Principles of Solar Cells, LEDs and Diodes · 5.4 Non-Radiative Recombination 221 5.5 Optical...

Principles of Solar Cells, LEDs and Diodes� e role of the PN junction

ADRIAN KITAI

RED BOX RULES ARE FOR PROOF STAGE ONLY. DELETE BEFORE FINAL PRINTING.

KITAIKITAI

Principles of Solar Cells, LED

s and Diodes

� e role of the PN junction

Principles of Solar Cells, LEDs and Diodes� e role of the PN junctionADRIAN KITAI, Departments of Engineering Physics and Materials Science and Engineering, McMaster University, Hamilton, Ontario, Canada

A textbook introducing the physical concepts required for a comprehensive understanding of p-n junction devices, light emitting diodes and solar cells.

Semiconductor devices have made a major impact on the way we work and live. Today semiconductor p-n junction diode devices are experiencing substantial growth: solar cells are used on an unprecedented scale in the renewable energy industry; and light emitting diodes (LEDs) are revolutionizing energy e� cient lighting. � ese two emerging industries based on p-n junctions make a signi� cant contribution to the reduction in fossil fuel consumption.

Principles of Solar Cells, LEDs and Diodes covers the two most important applications of semiconductor diodes - solar cells and LEDs - together with quantitative coverage of the physics of the p-n junction. � e reader will gain a thorough understanding of p-n junctions as the text begins with semiconductor and junction device fundamentals and extends to the practical implementation of semiconductors in both photovoltaic and LED devices. � e treatment of a range of important semiconductor materials and device structures is also presented in a readable manner.

Topics are divided into the following six chapters;

• Semiconductor Physics• Th e PN Junction Diode• Photon Emission and Absorption• Th e Solar Cell• Light Emitting Diodes• Organic Semiconductors, OLEDs and Solar Cells

Containing student problems at the end of each chapter and worked example problems throughout, this textbook is intended for senior level undergraduate students doing courses in electrical engineering, physics and materials science. Researchers working on solar cells and LED devices, and those in the electronics industry would also bene� t from the background information the book provides.

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Principles of Solar Cells,LEDs and Diodes

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Principles of Solar Cells,LEDs and Diodes

The role of the PN junction

ADRIAN KITAI

Departments of Engineering Physics and Materials Science andEngineering, McMaster University, Hamilton, Ontario, Canada

A John Wiley & Sons, Ltd., Publication

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This edition first published 2011© 2011 John Wiley & Sons, Ltd

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The publisher and the author make no representations or warranties with respect to the accuracy or completeness of the contentsof this work and specifically disclaim all warranties, including without limitation any implied warranties of fitness for aparticular purpose. This work is sold with the understanding that the publisher is not engaged in rendering professional services.The advice and strategies contained herein may not be suitable for every situation. In view of ongoing research, equipmentmodifications, changes in governmental regulations, and the constant flow of information relating to the use of experimentalreagents, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert orinstructions for each chemical, piece of equipment, reagent, or device for, among other things, any changes in the instructions orindication of usage and for added warnings and precautions. The fact that an organization or Website is referred to in this work asa citation and/or a potential source of further information does not mean that the author or the publisher endorses the informationthe organization or Website may provide or recommendations it may make. Further, readers should be aware that InternetWebsites listed in this work may have changed or disappeared between when this work was written and when it is read. Nowarranty may be created or extended by any promotional statements for this work. Neither the publisher nor the author shall beliable for any damages arising herefrom.

Library of Congress Cataloging-in-Publication Data

Kitai, Adrian, 1957–Principles of solar cells, LEDs, and diodes : the role of the PN junction / Adrian Kitai.

p. cm.Includes bibliographical references and index.ISBN 978-1-4443-1834-0 (hardback) – ISBN 978-1-4443-1833-3 (paper)

1. Diodes, Semiconductor. 2. Light emitting diodes. 3. Semiconductors–Junctions. 4. Solar cells. I. Title.TK7871.86.K48 2011621.3815′2–dc22

2011010956

A catalogue record for this book is available from the British Library.

HB ISBN: 9781444318340PB ISBN: 9781444318333ePDF ISBN: 9781119974550oBook ISBN: 9781119974543ePub ISBN: 9781119975236eMobi ISBN: 9781119975243

Set in 10/12pt Times by Aptara Inc., New Delhi, India.Printed in Malaysia by Vivar Malaysia

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http://www.wiley.com

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Dedicated to my wife Tomoko

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Contents

Introduction xiAcknowledgements xv

1 Semiconductor Physics 11.1 Introduction 21.2 The Band Theory of Solids 21.3 The Kronig–Penney Model 31.4 The Bragg Model 81.5 Effective Mass 81.6 Number of States in a Band 101.7 Band Filling 121.8 Fermi Energy and Holes 141.9 Carrier Concentration 151.10 Semiconductor Materials 251.11 Semiconductor Band Diagrams 261.12 Direct Gap and Indirect Gap Semiconductors 291.13 Extrinsic Semiconductors 351.14 Carrier Transport in Semiconductors 401.15 Equilibrium and Non-Equilibrium Dynamics 431.16 Carrier Diffusion and the Einstein Relation 451.17 Quasi-Fermi Energies 481.18 The Diffusion Equation 501.19 Traps and Carrier Lifetimes 531.20 Alloy Semiconductors 561.21 Summary 59

Suggestions for Further Reading 61Problems 63

2 The PN Junction Diode 692.1 Introduction 702.2 Diode Current 722.3 Contact Potential 752.4 The Depletion Approximation 782.5 The Diode Equation 852.6 Reverse Breakdown and the Zener Diode 97

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viii Contents

2.7 Tunnel Diodes 1002.8 Generation/Recombination Currents 1012.9 Ohmic Contacts, Schottky Barriers and Schottky Diodes 1042.10 Heterojunctions 1132.11 Alternating Current (AC) and Transient Behaviour 1152.12 Summary 117


3 Photon Emission and Absorption 1233.1 Introduction to Luminescence and Absorption 1243.2 Physics of Light Emission 1253.3 Simple Harmonic Radiator 1283.4 Quantum Description 1293.5 The Exciton 1323.6 Two-Electron Atoms 1353.7 Molecular Excitons 1413.8 Band-to-Band Transitions 1443.9 Photometric Units 1483.10 Summary 152


4 The Solar Cell 1594.1 Introduction 1604.2 Light Absorption 1624.3 Solar Radiation 1644.4 Solar Cell Design and Analysis 1644.5 Thin Solar Cells 1724.6 Solar Cell Generation as a Function of Depth 1764.7 Solar Cell Efficiency 1794.8 Silicon Solar Cell Technology: Wafer Preparation 1844.9 Silicon Solar Cell Technology: Solar Cell Finishing 1874.10 Silicon Solar Cell Technology: Advanced Production Methods 1914.11 Thin Film Solar Cells: Amorphous Silicon 1924.12 Telluride/Selenide/Sulphide Thin-Film Solar Cells 1994.13 High-Efficiency Multijunction Solar Cells 2004.14 Concentrating Solar Systems 2034.15 Summary 204


5 Light Emitting Diodes 2155.1 Introduction 2165.2 LED Operation and Device Structures 2175.3 Emission Spectrum 220

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Contents ix

5.4 Non-Radiative Recombination 2215.5 Optical Outcoupling 2235.6 GaAs LEDs 2255.7 GaAs1−xPx LEDs 2265.8 Double Heterojunction AlxGa1−xAs LEDs 2285.9 AlGaInP LEDs 2345.10 Ga1−xInxN LEDs 2365.11 LED Structures for Enhanced Outcoupling and Power Output 2445.12 Summary 247


6 Organic Semiconductors, OLEDs and Solar Cells 2536.1 Introduction to Organic Electronics 2546.2 Conjugated Systems 2556.3 Polymer OLEDs 2606.4 Small-Molecule OLEDs 2666.5 Anode Materials 2706.6 Cathode Materials 2706.7 Hole Injection Layer 2716.8 Electron Injection Layer 2726.9 Hole Transport Layer 2726.10 Electron Transport Layer 2756.11 Light Emitting Material Processes 2766.12 Host Materials 2786.13 Fluorescent Dopants 2796.14 Phosphorescent Dopants 2836.15 Organic Solar Cells 2836.16 Organic Solar Cell Materials 2896.17 Summary 292


Appendix 1: Physical Constants 301Appendix 2: Properties of Semiconductor Materials 303Appendix 3: The Boltzmann Distribution Function 305Index 309

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Introduction

Semiconductor devices have revolutionized the way we work and live. Transistors arethought of as one of the most important developments of the twentieth century and theyhave given rise to the computer age as well as to compact, reliable electronics found ineverything from televisions to cell phones.

An even more fundamental semiconductor device exists, however. It is the semiconductordiode or p-n junction diode. Diodes had been developed before the transistor and wereused for rectification whereby alternating current can be converted to direct current byemploying the unidirectional property of diodes: current normally only flows efficientlyin one direction through a diode, and current flow is blocked in the opposite direction.This property of diodes is exploited in power supplies as well as in many other circuitssuch as those found in radios and limiters. Since an understanding of diodes is required toexplain the principles of transistors, diodes are frequently presented as a stepping stone tothe transistor.

In the twenty-first century, however, two new major industries are undergoing very rapiddevelopments based directly on the p-n junction diode. Photovoltaic (PV) solar cells andlight emitting diodes (LEDs) are both p-n junctions that are designed and optimized toeither absorb or emit light. In both cases, an energy conversion process between photonsand electrons occurs within a p-n junction.

The consequences of this development constitute a revolution in two major industrialsectors:

1. Energy production has relied on hydrocarbons and nuclear power, and although thesewill continue to be important, the direct conversion of solar radiation into useful power isthe key to a long-term, sustainable energy supply. Ninety-seven percent of all renewableenergy on earth is in the form of solar radiation. The beginning of the twenty-first centuryhas seen the abrupt growth of a global solar photovoltaic industry in conjunction with theinvolvement of governments worldwide, and unprecedented growth in PV productionand deployment is now underway. The worldwide consumption of silicon semiconductormaterial for the entire microelectronics industry is being overtaken by its use for solarcells alone.

2. Electric lighting was achieved by the incandescent lamp in the early twentieth century.In the second half of the twentieth century a significant movement to fluorescent lampsand discharge lamps in which a gas is excited into a plasma that radiates energy moreefficiently than a tungsten filament occurred. The twenty-first century will give riseto the virtual displacement of both incandescent lamps and fluorescent lamps by LEDlamps. In preparation for this revolution, the world’s major lighting companies have

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xii Introduction

acquired LED research, development and production capabilities. LEDs have alreadyachieved lower power consumption and longer life in small, rugged packages than eitherincandescent or fluorescent lamps. They also eliminate heavy metals such as mercuryused in fluorescent lamps.

The purpose of this book, therefore, is to introduce the physical concepts requiredfor a thorough understanding of p-n junctions starting with semiconductor fundamentalsand extending this to the practical implementation of semiconductors in both PV andLED devices. The treatment of a range of important semiconductor materials and devicestructures is also presented.

The book is aimed at senior undergraduate levels (years three and four). An introduc-tory background in quantum mechanics is assumed, together with general knowledge ofjunior mathematics, physics and chemistry; however, no background in electronic mate-rials is required. As such this book is designed to be relevant to all engineering studentswith an interest in semiconductor devices and not specifically to electrical or engineeringphysics/engineering science students only. This is intentional since solar cells and LEDsinvolve a wide range of engineering disciplines and should not be regarded as belonging toonly one branch of engineering.

In Chapter 1, the physics of solid state electronic materials is covered in detail startingfrom the basic behaviour of electrons in crystals. The quantitative treatment of electrons andholes in energy bands is presented along with the important concepts of excess carriers thatbecome significant once semiconductor devices are either connected to sources of poweror illuminated by light. A series of semiconductor materials and their important propertiesis also reviewed. The behaviour of semiconductor surfaces and trapping concepts are alsointroduced since they play an important role in solar cell and LED device performance.

In Chapter 2, the basic physics and important models of a p-n junction device arepresented. The approach taken is to present the diode as a semiconductor device that can beunderstood from the band theory covered in Chapter 1. Various types of diode behaviour,including tunnelling, metal-semiconductor contacts and heterojunctions, are presented aswell as reverse breakdown behaviour.

Chapter 3 introduces the theory of photon emission and absorption, a topic that bookson semiconductor devices frequently pay less attention to. The standard description thata photon is created when an electron and a hole recombine, or a photon is absorbedwhen an electron and a hole are generated, is not adequate for a deeper understanding ofphoton emission and absorption processes. In this chapter the physics of photon creationis explained with a minimum of mathematical complexity, and these concepts are muchbetter understood by following radiation theory and describing the oscillating dipole bothclassically and using simple quantum mechanics. A section of Chapter 3 describes theexciton relevant to inorganic semiconductors as well as the molecular exciton for organicsemiconductors. In addition lineshapes predicted for direct-gap semiconductors are derived.Finally the subject of photometric units introduces the concepts of luminance and colourcoordinates that are essential to a discussion of organic and inorganic light emitting diodes.

Chapter 4 covers inorganic solar cells. The concepts regarding the p-n junction introducedin Chapter 2 are further developed to include illumination of the p-n junction and the simplestpossible modelling is used to illustrate the behaviour of a solar cell. Then a more realisticsolar cell structure and model are presented along with the attendant surface recombination

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Introduction xiii

and absorption issues that must be understood in practical solar cells. A series of solar celltechnologies are reviewed starting with bulk single and multicrystalline silicon solar celltechnology. Amorphous silicon materials and device concepts are presented. Solar cellsmade using other semiconductors such as CdTe are introduced followed by multijunctionsolar cells using layered, lattice-matched III-V semiconductor stacks.

Chapter 5 on inorganic LEDs considers the basic LED structure and its operating prin-ciples. The measured lineshape of III-V LEDs is compared with the predictions of Chapter3. LEDs must be engineered to maximize radiative recombination, and energy loss mech-anisms are discussed. The series of developments that marked the evolution of current,high-efficiency LED devices is presented starting from the semiconductors and growthtechniques of the 1960s, and following trends in succeeding decades that brought better ma-terials and semiconductor growth methods to the LED industry. The double heterojunctionis introduced and the resulting energy well is analysed on the basis of the maximum currentdensity that can be accommodated before it becomes saturated. LED optical outcoupling,which must also be maximized to achieve overall efficiency, is modelled and strategies tooptimize outcoupling are discussed. Finally the concept of spectral down-conversion usingphosphor materials and the white LED are introduced.

Chapter 6 introduces new concepts required for an understanding of organic semicon-ductors in general, in which conjugated molecular bonding gives rise to π bands andHOMO and LUMO levels in organic semiconductors. The organic LED is introduced bystarting with the simplest single active layer polymer-based LED followed by successivelymore complex small-molecule LED structures. The roles of the various layers, includingelectrodes and carrier injection and transport layers, are discussed and the relevant candi-date molecular materials are described. Concepts from Chapter 3, including the molecularexciton and singlet and triplet states are used to explain efficiency limitations in the lightgeneration layer of small-molecule OLEDs. In addition the opportunity to use phosphores-cent host-guest light emitting layers to improve device efficiency is explained. The organicsolar cell is introduced and the concepts of exciton generation and exciton dissociation aredescribed in the context of the heterojunction and the bulk heterojunction. The interest inthe use of fullerenes and other related nanostructured materials is explained for the bulkheterojunction.

All the chapters are followed by problem sets that are designed to facilitate familiaritywith the concepts and a better understanding of the topics introduced in the chapter. Inmany cases the problems are quantitative and require calculations; however, a number ofmore conceptual problems are presented and are designed to give the reader experience inusing the Internet or library resources to look up further information. These problems areof particular relevance in Chapters 4, 5 and 6, in which developments in solar cells andLEDs are best understood by referring to the recent literature once the basic concepts areunderstood.

Adrian Kitai

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Acknowledgements

I would like to acknowledge the many people who helped with this book, including AyseTurak for her advice, students Huaxiang Shen, Bo Li and Alexander Subotich, McMasterUniversity staff Laura Honda, Ginny Riddell, Janet Delsey and Regina Bendig, Wileystaff Rebecca Stubbs, Emma Strickland, Amie Marshall, Mohan Tamilmaran, Robert Hine,Sarah Tilley and John Peacock and Project Manager Shalini Sharma, Production HeadKamal Kishore, Manish Gupta of Aptara. I owe a special debt of gratitude to my wifeTomoko for her steady encouragement, her patience and her considerable help in obtainingthe permissions for figures.

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1Semiconductor Physics

1.1 Introduction 21.2 The Band Theory of Solids 21.3 The Kronig–Penney Model 31.4 The Bragg Model 81.5 Effective Mass 81.6 Number of States in a Band 101.7 Band Filling 121.8 Fermi Energy and Holes 141.9 Carrier Concentration 151.10 Semiconductor Materials 251.11 Semiconductor Band Diagrams 261.12 Direct Gap and Indirect Gap Semiconductors 291.13 Extrinsic Semiconductors 351.14 Carrier Transport in Semiconductors 401.15 Equilibrium and Non-Equilibrium Dynamics 431.16 Carrier Diffusion and the Einstein Relation 451.17 Quasi-Fermi Energies 481.18 The Diffusion Equation 501.19 Traps and Carrier Lifetimes 531.20 Alloy Semiconductors 561.21 Summary 59Suggestions for Further Reading 61Problems 63

Principles of Solar Cells, LEDs and Diodes: The role of the PN junction, First Edition. Adrian Kitai.© 2011 John Wiley & Sons, Ltd. Published 2011 by John Wiley & Sons, Ltd.

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2 Principles of Solar Cells, LEDs and Diodes

Objectives

1. Understand semiconductor band theory and its relevance to semiconductor de-vices.

2. Obtain a qualitative understanding of how bands depend on semiconductor mate-rials.

3. Introduce the concept of the Fermi energy.4. Introduce the concept of the mobile hole in semiconductors.5. Derive the number of mobile electrons and holes in semiconductor bands.6. Obtain expressions for the conductivity of semiconductor material based on the

electron and hole concentrations and mobilities.7. Introduce the concepts of doped semiconductors and the resulting electrical char-

acteristics.8. Understand the concept of excess, non-equilibrium carriers generated by either

illumination or by current flow due to an external power supply.9. Introduce the physics of traps and carrier recombination and generation.

10. Introduce alloy semiconductors and the distinction between direct gap and indirectgap semiconductors.

1.1 Introduction

A fundamental understanding of electron behaviour in crystalline solids is available usingthe band theory of solids. This theory explains a number of fundamental attributes ofelectrons in solids including:

(i) concentrations of charge carriers in semiconductors;(ii) electrical conductivity in metals and semiconductors;

(iii) optical properties such as absorption and photoluminescence;(iv) properties associated with junctions and surfaces of semiconductors and metals.

The aim of this chapter is to present the theory of the band model, and then to exploitit to describe the important electronic properties of semiconductors. This is essential for aproper understanding of p-n junction devices, which constitute both the photovoltaic (PV)solar cell and the light-emitting diode (LED).

1.2 The Band Theory of Solids

There are several ways of explaining the existence of energy bands in crystalline solids.The simplest picture is to consider a single atom with its set of discrete energy levels forits electrons. The electrons occupy quantum states with quantum numbers n, l, m and sdenoting the energy level, orbital and spin state of the electrons. Now if a number N ofidentical atoms are brought together in very close proximity as in a crystal, there is somedegree of spatial overlap of the outer electron orbitals. This means that there is a chance that

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Semiconductor Physics 3

any pair of these outer electrons from adjacent atoms could trade places. The Pauli exclusionprinciple, however, requires that each electron occupy a unique energy state. Satisfying thePauli exclusion principle becomes an issue because electrons that trade places effectivelyoccupy new, spatially extended energy states. The two electrons apparently occupy thesame spatially extended energy state.

In fact, since outer electrons from all adjacent atoms may trade places, outer electronsfrom all the atoms may effectively trade places with each other and therefore a set ofoutermost electrons from the N atoms all appear to share a spatially extended energy statethat extends through the entire crystal. The Pauli exclusion principle can only be satisfiedif these electrons occupy a set of distinct, spatially extended energy states. This leads toa set of slightly different energy levels for the electrons that all originated from the sameatomic orbital. We say that the atomic orbital splits into an energy band containing a set ofelectron states having a set of closely spaced energy levels. Additional energy bands willexist if there is some degree of spatial overlap of the atomic electrons in lower-lying atomicorbitals. This results in a set of energy bands in the crystal. Electrons in the lowest-lyingatomic orbitals will remain virtually unaltered since there is virtually no spatial overlap ofthese electrons in the crystal.

The picture we have presented is conceptually a very useful one and it suggests thatelectrical conductivity may arise in a crystal due to the formation of spatially extendedelectron states. It does not directly allow us to quantify and understand important details ofthe behaviour of these electrons, however.

We need to understand the behaviour in a solid of the electrons that move about in thematerial. These mobile charge carriers are crucially important in terms of the electricalproperties of devices. An electron inside an infinitely large vacuum chamber is a freeelectron, but a mobile electron in a solid behaves very differently.

We can obtain a more detailed model as follows. The mobile electrons in a crystallinesemiconductor are influenced by the electric potential in the material. This potential has aspatial periodicity on an atomic scale due to the crystal structure. For example, positivelycharged atomic sites provide potential valleys to a mobile electron and negatively chargedatomic sites provide potential peaks or barriers. In addition, the semiconductor is finite inits spatial dimensions and there will be additional potential barriers or potential changes atthe boundaries of the semiconductor material.

The quantitative description of these spatially extended electrons requires the use ofwavefunctions that include their spatial distribution as well as their energy and momentum.These wavefunctions may be obtained by solving Schrodinger’s equation. The followingsection presents a very useful band theory of crystalline solids and the results.

1.3 The Kronig–Penney Model

The Kronig–Penney model is able to explain the essential features of band theory.First, consider an electron that can travel within a one-dimensional periodic potential

V(x). The periodic potential can be considered as a series of regions having zero potentialenergy separated by potential energy barriers of height V0, as shown in Figure 1.1, forminga simple periodic potential with period a + b. We associate a + b also with the latticeconstant of the crystal. Note that the electric potential in a real crystal does not exhibit the

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V (x)

V0

–b 0 a xa + b

Figure 1.1 Simple one-dimensional potential V(x) used in the Kronig–Penney model

idealized shape of this periodic potential; however, the result turns out to be relevant inany case, and Schrodinger’s equation is much easier to solve starting from the potential ofFigure 1.1.

In order to obtain the electron wavefunctions relevant to an electron in the crystallinesolid, V(x) is substituted into the time-independent form of Schrodinger’s equation:

− �2

2m

d2ψ(x)

dx2+ V (x) = Eψ(x) (1.1)

where V(x) is the potential energy and E is total energy.For 0 ≤ x ≤ a we have V = 0 and the general solution to Equation 1.1 yields:

ψ(x) = AeiKx + Be−iKx (1.2a)

where

E = �2 K 2

2m(1.2b)

For −b ≤ x ≤ 0 we have

ψ(x) = CeQx + De−Qx (1.3a)

where

V0 − E = �2 Q2

2m(1.3b)

Boundary conditions must be satisfied such that ψ(x) and dψ(x)dx are continuous functions.

At x = 0, equating (1.2a) and (1.3a), we have

A + B = C + D (1.4a)

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and equating derivatives of (1.2a) and (1.3a),

i K (A − B) = Q(C − D) (1.4b)

An important additional constraint on the required wavefunctions results from the peri-odicity of the lattice. The solution to Equation 1.1 for any periodic potential must also havethe form of a Bloch function:

ψ(x) = uk(x)eikx (1.5)

Here, k is the wavenumber of a plane wave. There are no restrictions on this wavenumber;however, uk(x) must be a periodic function with the same periodicity as the lattice.

Consider two x-values separated by one lattice constant, namely x = −b and x = a. Now,Equation 1.5 states that ψ (x + a + b) = ψ(x)eik(a+b). At x = −b this may be written as:

ψ(a) = ψ(−b)eik(a+b) (1.6)

The boundary conditions to satisfy ψ(x) and dψ(x)dx being continuous functions at x = a

may now be written by substituting ψ from Equations 1.2 and 1.3 into Equation 1.6:

Aei K a + Be−i K a = (Ce−Qb + DeQb)eik(a+b) (1.7a)

and substituting the corresponding derivatives:

i K (Aei K a − Be−i K a) = Q(Ce−Qb + DeQb)eik(a+b) (1.7b)

Equations 1.4a, 1.4b, 1.7a and 1.7b constitute four equations with four unknowns A, B,C and D. A solution exists only if the determinant of the coefficients of A, B, C and D iszero (Cramer’s rule). This requires that

Q2 − K 2

2QKsinh QB sin Ka + cosh Qb cos Ka = cos k(a + b) (1.7c)

This may be simplified if the limit b → 0 and V0 → ∞ is taken such that bV0 is constant(see Problem 1.1). We now define

P = Q2ba

2

Since Q � K and Qb � 1 we obtain

cos ka = Psin Ka

Ka+ cos Ka (1.8)

Here k is the wavevector of the electron describing its momentum p = �k and

K = 1

�

√2m E (1.9)

which means that K is a term associated with the electron’s energy.Now, Equation 1.8 only has solutions if the righthand side of Equation 1.8 is between

−1 and +1, which restricts the possible values of Ka. The righthand side is plotted as afunction of Ka in Figure 1.2.

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iii ii i i ii iii

Ka

KaP

sinKacosKa

Solutions

+

existSolutions

+1

–1

–3π –2π –π π 2π 3π

exist

0

Figure 1.2 Graph of righthand side of Equation 1.8 as a function of P for P = 2

Since K and E are related by Equation 1.9, these allowed ranges of Ka actually describeenergy bands (allowed ranges of E) separated by energy gaps (forbidden ranges of E). Kamay be re-plotted on an energy axis, which is related to the Ka axis by the square rootrelationship of Equation 1.9. It is convenient to view E on a vertical axis as a variabledependent on k. Note that k = nπ

a for integer values of n at the edges of each energy bandwhere the left side of Equation 1.8 is equal to ±1. These critical values of k occur at theboundaries of what are called Brillouin zones. A sketch of E versus k is shown in Figure1.3, which clearly shows the energy bands and energy gaps.

Let us now plot the free electron graph for E versus k. Solving Equation 1.1 for a freeelectron with V = 0 yields the solution

ψ(x) = Aeikx + Be−ikx

3πa

0

E

–2πa

–2πa

3πk

aπa

–πa

Figure 1.3 Plot of E versus k showing how k varies within each energy band and the existenceof energy bands and energy gaps. The vertical lines at k = n π

a are Brillouin zone boundaries

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a b

c d

E

Eg2

Eg

Eg1

3πa

0–2πa

–2πa

3πa

πa

–πa

k

Figure 1.4 Plot of E versus k comparing the result of the Kronig–Penney model to the freeelectron parabolic result

where

E = �2k2

2m(1.10)

This parabolic E versus k relationship is plotted superimposed on the curves from Figure1.3. The result is shown in Figure 1.4.

Taking the limit P → 0, and combining Equations 1.8 and 1.9, we obtain:

E = �2k2

2m

which is identical to Equation 1.10. This means that the dependence of E on k in Figure1.4approaches a parabola as expected if the amplitude of the periodic potential is reducedto zero. In fact, the relationship between the parabola and the Kronig–Penney model isevident if we look at the solutions to Equation 1.4 within the shaded regions in Figure1.4 and regard them as portions of the parabola that have been broken up by energy gapsand distorted in shape. For a weak periodic potential (small P) the solutions to Equation1.4 would more closely resemble the parabola. We refer to Equation 1.10 as a dispersionrelation – it relates energy to the wavenumber of a particle.

At this point, we can draw some very useful conclusions based on the following result:The size of the energy gaps increases as the periodic potential increases in amplitude in acrystalline solid. Periodic potentials are larger in amplitude for crystalline semiconductorsthat have small atoms since there are then fewer atomically bound electrons to screenthe point charges of the nuclei of the atoms. In addition, periodic potentials increase inamplitude for compound semiconductors as the ionic character of the crystal bondingincreases. This will be illustrated in Section 1.10 for some real semiconductors.

To extend our understanding of energy bands we now need to turn to another picture ofelectron behaviour in a crystal.

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1.4 The Bragg Model

Since electrons behave like waves, they will exhibit the behaviour of waves that undergoreflections. Notice that in a crystal with lattice constant a, the Brillouin zone boundariesoccur at

k = nπ

a= 2π

λ

which may be rearranged to obtain

2a = nλ

The well-known Bragg condition relevant to waves that undergo strong reflections whenincident on a crystal with lattice constant a is

2a sin θ = nλ

Now, if the electron is treated as a wave incident at θ = 90◦ then we have

2a = nλ

which is precisely the case at Brillouin zone boundaries. We therefore make the followingobservation: Brillouin zone boundaries occur when the electron wavelength satisfies therequirement for strong reflections from crystal lattice planes according to the Bragg con-dition. The free electron parabola in Figure 1.4 is similar to the Kronig–Penney model inthe shaded regions well away from Brillouin zone boundaries; however, as we approachBrillouin zone boundaries, strong deviations take place and energy gaps are observed.

There is therefore a fundamental connection between the Bragg condition and the forma-tion of energy gaps. The electrons that satisfy the Bragg condition actually exist as standingwaves since reflections will occur equally for electrons travelling in both directions of thex axis, and standing waves do not travel. Provided electrons have wavelengths not closeto the Bragg condition, they interact relatively weakly with the crystal lattice and behavemore like free electrons.

The E versus k dependence immediately above and below any particular energy gap iscontained in four shaded regions in Figure 1.4. For example, the relevant shaded regionsfor Eg2 in Figure 1.4 are labelled a, b, c and d. These four regions are redrawn in Figure 1.5.Energy gap Eg2 occurs at k = ± 2π

a . Since this is a standing wave condition with bothelectron velocity and electron momentum p = �k equal to zero, Eg2 is redrawn at k = 0 inFigure 1.5. Since we are only interested in relative energies, the origin of the energy axisis moved for convenience, and we can arbitrarily redefine the origin of the energy axis.Figure 1.5 is known as a reduced zone scheme.

1.5 Effective Mass

We now introduce the concept of effective mass m* to allow us to quantify electronbehaviour. Effective mass changes in a peculiar fashion near Brillouin zone boundaries,and generally is not the same as the free electron mass m. It is easy to understand that theeffective acceleration of an electron in a crystal due to an applied electric field will depend

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E

Eg2

πa

–πa

0

k

a b

c d

Figure 1.5 Plot of E versus k in reduced zone scheme taken from regions a, b, c and d inFigure 1.4

strongly on the nature of the reflections of electron waves off crystal planes. Rather thantrying to calculate the specific reflections for each electron, we instead modify the massof the electron to account for its observed willingness to accelerate in the presence of anapplied force.

To calculate m* we start with the free electron relationship

E = 1

2mv2

g

where vg is the group velocity of the electron. Upon differentiation with respect to k,

dE

dk= mvg

dvg

dk(1.11)

Since p = �k = mvg we can write

vg

k= dvg

dk= �

m(1.12)

Combining Equations 1.11 and 1.12 we obtain

dE

dk= vg�

or

vg = 1

�

dE

dk(1.13)

Note that the group velocity falls to zero at the Brillouin zone boundaries where the slopeof the E versus k graph is zero. This is consistent with the case of a standing wave.

Now, using Newton’s law,

F = dp

dt= �

dk

dt(1.14)

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From Equations 1.13 and 1.14, we can write

dvg

dt= 1

�

d2 E

dkdt= 1

�

d2 E

dk2

dk

dt= F

�2

d2 E

dk2(1.15)

If we assign m* to represent an effective electron mass, then Newton’s law tells us that

dvg

dt= F

m∗

Upon examination Equation 1.15 actually expresses Newton’s law provided we define

m* = �2

d2 Edk2

(1.16)

Since d2 Edk2 is the curvature of the plot in Figure 1.5, it is interesting to note that m* will

be negative for certain values of k. This may be understood physically: if an electron thatis close to the Bragg condition is accelerated slightly by an applied force it may thenmove even closer to the Bragg condition, reflect more strongly off the lattice planes, andeffectively accelerate in the direction opposite to the applied force.

We can apply Equation 1.16 to the free electron case where E = �2k2

2m and we immediatelysee that m* = m as expected. In addition at the bottom or top of energy bands illustrated inFigure 1.5, the shape of the band may be approximated as parabolic for small values of k andhence a constant effective mass is often sufficient to describe electron behaviour for smallvalues of k. This will be useful when we calculate the number of electrons in an energy band.

1.6 Number of States in a Band

The curves in Figure 1.5 are misleading in that electron states in real crystals are discreteand only a finite number of states exist within each energy band. This means that the curvesshould be regarded as closely spaced dots that represent quantum states. We can determinethe number of states in a band by considering a semiconductor crystal of length L andmodelling the crystal as an infinite-walled potential box of length L with a potential of zeroinside the well. See Example 1.1.

Example 1.1

An electron is inside a potential box of length L with infinite walls and zero potentialin the box. The box is shown below.

V = 0

V = ∞ V = ∞

0 L x

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(a) Find the allowed energy levels in the box.(b) Find the wavefunctions of these electrons.

Solution

(a) Inside the box, from Schrodinger’s equation, we can substitute V (x) = 0 and weobtain

− �2

2m

d2ψ(x)

dx2= Eψ(x)

Solutions are of the form

ψ(x) = A expi√

2m E

�x + B exp

−i√

2m E

�x

In regions where V = ∞ the wavefunction is zero. In order to avoid discontinu-ities in the wavefunction we satisfy boundary conditions at x = 0 and at x = Land require that ψ (0) = 0 and ψ (L) = 0. These boundary conditions can bewritten

0 = A + B or B = −A

and

0 = A expi√

2m E

�L + B exp

−i√

2m E

�L

= A

(exp

i√

2m E

�L − exp

−i√

2m E

�L

)= Csin

√2m E

�L

where C is a constant. Now sinθ is zero provided θ = nπ where n is an integerand hence

√2m E

�L = nπ

A discrete set of allowed energy values is obtained by solving for E to obtain

En = n2π2�

2

2mL2

(b) The corresponding wavefunctions may be found by substituting the allowedenergy values into Schrodinger’s equation and solving:

− �2

2m

d2ψ(x)

dx2= n2π2

�2

2mL2ψ(x)

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now

d2ψ(x)

dx2= −n2π2

L2ψ(x)

and hence

ψn(x) = A sin(nπ

Lx)

From Example 1.1 we obtain

ψn(x) = A sinnπ

Lx (1.17)

where n is a quantum number, and

k = nπ

L, n = 1, 2, 3 . . .

As n increases we will inevitably reach the k value corresponding to the Brillouin zoneboundary from the band model

k = π

a

This will occur whennπ

L= π

a

and therefore n = La . The maximum possible value of n now becomes the macroscopic

length of the semiconductor crystal divided by the unit cell dimension, which is simply thenumber of unit cells in the crystal, which we shall call N. Since electrons have an additionalquantum number s (spin quantum number) that may be either 1

2 or − 12 , the maximum

number of electrons that can occupy an energy band becomes

n = 2N

Although we have considered a one-dimensional model, the results can readily be ex-tended into two or three dimensions and we still obtain the same result. See Problem 1.3.

We are now ready to determine the actual number of electrons in a band, which willallow us to understand electrical conductivity in semiconductor materials.

1.7 Band Filling

The existence of 2N electron states in a band does not determine the actual number ofelectrons in the band. At low temperatures, the electrons will occupy the lowest allowedenergy levels, and in a semiconductor like silicon, which has 14 electrons per atom, severallow-lying energy bands will be filled. In addition, the highest occupied energy band willbe full, and then the next energy band will be empty. This occurs because silicon has aneven number of valence electrons per unit cell, and when there are N unit cells, there will

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