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Physica Scripta

Virtual Process in Time-dependent Tunneling

Chao Zhang, Narkis Tzoar

Published Jan 1, 1989• 2 citations
Publisher

Abstract

The frequency-dependent transmission coefficient for a particle tunneling through a time-dependent barrier is calculated taking into account inelastic and virtual processes. We consider a barrier of arbitrary shape with a time-dependent modulation of single frequency ω. For small modulation amplitudes V 1 , our result is exact up to order V 1 2 . The result is obtained in the low transmission limit and is consistent with the exact calculation for a rectangular barrier. We find that the virtual processes can make a significant contribution to the tunneling current, especially at low ω. In the static approximation ωτ << 1, where τ is the barrier transit time, the contribution of the second order virtual process is equal to one half of the tunneling current from the first order inelastic processes.

DOI
10.1088/0031-8949/1989/t25/060

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The frequency-dependent transmission coefficient for a particle tunneling through a time-dependent barrier is calculated taking into account inelastic and virtual processes. We consider a barrier of arbitrary shape with a time-dependent modulation of single frequency ω. For small modulation amplitudes V 1 , our result is exact up to order V 1 2 . The result is obtained in the low transmission limit and is consistent with the exact calculation for a rectangular barrier. We find that the virtual processes can make a significant contribution to the tunneling current, especially at low ω. In the static approximation ωτ << 1, where τ is the barrier transit time, the contribution of the second order virtual process is equal to one half of the tunneling current from the first order inelastic processes.
The frequency-dependent transmission coefficient of a particle tunneling through a time-dependent barrier is calculated taking into account the quantum interference effect. We consider a rectangular barrier of height V0 with a time-dependent modulation of single frequency ω. For small modulation amplitude V1, our result is exact up to order V21 for any barrier width. In the low transmission region, our result is consistent with those obtained using a semiclassical method. We find that quantum interference (two modulation quanta process) makes a significant contribution to the tunneling current. In the static approximation, ωτ≪1, where τ is the barrier transit time, the contribution to the current from interference process is to cancel one-half of the tunneling current from first-order process (one modulation quantum process). To the lowest order in power series of V1/ℏω, the tunneling current is reduced due to interference.
One of several contradictory existing results for the time a tunneling particle interacts with its barrier is confirmed, by considering tunneling through a time-modulated barrier. At low modulation frequencies the traversing particle sees a static barrier. At high frequencies the particle tunnels through the time-averaged potential, but can do it inelastically, losing or gaining modulation quanta. The transition between the two regimes yields $\ensuremath{\int}\mathrm{dx}{[\frac{m}{2(V\ensuremath{-}E)}]}^{\frac{1}{2}}$ for the traversal time.
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.
We present a numerical investigation of quantum mechanical tunneling process in a time-dependent (fluctuating) barrier using a one dimensional model of Eckart barrier. The tunneling probability is calculated for two cases in which (1) the height of the barrier is undergoing harmonic oscillation with frequency ω and (2) the location of the barrier is undergoing harmonic oscillation with frequency ω. It is observed in both cases that the quantum mechanical tunneling probability exhibits a maximum as a function of the oscillating frequency ω between the low and high frequency limits. The physical origin and process underlying this resonantlike phenomenon are proposed in this paper based on the current model study.
Solutions to the time-dependent Schr\"odinger equation for a particle in a spatially uniform time-dependent field and some potentials of arbitrary form are proved to be like the time-independent eigenfunctions for an identical static potential. The field phase modulates the wave function and translates its envelope. This work applies directly to constant, linear, and parabolic static potentials and is self-consistent. Results for the modulation by a time-dependent field of the current through a rectangular barrier are relevant to the traversal time for tunneling and associated device limitations.
A real-time functional-integral method is used to derive an effective action that gives the transmission spectrum of a tunneling particle interacting with a bath of harmonic oscillators. The transmission spectum is expressed in terms of double functional integrals with respect to the coordinate of the particle which are evaluated by means of the stationary-phase approximation. The equations of motion for the stationary-phase trajectories are solved exactly for an arbitrary spectral density function of the bath, and the obtained solutions are used to find the transmission spectra for specific examples. For a bath with single frequency \ensuremath{\omega}, an analytic expression of the transmission spectrum is obtained which covers from the sudden tunneling (\ensuremath{\omega}T0{\mathit{T}}_{0}T0​\ensuremath{\ll}1) to the adiabatic one (\ensuremath{\omega}T0{\mathit{T}}_{0}T0​\ensuremath{\gg}1), where T0{\mathit{T}}_{0}T0​ is the time it would take a classical particle to traverse the inverted bare potential barrier. For a bath with an Ohmic spectrum, the differential tunneling conductance at low bias voltage V and for \ensuremath{\eta}T0{\mathit{T}}_{0}T0​\ensuremath{\ll}1 is found to obey a power law \ensuremath{\sim}(eVT0{\mathit{eVT}}_{0}eVT0​/\ensuremath{\Elzxh}${)}^{\mathrm{\ensuremath{\eta}}{\mathit{T}}_{0}{\mathit{S}}_{0}/2\mathrm{\ensuremath{\pi}}\mathrm{\ensuremath{\Elzxh}}}$, where \ensuremath{\eta} is the friction coefficient and S0{\mathit{S}}_{0}S0​ is the tunneling exponent in the absence of interaction. \textcopyright{} 1996 The American Physical Society.