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physica status solidi (b)

Resonant Delay in Tunnelling through Smooth Double and Triple Barriers

L. V. Chebotarev

Published Jul 1, 1998• 2 citations
Publisher

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 Schrödinger equation confirms the quality of the obtained expressions in wide energy ranges up to the barrier top.

DOI
10.1002/(sici)1521-3951(199807)208:1<69::aid-pssb69>3.0.co;2-8

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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.
In order to better understand the trapping and response time of resonant electron wave packets in a resonant tunneling structure, we examined the time-dependent transport of a spherically symmetric Gaussian wave packet through a model double-barrier potential. Within the effective mass picture, the wave packet is projected onto the complete eigenstates of the model potential. The wave packet is initially localized to the left of the barriers and allowed to propagate through the structure. At wave packet energies corresponding to the resonant energies, the probability transmitted to the right of the barriers exhibits a steplike behavior as a function of time. A computation is made of the quantum mechanical delay time, to~tclass, where to is the packet transit time and tclass is the transit time for a classical particle. An increase in this delay time is observed when the packet energy corresponds to the energy of the resonant levels, and a comparison is made to the decay time of a particle in the first resonant state of the well. The delay time increases with increasing barrier area and decreasing packet energy width.
Transient tunneling effects in triple barrier systems are investigated by considering a time-dependent solution to the Schr\"odinger equation with a cutoff wave initial condition. We derive a two-level formula for incidence energies E near the first resonance doublet of the system. Based on that expression we find that the probability density along the internal region of the potential is governed by three oscillation frequencies: one of them refers to the well known Bohr frequency, given in terms of the first and second resonance energies of the doublet, and the two others represent a coupling with the incidence energy E. This allows us to manipulate the above frequencies to control the tunneling transient behavior of the probability density in the short-time regime.
We calculate the tunneling process of a Dirac particle across two square barriers separated by a distance d, as well as the scattering by a double-cusp barrier, where the centers of the cusps are separated by a distance larger than their screening lengths. Using the scattering matrix formalism, we obtain the transmission and reflection amplitudes for the scattering processes of both configurations. We show that the presence of transmission resonances modifies the Lorentzian shape of the energy resonances and induces the appearance of additional maxima in the transmission coefficient in the range of energies where transmission resonances occur. We calculate the Wigner time delay and show how their maxima depend on the position of the transmission resonance.
The resonant tunneling of a wave packet is studied, for a one-dimensional double barrier, by solving the time-dependent Schrödinger equation. The transmission is determined as a function of the spatial width of the incident wave packet. It is shown that if the double barrier acts as an energy filter, then the spatial width of the transmitted wave packet is increased, due to the trapping of the particle between the two barriers for a time, ℏ/Γ, where Γ is the width of the resonance. The reflected wave packet consists of two components: one corresponding to reflection from the first barrier and the other due to the particle being trapped between the barriers for a time ℏ/Γ.
A high-precision numerical solution of the tunnelling problem for three different one-dimensional potentials is performed in order to verify the accuracy and the range of validity of the recently proposed general expressions for the phase of the transmission amplitude and the total delay time of a quantum particle tunnelling through smooth potential barriers. The results show those expressions, as well as Kemble's formula for the transmission rate, to be of high quality in wide ranges of the particle's energy that include the top of the barrier but not its base. Zusammenfassung. Eine hochpräzise Zahlenlösung des quanten-mechanischen Tunnelproblems ist für drei verschiedene eindimensionale Potentiale durchgeführt worden, die zum Zweck hatte, die Genauigkeit und den Gültigkeitsbereich der vor kurzem abgeleiteten grundlegenden Formeln für die Phase der Übertragungsamplitude und die gesamte Verzögerungszeit eines Teilchens zu überprüfen, das im Tunnelprozeß durch glatte Potentialsperren einbegriffen ist. Die Ergebnisse der ausgeführten Zahlenanalyse bestätigen die hohe Qualität dieser Fromeln, sowie auch die der Formel von Kemble für die Übertragungswahrscheinlichkeit, im breiten Bereich der Energie des Teilchens, einschließlich des Scheitels der Potentialsperre doch nicht des Sperrenfußes.