteten Formeln mittels einer hochpra È zisen Zahlenlo È sung der Schro È dinger-Gleichung besta È tigt die Qualita È t dieser Formeln in breiten Energiebereichen einschlieûlich des Scheitels der Barriere.

Arnold’s potentials and quantum catastrophes

“…For example, the necessity of having the weight "12" at the outermost shift-parameter δ 2 disproved a tentative "binomial-coefficients" extrapolation hypothesis inspired by Eq. (21).…”

“…In parallel, a remarkable methodical encouragement emerges also from the correspondence between the bound-state and scattering problems, especially due to an intimate relationship between the necessary matching formulae in the systems with and without scattering [20]. The analogies become particularly striking when one considers a particle passing through several separate barriers [21]. In this context, a technical inspiration might also result from certain mathematically analogous situations occurring in classical optics [22,23] or in classical electrodynamics [24].…”

“…The aim of such an analysis is to find an analogous optimal parametrization ansatzs at the higher odd degrees k > 7. At k = 7 itself the elimination of the fractional coefficients may be sought via the following generalized ansatz (21),…”

Explicit expression for the reflection and transmission probabilities through an arbitrary potential barrier

“…Quantum tunneling which is associated with a quantum particle through a potential barrier is a subject of profound importance in many areas of physics [1][2][3][4][5]. One of the reasons for an increased interest in the problem is that recent advances in molecular-beam-epitaxy (MBE) technology have opened up new possibilities of building next-generation nanometer-scale electronic devices [6]. On the other hand, a better fundamental understanding of quantum transport theory becomes an urgent challenge.…”