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Applied Physics Letters

Determination of barrier traversal time by modulation of the incident wave

Mark J. Hagmann

Published Jan 11, 1993• 11 citations
Publisher

Abstract

The semiclassical expression for barrier traversal time results from a variety of theoretical procedures, but it appears that an opaque barrier is required in each derivation. Following Büttiker and Landauer [IBM J. Res. Dev. 30, 451 (1986)], we use two interfering plane waves as the incident wave and determine the traversal time by attributing differences in the transmitted and incident waves to the ratio of dispersion in the transit time to the modulation period. Calculations are made for rectangular barriers with different sizes. There is essentially agreement with Büttiker and Landauer for opaque barriers which they considered, but the present solution does not diverge for energies near the top of the barrier.

DOI
10.1063/1.109314

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The semiclassical expression for barrier traversal time results from a variety of theoretical procedures, but it appears that an opaque barrier is required in each derivation. Following Büttiker and Landauer [IBM J. Res. Dev. 30, 451 (1986)], we use two interfering plane waves as the incident wave and determine the traversal time by attributing differences in the transmitted and incident waves to the ratio of dispersion in the transit time to the modulation period. Calculations are made for rectangular barriers with different sizes. There is essentially agreement with Büttiker and Landauer for opaque barriers which they considered, but the present solution does not diverge for energies near the top of the barrier.
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 time that a transmitted particle interacts with a tunneling barrier is evaluated in one of the principal existing approaches by considering a small oscillatory perturbation added to the static barrier. The modulation frequency, at which a serious deviation from quasistatic behavior appears, is indicative of the traversal time. This approach is extended here, to a long region including the barrier, and utilizing a modulation potential which is spatially uniform throughout this region, except in the incident region ahead of the actual barrier, where it is made sufficiently smooth to avoid reflections. The resulting traversal time is \ensuremath{\Elzxh}\ensuremath{\partial} ln[\ensuremath{\partial}t/\ensuremath{\partial}E]/\ensuremath{\partial}E, where the complex transmission coefficient t(E) is defined as the ratio of the wave function at the far end to that at the point of incidence. This answer applies to arbitrary potentials and to all energies, including both propagation and tunneling in a single framework. This result is closely related to, but not identical to, that obtained by another earlier approach in which the potential is fixed, but the incident amplitude is modulated.
A variety of different theoretical procedures have been used to determine tunneling times. These include (1) phase time; (2) the Stevens procedure; (3) Larmor times; (4) a complex “time”; (5) dwell time; (6) the Büttikker-Landauer time; (7) Feynman path-integrals; (8) scattering theory; (9) a stochastic formulation; (10) oscillatory perturbation of the barrier; (11) kinetic time; and (12) the semiclassical solution. Interest in tunneling times is not purely pedagogical, since correction for changes in image potential during transit is required in determining the conductance of several new high-speed semiconductor devices. A new procedure to evaluate transit time is presented, which assumes that energy fluctuations keep a tunneling particle above the barrier while transiting the classically forbidden region. For large rectangular barriers, the most probable fluctuations minimize the product of their magnitude and the transit time. This results in the semiclassical solution. For the case of very small barriers, there is a separate solution in which the transit time is indeterminate but bounded. The new solution is compared with 12 different results by others. Numerical values, limiting forms, and interpretations are presented for these various tunneling times. Difficulties in using time domain numerical solutions to determine the transit time for a wave packet will also be described. © 1992 John Wiley & Sons, Inc.
Exact analytical solutions of the time-dependent Schrodinger equation with the initial condition of an incident cutoff wave are used to investigate the traversal time for tunneling. The probability density starts from a vanishing value along the tunneling and transmitted regions of the potential. At the barrier width it exhibits, at early times, a distribution of traversal times that typically has a peakp and a width ��. Numerical results for other tunneling times, as the phase-delay time, fall within ��. The Buttiker traversal time is the closest top. Our results resemble calculations based on Feynman paths if its noisy behaviour is ignored.
Büttiker–Landauer traversal times in the radial direction of single and double barriers with cylindrical geometry are studied by modulating the incident wave and the barriers, respectively. The traversal times obtained by modulating the incident wave oscillate as a function of the incident wave energy. The oscillation is negligibly small in the traversal times obtained by modulating the barriers. The coherence of particle waves (Hankel functions) in cylindrical geometry is responsible for the oscillation.
Exact analytical solutions of the time-dependent Schrödinger equation with the initial condition of an incident cutoff wave are used to investigate the traversal time for tunneling. The probability density starts from a vanishing value along the tunneling and transmitted regions of the potential. At the barrier width it exhibits, at early times, a distribution of traversal times that typically has a peak τpτ_pτp​ and a width ΔτΔτΔτ. Numerical results for other tunneling times, as the phase-delay time, fall within ΔτΔτΔτ. The Büttiker traversal time is the closest to τpτ_pτp​. Our results resemble calculations based on Feynman paths if its noisy behaviour is ignored.