Delve into the mesmerizing world of quantum transport with this comprehensive exploration of electron transmission through an isolated quantum well, a critical area of nanostructure physics and microelectronics. This book unveils the secrets behind the oscillatory behavior of electrons traversing a quantum well in the non-tunneling regime, offering a detailed analysis of the factors governing their passage. Beginning with a solid foundation in quantum mechanics, including the Schrödinger equation and wave function properties, and progressing through essential microelectronics concepts like bandgap engineering and semiconductor heterostructures, the narrative seamlessly bridges theoretical underpinnings with practical applications. Discover how the interplay of quantum phenomena shapes the transmission characteristics of electrons in these nanoscale systems. Explore the intricacies of nanostructure physics, examining tunnel barriers and quantum well structures, laying the groundwork for understanding electron behavior at the quantum level. The book meticulously details the analytical calculation of the transfer matrix and transmission coefficient, providing a rigorous mathematical framework for understanding electron transport. Uncover the qualitative aspects of transmission through numerical investigations, revealing how quantum well depth and width influence electron behavior. The book further explores the parametric dependence of transmission maxima and minima through meticulous quantitative analysis. Essential reading for researchers, engineers, and students in quantum mechanics, microelectronics, and nanotechnology, this book offers a unique blend of analytical rigor and numerical insight, making it an indispensable resource for anyone seeking to master the complexities of electron transport in quantum structures. Unlock the potential of quantum well technology with this insightful guide, and explore advanced topics in nanostructure physics. Gain a comprehensive understanding of oscillatory transmission, its parametric dependencies, and the crucial role of quantum mechanical principles in shaping the future of microelectronic devices. This book is a vital addition to the libraries of those working in semiconductor physics, solid-state physics, and materials science, providing a detailed exploration of quantum transport phenomena.
Inhaltsverzeichnis (Table of Contents)
- Chapter I: Background on Quantum Mechanics
- 1.1 Wave equation of a free particle: Schrödinger equation
- 1.2 Schrödinger equation of a particle subject to a conservative mechanical force
- 1.3 Conservation of probability and probability current density
- 1.4 Time-independent Schrödinger equation and stationary state
- 1.5 Continuous and discontinuous function
- 1.6 Finite and infinite discontinuity
- 1.7 Admissibility conditions on wavefunction
- 1.8 Free particle: eigenfunctions and probability current density
- 1.9 Single rectangular tunnel barrier
- 1.9.1 Calculation of transfer matrix and investigation of its properties
- 1.9.2 Calculation of transmission coefficient
- 1.10 Further topics on Quantum Mechanics
- Chapter II: Background on Microelectronics
- 2.1 Insulator and its band model
- 2.2 Intrinsic semiconductor and its band model
- 2.3 Elemental and compound semiconductors
- 2.4 Alloy semiconductors: ternary and quaternary semiconductors
- 2.5 Bandgap engineering
- 2.6 Substrate and epitaxial layer
- 2.7 Semiconductor heterostructure and heterojunction
- 2.8 Further topics on Microelectronics
- Chapter III: Background on Nanostructure Physics
- 3.1.1 Tunnel barrier: structure and band model
- 3.1.2 Transport of electron or hole through tunnel barrier
- 3.2 Quantum Well (QW)
- 3.3 Symmetric double barrier
- 3.4 Further study of Nanostructure Physics
- Chapter IV: Transmission of electron through non-tunneling regime of isolated Quantum Well: Analytical calculation
- 4.1 Description of the problem
- 4.2 Calculation of transfer matrix of isolated QW for non-tunneling regime
- 4.3 Calculation of transmission coefficient of isolated Quantum Well for non-tunneling regime
- Chapter V: Numerical investigation of parametric dependence of transmission through non-tunneling regime of isolated Quantum Well: Qualitative aspects
- 5.1 The analytical expressions for transmission coefficient of isolated Quantum Well for non-tunneling regime
- 5.2 The typical T versus E curve
- 5.3 Dependence of oscillatory transmission coefficient on QW depth
- 5.4 Dependence of oscillatory transmission coefficient on QW width
- Chapter VI: Numerical investigation of parametric dependence of transmission peaks for non-tunneling regime of isolated Quantum Well: Quantitative aspects
- Chapter VII: Numerical investigation of parametric dependence of transmission minima for non-tunneling regime of isolated Quantum Well: Quantitative aspects
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
This work aims to investigate the oscillatory transmission of electrons through the non-tunneling regime of an isolated quantum well. The study focuses on the parametric dependence of both transmission maxima and minima. The research combines analytical calculations with numerical investigations to provide a comprehensive understanding of this phenomenon.
- Quantum mechanical principles governing electron transmission.
- The behavior of electrons in quantum wells and the impact of various parameters.
- Analytical and numerical methods for calculating transmission coefficients.
- Parametric dependence of transmission maxima and minima.
- Qualitative and quantitative analysis of oscillatory transmission.
Zusammenfassung der Kapitel (Chapter Summaries)
Chapter I: Background on Quantum Mechanics: This chapter lays the foundational groundwork in quantum mechanics necessary for understanding the subsequent analysis of electron transmission through quantum wells. It covers essential concepts like the Schrödinger equation for free and bound particles, probability density and current, the time-independent Schrödinger equation, and the properties of wave functions. The chapter delves into the characteristics of continuous and discontinuous functions, the conditions for admissibility of wavefunctions, and provides a detailed analysis of electron behavior in a single rectangular tunnel barrier, including the calculation of the transfer matrix and transmission coefficient. This foundational knowledge forms the basis for understanding the more complex systems explored later.
Chapter II: Background on Microelectronics: This chapter provides a detailed overview of the microelectronics concepts crucial for comprehending the behavior of electrons in semiconductor structures. The chapter starts by defining insulators and their band models and explaining intrinsic semiconductors, their band models, as well as elemental and compound semiconductors. It then covers alloy semiconductors, explaining ternary and quaternary semiconductors and the concept of bandgap engineering, essential for tailoring the properties of semiconductor materials. The chapter also discusses the concepts of substrates, epitaxial layers, semiconductor heterostructures, and heterojunctions – all critical components of the quantum well structures studied later in the text.
Chapter III: Background on Nanostructure Physics: This chapter establishes the fundamental principles of nanostructure physics, focusing on the quantum well systems central to the study. It begins by describing the structure and band model of a tunnel barrier and detailing the transport of electrons or holes through this barrier. The chapter then provides a comprehensive overview of quantum wells (QW), their properties, and the associated energy levels. The concepts of symmetric double barriers are also introduced. This chapter successfully bridges the gap between the general quantum mechanics and microelectronics principles established in the previous chapters and the specifics of nanostructured quantum wells that form the core of the study.
Chapter IV: Transmission of electron through non-tunneling regime of isolated Quantum Well: Analytical calculation: This chapter tackles the core problem analytically, detailing the process of calculating the electron transmission through a quantum well in the non-tunneling regime. The chapter explicitly describes the problem setting before focusing on the derivation and calculation of the transfer matrix for the isolated quantum well within the defined regime. It culminates in the calculation of the transmission coefficient, laying the theoretical foundation for the numerical investigation in later chapters. This chapter provides the analytical framework that will be compared and validated against the numerical results.
Chapter V: Numerical investigation of parametric dependence of transmission through non-tunneling regime of isolated Quantum Well: Qualitative aspects: This chapter shifts the focus to a numerical investigation of the transmission through the quantum well, focusing on qualitative aspects. The chapter begins by restating the analytical expressions for the transmission coefficient derived in Chapter IV. It analyzes the typical transmission (T) versus energy (E) curve, highlighting its characteristic oscillatory behavior. Importantly, this chapter investigates the qualitative dependence of this oscillatory transmission on key parameters: the quantum well depth and width. The observations here set the stage for the more rigorous quantitative analysis in the subsequent chapters.
Schlüsselwörter (Keywords)
Quantum mechanics, Schrödinger equation, microelectronics, nanostructure physics, quantum well, transmission coefficient, transfer matrix, non-tunneling regime, oscillatory transmission, parametric dependence, analytical calculation, numerical investigation.
Häufig gestellte Fragen
What is the main topic covered in the "Comprehensive Language Preview"?
The "Comprehensive Language Preview" focuses on the transmission of electrons through the non-tunneling regime of an isolated quantum well. It investigates the oscillatory behavior of the transmission coefficient and its dependence on various parameters.
What foundational knowledge is covered in Chapter I?
Chapter I, "Background on Quantum Mechanics," introduces key quantum mechanical concepts. These include the Schrödinger equation for free and bound particles, probability density and current, the time-independent Schrödinger equation, properties of wave functions, and the analysis of electron behavior in a single rectangular tunnel barrier.
What topics are covered in Chapter II?
Chapter II, "Background on Microelectronics," covers the basics of microelectronics. This includes insulators and their band models, intrinsic semiconductors, elemental and compound semiconductors, alloy semiconductors (ternary and quaternary), bandgap engineering, substrates, epitaxial layers, semiconductor heterostructures, and heterojunctions.
What does Chapter III discuss?
Chapter III, "Background on Nanostructure Physics," establishes the fundamentals of nanostructure physics, particularly focusing on quantum well systems. It covers the structure and band model of a tunnel barrier, electron/hole transport through the barrier, properties of quantum wells (QWs), energy levels, and symmetric double barriers.
What is the focus of Chapter IV?
Chapter IV, "Transmission of electron through non-tunneling regime of isolated Quantum Well: Analytical calculation," provides the analytical framework for calculating electron transmission through a quantum well in the non-tunneling regime. It includes the derivation of the transfer matrix and the calculation of the transmission coefficient.
What does Chapter V investigate?
Chapter V, "Numerical investigation of parametric dependence of transmission through non-tunneling regime of isolated Quantum Well: Qualitative aspects," investigates the qualitative aspects of electron transmission through the quantum well using numerical methods. It examines the dependence of the oscillatory transmission coefficient on the quantum well depth and width.
What are the main objectives of the work?
The primary objectives are to investigate the oscillatory transmission of electrons through the non-tunneling regime of an isolated quantum well, focusing on the parametric dependence of transmission maxima and minima using both analytical and numerical methods.
What are the key themes explored in the work?
The key themes include quantum mechanical principles of electron transmission, electron behavior in quantum wells, analytical and numerical calculation methods for transmission coefficients, parametric dependence of transmission maxima and minima, and the qualitative/quantitative analysis of oscillatory transmission.
What keywords are associated with this study?
The relevant keywords are: Quantum mechanics, Schrödinger equation, microelectronics, nanostructure physics, quantum well, transmission coefficient, transfer matrix, non-tunneling regime, oscillatory transmission, parametric dependence, analytical calculation, numerical investigation.
- Quote paper
- Sujaul Chowdhury (Author), Chaion Chandra Sarker (Author), 2006, Oscillatory transmission through non-tunneling regime of isolated Quantum Well, Munich, GRIN Verlag, https://www.grin.com/document/210828
