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Physical review. B, Condensed matter

Tunneling times for one-dimensional systems

V. Gasparian, M. Ortuño, J. Ruiz, Emilio Cuevas, M. Pollak

Published Mar 1, 1995• 26 citations
Publisher

Abstract

We derive a general expression for the tunneling time in layered systems based on the Larmor-clock approach. We show that our results are equivalent to those obtained by the Feynman path-integral technique. We establish a relation between functional derivatives of the barrier potential and partial derivatives with respect to the incident energy. In application, we computed the tunneling time and the reflection time for a rectangular barrier, for a double rectangular barrier, and for a Gaussian potential.

DOI
10.1103/physrevb.51.6743

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Journal of Applied Physics•Dec 1, 1962•821 citations
We derive a general expression for the tunneling time in layered systems based on the Larmor-clock approach. We show that our results are equivalent to those obtained by the Feynman path-integral technique. We establish a relation between functional derivatives of the barrier potential and partial derivatives with respect to the incident energy. In application, we computed the tunneling time and the reflection time for a rectangular barrier, for a double rectangular barrier, and for a Gaussian potential.
Tunneling phenomenon is one of the wondrous events that take place particularly in quantum mechanics and it has no counterpart in classical physics. We focus on its two important implications : tunneling time and tunneling probability. There has been no consensus on tunneling times among the tunneling time community for last seven decades. So far, most of the attempts to calculate tunneling times have apparently been centered on a simple rectangular barrier example. We implement the tunneling times with the most widely accepted approach, which is using physical Larmor clock, into a nonsymmetrical barrier and a double barrier. It results in different and unique insights into the usual understanding of tunneling time. Three characteristics tunneling time expressions are thoroughly studied in these quantum structures. The difference between dwell time and phase-delay time is once again approved. The unique characteristics of the phase-delay time, which was thought to be exactly similar to dwell time when we assign higher energies to the particles, is also revealed in the nonsymmetrical barrier that it should actually be regarded as a distinct feature from the dwell time. The singularity of reflection time in rectangular barrier is no longer valid in nonsymmetrical barrier. The finiteness of reflection time fairly shows that different quantum structures might bear some taboo breaking results. Since a double barrier involves much more complicated features and calculations, the calculation of tunneling times under this barrier is praiseworthy. The dwell time is the most crucial, troublesome and distinctive tunneling time element to look into for a double barrier. Besides the dwell and phase-delay time, we also calculate the three characteristic times as well as the reflection time for a double barrier. Another area that we study within the context of tunneling phenomenon is the tunneling probability. Recently, there has been ground shaking attempts in modifying the conventional WKB formulae. We follow one of the modified versions of WKB formulae in order to be able to calculate the tunneling probability through different quantum structures where conventional WKB formulae fail. These generalized WKB connection formulae are applied to some complex quantum potentials by seeking aid from the Green's function and the path integral approach. One of the biggest problems in quantum mechanics (visualization) is partially overcome through implementing the method of diagrammatic representations that make the life easier in understanding the events happening in the quantum world when we attempt to calculate the transmission and reflection probabilities. This part therefore has a pedagogical sense too in terms of interpreting the results in a diagrammatic way with the help of the path integral approach.
The desire to minimize the response time of tunneling devices has led to renewed interest in the unresolved fundamental question How much time, on average, does an incident electron that is ultimately transmitted spend inside a one-dimensional potential barrier? In addition to this transmission time ⊤t ·,there are two other times of interest: the reflection time ⊤r for electrons that are ultimately reflected and the dwell time TD, which is the average time spent inside the barrier regardless of the ultimate fate of the incident particle. An exact expression for TD and the time-modulated barrier, Feynman path, Larmor clock, and wave-packet evolution approaches for calculating ⊤t and ⊤r arc discussed. It is argued that none of these approaches is completely satisfactory because they all lead to transmission and reflection times that either are complex or, if real, do not satisfy: all of a few reasonable (but not universally accepted) criteria, such as consistency with the exact expression for the dwell time.
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.
The path integral formulation of the traversal time in tunneling utilizes a restricted Feynman amplitude. We present a derivation based on the Fisher Lee formula to compute the average time and the integrated distribution for arbitrary potentials. The (complex) tunneling time can be expressed as a derivative with respect to a potential which surrounds the region over which the particle is “detected”, and a correspondence with other tunneling time scales which depend on energy derivatives of the transmission coefficient is found. A semiclassical treatment confirms that there are two time scales associated with this measurement procedure. The imaginary component is related to the multiple instanton bounces in the inverted potential. The issue of positive times is found to depend crucially on the behavior of the transmission coefficient for asymptotic limits of the measurement potential. The calculation of higher moments of the distribution allows to define a positive barrier interaction time, without appealing to any regularization procedure.
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.
Analytic expressions are given for the time spent by a particle tunneling through a potential barrier. The expressions are derived for an incident wave packet which is initially Gaussian, centered about a point an arbitrary distance away from a rectangular potential barrier and moving toward the barrier with constant average velocity. Upon collision with the barrier, the packet splits into a transmitted and a reflected packet. The resultant transmission time is positive, nonzero and in principle measurable. Although the transmission time becomes quite large as the incident kinetic energy becomes very small, in general, for nonzero incident momentum and finite potential barriers which are neither very thick nor very thin, the transmission times are less than the time that would be required for the incident particle to travel a distance equal to the barrier thickness. The transmission times for metal-insulator-metal thin film sandwiches, given approximately by δt3=ℏ/(Efφ)12, where Ef is the Fermi energy of the metal, and φ the vacuum work function, are of the order of 10−16 sec, compared to RC time constants of about 10−13 sec.