Introduction to electronic and ionic materials 1999 gao sammes

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An Introduction to Electronic and

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World Scientific Publishing Co Pte Ltd P O Box 128, Farrer Road, Singapore 912805

USA office: Suite 1B, 1060 Main Street, River Edge, NJ 07661

UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Gao, Wei, Ph D

An introduction to electronic and ionic materials / Wei Gao, Nigel M Sammes

p cm

Includes bibliographical references and index ISBN 9810234732 (alk paper)

1 Electronics Materials 2 Ionic crystals I Sammes, Nigel M IL Title

TK7871.G36 1999

621.381 dc21 99-17898 CIP British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library First published 1999

Reprinted 2000

Copyright © 1999 by World Scientific Publishing Co Pte Ltd

All rights reserved This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented; without written permission from the Publisher

For photocopying of material in this volume, please pay a copying fee through the Copyright

Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA In this case permission to

photocopy is not required from the publisher

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Wei Gao

To my wife, Mary Anne

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Engineers in many areas, such as materials science and engineering, and electrical and electronic engineering, have felt frustration due to the lack of suitable texts on electronic and ionic conductivity This led us to write just such a book Most of the textbooks on the market cover the chemistry or the physics of electronic or ionic behaviour, usually titled “Solid State Physics or Chemistry”, and put great emphasis on the theoretical aspects of the solid state On the other hand, current materials science and engineering texts do not cover the area of electronic and ionic materials in enough detail, leaving the student to wonder how important these materials really are As the area of materials science expands, so do the topics that need to be taught in the field It is, therefore, very important that students feel that the topic under study is relevant, and not just a rehash of other topics in chemistry or physics The application of electronic and ionic materials has definitely taken a driving seat in today’s world This is shown by, for example, the need for particular materials in the electronics industry, the energy industry, and many other applications which are too numerous to be mentioned here This subject, therefore, now plays an increasingly important part in the curricula of most undergraduate engineering studies, including materials science and engineering and electrical and electronic engineering

This book is particularly aimed at undergraduate level as the text as- sumes a very limited knowledge of chemistry or physics (up to, perhaps, first-year level) It aims to convey a basic understanding of a wide range of electronic and ionic materials important to today’s world, and it emphasises the properties and applications of such materials A question commonly

asked is: “How much mathematics will I have to know?” Some mathematics

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is, of course, necessary in any science and engineering degrees However, we have tried to keep the mathematics needed to describe particular pro- cesses to a minimum We have also used mathematics as a descriptive tool and have considered all necessary formulae from first principles This will enable the student to understand the originating points

Chapter 1 gives a brief overview of materials science and engineering Chapters 2 to 4 introduce some basic theories covering the principles of elec- trical conductivity, electron energy levels, band structure and work func- tion The remainder of the book is divided into properties of particular materials and their applications Chapter 5 discusses semiconductor prop- erty, materials and device applications Chapters 6 to 9 discuss magnetic, dielectric, optical and thermal properties and materials, while Chapter 10 covers the relatively new area of superconductivity and superconductors Chapter 11 is devoted to the new topic (from an applications rather than a theoretical viewpoint) of ionic conductivity, while Chapter 12 discusses

mixed conductivity Chapter 13 discusses the techniques available for mea- suring ionic and mixed ionic/electronic conductivity The book concludes

with a case study describing the solid oxide fuel cell, and brings together much of the material covered in the previous chapters Throughout the book, we have attempted to maximise the number of examples where the particular property is used in today’s world We have also concluded most chapters with a case study covering at least one example of that particular property in more detail

In conclusion, we trust that the book will be entirely suitable for ma- terials science and engineering, and electrical and electronic engineering students at an undergraduate level It is anticipated that first-year post- graduates, who are new to the topic, may also find the subject matter of interest to them

We would like to thank the following individuals for their invaluable contributions Wei Gao would like to thank Professors R Sharp and J

Chen for their valued support and encouragement He also thank Mike Hodgson for proofreading the manuscript and Charlie Gao for scanning

and editing the figures and tables Nigel Sammes would like to thank

Dr H Nafe, who proofread the text, and M Keppeler, whose invaluable

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he spent his sabbatical year The authors appreciate the help from many

colleagues in the Department of Chemical and Materials Engineering in the University of Auckland and the Department of Technology in the University of Waikato

Finally, we would like to thank our family and friends, without whose

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Acknowledgements

The authors would like to thank the following for permission to reproduce

figures in the book:

1

The American Ceramic Society, Ohio, USA: Figs 11.12, 11.13, 13.6

American Chemical Society, Washington, USA: Figs 11.33, 11.34 SONATA WH 10 11 12 13 14 15 16

Addison-Wesley Longman, Essex, UK: Fig 6.27; Table 8.2

American Institute of Physics, MD, USA: Figs 6.1, 7.6

ASM International, Ohio, USA: Tables 3.1, 6.4

Butterworths Publishers, UK: Figs 5.16, 7.10, 7.12, 8.9 Business Books Ltd., London, UK: Table 7.4

Chapman and Hall, London, UK: Figs 6.31, 7.3, 8.1, 8.3, 8.4, 8.5, 8.6,

8.7, 8.8, 8.10, 8.11, 8.12, 8.13, 8.15, 8.19, 9.1, 9.2, 9.3, 9.5, 9.6, 11.17, 11.23(a), 11.24, 11.29, 11.32, 11.37(a), 14.2

CRC Press, Florida, USA: Table 8.3

The Electrochemical Society Inc., Pennington, USA: Figs 11.21,

11.23(b), 11.37(a), 14.2

Elsevier Science Publishers, Amsterdam, The Netherlands: Figs 11.18, 11.19, 11.20, 11.35, 11.36, 11.37(b), 11.39, 14.4

Commission of the European Communities, Brussels, Belgium: Fig 14.6 IBC Tech Services Ltd., UK: Figs 10.2, 10.9

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17 18 19 20 21 22 23 24, 25 26 27 28 Macmillan Ltd., London, UK: Figs 5.4, 5.10, 5.18, 6.30; Tables 2.1, 2.2, 2.3, 2.5, 5.4, 6.2, 8.3 Marcel Dekker Inc., New York, USA: Figs 9.7, 9.8, 9.9; Tables 7.1, 7.2, 7.3, 9.3, 9.10

Materials Research Society, PA, USA: Fig 6.28

McGraw-Hill Co Inc., New York, USA: Figs 5.27, 6.7, 6.8, 6.16, 6.19, 6.26

National Academy Press, Washington, USA: Fig 8.14

Pergamon Press Ltd., Oxford, UK: Fig 2.2

Plenum Publishing Corporation, New York, USA: Table 10.1 Proc Roy Soc of London, London, UK: Fig 2.7

Springer Verlag, Heidelberg, Germany: Figs 10.7, 11.10

Westinghouse Corporation, PA, USA: Fig 14.8

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Preface vii Acknowledgements xi Symbols and Units xix Constants xxiii 1 Introduction 1 1.1 Engineering Materials 1 1.2 Classification and Properties of Materials 2 1.3 Materials and Electrical/Electronic Engineering 5 1.4 Nature and Purpose of this Book 6 2 Classical Theory of Electrical Conduction and Conducting Materials 8 2.1 Resistivity and Temperature Coefficient of Resistivity (TCR), Matthiessen’s Rule 8 2.2 Traditional Classification of Metals, Insulators and Semiconductors 10 2.3 Drude’s Free Electron Theory 12

2.4 The Hall Effect 14

2.5 The Wiedemann-Franz Law 17

2.6 Resistivity of Alloys, Nordheim’s Rule 19 2.7 Resistivity of Alloys and Multiphase Solids 23

2.8 Materials for Electricity Transmission 26

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2.9 2.10 2.11 2.12

Materials for Electrical Resistors and Heating Elements

Case Study — Materials Selection for Electrical Contacts Summary Important Concepts 3 Electron Energy in Solids 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

Introduction and Schrodinger’s Equation Quanta and Waves

Atomic Energy Levels — The Electronic Structure of Atoms

The Lowest Energy Principle, Fermi Level and Fermi Distribution

Energy Band in Solids and Forbidden Energy Gaps The Zone Model and Energy Well

Band Structures and Electrical Conductivity Summary Important Concepts 4 Electron Emission 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Work Function Photoemission Photocathode Materials

Thermal Electron Emission and Emitters

Secondary Electron Emission

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5.0 5.6 5.7 5.8 5.9 5.10 9.11 9.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19

Conductivity in Extrinsic Semiconductors, Temperature Effect and Mass Action Law

Lattice Defects in Semiconductors

The p-n Junction and Rectifier Doping Processes

The Metal-Semiconductor Contacts Some Simple Semiconductor Devices Integrating Microelectronic Circuits

Crystal Growth and Wafer Production

Physical Vapour Deposition (PVD), Chemical Vapour Deposition (CVD) Processes and Metallisation

Lithography and Etching

Oxidation of Silicon Semiconductors Packaging of Semiconductor Devices Future Development of Semiconductors Summary Important Concepts 6 Magnetic Phenomena and Magnetic Materials 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16

Introduction and Brief History Magnetic Properties and Units

The Origin of Magnetism, Bohr Magneton

Types of Magnetism

Effect of Temperature on Ferromagnetism — Curie Point

Ferromagnetic Anisotropy and Magnetostriction Ferromagnetic Domains and Domain Movement Magnetic Energy abd Domain Structure — Types of Energy that Determine the Structure of Domains

Magnetisation and Demagnetisation of Ferromagnetic Materials, Hysteresis Loop

Energy Losses in Magnetic Materials

Soft Magnetic Materials Hard Magnetic Materials

Ferrites — Ceramic Magnetic Materials

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7 Dielectric Materials 7.1, 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9

Introduction — Dielectric Properties

Dipoles and Polarisation

Frequency, Temperature Dependence of €, and

Energy Loss

Application of Dielectric Materials Case Study — Capacitors

Ferroelectricity, Piezoelectricity, Electrostriction and Pyroelectricity Case Study — Materials for Transducers Summary Important Concepts 8 Optical Properties and Materials 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 Introduction — The Reaction between Radiation and Materials

Emission of Continuous and Characteristic Radiation

Applications of Photon Emission

Laser and Laser Materials

Case Study — Compact Disc (CD) System

Thermal Photoemission

Interaction of Photons with Materials

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10 Superconductivity and Superconductors 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8

Introduction and Brief History

Superconducting Properties and Measurements BCS Theory of Superconductivity Conventional (Low T,) Superconductors High T, Superconductors Application of Superconducting Materials Summary Important Concepts 11 Ionic Properties of Materials 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 11.11 11.12 11.13 11.14 11.15 11.16 11.17 11.18 11.19 11.20 11.21 Point Defects Line defects Planar Defects Diffusion Mechanisms Diffusion as a Random Walk Vacancy Mechanism Interstitial Mechanism Interstitialcy Mechanism Other Mechanisms Conductivity

Introduction to Solid Electrolytes

Oxygen Ion Conductors Alkali Ion Conductors

Halide Ion Conductors Ag-Ion Conductors Beta-AlaOa Proton Conductors lonic Glasses Case Study — Potentiometric Solid State Sensors Summary Important Concepts 12 Mixed Conductivity, 12.1 12.2 12.3 Introduction

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12.4 12.5 12.6 12.7 12.8 12.9

Electrolytic and Ionic Domains

Effect of Dopants on the Defect Equilibria

Ionic Transport Number

Three-Dimensional Representation

Summary

Important Concepts

13 Techniques for Studying the Conductivity and Transport Behaviour in Ionic and Mixed Ionic/Electronic Materials 13.1 13.2 13.3 13.4 13.5 13.6 DC Conductivity Measurements AC Techniques Current Interruption Technique Other Techniques Techniques for Measuring Partial (Mixed) Conductivity Summary 14 Case Study — The Solid Oxide Fuel Cell 14.1 14.2 14.3 14.4 14.5 Introduction

Characteristics of the SOFC

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Length Area Mass Density Time Temperature Kelvin (K) Force Stress Strain Young’s modulus Energy /work/quantity of heat Kinetic energy Energy loss Power Electric charge Current flow Current density Potential difference Electric field Electric resistance Electric resistivity Electric conductivity

Temperature coefficient of resistivity Electron drift velocity

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Hall constant Lorenz number

Fermi energy level

Band gap energy Work function Frequency Wavelength

Wave number

Bragg diffraction angle

Intrinsic carrier concentration Electron concentration Hole concentration Reverse saturation current Diffusion coefficient Activation energy eV/atom Magnetic field Magnetic induction (flux density) Magnetic permeability Relative permeability Magnetisation Magnetic susceptibility Curie temperature Magnetostriction Eddy current loss Saturation induction Remanent induction Coercive force Maximum energy product Capacitance Permittivity

Relative permittivity (dielectric constant) Tangent of loss angle

Relative loss factor

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Polarisation

Concentration of charge centres

Power loss of a dielectric material Light intensity Relaxation time Linear absorption coefficient of light Index of refraction Reflectivity

Incident and refracted angles

Heat capacity at constant volume Heat capacity at constant volume

Linear coefficient of thermal expansion Thermal conductivity

Thermoelectric (Seebeck) power

Superconducting transition temperature Critical superconducting current

Critical superconducting current density Critical magnetic field Persistent current Burgers vector Chemical potential Concentration of ionic defects Impedance

Real part of impedance Imaginary part of impedance

Distribution parameter in Nyquist plots

Ionic transport number

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An Introduction to Electronic and

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Introduction

1.1 Engineering Materials

Materials are substances from which something is composed or made from

Let us consider engineering materials used in building our material world:

building construction, roads, bridges, irrigation systems, pipelines, ma- chines, transportation equipment, electricity systems, tools, furniture, com- munication facilities, instrumentation, and various utilities and appliances

both at home and in the office

Materials are central to the growth, prosperity, security, and quality of human life Throughout history, the development of human civilisation has been closely tied to materials which have been produced and used in society The levels of involvement have been designated according to the materials used In early human civilisation, people used materials existing in nature such as stone, wood and clay This period became known as the “Stone Age” In time, techniques to produce materials having properties superior

to those occurring naturally were discovered With the Industrial Revolu-

tion, modern heavy industries were based largely on iron, steel and other

metals, and this became known as the “Metal Age” Only recently have sci- entists begun to study the relationships between the composition, structure and properties of materials, and the knowledge and understanding gained have enabled us to design and create the numerous materials necessary to

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Material scientists and engineers now have a rapidly evolving ability to tailor materials from the atomic scale upwards to obtain desired properties

A new age in materials, known as the “Tailored (or Designed) Materials Age”, has been used to describe the revolutionary changes in materials science and engineering (MSE), as well as their impact on society For

example, advanced composites have been developed to combine the prop- erties of high stiffness, strength, toughness and low density to meet special structural requirements Surface treatments, including the development of various coatings and surface modification techniques, provide a combina- tion of extreme hardness, wear, corrosion and high temperature oxidation

resistance on the top surface, combined with a tough, shock-absorbing body Artificial layered structures offer limitless possibilities for creating new elec-

tronic and semiconductor devices which can be produced using many meth- ods, including molecular beam epitaxy (MBE), chemical vapour deposition

(CVD), vacuum evaporation, sputter deposition, ion beam deposition, and

solid-phase epitaxy We now face exciting and dramatic changes in the materials world, giving our industries and society endless developmental opportunities

1.2 Classification and Properties of Materials

1.2.1 Classification of Materials

Engineering materials are classified using different methods The tradi- tional method is to classify then according to their nature:

(i) Metals and alloys are inorganic materials composed of one or more

metallic elements They may also contain a small number of non- metallic elements Metals usually have a crystalline structure and are

good thermal and electrical conductors Many metals are strong and

ductile at room temperature and maintain good strength at high and low temperatures

(ii) Ceramics are inorganic materials consisting of both metallic and non-

metallic elements bonded together chemically Ceramics can be crys- talline, non-crystalline, or a mixture of both Generally, they have

high melting points and high chemical stabilities They also have high

hardness and high temperature strength, but tend to be brittle Ce-

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(iii) Polymers are organic materials which consist of long molecular chains

or networks containing carbon Most polymers are non-crystalline,

but some consist of mixtures of both crystalline and non-crystalline re-

gions They typically have low densities and are mechanically flexible

Their mechanical properties may vary considerably Most polymers are poor electric conductors due to the nature of the atomic bonding

(iv) Composites are mixtures of two or more types of materials Usually, they consist of a matrix phase and a reinforcing phase They are

designed to ensure a combination of the best properties of each of the

component materials

There is also an increasing trend to classify engineering materials into two further categories: structural materials and functional materials Struc- tural materials, as the name indicates, are materials used to build struc-

tures, bodies and components For instance, in a car the body, frame,

wheels, seats, inside lining, engine and various mechanical transmission

parts are all constructed from structural materials The most important

consideration for this type of application are the mechanical properties The functional materials, on the other hand, are used for special pur- poses in equipment such as conductors, insulators or the storage of elec- tricity, the generation or conduction of light, the conversion of optical, mechanical or thermal signals into electrical voltages, or the provision of

a strong magnetic field The electronic devices in the control systems of

a car, for instance, are built with semiconductors, an important type of functional material

1.2.2 Properties of Materials

The properties of engineering materials can be classified into two main

groups: physical and chemical Dependent on the application, the physical

properties of materials can be further grouped into two categories, which correspond either to structural or functional materials

(i) Mechanical properties include Young’s modulus, tensile and shear strengths, hardness, toughness, ductility, deformation and fracture be- haviours, fatigue and creep strengths, wear resistance, etc

(ii) Physical properties include electrical and electronic properties, mag-

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The chemical properties of engineering materials generally include cor-

rosion, oxidation, catalysis properties and chemical stabilities

This text does not concentrate on the mechanical properties of materi-

als, but on the physical properties, such as electrical and electronic prop- erties, magnetic properties, and optical and thermal properties It must be remembered that mechanical properties are always important in the appli-

cations of functional materials This text emphasises the performance and applications of functional materials in modern industries

1.2.3 Four Elements of Materials Science and Engineering

Materials science is primarily concerned with the study of the basic know]l-

edge of materials: the relationships between the composition/structure, properties and processing of materials Materials engineering is mainly

concerned with the use of this fundamental knowledge to design and to produce materials with properties that will meet the requirements of soci- ety As subjects of study, materials science and materials engineering are

very often closely related The subject “materials science and engineer- ing” combines both a basic knowledge and application, and forms a bridge

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There are four essential elements in materials science and engineering: (i) processing/synthesis; (ii) structure/composition; (iii) properties; and (iv) performance/application (Fig 1.1) There is a growing realisation among scientists and engineers that in order to develop new materials and

provide materials efficiently for society, all four elements need to be con- sidered This gives materials science and engineering its interdisciplinary

nature Nowadays, it is common (and indeed preferred in many cases) for

people with different backgrounds (materials, physics, chemistry, metal-

lurgy, ceramics, electronics, etc.) to work together to solve materials prob-

lems and to make important contributions to this field

1.3 Materials and Electrical/Electronic Engineering

Functional materials play a very important role in almost all industries They are often the critical component of equipment or instrumentation determining the overall performance and efficiency of the whole system

The topic of functional materials has experienced rapid growth in the last

thirty years as it is among the most active discipline in modern science and engineering New developments in functional materials, such as semi- conductors, superconductors and optical fibres, are revolutionising modern society Without these materials, modern devices, computers, automatic machines, telecommunications systems, aircraft, etc., could not exist For example, in the automobile industry, a standard 1994 model car consisted

of electronics worth more than US$800 This is more than the value of steel used in its body, frame, engine and transmission system, which cost approximately US$675

The electronics industry is one of the most dynamic industries in the

global economy The worldwide electronic materials market, valued at

US$2.5 billion, was responsible for an equipment market valued at more

than US$400 billion in 1985 The electronics industry is also a leader in the

use of new materials Highly engineered materials are vital to the contin- ued progress of the electronics industry Strong interrelationships between

device design, materials science and engineering, and process chemistry

determine the performance of a device Semiconductors are the basic ma- terials in the electronics industry; most current devices are based on a

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process Oxygen levels for gathering impurities are controlled by com-

puter Epitaxial growth is widely used to form devices on Si wafers, and it

is also controlled by computer The processing steps, which include mask- ing, photolithography, etching, diffusion, ion implantation, metallisation and oxidation, largely determine the performance and quality of the de-

vices The development of materials and processing in the semiconductor

industry allows us to produce integrated circuits with a billion components contained in each chip, thus furthering the revolution in information tech- nology which has reshaped our society

The telecommunications industry is another example where functional materials play an extremely important role The shift from electronic to optical technology required the development of many new optical materials

Optical fibres have to be very transparent to transmit light signals over a

long distance The development of new process technologies has resulted in silica optical fibres with transmission losses of 100 orders of magnitude lower (x10—199) than ordinary optical glasses, and which approaches the theoretical minimum Materials based on new systems, including fluorides,

are being studied in an effort to further reduce the optical losses However, new light sources and detectors will be needed as transmission frequencies

make inroads into the infrared region

1.4 Nature and Purpose of this Book

Engineers in all disciplines should have some basic and applied knowledge

of materials in order to optimise their understanding and effectiveness This textbook provides an introductory course for electrical engineering and other engineering-technology students Its emphasis is on the applica-

tion of materials in electric, electronic and telecommunication industries A basic understanding of the relationships between processing, structure,

properties and performance provides the tools and, therefore, the main purposes are as follows:

(i) To appreciate the importance of materials in modern technology

(ii) To understand the basic principles of electronic properties of materials (iii) To familiarise the reader with the various groups of materials used in

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With the above goals in mind, we will first focus on an understand-

ing of a few basic principles of physical/electronic properties of materials These principles will be used to discuss a wide range of properties, includ- ing electrical and electronic properties, magnetic properties, and optical and thermal properties After examining such properties, the typical mate- rials found in this group and brief processing techniques will be introduced The applications of these materials will follow and will be emphasised and enhanced with case studies, where appropriate It is hoped that this text will provide engineering students with an understanding of the basic con-

cepts and some working knowledge of materials, and that it will be used as a general reference for reviewing electronic materials in the future

This book has also been designed as a reference textbook at the in-

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Classical Theory of Electrical Conduction and

Conducting Materials

In this chapter, the free electron conduction theory is described This description is then used to explain the conduction properties of materials Finally, materials which are used for electrical conduction in electrical and electronic industries are introduced

In classical electron conduction theory, an electron is treated as a very

small particle with certain mass and electric charge:

Electron mass me = 9.1 x 1073! kg Electron charge e = —1.6 x 10-9 C

Because electrons behave like particles in this theory, they obey Newton’s Laws of motion In Sections 2.3 to 2.5, we will apply this theory to describe the electron conduction behaviour in conductors

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or

R= pl/A, (2.2)

where | is the length, A is the area of the cross-section of the conductor

and p is called electrical resistivity Equation (2.2) indicates that p is the

resistance of a material in unit length and unit cross-section area 2.1.2 Matthiessen’s Rule and TCR

For pure metals, resistivity p is the sum of two items: a residual part, p,, and a thermal part, p; (see Fig 2.1) This is called Matthiessen’s rule:

P(total) = Pr + Pt (2.3)

p = pr(l + pe/ Pr) (2.4)

pt/pr = f(T) (2.5)

ø = ø[\ + ƒŒ)] (2.6)

For most metals and alloys, p is approximately proportional to temperature

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30 7 26 20 E S S lñt x a 10 5 foe 0 i 1 i 1 l l ) —273 ~200 —100 0 100 200 300 400 500 Temperature, °C

Fig.2.2 The effect of temperature on the resistivity of selected metals (After Zwikker, Physical Properties of Solid Materials, Pergamon, 1954, p 247)

2.2 Traditional Classification of Metals, Insulators and Semiconductors

Table 2.1 lists the resistivity and TCR for various materials Materials can be classified into three groups according to their electrical conduction prop- erties: conductors, insulators and semiconductors The general conduction properties of materials can be described as follows:

(i) Compared with other physical properties of materials (e.g., density

or Young’s modulus), electrical resistivity varies over a much greater

range (~ 1074) Metals and alloys have resistivity over the range

of 107° to 107® Q-m Insulators have resistivity over the range of 108-16 ( m The resistivities of semiconductors are around 107? Q-m (ii) For pure metals (99.9%), p = 10-® to 10-7 O-m A higher purity

(99.999%) only makes a small difference

(iii) TCR for various pure metals are similar in the range of ~ 0.004/K, regardless of their resistivity

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(v) Non-metals have much higher resistivity Purity has little effect on the

resistivity

(vi) Typical semiconductors have a high and negative TCR (see Fig 2.3)

2.2.1 Resistance (R) Measurement Methods

For a conductor, resistance is easy to measure using standard “four-point”

methods (see Fig 13.2) Resistance can be calculated directly from the

drop in voltage However, it is difficult to measure the resistance of an

Table 2.1 Resistivities and TCRs at 293K Material p (Q m) TCR(%/K) Silver 1.6x 1078 +0.41 Copper 1.7 x 1078 +0.43 Aluminium 2.7x 1078 +0.43 Sodium 5.0 x 1078 +0.4 Tungsten 5.7x 10-8 +0.45 Iron 97x10 8 +0.5 Platinum 10.5x 10-8 +0.39 Tantalum 13.5x 10-8 +0.38 Manganin (87Cul3Mn) 38x 107-8 +0.001 Constantan (57CU43Ni) 49x10-8 +0.002 Nichrome (80Ni20Cr) 112x108 +0.0085 SiC, commercial 1-2x10-& -0.15 Graphite, commercial c 1x 1075 —0.07 InAs, very pure 3 x 1073 —1.7 Tellurium, very pure 4x 1073 —2

Germanium, diode grade 1x10-% +40.4 Germanium, very pure 5x10"! -4

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Resistance > yo oe —_ ee Temperature —>

Fig 2.3 Typical “metallic” and “non-metallic” conduction behaviours

insulator accurately using the above methods This is because a very high

voltage is needed The measurements will be strongly influenced by a num- ber of factors, including contacting points, surface finishing, environmental

conditions (moisture in air) and the defects in the material

2.3 Drude’s Free Electron Theory

2.3.1 Drude’s Free Electron Theory

P K Drude developed a theory of electron conduction in 1900 In his

theory, electrons are taken to be particles which move through the metal lattice freely obeying Newton’s laws of motion and Maxwell-Boltzmann

statistics When an electric field E is applied to a metal, the force acting on an electron is

F=-eE (2.8)

According to Newton’s law,

a = F/m = —eE/m = 1.75 x 101!1E, (2.9)

where a is the acceleration, m is the mass of an electron, m = 9.1x10~3! kg,

and e = 1.6 x 107!9 C In time 7, the electron will obtain a velocity of vg: Ud =arT (2.10) vq is called drift velocity From (2.9) and (2.10),

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This can be rewritten as

vq = —pE (2.12)

h = er/m, (2.12a) where yp is called the mobility of the electron and can be defined as the drift velocity in the unit electrical field