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2021 | Book

Introduction Read chapter Read first chapter

Resonant Tunneling

Quantum Waveguides of Variable Cross-Section, Asymptotics, Numerics, and Applications

Authors: Prof. Dr. Lev Baskin, Prof. Dr. Pekka Neittaanmäki, Prof. Dr. Boris Plamenevskii, Prof. Dr. Oleg Sarafanov

Publisher: Springer International Publishing

Book Series : Lecture Notes on Numerical Methods in Engineering and Sciences

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book studies electron resonant tunneling in two- and three-dimensional quantum waveguides of variable cross-sections in the time-independent approach. Mathematical models are suggested for the resonant tunneling and develop asymptotic and numerical approaches for investigating the models. Also, schemes are presented for several electronics devices based on the phenomenon of resonant tunneling. Compared to its first edition, this book includes four new chapters, redistributes the content between chapters and modifies the estimates of the remainders in the asymptotics of resonant tunneling characteristics. The book is addressed to mathematicians, physicists, and engineers interested in waveguide theory and its applications in electronics.

Table of Contents

Frontmatter
Chapter 1. Introduction
Abstract
In the introduction, we briefly discuss the phenomenon of resonant tunneling and outline the content of the book.
Lev Baskin, Pekka Neittaanmäki, Boris Plamenevskii, Oleg Sarafanov
Chapter 2. Waveguides. Radiation Principle. Scattering Matrices
Abstract
In this chapter presents a radiation principle for the Helmholtz equation in waveguides, that is the solvability of a boundary value problem with radiation conditions, the asymptotics of solutions at infinity, and the scattering matrix definition.
Lev Baskin, Pekka Neittaanmäki, Boris Plamenevskii, Oleg Sarafanov
Chapter 3. Properties of Scattering Matrices in a Vicinity of Thresholds
Abstract
In this chapter, the scattering matrix S is defined at the thresholds of the continuous spectrum. It is shown that, at any threshold \(\tau \), there exist both one-sided limits of \(S(\mu )\) as \(\mu \rightarrow \tau \pm 0\) and, moreover, the scattering matrix S is continuous from the right at the threshold \(\tau \).
Lev Baskin, Pekka Neittaanmäki, Boris Plamenevskii, Oleg Sarafanov
Chapter 4. Method for Computing Scattering Matrices
Abstract
Section 4.1 is independent of Chap. 3. Section 4.2 is devoted to computing the scattering matrices in a neighborhood of a threshold and uses the results of Chap. 3. In fact, the scheme of the method in Sect. 4.2 is similar to that in Sect. 4.1; however, near a threshold we first calculate the augmented scattering matrix defined in a basis of waves stable at the threshold and then take into account its connection with the usual (not augmented) S-matrix.
Lev Baskin, Pekka Neittaanmäki, Boris Plamenevskii, Oleg Sarafanov
Chapter 5. Asymptotic and Numerical Studies of Resonant Tunneling in 2D-Waveguides for Electrons of Small Energy
Abstract
In this chapter, we consider a 2D-waveguide that coincides with a strip having two narrows of the same width \(\varepsilon \) symmetric about the waveguide axis. The resonant tunneling is discussed for electrons with energy between the first and the second thresholds, so only one incoming wave and one outgoing wave can propagate in every outlet of the waveguide; in other words, we deal with electrons of small energy. There are no external fields. We derive asymptotics for the resonant energy, for the transmission coefficient, and for the width of the resonant peak at its half-height as \(\varepsilon \) tends to zero. Then we compare the asymptotic results with those obtained by numerical calculation of the scattering matrix. Finally, we discuss the impact of a finite waveguide work function on the resonant tunneling and assess the mathematical model adequacy for the tunneling in quantum waveguides with narrows.
Lev Baskin, Pekka Neittaanmäki, Boris Plamenevskii, Oleg Sarafanov
Chapter 6. Resonant Tunneling in 2D-Waveguides with Several Resonators
Abstract
In this chapter, we consider a two-dimensional waveguide that coincides with a strip having \(n+1\) narrows of small diameter \(\varepsilon \). All narrows are of the same shape and are spaced from each other by equal distances. Parts of the waveguide between two neighboring narrows play the role of resonators. The wave function of a free electron satisfies the Dirichlet boundary value problem for the Helmholtz equation in the waveguide. Near a simple eigenvalue of the closed resonator there are n resonant peaks of height close to 1. We let \(\varepsilon \rightarrow 0\) and obtain asymptotic formulas for the resonant energies and for the widths of the resonant peaks at their half-height. The behavior of the transmission coefficient in a neighborhood of a resonance is described.
Lev Baskin, Pekka Neittaanmäki, Boris Plamenevskii, Oleg Sarafanov
Chapter 7. Resonant Tunneling of High-Energy Electrons in 2D-Waveguides
Abstract
The waveguide occupies a strip in \(\mathbb {R}^2\) having two identical narrows of small diameter \(\varepsilon \). An electron wave function satisfies the Helmholtz equation with the homogeneous Dirichlet boundary condition. The energy of electrons (spectral parameter) may be rather high, i.e. any (fixed) number of waves can propagate in the strip far from the narrows. The spectral parameter varies in the vicinity of a degenerate eigenvalue of the resonator and is separated from the thresholds, in other words, the number of scattering channels remains constant. The purpose is to describe an asymptotics of the transmission coefficient at the specified values of the spectral parameter.
Lev Baskin, Pekka Neittaanmäki, Boris Plamenevskii, Oleg Sarafanov
Chapter 8. Numerical Simulation of High Energy Electron Transport
Abstract
The chapter is devoted to the numerical simulation of resonant tunneling for electrons with energy E between the first and the fifth thresholds. Numerical simulations have shown that the resonances are of Fano type. The form of the transmission probability curve is conditioned by interference of the quantum states into which the electron wave is scattered by the narrows.The suggested interference model makes possible to find the resonance parameters with high precision and to link them to the closed resonator eigenvalues.
Lev Baskin, Pekka Neittaanmäki, Boris Plamenevskii, Oleg Sarafanov
Chapter 9. The Impact of a Finite Waveguide Work Function on Resonant Tunneling
Abstract
To describe electron transport in a waveguide, we assume that the electron wave functions vanish at the waveguide boundary. This means that, being in the waveguide, an electron can not cross the waveguide boundary because of the infinite potential barrier. In reality, the assumption has never been fulfilled: generally, electrons can penetrate through the waveguide boundary and go some distance away from the waveguide. Therefore, we have to clarify how this phenomenon affects the resonant tunneling.
Lev Baskin, Pekka Neittaanmäki, Boris Plamenevskii, Oleg Sarafanov
Chapter 10. Asymptotics of Resonant Tunneling in 3D Waveguides for Electrons of Small Energy
Abstract
In this chapter, we consider electron propagation in a waveguide with two cylindric outlets to infinity and two narrows of small diameters \(\varepsilon _1\) and \(\varepsilon _2\). The boundary of the waveguide is assumed to be smooth. The electron motion is described by the Helmholtz equation. The electron energy is supposed to be between the first and the second thresholds. We generalize and implement the asymptotic approach developed in Chap. 5. The basic results are presented by Theorem 10.4.5.
Lev Baskin, Pekka Neittaanmäki, Boris Plamenevskii, Oleg Sarafanov
Chapter 11. Resonant Tunneling in 2D Waveguides in Magnetic Field
Abstract
The presence of a magnetic field can essentially affect the basic characteristics of the resonant tunneling and bring new possibilities for applications in electronics. In particular, in the presence of a magnetic field, the tunneling phenomenon is feasible for producing spin-polarized electron flows consisting of electrons with spins of the same direction. In Chap. 14, we describe magnetic field sensors based on resonant tunneling in magnetic field.
Lev Baskin, Pekka Neittaanmäki, Boris Plamenevskii, Oleg Sarafanov
Chapter 12. Effect of Magnetic Field on Resonant Tunneling in 3D Waveguides of Variable Cross-Section
Abstract
In this chapter, we consider a three-dimensional waveguide that, far from the coordinate origin, coincides with a cylinder G containing the axis x. The cross-section of G is a two-dimensional domain (of an arbitrary form) with smooth boundary. The waveguide has two narrows of small diameter \(\varepsilon \). The waveguide part between the narrows plays the role of a resonator and there can arise conditions for electron resonant tunneling. We take \(\varepsilon \) as small parameter and obtain asymptotic formulas for characteristics of the resonant tunneling as \(\varepsilon \rightarrow 0\).
Lev Baskin, Pekka Neittaanmäki, Boris Plamenevskii, Oleg Sarafanov
Chapter 13. Asymptotic Analysis of Multichannel Resonant Tunneling
Abstract
In the chapter, we generalize, for electrons of high energy, the asymptotic theory exposed in Chap. 10. We present and justify the asymptotics of tunneling characteristics as the narrow diameters tend to zero.
Lev Baskin, Pekka Neittaanmäki, Boris Plamenevskii, Oleg Sarafanov
Chapter 14. Electronics Devices Based on Resonant Tunneling
Abstract
In this chapter, we present examples of electronics devices based on quantum waveguides with narrows: transistors controlled by external electric field and magnetic field sensors controlled by external magnetic field. Moreover, we also consider an electron flow switch for quantum nets. Unlike the transistors and the sensors, the switch has no relation to the phenomenon of resonant tunneling. However, the scattering matrix needed for analyzing the switch operation has been calculated by the method presented in Chap. 4.
Lev Baskin, Pekka Neittaanmäki, Boris Plamenevskii, Oleg Sarafanov
Backmatter
Metadata
Title
Resonant Tunneling
Authors
Prof. Dr. Lev Baskin
Prof. Dr. Pekka Neittaanmäki
Prof. Dr. Boris Plamenevskii
Prof. Dr. Oleg Sarafanov
Copyright Year
2021
Electronic ISBN
978-3-030-66456-5
Print ISBN
978-3-030-66455-8
DOI
https://doi.org/10.1007/978-3-030-66456-5