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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
Gain an introduction to the concepts behind semiconductor materials and devices in this advanced textbook Semiconductors are the foundation of the electronics industry, and are therefore embedded in virtually all modern technology. No engineer or materials scientist can be without an understanding of this essential field. Since semiconductors are also the foundation of solar cells, they play an increasingly critical role in the transition to sustainable technology and promise, as a result, to become even more central in global technological progress. Fundamentals of Semiconductor Materials and Devices is a textbook that presents the advanced principles underlying semiconductors in an accessible and comprehensive way. Combining material from both engineering and physics, it remains grounded throughout in practical applications of semiconductors. Its approach makes it ideal for readers looking to gain a thorough understanding of this ubiquitous technology. Fundamentals of Semiconductor Materials and Devices readers will also find: * Questions and exercises to stimulate learning and increase comprehension * Introductory chapters detailing the fundamentals of quantum and solid state physics, as well as the foundational principles of semiconductor tech * Detailed analysis of topics including flash memory, the quantum dot, two-dimensional semiconductor materials, and more Fundamentals of Semiconductor Materials and Devices is a valuable guide for students and researchers in any area of engineering, physics, or materials science.
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Veröffentlichungsjahr: 2023
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Fundamentals of Semiconductor Materials and Devices
Adrian Kitai
Departments of Engineering Physics and Materials Science and Engineering, McMaster University, Hamilton, ON, Canada
This edition first published 2023
© 2023 John Wiley & Sons Ltd.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.
The right of Adrian Kitai to be identified as the author of this work has been asserted in accordance with law.
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Hardback ISBN: 9781119891406; ePub ISBN: 9781119891420; ePDF ISBN: 9781119891413
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Set in 9.5/12.5pt STIXTwoText by Integra Software Services Pvt. Ltd, Pondicherry, India
Contents
Cover
Title Page
Copyright Page
Acknowledgments
Preface
About the Companion Website
CHAPTER 1 Introduction to Quantum Mechanics
1.1 Introduction
1.2 The Classical Electron
1.3 Two-Slit Electron Experiment
1.4 The Photoelectric Effect
1.5 Wave-Packets and Uncertainty
1.6 The Wavefunction
1.7 The Schrödinger Equation
1.8 The Electron in a One-Dimensional Well
1.9 The Hydrogen Atom
1.10 Electron Transmission and Reflection at Potential Energy Step
1.11 Spin
1.12 The Pauli Exclusion Principle
1.13 Operators and the Postulates of Quantum Mechanics
1.14 Expectation Values and Hermitian Operators
1.15 Summary
Problems
Note
Suggestions for Further Reading
CHAPTER 2 Semiconductor Physics
2.1 Introduction
2.2 The Band Theory of Solids
2.3 Bloch Functions
2.4 The Kronig–Penney Model
2.5 The Bragg Model
2.6 Effective Mass in Three Dimensions
2.7 Number of States in a Band
2.8 Band Filling
2.9 Fermi Energy and Holes
2.10 Carrier Concentration
2.11 Semiconductor Materials
2.12 Semiconductor Band Diagrams
2.13 Direct Gap and Indirect Gap Semiconductors
2.14 Extrinsic Semiconductors
2.15 Carrier Transport in Semiconductors
2.16 Equilibrium and Nonequilibrium Dynamics
2.17 Carrier Diffusion and the Einstein Relation
2.18 Quasi-Fermi Energies
2.19 The Diffusion Equation
2.20 Traps and Carrier Lifetimes
2.21 Alloy Semiconductors
2.23 Summary
Problems
Suggestions for Further Reading
CHAPTER 3 The p-n Junction Diode
3.1 Introduction
3.2 Diode Current
3.3 Contact Potential
3.4 The Depletion Approximation
3.5 The Diode Equation
3.6 Reverse Breakdown and the Zener Diode
3.7 Tunnel Diodes
3.8 Generation/Recombination Currents
3.9 Metal-Semiconductor Junctions
3.10 Heterojunctions
3.11 Alternating Current (AC) and Transient Behavior
3.12 Summary
Problems
Note
Suggestions for Further Reading
CHAPTER 4 Photon Emission and Absorption
4.1 Introduction to Luminescence and Absorption
4.2 Physics of Light Emission
4.3 Simple Harmonic Radiator
4.4 Quantum Description
4.5 The Exciton
4.6 Two-Electron Atoms and the Exchange Interaction
4.7 Molecular Excitons
4.8 Band-to-Band Transitions
4.9 Photometric Units
4.10 Summary
Problems
Note
Suggestions for Further Reading
CHAPTER 5 Semiconductor Devices Based on the p-n Junction
5.1 Introduction
5.2 The p-n Junction Solar Cell
5.3 Light Absorption
5.4 Solar Radiation
5.5 Solar Cell Design and Analysis
5.6 Solar Cell Efficiency Limits and Tandem Cells
5.7 The Light Emitting Diode
5.8 Emission Spectrum
5.9 Non-Radiative Recombination
5.10 Optical Outcoupling
5.11 GaAs LEDs
5.12 GaP:N LEDs
5.13 Double Heterojunction Al
x
Ga1-
x
As LEDs
5.14 AlGaInP LEDs
5.15 Ga1-
x
In
x
N LEDs
5.16 Bipolar Junction Transistor
5.17 Junction Field Effect Transistor
5.18 BJT and JFET Symbols and Applications
5.19 Summary
Problems
Further Reading
CHAPTER 6 The Metal Oxide Semiconductor Field Effect Transistor
6.1 Introduction to the MOSFET
6.2 MOSFET Physics
6.3 MOS Capacitor Analysis
6.4 Accumulation Layer and Inversion Layer Thicknesses
6.5 Capacitance of MOS Capacitor
6.6 Work Functions, Trapped Charges, and Ion Beam Implantation
6.7 Surface Mobility
6.8 MOSFET Transistor Characteristics
6.9 MOSFET Scaling
6.10 Nanoscale Photolithography
6.11 Ion Beam Implantation
6.12 MOSFET Fabrication
6.13 CMOS Structures
6.14 Threshold Voltage Adjustment
6.15 Two-Dimensional Electron Gas
6.16 Modeling Nanoscale MOSFETs
6.17 Flash Memory
6.18 Tunneling
6.19 Summary
Problems
Notes
Recommended Reading
CHAPTER 7 The Quantum Dot
7.1 Introduction and Overview
7.2 Quantum Dot Semiconductor Materials
7.3 Synthesis of Quantum Dots
7.4 Quantum Dot Confinement Physics
7.5 Franck-Condon Principle and the Stokes Shift
7.6 The Quantum Mechanical Oscillator
7.7 Vibronic Transitions
7.8 Surface Passivation
7.9 Auger Processes
7.10 Biological Applications of Quantum Dots
7.11 Summary
Problems
Recommended Reading
CHAPTER 8 Organic Semiconductor Materials and Devices
8.1 Introduction to Organic Electronics
8.2 Conjugated Systems
8.3 Polymer OLEDs
8.4 Small-Molecule OLEDs
8.5 Anode Materials
8.6 Cathode Materials
8.7 Hole Injection Layer
8.8 Electron Injection Layer
8.9 Hole Transport Layer
8.10 Electron Transport Layer
8.11 Light Emitting Material Processes
8.12 Host Materials
8.13 Fluorescent Dopants
8.14 Phosphorescent and Thermally Activated Delayed Fluorescence Dopants
8.15 Organic Solar Cells
8.16 Organic Solar Cell Materials
8.17 The Organic Field Effect Transistor
8.18 Summary
Problems
Notes
Suggestions for Further Reading
CHAPTER 9 One- and Two-Dimensional Semiconductor
Materials and Devices
9.1 Introduction
9.2 Linear Combination of Atomic Orbitals
9.3 Density Functional Theory
9.4 Transition Metal Dichalcogenides
9.5 Multigate MOSFETs
9.6 Summary
Problems
Recommended Reading
Appendix 1: Physical Constants
Appendix 2: Derivation of the Uncertainty Principle
Appendix 3: Derivation of Group Velocity
Appendix 4: Reduced Mass
Appendix 5: The Boltzmann Distribution Function
Appendix 6: Properties of Semiconductor Materials
Appendix 7: Calculation of the Bonding and Antibonding Orbital Energies Versus Interproton Separation for the Hydrogen Molecular Ion
Index
End User License Agreement
List of Tables
APPENDIX 05
TABLE A5.1 A list of all possible...
CHAPTER 01
TABLE 1.1 The allowed quantum...
TABLE 1.2 Physical observables and...
CHAPTER 04
TABLE 4.1 Luminescence types, applications...
TABLE 4.2 Possible spin states...
TABLE 4.3 Luminous efficiency values...
CHAPTER 06
TABLE 6.1 Showing channel lengths...
CHAPTER 07
TABLE 7.1 Relevant semiconductor materials...
List of Illustrations
APPENDIX 05
FIGURE A5.1 Plot of the probability...
FIGURE A5.2 Plot of the probability...
CHAPTER 01
FIGURE 1.1 Electron beam emitted...
FIGURE 1.2 Classically expected result...
FIGURE 1.3 Result of two-slit...
FIGURE 1.4 Davisson-Germer experiment...
FIGURE 1.5 The photoelectric experiment...
FIGURE 1.6 Wave-packet. The envelope...
FIGURE 1.7 Potential well with...
FIGURE 1.8 a) A beam of...
FIGURE 1.9 The Stern-Gerlach...
FIGURE 1.10 A magnetic dipole...
CHAPTER 02
FIGURE 2.1 The energy levels...
FIGURE 2.2 Periodic one-dimensional...
FIGURE 2.3 Graph of right-hand...
FIGURE 2.4 Plot of
E
...
FIGURE 2.5 Plot of
E
...
FIGURE 2.6 Plot of
E
...
FIGURE 2.7 The filling of...
FIGURE 2.8 Room temperature semiconductor...
FIGURE 2.9 Silicon atoms have...
FIGURE 2.10 Plot of the fermi–dirac...
FIGURE 2.12 Reciprocal space lattice...
FIGURE 2.13 The positive octant...
FIGURE 2.11 A semiconductor band...
FIGURE 2.14 A portion of...
FIGURE 2.15 Plot of commonly...
FIGURE 2.16 (A) the diamond...
FIGURE 2.17 Band structures of...
FIGURE 2.18 The substitution of...
FIGURE 2.19 The substitution of...
FIGURE 2.20 The band diagrams...
FIGURE 2.21 Carrier concentration as...
FIGURE 2.22 Current (
I
) flows...
FIGURE 2.23 Spatial dependence of...
FIGURE 2.24 Dependence of drift...
FIGURE 2.25 Plot of excess...
FIGURE 2.26 The energy bands...
FIGURE 2.27 The quasi-fermi levels...
FIGURE 2.28 A solid semiconductor...
FIGURE 2.29 Plot of excess...
FIGURE 2.30 Hole current density...
FIGURE 2.31 A trap level...
FIGURE 2.32 Surface traps at...
FIGURE 2.33 Surface traps at...
FIGURE 2.34 Bandgap versus lattice...
CHAPTER 03
FIGURE 3.1 The p-n junction...
FIGURE 3.2 Band model of...
FIGURE 3.3 Flow directions of...
FIGURE 3.4 A p-n junction...
FIGURE 3.5 Diode band model...
FIGURE 3.6 Diode band model...
FIGURE 3.7 Diode current as...
FIGURE 3.8 The equilibrium p-n...
FIGURE 3.9 Depletion occurs near...
FIGURE 3.10 A depletion region...
FIGURE 3.11 A gaussian surface...
FIGURE 3.12 The electric field...
FIGURE 3.13 The equilibrium electric...
FIGURE 3.14 Coordinates
x
p
and
x
n
...
FIGURE 3.15 Quasi-Fermi levels for...
FIGURE 3.16 Excess carrier concentration...
FIGURE 3.17 A) minority currents...
FIGURE 3.18 Measured current–voltage...
FIGURE 3.19 Increase in depletion...
FIGURE 3.20 Tunneling of valence-band...
FIGURE 3.21 In a tunnel diode...
FIGURE 3.22 Current–Voltage (
I–V
)...
FIGURE 3.23 The Quasi-Fermi levels..
FIGURE 3.24 Metal-Semiconductor contact...
FIGURE 3.25 Metal-Semiconductor junction...
FIGURE 3.26 A) vacuum “box”...
FIGURE 3.27 A) reciprocal space...
FIGURE 3.28 If the interface...
FIGURE 3.29 Example of an...
FIGURE 3.30 Example of heterojunction...
CHAPTER 04
FIGURE 4.1 Lines of electric...
FIGURE 4.2 Closed lines of...
FIGURE 4.3 Lines of electric...
FIGURE 4.4 Direction of magnetic...
FIGURE 4.5 A Time-Dependent plot...
FIGURE 4.6 The exciton forms...
FIGURE 4.7 Low-Temperature transmission as...
FIGURE 4.8 A depiction of...
FIGURE 4.9 Energy levels associated...
FIGURE 4.10 (A) parabolic conduction...
FIGURE 4.11 Photon emission rate...
FIGURE 4.12 Absorption edge for...
FIGURE 4.13 The eye sensitivity...
FIGURE 4.14 Color space chromaticity...
CHAPTER 05
FIGURE 5.1 Band diagram of...
FIGURE 5.2 The
I–V
characteristic...
FIGURE 5.3 The absorption of...
FIGURE 5.4 Indirect gap semiconductor...
FIGURE 5.5 Absorption coefficients covering...
FIGURE 5.6 Solar radiation spectrum...
FIGURE 5.7 Cross-Section of a...
FIGURE 5.8 Concentrations are plotted...
FIGURE 5.9 Operating point of...
FIGURE 5.10 Multijunction solar cell...
FIGURE 5.11 Efficiency of leds...
FIGURE 5.12 Forward-Biased led p-n...
FIGURE 5.13 Led packaging may...
FIGURE 5.14 Examples of led...
FIGURE 5.15 Emission spectra of...
FIGURE 5.16 Light generated in...
FIGURE 5.17 Band gaps of...
FIGURE 5.18 Electron and hole...
FIGURE 5.19 (Al
x
ga
1
−
x
)
Y
IN
1−Y
P bandgap...
FIGURE 5.20 Radiative efficiency as...
FIGURE 5.21 Dislocations in gan...
FIGURE 5.22 (A) band diagram...
FIGURE 5.23 Emission spectrum of...
FIGURE 5.24 Reverse-Biased p-n...
FIGURE 5.25 Characteristics of a...
FIGURE 5.26 Voltages and currents...
FIGURE 5.27 Base region of...
FIGURE 5.28 Plot of excess...
FIGURE 5.29 Plot of collector...
FIGURE 5.30 Diagram of n-channel...
FIGURE 5.31 Family of characteristic...
FIGURE 5.32 Detail of figure...
FIGURE 5.33 The tapered shape...
FIGURE 5.34 Circles show the...
FIGURE 5.35 Symbols for bjt...
CHAPTER 06
FIGURE 6.1 Cross section of...
FIGURE 6.2 Cross sections of...
FIGURE 6.3 The mos capacitor...
FIGURE 6.4 Four distinct biasing...
FIGURE 6.5 The semiconductor band...
FIGURE 6.6 a) Gaussian box starting...
FIGURE 6.7 Magnitude of space...
FIGURE 6.8 Band diagram, charge...
FIGURE 6.9 P-type silicon mos...
FIGURE 6.10 Mos capacitor series...
FIGURE 6.11 Charge per unit...
FIGURE 6.12 Based on numerous...
FIGURE 6.13 Electrostatic potential in...
FIGURE 6.14 Transverse potential along...
FIGURE 6.15 Drain current for...
FIGURE 6.16 Mosfet gate length...
FIGURE 6.17 Moore’s law showing...
FIGURE 6.18 Radiation exposes photoresist...
FIGURE 6.19 Single slit diffraction...
FIGURE 6.20 Photolithography light source...
FIGURE 6.21 Water-coupled lens.
FIGURE 6.22 A simple example...
FIGURE 6.23 A phase shift...
FIGURE 6.24 Double patterning by...
FIGURE 6.25 Grid design rules...
FIGURE 6.26 a) Development rate versus...
FIGURE 6.27 Sinusoidal roughness pattern...
FIGURE 6.28 Ion beam implanter...
FIGURE 6.29 Distribution of a)...
FIGURE 6.30 Mosfet fabrication steps.
FIGURE 6.31 Cmos circuit, the...
FIGURE 6.32 Cross section of...
FIGURE 6.33 Shows the inversion...
FIGURE 6.34 Three-dimensional reciprocal...
FIGURE 6.35 Density of states...
FIGURE 6.36 Computer modeling of...
FIGURE 6.37 A flash memory...
FIGURE 6.38 Charge is deposited...
FIGURE 6.39 An electron traveling...
FIGURE 6.40 Tunneling of electrons...
FIGURE 6.41 Equivalent oxide thickness...
CHAPTER 07
FIGURE 7.1 A) van der...
FIGURE 7.2 An atomic resolution...
FIGURE 7.3 Ligands have a...
FIGURE 7.4 Quantum dots formed...
FIGURE 7.5 A Box-Shaped semiconductor...
FIGURE 7.6 The conduction and...
FIGURE 7.7 The absorption spectrum...
FIGURE 7.8 Bond energy versus...
FIGURE 7.9 Bond energy versus...
FIGURE 7.10 Electron energy levels...
FIGURE 7.11 A set of...
FIGURE 7.12 The first five...
FIGURE 7.13 The Franck-Condon principle...
FIGURE 7.14 The oleic acid...
FIGURE 7.15 After bonding, the...
FIGURE 7.16 Core-Shell quantum dot...
FIGURE 7.17 Type I core-shell...
FIGURE 7.18 Type II heterojunction...
FIGURE 7.19 Expected single exciton...
FIGURE 7.20 Examples of auger...
FIGURE 7.21 The observed radiative...
FIGURE 7.22 The observed radiative...
FIGURE 7.23 Radial probability density...
FIGURE 7.24 Blue light from...
FIGURE 7.25 Biocompatible quantum dots...
CHAPTER 08
FIGURE 8.1 The molecular structure...
FIGURE 8.2 Polyacetylene is the...
FIGURE 8.3 Energy levels and...
FIGURE 8.4
1
Molecular structures of...
FIGURE 8.5 Poly para-phenylene vinylene...
FIGURE 8.6 Absorption and emission...
FIGURE 8.7 Structure of basic...
FIGURE 8.8 Energy diagram showing...
FIGURE 8.9 The upper
Π
*
FIGURE 8.10 Typical luminance–voltage...
FIGURE 8.11 Small organic molecules...
FIGURE 8.12 Small-molecule oled...
FIGURE 8.13 A more optimized...
FIGURE 8.14 Band diagram of...
FIGURE 8.15 Oled package includes...
FIGURE 8.16 Copper phthalocyanine, or...
FIGURE 8.17 Organo-metallic complexes...
FIGURE 8.18 Two further examples...
FIGURE 8.19 Phenylazomethines are formed...
FIGURE 8.20 Tpbi, atzl, and...
FIGURE 8.21 Host-guest energy transfer...
FIGURE 8.22 Electron transport hosts...
FIGURE 8.23 Hole transport hosts...
FIGURE 8.24 Coumarin-based green...
FIGURE 8.25 The red fluorescent...
FIGURE 8.26 Arylene family host...
FIGURE 8.27 Scheme illustrating internal...
FIGURE 8.28 Phosphorescent emitters: a)...
FIGURE 8.29 Left: Thermally-activated delayed...
FIGURE 8.30 Single-layer organic solar...
FIGURE 8.31 Energy level diagram...
FIGURE 8.32 Organic planar heterojunction...
FIGURE 8.33 Heterojunction solar cell...
FIGURE 8.34 Bulk heterojunction organic...
FIGURE 8.35 (a) Bulk heterojunction...
FIGURE 8.36 Molecular structures of...
FIGURE 8.37 Absorption spectra of...
FIGURE 8.38 Molecular structures of
FIGURE 8.39 Carbon nanotube. courtesy...
FIGURE 8.40 Perovskite solar cell...
FIGURE 8.41 Perovskite solar cell...
FIGURE 8.42 Spiro-meotad hole...
FIGURE 8.43 Since the 1980s...
FIGURE 8.44 a) C8-BTBT is a...
FIGURE 8.45 Top gate OFET
CHAPTER 09
FIGURE 9.1 Two protons separated...
FIGURE 9.2 Orbital wavefunction amplitudes...
FIGURE 9.3 Bonding orbital system...
FIGURE 9.4 H
2
+
correlation diagram...
FIGURE 9.5 Structure of a...
FIGURE 9.6 The band diagram...
FIGURE 9.7 Transition metal dichalcogenide...
FIGURE 9.8 Drain current versus...
FIGURE 9.9 A typical silicon...
FIGURE 9.10 Cross section of...
FIGURE 9.11 The Gate-all-around FET...
FIGURE 9.12 A method of...
Guide
Cover
Title Page
Copyright Page
Table of Contents
Acknowledgments
Preface
About the Companion Website
Begin Reading
Appendix 1: Physical Constants
Appendix 2: Derivation of the Uncertainty Principle
Appendix 3: Derivation of Group Velocity
Appendix 4: Reduced Mass
Appendix 5: The Boltzmann Distribution Function
Appendix 6: Properties of Semiconductor Materials
Appendix 7: Calculation of the Bonding and Antibonding Orbital Energies Versus Interproton Separation for the Hydrogen Molecular Ion
Index
End User License Agreement
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Acknowledgments
I would like to thank McMaster University colleagues Oleg Rubel and Randy Dumont and Wiley staff Jenny Cossham, Skyler Van Valkenburgh, Martin Preuss, Monica Chandrasekar, Dilip Varma and Christy Michael who played important roles in providing advice, expertise, general guidance, editing, and support.
Additional expertise from Mark Tuckerman (computational chemistry) and Nicolas Heumann (physics) is gratefully acknowledged.
I need to thank the students in the senior level course I teach at McMaster University for their invaluable assistance in finding errors in the manuscript.
Finally, this book would not have been possible without the patient support of my family.
Preface
The traditional, discrete silos of materials science, electrical engineering, engineering physics, physics, and chemistry are being increasingly blended due to the expanding needs and expectations of semiconductor materials and devices in both industry and academia.
This book is designed to introduce the topic of semiconductor materials and devices at a foundational level and to a broad audience. Only knowledge of first- and second-year-undergraduate-level math and science is assumed. This is a third- or fourth-year-undergraduate or graduate-level textbook.
The book begins with introductory chapters on quantum and solid-state physics followed by an introduction to practical semiconductor materials and basic p-n junction theory.
Next, to prepare for optoelectronic devices, radiation theory covering the very important concepts of the exciton, the exchange interaction, singlets/triplets, and the joint density of states is introduced. Associated coverage includes accelerating charges, a quantum treatment of the dipole radiator, the dipole matrix element, and Forster/Dexter radiative energy transfer processes.
The electrical and optical properties of photodiodes, solar cells, and light emitting diodes follow together with an introduction to bipolar junction transistors and junction field effect transistors.
The silicon MOSFET (metal oxide field effect transistor) is presented next in detail. Coverage starts with MOS band theory and the metal oxide capacitor. Coverage of the MOSFET is extended to the nanoscale and a series of new nanoscale concepts are introduced in context. These concepts include details of bulk and interface charge trapping, coulombic shielding and the Debye length, the two-dimensional regime for channel charge, effective work functions, surface mobility, electron tunneling, Fowler Nordheim current, and saturated carrier velocity.
Key nanoscale fabrication strategies for MOSFET devices are covered including ion implantation, self-aligned structures, deep UV lithography, rapid thermal annealing, high oxidation potential, high K dielectric material (HfO2), and high thermal budget electrode materials.
Content then moves into truly nanoscale concepts introduced via the quantum dot and progresses to the small organic molecule size range. Optical properties and the physics of quantum confinement in quantum dots are covered with an effective mass model to explain an increase in energy gap. Limitations of the effective mass model are explained as well as the need for a Rydberg term. The exciton radius relative to the quantum dot radius is a key new concept which can lead to an additional energy correction for the calculation of energy gap.
Vibrational aspects including the Stokes shift in quantum dot emission and absorption spectra are then understood by presenting the Frank-Condon principle. A configurational concept is introduced starting from a basic bi-atomic bonding model and is then generalized to the widely used configurational coordinate axis applicable to multi-atomic systems including quantum dots. The quantum harmonic oscillator is introduced in this context. Auger processes and biexcitons are treated in some detail and the biocompatibility of quantum dots is introduced.
Molecular electronics is treated next since the molecular length scale constitutes a natural progression toward smaller structures. This requires the introduction of organic semiconductor molecules and both oligomeric and polymeric materials. A range of well-known organic oligomers is presented. Sigma and pi bonds are introduced and the concepts of conjugated bonding leading to intramolecular electron transport through pi bands is covered together with intermolecular transport and the latter’s dependence on molecular packing in both crystalline and amorphous molecular materials. The highest occupied molecular orbital and the lowest unoccupied molecular orbital are introduced and shown to define the effective energy gap in molecular semiconductors.
Functional molecule side groups are discussed in terms of modifying emission and absorption spectra and as a means to allow for solubility and low-cost solution deposition of oligomers in contrast to vapor deposition via vacuum evaporation.
The carrier mobility and electron affinity values of important molecular materials are discussed and tabulated. The low carrier mobilities and their strong influence on organic device design are described. Band models of a series of both polymeric and oligomeric organic light emitting devices are presented showing the importance and relevance of electron affinity. Electron and hole transport within these devices leading to the formation of excitons is described.
The molecular exciton and the associated energy transfer processes are now applied to molecular optoelectronic devices. A good understanding of the exciton is needed for both organic light emitting diodes and organic solar cells. Additional concepts are introduced including phosphorescent organic molecules, organometallic molecule spin–orbit coupling and thermally stimulated delayed fluorescence. Both homojunction and heterojunction organic solar cells are introduced as well as the bulk heterojunction leading to substantial improvements in organic solar cell performance.
Finally, materials and devices are considered in which at least one dimension exists at length scales down to the angstrom range. At the smallest length scales, two-dimensional atomically thin semiconductors are introduced via the Transition Metal Dichalcogenides (TMDCs). These materials naturally form semiconductors having well-defined energy gaps in a single dichalcogenide layer. There are no dangling bonds since bonding between layers in analogous bulk materials is via van der Walls forces only. Details of known atomic structures are presented.
Due to the limitations of bulk band theory for the smallest structures, the linear combination of atomic orbitals and density functional theory are introduced and examples are provided. Data is presented including the progression in band gaps and band diagram dispersion curves from bulk to single layers of TMDCs exhibiting both indirect gap and direct gap behavior. Examples of prototype TMDC transistors and their characteristics are presented. Coulomb scattering is shown to limit carrier mobility in practice.
One-dimensional semiconductor devices are introduced firstly by a description of the finFET which is an extension of the MOSFET. The finFET is then further extended to the concept of a true nanowire FET that is a type of gate-all-around FET. Fabrication steps for gate-all-around FETs are presented.
This book is uniquely comprehensive compared to existing books. It covers concepts at an introductory level but also highlights their relevance to a wide range of semiconductor materials and devices from the bulk to the nanoscale. Sitting at the interface between engineering and physics, the book employs math and physics throughout, but always ensures that the presentation is relevant to real applications. The math is sufficiently simple to allow a wide spectrum of scientists and engineers to follow the material. This book contains pedagogically useful questions and exercises throughout.
About the Companion Website
This book is accompanied by a companion website which includes a number of resources created by author for students and instructors that you will find helpful.
www.wiley.com/go/kitai_fundamentals
The website includes the following resource for each chapter:
PowerPoint slides
CHAPTER 1 Introduction to Quantum Mechanics
CONTENTS
1.1 Introduction
1.2 The Classical Electron
1.3 Two-Slit Electron Experiment
1.4 The Photoelectric Effect
1.5 Wave-Packets and Uncertainty
1.6 The Wavefunction
1.7 The Schrödinger Equation
1.8 The Electron in a One-Dimensional Well
1.9 The Hydrogen Atom
1.10 Electron Transmission and Reflection at Potential Energy Step
1.11 Spin
1.12 The Pauli Exclusion Principle
1.13 Operators and the Postulates of Quantum Mechanics
1.14 Expectation Values and Hermitian Operators
1.15 Summary
Problems
Notes
Suggestions for Further Reading
Objectives
Review the classical electron and motivate the need for a quantum mechanical model.
Present experimental evidence for the photon as a fundamental constituent of electromagnetic radiation.
Introduce quantum mechanical relationships based on experimental results and illustrate these using examples.
Introduce expectation values for important measureable quantities based on the uncertainty principle.
Motivate and define the wavefunction as a means of describing particles.
Present Schrödinger’s equation and its solutions for practical problems relevant to semiconductor materials and devices.
Introduce spin and the associated magnetic properties of electrons.
Present the postulates of quantum mechanics.
Introduce the Pauli exclusion principle and give an example of its application.
1.1 INTRODUCTION
The study of semiconductor materials and devices relies on the electronic properties of solid state materials and hence a fundamental understanding of the behavior of electrons in solids.
Electrons are responsible for electrical properties and optical properties in metals, insulators, inorganic semiconductors, and organic semiconductors. These materials form the basis of an astonishing variety of electronic components and devices.
The electronics age in which we are immersed would not be possible without the ability to grow these materials, control their electronic properties, and finally fabricate structured devices using them which yield specific electronic and optical functionality.
Electron behavior in solids requires an understanding of the electron that includes the quantum mechanical description; however, we will start with the classical electron.
1.2 THE CLASSICAL ELECTRON
We describe the electron as a particle having mass
and negative charge of magnitude
If an external electric field is present in three-dimensional space and an electron experiences this external electric field, the magnitude of the force on the electron is
The direction of the force is opposite to the direction of the external electric field due to the negative charge on the electron. If is expressed in volts per meter () then will have units of newtons.
If an electron accelerates through a distance from point A to point B in vacuum due to a uniform external electric field it will gain kinetic energy in which
This kinetic energy gained by the electron may be expressed in joules within the Meters-Kilograms-Seconds (MKS) unit system. We can also say that the electron at point A has a potential energy U that is higher than its potential energy at point B. Since total energy is conserved,
There exists an electric potential defined in units of the volt at any position in three-dimensional space associated with an external electric field. We obtain the spatially dependent potential energy for an electron in terms of this electric potential as
We also define the electron-volt, another commonly used energy unit. By definition, one electron-volt in kinetic energy is gained by an electron if the electron accelerates in an electric field between two points in space whose difference in electric potential is one volt.
EXAMPLE 1.1
Find the relationship between two commonly used units of energy, namely the electron-volt and the joule.
Consider a uniform external electric field in which . If an electron accelerates in vacuum in this uniform external electric field between two points separated by one meter and therefore having a potential difference of one volt, then from Equation 1.1a it gains kinetic energy expressed in joules of
But, by definition, one electron-volt in kinetic energy is gained by an electron if the electron accelerates in an electric field between two points in space whose difference in electric potential is one volt, and we have therefore shown that the conversion between the joule and the electron-volt is
If an external magnetic field is present, the force on an electron depends on the charge on the electron as well as the component of electron velocity perpendicular to the magnetic field which we shall denote . This force, called the Lorentz Force, may be expressed as . The force is perpendicular to both the velocity component of the electron and to the magnetic field vector. The Lorentz force enables the electric motor and the electric generator.
This classical description of the electron generally served the needs of the vacuum tube electronics era and the electric motor/generator industry in the first half of the twentieth century.
In the second half of the twentieth century, the electronics industry migrated from vacuum tube devices to solid state devices once the transistor was invented at Bell Laboratories in 1954. The understanding of the electrical properties of semiconductor materials from which transistors are made could not be achieved using a classical description of the electron. Fortunately the field of quantum mechanics, which was developing over the span of about 50 years before the invention of the transistor, allowed physicists to model and understand electron behavior in solids.
In this chapter we will motivate quantum mechanics by way of a few examples. The classical description of the electron is shown to be unable to explain some simple observed phenomena, and we will then introduce and apply the quantum-mechanical description which has proven to work very successfully.
1.3 TWO-SLIT ELECTRON EXPERIMENT
One of the most remarkable illustrations of how strangely electrons can behave is illustrated in Figure 1.1. Consider a beam of electrons arriving at a pair of narrow, closely spaced slits formed in a solid. Assume that the electrons arrive at the slits randomly in a beam having a width much wider than the slit dimensions. Most of the electrons hit the solid, but a few electrons pass through the slits and then hit a screen placed behind the slits as shown.
FIGURE 1.1 Electron beam emitted by an electron source is incident on narrow slits with a screen situated behind the slits.
If the screen could detect where the electrons arrived by counting them, we would expect a result as shown in Figure 1.2.
FIGURE 1.2 Classically expected result of two-slit experiment.
In practice, a screen pattern as shown in Figure 1.3 is obtained. This result is impossible to derive using the classical description of an electron.
FIGURE 1.3 Result of two-slit experiment. Notice that a wavelike electron is required to cause this pattern. If light waves rather than electrons were used then a similar plot would result except the vertical axis would be a measure of the light intensity instead.
It does become readily explainable, however, if we assume the electrons have a wavelike nature. If light waves, rather than particles, are incident on the slits then there are particular positions on the screen at which the waves from the two slits cancel out. This is because they are out of phase. At other positions on the screen the waves add together because they are in-phase. This pattern is the well-known interference pattern generated by light traveling through a pair of slits. Interestingly we do not know which slit a particular electron passes through. If we attempt to experimentally determine which slit an electron is passing through we immediately disrupt the experiment and the interference pattern disappears. We could say that the electron somehow goes through both slits. Remarkably, the same interference pattern builds up slowly and is observed even if electrons are emitted from the electron source and arrive at the screen one at a time. We are forced to interpret these results as a very fundamental property of small particles such as electrons.
We will now look at how the two-slit experiment for electrons may actually be performed. It was done in the 1920s by Davisson and Germer. It turns out that very narrow slits are required to be able to observe the electron behaving as a wave due to the small wavelength of electrons. Fabricated slits having the required very small dimensions are not practical, but Davisson and Germer realized that the atomic planes of a crystal can replace slits. By a process of electron reflection, rows of atoms belonging to adjacent atomic planes on the surface of a crystal act like tiny reflectors that effectively form two beams of reflected electrons that then reach a screen and form an interference pattern similar to that shown in Figure 1.3.
Their method is shown in Figure 1.4. The angle between the incident electron beam and each reflected electron beam is . The spacing between surface atoms belonging to adjacent atomic planes is . The path length difference between the two beam paths shown is . A maximum on the screen is observed when
FIGURE 1.4 Davisson-Germer experiment showing electrons reflected off adjacent crystalline planes. Path length difference is .
or an integer number of wavelengths. Here, is an integer and is the wavelength of the waves. A minimum occurs when
which is an odd number of half wavelengths causing wave cancellation.
In order to determine the wavelength of the apparent electron wave we can solve Equations 1.2a and 1.2b for . We have the appropriate values of ; however, we need to know . Using X-ray diffraction and Bragg’s law we can obtain . Note that Bragg’s law is also based on wave interference except that the waves are X-rays.
The results that Davisson and Germer obtained were quite startling. The calculated values of were on the order of angstroms, where 1 angstrom is one tenth of a nanometer. This is much smaller than the wavelength of light which is on the order of thousands of angstroms, and it explains why regular slits used in optical experiments are much too large to observe electron waves. But more importantly the measured values ofactually depended on the incident velocityor momentumof the electrons used in the experiment. Increasing the electron momentum by accelerating electrons through a higher potential difference before they reached the crystal caused to decrease, and decreasing the electron momentum caused to increase. By experimentally determining for a range of values of incident electron momentum, the following relationship was determined:
This is known as the de Broglie equation, because de Boglie postulated this relationship before it was validated experimentally. Here is the magnitude of electron momentum, and is a constant known as Planck’s constant. In an alternative form of the equation we define , pronounced h-bar to be and we define , the wavenumber to be . Now we can write the de Broglie equation as
Note that is the magnitude of the momentum vector and is the magnitude of wavevectork. The significance of wavevectors will become clearer in Chapter 2.
EXAMPLE 1.2
An electron is accelerated through a potential difference volts.
Find the electron energy in both joules and electron-volts.
Find the electron wavelength.
Solution
Assume the initial kinetic energy of the electron was negligible before it was accelerated. The final energy is
To express this energy in electron-volts,
b) From
Equation 1.3b
1.4 THE PHOTOELECTRIC EFFECT
About 30 years before Davisson and Germer discovered and measured electron wavelengths, another important experiment had been undertaken by Heinrich Hertz. In 1887 Hertz was investigating what happens when light is incident on a metal. He found that electrons in the metal can be liberated by the light. It takes a certain amount of energy to release an electron from a metal into vacuum. This energy is called the workfunction, and the magnitude of depends on the metal.
If the metal is placed in a vacuum chamber, the liberated electrons are free to travel away from the metal and they can be collected by a collector electrode also located in the vacuum chamber shown in Figure 1.5. This is known as the photoelectric effect.
FIGURE 1.5 The photoelectric experiment. The current flowing through the external circuit is the same as the vacuum current.
By carefully measuring the resulting electric current flowing through the vacuum, scientists in the last decades of the nineteenth century were able to make the following statements:
The electric current increases with increasing light intensity.
The color or wavelength of the light is important also. If monochromatic light is used, there is a particular critical wavelength above which no electrons are released from the metal even if the light intensity is increased.
This critical wavelength depends on the type of metal employed. Metals composed of atoms having low ionization energies such as cerium and calcium have larger critical wavelengths. Metals composed of atoms having higher ionization energies such as gold or aluminum require smaller critical wavelengths.
If light having a wavelength equal to the critical wavelength is used then the electrons leaving the metal surface have no initial kinetic energy and collecting these electrons requires that the voltage must be positive to accelerate the electrons away from the metal and toward the second electrode.
If light having wavelengths smaller than the critical wavelength is used then the electrons do have some initial kinetic energy. Now, even if is negative so as to retard the flow of electrons from the metal to the electrode, some electrons may be collected. There is, however, a maximum negative voltage of magnitude for which electrons may be collected. As the wavelength of the light is further decreased, increases (the maximum voltage becomes more negative).
If very low intensity monochromatic light with a wavelength smaller than the critical wavelength is used then individual electrons are measured rather than a continuous electron current. Suppose at time there is no illumination, and then at time very low intensity light is turned on. The first electron to be emitted may occur virtually as soon as the light is turned on, or it may take a finite amount of time to be emitted after the light is turned on. There is no way to predict this amount of time in advance.
Einstein won the Nobel Prize for his conclusions based on these observations. He concluded the following:
Light is composed of particle-like entities or
wave-packets
commonly called
photons
.
The intensity of a source of light is determined by the photon flux density, or the number of photons being emitted per second per unit area.
Electrons are emitted only if the incident photons each have enough energy to overcome the metal’s workfunction.
The photon energy of an individual photon of monochromatic light is determined by the color (wavelength) of the light.
Photons of a known energy are randomly emitted from monochromatic light sources, and we can never precisely know when the next photon will be emitted.
Normally we do not notice these photons because there are a very large number of photons in a light beam. If, however, the intensity of the light is low enough then the photons become noticeable and light becomes
granular
.
If light having photon energy larger than the energy needed to overcome the workfunction is used then the excess photon energy causes a finite initial kinetic energy of the escaped electrons
By carefully measuring the critical photon wavelength as a function of , the relationship between photon energy and photon wavelength can be determined: The initial kinetic energy of the electron leaving the metal can be sufficient to overcome a retarding (negative) potential difference between the metal and the electrode. Since the kinetic energy lost by the electron as it moves against this retarding potential difference is we can deduce the minimum required photon energy using the energy equation . This experimentally observed relationship is
where is the velocity of light. Since the frequency of the photon is given by
we obtain the following relationship between photon energy and photon frequency:
where . Note that Planck’s constant is found when looking at either the wavelike properties of electrons in Equation 1.3b or the particle-like properties of light in Equation 1.4. This wave-particle duality forms the basis for quantum mechanics.
Equation 1.4 which arose from the photoelectric effect defines the energy of a photon. In addition, Equation 1.3b which arose from the Davisson and Germer experiment applies to both electrons and photons. This means that a photon having a known wavelength carries a specific momentum even though a photon has no mass. The existence of photon momentum is experimentally proven since light-induced pressure can be measured on an illuminated surface.
In summary, electromagnetic waves exist as photons which also have particle-like properties such as momentum and energy, and particles such as electrons also have wavelike properties such as wavelength.
EXAMPLE 1.3
Ultraviolet light with wavelength 190 nm is incident on a metal sample inside a vacuum envelope containing an additional collector electrode. The collector electrode potential relative to the sample potential is defined by potential difference as shown in Figure 1.5. Photoelectric current is observed if and ceases if . Identify the metal using the following table.
Metal
Aluminum
Nickel
Calcium
Cesium
Workfunction (eV)
4.1
5.1
2.9
2.1
Solution
190 eV photons have energy
Expressed in electron-volts, this photon energy is
The potential difference will decelerate photoelectrons released from the metal surface if . This results in an energy loss of the photoelectrons and if , the energy loss will be . The photons therefore supply enough energy to overcome both the metal workfunction and the energy loss due to deceleration. Hence the predicted metal workfunction is . The metal having the closest workfunction match is nickel.
1.5 WAVE-PACKETS AND UNCERTAINTY
Uncertainty in the precise position of a particle is embedded in its quantum mechanical wave description. The concept of a wave-packet introduced in Section 1.4 for light is important since it is applicable to both photons and particles such as electrons.
A wave-packet is illustrated in Figure 1.6 showing that the wave-packet has a finite size. A wave-packet can be analyzed into, or synthesized from, a set of component sinusoidal waves, each having a distinct wavelength, with phases and amplitudes such that they interfere constructively only over a small region of space to yield the wave-packet, and destructively elsewhere. This set of component sinusoidal waves of distinct wavelengths added to yield an arbitrary function is an example of a Fourier series.
FIGURE 1.6 Wave-packet. The envelope of the wave-packet is also shown. More insight about the meaning of the wave amplitude for a particle such as an electron will become apparent in Section 1.6 in which the concept of a probability amplitude is introduced.
The uncertainty in the position of a particle described using a wave-packet may be approximated as as indicated in Figure 1.6. The uncertainty depends on the number of component sinusoidal waves being added together in a Fourier series: If only one component sinusoidal wave is present, the wave-packet is infinitely long, and the uncertainty in position is infinite. In this case, the wavelength of the particle is precisely known, but its position is not defined. As the number of component sinusoidal wave components of the wave-packet approaches infinity, the uncertainty of the position of the wave-packet may drop and we say that the wave-packet becomes increasingly localized.
An interesting question now arises: If the wave-packet is analyzed into, or composed from, a number of component sinusoidal waves, can we define the precise wavelength of the wave-packet? It is apparent that as more component sinusoidal waves, each having a distinct wavelength, are added together the uncertainty of the wavelength associated with the wave-packet will become larger. From Equation 1.3b, the uncertainty in wavelength results in an uncertainty in momentum , and we write this momentum uncertainty as .
By doing the appropriate Fourier series calculation (see Appendix 2), the relationship between and can be shown to satisfy the following condition:
As is reduced there will be an inevitable increase in This is known as the Uncertainty Principle. We cannot precisely and simultaneously determine the position and the momentum of a particle. If the particle is an electron we know less and less about the electron’s momentum as we determine its position more and more precisely.
Since a photon is also described in terms of a wave-packet, the concept of uncertainty applies to photons as well. As the location of a photon becomes more precise, the wavelength or frequency of the photon becomes less well defined. Photons always travel with velocity of light in vacuum. The exact arrival time of a photon at a specific location is uncertain due to the uncertainty in position. For the photoelectric effect described in Section 1.4, the exact arrival time of a photon at a metal was observed not to be predictable for the case of monochromatic photons for which is accurately known. If we allow some uncertainty in the photon frequency the energy uncertainty of the photon becomes finite, but then we can know more about the arrival time. The resulting relationship which may be calculated by the same approach as presented in Appendix 2 may be written as This type of uncertainty relationship is useful in time-dependent problems and, like the derivation of uncertainty for particles such as electrons, it results from a Fourier transform: The frequency spectrum of a pulse in the time domain becomes wider as the pulse width becomes narrower.
A wave travels with velocity . Note that we refer to this as a phase velocity because it refers to the velocity of a point on the wave that has a given phase, for example the crest of the wave. For a traveling wave-packet, however, the velocity of the particle described using the wave-packet is not necessarily the same as the phase velocity of the individual waves making up the wave-packet. The velocity of the particle is actually determined using the velocity of the wave-packet envelope shown in Figure 1.6. The velocity of propagation of this envelope is called the group velocity because the envelope is formed by the Fourier sum of a group of waves.
When photons travel through media other than vacuum, dispersion can exist. Consider the case of a photon having energy uncertainty due to its wave-packet description. In the case of this photon traveling through vacuum, the group velocity and the phase velocity are identical to each other and equal to the speed of light . This is known as a dispersion-free photon for which the wave-packet remains intact as it travels. But if a photon travels through a medium other than vacuum there is often finite dispersion in which some Fourier components of the photon wave-packet travel slightly faster or slightly slower that other components of the wave-packet, and the photon wave-packet broadens spatially as it travels. For example photons traveling through optical fibers typically suffer dispersion which limits the ultimate temporal resolution of the fiber system.
It is very useful to plot versus for the given medium in which the photon travels. If a straight line is obtained then is a constant and the velocity of each Fourier component of the photon’s wave-packet is identical. This is dispersion-free propagation. In general, however, a straight line will not be observed and dispersion exits.
In Appendix 3 we analyze the velocity of a wave-packet composed of a series of waves. It is shown that the wave-packet travels with velocity
This is valid for both photons and particles such as electrons. For wave-packets of particles, however, we can further state that
This relationship will be important in Chapter 2 to determine the velocity of electrons in crystalline solids.
1.6 THE WAVEFUNCTION
Based on what we have observed up to this point, the following four points more completely describe the properties of an electron in contrast to the description of the classical electron of Section 1.2:
The electron has mass .
The electron has charge .
The electron has wave properties with wavelength .
The exact position and momentum of an electron cannot be measured simultaneously.
Quantum mechanics provides an effective mathematical description of particles such as electrons that was motivated by the above observations. A wavefunction is used to describe the particle and may also be referred to as a probability amplitude. In general, is a complex number which is a function of space and time. Using cartesian spatial coordinates, . We could also use other coordinates such as spherical polar coordinates in which case we would write .
The use of complex numbers is very important for wavefunctions because it allows them to represent waves as will be seen in Section 1.7.
Although is a complex number and is therefore not a real, measureable or observable quantity, the quantity where is the complex conjugate of , is an observable and must be a real number. is referred to as a probability density. At any time , using cartesian coordinates, the probability of the particle being in volume element at location will be . If a particle exists, then it must be somewhere is space and we can write
The wavefunction, therefore, fundamentally recognizes the attribute of uncertainty and simultaneously is able to represent a wave. We cannot precisely define the position of the particle; however, we can determine the probability of it being in a specific region. Equation 1.6 is referred to as the normalization condition for a wavefunction and a wavefunction that satisfies this equation is a normalized wavefunction.
In order to give the particle we are trying to describe the attributes of a wave, the form of is a mathematical wave expression such as the sinusoidal function used in Example 1.4.
EXAMPLE 1.4
A time-independent wavefunction is defined in one dimension along the x-axis.
Such that:
if or then
and if then
Normalize this wavefunction by determining the appropriate value of coefficient .
Solution
Hence
Now the normalized wavefunction is:
for and
and for
1.7 THE SCHRÖDINGER EQUATION
The most fundamental law in classical physics also applies to quantum physics. This is the conservation of energy. More specifically, this can be expressed as follows: The total energy of a closed system is fixed and is the sum of the internal potential and kinetic energies.
Building on the emerging understanding of particles we have outlined in this chapter and through the remarkable insights of Erwin Schrödinger, in 1925 the following wave equation, called the Schrödinger Equation, was postulated:
represents the potential energy in the electric field in which the particle of mass exists and the equation allows the particle’s wavefunction to be found. The first term is associated with the kinetic energy of the particle. The second term is associated with the potential energy of the particle, and the right-hand side of the equation is associated with the total energy of the particle. Once is known, the particle’s position, energy, and momentum can be determined either as specific values or as spatial distribution functions consistent with the uncertainty principle. For time-varying systems, a possible time evolution of the particle’s properties may also be described.
This equation is applicable to a range of particles including electrons and protons; we are most often interested in the electrical properties of materials and we will therefore focus now on the electron. In Chapter 7 it will be applied to the proton and to atomic nuclei.
By solving Schrödinger’s equation for an electron in a few simple scenarios we will be able to appreciate the utility of the equation. In addition, a better understanding will be gained about the quantum mechanical wavefunction-based description of particles.
Let us propose a solution to Equation 1.7 having the form
Note that we have separated the solution into two parts, one for spatial dependence and one for time dependence. Now substituting Equation 1.8 into Equation 1.7 and dividing by we obtain
Since the left side is a function of independent variables only and the right side is is a function of independent variable only, the only way for the equality to hold for both arbitrary spatial locations and for arbitrary moments in time is for both sides of the equation to be equal to a constant that we will call .
The resulting equations are
and
Equations. 1.9 and 1.10 are the result of the just-described method known as theseparation of variables applied to Equation 1.7. Equation 1.9 is easy to solve and has solution
If we now identify with the energy of the electron and use Equation 1.4 we obtain
and therefore
which represents the expected time-dependence of a wave having frequency .
Equation 1.10 is known as the Time-Independent Schrödinger Equation and it is useful for a wide variety of steady state (time-independent) situations as illustrated in Examples 1.5 to 1.8.
EXAMPLE 1.5
Consider a one-dimensional problem in which the only spatial coordinate is the -axis. Assume that the potential energy of the electron for all values of is zero.
Find the wavefunctions.
Find the probability density.
Can the wavefunctions be normalized?
Solution
This absence of potential energy for all values of implies that there is no net force acting on the electron. This situation may be applicable if no electric field is present. Now Equation
1.10
becomes
The general solution to this differential equation is
where
