High Speed Semiconductor Physics. Theoretical Approaches and Device Physics
©2015
Textbook
391 Pages
Summary
Solid state physics is a fascinating sub-genre of condensed matter physics - though some graduate students consider it a very boring and tedious subject area in Physics and others even call it a “squalid state”. Topics covered in this book are built on standard solid state physics references available in most online libraries or in other books on solid state physics. The complexity of high speed semiconductor physics and related devices arose from condensed solid state matter. The content covered in this book gives a deep coverage on some topics or sections that may be covered only superficially in other literature. Therefore, these topics are likely to differ a great deal from what is deemed important elsewhere in other books or available literature. There are many extremely good books on solid-state physics and condensed matter physics but very few of these books are restricted to high speed semiconductor physic though. Chapter one covers the general semiconductor qualities that make high speed semiconductor devices effect and includes the theory of crystals, diffusion and ist mechanisms, while chapter two covers solid state materials, material processing for high speed semiconductor devices and an introduction to quantum theory for materials in relation to density of states of the radiation for a black body and ist radiation properties. Chapter three discuss high speed semiconductor energy band theory, energy bands in general solid semiconductor materials, the Debye model, the Einstein model the Debye model and semiconductor transport carriers in 3D semiconductors while chapter four discuss effect of external force on current flow based on the concept of holes valence band, and lattice scattering in high speed devices. Chapter five briefly describes solid state thermoelectric fundamentals, thermoelectric material and thermoelectric theory of solids in lattice and phonons while chapter six scattering in high field effect in semiconductors in inter-valley electron scattering and the associated Fermi Dirac statistics and Maxwell-Boltzmann approximation on their carrier concentration variation with energy in extrinsic doping chapter seven covers p-n junction diodes, varactor diode, pin diode Schottky diode and their transient response of diode in multi-valley semiconductors. Chapter eight discusses high speed metal semiconductor field effect transistors.
Excerpt
Table Of Contents
Topics covered in this book are built in standard solid state physics references available in
most online libraries or in other books. Studies on the complexity high speed semiconductor
physics arises from condensed solid state matter. We believe that the content covered in this
book gives a deep coverage in some topics or sections and gloss coverage on others. These
topics are likely to differ a great deal from what is deemed important elsewhere in other books
or literature. We know that there are many extremely good books on solid-state physics and
condensed matter physics but very few books are restricted to high speed semiconductor
physics though there are many good resources online.
Chapter one covers the general semiconductor qualities that make high speed semiconductor
devices effect and includes the theory of crystals, diffusion and its mechanisms, while chapter
two covers solid state materials, material processing for high speed semiconductor devices
and an introduction to quantum theory for materials in relation to density of states of the
radiation for a black body and its radiation properties. Chapter three discuss high speed
semiconductor energy band theory, energy bands in general solid semiconductor materials,
the Debye model, the Einstein model the Debye model and semiconductor transport carriers
in 3D semiconductors while chapter four discuss effect of external force on current flow
based on the concept of holes valence band, and lattice scattering in high speed devices.
Chapter five briefly describes solid state thermoelectric fundamentals, thermoelectric material
and thermoelectric theory of solids in lattice and phonons while chapter six scattering in high
field effect in semiconductors in inter-valley electron scattering and the associated Fermi
Dirac statistics and Maxwell-Boltzmann approximation on their carrier concentration varia-
tion with energy in extrinsic doping chapter seven covers p-n junction diodes, varactor diode,
pin diode Schottky diode and their transient response of diode in multi-valley semiconductors.
Chapter eight discusses high speed metal semiconductor field effect transistors.
(Author)
Dr. Cliff Orori Mosiori
Department of Physics
Kenyatta University,
P. O. Box 43844 00100
Nairobi
Edited by:
Dr. Walter Kamande Njoroge
Department of Physics
Kenyatta University,
P. O. Box 43844 00100
Nairobi
DEDICATION
I wholeheartedly and most sincerely dedicate this book to my loving wife Alice Mwango
Nyamwange, the firstborn daughter to Mzee Johnson Nyamwange Nyakenyanya who
without fail gave me constant encouragement and the much love I desperately needed as I was
writing this book. I specially dedicate this book to all my loving children; James Ogoti Orori,
(now in Form two at Michinda High School Elburgon), John Okongo Orori (a standard
seven pupil at Workers Primary School Nakuru), Nehemiah Mienya Orori (in standard two
at Workers Primary school) and Joyce Kerubo Orori (at Mary Hill Academy- Nakuru), a
daughter in whom I see the love for my late mother replicated. A special dedication to my
loving father, Mr. Mosiori Maigo Ontiri and my late mother, Mrs. Ebisibah Nyaboke
Mosiori, without forgetting my brothers; James Twara Mosiori, Josphat Ombwera Mosiori,
Cedric Nyanguka Mosiori, Elikana Moenga Mosiori, Robert Omosa Mosiori, Peter Maigo
Mosiori and sisters; Jane Nyanchama Mosiori, Everlin Mosiori and Mary Nyamunsi Mosiori,
who gave me a lot of support. To my brother Mr. Cedric Nyanguka Mosiori who inspired
me without knowing it during his High School days at Kakamega Technical School in the mid
of 1985 when he made several attempts to publish a book in the form of a Novel but never
succeeded because he could not afford the cost of publishing a book in Kenya. I salute you all
my Teachers from high school (Kisii High School), University lecturers, Kenyatta University
and Academic Advisors, Dr. Walter Kamande Njoroge, Chairman, Department of Physics and
Prof John Okumu, DVC- Academic, Kenyatta University, Kenya. I will not forget Kenyatta
University that shaped my life. May God bless the support I received from Rift Valley
Institute of Science and Technology lectures who among them; Mr. David Maru, Mr. Sangi-
riaki, Mr. Chege, Mrs. Mary Maina, Mr. Mutai, the Rotich's and the chief workshop techni-
cian, Mr. Brian and all the others. To KITI SDA church that shaped my spiritual and moral
life, I dedicate this book to you too. I dedicate it to you all that know me.
My parents
Mr. Mosiori Maigo Ontiri Mrs. Ebisibah Nyaboke Mosiori
They gave me the only thing they did not have; Education
My Academic Advisers
Dr. Walter Kamande Njoroge Prof John Okumu - Kenyatta University
They gave me a Golden heart to work, learn and research in the Scientific World.
Alice Mwango Nyamwange
She is my brilliant and beautiful wife,
Without whom I would have been nothing.
She gave me Three Handsome Sons;
Eldest Ogoti, Okon'go and Mienya,
And a beautiful daughter, Joyce Kerubo
In place of My Mother and Her Mother too
Who she named after her humble but late mother
I admire her Brilliance, Beauty and above all, her Confidence
She always Comforts, Consoles and Encourages me.
Cares for me and NEVER Complains nor Interferes with my work,
She Asks Nothing, Endures all for ME to make me feel Full
And writes my Heart with Dedications, Encouragements
And lots of Love Songs but above all
Without Erasing its Loving Long lasting Effects
Oh Dear Alice... !!!
From the bottom of my Ears and Heart,
I love YOU
Waooh !!!!...!!!
TABLE OF CONTENTS
ABOUT THE AUTHOR ... 5
PREFACE ... 7
DEDICATION ... 11
CHAPTER ONE
CRYSTALS IN SEMICONDUCTOR MATERIALS ... 17
Crystals ... 17
Strength of Crystals ... 19
Plastic Behaviour ... 20
Shear Strength Crystals ... 21
Dislocation ... 23
Etching ... 29
Diffusion ... 35
Colour centers ... 46
CHAPTER TWO
THEORIES FOR MATERIALS PROCESSING ... 69
Material Technologies ... 69
Material processing ... 70
Quantum Theory ... 76
Black body ... 81
Bohr's Atomic Theory ... 87
Schrödinger Equation ... 92
CHAPTER THREE
ENERGY BAND THEORY ... 107
Energy Bands Theory in Solids ... 107
Bond and Structures ... 108
Energy band structure Theory ... 109
The pn junction ... 119
Semiconductors Models for Heat Capacity ... 123
The Debye Model ... 123
The Einstein Model ... 124
The Debye Model ... 125
Semiconductor transport carriers ... 131
3D semiconductors Crystal structures ... 134
CHAPTER FOUR
BRILLOUN ZONES ... 137
Bloch Theorem ... 139
Eigen Value Equation ... 143
Bloch parameter, k ... 144
Effect of External Force ... 145
Current Flow in Crystals ... 148
Concept of Holes ... 149
Valence Band ... 150
Conduction Band ... 151
Mobility ... 152
Lattice Vibrations ... 152
Actual vibration ... 158
Lattice Scattering for Mobility ... 160
CHAPTER FIVE
SOLID STATE THERMOELECTRIC FUNDAMENTALS ... 167
Thermoelectric Materials ... 167
The Absolute Scale or Thermodynamic Kelvin, Temperature Scale ... 167
Temperature-Dependent Effects ... 172
Thermal Equilibrium and the Zeroth Law of Thermodynamics ... 174
Thermodynamics Systems and Processes ... 177
Thermodynamic Equilibrium of state variables ... 183
Thermodynamic States and State Variables ... 185
Thermodynamic Processes for Pure Substances ... 189
Electrical Equilibrium ... 194
The thermal equation of state for a solid ... 201
The Equation of State for an Ideal Gas ... 201
The Equation of State for a Van Der Waal gas ... 206
Equations of State for Other Systems ... 208
System Parameters and Partial Derivatives ... 209
The Partial Derivatives of State Variables ... 211
Thermodynamics ... 215
The Thompson Coefficient ... 216
Thermoelectric Theory of Solids ... 217
Lattice and Phonons ... 217
Phonon Dispersion ... 219
CHAPTER SIX
CARRIER SCATTERING ... 221
Mobility ... 222
Field effect due to Scattering ... 223
Electron Scattering ... 224
Carrie Density ... 225
Effect of Fermi Dirac statistics in scattering ... 230
Distribution of Electron due to scattering ... 231
Maxwell-Boltzmann Approximation in carrier scattering ... 233
Extrinsic Doping and Carrier Scattering ... 236
Generation Recombination process ... 241
CHAPTER SEVEN
SOLID STATE CONTINUITY EQUATIONS ... 247
Einstein and Continuity Equations ... 247
Semiconductor Diodes ... 252
P-N junction Diodes ... 252
Varactor Diodes ... 270
The P-I-N Diodes ... 275
Schottky diodes ... 277
Multi-Valley semiconductor Diode Theory ... 284
Diode Diffusion Theory ... 284
CHAPTER EIGHT
HIGH SPEED FIELD EFFECT TRANSISTORS ... 287
FET operation ... 287
Mobility in FETs ... 292
Hetero-junction in FETs ... 295
FET Current transport ... 299
BJT Models ... 301
The BJT Current Model ... 303
Hetero-junction Bipolar Transistor (HBT) ... 304
Hetero-junction FET ... 311
CHAPTER NINE
SWITCHING IN HIGH SPEED BIPOLAR JUNCTIONS ... 321
Semiconductors and junctions ... 321
Drift currents in semiconductors ... 322
Metal-semiconductor junctions ... 327
Semiconductor-semiconductor junctions ... 329
p-n diode in high speed semiconductor devices ... 330
Band diagram of unbiased high speed junction ... 331
Long p-n diodes ... 336
Minority carrier variations ... 340
Switching delays in high speed pn-diodes ... 348
CHAPTER TEN
HIGH SPEED SWITCHING IN BIPOLAR TRANSISTORS ... 359
BJTs Principle of operation ... 359
Currents in a BJT ... 365
High Speed Switching of the BJT ... 373
Switching cycle ... 373
BJT Schottky diode clamp ... 382
Equivalent BJT circuit ... 382
APPENDIX ... 385
17
CHAPTER ONE
CRYSTALS IN SEMICONDUCTOR MATERIALS
Crystals
In ideal crystals atoms were arranged in a regular way but real crystals differ from that of
ideal ones. Real crystals always have certain defects or imperfections and the arrangement of
atoms in any volume of the crystal is imperfectly irregular. Real crystals always contain
defects in abundance because of the uncontrolled conditions under which they are formed.
These defects also affect the colour of a crystal making them their crystals valuable as gems.
Even crystal prepared in advanced laboratories also contains defects. The characteristics of
behavior of a material depend upon the material, type of defect and other properties. Proper-
ties like density and elastic constants are mostly proportional to the concentration of defects.
Small defect concentrations will have a very small effect. There are other properties like the
colour of an insulating crystal or the conductivity of a semiconductor crystal that are more
sensitive to the presence of small number of defects.
The term defect carries with it the connotation of undesirable qualities. Defects are responsi-
ble for many of the important properties of materials. Material science involves the study and
engineering of defects so that solids will have desired properties for specific applications.
Therefore defect free crystals are very few. An ideal silicon crystal is of little use in modern
electronics. The use of silicon in electronic devices is dependent upon the introduced small
concentrations of chemical impurities such as phosphorus and arsenic which give it desired
properties. Some properties of materials such as stiffness, mechanical strength, ductility,
density and electrical conductivity which are termed as structure-insensitive in solid state
physics because they are not affected by the presence of defects in crystals as shown in the
figure below.
18
Properties like hysteresis and dielectric strength condition in semiconductors are termed as
structure sensitive. They greatly affect minor changes in crystal structure and this are intro-
duced due to defects or imperfections. Therefore, crystalline defects are classified the based
on their geometry as follows:
(i) Point imperfections
(ii) Line imperfections
(iii) Surface and grain boundary imperfections
(iv) Volume imperfections
At dimensional level, the point defects are close to those found in interatomic space. Linear
defects have a length of several orders of magnitude greater than their width while surface
defects have a small depth with their width and length being several orders larger. Finally the
volume defects which are regarded as pores and cracks have substantial dimensions in all
measurements. All these defects contribute to the change of solid state properties of materials
used in high speed electronic devices. It is traditional to make a distinction between the
properties of the lattice and the properties of the electrons. The term lattice refers to the
positions of the atoms themselves. Atoms are not stationary but they are considered to move
slightly compared to the distances between the atoms. They vibrate about their average
position but they do not move throughout the crystal. Most of the electrons are considered to
be localized and thus they are always associated with the same particular atom. Localized
electrons do not carry any current even when a force is applied. This makes them to be
considered as part of the lattice. Although some of the electrons are essentially free to move
throughout the solid, these free electrons determine the ability of a material to carry an
electrical current.
19
Silicon and Germanium forms the diamond crystal structure and all the atoms are identical
while the PbTe and PbSe occur in the sodium chloride crystal structure. It is convenient to
speak of the properties of the lattice and the properties of the electrons, but this division of
properties is somewhat artificial. Sometimes it is important to remember that the lattice
influences the behavior of the electrons and the electrons influence the behavior of the lattice.
At the minimum, it is important to treat both systems on an equal footing for a reliable
description of thermoelectric effects.
Strength of Crystals
Mechanical strength of a crystal depends on the amount of dislocations available. Plastic
deformation in metals is caused by the formation of slip bands in which one portion of the
material is sheared with respect to the other. Since metals are crystalline it is evident slip
represent the shearing of one portion of a crystal with respect to the other upon a rational
crystal plane. Some of the deformation operations correspond to slip while others correspond
to a dislocation. The configuration below in; (a) shows the cylinder as originally cut (b) and
(c) correspond to edge dislocations while (d) corresponds to screw dislocation.
20
The intensity of X-ray beams reflected from actual crystals is about 20 times greater than that
expected from a perfect crystal and therefore in a perfect crystal intensity is very low because
of long absorption paths encountered by multiple internal reflections. This causes the width of
the reflected beam from an actual crystal to be about longer than that obtained from a perfect
crystal. This is because the actual crystal is small, roughly equiaxed crystallites and about 10
-4
to 10
-5
cm in diameter. This causes a slight disorientation with respect to each other resulting
into the boundaries amorphous material. The disorientation explains the width of the beam.
This is the Mosaic Block theory in which the size of the crystallites limits the absorption path
and increases the intensity. These boundaries are actually arrays of dislocation lines.
A crystal can be deformed by simply applying stresses on it. If this stress is very large in the
order of about 10
6
-10
7
dynes per cm
2
then a small deformation occurs and undergoes plastic
deformation and this requires postulating a new type of defect called dislocations. Poorly
prepared crystal are infested with dislocations and defects which interfere with each other's
motion and as the crystals are purified and improved, dislocation largely move out of the
crystal creating vacancies and interstitials. At low thermal equilibrium concentrations and the
unimpeded motion makes it possible for the crystal to deform and the crystal is very soft.
Plastic Behaviour
Plastic deformation occurs when a crystal slides over another crystal with respect to the other
and result in a slight increase in the length of the crystal ABCD under the effect of a tension
FF applied to it as shown in figure below.
21
The process of sliding is called slip and the direction and place in which the sliding takes
place are called the slip direction and slip plane respectively. The outer surface of the single
crystal is deformed and a slip band is formed which can be observed by means of an optical
microscope.
Shear Strength Crystals
The model given in the figure below which shows a cross-section through two adjacent
atomic planes separated by a distance d is used to calculate the theoretical shear strength of a
perfect crystal. In the figure, the full line circles indicate equilibrium positions of the atoms
without any external force and indicate shear stress in the direction shown. Consider a case
where all the atoms in the upper plane are displaced by an amount x from the original posi-
tions as shown by the dotted circles. Using the figure above, the shear stress
is plotted as a
function relative displacement of the planes from their equilibrium positions. The periodic
behavior of
is found to become zero for x = 0, a/2, a, ... etc. where a is the distance between
the atoms in the direction of the shear. It is assumed that this periodic function is given by;
a
x
c
2
sin
=
where the amplitude
c
is the critical shear stress. To calculate such that for x a, we have;
a
x
c
2
=
22
From the definition of shear modulus the force required to shear the two planes of atoms, we
have;
a
x
d
x
y
Strain
Stress
G
c
2
/
=
=
=
From the above formula G is the shear module such that
G
d
x
y
=
=
is the elastic strain to give;
¸
¹
·
¨
©
§
=
d
x
G
When the two equations are compared, we have;
¸
¹
·
¨
©
§
¸
¹
·
¨
©
§
=
¸
¹
·
¨
©
§
b
a
x
G
or
d
x
G
a
x
c
c
.
2
2
or
d
a
if
G
x
G
c
,
6
2
Based on this expression, the maximum critical stress above which the crystal becomes
unstable is about one sixth of the shear modulus. In a cubic crystal, G c
44
= 10
11
dynes per cm
for a shear in the 100 direction.
23
Dislocation
The Edge Dislocation
A dislocation is a region of a crystal in which the atoms are not arranged in the perfect crystal
lattice. There are two extreme types of dislocations which are the edge type and the screw
type. Any type of dislocation is a more complicated defect than any of the point defects. It is
also noted that any particular dislocation is a mixture of these two types. Edge dislocation is
the simplest as shown in figure below.
or
The part of the crystal above the slip plane at ABC has one more plane of atoms DB than the
part below it. The line normal to the paper at B is called the dislocation line and the symbol at
B is used to indicate the dislocation. All dislocations have intense near the dislocation line
where the atoms do not have the correct number of neighbours and this region is called the
core of dislocation. A few atom distances away from the center, the distortion is very small
and the crystal is almost perfect locally. Dislocations are produced when the crystal solidifies
from the melt. Plastic deformation of cold crystals also produces dislocations. Dislocations
are of importance in determining the strength of ductile metals. These dislocations can be
experimentally observed by many techniques. Electron microscopes can be used to study
dislocations in their specimens of the order of a few angstroms which may transmit 100 kV
electrons.
The Screw Dislocation
This type of dislocation called Burger's dislocation and was introduced by Burger in 1939.
Consider a sharp cut made through a perfect crystal and the crystal on one side of the cut be
moved down by one atomic spacing relative to the other as shown below.
24
A line BD of distortion exists along the edge of the cut called the screw dislocation. The pitch
of the screw may be left-handed or right-handed and one or more atom distances per rotation.
The distortion is very little in regions away from the screw dislocation of while atoms near the
center are in regions of high distortion so much so that the local symmetry in the crystal is
completely destroyed.
Motion of a Dislocation
Many dislocations move just like the point defects lattices. However they are more con-
strained in motion because a dislocation must always be a continuous line. Dislocation move
by a climbing or by a slipping or by a gliding as illustrated in the figure below. When the
upper half is pushed sideways by an amount b, as shown then under shear motion tends to
move the upper surface of the specimen to the right. Edge dislocations for which the extra half
plane DB lies above the slip plane are called positive and if it is below the slip plane it is
called negative edge dislocation. When the extra half plane BD reaches the right hand side of
the block, the upper half of the block has completed the slip or glide by an amount b. Climb
of a dislocation corresponds to its motion up or down from the slip plane. If the dislocation
absorbs additional atoms from the crystal, it moves downward by substituting these atoms
below B in the lattice. If the dislocation absorbs vacancies it moves up as the atoms are
removed one by one from above B from the lattice sites.
25
Dislocation Stress field
The Burger's Vector
The Burger's vector b above denotes the actually dislocation-displacement vector. Thus a
dislocation can be described by a closed loop surrounding the dislocation line and this loop is
called the Burger's circuit. Burger's circuit is formed by the undisturbed region surrounding a
dislocation in steps. These steps are integral multiples of a lattice translation. They are
theoretically completed by going an equal number of translations in a positive sense and
negative sense in a plane normal to the dislocation line. Such a loop must close upon itself by
an amount called a Burger's vector given by;
s = n
a
a + n
b
b + n
c
c
where n
a,
n
b,
n
c
are equal to integers or zero and a, b, c are the three primitive lattice translations.
The Burger's circuit demonstrated by S-1-2-3-4-F in the figure above show a screw disloca-
tion. Starting at some lattice point S, the loop fails to close on itself by one unit translation
parallel to the dislocation line and this is the Burger's vector pointing in a direction parallel to
the screw dislocation.
26
If the loop is continued, it will describe a spiral path around the Burger's dislocation just like
the thread of a screw. Thus, b is a vector giving both the magnitude and the vector of the
dislocation and must be in multiple of the lattice spacing so that an extra plane of atoms could
be inserted to produce a dislocation. The dilatation ¨ at a point near an edge dislocation can
now be described by;
sin
r
b
V
V =
=
where b is the Burger's vector.
This Burger's vector measures the strength of the distortion caused by the dislocation, in
which r is the radial distance from the point to the dislocation line and
is the angle between
the radius vector and the slip plane. The atoms on a sheared lattice in screw dislocation being
are displaced from their original positions according to the equation of a spiral ramp given by;
2
b
u
z
=
where the z-axis lies along the dislocation and u, is the displacement in that direction,
is
measured from one axis perpendicular to the dislocation. Thus when
increases by
2
the
splacement increases by a quantity b, the Burger's vector, which measures the strength of the
dislocation. The Burger's vector of a screw dislocation is parallel to the dislocation line while
that of an edge dislocation.
Stress Fields around Dislocations
The core of dislocations is a region within a few lattice constants of the centre of dislocation.
The regions outside the core are stable regions and the strains in these regions are elastic
strains. Consider a cylindrical shell of a material surrounding an axial screw dislocation.
27
Assume the radius of the shell be r and the thickness dr, then the circumference of the shell is
r
2
and let the it shear by an amount b, so that the shear strain is given by;
r
b
e
2
=
This will result into a corresponding shear stress in the good region given by;
r
b
G
e
G
2
.
.
.
=
=
where G is the shear modulus or modulus of rigidity of the material
Considering the dimensions shown above, a distribution of forces is exerted over the surface
of the cut for producing a displacement b result into work done by the forces to give an
amount of energy E
s
of the screw dislocation as
³
=
dA
b
F
E
S
.
.
The force, F is the average force per unit area at a point on the surface during the displace-
ment and the integral extends over the surface area of the cut and therefore this the average
force is obtained as;
)
2
/
1
(
=
=
F
is half the final value when the displacement is b i.e.,
r
b
G
F
4
=
Substituting this force in the energy expression, we get;
³
=
.
4
2
dA
r
Gb
E
S
28
Since the expression dA = dz sr as per the dimensions used, then for a dislocation of length l,
we have ;
0
2
0
2
0
log
4
1
4
r
R
Gb
dA
r
Gb
E
R
r
S
=
=
³
³
The total elastic energy per unit length of a screw dislocation is thus given by;
0
2
log
4
r
R
Gb
E
S
=
where R and r
0
the proper upper and lower limits of r. This energy is found to depends upon
the values taken for R and r
0
suitably and when it is equal to about the Burger's vector b or
equal to one or two lattice constants and the value of R is not more than the size of the crystal.
Stress field of an edge dislocation
Consider the cross-section of a cylindrical material of radius R whose axis is along the z-axis
and in which a cut has been in the plane y = 0, which becomes the slip plane. The portion
above the cut is now slipped to the left by an amount b, the Burger's vector along the x-axis so
that the new position assumes the shape shown dotted in figure below. The calculation of the
stress field is done on the assumption that the medium is isotropic having a shear modulus G
and Poisson's ratio . The positive edge dislocation is produced along the z-axis and if we
assume that
rr
is the radial tensile stress, along the radius r and
is the circumferential
tensile stress acting in a plane perpendicular to r. r will denote the shear stress acting in a
radial direction.
29
In isotropic elastic continuum,
rr
and
compression or tension acts in a plane perpendicu-
lar to r. It can be shown that the constants of proportionality in the stress vary as G and b of
the edge dislocation in terms of r and
are given;
r
v
Gb
rr
sin
.
)
1
(
4
-
=
=
and
r
v
Gb
r
cos
.
)
1
(
2
-
=
where the positive values of are for tension and negative values for compression. Above the
slip plane
rr
is negative giving a compression while below the slip plane corresponds to a
tensile stress. It may be noted that for r = 0, the stresses become infinite and so a small
cylindrical region of radius r
o
around the dislocation must be excluded. The value of r
o
, is
calculated by letting r
o
= b, so that the magnitude of the strain is in the order of 1/2 (1-v)§1/4
so that the energy of formation of an edge dislocation of unit length for edge dislocation will
be given by;
³
³
-
=
-
=
=
R
r
e
r
R
v
r
Gb
dr
v
r
Gb
strain
x
stress
E
0
0
2
2
log
)
1
(
4
)
1
(
2
2
1
2
1
The core energy of edge dislocation should be added to the elastic strain energy. For a screw
dislocation in the Z-direction in a cylindrical material, the stress field is given by a shear
stress, according to the expression;
r
Gb
z
z
2
=
=
Etching
Types of Etching
In order to form a functional Micro-Electro-Mechanical Systems structure on a substrate, it is
necessary to etch thin films deposited and also the substrate itself. There are two classes of
etching processes.
30
Wet etching
This is the simplest etching technology and only requires a container with a liquid solution
that will dissolve the material in question. One must find a mask that at least etches much
slower than the material to be patterned. Some single crystal materials such as silicon exhibit
anisotropic etching in certain chemicals. Anisotropic etching is in contrast to isotropic etching
means different etch rates in different directions in the material. This is a simple technology
that gives good results when the right combination of etchant and mask material to suit your
application and it also works very well for etching thin films on substrates, and can also be
used to etch the substrate itself. Anisotropic processes allow the etching to stop on certain
crystal planes in the substrate, but still results in a loss of space, since these planes cannot be
vertical to the surface when etching holes or cavities.
Dry etching
The dry etching technology can split in three separate classes called reactive ion etching
(RIE), sputter etching, and vapor phase etching.
Reactive ion etching
In RIE, the substrate is placed inside a reactor in which several gases are introduced and a
plasma material is struck in the gas mixture using an RF power source, breaking the gas
molecules into ions. The ions are accelerated towards, and react at, the surface of the material
being etched, forming another gaseous material. This is known as the chemical part of
reactive ion etching. There is also a physical part which is similar in nature to the sputtering
deposition process. A schematic of a typical reactive ion etching system is shown in the figure
below.
31
If the ions have high enough energy, they can knock atoms out of the material to be etched
without a chemical reaction. It is a very complex task to develop dry etches processes that
balance chemical and physical etching, since there are many parameters to adjust. By chang-
ing the balance it is possible to influence the anisotropy of the etching, since the chemical part
is isotropic and the physical part highly anisotropic the combination can form sidewalls that
have shapes from rounded to vertical.
Sputter etching
Sputter etching is essentially RIE without reactive ions. The systems used are very similar in
principle to sputtering deposition systems. The big difference is that substrate is now subject-
ed to the ion bombardment instead of the material target used in sputter deposition.
Vapor phase etching
Vapor phase etching is another dry etching method, which can be done with simpler equip-
ment than what RIE requires. In this process the wafer to be etched is placed inside a cham-
ber, in which one or more gases are introduced. The material to be etched is dissolved at the
surface in a chemical reaction with the gas molecules. The two most common vapor phase
etching technologies are silicon dioxide etching using hydrogen fluoride (HF) and silicon
etching using xenon di-flouride (XeF
2
), both of which are isotropic in nature. Usually, care
must be taken in the design of a vapor phase process to not have bi-products form in the
chemical reaction that condense on the surface and interfere with the etching process.
32
Point Defect in Crystals
The point imperfections occur due to the imperfect packing of atoms during crystallization.
Therefore they can be regarded as lattice errors at isolated lattice points. They also occur due
to vibrations of atoms when the material is exposed to very high temperatures. Point imper-
fections are completely local in effect and sometimes they are regarded as a vacant lattice site
and that is why point defects are always present in crystals. The presence of point defects
results in a decrease in free energy and therefore one can compute the number of defects at
equilibrium concentration at a certain temperature as,
n = N exp [-E
d
/ kT]
where, n - number of imperfections, N - number of atomic sites per mole, k - Boltzmann
constant, E
d
- free energy required to form the defect and T - absolute temperature. Theoreti-
cally, the value of E is of the order of l eV. This is when it is calculated on k = 8.62 X 10
-5
eV
/K, at T = 1000 K, n/N = exp[-1/(8.62 x 10
-5
x 1000)] § 10
-5
.
Vacancies
Vacancy is the simplest point defect known and refers to an empty or an unoccupied site of a
crystal lattice. It can also refer to a missing atom or vacant atomic site as shown in figure
below.
Vacancy defects may also arise from imperfect packing during original crystallization. When
the thermal energy due to vibration is increased, there is always an increased probability that
33
individual atoms will jump out of their positions of lowest energy. They can result also from
thermal vibrations of the atoms at higher temperatures where each temperature has a corre-
sponding equilibrium concentration of vacancies and interstitial atoms. An interstitial atom is
an atom transferred from a site into an interstitial position. For most crystals the vacancy
defect at any thermal energy change is of the order of I eV per vacancy as thermal vibrations
of atoms increases with the rise in temperature.
Single or two or more vacancies may condense forming a di-vacancy or tri-vacancy. Where
such a vacancy exists, the atoms surrounding a vacancy tend to be closer together, thereby
distorting the lattice planes. At thermal equilibrium, the vacancies exist in a certain proportion
in a crystal and at higher temperatures; these vacancies have a higher concentration and tend
to move from one site to another more frequently. Vacancies accelerate all processes associat-
ed with displacements of atoms that include diffusion, powder sintering and scattering.
Interstitial Imperfections
Interstitial imperfection occurs when an extra atom is lodged within the crystal structure of a
closed packed structure having low atomic packing factor. This vacant space is known as
interstitial position or voids. An extra atom can enter the interstitial space or void between the
regularly positioned atoms only when it is substantially smaller than the parent atoms and
produce an atomic distortion. The defect caused is known as interstitial defect.
In close packed structures especially in FCC and HCP, the largest size of an atom that can fit
in the interstitial void or space have a radius about 22.5% of the radii of parent atoms and thus
interstitial are classified as single interstitial, di-interstitials and tri-interstitials.
34
Frenkel Defect
Whenever a missing atom leaves it space and occupies an interstitial the defect caused is
known as Frenkel defect. Frenkel defect is therefore a combination of vacancy and interstitial
defects. These defects are less in number because energy is required to force an ion into new
position. This type of imperfection is more common in ionic crystals, because of their positive
ions that are smaller in size get lodged easily in the interstitial positions.
Schottky Defect
In Schotky defect, the defect occurs whenever a pair of positive and negative ions is missing
from a crystal. It is similar to vacancies as shown below. This type of imperfection maintains
charge neutrality. It is known that closed-packed structures have fewer interstitial and Frenkel
defects than those found for vacancies and Schottky defects. This is because additional energy
is required to force the atoms in their new positions.
Substitution Defect
This defect occurs whenever a foreign atom replaces the parent atom of the lattice and
occupies the position of parent atom. The defect caused is called substitution defect and in
35
this type of defect, the atom which replaces the parent atom may be of same size or slightly
smaller or greater than that of parent atom.
Phonon
When the temperature is raised, thermal vibrations increase in a material and this causes the
symmetry of the defect and deviation to change. This defect has much effect on the magnetic
and. electric properties. All kinds of point defects are important though they distort the crystal
lattice and cause a certain influence on their physical properties. In commercially available
pure metals, point defects increase electric resistance. In addition to point defects created by
thermal fluctuations, point defects can also be created by other means that include producing
an excess number of point defects at a given temperature is by quenching or quick cooling
from a higher temperature. Another method of creating excess defects is by severe defor-
mation of the crystal lattice which involves hammering or rolling. Excess point defects can be
created using external bombardment by using atoms or high-energy particles. This is applied
in the beam of the cyclotron or the neutrons in a nuclear reactor in which the first particle
collides with the lattice atoms and displaces them, thereby causing a point defect.
Diffusion
Many processes occurring in metals and their alloys or in semiconductors especially at
elevated temperatures are associated with self-diffusion or diffusion. Diffusion refers to the
transport of atoms through a crystalline or glassy solid. Diffusion processes play a crucial role
in many solid-state phenomena and also in the kinetics of micro structural changes during
metallurgical processing and applications. This can be seen in phase transformations, nuclea-
tion, recrystallization, oxidation, creep, sintering, ionic conductivity and intermixing in thin
36
film devices. Technological that uses diffusion include solid electrolytes for advanced battery
and fuel cell applications, semiconductor chip and microcircuit fabrication.
Types of Diffusion
Self- Diffusion
It is the transition of a thermally excited atom from a site of crystal lattice to an ad-
jacent site or interstice.
Inter Diffusion
This is observed in binary metal alloys such as the Cu-Ni system.
Volume Diffusion
This type of diffusion is caused due to atomic movement in bulk in materials.
Grain Boundary Diffusion
This type of diffusion is caused by atomic movement along the grain boundaries
alone.
Surface Diffusion
This type of diffusion is caused due to atomic movement along the surface of a
phase.
Diffusion Mechanisms
Various mechanisms of diffusion explain how diffusion occurs and all these explanations are
based on the theory of vibrational energy of atoms in a solid. They also confine themselves
onto direct-interchange or cyclic or interstitial or vacancy as diffusion based mechanisms.
Diffusion is defined as the transfer of unlike atoms when a change of concentration of the
components in certain zones of an alloy changes. From all the proposed mechanisms, the most
probable mechanism of diffusion usually and largely accepted involve the one that explain
diffusion from magnitude of energy barrier or activation energy used to be overcome by
moving atoms. Activation energy is known to depend on the forces of interatomic bonds and
crystal lattice defects. These defects are then known to facilitate diffusion transfer and
therefore vacancy mechanism of diffusion is the most probable. For elements which have a
small atomic radius like hydrogen, H, nitrogen N and carbon, C, it is believed that the
interstitial mechanism explains their diffusion better.
37
Vacancy Mechanism
Vacancy mechanism is a very dominant process for diffusion especially in FCC, BCC and HCP
metals and related solid solution alloys. In this mechanism, the activation energy comprises of the
energy required to create a vacancy and that required to move the vacancy. This type of diffusion
can occur when atoms move into adjacent sites that have a vacancy such that in a pure solid
during diffusion process, the atoms surrounding the vacant site shift their equilibrium positions to
adjust in tandem with the change in binding that accompanies the removal of a metal ion and its
valence electron. The vacancies move through the lattice and produce random shifts of atoms
from one lattice position to another by the atom jumping. Vacancies are continually being created
and destroyed at the surface or grain boundaries or at suitable interior positions. Therefore in a
pure solid, the vacancy mechanism diffusion can be illustrated by the below.
The rate of diffusion increases rapidly with increasing temperature as concentration changes
takes place due to diffusion over a period of time. It should be noted that if a solid is com-
posed of a single element or is a pure metal, the movement of thermally-excited atom from a
site of the crystal lattice to an adjacent site or is called self-diffusion though the moving atom
and the solid involved are of the same element. The self-diffusion in metals occurs mainly
through vacancy mechanism.
38
Interstitial Mechanism
This is a type of diffusion mechanism where an atom changes positions using an interstitial
site. In cases where a solid is composed of two or more elements diffusing elements whose
atomic radii differ significantly, interstitial solution diffusion is likely to occur. Therefore, the
large atoms occupy lattice sites while the smaller atoms fit into the voids which are also called
as interstices. It is favored when interstitial impurities are present because of the low activa-
tion energy and can be illustrated as shown in the figure below.
These interstices are created by the larger atoms and in such cases, the activation energy is
associated with interstitial diffusion based on the fact that for the diffusing atom to arrive at
the vacant site, it must squeeze past the neighbouring atoms using the energy supplied by the
vibrational energy of the moving atoms and that is why interstitial diffusion said to be a
thermally activated process. This mechanism is used at low temperatures, oxygen, hydrogen
and nitrogen can be diffused in metals easily.
Interchange Mechanism
In interchange mechanism, atoms exchange places through rotation about a mid-point and
therefore its activation energy is very high. This makes this type of mechanism is highly
unlikely in most systems.
39
It is also noted that two or more adjacent atoms can jump past each other and as a result
exchange positions while the number of sites remains constant. This interchange may involve
two-atom or four-atom forming a Zenner ring when in the BCC face and thus much more
energy is required. As a result of these displacements of atoms surrounding the jumping pairs
interchange mechanism cause severe local distortion. This is also termed as Kirkendall's
effect
. Kirkendall's effect is very important in diffusion. We may note that the practical
importance of this effect is in metal cladding, sintering and deformation of metals (creep).
Kirkendall was the first person to show the inequality of diffusion by using an brass/copper
couple. Kirkendall showed that Zn atoms diffused out of brass into Cu more rapidly than Cu
atoms diffused into brass. Due to a net loss of Zn atoms, voids can be observed in brass.
Fick's Laws of Diffusion
Some laws treated diffusion as a mass flow process by in which atoms or molecules change
their positions relative to their neighbours in a given phase under the influence of thermal
energy or a gradient variation. Gradient can be due a concentration gradient or an electric
field gradient or magnetic field gradient or due a stress gradient. In Solid State Physics, we
consider mass flow under concentration gradients in which thermal energy is necessary for
this mass flow as the atoms have to jump from site to site during diffusion. During diffusion,
thermal energy is in the form of vibrations of atoms about their mean positions in the solid
and thus classical laws of diffusion like that of Fick's laws only hold for weak solutions and
systems with very low concentration gradients of the diffusing substance, such that the slope
of concentration gradient is given by;
(
)
40
Fick's Firfst Law
Fick's first law describes the rate at which diffusion occurs and it can be states as;
[ ]
dt
a
dx
dc
D
dn
-
=
The quantity, dn of a substance diffusing at constant temperature per unit time t through unit
surface area a is proportional to the concentration gradient dc/dx and the coefficient of
diffusion (or diffusivity) D (m
2
/s). The 'minus' (-) sign implies that diffusion occurs in the
reverse direction to concentration gradient vector. This implies that it is from a higher
concentration zone to a lower concentration of the diffusing element reducing the equation
above to become;
a
dx
dc
D
dt
dn
-
=
So that
dx
dc
D
dt
dn
a
J
-
=
-
=
1
where J is the flux or the number of atoms moving from unit area of one plane to unit area of
another per unit time. The flux J is therefore the flow per unit cross sectional area per unit
time and is proportional to the concentration gradient. The negative sign (-) in this expression
means that flow occurs down the concentration gradient as shown in figure below.
From the above curve, a large negative slope corresponds to a high diffusion rate and there-
fore in accordance with Fick's first law, the atoms of B will diffuse from the left side towards
the right side and the net migration of B atoms to the right side means that the concentration
41
will decrease on the left side of the solid and increase on the right as diffusion progress till a
state where the concentration is uniform only at the center as shown in the figure below.
Fick's first law can also be used to describe mass flow under steady state conditions. The
explanation is identical in form to that given by Fourier's law for heat flow under a constant
temperature gradient or that given by Ohm's law for a current flow under a constant electric
field gradient. Under steady state flow, the flux is independent of time and remains the same
at any cross-sectional plane along the diffusion direction.
Fick's second Law
Frick's second law is an extension of Fick's first law to a non-steady flow. Frick's first law
allows the calculation of the instantaneous mass flow rate or Flux past any lattice plane in a
solid. It also provides no information about the time dependence of the concentration but the
commonly available situations in material engineering are for non-steady in which concentra-
tions of solute atom changes at any point with respect to time in non-steady diffusion. When
the concentration gradient of various with time and the diffusion coefficient is assumed to be
independent of concentration and this gives the diffusion process as described by Frick's
second law when derived from the first law as;
»
¼
º
«
¬
ª
¸
¹
·
¨
©
§
=
dx
dc
D
dx
dc
dt
dc
This equation of Fick's second law is for unidirectional mass flow under non steady condi-
tions and therefore a solution to it can be expressed as;
42
[
]
)
4
/
(
exp
)
,
(
2
Dt
x
Dt
A
t
x
c
-
=
In this solution, A is constant. Consider a self-diffusion due to radioactive nickel atoms in a
non-radioactive nickel block specimen. Using the solution of Fick's second law, the concen-
tration at x = 0 falls with time as r
-1/2
. As time increases, the radioactive penetrate deeper in
the metal block and at time t
1
the concentration of radioactive atoms at x = 0 is c
1
= A/(Dt
1
)
1/2
.
After some time at a distance x
1
= 0 is (Dt
1
)
1/2
the concentration falls to 1/e of c
1
. At another
later time t
2
the concentration at x = 0 is c
2
= A/(Dt
2
)
1/2
and this falls to 1/e and x
2
= 2 (Dt
2
)
1/2
.
This can be illustrated as shown in the figures below;
In figure, the diffusion of atoms is shown (i) for t= 0 (ii) for t
1
, and (iii) t
2
with t
2
t
1
and thus
1D is independent of concentration, Fick's second law is simplified to;
2
2
dx
c
d
D
dt
dc =
It should be noted that even if D may vary with concentration, solutions to the differential
above are quite simple for practical problems since it is for unidirectional diffusion from one
medium to another a cross a common interface. It can thus be given in a general form as.
Dt
x
erf
B
A
t
x
c
2
/
(
)
,
(
-
=
where, A and B are constant to be determined from the initial and boundary conditions of a
particular problem.
This equation applies for two media taken to be semi-infinite in which the interface of one
end is only defined while the other two ends are at an infinite distance. Thus the initial
uniform concentrations of the diffusing species in the two media are different causing an
abrupt change in concentrations at the interface. Therefore a solution in erf of Fick's second
law expressed with error function reduces to;
43
³
-
=
Dt
x
d
Dt
x
erf
2
/
0
2
)
exp(
2
2
Where
, is an integration variable that gets deleted as the limits of the integral are substitut-
ed and always the lower limits of the integral is zero while the upper limit of the integral is
function whose quantity is to be determined by
2
, which is a normalization factor. The
diffusion coefficient D (m
2
/s) is used to determine the rate of diffusion at a concentration
gradient equal to unity and depends on the composition of alloy's size of grains, and tempera-
ture. Therefore the solutions to Fick's second equations exist for a a large variety of boundary
conditions which help to evaluate D from c as a function of x and t. Based on the solution
given above, the time dependence of diffusion is shown in figure below.
A closer analysis of the curves shows that the curve corresponding to a concentration profile
at a start time, t
1
is marked by time, t
1
. At a later time t
2
, the concentration profile has changed
and this profile change is due to the diffusion of B atoms that has occurred in the time interval
t
2
- t
1
. After some time t
3,
the
concentration profile is marked by t
3
. Thus due to diffusion
process, atoms of B tend to get distributed uniformly throughout the solid salutation and this
causes the concentration gradient to become less negative as time increases.
Dependence of Diffusion Coefficient on Temperature
The diffusion coefficient D is known to determine the rate of diffusion at a concentration
gradient equal unity. However, it also depends on the composition of the alloy, the size of
grains and its temperature. The diffusion coefficient based on temperature dependency is
described by Arrhenius exponential relationship given by;
44
D = D
0
exp (-Q/RT)
where D
0
is a pre-exponential frequency factor which heavily relies on the bond force
between atoms of crystal lattice, Q is the activation energy of diffusion in which it is a sum of
the activation energies for the formation, Q
v
and motion of vacancies, Q
m
such that;
Q = Q
v
+Q
m
,
The experimental behavior of Q for the diffusion of carbon in
-Fe is about 20.1 kcal/mole
and that of D
0
is 2
×10
-6
m
2
/s when R is take as the gas constant.
Factors Affecting Diffusion Coefficient
Diffusion co-efficient is affected by concentration. The rate at which atoms jumped mainly
depends on their vibrational frequency, the crystal structure. The energy required by the atom
to overcome this energy barrier is called the activation of diffusion and it can be illustrated by
the figure below.
This activation energy is required to pull the atom away from its nearest atoms. Thus in
vacancy mechanism, this energy is also required to force the atom into closer contact with
neighbouring atoms as it moves along them in interstitial diffusion. When the normal inter-
atomic distance is increases or decrease, this energy has to be increased. Activation energy
depends on the size of the atom and it varies with the size of the atom, strength of bond and
the type of the diffusion mechanism. Activation energy is high for large- sized atoms for
strongly bonded materials like corundum and tungsten carbide.
45
Applications of Diffusion
Diffusion processes are the basis of Crystallization, recrystallization, phase transformation
and saturation of the surface of alloys by other elements are the few important applications of
diffusion. Other includes the following: Oxidation of metals; Doping of semiconductors;
Joining of materials by diffusion bonding. Production of strong bodies by sintering i.e.
powder metallurgy; Surface treatment. Diffusion is fundamental to phase changed. Some
practical applications of diffusion are discussed below:
Hardening steel
Solid state diffusion is used in surface hardening of steel. Steel is used in the manufacture of
gears and shafts. Steel parts made in low carbon steel are brought in contact with hydrocarbon
gas like methane (CH
4
) in a furnace atmosphere at about 927
0
C temperature. The carbon in
CH
4
diffuses into surface of steel part and the carbon concentration increases on the surface
and this is what hardness the surface of steel. The concentration of carbon is higher near the
surface and reduces with increasing depth as depicted in the figure below.
Ionic conductivity
Anion vacancy is the dominant lattice defect responsible for the ionic conductivity in pure and
doped lead chloride. Activation energy responsible for the migration of the anion vacancy
ranges from 0 - 48 eV to 0 - 24 eV in lead chloride crystals, however, in single crystals of
pure and doped lead chloride, the energy of formation of vacancies ranges from 1 - 66 eV and
that for migration of the anion vacancies is 0-35 eV. The roles of the various point defects in
pure and doped lead chlorides are yet understood.
46
Colour centers
Therefore a colour centre is a lattice defect, which absorbs light. A scientist known as
Becquerel discovered that when transparent NaCl crystals are placed near a discharge tube
they appear yellowish in colour. This discovery gave rise to the study of colour centres.
Rocksalt has an infrared absorption that is caused by the vibrations of its ions and also an
ultraviolet absorption that is due to the excitation of the electrons. A perfect NaCl crystal does
not absorb visible light. This makes the perfectly transparent in the visible region. The
colouration of crystals is therefore due to defects in the crystals.
Exposure of a coloured crystal to white light can result in bleaching of the colour and this is
used give a clue of the nature of the absorption done by crystals. Experiments show that
during the bleaching of the crystal the crystal becomes photoconductive. This is due to the
electrons that are excited into the conduction band. Further, photo-conductivities express the
quantum efficiency of a material. It shows the number of free electrons produced per incident
photon resulting from the colour centres. Insulators have very large energy gaps. As a result
many of them are transparent to visible light. Ionic crystals have the forbidden energy gap of
about 6 eV. This energy corresponds to a wavelength of about 2000A
0
which is found in the
ultraviolet region.
A study of the dielectric properties show that the ionic polarizability of a material resonates at
a wavelength of 60 microns in the far infrared region and at that resonance levels crystals are
expected to be transparent over a wide range of spectrum including the visible region.
Crystals of KCl, NaCl, LiF and other alkali halides are have such dielectric properties, that is
why they are used for making prisms, lenses and optical windows in optical and infrared
spectrometers. Due to different various reasons, absorption bands occur in the visible, near
ultraviolet and near infrared regions in some of these crystals. When the absorption band is in
the visible region and the band is quite narrow. This gives a characteristic colour to the crystal
and when the crystal gets coloured, it is then said to have colour centres. Therefore a colour
centre is a lattice defect, which absorbs light. This theory has been used greatly to colour
crystals in a number of applications as described below:
(i).
By the addition of suitable chemical impurities like transition element ions with
excited energy levels. Hence alkali halide crystals can be coloured by ions whose
salts are normally coloured.
47
(ii).
By introducing stoichiometric excess of the cation by heating the crystal in the
alkali metal vapour and then cooling it quickly. The colours produced depend upon
the nature of the crystals e.g., LiF heated in Li vapour colours it pink, excess of K
in KCl colours it blue and an excess of Na in NaCl makes the crystal yellow. Crys-
tals coloured by this method on chemical analysis show an excess of alkali metal
atoms, typically 10
16
to 10
19
per unit volume.
(iii).
By exposing them to high energy radiations like X-rays or -rays or by bom-
barding them with energetic electrons or neutrons. Crystals can also be coloured or
made darker.
Types of colour centers
They include the following types of colour centres:
F Centres
The simplest colour centre is an F centre and it is called so because its name comes from the
German word Farbe which means `colour'. The F centres are produced by heating a crystal in
the presences of an excess of an alkali vapour. It can also be grown by irradiating the crystal
by X rays. Such crystals with F centers include NaCl and they have the main absorption band
occurs at center at about4650A
0
hence called the F band as shown in the figure 10 below. This
absorption in the blue region is responsible for the yellow colour produced in NaCl crystal
and thus it is the F band that is characteristic of the crystal. The F band in KCI or NaCl will be
the same whether the crystal is heated in a vapour of sodium or of potassium as shown below.
48
Formation of F-Centres
Colour centres in crystals occur when crystals have an excess of one of its constituents due to
their non-stoichiometric properties. NaCl crystal can be coloured by heating it in an atmos-
phere of sodium vapour and then cooling it quickly so that the excess sodium atoms absorbed
from the sodium vapour causing it to split up into electrons and positive ions in the crystal as
shown in the figure below.
Through this process the crystal becomes slightly non-stoichiometric having excess of sodium
ions than chlorine ions. This creates CI
-
vacancies and therefore the valence electron of the
alkali atom is not bound to the chlorine atom. Instead it diffuses into the crystal to be bound to
a vacant negative ion site at F band because a negative ion vacancy in a perfect periodic
lattice has the effect of an isolated positive charge. The positive charge then traps the electron
in order to maintain local charge neutrality. The excess electron captured this way at a
negative ion vacancy in an alkali halide crystal is called an F centre. This model was first
suggested by De-Boer and was further developed by Mott and Gurney.
Change of Density
In a normal crystal of NaCl, some Cl
-
vacancies are always present in thermodynamic
equilibrium. Thus if any sort of radiation is applied, it will cause some electrons to be
knocked into the Cl
-
vacancies. The result will be the formation of F centres. The generation
of Cl
-
vacancies through the introduction of excess metal atoms can be demonstrated by a
decrease in the density of the crystal. This change of density is determined by X-ray diffrac-
tion measurements.
Details
- Pages
- Type of Edition
- Erstausgabe
- Publication Year
- 2015
- ISBN (PDF)
- 9783954899326
- ISBN (Softcover)
- 9783954894321
- File size
- 26 MB
- Language
- English
- Publication date
- 2015 (May)
- Keywords
- Physics semiconductor photonics high speed semiconductor energy band theory scattering in high field effect in semiconductors multi-valley semiconductor high speed metal semiconductor field effect transistor high speed processing device
- Product Safety
- Anchor Academic Publishing
