We have numerically investigated parametric variations of transmission peaks of symmetric rectangular double barrier in non-tunneling regime. We have compared the variations with those for tunneling regime. One of the three variations in non-tunneling regime is completely different from that for tunneling regime warranting rapid dissemination. The book contains 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 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
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 Numerical investigation of parametric variations of transmission peaks of symmetric rectangular double barrier for non-tunneling regime
4.1 Transmission through symmetric rectangular double barrier
4.2 Description of the problem
4.3 Tables and Figures showing parametric variations of energy of transmission peaks of symmetric rectangular double barrier for non-tunneling regime
4.4 Conclusions about the parametric variations
Research Objective and Scope
This work investigates the oscillatory transmission behavior and the parametric variations of energy transmission peaks for particles passing through a symmetric rectangular double barrier, specifically focusing on the non-tunneling regime.
- Quantum mechanical modeling of single and double tunnel barriers.
- Analysis of carrier transport through potential barriers in nanostructures.
- Numerical investigation of energy transmission peaks in non-tunneling regimes.
- Parametric study of barrier dimensions and material composition (Al content).
- Comparison of tunneling vs. non-tunneling transmission characteristics.
Excerpt from the Book
1.9.1 Calculation of transfer matrix and investigation of its properties (E < V0)
There are two finite discontinuities of V(x), one at x = −a and another at x = +a. There are three regions as shown. According to the choice of origin, V(x) is zero in two of the three regions and is constant (V0) in region II. We now proceed to obtain transfer matrix of the barrier and find its properties. We shall use waves and match them at the potential discontinuities using the boundary conditions described in section 1.7.
Solutions of time-independent Schrödinger equation [− (ħ^2 / 2m) d^2/dx^2 + V(x)] u(x) = E u(x) or, d^2u/dx^2 + (2m / ħ^2)(E − V(x))u = 0 in the three regions are u1 , u2 and u3 given by u1(x) = Ae^ikx + Be^−ikx where k^2 = 2mE / ħ^2, u2(x) = Ce^βx + De^−βx where β^2 = 2m(V0 − E) / ħ^2, u3(x) = Ge^ikx + He^−ikx. The expressions for u2 and β^2 imply that we are considering free electrons of kinetic energy less than V0 impinging on the barrier from the left.
Summary of Chapters
Chapter I Background on Quantum Mechanics: This chapter establishes the fundamental quantum mechanical principles, including the Schrödinger equation, operator formalism, and probability current density, necessary for analyzing particle transport.
Chapter II Background on Microelectronics: This chapter provides an overview of semiconductor physics, covering band models for insulators and semiconductors, bandgap engineering, and the properties of heterostructures.
Chapter III Background on Nanostructure Physics: This chapter applies quantum mechanical concepts to nanostructures, specifically analyzing tunnel barriers, quantum wells, and the physics of symmetric double barriers.
Chapter IV Numerical investigation of parametric variations of transmission peaks of symmetric rectangular double barrier for non-tunneling regime: This final chapter presents a comprehensive numerical study on how various parameters, such as barrier width and height, influence the transmission peaks of double barriers in the non-tunneling regime.
Keywords
Quantum Mechanics, Schrödinger Equation, Tunnel Barrier, Symmetric Double Barrier, Transmission Coefficient, Non-tunneling Regime, Nanostructure Physics, Bandgap Engineering, Resonant Tunneling, Semiconductor Heterostructure, Quantum Well, Parametric Variations, Electron Transport, Energy Peaks, Eigenfunctions.
Frequently Asked Questions
What is the primary focus of this work?
The work focuses on the numerical investigation of oscillatory transmission behavior and parametric variations of energy transmission peaks for particles incident on a symmetric rectangular double barrier in the non-tunneling regime.
What are the central themes discussed?
The central themes include basic quantum mechanics, semiconductor band theory, the physics of nanostructures like tunnel barriers and quantum wells, and the analysis of resonant tunneling phenomena.
What is the core research objective?
The objective is to calculate and analyze the transmission coefficients and energy transmission peaks of a symmetric double barrier system when incident particle energy E exceeds the barrier height V0.
Which scientific methodology is utilized?
The study utilizes quantum mechanical wave function matching at potential discontinuities to derive transfer matrices, followed by numerical simulations performed using Mathematica software.
What topics are covered in the main section?
The main sections detail the derivation of the Schrödinger equation for various potentials, the calculation of transfer matrices for tunnel barriers, and a systematic numerical evaluation of energy peaks based on variations in barrier width and Al composition.
What are the key terms characterizing this research?
Key terms include Quantum tunneling, symmetric double barrier, transmission coefficient, non-tunneling regime, and parametric variation analysis.
How does the transmission behavior change in the non-tunneling regime compared to tunneling?
In the non-tunneling regime (E > V0), transmission is oscillatory and exhibits peaks where the wave resonance condition is met, significantly differing from the exponentially decaying characteristics seen in the tunneling regime (E < V0).
What role does the Al content x play in the AlxGa1-xAs layers?
The Al content x determines the height of the tunnel barrier and the depth of the quantum well, which directly impacts the energy values at which transmission resonance peaks occur.
- Arbeit zitieren
- Dr Sujaul Chowdhury (Autor:in), Abidur Rahman (Autor:in), 2013, Oscillatory transmission through non-tunneling regime of symmetric rectangular double barrier, München, GRIN Verlag, https://www.hausarbeiten.de/document/230804
