Introduction to the electronic properties of materials 2nd edition

3 CONDUCTION ELECTRONS IN MATERIALS 3.2 Electrical properties and the classical free electron model 453.3 Thermal properties and the classical free electron model 48 4.4 Distribution of

Introduction to the Electronic

Properties of Materials

Introduction to the Electronic Properties

Department of Electrical and Computer Engineering

Iowa State University

Original illustrations © Nelson Thornes Ltd 1994, 2001

The right of David Jiles to be identified as author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording

or any information storage and retrieval system, without permission in writing from the publisher or under licence from the Copyright Licensing Agency Limited, of

90 Tottenham Court Road, London WIT 4LP.

Any person who commits any unauthorised act in relation to this publication may be liable to criminal prosecution and civil claims for damages.

First published in 1994 by:

Chapman & Hall

Second edition published in 2001 by:

Transferred to Digital Printing 2010

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

ISBN 0 7487 6042 3

Page make-up by Aarontype Limited, Easton, Bristol

Every effort has been made to contact copyright holders of any material

reproduced within the text and the authors and publishers apologise if

any have been overlooked.

The cover illustration represents the three-dimensional Fermi surface

of copper.

Publisher's Note

The publisher has gone to great lengths to ensure the quality of this reprint

but points out that some imperfections in the original may be apparent.

IN MATERIALS

1.1 Relationships between macroscopic properties of materials 3

3 CONDUCTION ELECTRONS IN MATERIALS

3.2 Electrical properties and the classical free electron model 453.3 Thermal properties and the classical free electron model 48

4.4 Distribution of electrons among allowed energy levels 714.5 Material properties predicted by the quantum

5.2 Solution of the wave equation in a one-dimensional

5.3 The origin of energy bands in solids: the tight-binding

PART Two: PROPERTIES OF MATERIALS

7 ELECTRONIC PROPERTIES OF SEMICONDUCTORS 130

8 ELECTRICAL AND THERMAL PROPERTIES OF MATERIALS 161

8.2 Quantum-mechanical description of conduction

8.4 Other effects caused by electric fields, magnetic fields

9.2 Interpretation of optical properties in terms of

9.3 Band structure determination from optical spectra 193

10.3 Microscopic classification of magnetic materials 206

PART THREE: APPLICATIONS OF ELECTRONIC MATERIALS

11 MICROELECTRONICS - SEMICONDUCTOR TECHNOLOGY 227

11.1 Use of materials for specific electronic functions 227

12 OPTOELECTRONICS - SOLID-STATE OPTICAL DEVICES 254

13 SUPERCONDUCTIVITY AND SUPERCONDUCTING MATERIALS 280

14.2 Magnetic recording heads and the recording process 316

15 ELECTRONIC MATERIALS FOR TRANSDUCERS:

PREFACE TO THE FIRST EDITION

The subject of electronics, and in particular the electronic properties of materials,

is one which has experienced unprecedented growth in the last thirty years Thediscovery of the transistor and the subsequent development of integrated circuitshas enabled us to manipulate and control the electronic properties of materials tosuch an extent that the entire telecommunications and computer industries aredependent on the electronic properties of a few semiconducting materials Thesubject area is now so important that no modern physics, materials science orelectrical engineering degree programme can be considered complete without asignificant lecture course in electronic materials Ultimately the course require-ments of these three groups of students may be quite different, but at the initialstages of the discussion of electronic properties of materials, the course require-ments are broadly identical for each of these groups Furthermore, as the subjectcontinues to grow in importance, the initial teaching of this vital subject needs tooccur earlier in the curriculum in order to give the students sufficient time later

to cover the increasing amount of material

It is with these objectives in mind that the present book has been written It isaimed at undergraduates who have only an introductory knowledge of quantummechanics The simplified approach used here enables the subject to be introducedearlier in the curriculum The goal at each stage has been to present the principles

of the behaviour of electrons in materials and to develop a basic understandingwith a minimum of technical detail This has resulted in a discussion in breadthrather than depth, which touches all of the key issues and which provides a securefoundation for further development in more specialized courses at a later stage.The presentation here should be of interest to two groups of students: those whohave a primary interest in electronic materials and who need an introductory text

as a stepping-stone to more advanced texts; and those whose primary interest lieselsewhere but who would nevertheless benefit from a broad, passing knowledge ofthe subject

As with the earlier textbook, Introduction to Magnetism and Magnetic Materials

(1991) the subject area under discussion here is truly multidisciplinary, spanningthe traditional subject areas of physics, electrical engineering and materialsscience In writing this book I have striven to keep this in mind in order tomaintain the interest of a wider audience Therefore some of the treatment willseem relatively easy for one group of students while relatively hard for another.Over the entire book, however, I think that the general mix of subject areas leads

to a text that is equally difficult for these three groups of students Chapters 1-5could easily be included in a traditional solid-state physics course and should bevery familiar to physicists However Chapters 6-10 will appeal more to materialsscientists since they will be more familiar with dealing with meso- and macro-scopic properties Finally Chapters 11-15 discuss the functional performance ofthese materials in technological applications which are likely to be of most interest

to electrical engineers These chapters provide a rapid introduction to five tant applications of electronic materials, each of which could be further developed

impor-in a separate advanced course Also, as impor-in Introduction to Magnetism and Magnetic Materials, the early chapters of this book contain a number of key exercises for the

student to attempt Completed worked solutions are given at the back of the book

It has been my experience that this is much more useful than simply giving anumerical answer at the back, since if you do not get the problem exactly rightunder those conditions, you cannot easily find out where you went wrong!

On completion of the text the reader should have gained an understanding ofthe behaviour of electrons within materials, an appreciation of how the electronsdetermine the magnetic, thermal, optical and electrical properties of materials and

an awareness of how these electronic properties are controlled for use in a number

of important technological applications I hope that the text will provide a usefulintroduction to more detailed books on the subject and that it will also provide thebackground for developing the interest of students in this fascinating subject at anearly stage in their careers

Finally, I would like to acknowledge the assistance of several friends andcolleagues who have helped me in writing this book In particular thanks go to

M F Berard, F J Friedlaender, R D Greenough, R L Gunshor, J Mallinson,

R W McCallum, R E Newnham, S B Palmer and A H Silver

DJ, Ames, Iowa

ACKNOWLEDGEMENTS

I am grateful to those publishers credited in captions for permission to reproducesome of the figures in this book

PREFACE TO THE SECOND EDITION

Electronic materials provide the basis for many of our 'high tech' industries such ascomputers, semiconductors, data storage, electronic devices, sensors and actuators

In particular, the range of available materials and their technological applications,have made enormous progress in the seven years since the first edition of this bookwas published So it is timely to bring the book up to date I have chosen tomaintain the same basic layout as in the first edition The early chapters contain thebasic concepts and are, in places, rather abstract and mathematical The laterchapters describe applications and are more descriptive and practical In this way,

I have deliberately sought to maintain a blend and balance between a need forbasic understanding of ideas and a description of how these are incorporatedinto applications

The underlying physics and physical descriptions of these materials change onlyslowly with time, so that the information contained in the early chapters of thefirst edition of the book has remained as relevant to today as it was seven yearsago Therefore in Chapters 1-10, I have chosen simply to expand the number ofexercises with complete worked solutions to offer a wider range of examples thatwill deepen knowledge of the underlying physical basis for understanding thesematerials These examples have been used over the last few years by students atIowa State University to consolidate concepts presented in lecture classes In a fewinstances additional information on topics such as anharmonicity and Gruneisenparameters, Bragg reflection at Brillouin zone boundaries and more detaileddescriptions of charge carriers in the conduction and valence bands at semi-conductor junctions have been added to increase the scope of the chapters where itseemed appropriate

The technological applications in the areas of semiconductor materials anddevices, computer technology, data storage and magnetics have all seen remark-able progress since the first edition Therefore in the later chapters of the book thatdeal with technological applications, Chapters 11-15, it was essential to providemore up-to-date information In microelectronics there has been an expansion ofthe number of materials that are available and now being used in technologicalapplications The continual reduction of device and feature sizes in microelec-tronics has allowed a dramatic increase in the number of components on a singlesemiconductor chip In optoelectronics, particularly the applications to data com-munication, there have been important developments both on the small scale (localcomputer communications) and the large scale (intercontinental telecommunica-tions) Once again, the introduction of new materials with improved performanceover previously available materials has been the enabling technology In somecases, optoelectronic materials with completely new characteristics (such as theoptical amplifier materials based on rare-earth doped silicon) have been intro-duced In superconductivity, after the surge of interest in the high-temperatureceramic superconducting materials prior to the first edition, the major advances

subsequently have been in applications, rather than in the identification of newmaterials So new sections on superconducting wires, superconducting energystorage devices, and superconducting transformers, motors and generators havebeen added In magnetic recording the rate of progress has accelerated since 1994

so that today, as a result of the availability of new multi-layered giant resistive materials, data storage densities are doubling every 9 months instead ofevery two years as they were in the early 1990s In 1994 a typical personal com-puter came with a hard disk drive that had 500 Mbytes of capacity In the year 2000

magneto-a typicmagneto-al computer cmagneto-ame with magneto-a hmagneto-ard disk drive thmagneto-at hmagneto-ad magneto-about 80 Gbytes of ory The availability of new magnetic materials has made these advances possible.Data storage technology has moved so far since the first edition that the wholechapter on magnetic recording had to be rewritten In the area of transducers,sensors and actuators new ferroelectric thin film materials have been developed,and applications have been found in dynamic random access memory (DRAM) andnonvolatile ferroelectric random access memory (FRAM) and microelectronicmechanical machines (MEMs)

mem-In selecting material for this new edition I have attempted to focus on theadvances in major technological areas Clearly, in order to keep any book on such

an important and diverse subject as electronic materials to a reasonable size, manyinteresting areas must necessarily be left to others However, I believe that theinformation contained in this second edition will provide a sound introduction tothis subject The essential concepts that allow understanding of these materials isincluded, together with a description of the most important materials and theirtechnological applications

DJ, Ames, Iowa

ACKNOWLEDGEMENTS

I would like to take this opportunity to thank various friends and colleagueswho have helped to make this second edition possible, either through helpful dis-cussions, indicating corrections to the original book or through suggestions fortopics to include I particularly would like to thank M J Sablik, J E Snyder,

F J Friedlaender, H Hauser, D L Atherton, R D Greenough, F Salas and

D P Cann

GLOSSARY OF SYMBOLS

a Lattice spacing

a Mean field constant

Optical attenuation coefficient

B Magnetic induction (magnetic flux density)

B R Rémanent magnetic induction

BS Saturation magnetic induction

C Capacitance

Curie constant

Specific heat or heat capacity

Ce Electronic specific heat

C1 Lattice specific heat

Cv Specific heat capacity at constant volume

Cp Specific heat capacity at constant pressure

c Velocity of light

X Magnetic susceptibility

XP Pauli paramagnetic susceptibility

D Electric displacement (electric flux density)

D(UJ) Vibrational density of states

Phonon density of states

D(E) Density of available energy states

Distance

6 Optical penetration depth

EF Fermi energy

£ Electric field strength

EB Elastic (bulk) modulus

Binding energy

EY Elastic (Young's) modulus

ES Elastic (shear) modulus

E Energy and electric field

EH Magnetic field energy (Zeeman energy)

f Hail Hall field

e Permittivity (dielectric constant)

e\ Real component of dielectric constant (polarization)

62 Imaginary component of dielectric constant (absorption)

£Q Permittivity of free space

Fint Internal, or interactive, force

Fapp Applied force

g Transducer generation coefficient

Spectroscopic splitting factor

Lande splitting factor

Rate of generation of charge carriers

Heff Effective magnetic field

I Magnetic polarization (intensity of magnetization)

/ Intensity of light

Electric current

JQ Thermal current density

/ Electric current density

Atomic angular momentum

/ Total atomic angular momentum quantum number

Exchange constant

; Total electronic angular momentum quantum number

/atom Exchange integral for an electron on an atom with electrons on several

nearest-neighbours

/ex Exchange integral; exchange interaction between two electrons

GLOSSARY OF SYMBOLS

K Anisotropy constant

Thermal conductivity

k Optical extinction coefficient

Interatomic force constant

Coupling coefficient of transducer

Wave vector

Xul First anisotropy constant for uniaxial system

K u2 Second anisotropy constant for uniaxial system

KI First anisotropy constant for cubic system

K 2 Second anisotropy constant for cubic system

Length

Electronic orbit length

Macroscopic length of lattice chain

Length of side of cubic specimen

/o Unstrained length

/ Orbital angular momentum quantum number

&(x) Langevin function of x

m e Orbital magnetic quantum number

mo Orbital magnetic moment of electron

MR Rémanent magnetization

MO Saturation magnetization

(spontaneous magnetization at 0 K)

Ms Spontaneous magnetization within a domain

m s Spin magnetic moment of electron

m tot Total magnetic moment of atom

m* Effective mass of electrons in bands

Mobility of charge carriers

JL¿B Bohr magneton

IJLQ Permeability of free space

N(E) Density of occupied energy states (= 2D(E)f(E)) 9 electron population

density

N Number of atoms per unit volume

Number of electrons per unit volume

Number of turns on coil or solenoid

Principal quantum number

Number of atoms

No(E) Total number of energy states between zero energy and energy E

N* Effective number of conduction electrons

v Frequency (u;/27r)

uj Angular frequency (2-Kv)

P(E) Probability of occupancy of state with energy £

P(x) Probability of electron being at location x

Angular momentum operator

PO Orbital angular momentum of electron

Ps Spin angular momentum of electron

Ptot Total angular momentum of electron

Spin wave function

\I> Total wave function

•0 Electron wave function

GLOSSARY OF SYMBOLS

r Interatomic separation

Radius

Radius of ionic cores of atoms in lattice

Electronic orbit radius

Rate of recombination of charge carriers

Charge carrier lifetime

rmax Maximum torque

U Internal energy

u Unit vector

u Displacement of an atom from equilibrium

v t Final velocity (terminal velocity) of electrons in Drude model

y Distance along y-axis

Atomic number

£ Number of nearest-neighbour atoms

Si UNITS, SYMBOLS AND DIMENSIONS

m

kg

SHzNPaJWCAVOOmS

TAm'1

Am-1

H m-1

Namemetrekilogramsecondhertznewtonpascaljoulewattcoulombampèrevolt

ohm

ohm mètresiemenssiemens/metrefarad

coulomb mètre"2volt mètre"1coulomb mètre"2farad/mètrehenryweber

teslaampere/metreampere metre"1henry/metre

MKSABase units

mkgss-1kgm$-2

kg m"1 s"2

AsA

kgm2A~1s-3kgm2A-2s-3kgm3A-2s-3kg~1m-2A2s3

A2s3kg-1m-3

A2s4kg-1m-2

Asm"2kgmA^s"3

Dimensions

LMTT-1MLT~2ML-!T-2

ML2T~2

ML2T~3

CTC

ML2C-JT-3

ML2C-2T~3

ML3C-2T-3M-1L-2C2T3

M-1L-2C2T4

CL-2TMLC-1!-3CL"2T

M~ 1 L~ 3 C 2 J 4

ML2C-2T-2

ML2C-1T-2

MC-!T-2CL-1CL-1MLC-2T-2

VALUES OF SELECTED PHYSICAL CONSTANTS

Avogadro's number

Boltzmann's constant

Gas constant

Planck's constant

Velocity of light in empty space

Permittivity of empty space

Permeability of empty space

Atomic mass unit

Properties of electrons

Electronic charge

Electronic rest mass

Charge to mass ratio

MB = 9.274 x 10-24 Am2 (^JT'1)

= 1.165 x lO-^JmA-1

/XN =5.051 x 10-27Am2(=JT-1)

O0 = 2.067 x 10-15 Wb (=Vs)

FOREWORD FOR THE STUDENT

The objective of this book is to present an introduction to the electronic properties

of materials that is broad in its coverage but not exhaustive The book focuses

on the understanding of a few basic principles of the behaviour of electrons inmaterials and uses them to provide a description of a wide range of phenomenaincluding magnetic, electrical, thermal and optical properties of materials I havealso given a number of historical references in the text, particularly in the earlychapters It seems to me that an appreciation of the historical development of asubject helps the overall understanding, apart from which it is interesting to knowwho originally developed the underlying ideas and even to re-read some of theselandmark papers

It has been my experience that, with the possible exception of the prospectivespecialist in solid-state physics, the majority of students do not benefit greatlyfrom being confronted with a mass of detailed results arising from the theory ofelectrons in solids This can come later for the intending specialist In introducingthis subject it seems more useful to present a few key results based on relativelysimple models, which give a general feel for the behaviour of electrons in materialsand how they contribute to the observed properties These models themselvesneed not be particulary complex to be useful For example, the basic premises ofboth the classical Drude model and the Sommerfeld model are quite far fromreality Yet the predictions that they make about the properties of the materialcontain some of the essential known results, for example the Wiedemann-Franzlaw and the electronic contribution to the heat capacity

Therefore, the general approach taken here has been to introduce and discussthe consequences of such simple models which can be used to guide our thinking

We begin on the level of a few electrons subjected to an electrostatic potential due

to the rest of the material Subsequently the bulk properties of materials are sidered and the phenomena are related to the earlier discussion of the behaviour

con-of electrons Finally several key applications are discussed, in which the tronic properties of materials play the central role in determining the suitability

elec-of materials for these applications In particular the areas elec-of microelectronics,optoelectronics, superconductivity, magnetism and piezoelectricity are examined

PART ONE

FUNDAMENTALS

OF ELECTRONS

IN MATERIALS

1 PROPERTIES OF A MATERIAL CONTINUUM

OBJECTIVE

The objective of this chapter is simply to remind ourselves of the macroscopic properties of materials and to point out that in uses of electronic materials we are mostly interested in these bulk properties, which are the ones that we usually measure In order to measure these properties it is necessary that we also give exact definitions of the various quantities The microscopic properties are of interest because they help us to explain the variation of the macroscopic prop- erties with external conditions, including any interrelationships which exist between the macroscopic properties Once we have achieved an understanding

of the relationship between macroscopic properties and the microscopic ture of a material it becomes possible to control the structure in order to produce materials with specific desired properties.

struc-1.1 RELATIONSHIPS BETWEEN MACROSCOPIC PROPERTIES

no indication what the underlying common mechanism might be

The unexplained relationships between the macroscopic properties of materialsform the starting point for our investigation of the electronic properties of mate-rials It seems that there must be some common underlying mechanism that isresponsible for all three properties, optical, electrical and thermal, and that thiscauses the close relationship between them In fact the correlation between thebehaviour of the various properties of the materials can not be explained withoutsome understanding of the structure of the materials and this involves thedevelopment of microscopic theories of the atomic and electronic structure insidethe materials The relationship between the structure of matter and its physical

properties has been treated in detail using only classical physics to describe thematerials in the excellent work of Landau et al [1].

We will begin with some simple definitions of macroscopic properties and thenconsider some of the well-known macroscopic laws obeyed by materials Our goalwill then be to provide a conceptual framework for understanding these propertiesand relationships

1.1.1 Measurable properties of materials

How do we characterize materials in terms of measurable quantities?

In order to measure the properties of a material we do not need to know anythingabout its internal structure The properties of interest depend, of course, on theapplication under consideration, but broadly we are usually interested in one ormore of the following categories: mechanical, electrical, optical, thermal andmagnetic properties In most cases the materials properties are obtained as a result

of measurements of two quantities, which by themselves do not representmaterials properties

Often, a measurement is made of the response of a material, in terms of a stateparameter (e.g strain, change in temperature or current density), to the influence

of an external effect or field parameter (e.g stress, amount of heat input or electricfield strength) The quotient of these two measurements is then the materialproperty (e.g elastic modulus, specific heat capacity or electrical conductivity).Compilations of the various macroscopic properties of materials have been made

by many authors, of which the most comprehensive is that by Lide [2]

1.1.2 Bulk properties of materials

How can these macroscopic properties be explained?

These bulk, continuum properties of materials are almost exclusively what we arereally concerned with in using the materials, because these are the propertieswhich can be directly measured However, explanation of the behaviour of theseproperties as a function of external conditions, such as temperature, field orfrequency of incident electromagnetic radiation, for example, requires a deeperinsight into the underlying physical mechanisms

Although these properties are often documented in great detail for materials,the macroscopic continuum picture gives no explanation of why, for example,copper is a better conductor than glass; why iron is ferromagnetic but manganese

is not; why aluminium conducts heat better than sulphur and so on In order toexplain these properties of materials we must look inside the material and try

to develop a better understanding of what is happening These explanations arefounded on a description of microscopic rather than macroscopic effects

1.1.3 Dependence of properties on the environment

Are the material 'constants 9 really invariant when the external conditions change?

The macroscopic properties of materials, such as Young's modulus, thermalconductivity and electrical conductivity and magnetic permeability, do not remain

MECHANICAL PROPERTIESconstant, however The optical parameters k and n are dependent on the wave-

length of incident electromagnetic radiation, permeability is dependent ontemperature, and so is electrical conductivity

The elastic modulus of gadolinium for example shows unusual behaviour close

to 293 K The variation of the reflectivity of silver with energy of incidentelectromagnetic radiation reveals a drastic change at about 4 eV The specific heat

of nickel reveals anomalous behaviour at around 600 K, and the magneticsusceptibility of manganese fluoride MnF2, shows an anomaly in the vicinity of

70 K All of these show variations in bulk properties that lie beyond explanation

on the basis of the continuum theory of matter

These examples show interesting features in some of the bulk properties of thesematerials In order to explain these observations it is necessary to consider theproperties of the elementary constituents of these materials, that is the atoms andparticularly the electrons Before doing this, however, we will look briefly at a fewdefinitions These are used to quantify the material properties in which we will beinterested and which we will need to refer to throughout this book

1.2 MECHANICAL PROPERTIES

How do we quantify the mechanical behaviour of materials?

The mechanical properties broadly encompass the elastic, plastic and acousticproperties of a material These may be quantified by the following: the bulk modu-lus EB, Young's modulus £Y and the shear modulus Es- (We use these symbols toavoid possible confusion between the elastic moduli, particularly Young's moduluswhich is often given the symbol E, and the energy which we use extensively later,and which also takes the symbol £.)

1.2.1 Elastic moduli

How does a material respond to stress?

In a material that is isotropic the elastic properties can be completely specified interms of two elastic moduli, the longitudinal (or Young's) modulus and thetransverse (or shear) modulus Other elastic properties, such as Poisson's ratio, can

be completely defined in terms of a combination of these two moduli

Young's modulus, is a material property obtained from measurement oftwo quantities; the applied longitudinal stress a and the resulting strain e in the

same direction Since by Hooke's law stress is proportional to strain for smalldisplacements,

The following table gives values of the elastic Young's modulus for variousmaterials.

Table I.I Elastic moduli of various materials.

7-50 7-38 7-26 7-14 7-02

Temperature (K)

Figure I.I Variation of the elastic modulus of the metal gadolinium with temperature [3].

1.3 ELECTRICAL PROPERTIES

How do we quantify the electrical behaviour of materials?

In the case of the electrical properties, we are often concerned with the tivity This determines, for example, whether we are dealing with an electrical

conduc-ELECTRICAL PROPERTIESconductor or insulator In some cases we may be concerned with the electricalpolarization as determined by the dielectric constant or permittivity, and in otherswith the dissipation of electrical energy under ac conditions (eddy currents) Theelectrical properties of principal interest are: the electrical conductivity a and

the dielectric constant e.

jH—mfLftrtft

oKunonu

mica aluminium oxide phosphorus glass

germanium (doped, transistor grade) ferries

germanium (tunnel diode grade) stainless steel, nfchrome, bismuth — co

•'• permaloy, siMcon-Jcon ^rf nickel, tungsten, iron £

•jO" 8 -L silver, copper, aluminium

Figure 1.2 Range of resistivities for various materials, including metals, semiconductors and insulators.

e

§

•o

8

We use £ here to distinguish the electric field from energy which is denoted later

by E and <j is the electrical conductivity Alternatively, if the voltage across a

material is V V and the current passing is /A then,

where R is the resistance of the material Figure 1.2 shows a range of resistivities

for various materials

1.3.2 Electrical conductivity

How is electric charge transmitted in a material?

The electrical conductivity is the amount of electric charge transferred per unittime Aq/àt across unit cross-sectional area A under the action of unit potential

gradient dV/dx

(dq/ai)

Table 1.2 Numerical values of electrical resistivities and conductivities

of various materials at room temperature.

ELECTRICAL PROPERTIESand from Ohm's law

/ current density

& = - =£ —;electric held:— r t j (1.6)

The electrical conductivities of materials exhibit probably the widest range

of variations of all material properties: 23 orders of magnitude between the ductivities of copper and sulphur, as shown in Table 1.2 Macroscopic continuumtheory gives no reason for this variation

con-1.3.3 Dielectric properties

How does a nonconducting material respond to the presence of an external tric field?

elec-The dielectric constant or permittivity e is a material property which relates

the amount of electric polarization (charge displacement) P of a material under the

action of an electric field f

The term P/£Q£ is known as the electric susceptibility xe- Materials with high mittivity, and hence high electric susceptibility, give a large electric polarizationfor a given field strength Values of the relative permittivity £r can be as high as

per-7000 in barium titanate, but in most cases are much lower, for example therelative permittivity of water is 80 The relative permittivity of a material canalso be determined from the capacitance C of a condensor with the material asdielectric, compared with that of the same condensor CQ with a vacuum in place ofthe material:

P / P\

e = eo+- = eo[l+—- = ee r

even among materials which are nominally identical Therefore, it is not reliable toquote values for particular materials, although it is typically in the range of

106 Vm"1 for dielectrics

1.4 OPTICAL PROPERTIES

How do we quantify the optical behaviour of materials?

The optical properties of a material tell us how the material interacts with incidentelectromagnetic waves These properties can be expressed in terms of two opticalconstants Often, the refractive index n and the extinction coefficient k are used,

both of which change with the wavelength of the incident light Alternatively, wecan define the optical properties using the reflectance R together with one of the

above We can also use instead the real and imaginary components of the dielectricconstant e These five quantities are the principal optical properties of interest,

and all five change with the frequency of the incident electromagnetic waves

1.4.1 Refractive index and Snell's law

How does the speed of light in a material determine its change of direction at

an interface?

The refractive index of a material is the ratio of wavelength, or phase velocity, oflight in a vacuum to that in the material It is a material property which can beobtained, in principle, solely from the measurement of the speed of light in amaterial, although this is never attempted in practice:

speed of light in vacuumspeed of light in material*

The refractive index of a transparent material is usually determined on the basis

of the measurements of two angles. 6\ is the angle of incidence of a light beam at

the surface of the material and 0r is the angle of refraction of the light beam insidethe material:

o î « A

(1.10)

In fact, the refractive index is frequency dependent, which is why a prism can

be used to split white light into different colours (dispersion on the basis offrequency)

1.4.2 Extinction coefficient k and the Lambert-Beer law

How is light energy absorbed by a material?

The optical extinction coefficient k is defined as the fractional rate of decrease of

light intensity d//J in a material per unit path length multiplied by A/4?r where A isthe wavelength

'-fiifV 4?r/ \oxj <""

??? ?

??? ?

How do we quantify the amount of light reflected at an interface?

The optical reflectance R is the fraction of incident light that is reflected from a

surface The value of R is dependent on both the frequency of the light and the

angle of incidence

reflected intensity

K = incident intensityr-¡ : :—•

It is usually measured using normal incidence of light

The optical constants shown in Table 1.4 are valid at an energy of 1 eV orequivalently at a wavelength of 1240 x 10~9 m

Table 1.4 Optical properties of various materials at an energy of I eV

1 2.464

5.86 8.48 8.03 4.79 5.74

0.9697 0.722 0.976 0.992 0.678 0.753

The optical reflectances of metals and semiconductors have very characteristicfeatures Metals have high reflectance at long wavelengths but at shorter wave-lengths the reflectance declines On the other hand semiconductors have lowreflectance at long wavelengths but beyond a threshold wavelength known as theband edge or absorption edge, the reflectance increases rapidly as the wavelengthdecreases This frequency or energy dependence of optical properties is demon-strated in Figs 1.3 and 1.4 The continuum model gives no reasons for thischaracteristic behaviour of the materials

0.1 0.05 0.01

Energy (eV)

Figure 1.4 Optical absorption spectrum a(£) of gallium arsenide [5] Reproduced with permission

from M R Sturge, Phys Rev 1962.

1.4.4 The Hagen-Rubens law

Is there a relationship between the electrical and the optical properties of a metal?

The optical reflectivity and the electrical conductivity of metals at 'low' quencies (v < \ x lO^s"1) or long wavelengths (A > 3 //m) are also related by

fre-an equation of the form,

where CTO is the dc electrical conductivity and R is the reflectance This is known as

the Hagen-Rubens relation Therefore the mechanisms underlying conductivityand reflectivity seem to be related The prediction of the reflectance on the basis ofthe Hagen-Rubens law is shown in Fig 1.5

ce

oc

y ^o

How do we quantify the thermal properties of materials?

In the case of thermal properties we are often concerned with the rate of flow ofheat through the material as measured by the thermal conductivity K This

determines whether the material is a thermal conductor or insulator Anotherquantity of interest is the amount of heat which must be supplied to raise thetemperature of unit mass by one degree, that is the specific heat or heat capacity C

1.5.1 Thermal conductivity

How is the thermal conductivity defined?

The thermal conductivity K of a material is the rate of transfer of heat per unit

time, per unit cross sectional area, per unit distance, per unit temperature gradient

where A is the cross-sectional area through which the heat passes, Q is the

heat energy transferred in time t between two locations a distance x apart, where

T2 and T! are the temperatures at the two locations An alternative, but equivalent

definition is that K is the quotient of the thermal flux density JQ with respect to

the temperature gradient dT/dx This equation only applies under steady-state

conditions

Table /.5 Thermal conductivities of various materials. Material

Silver Copper Gold Aluminium Nickel Tungsten Zinc Iron Silicon Platinum Glass

K (Wm-'K -•) 428 398 315 237 158 182 115 80 83 73 0.2

1.5.2 The Wiedemann-Franz law

7s there a relationship between the electrical and thermal properties of a metal?

In most cases good electrical conductors are also good thermal conductors.Quantitative investigation by Wiedemann and Franz revealed that for most metalsthe relationship between electrical conductivity and thermal conductivity K

obeyed the following law

This seems to imply that the underlying mechanisms behind electrical andthermal conductivity are related in some way The continuum model offers noexplanations

1.5.3 Specific heat capacity

What determines the increase in temperature of a material when it is heated?

The specific heat Cm of a material is the amount of heat required to raise unit mass

of the substance by one degree of temperature, while the heat capacity C is theamount of heat required to raise the temperature of an unspecified mass by onedegree of temperature

Here, M is the mass, dT is the change in temperature, U is the internal energy

and dQ is the heat energy absorbed The specific heat is itself dependent on

c »4f «-' 7 >

THERMAL PROPERTIEStemperature It is also dependent on whether the measurement is made underconstant-volume or constant-pressure conditions.

The heat capacity of some materials varies in a very characteristic way Forexample, the temperature dependence of the heat capacity of iron which is shown

in Fig 1.6 has an anomaly at 1040 K As we shall see later this also corresponds to

a magnetic phase transition

Temperature (K)

Figure 1.6 Heat capacity of iron showing anomalous behaviour at about 1040 K [6] Reproduced

fromj Phys Chem Solids, 1, J A Hoffmann et ol., p 52, copyright 1956, with kind

permission from Elsevier Science.

1.5.4 The Dulong-Petit law

Is there a relationship between the heat capacities of various materials?

The heat capacities of many materials are found to be linearly dependent on themolecular or atomic weight of the substance, at least at higher temperatures Thiscan be expressed as the 'molar heat capacity' This is the heat capacity of a fixednumber (N0 = 6.02 x 1023) of atoms or molecules of a substance For manymaterials this has a value close to 25 J mol"1 K"1, a result discovered by Dulongand Petit

This seems to imply that the heat capacity is dependent only on the number ofelementary entities, either atoms or molecules depending on the material How-ever, even this law only applies at high temperatures, since the heat capacity varies