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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
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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.
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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
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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
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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-
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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
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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
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18
Ch
1. The Band T h r
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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.
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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.
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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.
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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
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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
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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-
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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
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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.
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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
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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
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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
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Ch
2
Electrical Conductivity of Solids
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.
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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
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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
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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
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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
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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
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58
Ch
2
Electrical Conductivity of Solids
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
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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
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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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