An exact tunneling model and its application to transmission and reflection delay times

Abstract

Electron emission into a nanogap is not instantaneous, which presents a difficulty in simulating ultra-fast behavior using particle models. A method of approximating the transmission and reflection delay (TARD) times of a wave packet interacting with barriers described by a delta function, a metal–insulator–metal (MIM, rectangular) barrier, and a Fowler Nordheim (FN, triangular) barrier is given and has application to simulation. It is based on the superposition of a finite number of exact basis states obtained from Schrödinger’s equation, analogous to how quantum carpets are simulated. As a result, it can exactly and uniquely follow exponentially small tunneling currents. A Bohm-like trajectory is obtained from the time evolution of the density: it shows delay in both the transmitted and reflected packets that can be simply evaluated. The relations to prior studies of the analytic [Formula: see text]-function barrier and the Wigner distribution function (WDF) methods are described. A comparison of the TARD times is contrasted to alternate times in the Büttiker–Landauer (BL) and McColl–Hartman (MH) times; the MH approach is further reformulated explicitly in terms of Gamow factors to consider how the McColl–Hartman effect is to be related, particularly in the case of the FN barrier of field emission.

Electromagnetic analogs of quantum mechanical tunneling

“…In this paper, we have used simple, analytical examples to demonstrate the correspondence between quantum mechanics and electromagnetism with the intent of establishing a framework for the verification and vali-dation of our previously developed atomistic emission models 18,[24][25][26][27] . Beginning with the frequency-dependent EM S-parameters which can either be calculated using HFSS or obtained through laboratory measurement, the frequency-dependent refractive index vector of a wire array medium was calculated.…”

“…The McColl-Hartman effect, which characterizes tunneling delay as a function of barrier width, was re-formulated using the Gamow factor, which is a function of barrier shape and incident energy as well as barrier width. Next, an exact trajectory method based on the Schrödinger equation 27 was developed to handle systems that are difficult to simulate using the Wigner approach, such as the rectangular and Fowler-Nordheim barriers. The approaches described here model both above and below barrier emis-sion and delays on an atomistic level, yet the trajectorybased nature of these approaches enables statistical modeling of emission on the mesoscale and generates output that can be used to inform higher-order material and device models.…”

Tunneling Time of the Triangular Barrier and the Rectangular Barrier

“…incident on the barrier from the z < 0 region to the right. Where V0 and L are all greater than zero, V0 and L are the height and width of the triangular barrier respectively [17,18].…”

Does the Hartman effect exist in triangular barriers

“…[48] A recent study of the Harman effect through comparisons of tunneling in square and TBs can be found in Refs. [49,50]. Here we mainly compare the phase, dwell and Larmor times for a particle tunneling through a pair of MS ETBs, namely the positive-slope exact triangular barrier (PETB) and the negative-slope exact triangular barrier (NETB).…”