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1
The Crystal Structure of Solids
The Crystal Structure of Solids
This lectures deal with the electrical properties and characteristics of semiconductor materials and devices.
The semiconductor is in general a single crystal.
The electrical properties of a single-crystal material are determined not only by the chemical composition but
also by the arrangement of atoms in the solid.
The formation or growth of the single-crystal material is an important part of semiconductor technology.
This introductory chapter provides the necessary background in single-crystal materials for the basic understanding of the electrical properties of semiconductor materials and devices.
1.1
Semiconductor Materials
Semiconductor Materials
Semiconductors are a group of materials having conductivities between those of metals and insulators.
Two general classifications of semiconductors are the elemental semiconductor materials found in group IV
of the periodic table (see Fig. 1) and the compound semiconductor materials, most of which are formed from
special combinations of group III and group V elements.
PERIODIC TABLE OF THE ELEMENTS
PERIOD
GROUP
IA
1
1
1.0079
H
1
HYDROGEN
3
6.941
Li
Be
BERYLLIUM
11
3
22.990
SODIUM
19
39.098
K
POTASSIUM
37
85.468
MAGNESIUM
20
40.078
Ca
CALCIUM
38
87.62
Sr
RUBIDIUM
STRONTIUM
132.91
Cs
CAESIUM
87
7
10.811
RELATIVE ATOMIC MASS (1)
B
BORON
(223)
56
137.33
Ba
BARIUM
88
(226)
Fr
Ra
FRANCIUM
RADIUM
3
21
IIIB 4
22
44.956
Sc
88.906
Y
TITANIUM
40
YTTRIUM
57-71
Lanthanide
Actinide
92.906
Cr Mn
6
However three such elements (Th, Pa, and U)
do have a characteristic terrestrial isotopic
composition, and for these an atomic weight is
tabulated.
178.49
NIOBIUM
73
180.95
Ta
Hf
HAFNIUM
TANTALUM
(261)
105
(262)
43
(98)
Tc
IRON
44
101.07
Ru
MOLYBDENUM TECHNETIUM RUTHENIUM
74
183.84
W
TUNGSTEN
106
(266)
75
186.21
Re
RHENIUM
107
(264)
76
190.23
OSMIUM
108
(277)
Db
Sg
Bh
Hs
RUTHERFORDIUM
DUBNIUM
SEABORGIUM
BOHRIUM
HASSIUM
La
Ce
LANTHANUM
CERIUM
232.04
9
27
58.933
10
28
Co
102.91
192.22
PALLADIUM
78
Ir
(268)
107.87
Ag
SILVER
79
IIB
196.97
65.39
Zn
ZINC
COPPER
47
48
112.41
CADMIUM
200.59
(281)
GOLD
111
(272)
MERCURY
112
N
O
F
Ne
NITROGEN
OXYGEN
FLUORINE
NEON
26.982
ALUMINIUM
31
69.723
14
28.086
SILICON
PHOSPHORUS
32
72.64
GALLIUM
GERMANIUM
114.82
In
50
118.71
Sn
INDIUM
81
204.38
Tl
TIN
82
207.2
33
74.922
As
ARSENIC
51
121.76
Sb
16
32.065
S
SULPHUR
34
78.96
Se
SELENIUM
52
127.60
ANTIMONY
TELLURIUM
84
(209)
35.453
Cl
CHLORINE
35
79.904
Br
BROMINE
53
Te
83
208.98
17
126.90
I
IODINE
85
(210)
18
39.948
Ar
ARGON
36
83.80
Kr
KRYPTON
54
131.29
Xe
XENON
86
(222)
Pb
Bi
Po
At
Rn
LEAD
BISMUTH
POLONIUM
ASTATINE
RADON
THALLIUM
114
(285)
30.974
P
Ge
49
15
Si
Ga
Mt Uun Uuu Uub
MEITNERIUM UNUNNILIUM UNUNUNIUM
20.180
C
Cd
80
18.998
HELIUM
10
CARBON
Au Hg
PLATINUM
110
IB 12
30
63.546
VIIA
(289)
Uuq
UNUNBIUM
UNUNQUADIUM
Copyright © 1998-2002 EniG. ([email protected])
140.91
Pr
60
144.24
61
(145)
62
150.36
231.04
63
151.96
Nd Pm Sm Eu
PRASEODYMIUM NEODYMIUM PROMETHIUM SAMARIUM
91
195.08
Pt
IRIDIUM
109
106.42
Pd
RHODIUM
11
29
Cu
NICKEL
46
Rh
77
58.693
Ni
COBALT
45
Os
Rf
ACTINIDE
89 (227) 90
7
95.94
55.845
Fe
CHROMIUM MANGANESE
42
LANTHANIDE
57 138.91 58 140.12 59
(1) Pure Appl. Chem., 73, No. 4, 667-683 (2001)
Editor: Aditya Vardhan ([email protected])
41
VIB 7 VIIB 8
25 54.938 26
51.996
Nb Mo
ZIRCONIUM
72
89-103 104
Ac-Lr
Relative atomic mass is shown with five
significant figures. For elements have no stable
nuclides, the value enclosed in brackets
indicates the mass number of the longest-lived
isotope of the element.
VANADIUM
91.224
Zr
La-Lu
VB 6
24
50.942
V
Ti
SCANDIUM
39
IVB 5
23
47.867
VIA 17
15.999 9
B
Al
VIIIB
VA 16
14.007 8
BORON
13
ELEMENT NAME
IVA 15
12.011 7
IIIA 14
10.811 6
13
5
IIIA
13
5
SYMBOL
24.305
Rb
55
6
12
ATOMIC NUMBER
Na Mg
4
5
9.0122
He
GROUP NUMBERS
CHEMICAL ABSTRACT SERVICE
(1986)
GROUP NUMBERS
IUPAC RECOMMENDATION
(1985)
IIA
2
4
LITHIUM
2
18 VIIIA
2 4.0026
http://www.ktf-split.hr/periodni/en/
92
238.03
Ac
Th
Pa
U
ACTINIUM
THORIUM
PROTACTINIUM
URANIUM
93
(237)
Np
94
(244)
64
157.25
Gd
EUROPIUM GADOLINIUM
95
(243)
96
(247)
65
158.93
AMERICIUM
CURIUM
162.50
Tb
Dy
TERBIUM
DYSPROSIUM
97
(247)
Pu Am Cm Bk
NEPTUNIUM PLUTONIUM
66
98
(251)
Cf
67
164.93
Ho
HOLMIUM
99
(252)
Es
BERKELIUM CALIFORNIUM EINSTEINIUM
68
167.26
69
168.93
70
173.04
Er Tm Yb
ERBIUM
100
(257)
THULIUM
101
(258)
YTTERBIUM
102
(259)
Fm Md No
FERMIUM
MENDELEVIUM
71
174.97
Lu
LUTETIUM
103
(262)
Lr
NOBELIUM LAWRENCIUM
Figure 1: Periodic Tables of the Elements
Table 1 shows a portion of the periodic table in which the more common semiconductors are found.
III
B
Al
Ga
In
IV
C
Si
Ge
Sn
V
N
P
As
Sb
Table 1: A portion of the periodic table
Table 2 lists a few of the semiconductor materials.
1
Elemental semiconductors
Si
Silicon
Ge
Germanium
Elemental semiconductors
AlP
Aluminium phosphide
AlAs Aluminium arsenide
GaP
Gallium phosphide
GaAs Gallum Arsenide
InP
Indium Phosphide
Table 2: A list of some semiconductor materials
Semiconductors can also be formed from combinations of group II and group VI elements.
The elemental materials, those that are composed of single species of atoms, are silicon and germanium.
Silicon is by far the most common semiconductor used in integrated circuits.
The two-element, or binary compounds such as gallium arsenide or gallium phosphide, are formed by combining
one group III and one group V element.
Gallium arsenide is one of the more common of the compound semiconductors.
His good optical properties make it usefull in optical devices.
GaAs is also used in specialized applications in which, for example, high speed is required. We can also fom
a three-element, or ternary compound semiconductors
An example is Alx Ga1−x As in which the subscript x indicates the fraction of the lower atomic number element
component.
More complex semiconductors can also be formed that provide flexibility when choosing material properties.
1.2
Types of solids
Types of solids
Amorphous, polycrystalline, and single crystal are the three general types of solids.
Each type is characterized by the size of an ordered region within the material.
An ordered region is a spatial volume in which atoms or molecules have a regular geometric arrangement or
periodicity.
Amorphous materials have order only within a few atomic or molecular dimensions, while polycrystalline materials have a high degree of order over many atomic or molecular dimensions.
These ordered regions, or single-crystal regions, vary in size and orientation with respect to one another.
The single-crystal regions are called grains and are separated from one another by grain boundaries.
Single-crystal materials, ideally, have a high degree of order, a regular geometric periodicity throughout the
entire volume of the material.
The advantage of a single-crystal material is that, in general, its electrical properties are superior to those of a
nonsingle-crystal material, since grain boundaries tend to degrade the electrical characteristics.
Two-dimensional representations of amorphous, polycrystalline, and single-crystal materials are shown in Fig. 2.
1.3
Space lattices
Space lattices
Our primary concern will be the single crystal with it regular geometric periodicity in the atomic arrangement.
A representative unit, or group of atoms, is repeated at regular intervals in each of the three dimensions to form
the single crystal.
The periodic arrangement of atoms in the crystal is called the lattice.
2
Figure 2: Schematic, of three general types of crystals: (a) amorphous. (b) polycrystalline. (c) single crystal.
1.3.1
Primitive and Unit Cell
Primitive and Unit Cell
We can represent a particular atomic array by a dot that is called a lattice point. Figure 3 shows an infinite
two·dimensional array of lattice points.
The simplest means of repeating an atomic array is by translation.
Each lattice point in Figure 3 can be translated a distance a1 in one direction and a distance b1 in a second
noncolinear direction to generate the two-dimensinal lattice.
A third noncolinear translation will produce the three-dimensional lattice. T
The translation directions need not be perpendicular.
Since the three-dimensional lattice is a periodic
Figure 3: Two-dimensional representation of a single-crystal lattice.
repetition of a group of atoms. we do not need to consider the entire lattice, but only a fundamental unit
that is being repeated.
A unit cell is a small volume of the crystal that can be used to reproduce the entire crystal.
A unit cell is not a unique entity.
3
s not a Uni4uc emily. Figure 1.3 shows several possible unil
l lattice.
---- --
al
ystallauice.
-
n;
.. ------ -£- --I't
. ,,'" -
-
-
-
-
I
,,t'
' B
-
- - - - ., . ,.
_. J;L..!.:'
_- _ : _- _- _- _'"
Figure 4: Two-dimensional
of a single-crystal
latticeofshowing
various possible unit cells.
Figure 1.31 representation
Two-dimensional
l'ellresent<ltion
a single-crystnl
lallic.:e .showing various plll,sible unit cells.
Figure 4 shows several possible unit cells in a two-dimensional lattice.
The unit cell A can be translated in directions a2 and b1 , the unit cell B can be translated in directions a3 and
b3 , and the entire two-dimensional lattice can be constructed by the translations of either of these unit cells.
The unit cells C and D in Figure 4 can also be used to construct the entire lattice by using the appropriate
translations.
This discussion of two-dimensional unit cells can easily be extended to three dimensions to describe a real
single-crystal material. A primitive cell is the smallest unit cell that can be repeated to to form the lattice.
In many cases, it is more convenient to use a unit cell that is not a primitive cell.
PT.R 1Unit The
Crystal
Stn,JCiure
ofsides
So::ds
cells may
be chosen that
have orthogonal
whereas the sides of a primitive cell may be nonorthogonal.
A generalized three-dimensional unit cell is shown in Figure 5.
,,
,,,
,,,
,,
"
Figure 5: A generalized primitive unit cell.
Figure 1.4 rA gener:lIized
The relationship between this
cell and the
is characterized
primifi
velattice
unit
cell . by three vectors: a. b and c, which need not
be perpendicular and which mayor may not be equal in length.
Every equivalent lattice point in the three-dimensional crystal can be found using the vector
r = pa + qb + sc
(1)
The unit ce([ A can be translated 4in directions al and "1. the unit
where p, q and s are integers.
Since the location of the origin is arbitrary, we will let p, q and s be positive integers for simplicity.
1.3.2
Basic Crystal Structures
Basic Crystal Structures
Before we discuss the semiconductor crystal, let us consider three crystal structures and determine some of the
basic characteristics of these crystals.
Figure 6 shows the simple cubic (sc), body-centered cubic (bcc) and face-centered cubic (fcc) structures.
1 . 3 Space lattioes
\
I \
I \
I \
I
I
\
I
I
II
,
I
I
I
,.----I
I
I
I
I
I
(a)
(e)
(b)
Figure 1.5 1Three laltice Iypes: (a) simple cubic. (b) body·cenlcred cubic. (c) Facc·ccmcrcd cubic.
Figure 6: Three lattice Iypes: (a) simple cubic. (b) body·centered cubic. (c) Face·centered cubic.
Objective
EXAMPLE 1.1
To find the volume
of atoms in a crystal.
For these simple structures,
we may choose unit cells such that the general vectors a, b and c are perpendicular
Consider a singlc-crY5131 material that is a body-centered cubic with a lattice Constant
to each other and Qthe
= 5 Alengths
= 5 x LO- 8are
CIll . Aequal.
corner alom shared by eighl unit cells which meet at each corner
$0 that each comer atom effectively contribUles olle.eighth of its vt)lume to each unil cell. The
The simple cubic eight
(sc)comer
structure
has an atom located at each corner: the body-centered cubic (bcc) structure
atoms th en contribute an equivalent of one atom to the unil cell. If we add the. body-
has an additional centered
atom310m
attothe
center
cube;
and
the
face-centered
cubic (fcc) structure has additional
the comer
aU)flls .of
eachthe
unit cell
contains
an etlui
valcnt
of two tll(,uns•
atoms on each face
plane.
By
knowing
the
crystal
structure
of
a
material
and its lattice dimensions, we can
• Solution
determine several The
characteristics
of the
volume density of atoms
is thencrystal.
found as
.
2 atoms density of atoms. 1
For example, we can determine
the
volume
DensIty =
= 1.6 :x 10-- atoms per em'
<
n
(5 x 10- ,)·1
To find the volume
density of atoms in a crystal
.Comment
The \'olume densi ty of atoms j usl calculated repreSl!nts the order of magnitude of density fOf
Consider a single-crystal
material that is a body-centered cubic with a lattice
constant a = 5 Å = 5 × 10−8 cm.
most materials, The aC lUa] density is a function of tht crystal type and
since
A corner atom is theshared
by
eight
unit
cells
which
meet
at
each
corner
so
that
each comer atom effectively
packing density- numbel' uf ;:lIeum. per Ullil <.:el1--depends un <.:ryStaJ strUCllII'e .
contributes one-eighth of its volume to each unit cell. The eight comer atoms then contribute an equivalent of
TEST YOUR
one atom to the unit cell. If we add the body-centered
atomUNDERSTANDING
to the comer atoms, each unit cell contains an
equivalent of two atoms.
F:t.l The lauice cons{ant of il face"f..'C lltcred-eubit:: structure is 4.75 A. Determine Ihe vol·
ume den sity of atoms. (t_ tUJ ;:.0 I x £L'C 'su\, )
Solution
£1.2 The volume density of atoms fur 'l simple cubit laltice is 3 x 1022 cm- :', Assume Ihat
atoms areisnard
sphere\!.
with each
The volume density oftheatoms
then
found
asat()m lnuching it'S
the lattice constant and the radium of the atom. (y 19' 1 =
n=
2 atoms
.1
'Y
neigtlOOr. Oetermine
= Un 'SHV)
= 1.6 × 1022 cm−3
1.3.3 Crystal Planes and Miller
Indices
(5 ×
10−8 )3
Since real crystals are not intinitely large, they evenlually (crminate at a surrace.
Semiconductor devic,'s are fabricated at or near a surface, so the surface properties
The volume density of atoms just calculated represents the order of magnitude of density for most materials,
The actual density is a function of the crystal type and crystal structure since the packing density-number of
atoms per unit-cell depends on the crystal structure.
1.4
Crystal Planes and Miller Indices
Crystal Planes and Miller Indices
Since real crystals are not infinitely large, they eventually terminate at a surface.
Semiconductor devices are fabricated at or near a surface, so the surface properties may influence the device
characteristics.
We would like to be able to describe these surfaces in terms of the lattice.
5
MPLE 1.2
surfaces in terms of th e lattice. Surfaces. Or planes through the crystal, can be de·
scribed by first considering the intercepts of the plane along the ii , b, and axes used
(0 describe the laILicc.
c
Surfaces, or planes through the crystal, can be described by first considering the intercepts of the plane along
the a, b and c axes used to describe the lattice.
Objective
To describe the planes shown in Figure 7
To describe Iht JJ1 anc shown in Figure 1.6. (The lanice points in Figure 1.6 are shown
the ii. b. and caxes only.)
Ie
Figure 1.61 A representati ve crystal-
Figure 7: A representative crystal lattice plane.
Janice plane.
• Solution
From Equation
1 the intercepts
the planeofcorrespond
to p = 3, q =
theNow
reciprocals
From Equati(.\11
(1 .1). (heofintercepts
the plane correspond
to 2I' and
= 3.s q==1. 2.Now
andwrite
$" == I.
of the intercepts,
which
gives
gives wrice Ihe
of Ihe inlcrcepcs. which
1 1 1
, ,
3 2 1
Multiply by the lowest common denominator, which in this case is 6, to obtain (236).
The plane
in Figure
7 islowes!
then referred
to denominal0r.
as the (236) plane.
Multiply
by Ihe
Common
which in this case is 6. to obtain (2. 3. 6). The
Multiply
by Ihe
denominator,
this case
is 6,
to integers
obtain (236).
plane
in lowest
Figure common
1.6 is th en
referred towhich
as thein(236}
plune.
The
arc referred to as the
The plane
in Figure
7 is
then
as the (236)
Miller
Jndi(:es.
We
willreferred
refer 10toa general
planeplane.
as the (hkl) plane.
The integers are referred to as the Miller indices.
• Comment
We will refer to a general plane as the (hkl) plane.
We <:an show that the same Ihree Miller indices are obtained for ito)' pl ane that is paraJlel ( 0 lhe
1. 3shown
S;>ace in
Lattices
7
Three planes that are commonly considered in a cubic crystal are
Figure 8
Oile shown ill rigurc J.6. Any parallel plane is entirely ctI,uivalenl t() any other.
Three planes that are commonly considered in a cubic crystal arc shown in Figure ) .7. The plane in Figure 1.7a is parallel to the band axes so the intercepts arc
given as p = I, q = 00. and S 00 .II Takin· the reciprocal, we
,I II obtain the Miller in·
I
I
dices as (I. I 0, 0), so the plane shownI in Figure 1.7a is referred II to as the, (100) plane.
---- - - - - - , ;;:
, plane
, ----in Figure 1.7a
jj
b and separated
,,
Again, any
parallel to the one ,shown
by an integral
/
c
=
...
...
....
(
(
(
(
--
-
(
'- /
/
/
Ii
ii
(a)
.-------
",,.-
(h)
(e)
Figure 1.71 Three lauiee planes: (.)( 100) plane. (b) (110) plane. Ie) (I II ) plane.
Figure 8: Three lattiee planes: (a) (100) plane. (b) (110) plane. (c) (111) plane.
number of lattice constants is equivalent and is refe rred to as the (100) plane. One adto taking
reciprocal
I() obtain
indices isare
thatgiven
the as p = 1, q = ∞, and s = ∞.
The plane invancage
Figure
8a isthe
parallel
tooftlte
the bintercepts
and c axes
so the
theMiller
intercepts
use of infinity is avoided when describing. plane that is paraUel to an axis. If we were
to describe a plane passing through the origin of our system. we would obtai n infinity as one or nlore of the Miller indices after taking the reciprocal of the intercepts.
However. the location of the origin of our system is entirely
6 arbitmry and so, by translating the origin to anoth er equivalent lattice poinl , wecan avoid the use of infinity in
the set of Miller indices.
Taking the reciprocal, we obtain the Miller indices as (1, 0, 0), so the plane shown in Figure 8a is referred to as
the (100) plane.
Again, any plane parallel to the one shown in Figure 8a and separated by an integral number of lattice constants
is equivalent and is referred to as the (100) plane. One advantage to taking the reciprocal of the intercepts to
obtain the Miller indices is that the use of infinity is avoided when describing a plane that is parallel to an axis.
If we were to describe a plane passing through the origin of our system. we would obtain infinity as one or more
of the Miller indices after taking the reciprocal of the intercepts.
However, the location of the origin of our system is entirely arbitrary and so, by translating the origin to another
equivalent lattice point, we can avoid the use of infinity in the set of Miller indices.
For the simple cubic structure, the body-centered cubic, and the face-centered cubic, there is a high degree
of symmetry.
8
The axes can be rotated by 90◦ in each of the three dimensions and each lattice point can again be described
CHAPT.A 1 The Crystal Structure ot SolidS
by Equation 1.
Each face plane of the cubic structure shown in Figure 8a is entirely equivalent.
EXAMPLE These
1.3
planesObjective
are grouped together and are referred to as the [100] set of planes.
One characteristic
of a crystal
that density
can be determined
the distance
nearest equivalent parallel
To c3lculate
the surface
of atoms on aispanicular
planebetween
in a crystal.
planes.
C(msider the body·ccntcrcd cubic struclure and the (110) plane shown in Figure 1.83.
Another characteristic
the surface
concentration
of spheres
atoms, number
square
centimeter,
Assume \heisatoms
can be represented
as hurd
with the per
clnseSl
atoms
tom:hing that
each are cut by
a particular other.
plane.Assume the lanice constant is Cli -= 5 A. Figure J.Sb
how (he atoms arc Cut by the
Again, a single-crystal
(110) plane.semiconductor is not infinitely large and must terminate at some surface.
Theofulom
31each
j s shitrx:d by
similarin
equivalent
I:ltlice
pl anes.
so each
Comer such as an
The surface density
atoms
mayC,)rner
be important,
forfour
example,
determining
how
another
material,
insulator, will
”fit”
on
the
surface
of
a
semiconductor
material.
atom effecti vely contributes one-four1h of its area to this lanice plane as indicated in the
ure. The four corner alUms then effecti vely contribute one atom (0 this lattice plane. The atorn
in the center is comp letely enclosed in the Jauice plane. There is no other equivalent plane thai
Consider the body·centered cubic structure and the (110) plane shown in Figure 9a. Assume the atoms can be
eul S the ccolcr illom and the COMler aloms. so lhe entire center almll included in the number
represented as hard spheres with the closest atoms touching each other. Assume the lattice constant is a1 = 5Å.
of ,HOrns
(heatoms
crystalare
plane.
Jaui<.
.'C 1>lanc
in Fjgurc ).8b. then. contains two atoms.
Figure 9b shows
how in
the
cut The
by the
(110)
plane.
To calculate the surface density of atoms on a particular plane in a crystal.
III
I
!" ·...
Ibl
(a)
Figure 1.SI <a) The ( ltO) plane in a body-centered cubic and <b) the alom, cui by Ihe
Figure 9: (a) (The
in a body-centered
110)(110)
plane plane
ill a body-centered
cubic . cubic and (b) the atoms cut by the (110) plane in a bodycentered cubic .
• Solution
We find the surface
density by dividing the number of lattice atoms by the surface area, or in this case
We find the surface ltellSicy by dividing the number of lattice atoms by the surface area. Of in
2 atoms
2
Ihis case
√ =
√ =
Surface density =
(a1 )(a1 2)
(5 × 10−8 )2 2
2
= 5.66 × 1014 cm−2
Sunace de nsil y =
=
(a ,) (tI, ./2)
(5 x 10- ')' <,12)
In addition to describing crystal planes in a lattice, we may want to describe a particular direction in the
crystal.
which is
5.66 )(
• Comment
7
tion as distinct from the parentheses used for the crystal planes. The three basic
directions and the associated crystal planes for the simple cubic stRIcture are shown in
Figure 1.9. Note that in Ule simple cubic Ian ices. the ["HI direction is perpendicular to
the (hk!) plane. This perpendicularity may tlot be true in noncubic lattices.
The direction can be expressed as a set of three integers which are the componems of a vector in that direction.
1.3.4 The Diamond Structure
For example, the body diagonal in a simple cubic lattice is composed of vector components 1, 1 and 1.
As already stated. silicon is the most common semiconductor material. Silicon is reo
The body
diagonal
is then
described
thea [111]
direction.
ferred
to as a group
IV clement
andashas
diamond
crystal structure. Germanium is
also a group
IV to designate
and
the same diamond
stl'llct1.1
re.the
A utlit
cell of theused
dia· for the crystal planes.
The brackets
are used
direction
as distinct
from
parentheses
mond structure, shown in Figure 1.10. is more complicated than the simple cubic
The three basic directions and the associated crystal planes for the simple cubic structure are shown in Figure 10.
struclUres that we have considered up to this paint.
Note that We
in the
Iattices,
[hkl] direction
is perpendicular
to the (hkl) plane.
maysimple
begin tocubic
understand
thethe
diamond
latrice by conSidering
the tetrahedral
structure sholVn ill may
Figurenot
1.11.
re is basically"
This perpendicularity
beThis
truestructu
in noncubic
lattices.bOOy-centered cubic with
, .. .- - --::j;
l'tOI
(al
\b)
(el
Figure 1.9 1Three lallice direeti<>ns and plan",,, (a) (100) ptane and ll 00j direction, (b) ( I to) plane and II tOI direCli()n.
Figure IC)
10:1I1Three
direetions
1)pl,ne lallice
and \ 1111
dircc.ion. and planes: (a) (100) plane and (100) direction, (b) (100) plane and (100)
direction, (c) (111) plane and (111) direction.
1.4.1
The Diamond Structure
The Diamond Structure
Silicon is the most common semiconductor material.
Silicon is referred to as a group IV element and has a diamond crystal structure.
Germanium is also a group IV eIement and has the same diamond structure.
A unit cell of the diamond structure, shown in Figure 11 is more complicated than the simple cubic structures.
We may begin to understand the diamond latrice by considering the tetrahedral structure shown in Figure 12.
This structure is basically a body-centered cubic with four of the corner atoms missing.
Every atom in the tetrahedral structure has four nearest neighbors and it is this structure which is the basic
building block of the diamond lattice.
There are several ways to visualize the diamond structure.
One way is by considering Figure 13.
Figure 13a shows two body·centered cubic, or tetrahedral, structures diagonally adjacent to each other.
The shaded circles represent atoms in the lattice that are generated when the structure is translated to the
right or left one lattice constant a.
Figure 13b represents the top half of the diamond structure.
The top half again consists of two tetrahedral structures joined diagonally, but which are at 90◦ with respect
to the bottom half diagonal.
An important characteristic of the diamond lattice is that any atom within the diamond structure will have
four nearest neighborig atoms.
The diamond structure refers to the particular lattice in which all atoms are of the same species, such as silicon
or germanium.
The Zincblende structure differs from the diamond structure onIy in that there are two different types of
atoms in the lattice.
8
C HAP n R 1 The Crystal SIIUClule of Solids
10
UClule of Solids
Figu
struc
Figure 1.10 ITheFigure
diamond
structure.
11: The diamond structure.
in the
T
1
. 12
/
,
,
//
/
'
II
___ ___ __ _ . -_
L.
t _ _ __
F
" _ _ _- I
Figure 1.111 The '<l mhed",1
(a)
structure of closest
neighbors
Figure 12: The tetrahedral structure of closest neighbors in the diamond lattice
in the
di:lnlond
Figure 1.12
1Portions
of thelattice.
diamond lallice: (a) honom half and (b) top ha
Compound semiconductors, such as gallium arsenide, have the zincblende structure shown in Figure 14
The important feature of both the diamond and the zincblende structures is that the atoms are joined together
to form a tetrahedron.
Figure 15 shows the basic tetrahedral structure of GaAs in which each Ga atom has four nearest As neighbors
and each As atom has four nearest Ga neighbors.
four of the corner atoms missing. Every atom in the
nearest neighbor.; and it is this stTUcture whi ch is the b
mono lattice.
There9 are several ways to visuali ze the diamond st
ther under.;tanding of the diamond lattice is by conside
This figure also shows the interpenetration of two sublattices that can be used to generate the diamond or
zincblende lattice.
Figure 1.111 The '<l mhed",1
structure of closest neighbors
Figure 1.10 IThe diamond structure.
in the di:lnlond lattice.
I
;' /"
: / JI . . -
/
,
,
//
,'
//'
'
II
___ ___ __ _ . -_
L.
t _ _ __
" _ _ _- I
_
.......Y
F
(b)
(a)
t .4 Atom'c Son:rlO9
Figure 1.12 1Portions of the diamond lallice: (a) honom half and (b) top half.
Figure 13: Portions of the diamond lattice: (a) bottom half and (b) top half.
four of the corner atoms missing. Every atom in the tetrahedral <tructure has four
nearest neighbor.; and it is this stTUcture whi ch is the basic building bloc k of the dia·
mono lattice.
There are several ways to visuali ze the diamond structu re. One wa), to gain a fur·
ther under.;tanding of the diamond lattice is by considering Figure I. I 2. Figure 1.1 2a
shows two body·centcred cubic, or tctrahedral, structures diagonally adj acent to each
other. The shaded ci rcles rcpresem atoms in the lattice that are generated when the
stnlcturc
uanslated to the right or lefl . one lattice constant , tl. Figure 1. 12b repre-.... ... I
#
sents the top half of the diamond struc ture. The top half again consists of tWO tetra·
' e
'/
hedral structures joined diagonally. but which are at 90· with respect to the bottom·
'
1
•
1
half diagonal. An important characteristic of the diamond lattice
that any atom
wi thin the diamond struClure wiU have four ncarestneighboriog atoms. We will note
/
this char'lcteristic again in our di scussion of atomic bonding in the next section.
/
,
,
,, -.., - __ l _
\
',
/
Figure 1.J4 ITh
structure of
f'igure 1.13 IThe zillcblende (sphalerite) lanice or GaA,.
Figure 14: The zincblende lattice of GaAs
1.5
ziucblcndc l
Atomic bonding
The diamond structure refers to the particular latlice in which all atoms are of the
Atomic Bondingsame
species, such as silicon or germanium. The Zincblende (sphalerite) struclure
fromonethe
diamond
structure
onisIyfavored
in thatover
there
are for
two
differentassembly
types of atoms
The question arisesdiffers
as to why
particular
crystal
structure
another
a particular
of atoms.
in Ihe lattice. Compound semiconductors. such as galJium ar!\enide, have the zinc·
A fundamental lawblende
of nature
is that shown
the total
in thermal feature
equilibrium
tends
structure
in energy
Figure of1.a13.system
The important
of both
thetodiamond
reach a minimum value.
and the zincblende structures is that the atoms are joined together 10 form a tetra h,,·
The interaction that occurs between atoms to form a solid and to reach the minimum total energy depends on
1.14 shows the basic tetrahedral structure of GaAs in which each Ga
the type of atom ordron.
atomsFigure
involved.
atom
has fourbetween
nearestatoms.
As neighbors
andoneach
As atomatom
hasorfour
nearest
Ga neighbors.
The type of bond, or
interaction,
then, depends
the particular
atoms
in the crystal.
Thisbond
figure
also begins
show
lhe ”stick
interpenetration
of two
sublattices that can be used
If there is not a strong
between
atoms, to
they
will not
together” to create
a solid.
to generate
thebediamond
zincblende
lanice.
The interaction between
atoms can
described or
by quantum
mechanics
We can nevertheless obtain a qualitative
understanding of how various atoms interact by considering the valence electrons of an atom.
The atoms at the two extremes of the periodic table (excepting the inert elements) tend to lose or gain valence
electrons, thus forming ions.
TEST YOUR UNDERSTANDING
These ions then essentially have complete outer energy shells.
El.S The JaujC(: COIl.o.;l nlll of siJi{.xm
is
5.43 A. Calculare {he voJume density of silicon
The elements in group I of the periodic table tend to lose their one electron and become positively charged,
.toms.
(, _Wto'"O
x electron
5; 'suy) and become negatively charged.
while the elements in group
VII tend
gainIan
10
1
•
1
shells is cOlla/em bonding. an e
A hydrogenttatom has one electr
t .4 Atom'c Son:rlO9
est energy shell. A schematic o
,, molec ule with the covalent
,",ge!!
,
ing
results in ele<o!rons being s
enercAY s.hen of each atom is ful
Atoms in group IV of the
.... ... I
tend to form covalent bonds. E
' e
'/
-.. - - -needs
---1f-..Yfour more eJectrons to co
,, __
' ,
l _ __ _
,
\
,
', example, has four nearest neig
/
,,
lence electron to be shared, then
its outer shell. Figure 1.16a sc
Figure
1.J4
I
The
leuahedraJ
Figure 15: The tetrahedral structure of closets neighbors in the zincblendc lattice
with
the
structure of
neighbors
ill four valence electrons
#
/
/
lanice or GaA,.
ziucblcndc latti ce.
These oppositely charged ions then experience a coulomb attraction and form a bond referred to as an ionic
bond.
If the ions were to get too close a repulsive force would become dominant, so an equilibrium distance results
between these two ions.
articular latlice
in which all atoms are of the
In a crystal, negatively charged ions tend to be surrounded by positively charged ions and positively charged
tend to be surrounded
by negatively
charged ions.
ium. Theions
Zincblende
(sphalerite)
struclure
a periodic array of the atoms is formed to create the Iattice.
n that thereSo are
two different types of atoms
A classic example of ionic bonding is sodium chloride. Another atomic bond that tends to achieve closed-valence
s. such asenergy
galJium
have an
theexample
zinc·of which is found in the hydrogen molecule.
shells isar!\enide,
covalent bonding,
The important
feature
of one
both
theand
diamond
A hydrogen
atom has
electron
needs one more electron to complete the lowest energy shell.
schematic
of two noninteracting
toms are Ajoined
together
10 form hydrogen
a tetra atoms,
h,,· and the hydrogen molecule with the covalent bonding, are
shown in Figure 16.
edral structure of GaAs in which each Ga
ach As atom has four nearest Ga neighbors.
netration of two sublattices that can be used
nice.
(b)
(a)
@-
-@
I
-@)I
@ =®
TEST
YOUR UNDERSTANDING
Figure 16: Representation
of (a) hydrogen
valence
electrons and
.'igure
1.15
J Rcpre
. (b) covalent bonding
of in a hydrogen molecule
A. Calculare {he voJume density
silicon
(a)ofhydrogen
...alence
dectrons
ami (b) cov3lenl honding ill ;)
Figure
Atoms in group IV of the periodic table, such as silicon and germanium, also tend to form covalent bonds.
Each of these elements has four valence electrons and needs four more electrons to complete the clCClrH
bydrogen molecule.
valence energy shell.
Covalent bonding results in electrons being shared between atoms, so that in effect the valence energy shell
of each atom is full.
If a silicon atom has four nearest neighbors, with each neighbor atom contributing one valence electron to
be shared, then the center atom will in effect have eight electrons in its outer shell. Figure 17 a schematically
shows five noninteracting silicon atoms with the four valence electrons around each atom.
crystal structures. The question arises as to
ored over anolher for a particular assembly
lhallhe total energy of a
in them,.1
11
@
lence electron to be shared, then the center atom will in effect have eight electrons in
its outer shell. Figure 1.16a schematically shows five noninterilC:ting silicon atoms
with the four valence electrons around each atom. A (wo .. dimensional
I
-@)I
I
I
-@)I
@ =®
(b)
I
-@)I
I
- @)I
-@)II
I
-@)=@)=@)I
II
I
@)
I
I
I
-@)I
15 J Rcpre.
of
(a)
ib)
gen ...alence dectrons
ov3lenl hondingFigure
ill ;) 17: Representation
Figure
1.161 Representation of (a) silicon valence
of (a) silicon valence electrons and (b) covalent bonding in the silicon crystal.
clCClrHns and (b) covalent bOJlding in the si licon crystal.
molecule.
A two-dimensional representation of the covalent bonding in silicon is shown in Figure 17b.
The center atom has eight shared valence electrons.
A significant difference between the covalent bonding of hydrogen and of silicon is that, when the hydrogen
molecule is formed, it has no additional electrons to form additional covalent bonds, while the outer silicon
atoms always have valence electrons available for additional covalent bonding.
The si icon array may then be fomed into an infinite crystal, with each silicon atom having four nearest neighbors
and eight shared electrons.
The four nearest neighbors in silicon forming the covalent bond correspond to the tetrahedral structure and the
diamond lattice, which were shown in Figures 12 and 11, respectively.
Atomic bonding and crystal structure are obviously directly related.
The third major atomic bonding scheme is referred to as metallic bonding.
Group I elements have one valence electron.
If two sodium atoms (Z = 11), for example, are brought into close proximity, the valence electrons interact in
a way similar to that in covalent bonding.
When a third sodium atom is brought into close proximity with the first two, the valence electrons can also
interact and continue to form a bond.
Solid sodium has a body-centered cubic structure, so each atom has eight nearest neighbors with each atom
sharing many valence electrons.
We may think of the positive metallic ions as being surrounded by a sea of negative electrons, the
solid being held together by the electrostatic forces.
A fourth type of atomic bond, called the Van der Waals bond, is the weakest of the chemical bonds.
A hydrogen fluoride (HF) molecule, for example, is formed by an ionic bond.
The effective center of the positive charge of the molecule is not the same as the effective center of the negative
charge.
This nonsymmetry in the charge distribution results in a small electric dipole that can interact with the dipoles
of other HF molecules.
With these weak interactions, solids formed by the Van der Waals bonds have a relatively low melting temperature in fact, most of these materials are in gaseous form at room temperature.
1.6
Imperfections and Impurities in Solids
Imperfections and Impurities in Solids
In a real crystal, the lattice is not perfect, but contains imperfections or defects.
The perfect geometric periodicity is disrupted in some manner.
12
Imperfections tend to alter the electrical properties of a material and, in some cases, electrical parameters can
be dominated by these defects or impurities.
1.6.1
Imperfections in Solids
Imperfections in Solids
One type of imperfection that all crystals have in common is atomic thermal vibrations.
A perfect single crystal contains atoms at particular lattice sites, the atoms separated from each other by a
distance we have assumed to be constant.
The atoms in a crystal. however, have a certain thermal energy, which is a function of temperature.
The thermal energy causes the atoms to vibrate in a random manner about an equilibrium lattice point.
The random thermal motion causes the distance between atoms to randomly fluctuate, slightly disrupting the
perfect geometric arrangement of atoms.
This imperfection, called lattice vibrations or phonons, affects some electrical parameters, as we will see
later in our discussion of semiconductor material characteristics.
Another type of defect is called a point defect.
There are several of this type that we need to consider. In a real crystal, an atom may be missing from a
particular lattice site.
This defect is referred to as a vacancy;
it is
schematically
t4
1 The
Oysta\
Stl\)c'."re 01shown
So'.i(Js in Figure 18a.
--- - ---- - -(3)
..
I
I
I
I
I
I
.. -
I
...
I
/ -
I
/
r "..,J - ---t6
I
J
- . . . . :• •
I
I
..'
_
__
_
...
0-
"I
,'"
I
(b)
..
/
' "
I
I
J
J
I
I
-+ - --'!'-" - -?'
,. . , '
"
....
..
,/
i
.•
I
,I
,/
,
-- -I·
I
I
F1gure 1.17 ITwo-dimensional representation of a single-crystal lattice sho\\ting (a) if vacancy defect
Figureand
18: (b)
Two-dimensional
representation of a single-crystal lattice showting (a) a vacancy defect and (b) an
an interstitial defect.
interstitial defect.
In another situation, an crystal.
atom may
be located
lattice
sites.energy, whic h is a fu nction of temperature.
however,
havebetween
a certain
thennal
thermal
energyand
causes
the alOms toshown
vibrate
a mndom
This defect is referred toThe
as an
interstitial
is schematically
in in
Figure
18b. manner abo ut an equi.
librium
latticedefects,
point. Thb
random
thermal
motion
causes
the distance
atoms
In the case of vacancy and
interstitial
not only
is the
perfect
geometric
arrangement
of between
atoms broken,
to randomly
fluctuate,
slightl
disrupting
the perfect
geometric
arrangement
of atoms.
but also the ideal chemical
bonding between
atoms
is y
disrupted,
which
tends to
change the
chemical properties
This imperfection, caBed lattice vibratiOIlS, affects some electrical parameters, as we
of the material
willmay
seebe
later
ourenough
discussion
of semiconductor
characteristics.
A vacancy and interstitial
in in
close
proximity
to exhibit anmaterial
interaction
between the two point
Another type of defect is called a poim defect. There are several of this type that
defects.
we need to cOllsider. Again. in an ideal single-crystal Jaulce:. the atoms arc arranged
This vacancy-interstitial defect, also known as a Frenkel defect, produces different effects than the simple
in a perfect
periodic
arrangement.
However.
in a real crystal.
,an atom may be missing
vacancy or interstitial. The
point defects
involve
single atoms
or sjngle-atom
locations.
from a panicular lauice site. This defect is referred to as a VG,'IIllCY; it is schematically
shown in Figure I. I 7a. In another situation, an atom may be located between lattice
A line defect, for example,
when an
rowtoofasatoms
is missing.and is sc hematically shown in Fig·
sites.occurs
This defect
is entire
referred
an interstilial
Thi defect is referred to ure
as a 1.lib.
line dislocation
is shownand
in Figure
19 defects. not only is the perfect geoIn the caseand
of "<lcaney
interstitial
arrangemenL
of atoms
but geometric
also the ideal
chemical
bonding
As with a point defect, ametric
line dislocation
disrupts
both broken.
the normal
periodicity
of the
lattice between
and the
ideal atomic bonds in theatoms
crystal.
is disrupted, which tends to change the clecLrical properties of lhe material. A
interstitial
may of
be the
in close
enough
proximity
t() unpredictable
exhibit an interaction
This dislocation can alsovacancy
alter theand
electrical
properties
material.
usually
in a more
manner
In fomring single-crystal materials, more complex defects may occur.
than the simple point defects.
between the twO point defects. This vacancy-interstitial defect. also known ,lS ;l
Frenkel tlefeel. prodllces different effect - than the simple vacancy or interstitial.
The point defects involve singJe atoms or sjngle-atoll1
In tonning
single-crystal materials, more complex defects may occur. A line defect. for example,
occurs when an entire ['()w of13aLOrns is missing from
normal lattict: site. Thi$ de..
reCt is referred to as a line dislucation and shown in Figure I. I 8. As wi th a point
-
1 .5 Imperieclioos and Impuriti
...
,.
,
, '",
'"
... • • • • ..., ..., ... •
.....
-+-++ + , , .- +
• • • • •, ,
+, •, +..., , ...... ... .-, +
•, ..., •, , •,, ... •,
.-... .- '" ...... ,. ... ... ... •
'"
or, ...,, or, ,
I
I
I
I
I
I
I
\
\
\
I
I
I
\
\
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
1 .5 Imperieclioos and Impurities in SolidS
15
, , ,
... • • • • ..., ..., ... •
Figure 19:.- A two·dimensional
representation of a line dislocation
Figure
two·
+ +-+-, -+-1.18
+
, ... .-1A
• • •\ • .- •, ,representation
dimensional
1.6.2 Impurities in Solids+
, , ... ... .-I, +
, •, +
... •, ...,di•,,slocation
...
... •,I
of
a
line
.
•
,
,
Impurities in Solids
.- .- ... ... ... ...
or, ,, or, ,
I
I
I
I
I
I
I
\
\
\
I
I
I
\
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
•
Foreign atoms, or impurity atoms, may be present in a crystal lattice.
Figure 1.18 1A two·
Impurity atoms may be located at normal lattice sites. in which case they are called substitutional impurities.
dimensional representation
Impurity atoms may also be located
normal
sites, in which case they are called interstitial impurities.
of a linebetween
di slocation
.
Both these impurities are lattice defects and are schematically shown in Figure 20.
...
... ... ...
...
...
... ...- -. -,-. ...- - - ......
...
...
...
. , . - tnlpunty
tnlpunty
-----
(b)
(aJ
Figure 1.19 ITwo-dimensional representation of a
showing (a) a SubSlittltional impurity
Figureand
20:(b)Two-dimensional
representation of a single-crystal lattice showing (a) a substitutional impurity and
;\11 intersitilcd impurily.
(b) interstitial impurity.
\.5.2 Impurities in Solids
Some
impurities,
as oxygen
silicon,
tend toinbea essentially
inert;
however,
other impurities, such as gold
Foreign
atoms. orsuch
impurity
atoms.inmay
be present
crystal lattice.
Impurity
atoms
or may
phosphorus
drastically
electrical
of the material.
be locatedinatsilicon,
normalcan
lattice
sites. in alter
whic hthe
case
they areproperties
caned £ub.\'/i'I4I;ollu/
im·
Impurity
maythat,
alsoby
be adding
localedcontrolled
between normal
sites,
in which case
In purities.
the following
weatoms
will see
amounts
of particular
impurity atoms, the electrical
characteristics
a semiconductor
material
can be
favorablyare
altered.
they are calledofilllersr;r;al
impurities,
Both these
impurities
lattice defects and arc
schematically
in Fib"'JTC
1.1 9.atoms
Someto
impurilies,
such as oxygen
in silicon.
The
technique shown
of adding
impurity
a semiconductor
material
in ordertend
to change its conductivity is
to
be
essentially
inert;
however,
other
impurities,
such
as
gold
or
phosphorus
in sili·
called doping.
(aJ
ure 1.19There
Icon,
Two-dimensional
representation
ofionaimplantation.
can drastically
the electrical
properties
of the
material.and
are
two generalalter
methods
of doping:
impurity
diffusion
In Chapter 4 we will see that. by adding controlled amOuntS of panicular impu·
In general, impurity
diffusion occurs when a semiconductor crystal is placed in a high·temperature (≈ 1000 C)
(b) ;\11 intersitilcd
impurily.
rity atoms, the electrical
characleristics of a semiconductor material can be favorably
◦
gaseous atmosphere containing the desired impurity atom.
altered. The technique of adding impurity atoms to a semiconductor material in order
to change its conductivity is called dop;I/!:. There are two general methods of doping:
14
impurity diffusion and ion implantalion.
The actual
process depends
Impurities
indiffusion
Solids
lO
some extent on the material but. in gen-
showi
At this high temperature, many of the crystal atoms can randomly move in and out of their single-crystal lattice
sites.
Vacancies may be created by this random motion so that impurity atoms can move through the lattice by
hopping from one vacancy to another.
Impurity diffusion is the process by which impurity particles move from a region of high concentration near the surface, to a region of lower concentration within the crystal.
When the temperature decreases, the impurity atoms become permanently frozen into the substitutional lattice
sites.
Diffusion of various impurities into selected regions of a semiconductor allows us to fabricate complex electronic
circuits in a single semiconductor crystal.
Ion implantation generally takes place at a lower temperature than diffusion.
A beam of impurity ions is accelerated to kinetic energies in the range of 50 keV or greater and then directed
to the surface of the semiconductor.
The high-energy impurity ions enter the crystal and come to rest at some average depth from the surface.
One advantage of ion implantation is that controlled numbers of impurity atoms can be introduced into specific
regions of the crystal.
A disadvantage of this technique is that the incident impurity atoms collide with the crystal atoms causing
lattice displacement damage. However, most of the lattice damage can he removed by thermal annealing,
in which the temperature of the crystal is raised for a short time.
Thermal annealing is a required step after implantation.
1.7
Growth of Semiconductor Materials
Growth of Semiconductor Materials
The success in fabricating very large scale integrated (VLSI) circuits is a result of the development of and
improvement in the formation or growth of pure single-crystal semiconductor materials.
Semiconductors are some of the purest materials.
Silicon, for example, has concentrations of most impurities of less than 1 part in 10 billion. The high purity
requirement means that extreme care is necessary in the growth and the treatment of the material at each step
of the fabrication process.
The mechanics and kinetics of crystal growth are extremely complex.
A general knowledge of the growth techniques and terminology is valuable.
1.7.1
Growth from a Melt
Growth from a Melt
A common technique for growing single-crystal materials is called the Czochralski method.
In this technique, a small piece of single crystal material, known as a seed, is brought into contact with the
surface of the same material in liquid phase, and then slowly pulled from the melt.
As the seed is slowly pulled, solidification occurs along the plane between the solid-liquid interface.
Usually the crystal is also rotated slowly as it is being pulled, resulting in a more uniform temperature.
Controlled amounts of specific impurity atoms, such as boron or phosphorus, may be added to the melt so that
the grown semiconductor crystal is intentionally doped with the impurity atom.
Figure 21 shows a schematic of the Czochralski growth process and a silicon ingot grown by this process.
Some impurities may be present in the ingot that are undesirable.
Zone refining is a common technique for purifying material.
A high-temperature coil, or r-f induction coil, is slowly passed along the length of the ingot.
The temperature induced by the coil is high enough so that a thin layer of liquid is formed.
At the solid-liquid interface, there is a distribution of impurities between the two phases. The parameter that
describes this distribution is called the segregation coefficient: the ratio of the concentration of impurities
in the solid to the concentration in the liquid.
15
1 .8
Grov/th
SemiconduCiOt Malenals
Container
Chuck
Seed
Tube
Crystal -
Heaters
0
0
0
0
0
Md t
0
Cmciblc
(0)
Figure 21: Model of a crystal puller
If the segregation coefficient is 0.1, for example, the concentration of impurities in the liquid is a factor of 10
greater than that in the solid.
As thc liquid zone moves through the material, the impurities are driven along with the liquid.
After several passes of the r-f coil, most impurities are at the end of the bar, which can then be cut off.
The zone-refining technique, can result in considerable purification.
After the semiconductor is grown, the ingot is mechanically trimmed to the proper diameter and a flat is ground
over the entire length of the ingot to denote the crystal orientation.
The flat is perpendicular to the [110] direction or indicates the (110) plane.
his then allows the individual chips to be fabricated along given crystal planes so that the chip can be sawed
apart more easily.
The ingot is then sliced into wafers.
The wafer must be thick enough to mechanically support itself.
A mechanical two-sided lapping operation produces a flat wafer of uniform thickness.
Since the lapping procedure can leave a surface damaged and contaminated by the mcchanical operation. the
surface must be removed by chemical etching.
The final step is polishing.
(bJ
This provides a smooth surface on which devices may be fabricated or further growth processes may be carried
out.
Figure 1.20 I (0)wafer
Model
of a crystal
puller and material.
(b) photogmph of a silicon wafer with an
This final semiconductor
is called
the substrate
ilrray of integrilled. circuits. The ci l'Cuits arc tested on the wafer then sawed apart into chips
rba' are moonled into package..s. (Phow courresy of 100e) CorporatiOJl.)
1.7.2 Epitaxial Growth
Epitaxial Growth
A common and versatile growth technique that is used extensively in device and integrated circuit fabrication
is epitaxial growth.
Epitaxial growth is a process whereby a thin, single-crystal layer of material is grown on the surface of a
single-crystal substrate.
In the epitaxial process, the single-crystal substrate acts as the seed.
When an epitaxial layer is grown on a substrate of the same material, the process is termed homoepitaxy.
16
Growing silicon on a silicon substrate is one example of a homoepitaxy process.
work is being done with heteroepitaxy.
At present, a great deal of
In a heteroepitaxy process, although the substrate and epitaxial materials are not the same, the two crystal
structures should be very similar if single-crystal growth is to be obtained and if a large number of defects are
to be avoided at the epitaxial-substrate interface.
Growing epitaxial layers of the ternary alloy AIGaAs on a GaAs substrate is one example of a heteroepitaxy
process.
One epitaxial growth technique that has been used extensively is called chemical vapor-phase deposition
(CVD).
Silicon epitaxial layers, for example, are grown on silicon substrates by the controlled deposition of silicon atoms
onto the surface from a chemical vapor containing silicon.
A sharp demarcation between the impurity doping in the substrate and in the epitaxial layer can be achieved
using the CVD process.
Liquid-phase epitaxy is another epitaxial growth technique.
A compound of the semiconductor with another element may have a melting temperature lower than that of
the semiconductor itself.
The semiconductor substrate is held in the liquid compound and, since the temperature of the melt is lower
than the melting temperature of the substrate, the substrate does not melt.
As the solution is slowly cooled, a single-crystal semiconductor layer grows on the seed crystal.
A versatile technique for growing epitaxial layers is the molecular beam epitaxy (MBE) process.
A substrate is held in vacuum at a temperature normally in the range of 400 to 800 ◦ C, a relatively low
temperature compared with many semiconductor-processing steps.
Semiconductor and dopant atoms are then evaporated onto the surface of the substrate.
In this technique, the doping can be precisely controlled resulting in very complex doping profiles.
Complex ternary compounds, such as AIGaAs, can be grown on substrates, such as GaAs, where abrupt changes
in the crystal composition arc desired.
Many layers of various types of epitaxial compositions can be grown on a substrate in that manner.
2
Amorphous Solids
Amorphous Solids
Amorphous solid ia any noncrystalline solid in which the atoms and molecules are not organized in a definite
lattice pattern.
Such solids include glass, plastic, and gel.
Solids and liquids are both forms of condensed matter; both are composed of atoms in close proximity to each
other. But their properties are, of course, enormously different.
While a solid material has both a well-defined volume and a well-defined shape, a liquid has a well-defined
volume but a shape that depends on the shape of the container.
Externally applied forces can twist or bend or distort a solid’s shape, but (provided the forces have not exceeded
the solid’s elastic limit) it ”springs back” to its original shape when the forces are removed.
A liquid flows under the action of an external force; it does not hold its shape.
These macroscopic characteristics constitute the essential distinctions: a liquid flows, lacks a definite shape
(though its volume is definite), and cannot withstand a shear stress; a solid does not flow, has a definite shape,
and exhibits elastic stiffness against shear stress.
On an atomic level, these macroscopic distinctions arise from a basic difference in the nature of the atomic
motion.
Figure 22 contains schematic representations of atomic movements in a liquid and a solid.
Atoms in a solid are not mobile. Each atom stays close to one point in space, although the atom is not stationary
but instead oscillates rapidly about this fixed point (the higher the temperature, the faster it oscillates).
The fixed point can be viewed as a time-averaged centre of gravity of the rapidly jiggling atom.
17
Figure 22: Representation of the atomic motion in a liquid (left) and in a solid (right)
The spatial arrangement of these fixed points constitutes the solid’s durable atomic-scale structure.
In contrast, a liquid possesses no enduring arrangement of atoms.
Atoms in a liquid are mobile and continually wander throughout the material.
2.1
Crystalline and amorphous solids
Crystalline and amorphous solids
There are two main classes of solids: crystalline and amorphous.
What distinguishes them from one another is the nature of their atomic-scale structure.
The essential differences are displayed in Figure23.
The salient features of the atomic arrangements in amorphous solids (also called glasses), as opposed to crystals,
are illustrated in the figure for two-dimensional structures; the key points carry over to the actual threedimensional structures of real materials.
Figure 23: The atomic arrangements in (A) a crystalline solid, (B) an amorphous solid, and (C) a gas
Also included in the figure, as a reference point, is a sketch of the atomic arrangement in a gas.
For the sketches representing crystal (A) and glass (B) structures, the solid dots denote the fixed points about
which the atoms oscillate; for the gas (C), the dots denote a snapshot of one configuration of instantaneous
atomic positions.
Atomic positions in a crystal exhibit a property called long-range order or translational periodicity; positions
repeat in space in a regular array, as in Figure 23A.
In an amorphous solid, translational periodicity is absent.
As indicated in Figure 23B, there is no long-range order.
The atoms are not randomly distributed in space, however, as they are in the gas in Figure 23C.
18
In the glass example illustrated in the figure, each atom has three nearest-neighbour atoms at the same distance
(called the chemical bond length) from it, just as in the corresponding crystal. All solids, both crystalline and
amorphous, exhibit short-range (atomic-scale) order.
The well-defined short-range order is a consequence of the chemical bonding between atoms, which is responsible
for holding the solid together.
In addition to the terms amorphous solid and glass, other terms in use include noncrystalline solid and vitreous
solid.
Amorphous solid and noncrystalline solid are more general terms, while glass and vitreous solid have historically
been reserved for an amorphous solid prepared by rapid cooling (quenching) of a melt as in scenario 2 of Figure24.
Figure 24: The two general cooling paths by which a group of atoms can condense. Route 1 is the path to the
crystalline state; route 2 is the rapid-quench path to the amorphous solid state
Figure 24, which should be read from right to left, indicates the two types of scenarios that can occur when
cooling causes a given number of atoms to condense from the gas phase into the liquid phase and then into the
solid phase. The temperature Tb is the boiling point, Tf is the freezing (or melting) point, and Tg is the glass
transition temperature.
In scenario 1 the liquid freezes at Tf into a crystalline solid, with an abrupt discontinuity in volume.
When cooling occurs slowly, this is usually what happens. At sufficiently high cooling rates, however, most
materials display a different behaviour and follow route 2 to the solid state. Tf is bypassed, and the liquid
state persists until the lower temperature Tg is reached and the second solidification scenario is realized.
In a narrow temperature range near Tg , the glass transition occurs: the liquid freezes into an amorphous solid
with no abrupt discontinuity in volume. The glass transition temperature Tg is not as sharply defined as Tf .
2.2
Preparation of amorphous solids
Preparation of amorphous solids
It was once thought that relatively few materials could be prepared as amorphous solids, and such materials
(notably, oxide glasses and organic polymers) were called glass-forming solids.
It is now known that the amorphous solid state is almost a universal property of condensable matter.
The table 3 of representative amorphous solids presents a list of amorphous solids in which every class of
chemical bonding type is represented.
The glass transition temperatures span a wide range.
Glass formation is a matter of bypassing crystallization. The channel to the crystalline state is evaded by
quickly crossing the temperature interval between Tf and Tg .
19
glass
silicon dioxide
germanium dioxide
silicon, germanium
beryllium difluoride
arsenic trisulfide
polystyrene
isopentane
iron, cobalt, bismuth
bonding
covalent
covalent
covalent
ionic
covalent
polymeric
van der Waals
metallic
transition temperature (K)
1430
820
570
470
370
65
-
Table 3: Bonding types and glass transition temperatures of representative amorphous solids
Nearly all materials can, if cooled quickly enough, be prepared as amorphous solids.
The definition of ”quickly enough” varies enormously from material to material.
Four techniques for preparing amorphous solids are illustrated in Figure 25.
These techniques are not fundamentally different from those used for preparing crystalline solids; the key is
simply to quench the sample quickly enough to form the glass, rather than slowly enough to form the crystal.
The quench rate increases greatly from left to right in the figure.
Figure 25: Four methods for preparing amorphous solids. (A) Slow cooling, (B) moderate quenching, (C) rapid
splat quenching, and (D) condensation from the gas phase.
2.3
The radial distribution function
The radial distribution function
The absence of long-range order is the defining characteristic of the atomic arrangement in amorphous solids.
However, because of the absence in glasses of long parallel rows and flat parallel planes of atoms, it is extremely
difficult to determine details of the atomic arrangement with the structure-probing techniques (such as X-ray
diffraction) that are so successful for crystals.
For glasses the information obtained from such structure-probing experiments is contained in a curve called the
radial distribution function (RDF).
Figure 26 shows a comparison of the experimentally determined RDFs of the crystalline and amorphous forms
of germanium, an elemental semiconductor similar to silicon.
20
The significance of the RDF is that it gives the probability of neighbouring atoms being located at
various distances from an average atom.
The horizontal axis in the figure specifies the distance from a given atom; the vertical axis is proportional to
the average number of atoms found at each distance
Figure 26: Comparison of the atomic radial distribution functions of crystalline (c-Ge) and amorphous (a-Ge)
germanium. The value of the function at each distance r from a given atom is proportional to the number of
atoms found at that distance.
The curve for crystalline germanium displays sharp peaks over the full range shown, corresponding to welldefined shells of neighbouring atoms at specific distances, which arise from the long-range regularity of the
crystal’s atomic arrangement.
Amorphous germanium exhibits a close-in sharp peak corresponding to the nearest-neighbour atoms (there are
four nearest neighbours in both c-Ge and a-Ge), but at larger distances the undulations in the RDF curve
become washed out owing to the absence of long-range order.
The first, sharp, nearest-neighbour peak in a-Ge is identical to the corresponding peak in c-Ge, showing that
the short-range order in the amorphous form of solid germanium is as well-defined as it is in the crystalline
form.
2.4
Applications of amorphous solids
Applications of amorphous solids
In this sections we discuss technological applications of amorphous solids in connection with the properties that
make those applications possible.
Amorphous solids exhibit essentially the full range of properties and phenomena exhibited by crystalline solids.
There are amorphous-solid metals, semiconductors, and insulators; there are transparent glasses and opaque
glasses; and there are superconducting amorphous solids and ferromagnetic amorphous solids.
The atomic-scale disorder present in a metallic glass causes its electrical conductivity to be lower than the
conductivity of the corresponding crystalline metal, because the structural disorder impedes the motion of the
mobile electrons that make up the electrical current.
For a similar reason, the thermal conductivity of an insulating glass is lower than that of the corresponding
crystalline insulator; glasses thus make good thermal insulators.
Crystals and glasses also differ systematically in their optical spectra, which are the curves that describe the
wavelength dependence of the degree to which the solid absorbs infrared, visible, or ultraviolet light.
Although the overall spectra are often similar, crystal spectra typically exhibit sharp peaks and other features
that specifically arise as a consequence of the long-range order of the crystal?s atomic-scale structure.
These sharp features are absent in the optical spectra of amorphous solids.
The continuous liquid-to-solid transition near Tg , the glass transition, has a profound significance in connection
with classical applications of glasses.
21
While crystallization abruptly transforms a mobile, low-viscosity liquid to a crystalline solid at Tf , near Tg the
liquid viscosity increases continuously through a large range in the transformation to an amorphous solid.
Viscosity, expressed in units of poise, is used in the table of characteristics of oxide glasses to specify characteristic
working temperatures in the processing of the liquid precursors of various oxide glasses.
A poise is the centimetre-gram-second (cgs) unit of viscosity.
It expresses the force needed to maintain a unit velocity difference between parallel plates separated by one
centimetre of fluid: one poise equals one dyne-second per square centimetre.
Molten glass may have a viscosity of 1013 poise (similar to honey on a cold day), and it quickly gets stiffer when
cooled since the viscosity steeply increases with decreasing temperature.
The ability to ”tune” the viscosity of the melt (by changing temperature) allows glass to be conveniently
processed and worked into desired shapes.
The table4 below lists some important technological uses of amorphous solids.
In addition to the application, the general type of amorphous solid used, and the material properties that make
the application possible, the table also includes information about the chemical compositions of typical materials
employed in these techniques.
type
oxide glass
application
window glass
oxide glass
fibre-optic waveguides
organic polymer
structural materials, plastics
chalcogenide glass
copiers and laser printers
amorphous semiconductor
solar cells, copiers,
flat-panel displays
metallic glass
transformer cores
special property
transparency, solidity,
formability as large sheets
ultratransparency, purity,
formability as uniform fibres
strength, light weight,
ease of processing
photoconductivity,
formability as large-area films
photovoltaic optical properties,
arge-area thin films,
semiconducting properties
ferromagnetism, low power loss,
formability as long ribbons
Table 4: Some technological applications of amorphous solids
A significant theme of the table is the role of amorphous solids in applications calling for large-area sheets or
films.
Amorphous solids often have great advantages over crystalline solids in such applications, since their use avoids
the functional problems associated with polycrystallinity or the expense of preparing large single crystals.
Thus, while it would be prohibitively expensive to fabricate large windows out of crystalline SiO2 (quartz), it
is practical to do so using SiO2 -based silicate glasses.
2.4.1
Transparent glasses
Transparent glasses
The terms glass and window glass are often used interchangeably in everyday language, so familiar is this ancient
architectural application of amorphous solids.
Not only are oxide glasses, such as those characterized in the table, excellent for letting light in, they are also
good for keeping cold out, because (as mentioned above) they are efficient thermal insulators.
The second application in the table of technological applications of amorphous solids represents a modern
development that carries the property of optical transparency to a phenomenal level.
The transparency of the extraordinarily pure glasses that have been developed for fibre-optic telecommunications
is so great that, at certain wavelengths, light can pass through 1 kmof glass and still retain 95% of its original
intensity.
Glass fibres (transmitting optical signals) are now doing what copper wires (transmitting electrical signals) once
did and are doing it more efficiently: carrying telephone messages around the planet.
How this is done is schematically indicated in Figure 27
Digital electrical pulses produced by encoding of the voice-driven electrical signal are converted into light pulses
by a semiconductor laser coupled to one end of the optical fibre.
22
Figure 27: The use of ultratransparent glass fibres in telecommunications networks.
The signal is then transmitted over a long length of fibre as a stream of light pulses. At the far end it is
converted back into electrical pulses and then into sound.
The glass fibre is somewhat thinner than a human hair.
The simplest type, as sketched in the upper left of the figure, has a central core of ultratransparent glass
surrounded by a coaxial cladding of a glass having a lower refractive index, n.
This ensures that light rays propagating within the core, at small angles relative to the fibre axis, do not leak
out but instead are 100% reflected at the core-cladding interface by the optical effect known as total internal
reflection.
The great advantage provided by the substitution of light-transmitting fibres of ultratransparent oxide glass for
electricity-transmitting wires of crystalline copper is that a single optical fibre can carry many more simultaneous
conversations than can a thick cable packed with copper wires.
This is the case because light waves oscillate at enormously high frequencies (about 2 × 1014 cycles per second
for the infrared light generally used for fibre-optic telecommunications).
This allows the light-wave signal carrier to be modulated at very high frequencies and to transmit a high volume
of information traffic.
Fibre-optic communications have greatly expanded the information-transmitting capacity of the world?s telecommunications networks.
2.4.2
Polymeric structural materials
Polymeric structural materials
Polystyrene is a prototypical example of a polymeric glass.
These glasses make up a broad class of lightweight structural materials important in the automotive, aerospace,
and construction industries. These materials are also ubiquitous in everyday experience as plastic molded
objects.
The quantity of polymer materials produced each year, measured in terms of volume, exceeds the quantity of
steel produced.
Polystyrene is among the most important of the thermoplastic materials that, when heated (to the vicinity of
the glass transition temperature), soften and flow controllably, enabling them to be processed at high speeds
and on a large scale in the manufacture of molded products.
2.4.3
Amorphous semiconductors in electronics
Amorphous semiconductors in electronics
23
Amorphous semiconductors, in the form of thin films prepared by methods such as that shown in Figure 25D,
are important in applications requiring large areas of electronically active material.
The first electronic application of amorphous semiconductors to occur on a large scale was in xerography (or
electrostatic imaging), the process that provides the basis of plain-paper copiers.
The photoconductor, which is an electrical insulator in the absence of light but which conducts electricity when
illuminated, is exposed to an image of the document to be copied.
This process is also widely used in laser printers, in which the photoconductor is exposed to a digitally controlled
on-and-off laser beam that is raster scanned (like the electron beam in a television tube) over the photoconductor
surface.
Polymeric organic glasses, in the form of thin films, are now used in multilayer photoconductor configurations
in which the light is absorbed in one layer and electrical charge is transported through an adjacent layer.
Both layers are formed of amorphous polymer films, and these photoreceptors can be made in the form of
flexible belts.
Amorphous silicon thin films are used in solar cells that power handheld calculators.
This important amorphous semiconductor is also used as the image sensor in facsimile (?fax?) machines, and
it serves as the photoreceptor in some xerographic copiers.
All these applications exploit the ability of amorphous silicon to be vapour-deposited in the form of large-area
thin films.
Hydrogenated amorphous silicon also is used in high-resolution flat-panel displays for computer monitors and
for television screens.
In such applications the large-area amorphous-semiconductor thin film is etched into an array of many tiny
units, each of which forms the active element of a transistor that electronically turns on or off a small pixel
(picture element) of a liquid-crystal display.
2.4.4
Magnetic glasses
Magnetic glasses
The last entry in the table of technological applications of amorphous solids is an application of metallic glasses
having magnetic properties.
These are typically iron-rich amorphous solids.
They are readily formed as long metallic glass ribbons by melt spinning or as wide sheets by planar flow casting
Ferromagnetic glasses are mechanically hard materials, but they are magnetically soft, meaning that they are
easily magnetized by small magnetic fields.
Also, because of their disordered atomic-scale structure, they have higher electrical resistance than conventional
(crystalline) magnetic materials. The three attributes of ease of manufacture, magnetic softness, and high
electrical resistance make magnetic glasses extremely suitable for use in the magnetic cores of electrical power
transformers.
High electrical resistance (which arises here as a direct consequence of amorphicity) is a crucial property in
this application, because it minimizes unwanted electrical eddy currents and cuts down on power losses For
these reasons, sheets of iron-based magnetic glasses are used as transformer-core laminations in electrical power
applications.
Thin films of magnetic glass are finding use in many other applications.
These include magnetic recording media for audio and video digital recording, as well as recording heads used
with magnetic disks.
3
Insulators and dielectrics
Insulators and dielectrics
First we relate the applied electric field to the internal electric field in a dielectric crystal. The study of the
electric field within dielectric matter arises when we ask:
What is the relation in the material between the dielectric polarization P and the macroscopic electric field E
in the Maxwell equations?
24
What is the relation between the dielectric polarization and the local electric field which acts al the site of an
atom in the lattice? The local field determines the dipole moment of the atom.
3.1
Polarization
Polarization
Let su recall the Maxwell equations (in SI)
Maxwell equations
∂
(0 E + P)
∂t
∂B
rotE = −
∂t
div0 E = ρ
rotH = j +
divB = 0
The polarization P is defined as the dipole moment per unit volume, averaged over the volume of a cell.
The total dipole moment is defined as
p = Σqn rn
where rn is the position vector of the charge qn . The value of the sum will be independent of the origin chosen
for the position vectors, provided that the system is neutral.
The dipole moment of a water molecule is shown in Figure 28.
The electric field at a point r from a dipole
F"lgUre 1 Th
water hu th
rCdoo from
n.:ctlng the
j X 10",)
Figure 28: The permanent dipole moment of a molecule of water is directed from O−− ion toward the midpoint
of the line connecting the H+ ions.
moment p is given by a standard result of elementary electrostatics:
E(r) =
3(p · r)r − r2 p
4π0 r5
The lines of force of a dipole pointing along the z axis are shown in Figure29
3.2
Macroscopic Electric Field
Macroscopic Electric Field
One contribution to the electric field inside a body is that of the applied electric field, defined as
E0 = field produced by fixed charges external to the body
The other contribution to the electric field is the sum of the fields of all charges that constitute the body.
25
(2)
n.:ctlng the H ' kin" [To
C'O<!\'ert
to 51 units, nluiliply
j X 10",)
Figure 2 Ekctrostlltic pomltial alld lidd cornpo.:IC'I\IS in CCS at po5itloll ,., 11 b' • dipol
Figure 29: Electrostatic potential and field components in CGS at position (r, θ) for a dipole p directed along
dirttb:d aIaig the axis. For 11 - 0, we
E, - E" - 0 and E. - 2pI?; lOr 11 .. 'IIf2 ....1l h
the z axis.
£. =
E,, " 0 and E. " -pi? To convert 10 51, rcpIaa: P by
(Alter E M. Puroel\.)
If the body is neutral, the contribution to the average field may be expressed in terms of the sum of the fields
of atomic dipoles.
We define the average electric field E(r0 ) as the average field over the volume of the crystal cell that contains
the lattice point r0 :
Z
1
E(r0 ) =
dV e(r)
Vc
where e(r) is the microscopic electric field at the point r.
microscopic field e.
The field E is a much smoother quantity than the
We could well have written the dipole field in eq. (2) as e(r) because it is a microscopic unsmoothed field.
We call E the macroscopic electric field.
It is adequate for all problems in the electrodynamics of crystals provided that we know the connection between
E, the polarization P, and the current density j, and provided that the wavelengths of interest are long in
comparison with the lattice spacing.
To find the contribution of the polarization to the macroscopic field, we can simplify tihee sum over all the
dipoles.
By a famous theorem of electrostatics the macroscopic electric field caused by a uniform polarization is equal
to the electric field in vacuum of a fictitious surface charge density σ = n̂ · P on the surface of the body.
Here n̂ is the unit normal to the surface., drawn outward from the polarized matter.
We apply the result to a thin dielectric slab (Fig. 30a) with a uniform volume polarization P. The electric
field E1 (r) produced by the polarization is equal to the field produced by the fictitious surface charge density
σ = n̂ · P on the surface of the slab.
On the upper boundary the unit vector n̂ is directed upward and on the lower boundary n̂ is directed downward.
The upper boundary bear the fictitious charge σ = n̂ · P = P per unit area, and the lower boundary bears −P
per unit area.
The electric field E1 due to these charges has a simple form at any point between the plates, but comfortably
removed from their edges.
By Gauss’s law
E1 = −
|σ|
P
=−
0
0
26
(3)
"'
jill A uniformty pobrized
(I.)
Flgun:lFigure 30: (a) A uniformly polarized dielectric slab,
withwith
the the
poIariutlon
"cclOl"
nOl"mal
poIarization
vectorP P
normaltotothe
the plane of the
plane orslab.
Ihe (b) A (h)
of unilOnnly
po;mIllel
ptales
rise to Ihe
identlclll
pairAofpair
uniformly
chargedcbarged
parallel plates
which
givewh;..,h
rise to the identical
electric
field E1 , as in (a).
electric Beld E, IS In (a). TIle opper plale lli15 tIle
density q - +1', And the lower
1,"'le l..u
q -
- I'.
We add E1 to the applied field E0 to obtain the total macroscopic field inside the slab, with ẑ the unit vector
normal to the plane of the slab:
if = Ii, P on the surl:,cc of the body. Here Ii is the unit
P normal to the surfaCt'.,
E = E0 + E1 = E0 − ẑ
0
drawn outward from the polarized matter.
We define
We
apply the result Eto =a field
thinofdielectric
slab (Fig. 3a) with au nironn volume
the surface charge density n̂ · P on the boundary
1
polarization
P . The electric l'icld E.(r) produced by tlle polarization is equal to
This field is smoothly varying in space inside and outside the body and satisfies the Maxwell equations as
written
for the macroscopic
E.
the field
p roduced
by the field
fictitious
surface charge density u """ Ii' P on the
E is.aOn
smooth
viewed on
an atomic
scale is that
have replaced
the discrete lattice
surf.1.ceThe
ofreason
the slab
thefunction
upperwhen
boundary
the
unit vector
Ii iswe
directed
upwanl
or dipoles pj with the smoothed polarization P.
and on the 100vcr boundary Ii is d irected downward. The upper boundary bean
the flctitious
charge u = n'
P = P per unit area, and the lower boundary bears
3.2.1 Depolarization
field
- P per
unit area.
Depolarization field
111e
e lectric field E . due to these charges has a simple roml at allY point
The geometry in many of our problems is such that the polarization is uniform within the body, and then the
between
plates, but
removed
edges. By Gauss's law
onlythe
contributions
to theCOIllrortably
macroscopic field
are from Efrom
E1 :
0 and their
E = E0 + E1
applied =
field-4wP
and E1 ;is the field due
to the uniform polarization
(SQ
ICeS)Here£,E0=is the
- 4-lul
.
(4)
P
(4.)
The field E1 is called the depolarization field, for within the body <0
it tends to oppose the applied field E0 as
in Fig. 31.
Specimens in the shape or ellipsoids, a class that includes spheres, cylinders, and discs as limiting forms, have
We.
add EI toproperty:
the applied
field
Eo to produces
obtain athe,
total
macroscopic
fieldthe body.
an advantageous
a uniform
polarization
uniform
depolarization
field inside
inside the slab, with i the unit vector normal to the plane cf the slnb;
(ces)
(51)
We define
>
:E+
(5)
E. - field of the surface charge density Ii' P on the boundary
+ +
(6)
-
Cld E,
iJ field
OVI_ite
P.varying
TI", field
foclitioo.
surfaa:
lin'!
Indand
' aresatisfies
ted.
Figureis31:
The10
depolarization
is opposite
to and
P. The
fictitiousthe
surface
charge
indicated: the field of
nils
smoothly
in E
space
inside
outside
lxxIy
these
charges
is
E
within
the
ellipsoid.
\\ith'nlhe
eHipsoKl.
...,. E,. is
the Maxv.-ell
equations as written ror the macroscopic field E. The reason
1
1
a smooth function when viewed on an atomic scale is that we have replaced the
d iscrete lattice or d ipoles PJ with the smoothed
27 polari7.ation P.
cc,
"
3
If Px , Py and Pz are the components of the polarization P referred to the principal axes or an3 ellipsoid, then
the components of the depolarization field are written
E1x = −
:E+
Nx Px
Ny Py
E1y = −
0
0
>
E1z = −
Nz Pz
0
Here Nx , Ny Nz are the depolarization factors; their values depend on the ratios of the principal axes of the
ellipsoid.
The N ’s are positive and satisfy the sum rule Nx + Ny + N+z =+ 1 in SI.
Flgure"
11M!
depolari:r..:lOOn
flCld
E,
iJ OVI_ite 10 P. TI", foclitioo. surfaa:
lin'! Ind ' ted.
Values of N parallel to the figure axis of ellipsoids of revolution are plotted in Fig. 5.
the fldd
is E, \\ith'nlhe eHipsoKl.
...,. .
cc,
"
"
"
•"
"
<
'"
"
no
<
"
"
"
,,
'"
, ,'
,',
Figure S . Ot!pol.... ization factor N panlUd to the flgure
of ellipsoiIJs of revolution , llj; II fnncliOIl
of the ....Figure
ial ratio do .
32: Depolarization factor N parallel to the flgure axis of ellipsoids of revolution, as a function of the
axial ratio c/a.
A uniform applied field Eo will induce uniform polarization in an ellipsoid.
In limiting
N has the
values:
We introduce
thecases
dielectric
susceptibility
X such that the relations
(ces)
shape
axis
N (SI)
(51) I' sb EoXE.
(9)
sphere
any
1/3
thin slab
1
connect the macroscopic field E inside
the c llipsoid withnormal
the pola.·ization
P.
thin slab
in plane
0
Here X St = 4nxccs .
long circular cylinder longitudinal 0
If Eo is uniform and parallel to long
a principal
axis of thetransverse
ellipsoid, thcli
circular cylinder
1/2
p "" XE ;
(ces) E
of depolarizationNP
factor N .
= Eo + E. = Eo - NP ; Table 5: Values
(51) £ .... [ 0 - - (10)
<
by (8). We
whence
can reduce the depolarization field to zero in two ways, either by working with a long flne specimen or by
making an electrical connection between electrodes deposited on the opposite surfaces of a thin slab.
(CC5) A uniform P
applied
field- E:0'>orp)
will ;induce uniform
-X- Eo in an ellipsoid.
P _ - polarization
= x(Eo
(II)
1 + Nx
We introduce the dielectric susceptibility χ
(51)
P =
-
NP)
P = 0 χE
This quantity connects the macroscopic field E inside the ellipsoid with the polarization P.
The value of tlte polarb:ation depends on the
factor N.
If E0 is uniform and parallel to a principal axis of the ellipsoid, then
E = E0 −
We obtain
Polarization
28
NP
0
(5)
P = χ (0 E0 − N P ) P =
χ0
E0
1 + Nχ
The value of the polarization depends on the depolarization factor N .
3.3
Local electric field at an atom
Local electric field at an atom
The value of the local electric field that acts at the site of an atom is significantly different from the value of
the macroscopic electric field.
We can convince ourselves of this by consideration of the local field at a site with a cubic arrangement of
neighbors in a crystal of spherical shape. The macroscopic electric field in a sphere is
1
P
30
E = E0 + E1 = E0 −
where we have used eq. (5).
But consider the field that acts on the atom at the center of the sphere. If all dipoles are parallel to the z axis
and have magnitude p, the z component of the field at the center due to all other dipoles is, from (2),
Edipole =
p X 2zi2 − x2i − yi2
p X 3zi2 − ri2
=
4π0 i
ri5
4π0 i
ri5
The x, y, z directions are equivalent because of the symmetry of the lattice and of the sphere; thus
X z2
i
i
ri5
=
X x2
i
ri5
i
=
X y2
i
i
ri5
whence Edipole = 0. The correct local field is just equal to the applied fleld, Elocal = E0 . for an atom site with
a cubic environment in a spherical specimen.
Thus the local field is not the same as the macroscopic average field E.
We now develop an expression for the local field at a general lattice site, not necessarily of cubic symmetry.
The local field at an atom is the sum of the electric field E0 from external sources and of the field from the
dipoles within the specimen.
It is convenient to decompose the dipole field so that part of the summation over dipoles may be replaced by
integration.
We write
Elocal = E0 + E1 + E2 + E3
Here
E0 : field produced by fixed charges external to the body; E1 : depolarization field, from a surface charge density
n̂ · P on the outer surface of the specimen; E2 : Lorentz cavity field: field from polarization charges on inside of
a spherical cavity cut (as a mathematical fiction) out of the specimen with the reference atom as center, as in
Fig. 33; E1 + E2 is the field due to uniform polarization of the body in which a hole has been created; E3 : field
of atoms inside cavity The contribution E1 + E2 + E3 to the local field is the total field at one atom caused
by the dipole moments of all the other atoms in the specimen:
E1 + E2 + E3 =
1 X 3(pi · ri )ri − ri2 pi
4π0 i
ri5
Dipoles at distances greater than perhaps ten lattice constants from the reference site make a smoothly varying
contribution to this sum, a contribution which may be replaced by two surface integrals.
One surface integral is taken over the outer surface of the ellipsoidal specimen and defines E1 .
The second surface integral defines E2 and may be taken over any interior surface that is a suitable distance
(say 50 Å) from the reference site. We count in E3 any dipoles not included in the volume bounded by the
inner and outer surfaces.
It is convenient to let the interior surface be spherical.
29
3'"
E, tn."
"".1..
' '''e illlen,al
dl.:Clric
f",1d
011 an
....omfield
in of
a Cl"yslal
SUm isofthe
thesum
co:ten"
applied
Ileklfield E0
Figure
33: The
internal
electric
an atom is
in Ih.,
a crustal
of the•.1eternal
applied
the
field
due
to
the
other
atoms
in
the
crystal.
f lhe fleld due to the other .... oms in Ih.,
The standard m•."lhod of summing the
ld, oflhe other
is flnt to lium individually over a moderate "limber of ndghl>oring
ide an imagInary sphere conc.,ntric "'ith the reference atom : Ihis d(..fl"", Ih" Ileld E33.3.1
and of
Lorentz field
nishes at a reference site with rubic symmeiry. The atomli outsidto the sphere can be
Lorentz
field dielectric. TIleir contribution to Ihe field at II,c refcrence point is
a unifunn l),
po1.ari7.cd
E2 due to the polarization
charges with
on thethe
surface
of the
fictitious,"00
cavity
calculated
wlv.:re E, iiiThe
Ihefield
depoLariz:Jtion
field as.sociatcd
outcr
boundary
E"was
is Ihe
lidtl by Lorentz.
If θ is the oflhe
polar angle
(Fig. 34)cavil)'.
referred to the polarization direction, the surface charge density on the surface
d with Ihe surface
sfherlcal
of the cavity is −P cos θ.
The electric field at the center of the spherical cavity of radius a is (see Fig. 34):
Z π
1
1
(a−2 )(2πa sin θ)(adθ)(P cos θ)(cos θ) =
P
E2 =
4π0 0
30
field produced
fixed
edema]
the body;
This is the by
negative
of thecharges
depolarization
field E1 into
a polarized
sphere, so that E1 + E2 = 0 for a sphere.
depolari:r.ation fie ld, from a surface charge density 1\' P on the outer
of the spt!cimen;
Lorent-..r. cavity Held: field from polarization chargeli on inside of a
al cavily cut (as a mathematical fiction) out of the spl:cimen with the
ce atom as center, as in Fig. 6; EI + E2 is the field doe to unifonn
tion of the body in which a hole has bt..oen created;
field of atoms inside cavity.
e contribution E I +
+ E:t to the local field is lhe total field at one
Au>ed by the dipole moments of all the other atoms in the specimen:
."
• •<.:...".,..
..
!."-.';wf · .tl8 - 1',,,,/I
(15)
Figure 34: Calculation of the field in a spherical cavity in a uniformly polarized medium.
(9) is related to the dielectric constant by
SJ1we replace
PI by
3.3.2 Field
of dipoles inside cavity
P ten lattice ('Ollstants from the
greater
than perhaps
-".poles at distances
(S I)cavity
. (19)
Field of dipoles inside
,,E
ce site make a smoothly varying contribution to this sum, a contribution
30 surface integral is taken
may be replaced by two surface integrals. One
he
dielectdc
resPQnse
is dcscrilx:d
by the components
of a<; in Eq. (6).
e outer
surface
of the ellipsoidal
specimen
and defines [j.
The field E3 due to the dipoles within the spherical cavity is the only term that depends on the crystal structure.
We showed for a reference site with cubic surroundings in a sphere that E3 = 0 if all the atoms may be replaced
by point dipoles parallel to each other. The total local field at a cubic site is, from (187) and (193),
Lorentz relation
Elocal = E0 + E1 +
1
1
P=E+
P
30
30
(6)
This is the Lorentz relation: the field acting at an atom in a cubic site is the macroscopic field E of Eq. (4)
plus P/30 from the polarization of the other atoms in the specimen.
Experimental data for cubic ionic crystals support the Lorentz relation.
3.4
Dielectric constant and polarizability
Dielectric constant and polarizability
The dielectric constant of an isotropic or cubic medium relative to vacuum is defined in terms of the macroscopic
field E:
Dielectric constant
=
0 E + P
=1+χ
0 E
The susceptibility (9) is related to the dielectric constant by
Susceptibility
χ=
P
=−1
0 E
In a noncubic crystal the dielectric response is described by the components of the susceptibility tensor or of
the dielectric constant tensor:
Pµ = χµν 0 Eν µν = δµν + χµν
The polarizability α of an atom is defined in terms of the local electric field at the atom:
p = αElocal
where p is the dipole moment.
The polarizability is an atomic property, but the dielectric constant will depend on the manner in which the
atoms are assembled to form a crystal.
For a non-spherical atom α will be a tensor. The polarization of a crystal may be expressed approximately as
the product of the polarizabilities of the atoms times the local electric field:
X
X
P =
Nj pj =
Nj αj Elocal (j)
j
j
where Nj is the concentration and αj the polarizability of atoms j, and Elocal (j) is the local field at atom sites
j. We want to relate the dielectric constant to the polarizabilities.
The result will depend on the relation that holds between the macroscopic electric field and the local electric
field.
If the local field is given by the Lorentz relation (193), then
X
1
P =
Nj αj
E+
P
30
We solve for P to find the susceptibility
P
χ=
=
E
P
Nj αj
1 X
1−
Nj αj
30
31
(7)
By definition = 1 + χ; we rearrange (7) to obtain
Clausius-Mossotti relation
−1
1 X
=
Nj αj
+2
30
The Clausius-Mossotti relation relates the dielectric constant to the electronic polarizability, but only for crystal
structures for which the Lorentz local field (6) obtains.
3.4.1
Electronic Polarizability
Electronic Polarizability
The total polarizability may usually be separated into three parts: electronic, ionic, and dipolar, as in Fig. 35.
The electronic contribution arises from the displacement of the electron shell relative to a nucleus.
The ionic contribution comes from the displacement of a charged ion with respect to other ions.
The dipolar polarizability arises from molecules with a permanent electric dipole moment that can change
13 Didecfric. and Ferroelectric.
orientation in an applied electric field.
1,.,.,
"''' ''1
Figure 8
fre</ut""cr dt"pcndt"ncc of the 5e'\cral «>1ltriLutions to the pobrizabill1r.
Figure 35: Frequency dependence of the several contributions to the polarizability.
Table 1
Electronic
polarizabilities
In 10-«
em3
In heterogeneous materials
there
is usually alsoof
anions,
interfacial
polarization
arising from the accumulation of
charge at structural interface.’
I-Ie but it isLiof
This is of little fundamental Interest,
' considerable
Ik" practical
B" interest because
c" commercial insulating
materials are usually heterogeneous The dielectric constant at optical frequencies arises almost entirely from
Puulingthe electronic polarizability. 0. 201
0.0'29
0.006
0.003
0.0013
jS
0.029
The dipolar and ionic contributions are small
at high frequencies because of the inertia of the molecules and
0'FN,
ions.
A13+
Si H
Pauling
3.88
1.04
0.300
0.179
0
....
0.052
In the optical range (204) reduces to
0.0165
jS-(fKS)
(2.4)
Pallli'lg
JS·(TKS)
0.858
0.290
S' -
c,-
10.2
(5.5)
M
3.66
1.(J2
1 X",,,
n2 − K'
1
Sc3 +
=
Nj αj (electronic)
n2 +0.83
2
30
0.47
0.286
2.947
1.133
(1. 1)
2
here we have used the
n is the refractive
index.
D,- relation n
K, = , where
Rb '
Y"
Pauling
10.5
4.77
1.40
0.86
0.55
JS-(fKS)
(7.)
4.091
1.679
(1.6)
Te l X.
Dat >
W,.
Pauling
14.0
7. 10
32
3.99
2.42
1.55
1.().I
JS-{TKS)
(9.)
6. 116
2.743
,-
""
Tj'1+
0.185
(0. 19)
Zr4 +
0.37
Co"
0.73
ing probability function will determine the distribution of electrons among the avail.ble energy states. The energy band theory and the probability function will be used
extensively in the next chapter. when we develop the theory of the semiconductor in
equilibrium . •
4
Energy Bands
3.1 Bands
I ALLOWED
Energy
AND FORBIDDEN ENERGY BANDS
One In
of the
ourlast
goals
is towe
determine
theone-electron,
electrical properties
of a mom.
semiconductor
material, which we will use to
chapter,
treated the
or hydrogen.
That analysis
develop
the that
current-voltage
of semiconductor
devices.
showed
the energy ofcharacteristics
the bound electron
is quantized: Only
discrete values of
electron
allowed.
The radial
probability
density for of
Iheelectrons
eleclron in
was
also lattice, and to determine
We have
twoenergy
tasks are
in this
chapter:
to determine
the properties
a crystal
determined.
Th;s
function
gives
the
probability
of
tindillg
the
electron
at
a
particular
the statistical characteristics of the very large number of electrons in a crystal.
We can extrapolate the
distance results
from thetonucleus
andand
shows
thai the electron
is not
localized
at a given
single-atom
a crystal
qualitatively
derive the
concepts
of allowed
and forbidden energy bands.
radius.
We apply
can extrapolate
single-atom
[0 a crystal and qualitatively deWe can
then
quantumthese
mechanics
and Schrodinger’s
wave equation to the problem of an electron in a
rive
the
conceplS
of
allowed
and
forbidden
energy
bands.
We can then apply quansingle crystal.
lum mechanics and Schrodinger's wave equation to the problem of an electron in a
We find that the electronic energy states occur in bands of allowed states that are separated by forbidden energy
single crystal. We find that the electronic energy states occur ;11 hands of allowed
bands.
<Iales that are separated by forbidden energy bands.
4.1 3.1.1
Formation
bands
Formationofofenergy
Energy Bands
Figure 3.la
thebands
radi,,1 probability density function for Ihe lowest electron
Formation
of shows
energy
energy
of the
hydrogenfunction
atom, and
3.tb shows
the energy state of the single,
Figure
36astale
shows
thesingle,
radialnonintcracting
probability density
forFigure
the lowest
electron
same
probability
curves
for
two
atoms
that
are
in
close
proximity
to
each
other.
The
noninteracting hydrogen atom, and Figure 36b shows the same probability curves for two atoms that are in
functions
the two
atom electrons
overlap,
which
the two
electrons
close wave
proximity
toofeach
other.
The wave
functions
of means
the twothat
atom
electrons
overlap, which means that the
ilL
==_=_ ".
_"_= _I_=__
(a)
I
(e)
(b)
Figure 3.11 (a) Probllbility
function of an isolated hydrogen alUm . (b) O"erlapping probability density
Figurefunctions
36: (a)ofProbability
density function
isolated
hydrogen
atom. (b) Overlapping probability density
two adjacent hydrogen
alOnis . (c) of
Theansplitting
of the
II = l s lale.
functions of two adjacent hydrogenal atoms.(c)The splitting of the n = 1 state.
two electrons will interact.
This interaction or perturbation results in the discrete quantized energy level splitting into two discrete energy
levels. schematically shown in Figure 36c.
The splitting of the discrete stale into two states is consistent with the Pauli exclusion principle. Now, if we
somehow start with a regular periodic arrangement of hydrogen-type atoms that are initially very far apart,
and begin pushing the atoms together, the initial quantized energy level will split into a band of discrete energy
levels.
This effect is shown schematically in Figure 37 where the parameter r0 represents the equilibrium interatomic
distance in the crystal. At any energy level, the number of allowed quantum states is relatively small.
In order to accommodate all of the electrons in a crystal we must have many energy levels within the allowed
band.
As an example., suppose that we have a system with 1019 one-electron atoms and also suppose that, at the
equilibrium interatomic distance, the width of the allowed energy band is 1 eV.
For simplicity, we assume that each electron in the system occupies a different energy level and, if the discrete
energy states are equidistant, then the energy levels are separated by 10−19 eV.
This energy difference is extremely small, so that for all practical purposes, we have a quasi-continuous
energy distribution through the allowed energy band. Consider again a regular periodic arrangement of
atoms, in which each atom now contains more than one electron.
Suppose the atom in this imaginary crystal contains electrons up through the n = 3 energy level.
If the atoms are initially very far apart the electrons in adjacent atoms will not interact and will occupy the
discrete energy levels. If these atoms are brought closer together, the outermost electrons in the n = 3 energy
shell will begin to interact initially, so that this discrete energy level will split into a band of allowed energies.
33
rposes. we have a quasi-continuous energy distribution through the a
n$:
t
u
g
L-__
____
ro
____
Inter-nomic distance -----..
Figure 37: The splitting of an energy state into a band of allowed energies.
"'igure
3.2 1
splittjng of;m energy
state into a band of allowed energies.
If the atoms continue to move closer together,the electrons in the n = 2 shell may begin to interact and will also
split into a band of allowed energies. Finally, if the atoms become sufficiently close together, the innermost
electrons in the n = 1 level may interact, so that this energy level may also split into a band of allowed energies.
HAP T of
E Rthese
3 Intrcducoon
to levels
tl'..e Quantum
Theosyshown
of Sdds
The C
splitting
discrete energy
is qualitatively
in Figure 38.
t
If the equilibrium interatomic
S<=,.....--------- n
..
2
= I
ro
Inlcraf(lmic distauc-e ----..
3.31
Schemalic
showing
the splitting
of three
Figure 38: fijgure
Schemalic
showing
the splitting
of three
energy states
intoenergy
allowedstates
bands of energies.
into allowed bands of energies.
distance is r0 then we have bands of allowed energies that the electrons may occupy separatd by
bands of forbidden energies.
4N
This energy·band splitting and the formation of allowed and forbidden bands is the energy-band theory of
o
single-crystal
materials.
6N $tnte$
,
,,
t
'"
The actual band splitting in a crystal is much more complicated than indicated in Figure38.
A schematic representation of an isolated silicon atom is shown in Figure 39.
electrons occupy deep-lying energy levels close to the nucleus.
..>-----;::
)p
Ten of the fourteen
silicon atom
3,
The four remaining valence electrons are weakly bound and
are the electrons involved in chemical reactions.
4N Sl3tCS
S;, allowed le"cI,
Figure39b shows the band splitting of silicon. We need4N
only consider the n = 3 level for the valence electrons,
\
at .. me energy
2N electron$.
since the first two energy shells are completely full and are tightly bound to the nucleus.
The 3s state
tr
;r
1t =1
, P
Two allowed levels
At $:IMC energy
34
ro
Inlcraf(lmic distauc-e ----..
fijgure 3.31 Schemalic showing the splitting of three energy states
into allowed bands of energies.
o
,, ,
4N
6N $tnte$
t
'"
n =2
S electIOn$:
\
tr
;r
S;, allowed le"cI,
at .. me energy
..>-----;::
)p
3,
4N Sl3tCS
4N
2N electron$.
Two allowed levels
At $:IMC energy
, P
1t =1
2 clC(lrQns
'1 -
3
(aJ
(b)
,--
I
3.4 1(a) Schematic of an jsolated silicon
(b) The spliuiJlg of the 35 and 3p SlaleS of silicon into the
Figure 39:
(a) Schematic of an isolated silicon atom. (b) The splitting of the 3s and 3p states of silicon into the
aHowell and ft)Tb)dden e1)ergy bands.
allowed( frum
and SIJ{x:kley
forbidden
15/.)energy bands.
qualitatively shown in Figure 3.3. If the equilibrium interatomic distance is ro othen
corresponds to n = 3 and l =
and bands
contains
two quantum
per
atom. may occupy sep",atcd by bands
we0have
of ,tllowed
energiesstates
that the
electrons
of forbidden
This state will contain two electrons
at Tenergies.
= 0 K. This energy·band splitting and the formation of allowed and
forbidden
bands
the energy-band
of single-crystal
materials.
The 3p state corresponds to n = 3 and l = 1is and
contains sixtheory
quantum
states per atom.
The actual band sp1itting in a crystal is much more complicated than indicated
This state will contain the remaining
twoAelectrons
the individual
atom.
As 3tOm
the interatomic
distance
in Figure 3.3.
schematicinrepresentation
of silicon
an isolated
silicon
is shown in Figdecreases, the 3s and 3p states
interact
and
overlap.
ure 3.4a. Ten of the fourteen silicon atom electIons occupy deep-lying energy level,
close
to the nucleus.
The four
valence
rc)ari vcly
weak)},
At the equilibrium interatomic
distance,
the bands
haveremaining
again split,
but c)cctrOns
now fourarC
quantum
states
perbound
atom are
and are thestates
electrOns
chemical
reactions.
in the lower band and four quantum
perinvolved
atom arein in
the upper
band.Figure 3.4b shows the band split·
ting of silicon.
need
only energy
considerstate,
the n =
for states
the valence
electrons.
since (the
At absolute zero degrees, electrons
are inWe
the
lowest
so 3leve!
that all
in the
lower band
the first
energy
shells
are completely
and are tightly
bound
to the
nucleus. The
valence band) will be full and
all two
states
in the
upper
band (thefull
conduclion
band)
will
be empty.
The bandgap energy Eg between the top of the valence band and the bottom of the conduction band is the
width of the forbidden energy band.
4.2
The Kronig-Penney Model
The Kronig-Penney Model
In the previous section, we discussed qualitatively the spilling of allowed electron energies as atoms are brought
together to form a crystal.
The concept of allowed and forbidden energy bands can be developed more rigorously by considering quantum
mechanics and Schrodingers wave equation. The potential function of a single, noninteracting, one-electron
atom is shown in Figure 40a.
Also indicated on the figure are the discrete energy levels allowed for the electron.
Figure 40b shows the same type of potential function for the case when several atoms are in close proximity
arranged in a one-dimensional array. The potential functions of adjacent atoms overlap, and the net potential
function for this case is shown in Figure 40c. It is this potential function we would need to use in Schrodinger’s
wave equation to model a one-dimensional single-crystal material.
The solution to Schrodinger’s wave equation, for this one-dimensional single-crystal Iattice, is made more
tractable by considering a simpler potential function.
Figure 41 is the one-dimensional Kronig-Penney model of the periodic potential function, which is used to
represent a one-dimensional single-crystal lattice.
We need to solve Schrodinger’s wave equation in each
region.
As with previous quantum mechanical problems, the more interesting solution occurs for the case when E < V0
which corresponds to a particle being bound within the crystal.
The electrons are contained in the potential wells, but we have the possibility of tunneling between wells.
The Kronig-Penney model is an idealized periodic potential representing a one·dimensional single crystal, but
the results will illustrate many of the important features of the quantum behavior of electrons in the periodic
lattice.
35
v=o
I
I
I
cHAPT a A 3 Introduction to the Ouantum Tt)ooryol Solids
82
lnlnlr "
Atom
X
Atom
ACorn
Atom
Atom
(a)
(e )
v=o
v
Figure 3.5 1(I ) Potential functiun of a single
alOIn. (b) Overlapping potential fUilCtions of adjacent
AIOm
Atom
Atom
Alom
(b)
atoms. (c) Net pOlemial
function of a one-<limensional
v=o
single crystal.
I
I
I
Atom
lnlnlr "
ACorn
Atom
Atom
(e )
Figure 3.5 1(I ) Potential functiun of a single
(b) Overlapping
potential
fUilCtions
of adjacent
Figure 40: (a) Potential function of alOIn.
a single
isolated
atom.
(b)
Overlapping potential functions of adjacent
atoms. (c) Net pOlemial function of a one-<limensional
atoms. (c) Net potential function of asingle
one-dimensional
single crystal.
crystal.
V{X)
V{X)
II
II
- (n
II
If
o
+ h ) -h
II
a
(a
II
+ bJ
If
II
Figure 3.6 IThe cme·dirnensional periodic potential
funct ion of the Kronig- Penney model.
- (n
+ h ) -h
o
a
(a
+ bJ
Figure
3.6
IThe cme·dirnensional
potential
Figure
41: The
one·dimensional
periodic potential periodic
function of the
Kronig-Penney model.
funct ion of the Kronig- Penney model.
To obtain the solution to Schrodinger’s wave equation, we make use of a mathematical theorem by Bloch. The
theorem states that all one-electron wave functions for problems involving periodically varying potential energy
functions, must be of the form :
ψ(x) = u(x)ejkx
(8)
The parameter k is called a constant of motion and will be considered in more detail as we develop the theory.
The function u(x) is a periodic function with period (a + b). The total solution to the wave equation is the
product of the time-independent solution and the time-dependenl solution, or
Ψ(x, t) = ψ(x)φ(t) = u(x)ejkx e−j(E/~)t = u(x)ej[kx−(E/~)t]
(9)
This traveling-wave solution represents the motion of an electron in a single-crystal material. The amplitude of
the traveling wave is a periodic function and the parameter k is also referred to as a wave number. We can
now begin to determine a relation between the parameter k, the total energy E, and the potential V0 .
If we consider region I in Figure 41 (0 < x < a) in which V (x) =, take the second derivative of Equation (8),
and substitute this result into the time-independent Schrodinger’s wave equation we obtain :
du1 (x)
d2 u1 (x)
+ 2jk
− (k 2 − α2 )u1 (x) = 0
dx2
dx
36
(10)
The function u1 (x) is the amplitude of the wave function in region I and the parameter a is defined as
α2 =
2mE
~2
(11)
Consider now a specific region II (−b < x < 0) in which V (x) = V0 , and apply Schrodinger’s equation. We
obtain :
du2 (x)
2mV0
d2 u2 (x)
2
2
+ 2jk
u2 (x) = 0
(12)
− k −α +
dx2
dx
~2
where u2 (x) is the amplitude of the wave function in region II. We may define :
2m
2mV0
(E − V0 ) = α2 −
= β2
2
~
~2
(13)
d2 u2 (x)
du2 (x)
+ 2jk
− k 2 − β 2 u2 (x) = 0
2
dx
dx
(14)
so that Equation 12 may be written as
Note that from Equation (13), if E > V0 , the parameter β is real, whereas if E < V0 , then β is imaginary. The
solution to Equation (10), for region I, is of the form
u1 (x) = Aej(α−k)x + Be−j(α+k)x for (0 < x < a)
(15)
and the solution Equation (14), for region II, is of the form
u2 (x) = Cej(β−k)x + De−j(β+k)x for (−b < x < 0)
(16)
Since the potential function V (x) is everywhere finite, both the wave function ψ(x) and its first derivative
∂ψ(x)/∂x must be continuous.
This continuity condition implies that the wave amplitude function u(x) and its first derivative ∂u(x)/∂x must
also be continuous.
If we consider the boundary at x = 0 and apply the continuity condition to the wave amplitude, we have
u1 (0) = u2 (0)
(17)
Substituting Equations (15) and (16) into Equation (17), we obtain
A+B−C −D =0
(18)
du2 du1 =
dx x=0
dx x=0
(19)
(α − k)A − (α + k)B − (β − k)C + (β + k)D = 0
(20)
Now applying the condition that
we obtain
We have considered region I as 0 < x < a and region II as −b < x < 0.
The periodicity and the continuity condition mean that the function u1 , as x → a, is equal to the function u2 ,
as x → −b.
This condition may be written as
u1 (a) = u2 (−b)
(21)
Applying the solutions for u1 (x) and u2 (x) to the boundary condition in Equation (21) yields
Aej(α−k)a + Be−j(α+k)a − Ce−j(β−k)b − Dej(β+k)b = 0
(22)
The last boundary condition is
du2 du1 =
dx x=a
dx x=−b
(23)
which gives
(α − k)Aej(α−k)a − (α + k)Be−j(α+k)a
− (β − k)Ce−j(β−k)b + (β + k)Dej(β+k)b = 0
37
(24)
We now have four homogeneous equations, Equations (18), (20), (22), and (24), with four unknowns as a result
of applying the four boundary conditions.
In a set of simultaneous, linear, homogeneous equations, there is a nontrivial solution if, and only if, the
determinant of the coefficients is zero.
In our case the coefficients in question are the coefficients of the parameters A, B , C and D. The evaluation
of this determinant is extremely laborious and will not be considered in detail.
The result is
−(α2 + β 2 )
(sin αa)(sin βb) + (cos αa)(cos βb) = cos k(a + b)
2αβ
(25)
Equation (25) relates the parameter k to the total energy E (through the parameter α) and the potential
function V0 (through the parameter β).
As we mentioned, the more interesting solutions occurs for E < V0 , which applies to the electron bound within
the crystal.
From Equation (13), the parameter β is then an imaginary quantity. We may define
β = jγ
(26)
γ 2 − α2
(sin αa)(sinh γb) + (cos αa)(cosh γb) = cos k(a + b)
2αγ
(27)
where γ is a real quantity.
Equation (25) can be written in terms of γ as
Equation (27) does not lend itself to an analytical solution, but must be solved using numerical or graphical
techniques to obtain the relation between k, E and V0 .
The solution of Schrodinger’s wave equation for a single bound particle results in discrete allowed energies.
The solution of Equation (27) will result in a band of allowed energies.
To obtain an equation that is more susceptible to a graphical solution and thus will illustrate the nature of the
results, let the potential barrier width b → 0 and the barrier height V0 → 0, but such that the product bV0
remains finite.
Equation (27) then reduces to
mV0 ba
~2
sin αa
+ cos αa = cos ka
αa
(28)
We may define a parameter P 0 as
P0 =
mV0 ba
~2
(29)
The, finally, we have the relation
P0
sin αa
+ cos αa = cos ka
αa
(30)
Equation (30) again gives the relation between the parameter k, total energy E (through the
parameter α), and the potential barrier bV0 .
We may note that Equation (30) is not a solution of Schrodinger wave equation but give the
condition for which Schrodinger wave equation will have a solution.
If we assume the crystal is infinitely large, then k in Equation (30) can assume a continuum of values and must
be real.
4.3
The k-space diagram
The k-space diagram
To begin to understand the nature of the solution, initially consider the special case for which V0 = 0.
In this case P 0 = 0, which corresponds to a free particle since there are no potential barriers.
From Equation (30), we have that
cos αa = cos ka
38
(31)
1"
k' /;'
£= - = 2m
2m
(3.2
or
α=k
(32)
gure 3.7 shows th e parabolic relation of
Equation (3.28) between the energy
£a
mentulllSince
p the
forpotential
the isfree
particle. Since the momentum and wave number are l
equal to zero, the total energy E is equal to the kinetic energy, so that, from Equation (11),
Equation (32) may be written as
ly related,
Figure 3.7 is also the versus
k curve for the free particle.
s
r
2m
2mE
p e and k from Equati on (3.24)
We nOw want to consider the
relation
bemv
tween
=
= =k
α=
(33)
~
~
~
particle in the Single·crystal lanice. As the paramete r P' incrca$es, th e partic
where p is the particle momentum.
comes more
tightly
bound
to the potential
or atom.
Weformay
define
the left si
The constant
of the
motion parameter
k is related towell
the particle
momentum
the free
electron.
parameter to
k is be
also referred
to as on
a wave
number.) . so that
EquationThe(3.24)
a fu ncti
f(aa
e
1
2
2
2
2
We can also relate the energy and momentum as
p
~ k
,sinew
=
2
= P - exaE=
f(aa)
2m
2 2
COs ex(/
2m
(34)
Figure 42 shows the parabolic relation of Equation (34) between the energy E and momentum p for the free
particle.
I
I
I
I
I
I
,,,
I
,,
I
\
\
\
,
,
,
I
I
I
I
\
I
'"
--
p=o
.'
.-
Figure 42: The parabolic E versus k curve for the free electron.
Figure 3.7 I The parob")ic Ii versus k
curve for Ute free electron.
Since the momentum and wave number are linearly related, Figure 42 is also the E versus k curve for the free
particle.
We now want to consider the relation between E and k from Equation (30) for the particle in the single-crystal
lattice.
As the parameter P 0 increases, the particle becomes more tightly bound to the potential well or atom.
We may define the left side of Equation (30) to be a function f (αa), so that
f (αa) = P 0
sin αa
+ cos αa
αa
(35)
Figure 43a is a plot of the first term of Equation (35) versus αa. Figure 43b shows a plot of the cos αa and
Figure 43c is the sum of the two terms, or f (αa).
Now from Equation (30), we also have that
f (αa) = cos ka
39
(36)
(3.2
3 • 1 Allowed and Fo.b:dden Energy Bands
67
-41l'
(a)
(b)
j«(rtI)
(e)
Figu", 3.81 A plol of <a) Ihe firsl lenn in Equalion (3.29). (h) Ihe second lerm in Equalion
(3.29),
and term
(c) the entire
f(fXa) function.
The shaded
show the allowed
of
Figure 43: A plot of (a) the
first
in Equation
(35),
(b) the second
term va
inlues
Equation
(35), and (c) the entire
(era) corresponding to real vaJues of k.
f (αa) function. The shaded
areas show the allowed values of (αa) corresponding to real values of k.
figure 3.8. is. plol of the first term of Equation (3.29) verslls 'f{f. Figure 3.Sb shows
a plot of Ihe cos aa len" and Figure 3.8e is the sum of lhe lwo lemlS, or /(au ).
Nowthe
fromallowed
Equation (3.24).
we of
alsothe
havefthat
valid,
values
(αa) function must be bounded
For Equation (36) to be
between +1 and −1.
Figure 43c shows the allowed values of f (αa) and/ (aa)
the allowed
shaded areas.
cosku values of αa in the(3.30)
Equalion
10 be
Ihe allowed
Ihe j'(aa
) funClion mUsl
be which correspond to the
Also shown on the figureForare
the (3.30)
values
of valid,
ka from
the values
right ofside
of Equation
(36)
allowed values of f (αa). bounded belween + 1 !lnd -I. Figure 3.8c shows the allowed values of l(aa) and
!he allowed values of aa in the shaded areas. Also shown on the figure are the values
right side
of EqE
uation
(3.30)particle
which correspond
Ihe allowed values
The parameter α is relatedof ka
tofrom
thethetotal
energy
of the
throughto Equation
(11), which is α2 = 2mE/~2 .
of [(aa) .
A plot of the energy E of the
function
the
wave
number
k can
beEqua.
generated from Figure 43c.
Theparticle
parameter as
a isarelated
to the of
lOtal
energy
E of
Ihe particle
through
(3.5). which is a' = 2m E/ Ii'. A plot of Ihe energy E of Ihe particle as a f"nelion
Figure 44 shows this plot lion
and
shows the concept of allowed energy bands for the particle propagating in the
of Ihe wave number k can be generaled from Figure 3.8e. Figure 3.9 shows this plot
crystal lattice.
and shows Ihe coneepl of allowed energy bands {or Ihe parlide propagating in Ihe
Since the energy E has discontinuities, we also have the concept of forbidden energies for the particles in the
crystal.
Consider again the right side of Equation (30), which is the function cos ka.
The cosine function is periodic so that
cos ka = cos(ka + 2nπ) = cos(ka − 2nπ)
(37)
where n is a positive integer. We may consider Figure 44 and displace portions of the curve by 2π.
Mathematically, Equation (30) is still satisfied.
Figure 45 shows how various segments of the curve can be displaced by the 2π factor.
Figure46 shows the case in which the entire E versus k plot is contained within −π/a < k < π/a.
This plot is referred to as a reduced k·space diagram. Or a reduced-zero representation.
We noted in Equation (33) that for a free electron, the particle momentum and the wave number k
are related by p = ~k.
Given the similarity between the free electront solution and the results of the single crystal shown in Figure 44,
the parameter ~k in a single crystal is referred to as the crystal momentum.
This parameter is not the actual momentum of the electron in the crystal, but is a constant of the motion that
includes the crystal interaction.
We have been considering the Kronig-Penney model, which is a one- dimensional periodic potential function
used to model a single-crystal lattice.
40
(roollhis Kronig-Penoey mOdel.
8
CHAPT ER 3 Intmcrucfon to the OUamum Thecry Of So.XJs
,,TEST YOUR UNDERSTANDING
, given in Example 3.2. dec,../ennine the width (i n eV) oftht:
E3.1 Using the parameters
'i forbidden energy,, band that exists at ka = rr (sec
Figure 3.8c). (1\0 6Ct = '3"1 'stry)
,,
,
,,
,,
I
,,
,
,
Consilier again the right side of Equation (3.24), which is the fu nction cos ka.
, periodic so that
The cosine function is
,
,
'N]
I
,, ) = cos (kG - 211rr)
(3.3 1)
,, cos
,, k. = cos (ka + 2nrr
,
energy
,
,integer. \Vc
bandInay consider Figure 3.9 and displa<.:c portions Of the
,,,
where n is a positive
,
I
I
I
I
I ,
, .24) is still satisfied. Figure 3.10 shows
I
CUrve by 2". I Mathematically.
Equation
<3
I
how various segments of the cu,ve can be di.<placed by the 211" fac tor. Figure 3. 11
I
: Forbidden
I
shows the case
in which the entire
k plot is contained within -n/a <
lenergy"e"us
banlt
I
k < ,,/a. ThisI plot is referred to as a reduced k·space diagram. Or a reduced-zero
JlI'
Z·ff
) :1'
>
<
()
-7
u
representation.
•
"
-+We noted in Equation (3.27) that fo r a free kelectron.
the panicle momentum ,mel
<he
wave3.9number
are krelated
p = lik.
Give n the Similarity between the free
Figure
1Tht: £ kVCf$\lS
diagramby
generated
from
-,
,
It:
"
,
e
Figure 44: The E versus k Figure
diagram
generated
fromenergy
Figurebands
43. The
energy bands and forbidden energy
3,8.
The allowed
and allowed
forbidden
bandgaps are indicated.
energy bandgaps 3fe indicated.
.' ,
,,
" the energy
, also have the concepl of
crystal lattice. Since
E has Ediscontinuities, we
1
,
,
forbidden energies for the panicles in Ihe c rystal.
'i
XAM PLE 3.2
:
N
1
I
I
,I
I
J
I
Objective
I
I
I
\
I
\
I
I
\
I
\
:
f'
I
I
I
I
I
:./
I
,,
E
t\
I
To determine the lowest
allowed
energy bandwi
dth ,
I
I
\,
I
\
I
Assume that theJ(;'()cft'icicol
P' = 10 and
that
the Ipolenti;t! width a = 5 A.
• Solution
:
'1'----:- £'--
, _ , -- -1. £,, --
Ii':
To find the lowest allowed
energy Bbandwidth.
we" \need to find the di fference in aa values as
,,
I
I
ka changc,s from 0 to 1C (see Figure 3.8e). For ka =: O. Equation <3.29) becomes
:
,,,
,,
,fi nd
By trial and error. we,
For eta
/
] = I O aa
(w
= rr . we have
I
\
Sinaa
cosO'a
== 2.628 (ad. We see that for ka =
-,,.
-.
o
,..
It, IXlI :=
if.
_
•
"
m
or
3.101showing
The £ versus
k diagram showing
2" sections of allowed energy bands.
Figure 45: The E versusFi&ure
k diagram
displacements
of several
054the
x 10-")'
Figure 3.11 I The
of several
sections of allOWed
=
'
2.407energy
)( 10- '<' ) 1.50eV
2(9. 11 x 10-")(5 x 10-'0 )'
in the reduced-lo
bands.
,,'(I
=
=
For CUI
= 2.628.
we analysis
find thal E,
= 1.68
X 10- 1" J
1.053
eV. The
allowed
energyallowed
band- energy
The principle
result
of this
is that
electrons
in=the
crystal
occupy
certain
wid
th isexcluded
then
bands and
are
from the forbidden energy bands.
=
=
=
For real three-dimensionl single-crystal
energy-band
6E materials,
t:, - E, a similar
1.50 - 1.053
0.447theory
eV exists.
5
Electrical Conduction in Solids
Electrical Conduction in Solids
We are interested in determining the current-voltage characteristics of semiconductor devices.
41
by p = lik. Give n the Similarity between the free
We IUlve been consideriug 'he Krrmig-Pemrey model, which is a onedimensional periodic pOlential junction used fO model a single-crystal lattice. The
,
principle result of this analysis. so jar. is thar electrons jll the crystal OCCup)' cCI"loin
,, E from ,,Ille forhidden ellergy ba"d". For real
aI/owed
,
,, elterg)' b(lllds alld are excillded
,
f'
I
I
:./
three-dimension'll
materials, a'-!
similar encr<Jy-band theory ex ists. We
:
will
obtain additional electron properties from the Kronig-Penney model in the
I
I
I t\
sections.
--:- £'-- -1. £,, --3.2 I EI"ECTRICAL CONDUCTION IN SOLIDS
I
Ii':
\
•
"
Again, we are evenmall y illlerested in determining th e current-voltage characteristics of semiconductor devices. We will need , to consider electrical conduction in
solids as it relates to the band theory we have just developed . Let uS begin by cOnsidering th e motion of
in the various allowed energy hands.
.
,.
3_2.1
_
m
The Energy Band and the Bond Model
In Chapter I, we di scussed Ihe cov.knl oonoit'g of silicon. Figure 3. 12 shows a two-
m showing 2"
dimensional
representation 01"
the 3.11
covalent
bonding
in a si ngle-crystal silicon lallice.
Figure
I The £ versus
k diagram
f allOWed
energy
Figure 46: TheinE the
versus
k diagram in
the reduced-zone representation.
reduced-lone
represe1uation.
This figure represents silicon at T = 0 1< in which eac h silicon atom is surrounded by
eight valence electrons that are jn their lowest e nergy Slale cUld arc tiireclly involved
We will need to consider electrical conduction in solids as it relates to the band theory we have just developed.
in the
covalent bonding. Figure 3.4b represented the splilling of the di screte silicon
Let us begin by considering the motion of electrons in the various allowed energy bands.
e nergy states into bands of allowed e nergies as the silicon crystal is formed. At
T =5.10 K,
theEnergy
4N states
the the
lower
bane\.
the valence band. are filled with the vaThe
Bandinand
Bond
Model
lence
electrons. All of the valence electrons schematically s\town in FiguI"C 3.12 are
The Energy Band and the Bond Model
in the
valence
The upper
energy
band,
the
conducti
on band.silicon
is completely
Figure
47 shows aband.
two-dimensional
representation
of the
covalent
bonding
in a single-crystal
lattice.
This at
figure
empty
T represents
0 K. silicon at T = 0 K in which each silicon atom is surrounded by eight valence electrons
=
that are jn their lower energy state and are directly involved in the covalent bonding.
II
II
II
II
11
II
II
II
II
II
II
II
11
===0=0=0=0=='=
II
===0=0=0=0===
II
II
II
I)
..
II
II
II
II
II
..
II
===0=0=0=
0=
==
I
II
II
II
Figure
3.12 I of the covalent bonding in a semiconductor at T = 0 K.
Figure 47: Two-dimensional
representation
represe nt3(ion of the c()Va lenl bondin g
Figure 39b represented the splitting
of the discrete silicon
states into bands of allowed energies as the
in a semiconductor
at Tenergy
:: 0 K.
silicon crystal is formed.
42
effect and Pigu", 3.l3b, a simple line represent'ltion of the e"ergy-band model,
shows the same effect.
The semiconductor is neutrall y charged. This means thaI. as the negatively
charged electron breaks away from its covalent bonding position. a positively
At Tch.rged
= 0 K, the
4N states
thecreated
lower band,
theovalence
are filled
with thepoSition
valence electrons.
"empty
state"in is
in the
riginalband,
covalent
honding
in the vaAll of
the
valence
electrons
schematically
shown
in
Figure
47
are
in
the
valence
band.
lence band. As the temperature funhcr increases. more covalent ho nds are b)'Oken,
The more
upperelectrons
energy band,
the to
conduction
band, is completely
empty
at Tpositive
= 0 K. "empty states" ore
jump
the conduction
band, and
more
As the
temperature
increases
l.'reatcd
in the valence
band.above 0 K, a few valence band electrons may gain enough thermal
energy to break the covalent bond and jump into the conduction band.
We can also relate this bond breaking to the E verSuS k e nergy bands.
Figure 48a shows a two-dimensional representation of this bond-breaking effect and Figure 48b, a simple line
Figure 3.14a
shows
lhe E versus
k diagram
ofeffect.
the conduction and valence bands at
representation
of the
energy-band
model, shows
the same
1/
It
JI
IJ
"
II
II
II
"
II
II
II
='=0=0-0-0===
=
=
=!=!at4"
6==
=
I C I I
"=0-0-0-0==
=
It
II
II
II
tI
11
II
II
II
II
f'
II
(b)
(a)
.t
3.131
(a) Two-dimensional
representation
of the ofbreaking
of bond.
cova lent
bond.
FigureFigure
48: (a)
Two-dimensional
representation
of the breaking
a covalent
(b) Corresponding
line
representation
of the energy
and the 0n
generation
of a negative
and the
positive
charge wlth
(b) Corresponding
Ii lieband
fcpreStllla'1
of (he enetgy
band and
generarion
of :a the breaking of a
covalent
bond. and positi ve charge. wlth the brcflking of a covalent bond.
negative
The semiconductor is neutrally charged.
I
£
I electron breaks away from its covalent bonding position,
This means that,
as the negatively
charged
a positively charged ”empty state” is created in the original covalent bonding position in the
valence band.
As the temperature further increases, more covalent bonds are broken, more electrons jump to
the conduction band, and more positive ”empty states” are created in the valence band.
We can also relate this bond breaking to the E versus k energy bands.
I
I
Figure 49a shows the E versus k diagram of the conduction and valence bands at T = 0 K.
The energy states in the valence band are completely full and the states in the conduction band
k
k
are empty.
Figure 49b shows these same bands for T > 0 K, in which some electron, have gained enough energy to jump
to the conduction band and have left empty states in the valence band.
We are assuming at this point that no external forces are applied so the electron and ”empty
state” distributions are symmetrical with k.
(0)
5.2
(b)
Drift Current
figure 3.14 i The £ \'e ..
k diagralll Of lhc. conduclion and valence bands of a
semiconductor at (a) T '" 0 K and (b) T ,. 0 K.
Drift Current
Current is due to the net flow of charge.
If we had a collection of positively charged ions with a volume density N (cm−3 ) and an average drift velocity
vd (cm/s), then the drift current density would be
2
J = qN vd (A/cm )
(38)
If, instead of considering the average drift velocity, we considered the individual ion velocities. then we could
write the drift current density as
N
X
J =q
vi
(39)
i=1
43
Figure 3.131 (a) Two-dimensional representation of the breaking of .tcova lent bond.
(b) Corresponding Ii lie fcpreStllla'10n of (he enetgy band and the generarion of :a
negative and positi ve charge. wlth the brcflking of a covalent bond.
I
£
I
I
I
k
k
(0)
(b)
Figurefigure
49: The
E versus
the conduction
valence bands
a semiconductor
at a
(a) T = 0 K
3.14
i The k£ diagram
\'e .. kof diagralll
Of lhc.and
conduclion
andofvalence
bands of
and (b)
T
>
0
K.
semiconductor at (a) T '" 0 K and (b) T ,. 0 K.
where vi is the velocity of the ith in.
The summation in Equation (39) is taken over a unit volume so that the current density J is still in units of
A/cm2 .
Since electrons are charged particles, a net drift of electrons in the conduction band will give rise to a current.
The electron distribution in the conduction band, as shown in Figure 49b, is an even function of
k when no external force is applied.
Recall that k for a free electron is related to momentum so that, since there are as many electrons with
a +|k| value as there are with a −|k| value, the net drift current density due to these electrons is
zero.
This result is certainly expected since there is no externally applied force.
and the particle moves, it must gain energy.
If a force is applied to a particle
This effect is expressed as
dE = F dx = F vdt
(40)
where F is the applied force, dx is the differential distance the particle moves, v is the velocity, and dE is the
increase in energy.
If an external force is applied to the electrons in the conduction band, there are empty energy states into which
the electrons can move: therefore, because of the external force, electrons can gain energy and momentum.
The electron distribution in the conduction band may look like that shown in Figure 50, which implies that the
electrons have gained a net momentum. We may write the drift current density due to the motion of electrons
as
n
X
J = −e
(41)
i=1
where e is the magnitude of the electronic charge and n is the number of electrons per unit volume in the
conduction band.
Again, the summation is taken over a unit volume so the current density is A/cm2 .
We may note from Equation (41) that the current is directly related to the electron velocity;
that is, the current is related to how well the electron can move in the crystal.
44
3.2
8ectrical Ccoducti
E
k
Figure 3.1S I The nsymrnclric distribution
of eJeclrons in Ihe 1;.' versus k diagram
5.3 Electron Effective Mass
when :11\ eXlcmal force is applied.
Electron Effective Mass
Figure 50: The asymmetric distribution of electrons in the E versus k diagram when an external force is applied.
The movement of an electron in a lattice will in general, be different from that of an electron in free space.
In addition to an externally applied force, there are internal forces in the crystal due to positively charged ions
or protons and negatively charged electrons, which will influence the motion of electrons in the lattice.
can write
e SO theWecurrent
density is Alem'.
may
note from Equation
(3.35
F
= F We
+F =
ma
(42)
where F
, F and F are the total force, the externally applied forces and the internal forces, respectively,
is directly
related
the electron velOCity; that is, the current is relat
acting on
a particle in ato
crystal.
Since it is difficult to take into account all of the internal forces, we will write the equation
e electron
can move in the crystal.
F =m a
(43)
total
total
ext
ext
int
int
∗
ext
where the acceleration a is now directly related to the external force.
Electron Effective Mass
The parameter m∗ , called the effective mass, takes into account the particle mass and also takes
into account the effect of the internal forces. We can also relate the effective mass of an electron in a
crystal to the E versus k curves, such as was shown in Figure 49.
In a semiconductor’ material, we will be dealing with allowed energy bands that are almost empty of electrons
other
bands that arein
almost
of electrons.
the case ,ofbe
a freedifferent
electron whose
ovementandof
anenergy
elec[ron
a full
lattice
wilTo l!begin,
inconsider
genera1
from
E versus k curve was shown in Figure 43.
n in freeRecalling
space.
to an areexternally
applied
there ar
EquationIn
(34),addition
the energy and momentum
related by E = p /2m
= ~ k /2m, force,
where m is the
mass of the electron.
The momentum
wave number k are related
by p = ~k. If ions
we take the
of Equation
(34) negativel
with
in the crystal
dueandto
charged
orderivative
protons
and
respect to k, we obtain
dE
~ k of
~p
ns, which will influence the motion
=
= electrons in 111. laltice.
(44) We can
dk
m
m
2
2 2
2
Relating momentum to velocity, Equation (44) can be written as
1 dE
p
=
=v
~ dk
m
(45)
where v is the velocity of the particle.
and F;" arc the total force, the externall y applied f()ree. a
forces, respectively. acti ng on ad particle
in a crystal. The paramete
E
~
=
(46)
dk
m
ation and ttl is the rest mass of the panicle.
45
nce it is difficult to take into aCCOUnt all of the internal
we wil
F,,,,t.
The first derivative of E with respect to k is related to the velocity of the particle. If we now take
the second derivative of E with respect to k, we have
2
2
2
We may rewrite Equation (46) as
1 d2 E
1
=
~2 dk 2
m
(47)
The second derivative of E with respect to k is inversely proportional to the mass of the particle.
For the case of a free electron, the mass is a constant (nonrelativistic effect), so the second derivative function
is a constant.
We may also note from Figure 42 that d2 E/dk 2 is a positive quantity, which implies that the mass of the electron
is also a positive quantity. If we apply an electric field to the free electron and use Newton’s classical equation
of motion, we can write
F = ma = −eE
(48)
where a is the acceleration, E is the applied electric field, and e is the magnitude of the electronic charge.
Solving for the acceleration, we have
a=
−eE
m
(49)
The motion of the free electron is in the opposite direction to the applied electric field because of the negative
charge.
We may now apply the results to the electron in the bottom of an allowed energy band.
Consider the allowed energy band in Figure 51a.
The energy near the bottom of this energy band may be approximated by a parabola, just as that of a free
particle.
We may write
E − Ec = C1 k 2
3.2
8ectric Conduction in Solids
(50)
P:I.tClbolic
P;'Irabolic
,I
s.pproximation :
k =O
k_
k =O
(al
k_
(hi
(a) The conduccion
bandk in
reduccd
Ii sIX,ce.
andapproximation.
the parabolic(b) The valence
Figure Jiigure3.t61
51: (a) The conduction
band in reduced
space,
and the
parabolic
band inapproximnlion.
reduced k space,(b)
and
the
parabolic
approximation.
The \'illence billld in reduced k space, and the parabolic
approximation.
The energy Ec is the energy at the bottom of the band.
Since E > Ec , the parameter C1 is a positive quantity.
The energy E,. is the energy at the bollom of the band. Since E > Ec. the parameter
C, is a positive quantity.
2
E
Taking the second derivative of £d with
to k from Equation (3.44), we
= 2Crespect
(51)
1
dk 2
obtain
Taking the second derivative of E with respect to k from Equation (50), we obtain
We may put Equation 51 in the form
,
1 d2 E
2C1
= 2
~2 dk 2
~
(52)
(3A5)
Comparing Equation (52) with Equation (47), we may equate ~2 /2C1 to the mass of the particle.
We may put Equation (3 A5) in the fonn
I d'£ 46 2C,
Ii' dk' = h'
(3.46)
C HAP T. R 3
76
3.2.4
Introduction to the Quantum Theory of Solids
Concept of the Hole
In considering the two-dimensional repre.'>Cntation of the covalent bonding shown in
However, the curvature of
the curve
Figure 51 charged
will not,"empty
in general,
thecrcmcd
same as
the acurvature
of the
Figure
3. I3a,ina positively
state"bewas
when
valence electron
free)particle curve. We may write
was elevated into the
conduction
band. 1For T > 0 K. all valence electrons may gain
d2 E
2C1
1
=
=
= ∗a small amount of thermal energy, il may(53)
thennal energy; if a~2valence
electron
gains
hop
dk 2
~2
m
into the empty state. The movement of a valence eleclron into Ihe empty Slate is .quivwhere m∗ is called the effective mass.
atem 10 the movement of lhe posilively charged emply slate itself. Figure 3. I 7 shows Ihe
movement of valence electrons in the crystal alternalely filling olle empty Siale and en.The effective mass is a parameter
that
relates
theaquantum
mechanical
to the classical
force
cuing a new
empty
state.
motion equivalent
to results
a positive
moving
in equations.
the valence
In most instances, the electron
in crysl<ll
the bottom
of the
conduction
can be
thought
of as
classical
parband. The
now has
a s<->cond
equallyband
important
charge
carrier
thaIa can
give riS(l
to
ticle whose motion can be
modeledThis
by charge
Newtonian
mechanics,
thewill
internal
forces
quantum
a current.
carrier
is called aprovided
!wle and,that
as we
see, can
alsoand
be Ihoughl
of
mechanical properties areastaken
into account
the effective
a classical
particlethrough
whose mOlion
can bemass.
modeled using Newtonian mechanics.
If we apply an electric field The
to the
the bottom
the allowed
band,
may write
driftelectron
currentindensity
due to of
electrons
in theenergy
valence
band.wesuch
shownthein
acceleration as
Figure 3. 14b, can be writlen as
Since C1 > 0, we have that m∗ > 0 also.
Electron effective mass
−eE
m∗n
J=
L
- i!
(3.49)1
U;
;(liIIi.'dJ
(54)
where the summation extends over all filled staleS. Thi summation IS II1convenient
a=
since it extends over a nearly full valence band and takes into account a very large l
number of staces. We may rewri te Equation (3.49) inlhc form
where m∗n is the effective mass of the electron.
The effective mass m∗n of the electron near the bottom of the conduction band is constant.
J= - e
L
U;
+e
5.4
L
Vi
i (empey)
i(!o::tlJ
Concept of the Hole
If we consider a band that is tOlally full , all available states are occupied by eloc-.
Concept of the Hole troos. The individual electrons can be thought of as moving wich a ve]ociLy as given;
by Equationrepresentation
(3.39):
In considering the two-dimensional
of the covalent bonding shown in Figure 49a, a positively
charged ”empty state” was created when a valence electron was elevated into the conduction band.
(dE)
=
(3.39)
veE) if a valence electron gains a small amount
For T > 0 K, all valence electrons may gain thermal energy;
of
/I.
dk
thermal energy, it may hop into the empty state. The movement of a valence electron into the empty state is
equivalent to the movement
the positively
charged
empty
state
Theofband
is symmetric
in k and
each
stateitself
is occupied so Ihat, for every e1eclron with
a velocily
lvi, there
is a corresponding
with
a velocity
-l vi,
Since
band is
Figure 52 shows the movement
of valence
electrons
in the crystal eleclron
alternately
filling
one empty
state
andthe
creating
a new empty state, a motion
to a positive
charge moving
in the valence
band. be changed wilh an
full, equivalent
the dislribution
of eleclrons
wilh respect
10 k cannol
extemally
applied
force.charge
The nel
drift current
generdled
from a completely full
The crystal now has a second
equally
important
carrier
that candensily
give rise
to a current.
'i
Ii
\I
II
"
II
II
II
II
II
II
II
"
II
II
II
II
II
..
II
II
II
II
II
II
II
II
II
tI
II
II
II
II
It
?=?== ==r=?___ .?=?!==
=
=?=?-?=r==
==9=9=9=9====9=9=9=9==
II
(b)
(a)
Figure 3.17 IVisualization ofrhc
IlII1\'t menl
II
(e)
of a hole in a semiconductor.
Figure 52: Visualization of the movement of a hole in a semiconductor.
This charge carrier is called a hole and, as we will see, can also be thought of as a classical particle whose motion
can be modeled using Newtonian mechanics.
The drift current density due to electrons in the valence band, such as shown in Figure 49b, can be written as
X
J = −e
vi
(55)
i(filled)
where the summation extends over all filled states.
47
I
(D
v( E ) =
This summation
inconvenient
over a (3.52)
nearly fullis
valence
takes into account
a very
associated
with isthe
empty since
state.it extends
Equation
enti band
relyand
equi,'alent
to placing
a
large number of states.
positively
charged panicle in the empty states and assuming all other states in the band
We may rewrite Equation (55) in the form
.,re empty, or neutrally charged. This concept
isX
shown in Figure 3.l 8. Figure 3. l 8"
X
J = −e
vi + e
vi
shows the \'alcnee band with the conven
tional e1ectron·filled
states and empty (56)
states,
i(total)
i(empty)
while Figure 3.18b shows the new concept of positive charges occupying the original
If we consider a band that is totally full, all available states are occupied by electrons.
empty
states. This concept is consistent with the discussion of the positively charged
The individual electrons can be thought of as moving with a velocity as given by Equation (45):
"empty state" in the valence band as shown in Figure 3.17 .
1 dE
v(E) (3.52)
=
(57)
lbe Vi in the summation of Equation
~ dk is related to how well this positi vely
charged
particle moves in the semiconductor. Now consider an electron neal' the top of
The band is symmetric in k and each state is occupied so that, for every eleclron with a velocity |v|, there is a
electron
with shown
a velocity in
−|v|.
the corresponding
allowed energy
band
Figure 3. 16b. The energy near the top of the allowed
Since
the
band
is
full,
the
distribution
of
electronsby
witha respect
to k cannot
be we
changed
with
an externally
energy
band may again be approximated
parabola
so that
may
",'fite
applied force.
=
The net drift current density generated
from
completely- full
(£
- a E,)
C 1band
(k)2is zero or
X
−e
vi = 0
(3.53)
where the vi in the summation is the
(3.54)
(58)
The energy E,. is the energy at the top of the
energy band. Since £ < E. for electrons
total
in this band. then (he parameter G; mWil be a
quantity_
We can now write the drift current density from Equation (56) for an almost full band as
Taking the second derivati ve of E withXrespec t to k (rom Equation 0.53), we
J = +e
vi
(59)
obtain
empty
1 dE
~ dk
associated
with
the
empty
state
Equation
(59)
is
entirely
equivalent to placing a positively charged particle in
We
the empty states and assuming all other states in the band are empty, or neutrally charged.
v(E) =
may rearmnge this equation so that
This concept is shown in Figure 53.
I d'E
- 2C,
(3.55)
Ii'
Figure 53a shows the valence band with the conventional electron-filled states and empty states, while Figure 53b
shows the new concept of positive charges occupying the original empty states. The vi in the summation of
Ii' dk? =
E
I
I
k
°••
•
k
I
00
I
I
I
00'
I
(.J
(b)
Figure
53: 3.181
(a) Valence
band withbttnd
conventional
electron-filled
states
and empty states.
(b) Concept
of positive
Figure
(a) Valence
wi th conve
ntional
electron-filled
states
and empty
charges occupying the original empty states.
states. (b) Concept of positive charges OCC\lpyi1l.g the ()tiginal empty states.
Equation (59) is related to how well this positively charged particle moves in the semiconductor.
Now consider an electron near the top of the allowed energy band shown in Figure 51b.
The energy near the top of the allowed energy band may again be approximated by a parabola so that we may
write
(E − Ev ) = −C2 k 2
(60)
48
The energy Ev is the energy at the top of the energy band.
Since E < Ev for electrons in this band, then the parameter C2 must be a positove quantity.
Taking the second derivative of E with respect to k from Equation (60), we obtain
d2 E
= −2C2
dk 2
(61)
We may rearrange this equation so that
1 d2 E
−2C2
=
2
2
~ dk
~2
Comparing Equation (61) with Equation (47), we may write
1 d2 E
−2C2
1
=
= ∗
~2 dk 2
~2
m
(62)
(63)
where m∗ is again an effective mass.
We have argued that C2 is a positive quantity which now implies that m∗ is a negative quantity.
An electron moving near the top of an allowed energy band behaves as if it has a negative mass.
We must keep in mind that the effective mass parameter is used to relate quantum mechanics and classical
mechanics. ’
The attempt to relate these two theories leads to this strange result of a negative effective mass.
However, we must recall that solutions to Schrodinger’s wave equation also led to results that contradicted
classical mechanics.
The negative effective mass is another such example.
If we again consider an electron near the top of an allowed energy band and use Newton’s force equation for an
applied electric field, we will have
F = m∗ a = −eE
(64)
However, m∗ is now a negative quantity, so we may write
a=
−eE
+eE
=
∗
−|m |
|m∗ |
(65)
All electron moving near the top of an allowed energy band moves in the same direction as the applied electric
field.
The net motion of electrons in a nearly full band can be described by considering just the empty states, provided
that a positive electronic charge is associated with each state and that the negative of m∗ from Equation (63)
is associated with each state.
We now can model this band as having particles with a positive electronic charge and a positive
effective mass. The density of these particles in the valence band is the same as the density of
empty electronic energy states.
This new particle is the hole.
The hole, then, has a positive effective mass denoted by m∗p and a positive electronic charge, so
it will move in the same direction as an applied field.
5.5
Metals, Insulators, and Semiconductors
Metals, Insulators, and Semiconductors
Each crystal has its own energy-band structure.
We noted that the splitting of the energy states in silicon, for example, to form the valence and conduction
bands, was complex.
Complex band splitting occurs in other crystals, leading to large variations in band structures between various
solids to a wide range of electrical characteristics observed in these various materials. We can qualitatively
begin to understand some basic differences in electrical characteristics caused by variations in band structure
by considering some simplified energy bands.
There are several possible energy-band conditions to consider.
49
trons in theeonduction band and the vllienee band remains completely full. TIlere",e
very few thcnnaUy generated efoclrons and holes ill an insulator.
Figure 3.20a shows an energy band with relatively few eleelrons near the bottom
of the band. Now. if an electric field is applied. the electrons can gain energy. move to
Allowed
J\lIowt.t1
Cnt1gy
energy
band
(almost
band
(empty)
empty)
(a)
I")
i\l1owcd
Allowed
energy
bUild
(almost
band
(full)
full)
(b)
(b)
Conduction
- - - - - - - - - - Conduction
band
band
- - - - r - - - - --
(.mp'Y)
IE,
- -
(almost
empty)
Electrons
E, (
Empty
s lates
I
--!..}''''
............... . .,. ... Valencc
Valence
b.md
_ _ __ _ __ _-'-'
: , (full)
Ie)
Figure 3.19 I AllOwed energy blinds
Figure 3.20 I Allowed energy bands
showing
ft ll almost
empty
band,
(b)bandgap
an
an empty band.
(b) a(b) a completely
Figure 54: Allowed energy bands showing 'howing
(a) an(a) empty
band,
full(a)band,
and
(c)
the
oJmOSllulJ b.nd. and (e) Ihe bandgap
energy between the two allowed bands. completely full band. and (e) Ihe bandgap
energy belween the tWOaUo\\,,<ed bands.
energy between the 1wo allowoo bands.
Figure 54a shows an allowed energy band that is completely empty of electrons.
If an electric field is applied, there are no particles to move, so there will be no current.
another allowed energy band whose energy states are completely full of electrons.
Figure 54b shows
We argued in the previous section that a completely full energy band will also not give rise to a current.
A material that has energy bands either completely empty or completely full is an insulator.
The resistivity of an insulator is very large or, conversely, the conductivity of an insulator is very small. There
are essentially no charged panicles that can contribute to a drift current.
Figure 54c shows a simplified energy-band diagram of an insulator.
The bandgap energy Eg of an insulator is usually in the order of 3.5 to 6 eV or larger, so that
at room temperature, there are essentially no electrons in the conduction band and the valence
band remains completely full.
There are very few thermally generated electrons and holes in an insulator. Figure 55a shows an energy band
with relatively few electrons near the bottom of the band.
Now, if an electric field is applied, the electrons can gain energy, move to higher energy states, and move through
the crystal.
The net flow of charge is a current.
Figure 55b shows an allowed energy band that is almost full of electrons, which means that we can consider the
holes in this band.
If an electric field is applied, the holes can move and give rise to a current.
Figure 55c shows the simplified energy-band diagram for this case.
The bandgap energy may be on the order of 1 eV.
This energy-band diagram represents a semiconductor for T > 0 K.
The resistivity of a semiconductor can be controlled and varied over many orders of magnitude.
The characteristics of a metal include a very low resistivity.
The energy-band diagram for a metal may be in one of two forms.
Figure 56a shows the case of a partially full band in which there are many electrons available for
conduction, so that the material can exhibit a large electrical conductivity.
Figure 56b shows another possible energy-band diagram of a metal.
The band splitting into allowed and
forbidden energy bands is a complex phenomenon and Figure 56b shows a case in which the conduction and
valence ban s overlap at the equilibrium interatomic distance.
50
ord"rof 3.5 10 6cY or larger, so Ihat al room temperature. [here are esscnlially no electrons in theeonduction band and the vllienee band remains completely full. TIlere",e
very few thcnnaUy generated efoclrons and holes ill an insulator.
Figure 3.20a shows an energy band with relatively few eleelrons near the bottom
of the band. Now. if an electric field is applied. the electrons can gain energy. move to
Allowed
J\lIowt.t1
Cnt1gy
energy
band
(almost
band
(empty)
empty)
(a)
I")
i\l1owcd
Allowed
energy
bUild
(almost
band
(full)
full)
(b)
(b)
- - - - - - - - - - Conduction
Conduction
band
band
- - - - r - - - - --
(almost
empty)
(.mp'Y)
IE,
- -
Electrons
E, (
Empty
s lates
Valence
I
--!..}''''
............... . .,. ... Valencc
b.md
_ _ __ _ __ _-'-'
: , (full)
Ie)
Figure 3.20 I Allowed energy bands
Figure 3.19 I AllOwed energy blinds
almost empty
band, (b) band,
an
'howing (a) an empty
band. (b)
a
Figure 55: Allowed
energy
bands
showingshowing
(a) (a)
anft llalmost
empty
(b)an almost full band, and (e) the
completely full band. and (e) Ihe bandgap
oJmOSllulJ b.nd. and (e) Ihe bandgap
bandgap
energy
between
the
two
allowed
bands.
energyRbelween
tWOaUo\\,,<ed bands.to the QuanluO'l
energy betweenTheolY
the 1wo allowoo
bands.
C HAP T.
3 theIntroduction
of Solids
Partially
•
.. .. • • • .
• ,, _ band
••••• - •••••••••• • )
/
Full
band
Lower
band
\
_
•••• • ••• •••••••• .)
• ' • . - ,-
----------- -
Upper
b3nd
EicClfOtlS
(b)
(a)
Two
possible
bands
of. metal
showingfilled
(a) band
a panially
band allowed
FigureFigure
56: Two3.211
possible
energy
bandsenergy
of a metal
showing
(a) a partially
and (b)filled
overlapping
energyand
bands.
(b) overlapping allowed energy b.,ods.
I
higher
energy
and56a,
move
crystal.of'The
nct now
churge
is a numbers
current. of
As in
the case
shownstates.
in Figure
therethrough
are largethe
numbers
electrons
as of
well
as large
empty
energy
states
into
which
the
electrons
can
move,
so
this
material
can
also
exhibit
a very
Figure 3.20b shows an allowed energy band that is almost full of electrons. which
high electrical conductivity.
means that we can consider the holes in this band. If an electric field is applied. the
holes can mOve and give rise to a CUrrent. Figure 3.2Oc shows the simplified energy·
5.6 Extension to Three Dimensions
band diagram for this case. The bandgap energy may be on the order of I cV. This
energy·band
diagram
represents a semiconductor for T > 0 K. The resistivity of a
Extension
du Three
Dimensions
as inwe
will see
the next
Chapter.
can be contrOlled
andis varied
Onesemiconductor,
problem encountered
extending
theinpotential
function
to a three-dimensional
crystal
that theover
distance
between
atoms
varies
the direction through the crystal changes.
many
orders
ofasmagnitude.
Figure 57The
shows
a face-centered cubic
with thea [100]
directionsThe
indicated.
characteristics
of a structure
tTIctal include
veryand
low[110]
resistivity.
energy·band
di·
Electrons
in different
encounter
patterns
and therefore
different
k-space
agramtraveling
for a metal
maydirections
be in one
of two different
forms. potential
Figure 3.2
1a shows
the case
of a par·
boundaries.
tially full band in which there are many electrons avai lable for conduction, so that tltt
The E versus k diagrams are in general function of the k-space direction in a crystal. Figure 58
material
Can kexhibit
ciectrical
conductivity.
shows
an E versus
diagramaoflarge
gallium
arsenide and
of sillicon. Figure 3.21b shows another pos·
sible
energy·band
diagram
of aproperties
melal. The
band insplitting
allowed
forbidden
These
simplified
diagrams show
the basic
considered
this text,into
but do
not showand
many
of the details
moreenergy
appropriate
for is
advanced-level
bands
a complex courses.
phenomenon and Figure 3.21 b shows a case in which the
bands
overlap
librium
distance.
All in
Noteconduction
that in place and
of thevalence
usual positive
and
negativeatk the
axes,equi
we now
show interatomic
two different crystal
directions.
The the
E versus
k diagram
the one-dimensional
model
wasnumbers
symmetricof
in electrons
k so that no
case shown
in for
Figure
3.21 a, lhere are
large
as new
wellinformation
as large is
obtained
by
displaying
the
negative
axis.
numbers of empty energy states: into which the electrons can move, $0 thi s material
can also exhibit a very high electrical conductivity.
51
Extension to Three CXtnensions
11 1013 • 3
di rection
-0
,
,
'
,11
,
,
,
,
11 101
I
di rection
I
-0
,
,
'
,11
,
(100)
Idirection
,
,
,
I
Figure 3.22 1The (100) plane of a
face-ccntcrea cubic erystaJshowing
Ihe 11001nnd 111 01 directions.
(100)
3.3.1 The k-Space Diagrams of 5i and GaAs
direction
Figure 3.23 shows an E versus k diagram of gallium arsenide and of silkon. These
simplified diagrams show the basic propenies cullsidered in this text. but do not
show many of the details more app ropriate for advanced-level courses.
Note that in place of the usual positive and negative k axes, we now show two
Figure
1The (100)
plane
of a the [100] [110]model
Figuredifferent
57: The
(100)directions.
plane of a3.22
face-centered
crystal
directions.
crystal
The
E versus kcubic
diagram
for showing
the one-dimensional
face-ccntcrea cubic erystaJshowing
Ihe 11001nnd 111 01 directions.
4
4
GaAs
Conduction
halld
Si
band
3
3
3.3.1 The k-Space Diagrams of 5i and GaAs
2
Figure 3.23 shows2 an E versus k diagram of gallium
arsenide and of silkon. These
>
> show the basic propenies cullsidered in this
.... text. but do not
simplified diagrams
>.[b
e!'
show many of the
•c details more app ropriate Jl"for advanced-level courses.
w
I' we now show two
Note that in place
of the usual positive and
negative k axes,
0
0
different crystal directions. The E versus k diagram for the one-dimensional model
,
"
- I
- I
4
V:;i!cnc(!
Valence
band
-2
GaAs
-2
(111 1 Conduction
u
11001
k halld
1111]
Si
0
11001
band
k
(a)
3
4 ban<.l
3
(b)
Figure 3.23 1Energy band SlrUClUres of (a) G.As nnd (0) Si.
Figure 58: Energy bands structures of (a) GaAs and (b) Si.
(FromS,, /lI ).)
2
2
It is normal practice to plot the [100] direction along the normal +k axis and to plot the [111] portion of the
>
diagram so the +k points to the left.
>
....
.-
,
"I'
In the case of diamond or zincblende lattices, the maxima>in the valence band energy and minima in the
conduction band energy occur at k = 0 or along one of these e!'
two directions.
c 58a shows the E versus k diagram for GaAs.
Figure
[b
•
w
Jl"
The valence band maximum and the conduction band minimum both occur atk = 0.
0
0
The electrons in the conduction band tend to settle at the minimum conduction band energy which is at k = 0.
Similarly, holes in the valence band tend to congregate at the uppermost valence band energy.
In GaAs, the minimum conduction band energy and maximum valence band energy occur at the
- I
I
same k-value.
A semiconductor with this property is said to be a direct bandgap semiconductor;
-2
V:;i!cnc(!
Valence
band
(111 1
52
u
11001
-2
ban<.l
1111]
0
11001
transitions between the two allowed bands can take place with no change in crystal momentum.
This direct nature has significant effect on the optical properties of the material, GaAs and other
direct bandgap materials are ideally suited for use in semiconductor lasers and other optical
devices. The E versus k diagram for silicon is shown in Figure 58b.
The maximum in the valence band energy occurs at k = 0 as before.
The minimum in the conduction band energy occurs not at k = 0, but along the [100] direction.
The difference between the minimum of conduction band energy and the maximum valence band energy is
still defined as the bandgap energy Eg . A semiconductor whose maximum valence band energy and
minimum conduction band energy do not occur at the same k value is called an indirect bandgap
semiconductor.
When electrons make a transition between the conduction and valence bands, we must invoke
the law of conservation of momentum.
A transition in an indirect bandgap material must necessarily include an interaction with the
crystal so that crystal momentum is conserved.
Germanium is also an indirect bandgap material whose valence band maximum occurs at k = 0 and whose
conduction band minimum occurs along the [111] direction.
GaAs is a direct bandgap semiconductor but other compound semiconductors such as GaP and AlAs, have
indirect bandgaps.
The curvature of the E versus k diagrams near the minimum of the conduction band
energy is related to the effective mass of the electron.
We may note from Figure 58 that the curvature of the conduction band at its minimum value for GaAs is larger
than that of silicon, so the effective mass of an electron in the conduction band GaAs will be smaller
than that in silicon. The curvature of the E versus k diagram at the conduction band minimum may not
be the same in the three k directionq.
We will not consider the details of the various effective mass parameters here.
In later sections and chapters, the effective mass parameters used in calculations will be a kind of statistical
average that is adequate for most device calculations.
6
Density of States
Density of States
Since current is due to the flow of charge, an important step in the process is to determine the number of
electron, and holes in the semiconductor that will be available for conduction.
The number of carriers that can contribute to the conduction process is a function of the number of available
energy or quantum state, since, by the Pauli exclusion principle, only one electron can occupy a given quantum
state.
When we discussed the splitting of energy levels into bands of allowed and forbidden energies, we
indicated that the band of allowed energies was actually made up of discrete energy levels.
We must determine the density of these allowed energy states as a function of energy in order to calculate the
electron and hole concentrations.
6.1
Mathematical derivation
Mathematical derivation
Electrons are allowed to move relatively freely in the conduction band of a semiconducto, but are confined to
the crystal.
As a first step, we will consider a free electron confined to a three-dimensional infinite potential well, where the
potential well represents the crystal.
The potential of the infinite potential well is defined as
V (x, y, z) = 0 for 0 < x < a, 0 < y < a, 0 < z < a
(66)
V (x, y, z) = ∞ elsewhere
(67)
and
where the crystal is assumed to be a cube with length a.
53
Schrodinger’s wave equation in three dimensions can be solved using the separation of variables technique.
Extrapolating the result from the one-dimensional infinite potential well, we can show that
2
π
2mE
2
2
2
2
2
2
2
=
k
=
k
+
k
+
k
=
(n
+
n
+
n
)
x
y
z
x
y
z
~2
a2
(68)
where nx , ny and nz are positive integers.
Negative values of nx , ny and nz yield the same wave function, except for the sign, as the positive integer
values, resulting in the same probability function and energy, so the negative integers do not represent a
different quantum slate. We can schematically plot the allowed quantum states in k space.
Figure 59 shows a two-dimensional plot as a function of kx and ky .
Each point represents an allowed quantum state corresponding to various integral values of nx and ny .
CHAPTER 3
trcdlJction to lhe Ovamum Tt'.eOty ofSctds
Positive and negative values of kx , ky , or kz have the same energy and represent the same energy state.
t
-c"
k
Since
•
•••••••••
•••
• •
• ••
·• .• -.•(: .• ,
•
,
..••••.... .
k
•
k.• .. I
(a)
(b)
Figure 59:
(a) A 3.24
two-dimensional
array of allowed quantumof
states
in k space.
(b) The
positive
Figure
I (a) A two-dimensional
aUO\ved
<.)uantum
SHUtS
in one-eighlh of
the spherical k space.
k space. (b) The posi ti ve one.eighlh of 1he spheri cal k spate.
negativa values of kx , ky , or kz do not represent additional quantum states, the density of quantum states will
be determined by considering only the positive one-eighth of the spherical k space as shown in Figure 59b.
energy
state.between
Sincetwo
negmive
values
ofkk",
ky, Orfor
k:example,
do nOtisrepre
sent additional quan·
The distance
quantum states
in the
given by
x direction,
π
π π
tum states. the densit), of quantum
states will
be
determined
by considering only(69)
the
kx+1 − kx = (nx + 1)
− nx
=
a
a
a
positi ve one-eighth of the spherical k space as shown in Figure 3.24b.
Generalizing this result to three dimensions, the volume Vk of a single quantum state is
The distance between two quantum slates in the k.,( direction, for example, is
π 3
given by
Vk =
(70)
a
I}(::)
(:!.)
We can now determine the density of quantum states in k space. A differential volume in k space is shown in
Figure 59b and is given by 4πk 2 dk, so the differential density of quantum states in k space can be written as
(3.61)
kX+ 1 - k., = (II.,
= ::
- 2 II .•
a
a
1II 4πk dk
gT (k)dk = 2
(71)
π 3
8
a
Generalizing this resullto Ihree dimensions, the volume
V( of a single quantum state is
+
The first factor, 2, takes into account the two spin states allowed for each quantum state; the next factor,1/8,
takes into account that we are considering only the quantum states for positive values of kx , ky , and kz .
The factor 4πk 2 dk is again the differential volume and the factor (π/a)3 is the volume of the quantum (3.62)
state.
Equation 71 may be simplified to
π 2 kdk 3
gT (k)dk =
a
(72)
π 3 states in k space_ A dilicrential volWe can now delermine the density of quant um
ume in k spnce is shown in Figure 3.24b and
is given by 417 k2 dk. so the differential
54
density of quantum stales in k space can be written as
Equation 72 gives the density of quantum states as a function of momentum, through the parameter k.
We can now determine the density of quantum states as a function of energy E.
For a free electron, the parameters E and k are related by
2mE
~2
(73)
1√
2mE
~
(74)
k2 =
or
k=
The differential dk is
dk =
1
~
r
m
dE
2E
(75)
Then, substituting the expressions for k 2 and dk into Equation (72), the number of energy states between E
and E + dE is given by
r
πa3 2mE 1 m
gT (E)dE = 3
dE
(76)
π
~2
~ 2E
Since ~ = h/2π, Equation (76) becomes
gT (E)dE =
√
4πa3
(2m)3/2 EdE
3
h
(77)
Equation (77) gives the total number of quantum states between the energy E and E + dE in the crystal space
volume of a3 . If we divide by the volume a3 then we will obtain the density of quantum states per unit volume
of the crystal.
Equation (77) then becomes
Density of states per unit volume in a three-dimensional infinite potential well
4π(2m)3/2 √
g(E) =
E
h3
(78)
The density of quantum states is a function of energy E.
As the energy of this free electron becomes small, the number of available states decreases.
This density function is really a double density, in that the units are given in tems of states per unit energy per
unit volume.
6.2
Extension to Semiconductors
Extension to Semiconductors
In the last section we derived a general expression for the density of allowed electron quantum states using the
model of a free electron.
We can extend this same general model to a semiconductor to determine the density of quantum states in the
conduction band and the density of quantum states in the valence band.
Electrons and holes are confined within the semiconductor crystal so we will again use the basic model of the
infinite potential well. The parabolic relationship between energy and momentum of a free electron was given
in Equation (34) as E = p2 /2m = ~2 k 2 /2m.
Figure 51a showed the conduction energy band in the reduced k space.
The E versus k curve near k = 0, the bottom of the conduction band, can be approximated as a parabola, so
we may write
~2 k 2
(79)
E = Ec +
2m∗n
where Ec is the bottom edge of the conduction band and m∗n is the electron effective mass.
Equation (79) may be rewritten to give
E − Ec =
55
~2 k 2
2m∗n
(80)
The general form of the E versus k relation for an electron in the bottom of a conduction band is the same as
the free electron, except the mass is replaced by the effective mass.
We can then think of the electron in the bottom of the conduction band as being a ”free” electron with its own
particular mass. The right hand side of Equation (80) is of the same form as the right side of Equation (34),
which was used in the derivation of the density of states function.
Because of this similarity, which yields the ”free” conduction electron model, we may generalize the free electron
results of Equation (78) and write the density of allowed electronic energy states in the conduction band as
Electrons density of states per unit volume in the conduction band
4π(2m∗n )3/2 p
E − Ec
gc (E) =
h3
(81)
Equation (81) is valid for E ≥ Ec .
As the energy of the electron in the conduction band decreases, the number of available quantum states also
decreases.
The density of quantum states in the valence band can be obtained by using the same infinite potential well
model, since the hole is also confined in the semiconductor crystal and can be treated as a ”free” particle.
The effective mass of the hole is m∗p .
Figure 51b showed the valence energy band in the reduced k space. We may also approximate the E versus k
curve near k = 0 by a parabola for a ”free” hole, so that
E = Ev −
~2 k 2
2m∗p
(82)
~2 k 2
2m∗p
(83)
Equation Equation (82) may be rewritten to give
Ev − E =
Again, the right side of Equation (83) is of the same form used in the general derivation of the density of states
function.
We may then generalize the density of states function from Equation (78) to apply to the valence band, so that
Holes density of states per unit volume in the conduction band
4π(2m∗p )3/2 p
gv (E) =
Ev − E
h3
(84)
Equation (84) is valid for E ≤ Ev . We have argued that quantum states do not exist within the forbidden
energy band, so g(E) = 0 for Ev < E < Ec .
Figure 60 shows the plot of the density of quantum states as a function of energy.
If the electron and hole effective masses were equal, then the functions gc (E) and gv (E) would be symmetrical
about the energy midway between Ec and Ev or the midgap energy, Emidgap .
7
Statistical Mechanics
Statistical Mechanics
In dealing with large numbers of particles, we are interested only in the statistical behavior of the group as a
whole rather than in the behavior of each individual particle.
In a crystal, the electrical characteristics will be determined by the statistical behavior of a large number of
electrons.
56
CHAPTER 3
Introduction to the Quaflluln Theory 01Solids
I
8v(E}
Figure 3.25 IThe density of energy
Figure 60: The density of energy states in the conduction band and the density of energy states in the valence
states in the co nduction band and the
band as a function of energy.
denl'icy l)f energy Slilles in the \'ulence
J
band as a function of energy.
7.1
Statistical Laws
Statistical
Laws
TEST
YOUR UNDERSTANDING
In determining the statistical behavior of particles, we must consider the laws that the panicles obey.
E3.2 Dctennine (he tocal number of energy
in sili con between £( and E•. + k T at
There are three Tdistribution
distribution of particles among available energy states. One
= 300 K. laws determining
",01 x l (" lthe
'suV)
distribution law is the Maxwell-Boltzmann probability function.
F,J.3
Oct'cnninc th e IOU1) number of energy state s in silicon between £ ,. and E, - kT at
=
(,_'"O
In this case, the particles are considered to be distinguishable by being numbered, for example, from 1 to N
T
300 K.
,,01 x Z6'CSllV)
with no limit to the number of particles allowed in each energy state.
The behavior of gas particles in a container at fairly low pressure is an example of this distribution distribution.
A second distribution law is the Bose-Einstein function.
3.51in STATISTICAL
MECHANICS
The particles
this case are indistinguishable
and, again, there is no limit to the number of particles permitted
in each quantum state.
In deali ng with large numbers of particles, we are intcrested only in the Slmistical be· I
The behavior
photons,
or black
is aninexample
of this of
law.
The
third distribution
ha"iorofof
the group
as a body
wholeradiation,
rather than
the behavior
each
individual
particle, ilaw is the
Fermi-Dirac probability function.
For example, gas within a container will exert an average pressure on the walls of the
In this case, the particles are again indistinguishable, but now only one particle is permitted in each quantum
vessel. The pressure is actually due to the collisions of the individual gas molecules
state.
wi th thc walls, but we do not fo\low each individual molecule as it collides wi th the
wall. Likewise in a crystal. the electrical characteristics will be determined by the
Electrons in a crystal obey this law.
In each case,
thecal
particles
are of
assumed
be noninteracting.
stati sti
behavior
a largetonumber
of e1ectrons.
7.2
•
The
Function
3.5.1Fermi-Dirac
Statistical Probability
Laws
The Fermi-Dirac
Probability
Function
In detennining
the statistical
behavior of particles, we must consider the laws that lbe
Figure 61panicles
shows the
ith energy
with gdistribution
i quantum states.
obey.
There level
are three
1,l\v,
determining the distribution of par·
A maximum
ofamong
one particle
is allowed
in each
quantum state by the Pauli exclusion principle.
ti cles
availabJe
energy
Slates.
There are gi ways of choosing where to place the first particle, (gi − 1) ways of choosing where to place the
second particle, (gi − 2) ways of choosing where to place the third particle, and so on. Then the total number
of ways of arranging Ni , particles in the ith energy level (where Ni < gi ) is
(gi )(gi − 1) · · · (gi − (Ni − 1)) =
gi !
(gi − Ni )!
This expression includes all permutations of the Ni , particles among themselves.
57
(85)
ilhener
gy l· l·2 f 3 I
I<:"el I
------.
QU3.llIum
Figure 61: The ith energy level with g quantum states.
Figure 3.261
The. ilh tncrgy
wilh 8,
quantum
However,since
the particles arestates.
indistinguishable, the N ! number of permutations that the particles have among
i
i
themselves in any given arrangement do not count as separate arrangements.
The interchange of any two electrons, for example. does not produce a new arrangement.
Therefore, the actual number of independent ways of realizing a distribution of Ni particles in the ith level is
Wi =
gi !
Ni !(gi − Ni )!
(86)
Example 1. To determine the possible number of ways of realizing a particular distribution. Let gi = Ni = 10.
Then (gi − Ni )! = 1.
10!
gi !
=
=1
Ni !(gi − Ni )!
10!
Example 2. To determine the possible number of ways of realizing a particular distribution. Let gi = 10 and
Ni = 9.
gi !
10!
=
= 10
Ni !(gi − Ni )!
9!1!
Equation (86) gives the number of independent ways of realizing a distribution of Ni particles in the ith level.
The total number of ways of arranging (N1 , N2 , N3 , · · · Nn ) indistinguishable particles among n energy levels is
the product of all distributions or
n
Y
gi !
(87)
W =
Ni !(gi − Ni )!
i=1
The parameter
W is the total number of ways in which N electrons can be arranged in this system, where
Pn
N = i=1 Ni is the total number of electrons in the system.
We want to find the most probable distribution, which means that we want to find the maximum W .
The maximum W is found by varying Ni among the Ei levels, which varies the distribution, but at the same
time, we will keep the total number of particles and total energy constant. We may write the most probable
distribution function as
Fermi-Dirac distribution
N (E)
= fF (E) =
g(E)
1
E − EF
1 + exp
kT
(88)
where EF is called the Fermi energy.
The number density N (E) is the number of particles per unit volume per unit energy and the function g(E) is
the number of quantum states per unit volume per unit energy.
The function fF (E) is tailed the Fermi-Dirac distribution or probability function and gives the probability
that a quantum state at the energy E will be occupied by an electron.
Another interpretation of the distribution function is that fF (E) is the ratio of filled to total quantum
states at any energy E.
58
+00. The resulting Fermi- Dirac distr
IF(E> EF ) = O.
exp(+oo)
The Fenni-Dirac distribution function for T = Il
To begin to understand
the meaning
distribution
Fermithe
energy,electrons
we can plot the distriresult
showsof thethat.
forfunction
r =and0theK.
are in the
bution function versus energy.
Initially, let T = 0The
K and probability
consider the case when
.
ofE <aEquantum
state being occupied is u
The exponential term in Equation (88) becomes exp[(E − E )/kT ] → exp(−∞) = 0.
bility function
of a state
2ero
for
E when
> EF • A
The resulting distribution
is f (E <being
E ) = 1. occupied
Again let T = 0is
K and
consider
the case
E>E .
theinFc-mU
energy
T exp[(E
= 0− EK.)/kT ] → exp(+∞) = +∞.
The exponential term
the distribution
function at
becomes
The resulting Fermi- Dirac distribution function now becomes f (E > E ) = 0.
Figure 3.28 shows discrete energy fevers of a p
The Fermi-Dirac distribution function for T = 0 K is plotted in Figure 62.
This result shows number
that, for T = 0of
K, the
electrons
in their lowest possible
energy
avai
lableare quantum
states
atstates.
each energy. I
7.3
The Distribution function and the Fermi Energy
The Distribution function and the Fermi Energy
F
F
F
F
F
F
F
F
The probability of a quantum state being occupied is unity for E < EF and the probability of a
state being occupied is zero for E > EF .
All electrons have energies below the Fermi energy at T = 0 K.
--
G
::!'; LO f - - - - - - - - ,
rlllure 3.271 The Feooi probability
Figu
and q
funclion versusenergy for T = 0 K.
!:;yst¢
Figure 62: The Fermi probability function versus energy for T = 0 K.
Figure 63 shows discrete energy levels of a particular system as well as the number of available quantum states
at each energy.
If we assume, for this case, that the system contains 13 electrons, then Figure63 shows how these electrons are
distributed among the various quantum states at T = 0 K.
The electrons will be in the lowest possible energy state, so the probability of a quantum state being occupied
in energy levels E1 through E4 is unity, and the probability of a quantum state being occupied in energy level
E5 is zero.
The Fermi energy, for this case, must be above E4 but less than E5 .
The Fermi energy determines the statistical distribution of electrons and does not have to correspond to an
allowed energy level. Now consider a case in which the density of quantum states g(E) is a continuous function
of energy as shown in Figure 64.
If we have N0 electrons in this system, then the distribution of these electrons among the quantum states at
T = 0 K is shown by the dashed line.
The electrons are in the lowest possible energy state so that all states below EF are filled and all states above
EF are empty.
59
crete energy fevers of a parl'icular system
as.the
well
as (he
Consider
situation
when
um states at each
-,
'ft!
--"''--
'"
i,(E = E,..) =
I
I
= - I ,. exp (0)
I+
The probability of a state being occupied at £ = EF is
Figure 3.28 I Discre(c energy
for several
Fermi efor
nergy
is independent of temperature.
and quantum states
a particular
!:;yst¢m
at T ==system,
0 K.then the Fermi energy E can be determined. Consider
are known
for this particular
SHIIcs system
Fermi-Dirac
function
Figure 63: Discrete energy levels and
quantum states distribution
for a particular
at T =ploued
0K.
ability
= 0 K.
the temperature incrca
tron s gain
ccrtain
amount
energy. If we assume,
fora this
case.
thatof thermal energy so that so
higher energy levels. which means that the distributi on or
able energy states wi11 change. Flgu1e 3.30 shows the same
quantum state. as in Figure 3.28. The di stribution of ele
states has c hanged from the T = 0 K case. Two elec tro
gai ned enough energy to£.,
jump to Es . and one electron fro
--'''-'''-''=<--=-=-='-"" ""the""temperature
'ai "" ""changes, the distribution of electrons vers
The change in the electron distributi on among energy
seen by plotting the Fermi-Dirac distributi on functi on.lrw
then Equation (3.79) becomes
If g(E) and N0
F
g(E )
t
-
HE
lgure 3.29
1
quantum
and elccrrons
in a at T = 0 K.
Figure 64:JI'Density
of quantum
statesofand
electronsStales
in a continuous
energy system
continuous energy system at T
=
0 K.
Figure 3.30 I
quantum stale
shown in Figu
the situation when the temperature increases above T = 0 K.
Electrons gain a certain amount of thermal energy so that some electrons can jump to higher energy levels.
which means that the distribution of electrons among the available energy states will change.
Figure 65 shows the same discrete energy levels and quantum state as in Figure 63.
electrons among the quantum states has changed from the T = 0 K case.
The distribution of
Two electrons from the E4 level have gained enough energy to jump to E5 and on electron from E3 has jumped
to E4 .
As the temperature changes, the distribution of electrons versus energy changes. The change in the electron
distribution among energy levels for T > 0 K can be seen by plotting the Fermi-Dirac distribution function.
If we let E = EF , then Equation (88) becomes
fF (E = EF ) =
1
1
1
=
=
1 + exp(0)
1+1
2
The probability of a state being occupied at E = EF is 1/3.
60
(89)
)
in a
...
- - " ' ' - - 1-:,
Figure 3.30 I Discrete fnergy state.!) :tnd
quantum staleS for the snme system
shown in Figure 3.28 for T > 0 K.
Figure 65: Discrete energy states and quantum states for the same system show
in Figure
63 for TMechaniCS
> 0 K.
3.5
StatiStical
I
)'0
§
,
I
n
E
Figure
66: The
Fermi
probability
function versusfunction
energy forversu,
differentenergy
temperatures.
Figure
3.31
IThe
Fermi probability
foc differ!!llt temperarures.
Figure 66 shows the Fermi-Dirac distribution function plotted for several temperatures, assuming the Fermi
energy is independent of temperature
We We
can see
temperatures
above absoluteabove
zero there
is a nonzero
probability
energy probastates
canthat
seeforthat
for
absolute
Zero.
there that
is asome
nonz.ero
above EF will be occupied by electrons and some energy states below EF will be empty.
EF
bility
thai SOme energy slates above
will be occupied by eleclrOns and some
This result again means that some electrons have jumped to higher energy levels with increasing thermal energy.
energy
EFthatwill
be empty.
This
result
again
that some electrons
We canslates
sec frombelow
Figure 66
the probability
of an
energy
above E
F being occupied increases as the temperaturejumped
increases to
andhigher
the probability
of alevels
state below
being empty increases
the temperature increases.
have
energy
wi thEFincreasing
thermalas energy.
Example 3. To calculate the probability that an energy state above EF is occupied by an electron. Let T =
300 K. Determine the probability that an energy level 3kT above the Fermi energy is occupied by an electron.
fF (E) =
1
1
=
E − EF
3kT
1 + exp
1 + exp
kT 3Dove £, is lX'cupicd
kT
energy SI<tlC
Objective
To calculate (he prob"bility rhal 3n
by an eleclron.
which
Letbecomes
T = 300 K.l)etennine (he prubability
that an energy level 3kT above Ihe Fermi en1
fF (E) =
= 0.0474 = 4.74%
ergy is occupied by an elcc:tron .
1 + 20.09
• Solution
From Equ:lIion (3.79), we can write
61
Example 4. Assume that the Fermi energy level for a particular material is 6.25 eV. Calculate the temperature
at which there is a 1 percent probability that a state 0.30 eV below the Fermi energy level will not contain an
electron.
1 − fF (E) = 1 −
1
E − EF
1 + exp
kT
Then
0.01 = 1 −
1 + exp
1
5.95 − 6.25
kT
We find kT = 0.06529 eV, so that the temperature is T = 756 K.
We may note that the probability of a state a distance dE above EF being occupied is the same as the probability
of a state a distance dE below EF being empty.
The function fF (E) is symmetrical with the function 1 − fF (E) about Fermi energy, EF .
3 .5
This symmetry effect is shown in Figure 67.
Statistical Mect'anics
,!
£-
B,
Figure 3.32
of faFstate
being
Figure 67: The probability
of aIThe
state probability
being occupied,
(E), and
the occupied.
probability of a state being empty,
1 − fF (E).
/F(e). <Uld th e probabilit.y of a Slate being elUJHy, 1- ,/j<£) .
,,
We may neglect the 1 in the denominator, so the Femi-Dirac
function(uncliQlI
becomes
Fenni-Oirn.:
, distribution
1.01-----_...
Maxwell-Boltzmann
distribution function
,
−(E − E )
f (E) ≈ exp
'VkT BoltzmOlnll 3ppro)(.i matiun
Consider the case when E − EF kT , where the exponential term in the denominator of Equation (88) is
much greater than unity.
I
t
I
F
F
I
(90)
'.,
Equation (90) is known
as-the
Maxwell-Boltzmann
or simply the Boltzmann approximation, to
----- -- --- --approximation,
-the Fermi-Dirac distribution function.
Figure 68 shows the Fermi-Dirac probability function and the Boltzmann approximation.
This figure gives an indication of the range of energies over which the approximation is valid.
7.4
Planck’s law
Planck’s law
E
E,.
Figure 3.331 The Fenni- Dirat probability fllncfiUll and rhe
A black bodyMa.xwellis an idealized
physicalapproximation.
body that absorbs all incident electromagnetic radiation.
Boltzmann
Consider the case when £ - CF » kT62. where the exponential term in the denominator of Equation (3.79) is much greater Ihan Dnily. We may neglect the I in the
Figure 3.32 IThe probability of a state being occupied.
/F(e). <Uld th e probabilit.y of a Slate being elUJHy, 1- ,/j<£) .
,,
,
,
'V
I
t
1.01-----_...
Fenni-Oirn.: (uncliQlI
I
I
-- - --- -- -- --- -- --
E
BoltzmOlnll 3ppro)(.i matiun
'.,
E,.
Figure
3.331 The probability
Fenni- Dirat
probability
fllncfiUll and rhe approximation.
Figure 68:
The Fermi-Dirac
function
and the Maxwell-Boltzmann
Ma.xwell- Boltzmann approximation.
»
Because of this perfect absorptivity at all wavelengths, a black body is also the best possible emitter of thermal
Consider
case when
£ - CF
kT . wherecontinuous
the exponential
term
in the
de-body’s
radiation,
whichthe
it radiates
incandescently
in a characteristic,
spectrum that
depends
on the
temperature.
nominator
of Equation (3.79) is much greater Ihan Dnily. We may neglect the I in the
At Earth-ambient
temperatures
this emission
is in the infrared
region
of the electromagnetic spectrum and is
denominator,
so the
Femli-Dirdc
distribution
function
becomes
not visible. The object appears black, since it does not reflect or emit any visible light. The thermal radiation
from a black body is energy converted electrodynamically from the body’s pool of internal thermal energy at
any temperature greater than absolute zero.
- (I:." - £F) ]
(3.80)
kT
It is called blackbody radiation fF(E)
and has ""exp
a frequency
distribution
with a characteristic frequency of
[
maximum radiative power that shifts to higher frequencies with increasing temperature. As
the temperature increases past a few hundred degrees Celsius, black bodies start to emit visible wavelengths,
appearing red, orange, yellow, white, and blue with increasing temperature.
Equation (3.80) is known as the Maxwell- Boltzmann approximation. or simply the
When an object is visually white, it is emitting a substantial fraction as ultraviolet radiation.
Boltzmann appro.xim,uion. IO the Fcmli- Din}c dislribUlion function. Figure.3.33 shows
the Fermi- Dirac probability function and the Bolrzmann approximatioll. This figure
givcs an indication of the range of energies over which the approximation is valid.
Objective
To detennine Ihe energy at which the Doi17.mann approximation may be
valid.
Calculate the energy, in terms t')f kT
EF • at which the difference belwe<:n rhe
Bohlmann approximatioll .lIld 1he rermi-Dirac functiOil is 5 percent of the Penni function.
Figure 69: As the temperature decreases, the peak of the blackbody radiation curve moves to lower intensities
and longer wavelengths. The blackbody radiation graph is also compared with the classical model of Rayleigh
and Jeans.
63
£XAM
All matter emits electromagnetic radiation when it has a temperature above absolute zero. The radiation
represents a conversion of a body’s thermal energy into electromagnetic energy, and is therefore called thermal
radiation. It is a spontaneous process of radiative distribution of entropy.
Conversely all matter absorbs electromagnetic radiation to some degree.
An object that absorbs all radiation falling on it, at all wavelengths, is called a black body.
When a black body is at a uniform temperature, its emission has a characteristic frequency distribution that
depends on the temperature.
Its emission is called blackbody radiation.
The concept of the black body is an idealization, as perfect black bodies do not exist in nature.
Experimentally, blackbody radiation may be established best as the ultimately stable steady state equilibrium
radiation in a cavity in a rigid body, at a uniform temperature, that is entirely opaque and is only partly
reflective.
A closed box of graphite walls at a constant temperature with a small hole on one side produces a good
approximation to ideal blackbody radiation emanating from the opening.
In the laboratory, blackbody radiation is approximated by the radiation from a small hole in a large cavity, a
hohlraum, in an entirely opaque body that is only partly reflective, that is maintained at a constant temperature.
Any light entering the hole would have to reflect off the walls of the cavity multiple times before it escaped,
in which process it is nearly certain to be absorbed. Absorption occurs regardless of the wavelength of the
radiation entering (as long as it is small compared to the hole).
The hole, then, is a close approximation of a theoretical black body and, if the cavity is heated, the spectrum
of the hole’s radiation (i.e., the amount of light emitted from the hole at each wavelength) will be continuous,
and will depend only on the opacity and partial reflectivity of the walls, but not on the particular material of
which they are built nor on the material in the cavity (compare with emission spectrum).
Calculating the blackbody curve was a major challenge in theoretical physics during the late nineteenth century.
The problem was solved in 1901 by Max Planck in the formalism now known as Planck’s law of blackbody
radiation.
He found a mathematical expression fitting the experimental data satisfactorily.
Planck had to assume that the energy of the oscillators in the cavity was quantized, i.e., it existed in integer
multiples of some quantity.
Einstein built on this idea and proposed the quantization of electromagnetic radiation itself in 1905 to explain
the photoelectric effect.
These quanta were called photons and the blackbody cavity was thought of as containing a gas of photons.
In addition, it led to the development of quantum probability distributions, called Fermi-Dirac statistics and
Bose-Einstein statistics, each applicable to a different class of particles, fermions and bosons.
Planck’s law states that
Planck’s law
I(ν, T ) =
2hν 3
c2
exp
1
hν
−1
kT
(91)
where I(ν, T ) is the energy per unit time (or the power) radiated per unit area of emitting surface in the normal
direction per unit solid angle per unit frequency by a black body at temperature T Wien’s displacement law
shows how the spectrum of black body radiation at any temperature is related to the spectrum at any other
temperature. If we know the shape of the spectrum at one temperature, we can calculate the shape at any
other temperature.
A consequence of Wien’s displacement law is that the wavelength at which the intensity of the radiation produced
by a black body is at a maximum, λmax , it is a function only of the temperature
λmax =
64
b
T
(92)
where the constant, b, known as Wien’s displacement constant, is equal to 2.898 × 10−3 K m. The StefanBoltzmann law states that the power emitted per unit area of the surface of a black body is directly proportional
to the fourth power of its absolute temperature:
j ∗ = σT 4
(93)
where j ∗ is the total power radiated per unit area, T is the absolute temperature and σ = 5.67×10−8 Wm−2 K−4
is the Stefan-Boltzmann constant.
65