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American Journal of Physics

Time-dependent tunneling through thin barriers: A simple analytical solution

W. Elberfeld, M. Kleber

Published Feb 1, 1988• 56 citations
Publisher

Abstract

Analytical solutions for the time-dependent tunneling of wave trains and wave packets through delta-function barriers are presented. Tunneling currents and density distributions are calculated with no approximations. A standby mechanism is demonstrated: The particle ‘‘waits’’ some time in front of the barrier before it tunnels through the barrier.

DOI
10.1119/1.15695

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American Journal of Physics•Mar 1, 1996•19 citations
Analytical solutions for the time-dependent tunneling of wave trains and wave packets through delta-function barriers are presented. Tunneling currents and density distributions are calculated with no approximations. A standby mechanism is demonstrated: The particle ‘‘waits’’ some time in front of the barrier before it tunnels through the barrier.
A formal analytic solution of the time-dependent Schr\"odinger equation is obtained for cutoff Gaussian wavepackets incident on finite range potentials of arbitrary shape. We derive closed analytical expressions for transmitted wavepackets with a small cutoff and the solutions are applied to investigate the dynamics of wavepacket tunneling. We showed that the analytic solutions satisfy a useful rescaling property that was applied to study and characterize the delay time in delta potential barriers.
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.
Quantum tunneling in one spatial dimension in the presence of time-dependent potentials is investigated theoretically. First, a general multichannel formulation of the problem is given for harmonic time-dependence. The case of an oscillatory delta-function potential of constant strength is discussed at length and specific numerical calculations are presented for the reflection and transmission coefficients. Then other exactly solvable time-dependent potentials are obtained by carrying out successive unitary transformations. As an example of this class, a delta-function potential with time-dependent strength and boundary conditions is studied. Finally tunneling time-delay for time-dependent potentials is formulated in the multichannel situation, and also the concept of first passage time for the decay of a wave packet confined to one well of a double well potential is generalized for the case of time-dependent barrier and boundary conditions.
We study the time required for a wave packet to tunnel beyond a square barrier, or to be reflected, by envisaging a physical clock which ticks only when the particle is within the barrier region. The clock consists in a magnetic moment initially aligned with the xxx axis which in the barrier region precesses around a constant magnetic field aligned with the zzz axis, the motion being in the yyy direction. The values of the xxx and yyy components of the magnetic moment beyond or in front of the barrier allow to assign a tunneling or reflection time to every fraction of the packet which emerges from the barrier and to calculate tunneling times τT,xτ_{\rm T,x}τT,x​ and τT,yτ_{\rm T,y}τT,y​ and reflection times τR,xτ_{\rm R,x}τR,x​ and τR,yτ_{\rm R,y}τR,y​. The times τT,xτ_{\rm T,x}τT,x​ and τT,yτ_{\rm T,y}τT,y​ (τR,xτ_{\rm R,x}τR,x​ and τR,yτ_{\rm R,y}τR,y​) are remarkably equal, and independent of the initial position (in front of the barrier) of the packet.
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.