We have investigated Nanostructure Physics of a Quantum Well (QW) adjacent to a tunnel barrier. On the other side of the QW, we have a barrier layer of semi-infinite thickness. The book contains thorough and complete analytical calculations leading to two transcendental equations obeyed by quasi-bound energy levels of the QW. The book also contains numerical investigation of parametric variations of the quasi-bound energy levels. The variations are quite unexpected. For some values of tunnel barrier width and height, each quasi-bound energy level yields two allowed energy levels. The two equations reduce to those for isolated QW in proper limiting values of parameters. The book contains necessary background on Quantum Mechanics, Microelectronics and Nanostructure Physics to enable readers assimilate the book completely.
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 Allowed values of an observable
1.4 Eigenvalue equation, eigenfunction and eigenvalue
1.5 Time-independent Schrödinger equation and stationary state.
1.6 Continuous and discontinuous function
1.7 Finite and infinite discontinuity
1.8 Admissibility conditions on wave function
1.9 Calculation of confined energy levels of isolated quantum well (QW)
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
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
Chapter IV Analytical calculation of transcendental equation obeyed by quasi-bound energy levels of the Quantum Well
4.1 Introduction to the Quantum Well
4.2 Analytical calculations of transcendental equations obeyed by quasi-bound energy levels of the Quantum Well
4.3 Recovering results for isolated QW
Chapter V Analytical calculation of transcendental equation obeyed by quasi-bound energy levels of the Quantum Well using another approach
5.1 Analytical calculations of transcendental equation obeyed by quasi-bound energy levels of the Quantum Well: using another approach
Chapter VI Numerical investigation of parametric dependence of quasi-bound energy levels of the Quantum Well
6.1 The transcendental equations obeyed by quasi-bound energy levels of the Quantum Well
6.2 The numerical investigation
6.3 Conclusions about the parametric variations
Research Objective and Topics
This work focuses on the nanostructure physics of quantum wells adjacent to tunnel barriers, specifically investigating the analytical calculation and numerical determination of transcendental equations that define quasi-bound energy levels within these systems. The research aims to explore how parametric variations—such as well width, barrier width, and material composition—affect the quantized energy states and transmission characteristics of carriers in such heterostructures.
- Quantum mechanics and operator formalism in microelectronics.
- Theoretical modeling of tunnel barriers and carrier transport mechanisms.
- Analytical derivation of transcendental equations for quasi-bound states in quantum wells.
- Numerical investigation of parametric dependencies in nanostructure heterojunctions.
- Application of quantum well structures in resonant tunneling devices.
Excerpt from the Book
1.2 Schrödinger equation of a particle subject to a conservative mechanical force
1) A comparison of < x > = ∫ Ψ * x Ψ dx and < p > = ∫ Ψ * (ħ / i ∂x) Ψ dx shows that expectation value of momentum < p > of a particle having wavefunction Ψ associated with it can be computed in the same way as that of position < x > if the operator (ħ / i ∂x) is substituted in place of x. This statement introduces operator formalism in quantum mechanics. Thus the operator of x is x, operator of p is (ħ / i ∂x).
2) The Schrödinger equation of a free particle is - (ħ² / 2m) (∂²Ψ / ∂x²) = iħ (∂Ψ / ∂t). This equation can be obtained from the classical equation (p² / 2m) = E by the operator correspondence of p and E as (ħ / i ∂x) and iħ (∂ / ∂t) respectively, and letting the operators operate on the wavefunction Ψ. Thus the operator of E is iħ (∂ / ∂t).
3) If a conservative force acts on a particle, the total energy E = (p² / 2m) + V, where V is potential energy and hence V is a function of position only. Hence < V(x) > = ∫ Ψ * V(x) Ψ dx = ∫ Ψ * V(x) Ψ dx. Thus the operator of V(x) is V(x).
4) Using the operator correspondences of E, p and V, (p² / 2m) + V = E gives [ (1 / 2m) (ħ / i ∂x)² + V ] = iħ (∂ / ∂t) or, - (ħ² / 2m) (∂² / ∂x²) + V = iħ (∂ / ∂t) or, - (ħ² / 2m) (∂²Ψ / ∂x²) + VΨ = iħ (∂Ψ / ∂t) or, [ - (ħ² / 2m) (∂² / ∂x²) + V ] Ψ = iħ (∂Ψ / ∂t). This is the Schrödinger equation of a particle moving in a potential V. In three dimensions, [ - (ħ² / 2m) ∇² + V ] Ψ = iħ (∂Ψ / ∂t) is the Schrödinger equation of a particle moving in a potential V. The operator [ - (ħ² / 2m) ∇² + V ] is the operator of total energy of a conservative system and is called Hamiltonian operator.
Summary of Chapters
Chapter I Background on Quantum Mechanics: Provides the foundational principles of quantum mechanics, including the Schrödinger equation and the definition of wave functions under conservative forces.
Chapter II Background on Microelectronics: Discusses the band models of insulators and semiconductors, covering the properties of elemental and compound semiconductors and bandgap engineering.
Chapter III Background on Nanostructure Physics: Details the structural and electronic properties of tunnel barriers and quantum wells, including resonant tunneling phenomena.
Chapter IV Analytical calculation of transcendental equations obeyed by quasi-bound energy levels of the Quantum Well: Derives the analytical expressions for energy levels using boundary conditions in a piece-wise constant potential model.
Chapter V Analytical calculation of transcendental equation obeyed by quasi-bound energy levels of the Quantum Well using another approach: Presents an alternative analytical derivation using transfer matrix methods for the potential structure.
Chapter VI Numerical investigation of parametric dependence of quasi-bound energy levels of the Quantum Well: Provides numerical results and parametric studies on how physical dimensions and barrier heights influence the energy levels.
Keywords
Quantum Well, Tunnel Barrier, Schrödinger Equation, Quasi-bound States, Transcendental Equations, Heterostructure, Bandgap Engineering, Resonant Tunneling, Semiconductor Physics, Hamiltonian Operator, Energy Quantization, Effective Mass, Microelectronics, 2DEG, Parametric Dependence
Frequently Asked Questions
What is the core subject of this publication?
The work focuses on the nanostructure physics of quantum wells located adjacent to tunnel barriers, utilizing both analytical and numerical methods to solve for quasi-bound energy levels.
What are the primary thematic areas covered?
The book spans from foundational quantum mechanics and semiconductor band theory to advanced nanostructure modeling, including resonant tunneling diodes and bandgap engineering.
What is the central research question?
The research primarily aims to derive and solve the transcendental equations that govern the quasi-bound energy states in a quantum well, and to analyze how these states depend on structural parameters like width and potential height.
Which scientific methodologies are applied?
The authors employ the Schrödinger equation, operator formalism, and transfer matrix methods to analytically describe the system, supplemented by numerical investigations using Mathematica software.
What does the main body address?
The main sections move from theoretical backgrounds in quantum mechanics and microelectronics to the specific derivation of energy level equations for non-isolated quantum wells and their numerical parametric analysis.
Which keywords best characterize the work?
Key terms include Quantum Well, Tunnel Barrier, Schrödinger Equation, Resonant Tunneling, and Bandgap Engineering.
How does the tunnel barrier width affect resonant transmission?
The research shows that increasing the tunnel barrier width decreases the full-width-at-half-maximum (FWHM) of resonance peaks, as it increases the carrier lifetime in the quantum well.
Does the numerical investigation validate the analytical results?
Yes, the numerical results using the programs provided in the chapters confirm the validity of the transcendental equations and demonstrate their convergence to isolated quantum well results in limiting cases.
- Quote paper
- Dr Sujaul Chowdhury (Author), Zuned Ahmed (Author), Emrul Hasan (Author), 2014, Nanostructure Physics of a Quantum Well adjacent to a tunnel barrier, Munich, GRIN Verlag, https://www.grin.com/document/273772
