The book deals with pure Nanostructure Physics; it contains thorough and complete analytical calculations leading to equation(s) obeyed by quasi-bound energy levels of non-isolated Quantum Well of symmetric rectangular double barrier as well as of symmetric double barrier of general shape. Methodology and Physics involved with the calculations is also clearly described. WKB method has been used in the case of symmetric double barrier of general shape. The book also contains necessary background on Quantum Mechanics, Microelectronics, Nanostructure Physics and WKB method to enable readers assimilate the book completely. The book explores and makes well documented, with thorough and complete calculation and discussion, pure Physics of semiconductor nanostructures.
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 (E < V0)
1.9.2 Calculation of transmission coefficient
1.10 Further topics on Quantum Mechanics
Chapter II Background on Microelectronics
2.1 Intrinsic semiconductor
2.2 Semiconductors: elemental and binary
2.3 Alloy semiconductors (ternary and quaternary)
2.4 Bandgap engineering
2.5 Semiconductor heterojunction and heterostructure
2.6 Effective mass
2.7 Further topics on Microelectronics
Chapter III Background on Nanostructure Physics
3.1 Single rectangular tunnel barrier
3.2 Transmission coefficient of a single rectangular tunnel barrier
3.3 Quantum well (QW)
3.4 Double potential barrier
3.5 Transmission coefficient of double potential barrier
3.6 Transmission coefficient of double barrier if two barriers are identical
3.7 Profile of transmission peak
3.8 T versus E curve of symmetric rectangular double barrier
3.9 Further study of double barrier structures
Chapter IV Analytical calculation of transcendental equation obeyed by quasi-bound energy levels of the non-isolated Quantum Well of symmetric rectangular double barrier
4.1 Description of the problem
4.2 Methodology and Physics of calculation of the transcendental equation
4.3 Calculation of transfer matrix of the left hand rectangular tunnel barrier
4.4 Calculation of inverse transfer matrix of the right hand rectangular tunnel barrier
4.5 Calculation of transcendental equation obeyed by quasi-bound energy levels
4.6 Resonant transmission peaks obey the same condition
Chapter V Taking effective mass inequality into account: analytical calculation of transcendental equation obeyed by quasi-bound energy levels of the non-isolated Quantum Well of symmetric rectangular double barrier:
5.1 Introduction
5.2 Calculation of transfer matrix of the left hand rectangular tunnel barrier: taking effective mass inequality into account
5.3 Calculation of inverse transfer matrix of the right hand rectangular tunnel barrier: taking effective mass inequality into account
5.4 Calculation of transcendental equation obeyed by quasi-bound energy levels: taking effective mass inequality into account
5.5 Resonant transmission peaks obey the same condition: taking effective mass inequality into account
Chapter VI Derivation of WKB solution of Schroedinger equation and introduction to WKB connection formulae
6.1 WKB approximation
6.2 WKB solution of (1 dimensional) Schroedinger equation
6.3 Classical turning point
6.4 WKB connection formula: case I
6.5 WKB connection formula: case II
Chapter VII Analytical calculation of transfer matrix and transmission coefficient of single tunnel barrier of general shape using WKB method
7.1 Calculation of transfer matrix of single tunnel barrier of general shape using WKB method
7.2 Calculation of transmission coefficient of single tunnel barrier of general shape using WKB method
Chapter VIII Symmetric double barrier of general shape: analytical calculation of condition or equation obeyed by quasi-bound energy levels of non-isolated Quantum Well using WKB method
8.1 Description of the problem
8.2 Methodology and Physics of calculation of the condition or equation
8.3 Calculation of inverse transfer matrix of right hand tunnel barrier
8.4 Calculation of transcendental equation obeyed by quasi-bound energy levels using WKB method
8.5 Resonant transmission peaks obey the same condition
Research Objective and Focus Areas
The primary research objective of this work is to derive the analytical expressions for the transcendental equations that govern the quasi-bound energy levels within a non-isolated quantum well formed by a symmetric rectangular double barrier structure, considering both standard and effective mass inequality conditions. Additionally, the work investigates the resonant transmission characteristics and applies the WKB approximation to generalize these findings to barriers of arbitrary shapes.
- Quantum mechanical modeling of single and double potential barrier structures.
- Analytical derivation of transfer matrices and transmission coefficients for tunnel barriers.
- Investigation of quasi-bound state energy levels in non-isolated quantum wells.
- Incorporation of effective mass inequality effects into the quantum mechanical boundary conditions.
- Application of the WKB (Wentzel-Kramers-Brillouin) approximation for generalized barrier shapes.
Excerpt from the Book
1.9.1 Calculation of transfer matrix and investigation of its properties (E < V0)
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.
Using the boundary condition u1 = u2 at x = −a, we get Ae^−ika + Be^ika = Ce^−βa + De^βa. Again, the boundary condition du1/dx = du2/dx at x = −a gives ikAe^−ika − ikBe^ika = βCe^−βa − βDe^βa at x = −a.
From equation (1.20) + (1.21), we have 2Ce^−βa = (1 + ik/β)Ae^−ika + (1 − ik/β)Be^ika => C = (1/2)(1 + ik/β)Ae^−ika+βa + (1/2)(1 − ik/β)Be^ika+βa.
From equation (1.20) − (1.21), we have 2De^βa = (1 − ik/β)Ae^−ika + (1 + ik/β)Be^ika => D = (1/2)(1 − ik/β)Ae^−ika−βa + (1/2)(1 + ik/β)Be^ika−βa.
Chapter Summaries
Chapter I: Provides the foundational principles of quantum mechanics, including Schrödinger equations, probability conservation, and boundary conditions for wavefunctions.
Chapter II: Covers microelectronic backgrounds, focusing on semiconductors, energy bands, bandgap engineering, and the concept of effective mass.
Chapter III: Introduces nanostructure physics, analyzing single rectangular barriers, quantum wells, and double barrier structures.
Chapter IV: Derives the transcendental equation for quasi-bound energy levels in a symmetric rectangular double barrier structure.
Chapter V: Extends the analytical calculations of the previous chapter by incorporating effective mass inequality effects.
Chapter VI: Explains the derivation of the WKB solution to the Schrödinger equation and introduces the essential WKB connection formulae.
Chapter VII: Applies the WKB method to calculate transfer matrices and transmission coefficients for single tunnel barriers of general shape.
Chapter VIII: Utilizes the WKB method to determine the conditions and equations for quasi-bound energy levels in symmetric double barriers of general shape.
Keywords
Quantum Mechanics, Nanostructure Physics, Schrödinger Equation, Double Barrier, Quantum Well, Transmission Coefficient, Transfer Matrix, WKB Approximation, Effective Mass, Transcendental Equation, Quasi-bound States, Semiconductor, Bandgap Engineering, Tunneling, Resonant Transmission.
Frequently Asked Questions
What is the core subject of this publication?
The work focuses on the physics of nanostructures, specifically investigating quantum wells and double barrier structures using analytical mathematical models.
What are the primary themes discussed?
Central themes include quantum tunneling, the derivation of transcendental equations for energy levels, the impact of effective mass variations, and the application of the WKB approximation.
What is the research goal?
The goal is to analytically calculate and demonstrate the conditions under which electrons exhibit resonant tunneling and to determine the quasi-bound energy levels in non-isolated quantum wells.
Which scientific methodology is primarily employed?
The work utilizes quantum mechanical operator formalism, transfer matrix methods, and the WKB (Wentzel-Kramers-Brillouin) approximation to solve the time-independent Schrödinger equation.
What topics does the main part of the book cover?
The main sections cover the background of quantum mechanics and microelectronics, followed by a rigorous derivation of energy conditions for symmetric double barriers under various physical assumptions.
How is the work characterized?
The publication is characterized by a high level of mathematical rigor in deriving analytical conditions for non-isolated quantum well systems in nanostructure physics.
How does effective mass inequality change the physical results?
Taking effective mass inequality into account requires a modification of the standard boundary conditions, which leads to different transmission coefficients and energy level conditions compared to the equal effective mass assumption.
Why is the WKB approximation used for general barrier shapes?
The WKB approximation is used because it provides a reliable analytical tool for solving the Schrödinger equation when the potential barrier is a slowly varying function of position, making it applicable to arbitrary barrier profiles.
- Citation du texte
- Sujaul Chowdhury (Auteur), Abdus Samad (Auteur), Dipak Dash (Auteur), 2011, Nanostructure Physics of non-isolated Quantum Well of symmetric double barrier, Munich, GRIN Verlag, https://www.grin.com/document/211440
