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arXiv:cond-mat/9210013 · Physical review. B, Condensed matter

Numerical studies of tunneling in a nonharmonic time-dependent potential

J. A. Sto vneng, Antti‐Pekka Jauho

Published Apr 15, 1993• 7 citations
arXivHTMLPDF

Abstract

Tunneling through time-dependent potentials is of relevance to a number of physical problems. Using a WKB analysis, Azbel' has recently studied the effects of a nonharmonic time-dependent perturbation embedded in an opaque potential barrier. He suggests the existence of three different regimes for transmission in such systems: direct tunneling, activation-assisted tunneling, and elevator resonant activation. We address the same problem with a numerical technique based on wave-packet simulations. Our numerical results are in qualitative agreement with the picture of Azbel'. The outcome depends on the characteristic time T of the nonharmonic potential. There is a transition from direct tunneling to the activation regime around a ``crossover'' value ${\mathit{T}}_{\mathit{c}}$ which is determined by the B\"uttiker-Landauer time, the distance between the initial energy and the lowest resonant level, and the amplitude of the time-dependent perturbation. The total transmission probability is strongly enhanced when entering the activation regime.

DOI
10.1103/physrevb.47.10446

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Tunneling through time-dependent potentials is of relevance to a number of physical problems. Using a WKB analysis, Azbel' has recently studied the effects of a nonharmonic time-dependent perturbation embedded in an opaque potential barrier. He suggests the existence of three different regimes for transmission in such systems: direct tunneling, activation-assisted tunneling, and elevator resonant activation. We address the same problem with a numerical technique based on wave-packet simulations. Our numerical results are in qualitative agreement with the picture of Azbel'. The outcome depends on the characteristic time T of the nonharmonic potential. There is a transition from direct tunneling to the activation regime around a ``crossover'' value Tc{\mathit{T}}_{\mathit{c}}Tc​ which is determined by the B"uttiker-Landauer time, the distance between the initial energy and the lowest resonant level, and the amplitude of the time-dependent perturbation. The total transmission probability is strongly enhanced when entering the activation regime.
The authors analyse tunnelling through time-modulated barriers. The numerical results for the transmitted sideband intensities, obtained with wave-packet simulations, are in quantitative agreement with the analytical results of Buttiker and Landauer (1982, 1985). For opaque barriers the asymmetry of the sideband intensities can be characterized with a single frequency whose inverse Buttiker and Landauer interpret as the traversal time. This simple behaviour does not hold in general, and the authors discuss some of the difficulties one meets in an attempt to generalise the one-parameter description to arbitrary barriers. Furthermore, they examine analytically the transmitted wavefunction in the adiabatic limit, and show that the related density and current involve both the dwell time and the Buttiker time, defined in the context of a Larmor clock. This suggests the possibility that experiments probing different aspects of the transmitted wavefunction may yield different, but complementary, information about the tunnelling process.
The tunneling through an opaque barrier with a strong oscillating component is investigated. It is shown, that in the strong perturbations regime (in contrast to the weak one), higher perturbations rate does not necessarily improve the activation. In fact, in this regime two rival factors play a role, and as a consequence, this tunneling system behaves like a sensitive frequency-shifter device: for most incident particles' energies activation occurs and the particles are energetically elevated, while for specific energies activation is depressed and the transmission is very low. This effect is unique to the strong perturbation regime, and it is totally absent in the weak perturbation case. Moreover, it cannot be deduced even in the adiabatic regime. It is conjectured that this mechanism can be used as a frequency-dependent transistor, in which the device's transmission is governed by the external field frequency.
The tunnelling through an opaque barrier with a strong oscillating component is investigated. It is shown that in the non-weak perturbations regime (in contrast to the weak one), higher perturbations rate do not necessarily improve the activation. In fact, in this regime two rival factors play a role, and, as a consequence, this tunnelling system behaves like a sensitive frequency-shifter device: for most incident particles' energies activation occurs and the particles are energetically elevated , while for specific energies activation is depressed and the transmission is very low. This effect is unique to the strong perturbation regime, and it is totally absent in the weak-perturbation case. Moreover, it cannot be deduced even in the adiabatic regime. It is conjectured that this mechanism can be used as a frequency-dependent transistor, in which the device's transmission is governed by the external field frequency .
The usual WKB analysis for quantum tunneling applies when the tunneling action is large, as it is for tall, wide potential barriers. In contrast we analyze tunneling when the action is small, as it is for tunneling across a tall, thin barrier. We develop a perturbative analysis where the control parameter is the inverse of the area under the potential barrier and apply our technique to several examples in 1+11+11+1 dimensions. In resonant situations for bound particles we find that the tunneling probability grows with time as ∝t2\propto t^2∝t2, while in non-resonant situations it grows linearly with time. We evaluate not only the tunneling probability but also the time-dependent tunneling wavefunction for a particle that escapes to infinity, {\it i.e.} from a quasi-bound state to the continuum.
The physics of dynamic resonant tunneling is investigated.First, the resonant tunneling effect through an opaque barrier via a delta-function well is illustrated.Then, it is shown that, even in the adiabatic regime, where the dynamics can be governed by an analytic solution, the particle can be activated to higher energies.If the well varies quickly enough that the particle cannot escape from the well during the energetic elevation, the activation can be enhanced, as was anticipated by Azbel.However, and this is the main result of this work, the quasi-bound state of the well can even "reduce" the activation.In fact, because the resonant energy of the well matches twice the incoming particle's energy, and if the contribution to the wave function from both parts destructively interferes, then the particle cannot dwell in the well and activation is suppressed.This effect can be utilized in frequency-controlled transistors, and it is even speculated that it may explain the reason that humans can distinguish between tens of thousands of different odors with merely few hundreds of odor receptors.Lastly, the short time dynamics of a very fast perturbative well is also discussed.