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Journal of Physics A Mathematical and General

Reflection and transmission behaviour of a particle in a resonant tunnelling barrier

Ph. Grossel, Françoise Depasse, J. M. Vigoureux

Published Nov 7, 2002• 11 citations
Publisher

Abstract

Stationary states are used to describe the transmission and the reflection of a wave packet through a resonant tunnelling barrier. The transmitted and the reflected spectra are studied and different links between numerical evaluations and the important characteristics of the resonant regime are underlined. The differences between the behaviours of wide and narrow near-resonant wave packets are delineated: two time scales appear in the case of quasi-bound states.

DOI
10.1088/0305-4470/35/46/305

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Stationary states are used to describe the transmission and the reflection of a wave packet through a resonant tunnelling barrier. The transmitted and the reflected spectra are studied and different links between numerical evaluations and the important characteristics of the resonant regime are underlined. The differences between the behaviours of wide and narrow near-resonant wave packets are delineated: two time scales appear in the case of quasi-bound states.
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.
The reflection of wave packets from a quantum well bounded by rectangular potential barriers, and which has a tunneling transmission resonance, is discussed. Numeric solutions to Schrödinger’s time-dependent equation show that the shape of the probability function ψψ* for a reflected wave packet can be quite different from that of the incident packet. Either a symmetric double hump or an asymmetric shape are possible for the reflected ψψ*, depending on the sharpness of the resonance. These shapes are explained analytically in terms of the reflection coefficient.
In this paper, we discuss the tunneling time of a quantum particle through a rectangular barrier. The reflection and transmission times associated with the wave packets representing the particle are discussed. By using an initial Gaussian momentum distribution, we carry out a comparative analysis of the stationary phases of the incident, reflected, and transmitted wave packets leading to the reflection and transmission times $\ensuremath{\Delta}{t}_{R}$ and $\ensuremath{\Delta}{t}_{T},$ respectively. In the present treatment of this old and very known problem we take into account the deformations of the reflected and transmitted momentum distributions. These deformations produce a dependence of the reflection and transmission times on the location of the initial wave packet. In a parallel calculation, by numerically monitoring the time evolution of the system, we characterize a reflection and a transmission time. Such times agree with the ones obtained via the stationary phase method.
The proper basis for the calculation of transmission and reflection times for wave packets scattered off arbitrary tunneling structures in one dimension is considered. With packets narrow in wave-number space, we demonstrate that the classic phase times are indeed correct to lowest order. Explicit, general expressions for the leading correction terms for finite wave packets are given. The physics associated with these corrections is discussed. We also consider the dwell time, as it is currently defined, and derive a general relation between this dwell time and the phase times. This relation shows when the dwell time can and cannot be used. Finally, we discuss wave packets transmitted from narrow resonances, and derive an explicit, exact formula for the tunneling time with resonance transmission from a symmetric double barrier. Comparison with earlier approximate results is made.
It is shown that in the case of the one-particle one-dimensional scattering problem for a given time-independent potential, for each state of the whole quantum ensemble of identically prepared particles, there is an unique pair of (subensemble's) solutions to the Schrödinger equation, which, as we postulate, describe separately transmission and reflection: in the case of nonstationary states, for any instant of time, these functions are orthogonal and their sum describes the state of all particles; evolving with constant norms, one of them approaches at late times the transmitted wave packet and another approaches the reflected packet. Both for transmission and reflection, 1) well before and after the scattering event, the average kinetic energy of particles is the same, 2) the average starting point differs, in the general case, from that for all particles. It is shown that for reflection, in the case of symmetric potential barriers, the domain of the motion of particles is bounded by the midpoint of the barrier region. We define (exact and asymptotic) transmission and reflection times and show that the basic results of our formalism can be, in principle, checked experimentally.