Semiconductors Made Simple Polyakov

8/11/2019 Semiconductors Made Simple Polyakov

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.sJ;,J,

p 5

Fi rs t published

1985

~ e v i s e d r om th e

1981

Russ ian ed i t ion

~ ~ T ~ J I ~ C T B O~ ~ O C B ~ Q ~ H E ~ D ,

981

English translation, Mir Publishers ,

1985

Contents

Chapter 1.

T H E B A ND T H E O R Y O F

SOLIDS

Sec. 1. Str uct ure of Atoms. Hydrogen

Atom

Sec. 2. Many -Electron Atoms

Sec.

3

Degeneracy of Ener y Levels in

F ree A tom s. ~ e m o v a fof Degener-

acy of Ex tern al Effects

Sec.

4.

Formation of Ene rgy ~ a n d sir;

Crystals

Sec.

5

Fil ling of Ene rgy ~ a n h s y Elec-

trons

Sec. 6 Division of Solid s in to Conductors,

Semiconductors, and Dielectrics

Chapter

2 ELECTRICA L CONDUCTIVITY

OF SOLIDS

Sec. 7. Bonding Forces in a Crystal La ttic e

Sec.

8

Electrical Conductivity of Metals

Sec. 9 Conductivity of Semiconductors

Sec.

10

Effect of T emp erature on th e

Charge Carrier Con centra tion in Se-

miconductors

Sec. 11. Tem peratu re Dependence o i Elec:

trical Conductivity of Semiconduc-

tors

Chapter

3 NONEQUILIBRIUM PROCES-

SE S IN SEMICONDUCTORS

Sec. 12. Generation and Recom bination of

Nonequilibrium Charge Carriers

Sec. 13. Diffusion Phenom ena in Semicon-

ductors

Sec.

14.

~hoto cond uct ion and Absorp tion

of Light

Sec. 15. Luminescence


5/122

Contenta

Chapter

4. CONTACT PHENOMENA 121

Sec. 16. Work Function of Metals 121

Sec.

17.

The Fermi Level in Metals anh ;he

Fermi-Dirac Distribution Function

127

Sec. 18.The Fermi Level in Semiconductors 133

Sec.

19.

The Contact Po ten ti a1 Difference

143

Sec.

20.

Metal-to-Semiconductor Contact

147

Sec. 21. Rec tifier Prop erties of the Metal-

Semiconductor Junction

152

Sec.

22.

p-n Junction

161

Sec.

23.

Rec tifying Effect of the p-n junc-

tion

175

Sec.

24.

Breakdown of the p-n ~un cti on

186

Sec. 25. Elec tric Capacitance of the p-n

Junction 190

Chaptel 5. SEMICONDUCTOR DEVICES 193

Sec. 26. Hall Effect and Hall Pickups 193

Sec.

27.

Semiconductor Diodes

205

Sec.

28.

Tunnel Diodes

211

ec.

29

Transistors

223

Sec.

30.

Semiconductor Injection (~ io ie j

Lasers

233

Sec.

31.

Semiconductors at Present and in

Future

24L

hapter

The Band Theory of Solids

W he n researchers f i rs t come across serniconduc-

to rs , there was a clear-cut division of al l sol id s

in to two la rge groups , v iz . conductors ( inc luding

a l l meta ls ) and insula tors (or d ie lect r ics ) and

these d i f fered in pr inc ip le in the i r proper t ies .

These new semiconductor mater ia l s could not be

inc luded in e i ther of these groups . On the one

ha nd , they conducted e lec t r ic c ur rent , a l though

t o a m uch l e sser ex t en t t h an me ta l l i c conduc to r s ,

and on t he o the r , t hey d id no t a lways conduc t .

Never the less , they d id conduct e lec t r ic i ty and so

were named semiconductors (or half conductors) .

La t e r , i t was d i s cove red t h a t s emiconduc tor s

di ffer f rom m eta ls both i n the way th ey conduct

and in the way externa l fac tors inf luence the i r

con duc tion. For exa mp le, the effect of tempe ra-

tur e on conduc t iv i ty of meta l l ic conductors an d

semiconductors i s qui te oppos i te . In meta ls , an

increase in tem pera ture causes a gradu al decrease

in con duct iv i ty , w hi le the hea t ing of semiconduc-

tors resul t s in a sharp increase in conduct iv i ty .

The introduction of impuri t ies also has different

ef fec ts on co nduct iv i ty of me ta l l ic conductors and

semiconductors . In meta ls , as a ru le , impur i t ies

worsen conduct iv i ty , whi le in semiconductors

the in t rodu ct ion of a negl ig ib ly smal l am ount of

ce r t a in impur i t i e s can r a i s e t he conduc t iv i t y by

ten s or even hundre ds of tho usan ds of t im es.


6/122

Ch

i

The Band Theory of Solids

Finally, if we send a beam of light of a flux of

some par t ic les on to a conduc tor , i t wi l l have

prac t ica l ly no e f fect on i t s conduc t iv i ty . On the

other han d, i r radiat ion or bom bard men t of a se-

miconduc tor causes a d ras t ic inc rease in i t s

c o n d u c t i v i t y .

I t i s in te res t ing to no te t ha t these p roper ties

of semiconductors are , to a considerable extent ,

typic al of d ielectr ics , hence i t wo uld be much

more correct to cal l semiconductors semi- insula-

tors or semidielectr ics .

In order to explain the behaviour of semicon-

duc tors in var ious condi t ions , to account fo r

their propert ies and to predict new effects , we

must cons ider the i r s t ruc tura l pecu l iar i t ie s . Th a t

i s why we s ha l l s t a r t wi th th e d i scuss ion of the

a tomic s t ruc ture o f mat te r .

Sec

1

St ru ctu re of Atoms.

Hydrogen Atom

From the course of phys ics you should know th a t

an a tom cons i st s of a nuc leus and e lec trons ro ta t -

ing a round i t . Th is model of an a tom was p ro-

posed by the En glish physicis t Ruther ford. Inl l913,

the Danish physicis t Niels Bohr , one of the

founders of qu ant um m echanics , used the model

for the f irs t correct calcula t ions of hydro gen ato m

t h a t a g r e e d w e l l w i t h e x p e r i m e n t a l d a t a . H i s

theory of th e hydrogen a tom has p layed an ex t reme-

ly im por ta n t ro le i n the deve lopment of quan-

tum mechanics , though i t underwent cons iderab le

changes la ter .

Hydrogen Atom. Bohr's Postulates. Accord-

ing to the Ruther ford-Bohr model , hydrogen

1

Structure of Atoms. Hydrogen Atom

atom con sis ts of a s ingly charged posi t ive nucle us

and one e lec t ron ro ta t ing a round i t . To a f i r s t

a p p r o x i m a t i o n , i t c a n b e a s su m e d t h a t t h e

e lec t ron moves a long the t ra jec to ry which i s

a c i rc le wi th the f ixed nuc leus a t i t s cen t re .

According to the laws of c lassical e lectrodynam-

ics , an y accelerated mo tion of a charged bo dy

( inc lud ing the e lec t ron) mus t be accompai~ ied y

the emission of e lectrom agnet ic waves. In the

model under cons idera t ion , the e lec t ron moves

wi th a t remendous cen t r ipe ta l acce le ra t ion , and

there fore i t shou ld con t inuous ly emi t l igh t .

S h o u l d i t d o s o , i t s e n e r g y w o u l d g r a d u a l l y

decrease and the electron would come closer

and c loser to the nuc leus . F ina l ly , the e lec t ron

would un i te wi th th e nucleus ( fal l on i t ) .

Noth ing of th i s k ind occurs in rea l i ty , a nd a t om s

c lo no t emi t l igh t in the i r unexc i ted s ta te . In

ort1r.r to expla in th is fac t , Bohr fo rmu lated

two postulates .

According to Bohr 's f i rs t postulate , an electron

can on ly be in an o rb i t fo r which i t s angula r

momentum ( i .e . the product of the electron

m o m e n t u m

u

b y t h e r a d i u s

r

of t h e o r b i t ) i s

a mult iple of

h/2n

(where h i s P lanck ' s con-

s tan t )* . Wh i le the e lec t ron i s in one of these

orb i t s , i t does no t emi t energy . Each a l lowed

Planck's constant i s a universal physical constant and

has the meaning of the product of energy and time, which

is called

act ion

in mechanics . Since the quan t i ty h i s so to

say an elementary act ion, Planck's constant i s cal lkd the

q u a n t u m

(portion)

of act ion

The introduction of the

quantum of act ion laid the basis for the most important

theory of the 20th century physics , viz. the qu an tum

theory

The

magni tude of the quantum of act ion is very

small: h

6.62 X 10 94 J

set


7/122

1

Ch. i The Band Theory of Solids

elec t ron orbi t cor responds to a cer t a in energy, o r

cer ta in energy s tate of the atom , wh ich is cal led

a

s t a t ionary s t a t e . Atoms do not emi t l ight in

a

s t a t ionary s t a t e . The analy t i c expres s ion of

Bohr ' s f i r s t pos tu la t e i s

where n

=

1 ,

2 , 3

i s an i n t ege r ca l l ed t he

pr inc ipal quantum number .

Bohr ' s s econd pos tu la t e s t a t es tha t absorpt ion

or emis s ion of l ight b y an a tom occurs dur ing

t rans i t ions of the a to m f rom one s t a t io nary s t a t e

t o ano t he r . T he ene r gy is abs o r bed o r em i t t ed

upon t r ans i t i on i n ce r t a i n am oun t s , ca l led quan -

ta , whose va lue hv i s de termine d by the d i ff erence

i n energies cor responding to the i n i t i a l an d f ina l

s t a t io na ry s t a t es of th e a tom:

where Wm i s the energy of th e in i t i a l s t a t e of th e

a tom , W, th e energy of i t s f ina l s t a t e , an d

v

t h e

f r equency o f l i gh t em i t t ed o r abs o r bed by t he

atom. If W,

>

W,, the a tom emi t s energy ,

an d if Wm W,, the energy i s absorbed. Qu anta

of l ight are cal led photons .

Thu s , acco r d ing t o Bohr ' s t h eo r y , t h e e l ec tr on

i n an a t om canno t change i t s tr a j ec t o ry g r ad ua l l y

(cont inuous ly) but can only jump f rom one

s t a t i ona r y o r b i t t o ano t he r . L i gh t i s em i t t ed

jus t when the e l ec t ron goes f rom a more d i s t ant

s t a t i o n a r y o r b i t t o a n e a re r s t a t i o n a r y o r b i t.

A t om i c Rad i i

of

Orbi ts and Energy Levels .

T h e r adi i of a l lowed e lec t ron orb i t s can be found

b y using Coulomb 's law , the relat io ns of class i -

1

Structure of

Atoms

Hydrogen

Atom

11

c a l mechanics , and Bohr ' s f ir st pos tu la te . T hey

ar e g iven by the fo l lowing expres sion:

h

~ = ~ 2 -

4n me

The neares t to the nucleus a l lowed orb i t i s char -

ac t e r i zed by

n

= I

U s i ng t he expe r i m en t a l ly

Me-.

/ \

\

\

Fig

1

ob tain ed v alue s of

m,

e a n d

h

we f ind for the

radiu s of t h i s orb i t

r

=

0.53

x

loo8

cm.

Th i s va l ue i s t aken f o r t he r ad i us of t he hydr o -

gen a t om . A ny o t he r o r b i t w i t h a quan t um num -

ber

n

h a s t h e r a d i u s

Hence , th e r ad i i of success ive e l ec t ron orb i t s

increase as n2 (Fig. 1 ) .

The to ta l energy of a n a tom w i th an e l ec t ron

i n

t he n t h o r b i t i s g i ven by t he f o rm ul a


8/122

2

Ch 1 The Band Theory of Solids

These energy values are cal led atomic energy

levels. If we plot the possible values of energy

of an a to m a long the ver t i ca l ax i s , we sha l l

obta in th e energy spectrum of the al lowed sta te s

of a n ato m (Fig. 2).

I t c a n b e s ee n t h a t w i t h i n c re a s in g n , t h e s e p a -

ra t ion be tween success ive energy leve ls rap id ly

Fig

2

decreases . This c .an be easi ly explained: an

increase in the energy of an a tom (due to the

energy absorbed by the a tom f rom outs ide) s

accompanied by a t ransi t ion of the electron to

more remote orb i t s where the in te rac tion be tween

the nucleus and the electron becomes weaker .

For this reason, a t ran si t ion between n eighbour-

ing fa r o rb i t s is assoc ia ted wi th a very smal l

1.

Structure of Atoms Hydrogen

Atom

3

change in energy . Th e energy leve ls cor responding

to remote orb i t s a re so c lose tha t the spec t rum

becomes prac t ica l ly con t inuous . I n the upper pa r t

the con t inuous spec t rum i s bounded by the

ionizat ion level of the atom (n

=

m which

cor responds to the comple te separa t ion of the

electron from the nucleus ( the electron becomes

free).

The minus s ign in the express ion for the

to ta l energy of an a tom ind ica tes th a t a tomic

energy i s the lower the c loser i s the e lec t ron to

the nuc leus. In o rder to remove th e e lec t ron

f r o m t h e n u d e u s , w e m u s t e x p e n d a c e r t a i n

am oun t of energ y, i.e . sup ply a defini te am oun t

of energy to the a tom f rom outs ide . For n

=

CQ,

i .e . w hen th e atom is ionized, the energy of a n

a tom i s t ake n equa l to ze ro . This i s why nega t ive

values of energy correspond t o n CQ The leve l

w i t h n

=

i s c h a r a c t e r i z e d b y t h e m i n i m u m

energy of the a tom and the min imum rad ius o f

the al lowed electron orbi t . This level is cal led

the ground, or unexcited level. Levels with n

=

=

2,

3

4

are calle d excitation levels.

Quantum

Numbers

Accord ing to Bohr s theo-

ry e lec t rons move in c i rcu la r o rb i t s . Th is theory

provided good resu l t s on ly for the s imples t a to m,

v iz. the hydrogen a tom . Bu t i t cou ld no t p rov ide

qua nt i t a t ive ly cor rect resu l t s even for the he l ium

a t o m . T h e n e x t s t e p w a s t h e p l a n e t a r y m o d e l

of an a tom . I t was assumed tha t e lect rons, l ike

the p lan e ts of the so la r sys tem, move in e l l ip -

t i ca l o r b i t s wi th th e nuc leus a t one of th e foc i .

However , th i s model was a l so soon exhaus ted

s ince i t f a i l ed to answer many ques t ions .

T h i s i s c o n n e c t e d w i t h t h e f a c t t h a t it

is

im-


9/122

4 Ch

1 The Band Theory of S olid s

possible in principle to determine the nature

of the motion of a n electron in an atom. There are

no analogues of this motion in the macroworld

accessible for observation. We are not only

unable to trace the motion of an electron but

we cannot even determine exactly its location

at a particular instant of time. The very concept

of an orbit or the trajectory of the motion of a n

electron in an atom has no physical meaning.

It is impossible to establish any regularity in

the appearance of an electron a t different po ints

of space. The electron is smeared in a certain

region usually called the electron cloud.

For

an unexcited atom, for example, this cloud has

a spherical shape, but it s density is not uniform.

The probability of detecting the electron i s

highest near the spherical surface of radius

r

corresponding to the radius of the first Bohr

orbit. Henceforth, we shall assume that the

electron orbit is a locus of points which are

characterized by the highest probability of

detecting the electron or, in other words, the

region of space with the highest electron cloud

density.

The electron cloud will be spherical only for

the unexcited st ate of the hydrogen atom for

which the principal quantum number is n

(Fig. 3a). When n 2, the electron, in addition

to a spherical cloud whose size is now four times

greater , may also form a dumb-bell-shaped cloud

(Fig. 3b). The nonsphericity of the region of

predominant electron localization (electron cloud)

is taken in to account by introducing a second

quantum number

1

called the arbital quantum

number. Each value of the principal quan tum

1 Struc ture of Atoms Hydrogen Atom

5

number n has corresponding positive integral

values of th e quarl tum number 1 from zero to

(n ):

For example, when n 1 ,

1

has a single value

equal to zero. If n

3

1 may assume

the

values

Fig 3

0, 1, and

2.

For n the only orbit is spherical,

therefore 0. When n

2

both the spherical

and the dumb-bell-shaped orbits are possible,

hence 1 may be equal either to zero or unity.

For n 3,

1

0, 1,

2.

The electron cloud corre-

sponding to the value

1

2 has quite a com-

plicated shape. However, we are not interested

in t he shape of the electron cloud but in the

energy of the atom corresponding to i t.

The energy of the hydrogen atom is only deter-

mined by the value of the principal quant um

number n and does not depend on the value of

the orbi ta l number 1. I n other words, if n 3

the atom will have the energy

W

regardless of

the shape of the electron orbit correspondingto


10/122

6

Ch. 1 The Band Theory

of

Solids

the given value of n and various possible values

1. This means that upon a transition from the

excitation level to the ground level, the atom

will emit photons whose energies are indepen-

dent of the value of 1.

While considering the spat ia l model of an

atom, we must bear in mind that electron clouds

have definite orientations in it. The position of

an electron cloud in space relative to a certain

selected direction is defined by the magnetic

quantum number m, which may assume integral

values from -1 to +l, including 0. For a given

shape (a given value of l), the electron cloud may

have severa l different spat ial orientations. For

1, there will be three, corresponding to

the -1, 0, and +I values of the magnetic quan-

tum number m. When 1 2, there will be five

different orientat ions of the electron cloud corre-

sponding to m , -1, 0 +I, and +2.

Since the shape of the electron cloud in a free

hydrogen atom does not influence the energy

of the atom, the more so i t applies to the spat ial

orientation.

Finall y, a more detailed analysis of experi-

melital results revealed that electrons in the

orbits may themselves be in two different states

determined by the direction of the electron spin.

But what is the electron spin?

In 1925, English physicists G Uhlenbeck and

S 'Goudsmit put forward a hypothesis to explain

the fine structure of the optical spectra of some

elements. They suggested that each electron

rotates about its axis like a top or a spin. In

this rotation the electron acquires an angular

momentum called the spin. Since the rotation

I\

1.

Structare o f A t o m s . ~ y d r o ~ e n 'tom

i7

can be either clockwise or anticlockwise, the

spin (in other words, the angular momentum

vector) may have two directions. In

h 2n

units,

the spin is equal to 112 and has either

+'I

or

- sign depending on the direction. Thus, the

electron orientation in the orbit is determined by

the spin quantum number a equ al to 12.' I t

should be noted that the spin 'orientation;" like

the orienta tion of the electron orbi t, does not

affect the energy of hydrogen a tom tin a free

state.

Subsequent investigations and calculations

have shown tha t it is impossible to explaint the

electron spin ,simply by it s rotation about the

axis. When the angular velocity of >theelectron

was calculated, it was found that the linear,

velocity of poi.nts on the electron equator (K w

assume that the electron has .the Spherical shape,)

would be higher than the velocity of light,

which is impossible. The spin is an insdparablki

character istic of the electron like it s mas? or

charge.

Quantum Numbers

as

the

Electron Address

in an Atom. Thus, we have learned that ,in

order to describe the motion of t he electron i n

an

atom or, as physicists say, to define the state of

an electron in an atom, we must define, a set of

four quantum numbers:

n,

1, m, and a

Roughly speaking, the principal qupn um,

number n defines the size of the dlect~on'Qrbit,

The larger n, the greater region of space is em-

braced by the corresponding electron cloud. By

setting the value of n, we define the number of

the electron shell of the atom. The number

n

itself can acquire any integral value from' 1


11/122

18

Ch

1. The Band T h r


12/122

2

Ch. 4 The Bahd

Theory of

Solids

2. Many-Electron Atoihs

21

The second shell n 2 consists of two sub-

shells since

I

can betkit her or 1 In atomic

physics,

let ter symbols 'instead of the numeri-

oal values of

I

arelused for describing subshells.

For example, regardless of thevalue of the

principal. quantum number n, all subshells with

I 0 are denoted

by

8 subshe~lswith I

=

1

are

denoted by p, for

I

= 2 the symbol is used,

and so on. In this connection, it is said that'the

second shell consists of the s- and p-subshells.

The s-subshell

I

=

0

consists of one circular

orbit and may contain only' two electrons, while

the p-subshell consists of three orbits (m may be

equal to -1, 0, and +1) and may contain six

electrons. The total n u a e r of electrons in the

second shell is equal to eight.

Similarly, we can calculate the possible num-

bar of electrons in any shell and subshell. For

example, there can be 10 electrons in the 3d

subshell (n = 3

1= 2), viz. two electrons in

each of the five orbits characterized by different

values of the quantum number m. The maximum

number of electrons in any subshell is equal to

2 (21 1). In spectroscopy, letter symbols (terms)

are ascribed to different shells: the first shell

is denoted by K the second by L the third by M

and so on.

The single electron in a hydrogen atom is in

a centrally symmetric field of the atomic nucleus;

its energy is determined solely by the value of

the principal quantum number n and does not

depend on the values of the other quantum num-

bers. On the other hand, in many-electron

atoms each electron is in -the field created both

by the nucleus and by the other electrons.


13/122

Ch. 1. The Band Theory of Solids

Consequent ly , the energy of an e lec t ron in m any-

e l ect ron a tom s tu rns ou t t o depend bo th on t he

p r inc ipa l qua n tum number

n

and on t he o rb i t -

a l n u m b e r I t hough r ema in ing i ndependen t

of t h e va lu es of m a n d a.

Th is fea ture of many-e lec t ron a tom s leads to

cons iderable d i f ferences be tween the i r energy

Fig.

spec t rum an d the spec t rum of hydrogen a tom .

Figure 4 shows a pa r t of t he spec t rum for many-

electron atom (the energy levels of the f irs t three

a tomic she l l s ) . Dark c i rc les on th e leve ls indica te

the max imu m num ber of e lec t rons which can

occupy the cor responding subshel l .

I t i s we ll known th a t a sys t em no t sub j ec ted

to ex t e rna l e f f ec t t ends t o go i n to t he s t a t e

wi th the lowes t energy. Atom

is

no t an except ion

2.

Many Electron Atoms

23

in th i s respec t . As the a to mic she l l s a re f i l led , the

e lec t rons tend to occupy the lowes t leve ls and

would al l occupy th e f irs t level if the re were no

l imi ta t ions imposed b y P aul i s exc lus ion pr in-

c ip l e . The on ly e l ec t ron i n t he hydrogen a tom

occupies the lowes t orb i t be longing to the

Is

level . irl the he l ium a tom, t he s ame o rb i t con -

ta ins a l so th e second e lec t ron , and the f i rs t a tom-

ic she l l i s f i l led . I t should be noted th a t he l ium

is an i ne r t ga s, and i t s g r ea t s t ab i l i t y i s due t o

the comple te outer she l l .

I n t he l i t h ium a tom, t he re a r e on ly t h r ee el ec -

t rons . Tw o of them occup y the fi r st she l l , an d

the t h i rd i s i n t he s econd she l l w i th

n 2

( i t

cannot occupy th e f i rs t sh e l l due to Pau l i s

exc lus ion p r inc ip l e ) . L i t h ium i s an a lka l i me ta l

whose va l ency equa l s un i t y . Th i s means t h a t

t he e l ec t ron i n t he s econd she l l i s weak ly bound

to the a tomic core and can be eas i ly de tached

from i t . This can be judged f rom the ioniza t ion

po ten t i a l wh ich fo r l i t h ium is o n l y e q u a l t o

5.37 V , whi le for he l ium i t i s equal t o 24 .45 V.

As th e numbe r of e lec t rons in an a tom increases ,

the outer subshel ls an d she l l s a re f il led . For

exam ple , s ta r t in g wi th boron, which has 5 e lec-

trons , th e 2p-subshell is f i l led. Thi s process is

comple t ed i n i ne r t ga s neon wh ich has a fu l l y

fi l led second shel l and is thus characterized by

the g rea t s t ab i l i t y . The e l even th e l ec tron i n t he

sod ium a tom s t a r t s popu la t i ng t he t h i rd she l l

(3s-subshell) , and so on.


14/122

4

Ch

1

The Band Theory of S olids

3

Degeneracy of Energy Levels in Free Atoms

25

Sec. 3 Degeneracy of Energy Levels

in Free Atoms, Removal of Degeneracy

by External Effects

b

~egenera te states.We have already noted that

in many-electron atoms the energy of electrons

i3 oqly determined by the values of t he quantum

aumbers n and I and does not depend on the

values of m and

a

This can be illustrated by the

energy spectrum shown in Fig.

4.

Indeed, a ll six

electrons in the 3p-subshell, for example, have

thersame energy, although they have different

values of m and a States described by different

sets of quantum numbers but having the same

en esgy are called degenerate. Similar ly; the

energy levels corresponding to these states are

also.called degenerate. The levels are degenerate

while the atoms are in the free sta te. If, however,

the atoms .are placed in a strong magnetic or

electric field, the degeneracy is partially or com-

pletely removed. Let us ill ustr ate th is removal of

degeneracy with respect to the quantum num-

ber

m

Degeneracy, Removed by an External Field.

Rifl eren t values of the quantum number m

correspond to different sp at ia l. orientat ions of

simi lar electron orbits. In the absence of an

external field, different orientations of the orbi ts

do not affect the energy of the electrons. If,

however, we place an atom in an external field,

the field will act differently on the electrons in

orbits oriented in different ways with respect to

th e direction of th is field. As a result, changes in

energies of electrons in simi lar ly shaped but

differently oriented orbits will be different both

in magnitude and in sign: energies of some.

electrons will increase while those of others

will decrease. The energy levels for different

electrons in the spectrum will also change their

arrangement. Moreover, ins tead of one energy

level corresponding to all electrons in similar

External

field is

External ~wl hed

field

is

swltched

= 2

Y

21 1

i n total

Fig 5

orbits several sublevels appear in the spectrum,

the number of sublevel being equal to the number

of differently oriented simil ar orbits , i.e. to the

number of possible values of the quantum number

m Figure

5

shows the result of an externa l electric

field acting on the 3d-level, for which n

3

and

1 2 I t can be seen that spli tti ng of the level

into sublevels and the displacement of sublevels

occur simultaneously.

The process in which previously indistinguish-

able (from the point of view of energy) degenerate

levels become distinguishable is called the

removal

of

degeneracy. Let us illustrate degener-

acy removal with another example.


15/122

6

Ch. 1. The Band Theory of Solids

We cons ider an e lec t ron hav ing a ce r ta in

energy

W

in a one-dimensional space character-

ized by the coord ina te x (Fig .

6 .

I n t h e a b se n ce

of an external f ie ld, the s ta te of this e lectron is

described by one energy level W i r respect ive

of the d i rec t ion of i t s mot ion . In o t l ie r words , in

th e absence of a n extern al f ield the energy level

Fig.

W

i s doubly degenera te . I f we apply an ex te r -

na l e lec t r ic f ie ld , say , a long the x-ax is , the

energy of the e lec t ron becomes depe ndent on th e

d i rec t ion of i t s mot ion . I f the e lec t ron moves

a long the x-ax is , i t wi l l be dece le ra ted by the

ex te r na l f i e ld , i t s energy becoming W E x

(where x i s the d i s tance covered b y th e e lec t ron).

If the e lec t ron moves in the oppo s i te d i rec t ion ,

i t s energy becomes W eEx. Cor responding ly ,

the a ppearanc e of two d i f fe ren t s ta t es i s mani -

fes ted in the energy spec t rum by the s l i t t in g of

the degenera te l eve l W in to two nondegenera te

levels W E x a n d W e E x . I n o t h e r w o rd s ,

the degeneracy i s removed under th e e f fec t of th e

ex te rna l f i e ld ,

4.

Formation of Energy Rands in Crystals

Sec. 4. Form ation of En erg y Band s

in C rys ta l s

Spli t t ing of Ene rgy Levels in a Crystal . Le t

u s do t h e f o l lo w i n g m e n t a l e x p e r i m e n t . T a k e N

a tom s of a subs tance a nd a r range them a t a suf fi -

c i e n t l y l a r g e d i s t a n c e fr o m e a c h o t h e r b u t i n

s u c h a w a y t h a t t h i s a r r a n g e m e n t r e pr o d u ce s

the c rys ta l l ine s t ruc tur e of th e mate r ia l . S ince

t h e s e p a r a ti o n b e tw e e n t h e a t o m s i s l a rg e , w e

can ignore the i r in te rac t ion and cons ider them

free. I n each of these ato ms , ther e are degenerate

levels with degeneracies equal to the number of

d i f fe ren t ly o r ien ted s imi la r o rb i t s in correspond-

ing subshe l ls . Le t us now s t a r t b r ing ing the

a toms c loser , re ta in ing the i r mutua l a r range-

ment . As the atoms converge, come closer , they

begin to experience the influence of their ap-

proach ing ne ighbours, which i s s imi la r to th e

influence of a n ex tern al elect ric field. T he

smal le r the separa t ion be tween the a toms ,

t h e

s t ronger i s the in te rac t ion be tween them. Owing

to this in terac t ion, degeneracy of t he energy

leve ls charac te r iz ing the f ree a toms i s removed:

each degenerate level spl i ts into

(21

1 nonde-

generate levels . All the atom s in a crys tal gen-

e ra l ly ex is t under the sam e condi t ions (excep t

for those which form th e extern al bou ndary of

the c rys ta l ) . I t cou ld seem there fore tha t each

a tom should cont r ibu te the same se t of nondegen-

era te sub leve ls in to the energy spec t rum th a t

character ized the crystal as a whole viz . one Is

sublevel, three 2p-sublevels, five 3d-sublevels,

and so on . Each sub leve l ma y conta in two e lec -

t rons wi th oppos ite sp ins . Al though th i s sp l i t t in g


16/122

8

Ch 1 The

Band

Theory of Solids

actually occurs, the corresponding sublevels ob-

tained from similar atomic levels differ from each

other in energy, some of them are higher in the

energy spectrum of the crystal than the initia l

levels of the individual atoms, while others lie

somewhat lower. This difference can be explained

by Pauli's exclusion principle generalized

for the entire crystal as a single entity. According

to this principle, no two nondegenerate sublevels

in a crystal may have the same enegy. Therefore,

when the crystal is formed, each energy level

spreads into an energy band consisting of N (21

I ) nondegenerate sublevels differing in energy.

For example, the Is-level spreads into Is-band

consisting of N sublevels which may contain 2 N

electrons, the 2p-level spreads into 2p-band con-

sisting of 3 N sublevels which may contain 6N

electrons, and so on.

The formation of energy band in a crystal from

discrete energy levels of individual atoms is

shown schematically in Fig. 7 The shorter the

distance r, the stronger the effect of the neighbour-

ing atoms and the more the levels are smeared .

The energy spectrum of a crystal is deter-

mined by the smearing of the levels correspond-

ing to the interatomic distance a, typical of

a given crystal.

The degree of smearing of levels depends on

their depth in an atom. The inner electrons are

strongly coupled to their nuclei and are screened

from external effects by the outer electron shells.

Therefore the corresponding energy levels are

weakly smeared. Naturally, the electrons in the

outer shells are most strongly affected by the

field of the crystal lattice, and the energy levels

4 Formation of Energy Bands in Crystals

29

corresponding to them are smeared the most.

I t should be noted that smearing of levels in to

energy bands does not depend on whether there

are electrons on these levels or whether they are

empty. In the lat ter case, the smearing of levels

0 10

Fig

7

indicates the broadening of the range of possible

energies which the electron may acquire in the

crystal.

Allowed and Forbidden Bands. From what

has been said above, it follows that there is an

enti re band of allowed energy values correspond-

ing to each allowed energy level in a crystal,

i.e. there is an allowed band. Allowed bands

alternate with the bands of forbidden energy, or

forbidden bands. Electrons in a pure crystal

cannot have an energy lying in the forbidden

bands. The higher the allowed atomic level on

the energy scale, the more the corresponding band


17/122

30 Ch. 1 T l ~ cRand

Theory

of

Solids

is smeared. As the energy increases , the forbid-

den bands beconie narrower.

The sep ara t ion of sub leve ls in an a l lowed ban d

is very smal l . In rea l c rys ta l s rang ing f rom 1 t o

100 cm3 in s ize , the sub leve ls a re separa ted by

10-22-10-24 eV. Th is difference in energy is so

smal l tha t the bands a re cons idered t o be cont in -

uous. N evertheless , the fact th at sublevels in

the bands a re d isc rete and th e numb er of sub leve ls

in the ban d i s a lways fin i te p lays a dec is ive

role in crystal physics , s ince depending on the

fi l ling of the ban ds by elec trons, a l l sol ids can be

divided into conductors , semiconductors , an d

dielectrics.

Sec 5 Filling of Energy Bands

by Electrons

Filled Levels Create Filled Bands While Empty

Levels Form Empty Bands Since the energy

bands in sol ids are formed from the levels of

i n d i v i d u a l a t o m s , i t i s q u i t e o b v i o u s t h a t t h e i r

f il ling by e lec t rons wi l l be de te rm ined above a l l

by the occupancy of the corresponding atomic

levels by electrons.

Le t us cons ider by way of an example the l i th -

i u m c r y st a l. I n t h e f re e s t a t e , t h e l it h i u m a t o m

has three electrons. Two of these are in th e

Is-

she l l , which i s th us comple ted . The th i rd e lect ron

belongs to th e 2s-subshell, wh ich i s half-filled.

Consequent ly , when a c ry s ta l i s fo rmed, the I s -

band turns out to be f i l led completely, the 2s-

band is half-filled, while the 2p-, 3s-, 3p-, etc.

b a n d s i n a n u n e x c i te d l i t h i u m c r y s t a l a r e e m p t y ,

5 Filling o Energy Bands

by Electrons

31

s ince the leve ls f rom which th ey a r e fo rmed a re

unoccupied.

The sa me i s t rue for a l l a lka l i meta l s . For exam-

ple , when a sod ium crys ta l i s fo rmed, the

Is- , 2s-, a nd 2p -bands a re comp letely f i l led, s ince

the cor responding leve ls in sod ium a toms a re

comple te ly packed by e lec t rons two e lec t rons

in the Is- level , two electrons in the 2s- level ,

and s ix e lec t rons in the 2p- leve l ) . The e leven th

e lec t ron in the sod ium a tom only ha l f - f i l l s the

3s- level, hence the 3s-band too i s half-f i lled w ith

electrons.

When c rys ta l s a re fo rmed by a toms wi th com-

pletely f i l led levels , the created bands in general

are also f i l led completely. For example, i f we

constructed a crystal f rom neon atoms, the Is- ,

2s- , and 2p-bands in the energy spectrum of

such a crys tal wou ld be com pletely fi l led each

neon at om has 10 electrons which fi l l the cor-

responding energy levels) . The remaining upper-

ly ing bands 3s, 3p , e tc . ) would tu rn ou t to be

empty .

Overlapping of Energy Bands in a Crystal

In some cases the problem of f i l l ing the energy

bands by e lec t rons i s more compl ica ted . This

refers to crystals of rare-e ar th elem ents and those

wi th a d iam ond- type la t t i ce , among which the

most inte rest in g for us are the crystals of typical

semiconductors viz . ge rmanium and s i l i con .

At a first glance, the crystals of rare-earth ele-

ments mu s t on ly hav e comple te ly f i ll ed an d emp-

ty bands in the i r energy spec t rum. Indee d ,

t h e b e r y l l i u m a t o m s , f o r e x a m p l e , w h i c h h a v e

four e lec t rons each , a re charac te r ized by two

com pletely f i l led levels , Is an d 2s levels . In m ag-


18/122

32

Ch

1 The

Band

Theory of Solids

nesium atom, which has 12 electrons, the levels

Is, 2s, 2p and 3s are also completed. However,

the upper energy bands in crystals of the rare-

earth elements, which are created by completely

filled atomic levels, are in fact only partially

filled. This can be explained by the fact that the

level

I

0

egion fil led with electrons

mpty region

Fig 8

energy bands corresponding to the upper levels

are smeared so much in the process of crystal

formation that the bands overlap. As a result

of this overlapping, hybrid bands are formed,

which incorporate both filled and empty levels.

For example, a hybrid band in a beryllium crys-

tal is formed by the completed 2s-levels and

the empty 2p-levels Fig. 8 , while in the magne-

~ i u m rystal, by the filled 3s-levels and empty

5

Filling

of

Energy Bands

by

Electrons

33

3p-levels. I t is due to the overlapping that the

upper energy bands in rare-earth crystals are

filled only partially.

In semiconductor crystals with diamond-type

lattices band overlapping leads to quite the

opposite result. In silicon atoms, for example,

the 3p-level 3p-subshell) contains only two

Fig

electrons, though this level may be occupied by

six electrons. It is natural to expect that during

the formation of a silicon crystal, the upper

energy band the 3p-band) will only be filled

part ially, while the preceding band the 3s-band)

will be filled completely since it is formed by

the completely filled 3s-level). Actually the over-

lapping during the formation of a silicon crysta l

not only leads to the appearance of a hybrid

bands composed of the 3s- and 3p-sublevels, but

also to a further splitting of the hybrid band


19/122

4

Ch.

1. The Band Theory of Solids

into two sub-bands separated by the forbidden

energy gap W

Fig.

9 .

In all, the 3s

-

3p

hybrid band must have 8 electron vacancies

per atom 2 vacancies in the 3s-subshell and

6

in

the 3p-subshell).

After the split ting of the hy-

brid band,

4

vacancies per atom are found to be

in each sub-band. Trying to occupy the lower

energy levels, electrons of the third shells of sili-

con atoms there are four of them-two in the

3s-subshell and two in the 3p-subshell) just fill

the lower sub-band, leaving the upper sub-band

empty.

Sec. 6. Division of Solids into Conductors,

Semiconductors, and Dielectrics

Physical properties of sol ids, and first of a ll

their electric properties, are determined by the

degree of filling of the energy bands rather than

by the process of their formation. From thi s

point of view all crystalline bodies can be di-

vided into two quite different groups.

Conductors. The first group includes substances

having a partially-filled band in their energy

spectrum above the completely filled energy

bands Fig. 10a). As was mentioned above,

a partially filled band is observed in alkali

metals whose upper band is formed by unfilled

atomic levels, and in alkali-earth crystals with

a hybrid upper band formed as a result of the

overlapping of filled and empty bands. All

substances belonging to the first group are

conductors.

Semiconductors and Dielectrics. The second

group comprizes substances with absolutely emp-

6. Conductors, Semiconductors, and Dielectrics

5

ty bands above completely filled bands Fig. lob ,

c). This group also includes crystals with dia-

mond-type structures, such as silicon, germanium,

grey tin , and diamond itself. Many chemical

compounds also belong to this group, for ex-

ample, metal oxides, carbides, metal nitr ides,

corundum Also,) and others. The second group

of solids includes semiconductors and dielectrics.

Fig.

1

The uppermost filled band in this group of crys-

tals is called the valence band and the first empty

band above it, the conduction band. The upper

level of the valence band is called the top of the

valence band and denoted by W . The lowest

level of the conduction band is called the bot tom

of the conduction band and denoted by W .

In principle, there is no difference between

semiconductors and dielectrics. The division in

the second group into semiconductors and dielec-

trics is quite arbitrary and is determined by the

width W of the forbidden energy gap separating

the completely-filled band from the empty band.

Substances with forbidden band widths W 5 eV),

boron nitride (W, 114.5 eV), and others.

The arbitrary nature of t he division of second-

group solids into dielectrics and semiconductors

is illustrated by the fact that many generally

known dielectrics are now used as semiconductors.

For example, silicon carbide with its forbidden

band wid th of about

3

eV is now used in semicon-

ductor devices. Even such a classical dielectric

as diamond is being investigated for a possible

application in semiconductor technology.

Energy Band Occupancy and Conductivity of

Crystals. Let us consider the properties of a

crystal with the partially filled upper band at

absolute zero

T

0). Under these conditions

and in the absence of an external electric field, a ll

the electrons wil l occupy the lowest energy levels

in the band, with two electrons in a level, in

accordance with Paul i s exclusion principle.

Let us now place

the crystal in an external

electric field with intensity

h .

The field acts on

each electron with a force

F - h

and accele-

rates it. As a result, the electron s energy increases,

and it will be able to go to higher energy

levels. These transitions are quite possible, since

there are many free energy levels in the par tial ly

filled band. The separation between energy

levels is very small, therefore even extremely

weak electric fields can cause electron transitions

6.

Conductors Semiconductors and Dielectrics

7

to upper-lying levels. Consequently, an external

field in solids with a partially filled band accele-

rates the electrons in the direction of the field,

which means that an electric current appears.

Such solids are called conductors.

Unlike conductors, substances with only com-

pletely filled or empty bands cannot conduct

electric current at absolute zero. In such solids,

an external field cannot create a directional mo-

tion of electrons. An additional energy acquired

by an electron due to the field would mean its

transition to a higher energy level. However,

all the levels in the valence band are filled. On

the other hand, there are many vacancies in the

empty conduction band but there are no electrons.

Common electric fields cannot impart sufficient

energy for electron to transfer from

the valence

band to the conduction band (here we do not

consider fields which cause dielectric breakdown).

For all these reasons an external field at absolute

zero cannot induce an electric current even in

semiconductors. Thus, at this temperature a

semiconductor does not differ at all from a dielec-

tri c with respect to electrical conductivity.


21/122

Chapter

Electrical Conductivity

of

Solids

Sec.

7

Bonding Forces in

a Crystal Lattice

Crystal as a System of Atoms in Stable Equilib-

rium State. How is a strictly ordered crystal

lattice formed from individual atoms? Why

cannot atoms approach one another indefinitely

in the process of formation of a crystal? What

determines a crystal s strength?

In order to answer these questions, we must

assume that there are forces of attraction Fat

and repulsive forces F which act between

atoms and which attain equilibrium during the

formation of a crystal structure. Irrespective of

the nature of these forces, their dependence on

the interatomic distance turns out to be the same

(Fig. Ila). At the distance r

>

a,, attractive

forces prevail; while for

r

a,, repulsive forces

dominate. At a certain distance

r

a,, which is

quite definite for a given crystal, the attractive

and repulsive forces balance each other, and the

resultant force F (which is depicted by curve

3

becomes zero. In this case, the energy of interac-

tion between particles at tains the minimum value

W (Fig. Ilb). Since the interaction energy is

at its minimum at

r

a,, atoms remain in this

position (in the absence of external excitation),

because removal from each other, as well as any

further approach, leads to an increase in the

energy of interaction. This means th at a t r

=

a,,

7 Bonding Forces in a Crystal Lattice

b)

Fig


22/122

4 Ch

2

Electrical Conductivity of Solids

th e system of atoms under consideration is in

stable equilibrium. This is the sta te which corre-

sponds to the formation of a solid with a strictly

definite structure, viz. a crystal.

Repulsive and Attractive Forces. Curve 2 in

Fig.

I l a

shows that repulsive forces rapidly

increase with decreasing distance r between the

atoms. Large amounts of energy are required in

order to overcome these forces. For example,

when the distance between a proton and a hy-

drogen atom is decreased from r

2a

to

a12

(where

a

is the radius of the first Bohr orbit) ,

the energy of repulsion increases 300 times.

For light atoms whose nuclei are weakly screened

by electron shells, the repulsion is primarily

caused by the interaction between nuclei.

On the other hand , when many-electron atoms get

closer, the repulsion is explained by the interac-

tion of the inner, filled electron shells. The

repulsion in thi s case is not only due to tlie

similar charge of the electron shells but also due

to rearrangement of the electron shells. At very

small distances, the electron shells should over-

lap, and orbits common to two atoms will appear.

However, since the inner, filled orbits have no

vacancies, and extra electrons cannot appear in

them due to the Paul i exclusion principle, some

of these electrons must go to higher shells.

Such a transi tion is associated with an increase in

the to tal energy of the system, which explains

the appearance of repulsive forces.

Obviously, the nature of repulsive forces is

the same for all atoms and does not depend on

the structure of outer, unfilled shells. On the

contrary, forces of at traction which act between

7

Bonding Forces in a Crystal Lattice

4i

atoms are much more diverse in nature, which

is determined by the structure and degree of

filling of the outer electron shells. Bonding

forces acting between atoms are determined by th e

nature of at tractive forces. When considering the

structure of crystals, the most important bonds

are the ionic, covalent, and metallic, and these

should be well known to you from the course of

chemistry. Here, we shall only consider the

covalent bond, which determines the basic prop-

erties of semiconductor crystals.

Covalent bond is the main one in the formation

of molecules or crystals from ident ical or similar

atoms. Natural ly, during the interact ion of iden-

tical atoms, neither electron transfer from one

atom to another nor the formation of ions takes

place. The redist ribu tion of electrons, however,

is very important in this case as well. The pro-

cess is completed not by the transfer of an elec-

tron from one atom to another but by the collectiv-

ization of some electrons: these electrons simul-

taneously belong to several atoms.

Let us see how the covalent bond is formed

in the molecule of hydrogen, H . Whilst the

two hydrogen atoms are far apart , each of them

L'p~sse~sests own electron, and the probability

of detecting foreign electrons within the limi ts

of a given atom is negligibly small. For example,

when the distance between the atoms is r 5 nm,

an electron may appear in the neighbouring atom

once in

1012

years. As the atoms come closer, the

probability of foreign electron appearing sharp-

ly increases. For r

0 2

nm, the transition

frequency reaches I O l sec-l, and at a further

approaching the frequency of electron exchange


23/122


24/122

Ch. 2. Electrical Conductivity of Solids 8. Electrical Conductivity of Metals

45

I

our electrons in the outer shell, each of which

forms a covalent bond with four nearest neigh-

bours (Fig.

23 .

In this process, each atom gives

it s neighbour one of i ts valence electrons for

partia l possession and simultaneously gets an

electron from the neighbour on the same basis.

Fig. 3 Fig.

14

As a result, every atom forming the crystal

fills up its outer shell to complete the popula-

tion (8 electrons), thus forming a stable struc-

ture, which is similar to tha t of the inert gases

(in Fig.

23,

these 8 electrons are conventionally

placed on the circular orbit shown by the dashed

curve). Since the electrons are indistinguishable,

and the atoms can exchange electrons, all the

valence electrons belong to all the atoms of the

crystal to the same extent. A semiconductor

crystal thus can be treated as a single giant

molecule with the atoms joined together by

covalent bonds. Conventionally, these crystals

are depicted by a plane structure (Fig.

14 .

where each double line between atoms shows a co-

valent bond formed by two electrons.

Sec.

8.

Electrical Conductivity

of Metals

The best account of this phenomenon is given

by the quantum theory of solids. But to elucidate

the general aspects, we can limit ourselves to

a consideration based on the classical electron

theory. According to this theory, electrons in

a crystal can, to a certain approximation, be

identified with an ideal gas by assuming that

the motion of electrons obeys the laws of classi-

cal mechanics. The interaction between electrons

is thus completely ignored, while the interaction

between electrons and ions of the crystal lat tice

is reduced to ordinary elastic collisions.

Metals conta in a tremendous number of free

electrons moving in the intersti tial space of

a crystal. There are about OZ3 atoms in cmS

of a crystal. Hence, if the valence of a metal is

Z

the concentration (number density) of free

electrons (also called conduction electrons) is equal

to Z x

l oz

~ m - ~ .hey are all in random

thermal motion and travel through the crystal

at a very high velocity whose mean value

amounts to 108cm/sec. Due to the random nature of

this thermal motion, the number of electrons

moving in any direction is on the average always

equal to the number of electrons moving in the

opposite direction, hence in the absence of an

external field the elect ric charge carried by

electrons is zero. Under the action of an external

field, each electron acquires an additional veloci-

ty and so all the free electrons in the metal move

in the direction opposite to the direction of t he

applied field intensity. The directional motion of


25/122

6

Ch

2


e lec t rons means tha t an e lec t r ic cur ren t appears

in the conduc tor .

I n an e lectr ic f ie ld of in tens i ty E

each electron

experiences a force F eE. Under the ac t ion

of th is force, th e electron acquires th e accelera-

t ion

where e is the charge of an elec tron and

rn is

i t s

mass.

Acco rding to th e laws of clas sical mechanics,

the v eloc i ty of e lectrons in free space would

increase indefinite ly. The sam e w ould be ob-

se rved dur ing the i r mot ion in a s t r i c t ly per iod ic

f ie ld ( fo r example , in an idea l c rys ta l wi th the

a toms f ixed a t the la t t i ce s i t es ) .

Actual ly, however , the direct ional motion of

electrons in a crystal is qui te insignif icant due

to imper fec t ions in the la t t i ce s po ten t ia l f i eld.

These imperfect ions are most ly associated with

therm al v ibra t ion s of the a to ms ( in the case

of meta l s , a tom ic cores ) a t the la t t i ce s i t es , the

v i b r a t i o n a l a m p l i t u d e b e in g t h e l ar g er t h e

higher th e temp eratu re of t h e crystal . Moreover ,

there a re a lways var ious defec t s in c rys ta l s

caused by im pur i ty a toms , vacanc ies a t the la t t i ce

si tes , inters t i t ia l a toms, and dis locat ions. Crystal

block boundaries , cracks, cavi t ies and other

macrodefects a lso affect the electr ic current .

In these condi t ions , e lec t rons a re con t inuous ly

co l l id ing and lose the energy acqui red in the

electr ic f ie ld. Therefore, the electron veloci ty

increases under th e effect of th e ext ern al field only

on a segment between two col l is ions. The mean

8 Electrical Conductivity of M etals

7

length of thi s segment is cal led the mean free

pa th of th e electron and is denoted by A

Thus , be ing acce le ra ted over

the

mean f ree

pa th , th e e lec tron acqui res th e add i t iona l ve-

loci ty of direct ional mo tion

where

Z

i s t h e m e an f r e e t i m e , o r t h e m e a n t i m e

between tw o successive collisions of th e electron

with defects . I f we know the mean free path h,

t h e m e a n f r e e t i m e c a n b e c a l c u l a t e d b y t h e

formula

where v, is the velocity of rando m th erm al motion

of th e electron. U sual ly, th e mean free path h of

the electron is very short and does not exceed

10 cm. Consequent ly , the mean f ree t ime

Z

and the increment of veloci ty Av are also small .

Sin ce Av< v,, we hav e

Assuming tha t upon co l l i s ion wi th a defec t

the electron loses pract ical ly the veloci ty of

direct ional motion, we can express the mean

velocity, called the drift velocity, as follows:

The propor t iona l i ty fac tor

e h

u=

m

v

be tween th e dr i f t ve loci ty

an d th e f ield intensi-

t y E is called the electron mobility.


26/122

8

Ch 2 Electr ical Con ductiv ity of Solids

Th e nam e of this q ua nt i ty ref lects i ts pl lysical

meaning : the mobi l i ty i s the d r i f t ve loc i ty

acquired by e lectrons in an electr ic f ie ld of u ni t

in tens i ty . A more r igorous ca lcu la t ion tak ing

i n t o a c c o u n t t h e f a c t t h a t i n r a n d o m t h e r m a l

motion electrons have different veloci t ies ra ther

t h a n t h e c o n s t a n t v e l o c i t y v gives a double

va lue for the e lec t ron mobi l i ty :

Accordingly, a more correct expression for the

dr i f t ve loc i ty i s g iven by the formula

E

=

moo

Let us now f ind the expression for the current

dens i ty in meta l s . S ince e lec t rons acqui re an

addit ional dr i f t veloci ty under the act ion of

an ex te rna l e lec t ric f ie ld , a l l the e lec t rons tha t a re

a t a d i s tance no t exceed ing f rom a ce r ta in a rea

element no rma l to the direct ion of th e i ie ld inten-

s i ty wi l l pass th rough i t in a u n i t of t ime . If the

area of th is e lem ent

is

S, al l the electrons con-

tained in th e paral le lepiped of length i l l pass

through i t in a un i t of t ime F ig .

15 .

If the

concentrat ion of f ree electrons in the metal is n ,

the number of e lectrons in this volume wil l be

nES The cur ren t dens i ty , which i s de te rmined

by th e charge ca r r ied by these e lec trons th rough

uni t a rea, ca n be expressed a s fol lows:

8. Electrical Conductivity of Metals

9

The ra t io of the cur ren t dens i ty to the in ten-

si ty of the f ie ld inducing the current is cal led

electrical conductivity an d i s deno ted by o Obvi-

ously, we get

The reciprocal of e lectr ical condnct ivi ty is

called resistivity, p:

Note th at th e appearance of a n electr ic curre nt

i n a c o n d u ct o r i s c l e a r ly c o n n e c t e d , w i t h t h e

Fig. 5

e lec t ron dr i f t . The dr i f t ve loc i ty tu rns ou t to be

qui te low and in rea l e lec t r ic f i e lds i t usua l ly

does no t exceed the veloci ty of a pedestr ian. A t

the same t ime , cur ren t p ropaga tes th rough wires

a lmos t ins tan taneous ly and can be de tec ted in

every part of a c losed circui t pract ical ly at the


27/122

5

Ch

2

Electrical onductivity

of Solids

same time. T his can be explained by the extreme-

ly high velocity of p ropaga tion of the electric

field itself. When a voltage source is connected

to a circuit, the electric field reaches the more

remote sections of the circuit a t the velocity of

light and causes the drift of all the electrons at

once.

Sec.

9

Conductivity of Semiconductors

As in th e case of m etals, elect ric current in semi-

conductors is related to the drift of charge car-

riers. In metals the presence of free electrons in

a crystal is due to the nature of th e metallic

bond itself, while in sem iconductors the appear-

ance of charge carriers depend s on ma ny factors

among which th e pur ity of a semiconductor and

its temperature are the most important .

Semiconductors are classified as intrinsic, and

imp urity extrinsic), or doped. Im pu rity semi-

conductors, in turn , can be d ivid ed int o electron-

ic, or n-type semiconductors and hole, or p-type

semiconductors depending on the type of impu-

ri ty introduced into

it

Let us consider each of

these groups separately.

Intrinsic Semiconductors

Intrinsic semiconductors are those that are very

pure. T he properties of th e whole crys tal are

thu s determined only by the properties of atoms

of th e semiconductor m ateria l itself.

Electron Conductivity. At temp eratures close

to absolute zero, all the atom s of a crystal are

connected by covalent bonds which involve all

the valence electrons. Although, as we mentioned

above, all valence electrons belong equally to

all the a toms of the crystal and m ay go from one

atom to another, the crystal does not conduct.

Every electron transition from one atom to

another is accompanied by a reverse transition.

These two transitions occur simultaneously, and

the ap plica tion of an exte rnal field cannot create

an y dire ction al motion of charges. On the other

hand,

there are no free electrons at such low

temperatures.

From the point of view of band theory, this

situation corresponds to the completely filled

valence band and an empty conduction band.

the temperature increases, the thermal

vibrations of the crystal lattice impart an addi-

tional energy to electrons. Under certain condi-

tions, the energy of an electrons becomes higher

than the energy of the covalent bond, and the

electron ruptures this bond and travels to the


28/122

Ch. 2

~lectricnlConductivity

of

Solids

crys ta l in te r s t i ce , thus becoming free . Such an

e lec t ron can f ree ly move in the in te r s t i t i a l space

of the crystal independent of the movements of

other e lectrons (electron in Fig. 16) .

On the energy levels d iagram , the l ibera tion

of a n electron means the electron t ransi t io n

f rom the va lence band to the conduc t ion band

(Fig. 17) . The energ y of ru ptu re of th e covalen t

bond in a c rys ta l i s exac t ly equa l to the fo rb id-

den band wid th Wg, i . e . the energy requ i red

for an electron to change from a valence elec-

t r o n t o a c on d u c t io n e l e c tr o n. I t i s c l e a r t h a t t h e

nar rower the fo rb idden gap fo r a c rys ta l , the

lower the tempera ture a t which f ree e lec t rons

beg in to appear . In o ther words , a t the same

tempe ra ture , c rys ta l s wi th a nar rower fo rb idden

band wi l l hav e h igher condu c t iv i ty due t o a h igh-

e r e lec t ron number dens i ty in the conduc t ion

band . Tab le 2 p r es e nt s t h e d a t a o n W g a n d f o r

some mate r ia l s a t room tempera ture .

I f , f o r e x a m p l e , w e h e a t d i a m o n d t o

600

K ,

the num ber dens i ty of f ree e lec t rons in i t wi l l

inc rease so much tha t becomes comparab le wi th

9.

Conductivity of Semiconductors

able

tha t o f the conduc t ion e lec t rons in germanium

a t r o o m t e m p e r a t u re . T h i s i s a n o t h e r r e a so n w h y

th e divis ion of sol ids int o dielectr ics an d semi-

conduc tors i s a rb i t ra ry .

Hole Conduct ivi ty . A great number of f ree

e lec t rons appear ing wi th increas ing tempera ture

i so n ly oneof the causes of in t r ins ic conduc t iv i ty

of a semiconductor . Another cause is associated

M a t e r i a l

g

eY) M-~ )

wi th a change in the s t r uc tu re of th e va lence

b o n d s i n t h e c r y s t a l, a n d t h i s i s d u e t o a t r a n s fe r

of va lence e lec t rons to the in te r s t i t i a l space .

Each e lec t ron which moves in to in te r s ti ces and

becomes a conduc t ion elec t ron leaves a vacanc y , o r

hole , in th e system of v alen ce bonds of th e

crys ta l ( in F ig . 1 6 the ho le i s shown as a l igh t

c i rc le ; the c ross ind ica tes the rup ture of the

bond caused by th i s t r ans i t ion) . Th is vacancy

may be occupied by a va lence e lec t ron f rom any

ne ighbour ing a tom. The vacancy fo rmed as

a resul t of this process may in turn be occupied

by an e lec t ron f rom a ne ighbour ing a tom, and so

on . Such e lec t ron t rans i t ions to vacan t p laces do

not requ i re reverse t rans i t ions (as i t was in the

case of a co mp letely filled sys tem of val enc e

bonds i n a crystal) , and th e possibi l i ty of a direc-

Indium antimonide

Germanium

Diamond

0 2

0 7

5 0

10l8

loi

102


29/122

5

Ch 2 Electrical Conductivity of Solids

tional charge transfer appears in the crystal.

In the absence of an external field, these transi-

tions are equally possible in all directions,

hence the total charge carried through any area

element in the crystal is zero. However, when

Fig

8

the external field i s switched on, these transi-

tions become directional: electrons in the system

of valence bonds move in the same direction as

free conduction electrons. The movement of

electrons in such a transition chain occurs con-

secutively, as if each electron in turn moves

into the vacancy left by its predecessor. If we

analyze the result of this consecutive process,

i t can be treated as the movement of the vacancy

itself in the opposite direction.

For the sake of i llus trat ion, le t us consider

a chain of checkers with a vacancy Fig. 18a).

The consecutive motion of four checkers from

left to right Fig.

18b

can be considered as the

motion of the vacancy itself by four steps in the

opposite direction. Something of this kind takes

place in a semiconductor. The consecutive transi-

tion of electrons

2

and Fig.

16

into the vacancy

9 Conductivity of Semiconductors

55

left by electron is equivalent to the transition

of the vacancy in the opposite direction, as

shown by the dashed line.

In semiconductor physics, these vacancies are

called holes. Each hole is ascribed a positive

charge +e, which is equal numerically to the

electron charge. This approach allows us to

consider a series of transit ions of a single hole

instead of describing the consecutive transitions

of a chain of electrons each to the neighbouring

atom), and this considerably simplifies our

calculations.

The hole conductivity in an intrinsic semicon-

ductor can be explained by the band theory.

A

transfer of electrons to the conduction band

see Fig. 17) is accompanied by the formation of

vacancies holes) in the valence band, which

previously was completely filled. Therefore, elec-

trons remaining in the valence band now can

move to vacant higher energy levels. This means

tha t in an external electric field they may acquire

an acceleration and thus take part in the direc-

tional charge transfer, viz. in creating electric

current.

The Number of

Holes

Equals the Number of

Free Electrons. In an intrinsic semiconductor,

there are two basic types of charge carriers:

electrons which carry negative charge) and holes

carrying positive charge). The number of holes is

always equal to the number of electrons because

the appearance of an electron in the conduction

band always leads to a hole appearing in the

valence band. Hence, electrons and holes are

equally responsible for the conductivity of an

intrinsic semiconductor. The only difference is


30/122

56

Ch.

2.

Electrical Conductiv~tyof Solids

tha t e lec t ron conduct iv i ty i s due to the mot ion

of free electrons in the interst i t ial space of the

crystal i .e. the motion of electron s trav ell in g

to the conduc t ion band) , w h i le ho le conduc t iv i ty

is associated with a transfer of electrons from

atom to ato m in the system of cova lent bonds of

the cry stal i .e . t ran sit io ns of electrons re-

maining in the valence bands) .

Since the re are two typ es of charge carriers

in intr insic semiconductors, the expression for

i t s e lec t r ica l conduct iv i ty can thus be repre-

sented a s the sum of two terms:

where n i s the e lec t ron number dens i ty in an

int r ins ic semiconductor , p i the hole concentra-

t ion, and

u,

and

up

he mobilities of electrons

and holes , respect ive ly .

In spi te of the apparen t

equivalence of elec-

tron s and holes an d the eq ua li ty of their concen-

tra t io ns , the contribu tion of electron conductiv-

i ty to the cond uct iv i ty of an in t r ins ic semicon-

ductor i s much larger tha n tha t of hole conduct iv-

i ty . Th is is because of the higher mobil i tyo f

electrons in comparison with holes. For example,

the e lec t ron mobi l i ty in germanium is a lmost

twice the mo bil i ty of holes, while in indium

ant imonide

InSb the ra t io between the e lec t ror~

and hole mobi l i t ies i s as much as 80

Although we shal l cover the topic la ter , note

In semiconductor technology, letter n denotes elec-

trons, their density, or

s

used as a subscript to indicate

that a physical quantity refers to electrons nega tive),

letter p (posit ive) is used in the same way for oles, and

the subscript means intripsic ,

I

9. Conductivity of Semiconductors 57

t ha t conduc t iv i ty i n a s em iconduc to r m ay be

caused not only by an increase in temperature

but a lso by other external ef fec ts such as i r radia -

t ion by l igh t or bombardm ent by fas t e lec t rons .

What i s necessary , i s tha t an external ef fec t

causes a transit ion of electrons from the

valence

band to the conduct ion band or , in o ther words ,

there must be condi t ions for genera t ing f ree

charge carriers in the bulk of the semiconductor.

In t r ins ic conduc t iv i ty w i th the s t r i c t equa l i t y

of num ber densit ies of un like charge carriers

ni pi) can only be rea l ized in superpure .

ideal semiconductor crys ta ls . In rea l i ty we a l -

ways have to deal wi th crys ta ls contaminated

to some extent . Moreover, i t i s doped semiconduc-

tors tha t a re most impor tant for semiconductor

technology.

Doped (Impurity) Semiconduclors

Donor Tmpurities. T he presence of imp urity atom s

in the bulk of a n in t r ins ic semiconductor leads to

certain changes in the energy spectrum of the

crys ta l . Wh i le the valence e lec t rons in an in-

t r ins ic semiconductor ma y only hav e energy in

the a l lowed band region wi thin the valence

band or the conduct ion ban d) and the i r presence

in the forbidden band i s ru led out , the e lec t rons

of some impu r i ty a tom s may have energies

ly ing w i th in the fo rb idden band . Thus , add i -

t i ona l llowed

impurity

levels appear in the energy

spec t rum in the fo rb idden band be tween the

t o p

W

of the valence band and the bottom W

of the conduction band.

Le t us

first

consider how impu r i ty levels ap pear


31/122

58

Ch

2


by using a n electronic n-type ) semiconductor

as an example . This i s obta ined when pentavalent

arsenic a to ms are in t roduced as a n im pur i ty

in to a te t rava len t germanium crys ta l F ig . 19).

Four of the five valence electrons of arsenic take

par t in the format ion of covalent bonds wi th th e

four neares t ne ighbour ing germanium a toms,

Fig.

9

thu s par t ic ipa t ing in the cons t ruc t ion of a c rys-

ta l l a t t ice . These e lec trons are in the same

condit ions as the electrons of the atoms of the

pa ren t ma te r i a l ge rman ium) , and t hus have t he

same energy va lues as the e lec t rons of germanium

atoms and l ie wi th in the va lence band of the

energy spectrum. Consequently, these electrons

of arsenic ato ms do not change th e energy spec-

tru m of germ anium . T he f ifth electron , however,

does not part icipate in the formation of covalent

bonds. S ince i t s t i l l be longs to the arsenic a tom ,

i t cont inues to move in the f ie ld of the a tomic

core . The in terac t ion be tween the e lec t ron and

9 Conductivity of Semicondu ctors 59

the a tom ic core i s , however , cons iderably weak-

ened l ike t he Coulomb force of inte rac t ion

between two charges placed in a dielectr ic . The

die lec t ric cons tant for germanium is 16 ,

hence th e force of intera ct io n between the arsen-

ic a tom ic radica l and the f if th va lence e lec t ron

Fig.

20

i s weakened 16 t imes and the energy of the i r

bond becomes almost 250 t imes less. Owing to

th is , th e o rb i ta l rad ius of the f if th e lec tron in-

creases 16 t imes , and i n order to de t ach i t f r om

the a tom and make i t i n to a conduc t ion e lec-

t ron , only 0 .01 eV energy is required.

In te rm s of th e band theo ry , th i s s i tua t ion jus t

means th a t an addi t ion al al lowed leve l corre-

spon ding the e nergy of the f if th valence electron

of the arsenic a tom has appeared in the energy

spect rum of the crys ta l . T his leve l li es near th e

bot tom of th e conduct ion band Fig . 20) and i s

s epa ra t ed f rom i t by

d

0.01

eV

.

The subscript d is the abbreviation of the word donor .

Correspondingly, the impurities and the energy levels

formed upon their introduction are called o n o r i m p u r i t i e s

and donor levels


32/122

60

Ch 2 Electrical Conriucbvity of Solids

At t empera tures c lose to absolu te zero , a l l t he

f i f th elect ron s of the arsenic ato ms re ma in bond-

ed t o t he i r a t om i c co r es , i n o t he r w or ds , a r e on

thei r donor l evel s . The conduct ion band i s

t he r e f o r e em pt y , and a t T an e l ec t r on i c

semiconductor s does not d i f f er f rom a typica l

d ie l ec t r i c as was the case for an in t r ins i c s emi -

conductor . However , g iven a s l ight increase in

t em per a t u r e s o t ha t t he ene r gy of t he r m a l v i b ra -

t ions of the l a t t i c e becomes comparable wi th th e

bond energy d X 0.01 eV, the f i f th elect rons

a r e de t ached f r om t he i r a r s en i c a t om s and go t o

the conduct ion band. The e l ec t ronic semiconduc-

tor ac qui res con duct iv i ty due to f r ee e l ec t rons

appear ing in the in te r s t i t i a l space of the crys ta l .

I t should be emphas ized th a t po s i t ive charges

th a t r ema in af t er e l ec t rons hav e l ef t the donor

level s d if f er in pr inc ip le f rom the holes of in t r in-

s i c s emicondu ctor s . The escaping e l ec t rons of

i m p u r i t y a t o m s d i d n o t t a k e p a r t i n t h e f o r m a ti o n

of covalent bonds in the crys ta l nor d id they

belong to the va lence band. Therefore , the r e-

maining pos i t ive charges are po s i t ive ly charged

ions of the donor im pu r i ty ( ar senic in the case

under co ns idera t ion) , f ixed in the crys ta l l a t t i ce

and m ak i ng n o co n t r i bu t i on t o t he conduc t i v i t y

of

the crys ta l .

S i nce e l ec tr on cond ~ i c t i v i t y s t he m a j o r t ype

of co nduc t i v i t y i n c r ys t a l s w i t h a donor i m pu r i l y ,

s em i conduc t o r s con t a i n i ng t h i s i m pur i t y a r e

called electronic, or n-type semiconductors.

In n- type sem iconductor s a t low tempe ra tures ,

e l ec tr on con duc t i v i t y i s p r edom i nan t .

At

elevat -

ed t em per a t u r e s , s ay , a t r oom t em per a t u r e , the

conduct ion band a l so conta ins e l ec t rons coming

f rom the va lence band due to the rup ture of va lence

bonds as wel l as e l ec t rons f rom the donor

level. T hese t r ans i t ions a re , as we kno w, accom-

pan i ed by ho l e s appea r i ng i n t he va l ence band

and by consequen t hole con dnct iv i t y . Never-

the les s , t he e l ec t ron co nd uct i v i ty exceeds the hole

conduc t i v i t y b y m a ny t i m es .

For examp le , if t here

is

on l y one a r s en i c

a tom per

l o

g e rm a n i u m a t o m s , i n a g e r m a n i u m

Fig 21

crys ta l , t he concent ra t ion of conduct ion e l ec t rons

a t r oom t em per a t u r e i s 2000 t i m e s h i g h e r t h a n

the hole concent ra t ion .

Charge car r i er s whose concent ra t ion dominates

i n a s em i conduc t o r unde r cons i de r a t i on a r e

cal led major i ty carr iers ; charge carr iers of the

oppos i t e s ign are ca l l ed minor i ty car r i ers . Natura l -

l y , e l ec t r ons a r e m a j o r i t y ca r r i e r s i n an e l ec t r on ic

semiconductor , whi l e holes are minor i ty car r i -

ers.

Hole Semiconductor s . Let us now cons ider

t he cas e w hen a ge r m an i um c r ys t a l con t a i ns

a t r i va l en t i nd i um a t om ( F i g . 21 instead of


33/122

6

Ch

2

Electrical Conductivity of Solid s

a pentavalent arsenic atom. The indium atom

lacks one electron to create covalent bonds with

its four nearest germanium atoms; in other

words, in the germanium crystal lattice one

double bond is not satisfied. In principle, a satu-

rated covalent bond with the fourth neighbour

can be ensured by a transition of an electron from

Fig

another germanium atom to the indium atom,

but, the electron will need some additional ener-

gy to do this. Hence, at temperatures close to

T 0, when there is no source of th is additional

energy, valence electrons of the germanium re-

main with their atoms, and the indium impurity

atoms remain neutral with unsatisfied fourth

bonds. However, the presence of indium atoms

in the crystal makes possible in principle transi-

tions of electrons which have acquired a certain

additional energy Wa, to the higher energy

levels required to form additional bonds with

indium atoms Fig. 22). Obviously, at T 0 our

semiconductor does not conduct electricity be-

cause there are no free charge carriers in i t nei-

9 Conductivity of Semiconductors 63

ther:electrons in the conduction band nor holes in

the valence band).

s the temperature rises, electrons acquire

addit ional energy of the order of Wa due to

thermal lattice vibrations in the case under

consideration, Wa 0.01 eV) and may go from

germanium atoms to indium atoms. vacancy

hole) is left where the electron moved from.

Naturally, a reverse transition is also possi-

ble, i.e. the electron may return to the ger-

manium atom. If another valence electron occu-

pies the vacancy while the original electron is

at the indium atom, the original electron will

have to remain there, thus converting the indium

atom to a negatively charged ion bonded with

the la ttice and hence immobile. The vacancy in

the system of valence bonds, formed after the

departure of the electron Fig. 23), thus becomes

a free hole. The formation of holes in the valence

band see Fig. 22) signifies that the hole-type

conductivity has become possible in the crystal.

This type of conductivity determined the name


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6

Documents

Semiconductors Made Simple Polyakov