OJIeKTpOHLI 1I KpHCTaJIJILI

l'I3AaTeJILCTBO cHayHa», MOCKBB

Th. Wolkenstein

Electrons and Crystals

Translated from the Russianby Michael Burov

English Translation Editedby R. N. Hainsworth

MirPublishersMoscow

First published 1985Revised from the 1983 Russian edition

Ha aneAufJcnoM .R8b1Xe

@ HaAaTeJIbCTBO sHayaa». rJIaBSaH peAaKI\IIJlepll3UKo-MaTeMaTlItleCKOD nareparypsr, 1983

@ English translation, Mir Publishers, t985

Instead of Preface:On the Laws of Popular Science

Any scientist or scholar knows that writlng ascientific monograph is much easier than writ.inga popular science book on the same topic. It iseasier to present a paper in front of a group ofcolleagues than to give a popular science lectureto the public.

This is because scien tists speak their ownlanguage, in fact each branch of knowledge hasits own, Hence even experts in other fields ofscience will not understand them, the more sopeople without a scientific education.

A popular science writer must translate fromthe scientific language into everyday language.The popularization of science like the translationof verse into another language is an art in itselfrequiring special skills.

I would like to propose some basic rules (a'legal code') any popular science writer shouldadhere to.

First, a popular science writer has to be abso­lutely accurate in his presentation.

The second rule is that each new term or con­cept should be explained immediately, otherwisethe text will become unclear for the reader, andthe most terrible thing for a writer wi ll happen:the reader will put his book aside.

An author (be he a professional writer or ascientist) should always 'see' his reader as noone can write into a void. The reader is alwaysin front of him, and an author should be aware

6 Instead of Preface

of what the reader knows and what he does not"This defines the language the author uses to com­municate with the reader, and will define theboundary between what is clear and what isnot.

The whole book (except perhaps certain points)has to be written at the same level. This is thethird (quite obvious) rule of popular sciencewriting.

Many science writers think it necessary toinsert some fictional digressions. Metaphors, ana­logies, assoc iations are warranted and even desir­able, but fictional excursions that are not relat­ed to the subject-matter do not make a text clear,they only dilute it, like a glass of water poured.into a bowl of soup. ",:

There is no harm in repeating the material:several times in different parts of the book. Ofcourse, if the light is thrown differently on thematerial, like theatre floodlights can colour thesame stage differently, it is more deeply in­grained on the mind of the reader.

When a reader opens a book, he puts his handin the hand of the writer, and the writer leadshim through the thick forest called 'science'.So as not to lose his way, the reader always hasto remember, and the author always has to re­mind him, where he is going and the destinationof his journey. In other words, a reader has to keepbefore him the plan of the book, like a touristkeeps a map of his route in his hands. When hestarts reading a book, he begins with its 'Con­tents' which provides information about thetopics and the structure of the book. The authorshould lead his readers to their destination by

On the Laws of Popular Science 7

the shortest route possible and without zigzags.Let us mention one more rule that is pos­

sibly the most essential in this 'legal code' forpopular science. The hook has to be interesting.What is "interesting' and 'uninteresting' issubjective. Everything depends on the way youlook at things. A writer may make 'interesting'things seem boring, or 'uninteresting' things fas­cinating. The art of the popular science writeris to turn the 'uninteresting' into the 'interesting' .

The aim is not so much to supply the readerwith knowledge about a subject, but to kindlehis interest in it. If a reader, having read the book,reaches for another book on the same or a similartopic, then the popular science wrrter has ful­filled his task.

This book tackles some of the problems ofsolid-state physics. We are going to discuss thebehaviour of electrons in metals, semiconductors,and dielectrics (insulators), and some of the prop­erties of solids affected by this behaviour. Wedo not pretend this is an exhaustive review ofthe latest achievements in solid-state physics, butwe hope we have described some of the fundamen­tal concepts in the physics of metals, semiconduc­tors, and dielectrics. The book is a supplementto appropriate parts of a physics course in second­ary schools and is intended for someone who isfinishing secondary school or who has finished itand remembers his school physics.

We do not expect a reader to know anythingbeyond secondary school physics. Moreover, some­times a topic is tackled somewhat differentlythan it might be treated in school. The mathemat­ics used in the book does not go beyond ele-

8 Instead of Preface

mentary algebra and the mathematical analysisthat should be familiar to anyone finishingsecondary school.

\\7"e have attempted to present the materialin terms of models and have not departed, asfar as is possible, from classical concepts. How­ever, modern solid-state physics is based on quan­tum mechanics, which is what governs the worldof atoms and electrons. This makes it hard forus since we cannot count on the reader knowingvery much about quantum mechanics. Thereforesometimes we have to be content with present­ing the results of solid-state theory (taking careto make these results understandable) withoutshowing how they were obtained. The reader willjust have to believe them, but this is, alas, un­avoidable.

\i\Te do not undertake to judge how far thisbook meets our popular science 'code'. At anyrate, it was a guide to us when we wrote this book.

A well-known French physicist, and one ofthe founders of quantum mechanics, Louis deBroglie, once said that 'science is the daughterof astonishment and curiosity'. The mission ofa popular science writer is to reawaken in thereader his feelings of astonishment and curiosity.

Contents

Instead of Preface: On the Laws ofPopular Science 5

Introduction 111.1 ~ Which Electrons? 111.2. Which Crystals? 15-

Chapter 1. Electrons in Metals 231.1. 'Free' and 'Bound' Electrons 231.2~ 'Electron Gas' in :Vletals 30f .3. The Successes and Failures of the

Classical Theory of Metals 401.4. Electron Emission from Metals 471.5. Electrons in the Periodic Field. Con-

ductors and Insulators 55

Chapter 2. Electrons in Semiconductors 642.1. 'Order' and 'D isorder ' in Crystals 642.2. Free Electrons and Free Holes in Semi-

conductors 692.3. 'Energy Bands' and 'Localized Levels' 76-2.4. Semiconductor Conductivity 822.5. Electrons and Quanta 91

Chapter 3. Electrons on a SemiconductorSurface 99

3.'1. Semiconductor Surface Phenomena 993.2. Adsorption on a Semiconductor Surface 1043.3. The Role of Electrons and Holes in

Adsorption 110

iO Contents

3.4. Interaction of the Surface wi th theBulk 115

3.5. Chemical Reactions on a Semicon-ductor Surface 120

Chapter 4. Electrons in Dielectrics 1274.1. Dielectric Conductivity 1274.2. Dielectric Breakdown 1334.3. Crystal Colouration 1414.4. Crystal Lurn inescence 1474.5. Electrets 1544.6. Dielectric Constant 1584.7. Ferroelectrics and P iezoelectrics 166

Remark in Conclusion: Theory andExperiment 173

Introduction

1.1. ·Which Electrons?

Any matter, as is generally known, can occur asa solid, a liquid, or a gas. Modern physics has.added another to these three states, which may'be referred to as the 'classical' states, and thatis the plasma state. A material is a plazma when.all its atoms are ionized, i.e. all the electron.shells have been partially or completely strippedaway from the atoms. \Ve are not going to discussgases, liquids or plasma. We shall focus entirely-on solids.

Modern physics has two branches which arethe physics of atomic nuclei and elementary par­ticles on the one hand and solid-state physics onthe other. At present these two disciplines areindependent and almost do not meet. Sometimes.a physicist studying atomic nuclei knows veryIittle about solid-state physics, and vice versa.

Speaking of solids, we have to distinguishbetween two kinds, i.e, crystalline and amorphoussolids. An example of a crystalline solid is thecommon (or rock) salt we see every day at the.dirmer table. A typical example of an amorphous.sol id is glass. Perhaps one of the basic differencesbetween an amorphous solid and a crystalline-one is the absence of a distinct melting pointin amorphous solids. Instead, there is a soften­ing range in which the material gradually turns

12 Introduction

from solid to liquid. We are not going to discussamorphous solids in this book and shall onlyconsider crystalline solids.

Crystalline solids are much more common innature than it seems at first sight. A crystalline'solid may have a regular geometric shape withsome symmetry. Such materials are called mono­crystals. At the same time there are what we callpolycrystals which are aggregates of vast numbers.of tiny monocrystals sticking together. A grainof common salt is an example of a monocrystal ;while a piece of copper wire is an example of a·polycrystalline solid. As is often true, the tinymonocrystals of copper cannot be seen with the-naked eye. _

In physics, the word 'crystal' has a wi delmeaning than we usually give it. Commonly,what we call 'crystals' a physicist calls 'mo uo­crystals'. This book will basically deal with rll01l0-··

crystals, but when we cover polycrystals, weshall see that all the properties of monocrvstalsare retained. However, some new proport iesappear that are caused by the joints between sepa­rate tiny monocrystals. The joints can, for in­stance, increase the resistance to an electric cur­ren t flowing through the polycrystal.

From now on we shall use the word 'crystal"instead of 'monocrystal'.

Any crystal consists of individual particles',(they may all be the same or there may be severaldifferent kinds) arranged in a regularly repr-at.ingpattern. A crystal is like a honeycomb of cel ls inan indefinite sequence. Each of these unit (orstructural) cells has the same configuration of the­particles the material consists of. The assembly

1.1. Which Electrons? 13

of unit cells is what is called a crystal lattice.Each unit cell has faces which intersect to formedges. The intersections of the edges are calledlattice points. Consequently, a crystal is likea house built of uniform bricks. This strict spa­tial regularity of a crystal's structure is its char­acteristic feature, and can moreover be conside­red to be a definition of a crystal. A crystal is asymbol of order, while its opposite, a symbol ofchaos, is a gas for its particles rush about, runinto each other, and change direction after eachcollision.

Crystals are classified as molecular, atomic,or ionic ones, according to the particles compris­ing them. In molecular crystals the 'structuralelements of the crystal are individual molecules,each of which is a group of closely located atomsthat are strongly bound to each other and definean entity. Atomic crystals are composed of atoms,each of which consists of a positively chargednucleus containing almost all the mass of theatom, and a collection of electrons whose totalnegative charge matches the positive charge ofthe nucleus. Atomic crystals may be made up ofeither one kind of atoms or atoms of several dif­ferent kinds each possessing a place of its ownin the crystal lattice. Ionic crystals are composedof atoms that may have either too many or toofew electrons, i.e, atoms from whose electronshells one or more electrons have been withdrawnor, vice versa, into whose electron shells one ormore electrons have been introduced (as everyoneknows electrically charged atoms are called ions).An ionic crystal must clearly contain at leasttwo kinds of oppositely charged ions.

14 Introduction

Further on we shall only discuss atomic and.ionic crystals. Molecular crystals have much incommon with them, but at the same time theyhave some features of their own that we shall nottouch upon.

The atoms or ions are located at particularpoints in the crystal lattice and oscillate aroundthe lattice points. The amplitude of the oscilla­tions increases with temperature.

Consequently, a crystal contains electrons(a few from each lattice point) that are initiallybound to their atoms or ions, but which can befairly easily separated from them and start trav­elling around the crystal. This concerns mainlythe electrons belonging to the outer electron shells.Sometimes the electrons are released easier whenthe crystal is affected in some way, for instance,heated or irradiated with light of a certain wave­length (i.e, light of a certain colour). The behav­iour of these electrons is governed by certainlaws, and their behaviour, in its turn, gives riseto many properties of the crystal. For instance,the electrical and thermal conductivity, heatcapacity, optical absorption, and many otherswhich we are going to discuss below are allproperties dependent on the free electrons. Theelectrons behave differently in metals, semicon­ductors, and dielectrics (we shall show what theseterms mean exactly in the next section). I t ismore accurate to say that a crystal is a metal, asemiconductor, or a dielectric depending on thebehaviour of its electrons.

The electrons populating a crystal are themain dramatis personae in this book. We shallalso deal with the behaviour of these electrons

1.2. Which Crystals? 1&

and discuss -how the properties of crystals aredetermined by their behaviour.

1.2. Which Crystals?

One of the basic properties of a crystal is itselectrical conduction, Le, its ability to carry anelectric current. Some crystals have very highconduction rate and some crystals have negli­gible conductivities.

I t is a very important fact that in nature thereare solids with high conductivities (conductors)and solids with conductivities that are practicallynil (insulators). Our whole civilization is actuallybased on this fact. Indeed, every electrical deviceis a combination of conductors and insulators.What would happen if one fine day every insulatorbecame a conductor? A catastrophe would happen:every electric lamp would go out, every trolleyand motor car would stop, and telephones wouldnot ring. Electric current would go into the earth,which is a bottomless pit for electric charge.Now let us imagine that, conversely, every con­ductor became an insulator. The current would'die' in the wires. Electric charge would freeze,having lost its ability to run along a wire. Thesame catastrophe would occur and the civilizedworld would be plunged into darkness and be­come paralyzed.

How large is the conductivity of a conductor(a typical example being a metal), and how largeis the conductivity of an insulator (a dielectric;an example is a crystal of rock salt)? How manytimes is the conductivity of an insulator lessthan the conductivity of a conductor? In order

1.6 Introduction

to answer these questions we should rememberthe units conductivity is measured in.

Let us consider Ohm's law. Suppose we havea piece of wire with a length L and a cross-sec­tional area S and suppose a potential difference Vis applied across the ends of the wire (V is alsocalled a voltage). Let us designate the currentthat flows through the wire I and the resistanceof the wire R. According to Ohm's law we have

v VI = If' whence "r r (1.1)

If V is measured in volts and I in amperes, thenR is measured in ohms. The reciprocal of R(i.e. 1/R) is called the conductance. I t is "mea-sured, therefore, in reci procal ohms or mhos (Q-1).We know from experience that a wire's conduc­tance is lower, the longer it is and the smallerits cross-sectional area, i.e,

(1.2)

The factor o here depends on what the wire ismade of and is called the electrical conduct ivity,or specific conductance of the material. I t followsfrom (1.2) that

L0= SR' (1.3)

If L is measured in centimeters, S in square centi­meters, and R in ohms, then we obtain for the

1.2. Which Crystals?

Cdimension' of 0':

17

The International System of Units (81) isused by scientists at present, and its unit ofconductance is the siemens (8), 1 S = 1 Q-l.Therefore the SI unit for conductivity a isg-m-I _

Note, by the way, that Ohm '8 law can beviewed differently on the basis of (1.2). Substi­tuting (1.2) into (1.1), we get

V I VI ~aS L' whence S =(1 y.

If we introduce the following notation:

i= ~ and E= ~ ,

we get instead of (1.4)

i = aE.

(1.4)

(1.5)

This is nothing more than Ohm's law writtendifferently _ Here i is current density (the currentper unit cross-sectional area of the conductor),while E is the strength at electric field (the changeof potential per unit length). It follows fromOhm's law given in form (1.5) that the current den­sity is directl y proportional to the electric fieldstrength (and this assertion is Ohm's law) withthe conductivity a being the proportionalityconstant. 1t is different for different crystals.

2-01536

18 Introduction

For metals,ais about 104 Q-l·cm-1 = 106Sx

xm- 1 • For dielectrics, a is about 10-10 Q-l~<

xcm-1 == 10-13 S·m-1. Thus the conductivity ofa dielectric is 1019 times less than that of a metal.This is an enormous number (a one followed bynineteen zeros).

We shall pause here to make an importantcomment. There is a vast group of solids whoseconductivities are essentially less than thoseof metals, but at the same time considerably great­er than those of dielectrics. These solids havebecame objects of study because of some of theremarkable properties they have. These solidsare called semiconductors (they could as well have

a

....,...---~-T

a

T~) ~)

Fig. 1. onductivity as a function of temperature: (a)jin metals, (b) in dielectrics and some semiconductors'

been called semidielectrics). The conductivitiesof semiconductors vary widely from a = 102 toa = 10-10 Q-l. em -lor, what is the same, froma = 104 to a = 10-8 S·m-1 • However, the difference between metals, semiconductors, and. dielectrics is not just that they have different conduct.ivities, but mainly that the conductivitie

1.2. Which Crystals? 19

of the materials in these groups respond dif­ferently to the same external factors.

The simplest factor is heat. When a metal isheated, its conductivity decreases slowly, asshown in Fig. ta. A dielectric behaves the otherway round, with its conductivity increasingrapidly as the temperature is increased. This ispresented diagrammatically in Fig. 1b. As far assemiconductors are concerned, some behave likemetals, while others, conversely, behave likedielectrics. A rise in the temperature of some(very many) semiconductors can increase theirconductivities by several million times.

Another factor to which the conductivity ofa crystal is usually sensitive is the introductionof an impurity. An impurity may be a foreign

Ag Ag Ag Ag Ag Ag Ag Ag Ag Ag Ag AgAg

Ag K Ag AgAg Ag Ag Ag Ag Ag Ag

Ag Ag Ag Ag Ag Ag Ag Ag Ag Ag Ag Ag

Ag Ag Ag Ag Ag Ag Ag Ag Ag Ag Ag Ag

(a) (b) (c)

Fig. 2. Structural defects in a crystal' of silver: (a) va­caney, (b) interstitial atom, (c) foreign atom substitutinga proper a tom of the lattice

atom (or ion) which is introduced into an inter­stitial space in the crystal lattice, or which sub­stitutes an atom (or ion) in the lattice itself.Typically, a real crystal contains a certainconcentration of impurities if the crystal

2*

20 Introduction

has not been subjected to special treatment. Anatom (ion) of the lattice itself that is not locatedwhere it should be, i.e. not at a lattice point butin the interstice, and a vacant point in the lattice,Le, a point from which an atom (or ion) hasbeen removed (it is called a vacancy) also fallinto ..the category of impurities. In the physicsof crystals, the notion of 'impurity' is widerthan what we attribute to it in our daily life.An impurity is a defect that spoils the regularpattern of a crystal lattice (Fig. 2).

The conductivity of a metal decreases some­what when an impurity of any kind is introduced,and decreases further the more impurity thereis. As a rule, the conductivity of a dielectric is,on the contrary, somewhat increased by an im­purity. Some semiconductors behave like metals,but in most cases semiconductors are extremelysensitive to impurities and the introduction ofeven tiny amounts of impurity results in largegain in their conductivities. Thus, one foreignatom per thousand proper atoms in a lattice canchange its conductivity by many hundredsof times. This is indeed a 'homeopathic'effect.

Let us note one more factor on which the con­ductivity of a crystal can depend. This is the ap­plication of an electric field to the crystal. Allmetals, however strong the field may be, obeyOhm's law (remember (1.5)). It states that thecurrent in the metal is directly proportional tothe strength E of the field, i ,e, a stays constantand does not depend on E. In nonmetal crystals,Ohm's law is only observed for small values ofE. If E increases further Ohm's law is broken and

1.2. Which Crystals? 2t

the current either starts increasing faster thanthe strength of the field or sometimes more slowlythan is required by Ohm's law (Fig. 3), i.e. (J

stops being constant and changes with the increaseof E. At large enough E the crystal is destroyedor breakdown occurs.

When we speak of electric current, we meanthe transfer of an electric charge in one direction.The charges in crystals are mainly the electronsthat leave the atoms or ions they belong to andstart wandering round the crystal. This kind ofcurrent is called electron current .. In ionic crystalsthe charge carriers are the ions themselves asthey can leave their lattice points for the inter­stitial space and thus they acquire the abilityto travel through the crystal. In this case we saythere is an ion current in the crystal. Electroncurrent occurs in metals and semiconductors,

-----~E

(a)

-=::::;;..----~~E

(b)

Fig. 3. Current density i as a function of strength E ofthe field: (a) in metals, (b) in nonmetallic crystals

while in dielectrics ion current (at moderate E)is the rule which gives way to electron currentat large E. Later we shall describe in more detail

Introduction

how current flows in the crystal, and the varioustypes of electron and ion currents.

In this section we described some of the fea­tures and properties of metals, semiconductors,and dielectrics which occur because of the behav­iour of the electrons within the crystals. Elec­trons govern the properties of crystals. We shalltry now to explain how they do it.

Chapter 1

Electrons in Metals

1.1. 'Free' and 'Bound' Electrons

In this chapter we shall discuss the electrons popu­lating metals, their behaviour and the laws theyobey. Our aim here is to explain some of theproperties of metals.

Let us consider a metal crystal made of atomsof one kind. For simplicity's sake, suppose theyare monovalent atoms (e..g. Ag or Na). A dis­cussion of metals with higher valences (s.g,bivalent ones like Zn or Ni) will add nothingfundamentally new to our picture.

Let us take only one atom in the lattice, ima­gine all the others around it-are at infinity andsee what happens when we move them back intoplace to form a crystal lattice.

Our atom consists of a positively chargedatomic core and a valence electron at some dis­tance from it. The atomic core can be approxi­mated as a point charge (an analogue of a hydro­gen atom). Let us assume that it is positioned at areference point (origin) O. Thus, the valence elec­tron travels in the Coulomb field of a point charge.The potential energy U of the electron 'dependsdirectly on 1 z ], the distance of the electronfrom the reference point (the U versus x curveis given in Fig. 4). We shall discuss a unidimen­sional model, I.e, we consider the crystal to be

Electrons in Metals

a chain of atoms, therefore

(1. 1)

where e is the charge of the electron. Expressionsfor potential energy always require an extraadditive constant which fixes the origin of theenergy scale. In (1.1) this constant has been chosenso tha t the potential energy U is zero at x == 00, i.e. when the electron is at infinity. If

uH/2-----+-----

x

Fig. 4. Potential crater of an isolated atom

the origin is chosen this way the potential energyU is negative for all finite a;

Although the potential energy U vs. x isshown in Fig. 4, we can also show there the totalenergy of the electron. Let us designate it W.Obviously W does not depend on x, i.e. it isshown by a horizontal straight line. It can havea specified value, but it should be constant regard­less of the motion of the electron. The horizontallines. W1 and Wa ~re \WQ arbitrary values of W.

t.f. 'Free' and 'Bound' Electrons 25

The kinetic energy K of the electron can beshown in the same figure. Clearly

W == K + U and therefore K = W - U.

(1.2)

If the electron has a total energy WI and is Xl

away from the origin, then its kinetic energy isgiven in Fig. 4 (see (1.2» as AB. But if the elec­tron is at a distance X 2, then its kinetic energyis given as CD. In this case W < U and henceK < 0, i.e. the kinetic energy turns to be nega­tive which is impossible. Indeed, in classicalmechanics

1K="2 mv2 ,

where m is the mass and v is the velocity of theelectron. Evidently, the kinetic energy K canonly be negative if the mass of the electron isnegative, which is absurd, or if the velocity vof the electron is an imaginary number, whichis no less absurd. It follows that an electron witha total energy of WI cannot be farther from theorigin than Xo. The area that is cross-hatched inFig. 4 is forbidden for the electron from the view­point of classical mechanics. Obviously, thehigher the total energy W of the electron, i.e.the higher the line WI in Fig. 4 is, the broader isthe area allowed for the electron. The inferenceis that this is a bound electron, i.e, an electronattached to an atom and incapable of leaving it ..

Conversely, if the total energy of the electronis given in Fig. 4 by the horizontal line W 2' whichis above U everywhere, then K > 0 for any X,

and the electron can go anywhere, Thi~ implies

26 Eleetrons in Metals

that this electron is free and its bond with theatom has broken; it has the right to travel asfar from the atom as need be.

Consequently, a bound electron has a negativetotal energy W (W < 0, see the horizontal lineW == WI in Fig. 4, which is below the abscissa),

u

I II II 1

LJa (b)

-----+---w3

x

(a)

Fig. 5. Potential energy diagram for an electron: (tl)in an isolated atom, (b) in a crystal

and a free electron has a total energy W that ispositive (W > 0, see the horizontal line W == W 2 ,

which is over the abscissa). An electron in abound state can be transferred in to the free stateif it is supplied with the energy needed for thetransition from level WI to level W 2 - This act ofrelease of a bound electron is called the ionizationof the atom.

So far we have discussed an isolated atom.Now let us consider a chain of atoms spaced outregularly with an interval a between them (seeFig. 5b), i.e. the unidimensional model of a crys­tal. This is a system of atomic cores and a systemof electrons. Each electron interacts with its

1.1. 'Free' and 'Bound' Electrons 27

own atomic core, with all the other cores andwith all the other electrons. We are going toneglect the last category, although there are nological grounds to do so. Indeed, the interactionsbetween the electrons are of the same order ofmagnitude as their interactions with the atomiccores. There is no real reason for neglecting theelectron-electron interactions while taking intoaccount the electron-core ones, but we shall do so,and shall consider each electron to be independentof the others, as it were. This simplifies the prob­lem appreciably and then we can return to ac­count for the electron-electron interactions.

Let us consider a model in which each elec­tron moves in the common field of all the atomiccores. The electron's potential energy in sucha force field can be given by a curve that is thesum of several curves like the one shown inFig. 4, but shifted over a distance a. The result­ant curve (the potential energy U of the elec­tron as a function of x) is plotted in Fig. 5b.This is a system of potential craters alternatingwith potential barriers and fringed on both theright and the left by potential thresholds. Thethresholds correspond to the end atoms (the firstand the last) of the chain. The potential craterof a single atom is shown once again on the left(Fig. 5a). The essential point about Fig. 5b isthat the potential energy U of the electron withinthe crystal lattice is periodic with a period equalto the lattice constant a.

Let us ask ourselves what the behaviour ofour electrons is now. Have they acquired theability to travel around the crystal, or will theyremain attached to their respective atoms? The

28 Eleetrons in Metals

answer depends on the total energy W of theelectron and there are three cases:

(1) If the electron is on level W1 , i.e. if ithas a total energy shown by a horizontal linethat cuts the tops of the barriers, then accordingto classical mechanics the electron is locked with­in its potential crater, i.e , it cannot get into theadjacent potential crater. In fact for an electronat Wl' to get into the next crater, it must piercethe potential barrier, i,e. the region in which itstotal energy is more than its potential energy andwhich therefore is forbidden to it. Consequently,the crystal turns to be, as it were, divided bywalls that are impenetrable for the electron.

(2) If the electron is on level W 2' which passesover the tops of the potential barriers (as shown'in Fig. 5b), then it can travel freely within thecrystal, passing from atom to atom. However itcannot go beyond the crystal as the space in whichthe electron is permitted to travel is bounded bythe potential barriers on the left and on the right.

(3) And lastly, if the total energy of electronin Fig. 5b is given by Wa, then the electron cannot only travel without hindrance within thecrystal, but it can also go beyond the crystal,however far.

So far we have tacitly assumed that the elec­tron, whose behaviour we discussed, is a classicalparticle, i.e. it obeys the laws of Newton's clas­sical mechanics. But in reality electrons arequantum particles and obey quantum mechanics,which dominates the microcosm. How is ourpicture corrected by quantum mechanics?

From the viewpoint of quantum mechanics,there is a nonzerc ptoba.btll\J ~ll"t ,p ~l~~~f~~

1.t. 'Free' and 'Bound' Electrons 29

may exist where it cannot occur from the purelyclassical viewpoint. In other words, a quantumparticle can penetrate the area forbidden to a clas­sical particle. Accordingly, there is a nonzeroprobability that an electron with a total energyW1 (Fig. 5b) can penetrate the area W1 < U,i.e. inside the potential barrier, and thereforeescape through it. The probability of such anescape increases the narrower and lower thepotential barrier becomes, i.e. as the cross­hatched area in Fig. 5b decreases. An electronon one side of the barrier can occur on the otherside owing to a 'tunnel' transition through thebarrier even though it possesses a total energy W1 •

This 'tunnel' transition (tunnelling. effect) ischaracteristic for modern solid-state quantumtheory and we shall discuss it in this book morethan once.

Thus, quantum mechanics results in an 'ab­solutely' bound electron becoming less bound.The use of quantum mechanics in solid-state theo­ry thus produces a certain 'liberation' of boundelectrons.

Let us make one more remark in conclusion.Perhaps you remember that at the beginning ofthis section we assumed the simplifying butgroundless approximation that there were noelectron-electron interactions. This type of inter­action can be taken into account by introducingwhat is called a self-consistent field. This field isthe one produced by the averaged charge of allthe electrons, and each electron is considered tomove both in the field of the atomic cores andin the self-consistent field of the electrons. Thefield makes us include another summand into

30 ElectroDs in Metals

the expression for the potential energy. However,this does not interfere with the potential energy'sbasic property, viz. its periodic nature, and wemay still use Fig. 5b. Accounting for the inter­actions between electrons is an involved prob­lem and we shall not discuss it anymore. Let usonly note that in modern solid-state quantumtheory a system of interacting electrons is con­sidered to be a system of noninteracting 'quasi­particles' (or elementary excitations).

1.2. ·Electron Gas' in Metals

Let us return to Fig. 5b, which is a diagram ofthe potential energy of an electron in a metal.The diagram can be substantially simplified ifthe periodic function of the potential energy isreplaced by a value averaged over the entirecrystal, and if the potential thresholds at theboundaries of the crystal are assumed to be ver­tical. Fig. 5b thus turns into Fig. 6. There isnow no difficulty in going from our unidimensionalcrystal model (a chain of atoms) to a three-dimen­sional model, in which the crystal is representedas a cube (Fig. 7) with an edge L and containingN atoms, so that

L = aN, (1.3)

where a is the distance between adjacent atoms(the lattice constant). The z-axis in Fig. 6 canbe replaced by the y- or a-axes,

This simplified model of the crystal, whichis a potentia! box with a nat bottom and verticalwalls, is the basis of Drude's classical theory ofmetals and Sommerfeld's quantum theory of

1.2. 'Electron Gas' in Metals 3t

metals. All the electrons in the system 'live' inthe potential box. To take a definite example forthe metals we shall still consider them to con­sist of monovalent atoms. Thus the number of

u

xL

Fig. 6. Potential box

(

IIIJ

.J----./

,/

L

y

Fig. 7. Cubic crystal

electrons in the system is equal to N3, and theconcentration of electrons No, will according to(1.3) be

(1.4)

32 Electrons in Metals

This is a very large number. There are about 1028

electrons in a cubic centimeter of a metal crystal(a one followed by 23 zeros!). All of these elec­trons can be considered to be free since theymove in the absence of any force field (the poten­tial energy of an electron within a crystal isconstant), i.e. they behave like the moleculesof an ideal gas. A metal in this model is a frame­work of atomic cores immersed in a gas ofelectrons.

Each electron at a given moment in time occu­pies a definite position in space or, in other words,it has three definite coordinates x, y, Z, and afinite velocity v. Thus it can also be described bythe three components of its velocity vx , vlI ' vz,or it occupies a definite position in velocity space,­i. e. in the space where the coordinates are notx, y, Z, but the velocities vx , vy, V z•

Now let us recall that our electrons are quan­tum particles. Each velocity component provesto be quantized, i.e. it can not have just anyvalue, it must have one of a set of distinctvalues, i.e.

h h hvx == maN n x , VlI = maN ny, vz = maN nze (1.5)

Here h is Planck's constant (it is always presentin quantum-mechanical formulas) and n;c, ny, n zare arbitrary, but always integer-valued numbers(including zero):

n:x, ny, n z = 0, ±1, ±2, ±3, . . . (1.6)

The numbers nx , n , n; are called quantum num­bers. We see that tte velocities V X ' VII' V~ are Dotcontinuous, but are a discrete sequence of values.

f.2. 'Electron Gas' in Metals 33

It follows that velocity space consists of uniformcells, ~y in volume (Fig. 8), i.e.

liy= ( m:N r·In agreement with (1.5) and (1.6), an electronmay not therefore exist anywhere in velocity

Vy

1 m

~

h1110 N'

Fig. 8. Quantized spaceof veloci ties

(1.7)

space but may only exist at the points of thelattice shown in Fig. 8.

According to (1.5), the energy W of the elec­tron is quantized as well. Indeed, the potentialenergy U of the electron in our model is constantthroughout the crystal and hence it can be as­sumed to be zero without loss of generality. Thetotal energy of an electron is therefore equal toits kinetic energy:

1-" _ 1 2 _ 1 (2 ~ 2 2)n -2" mv -2 m Vxi-Vy-t-Vz •

Substituting (1.5) into this, we geth2

w:=:: 2ma2N2 (lli + n~ + n;). (1.8)

3-01536

34 Electrons in Metals

We see that the total energy W, like the threevelocity components v-. "s: v, can only assumea discrete sequence of values. Both Wand vx ,vy, V z are shown in Fig. 9 as functions of thequantum numbers nX., ny, n z• In Figs. 9a and 9bthe values of n x (or ny or n z) are plotted along the

-4 -2 0 2 4

(a)

Fig. 9. Quantization of energy (a), quantization of .'thethree components of velocity (b)

abscissa, and the values of W in Fig. 9a and thevalues of V x (or vy or V z respectively) in Fig. 9balong the ordinate.

Now we have to take an important step inour presentation and introduce what is calledthe Pauli principle which a pool of electronsobey. This principle states that only one elec­tron can possess a given set of three quantum num­bers. In other words, each point in velocity spacecan only be occupied by one electron. The Pauliprinciple is the principle of impenetrabilitycarried over from usual coordinate space into

f.2. 'Electron Gas' in Metals 35

velocity space. The impenetrability principleforbids two or more electrons to occupy the samepoint in space at the same time. This is why it isoften called the Pauli exclusion principle. Thereare some essential consequences of the Pauli prin­ciple that we shall come across more than once.

I t is necessary to make a correction here.The three quantum numbers are not an exhaustivedescription of the state of an electron. There isa fourth parameter called spin. The presence ofspin is just as essential and inalienable a prop­erty of an electron as its mass or electric charge.

We can .imagine electron spin best of all ifthe electron is imagined to be a small ball spin­ningaround its axis. It can spin around an arbi­trary axis both clockwise and anticlockwise.That is why two values of spin are possible. Ac­cording to the Pauli principle not one, but twoelectrons can possess a given set of three quan­tum numbers if the electrons possess oppositespins. In other words, if electron spin is takeninto account, each point in velocity space canaccomodate two electrons.

Obviously, all electrons of our system shouldsomehow be distributed in velocity space. Thetotal summed energy of our system (i. e. thesum of the energies of all the electrons) dependson this distribution. Let us consider which dis­tribution satisfies the requirement of minimumenergy, or, what is the same, the condition ofabsolute zero temperature. (Recall that absolutezero is the temperature at which the energy ofa system is at the minimum.)

But for the Pauli principle, every electronwould occupy the point V x = Vu = V z = 0, i.e,

a-

36 Electrons in Metals

the origin of the velocity space. Every electronwould thus be at rest and the energy of the systemwould be zero. However, this is forbidden by thePauli principle. The energy of the system is atthe minimum, albeit nonzero, if they-'electrons

~

0 00 0

0 •0 •

VI VI0 0

o 0 0

000 0

0 o 000 0

(tt) • Electrons (b)

Fig'. 10. Distribution of electrons in velocity space:(a) for absolute zero, (b) for a temperature above absolutezero

are distributed as close to the origin as possiblewhile obeying the Pauli principle, i.e. if theyfill a sphere containing half as many cells as thereare electrons (each cell contains two electrons).The surface of this sphere is called the Fermisurface and the sphere itself is called the Fermisphere. This distribution is shown in Fig. fOa,where vl and V 2 are any two numbers out of theset of three numbers vx , vy, V z• It is clear becauseof this that the motion of electrons cannot stopeven at absolute zero. This follows from the ap­plication of the Pauli exclusion principle to thepool of eleettrons.

1.2. 'Electron Gas' in Metals 37

(1.9)

When the temperature rises, some of theelectrons in velocity space escape the sphereshown in Fig. 10a leaving vacant places. For thatreason we go from Fig. 10a to Fig. 10b. We cansay that heating loosens the dense packing of theelectrons in velocity space. The higher the tem­perature, the looser the packing gets, and thesummed energy of the electrons rises.

It is essential that there is a symmetry in thedistribution of the electrons in velocity space.In other words, each electron with a velocity vhas a counterpart with a velocity -v, so thatthe average vector velocity of all electrons isstill zero at any temperature. This means thatthere is no electric current in the crystal.

That is the way it is until an extraneous forceis applied to the electrons. If the crystal is placedin an external electric field by applying adifference of potentials to it, the symmetric dis­tribution of electrons is upset. A predominantdirection then appears in which the number ofoccupied places is greater than in the oppositedirection. The average velocity of the electronsthen becomes nonzero. This means that there isan electric current in the crystal.

We have already noted that the degree of loos­ening of the electron packing in velocity spaceincreases with a rise of temperature. Commonlythis degree of loosening is characterized by afunction F (W) that we shall call the energydistribution function. To define it consider a spherein velocity space with a layer of a radius vand a thickness dv. It follows from (1.7) that

.. /»'v== V 2 m

38

and

Electrons in Metals

dv= _i_ dW (1.10)V2m.W •

The number of electrons in this layer, i.e. thenumber possessing energies from W to W + dW,is designated X (W) dW. The number of pointsavailable for the electrons (twice the number ofcells in the velocity space) that the layer con­tains is designated Y (W) dW. The distributionjunction is the ratio of these two numbers:

X (W)dW 1F (W) = Y (W) dW • (1. 1)

This function has for our electron gas the form

fF (W) == W-WF. (1.1.2)

1+exp kT

Here T is the absolute temperature, k is Boltz­mann's constant, which should be familiar tothe reader from the kinetic theory of gases; andW F is the energy corresponding to the Fermisurface. The energy W F is called the Fermi energyor Fermi level, and the function given in (1.12)is the Fermi distribution function. Particles de­scribed by the Fermi function are said to satisfythe Fermi-Dirac statistics.

The function in (1.12) is illustrated in Fig. 11.The step shown by the thin line corresponds tothe function's shape at absolute zero. We seethat this is the case when all the cells in velocityspace that are within the Fermi sphere are filledby electrons (F = 1.), while all cells that arebeyond this sphere are vacant (F == 0). The smooth

1.2. 'Electron Gas' in Metals 39

bold curve in Fig. 11 corresponds to a temperature,ahove absolute zero. This curve F .- F (W) pas­ses through point A (see Fig. 11) for which W == W F, F = 1/2 at any temperature.

To conclude this section, let us return to whatwe noted in the beginning. Sommerfeld's quan­tum theory of metals deals with the same model

F(W)

1"2o~-----~-~~

Fig. 11. Fermi distribution function

as Drude's classical theory: that is an electrongas locked in a potential box with a flat bottom.But the electron gas Sommerfeld considereddiffers from the electron gas of the classical theory'in two respects.

(a) Each electron in Drude's theory obeysclassical mechanics, while in Sommerfeld's theoryit is considered to be a quantum particle gov­erned by the laws of quantum mechanics.

(b) When describing the behaviour of a poolof electrons, Drude's theory relied on the Max­well-Boltzmann classical statistics, i.e. the sta­tistical laws that describe an ordinary gas con­sisting of independent molecules. Sommerfeld'stheory applies the Fermi-Dirac quantum sta­tistics to the electron gas, i.e. the statistical

40 Electrons in Metals

laws that govern particles which are, first, iden­tical and, second, controlled by the Pauli prin­ciple. However, it is essential to point out andemphasize that all the formulas of the quantumstatistics are transformed under certain condi­tions (which are going to be discussed below)into the classical formulas. The electron gaswould then be called nondegenerate.

1.3. The Successes and Failures of the ClassicalTheory of Metals

Metals, unlike nonmetallic crystals, have veryhigh electrical and thermal conductivities. Wecan use our earlier notation, i.e, electrical con­ductivity 0', and thermal conductivity x, and­further designate the current density in metals i.This is directly proportional to the potentialgradient dVldx. Let us designate the thermalcurrent density [, It is directly proportional tothe temperature gradient dTldx. (The derivativesdVldx and dTldx show how fast the potential andtemperature, respectively, change along the x­axis.) We have

i= -(J~ , (1.13)

. dTJ~ --x,dX". (1.14)

Equation (1.14) holds true if there is no ex­ternal electric field. The minus sign on the right­hand side of equation (1.14) means that the ther­mal flux (Le. the kinetic energy carried by elec­trons) is in the opposite direction to the temper­ature gradient. In other words, heat is transferred

1.3. Classical Theory of Metals 41

from where the crystal is hotter to where thetemperature is lower.

Equation (1.13), on the contrary, refers to thecase in which the metal is in an electric field,but the temperature is constant throughout thecrystal, Le. the temperature gradient is every­where zero (dTldx = 0). Recall that

_dV==E (1.15)dx '

where E is electric field strength, and substitute(1.15) into (1.13) to obtain equation (1.5):t == aEwhich is Ohm's law (for a independent of E).

The constants a and x in (1.13) and (1.14)can be calculated. The results of these calcula­tions we shall cover below. Let us note that theclassical and quantum theories bring about differ­ent results even though both theories proceedfrom the same model.

According to both the classical and quantumtheories the electrons in metals are in constantchaotic thermal motion colliding with the atoms(with the atomic cores, to be more exact) of thelattice. The atoms (atomic cores) are arranged ina regular pattern and oscillate about their pointsof equilibrium. The amplitude of these oscilla­tions depends on the temperature; the higherthe temperature the larger the oscillations.According to the quantum theory, these oscilla­tions do not stop even at absolute zero.

When there is no electric field, the electron'smotion between two collisions is linear and uni­form. But when a field is present, each electronmoves between the collisions with an acceleration

42 Electrons in Metals

(1.16a)

(1.17a)

proportional to the field strength E and accumu­lates a kinetic energy that is then transferredto the lattice via collisions. This is the source ofthe resistance that the lattice offers to current,and the source of the heat that is evolved whena current flows.

However, there is an essential differencebetween the classical and quantum theories here.In the classical theory, the resistance is causedby the very presence of the crystal lattice, whichresists the freely moving electrons. Accordingto the quantum theory, the lattice does not itselfresist the flow and is transparent to the electrons,they do not notice it. The resistance is due to theoscillations of the lattice. The greater the temper­ature, i.e. the greater the amplitude of the os­cillations, the greater the resistance. It is theatoms, deviating from their equilibrium positions,that scatter the free electrons. The foreign atomsof an impurity, which are outside or inside theregular pattern of the lattice, are additional cen­ters of scattering and offer additional resistance.

The mean distance an electron travels be­tween two successive collisions with the latticeatoms is called the mean free path. We shall de­signate this value by the letter l. It decreases asthe temperature increases.

The classical and the quantum theories pro­duce the following expressions for (J and x ,We have in the classical theory

4 Nole2

a zx; 3 -y-----:--2-nm-k-T- ,

4 .. /2k3Tx=3 Nol V nm'

1.3. Classical Theory in Metals 43

where No is the concentration of free electrons asbefore, e is the electron charge, and m is themass of an electron. The quantum theory pro­duces

(J==~ e21 (3No )218

3 h n ,

X == 2n8 lk2T ( 3N0 ) 2189 h n •

(1. 16b)

(1. 17b)

Apart from the universal constants h, e, and k,these expressions include No and l, which are differ­ent for different metals and must be determined.

When dealing with electrical and thermal con­ductivities we cannot avoid the Wiedemann­Franz law. This law was discovered experimen­tally and it states that the ratio of the thermalconductivity to the electrical conductivity ofa metal is a constant independent of the metaland is directly proportional to the absolute tem­perature. Both the quantum and the classicaltheories of metals explain this law. Dividing(1.17a) into (1.16a) we obtain for the classicaltheory

(1.18a)

And dividing (1.17b) into (1.16b) we get for thequantum theory

x _ '11 (k )--_.- - T.

(J 3 e I(1.18b)

Evidently, this is a case when both the classicaland quantum theories bring about almost the

44 Electrons in Metals

same results. The only difference is in the factorof 1(2/3 "J 3.3 in the quantum theory instead of2 in the classical one.

The deduction of the Wiedemann-Franz lawis a great achievement of the classical theory ofmetals. We could also cite some other inferencesfrom the classical theory that are in good agree­mentwith experiment. However, the developmentof the classical theory came to a dead end. Thiswas the problem of the heat capaci ty. The classicaltheory gave results that were in stark contrast toexperimental results. Let us discuss this problemin some detail.

The heat capacity of a metal is the sum of twocomponents: the heat capacity that is due to thethermal oscillations of the lattice atoms, andthe heat capacity of the electron gas. Let usdesignate the electron heat capacity per unitvolume c and designate the total energy of theelectron gas per unit volume w. W is the sumof the kinetic energies of all the electrons in acubic centimeter of a metal. By definition

(1 .19)

For that reason the specific heat capaci ty of theelectron gas indicates the rate at which the totalenergy of the electron gas ri ses wi th tOlTI perat ure.

According to the classical theory, which attrib­utes to the electron gas all the properties of anordinary ideal gas, we have

- 3»T~2 NokT. (1.20)

1.3. Classical Theory in Metals 45

and therefore, according to (1.19)

d 3 ) 3c= dT {2"NokT ="2 Nok. (1.21)

Fig. 12. Total energy W ofelectron gas as a functionof temperature TT

Nondege­nerate

gas

o

However, the experimental data show thatthe heat capacity of the lattice is almost equalto the total-heat capacity of the metal, so thatalmost nothing is left for the electron gas. This

W

means that No in (1.21) must be small but thenthe conductivity a, the expression for which alsocontains No, according to (1.16a) and (1.16b),turns to be too small. TIle way out of this dif­ficulty was the use of the quantum theory insteadof the classical one. We now obtain another ex-pression for ·W instead of (1.20).

We are not going to give this expression hereand naturally, we are not going to deduce it.Instead, we shall present Fig. 12 in which thetotal energy W of the electron gas is shown as afunction of temperature T. The bold curve com-

46 Ilectrons in Metals

plies with the quantum theory of metals, whilethe thin straight line follows from the classicaltheory. The steepness of the curve at each givenpoint characterizes the heat capacity of the elec­tron gas at the respective temperature. We see thataccording to the classical theory the heat capac­ity does not depend on temperature. But theheat capacity in the quantum theory increases asthe temperature rises: it grows from zero (at abso­lute zero) to its classical values (1.21) (at hightemperatures). The temperature To is called thedegeneration temperature and an electron gasbehaves like a classical one at temperatures thatare substantially greater (T ~ To). Such a gasis called nondegenerate while the electron gas issaid to be degenerate at temperatures less thanthe temperature of degeneration. The heat capac­ity of a degenerate gas is lower than its classicalvalue and tends to zero as the temperature falls.The degeneration temperature To can be deter­mined thus3 -2 N okTo== W o'

whence

T 2 Wo0=="3 Nok'

(1.22)

where Wo corresponds to the Fermi energy (seesection 1.2). By deducing Wo and substitutingit into (1.22), it can be shown that the lower Nothe lower To. For metals, in which, as we havealready noted, No == 1023 em-s , the degenerationtemperature proves to be higher than the meltingpoint, That is why the electron gas in metals is

1.4. Electron Emission from Metals ·47

in practice degenerate at all temperatures. Thisaccounts for the fail ures of Drude's classical theorybecause it assumed a non degenerate gas.

1.4. Electron Emission from Metals

As we mentioned before the potential in a metalcan be represented as a potential box or potentialwell containing electrons. I t is thus natural to askwhether there is any way of extracting the elec­trons from a metal, which is a practically in­exhaustible source of them.

We shall consider three of such methods inthis section. A metal can be made to emit elec-trons as the result of: .

(1) heating it to a high temperature (to inducethe thermionic emission);

(2) applying a strong electric field (to inducethe field or cold emission);

(3) radiating it with high frequency light (toinduce the photoelectric effect).

We shall discuss each of these methods.1. Only those electrons filling the metal that

are capable of overcoming the potential thresholdW m shown in Fig. 13a can leave the metal, i.e.go beyond its surface. This figure is very muchlike Fig. 6 (only W is the ordinate instead of U;Fig. 13a shows the energy levels occupied byelectrons at absolute zero). Obviously, only thoseelectrons that possess sufficient velocity in thedirection at right angles to the surface of metalcan make it. If we choose a coordinate systemsuch that the x-axis is perpendicular to the sur­face of metal, then there will be a lower limit V:to

for the velocity component V:t such that only elec-

48 Electrons in Metals

trans for which V x > vx o will be able to over­come the potential threshold.

The velocity V x o can be determined from

~ m (vxo)2= W F

(see Fig. 13a). It is clear from Fig. 13b, whichshows the Fermi distribution function (as in

1 F(b)

o

w

(a)

w

- ... -- -<f"

J¥r

L -:::

It,

Fig. 13. Metal as a potential well (4), Fermi distributionfunction (b)J

Fig. 11), that the electrons under discussion be­long to the 'tail' of the Fermi distribution func­tion.

Calculations of the thermionic current den­sity i, i.e. the flux of electrons liberated from aunit surface, produce

t = A (kT)2 exp ( - k~ ) , (1.23)

which is in good agreement with experimentaldata. The quantity cp (see Fig. 13a as well) is calledthe work function. I t is a constant that is difier-

t4. Electron Emission from Metals 49

ent for different metals and is one of their char­acteristics. The work function is the smallestamount of energy an electron must spend to leavethe metal at absolute zero.

As can be seen from (1.23), the way the ther­mionic current i depends on temperature T isdetermined both by the exponent and the pre­exponent factor. They both act in the same di­rection hence i increases with the rise of T. How­ever, the pre-exponent factor changes with thetemperature very slowly (compared with therapid change due to the exponent), and thereforeit can be assumed to be constant at a first ap­proximation. Thus

t = C exp ( - k~ ) ,

or

Ini=lnC- ~ • (1.24)

If we plot In t and k~ along two coordinate axesthenequation (1.24) will be a straight line whoseslope ex gives the work function q> (see Fig. 14).Commonly this value is several electron volts*.

The phenomenon of thermionic emission, theoutflow of electrons from a heated metal, is alsoknown as the Richardson effect.

2. A metal can be made to emit electrons with­out heating it. All that is necessary is to put themetal into a strong external electric field. Sup­pose there are two flat electrodes facing each other,

• An electron volt is a unit of energy and equals the ener­gy acquired by an electron when it passes through a po~

tential ditlerence of one volt.

l-Itlll

50 Electrons in Metals

and let us consider the metal is the cathode. Theelectric field with a strength E is at rightlanglesto the metal surface. The bold curve in Fig. 15

In i

IkT

Fig. 14. Thermionic current as a function of temperature(tan ex = <p)

wWI

J§_~_-~~'-,--J,J!:!

J~!-,

~~~~~~

.....L ..L- ""-'-"-L..L..L.."j'-'---__~ X

o

Fig. 15. Potential energy diagram of an electron ill anexternal electric field

shows the potential energy of an electron as afunction of x (bear in mind that the z-axis isperpendicular to the surface of the metal). .Thethin curve in Fig. 15 shows the potential energyin the absence of the field (E = 0). We see thatthe field transforms the potential threshold on

1.4. Electron Emission from Metals 51

the metal surface into a potential barrier. Thereare two possible mechanisms for the appearanceof the electron emission current in this case.

First of all, the external field decreases thework function cp by Llcp .. This facilitates thermionicemission, which is extremely sensitive to changesin cp (according to (1.23) cp is in the exponent).Thus when the external electric field is applied,thermionic emission becomes noticeable at lowertemperatures than in the absence of the field.When there is no field, only the electrons forwhich W > Wl could be emitted (see Fig. 15,W, as before, is the total energy of an electron).When the field is present, electrons for whichW~ < W < W l can also be emitted.

Another mechanism for electron emission ispossible, in which slower electrons can take part.We refer to electrons whose energy W is shown inFig. 15 by a line below the top of the barrier(W < W 2) . These electrons (recall that electronsare quantum particles) can penetrate inside thebarrier with a nonzero probability. The proba­bility for a classical particle to do this is zero.We have already come across the tunnel effect(see Section 1.1). The probability an electron hasof being able to penetrate through the barrier de­pends on the energy level W a at which the electronis residing. The probability that an electron withina metal occurs to the right of the barrier in Fig. 15,i.e, leaves the metal, depends on the height andwidth of the barrier, the narrower and the lowerit is, the greater the probability, or to put it inanother way, the larger the cross-hatched regionin Fig. 15, the smaller the probability of a tunneleffect.

52 Electrons in Metals

Consequently, an electron can leave a metal,to which an external electric field is applied eitherby passing over the potential barrier (thermionicemission) or by passing through the barrier (coldemission). The first mechanism prevails at mod­erate fields, while the second at strong fields.

A mathematical treatment has shown thatin cold emission the flux of electrons i (i.e. emis­sion current density) depends on the strength ofthe applied field E in the following way:

i = aE exp (-bIE). (1.25)

Both a and b include the universal constants h,e, and m, as well as tp, Therefore i does not de­pend on temperature T, but depends OIl cp, i.eon the nature of the metal. The current denslty-zat a given E is the same at different temperatures,but it is different for different metals.

Formula (1.25) is similar in form to formula(1.23) where temperature T is replaced by fieldstrength E. However, the mechanism of coldemission is not at all similar to the mechanismof thermionic emission. But the current densityin cold emission is as sensitive to the externalfield E as it is sensitive to temperature T inthermionic emission.

Calculations show that according to formula(1.25) cold emission must become noticeable atfield strengths of a thousand million voltsipormeter (109 VIm), but in reality it can be observedin weaker fields. This is because of inhomogeneitieson the surface which cause very strong local fieldsin some points.

3. Let us discuss one more method of liber­ating electrons from metals. Suppose a" metal

1.4. Electron Emission from Metals 53

is exposed to light of a certain frequency. Thismeans that the metal surface is bombarded bylight quanta (photons) carrying a certain amountof energy. Penetrating into the metal, the quantapass their energy to electrons which can thenuse it to overcome the potential threshold andleave the metal. This phenomenon is called theexternal photoelectric efJect or just the photoeffect,The electrons emitted from a metal due to theaction of light are called photoelectrons.

The photoelectrons leave the metal with abroad spectrum of velocities, i.e. there are elec­trons with several different velocities in the pho­toelectron stream at a given frequency 'V of theincident light. The velocity of a photoelectron ata given 'V depends on the energy level from whichthe electron ·was displaced by the light.

",The highest energy level an electron can occu­py at absolute zero is the Fermi level W F (Fig. 16).

w

Fig. 16. Energy diagram illustrating theappearance of fast and slow photoelec­trons (the energy range occupied bre.l~ctrons is cross-h~tched)

54 Electrons In Metals

(1.26)

Every level higher than W F is vacant at T = 0,and every level below W F is occupied. Let usdesignate the velocity of an electron when ithas been removed from level W F by the notationv F. Clearly (see Fig. 16) we get

1 2 h2" mVF= v-cp,

where m is the electron's mass, hv is the energyof the incident quantum, and cp is the work func­tion we discussed while speaking of thermionicemission. Equation (1.26) is an expression ofthe law of the conservation of energy. The fre­quency Vcr for which the energy hVer of an incidentquantum is equal to the work function q> is cal-led the red limit of the photoeffect: .-

hVcr == rp. (1.27)

The velocity v F is the fastest at which an elec­tron can travel at T = O. Lower velocity photo­electrons are removed from deeper levels (e.g,from level WI in Fig. 16).

At T > 0 electrons with the energy greaterthan W F (e.g. with energy W 2 , see Fig. 16) ap­pear. Energies less than the work function (pare required to remove them from the metal' R

surface. We can write v > VF for such electrons.Although the red limit at T == 0 is sharp, thered limit at T > 0 turns to be somewhat blurred(depending on the temperature).

Note that if the metal is put into an externa1electric field, the red limit for the photoeffectbecomes shifted in the red direction, i.e. in thedirection of lower frequencies. This is because theelectric field decreases the potential threshol d

t.5. Electrons in Periodic Field S5

at the metal surface, as can be seen in Fig. 15,and therefore the work function <1', and also (ascan be seen from (1.27)) the red limit "cr arelowered.

I t should be emphasized in conclusion that theoptics of metals cannot be based on the model offree electrons that we used so far. It can be shownthat free electrons, i.e. electrons filling a poten­tial box with a flat bottom (Fig. 6), are unableto absorb photons. However, electrons in a period­ic field (Fig. 5b) can do so. Nonetheless, our argu­ment remains valid since the electrons in a pe­riodic field possess, as we shall see in the nextsection, a practically continuous energy spect­rum, which is like free electrons do. It consistsof very many closely arranged, or almost over­lapping energy levels.

1.5. Electrons in the Periodic Field.Conductors and Insulators

To conclude this chapter dealing with metals,it is natural to ask why some crystals are con­ductors and others insulators. If an external elec­tric field is applied, an electric current appearsin some crystals while no current appears in othercrystals (more exactly, the current that does ap­pear is many thousands of millions of times lessthan a current in a conductor). We shall try toexplain why in this section. We shall, however,have to put aside the simple model of the crystalwe have used so far. Instead of considering elec­trons to be locked in a potential box with aflat bottom, we must consider the periodic natureof an electron's potential energy within the crys-

56 Electrons in Metals

tal. In other words, we shall have to return fromFig. 6 to Fig. 5.

Let us proceed from a single isolated atom,and build up a crystal lattice from similar atoms

a

Isolatedatom

«(I)

Crystallattice

(b)

Fig. 17. Origination of an energy band: (a) isolated atom,(b) crystal lattice

as we did it in section 1.1. An atom of a chemicalelement differs from an atom of another elementby its energy spectrum, i.e. by the distribution ofenergy levels on the energy scale. The energy spec­trum of an atom is the set of all the values ofenergy that the atom may possess. The energyspectrum of an atom is a system of discrete energy.levels that become more concentrated together athigher energies. In case of a monovalent atom theenergyspectrum of an atom is the e~ergy spectrum

1.5. Electrons in Periodic Field 57

of its valence electron. The electron can reside onany of the energy levels. If it is on the lowest level,the energy of the atom is at the minimum andthe atom is said to be in its ground state. Thetransfer of an electron to any of the levels abovethis one means that the atom has been excited.

Now let us go on from an isolated atom to acrystal lattice. Every level W s in an atomic energyspectrum can shift and split into n closely posi­tioned levels to form an energy band; this is shownschematically in Fig. 17. Here n == N3 is thenumber of atoms in the crystal. Like we did be­fore (see Section 1.2), we shall imagine for thesake of certainty that the crystal is a cube, whereN is the number of atoms along the edge of thecube. Consequently, the electron's energy spec­trum in the crystal becomes a system of energybands that are separated by forbidden bands oroverlapping.

As in Sommerfeld's theory (the theory of freeelectrons, see Section 1.2) the state of each elec­tron is characterized by a set of three quantumnumbers n x , ny, n z that can assume the values

Nn:x:, ny, nz==O, +1, +2, ... , ± 2-

Instead of (1.5), we shall have the following foran electron's velocity

4na A • 2nv; =-h- p sin N nx '

4na · 2n 1VY=-h-~ SIn N nfl, ( .28)

4na . 2nV,= T~ SIn N n~.

58 ElectroDS in Metals

Instead of (1.6), we shall obtain the following foran electron's energy W at the lowest energy band

(2n 2n

W = W.+a-2~ cos Nnx+cosNny

+eos ~ nz ) . (1.29)

Here W s is the lowest energy level of an isolatedatom, a and ~ are parameters (a > 0, ~ > 0),and a, as before, is the lattice constant (the dis­tance between adjacent atoms). The upper limitWmax of the band occurs for values of the quan-tum numbers nx , ny, n% at which the cosines on theright-hand side of (1.29) are equal to -1. '¥heneach of the cosines equals +1, we get the lower ·limit Wmin of the band. Therefore, accordingto (1.29)the upper limit of the band is

Wmax = W.+a+6~,

the lower limit of the band is

Wmln = W.+a-6~,

the width of the band is

Wmax - Wm1n = 12~.

The only thing left is to distribute all elec­trons in the collection including the valence elec­trons of all atoms, among the levels of the bandkeeping the Pauli principle in mind. This prin­ciple requires that a given set of three quantumnumbers can only belong to two electrons withopposite spins. That is why the band has a cer­tain 'capacity' with respect to electrons, i.e,

t.5. Electrons in Periodic Field 59

it can only accomodate a limited number ofelectrons.

Now we can come back to the question we raisedat the beginning of the section, as to why somecrystals are conductors and some insulators. Fromthe viewpoint of the classical theory, the differ­ence between a conductor and an insulatorreduces to the presence or absence in the crystalsof free electrons, i.e. electrons capable of trav­elling throughout the crystal without hindrance.From the viewpoint of the quantum theory, everyelectron in the crystal (see Section 1.1) is free inthis sense. Although it gives electrons a certainfreedom of motion, the quantum theory imposesother constraints on their movement. The differ­ence between a conductor and an insulator isrelated to these new constraints.

We should note that the ability of electronsto travel is a necessary, but not a sufficient con­dition for the lattice to be a conductor. Indeed, ifthe electrons travel randomly in all directions,there is no electric current. For conduction toappear, it is necessary that the external electricfield regulates the random motion to a certaindegree, causing the requisite asymmetry in thevelocity distribution of the electrons. This secondcondition is a necessary complement to the firstone. In the classical theory, the first of thesetwo conditions was the prerequisite of conduction,while in the quantum theory the second conditionis prerequisite. In fact, the laws governing themotion of electrons are such that the symmetryof the velocity distribution of the electrons cannotbe disturbed for every crystal, i.e. can be dis­turbed for some crystals but not for others. For

60 Electrons in Metals

this reason not every crystal is a conductor,though every crystal contains electrons that trav­el freely over the lattice. Let us discuss the ori­gin of the new restrictive law that is inherent inthe quantum theory.

Let us return to the energy spectrum of thecrystal and consider the lowest energy band. Atabsolute zero the electrons are distributed amongthe levels of this band as densely as possible,starting with the lowest levels of the band, whichcorrespond to the lowest total energy of thesystem, But the band may be either partially orcompletely filled with electrons. These two casesare different. Let us consider each case separately.

(1) Imagine that the total number of electrons.is less than the number of places within the band.At absolute zero they will thus be distributed atth.e bottom of the band with the upper part ofthe band unoccupied. This is shown in Fig. 18awhere the energy Wand one of the velocity Com­ponents, vx , are shown as functions of the respec­tive quantum number n x • The range of n x thatis filled with electrons is double-hatched. We seethat the Vx velocity distribution of the electronscorresponding to the densest packing is symmet­rical. Thus the current in the crystal is zero be­cause for each electron with a velocity V x thereis an electron with a velocity -Vx • Note that wecan put vy or V z into Fig. 18 instead of vx , and nyand n z , respectively, instead of nx •

When. fin electric field is applied, the electronsare redistributed. among the levels so that thenumber of electrons with positive n x exceeds thenumber of electrons with negative n:c (or viceversa). This implies that there are more electrons

1.5. Electrons in Periodic Field 6i

travelling in one direction than in the oppositeone. We shall have to replace Fig. 18a with Fig.18b, which shows the distribution of the electronsin the presence of an electric field. The role ofthe field islto liberate some of the occupied levels

_fY...2

2

((1)

+­2

+J:{2

(IJ) (c)

Fig. 18. Distribution of electrons by states nx: (a) bandpartially filled with electrons (in the absence of an exter­nal field), (b) band partially filled wi th electrons (in thepresence of a field), (c) completely filled band

and to fill some of the vacant ones thereby intro­ducing some asymmetry into the velocity distrib­ution of electrons, as it is seen by comparingFigs. 18a and 18b.

This shift of electrons between levels is onlypossible, however, when there are vacant levelsin the vicinity of the occupied ones. This is pos­sible in this case since the band is only partiallyfull of electrons.

(2) Now imagine that the number of electronsin the lattice is equal to the number of levels in

62 Electrons in Metals

the band. This is the case when the densest pack­ing of the electrons results in every level in theband being occupied. This case is illustrated inFig. 18c where the whole range of n$ is double­hatched. I t is also essential in the case under con­sideration to know whether the filled band isadjacent to the vacant band lying above it orseparated from it by a forbidden band.

(a) If the band occupied by electrons is incontact with or overlaps the vacant band, thenthe vacant band can be considered to be an exten­sion of the filled one. This is the case when thevelocity redistribution of the electrons can becarried out at the expense of this overlying band.The two contacting or overlapping bands, thelower of which being occupied by the electrons.and the upper being vacant, can in reality beconsidered to be a single partially occupied band.

(b) If the band occupied by the electrons isseparated from the vacant band above it by a for­bidden zone, then a velocity redistribution ofthe electrons within the occupied band becomesimpossible. The only thing possible is for theelectrons to exchange places, but this will notbring about the desired asymmetry in the distrib­ution of the electrons (see Fig. 18c). An externalfield will then be incapable of producing anyeffect on the motion of electrons within the occu­pied. band, and therefore the lattice will beinsensitive to the influence of the external field.

Consequently, we come to the following cri­teria for distinguishing between a conductor andan insulator (see Fig. 18):

(1) A material is a conductor if the band ~con­

taining the electrons has a number of unoccupied

t5. Electrons in Periodic Field 63

levels, or if it is in contact with or overlaps thevacant band above it.

(2) A material is an insulator if the band iscompletely filled with the electrons and is sepa­rated from the vacant band above it by a forbid­den zone.

The forbidden zone between the bands can beovercome as the result of external effects, anda certain number of electrons can be removedfrom the underlying occupied band to appear inthe vacant band. This results in an insulator acquir­ing the ability to conduct electricity and thetopic will be considered in more detail in laterchapters that deal with nonmetallic crystals.

Chapter 2

Electrons in Semiconductors

2.1. 'Order' and 'Disorder' in Crystals

In this chapter we shall discuss the solids thatare called semiconductors. As we already know(see 'Introduction', section 1.2), they are a re­markable group of solids that are intermediate intheir properties between metals and dielectrics.Metals can be regarded as an extreme case ofsemiconductors with dielectrics at the other ex­treme.

Note that all the macroscopic (i.e. directlyobservable) properties of crystals (including semi­conductors) can be put into two classes. Thefirst contains all the properties that are deter­mined by the periodic structure of the crystallattice and for which deviations from the regu­larity of the structure (called defects) which areinevitably present in any real lattice, do notplaya significant role. These properties are calledstructure-stable. The other class includes all theproperties that are determined by the local ir­regulari ties in the periodic structure of the Ia t­tice. The defects in this case are very important.The properties belonging to this second classare commonly called structure-sensitive.

Theoretical interpretations of structure-stableand structure-sensitive properties require dif­ferent approaches. In the first case we can startfrom the theory of ideal crystal lattice, while

2.~. 'Order' and 'Disorder' in Crystal! 65

in the second case we have to deal with conceptsof real crystals. The theory of ideal crystals cannot handle any structure-sensitive property. Thetheory of semiconductors does not deal withideal crystals, but with real ones.

A real crystal differs from an ideal one inthat it has defects, Le, local irregularities inthe periodic structure of the lattice. We shoulddistinguish between the macroscopic and micro­scopic defects which are inherent in any reallattice. A macroscopic defect is a disturbancein the periodic structure embracing a region whosedimensions are substantially greater than thelattice constant. These include fractures, pores,and macroscopic inclusions. We are not goingto discuss defects of this kind here. A microscopicdefect is an irregularity whose dimensions areof the same order as the dimensions of an individ­ual crystallographic cell. Let us list the mainmicroscopic defects:

(1) an empty lattice point (a vacancy), theresult of removing an atom or ion from an ideallattice (Fig. 19a);

(2) an atom or ion of the lattice in an inter-stice (Fig. 19b);

(3) a foreign atom in an in terstice (Fig. 19c);(4) a foreign atom at a lattice point (Fig. 19d).The last two types of defects can be called

chemical defects. This is an 'impurity' in thenarrow sense of this word, an irregularity inthe chemical composition of the crystal. Chemi­cal defects come into a crystal from without:i.e. they appear due to its processing.

The first two types of defects can be calledstructural defects. In case .of a monatomic lattice

i-01536

66 Electrons in Semiconductors

(I,e. a lattice composed of atoms of only onekind), such defects do not distort the chemicalcomposition of the crystal. If the lattice is multi­component, they can distort the crystal's stoi­chiometry, i.e. change the relative quantities

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 •

0 0 0 0 0 0 0 0 0 0 0 0 0 • 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

(a) (b) (c) tdl

Fig. 19. Microdefects in a semiconductor crystal (ablank circle is a proper atom of the lattice, a filled circleis a foreign atom): (a) vacancy, (b) proper atom of thelattice in an interstice, (c) foreign atom in an interstice,(d) foreign atom substituting a proper atom of the lat­tice

of the component atoms that is a characteristicof an ideal lattice*. Structural defects can beintroduced from without and can appear" whenthe lattice is heated.

Note that each microdefect deforms somewhatthe lattice around it. Strictly speaking, thewhole region within which the lattice is deformed

• For instance, if the crystal is of cuprous oxide Cu20,then tbere should be 200 atoms of copper for every 100oxygen atoms, in which case the stoichiometric relation­ship is said to he correct. But if there are 205 or 195 cop­per a toms for every 100 oxygen atoms, then the stoichio­metry of the crystal is said to be distorted, in the formercase towards an excess of copper (or deficiency of oxygen),in the latter case towards a hyperstoichiometric pre­valence of oxygen (or deficiency of copper).

2.t. 'Order' and 'Disorder' in Crystals 67

should be considered to be a defect. Irrespec­tive of their actual nature, microdefects havea number of common properties.

1. Microdefects have a certain mobility thatincreases with temperature (i.e. they have anactivation energy). In fact, as a defect movesin any direction the system's energy changesperiodically, i.e. the energy passes throughalternate minima and maxima. Defect move­ment, therefore, is connected with overcomingenergy barriers whose levels are determinedby the nature of the defect, the lattice structure,and direction of travel. Defect movement requires,therefore, activation energy that is generallydifferent for different crystallographic directions.Defects can only be assumed to be fixed withina crystal at moderate temperatures.

2. Another common feature of microdefectsis that they interact when they approach eachother. The system's energy generally dependson the relative distribution of the defects, whichis the evidence of the interaction between them.Defects can attract or repel each other. Forinstance, in an ionic lattice MR, where M isa metal and R is a nonmetal, and which consistsof the ions M+ and R -, nonme tallic vacanciesrepel each other, but attract metal vacancies.

3. Having met, defects can produce compoundsby forming groups that should be considerednew defects and which generally possess otherfeatures. For instance, when a nonmetallic anda metallic vacancy join in an MR lattice, theyproduce a different sort of formation that hasproperties which are different from those itscomponents have when considered separately.

5*

68 Electrons in Semiconductors

The 'reactions' between defects, can, like ordi­nary reactions, be both exothermic and endo­thermic, and can occur both with and withoutactivation, depending on the nature of thereacting defects.

4. Defects of each kind, while 'reacting'with other defects, can thereby appear anddisappear. A mean lifetime can thus be attrib­uted to them in equilibrium conditions. Theycan in addition be absorbed or generated bythe lattice itself. An example of this is movementof an atom (or ion) from a lattice point into aninterstice which results in the appearance of twokinds of defects: interstitial atoms (or ions) andvacancies. Another example is, vice versa, therecombination of an interstitial atom (or ion).and a vacancy to result in the disappearance oftwo defects which, as it were, 'absorb' eachother.

The presence of defects in a crystal testifiesthat the order characteristic of an ideal crystalis disturbed in a real crystal. Individual crystalcells in such a crystal are 'spoiled'. The ratioof the number of spoiled cells to the overallnumber of cells is what can be called the degreeof disorder in the crystal.

The degree of disorder is determined prima­rily by the 'biography' (history) of the crystal,i.e, it depends on how the crystal was producedand what treatment it was subjected to. Thedegree of disorder also generally depends ontemperature. Heating actually tends to removelattice atoms (ions) from their lattice points,pushing them into interstices, and thereforeresults in more disorder in the lattice. Thus,

2.2. Free Electrons and Holes in Semiconductors 69

disorder in the lattice can be either 'biographi­cal' or thermal in origin. Biographical disorderis irreversible and remains at absolute zero,i.e, it is, as it were, a congenital disorder of thelattice. Disorder thermal in origin is superim­posed on biographical disorder.

In some cases thermal disorder prevails overbiographical disorder, it then being possible toneglect biographical defects compared with de­fects of thermal origin. But in other cases it ispossible to neglect thermal disorder comparedwith biographical disorder and this happens tocrystals that have many defects and at moderatetemperatures. This is the case when we dealwith semiconductors.

Consequently, a crystal lattice with itsdefects is an indivisible system whose propertiesare governed by two competing factors: thefactor of order and the factor of disorder. Theorder controls all the stable properties, and thedisorder all the structure-sensi tive properties ofthe crystal.

2.2. Free Electrons and Free Holesin Semiconductors

What are free electrons and free holes? How dothey behave within semiconductors? These arethe questions we shall try to answer in thissection.

Imagine the crystal lattice of a semiconduc­tor. Suppose that an extra electron, in additionto the normal set, is assigned to an atom (or ion)of the crystal lattice. 'I'his redundant electronis called a free electron. since it is capable of

70 Electrons in Semiconductors

travelling from one atom (ion) to a neighbouringatom (ion) and therefore of wandering throughoutthe lattice. However, it only needs a potentialdifference to be applied to the crystal for a predom­inant direction in the chaotic movement of the

+

Fig. 20. Electronic and hole currents (arrows show thedirection of the motion of an electron and a hole)

electrons to become apparent. On the average,they all begin travelling in one direction (fromthe negative to the positive electrode), and thisis illustrated diagrammatically in Fig. 20.Current appears in the semiconductor. We saythat there is electronic conduction in the semi­conductor since the current is due to electronstravelling. A semiconductor containing a con­centration of free electrons i ,e. possessing electro­nic conduction, is called an electronic semicon­ductor or n-type semiconductor (n is for 'negative'because negative charges are the current car­riers).

2.2. Pree Electrons and Holes In Semiconductor! 7f

Now suppose we do not add an extra electronto an atom (ion) of the-lattice, but instead removean electron from the normal set. We obtainan atom (or ion) that differs from all otheratoms (ions) by its lack of an electron. This iscalled a hole. We call it free because it cantravel around the crystal without hindrance,wandering from atom (ion) to atom (ion) of thesame element. When an external electric fieldis a-pplied, a hole current a-ppears in the semi­conductor (see Fig. 20). We say that there ishole conduction in the semiconductor, and thesemiconductor itself is called a hole, or p-type,semiconductor (p is for 'positive' since, as itwere, positive charges are the current carriers).

Take for example zinc oxide ZnO whoselattice we shall consider to be an ionic latticebuil t up of the Zn ++ and 0-- ions. A free electroncorresponds to the Zn" state, while a free holecorresponds to the 0- state. As another example,consider the lattice of cuprous oxide Cu 20,which is composed of the Cu" and 0-- ions.A free electron corresponds to the Cu state,while a free hole corresponds to the Cu"" state,both wandering among the regular Cu" ions.In a monatomic lattice, that of germanium forinstance, which is built up of the neutral Geatoms, a free electron or a free hole implies thepresence of a Ge- ion or a Ge+ ion, respectively,among the neutral Ge atoms.

The difference between electronic and holecurrents is illustrated by the following analogy.Suppose we have a school classroom with onlyone schoolboy in the first row, and all the placesin this row are vacant. The boy changes the

72 Electrons in Semiconductors

seats, say, from left to right and therefore moveswithin the row. This is an analogy of an electron­ic current flowing through a semiconductor.Now suppose all but one of the places in thefirst row are occupied by schoolboys, and the boyon the left of the unoccupied seat moves to takeit. The vacant place thus moves along the rowfrom right to left. This is an analogy of a holecurrent.

Note that a free electron (like a free hole)produces an electric field around it. The fieldbecomes less intense with the distance from thesource. If the semiconductor's lattice is ionic,then the ions surrounding the free electron (orhole) are within the field, and it tends to displacethem from their equilibrium positions. Ions ofone sign are attracted to the electron (or hole)while those with other sign are repelled from theelectron (or hole). The crystal lattice around theelectron (or hole) is thus somewhat deformed, thedeformation however decreases rapidly with thedistance. Thus it only covers the immediatevicinity of the electron (or hole). This formation(the electron or hole and the surrounding defor­mation of the lattice) is called a polaron. A pola­ron can travel across the lattice. A movingpolaron is a moving electron (or hole) pullingbehind it the deformation of the lattice it pro­duces. Naturally, it is more difficult for a pola­ron to move across a crystal than it is for just anelectron (or hole) if we neglect the distortion.

A semiconductor can under certain conditionshave at the same time both free electrons andfree holes in any concentrations. A semiconduc­tor is said to be intrinsic if the concentrations

2.2. Free Electrons and Holes in Semiconductors 73

are the same and both electronic and hole cur­rents appear if an external field is applied.We then say that there is mixed conductionin the semiconductor.

What is the source of these free electrons andholes? This is where the structural defects orimpurity atoms in a crystal come into play.

+

Fig. 21. Origina tlon of free electrons and holes (arrowsshow the motion of an electron and a hole). A cross­hatched circle is a proper atom of the lattice.ja blank circleis a donor im purity a tom, a filled circle is an acceptor im­purity atom

They can be divided into two groups accordingto the role they play in the electronic set-upof the crystal and we distinguish between donorand acceptor defects.

Donor defects are the suppliers of free elec­trons. An electron escaping from such a defectstarts travelling around the lattice. It leavesa hole that remains localized (Fig. 21). Thusdonor defects are the reservoir from which thesemiconductor draws free electrons. There are

74 Electrons In Semiconductor!

as many free electrons in the semiconductor asthere are ionized donor defects.

Clearly, the concentration of free electrons,and thus the electronic conduction of the semi­conductor, increases when the concentration ofa donor impurity (or donor defects) is increased.The concentration of free electrons also increaseswith the rise of temperature, because moreimpurity atoms (defects) are ionized at highertemperatures.

Donor defects play the opposite role withrespect to free holes. They do not supply freeholes but rather trap them. An electron froma donor defect can recombine with an approach­ing free hole, as a result of which the freehole disappears and a localized hole appears"in the defect. The larger the concentration ofdonor defects the less the concentration of freeholes becomes. That is why adding donor defectsto hole semiconductors reduces rather than raisestheir conductivity..

Let us consider now acceptor defects. Theyplay the role that is opposite to that of donordefects. Acceptor defects are traps for free elec­trons and suppliers of free holes. They catchelectrons, and therefore the introduction ofacceptor defects brings about a decrease inelectronic conduction and an increase in holeconduction.

Hyporstoich lornetrlc atoms of zinc in a crys­tal of zinc oxide ZnO, which is a typical n-typesemiconductor, are an example of donor defects.An example of acceptor defects are vacanciesof copper atoms in a crystal of cuprous oxideCU20, which is a p-type semiconductor.

2.2. Free Electrons and Holes in Semiconductors 75

A crystal of germanium can be made eitherto be an n-type or a p-type semiconductor depend­ing on whether acceptor or donor impuritiesare added. Let us have a look at Mendeleev'speriodic table of elements. Germanium (Ge) isin group 4. This means that a germanium atomis tetravalent, i.e. it has four valence electrons.Suppose we change the crystal lattice of germa­nium by adding as an impurity some atoms ofa fifth-group element, for instance, arsenic (As)or antimony (Sb), so that they substitute germa­nium atoms. Unlike germanium atoms, theseatoms possess fi. ve valence electrons not four,i.e. one electron more than germanium. Theseimpurities in a germanium crystal will act asdonors, so a germanium crystal with an arsenicor antimony impurity is an n-type semiconductor.

By contrast, if we include third-group atomsin a germanium crystal (for instance, gallium(Ga) or indium (In) atoms which possess threevalence electrons, i.e. one electron less thangermanium atoms) then we have added an ac­ceptor impurity. A crystal of germanium withan impurity of gallium or indium is a p-typesemiconductor.

Note that a semiconductor can draw freeelectrons and holes from the lattice itself athigh temperatures or low defect concentrations.By moving an electron from an atom (or ion)of the lattice to another atom (or ion) of thelattice, we get an electron-hole pair. Havingpulled the electron and the hole far enoughapart so that any oloct.rostat.ic in ternction be­tween them can he neglected, we get a froe elec­tron and a free hole, independent of each other.

76 Electrons in Semiconductors

2.3. 'Energy Bands' and 'Localized Levels'

Until now we have described things in terms ofmodels, but it is more convenient to describethe electronic processes in semiconductors interms of energy.

The energy spectrum of an electron in anideal crystal lattice, as we have already men­tioned, is a system of energy bands that either areseparated by forbidden zones or overlap. A finitenumber of electrostatic states corresponds toeach band, so that each band (recall the Pauliprinciple) has a certain capacity with respectto the electrons, i.e. it can accommodate onlya finite number of them.

At absolute zero, when the system energyis at the minimum, all the lower bands arefilled with electrons to capacity, while all thehands above them are totally vacant. We shalldiscuss the uppermost of the occupied bands,which we shall call the valence band, and thelowermost of the unoccupied bands, which weshall call the conduction band. When the widthof the forbidden zone between these two bandsis zero, it is a metal. If the width is not zero,it is a nonmetallic crystal (a semiconductor orinsulator). The appearance of an electron in theconduction band means a free electron hasemerged. The appearance of a hole in the valenceband means a free hole has emerged.

Let us consider an electron in the conductionband. Suppose its state is characterized by a setof three integer quantum numbers n x , nIL' n ,and a spin that can have either of two values.The energy W of the electron (as in the theory of

2.3. 'Energy Bands' and 'Localized Levels' 17

(2.1)

(2.2)

metals, Chapter 1) can be presented in the simpl­est case as (1.29). Let us write this formula onceagain:

(2n 2n

W = W 8 +ex - 2~ cos N n% + cos N nll

21t )+cos N nz ,

where N is the number of atoms along the edgeof the cube (we assume, as in the preceding chap­ter, that the crystal is cubic). Near the bottomof the conduction band, where the absolutevalues of the numbers nx , n ll , nz are small, eachof the cosines on the right-hand side of (2.1)is close to unity, and so we can use ap.proximateformulas that are the first expansion terms:

2n 1 (2n )2cos IT nx = 1- '2 N n« ,

2n 1 (2n )2cosNnll=1- 2 N nll ,

2n . 1 (2n )2cos IT n z = 1 - 2" N nz •

Substituting (2.2) into (2.1), we obtain

w= ~: ~(ni+n~+n~)+ Wo• (2.3)

where Wo is a constant, Wo = W" + a - 6~.We come to the familiar formula (1.8) fromSection 1.2 for a free electron gas

(2.4)

Compare (2.3) with (2.4) and note that theseformulas are the same correct to the additive

18 Electrons in Semiconductors

constant W 0 if m in (2.4) is replaced withh2

m*---~- 8a2n2~ ,

i.e. if the electron is ascribed an effective mass m*instead of its true mass m. I t is clear that ourelectron in the semiconductor conduction bandbehaves like a free electron of the Sommerfeldgas, if an effective mass m* is ascribed to itinstead of its true mass m, As for the additiveconstant W 0 that distinguishes (2.4) from (2.3),its presence is inessential, because it only meansa change in the reference point for energy.

At temperatures above absolute zero someof the electrons are moved from the valenceband into the conduction band. Let us call theconcentrations of free electrons and free holesnand p respectively. We shall determine nand pfor the case when the equilibrium in the semi­conductor is established. The probability for anelectron to be driven from the valence bandinto the conduction band is proportional to

exp (- Wc~Wv), (2.5)

where W c - Wv is the width of the forbiddenzone between the bands (Fig. 22a). The proba­bility that a free electron travelling in theconduction band will recombine with a free holetravelling in the valence band is proportionalto np_ Let us note that in our case

p == n,i.e. the number of electrons moved to the conduc­tion band equals the number of holes left in thevalence band. Thus, under equilibrium condi-

2.9. 'Energy Sands' and 'Localized Levels' 19

tions

C (WC-WV)\ 2n = exp - 2kT ' ( .. 6)

where C is a proportionality factor whose expres­sion is inessential here. Formula (2.6) refers toan ideal lattice, i.e, a semiconductor in which

~ -~ ~)$WweWD'~ -.- ..........~

N,AND~

-,- --w,A

Wv~~ --(a) (b) (c)

Fig. 22. Energy diagrams for semiconductors. It'c isthe bottom of the conduction band, Wv is the ceiling ofthe valence band, W D is a donor localized level, WAis an acceptor localized level

impurities and structural defects can be neglect­ed. This is called an intrinsic semiconductor(i-type semiconductor).

Now let us deal with a real lattice, i ,e ..a lattice with either donor or acceptor defects,or both at the same time. The presence of defectsinfluences the energy spectrum, and localizedlevels appear in the energy spectrum along withthe energy bands. The energy bands can beconsidered to be spread over the whole crystalin the sense that an electron belonging to a bandcan occur with equal probability anywhere

80 Electrons in Semlconductors

within the crystal. An electron belonging toa localized level is localized to a fairly limitedspace around a defect.

We should distinguish between two classesof localized levels: donor and acceptor levels.A donor level is a level occupied by an electronat absolute zero. By definition, an acceptor levelis a level vacant at absolute zero. With a risein temperature, acceptor levels become partiallyfilled with electrons while donor levels losesome of their electrons. Donor levels can releasetheir electrons to the conduction band, thusenriching the pool of free electrons. On theother hand, an electron from a donor level canrecombine with a free hole, thus reducing thepool of free holes. At the same time, an acceptor'localized level can accommodate an electronfrom the conduction band, thus reducing thepool of free electrons in the crystal. On theother hand, an electron from the valence bandcan be drawn to a vacant acceptor level leavinga hole instead. The crystal can be enriched withholes in this manner. Clearly, the roles of accep­tors and donors are quite different and in a senseoppose each other.

We shall determine now the concentration nof free electrons in an electronic semiconductorwith donor defects of only one kind (see Fig. 22b).Suppose the localized levels of these defects arenot too far below (at a distance W c - W D) theconduction band and suppose N D is the COIl­

centration of defects. At a steady electronicequilibrium, the number of electrons jumpingdue to thermal motion from the localized levelsto the conduction band per unit time per unit

2.3. 'Energy Bands' and 'Localized Levels' 81

volume should be equal to the number of reversetransitions of electrons from the conduction bandto the localized levels, i.e. to the number ofrecombinations between free electrons and thebound holes that stay at the localized levelswhen their electrons leave them. Thus we get

C 1\, (WC-WD ) (2 7)np = D D exp - kT' •

where W c - W D is the depth of the donor levelsbelow the conduction band, C D is a proportional­ity factor, and p is the concentration of vacantlocalized levels (holes). Since p === n, we get

"l/C "-T (WC-WD )n===- f DllDexp - 2kT' (2.8)

Obviously, the concentration n of free electronsincreases as more donor impurities are added tothe semiconductor (N D increases), and as thetemperature T rises.

Now let us tackle a p-type semiconductor(Fig. 22c) .. Let us determine concentration p offree holes in the valence band at electronic equi­librium. Just as in the determination of theconcentration n of free electrons in an n-typesemiconductor, we shall get

p=VCANA exp ( - WA2~Wv ), (2.9)

where N A is the concentration of acceptors inthe crystal, W A - W v is the distance fromacceptor localized levels to the ceiling of thevalence band, and CA is a proportionalityfactor.

For.mulas (2.8) and (2.9) correspond to thesimplest case when an n-type semiconductor

6-01536

82 Electrons in Semiconductors

contains only one kind of donors, and a p-typesemiconductor contains only one kind of accep­tors. In reality the same crystal can possessseveral kinds of donors and several kinds ofacceptors, and then the expressions for nand pare more complicated.

2.4. Semiconductor Conductivity

Electrical conductivity or specific conductanceis one of the most interesting and importantcharacteristics of a semiconductor. As before,we are going to designate it by the letter a.Conductivity can be expressed as

a == eun (2.19)

where e is the absolute value of the electroncharge; u is the mobility of the current carriers(electrons or holes), i ,e, the average drift veloc­ity of the current carriers in the direction ofelectric field E (at E == 1 V/cm); and n is theconcentration of carriers in the semiconductor.Formula (2.10) is quite general in nature andthe conductivity of a metal can also be expressedthis way_

A semiconductor differs from a metal aboveall by the fact that the concentration of carriersin a semiconductor is several hundred thousandtimes less than that in a metal. Metals contain1023 (or more) carriers per cubic centimeter.Besides, n in semiconductors is, as a rule, verysensitive to temperature, increasing hundreds ofthousands of times as the temperature risesmoderately. And, finally, n in .semtconductorsis usually very sensitive to impurities (to their

2.4. Semiconduc tor Conductivity 83

nature and concentration), while n in metalsis a constant that depends solely on the natureof the metal and does not depend at all on thetemperature or the impurity concentration.

The mobility u that appears in (2.10) poorlydepends on temperature. This poor dependenceof u on T, compared with the strong dependenceof n on T, allows us in many cases to assumethat u in formula (2.10) is constant and does notdepend on T at all.

Let us see how a in an electronic semiconductordepends on T and N D. We substitute (2.8) into(2.10) to get

0= 00 exp (

where

Wc-Wn)2kT '

(2. 11)

00 = eu y GDND- (2.12)

(2. 13)

Taking logarithms of both sides of (2.11) weobtain

I 1 WC-WDnO'=: n 00 - 2kT ·

By plotting the logarithm of conductivity alongthe ordinate, and the reciprocal of the absolutetemperature along the abscissa, equation (2.13)is shown by a straight line (Fig. 23). The angle abetween this line and the abscissa rises as thelocalized donor level that provides the conduc­tivity sinks below the conduction band:

tan ex -== Wc-WnkT

6'*

84 Electrons in Semiconductors

(2.14)

NO\V let us take a hole semiconductor. Assum­ing p ~ n in (2.10) and substituting (2.9)

In 0

IT

Fig. 23. Semiconductor conductivity as a function oftemperature

into (2.10), we obtain an expression similarto (2.13), i.e.

I I WA-WVn a = n a0 - 2kT '

where

ao=euVCANA -

In this case the straight line in Fig. 23 getssteeper the higher above the valence band thelocalized acceptor level W A is, i ,e.

WA-WVtana= kT -

Now let us take, at last, an intrinsic semi­conductor (i-type semiconductor). This type ofmaterial conducts in both ways and so insteadof (2.10) we have

a = e (unn + upp), (2.15)

where Un and up are the mobilities of electronsand holes respectively, nand p are the concentra-

2.4. Semiconductor Conductivity 85

tions of free electrons and free holes in theconduction band and the valence band respec­tively. If all the electrons in the conduction bandare borrowed from the valence band, formu­la (2.15) transforms into (2.10), where U is the

-7

log c

-3

-5-4-6-8-10

-""""_""""---...L...-~_-L103 ~ 1034 6 8 lOT 0 2.5 5.0 7.5 T

(a) (I})

Fig. 24. Conductivity as a function of temperature:(a) for a crystal of CuO, (b) for a crystal of W0 3

log (J

sum of mobilities for carriers of two types:

U = Un + Up'

In this case the slope of the line in Fig. 23 isthe more, the wider the forbidden zone betweenthe bands:

wc-wvtan a == kT •

Some typical experimental curves are shownin Fig. 24. Obviously, they are in good agree­ment with theory and conductivity increaseswith the rise of temperature very sharply. InFig. 24b, a doubling of temperature from 1IT === 5 X 10-3 to 1IT == 2.5 X 10-3 raises theconductivity about 10'1000 times (from a == 10-7

to (T == 10 -3 Q -1 • cm -1 -== 10-1 S. m -1) .

The straight line in Fig. 23 corresponds to

86 Electrons in Semicond uctors

the simplest cases. This line sometimes hasa bend, like in Figs. 25a, 25b. These cases alsocomply with the simple theory if the concentra­tions of acceptor and donor impurities, as well

log a

-1

-3

0.8 1.0 1.2

log (1

-2

-4

-6

-8

-10 L..--------L-_....L...-.--....l..-_

o 2.5 S.O 7.5

(a) (b)

Fig. 25. Bends of the conductivity versus temperaturediagrams: (a) for a crystal of ZnO, (b) for a crystal of CuO

as the respective localized levels in the forbid­den zone between the energy bands, are selectedproperly.

Let us consider Fig. 26. This shows a familyof curves for the same semiconductor but" withdifferent concentrations of an introduced impu­rity. The impurity concentration correspondingto the A· 3 B3C curve is greater than that corre­sponding to the A2B 2C curve, which in turn isgreater than that corresponding to the A1B1Ccurve. Obviously, the conductivity of a crystalat moderate temperatures is the greater, thegreater the concentration of impurity in it.Conduction in this case is due to electron transi­tions between localized levels and a band.This can be either electronic or hole conduction.At high temperatures the conductivity is tlrosame for all crystals (i.e. it does not depend 0 n

2.4. Semiconductor Conductivity 87

impurity concentration) since the conduction jsdue to electron transitions from band to band.

log a

IT

Fig. 26. Tempera ture diagram of conductivity for speci­mens with different impurity concentrations,

2 3(b)

o

.- 6

-2

-4

(a)

+2

+4

This is a case of mixed conduction. To illustratethe point, Fig. 27 shows experimental curves

Fig. 27. Temperature di-agram of conductivityfor Cu20 (a) and T12S (b) log (1

specimens with varyingdivergence from stoichio- +6metry

-5

-7

log (1

88 Electrons in Semiconductors

for cuprous oxide CU20 and thallium sulphideT'l2S. Fig. 27a gives curves for crystals of CU20(a hole semiconductor) with different concentra­tions of hyperstoichiometric oxygen, while the

z

Fig. 28. Transverse field E H caused by the Hall effect

curves in Fig. 27b are for Tl 2S (an electronicsemiconductor) with different concentrations ofh yperstoichiometric thallium.

A temperature diagram of conductivity ·doesnot allow us to infer the nature of conduction,i ,e. whether there is electronic or hole conduction.To do this we need some additional measure­ments, and those pertaining to the Hall effectare suitable.

To understand the Hall effect suppose thereis an electric current of density i flowing alongthe y-axis of a specimen (Fig. 28). Then supposethat the specimen is placed in a magnetic fieldwi th a strength H directed along the z-axis,i ,e. perpendicular to the current flow. A poten­tial difference arises between the sides of thespecimen," i.e. an electric field is set up directed

2.4. Semiconductor Conductivity 89

along the z-axis and we shall designate itsstrength E H. This is called the Hall field. I t isperpendicular both to the direction of the currentand to the direction of the magnetic field. Theappearance of an electric field due to a magneticfield is the Hall effect.

E H, H, and i are related thus:

EH=RiH, (2. 16)

(2.17)

where R is a proportionality constant called theHall coefficient or llall constant. A mathematicaltreatment produces the following expression forR:

pu 2 -nu2R-r.x 11 n

- (nun+pu p )2 ·

Here ex, is a scalar constant that is of no interestfor us. The rest of the notation is the same asbefore.

We have p == 0 for electronic semiconductors,and hence (2.17) becomes

R--~- n:

Meanwhile n == 0 for hole semiconductors, andhence (2.17) beCOlnes

R==.!!:...p.

Therefore

R > 0 for p-type semiconductors, and

R < 0 for n-type semiconductors.

90 Electrons in Semiconductors

In the. case of intrinsic semiconductors, whenn == p, we obtain from (2.17)

N u2 -u2

R == .z: P n (2 18)ni (up + un)2 •

where we introduced the notation n, = n = p.Since

u; - u~ = (up + un) (up - un),

we can rewrite (2.18) as

(2.19)

If the electrons and holes are equally mobile,i.e. up = Un' then it follows from (2.19) thatthe Hall constant becomes zero (R = 0). •

Note in conclusion that along with the deter­mination of the sign of the Hall constant thereis another way of establishing the sign of thecharges that carry the current in semiconductors.It consists in the study of the thermal electro­motive force (thermo-emf) that appears in any semi­conductor if there is a drop of temperaturewithin it. Suppose a semiconductor is a rod oneend of which is heated and the other end ofwhich i.s cooled. Since the concentration ofcarriers in a semiconductor is usually verysensitive to temperature, the concentration ofcarriers at the hot end of the specimen will begreater than that at the cool end. There willthus be diffusion, i.e. carriers from the hot endwill travel to the cool one to try to equalize theconcentrations. If the current carriers are elec­trons, then the cool end will receive an excessof electrons, and SQ will become negatively

2.5. Electrons and Quanta 91

charged, while the hot end will be left withoutsome of its electrons and so will become posi­tively charged. But if the current carriers areholes, then by contrast the cool end will becomepositively charged and the hot end negativelycharged. The data we obtain from the signin the Hall effect usually coincide with the datafrom the sign obtained using the thermo-emftechnique.

2.5. Electrons and Quanta

What will happen in a semiconductor if it isexposed to visible or ultraviolet light or X-rays?We shall try to answer this question in thissection.

When speaking of the exposure of a semicon­ductor, we mean the bombardment of a specimenwith light quanta with a given frequency 'V

or, in other words, by quanta with a mass m,because 'V and m are related, i.e.

(2.20)

where hand c are universal constants, h beingPlanck's constant (h := 6.62 X 10-27 erg-s), andc being the speed of light (c == 3 X 108 m/s).Therefore the smaller 'V is, i.e. the 'redder' thelight, the lower m is, i.e. the 'lighter' the quan­tum.

Not every incident quantum is absorbed bya crystal. The absorption spectrum is a systemof bands that more or less overlap. Some handsare structure-stable, i.e. they are not sensitiveto the way a specimen .has been processed (in-

92 Electrons in Semiconductors

trinsic absorption), other bands are structure­sensitive, i.e. they depend on the concentrationand the nature of impurities the crystal contains(impurity absorption). The absorption of a quan­tum by a crystal is the evidence that the systemhas become excited, i.e. its energy has beenincreased. Usually, this excitation is related to

~(a) (b) (c)

Fig. 29. Origination of: (a.) photoelectron, (0) photohole,(c) pair: photoelectron plus photohole

a transition of an electron from a lower to a higherenergy level. When a quantum is absorbed, anelectron-hole pair appears. There are three pos­sible cases:

(1) one of the partners (the electron or thehole) becomes free, and the other stays bound(localized) following the absorption of the quan­tum (Figs. 29a, 29b);

(2) both partners (the electron and the hole)become free and independent of each other(Fig. 29c);

(3) both partners stay bound to each otherand behave as an entity.

To begin with, let us take the first case.Suppose the crystal contains donor impurities

2.5. Electrons and Quanta 93

(or donor structural defects). The absorption ofa quantum ionizes a defect. An electron leavesa donor localized level and goes to a level in theconduction band, i.e. becomes free. This elec­tron, which we shall call a photoelectron, becomesable to travel over the lattice. The hole that isleft remains localized at the site the electron left.

Suppose that the crystal contains acceptorlocalized levels. When a quantum is absorbed,an electron leaves a level in the valence bandand goes to an acceptor localized level, i ,e.a free hole (a photohole) appears in the valenceband, and a localized electron appears on anacceptor localized level.

Let us remark that the donor aud acceptorlocalized levels responsible for the appearanceof photoelectrons and photoholes may be notthe same donor and acceptor levels that bringabout conduction in the dark.

Now let us take the second case, when theabsorption of a quantum results in the appearanceof two free carriers: an electron and a hole.

While impurity absorption was the pointin the preceding case, now the point is intrinsicabsorption, when an electron leaves a level inthe valence band (the emergence of a free hole)and goes to a level in the conduction band (theemergence of a free electron).

Let us note that the appearance in a crystalof either photoelectrons or photoholes, or bothis called the internal photoelectric effect, asopposed to the external photoelectric effect whichis when electrons are emitted due to radiationboth from semiconductor and metal crystals (seeSection 1.4).

94 Electrons in Semiconductors

Now let us take the third case. Suppose theabsorption of a quantum results in an electrongoing from one atom or ion of the lattice to anadjacent atom or ion. The electron and thehole that appear remain close together and inter­act electrostatically. This formation is calledexciton (from the word 'excitation'), and canwander about the lattice as an entity that isnot a current carrier. We shall return to themechanisms by which excitons appear anddisappear at the end of this section.

Meanwhile we shall discuss the destiny ofthe photoelectron and the photohole that appearedin the crystal as the result of light absorption.

The photoelectron can be moved by a lightquantum high into the conduction band. It.then has a reserve of energy. Any electron trav­elling around a lattice collides with the atoms(ions) of the lattice, thus spending its energy.While it stays in the conduction band, theelectron gradually moves down from its higherto its lower levels due to the collisions. Whenit reaches the bottom of the conduction band,the electron joins the pool of thermal electrons,i.e. the electrons that are present in some con­centration in any crystal lattice without anyincident radiation. A photoelectron is identicalto a thermal electron, only its origin is different.It has the same mobility as a thermal electronand at the end of its life it recombines witha hole in the crystal. The mean lifetime ofa photoelectron is estimated to be 10-3-10-7 s.

At the start of its life, when the energy of thephotoelectron is quite large, it can collide withan atom (ion) of the lattice and ionize it, i.e.

2.5. Electrons and Quanta 95

bring about a secondary free electron. Thissecondary electron, if its energy is sufficient,can bring about a tertiary electron. In this waythe electrons can multiply. We see that oneabsorbed quantum can give rise to severalphotoelectrons and not just one. The destiny ofa photohole is similar.

When there is a steady incident radiation,a stationary process can be established, whenthe number of appearing photoelectrons (photo­holes) equals the number of recombining onesfor the same time period. I t turns out that theconcentration of carriers (electrons or holes) isgenerally different from that in the absence ofradiation. Therefore the conductivity of a crystalis changed. The difference between the conduc­tivities of a crystal due to irradiation and thatwhich can appear in the dark is called photo­conductivity.

Photoconductivity depends on the intensityof the absorbed light. We shall discuss only thesimplest case when the number of thermal tran­sitions of electrons into the conduction band isnegligible compared with the number of photo­transitions. Let us call the latter a l • Clearly,a l is proportional to the number of absorbedquanta, which we shall designate J, i.e.

(2.21)

where ~ is a proportionality factor. The number(X2 of recombining electrons is proportional tothe concentration n of electrons and that ofholes, also equal to n:

(2.22)

96 Electrons in Semiconductors

where y is another proportionality factor. From(X,l == (X,2 we obtain, according to (2.21) and(2.22)

~J = yn 2 ,

whence

(2.23)

We see that the concentration n of photoelectronsthat bring about photoconduction is directlyproportional to the square root of the in tensityof absorbed light.

As a rule, photoconduction is positive, i.e.light causes an increase in the number of freecarriers. However, there are known cases when.light brings about not an increase, but a decreasein the number of carriers. I t can be shown thatthis may be related to the fact that light notonly changes the conditions of the generation,but also the conditions of the recombination ofcarriers.

In conclusion we shall return to the photo­electrically nonactive light absorption whichwe mentioned at the beginning of this section,that is when the absorption of a quantum doesnot cause either a free electron or a hole toappear. This is the exciton mechanism of lightabsorption. An exciton cannot live forever andsooner or later it is extinguished. There are twoways for this to happen.

First, the electron and the hole of which theexciton is composed can annihilate each other,i.e. the electron recombines with the hole.The electron is like a billiard ball that falls

2.5. Electrons and Quanta 97

into a pocket leaving neither a free electron nora free hole. This sort of disappearance is accom­panied by a release of energy.

The other way is for the exciton to dissociateinto a free electron and a free hole. The electronand the hole making up the exciton move farenough apart for the electrostatic attractionbinding the exciton together to fall practicallyto zero. An exciton requires some energy todisappear in this manner.

An exciton can disappear in a collision witha donor or acceptor structural defect in thelattice. In case of a collision with a donor defect,the energy released in the annihilation is usedto ionize the defect. A free electron and a holebound to the defect appear. In case of collisionwith an acceptor defect, a free hole and a local­ized electron appear. In both cases the excitondisappears and a free electron or a free holeappears instead. In these cases the absorptionof a quantum results, in the long run, in theappearance of a photoelectron or a photohole ,but these are the secondary not the primary con­sequence. The exciton is an intermediate stage.

I t is very interesting to look into a possiblemechanism for the collision between two exci­tons, when both excitons destroy each other.One of the excitons is annihilated, and itsannihilation energy is used to dissociate theother exciton. Thus instead of two excitons,we get a free electron and a free hole.

The exciton we have discussed so far, i.e. anelectron and a hole close together and boundto each other electrostatically, is called a Wan­nier-Mott exciton. It is possible (in an atomic

7-01536

98 Electrons in Semiconductors

crystal) for an electron and a hole to coexiston the same atom. This should be imagined asfollows. Suppose, for the sake of simplicity,there is a monovalent atom in which the valenceelectron is not in a normal internal orbit (whichis occupied by a hole in this case), but is in oneof the outer orbits. The atom is in an excitedstate and as such is conventionally called a Fren­kel exciton. A Frenkel exciton, like a Wannier­Mott exciton, can travel anywhere within thecrystal without carrying a charge.

So far we have discussed the excitation ofa crystal due to the absorption of a light quan­tum and resulting in the appearance of a freeelectron, hole, or exciton. Another kind ofexcitation which results in the crystal glowing­is possible. This means that a crystal, havingabsorbed a quantum of a given frequency, canemit quanta of the same or another frequency.The crystal glow caused by light is called photo­luminescence. We shall return to this phenomenonin Chapter 4 when we shall discuss the lumines­cence of nonmetallic crystals (semiconductorsincluded).

If a semiconductor that can luminesce is putinto an external electric field, i .e. we put a po­tential difference across it, the semiconductormay either glow less or, by contrast, glow more,depending on the direction of the field. Thisis a recently discovered effect called field lumines­cence.

Chapter 3

Electrons on a SemiconductorSurface

3.1. Semiconductor Surface Phenomena

We have operated so far with an infinitelyextensive crystal that exists only in the imagi­nation of the theorist. In reality, a crystal isbounded by a surface, and this is going to be atthe focus of our discussion in this chapter. As inthe preceding chapters, the dramatis personaein what takes place on the surface of the semi­conductors, are the electrons and holes of thecrystal lat ticc.

Semiconductor surfaces are becoming theobject of the research of scientists from manydifferent disciplines. First and foremost, thephysicists and engineers engaged in the physicsand production of semiconductors are mostinterested because modern semiconductor-deviceproduction rests upon the problem of surface.The quality of a semiconductor device cruciallydepends on the properties of its surface. Anyinstability in these properties, such as uncon­trollable changes with temperature or under theinfluence of the environment, disrupts the opera­tion of a semiconductor device. Hence, the highpercentage of production rejects. Learning tomanage the surface properties is one of the toppriorities of the semiconductor industry ..

Semiconductor surfaces are interesting forother sorts of researchers such as physical chem-

7*

too Electrons on a Semiconductor Surface

ists and chemists studying adsorption andcatalysis. Semiconductor surfaces are where theadsorption and catalytic processes take place.Most semiconductors can catalyse chern ical reac­tions. Semiconductor surfaces appear in catalysisstudies more often than it may seem at firstsight. The point is that many metals often havea semiconductor sheath (oxide film), so that theprocesses apparently occurring on a metal'ssurface are in reality occurring on the surfaceof a semiconductor. Industrial chemistry isfaced with the problem of producing catalyststhat are active for particular reactions. Learningto manage the adsorption and catalytic proper­ties of a surface is a top priority in the chemicalindustry.

Consequently, semiconductor surfaces are in­teresting from the viewpoint of semiconductorphysics and semiconductor-device production,and from the viewpoint of the studies in adsorp­tion and catalysis and the chemical industryemploying catalytic processes.

A semiconductor surface is the boundarybetween two phases (two media). It is the Iron­tier where our dramatis personae meet. Someof them come to the surface from the gas phase:these are the gas molecules. Others come fromthe depth of the semiconductor: these are freeelectrons and holes. The semiconductor surfaceis the boundary of both phases and interactswith one on one side and with the other on theother side. This is why surface effects are nottwo-dimensional , as it may seem to be at thefirst glance, they are three-dimensional.

A surface, like any boundary, can be tackleu.

3.1. Semiconductor Surface Phenomena 101

from two sides. Semiconduc tor physicists dealwith the surface from the side of the solid.They come, we might say, to the surface fromthe bulk of the semiconductor and study surface­bulk effects ignoring the interaction between thesurface and the environment. Catalytic chemistsstudy the surface in another way, from the sideof the gas and study the interaction between thesemiconductor surface and the environment,though often forgetting the surface-bulk interac­tions.

Basically, these two competing approachesdo not deal with the surface as it is, that is itsbehaviour and features escape our minds. Inorder to understand a surface, we have to consid­er it together with both phases between whichit is the boundary.

A semiconductor's surface is also at a bound­ary between two sciences: physics and chemistry.As Lomonosov* would say, 'physics and chemis­try are so intertwined that one cannot existwithout the other'. Modern technology oftenfocuses our attention on problems at the junctionof two sciences. These are the most promisingtopics both in their scientific and applied aspects.Semiconductor surfaces being a stumbling-blockfor semiconductor-device production on the onehand, and useful for catalytic chemistry on theother hand, is an example of such a problem.A surface is a two-faced J anus looking in dif­ferent directions: chemistry and physics,

We have so far spoken about a semiconductorin a gaseous environment. We called the semi-

... M. V. Lomonosov, an outstanding Russian scientistand scholar (1711-1765)..

102 Electrons on a Semiconductor Surface

conductor surface the interface between the solidand gas phases, but it can be used in a wider way.A surface does not necessarily imply the in ter­face between the semiconductor and a gas, itcan be also the interface between the semiconduc­tor and a liquid (an electrolyte), or between thesemiconductor and a metal, or between twosemiconductors with differing chemical compo­sitions. In every case, the surface brings intothe system its own features.

We shall return to the semiconductor contact­ing with the gas phase below. Here we shalldiscuss brief] y a semiconductor film in con tactwith a metal and the processes that occur on thesurface of a film on the metal. Then we shallconsider two semiconductors in contact, onebeing an n-type semiconductor, the other a p-typesemiconductor, and consider some of the im­portant effects that occur at such an interface.

Suppose that a metal is covered with a filmof its oxide (a semiconductor), so that the inter­nal surface of the semiconduc tor touches thometal and the external surface is in contactwith a gas phase containing oxygen. Moleculesof oxygen settling (adsorbing) on the surface ofthe semiconductor film dissociate into atoms.These oxygen atoms have a great affinity forelectrons and are typical acceptors capturing andretaining lattice electrons. Tho result is thatthe external surface of the semiconductor becomesnegatively charged and attracts the positivemetal ions both from the film and from themetal substrate. The external surface then hassome atoms of oxygen, borrowed from the gasphase. and metal atoms supplied by the metal

3.1. Semiconductor Surface Phenomena f03

substrate. The oxide film (the semiconductor)becomes thicker, and the metal phase less com­pact. Thus the metal is destroyed due to thegrowth of the semiconductor film on its surface.This phenomenon is called gas corrosion. It isa deadly disease that affects thousands of tonsof metal every year and has an army of scientistsfighting it. We see that surface effects attractthe attention of not only semiconductor physi­cists and catalysis chemists, but also corrosiontechnologists.

Let us in conclusion consider the case of twosemiconductors in contact, one of which is anelectronic semiconductor and the other a holesemiconductor. Bear in mind that· the samesemiconductor can become either an electronicor a hole one, depending on the nature of theimpurity it contains. Thus, for instance, weshould distinguish between n-germanium andp-germanium. Suppose that a p-type semicon­ductor is to the left of the interface, and ann-type semiconductor is to the right, as shownin Fig. 30. This interface is called a p-n junction.Now suppose that a potential difference isapplied across our semiconductors. It can beapplied so that the electric field is either directedfrom left to right (as shown in Fig. 30a) ordirected from right to left (as shown in Fig. 30b).In the first case the holes and the electrons traveltowards each other and recombine when meetingat the interface. An electric current thus appearsbetween the electrodes. If the cathode and theanode are exchanged, i.e. the direction of theelectric field is reversed and we go from Fig. 30ato Fig. 30b, then the holes and the electrons

104 Electrons on a Semiconductor Surface

appear and travel in the opposite directionwithout providing any electric current.

This is the phenomenon of unipolar conduc­tion, i.e. the current is only carried in one direc­tion and is absent in the opposite one. When the

(a)

++

'-1- +-g- +sz>- +~- +v- +

+~E +(h)

Fig. 30. Rectifying effect of a contact between p-type andn-type semiconductors

alternating current is passed, a contact thatallows 'the current to flow in one direction only,plays the role of a rectifier.

3.2. Adsorption on a Semiconductor Surface

Suppose a semiconductor' is placed in a gaseousenvironment and the gas molecules str ike thesurface of the SQ-lid aad adhere to it. 1?his is, the

3.2. A.dsorption on a Semiconductor Surface 105

process of adsorption. The molecules pass sometime on the surface and sooner or later are drawnaway from the surface and return to the gas.This is called desorption. Consequently, a gasmolecule passes a period of time in an adsorbedstate on the surface of the solid. The solid,therefore, captures the gas molecules and retainssome of them on its surface. The solid on whichadsorption occurs is called the adsorbent, whilethe substance that is adsorbed is called theadsorbate. In this section and in the followingone we are going to discuss adsorption and therole played by the electrons and holes of a semi­conductor.

Depending on the nature of the forces retain­ing an adsorbed molecu le on the surface of theadsorbent, we distinguish between physical ad­sorption and chemical adsorption (chemisorption),In physical adsorption, the forces are the samein nature as those between the molecules of a gas(Van der Waals forces). In chemisorption, theforces are chemical in nature between the atomsin the molecule (exchange forces).

Chemisorption differs from physical adsorp­tion in a number of ways. Firstly, the distancebetween an adsorbed particle and the surfacein physical adsorption is greater than in chemi­sorption, while a chemisorbed particle can beassumed to be pressed into the surface. Secondly,a particle is attached to the surface strongerin chemisorption than in physical adsorption.And lastly, in physical adsorption the adsorptionrate is the slower, the higher the temperature.,while in chemisorption the adsorption rateincreases as the temperature rises. If we designate

106 Electrons on a Semiconductor Surface

the adsorption rate dN/dt where dN is the numberof molecules captured per unit surface duringtime dt, then in chemisorption, as a rule, we have

dN (E )*dt=a, exp - kT ' (3.1)

where a, is a factor that slowly decreases withtemperature, and e is a constant that is calledthe adsorption activation energy. Adsorption thatobeys (3.1) is called activated adsorption. Asusual, in (3.1) the k is Boltzmann constant, andT is absolute temperature.

The process of adsorption is finished when thenumber of molecules adsorbing on a surface ina given time equals the number of moleculesdesorbing from the same surface. This means thata balance is established between the surface andthe gas (the adsorption equilibrium), and this ischaracterized by a certain coverage of the surfacewith the gas molecules.

This coverage of the surface, i.e , the numberof molecules retained by the surface in a steadyadsorption equilibrium, depends above all OIl

the temperature and pressure. I t decreases withtemperature at a given pressure and increaseswith pressure at a given temperature. A curveshowing how the coverage depends 011 pressure(at a given temperature) is called an adsorptionisotherm. Some adsorption isotherms are shownin Fig. 31. Here N is the number of moleculesadsorbed per unit surface in conditions of anadsorption equilibrium, and P is the pressure inthe gas phase. Curves 1 and 2 refer to two dif­ferent temperatures, curve 2 corresponding to thegreater temperature, We can see that the cover-

3.2. Adsorption on a Semiconductor Surface 107

age of the surface first rises with pressure, butwhen the pressure becomes great enough, itreaches saturation, i.e. it does not depend onpressure anymore.

The most molecules that can be retained ona surface at a given temperature, i.e. the numberN* that corresponds to the horizontal portionof the isotherm (at high pressures, see Fig. 31)is what we call the adsorptive capacity of thesurface. The adsorptive capacity of a surface isits holding capacity with respect to gas mole­cules. Naturally, the same surface can have dif­ferent adsorptive capacities for different kinds ofmolecules.

The adsorptive capacity of a surface dependson the nature of the surface and how it has beenprocessed. Adsorptive capacity can be changedby external effects. I t is interesting that a surfaceoften also responds to what is going on withinthe adsorbent. Thus, introduction of impuritiesinside the adsorbent affects the adsorptive capac­ity of its surface. Certain trace impuritiesnoticeably increase adsorptive capacity, whileother impurities decrease it. We shall returnto this subject when we discuss the interactionbetween the surface and the bulk.

N

N* ------~-~----

~--2

Fig. 31. Adsorption iso­therms for two differentte~peratures {curves 1

P and 2)

108 Electrons on a Semiconductor Surface

In particular, sometimes the adsorptive capac­ity changes when the adsorbent is exposed tolight. This is called the photoadsorptive effectand the adsorptive capacity can either increaseor decrease. The positive and negative photo­adsorptive effects are distinguished respectively.The sign of the effect is determined by the natureof the adsorbate molecules and by the nature ofthe adsorbent, and what is more it is determinedby history. Some recent experiments have shownthat the sign of the photoadsorptive effect canbe changed if the adsorbent has been treatedin a certain way.

An adsorbent is placed in a closed vesselfilled with a gas at constant pressure. Thismeans that the number of gas molecules per unit ·volume (i.e, the population density in the gas)is kept stable. Naturally, a number of gas mole­cules are attached to the adsorbent's surface.

Now if the adsorbent is exposed to photo­adsorption-active light, we may observe a dropor sometimes a gain in the gas pressure. Switchoff the light and the pressure returns to itsinitial level. This is the evidence that the expo­sure removes some of the adsorbed moleculesfrom the surface and returns them into the gasphase (this is photodesorption, the negati vephotoadsorptive effect) or, vice versa, it makesmore gas molecules stick to the surface thusincreasing the surface population (this is photo­adsorption, the positive photoadsorptive effect).

Sometimes the photoadsorption is irreversible.This means that the additional molecules ad­sorbed under the influence of the incident light arenot removed from the surface when the light is

3.2. Adsorption on a Semiconductor Surface 109

turned off. The temperature must be increasedin order to remove them.

I t was also found that adsorptive capacity canbe changed by an external electric field. Theadsorbent was used as one plate of a capacitoracross which a potential difference was applied.The adsorptive capacity turned out to be dif­ferent depending on whether there was an electricfield or not. I t was noticeably increased if thefield was in a certain direction, and decreasedif the field was in the opposite direction. I t iscurious that this effect is not symmetric withrespect to the direction of the field.

Let us remark that molecules adsorbed bya solid can creep over its surface while remaining'attached' to it. They can ehcounter otheradsorbed molecules until they manage to breakfree from the surface.

When a molecule collides with the surfaceit can often break in the very act of adsorption,so that it is separate parts of the molecule thatadhere to the surface rather than the moleculeitself. Molecules or their separate constituentsthat exist independently on a surface can jointogether into new combinations to form newmolecules.

Consequently, one kind of molecules mayarrive at a surface from a gas and another kindleave the surface. The composition of the gaschanges gradually, and its chemical transfor­mation occurs. The surface of the solid is wherethe molecules are restructured.

Note that molecules which settle on thesurface of a solid and are attached to it haveproperties different from those of the same mole-

iiO Electrons on a Semiconductor Surface

cules in the free state. Relationships between twomolecules can change, antagonism being changedfor sympathy. Molecules that will not reactin the free state will often react when they areprisoners on a surface.

Chemical reactions between gas molecules onsemiconductor surfaces will be discussed againin Section 3.5, which deals with the catalyticeffect of semiconductors.

3.3. The Role of Electrons and Holesin Adsorption

Gas molecules can adsorL on the surface ofa semiconductor. Then they can desorb andreturn to the glas phase again. An essential rolein adsorption and desorption is played by elec­trons and holes of the semiconductor becausethey control to a considerable extent the proper­ties and behaviour of the molecules that settleon the surface.

As they wander about a semiconductor,electrons and holes can come out onto the surface.They can then encounter adsorbed molecules thatarrive at the surface from the gas to pass sometime on the surface, 'rest', and then return to thegas phase. The adsorbed molecules play thesame role with respect to electrons and holesas foreign impurity atoms in the semiconductor;they capture free electrons and holes. (Onceagain: capturing a hole is the same as givingaway an electron. These are just two differentexpressions of the same thing.)

The adsorbed molecules and atoms are surfaceimpurities that interfere with the regular struc-

3.3. Electrons and Holes in Adsorption iii

ture of the surface. Much like the usual bulkimpurity, adsorbed atoms and molecules canplay the role of acceptors and donors. Thus, forinstance, adsorbed atoms or molecules of oxygenor chlorine are typical acceptors, while hydro­carbon molecules (e.g. C2H 4 ) are an exampleof donors. As a rule, hydrogen atoms are alsodonors.

An adsorbed molecule (or atom) that managedto capture and hold an electron or a hole becomeselectrically charged. The more current carriers(electrons or holes) there are in the semiconduc­tor, the greater proportion of the adsorbed par­ticles that are charged. The semiconductor sur­face thus gets charged as a result of adsorption.There are substantial consequences of this whichwe shall cover in the next section.

When the surface adsorbs acceptors, it getscharged negatively. Adsorption of donors, on thecontrary, results in a positive charge on thesurface. That is why the sign of the surface chargedepends on the nature of the adsorbate par­ticles.

We mentioned that at a given instant a cer­tain fraction of the adsorbed particles are charged.The same can be said in other words: each ad­sorbed particle passes some time in a charged stateduring its lifetime in the adsorbed state. There isthus always a certain probability that a neutraladsorbed particle will become charged and acharge d one will become neutral. The transition ofan adsorbed particle from the neutral state tothe charged one or vice versa implies the localiza­tion or the delocalization of a free electron orhole on the adsorbed particle.

112 Electrons on a Semiconductor Surface

The fact that an electron or a hole is capturedby an adsorbed particle interferes with thenature of its hond with the surface. The hondbecomes stronger. In other words, an adsorbedparticle in its charged state is attached morestrongly to the surface than it is in the neutralstate. I t is clear why. I t is more difficult totear a molecule with an attached electron orhole away from the surface than it is to removea neutral molecule. Indeed, the molecule withthe extra charge needs to turn its electron orhole over to the semiconductor before it can gohack into the gas. This in itself requires energy.

Consequently, we should distinguish betweenthe two ways an adsorbed particle can be at­tached to the surface; these are conventionallycalled weak and strong bonding. The free electronsand holes of the semiconductor do not participatein a weak bond, and only the electrons belongingto the adsorbed molecule (atom) or those belong­ing to atoms of the semiconductor's crystallattice participate in the bond. These electronsare drawn away from the adsorbed particleinto the lattice or from the lattice to the adsorbedparticle to form the hondo A strong hond isformed by the free electron or hole being cap­tured by the adsorbed particle.

Thus, there is weak and strong adsorption.Weak adsorption is electrically neutral, strongadsorption is electrically charged. The averagestrength of the bond between the adsorbed parti­cles and the surface is determined by the relativespread of the weak and strong bonds on thesurface, or, in other words, by the ratio of theprobabilities that the particle can be weakly or

3.3. Electrons and Holes in Adsorption 113

strongly bonded. The adsorptive capacity of thesurface, i .. e. the possible population of adsorbedparticles on the surface at given pressure andtemperature, depends on this parameter. Therelative spread of weak and strong bonds inturn is determined by the concentration of freeelectrons and holes on the semiconductor surface.This is why it is the free electrons and holes ofthe semiconductor that control its adsorptionproperties in the last analysis. They are themasters of the surface.

Obviously, the adsorptive capacity of thesurface with respect to acceptor molecules ishigh, and with respect to donor molecules islow, if the concentration of free electrons in theplane of the surface is also high (or the con­centration of free holes is low). The experimen­tally observed influence of incident light on theadsorptive capacity of a surface becomes clear now ..

Photoelectrically active light, i.e. light thatbrings about photoconduction, increases theconcentration of free electrons and free holesin the semiconductor. The additional electronsknocked out of the semiconductor by the light,come to the surface and partially settle there.If the surface is negatively charged, its negativecharge increases, or if it is positively charged,its positive charge decreases.. The extra holesthat also appear in the semiconductor due to theexposure to the light come to the surface as well,but they, by contrast, remove some electronsfrom it. The charge of a negatively chargedsurface decreases, while the charge of the posi­tively charged surface increases. The result ofthese two processes, that act simultaneously but

8-01536

i14 Electrons on a Semiconductor Surface

in opposite directions is that the charge of thesurface is changed.

The number of adsorbed particles in a chargedstate on the surface changes under the influenceof light. Hence the average strength of theirbond to the surface also changes. It becomes onthe average either harder or easier to tear a mole­cule away from the surface. In other words, theadsorptive capacity of the surface with respectto these molecules is affected. The result isphotoadsorption or photodesorption, or positiveor negative photoadsorptive effect.

When speaking of a semiconductor surface,we have so far had an idealized picture in mind.We supposed the surface was a plane with a reg­ular structure. Any real surface differs from theideal one in that there is a 'disorder' which dis­rupts the regular pattern. A real surface hasa variety of steps, peaks, atoms knocked fromtheir lattice positions to the surface, and vacan­cies, i.e. empty lattice points, and other macro­and microdefects of structure. These are con­venient sites for the adsorption of gas molecules,and any adsorption is focused around thesedefects which are called centers of adsorption.Adsorption can occur at these centers, as itdoes on an ideal surface, with or without theparticipation of free electrons and holes ('strong'and 'weak' bonds). It is clear that the natureand concentration of the adsorption centers onthe surface depends on the 'biography' of thosurface, i.e. on the treatment it has been sub-­jected to.

Let us remark in conclusion that defects ofthe surface, in particular surface impurities,

.3.4. Interaction of the Surface with the Bulk ii5

play in adsorption a double role. On the onehand, they regulate the concentration of freeelectrons and holes which, in turn, regulate theadsorptive properties of the surface. On theother hand, surface defects can themselves actas adsorption centers.

3.4. Interaction of the Surface with the Bulk

When a semiconductor is in a gaseous environ­ment, its surface always has some gas moleculesstuck to it. When gas molecules are attached tothe surface (adsorbed on the surface) there is,as we now know, some charging of the surface.This extra surface charge has important conse­quences which we shall discuss in thissection.

Let us note to begin with that the electronspopulating a semiconductor are locked inside it.To knock an electron out of the semiconductorrequires energy which, as is the case with metals,is called the work function. The work functionis the bill an electron has to pay to get beyondthe semiconductor. The work function character­izes how strongly the electrons of the semicon­ductor are attached to it. It is obvious that thework function depends on the nature of thesemiconductor. We shall now see that it alsodepends on the nature of the gas surroundingthe semiconductor.

Consequently, only those semiconductor elec­trons that possess sufficient energy can leave it.If we supply the electrons with energy, we caninduce their emission, i.e, make the semiconduc­tor release some of its electrons. Let us discuss

1*

fi6 Electrons on a Semiconductor Surface

the external forces on a semiconductor that cancause it to do so.

Electron emission can be produced, first ofall, by heating the semiconductor up to a highenough temperature. This is called thermionicemission, or Richardson effect in semiconductors,and is much like the same effect in metals. Themain point is that the mean kinetic energy ofthe free electrons in the semiconductor increaseswith the rise of temperature. More and morehigh-velocity electrons appear that are 'rich'enough to 'pay' for the 'right' to be liberatedfrom the semiconductor.

Electron emission can be also induced byexposing the semiconductor to light. The energyof the electrons in this case is increased at theexpense of the absorbed light. Here the Iight'is the source of electrons' energy. This is calledphotoelectron emission, or just photoemission; orthe external photoelectric effect that we discussedin the preceding chapter.

Bombarding a semiconductor with fast parti­cles, such as ions or electrons, can also give riseto electron emission. The striking particlestransfer their energy to the electrons of thesemiconductor. If this energy is sufficient forthe work function, electrons are emitted fromthe semiconductor. This is called secondary(electron) emission.

And finally, electron emission can be producedby an external electric field. A strong enoughfield will provide a force that can extract elec­trons from the semiconductor. An electron, pulledby this force, can overcome the energy barrierthat separates the semiconductor from the exter-

3.4. Interaction of the Surface with the Bulk 117

nal space and go beyond the semiconductor.We speak of cold emission in this case.

The work function of the electron, whichdescribes the height of the energy barrier sur­rounding the semiconductor, can be changed bysome effects. For instance, the work function canbe changed due to adsorption. The surface chargethat appears due to adsorption can either enhanceor hamper the emission of electrons from thesemiconductor. This depends on whether thesurface becomes positively or negatively charged,i.e. depends on the nature of the adsorbing gas.In case of negative charging the work functionincreases, in case of posi ti ve charging it' decreases.

Thus, the adsorption of oxygen (an acceptor,hence the surface is negatively charged) alwaysproduces an increase in the work function, whilethe adsorption of hydrogen (a donor, hence thesurface is positively charged) results in a decreasein the work function. In other words, the pay­ment an electron has to make to receive theright to leave the semiconductor depends both011 where it is going from and where it is goingto, i.e. both on the nature of the semiconductor,and on the nature of the surrounding gas.

We can often evaluate the composition ofa gas by the manner in which a work function ischanged by its adsorption. At higher gas pres­sures, i ,e. at larger gas densities, more moleculesaccumulate on a semiconductor's surface andhence the influence of the gas on the work func­tion is greater.

The gas molecules that are adsorbed on thesurface affect the behaviour of the electrons andh-Ql~~ not only when they arrive at the surface,

118 Electrons on a Semiconductor Surface

Electrons and holes deep within a semiconductoroften feel what occurs on the surface, just asthe population of a country responds to eventsat its fron tiers.

An example is the effect adsorption can haveon the conductivity of a semiconductor. Supposea gas is introduced into a closed vessel containinga semiconductor. The gas is adsorbed and theconductivity of the semiconductor changes mono­tonically with adsorption un til it reaches a cer­tain constant value. Often the process of adsorp­tion can be followed by observing the changein the conductivity.

If a semiconductor surface becomes positivelycharged due to adsorption, then the concentration:of free electrons near the surface increases. 'Theelectronic conductivity near the surface alsoincreases, and this may affect the conductivityof the whole specimen if it was not initiallyparticularly high. If the surface becomes chargednegatively due to adsorption, the surface layeracquires a surfeit of holes, and there is au in­crease in hole conductivity.

Consequently, adsorption produces a rise ora drop in the conductivity of the semiconductordepending on what kind of gas (acceptor or donor)is adsorbed and on the kind 0/ semiconductor(electronic or hole). Thus, the adsorption o.foxygen onto ziuc oxide (zinc oxid o is an electron­ic semiconductor) results in a decrease of theconductivity of zinc oxide, wh ile the conductivi­ty of nickel oxide (a hole semiconductor) isincreased by the adsorption of oxygen. Hydrogenusually produces the opposite effect. III theseexamples, the ~as molecules act like impurity•.• • • • I. t

3.4. Interaction of the Surface with the Bulk 119

atoms added to a crystal in that they eithertake away the current carriers from the semicon­ductor, or supply it with them.

The appearance on a semiconductor surface ofadsorbed molecules not only affects the workfunction and conductivity of the semiconductor,it also affects its typically bulk (and not surface)properties such as its luminescent properties.In fact, 3 semiconductor's luminescence can bereduced and sometimes extinguished altogether,although in other cases (depending on what kindof gas is adsorbed onto the semiconductor) theluminescence can be increased.

This all becomes clear if we recall how anexternal electric field affects the luminescenceof a semiconductor (the luminescent field effect).The appearance of adsorbed particles on a surfacemeans that the surface becomes charged. Theelectric field of the charged surface penetratesthe semiconductor and plays the same role asan external field applied to the semiconductor.The field is directed according to the nature ofthe adsorbed gas. In the last analysis, this isa case of the luminescent field effect.

So far we have considered the influence of thesurface on the bulk properties of a semiconductor.In conclusion, let us discuss the influence of thebulk on the surface. It is amazing how muchthe surface is sensitive to what goes 011 withinthe semiconductor. Thus, the adsorptive capacityof the surface is not only sensitive to impuritieson the surface, but also to impurities and theirconcentrations in the semiconductor. We canchange the adsorptive capacity of a semicon­ductor surface manyfold by introducing even tinyamounts of impurity to the hulk.

120 Electrons on a Semiconductor Surface

As we saw in Section 3.3, the adsorptivecapacity of a surface depends, all things beingequal, on the concentration of free electrons andfree holes in the semiconductor (to be more exact,in the plane of the semiconductor surface).At the same time we saw that the concentrationof free electrons and holes can be regulated byan impurity in the semiconductor. Thereforean impurity inside the semiconductor controlsthe adsorptive properties of its surface by affec­ting the pool of free electrons and holes popula­ting the semiconductor. This is a long rangeeffect. The impurity atoms are not in immediatecontact with the adsorbed gas molecules. Freeelectrons and holes are the intermediaries be~."

tween them.

3.5. Chemical Reactions on aSemiconductor Surface

We shall consider in this section what happenswhen a semiconductor is placed into a mixtureof different gases, i.e. a mixture of moleculesof different kinds. Suppose that a chemicalreaction can occur in the gas resulting in thedisappearance of some molecules and the appear­ance of new ones. Substances that participatein a chemical reaction are called reagents, anda substance that appears as a result of a chemicalreaction is called a product. The quantity of theproduct that appears per unit time is called therate of the reaction.

When a semiconductor is placed into an envi­ronment of reacting gases, gas molecules canadsorb onto its surface and react with other

3.5. Chemical Reactions on a Semiconductor Surface 121

adsorbed molecules or with molecules in the gasphase. In this way the reaction is transferredfrom the gas phase onto the solid's surface.The rate of the reaction changes, and if it isincreased, the semiconductor acts as a catalyst(an accelerator) to the reaction. Sometimesa reaction can be accelerated thousandfold.

For instance, the reaction of oxidation ofcarbon monoxide CO proceeds according to theequation

2CO + O2 ~ 2C02",---'---.... ~~--....

two molecules a molecule two moleculesof carbon of oxygen of carbonmonoxide dioxide

in the gas phase. When there is just a mixture ofcarbon monoxide and oxygen 'on its own", thereaction proceeds very slowly (or practically notat all). However, if a solid, such as manganesedioxide .Mn0 2 · or silver oxide Ag 20 (they aresemiconductors), is added to the mixture thereagents react rapidly to produce carbon dioxide.Manganese dioxide or silver oxide catalyze thisreaction. And if the catalyst is removed thereaction practically comes to a halt.

The relative increase in the rate of reactionbrought about by a catalyst is called its activity.Naturally, a catalyst that is active with respectto one reaction may be inactive with respectto another. Catalytic activity depends on a num­ber of conditions and in particular, it dependson the impurities in the catalyst or on its surface.Some impurities promote the activity, and soare called catalyst promoters? while some can

122 Electrons on a Semiconductor Surface

inhibit a reaction and are called catalyst poisons.The activity of a catalyst always rises withtemperature.

Now let us discuss the mechanism of thecatalytic activity of semiconductors and the roleof the free electrons and holes in the catalytic

(a) H-O-H (d) O=C=O

(b) 1-l-0-0-H (e) C === 0

H", /11(c) c==c U} tI-o--

Jr..../ ~II I

._ ". • ·.·_M · ._. .. . JFig. 32. Structural formulas for some simple molecules:(0) wa ter H20, (b) hydrogen peroxide 1-1 2° 2 ' (c) ethyleneC2Ii 4 , (d) carbon dioxide -C0 2 , (e) carbon monoxide CO,(f) hydroxyl group OR

reactions that occur on a semiconductor surface.These problems are covered by the electronictheory of chemisorption and catalysis.

Recall that individual atoms and groups ofatoms in a molecule are linked to each other byvalence bonds. These are shown in structuralchemical formulas by dashes. The structuralformulas for some simple molecules are given inFig. 32. A water molecule H 20 (see Fig. 32a)consists of two atoms of hydrogen each of whichis linked to the oxygen atom, i.e. each hydrogenatom is monovalent and the oxygen atom isbivalent. The same valences are observed ina molecule of hydrogen peroxide H 20 2 shown[n Fig. 32b. In ~ molecule of ethylene C~H,

3.5. Chemical Reactions on a Semiconductor Surface 123

(Fig. 32c), each carbon atom C is linked bya single bond to two hydrogen atoms H and bya double bond to the other C atom, i.e. the Catoms in an ethylene molecule are tetravalent.The same occurs in a CO2 molecule where theC atom is linked to two oxygen atoms ° eachof which is bivalent (see Fig. 32d). Note thatsome atoms can have a variable valence, Le.when they belong to different molecules theycan have different valences. Thus a C atom istetravalent in the C2H 4 and CO2 molecules,while it is bivalent in CO molecules (see Fig. 32e).

The valence of an atom has a remarkablefeature: it tends to 'saturation'. An 'unsaturated'bond always tends to become 'saturated' at theexpense of another 'unsaturated' link. In otherwords, a valence dash tries to link an atom (ora group of atoms) with another atom (or anothergroup of atoms). All the valence links in a stablemolecule are always 'saturated', i.o. none ofthe valence dashes dangles as shown in Fig. 32/.If a 'valence-saturated' molecule is split intotwo parts, the result is two molecules with 'un­saturated' ('free') links. Such molecules may becalled radicals. The OR or hydroxyl groupwhich can appear if an H atom is separated froman H 20 molecule, is an example of a radical(Fig. 32/).

Two radicals are two molecules that holdout their hands to join each other. A moleculewith one, two or more 'unsaturated' links maybe conventionally called a monovalent, bivalent,or polyvalent radical, respectively. It is clearthat a molecule in the radical state is alwaysmore reactive, i.e. its relative capacity to form.". '. _, • '..," ,J'

124 Electrons on a Semiconductor Surface

chemical compounds is greater than that ofa valence-saturated molecule.

The secret of the semiconductor catalysts,from the point of view of the electronic theoryof catalysis, is that gas molecules become radicals(at least partially) on the semiconductor's surface.This can be explained by saying that a semi­conductor crystal can be considered to be a largemolecule (a macromolecule) with some 'free'bonding sites or a kind of a polyradical. Thisbrings about its catalytic properties, accordingto the electronic theory. It is remarkable thatthe role of the 'free' links of the catalyst isplayed by free electrons and holes of the semi­conductor.

Considering free electrons and holes of a semi­conductor as 'free' links, we can attribute tothem the following specific properties. Generally,they are not localized and can roam freely overthe semiconductor, emerge on its surface, adhereto impurity atoms and become detached again,appear and disappear. They may encounter gasmolecules adsorbed on the surface, break thebonds within these molecules and become 'satu­rated' at the expense of these bonds, at thesame time transforming the molecules intoradicals, and turning radicals into 'valence­saturated' formations. This is the way the 'free'links of the catalyst come into play and carryout reactions.

As an example, let us take a catalytic reac­tion of oxidation of carbon monoxide CO. Sup­pose that an oxygen atom is adsorbed on the sur­face, and this atom is 'strongly' bonded to thesurface, i. e. it ~s attached to an electron localized

3.5. Chemical Reactions on a Semiconductor Surface 125

around it. This state is shown in Fig. 33a usingbond dashes. We now have a surface radical, anda CO molecule from the gas phase can now bondonto the oxygen's free electron pair. The C atombecomes tetravalent, and the surface formationshown in Fig. 33h appears. Now if the electronbinding this formation to the surface becomes

o=c

Io

((I)

1I

o==cIo

(IJ)

2 o=cIIo

/~(c)

Fig. 33. Possible mechanism of carbon monoxide oxida­tion. Circles with 'plus' and 'minus' signs and verticaldashes are 'free' valence links of the surface (electrons,holes)

delocalized or annihilated by an approachinghole, a CO2 molecule" the product of the reac­tion, is liberated into the gas phase (Fig. 33c).

According to Fig. 33, this reaction can beregarded as consisting of two stages, the first(arrow 1 in Fig. 33) proceeding the faster, thegreater the number of oxygen atoms 'strongly"adsorbed on the surface, or, in other words, thehigher the concentration of free electrons on thesurface. The rate at which the second stageproceeds (arrow 2 in Fig. 33) depends on the

126 Electrons on a Semiconductor Surface

probability that a localized electron is annihi­lated by a free hole, i. e. the rate is the fasterthe more the concentration of free holes. We seefrom this example that the rate of reaction iscontrolled by the concentration of free electronsand holes on semiconductor surface. They arethe true masters of the catalytic process (fromthe standpoint of the electronic theory).

Chapter 4

Electrons In Dielectrics

4.·j. Dlelectrle Conductivity

Typical dielectrics are alkali halide crystals, I.e,crystals composed of the positive ions of an alka­line metal (for instance, Na +, K +, or Li +) andthe negative ions of a halogen (for instance,CI-, Br", or 1-). The conductivity a of suchcrystals is extremely low. As we mentioned inSection 1.2, a dielectric has a conductivity ofthe order of 10-1°-10-16 Q -1· em -1(10-13-fO-14S .m-1)i.e. 1019-1020 times less than that of metals.

Dielectrics are not only unlike metals andsemiconductors as regards the value of theirconductivity, they are also unlike metals andsemiconductors because the current in them isnot electronic in nature, as it is in metals, andnot electronic or hole in nature, as in semicon­ductors, it is rather ionic, as it is in liquidelectrolytes. That is why alkali halide ioniccrystals are often called solid electrolytes. Elec­tric charge is carried by 'heavy' particles thatare several thousand times heavier thanelectrons.

Any real ionic crystal has ions at the latticepoints and at least a few in the interstices.These interstitial ions, squeezing through theregular ions of the lattice, move from one inter­stitial position to another. When they movein a prevailing direction, as happens when the

Electrons in Dielectrics

crystal is in an external electric field, we observean electric current.

The electric current in an ionic crystal is notonly carried by interstitial ions. Any real crystalhas at least a small number of empty latticepoints, i.e. lattice points from which the ionsthat should occupy them have been withdrawn.These are called vacancies. A vacancy can befilled by an ion from a neighbouring latticepoint, which is tantamount to the motion ofthe vacancy. The motion of vacancies causedby an external electric field is also an electriccurrent in a crystal. Vacancies resulting fromremoval of positive ions (for instance, due to Na+ions removed from a NaCI lattice) move in thedirection of the anode, i.e. they behave likenegative charges. Vacancies of negative ions (for'instance, due to CI- ions removed from a NaCIlattice) move to the cathode, i.e. behave likepositive charges. Accordingly, the current in anionic crystal can be due to either interstitialions or vacancies, or both.

If the crystal is heated, the regular ions inthe lattice points oscillate about their equilib­rium positions with some amplitude, whichincreases as the temperature rises. They can tearaway from their lattice points and jump intoan interstice at high temperatures. Owing tothis, heating promotes the appearance of bothinterstitial ions and vacancies, the number ofwhich rises with temperature. Now if the tem­perature is rapidly decreased, the interstitialions do not have time enough to return intotheir lattice points and they get stuck in theirinterstices. The defects of the lattice, the con-

-1.1. Dielectric Conductivity 129

centration of which is more' than the concentra­tion that should exist at the given temperature,are said to be frozen.

In order to push ions from their lattice pointsinto interstitial positions, energy is required andthis is called the dissociation energy. As thetemperature is increased the positive ions (thecations) dissociate at moderate temperatures, andthen, at higher temperatures, the negative ions

in 0

B A1

T

Fig. 34. Temperature diagram of ionic conductivity:AB-cationic conductivity, Be-anionic conductivity

(the anions) join the process. The anions arelarger and so require greater dissociation ener­gies. Thus, if we take crystals of common saltNaCl at room temperature, all the current iscarried by sodium ions Na+, and the migrationof chlorine ions CI- only becomes noticeable attemperatures exceeding 600°C. The addition ofimpurity ions always raises the conductivity.

As the temperature rises more ions dissociateand therefore the conductivity of the crystalincreases. As was the case with semiconductors,the conductivity (J of a dielectric is very sensitiveto temperature. This is presented diagrammati­cally in Fig. 34, where AB corresponds to thecationic conductivity, and Be represents the9-U1536

130 ElectroDS in Dielectrics

anionic conductivity. The slope of these straightlines yields the respective dissociation energies.

It is a feature of dielectric crystals that whena potential difference is applied across one thecurrent appearing in the crystal is not constant,but gradually decreases with time to a constantminimum value, even if the potential diHerenceis kept up constant. This is because as the ioncurrent passes through the crystal, the ions get

+ + +

(a) (b) (c)

Fig. 35. Potential diagrams in dielectrics: (a) initially,(b) in a dielectric with cationic conductivity, (c) in a di­electric with mixed ionic conductivity

stuck near one of the electrodes (or both) produc­ing a bulk charge. The field of this bulk chargeis in the opposite direction to the external field,thus counteracting it. This phenomenon is calledthe high-voltage polarization of a dielectric.The motion of an ion through a crystal is hinde­red by the high-voltage polarization, and so thecurrent drops.

The distribution of potential within a dielect­ric crystal is shown in Fig. 35. The dashed linerepresents the potential diagram at the initialinstant, when there is no bulk charge and thecurrent within the crystal obeys Ohm's law:

VI=R' (4.1)

4.1. Dielectric Conductivity 13f

where V is the potential difference, and R isthe resistance of the crystal. The solid line inFigs. 35b and 35c shows the potential distri­bution in the presence of the bulk charge in thecrystal. Fig. 35b refers to the case when the bulkcharge is produced by positive ions condensednear the cathode. Instead of (4.1), we have inthis case

v-pI-==-R-' (4.2)

where P is the potential caused by the polari­zation. Now if we short-circuit the crystal, weshall observe a reverse current passing from thecathode to the anode without any external poten­tial difference applied. And the amount ofelectricit y passed in this case will be equal tothe amount of electricity passed in the rightdirection. Fig. 35c refers to the case of mixedconduction in a crystal, when cations concentratenear the cathode and anions near the anode.

Let us remark in conclusion of this sectionthat ionic conduction in dielectrics ca.n be trans­formed into electron conduction, and the conductiv­ity can then be raised by several hundred thou­sand times.

A factor which can do this is exposure tolight. Light quanta of certain frequencies can beabsorbed by the lattice ions and can knockelectrons from them in the process so that thecrystal is filled with free electrons. This is theinternal photoelectric effect, which is identicalto that which occurs in semiconductors andwhich we discussed above.

V·

t32 Electrons in Dielectrics

Another factor is an external electric fieldapplied to the crystal. As before, let us designatethe strength of this Held E. The conductivities aof both semiconductors and dielectrics do notdepend on E if it is not too great, and we saythat Ohm's law holds true:

0' ~ const, (4.3)

Ohm's law is violated above a critical fieldstrength. This happens earlier in semiconductorswith electronic (or hole) conduction than indielectrics, for which ionic conduction is char­acteristic. When E rises above the criticallevel, there is a region of instability. In a nnm­ber of cases Poole's law starts being obeyed athigh field strengths, viz.

In (J = aE + ~, (4.4)

where a and ~ are constants. This law describesa rapid growth of the conductivity with theincrease of the field. Figure 36 is a diagrammaticrepresentation. of the dependence of (J on thofield strength E for the case of electronic or holeconduction (curve ABC) and for the case orionic conduction (curve A'B'). The electronicconductivity in crystals is small compared withionic conductivity at E < E1• By contrast, theelectronic component of conductivity becomes

c

/

·1.2. Dielectric Breakdown 133

predominant and starts increasing at E > E 2•

This increase in electric conductivity bringsabout, in the long run, a breakdown effect, whichwe shall discuss in the next section.

4.2. Dielectric Breakdown

When the strength of the electric field appliedto a dielectric is increased, the current in thedielectric rises. This rise, however, cannot beunlimited. Sooner or later a catastrophe hap­pens. The dielectric loses its insulating proper­ties an d is destroyed. This is called a dielectricbreakdown, and the electric field strength atwhich the breakdown occurs is called the break­down strength.

There are different. dielectric breakdownmechanisms. Let us consider them. :

The passage of current through any body(a conductor, a semiconductor, or a dielectric)is always accompanied by the evolution of heat.This is called Joule's heat, i. e. the beat evolvedin compliance with Joule's law:

V2Q --- c ­--- R' (4.5)

where Q is the amount of heat evolved in thespecimen per unit t.ime, R is the resistance ofthe specimen, ·V is the potent ial difference ap­plied across the specimen, and~c is a constantwhich depends on the system of 11 nits heingused.

When the heat is evolved, it is transferredfroIH the hot parts of the dielectric to coolerones. If a balance between the evolved and

iM Electrons in Dielectrics

abstracted heat is established at some stage,then the dielectric's temperature will not rise.The specimen will remain heated to a certainconstant temperature.

However, if there is no balance between theevolved and abstracted heat, the temperature ofthe dielectric will rise until the dielectric meltsand loses its dielectric properties. This is a caseof dielectric breakdown called thermal break­down.

The most essential and characteristic featureof a thermal breakdown is its temperaturedependence. When a certain temperature isreached, any further growth in temperature isaccompanied by a drop in the breakdown voltage."If we designate the breakdown voltage Vbr andthe amount of evolved heat required for thebreakdown to occur Qbr, then, according to (4.5),

vbrQbr:=: c-n,and hence the breakdown voltage Vhr and resis­Lance R are related, i.e.

Vhr == A Y·R,where A is a coefficient we are not. interested inat present.

Since the resistivity (the reciprocal of con­ductivity) and hence the resistance R of di­electrics drops very rapidly with temperature(i.e, the conductivity rises very rapidly, seeSection 4.1), the breakdown voltage Vhr mustdecrease considernhly due to the heating. Thereis no doubt at all that the breakdown at hightemperatures is purely thermal in origin ..

4.2. Dielectric Breakdown i35

Ifwe take a transparent specimen (for instance,a crystal of rock salt), then the process ofreaching a breakdown can be followed witha naked eye when the specimen is thick enough.The experiment is especially demonstrative ifit is carried out at a temperature for which thespecimen does not glow. Heat it a little, and itbecomes dark-red and incandescent. First, whenthe voltage is just applied, the specimen startsglowing with a weak violet light, and some timelater part of it becomes dark-red. This red glow,which characterizes the temperature of the moreheated portion of the dielectric, becomes moreand more intense, while the violet glow becomesweaker. Then all of a sudden there 'is a brightflare: this is the breakdown, and it producesa white-hot channel in the dielectric.

Another characteristic feature of thermalbreakdown is that there is an appreciable depen­dence of the breakdown voltage on the durationof the voltage application. The process of heat­ing and melting the entire dielectric is so slowthat it is only complete after several minutes.That is why considerably greater breakdownvoltages can be withstood in short pulses thanwhen the potential difference is applied fora long time.

There is another form of breakdown whichcan be called electrical breakdown.

A dielectric always contains one or more freeelectrons. They are a 'seed charge' that initiatethe development of a breakdown. These electronsroam randomly over the crystal, colliding withthe atoms or ions of the lattice. When an externalelectric field E is applied , the electrons are

136 Electrons in Dielectrics

accelerated by the field and they accumulatekinetic energy. Suppose T is the kinetic energythat is accumulated between two consecutivecollisions. Obviously

T = eEl,

where e is the absolute value of the electroncharge and l is the distance between two colli­sions.

If this energy is sufficient to ionize an atom(ion), Le.

T ~ I,

where I is the ionization potential, then one ofthe electrons belonging'Tto the atom (ion) willbe knocked out as a.....resultfof a collision. This

Fig. 37. Electric breakdown:the mechanism of collisionionization

new free electron will in turn ionize the atoms(ions) it encounters, given that the energy itwill have accumulated by moving in "the fieldis large enough to do so. In this way the numberof free electrons and therefore the electric currentin the dielectric can build up like an avalanche:it is diagrammatically presented in Fig. 37.After some time the current can be great enoughto cause a breakdown-

Consequently, the basic parameter! Jaffectingan electric breakdown is not the potential dif-

4.2. Dielectric Breakdown 137

ference V across the specimen, but the strengthof the electric field. When the electric field isstrong enough to give an electron the energyto ionize an atom while it is travelling betweentwo collisions, a breakdown will happen. This iscalled the collision mechanism of ionization.

Breakdown voltage Vbr and breakdown strengthEbr in a uniform field are related as follows:~ ...

E Vbrbr == -d-'

where d is the distance between the electrodes.Hence

Vbr == Ebr d.~

Accordingly, the breakdown voltage for anelectric breakdown should be proportional to thethickness of the specimen. ~

""",If the field is not uniform, and there may besites within a crystal where the strength of thefield is high, a breakdown can occur if the fieldstrength at a site equals the breakdown strength.We can thus assert that the breakdown voltagedepends on the heterogeneity of the dielectric.

On the other hand, the electric breakdownvoltage, unlike the thermal sort, should not benoticeably dependent on temperature, nor shouldit be dependent on the duration of the voltage.This is because lattice ionization is instanta­neous.fso the" development of an electron ava­lanche is an~extremely rapid process. Thereforeboth short and long duration applications ofa" potential differencc'<should'[result in more orless the same breakdown voltage.

138 Electrons in Dielectrics

Now let us consider collision ionization interms of energy bands, the terms we used whilediscussing the conductivity of semiconductors(Sections 2.3 and 2.4).

Suppose W is the total energy of an electronin a crystal lattice in the absence of an externalelectric field. When a field is applied, it willbecomeW + eEx,

where eEx is an additional term due to the field.As before, e is the absolute value of the electroncharge, E is the electric field strength, and xis the distance between the electron and anorigin where the potential energy of the electronin the field is assumed to be zero. The x-axisis assumed to be at right angles to the flat elec­trodes between which the specimen is placed, andtherefore the axis is in the direction of the field.

When the field is absent, the energy bandsin (W, x) coordinates are represented by hori­zontal strips as shown in Fig. 38a. When thefield is applied, the bands become sloped, asshown in Fig. 38b, the gradient being the steeper,the stronger the field. If in the absence of theelectric field the forbidden energy band is be­tween W = WI and W:;= W2 for all x (seeFig. 38a), then in the presence of the electric.field it is between

W == WI + eEx

andW:-:= W2 + eEx,

i.e. it is different for different x, in other wordsit is different at different sites in the crystal

4.2. Dielectric Breakdown 139

(see Fig. 38b). On the other hand, a given energyW is forbidden in the crystal within the intervalfrom x == Xl to X == X 2 (Fig. 38b) and is allowedoutside this interval. The conduct ionj'[band is

w ------w%

Jj,'2 W"l

W

IVI

~- JVI

... x

(a)

(I»

Fig. :-38. Energy bands in a crystal: (a) in the absence ofan electric Held, (b) in tho presence of an electric Iield

hatched while the valence band is double hatchedin Figs. 38a and 38b.

Suppose there is an occasional free electronthat moves in the conduction band in directionAB (Fig. 39). While moving, the electron accu­mulates a kinetic energy that can be shown inFig. 39 as a segment T for each x. When T isequal or greater than the energy U (U is thewidth of the forbidden range between the bands),a free electron in collision with a bound electronin the valence band can push the bound electroninto the conduction hand by transferring anenergy equal to or more than U. 'There are two

140 Electrons in Dielectrics

electrons in the conduction band now, each ofwhich moviug in the conduction band, accumu­lating kinetic energy, and able to ionize, i. e.push bound electrons into the free state. Thefree electron in Fig. 39 moves along the pathABeD, and the bound. one moveslalong B'C'D'.

w

Fig. 39. Mechanism s ofcollisionionizntion a mlcold ern ission presented

---------:l-.-x in a band diagram

This is the collision ionization mechanism U:'5

described in terms of the energy band diagram.Besides collision ionization, there is also

what is called field ionization. This is when anelectron from the valence band passes into thoconduction band directly due to the tunnel ef­fect: this is shown by the arrow B' F in Fig. 3fLThis mechanism is equivalent to cold emission.which we discussed in detail in the chapter 011

metals (see Section 1.4). However, and thisshould be emphasized, the field strengths (11which this effcc.(", produces nnt iceable currentscommonly exceed t.he strengths at which break­downs occur, i.e. an electrical breakdown happensbefore a cold emission breakdown can occur.

4.:3. Crystal Colonration 141

4.3. Crystal Colouration

Suppose we have a transparent crystal , a crystalof rock salt for example, By exposing the crystalto light or putting it into an electric field, wecan induce colouration in the crystal, The colour­ation is an evidence that electrons in the crystalhave transferred from one energy level to another.A transition from a lower level to a higher onecan take place if a light quantum is absorbed.And vice versa, a transition from a higher levelto a lower one is accompanied by the emissionof a quantum. The transfer of an electron froma donor localized level into the conduction bandmeans the electron has gone from a bound stateinto a free state. The transition of an electronfrom the valence band to an acceptor localizedlevel means a localized bound electron has appea­red accompanied by the appearance of a free hole.

The frequencies absorbed by a crystal can berepresented as a system of bands in the spectrum.The structure-stable and structure-sensitive bandsshould be distinguished (see Section 2.1).

Alkali halide crystals strongly absorb lightin the remote ultraviolet, i.e, where the frequen­cies are very high (or the wavelengths are short).This is called intrinsic absorption and the fre­quencies are gathered into an intrinsic absorptionband. This is a structure-stable band. Its intensi­ty and position in the spectrum do not dependon any external factor.

It is typical that intrinsic absorption isphotoelectrically inactive, i. e. light absorptionin an intrinsic band does not induce photocon­duct.ion. When a crystal is exposed to frequen-

142 Electrons in Dielectrics

cies from its intrinsic band, the crystal remainsan insulator as before.

Photoelectric inactivity seems to occur becauseabsorption in the intrinsic band is exciton inorigin. The absorption of a light quantum resultsin an electron being moved from a CI- ion to anadjacent Na " ion (for the sake of certainty, letus consider a crystal of rock salt NaCl, whichis built of Na" and Cl ' ions). The result is twoneutral atoms, Na and Ct. that are~in'the vicin­ity of each other and related by Coulombforces. This is the Wannier-Mott exciton andis an electrically neutral formation that movesthrough a crystal as an entity without trans­porting electric charge.

There is another explanation for the photo­electric inactivity of the intrinsic band. Theintrinsic band can absorb so much that all theincident light may be absorbed by the surfacelayer of the crystal without penetrating farinside.

Now let us take the structure-sensitive absorp­tion bands. Lattice structure defects that act asabsorption centers are responsible for the bands.We shall consider two main absorption bands:the U-band and F-band, which correspond toU-centers and F-centers (from the German wordsUltravioletzentren and Farbenzentren, ultravio­let and colour centers). The U-band is in thenear ultraviolet, at the 'tail' of the intrinsicband, and the F-band is in the visible part ofthe spectrum (Fig. 40). A crystal with an F-bandis coloured in the literal sense of the word. Thus,the F-band in crystals of rock salt is responsiblefor their light blue colour.

.'f.3. Crystal Colouration 143

There are various techniques to colour a crys­tal. One is to heat it for a long time at a hightemperature in the vapour of an alkali metal.This is called additive colouration. The resultingcolour depends on the composition of the hot crys­tal and does not depend on the alkali metal.

J

Fig. 40. Absorption bands in a crystal of NaCl: }-_.intrin­sic band, 2- U-band, 3-F-band. Absorption constantx (mrn -1) is plotted along the ordinate axis, energy 11",of the incident Quanta (in electron volts) is along the ab­scissa axis

Another method is to expose a transparent crystalto X-rays or ultraviolet light. This is calledsubstractive colouration. The same F-band appearsin the spectrum of the crystal. If exposed toU-band frequencies, the U-centers are destroyedand F-centers appear. Remarkably the reversephenomenon also occurs, i.e, exposing the crystalto the F-band frequencies attenuates or destroysthe F-band, creating or amplifying the U-band.

It is possible to colour a crystal by puttingit into an external electric field. At high temper-

144 Electrons in Dielectrics

atures, F-centers and U-centers appear near thecathode and move gradually towards the anode.This is observed directly when a blue colour (inrock salt) is seen to move from the cathode tothe anode spreading over a greater and greaterportion of the crystal. If the direction of thefield is reversed, the F-centers will gradually"return' into the cathode, i.e. the blue colourwill be <drawn' back into the cathode. As to theU-centers, they stay at their places, gettingstuck, as it were, in the crystal.

When a coloured strip appears followingadditive or substractive colouration , and then anelectric field is applied, the strip starts movingtowards the anode. At high temperatures itmoves as a single entity, without being deformedand leaving no colour behind it, while at lowtemperatures the trailing edge moves more slowlythan the front one, and the strip graduallyspreads. Thus as it moves the intensity of thewhole cloud progressively declines due to F­centers being transformed into U-centers. Thecurrent that passes through a crystal in theprocess increases gradually and becomes constantwhen the coloured cloud reaches the anode.If the current flows for a long time, the colour ofthe crystal will gradually disappear uniformlyover the entire bulk of it.

If a crystal contains an impurity, for instancea crystal of rock salt includes foreign atoms ofgold or copper, then the blue cloud, when itreaches the area containing the impurity, coloursthe area depending on the impurity (for instance,red or brown). It is remarkable that if the direc­tion of the electric field is reversed, then although

4.3. Crystal Colouration 145

the blue cloud is drawn back into the cathode,the colour brought about by the impurity remainsfixed. The crystal can only be discoloured com­pletely by heating it. That there is an impurityin a crystal and what its distribution is overthe bulk of the crystal can be determined withthe naked eye.

Let us note the influence of temperature onthe position and structure of the F-band. Whena crystal is heated, the F-band is deformed sothat its maximum shifts towards longer wave­lengths and the whole band spreads out. This isillustrated in Fig. 41 for crystals of KBr. Theprocess can be reversed: when the crystal iscooled, the band becomes narrower and itsmaximum shifts towards the shorter wavelengths.

It is characteristic that the U-band, likethe intrinsic band, is photoelectrically inactive.Meanwhile, exposing the crystal to the frequen­cies belonging to the F-band influences pro­nounced photoconduction and the crystal be­comes a good electronic conductor. This happensbecause. electrons go from the F-Ievel to theconduction band (see Fig. 42).

Let us discuss now the nature of colourcenters. An F-center in an alkali halide crystalis a halogen vacancy with an electron localizedeither in or near it, thus maintaining the crystalas a whole electrically neutral.. Experience hasshown that this combination has an affinity forfree electrons. A free electron travelling acrossthe crystal and meeting an F-center adheres toit and forms what is called an F ' -center, Accord­ingly, an F' -center is a halogen vacancy withtwo electrons localized around it.10-01536

146 Electrons in Dielectrics

A U-center is a negative hydrogen ion H­which has substituted a halogen vacancy ina lattice point. If the H - ion is drawn into aninterstice, it is called a Ul-center. Then if theion is neutralized, it becomes a U2-center.Consequently, a U2-center is an H atom in aninterstice.

There are many possible colour centers otherthan these centers. We shall only mentionV-centers, which are similar to the F-centers

1

Fig. 41. Deformation of the F-band ora KBr crystal with heating: 1- - 245°C,2- +200°C, 3 - +600°C. The coor-dinate axes are the same as in Fig. 40

Fig. 42. Origination of photoconduc­tion in the F-band. C-conductionband, V-valence band, F-level ofan F-center

4.4. Crystal Luminescence i47

though a V-center is a metal vacancy combinedwith a hole localized nearby.

Many features of crystal colouration have nowbeen established experimentally. However, so farthere is no theoretical background that allowsus to consider them all from a single standpoint.

4.4. Crystal Luminescence

We are going to discuss in this section the lumi­nescence of dielectrics and semiconductors, whichis the glow or light emitted from crystals. Inorder to produce the glow, the crystal has to beexcited, i.e, supplied with additional energythat can later be released from the crystal inthe form of light quanta. There are variousways of exciting crystals, and so there are thefollowing kinds of luminescence:

(a) photoluminescence (light excitation),(b) electroluminescence (electric field excita­

tion),(c) cathodoluminescence (excitation by an

electron flux),(d) chemiluminescence (excitation at the ex­

pense of energy produced in a chemical reaction).Crystals that can luminesce are called lumi­

nophors. The ability to luminesce is broughtabout by impurities in the crystals which arecalled activators. Crystal luminophors are calledcrystal phosphors, or just phosphors. The spectralcomposition of the luminescence depends on thenature of the activators, and the radiationintensity depends on their concentration. Weshould point out that an activator is an impurityin the wide sense of the word, i.e. in the sense10·

148 Electrons in Dielectrics

attributed to the term in solid-state physics.Hence activators are not necessarily chemicallyforeign particles in the crystal. They may alsobe structural defects, i.e. local imperfections ofthe regular lattice structure. These include, inparticular, vacancies and the lattice's own atomsor ions but existing in the interstices. The structur­al defects responsible for luminescence areconventionally called activator atoms or lumi­nescence centers.

A crystal phosphor can be excited by eitherexciting the luminescence centers or ionizingthem.

Let us consider the first case. Suppose a centerof luminescence, as often happens, is an acceptor­donor pair in the crystal, i.e. a combination of anacceptor and a donor defect in the immediatevicinity of each other. Suppose further that theenergy of electron affinity for an acceptor defectis greater than the ionization energy of a donordefect. Then normally an electron will be boundto the acceptor and a hole bound to the donor.Transition of the electron from the acceptor tothe donor means that the pair is excited, and thereverse transition of the electron (from the donorto the acceptor), i.e. the return of the pair intoits normal state, results in luminescence. There­fore, the excitation and the luminescence aretransitions of an electron within the pair fromone partner to the other.

Now let us consider a second case, when theexcitation of a crystal phosphor is due to ioni­zation of a luminescence center. The crystalstructure of a phosphor permits us to describethe ionization and neutralization of a lumines-

4.4. Crystal Luminescence 149

cence center in terms of energy bands (see Sec­tions 2.3 and 4.3).

The ionization of an activator during a photo­luminescent event can occur due to the absorp­tion of a light quantum directly by a center ofluminescence (this is the impurity absorptions.An electron is thus knocked out of the center(if it is a donor defect) into the conduction bandor it is knocked from the valence band to theluminescence center (if it is an acceptor defect).The reverse transitions, which result in theneutralization of the activator, result in theemission of a light quantum i.e. a luminescentevent.

The ionization of activators and their subse­quent luminescence not only result from lightabsorption by the luminescence centers directly,but also because of light absorption by the latticeitself, i.e. by the regular atoms (ions) of thelattice (this is called intrinsic absorption). Anelectron is then transferred from the valenceband to the conduction band (see Fig. 43, ar­row 1). A hole in the valence band and an elec­tron in the conduction band appear in the process"The return of the crystal to the normal statecould then be the result of two consecutive acts,i.e. transition 2 and then transition 3 (see Fig. 43a)if the activator is an acceptor, or transition 2'and then transition 3' if the activator is a donor(see Fig. 43b). Each transition (or all of themtogether) can be radiative, i.e. can be accom­panied by the emission of a light quantum, orluminescence,

A crystal phosphor can have what are calledadhesion centers in addition to luminescence

150 Electrons in Dielectrics

centers. These are defects that trap electrons orholes released as the crystal is excited. Thecenters do not participate directly in lumines­cence, nevertheless they can play an essentialrole in the process. They correspond in the energyband spectrum to .whatf7 are called adhesion

1 2

(a) (h)

Fig. 43. Appearance of luminescence in intrinsic absorp­tion. V-valence band, C-conduction band. A-acceptorcenter of luminescence, D-donor center of luminescence.Transition 1 is excitation, transitions 2, 2', 3, 3'represent luminescence

levels. They can be acceptor levels (adhesionlevels for electrons) or donor ones (adhesionlevels for holes). The former are found immediate­ly beneath the bottom of the conduction band,and the latter are immediately above the ceilingof the valence band (see levels T in Fig. 44).

An electron from the conduction band ora hole that appeared in the valence band dueto the excitation of the phosphor can be capturedby adhesion centers before they manage torecombine, i.e. unite with each other. The accu­rnulation of electrons or holes on the adhesion

4.4. Crystal Luminescence 151

levels means transition of the system into anexcited energy state (a metastable state). Leavingit requires what is called an activation energy.Indeed, the return of an electron to the conduc­tion band or a hole to the valence band can

~((I)

T-.. ....

(b)

Fig. 44. Adhesion levels T: (a) for electrons, (b) for holes.Ea- activa tion energy: filled circles-electrons, blankcircles-holes

only happen at the expense of heat or some exter­nal effect, for instance, due to exposure to infra­red radiation. The location of adhesion levelsin the energy band determines the duration of theafterglow of the phosphor, i.e. phosphorescence,or luminescence that persists after removal ofthe exciting source. The concentration of theselevels determines the amount of stored light,Le. the energy accumulated by the phosphorwhen it was being excited.

Besides luminescence centers and adhesioncenters, a phosphor contains, as a rule, somedefects that are centers of nonradiative recombi­nation, i.e. events when electrons join holeswithout radiation emission. These centers areshown in the energy spectrum of a luminophor

152 Electrons in Dielectrics

as localized levels (see level R in Fig. 45) thatcan be either donors or acceptors depending onthe nature of the centers. Recombination levels(R) are an intermediate stage in the recombina­tion of electrons from the conduction band andholes from the valence band (transitions 1 and 2in Fig. 45).

Consequently, the free electrons and holesthat appeared as a result of transition 1 inFig. 43 can either recombine through A or Datoms of the activator (transitions 2-3 or 2'-3'in Fig. 43) and emit quanta, or recombinethrough R-centers (transitions 1-2 in Fig. 45)without emitting quanta. The two channelscompete with each other. Obviously, recombina­tion through R-centers results in luminescentdecay (called extrinsic decay). Let us remark thatrecombination through activator atoms (A and Din Fig. 43) is not necessarily radiative. Generally,there is a nonzero probability for a recombinationto be nonradiative. In other words, a certainfraction of the recombinations that occur throughluminescence centers do not produce radiation(this phenomenon is called intrinsic decay).

R

2

fig. 45. Recombination level R

4.4. Crystal Luminescence i53

So far we have discussed photoluminescence.Now let us turn to electroluminescence forwhich the mechanism of excitation is quitedifferent. Suppose that a free electron in a crystalhas been accelerated in a strong electric fieldand has accumulated a large kinetic energy.If it collides with an activator atom, it knocksout one of the activator's electrons (the activatoris ionized). This is the excitation mechanism inelectroluminescence. A liberated electron thenfalls from the conduction band into the same orsimilar vacant place in the activator atom (it isneutralized) and a light quantum is emitted inthe process (a luminescent event). Therefore,a crystal phosphor is excited in the case ofelectroluminescence by the mechanism of colli­sion ionization, which we considered in Section 4.3.

In conclusion, let us take chemiluminescence.\ArPe shall only discuss a particular case of chemi­luminescence, i.e. radical recombination lumi­nescence, which was discovered and studied onlyrecently. Suppose two atoms or two radicalshave arrived at the surface of a crystal from thegas environment of the phosphor, and they meet.Recall that a radical is a molecule with an'unsaturated' ('free') .link. When the two atomsor two radicals join, they produce a 'valence­saturated' molecule. As a rule, a certain amountof energy is released in the process which can bespent to excite the phosphor and which sub­sequently appears as chemiluminescence. Conse­quently, this phosphor excitation and lumines­cence occur by utilizing the energy released ina chemical reaction that takes place on thecrystal surface,

f54

4.5. Electrets

Electrons in Dielectrics

Some dielectrics can have a special state calledthe electret state. This is characterized by thepresence of an internal electric polarizationcaused by an external electric field and persist­ing for a long time after the field is removed.A dielectric in the electret state is called anelectret. An electret is a charged dielectric withthe poles apart: the electric analogue of a magnet.Naturally, this analogy is limited, since freemagnetic charges have not so far been discovered,while free electric charges do exist.

If there is a negative charge on the surface ofan electret facing the anode and a positive chargeon the surface facing the cathode, the charge is'called a heterocharge. In case of the oppositepolarity, when the positive charge faces theanode and the negative charge faces the cathode,we speak of there being a homocharge in thedielectric. It is remarkable that in some casesan electret can undergo a spontaneous reversalof its polarization. After the polarizing field isremoved, a heterocharge sometimes graduallydrops to zero and then transforms into a homo­charge that gradually increases to its maximumvalue. The lifetime of an electret state is limitedthough it varies over a wide range. Sooner orlater the dielectric becomes depolarized, and theelectret state disappears. However, the electretstate can last several years.

An electret state that appears after the die­lectric has been heated and then cooled ina strong electric field is called a thermoelectretstate, and the respective electrets are called the

4.5. Electrets i55

thermoelectrets. I t has been established thatthe electret effect is a bulk and not a surfaceeffect. Hence cutting or scraping off layers fromthe surface of an electret does not change itsproperties.

The photoelectret state deserves special atten­tion. It appears when a photoconducting dielectricis put in a strong electric field and exposed tolight. A bulk charge accumulates in the dielectricdue to conduction and after both the field andlight have been removed, the bulk charge persistsin the dielectric for a long time. Dielectricsthat can be in a photoelectret state are calledphotoelectrets.

The photoelectret state can be observed bothin polycrystals and in a large number of mono­crystals. Naturally, the mechanism leading toa stable polarization in photoelectrets is differ­ent from that in thermoelectrets. A prerequisitefor the photoelectret state to appear is the pres­ence of photoconduction in the dielectric. Weshall only discuss photoelectrets below.:- The appearance of a stable heterocharge ischaracteristic for photoelectrets. As for a homo­charge, it is not usual in photoelectrets. Theevolution of a heterocharge starts when a lightquantum pushes an electron from the valenceband or from an activator level to the conductionband. Recall that an activator level is a donorlocalized level in the forbidden range between theenergy bands. If an electron is taken fromr.'anactivator level, the activator atom remainswithout its electron, i.e. a hole is localizedConthe activator atom. If an electron is pushed frOIDthe valence band, a free hole appears that can

156 Electrons in Dielectrics

roam over the crystal and meet an activator atomwith whose electron it recombines. In the longrun, the result is the same: a hole localized on anactivator atom.

As for the free electron that appears in theconduction band, it moves somewhat towards theanode because of the external electric field'sinfluence and then gets stuck on the adhesionlevel, which is an acceptor localized level underthe conduction band but above the activatorlevel. We can obtain crystals in which the lowerlevels (activator levels) are devoid of electronsand the higher levels (adhesion levels) are full.This crystal is said to be in a metastable state,i.e. a state from which a transition to the normalstate requires energy (activation energy) (seeSection 4.4). The adhesion levels correspond tolattice defects that are found in a narrow regionclose to the electrodes. Those that are close tothe anode are partially filled by electrons andyield a negative bulk charge. The crystal canthus be regarded as a dipole with oppositelycharged poles, one of which contains an excess ofelectrons and the other an excess of holes, or, toput it better, too few electrons.

The depolarization of a photoelectret, i.e. theremoval of the photoelectret state, can be broughtabout in many ways.

For instance, it is possible to remove thephotoelectret state by exposing the dielectric tolight quanta that knock electrons from thevalence band to the vacant activator levels. Theresult is that the holes localized on the activatorsare transferred to the valence band, Le, intothe free state. Then these free holes recombine

4.5. Electrets 157

with electrons on the adhesion levels, i.e. theelectrons responsible for the photoelectret state.The result is that these electrons leave the adhe­sion levels, and the dielectric is depolarized ..

Another way of depolarizing an electret is toexpose it to light that will remove electronsdirectly from the adhesion levels and transferthem to the conduction band. Sometimes thisprocess occurs with a transitory stage. An adhe­sion center is excited by the light and then theelectron is knocked out of the excited state intothe conduction band. The 'extra excitation'takes place when the photoelectret is heated.Depolarization in this case requires both lightand heat at the same time. Let us remark thatheating can be replaced by infrared radiation.

The adhesion levels responsible for the pho­toelectret state are, in fact, localized levels thatare well below the conduction band. Some shallowadhesion levels can also exist in the crystalimmediately below the conduction band and, verymuch like the adhesion levels, they are involvedin photoluminescence (see Section 4.4). Theselevels also can play a role in electret phenomena,being responsible for what is called dark polari­zation, l.e. the polarization that appears ina dielectric when it is placed in an electricfield although not exposed to light. While in theabsence of a field the adhesion levels are filleduniformly, a field, displaces the conductionelectrons which, in turn, results in their beingunevenly distributed among the adhesion levels.It is this uneven distribution of localized elec­trons that is seen experimentally as dark polari­zation ..

168 Electrons in Dielectrics

Photoelectret excitation is when short wave­length light pushes electrons from the valenceband or from activator levels to the conductionband, and then the electrons start filling theadhesion levels, both the deep ones, which areresponsible for the photoelectret state, and theshallow ones, which are responsible for darkpolarization.

4.6. Dielectric Constant

The dielectric constant E is another essentialparameter of dielectrics, alongside electricalconductivity (or specific conductance) (J whichwe discussed in Section 4.1. The dielectricconstant is the ratio of the forces between chargesin a vacuum to the forces between the samecharges, at the same distance apart, in thedielectric. Its value differs for different dielec­trics and usually varies from 1 to 100.

However, some dielectrics have unusuallyhigh values of 8. One such dielectric is Seignettesalt, or potassium sodium tartrate. Its dielectricconstant at room temperature is 10,000. Dielectricswith unusually large values of E are calledjerroelectrics, or seignette-electrics and we shalldiscuss their properties in the next section.

The dielectric constant E of most dielectricschanges little upon heating and is practicallyindependent of the electric field strength.

The e-fold decrease of the forces betweencharges in a dielectric and those in a vacuum iscaused by what is called dielectric polarization.This phenomenon is characteristic for dielectrics,-unlike electric conduction which occurs in crys

4.6. Dielectric Constant f59

tals of all three types, i.e. metals, semiconduc­tors, and dielectrics. As mentioned in Section 4.1,the conductivity of a dielectric is much less thanthat of a semiconductor or a metal becausedielectrics do not contain significant numbers offree charged particles whose directed motion inan electric field constitutes the electric current.

Let us discuss dielectric polarization indetail. Suppose we have a dielectric in the formof a rectangular parallelepiped. Suppose it isin the uniform field of a flat condenser, one ofwhose plates is charged positively, the othernegatively, and the base of the parallelepipedis parallel to the condenser plates. Thus theelectric field in the condenser is perpendicularto the base of the parallelepiped.

The positive and negative charges in themolecules move apart a small distance dueto theelectric field. Each molecule thus turns into anelectric dipole that can be represented as twopoint charges, +q and -q, which are rigidlyconnected to each other at a distance of d.

The electric moment of the dipole is a vector pequal to the product of the charge q and thedisplacement vector d, which points from thenegative charge to the positive one, i.e.

p = qd.

If there are already polarized molecules ina dielectric, they are turned and oriented bythe field. .

The displacement of charges within the die­lectric molecules and the orientation of thedipole molecules in the electric field bring aboutthe appearance of bound surface charges. The

160 Electrons in Dielectrics

dielectric face touching the positive plate carriesa charge -Q, and the one touching the negativeplate carries a charge +Q*. The positive andnegative charges compensate each other withinthe dielectric.

The process leading to the appearance ofbound charges with a dielectric is called polari­zation.

Polarization is quantified by its polarizationvector P, which is the electric moment of a die­lectric per unit volume. The vector points inthe same direction as the electric field, and itsabsolute value equals the sum of the projectionsof the electric moments of all the elementarydipoles per dielectric unit volume onto thefield direction.

The electric moment M of the entire dielec­tric is the product of P and the volume of thedielectric:

M = PSh,

where S is the area of the base of the parallelepi­ped, and h is its height.

On the other hand, M is the product of thesurface bound charge Q and the height of theparallelep iped:

M=Qh.

Therefore

PSh=Qh, p=~.

• Do not mix this up with the Joule heat designa ted bythe same letter Q in Section 4.2.

4.6. Dielectric Constan L 1G1

. 'ponsequently, the modulus of the dielectricpolarization is equal to the surface charge den­sity QIS

The charge that appears on the surface ofa dielectric due to its polarization produces inthe dielectric a field E1 that is opposite in itsdirection to the external field and equals in theSI units

E1 = l = !:-Seo e.o I

where the electric constant 8 0 = 8.85 XX 10-12 C2/(N ·m2) ::= 8.85 X 10-12 F/m.

The field strength E in a dielectric is equalto the difference between the field strength Eoin the absence of the dielectric and the strengthE1 of the polarization field, i.e.

pE=Eo-E1=Eo- s ·

o

The s-Iold reduction in the force betweencharges in a dielectric is tantamount to ans-Iold reduction in the electric field strengthwhen the dielectric is placed in the field:

(4.7)

Substituting (4.7) into (4.6), we can find thedependence between the polarization P and theelectric field strength E in a dielectric:

P = eo (s - 1) E. (4.8)

Hence, the modulus of the dielectric polari­zation is proportional to the strength of theelectric field. The proportionality factor in (4.8)

11-01536

162 Electrons in Dielectrics

is commonly designated as x:

x == £0 (8 -- 1), (4.9)

and is called the dielectric susceptibility (electricsusceptibility). I t does not in practice dependon E and can be considered a property of thematerial.

Equation (4.9) relates two of the parametersof a dielectric: x and 8.

We have only discussed the simplest mecha­nisms of polarization, but in real dielectricspolarization is more complex. It occurs due toa limited displacement of the charged particlesin dielectrics due to the effects of both externaland internal fields. A result is that each elementof the dielectric's bulk, within which the displace­ment took place, aquires a nonzero electricmoment. There are different types of polarizationdepending on the particles involved and theactual displacements.

The particles that can be displaced or orientedin an electric field fall into two categories, i.e.elastically bound and weakly bound particles. Thedisplacement of an elastically (strongly) boundparticle is opposed by an elastic force that tendsto return it to the equilibrium position aroundwhich the particle oscillates thermally. Anelastically (strongly) bound particle is shifteda small distance from the equilibrium positionby an electric field. A weakly bound particlehas more than one equally probable equilibriumpositions in the absence. of an electric field.Thus while it oscillates around one equilibriumposition it can be moved to another position dueto thermal fluctuation. When an electric field

4.6. Dielectric Constant 163

is applied, the probability of the transitionincreases: for a positively charged particle inthe direction of the field, and for a negativelycharged particle in the opposite direction. Oppo­sitely charged particles are shifted farther apartwhen they are weakly bound than when elasti­cally bound. Two types of polarization correspondto these two kinds of particles, i. e. elastic andrelaxation (or thermal) polarization.

The polarization does not occur the momentan electric field is applied, it takes some timeto be completed. The time for relaxation polari­zation to build up is essentially longer than forelastic polarization to build up. The followingtypes of polarization are commonly distinguished,

Electronic elastic polarization. This is thepolarization that results from the displacementof the electron shells of atoms, molecules, andions with respect to heavy 'fixed' nuclei toa distance that is less than the dimensions ofthe atoms, molecules, or ions. It occurs in alldielectrics for any state of aggregation, and takes10-1~-10-15 s, to build up.

Ionic elastic polarization. This occurs in ioniccrystals. It is brought about by the elasticdisplacement of positive and negative ions fromtheir equilibrium positions to a distance lessthan the distance between adjacent ions. It takes10-11_10-1~ s to build up.

Dipole elastic polarization. This occurs inmolecular crystals which consist of dipole mole­cules. Since these molecules are fixed and cannotrotate freely, they can only turn through a smallangle due to the effect of an electric field. Thistype of polarization builds up relatively slowly.

tl*

1M Electrons in Dielectrics

Electronic relaxation polarization. This occursin certain crystals in which the electrons canmove in the vicinity of the defects to which theyare attached. They move a small distance of theorder of several atomic spacings and the polari­zation builds up relatively slowly.

Ionic relaxation polarization. This occurs ina number of ionic crystals. It is brought aboutby the displacement of weakly bound ions fromtheir equilibrium positions to a distance of theorder of several atomic spacings. The polariza­tion builds up relatively slowly.

Dipole relaxation polarization. This occurs indipole dielectrics in their liquid and gaseousstates. The molecules can then rotate freelyaround their equilibrium positions, They areoriented randomly in the absence of an electricfield, but polarization appears in a field due tothe existence of a prevailing orientation of thedipoles. The build-up time is of the order of10-10 s or more.

Structural polarization. This includes certainkinds of polarization that are due to the appearanceof bulk charges in the dielectric and is closelyrelated to the process of electrical conduction.It occurs when free current carriers move underthe influence of an electric field, but are hinderedby some obstacles and so not all the carriersreach the electrodes or those that do approachthem cannot be fully discharged. Bulk chargesthus appear near the electrodes in homogeneousdielectrics (the high-voltage polarization we havealready discussed in Section 4. f). The advanceof the free charges can be hindered by defects ofthe crystal lattice that capture the charge car-

4.8. Dielectric CoDStaDt 165

riers. In heterogeneous dielectrics, the chargescan accumulate at the interfaces between phases(tnterlayer polarization). It is characteristic ofstructural polarization that the charges aredisplaced much farther than in other types ofpolarization., and the build-up time is very great,sometimes a dozen minutes or more.

Spontaneous polarization. This occurs in somedielectrics in the absence of an electric field.Ferroelectrics are examples.

Persistent polarizatlon. Given certain condi­tions., the polarization persists for a long timeafter the removal of the electric field in somedielectrics. These are called electrets and havelow electrical conductivity (see Section 4.5).

As we mentioned above, after the electricfield has been set up not every type of polariza­tion builds up at the same rate. If a dielectricis in an alternating electric field, only thosetypes of polarization whose build-up time 't isless than half the period T of the electric fieldcan be established. As a result the dielectricconstant e becomes dependent on the electricfield's frequency, the dielectric constant falling8S the frequency increases.

A phase difference between polarization andthe electric field occurs at 't > T. This phasedifference is brought about because the movingparticles lag behind the forces that cause themotion. Particle movement in the dielectric isaccompanied, as it were, by 'friction'. Thisresults in the dielectric becoming heated andtherefore in a loss of energy. The energy dissipatedper unit time in a dielectric in an electric fieldis- called the dielectric loss, or dielectric absorp-

166 Electrons in Dielectrics

tion. The loss due to electrical conduction, whichis accompanied by the evolution of Joule heat,is distinguished from the loss due to a lag inpolarization, which is called the relaxationdielectric loss.

4.7. Ferroeleetrics and Plezoeleetrles

Ferroelectrics are crystal dielectrics that possess,in a certain temperature range, a number ofunusual properties.

These properties were discovered for the firsttime in Seignette salt, and hence ferroelectricsare sometimes called seignette-electrics. Crystalsof Seignette salt have clearly pronounced anisotro­py, i.e. their properties depend on the orienta­tion of the crystals with respect to the directionof the electric field. The unusual properties canonly be observed when the crystal is in a special'Position with respect to the condenser plates.In any other position the crystal will have thesame properties as ordinary dielectrics.

Ferroelectrics include such dielectrics as po­tassium phosphate, potassium niohate, and ba­rium metatitanate. Over a hundred ferroelectricsare known at present.

The primary feature of a ferroelectric is theanomalous dependence of its dielectric constanton temperature. We noted in Section 4.6 thatthe dielectric constant E of common dielectricschanges very little with heating. The dielectricconstant temperature diagrams for 8 ferroelectrichas one or more very sharp maxima where"e canreach several thousands. The temperatures atwhich these maxima occur are called Curie

4.7. Ferroelectrics and Piezoelectrics f67

temperatures, or Curie points. For instance,Seignette salt has two Curie temperatures(+22.5°C and -15°C), while barium metati­tanate has only one Curie temperature (+80°C).If a ferroelectric has just one Curie temperature,its anomalous properties show up at temperaturesbelow it, whereas for one with two Curie tem­peratures, its properties are anomalous in thetemperatures between the Curie points.

The second feature of a ferroelectric is a non­linear dependence of its polarization on thestrength Eo of the external electric field. Itfollows from Equations (4.7) and (4.8), that thepolarization P is proportional to Eo in ordinarydielectrics. The relationship in ferroelectrics ismore complex in nature, being different indifferent ferroelectrics. The dielectric parameter eis then clearly not a constant but depends on Eo.

The third and the most remarkable feature offerroelectrics is that the dependence of the pola­rization on Eo has the appearance of a hysteresisloop, like the magnetization curve of a ferro­magnet. Suppose a ferroelectric that is notpolarized is put into the electric field betweentwo capacitor plates and the strength of thefield is increased. The polarization P will risenonlinearly and reach saturation at a certainvalue of P 8- The polarization will not changeeven if Eo is increased further. If the field in thecapacitor then decreases, the P versus Eo graphwill not be the same as the curve obtained forrising Eo: the new curve will lie above the oldone. When Eo is down to zero, the polarizationwill be nonzero and equal to a certain value Pothat is called the residual polarization. The resi-

Electrons .in Dlelectrics

dual polarization persists in the ferroelectric evenin the absence of an external electric field. Toreduce the polarization to zero. a field E; must beapplied in the opposite direction; this field iscalled an electric coercioe force.

There is a clear similarity between the electricproperties of Ierroelectrics and the magneticproperties of Ierromaznets, which is exactly whythe dielectrics are called ferroelectrics.

The behaviour of ferroelectrics can be explainedas follows. Strong interaction between theparticles making up the crystal results in theappearance of separate domains within the ferro­electric, domains being regions with nonzeropolarization. These domains were first discoveredin barium metatitanate, and they can be seenthrough a polarizing microscope.

The polarization vector has different direc­tions in different domains, so that the electricmoment of the whole specimen is close to zero.This corresponds to ths lowest energy state,since otherwise, polarization charges would ap-'pear on the surface of the specimen, and a fieldwould be induced around it containing additionalenergy.

The polarization directions of individual do­mains change in the external electric field to thedirection of the fi.eld. In addition, the mostadvantageously oriented domains can grow bypushing their boundaries into neighbouring do­mains. The resulting electric moment of theferroelectric becomes nonzero. The presence ofpolarized domains in Ierroelectrics has an essen­tial effect on their electric properties.

The domain! only exist within a certain tem-

4.7. Ferroelectrics and Piezoelectric! 169

perature range where the ferroelectric propertiesappear.

So far we have only considered cases whenthe polarization of a dielectric was brought aboutby an asymmetry in the localization of positiveand negative charges and was caused by an exter­nal electric field. This asymmetry can be producedin some crystals by deforming them mechanically.If such a crystal is compressed or stretched ina certain direction, it becomes polarized, andpolarization charges appear on its surface. Theycan then be registered, for instance, by anelectrometer. This phenomenon is called theplezoelectric effect. The electric moment thatappears in the crystals due to mechanical defor­mations depends on the crystal lattice directionalong which the deformation occurs and on thecrystal lattice type. The piezoelectric effectonly occurs in crystals whose structure does notpossess a center of symmetry.

Crystals that exhibit the piezoelectric effectare called piezoelectrics. They include Seignettesalt, ammonium phosphate, potassium phosphate,ethylenediamine tartrate, quartz, and tour­maline.

Let us discuss the piezoelectric effect in somedetail, taking quartz as an example. Quartzcrystals have four symmetry axes. An axis isa symmetry axis when a rotation of the crystalaround it through an angle a = 36001n bringsthe crystal into 8 position indistinguishablefrom its original position. If n = 2, the sym­metry axis is said to be a two-fold axis, if n = 3,it is 8 three-fold axis, etc. Quartz crystals haveone three-fold symmetry axis called the optt«

f70 Electrons in Dielectrics

axis and three two-fold axes called the electricalaxes. The latter are at right angles to the opticaxis and at 1200 to each other.

Suppose we have a specimen in the Iorm ofa parallelepiped shown in Fig. 28 cut froma quartz crystal so that its z-axis is the opticaxis of the crystal, and its x-axis is one of theelectric axes.

If we apply a force f x to the specimen compres­sing it in the direction of the z-axis, polarizationcharges +Q and -Q will appear on the facesperpendicular to this axis. The value. of Q isproportional to the compressing force f x anddoes not depend on the specimen's dimensions,I.e,

(4.10)

where k is the piezoelectric constant of the material.This is called longitudinal piezoelectric effect.

However, if we apply a compressive (or ten­sile) force f z to the crystal in the direction ofthe z-axis, no polarization will occur.

If a compressive force f y is applied in thedirection of the y-axis, polarization charges willappear like those in the first case and on thefaces perpendicular to the x-axis. However, thesigns of the charges will be opposite to the onesin the first case. The charges will also be pro­portional to the value of compressive force f y'

but now they also depend on the dimensions aand b of the specimen along the x- and y-axesrespectively, i. e.

bQ=k-fy,a

where k is the same constant as in (4.10).

4.7. Ferroelectrics and Piezoelectrics t7t

This is called transverse piezoelectric effect.It appears because the force compressing thespecimen along the y-axis causes a secondaryexpansion of the specimen along the x axis.

If the force applied to the crystal is notcompressive but tensile, the signs of the polari­zation charges change.

The reverse piezoelectric effect occurs as well.I f an electric field is applied to metal electrodesfixed onto opposite faces of a crystal, the dimen­sions of the crystal change a little, the crystalbecoming longer or shorter depending on thedirection of the field.

The inevitability of the reverse piezoelectriceffect can easily be shown to follow from the lawof the conservation of energy. Suppose a compres­sive force f is applied to a dielectric specimen.When there is no piezoelectric effect, the workof the external forces is equal to the potentialenergy of the specimen that is elastically com­pressed. If there is a piezoelectric effect, polari­zation charges appear on the specimen faces,and an electric field appears that contains anadditional energy. According to the law of theconservation of energy, the work performed whenthe specimen is compressed will be greater, andtherefore an additional force counteracting thecompression will appear. This is the force ofthe reverse piezoelectric effect. It acts in a direc­tion opposite to that of the force of the directpiezoelectric effect, i.e. if the charges +Q and-Q appear on the specimen faces in compression,the specimen will expand if the same chargesappear due to an applied electric field, and viceversa.

t72 Electrons in Dielectrics

It follows from what we have said in this andall the preceding sections that the properties ofcrystals and their behaviour under the influenceof external factors are determined by the behav­iour of the electrons in the crystals. Theseelectrons are the true masters of crystals.

Remarks in Conclusion:Theory and Experiment

To conclude this book, let us glance at the groundwe have covered. We have presented a varietyof experimental facts and patterns and inter­preted some of these patterns theoretically. Theunion of theory and experiment is the wonder­working alloy of which the modern physics isconstructed.

Experiment is the culture medium in whichtheory sprouts. The theory is like theatre flood­lights for the experiment. A theory divorcedfrom experiments is nonsense. An experimentwithout a theory is a blind alley. Experimentis the origin we always come back to, and it isthe natural judge of the theory. The theoristalways follows the experimenter while at thesame time showing him the path. But we mustnot forget that nature is far more complicatedthan any theory and it will never tire of chal­lenging our perception.

Any physicist, like any researcher, is foreverconfronted by 'how' questions and 'why' ques­tions. How does a physical phenomenon develop,and what are the laws it obeys? Why does itdevelop the way it does, and not in another way?To answer a 'how' question is to describe thephenomenon. To answer a 'why' question is toexplain it. The first sort of questions is answeredby the experimental physicist, and the theoreti­cal physicist is called in to answer the secondone. In the course of time some of 'how' and

174 Theory and Experiment

'why' questions are answered but then new'how' and 'why' questions emerge. This is thecause of the continuous evolution of science,'the daughter of astonishment and curiosity'.I will be very glad if the feelings of astonish­ment and curiosity are aroused in you as youread this book.

Mir Publishers would he grateful for your comments onthe content, translation, and design of this book.We would also be pleased to receive any other suggestionsyou may wish to make.

Our address is:Mir Publishers2 Pervy Rizhsky Pereulok1-110, GSP, Moscow, 129820USSR

Printed in the Union of Soviet Socialist Republics

The increasingly important field of solid-state physics concerns the behavior of electrons in various crystals. Problems of solid-state physics, which include specific differences between metals and dielectrics and the remarkable properties of semiconductors, are particularly topical in todays `electrnic’ society.

Electrons and Crystals by Dr. Theodore Wolkenstein covers fundamentals of solid state physics in an engaging way. Written in an easy, readable style, the book is intended as a supplement to textbook in secondary school physics courses, and the approach to certain topics in the colume is, therefore, unique. The material is presented in terms of models and required no special additional knowledge.

Suitable for general reader with a good command of elementary physics and mathematics, this book can also serve as a useful study guide for high-school students.