Transmission and reflection tunneling times

Donald H. Kobe, Hisakazu IWAMOTO, Masatoshi Goto +2 more

Gastón García‐Calderón, Alberto Rubio

Mário Goto, Hiromi Iwamoto, Verissimo Manoel de Aquino +2 more

The transmission tunnelling time through a barrier is often defined as the probability that the particle is in the barrier divided by the transmission probability current density. The dwell time, the time the particle stays in the barrier, is commonly defined as the probability that the particle is in the barrier divided by the incident probability current density. We define the reflection tunnelling time in a similar way by dividing the probability that the particle is in the barrier by the reflection current density. By conservation of probability, the reciprocal of the dwell time is equal to the sum of the reciprocals of the transmission and reflection times. This relation is illustrated by calculating the different tunnelling times for a rectangular barrier.

Mário Goto, H. Iwamoto, Verissimo Manoel de Aquino +1 more

Kevin L. Jensen, Joel L. Lebowitz, Jeanne M. Riga +2 more

A numerical solution to the time evolution equation of the Wigner distribution function (WDF) with an accuracy necessary to simulate the passage of a wave packet past a barrier is developed, where quantum effects require high accuracy and fine discretization. A wave packet incident on a barrier, a portion of which tunnels through, demonstrates behavior that can define various characteristic transmission and reflection delay (TARD) times useful in the simulation of electron emission. A model for the TARD times is proposed that relies only on the asymptotic maxima of the position $\ensuremath{\rho}(x,t)$ and wave number $\ensuremath{\rho}(k,t)$ densities given by the WDF and applied to a ballistic trajectory model for the (faster) transmitted and (slower) reflected parts. The dependence of the TARD times on barrier width, symmetry, and abruptness is analyzed. For symmetrical barriers with characteristics similar to field emission barrier heights and widths, TARD times are on the order of a fraction of a femtosecond. The TARD times for when tunneling predominates are contrasted to tunneling times in the literature. Use of the TARD times in simulations of field emission in nanogap devices or to model ultrashort pulses generated under rapidly changing conditions for electron sources are proposed.

Nikolay Leonidovich Chuprikov

As was shown in quant-ph/0405028, the state of a tunneling particle can be uniquely presented as a coherent superposition of two states to describe alternative sub-processes, transmission and reflection. In this paper, on the basis of the stationary wave functions for these sub-processes, we give new definitions of the dwell times for transmission and reflection. In the case of rectangular potential barriers the dwell times are obtained explicitly. In contrast with the well-known Büttiker's dwell-time, our dwell time for transmission increases exponentially, for the under-barrier tunneling, with increasing the barrier's width. By our approach the well-known Hartman effect is rather an artifact resulted from an improper interpretation of the wave-packet tunneling and experimental data, but not a real physical effect accompanying the tunneling phenomenon.