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International Vacuum Nanoelectronics Conference

A Delta Barrier in a Well and its Generalization for Emission Studies

Kevin L. Jensen, Jeanne M. Riga, Andrew Shabaev, M. S. Osofsky, Joseph Prestigiacomo, John J. Petillo

Published Jul 10, 2023• 0 citations
Publisher

Abstract

The transmission coefficient for a 6-function barrier is a convenient model for many technologically important applications relying on photoemission, simulations of wave packets, or modeling the narrow barrier of a normal- superconducting point contact. We examine an extension of the model to treat instead a function sequence (a rectangular barrier that approaches the behavior of a function in the limit of vanishing width). It is shown how the eigenstates of the sequence converge on the function barrier eigenstates, but more importantly, how the even and odd parity states depart from the 6-function limiting case. The exact eigenstates enable the time evolution of exponentially attenuated tunneling to be exactly evaluated, in contrast to numerical methods. The application is the inclusion of tunneling time effects in simulations of time-varying electron emission.

DOI
10.1109/ivnc57695.2023.10188982

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The transmission coefficient for a 6-function barrier is a convenient model for many technologically important applications relying on photoemission, simulations of wave packets, or modeling the narrow barrier of a normal- superconducting point contact. We examine an extension of the model to treat instead a function sequence (a rectangular barrier that approaches the behavior of a function in the limit of vanishing width). It is shown how the eigenstates of the sequence converge on the function barrier eigenstates, but more importantly, how the even and odd parity states depart from the 6-function limiting case. The exact eigenstates enable the time evolution of exponentially attenuated tunneling to be exactly evaluated, in contrast to numerical methods. The application is the inclusion of tunneling time effects in simulations of time-varying electron emission.
The analytic nature of the transmission coefficient for a δ-function barrier makes it a useful tool to examine a variety of technologically important applications, such as photoemission from semiconductors with an alkali coating, the examination of tunneling times for wave packets incident on a barrier, and for parameterizing tunneling through the narrow barrier of a normal-superconducting point contact. The analytic model of a δ-function barrier inside a confining well is extended to the finite height and width rectangular barrier (a delta-function sequence). Methods to exactly evaluate the eigenstates are given and their dependencies are examined. The time evolution of a superposition of the lowest eigenstates is considered for barriers having comparable Gamow tunneling factors so as to quantify the impact of barrier height and shape on time evolution in a simple and exact system and, therefore, serve as a proxy for tunneling time. Last, density profiles and associated quantum potentials are examined for coupled wells to show changes induced by weaker and wider barriers.
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.
The time evolution of quantum mechanical states in a square well of infinite depth with a Dirac δ function at its center has been examined for cases where the initial state was localized in one of the wells. Let well A denote the well in which the initial state is localized, and let P(t) denote the integrated probability density Ψ*Ψ in well A. For opaque barriers the time-dependent system is adequately described by a two-state model in which only the pair of stationary states of even and odd parity are considered whose wave functions in well A, apart from an arbitrary phase factor, are largely identical with the wave function of the initial state. For P(t) a harmonic oscillation is observed whose frequency νt is well approximated by the well-known formula for the tunneling frequency νt≂ΔE/h, where ΔE represents the energy separation among the pair of states in the model. For the present model of a symmetric double well it has been shown that for highly transparent barriers a three-state model can describe the time evolution adequately. The three stationary states involved in this model are a state of odd parity whose wave function in well A is largely the same as the wave function of the initial state and a pair of stationary states of even parity which on the energy scale are immediately above and below the first state. In this three-state model the function P(t) is a superposition of two sinusoidal functions with nearly identical amplitudes and frequencies plus a constant. As a consequence, the amplitude of P(t) changes harmonically. In the present model a δ function has been used as a barrier in order to minimize the mathematical detail involved in the time-dependent treatment. It is to be expected that the beating in P(t) can be observed also in the time evolution of a state localized in one of the wells of a symmetric double-minimum potential with a more realistic ‘‘low’’ barrier if the density of energy levels near the energies of the levels to be considered in the corresponding three-state model varies slowly with energy.
The Wigner function is assembled from analytic wave functions for a one-dimensional closed system (well with infinite barriers). A sudden change in the boundary potentials allows for the investigation of time-dependent effects in an analytically solvable model. A trajectory model is developed to account for tunneling when the barrier is finite. The behavior of the density (the zeroth moment of the Wigner function) after an abrupt change in potential shows net accumulation and depletion over time for a weighting of energy levels characteristic of the supply function in field emission. However, for a closed system, the methods have application to investigations of tunneling and transmission associated with field and photoemission at short time scales.
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.
In this paper, we examine transmission through one-dimensional potential barriers that are piecewise constant. The transfer matrix approach is adopted, and a new formula is derived for multiplying long matrix sequences that not only leads to an elegant representation of the wave function but also results in much faster computation than earlier methods. The proposed method covers a broad spectrum of potentials, of which multi-barrier systems are special cases. The procedure is illustrated with a finite lattice of nonuniform rectangular barriers—non-uniformity being a novel feature, as the uniform case has been solved exactly by Griffiths and Steinke. For the nonuniform multi-barrier problem, the intervening wells strongly influence the transmission probability. Surprisingly, we find that the wells act “individually,” i.e., their influence is a function only of their width and is independent of their exact locations in a multi-barrier system. This finding leads to an observation that we have termed the “alias effect.” The exact solutions are supplemented with asymptotic formulas.