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

Preface
Contents
1 Introduction
2 Waveguides. Radiation Principle. Scattering Matrices
	2.1 Boundary Value Problem in a Cylinder
		2.1.1 Statement of the Problem. Operator Pencil
		2.1.2 The Solvability of the Problem in a Cylinder
		2.1.3 Asymptotics of Solutions
	2.2 Problem in a Domain G with Cylindrical Ends
		2.2.1 Statement and Fredholm Property of the Problem
		2.2.2 Asymptotics of Solutions
		2.2.3 Properties of the Index Ind mathcalAβ(µ) and of the Spaces ker mathcalAβ(µ) and coker mathcalAβ(µ)
		2.2.4 Calculation of the Coefficients in the Asymptotics
	2.3 Waves and Scattering Matrices
		2.3.1 Waves
		2.3.2 Continuous Spectrum Eigenfunctions. The Scattering Matrix
		2.3.3 The Intrinsic Radiation Principle
3 Properties of Scattering Matrices  in a Vicinity of Thresholds
	3.1 Augmented Space of Waves
		3.1.1 Waves in a Cylinder
		3.1.2 Waves in Domain G
	3.2 Continuous Spectrum Eigenfunctions. Scattering Matrices
		3.2.1 Intrinsic and Expanded Radiation Principles
		3.2.2 Analyticity of Scattering Matrices with Respect to Spectral Parameter
	3.3 Other Properties of the Scattering Matrices
		3.3.1 The Connection Between mathcalS(µ) and S(µ) for τ'<µ<τ
		3.3.2 The Connection Between mathcalS(µ) and S(µ) for τ<µ<τ''
		3.3.3 The Limits of S(µ) as µtoτpm0
4 Method for Computing Scattering Matrices
	4.1 A Method for Computing Scattering Matrices Outside Thresholds
		4.1.1 Statement of the Method
		4.1.2 The Problem in GR
		4.1.3 Justification of the Method for Computing the Scattering Matrix
	4.2 A Method for Computing Scattering Matrices in Vicinity of Thresholds
5 Asymptotic and Numerical Studies  of Resonant Tunneling in 2D-Waveguides for Electrons of Small Energy
	5.1 Statement of the Problem
	5.2 Limit Problems
		5.2.1 First Kind Limit Problems
		5.2.2 Second Kind Limit Problems
	5.3 Special Solutions to the First Kind Homogeneous Problems
	5.4 Asymptotic Formulas
		5.4.1 Asymptotics of the Wave Function
		5.4.2 Formulas for widetildeS11, widetildeS12, and C1j
		5.4.3 Formulas for Resonant Tunneling Characteristics
	5.5 Justification of the Asymptotics
	5.6 Comparison of Asymptotic and Numerical Results
		5.6.1 Problems and Methods for Numerical Analysis
		5.6.2 Comparison of Asymptotic and Numerical Results
6 Resonant Tunneling in 2D-Waveguides with Several Resonators
	6.1 Statement of the Problem and Preliminaries
		6.1.1 Waveguide and the Radiation Conditions
		6.1.2 First Kind Limit Problems
		6.1.3 Second Kind Limit Problems
	6.2 Asymptotics of the Wave Function and the Scattering Matrix
		6.2.1 Asymptotics of the Wave Function
		6.2.2 Formulas for C0,  …,  C2n+1
		6.2.3 Asymptotics of the Scattering Matrix
	6.3 Asymptotics of the Resonant Tunneling
		6.3.1 Amplitude of the Transmitted Wave
		6.3.2 Poles of the Scattering Matrix
		6.3.3 Transmission Coefficient
	6.4 Comparison of Asymptotic and Numerical Results
7 Resonant Tunneling of High-Energy Electrons in 2D-Waveguides
	7.1 Introduction
	7.2 Statement of the Problem
	7.3 Limit Problems
		7.3.1 Limit Problems of the First Kind
		7.3.2 Limit Problems of the Second Kind
	7.4 Asymptotic Formulas
		7.4.1 Asymptotics of Wave Functions
		7.4.2 Formulas for Cml
		7.4.3 Asymptotics of Scattering Matrix
		7.4.4 Asymptotics for Poles of Scattering Matrix
		7.4.5 Asymptotics of Resonant Tunneling
	7.5 Comparison of Asymptotic and Numerical Results …
		7.5.1 Asymptotic Formulas
		7.5.2 Calculation of the Scattering Matrix
		7.5.3 Comparison of Asymptotics and Numerical Results
8 Numerical Simulation of High Energy Electron Transport
	8.1 Numerical Simulation of Multichannel Resonant Tunneling
		8.1.1 Closed Resonator
		8.1.2 The Method for Computing Scattering Matrix
		8.1.3 Discussion of Numerical Results
	8.2 Fano Resonances
9 The Impact of a Finite Waveguide Work Function on Resonant Tunneling
	9.1 Preliminaries
	9.2 A Qualitative Analysis of a Finite Work Function Impact on Electron Transport
	9.3 Numerical Simulation of Resonant Tunneling with Regard to the Waveguide Work Function
	9.4 Impact of Finite Work Function on High Energy Electron Transport
10 Asymptotics of Resonant Tunneling in 3D Waveguides for Electrons of Small Energy
	10.1 Statement of the Problem and Outline of the Results
	10.2 Limit Problems
		10.2.1 First Kind Limit Problems
		10.2.2 Second Kind Limit Problems
	10.3 Tunneling in a Waveguide with One Narrow
		10.3.1 Special Solutions to the First Kind Homogeneous Problems
		10.3.2 Passing Through the Narrow
		10.3.3 Formal Asymptotics
		10.3.4 The Estimate of the Remainder
	10.4 Tunneling in a Waveguide with Two Narrows
		10.4.1 Special Solutions to the Problem in the Resonator
		10.4.2 Formal Asymptotics
		10.4.3 The Estimate of the Remainder
11 Resonant Tunneling in 2D Waveguides in Magnetic Field
	11.1 Statement of the Problem
	11.2 The Limit Problems
		11.2.1 First Kind Limit Problems
		11.2.2 Second Kind Limit Problems
	11.3 Special Solutions to Homogeneous First Kind Limit Problems
	11.4 Asymptotic Formulas
		11.4.1 Asymptotics of the Wave Function
		11.4.2 Formulas for widetildeS11, widetildeS12, and C1j
		11.4.3 Formulas for Resonant Tunneling Characteristics
	11.5 Justification of the Asymptotics
	11.6 Comparison of Asymptotic and Numerical Approaches
		11.6.1 Problems and Methods for Numerical Analysis
		11.6.2 Comparison of Asymptotic and Numerical Results
		11.6.3 Dependence of Resonant Tunneling on the Magnetic Field Location in the Resonator
12 Effect of Magnetic Field on Resonant Tunneling in 3D Waveguides of Variable Cross-Section
	12.1 Introduction
	12.2 Statement of the Problem
	12.3 Limit Problems
		12.3.1 First Kind Limit Problems
		12.3.2 Second Kind Limit Problems
	12.4 Special Solutions of Limit Problems
	12.5 Asymptotic Formulas
		12.5.1 The Asymptotics of a Wave Function
		12.5.2 Formulas for widetildeS11, widetildeS12, and C11,…,C14
		12.5.3 Asymptotics for Resonant Tunneling Characteristics
	12.6 Justification of the Asymptotics
13 Asymptotic Analysis of Multichannel Resonant Tunneling
	13.1 Statement of the Problem and Limit Problems
	13.2 Tunneling in a Waveguide with One Narrow
		13.2.1 Special Solutions to the First Kind Homogeneous Problems
		13.2.2 Asymptotic Formulas
	13.3 Tunneling in a Waveguide with Two Narrows
		13.3.1 Formal Asymptotics
		13.3.2 The Estimate of the Remainder
14 Electronics Devices Based on Resonant Tunneling
	14.1 Magnetic Field Sensors Based on Quantum Waveguides
	14.2 Transistors Based on Quantum Waveguides
	14.3 Electron Flow Switch for Quantum Nets
Appendix  Bibliographical Sketch
Appendix  References