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Wavelet solution of the Schrödinger equation for a triangular potential barrier and applied results on a MOM tunnel junction

Emilia-Simona Malureanu, Florea Craciunoiu

Published Oct 1, 2013• 0 citations
Publisher

Abstract

Quantum tunneling has many important applications in electronics. In thin layers structures, when the tunneling phenomenon occurs at the metal-insulator contact barrier, as for the MIM or MOS structures, we are dealing with the Fowler-Nordheim field emission. In these structures we have to calculate the currents densities that go through. The current density is calculated based on the transmission coefficient through the barrier, which is determined by solving the Schrödinger equation. The calculation methods currently used lead to values that get predictions of emission current density too low. The transmission coefficient is calculated by wavelet approximation and the results are validated experimentally on a lateral MOM tunnel junction.

DOI
10.1109/smicnd.2013.6688091

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Quantum tunneling has many important applications in electronics. In thin layers structures, when the tunneling phenomenon occurs at the metal-insulator contact barrier, as for the MIM or MOS structures, we are dealing with the Fowler-Nordheim field emission. In these structures we have to calculate the currents densities that go through. The current density is calculated based on the transmission coefficient through the barrier, which is determined by solving the Schrödinger equation. The calculation methods currently used lead to values that get predictions of emission current density too low. The transmission coefficient is calculated by wavelet approximation and the results are validated experimentally on a lateral MOM tunnel junction.
Electronics industry is a major user of Metal-Insulator-Metal (MIM) structures. When the tunneling phenomenon occurs at the metal-insulator contact barrier, as for the MIM structures, we are dealing with the Fowler-Nordheim field emission. In MIM structures we have to calculate the currents densities that go through. Unfortunately, the calculation methods currently used lead to values that get predictions of emission current density too low. The current density is calculated based on the transmission coefficient through the barrier, which is determined by solving the Schrödinger equation. One of the oldest and most efficient methods of solving the Schrödinger equation is the WKB method. This paper proposes the wavelet method for solving the 1D Schrödinger equation, that is: determining the wave function using the harmonic multiresolution analysis.
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.
At the turn of the twentieth century, the establishment of quantum theory propelled rapid advancements, particularly in the understanding of quantum tunnelling—a fundamental phenomenon in quantum mechanics crucial for various physical processes. The quantum phenomenon of particle passes through potential barriers is of great importance. In classical physics, when the energy of a particle is less than the height of a double barrier structure, it is impossible for it to pass through. However, quantum mechanics allows a particle to penetrate the barrier and emerge on the other side. This paper explores the quantum tunnelling effect, focusing on the single potential barrier model in one dimension and subsequently extending to the double potential barrier model. The Schrödinger equation provides the foundational framework for elucidating the motion of microscopic particles, emphasizing wave-particle duality inherent in quantum mechanics. The analysis of the single potential barrier model involves solving the Schrödinger equation in different regions, determining wave functions and coefficients through boundary conditions. The transmission coefficient is derived, representing the probability of a particle passing through a barrier. In the case of a “thick” barrier, an approximate form for transmission coefficient is provided, demonstrating the exponential decrease in transmission probability with increasing barrier thickness.
A new numerical algorithm for tunneling is presented. This algorithm can be used to analyse an onedimensional\npotential barrier with an arbitrary profile. It is able to calculate not only the transmission coefficient\naccurately, but also the phases of the transmitted and reflected waves. Using the Schrodinger equation and plane waves,\nwe introduce a tunneling matrix to relate the transmitted and incident wave functions. Thus the transmission coefficient\nand phase can be calculated from the tunneling matrix.To test the accuracy of this algorithm,we calculate the transmission\ncoefficients and phases for a double -barrier (resonant tunneling) and a triangular barrier (field emission), and\nmake comparisous with corresponding analytical values. It shows that this algorithm is fast, accurate and efficient.
This research paper explores the theoretical foundations of quantum tunneling, explaining how particles can pass through potential barriers despite lacking the classical energy to do so. It develops the mathematical framework using the Schrödinger equation, derives the transmission coefficient for a rectangular potential barrier, introduces the WKB approximation for more realistic barriers, and examines key applications such as alpha decay, scanning tunneling microscopy (STM), and the Josephson effect. The paper also reviews the historical development, experimental verification, and limitations of quantum tunneling, highlighting its significance as one of the most fundamental and experimentally validated predictions of quantum mechanics.
This paper focuses on the quantum tunneling effect, which describes how a particle may penetrate barriers or obstructions using the Schrödinger equation. If a particle makes a barrier tunneling, it means that a particle with low energy goes through the barrier whose energy level is much higher. It is a basis or background for the entire theory of this part. The particle transport probability is explained using both single- and multi-barrier system models. The paper studies the relationship between barrier parameters and particle transmission probability for both single- and multiple-barrier system models.The aim of the study is the provision of the theoretical devices and algorithms via a quantum operational efficiency.Particularly, the paper deals with the multipath barrier tunneling effect, and it connects the transmission coefficients with the system parameters. The goal of this article is to come up with the new strategies and procedures that one can use the quantum phenomenon of the tunneling effect in the contemporary physical th applications, cutting the gap between theoretical physics and technology.This will be done by studying and analyzing theoretical processes and mathematics of this phenomenon along with the technological applications it has.