M.Sc. (FINAL)

PAPER II

BLOCK –I

SOLID STATE PHYSICS

AND

MATERIAL SCIENCE

Writer: Dr. Meetu Singh

Editor: Dr. Purnima Swarup Khare

SOLID STATE PHYSICS

AND

MATERIAL SCIENCE

Unit -01 Lattice dynamics and polarization

Unit-02 Band theory of solid

Unit-03 Magnetism

BLOCK -I

PAPER II

SOLID STATE PHYSICS AND MATERIAL SCIENCE

CONTENTS

UNIT-01 Lattice Dynamics and Polarization

Page No

1.0 INTRODUCTION 03

1.1 OBJECTIVE 03

1.2 POLARIZATION 03

1.3 LORENTZ RIELD 05

1.4 IONIC POLARIZABILITY 06

1.5 ORIENTATION POLARIZABILITY 07

1.6 DEBYE EQUATION FOR GASES 10

1.7 THE COMPLEX DIELECTRIC CONSTANT 11

1.8 DIELECTRIC LOSSES 13

1.9 DIELECTRIC RELAXATION TIME 16

1.10 SUMMARY 18

1.11 CHECK YOUR PROGRESS 19

Unit-02 Band Theory of Solid

2.1 INTRODUCTION 20

2.2 OBJECTIVE 20

2.3 KRONIG-PENNEY MODEL 20

2.4 EFFECTIVE MASS OF AN ELECTRON 22

2.5 QUANTUM FREE ELECTRON THEORY 25

2.6 FERMI – DIRAC STATISTICS, FERMI FACTOR AND FERMI ENERGY 25

2.7 CLASSIFICATION OF SOLIDS ON THE BASIS OF BAND THEORY 26

2.8 HALL EFFECT 28

2.9 BLOCH THEOREM 31

2.10 SUMMARY 33

2.11CHECK YOUR PROGRESS 33

Unit-03 Magnetism

3.1 INTRODUCTION 34

3.2 OBJECTIVE 34

3.3 MAGNETIC FIELD AND ITS STRENGTH: 34

3.4 MAGNETIC DIPOLE MOMENT: 36

3.5 ELEMENTARY IDEAS OF CLASSIFICATION: 37

3.6 QUANTUM THEORY OF PARA MAGNETISM: 39

3.7 THEORY OF FERROMAGNETISM: 42

3.8 QUANTUM THEORY OF FERROMAGNETISM: 43

3.9 DOMAIN THEORY OF FERROMAGNETISM: 44

3.10 MAGNETIC RESONANCE 47

3.11 SUMMARY 59

3.12CHECK YOUR PROGRESS 60

UNIT-01

Lattice Dynamics and Polarization

1.0 INTRODUCTION

A crystal lattice is consisting of a special long range order. This yield a sharp direction patterns in 3-d.

lattice vibrations are important. They contribute in many things like, the thermal conductivity of

insulators is due to dispersive lattice vibrations, and it can be quite large (in fact, diamond has a

thermal conductivity which is about 6 times that of metallic copper). In scattering they reduce of the

intensities, and also allow for inelastic scattering where the energy of the scattered (i.e. a neutron)

changes due to the absorption of a phonon in the target. Electron-phonon interactions renormalize the

properties of electrons.

1.1 OBJECTIVE

Lattice deformation can be studied in detail if one has the knowledge of the dielectric constant. For

that, the basic starting point is the Maxwell equations.

1.2 POLARIZATION

According to the dielectric properties, we deal more often with dipoles instead of isolated charges. In

electrostatics, as we know that a dipole with charges +e and –e displaced by distance d has the dipole

moment as

dep

(3.1)

and the electric field

due to this dipole at a point

r is

5

0

2

4

).(3)(

r

prrrpr

(3.2)

In case of insulators, under the influence of an electric field the forces acting upon the charges bring

about a small displacement of the electrons relative to the nuclei, as the field tends to shift the positive

and the negative charges in opposite directions. This is the state of electric polarization, in which a

certain amount of charge is transported through every plane element in the dielectric. This transport is

called the displacement current. After reaching the state of equilibrium in an applied field, every

volume element of the dielectric has acquired an induced dipole moment. The induced dipole moment

in a volume element V will be given by

VNdeVP iii

(3.3)

Where VNi is the average number of charges ie with displacement id . This gives the electric

polarization as

i

iii deNP

(3.4)

Alternatively, one can calculate the charge densityP induced at the ends of the dielectric specimen

by the displacement

id . This is simply the amount of charge per unit area which is separated by the

displacement from charge of the opposite sign or

i

iiiP deN

(3.5)

Comparison of these two equations gives

llP P

(3.6)

The sign of the polarization surface charge is positive, where

P is directed out of the body and

negative where it is directed in ward. In fact,

PnP

(3.7)

Where

n is the unit normal to the surface, drawn outward from the dielectric into the vacuum. The

electric field )(

rP produced by the polarization is equal to the field produced by the fictitious

charge density on the surface of the specimen as shown in Fig. 12.1 (a). The total macroscopic

field inside the specimen is then

P 0

(3.8)

Where

0 is the applied electric field.

1.3 LORENTZ FIELD

The field

s due to the polarization-induced surface charges on the surface of the fictitious

spherical cavity around the point A (where the field

s is to be. Calculated) was investigated by

Lorentz. The apparent surface charge density on the part of the spherical surface between and

d is

cosP

(4.1)

Choose the x-axis in the direction of the electric field. The contribution of all the surface charges

between and d to the y-and z-contribution of local field

loc cancel each other for reasons

of symmetry. The contribution of a surface charge to the x-component of local field is

2

cos

a

and the total surface charge between and d is da sin2 2 . Combining these

results, the electric field at the centre of the spherical cavity of radius a is

0

2

0

2

2

0 3sin2.

cos

4

1

Pda

a

Ps

(4.2)

Fig.1.1. Geometry for the determination of Lorentz fields.

The Lorentz field of dipoles inside the spherical cavity depends on the crystal structure. In order to

evaluate this contribution, we consider a cubic lattice structure and divide the sphere into a very

large number of small volume elements of equal size. If all dipoles are parallel to the x-axis and

have dipole moment p, then the Lorentz field of dipoles inside the spherical cavity

d is given by

i i

ii

i

z

i

ii

i i

iixd

r

zxP

r

yxPy

r

rxP5

0

5

0

5

22

0

3

4

3

4

3

4

(4.3)

Lorentz showed that by summing over all points of a sphere that are distributed symmetrically over a

sphere,

;03

4;0

3

4;0

35

0

5

0

5

22

i i

ii

i

z

i

ii

i i

ii

r

zxP

r

yxPy

r

rx

(4.4)

This gives

0d

Thus, in a cubic lattice the local field according to the Lorentz method of evaluation is

00

033

PP

Ploc

(4.5)

For non-cubic lattices, the procedure for evaluation of the local field is not that straight forward.

Mueller has worked out d tetragonal and hexagonal lattices.

1.4 IONIC POLARIZABILITY

Ionic Polarizability ia is due to the displacement of adjacent ions of opposite sign and is only found

in ionic substances. In an electric field the resultant torque lines up the dipole parallel to the field at

the absolute zero of temperature. The field produces forces on the charges of opposite sign so that the

distance between them is changed by some amount as shown in Fig. 1.2. The balance between the

electrostatic force and the inter atomic force due to stretching or compressing gives the value of the

change in the distance between two ions of Opposite charge.

Fig. 1.2 Ionic Polarization, the field distorts the lattice.

From this change in the distance, the ionic-dipole moment and hence the ionic Polarization i can be

determined. Like electronic Polarization, ionic Polarization is also independent of temperature at

moderate temperatures.

1.5 ORIENTATION POLARIZABILIY

Orientation Polarization i occurs in liquids and solids which have asymmetric molecules whose

permanent dipole moments can be aligned by the electric field, molecules whose permanent dipole

moments can be aligned by the electric field, as shown in Fig.1.3. Let us consider a system of N

permanent dipoles, with dipole moment

PP , subjected to an external field

Parallel to the x-axis.

The work required to bring one of the dipole molecules into a position where

PP makes an angle

with , as shown in Fig. 1.4, is given by

cosPP PP

(5.1)

According to Boltzmann‟s energy distribution law, the various positions of the dipoles are not equally

probable when the uniform field is applied. Without this field the number of dipoles, inclined to x-

axis between and d is equal to

Fig.1.3 Orientation polarization, the field orients the orients the permanent dipoles.

dA

a

adaAdN sin2

sin2)(

2

(5.2)

Where the constant A is determine from the total number of dipoles.

If an external uniform electric field is applied, then Boltzmann‟s law introduces a factor of T

Bke

/,

changing (5.3)into

Fig. 1.4 The couple produced on the dipole due to applied field.

TkP BPedAdN

/cossin2)(

(5.4)

The x component of each dipole, making an angle with the x-axis, will be cos

PP and therefore the

x component of all the dipoles within the range and d will be .)(cos dNPP The net x

component 0P due to all N dipoles will be the sum of equation (5.5) over all angles .

Fig.1.5. To calculate number of dipoles in range d as a function of .

0

cos

0 cossin2 dePAPTlkP

P

BP

(5.6)

The total number of dipoles N is

0

)( dNN

(5.7)

Which provides the value of the constant A. Substituting this un equation (5.6) we get

0

cos

0

cossin

0

sin2

2cos

Tlk

ePp

B

TBlk

ePpd

dPpN

P

(5.8)

Use the abbreviations yTk

P

B

P cos

and Tk

Px

B

P

xe

xye

x

NPp

dyex

dyyeNPp

Pxy

xy

x

x

y

x

x

y

0

xxNPp

xee

eeNPp

xx

xx 1coth

1

)(xNPpL

(5.9)

L (x) is called the Langevin function, since this formula was first derived by Langevin in 1905. A plot

of L (x) is shown in Fig. 1.6. It is obvious that L (x) has a limit unity. If x is very small, the value of L

(x) is nearly equal to3

x. In fact, the expansion is given by

.

Fig.1.6 Langevin function

...)9450

2

945

2

45

1

3

1( 753 xxxxxL

(5.10)

Therefore, using the approximation ,3

)(x

xL the orientation polarization is

Tk

NPpP

B3

20

(5.11)

This gives the orientation l polarization per molecule 0 as

Tk

P

B

p

3

2

0

(5.12)

At room temperature, the orientation polarization is of the same order as the electronic polarization

1.6 DEBYE EQUATION FOR GASES

The total polarization for dilute gas can now be written as the sum of the above discussed three

components as

Tk

P

B

p

ie3

2

(6.1)

Fig.1.7. Curve between )1( r and (1/T) for a polar gas.

If this equation (12.40) is substituted in the Clausius -Mossotti relation (12.26) one obtains

jB

p

ie

j

j

r

r

Tk

PN

33

1

2

12

0

(6.2)

This is the Debye equation for the determination of dipole moments and polarization from

measurements on gases. There is a very slight difference between 2r and 3 and therefore, a plot

between )1( r and (1/T) for a gas will be straight line as shown in Fig. 1.7. The intercept of the

line for 01

T provides the value of )( ie and the slope often line yields Pp and hence .0 this

unit usually used for the dipole moment is Debye 3.33103coulomb.

1.7 THE COMPLEX DIELEX DIELECTRIC CONSTANT

Till now, we were concerned with the dielectric constant when a dielectric was subjected to a static

electric field. Let us now consider the dielectric under an alternating electric field. Two situation exist:

(I) when there is no measurable phase difference between

D and

then polarization is in phase with

the alternating electric field and

D is a valid relation and (II) when there is a phase difference

between

D and

, then polarization

P is not in phase with the alternating field and

D is not a

valid relation. The basic difference between these two situations is that in the first possibility no

energy is absorbed by the dielectric from the electric field, whereas in the second possibility energy is

absorbed by the dielectric, which is known as dielectric loss. In order to see this, let us apply an

alternating voltage to dielectric between the plates f plane capacitor

t cos0

(7.1)

The true surface charge density on the capacitor plates, which is equal to ,

D gives the current

density as

t

DJ

(7.2)

In the first case, when the electric displacement

D is in phase with

, then

tDD cos0

(7.3)

giving tDJ sin0

(7.4)

Thus, the electric current density is out of phase by 2

from

. The dissipated energy per unit

volume per second of the dielectric is

/2

02

JW

Substituting

j and

from (12.45) and (12.42) respectively, we get

0)cos()sin(2

0

/

0

0

2

dtttDW

(7.5)

Thus there is no dissipation of energy when

D and

are phase. When they are out of phase by

, the electric displacement will then be

)cos(0

tDD

tDtD sinsincoscos 00

(7.6)

Substituting in (12.43) gives current density

sincoscos 00 tDtDJ

(7.7)

This will give the energy dissipated per unit volume per second as

dttDtDtW ]cossinsincos[cos2

00

/

0

0

2

sin2

0

D

(7.8)

This is the dielectric loss and therefore the term sin is called the loss factor (or power factor) and

the loss angle (or phase angle). In case of phase difference between

D and

, it is useful to useful

to use complex notation where D is the real part of )(

0

tieD and , is the real part of tie 0 . When

phase pg , then the ratio between the complex quantities )(

0

tieD and tie 0 will give a complex

dielectric constant as

ieD

0

0*

(7.9)

If the real and imaginary parts of are and respectively, then equation (12.51) will give,

on comparison of real and imaginary parts

cos0

0D and

sin

0

0D

(7.10)

The equation (7.10) will give

tan and

0

022

D

(7.11)

Substituting equation (7.10) in equation (7.8) will give dissipated energy per unit volume per second

as

2

02

W

(7.12)

1.8 DIELECTRIC LOSSES

In this section we show that the energy absorbed per second per unit volume (or the energy loss) in a

dielectric medium is proportional to the imaginary part " of the dielectric constant. The

relationship among vectors E, P and D clearly indicates that on the application of an alternating

electric field in a dielectric, relative to that of E. Defining vectors magnitudes as

tiEtE exp0 (8.1)

tiDtD exp0 (8.2)

where is the phase angle, giving the measure of phase lag.

In view of (8.1) and (8.2) we express the dielectric function in the following form, being

aware that it is a complex quantity in the present situation:

tE

tDi

0

"'

(8.3)

On substituting E and D form (8.1) and (8.2), respectively, in (8.3) and then rationalizing the obtained

relation for , we get

00

0 cos'

E

D

(8.4)

00

0 sin"

E

D

(8.5)

'

"tan (8.6)

Relation (8.6) establishes the frequency dependence of the phase angle.

Let us now take the example of a parallel plate capacitor filled with a dielectric material and

bearing a surface charge density t on its plates at any time t. Then the current density in the

capacitor at that moment of time is

tDtD

dt

tdD

dt

tdtj

cossinsincos 00

(8.7)

Since j(t) is real physical quantity, only the real part of D(t) is considered in (8.7)

The energy dissipated per unit time in one cubic meter of the dielectric is equal to

/2

02

dttEtjW (8.8)

Using (8.6) and (8.1) (taking the real part as W is real), we obtain

"2

1 2

00EW (8.9)

Showing thereby that the energy losses in the dielectric are proportional to " .

Relation (8.9) can also be put in the form

sin2

100DEW (8.10)

The tan , given by (8.7), is often referred to as the loss factor, But this terminology is relevant only

when is small, so that tan sin , and the usage may thus be held justified. In the

interpretation of optical phenomena it is a practice to use the complex index of refraction n instead

of . Therefore, a brief discussion in this regard is very much in order. The development of

Maxwell‟s equation for the electromagnetic field shows that the velocity of electromagnetic waves in

a medium is given by

2/1

00

v (8.11)

In free space, and are both equal to unity and, therefore,

2

1

00

c (8.12)

If medium is non-magnetic, I and then.

2/1v

c (8.13)

Since the ratio c/v by definition equals the index of refraction n .

n (8.14)

Further the electromagnetic waves in a dielectric medium are described by an electric field.

cxntiEE /exp0 (8.15)

Where the index of refraction n is complex function

iknn (From 10.82) (8.16)

Hence, we get

22' kn (8.17)

nk2" (8.18)

The signs of the exponent in (8.18) and in the decomposition of and n are chosen such that

" and k (the extinction coefficient0 have positive signs, i.e. that wave amplitude decreases in the +x-

direction. Had we taken a positive sign in the exponent, we would have been required to write

0' ikwithknnand

There is nothing sacred about the choice of sign in question as both representations are in common

use.

.

1.9 DIELECTRIC RELAXATION TIME

It has already been that the total polarization P in a static field comes from electronic, ionic (together

called atomic) and orientation polarization. When a dielectric is subjected to an external static field, a

certain time is required for polarization to reach its final value. It is observed that the electronic and

the ionic polarization is attained instantaneously, if we consider high frequencies

)1sec1010( 117 and not the optical frequencies. At these frequencies, the dielectric loss is mainly

due to the relaxation effect of the permanent dipoles. Therefore, first we will consider the transient

effects, in which this relaxation effect of the permanent dipoles is characterized by a relaxation time

and then we go on further to discuss the situation of applying an alternating field.

Let suffix s denote the static electric field case, so that equation (8.2)

)1(, 000 rsssrss DPD

(9.1)

The total polarization can be written as the sum of only two terms, one in which the polarization is

attained instantaneously, denoted by P and the other in which relaxation effects are important,

denoted by osP .

oss PPP

(9.2)

)1(0 P

(9.3)

Instantaneously on the application of static field, the polarizations is denoted by P and then let us

consider that in time sPt, part out osP is build up. So that at certain time t, we have

st PPP

(9.4)

In general, in relaxation processes, one assumes that the increase )( sPd /d t is proportional to the

difference between the final value osP and the actual value sP i.e.

)(1

soss PP

dt

dP

(9.5)

Where is a constant, known as relaxation time, a measure of the time log. Integration of (12.59),

using the initial boundary condition that at 0t , 0sP , we obtain

)1( /t

oss ePP

(9.6)

In case of an alternating field also, it is assumed that equation (12.59) is valid. Denoting for a complex

quantity, the super script *, we have

][1 ***

soss PPdP

(9.7)

With the help of equations (9.5), (9.6) and (9.7), ti

rssos ePPP 00

** )(*

(9.8)

Substituting (9.3) in (9.4) and integrating, we obtain

00/*

1

)(

iCP rst

es

tie

(9.9)

After some time, the first term on the right hand side is small that it can be neglected. Now, the total

polarization in an alternating field is

E 00

0

*

1

)()1(

iP rs tie

(9.10)

This will give the displacement as

001

)(***

i

PD rs ie

(9.11)

thus, the complex dielectric constant in case of alternating field, is

i

rs

1

)(* 00

0

(9.12)

Separation into real and imaginary parts provide

)1(

)()(

22

000

rs

(9.13)

)1(

)()(

22

00

rs

(9.14)

These equations are often referred to as Debye equations. These can be rewritten as

)1(

1

)(

))((22

00

0

rs

(9.15)

and )1()(

)(22

00

rs (9.16)

The right hand side of these equations is plotted as a function of in Fig. 1.8. It should be

mentioned that these relations are in satisfactory agreement with the experimental observations.

Fig.1.8. Frequency dependence of the real and the imaginary part of the dielectric constant.

1.10 SUMMARY:-

This chapter explain the concept of Polarization, which is a property of certain types of waves that

describes the orientation of their oscillations and show that Ionic polarization occurs when an electric field is

applied to an ionic material then cations and anions get displaced in opposite directions giving rise to a net

dipole moment. The Debye model is a solid-state equivalent of Planck's law of black body radiation, where one

treats electromagnetic radiation as a gas of photons in a box. This chapter describe the Debye equation for the

determination of dipole moments and polarization and derive the formula for dissipated energy per unit volume

per second. Next part shows that the energy loss in a dielectric medium is proportional to the imaginary part

" of the dielectric constant. When a dielectric is subjected to an external static field, a certain time is

required for polarization to reach its final value. At high frequencies, the dielectric loss is mainly due to the

relaxation effect of the permanent dipoles.

1.11 CHECK YOUR PROGRESS

Q. 1. Define Polarization and Lorentz field

Hint: - Refer to topic no. 1.2 & 1.3

Q. 2. Explain Orientation Polarizability

Hint: - Refer to topic no. 1.5

Q. 3. Explain Debye Equation for Gases

Hint: - Refer to topic no. 1.6

Q. 4. Drive the Complex Dielectric Constant

Hint: - Refer to topic no. 1.7

Unit-02

Band theory of solid

2.1 Introduction During the discussion of the free electro theory of metals, the conduction electrons behave like a

classical free particle of a gas obeying Fermi-Dirac statistics. But this could not be made clear that

why in metals the electrical conductivity is quite low. Band theory describes the behavior of electrons

in solids, by postulating the existence of energy bands. It uses a material's band structure to explain

many physical properties of solids, such as electrical resistivity and optical absorption. A solid creates

a large number of closely spaced molecular orbital, which appear as an energy band.

2.2 Objective It is the concept of electronic energy bands which provides the basis for the classification of solids as

good conductors, semiconductors and poor conductors of electricity. 2.3 Kronig-Penny Model The free electron model of solids considered the electrons to be free inside the solid. It was able to

explain the electrical and thermal conductivity of metals but could not explain the same for

semiconductors and insulators. Kronig and Penney considered the electrons to be moving in a variable

potential region in the crystal instead of being free. The potential was approximated by a square well

periodic potentials as shown in Fig. 2.1.

Fig. 2.1 The time independent Schrodinger wave equation in one dimension is,

Or

02

02

2

22

2

2

2

VEdx

d

m

VEm

dx

d

In region 0 < x < a, where v =0, the general solution for wave equation is, iKxiKx BeAe 1 …… (3.1)

and the energy is,

m

KE

2

22 …… ….(3.2)

in the region – b < x < 0, V = V0. The solution of wave equation is, QxQx DeCe 2 …..(3.3)

The continuity of wave function at x = 0 requires that the values of and be equal at this at this

point.

0201 xx …..(3.4)

DCBA

dx

d is also continuous at x = 0

0

2

0

1

xx dx

d

dx

d

DCQBAiK

...(3.5)

By Bloch theorem,

baikexbbaxa 0

Where k is the wave vector.

The condition for continuity of at x = a with Bloch theorem is

baik

bxx e

201

baikQbQbiKaika eDeCeBeAe …(3.6)

For continuity of dx

dat x = a,

baik

bxx

edx

d

dx

d

.2

0

1

baikQbQbiKaiKa eDeCeQBeAeiK .

.(3.7)

Equation (3.4), (3.5), (3.6) and (3.7) have solutions only if the determinant of coefficients of A, B, C

and d vanishes. This leads to the condition.

kaKakaKa

Pcoscossin ..

(3.8)

Where 2

2baQP

The R.H.S of equation (3.8) is cos ka which lies between – 1 and + 1. Hence solutions are obtained

only when the L.H.S lies between – 1 and +1. The graph of kaKakaKa

Pcoscossin plotted for

different values of Ka is shown in Fig. 2.2. In regions of solution for do not exist. The

corresponding energies which are forbidden can be obtained using equation (3.2).

Thus, there are bands of allowed energies, which are separated by energies which are not

allowed and hence known as forbidden bands.

Fig. 2.2

The graph of energy E for different wave vectors k is shown in Fig. 2.3. There are discontinuities at

...,.........2

,aa

k

which correspond to the condition that .1cossin KaKaKa

P

Thus according to Kronig-Penney model, the motion of electrons in a periodic potential in

crystals gives rise to certain allowed energy bands separated by forbidden energy bands.

Fig. 2.3

2.4 Effective Mass of an Electron

An electron in a crystal is not free. When an external field is applied, the electron in a crystal

behaves as if it had a mass different from its actual mass. Consider an electron described as a wave

packet having wave function in the region of wave vector k. Let the electron be in a crystal to which

an electric field is applied. The group velocity of the wave packet will be

dk

dvg

(4.1)

If E is the energy of the wave packet,

dt

dk

dk

Ed

dt

dv

dtdk

Ed

dt

dv

dk

Edv

dk

dv

E

E

vh

hvE

g

g

g

g

2

2

2

2

2

1

1

1

2.2

(4.2)

The work done dE by the electric field E in a time interval dt is

dE = - e E vg dt

(The force on the electron is – e E and the displacement is vg dt)

As dE = dkdk

dE

(4.3)

and gvE

From equation (4.1)

dkvdE g

(4.4)

From equations (4.3) and (4.4),

dt

dke

dtedk

(4.5)

The above equation describes the force –e E due to an external field E on an electron in terms of the

rate of change of wave vector k. Hence we can write

dt

dkF

(4.3)

F

dt

dk

Substituting in equation (4.2)

F

dk

Ed

dt

dvg.

12

2

dt

dv

dk

EdF

g

2

2

2

As dt

dvghas dimensions of acceleration, the quantity

2

2

2

dk

Ed

is defined as the effective mass (m

*) of

an electron,

2

2

2

*

dk

Edm

(4.7)

Thus the curvature

2

2

dk

Edof the energy band decides the effective mass of an electron in a crystal.

The curvature 2

2

dk

Edis negative for an electron at the top of the valence band and positive at the

bottom of conduction band as shown in Fig. 2.5. Hence the effective mass of an electron is negative

near the top of valence band and positive near the bottom of conduction band. The motion of valence

band electrons with negative charge and negative mass can be equivalently described by motion of

holes having positive charge and positive effective mass in same direction.

Fig. 2.5

For a free electron,

2

2

2

2

hhP

m

PE

kP

m

kE

2

22

mdk

Ed 2

2

2

From equation (4.7)

m* = m

i.e., the effective mass is same as its mass for a free electron.

2.5 Quantum free Electron Theory

The quantum free electron theory developed by Summerfield retained some of the assumptions of the

classical free electron theory and introduced a few new assumptions. The quantum free electron

theory was successful in eliminating certain drawback of the classical theory. The assumptions in

quantum free electron theory are listed below.

Assumptions

1) The valence electrons are free to move inside the metal.

2) The electrons are confined to the metal by potential barrier at the boundaries. The

potential is constant inside the metal.

3) The electrostatic forces of attraction between the free electrons and the ion cores are

negligible.

4) The electrostatic forces of repulsion amongst the free electrons are negligible.

5) The energies of electrons are quantized and the distribution of electrons in the

allowed discrete energy levels is according to Pauli‟s exclusion principle which

prohibits more than one electron in single quantum state.

2.6 Fermi – Dirac Statistics, Fermi Factor and Fermi Energy Different types of particles have different probabilities of occupying the available energy

states. Statistically, there are three different types of particles:

i) Identical particles which are so far apart that they can be distinguished and their wave

function do not overlap. The Maxwell – Boltzmann distribution function is applicable to

such particles. For example, molecules of a gas.

ii) Identical particles with 0 or integer spins with overlapping wave functions which cannot

be distinguished. Such particles are called „bosons‟ and obey the Bose – Einstein

probability distribution for energy. For example, photons.

iii) Identical particles for which the spin is an odd integer multiple of half

....

2

5,

2

3,

2

1which cannot be distinguished form one another. These particles are called

fermions and obey the Fermi - Dirac probability distribution function. Electrons are

example of this type.

The Fermi – Dirac probability distribution function, also known as Fermi function, is

kTe

EfFEE

/1

1)(

)(

(6.1)

Where )(Ef Probability of an electron occupying the energy state E

EF = Fermi energy

k = Boltzmann constant

and T = Absolute temperature

For T = 0 K, if E > EF

0)(

1

1

1

1)(

Ef

eEf

i.e. no electron can have energy greater than the Fermi energy at 0 K, It means that all energy states

above the Fermi energy are empty an 0 K.

For T = 0 K, if E < E F,

e

Ef1

1)(

01

1

1)( Ef

i.e. all electrons occupy energy states below the Fermi energy at 0 K.

Thus, all energy states below Fermi energy are filled and energy states above Fermi energy are empty

at 0K. Hence Fermi energy as the highest occupied energy state at 0K.

For T > 0 K, if E = EF,

f (EF) = 11

1

1

10

e

f (EF) = 2

1

i.e. the Fermi energy level represents the energy

state with a 50 % probability of being filled if

forbidden gap does not exist as in the case of good

conductors.

The Fermi functions described by equation (6.1) is

shown for different temperatures in Fig. 2.6

Fig. 2.6

Valence band: it is an energy band which contains the outermost valence electrons.

Conduction band: it is an allowed energy band next to the valence band which contains

free electrons that take part in conduction.

Forbidden band: it is an energy band between the valence and conduction band. The energies in

this band are forbidden, i.e. not allowed, for the electron. To raise the electrons from valence band to

conduction band, energy equivalent to the forbidden energy gap has to be supplied to the electrons.

2.7 Classification of Solids on the Basis of Band Theory Solids can be classified into conductors, insulators and semiconductors based on their energy

band structure

1) Conductors: In conductors, the valence band and the conduction band overlap. There is no

forbidden band. The electrons can be made to move and constitute a current by applying a small

potential different. The resistivity of conductors is very low and increases with temperature. Hence

the conductors are said to have a positive temperature coefficient of resistance. The energy band

structure is shown in Fig. 2.7 Metals like copper, silver, gold, aluminum etc. are good conductors of

electricity.

Fig. 2.7

2) Insulators: insulators have a completely filled valence band and an empty conductions band

which are separated by a large forbidden band. The band gap energy is large (of the order of 5 eV).

Hence large amount of energy is required to transfer electrons from valence band to conduction band.

The insulators have very low conductivity and high resistivity. Diamond, wood glass etc. are

insulators.

3) Semiconductors: In semiconductors, the valence band is completely filled and the conduction

band is empty at absolute zero temperature. The valence band and conductions band are separated by

a small forbidden band of the order of 1 eV. Hence, compared to insulators, smaller energy is

required to transfer the electrons from valence band to conduction band. Hence the conductivity is

better than insulators but not as good as the conductors. Silicon and germanium are semiconductors

having band gap energies of 1.1 eV and 0.7 eV respectively. Some compounds formed between group

III and group V elements like gallium arsenide (GaAs) are also semiconductors.

As temperature is increased, elements from valence band jump to conduction band leaving a vacancy

in valence band which is known as hole. The free elements in conduction band and the holes in

valence band take part in conduction. Hence conductivity increase and resistivity decreases with

increase in temperature. The semiconductors are said to have a negative temperature coefficient.

Conductivity in a Semiconductor In a semiconductor, the current is due to free electrons as well as holes. The current due to

electrons can be written as,

Ie = n e a v e

where ve = Drift velocity of holes

n = number density of holes

Similar the current due to holes is

peaI

Where = Drift velocity of holes

and p = number density of holes

The total current is

pneaI

III

e

e

The current density

pneJ

aJ

e

1

Also, EJ

Ep

Ene e

,pe

E

the mobility of electrons

and ,pE

the mobility of holes.

pe pne

For intrinsic semiconductors, n = p = ni is called the density of intrinsic charges carries.

peien

For n-type semiconductors, n >> np

ene

As each donor atom contributes one free electron, n is also the density of donor impurity atoms. For

p-type semiconductors, p >> n

ppe

Where hn is the density of holes which is same as the density of acceptor impurity atoms.

2.8 Hall Effect

When magnetic field is applied perpendicular to direction of current in a conductor, a potential

different develops along as axis perpendicular to both current and magnetic field. This effect is known

as hall effect and the potential difference developed is known as Hall voltage.

Force on a charge „q‟ moving with velocity

due to a magnetic field B

is given by,

BqF

(8.1)

for an electron, q = - e

BeF

BeF (8.2)

The forces on positive and negative charge carriers and the corresponding Hall voltages developed are

shown in Fig. 6.14.1 (a) and (b) respectively. The magnetic field is directed into the plane of the paper

and the current is flowing upwards.

From Fig.6.14 (a) and (b) it is clear that opposite polarity of hall voltage will be developed for the

types of charge carries for the same direction s of current and magnetic field. Therefore this effect can

be used to find the polarity of charge carriers and hence to find whether a given semiconductor is p-

type or n-type.

Hall voltage and Hall coefficient Consider a conductor of rectangular cross section of dimensions w × d in which current I flows along

x-axis, magnetic field is applied along z-axis and Hall voltage develops along y-axis which is

measured across terminals 1 and 2 as shown in Fig.2.8.

Fig. 2.8

The dimension „w‟ is parallel to the direction of magnetic field and „d‟ is parallel to the axis along

which hall voltage develops.

Let VH = Hall voltage and EH, the corresponding electric fields

and = Drift velocity of charges

Under equilibrium conditions, force on charge carriers due to magnetic field will be balanced by the

force on them due to EH.

BdV

Bd

V

d

VE

BE

BqqE

H

H

HH

H

(8.3)

From equation (8.1)

nqa

I

nqaI

Substituting is equation (8.3),

nqw

IBV

da

nqa

IBdV

H

H

The quantity nq

1is the reciprocal of charge density and is defined as the Hall coefficient „RH‟,

nqRH

1 (8.6)

From equation (8.6)

aR

IBdV

H

H (8.7)

As VH, B, d and a are measurable quantities, RH and hence charge density nq can be determined using

equations (8.7)

Once charge density is known, we can determine mobility of charge carriers using nq

Conductivity can be determined using

Ra

l

1

Thus the Hall Effect can be used to determine

i) Whether charge carriers are positive or negative which in turn determines whether

semiconductor is n-type or p-type.

ii) Density of charge carriers

iii) Mobility of charge carriers.

2.9 BLOCH THEOREM

In the quantum mechanical description of an electron in a crystal, a realistic view is of a single

electron in a perfectly periodic potential which has the periodicity of the crystal. The Bloch theorem

defines the form of the one electron wave functions for this perfectly periodic potential. For

simplicity, we consider one dimensional crystal of lattice parameter a, shown in fig 2.9, with the

potential energy of the electron (x) being periodic with period a i.e.

)()( axx (9.1)

The Schrodinger equation of an electron moving in one dimensional electrostatic potential field with

potential energy (x) is

0](x) [2

22

2

Em

dx

d

(9.2)

Since (x) is periodic, the solution of equation (9.2) can be easily written if we solve a general

differential equation

0)()(2

2

xxfdx

d

(9.3)

Where f (x) has a period a i.e.

)()( axfxf (9.4)

Since equation (9.3) represents a second order differential equation, it will have the general solution

as

Fig. 2.9 Potential in a perfectly periodic crystal

Surface potential barrier is shown at the ends.

)()()( xDhxCx g

( 9.5)

Where g (x) and h (x) are solution of equation (9.3), Also g (x + a) and h (x + a) will be solutions of

equation (8.3) because f (x )= f (x + a). These solutions g(x + a) and h (x + a) also can be expressed as

a linear combination of g (x) and h (x) equation (9.5), as

)()()( 11 xhBxgAaxg

)()()( 22 xhBxgAaxh (9.6)

Substitution in equation (9.5) will give

)()()()()( 2121 xhBDCBxgDACAax (9.7)

Since )( ax can always be expressed in form

)()( xax (9.8)

Where is a constant, Comparing (9.7) and (9.8), we get

0)( 21 DAAC

0)( 21 BDCB

(9.9)

Solution of equation (9.9) is the solution of the determinant

E

or 0)()( 122121

2 BABABA

(9.10)

This quadratic equation (9.10) gives two values of as 1 and 2 Now if these constants 1

and 2 are taken as a

ike 1

1 and aik

e 2

2

(9.11)

and let us define 1u (x) and 2u (x) as

)()( 1

1 xexuxk

)()( 2

2 xexuxk

(9.12)

then, use of equation (9.11) and (9.8) yields

)()()( 1

)()(

111 xeaxeaxu

axkaxk

)()()( 1

)( 111 xuxexeexkaikaxk

(9.13)

Similarly, )(2 xu will be periodic with period a. equation (9.12) can be rewritten in the form

ikx

kk exux )()(

(9.14)

Where )(xuk has the same periodicity as the )(x . This is Bloch‟s function, which on extension to

three-dimensional case is

rki

kk erur )()(

(9.15)

and the Bloch theorem can be stated that has the same form as a plane wave of vector

k modulated

by a function )(

ruk that depends on

k and has the periodicity of crystal potential.

Let us now try to find the probability density * using the Bloch function given by equation

(9.14). In the thN unit cell,

)()( )( NaxueNax k

Naikx

k

(9.16)

)(xuee k

ikxikNa [From equation (9.13)]

)(xe k

ikNa [From equation (9.16)]

Similarly

)()( ** xeNax k

ikNa

k

(9.17)

This gives )()()()( ** xxNaxNax kkkk

(9.18)

So we obtain the same probability density in each unit cell of the crystal. The same is true for a three

dimensional wave function.

If the crystal is finite, as the practical case is, then suitable boundary conditions must be satisfied at

the surfaces. For example, in a crystal of N atoms, if the wave function has to be single valued,

then we must have from equation (9.16)

or )()()( xxeNax kk

ikNa

k

1ikNae

or nNa

nkn ,

20, 1, 2,…N

(9.19) So the solutions, which satisfy the Schrodinger equation, are found only for certain discrete energy

Eigen values corresponding to values of nk given by equation (9.19). Since N is large, there will be

many allowed values of nk and they may be thought of forming a quasi-continuous range, hence the

notion of bands of energy Eigen values in solids.

2.10 SUMMARY:-

This chapter describes the behavior of electrons in solids. Kronnig penny model describe that the

motion of electrons in a periodic potential in crystals gives rise to certain allowed energy bands

separated by forbidden energy bands. Also show the concept of effective mass according to which

mass of the electrons changes inside the solids due to interaction of electron with atoms. This chapter

includes the classification of solids on the basis of band theory, which explains how some solids are

insulator some are semiconductor and other are metals. Hall Effect can be used to find the polarity of

charge carriers and also helped on finding which type of semiconductor we have used. Bloch theorem

defines the form of one electron wave function for perfectly periodic potential.

2.11 CHECK YOUR PROGRESS

Q. 1. Explain Kronig-Penney Model

Hint: - Refer to topic no. 2.3

Q. 2. Explain the concept of Effective Mass of an Electron

Hint: - Refer to topic no. 2.4

Q. 3. Explain Quantum Free Electron Theory

Hint: - Refer to topic no. 2.6

Q. 4. Explain what do you understand by Hall Effect in detail and what is Hall Coefficient.

Hint: - Refer to topic no. 2.8

Q. 5. Explain Bloch Theorem

Hint: - Refer to topic no. 2.9

UNIT 03

MAGNETISM

3.1Introduction

The phenomenon of magnetism attracts everybody. The following aspects of magnetism are generally

familiar to you-

A compass needle always points north, an observation reportedly made around 2500 BC by the

Chinese.

The stickers or alphabets with magnet sticks on the iron fridge or cupboard but falls down from

the aluminum window frames or copper, stainless steel objects.

Magnets have south and north poles. The like poles repel and unlike poles attract.

The magnetism is produced by the „electrical current‟ in a solenoid or by an „electronic‟

revolution in a permanent magnet i.e. always due to charge in motion.

The magnets have wide range of applications starting from a minute magnetism generated by our

brain, heart waves to huge magnets used in dock yards or particle accelerators. In our day-to-day

life we encounter with audio-video tapes, computer disks, motors, generators etc.

3.2 Objective

Define or explain the magnetism, you will find it difficult to put in proper words. St the post-graduate

level, let us just review some of the basic concepts learned by you during the college courses.

3.3 Magnetic field and its strength:

One of the most fundamental ideas in magnetism is the concept of magnetic field. A field is

generated whenever there is a change in the energy within a volume of space. In a most familiar way

the presence of the field is sensed by the forces (attractive or repulsive) or by the torque. Thus, the

attractive force on magnetic stickers and the torque on compass needle are the manifestation of the

magnetic field. The region of space where the force or torque is experienced is known as magnetic

field. A magnetic field is produced whenever there is electrical charge in motion. It was first observed

by H. C. Oersted in the year 1819 that the electric current flowing in a conductor produces certain

force. In case of permanent magnets, there is no conventional current. However, the electrons orbiting

around nucleus and spinning around them create so called „Ampere currents‟. These currents are

responsible for the magnetism therein. Although the electrons are mandatory constituents of all

materials, the magnetism is not exhibited by all of them. Few materials have ‟adhoc‟ magnetism very

few have „permanent‟ magnetism. The reasons for this variation will be clear to you as we proceed

through this course. The magnetic force is expressed in terms of the magnetic field strength (H). Its

magnitude, obviously, depend on the current, length of current carrying conductor and the distance at

which it is measured. Thus, for elemental conductor, the magnetic field strength is given by

ulir

H

.4

12

(3.1)

Where i is the current in ampere flowing through an elemental length 1 of a conductor, r is the radial

distance and u is the unit vector. There for the field strength is A/m.

Magnetic Flux :)(

In a conventional way, the presence of magnetic field is indicated by the magnetic flux lines, as

shown in Fig. 3.1.

Fig. 3.1

The flux lines are closed loops i.e. there is no source or sinks of magnetic flux. The magnetic flux is

measured in terms of Weber. The way magnetic field is created by the current, the changing magnetic

flux can generate e.m.f. Thus the Weber is defined as the amounts of magnetic flux which, when

reduced to zero in one second produces an e.m.f. of 1 volt in a turn of coli.

Magnetic Induction (B)

Whenever magnetic field is generated in a medium, it responds in a certain way. As a result some

induction is shown by the medium. The magnetic induction can be defined in terms of flux density.

According, the flux density of one Weber per Square meter is equivalent to the magnetic induction of

one Tesla. Alternatively, the magnetic induction is said to be one Tesla, when a force of one Newton

per meter is generated by one ampere current in a perpendicular direction. Generally, for a

nonmagnetic media the induction is proportional to the applied field strength. i.e.

B= H

(3.2)

Where, is known as the permeability. The permeability of a free space )( 0 is a universal constant

having value7

0 104 Henry/m. For the magnetic media, the equation (3.2) is not valid as the

response of the material is modified through a quantity called Magnetization.

3.4 Magnetic Dipole Moment:

The electric charge is the fundamental unit of electricity. We conveniently indicate the flow of charge

through a completed circuit, where we assume a source and link of charge. In case of magnetism, we

adopt a „pole‟ view. Note that the „pole‟ is a fictitious just conceived for the simplicity. Any smallest

magnet has a south and a north poles. Thus we cannot have a monopole like the charge. Instead, the

dipole is the fundamental unit of magnetism. A closed current loop having area a and current I,

generates magnetic dipole moment given by-

m =i .A

(4.1)

The dipole moment is always directed perpendicular to the plane of

loop as shown in Fig.3.2. The unit of magnetic dipole moment is A.m2.

In individual atoms, the magnetic dipole moments are due to angular,

spin motions of electron as well as spin motion of nucleus. Unless

these moments cancel each other, each atom will behave as a magnetic

dipole.

Magnetization (M):

In general, the magnetic dipoles inside a material are oriented randomly and there is no (or very less)

net magnetic moment. When external magnetic field is applied, these dipoles respond by aligning

themselves along the field direction. Then there can be bet magnetic dipole moment. The number of

such magnetic moments per unit volume is termed as magnetization. Thus,

M=N m /V

(4.2)

From equation (4.2) the unit of magnetization is A/m. Now, the total number of magnetic flux lines

will have two contributions: one from applied field (H) and second from magnetization (M). The

magnetic induction in a free space as per equation 4.2 is H0 . Similarly the induction due to the

magnetization will be M0 . Therefore: the net magnetic induction is-

)(000 MHMHB

(4.3)

The quantity M0 = 1 is often termed as magnetic polarization or intensity of magnetization. It is

noteworthy that the units H and M are the effect of magnetic field on magnetizations whereas B is

more convenient for the effect on currents. The distinction between B and H is really important hen

magnetic materials are present.

Magnetic Susceptibility:

In the presence of the magnetic field, different materials respond differently. It is mostly depends on

the presence and alignment of the magnetic dipole moments within. As we increase the strength of

applied magnetic field, more dipoles will be aligned or even some more will be created. It means.

HM

HM i.e. ./ HM

The proportionality constant )( or the ration of magnetization to the magnetic field strength is

knows as magnetic susceptibility. Since, M and H have the same unit is a unit-less quantity. It is

the basic parameter on the basis of which the materials are classified.

3.5 Elementary ideas of classification:

According to the classification of magnetic materials diamagnetic, Paramagnetic and ferromagnetic is

based on how the material reacts to a magnetic moment induced in them that opposes the direction of

the magnetic field. This property is now understood to be a result of electric currents that are induced

in individual atoms and molecules. These currents produce magnetic moments in opposition to the

applied field. Many materials are diamagnetic: the strongest ones are metallic

Bismuth and organic molecules, such as benzene, that have a cyclic structure, enabling the easy

establishment of electric currents.

Paramagnetic behavior results when the applied magnetic field lines up all the existing magnetic

moments of the individual atoms or molecules that makes up the material. This results in an overall

materials moment that adds to the magnetic field. Paramagnetic materials usually contain transition

metals or rare earth elements that possess unpaired electrons. Para magnetism in non-metallic

substances is usually characterized by temperature dependence; that is, the size of an induced

magnetic moment varies inversely with the temperature. This is a result of the increasing difficulty of

ordering the magnetic moments of the individual atoms along the direction of the magnetic field as

the temperature is raised.

A ferromagnetic substance is one that, like iron, a magnetic moment even when the external magnetic

field is reduced to zero. This effect is a result of a strong interaction between the magnetic moments

of the individual atoms or electrons in the magnetic substance that causes them to line up parallel to

one another. In ordinary circumstances, ferromagnetic materials are divided into regions called

domains; in each domain, the atomic moments are aligned parallel to one another. Separate domains

have total moments that do not necessarily point in the same direction. Thus, although an ordinary

piece of iron might not have an overall magnetic field. Therefore aligned the moments of all the

individual domain. The energy expended in reorienting the domains from the magnetized back to the

demagnetized state manifests itself in a lag in response, known as hysteresis. Ferromagnetic materials,

when heated, eventually lose their magnetic properties. This loss becomes complete above the Curie

temperature, named after the French physicist Pierre Curie, who discovered it in 1895. (The Curie

temperature of metallic iron is about 770o C/1418

o F.)

In recent years, a greater understanding of the atomic origins of magnetic properties has resulted in

the discovery of types of magnetic ordering. Substances are known in which the magnetic moments

interact in such a way that it is energetically favorable for them to line up anti-parallel; such materials

are called anti ferromagnetism. There is a temperature analogous to the Curie temperature called the

Neel temperature, above which anti ferromagnetic order disappears.

Other, more complex atomic arrangements of magnetic moments have also been found.

Ferromagnetic substances have at least two different kinds of atomic magnetic moment, which are

oriented anti-parallel to one another. Because the moments are of different size, a net magnetic

moment remains, unlike the situation in an anti ferromagnetic, where all the magnetic moments cancel

out. Interestingly, lodestone is a ferromagnetic rather than a ferromagnetic; two types of iron ion, with

different magnetic moments, occur in the material. Even more complex arrangements have been

found in which the magnetic moments are arranged in spirals. Studies of these arrangements have

provided much information on the interactions between magnetic moments in solids.

A representative list of various types of magnetic materials is given in Table 3.1

Table 3.1 Types of magnetic materials

Theory of Paramagnetism:

Atoms and ions with unfilled shells have non-zero magnetic moments, which, may be aligned by a

magnetic field. This alignment is off-set by the randomizing action of thermal agitation and the

analysis of these competing processes leads to an expression for magnetic susceptibility as a function

of temperature.

Before the advent of the quantum theory Langevin analyzed this problem classically, this entails

considering that all orientations are possible in an applied field. This Langevin analysis is applicable

to the description of the magnetic behavior of systems

containing units, which large values of magnetic moment. In fact, there are number of possible

explanations for the paramagnetic behavior. These are mainly,

1. Langewin‟s theory of non-interacting magnetic moments.

2. Van-vlack model of Localized moment.

3. Weiss theory of molecular field.

4. Pauli‟s model of paramagnetic.

5. Quantum theory of paramagnetic.

Besides, there are some laws based on the experimental observations like Curie law and Curie-Weiss

law, which indicate the temperature dependence of the susceptibility. Here, we will discuss only the

quantum theory of paramagnetic.

3.6 Quantum theory of Para magnetism:

Unlike the classical theories, the quantum theory of par magnetism is based on the assumption that the

permanent magnetic dipole moments are not free rotating but are restricted to a finite set of

orientations relative to the applied field.

Let N be the number of atoms per unit volume and J be the total angular momentum quantum number

such that J = L + S with L, S as orbital and spin quantum numbers respectively.

The magnetic moment of an atom is proportional to the total angular momentum J i.e.

JJ

(6.1)

Where, is called gyro magnetic ratio and is given by-

Bg

(6.2)

)1(2

)1()1()1(1

JJ

LLSSJJg

(6.3)

Where, g is Lande‟s g factor or spectroscopic splitting factor given by, and B =eh/2m is the Bohr

Magnetron

Thus, Jg BJ

(6.4)

In the presence of magnetic field H, the magnetic moment J will presses about the field direction

such that the resolved component of the magnetic moment in field direction is Mjg B where MJ is

magnetic quantum number having values MJ=-J,-J-1,-(J-2),…0,1,2,…J-2,J-1,J. The potential energy

will be-

HgME BJ 0

(6.5)

The average value of the magnetic moment in the field direction is given by-

kTE

kTE

j

j

j

j

j

ava

/exp

/exp

(6.6)

J

j

Bj

Bj

J

j

BJ

kTHgM

kTHgMgM

)/exp(

)/exp(

0

0

(6.7)

At normal temperature, TKHgM BBJ 0 i.e. 1/0 TKHgM BBJ

Or, )/exp( 0 TKHgM BBJ =1+ TKHgM BBJ /0

Therefore,

J

J

TkgM

J

J

TkgM

BJ

ava

B

BJ

B

BJ

gM

)1(

)1(

0

0

J

J

J

J

J

B

B

J

J

J

J

J

B

BJB

MTk

Hg

MTk

HgMg

0

20

1

But,

J

J

J

J

J

J

JJ

JJJMMJ

3

)12)(1(;0;121 2

Tk

JJHg

B

Bava

3

)1(.0

22

(6.8)

Therefore, total magnetization due to N number of atoms is M=N ava

Tk

JJHNM

B

Bg

3

)1(.0

22

(6.9)

The paramagnetic susceptibility will be

Tk

JJN

H

M

B

Bg

3

)1(.0

22

(6.10)

or T

C

Tk

N

B

B 3

.2

0

2

(6.11)

where )1(.22 JJg is known as Effective Bohr Magnetron number

Thus, the susceptibility has form C/T and C= BB KN 3/2

0

2 is known as the Curie constant.

The equation (3.17) is found true in the cases of monatomic gases. However, distinct discrepancies

arise for the transition group elements. According to van Vleck, it may be due to the fact that all

atoms may not have the same values of L, S and J.

At low temperature or strong fields, the situation will be rather different. In this case the

magnetization will be given by-

j

j

BJ

j

j

BJBJ

kTHgM

kTHgMgMN

M

/exp

/exp

0

0

(6.12)

Now let ,/0 TKHg BB Then

J

J

J

J

j

j

JB

xM

xMMNg

M

)exp(

)exp(

J

J

JeB xMdx

dNg )exp(log)(

Using the values of MJ=J,J-1,J-2,….,-(J-1),-J

JxxJJ

eB eeedx

dNgM x .......(log)( )1(

).......1(log)( 2JxxJx

eB eeedx

dng

x

xJJx

eBe

ee

dx

dNg

1

1log)(

)12(

x

xJxJx

eBe

eee

dx

dNg

1

.log)(

)2/sinh(

2

12sinh

log)(x

xJ

dx

dNg eB

)2/sinh(log

2

12sinhlog)( xx

J

dx

dNg eeB

)2/coth(

2

1

2

12coth

2

12)( xx

JJNg B

)2/coth(

2

1

2

12coth

2

12)( Jy

Jy

J

J

J

JJNg B

Jxywhere ...

)(yBNgJ jB

Where

)2/coth(

2

1

2

12coth

2

12)( Jy

Jy

J

J

J

JyB j is called Brillouin function and

y=Jx=Jg B 0H/KBT . This is a general equation for the par magnetism and the equation (6.12) is a

special case for low field and normal temperature. The Brillouin function varies from zero when the

applied field is zero to unity when the field is infinite. The saturation value of the magnetization is

Ms=NgJ B

3.7Theory of Ferromagnetism:

In diamagnetic materials, the magnetic moments are induced by the application of external field

whereas in paramagnetic materials, already exist ion dipoles are aligned in the field direction. In

ferromagnetic materials, the dipoles exist and oriented even in the absence of external field. The

spontaneous existence of magnetic dipoles can be attributed to the uncompensated electron spins. For

example, Fe with atomic number 26 has electronic configuration- 1s2

2s2

2p6

3s2

3p6 3d

6 4s

2. These

electrons are arranged in various orbitals in accordance with the Hand‟s rule as follows-

Note that in 3d orbital 6 electrons are arranged in such a way that two electrons are paired with spin

up and down while the other four electrons are in spin up configurations. The paired electrons cancel

magnetic moments of each other. However, net spin magnetic moments of 4 Bohr magnetrons is

always present due to the 4 unpaired electrons. In the bound states of atoms, the net spin magnetic

moments are affected due to the proximity of other atoms. As a result, the average spin moment is

reduced to 2.22 Bohr magnetrons. This magnitude of the magnetic moment is of the same order of the

paramagnetic materials. It means that the large magnetization of ferromagnetic substance is not only

due to the moments of individual atoms.

There are various theories of ferromagnetism based on two mutually exclusive approaches-

1. Localized moment model

2. Itinerant electron model.

The localized moment model assumes that the magnetic moments of atoms are due to electrons

localized to that particular atom and the magnetic properties of the solids are merely the perturbation

of the magnetic properties of the individual atoms. The theories based on this approach include Weiss

Mean Field theory, Weiss Domain theory, Heisenberg‟s model of Exchange interaction and Quantum

theory of Ferromagnetism. The approach works well for the rare earth metals. However, for the

elements of „3d‟ series eg. Fe, CO,Ni) the outer electrons are relatively free to move through the

solid. In such cases, the itinerant electron model is more realistic. The Pauli‟s free electron theory and

Slater‟s Band theory are examples of this second approach. The fundamental calculations are

extremely difficult with the itinerant electron theories. therefore, in spite of its realistic nature, they

are less preferred and the interpretations of magnetic properties are more conveniently made on the

basis of localized moment models.

3.8 Quantum theory of Ferromagnetism:

A paramagnetic material can behave as a ferromagnetic, if there is some internal interaction to alight

the magnetic moment. Weiss proposed such internal field that couples the magnetic moment of

adjacent atoms. Such interaction is called the exchange or Molecular or Weiss Field (BE). The

orientation effect of this field is opposed by the thermal agitation. At elevated temperature the

alignment is destroyed completely and the material becomes paramagnetic. According to Weiss Mean

Field approximation, exchange field is proportional to the magnetization.

MBE or MBE (8.1)

Where is known as Weiss constant, which determines the strength of interaction between magnetic

dipoles and it is temperature independent. Thus, each magnetic moment experiences a field due to

magnetization (alignment) of all other magnetic moments. Therefore if B is the applied magnetic

field, then the total field will be

BT = B + BE or HT = 0 (H+ M) (8.2)

Now, the quantum theory of ferromagnetism can be derived from the quantum theory of par

magnetism. A perturbation in the form of exchange field M has to be introduced in this case.

According to the quantum theory of paramagnetic, the energy of electron in the magnetic field BT will

be E = -Mjg B BT. Thus, with the perturbation term of the exchange field the energy is,

E = -Mjg B 0 (H+ M) (8.3)

Moreover, the magnetization at normal temperature i.e. in the limit E<< KBT will be

).(3

)1(.22

MHTK

JJNgM

B

B

(8.4)

TK

HJJNgM

TK

JJNg

B

B

B

B

3

).1(.

3

)1(.1

2222

Therefore the ferromagnetic susceptibility,

eBB

B

TT

C

JJNgTK

JJNg

H

M

)1.(2

)1.(2

0

2

22

(8.5)

This equation is similar to the Curie Weiss Law with

BB KJJNgC 3/)1.(2

0

2

and BB KJJNgT 3/)1.(2

0

2

Thus, the quantum theory also leads qualitatively the similar results to the classical theory.

3.9 Domain Theory of Ferromagnetism:

One of the most celebrated theories of ferromagnetism is the domain theory. It was originally

proposed by Weiss in the year 1906-07 and was based on the ideas of Ampere, Weber and Ewing

about magnetism. It can be understood through the concept of domains the origin of domains.

The concept of Domains:

According to Weiss proposal, the ferromagnetic solids are divided into a large number of small

regions termed as „Domains‟. The dimensions of these domains can be from few microns to the size

of the crystal and typically it consists of 1012

to1015

magnetic dipole moments aligned in a single

direction. It means, different domains have different directions of magnetization so that net magnetic

moment is zero. Thus, the immediate consequences of the domain theory are:

1. The magnetic dipole moments exist permanently.

2. There is alignment of these moments (ordered state) even in the demagnetized state.

3. The demagnetized state is characterized by the random alignments of the domains only.

4. The process of magnetization consists of reorientation of the domains. In weak applied field,

the volume of domain having magnetization in the field direction increases whereas in strong

applied field the magnetization of the domain is rotated in the field direction.

Origin of Domains:

We know that the Ferro magnets do not get magnetized spontaneously. Instead, the magnetization has

to be done by the application of external magnetic field. The empirical explanation for this fact was

given by Weiss through the postulation of the domains. The existence of the domains was further

confirmed several experiments like Barkhausen effect, Bitter patterns, faraday Effect, Kerr effect and

also through

Magneto-optic and transmission electron microscopy (TEM)

techniques. One such typical domain pattern observed through Kerr

Effect is shown in Fig.3.3

The first explanation for the origin of domains was given by Landau

and Lifschitz in 1930. They showed that the existence of the domains

is the consequence of the energy minimization. There are mainly

three contributions to the potential energy viz-

1. Magneto static or exchange energy

2. Anisotropy energy

3. Magneto striation energy

The magneto static energy ideas to the interaction of the magnetic dipole moments, which keep them,

aligned. The anisotropy energy is the natural consequence of the preferred directions of

magnetization. It is found that the ferromagnetic crystals have easy and hard directions of

magnetization i.e. higher fields are required to magnetize the crystal in a particular direction. E.g. for

iron crystal (100) is easy and (111) is hard direction whereas for Nickel (111) is easy and (100) is hard

direction. The excess of energy required for the magnetization along hard direction is called the

anisotropy energy. The process of magnetization can induce a slight change in the dimensions of the

samples. This change is obtained by the work done against the elastic restoring forces. The associated

energy is known as magnetostrictive energy. The origin of domains can be clearly understood by

considering the domain structures of a single crystal as shown in Fig. 3.4

In Fig 3.4a), the entire specimen has a single magnetic domain with the magnetic poles (S, N) formed

on the surfaces of the crystal. The magneto static energy of such configuration is 2)8/1( B dv. Its

value is quite high of the order of 106erg/cm

3. This much energy is required to assemble the atomic

magnets into single domain. This energy is reduced by approximately one half, if the crystal is

divided into two domains as shown in Fig 3.4b). In this case the two domains are magnetized in

opposite directions and the flux lines are completed on the same surfaces. The subdivision of

domains, then the magneto static energy will be reduced approximately by the factor 1/N. Further ,

there is another possible configuration as shown in Fig 3.4d). In this case, there are triangular domains

near the end faces of crystals. The magnetizations in the vertical and the triangular domains are at an

angle of 90o and the boundaries of the domains bisect this angle by making equal angles of 45

o with

both directions of magnetization. The surface domains complete the flux circuit and therefore are

referred as domains of closure. In such configuration, there are no free poles and the magneto static

energy is zero. The domains of closure are nucleated at the boundary of the specimen or at certain

defects inside. During magnetization processes, those domains are swept out certain defects inside.

During magnetization processes, those domains are swept out at higher fields only.

Thus, the origin of the domain structure is attributed to the possibility of lowering the energy of the

system by going from a saturated configuration of high energy (Fig 3.4a) to a domain configuration of

the lowest energy (Fig. 3.4d). The introduction of a domain raises the overall energy of the system,

therefore the division into domains only continues while the reduction in magneto static energy is

greater than the energy required to form the domain wall. The energy associated a domain wall is

proportional to its area. The schematic representation of the domain wall, is shown in Fig 3.5.

It illustrates that the dipole moments of the atoms within the wall are not pointing in the easy direction

of magnetization and hence are in a higher energy state. In addition, the atomic dipoles within the wall

are not at 180o

to each other and so the exchange energy is also raised within the wall. Therefore, the

domain wall energy is an intrinsic property of a material depending on the degree of magneto-

crystalline anisotropy and the strength of the exchange interaction between neighboring atoms. The

thickness of the wall will also vary in relation to these parameters as a strong magneto-crystalline

Anisotropy will favor a narrow wall, whereas strong exchange interaction will favor a wider wall. A

minimum energy can therefore be achieved with a specific number of domains within a specimen.

This number of domains will depend on the size and shape of the sample (which will affect the

magneto static energy) and the intrinsic magnetic properties of the material (which will affect the

magneto static energy and the domain wall energy).

3.10 Magnetic Resonance The course material, so far, is related to the response of materials to the static magnetic field.

However, there are many dynamical magnetic effects, which as frequency dependent. These effects

are particularly associated with the spin angular momentum of the electrons and the nuclei. The wide

known such phenomena can be identified as follows-

Nuclear Magnetic Resonance (NMR)

Electron Paramagnetic (ESR)

Nuclear Quadric pole Resonance (NQR)

Ferromagnetic Resonance (FMR)

Spin Wave Resonance (SWR)

Anti ferromagnetic Resonance (AFMR)

The first observation of the magnetic resonance was made by E. Zaviosky kin 1945 through electron

spin resonance absorption in the paramagnetic salt MnSo4 using 2.75 Ghz field. The magnetic

resonance can provide significant information about the samples. It can be categorized as follows.

1. The fine structure of absorption can reveal the electronic structure of defects.

2. The changes in line width of absorption pattern indicate the spin motion.

3. The position of resonance line reveals the internal magnetic field.

4. It can elaborate the collective spin resonance.

i. Nuclear Magnetic Resonance:

Theory: The atomic nuclei have an angular momentum due to the4 „nuclear spin‟ in the case of

electrons, the total angular momentum is the result of spin and orbital quantum number (I) its total

angular momentum is Ih. The „spinning „ nuclei will give rise to nuclear magnetic moment.

hl (10.1)

Where is called the gyro magnetic ratio.

In the presence of applied magnetic field (Ba) along direction the magnetic moment will process

about the field direction with resolved component

hmlz (10.2)

Where the allowed values of ml are I, I-1, I-2,…….-I

The potential energy of this interaction will be give by

U= aIaz BhmB . (10.3)

The nucleus with mI =2

1 will have two energy level viz., uI =

2

rhBa and

.2

2

Baru

The splitting of energy levels of nucleus is shown in Fig 3.6.

Fig.3.6 Nuclear energy levels

The energy levels of these two levels can be denoted in terms of frequency such tatt,

12 UU

e.i. aB

aB (10.4)

The equation (10.5) is the fundamental condition for magnetic resonance absorption.

It means that the resonance can be observed only if an alternating magnetic field of frequency is

applied.

For Proton, 810675.2 x (s

-1tesla

-1)

= 2.675 x 108. B (s

-1)

Orv =w/2 =42.58 x 106 B (s

-1)

Thus, the frequency (v) is of the order of few MHZ, which is in radio frequency range.

Experimental: When a sample of magnetically active nuclei is placed into an external magnetic field, the magnetic

fields of these nuclei align themselves with the external field into various orientations. Each of these

spin-states will be nearly populated with a slight excess in lower energy levels. During the

experiment, electromagnetic radiation is applied to the sample with energy exactly equivalent to the

energy separation id two adjacent spin states. Some of the energy is absorbed and the alignment of

one nucleus‟ magnetic field reorients from a lower energy to a higher energy alignment (spin

transition). By sweeping the frequency, and hence the energy, of the applied electromagnetic

radiation, a plot of frequency versus energy absorption can be generated. This plot is the NMR

spectrum as shown in Fig 3.7.

Fig.3.7

In a homogeneous system with only one kind of nucleus, the NMR spectrum will show only a single

peak at a characteristic frequency. In real samples the nucleus is influenced by its environment.

Some environments will increase the energy separation of the spin-states giving a spin transition at a

higher frequency. Others will lower the separation consequent lowering the frequency at which the

spin transition occurs. These changes in frequency are called the chemical shift of the nucleus and

can be examined in more detail. By examining the exact frequencies (chemical shift) at which the

spin transitions occur conclusions about the nature of the various environment can be made.

This simply type of experiment, where the frequency is swept across a range, is know as a

continuous wave (CW) experiment. One simple variation on this experiment is to hold the frequency

of the electromagnetic radiation constant and to sweep the strength of the applied magnetic field

instead. The energy separation of the spin states will increase as the external field becomes stronger.

At some point, this energy separation matches the energy of the electromagnetic radiation and

absorption occurs. Plotting energy absorption versus external magnetic field strength produces the

identical NMR spectrum as shown in Fig. 3.8.

Fig. 3.8

In fact, the MNR spectrum obtained by plotting magnetic field increasing to the right will be a mirror

image of the spectrum where frequency is plotted increasing to the right. Low energy transitions (to

the left) in a frequency swept experiment will not occur until very high magnetic fields (to the right)

in a magnetic field swept experiment. Early NMR spectrometers swept the magnetic field since it

was too difficult to build the very stable swept RF sources that NMR required. Even today where this

is no longer required, NMR spectra are still plotted with magnetic field increasing to the right.

Technological advances have made the CW experiment obsolete and today virtually all NMR

experiments are conducted using pulse methods. These methods are inherently much more sensitive

and this explains part of their popularity.

A simplified block diagram of the NMR apparatus is shown in Fig. 3.9. The diagram does not show

all the functions of each module, but it does represent the most important functions of each modular

component of the spectrometer. The Pulse Programmer creates the pulse stream that gates the

synthesized oscillator into radio frequency pulse bursts, as well triggering the oscilloscope on the

appropriate pulse. The pulse bursts are amplified and sent to the transmitter coils in the sample

probe. The current bursts in these coil produce a homogeneous 12 Gauss rotating magnetic field at

the sample. These

are the time dependent B, fields that produce the precession of the magnetization, referred to as the

90o or 180

o pulses. The transmitter coils are wound in a Helmholtz configuration to optimize rf

magnetic field homogeneity.

Nuclear magnetization processing in the direction transverse to the applied constant magnetic field

(the so called x-y plane) induces an EMF in the receiver coil, which is then amplified by the receiver

circuitry. This amplified radio frequency (15 MHZ) signal can be detected (demodulated) by two

separate and different detectors. The Amplitude Detector rectified the signal and has an output

proportioned to the peak amplitude of the processionals signal. This detector is used to record both

the free induction decays and the spin echoes signals.

The second detector is a Mixer, which effectively multiplies the precession signal from the sample

magnetization with the master oscillator. Its output frequency is proportional to the difference

between the two frequencies. This Mixer is essential for determining the proper frequency of the

oscillator. The magnet and the nuclear

Magnetic moment of the protons uniquely determined the processionals frequency of the nuclear

magnetization. The oscillator is tuned to this precession frequency when a aero-beat output signal of

the mixers obtained. A dual channel scope allows simultaneous observations of the signals from both

detectors. The field of the permanent magnet is temperature dependent so periodic adjustments in the

frequency are necessary to keep the spectrometer on resonance.

There are both an analog and a digital (sampling) oscilloscope for the measurements. Both have their

strengths and weaknesses. The digital oscilloscope samples its input signals at a fixed high

frequency, which, if the signal you are measuring has similar frequency components, can lead to

spurious displays. The analog oscilloscope samples continuously, but it is slightly less convenient for

making numerical measurements of pulse heights or widths.

The NMR apparatus are widely used in the hospitals with a common name as Magnetic Resonance

Imaging (MRI). It can detect the minute magnetic signals generated by organs. By using

sophisticated instrumentation and image processing software, it can produce a three dimensional

color. A schematic view of a typical MRI scanner and a brain scan is shown in Fig.3.10.

ii Electron Paramagnetic Resonance:

Electron paramagnetic resonance (EPR) and/or electron spin resonance (ESR) is defined as the form

of spectroscopy concerned with microwave-induced transitions between magnetic energy levels of

electrons having a net spin and orbital angular momentum. The term electron paramagnetic resonance

and the symbol EPR are

Preferred and should be used for primary indexing. The correspondence between NMR and ESR is

very close, of ESR it is necessary to have an unpaired electron instead of an unpaired nuclear spin (as

in NMR). Further, it is also necessary to provide an external static magnetic field to generate the

ground and excited state energy levels. The major difference is that ESR spectroscopy has a higher

absorption frequency than NMR spectroscopy. Consequently, the sensitivity of EPR is considerably

higher. However, the absorptions lines are also significantly broader.

The electron spin resonance spectrum of a free radical or coordination complex with one unpaired

electron is the simplest of all forms of spectroscopy. The degeneracy of the electron spin states

characterized by the quantum number, ,2/1sm is lifted by the application of a magnetic field and

transitions between the spin levels are induced by radiation of the appropriate frequency, as shown in

Fig. 3.11. If unpaired electron in radicals were indistinguishable from free electrons, the only

information content of an ESR spectrum would be the integrated intensity, proportional to the radical

concentration.

Fig.3.11

An unpaired electron interacts with its environment, and the details of ESR spectra depend on the

nature of those interactions. There are two kinds of environmental interactions which are commonly

important in the ESR spectrum of a free radical: (i) To the extent that the unpaired electron has

unquenched orbital angular momentum, the total magnetic moment is different from the spin-only

moment (either larger or smaller, depending on how the angular momentum vectors couple). It is

customary to ump the orbital and spin angular moment together in an effective spin and to treat

the effect as a shift in the energy of the spin transition. (ii) The electron spin energy levels are split by

interaction with nuclear magnetic moments –the nuclear hyperfine interaction.

Each nucleus of spin I splits the electron spin levels into )12( I sublevels. Since transitions are

observed between sublevels with the same values of mI, nuclear spin splitting of energy levels is

mirrored by splitting of the resonance line.

When an electron is placed magnetic field, the degeneracy of the electron spin energy levels is lifted

as shown in Figure 3.11 and as described by the spin

Hamiltonian:

zBs SBgH . (10.5)

In equation (10.5), g is called the g-va;ie(g=2.00232 fro a free electron), B is the Bohr magnetron

(9.274x10-28

JG-1

), B is the magnetic field strength in Gauss, and Sz is the z-component of the spin

angular momentum operator (the field defines the z direction). Energy level splitting in a magnetic

field is called the Zeeman effect, and the Hamiltonian of equation (10.5) is sometimes referred to as

the electron Zeeman Hamiltonian. The electron spin energy levels are easily found by application of

HS to the electron spin Eigen functions corresponding to :2/1sm

)(2/1 BgE B (10.6)

The difference in energy between the two levels is.

BgEEE B (10.7)

It corresponds to the energy of a photon required to cause a transition, i.e.

Bghy B (10.8)

or hBgv B / (10.9)

where hg B / =0.9348 x10-4 cm-1

G-1

. Magnetic fields of up to 15 KG are easily obtained with an

iron-core electromagnet; thus we could use radiation with up to 1.4 cm-1

(y < 42 GH or > 0.71cm).

Radiation with this kind of wavelength is in the microwave region. Microwaves are normally handled

using wave guides designed to transmit over a relatively narrow frequency range. Wave guides look

like rectangular cross-section pipes with dimensions on the order of the wavelength to be transmitted.

The ESR sensitivity (net absorption) increases with decreasing temperature and with increasing

magnetic field strength. Since field is proportional to microwave frequency, in principle sensitivity

should be greater for K-band or Q-band spectrometers than for X-band. However, since the K- or Q-

band waveguides are

Smaller, samples are also necessarily smaller, usually more than canceling the advantage of a more

favorable Boltzmann factor. Under ideal conditions, a commercial X-band spectrometer can detect the

order f 1012

spins (10-12

moles) at room temperature. The ESR is a remarkably sensitive technique,

especially compared with NMR,

Because the spin levels are so nearly equally populated, magnetic resonance suffers from a problem

not encountered in higher energy forms of spectroscopy: An intense radiation field will tend to

equalize the populations, leading to a decrease in net absorption; this effect is called “saturation”. A

spin system returns to thermal equilibrium via energy transfer to the surroundings, a rate process

called spin-lattice relaxation, with a characteristic time, T1, the spin-lattice relaxation time (rate

constant = 1/T1). Systems with a long T1(i.e., spin systems weakly coupled to the surroundings) will

be easily saturated; those with shorter T1 will be more difficult to saturate.

Experimental

Although many spectrometer designs have been produced over the years, the vast majority of

laboratory instruments are based on the simplified block diagram shown in figure 3.12. Microwaves

are generated by the Klystron tube and the power level is adjusted with the Attenuator. The Circulator

behaves like a traffic circle: microwaves entering from the Klystron are routed toward the Cavity

where the sample is mounted. Microwaves reflected back from the cavity (less when power is being

absorbed) are routed to the diode detector, and any power reflected from the diode is absorbed

completely by the Load. The diode is mounted along the E-vector of the plane-polarized microwaves

and thus produces a current proportional to the microwave power reflected from the cavity. Thus in

principle, the absorption of microwaves by the sample could be detected by noting a decrease in

current in the micro ammeter.

Fig.3.12

In practice, of course, such a d.c. measurement would be far too noisy to be useful. The solution to the

signal-to-noise ratio problem is to introduce small amplitude field modulation. An oscillating

magnetic field is superimposed on the d.c. field by means of small coils, usually built into the cavity

walls. When the field is in the vicinity of a resonance line, it is swept back and forth through part of

the line, leading to an a.c. component in the diode current. This a.c. component is amplified using a

frequency selective amplifier, thus eliminating a great deal of noise. The modulation amplitude is

normally less than the line width.

Thus the detected a.c. signal is proportional to the change in sample absorption. As shown in Figure

3.13, this amounts to detection of the first derivative of the absorption curve. It takes a little practice

to get used to looking at first-derivative spectra, but there is a distinct advantage; first derivative

spectra have much better apparent resolution than do absorption spectra. Indeed, second-derivative

spectra are even better resolved (though the signal-to-noise ratio decreases on further differentiation).

Fig. 3.13

b. Mossbauer spectroscopy:

The Mossbauer spectroscopy is a versatile technique that can be used to provide information in many

areas of science such as Physics, Chemistry, Biology and Metallurgy. It can give very precise

information about the chemical, structural, magnetic and time-dependent properties of a material. The

discovery of recoilless gamma ray emission and absorption, is referred as the Mossbauer Effect‟, after

its discoverer Rudolph Mossbauer, who first observed the effect in 1957 and received the Nobel Prize

in Physics in 1961 for his work.

ii. The Mossbauer Effect:

In a free nucleus during emission or absorption of a gamma ray it recoils due to conservation of

momentum, just like a gun recoils when firing a bullet, with a recoil energy ER greater than the

transition energy due to the recoil of the absorbing nucleus. To achieve resonance the loss of the

recoil energy must be overcome in some way.

As the atoms will be moving due to random thermal motion the gamma-ray energy has a spread of

values ED caused by the Doppler effect. This produces a gamma-

ray energy profile as

shown in Fig 3.15. To produce a resonant signal the two energies

need to overlap and this is shown in the shaded area. This area is

shown exaggerated as in reality it is extremely small, a millionth

or less of the gamma-rays are in this region, and impractical as a

technique.

What Mossbauer discovered is that when the atoms are within a solid matrix the effective mass of the

nucleus is very much greater. The recoiling mass is now effective the mass of the whole system.

Making ER and ED very small. If the gamma-ray energy is small enough the recoil of the nucleus is

too low to be transmitted as a phonon (vibration in the crystal lattice) and so the whole system recoils,

making the recoil energy practically zero: a recoil-free event. In this situation, as shown in Fig 3.16,

the emitted and absorbed gamma-ray have the same energy: resonance!

If emitting and absorbing nuclei are in identical, cubic environments then the transition energies are

identical and this produces a spectrum as shown in Fig 3.17- a single absorption line.

Now for achieving resonant emission and absorption can we use it to probe the tiny hyperfine

interaction between an atom‟s nucleus and its environment? The natural line width of the excited

nuclear state is related to the average lifetime of the excited state before it decays by emitting the

gamma-ray. For the most common Mossbauer isotope, 57

Fe, this line width is 5x10-9

eV. Compared to

the Mossbauer gamma-ray energy is 14.4 KeV this gives a resolution of 1 in 1012

or the equivalent of

one sheet of paper in the distance between the Sun and the Earth.

As resonance only occurs when the transition energy of the emitting and absorbing nucleus match

exactly this effect is isotope specific. The relative number of recoil-free events (and hence the strength

of the signal) is strongly dependent upon the gamma-ray energy and so the Mossbauer effect is only

detected in isotopes with very low lying excited states. Similarly the resolution is dependent upon the

lifetime of the excited state. These two factors limit the number of isotopes that can be used

successfully for Mossbauer spectroscopy e.g. Fe, Ru, Sn, Sb, Te,I, W, Au, Eu, Gd. Dy, etc. The most

used is 57

Fe, which has both a very low energy gamma-ray and long-lived excited state.

iii. Fundamentals of Mossbauer Spectroscopy

The energy changes caused by the hyperfine interactions are very small, of the order of billionths of

an eV. Such miniscule variations of the gamma-ray are quite easy to achieve by the use of the Doppler

effect i.e. by moving the gamma-ray source towards and away from the absorber. This is most often

achieved by oscillating a radioactive source with a velocity of a few mm/s and recording the spectrum

in discrete velocity steps. Fractions of mm/s compared to the speed of light (3x1011

mm/s) give the

minute energy shifts necessary to observe the hyperfine interactions. For convenience the energy scale

of a Mossbauer spectrum is quoted in terms of the source velocity, as

With an oscillating source we can now modulate the energy of the gamma-ray in very small

increments. With the modulated gamma-ray energy matches precisely the energy of a nuclear

transition in the absorber the gamma-rays are resonantly absorbed and we see a peak. As we‟re seeing

this in the transmitted gamma-rays the sample must be sufficiently thin to allow the gamma-rays to

pass through, the relatively low energy gamma-rays are easily attenuated. In Fig 3.18 the absorption

peak occurs at 0mm/s, where source and absorber are identical. The energy levels in the absorbing

nuclei can be modified by their environment in three main ways: by the Isomer Shift, Quadruple

Splitting and Magnetic Splitting.

The isomer shift arises due to the non-zero volume of the nucleus and the electron charge density due

to s-electrons within it. This leads to a monopole (Coulomb)

interaction, altering the nuclear energy levels. Any difference in the s-electron environment between

the source and absorber thus produces a shift in the resonance energy of the transition. This shifts the

whole spectrum positively or negatively depending upon the s-electron density, and sets the spectrum.

The isomer shift is useful for determining valence states, bonding states, electron shielding and the

electron-drawing power of electronegative groups. The nuclei in states with an angular momentum

quantum number I>1/2 have a non-spherical charge distribution. This produces a nuclear quadruple

moment. In the presence of an asymmetrical electric field (produced by an asymmetric electronic

charge distribution or ligand arrangement) this splits the nuclear energy levels. The magnitude of

splitting is related to the nuclear quadruple moment and electronic charge distribution. Thus,

additional energy levels are available for the absorption spectra. In the presence of a magnetic field

the nuclear spin moment exp experiences a dipolar interaction with the magnetic field ie Zeeman

splitting. The magnetic field splits nuclear levels with a spin of I into (2I+1) sub-states. The

transitions between the excited state and ground state can only occur wherem1 changes by 0 or1. The

line positions are related to the splitting of the energy levels, but the line intensities are related to the

angle between the Mossbauer gamma-ray and the intensities can give information about moment

orientation and magnetic ordering. These interactions, Isomer Shift, Quadruple Splitting and Magnetic

Splitting, alone or in combination are the primary characteristics of many Mossbauer spectra.

iv. Application to impure crystal

The impurities, which can be displaced from their regular positions, in a crystal lattice are termed as

off-center impurities. They can be considered as existing in an asymmetric double potential well.

Such atoms can change their position as the temperature changes. Unfortunately there are often many

other phenomena in such systems that can mask the off-centering effect. The Mossbauer spectroscopy

provides a good tool for observing this effect. Firstly the movement of the off-center atom within the

lattice will change the symmetry of the electric field. Mossbauer spectroscopy is also isotope and site

specific, meaning we can observe the off-center single component without any masking from other

elements or effects.

A compound which was thought to exhibit off-centering is Pb0.8 Sh0.2, with tin as an off-center atom.

The typical spectra of this sample at 200K and 2K are shown in Fig 3.19. There are two components:

one from an off-center site and one from a normal single-potential site. It can be seen in the

highlighted region that the small component develops from a single line to a (broad) doublet. The

quadrupole splitting is increasing, indicating the electric field environment around these particular

atoms has become more asymmetrical. This is consistent with atom moving within an asymmetric

potential well. The other component shows no variation in quadruple splitting. A series of spectra

were taken in a temperature cycle and a hysteresis was observed in the values of quadruple splitting.

These results show that tin is an off-center atom in this compound and that there are two tin sites

within it: one normal and one off-center.

3.11SUMMARY:-

This chapter describes the concept of magnetism, according to which magnetic field is produced due

to motion of electic charge and define the various type of magnetic materials like diamagnetic,

Paramagnetic and ferromagnetic, which are based on how the material reacts to a magnetic moment

induced in them that opposes the direction of the magnetic field. This chapter also explains the theory

of Para magnetism, in which we have derived the formula for magnetization. It also cover the

quantum theory of ferromagnetism Which explain the origin of domain, which are formed when

ferromagnetic materials are divided in to small region Magnetic resonance is related to the response of

materials to the static magnetic field. It elaborates the fundamental condition for magnetic resonance

absorption and experimental verification and their application.

3.12CHECK YOUR PROGRESS

Q.1. Explain Magnetic Field And Its Strength.

Hint: - Refer to topic no. 3.3

Q.2. Explain Magnetic Dipole Moment.

Hint: - Refer to topic no. 3.4

Q.3. Explain the Theory of Ferromagnetism.

Hint: - Refer to topic no. 3.7

Q.4. Explain Quantum Theory of Ferromagnetism.

Hint:- Refer to topic no. 3.8

Q.5. Define Magnetic Resonance

Hint: - Refer to topic no. 3.10

Books Recommended

1. Solid State Physics : A.J. Dekker

2. Introduction to Solide State Physics : C. Kittel

3. Solid State Physics : Azaroff.

4. Thin Film Technology : K.L. Chopra

UNIT –IV DEFECTS IN CRYSTALS

Structure

I. 4.0Introduction 4.1 Objectives

4.2 Point Defect in ionic crystals and metals

4.3 Diffusion in solids

1.3.1 Type of Diffusion

1.3.2 Diffusion Mechanisms

1.3.3 Diffusion Coefficient

1.3.4 Applications

4.4 Ionic Conductivity

4.5 Colour Centres

1.5.1 F- Centres

1.5.2 V-Centres

4.6 Excitions

4.7 General Idea of Luminescence

4.8 Dislocations & Mechanical Strength of Crystals

4.9 Plastic Bahaviour

4.10 Type of Dislocations

4.11 Stress field of Dislocations

4.12 Grain Boundaries

4.13 Etching- Types of Etching

4.14 Let Us Sum Up

4.15 Check Your Progress: The Key

II. 4.0 INTRODUCTION

Up to now, we have described perfectly regular crystal structures, called ideal crystals and

obtained by combining a basis with an infinite ·space lattice. In ideal crystals atoms were

arranged in' a regular way. However, the structure of real crystals differs from that of ideal

ones. Real crystals always have certain defects or imperfections, and therefore, the

arrangement of atoms in the volume of a crystal is far from being perfectly regular.

Natural crystals always contain defects, often in abundance, due to the uncontrolled

conditions under which they were formed. The presence of defects which affect the

colour can make these crystals valuable as gems, as in ruby (chromium replacing a

small fraction of the aluminium in aluminium oxide: Al203). Crystal prepared in

laboratory will also always contain defects, although considerable control may be

exercised over their type, concentration, and distribution.

The importance of defects depends upon the material, type of defect, and properties, which

are being considered. Some properties, such as density and elastic constants, are

proportional to the concentration of defects, and so a small defect concentration will have a

very small effect on these. Other properties, e.g. the colour of an insulating crystal or the

conductivity of a semiconductor crystal, may be much more sensitive to the presence of

small number of defects. Indeed, while the term defect carries with it the connotation of

undesirable qualities, defects are responsible for many of the important properties of

materials and much of material science involves the study and engineering of defects so that

solids will have desired properties. A defect free, i.e. ideal silicon crystal would be of little

use in modern electronics; the use of silicon in electronic devices is dependent upon small

concentrations of chemical impurities such as phosphorus and arsenic which give it desired

properties. Some simple defects in a lattice are shown in Fig. 1.

There are some properties of materials such as stiffness, density and electrical conductivity

which are termed structure-insensitive, are not affected by the presence of defects in

crystals while there are many properties of greatest technical importance such as

mechanical strength, ductility, crystal growth, magnetic

Defects in crystals and Elements of Thin Films

Key

a = vacancy (Schottky defect)

b = interstitial

c = vacancy – interstitial pair (Frenkel defect)

d = divacancy

e = split interstitial

= vacant site

Fig. 1 Some Simple defects in a lattice

Hysteresis, dielectric strength, condition in semiconductors, which are termed structure

sensitive are greatly affected by the-relatively minor changes in crystal structure caused by

defects or imperfections. Crystalline defects can be classified on the basis of their geometry

as follows:

(i) Point imperfections

(ii) Line imperfections

(iii) Surface and grain boundary imperfections (iv) Volume imperfections

The dimensions of a point defect are close to those of an interatomic space. With linear

defects, their length is several orders of magnitude greater than the width. Surface defects

have a small depth, while their width and length may be several orders larger. Volume

Defects in Crystal

defects (pores and cracks) may have substantial dimensions in all measurements, i.e. at least

a few tens of A0. We will discuss only the first three crystalline imperfections.

4.1 OBJECTIVES

III. The Main aim of this unit is to study defect in crystals after

going through the unit you should be able to Describe the type of defects

Explain the diffusion in crystal

Explain the color center and excitations

Explain the type of dislocation

IV. 4.2 POINT DEFECT IN IONIC CRYSTALS AND METALS

The point imperfections, which are lattice errors at isolated lattice points, take place due to

imperfect packing of atoms during crystallisation. The point imperfections also take place

due to vibrations of atoms at high temperatures. Point imperfections are completely local in

effect, e.g. a vacant lattice site. Point defects are always present in crystals and their present

results in a decrease in the free energy. One can compute the number of defects at

equilibrium concentration at a certain temperature as,

n = N exp [-Ed / kT] (1)

Where n - number of imperfections, N - number of atomic sites per mole, k - Boltzmann

constant, Ed - free energy required to form the defect and T - absolute temperature. E is

typically of order l eV since 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, or 10 parts per million. For many purposes, this fraction would be intolerably

large, although this number may be reduced by slowly cooling the sample.

(i) Vacancies: The simplest point defect is a vacancy. This refers to an empty

(unoccupied) site of a crystal lattice, i.e. a missing atom or vacant atomic site [Fig. 2

(a)] such defects may arise either from imperfect packing during original

crystallisation or from thermal vibrations of the atoms at higher temperatures. In the

latter case, when the thermal energy due to vibration is increased, there is always an

Defects in crystals and Elements of Thin Films

increased probability that individual atoms will jump out of their positions of lowest

energy. Each temperature has a

Fig. 2 Point defects in a crystal lattice

corresponding equilibrium concentration of vacancies and interstitial atoms (an interstitial

atom is an atom transferred from a site into an interstitial position). For instance, copper can

contain 10-13 atomic percentage of vacancies at a temperature of 20-25°C and as many as

0.01 % at near the melting point (one vacancy per 104 atoms). For most crystals the-said

thermal energy is of the order of I eV per vacancy. The thermal vibrations of atoms increases

with the rise in temperature. The vacancies may be single or two or more of them may

condense into a di-vacancy or trivacancy. We must note that the atoms surrounding a

vacancy tend to be closer together, thereby distorting the lattice planes. At thermal

equilibrium, vacancies exist in a certain proportion in a crystal and thereby leading to an

increase in randomness of the structure. At higher temperatures, vacancies have a higher

concentration and can move from one site to another more frequently. Vacancies are the

most important kind of point defects; they accelerate all processes associated with

displacements of atoms: diffusion, powder sintering, etc.

(ii) Interstitial Imperfections: In a closed packed structure of atoms in a crystal if the atomic

packing factor is low, an extra atom may be lodged within the crystal structure. This is

known as interstitial position, i.e. 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

Defects in Crystal

atoms [Fig. 2(b)], otherwise it will produce atomic distortion. The defect caused is known as

interstitial defect. In close packed structures, e.g. 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. Interstitialcies may also be single interstitial, di-interstitials, and tri-interstitials. We

must note that vacancy and interstitialcy are inverse phenomena.

(iii) Frenkel Defect: Whenever a missing atom, which is responsible for vacancy

occupies an interstitial site (responsible for interstitial defect) as shown in Fig. 2(c),

the defect caused is known as Frenkel defect. Obviously, Frenkel defect is 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 the positive ions, being smaller in size, get

lodged easily in the interstitial positions.

(iv) Schottky Defect: These imperfections are similar to vacancies. This defect is caused,

whenever a pair of positive and negative ions is missing from a crystal [Fig. 2(e)]. This type of

imperfection maintains charge neutrality. Closed-packed structures have fewer

interstitialcies and Frenkel defects than vacancies and Schottky defects, as additional energy

is required to force the atoms in their new positions.

Check Your Progress 1

Notes : (i) Write your answer in the space given below

(ii) Compare your answer with those given at the end of the unit

Explain Frenkel and Schottky defects?

……………………………………………………………………………………….

………………………………………………………………………………………….

………………………………………………………………………………………….

……………………………………………………………………………………….

(v) Substitutional Defect: Whenever a foreign atom replaces the parent atom of the lattice

and thus occupies the position of parent atom (Fig. 2(d)], the defect caused is called

substitutional defect. In 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.

(vi) Phonon: When the temperature is raised, thermal vibrations takes place. This results in

the defect of a symmetry and deviation in shape of atoms. This defect has much effect on

the magnetic and. electric properties.

All kinds of point defects distort the crystal lattice and have a certain influence on the

physical properties. In commercially pure metals, point defects increase the electric

resistance and have almost no effect on the mechanical properties. Only at high

concentrations of defects in irradiated metals, the ductility and other properties are reduced

noticeably.

In addition to point defects created by thermal fluctuations, point defects may also· be

created by other means. One method of producing an excess number of point defects

at a given temperature is by quenching (quick cooling) from a higher temperature.

Another method of creating excess defects is by severe deformation of the crystal

lattice, e.g., by hammering or rolling. We must note that the lattice still retains its

general crystalline nature, numerous defects are introduced. There is also a method of

creating excess point defects is by external bombardment by atoms or high-energy

particles, e.g. from the beam of the cyclotron or the neutrons in a nuclear reactor. The

first particle collides with the lattice atoms and displaces them, thereby causing a

Defects in crystals and Elements of Thin Films

point defect. The. number of point defects created in this manner depends only upon

the nature of the crystal and on the bombarding particles and not on the temperature.

Check Your Progress 2

Notes : (i) Write your answer in the space given below

(ii) Compare your answer with those given at the end of the unit

What are crystal defects and how are they classified?

……………………………………………………………………………………

……………………………………………………………………………………

……………………………………………………………………………………

……………………………………………………………………………………

4.3 DIFFUSION

Diffusion refers to the transport of atoms through a crystalline or glassy solid. Many

processes occurring in metals and alloys, especially at elevated temperatures, are associated

with self-diffusion or diffusion. Diffusion processes play a crucial 'role in many solid-state

phenomena and in the kinetics of micro structural changes during metallurgical processing

and applications; typical examples include phase transformations, nucleation,

recrystallization, oxidation, creep, sintering, ionic conductivity, and intermixing in thin film

devices. Direct technological uses of diffusion include solid electrolytes for advanced battery

and fuel cell applications, semiconductor chip and microcircuit fabrication and surface

hardening of steels through carburization. The knowledge of diffusion phenomenon is

essential for the introduction of a very small concentration of an impurity in a solid state

device:

V. 4.3.1 Types of Diffusion (i) Self Diffusion: It is the transition of a thermally excited atom from a site of

crystal lattice to an adjacent site or interstice.

(ii) Inter Diffusion: This is observed in binary metal alloys such as the Cu-Ni

system.

iii) Volume Diffusion: This type of diffusion is caused due to atomic movement

Defects in Crystal

in bulk in materials. (iv) Grain Boundary Diffusion: This type of diffusion

is caused due to atomic movement along the grain boundaries alone.

(v) Surface Diffusion: This type of diffusion is caused due to atomic movement

along the surface of a phase.

VI. 4.3.2 Diffusion Mechanisms

Diffusion is the transfer of unlike atoms which is accompanied with a change of

concentration of the components in certain zones of an alloy. Various mechanisms have

been proposed to explain the processes of diffusion. Almost all of these mechanisms are

based on the vibrational energy of atoms in a solid. Direct-interchange, cyclic, interstitial,

vacancy etc. are the common diffusion mechanisms. Actually, however, the most probable

mechanism of diffusion is that in which the magnitude of energy barrier (activation energy)

to be overcome by moving atoms is the lowest. Activation energy depends on the forces of

interatomic bonds and crystal lattice defects, which facilitate diffusion transfer (the

activation energy at grain boundaries is only one half of that in the bulk of a grain). For metal

atoms, the vacancy mechanism of diffusion is the most probable and for elements with a

small atomic radius (H, N and C), the interstitial mechanism. Now, we will study these

mechanisms.

(i) Vacancy Mechanism: This mechanism is a very dominant process for diffusion in FCC, BCC

and HCP metals and solid solution alloy. The activation energy for this process comprises the

energy required to create a vacancy and that required to move it. In a pure solid, the

diffusion by this mechanism is shown in Fig. 3(a). Diffusion by the vacancy mechanism can

occur by atoms moving into adjacent sites that are vacant. In a pure solid, during diffusion

by this mechanism, the atoms surrounding the vacant site shift their equilibrium positions to

adjust for the change in binding that accompanies the removal of a metal ion and its valency

electron. We can assume that the vacancies move through the lattice and produce random

shifts of atoms from one lattice position to another as a result of atom jumping.

Concentration changes takes place due to diffusion over a period of time. We must note that

vacancies are continually being created and destroyed at the surface, grain boundaries and

suitable interior positions, e.g. dislocations. Obviously, the rate of diffusion increases rapidly

with increasing temperature.

Defects in crystals and Elements of Thin Films

Fig. 3. Various Diffusion mechanism

(a) Vacancy mechanisms (b) Interstitial mechanisms (c) Two atoms interchange mechanisms (d) Four atoms interchange mechanism

Defects in Crystal

If a solid is composed of a single element, i.e. pure metal, the movement of thermally

excited atom from a site of the crystal lattice to an adjacent site or interstice is called self

diffusion because the moving atom and the solid are the same chemical-element. The self-

diffusion in metals in which atoms of the metal itself migrate in a random fashion

throughout the lattice occurs mainly through this mechanism.

We know that copper and nickel are mutually soluble in all proportions' in solid state and

form substitutional solid solutions, e.g., plating of nickel on copper. For atomic diffusion, the

vacancy mechanism is shown in Fig. 4.

Fig. 4. Vacancy mechanism for atomic diffusion

(a) Pure solid solution, and

(b) Substitutional solid solutions.

(ii) The Interstitial Mechanism: The interstitial mechanism where an atom

changes positions using an interstitial site does not usually occur in metals for elf-

diffusion but is favored when interstitial impurities are present because of the low

activation energy.

When a solid is composed of two or more elements whose atomic radii differ

significantly, interstitial solutions may occur. The large size atoms occupy lattice

sites where as the smaller size atoms fit into the voids (called as interstices)

created by the large atoms. We can see that the diffusion mechanism in this case is

similar to vacancy diffusion except that the interstitial atoms stay on interstitial

sites (Fig. 3(b)). We must note that activation energy is associated with interstitial

diffusion because, to arrive at the vacant site, it must squeeze past neighbouring

atoms with energy supplied by the vibrational energy of the moving atoms.

Obviously, interstitial diffusion is a thermally activated process. The interstitial

mechanism process is simpler since the presence of vacancies is not required for

the solute atom to move. This mechanism is vital for the following cases:

(a) The presence of very small atoms in the interstices of the lattice affect to a

great extent the mechanical properties of metals.

(b) At low temperatures, oxygen, hydrogen and nitrogen can be diffused in metals

easily.

(iii) Interchange Mechanism: In this type of mechanism, the atoms exchange

places through rotation about a mid point. The activation energy for the process is

very high and hence this mechanism is highly unlikely in most systems.

Two or more adjacent atoms jump past each other and exchange positions, but the

number of sites remains constant (Fig. 3 (c) and (d)). This interchange may be

two-atom or four-atom (Zenner ring) for BCC. Due to the displacement of atoms

surrounding the jumping pairs, interchange mechanism results in severe local

distortion. For jumping of atoms in this case, much more energy is required. In

this mechanism, a number of diffusion couples of different compositions' are

produced, which are objectionable. This is also termed as Kirkendall's effect.

Kirkendall was the first person to show the inequality of diffusion. By using an ά

brass/copper couple, Kirkendall showed that Zn atoms diffused out of brass into

Defects in crystals and Elements of Thin Films

Defects in Crystal

Cu more rapidly than Cu atoms diffused into brass. Due to a net loss of Zn atoms,

voids can be observed in brass.

From theoretical point of view, 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).

4.3.3 Diffusion Coefficient: Fick’s Laws of Diffusion

Diffusion can be treated as the mass flow process by which atoms (or molecules)

change their positions relative to their neighbours in a given phase under the

influence of thermal energy and a gradient. The gradient can be a concentration

gradient; an electric or magnetic field gradient or a stress gradient. We shall

consider mass flow under concentration gradients only. We know that thermal

energy is necessary for mass flow, as the atoms have to jump from site to site

during diffusion. The thermal energy is in the form of the vibrations of atoms

about their mean positions in the solid.

The classical laws of diffusion are Fick's laws which hold true for weak solutions

and systems with a low concentration gradient of the diffusing substance, dc/dx

(= C2 – C1/X2 – X1), slope of concentration gradient.

(i) Fick's First Law: This law describes the rate at which diffusion occurs. This

law states that

dtadx

dcDdn (2)

i.e. 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 (m2/s). The 'minus' sign implies

that diffusion occurs in the reverse direction to concentration gradient vector, i.e.

from the zone with a higher concentration to that with a lower concentration of

the diffusing element.

The equation (2) becomes:

adx

dcD

dt

dn

dx

dcD

dt

dn

aJ

1 (3)

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, i.e. flux J is flow per unit cross sectional

Defects in crystals and Elements of Thin Films

area per unit time. Obviously, J is proportional to the concentration gradient.

The negative sign implies that flow occurs down the concentration gradient.

Variation of concentration with x is shown in Fig. 5. We can see that a large

negative slope corresponds to a high diffusion rate. In accordance with Fick's

law (first), the B atoms will diffuse from the left side. We further note that the

net migration of B atoms to the right side means that the concentration will

decrease on the left side of the solid and increase on the right as diffusion

progress.

This law can be used to describe flow under steady state conditions. We find that it is

identical in form to Fourier's law for heat flow under a constant temperature gradient

and Ohm's law for current flow under a constant electric field gradient. We may see

that under steady state flow, the flux is independent of time and remains the same at

any cross-sectional plane along the diffusion direction.

Diffusion coefficient (diffusivity) for a few selected solute solvent systems is given in

Table.1.

Fig.5 Model for illustration of diffusion : Fick’s first law. We note that the concentration of B

atoms in the direction indicates the concentration profile

Defects in Crystal

Parentheses indicate that the phase is metastable

(ii) Fick’s second Law: This is an extension of Fick’s first law to non steady flow. Frick’s first

law allows the calculation of the instaneous mass flow rate (Flux) past any plane in a solid

but provides no information about the time dependence of the concentration. However,

commonly available situations with engineering materials are non-steady. The concentration

of solute atom changes at any point with respect to time in non-steady diffusion.

If the concentration gradient various in time and the diffusion coefficient is taken to be

independent of concentration. The diffusion process is described by Frick’s second law

which can be derived from the first law:

dx

dcD

dx

dc

dt

dc

(4)

Equation 4 Fick’s second law for unidirectional flow under non steady conditions. A solution

of Eq. (4)given by

)4/(exp),( 2 DtxDt

Atxc (4a)

Where A is constant Let us consider the example or self diffusion or radioactive nickel atoms

in a non-radioactive nickel specimen. Equation (4a) indicates that the concentration at x = 0

falls with time as r-12 and as time increases the radioactive penetrate deeper in the metal

block [Fig.6 ] At time t1 the concentration of radioactive atoms at x = 0 is c1= A/(Dt1)1/2. At a

distance x1 = 0 (Dt1)1/2 the concentration falls to 1/e of c1. At time t2 . the concentration at x =

0 is c2 = A/(Dt2)1/2 and this falls to 1/e and x2 = 2 (Dt2)

1/2 . These results are in agreement with

experiments.

If D is independent of concentration, Eq. (4) simplifies to

2

2

dx

cdD

dt

dc (5)

Even though D may vary with concentration, solutions to the differential Eq. 5 are quite

commonly used for practical problems, because of their relative simplicity. The solution to

Eq.5 for unidirectional diffusion from one medium to another a cross a common interface is

of the general form.

DtxerfBAtxc 2/(),( (5a)

Where A and B are constant to be determined from the initial and boundary conditions of a

particular problem. The two media are taken to be semi-infinite i.e. only one end of each of

them, which the interface is defined. The other two ends are at an infinite distance The

initial uniform concentrations of the diffusing species in the two media are different, with an

Fig.6. The radioactive sheet of Nickel (shown by shaded section) is kept in contact with a block of

nonradioactive nickel. Radioactive atoms diffuse from the sheet to the bulk metal and can be

detected as a function of time. In figure, the diffusion of atoms is shown (i) for t= 0 (ii) for t1, and (iii)

t2 with t2> t1

Defects in crystals and Elements of Thin Films

abrupt change in concentrations at the interface erf in eqn.5 (a) stands for error function,

which is

Dtx

dDt

xerf

2/

0

2 )exp(2

2

(5a)

is an integration variable, that gets deleted as the limits of the integral are substituted.

The lower limits of the integral is always zero, while the upper limit of the integral is the

quantity, whose function is to be determined 2 is a normalization factor. The diffusion

coefficient D (m2/s) determines the rate of diffusion at a concentration gradient equal to

unity. It depends on the composition of alloy, size of grains, and temperature.

Solutions to Fick’s equations exist for a wide variety of boundary conditions, thus permitting

an evaluation of D from c as a function of x and t.

A schematic illustration of time dependence of diffusion is shown in fig7. The curve

corresponding to the concentration profile at a given instant of time t1 is marked by t1. We

can see from fig.7 at a later time t2, the concentration profile has changed. We can easily

see that this changed in concentration profile is due to the diffusion of B atoms that has

occurred in the time interval t2-t1 The concentration profile at a still later time t3 is marked

by t3 . Due to diffusion, B atoms are trying to get distributed uniformaly throughout the

solid salutation. From Fig. 7 Its is evident that the concentration gradient becoming less

negative as time increases. Obviously, the diffusion rate becomes slower as the diffusion

process progress.

Defects in Crystal

Fig. 7. Time dependence of diffusion (Fick’s second law)

Dependence of Diffusion Coefficient on Temperature

The diffusion coefficient D (m2/s) determines the rate of diffusion at a oncentration gradient

equal unity. It depends on the composition of alloy, size of grains, and temperature.

The dependence of diffusion coefficient on temperature in a certain temperature range is

described by Arrhenius exponential relationship

D = D0 exp (-Q/RT) (6)

Where D0 is a preexponential (frequency) factor depending on bond force between atoms of

crystal lattice Q is the activation energy of diffusion: where Q = Qv+Qm, Qv and Qm are the

activation energies for the formation and motion of vacancies respectively, the experimental

value of Q for the diffusion of carbon in -Fe is 20.1 k cal/mole and that of D0 is 2 10-6m2/s

and R is the gas constant.

Factors Affecting Diffusion Coefficient (D)

We have mentioned that diffusion co-efficient is affected by concentration. However, this

effect is small compared to the effect of temperature. While discussion diffusion

mechanism, we have assumed that atom jumped from one lattice position to another. The

rate at which atoms jumped mainly depends on their vibrational frequency, the crystal

structure. Activation energy and temperature we may note that at the position. To

overcome this energy barrier, The energy required by the atom is called the activation of

diffusion (Fig. 8)

Defects in crystals and Elements of Thin Films

Fig. 8. Activation energy for diffusion (a) vacancy mechanism (b) interstitial mechanism

The energy is required to pull the atom away from its nearest atoms in the vacancy

mechanism energy is also required top force the atom into closer contact with neighbouring

atoms as it moves along them in interstitial diffusion. If the normal inter- atomic distance is

either increases or decrease, addition energy is required. We may note that the activation

energy depends on the size of the atom. i.e. it varies with the size of the atom, strength of

bond and the type of the diffusion mechanism. It is reported that the activation energy

required is high for large- sized atoms, strongly bonded material , e.g. corundum and

tungsten carbide (since interstitial diffusion requires more energy than the vacancy

mechanism.)

4.3.4 Applications of Diffusion

Diffusion processes are the basis of crystallization recrystallization, phase transformation

and saturation of the surface of alloys by other elements, Few important applications of

diffusion are :

(i) Oxidation of metals

(ii) Doping of semiconductors.

(iii) Joining of materials by diffusion bonding, e.g. welding, soldering, galvanizing, brazing

and metal cladding

(iv) Production of strong bodies by sintering i.e. powder metallurgy.

(v) Surface treatment , e.g. homogenizing treatment of castings , recovery,

recrystallization and precipitation of phases.

(vi) Diffusion is fundamental to phase changed e.g. y to -iron.

Now, we may discuss few applications in some detail. A common example of solid state

diffusion is surface hardening of steel, commonly used for gears and shafts. Steel parts made

in low carbon steel are brought in contact with hydrocarbon gas like methane (CH4) in a

furnace atmosphere at about 9270C temperature. The carbon from CH4 diffuses into surface

of steel part and theory carbon concentration increases on the surface. Due to this, the

hardness of the surface increase. We may note that percentage of carbon diffuses in the

surface increases with the exposure time. The concentration of carbon is higher near the

surface and reduces with increasing depth Fig. (9)

Defects in Crystal

Fig. 9. C gradient in 1022 steel carburized in 1.6% CH4, 20% CO and 4%H.

Check Your Progress 3

Notes : (i) Write your answer in the space given below

(ii) Compare your answer with those given at the end of the unit

What is diffusion and on what variable it depends?

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4.4 IONIC CONDUCTIVITY

It is known that the dominant lattice defect responsible for the ionic conductivity in pure

and doped lead chloride is the anion vacancy (Jost 1952). The activation energy for

migration of the anion vacancy has been measured by Simkovich (1963), Seith (De Vries

1965) and Gylai (De Vries 1965) in powder samples and is found to range from 0-48 eV to 0-

24 eV. The measurements on single crystals of pure and doped lead chloride, however,

show that the energy of formation of vacancies is 1-66 eV and that for migration of the

anion vacancies is 0-35 eV (De Vries and Van Santen 1963; De Vries 1965).

Theroles of various point defects in this material are not yet clearly understood. Simkovich,

fox example, concluded that in the extrinsic region half of the anion vacancies are associated

with cation vacancies to form charged pairs. Barsis and Taylor (1966), on the other hand,

proposed that appreciable number of inteistitials, i.e., unassociated Frenkel defects, are

present in the extrinsic region as seen from the analysis of isotherms obtained by them from

the data of De Vries and Van Santen. The recent experiments by Van den Brom etal (1972)

Defects in crystals and Elements of Thin Films

on the dielectric relaxation in pure lead chloride suggest that in this region dipole species

such as anion vacancy-impurity associates are piesent.

In this paper, we shall present the results of self-diffusion and ionic conductivity

measurements made on pure crystals of lead chloride, and show that ir this material

Schottky defects are mainly responsible for the observed ionic transport and that the

impurity anion vacancy associates, particularly the oxygen ions, influence it markedly in the

extrinsic region.

4.5 COLOUR CENTRES

Colour centres: Becquerel discovered that a transparent NaCl crystal was coloured yellowish

when it was placed near a discharge tube. The colouration of the NaCl and other crystals was

responsible for the study of colour centres. Actually, rocksalt should have an infrared

absorption due to vibrations of its ions and an ultraviolet absorption due to the excitation of

the electrons. A perfect NaCl crystal should not absorb visible light and so it should be

perfectly transparent. This leads us to the conclusion that the colouration of crystals is due

to defects in the crystals. It is also found that exposure of a coloured crystal to white light

can result in bleaching of the colour. This gives further clues to the nature of absorption by

crystals. Experiments show that during the bleaching of the crystal the crystal becomes

photoconductive. i.e., electrons are excited to the conduction band. Photoconductivity tells

us about the quantum efficiency (number of free electrons produced per incident photon) of

the colour centres.

It is known that insulators have large energy gaps and that they are transparent to visible

light. Ionic crystals have the forbidden energy gap of about 6eV which corresponds to a

wavelength of about 2000A0 in the ultraviolet region. From dielectric properties we know

that the ionic polarizability resonates at a wavelength of 60 microns in the far infrared

region. It is why these crystals are expected to be transparent over a wide range of spectrum

including the visible region. Due to such a good transparency, the crystals of KCl, NaCl, LiF

and other alkali halides are used for making prisms, lenses and optical windows in optical

and infrared spectrometers. However, due to different reasons, absorption bands may occur

in the visible, near ultraviolet and near infrared regions in these crystals. If the absorption

band is in the visible region and the band is quite narrow, it gives a characteristic colour to

Defect in Crystal

the crystal. When the crystal gets coloured, it is said to have colour centres. Thus a colour

centre is a lattice defect, which absorbs light.

It is possible to colour the crystals in a number of different ways as described below:

(i) Crystals can be coloured 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.

(ii) The crystals can be coloured 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. Crystals

coloured by this method on chemical analysis show an excess of alkali metal atoms, typically

1016 to 1019 per unit volume.

(iii) Crystals can also be coloured or made darker by exposing them to high energy

radiations like X-rays or ϒ-rays or by bombarding them with energetic electrons or

neutrons.

4.5.1 F Centres: The simplest and the most studied type of colour centre is an F centre. It is called

an F centre because its name comes from the German word Farbe which means colour. F

centres are generally produced by heating a crystal in an excess of an alkali vapour or by

irradiating the crystal by X rays, NaCl is a very good example having F centres. The main

absorption band in NaCl occurs at about

Defects in crystals and Elements of Thin Films

Defects in Crystal

4650A 0 and it is called the F band. This absorption in the blue region is said to be

responsible for the yellow colour produced in the crystal. The F band is characteristic of the

crystal and not of the alkali metal used in the vapour i.e., the F band in KCI or NaCl will be

the same whether the crystal is heated in a vapour of sodium or of potassium. The F bands

associated with the F centres of some alkali halide crystals are shown in fig. 10, in which the

optical absorption has been plotted against wavelength or energy in eV

Formation of F-Centres: Colour centres in crystals can be fanned by their non-stoichiometric

properties i.e., when crystals have an excess of one of its constituents. NaCl crystal can

therefore be coloured by heating it in an atmosphere of sodium vapour and then cooling it

quickly. The excess sodium atoms absorbed from the vapour

Split up into electrons and positive ions in the crystal (fig. 11). The crystal becomes slightly

non-stoichiometric, with more sodium ions than chlorine ions. This results in effect in CI-

vacancies. The valence electron of the alkali atom is not bound to the atom, it diffuses into

the crystal and becomes bound to a vacant negative ion site at F because a negative ion

vacancy in a perfect periodic lattice has the effect of an isolated positive charge. It just traps

an electron in order to maintain local charge neutrality. The excess electron captured in this

way at a negative ion vacancy in an alkali halide crystal is called an F centre. This electron is

shared largely by the six positive metal ions adjacent to the vacant negative lattice site as

shown in 2-dimensions by the dotted circle in fig. 11. The figure shows an anion vacancy and

an anion vacancy with an associated electron, i.e., the centre. This model was first suggested

by De-Boer and was further developed by Mott and Gurney.

Change of Density: Since some Cl- vacancies are always present in a NaCl crystal in

thermodynamic equilibrium, any sort of radiation which will cause electrons to be knocked

Defect in Crystal

into the Cl- vacancies will cause the formation of F centres. This explains Becquerel's early

results also. With that the generation of vacancies by the introduction of excess metal can

be experimentally demonstrated by noting a decrease in the density of the crystal. The

change of density is determined by X-ray diffraction measurements.

Energy Levels of F -centres: Colour centres are formed when point defects in a crystal trap

electrons with the resultant electronic energy levels spaced at optical frequencies. The

trapped electron has a ground; state energy determined by the surroundings of the vacancy.

These energy levels lie in the forbidden energy gap and progress from relatively widely

spaced levels to an almost continuous set of levels just below the bottom of the conduction

band. When the crystal is exposed to white light, a proper component of energy excites the

trapped electron to a higher energy level, it is absorbed in the process and a characteristic

absorption peak near the visible region appears in the absorption spectrum of the crystal

having F-centres. The peak does not change when an excess of another metal is introduced

in the crystal if the foreign atoms get substituted for the metal atoms of the host crystal.

This justifies the assumption that the absorption peak is due to transitions to excited states

close to the conduction band-determined by the trapped electron. Fig. 12

shows the energy level diagram for an F centre. It also shows that the F absorption

band is produced due to a transition from the ground state to the first excited state

below the conduction band.

Effect of temperature on F-band: We have seen above that the energy levels of an F-

centre depend upon the atomic surroundings of vacancy. This means that the

absorption peak should shift to shorter wavelengths i.e., higher energies when the

interatomic distances in the crystal are decreased. This shift is actually observed on

varying the temperature of the crystal. The absorption maximum has a finite breadth

even at very low temperature, which increases on increasing the temperature of the

crystal. It can be explained by studying the dependence of the energy of-a colour

Defects in crystals and Elements of Thin Films

centre on temperature. Fig. 13. Shows a graph plotted between the changes in energy

of an electron in F centre and the coordination of a vacancy

i.e., the distance from centre of vacancy to nearest ions surrounding it E denotes the

excited state of the electron bound to a CI- vacancy and G is for the ground

state of that electron.

At any finite temperature the ground state is not at 0, the minimum of curve G but

lies above it by about kE because the coordinating ions vibrate between A and B due

to thermal energy. Hence the energy of the absorbed radiation can range between that

of transition A → A` or B → B`. The difference between energies, corresponding to

A` and B` gives the width of the absorption peak. AB represents the amplitude of

vibration of ions at a lower temperature but as the temperature rises it moves to a

higher energy position so that CD represents the amplitude of vibration at the higher

temperature and thus the width of the absorption peak- the F band increases.

Klcinschord observed that the F band instead of being exactly like a bell, ossesses a

shoulder and a tail on the short wavelength side. Seitz called the shoulder as a K-band

and it may be considered to be due to transitions of the electron to excited states,

which lie between the first excited state and the conduction band. The tail may be

supposed to be due to the transition from the ground state of F-centre to the

conduction band.

Magnetic Properties of F-Centres: In fig. 13, the upper curve E is determined by the

change in the surroundings of a vacancy when the trapped electrons is in the excited

state. This is usually expressed by a change of the effective dielectric constant in the

neighbourhood of such a vacancy. An alkali halide crystal is normally diamagnetic

because the ions have closed outer shells. Since an F-centre contains an unpaired

trapped electron, crystals additively coloured with a metal have some paramagnetic

Defects in crystals and Elements of Thin Films

Defect in Crystal

behavior. Thus the structure of F-centres can be studied by electron paramagnetic

resonance experiments which tell us about the wave-functions of the trapped electron.

4.5.2 V Centres : .Till now we had been considering the electronic properties

associated with an excess of alkali metal. It is, however, quite natural to think what

will happen if we have an excess of halogen in alkali halides. Thus if an alkali halide

crystal is heated in a halogen vapour, a stoichiometric excess of halogen ions is

introduced in it, the accompanying cation vacancies trap holes just as the anion

vacancies trap electrons in F centres. Thus we should expect a whole new series of

colour centres, which are produced by excess alkali metal atoms. The new centres

have holes in place of electrons. The colour centres produced in this way are called V

centres and the crystals having these centres show several absorption

maxima which are called as V1, V2 bands and so on. Mollwo was able to introduce access

halogen into KBr and KI and found that it is was not possible in case of KCl. He shows that by

heating KI in iodine vapour ,new absorption bands are obtain in the ultraviolet .the bands

obtain by Mollwo for KBr when heated in Br2 vapour are shown in fig. 14, having V1, V 2 and

V3 bands.

The formation of V centres can be explained on the same lines as for F centres. The

excess bromine enters the normal lattice positions as negative ions. Positive holes are

thus formed which are situated near a positive ion vacancy where they can be trapped.

A hole trapped at a positive ion vacancy forms a V centre as shown in fig. 15. The

optical absorption associated with a trapped hole may be due to the transition of an

electron from the filled band into the hole.

Defects in crystals and Elements of Thin Films

Fig.15 Proposed models of V centres after Seitz Nagamiya

It can be understood that the strong peak observed by Mollwo in KBr as shown in fig. 14 is

however, not of the above type. Mollwo's experiment proves that the saturation density of

colour centres is proportional to the number of bromine molecules at a particular

temperature. By the law of mass action, we know that one colour centre should be

produced by each molecule absorbed from the vapour. Hence it was proposed by F. Seitz

that the centres associated with the strong peak are of molecular nature, i.e., two holes are

trapped by two positive ion vacancies. Such a centre is called a V2 centre and is shown in fig.

15.

As is evident from the figures 13 and 15, the V1 centre is the counterpart of the F-centre, V2

and V3 are those of the R centres and V 4 is the counterpart of the M centre. However, the

identification of the V1 centre with the V 1 band is uncertain because the spin resonance

results of Kaenzig suggest that a centre having the symmetry of the V3 centre produces the

V1 band. The detailed properties of V centres have not yet been properly understood.

Production or Colour Centres by X-rays or Particle Irradiation: The colour centres

can also be produced in crystals by irradiating them with very high energy radiation

like X -rays or ϒ rays. An X-ray quantum when passes through an ionic crystal

produces fast photo electrons having the energy nearly equal to that of the incident

quantum. These high energy electrons interact with the valence electrons in the crystal

and lose their energy by producing free electrons and holes, excitons (electron hole

pairs) and phonons. These free electrons and holes diffuse into the crystal and come

across vacancies present in the crystal where they may be caught producing trapped

electrons and holes. In this way both F and V types of colour centres are produced in

crystals irradiated with high energy radiations. However, these are not permanent like

those produced in non stoichiometric crystals in which there is an internally produced

excess of electrons and holes. Their colours cannot be removed permanently without

changing them chemically. The colour centres produced by X-ray radiation are easily

bleached by visible light or by heating because the excited electrons and holes

ultimately recombine with each

Defects in Crystal

other. The F and V centres produced by irradiation with 30 keV X -rays at room temperature

(20°C) have been shown in fig. 16 in the absorption spectrum of KCl taken by Dorendorf and

Pick.

Check Your Progress 4

Notes : (i) Write your answer in the space given below

(ii) Compare your answer with those given at the end of the unit

What are color centers and how do they affect electric conductivity of solids?

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4.6 EXCITIONS

Defects in crystals and Elements of Thin Films

The most obvious point defects consist of missing ions (vacancies), excess ions

(interstitials), or the wrong kind of ions (substitution impurities). A more subtle

possibilitials is the case of an ion in a perfect crystal, that differs from its colleagues only by

being in an excited electronic state. Such a “defect” is called a Frenkel exciton. Since any

ion is capable of being so excited, and since the coupling between the ions’ outer

electronic shells is strong, the excitation energy can actually be transferred from ion to ion

. Thus the Frenkel exciton can move through the crystal wit\hout the ions themselves

having to change places, as a result of which it is (like the polaron) for more mobile than

vacancies, interstitials, or substitutional impurities. Indeed, for more accurate to describe

the electronics structure of a crystal containing an exciton, as a quantum mechanical

superposition of states, in which it is equally probable that the excitation is associated with

any ion in the crystal. This latter view bears the same relation to specific excited ions, as

the Bloch tight – binding levels (Chapter 10) bear to the individual atomic levels, in the

theory of band structures.

Thus the exciton isprobably better regarded as one of the more complex manifestions of

electronic band structure that as a crystal defect. Indeed, once one recognizes that the

proper description of an exciton is really a problem in electronic band structure, one can

adopt a very different view of the same phenomenon:

Suppose we have calculated the electronic ground state of an insulator in the independent

electron approximation. The lowest excited state of the insulator willevidently be given by

removing one electron from the highest level in the highest occupied band 9the valence

band) and placing it into the lowest – lying level of the lowest unoccupied band

(conduction band). Such a rearrangement of the distribution of electrons does not alter the

self- consistent periodic potential in which they move. This is because the Bloch electron

are not localized (since nk(r)2 is periodic), and therefore the change in local charge

density produced by changing the level of a single electron will be of order 1/N (since only

an Nth of the electron's charge will be in any given cell) i.e. negligibly small. Thus the

electronic energy levels do not have to be recomputed for the excited configuration and

the first excited state will lie an energy c -v above the energy of the ground state, where

c is the conduction band minimum and v the valence band maximum.

However, there is another way to make an excited state. Suppose we form a one-electron

level by superposing enough level near the conduction band minimum to form a well-

localized wave packet. Because we need levels in the neighborhood of the minimum to

Defects in Crystal

produce the wave packet, the energy c of the wave packet will be somewhat grater than

c. Suppose in addition that the valence band level we depopulate is also wave packet. ,

formed of levels in the neighborhood of the valence band maximum (so that its energy v is

somewhat less than v) and chosen so that the center of the wave packet is spatially very

near the center of the conduction band wave packet. If we ignored electron – electron

interactions, the energy required to move an electron from valence to conduction band

wave packet. If we ignored electron- electron interactions, the energy required to move an

electron from valence to conduction band wave packets would be c - v > c - v, but

because the levels are localized, there will, in addition, be a non – negligible amount of

negative Coulomb energy due to the electrostatic attraction of the (localized) conduction

band electron and (localized) valence band hole.

This additional negative electrostatic energy can reduce the total excitation energy to an

amount that is less than c - v, so the more complicated type of excited state, in which the

conduction band electron is spatially correlated with the valence band hole it left behind,

is the true lowest excited state of the crystal. Evidence for this is the onset of optical

absorption at energies below the inter band continuum threshold the following

elementary theoretical argument, indicating that one always does better by exploiting the

electron hole attraction:

Let us consider the case in which the localized electron and hole levels extend over many

lattice constants. We may then make the same type of semi classical argument that we

used to deduce the form of the impurity levels in semiconductors. We regard the electron

and hole as particles of mass mc and mv (the conduction and valence band effective

masses, which we take, for simplicity, to be isotropic). They interact through an attractive

Coulomb interaction screened by the dielectric constant of the crystal. Evidently this is

just the hydrogen atom problem, with the hydrogen atom reduced mass (1/ = 1/Mproton

+ 1/melectron 1/melectron) replaced by the reduced effective mass m* (1/m* = 1/mc + 1/mv ) ,

and the electronic charge replaced by e2/. Thus there will be bound states, the lowest of

which extends over a Bohr radius given by:.

02

2

*)/(*a

m

m

emaex

Defects in crystals and Elements of Thin Films

Defects in Crystal

the energy of the bound state will be lower than the energy (c - v ) of the non-interacting

electron and hole by

eVm

m

a

e

m

m

a

eEex

)6.13(1*

2

1*

2

)/(

2

0

2

2*

0

2

The validity of this model requires that aex be large on the scale of the lattic (i.e., aex >>a0),

but since insulators with small energy gaps tend to have small effective masses and large

dielectric constants, that is no difficult to achieve, particularly in semiconductors. such

hydrogenic spectra have in fact been observed in the optical absorption that occurs below

the inter and threshold.

The exciton described by this model is known as the Mott- Wannier exciton Evidently as

the atomic levels out of which the band levels are formed become more tightly bound

will decrease m* will increases, a0* will decrease, the exciton will become more localized,

and the Mott- Wannier picture will eventually break down. The Mott- Wannier exciton and

the Frenkel exciton are opposite extremes of the same phenomenon. In the Frenkel case,

based as it is on a single excited ionic level, the elelctron and hole are sharply localized on

the atomic scale. The exciton spectra of the solid range gases fall in this class.

4.7 GENERAL IDEA OF LUMINESCENCE

When a substance absorbs energy in some form or other, a fraction of the absorbed

energy may be re-emitted in the form of electromagnetic radiation in the visible or

near-visible region of the spectrum. This phenomenon is called luminescence, with

the understanding that this term does not include the emission of blackbody radiation,

which obeys the laws of Kirchhoff and Wien. Luminescent solids are usually referred

to as phosphors. Luminescence is a process, which involves at least two steps: the

excitation of the electronic system of the solid and the subsequent emission of

photons. These steps may or may not be separated by intermediate processes.

Excitation may be achieved by bombardment with photons (photoluminescence: with

electrons (cathodo luminescence), or with other particles. Luminescence can also be

induced as the result of a chemical reaction (chemi luminescence) or by the

application of an electric field (electro luminescence)

When one speaks of fluorescence, one usually has in mind the emission of light during

excitation; the emission of light after the excitation ha ceased is then referred to as

phosphorescence or afterglow. These definitions are not very exact since strictly speaking

there is always a time la between a particular excitation and the corresponding emission of

photon, even in a free atom. In fact, the lifetime of an atom in an excite state for which the

return to the ground state is accompanied by dipole radiation is 10-8 second. For forbidden

transitions, involving quadrupole or higher-order radiation, the lifetimes may be 10-4 second

or longer. One frequently takes the decay time of ~10-8 second as the demarcation line

between fluorescence and phosphorescence. Some authors define fluorescence as the

emission of light for which the decay time is temperature independent, and

phosphorescence as the temperature-dependent part .In many cases the latter definition is

equivalent to the former, but these are exceptions.

One of the most important conclusions reached already in the early studies of

luminescence, is that frequently the ability of a material to exhibit luminescence is

associated with the presence of activators. These activators may be impurity atoms

occurring in relatively small concentrations in the host material, or a small

stoichiometric excess of one of the constituents of the material. In the latter case one

speaks of self-activation. The presence of a certain type of impurity may also inhibit

the luminescence of other centers, in which case the former are referred to as "killers."

Since small amounts of impurities may play such an important role in determining the

luminescent properties of solids, studies aimed at a better understanding of the

mechanism of luminescence must be carried out with materials prepared under

carefully controlled conditions. A great deal of progress has been made in this respect

during the last two decades.

A number of important groups of luminescent crystalline solids may be mentioned here.

(i) Compounds which luminesce in the "pure" state. According to Randall, such

compounds should contain one ion or ion group Per unit cell with an incompletely

filled shell of electrons which is well screened from its surroundings. Examples are

probably the manganous halides, samarium and gadolinium sulfate, molybdates, and

platinocyanides.

Defects in Crystal

Defects in crystals and Elements of Thin Films

(ii) The alkali halides activated with thallium or other heavy metals.

(iii) ZnS and CdS activated with Cu, Ag, Au, Mn, or with an excess of one of their

constituents (self-activation).

(iv) The silicate phosphors, such as zinc orthosilicate (willernite, Zn2Si04) activated

with divalent maganese, which is used as oscilloscope screens.

(v) Oxide phosphors, such as self-activated ZnO and Al203 activated with transition metals.

(vi) Organic crystals, such as anthracene activated with naphtacene these materials

are often used as scintillation counters.

A. 4.8 DISLOCATIONS & MECHANICAL STRENGTH OF CRYSTALS

The first idea of dislocations arose in the nineteenth century by observations that the

plastic deformation of metals was caused by the formation of slip bands in which one

portion of the material sheared with respect to the other. Later with the discovery that

metals were crystalline it became more evident that such slip must represent the

shearing of one portion of a crystal with respect to the other upon a rational crystal

plane. Volterra and Love while studying the elastic

behaviour of homogene-ous isotropic media considered the elastic properties of a

cylinder cut in the forms shown in Figs. 17 (a) to (d), some of the deformation

operations correspond to slip while some of the resulting configurations correspond to

dislocation. The work on crystalline slip was then left out till dislocations were

postulated as crystalline defects in the late 1930's. The configuration (a) shows the

cylinder as originally cut (b) and (c) correspond to edge dislocations while (d)

corresponds to screw dislocation.

Defects in crystals and Elements of Thin Films

After the discovery of X-rays, Darwin and Ewald found that the intensity of X-ray beams

reflected from actual crystals was about 20 times greater than that expected from a perfect

crystal. In a perfect crystal, the intensity is low due to long absorption path given by multiple

internal reflections. Also, the width of the reflected beam from an actual crystal is about 1 to

30 minutes of an are as compared with that expected for a perfect crystal which is only

about a few seconds. This discrepancy was explained by saying that the actual crystal

consisted of small, roughly equiaxed crystallites, 10-4 to 10-5 cm. In diameter, slightly

misoriented with respect to one another, with the boundaries between them consisting of

amorphous material. This is the "mosaic block" theory in which the size of the crystallites

limits the absorption path and increases the intensity. The misorientation explains the width

of the beam. It was however found recently that the boundaries of the crystallites are

actually arrays of dislocation lines.

The presence of dislocation lines is also proved by the study of crystal growth.

Volmer's and Gibbe's theoretical study on nucleation of new layers showed that the

layer growth of perfect crystals is not appreciable until supersaturation of about 1.5

were attained. However, experimental work of Volmer and Schultze on iodine

showed that crystals grew under nearly equilibrium conditions. Frank removed this

discrepancy by saying that the growth of crystals could take place at low

supersaturations by the propagation of shelves associated with the production of a

dislocation at the surface.

The development of the theory of dislocations was given a great impetus by the

consideration of the strength of a perfect crystal. A crystal can be deformed elastically by

applying stresses on it but it can regain its original condition when the stresses are removed.

If the stresses applied be very large, of the order of about 106 -107 dynes per cm2 then a

small amount of deformation will be left on removing these stresses and the crystal is said to

suffer a plastic deformation. It will be seen that the atomic interpretation of the plastic flow

of crystals requires the postulation of a new type of defect called dislocations.

Mechanical Strength of Crystal : The weakness of good crystals was a mystery for many

years, in part, no doubt, because the observed data easily led one to the wrong conclusion.

Relatively poorly prepared crystal were found to have yield strengths close to the high value

we first estimate for the perfect crystal. However, as the crystals were improved (for

example, by annealing) the yield strengths were found to drop drastically, falling by several

orders of magnitude in very well prepared crystals. It was natural to assume that the yield

strength was approaching that of a perfect crystal as specimens were improved, but, in fact,

quite the opposite was happening.

Three people independently came up with the explanation in 1943, inventing the dislocation

to account for the data. They suggested that almost all real crystals contain dislocations, and

that plastic slip occurs through their motion as described above. There are then two ways of

making a strong crystal. one is to make an essentially perfect crystal, free of all dislocation.

This is extremely difficult to achieve. Another way is to arrange to impede the flow of

dislocations, for although dislocations move with relative ease in a perfect crystal, if they

work required to move them can increase considerably.

Thus the poorly prepared crystal is hard because it is infested with dislocations and defects,

and these interfere so seriously with each other's motion that slip can occur only by the

more drastic means described earlier. However, as the crystal is purified and improved,

dislocation largely move out of the crystal, vacancies and interstitials are reduced to their

(low) thermal equilibrium concentrations, and the unimpeded motion of those dislocations

that remain makes it possible for the crystal to deform with c\ease. At this point the crystal

is very soft. If one could continue the process of refinement to the point where all

dislocations were removed, the crystal would again become hard.

Defects in Crystal

4.9 PLASTIC BEHAVIOUR

Plastic deformation takes place in a crystal due to the sliding of one part of a crystal

with respect to the other. This results in slight increase in the length of the crystal

ABCD under the effect of a tension FF applied to it as shown in fig. 18. The 'process

of sliding is called slip. The direction and place in which the sliding takes place are

called respectively the slip direction as shown by the arrow P and slip plane. The

outer surface of the single crystal is deformed and a slip band is formed, as is seen in

the figure, which may be several thousand Angstroms wide. This can be observed by

means of an optical microscope, but when observed by an electron microscope a slip

band is found to consist of several slip lamellae. The examination of slips by an

electron microscope reveals that these extend over several tens of lattice constants.

The slip lines do not run throughout the crystal but end inside it, showing that slips do

not take place simultaneously over the whole Slip planes but occur only locally. The

study of slips in detail tells us that plastic deformation is inhomogeneous i.e., only a

small number of those atoms take part in the slip which form layers on either side of a

slip plane. In the case of elastic deformation all atoms in the crystal are affected and

its properties can be understood in terms of interatomic forces acting in a perfect

lattice. On the other hand, plastic deformation cannot be studied by simply extending

elasticity to large stresses and strains or on the basis of a perfect lattice. We will now

prove below that for plastic flow in a perfectly periodic lattice, we have to apply very

much larger stresses (~ 1010

dynes per cm2) than those required for the normal plastic

flow observed in actual crystals (~106

dynes per cm2).

Shear Strength Crystals: J. Frenkel in calculating the theoretical shear strength of a perfect

crystal. The model proposed by him is given in fig. 19, showing a cross-section through two

Defects in crystals and Elements of Thin Films

adjacent atomic planes separated by a distance d. The full line circles indicate the

equilibrium positions of the atoms without any external force.

Let us now apply a shear stress Ʈ in the direction shown in fig.19 (a). All the atoms in the

upper plane are thus displaced by an amount x from the original positions as shown by the

dotted circles. In fig. (b), the

Fig. 19

shear stress has been plotted as a function of the relative displacement of the planes from

their equilibrium positions and this gives the periodic behavior of as supposed by Frenkel. '

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. Frenkel assumed that this periodic function is given by

Defects in Crystal

a

xc

2sin (7)

where the amplitude c denotes the critical shear strees which we have to calculate For x

<< a, we have as usual,

a

xc

2 (8)

In order to calculate the force required to shear the two planes of atoms, we from the

definition of shear modulus

a

x

dxyStrain

StressG c

2

/

where G is the shear module and Gd

xy

is the elastic strain

d

xG (9)

Comparing it with equation (12), we have

b

a

x

Gor

d

xG

a

xcc .

2

2

or daifG

x

Gc ,

62 (10)

This gives the maximum critical stress above which the crystal becomes unstable. It is about

one sixth of the shear modulus. In a cubic crystal, G c44 = 1011 dynes per cm. for a shear in

the <100> direction. Hence the theoretical value of the critical shear stress on Frenkel's

model is c = 1010 dynes per sq. .cm. which is much larger than the observed values for pure

crystals. However., the experimental values for the maximum resolved shear stress required

to start the plastic flow in metals were of the order of 10-3 to 10-4 G at that time and it was

not a agreement with the results of eqn.. (10)

Later it was considered that eqn (10) gave a higher value as the different semi-inter –atomic

force of Fig.19 (b) as taken by Frenkel. The above disagreement may also be due to other

special configuration of mechanical stability which the lattice may develop when it is

sheared. Mackenzie in 1949, using central forces in the case of close packed lattices found

that c could be reduced to a value G/30 , corresponding to a critical shear strain of about 20

Defects in crystals and Elements of Thin Films

This value, however, is supposed to be an underestimate due to the neglect of the small

directional force which are also present in such lattices. The contributions of thermal

stresses also reduce c below G/30 only near the melting point. Thus at room temperature

we should have G/5 > c > G/30 i.e. = G/15. In the case of whiskers only,. the experimental

value of c for various metals has been found to be of this order which is in excellent

agreement with the theoretical result. Recent experimental work on bulk copper and zinc

has shown that plastic deformation being at stresses of the order of 10-9 G. Hence, except

for whiskers the disagreement is even larger than before..

It is therefore clear that agreement between theory and experiment be obtained on the

basis of Frenkel's model where atomic plant glide past each other assuming fig. 19 (a) that

the atoms of the upper atomic plane move simulantaneously relative to the lower plane.

This assumption is based on the supposition of a perfect lattices, and that is the main cause

of difficulty. We have , therefore, to consider the presence of imperfections which act as

sources of mechanical weakness in actual crystals and which may proud a slip by the

consecutive motion of the atoms but not by simultaneous motion of the atoms of one plane

relative to another. After Frenkel theory Masing and Polanyi, Pradtl and Dehlinger proposed

different defects but in 1934, Orowan, Polanyi, and Taylor proposed edge dislocation, while

in

1939 Burgers gave the description of screw dislocation to explain the discrepancy between

c theoretical and experimental. It has now been established that the new type of defect

called dislocation exists in almost all crystals and it is responsible for producing slip by the

application of small stresses only

4.10 TYPE OF DISLOCATION

The Edge Dislocation: Dislocation is a more complicated defect than any of the point

defects. 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 viz., the edge

type and the screw type. Any particular dislocation is usually a mixture of these two

types. An edge dislocation is the simplest one and a cross-sectional view of the atomic

arrangement of atoms in it and the distortion of the crystal structure is shown in fig.

20. 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

Defects in Crystal

line and the symbol ┴ at B is used to indicate the dislocation. The distortion is mostly

present about the lower edge of the half plane of extra atoms and so the dislocation is

that line of distortion which is near the end of the half plane. Hence a dislocation is a

line imperfection as compared to the point imperfections considered before.

In all the dislocations, the distortion is very intense near the dislocation line where the

atoms do not have the correct number of neighbours. This region is called the core of

dislocation. A few atom distances away from the centre, the distortion is very small and the

crystal is almost perfect locally. At the core, the local strain is very high whereas it is so small

at distances away form the core that the elasticity theory can be applied and it is called the

elastic region. Another characteristic of the distortion of atomic arrangement in an edge

dislocation is that the atoms just above the end of the extra half plane are in compression

but just below the half plane the two rows of atoms to the right and left of the extra plane

BD are farther apart from each other and the structure is expanded. This local expansion

round an edge dislocation is called a dislocation. Besides the expansion and contraction near

the dislocation, the structure is sheared also and this shear distortion is quite complicated.

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. It can be studied by the precipitation of impurities because

the region of dilatation along a dislocation line is very suitable for their precipitation.

Optical microscopes can be used to study them if the dislocations are first decorated

by precipitating metallic impurities along the dislocation lines e.g., silver decoration

of alkali halides. The intersection of dislocation lines with the surface of a crystal can

be revealed by the etch pitch technique.

Defects in crystals and Elements of Thin Films

Fig. 20

Fig. 21

The Screw Dislocation: The screw dislocation was introduced by Burger in 1939. It is also

called Burger's dislocation. To understand this, let a sharp cut be made part way through a

perfect crystal and let the crystal on one side of the cut be moved down by one atomic

spacing relative to the other so that the rows of atoms are placed back into contact as

shown in fig. 21. A line BD of distortion exists along the edge of the cut, which is called the

screw dislocation. In this case complete planes of atoms normal to the dislocation do not

exist any longer but all atoms lie on a single surface which spirals from one end of the crystal

to the other and so it is called 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 centre are in regions of

high distortion so much so that the local symmetry in the crystal is completely destroyed. In

Defects in Crystal

this case, the atoms near the centre of the screw dislocation are not in a dilatation as in

edge dislocation but are on a twisted or sheared lattice.

Motion of a Dislocation: Dislocations can move just like the point defects move in the lattice

but these are more constrained in motion because a dislocation must always be a

continuous line. Motion of a dislocation is possible either by a climb or by a slip or by a glide.

The motion of dislocation can give rise to a slip by a mechanism shown in fig. 22. When the

upper half is pushed sideways by an amount b, then under the shear the motion of a

dislocation 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. If it is below

the slip plane it is called negative edge dislocation. When an edge dislocation moves from

one lattice site to another on the Fig. 22 slip plane, the atoms in the core move slowly so

that the extra half plane at one lattice position becomes connected to a plane of atoms

below the slip plane and the nearby plane of atoms becomes the new extra half plane.

When finally 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.

Fig. 22

Defects in crystals and Elements of Thin Films

4.11 STRESS FIELD OF DISLOCATION

The Burger's Vector: The Burger's vector b denotes actually the dislocation-displacement

vector. A dislocation can be very well described by a closed loop surrounding the dislocation

line. This loop, called the Burger's circuit is formed by proceeding through the undisturbed

region surrounding a dislocation in steps which are integral multiples of a lattice translation.

The loop is completed by going an equal number of translation in a positive sense and

negative sense in a plane normal to the dislocation line. Such a loop must close upon itself if

it does not enclose a dislocation, or fail to do so by an amount called a Burger's vector

s = naa + nbb + ncc

Where na, nb, nc are equal to integers or zero and a, b, c are the three primitive lattice

translations.

Fig. 23

The Burger's circuit S1234F is shown by dark line in fig. 23 for a screw dislocation. Starting at

some lattice point S at the front of the Fig. 23 crystal, the loop fails to close on itself by one

unit translation parallel to the dislocation line. This is the Burger's vector which always

Defects in Crystal

points in a direction parallel to the screw dislocation. If the loop is continued, it will describe

a spiral path around the Burger's dislocation just like the thread of a screw. In the figure, the

height of the step on the top surface is one lattice spacing i.e., b, thus b is a vector giving

both the magnitude and the vector of the dislocation. It must be some 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 to be given by

sinr

b

V

V

where b is the Burger's vector which measures the strength of the distortion caused by the

dislocation, r is the radial distance from the point to the dislocation line and is the angle

between the radius vector and the slip plane .as shown in fig. 20. Similarly, the atoms which

are on a sheared lattice in a screw dislocation being on a spiral ramp, are displaced from

their original positions in the perfect crystal according to the equation of a spiral ramp i.e.

2

bu z

where the z-axis lies along the dislocation and u, is the displacement in that direction. The

angle is measured from one axis perpendicular to the dislocation. Thus when increases

by 2 the displacement 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, it is Perpendicular to the dislocation line

and lies in the slip plane. In general cases, the Burger's vector may have other directions

with respect to the dislocation and for these cases the dislocation is a mixture of both edge

and screw types. Thus the mixed dislocation is defined in terms of the direction of the

Burger's vector.

Stress Fields around Dislocations : Stress field of a screw dislocation: We know that the core

of dislocations is a region within a few lattice constants of the centre of dislocation and that

it is a "bad" region where the atomic arrangement of the crystal is severely changed from

the regular state. The regions outside the core are "good" regions and the strains in these

regions are elastic strains and so these can be treated by the theory of elasticity as an elastic

continuum and the core region can be added later as a proper correction term. The

Defects in crystals and Elements of Thin Films

calculation of dislocation energy is simple for a straight screw dislocation but similar results

are obtained for edge dislocation.

Let us have a cylindrical shell of a material surrounding an axial screw dislocation. Let the

radius of the shell be r and the thickness dr, The circumference of the shell is r2 and let it

be sheared by an amount b, so that the shear strain

Fig. 24

r

be

2 (11)

and the corresponding shear stress in the good region is,

r

bGeG

2

... (12)

where G is the shear modulus or modulus of rigidity of the material. A distribution of forces

is exerted over the surface of the cut for producing a displacement b and the work done by

the forces to do it gives the energy Es of the screw dislocation.

Hence,

dAbFES .. (13)

where F is the average force per unit area at a point on the surface during the displacement

and the integral extends over the surface area of the cut. The average value is to be taken

because the force at a point builds up linearly from zero to a maximum value as the

displacement is produced. Thus the average force

)2/1(F

is half the final value when the displacement is b i.e.,

r

bGF

4 (14)

Putting it in (13), we get

.4

2

dAr

GbES

But dA = dz sr and so for a dislocation of length l, we have

)16(log4

1

)15(4

0

2

0

2

0

r

RGb

dAr

GbE

R

rS

Thus, total elastic energy per unit length of a screw dislocation is given by

0

2

log4 r

RGbES

(17)

where R and r0 the proper upper and lower limits of r. The energy depends upon the values

taken for R and r0 is suitable 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. Actually,

however in most cases K is very much smaller than the size of the crystal. The value of R/r0 is

not important as it occurs in the logarithmic term.

Stress field of an edge dislocation: 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 ʋ Let

us consider the cross-section of a cylindrical material of radius R whose axis is along the z-

Defects in Crystal

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 fig. 25. Thus, a

positive edge dislocation has been produced along the z-axis. Let σrr be the radial tensile

stress, ie, compression or tension along the radius r and let σθθ be the circumferential tensile

stress i.e., compression or tension acting in a plane perpendicular to r. Let τ rθ denote the

shear stress acting in a radial direction. As seen from fig. 20, it is an odd function of x,

considering the plane y = 0 and is found to be proportional to (cos /r). In an isotropic

elastic continuum σrr and σθθ compression or tension acting in a plane perpendicular to r.

are found to be proportional to (sin /r) because we require a function which varies as 1/r

and which changes sign when y changes sign. Also it can be shown dimensionally that the

constants of proportionality in the stress vary as G and b.

Without giving the details of calculations here, the stress field of the edge dislocation in

terms of r and are given by the following : .

rv

Gbrr

sin.

)1(4 ……(18)

and rv

Gbr

cos.

)1(2 …..(19)

where the positive values of σ are for tension and negative values for compression. Above

the slip plane σrr is negative giving a compression, below the slip plane, it 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 ro around the dislocation must be excluded. This is necessary

because in the bad region, the theory of elasticity

Defects in crystals and Elements of Thin Films

Fig. 25

does not hold as the stresses near a dislocation are very large. To know the value of ro, let us

put ro = b, the magnitude of the strain there is then of the order of 1/2π(1-v)≈1/4 which is

too much large to be treated by Hooke's law. We shall now calculate the energy of

formation of an edge dislocation of unit length. The final shear stress in the plane y = 0 is

given by (19) by putting θ = 0. For a cut along z-axis in a unit length, the strain energy for

edge dislocation will be given by

R

re

r

R

vr

Gbdr

vr

GbstrainxstressE

00

22

log)1(4)1(22

1

2

1

….(20)

This shows that the energy of formation becomes infinite if R becomes infinite. But even in

large crystals the stress field are actually displaced some distance by other dislocation so

that R = 10-3 cm.

Assuming r0 = 5 10-8 cm. for a dislocation in copper

Ee = 310-4 erg/cm = eV/atom plane

Since G = 4 1011 dynes/cm2

b = 2.510-8 cm. and v = 0.34

In the case of screw dislocations its value is about (2/3) of this. The core energy of edge

dislocation should be added to the elastic strain energy but it is of the order of 1eV per

atom plane which is much less than the elastic strain energy and can be neglected. For a

screw dislocation in the Z-direction in a cylindrical material, the stress field is given by a

shear stress, according to (12).

r

Gbzz

2 …….(21)

Defects in Crystal

Fig. 26 Low angle grain boundary (a) Two crystals Joined Together

(b) Grain boundary formed with 2 rows of dislocations.

There is no tensile and compress ional stress in this expression and this is perhaps due

to the fact that there is no extra half plane in a screw dislocation. Also in this case the

stresses are independent of θ expecting thereby that the stress field is cylindrically

symmetric.

We can also explain the free energy of a dislocation. The contribution to the free

energy by entropy, in a dislocation, is very small as compared to the strain energy and

so the free energy in crystals of ordinary size at room temperature can be assumed to

be nearly equal to the strain energy. Since the strain energy is positive, the free energy

increases by the formation of dislocation. Hence no dislocation can exist as a

thermodynamically stable lattice defect

4.12 GRAIN BOUNDARIES

Burger suggested that the boundaries of two crystallites or crystal

grains at a low angle inclination with each other can be . Considered to be a regular

array of dislocations. Two such crystallites placed close together at a small angle θ

have been shown in fig. 26 (a). There are simple cubic crystals with

Defects in crystals and Elements of Thin Films

their axes perpendicular to the plane of the paper and parallel. The crystals have been

rotated by θ /2 left and right of these axes. The results of joining the two crystals

together is shown in fig. 26 (b). A grain

boundary of the simple example of Burger's model is formed. The boundary plane contains a

crystal axis common to the two crystals. Such a boundary is called a pure tilt boundary.

Crystal orientations on both sides of the boundary plane are symmetric with each other such

a boundary has a vertical arrangement of more than two edge dislocations of same sign. This

arrangement is also stable as that for two dislocations. From the figure it is seen that the

interval D between the dislocations so formed is given by

bDor

D

b

22tan …….(22)

Where b is the Burger’s vector of the dislocations and is small

Burger’s model of low angle grain boundary has been was confirmed experimentally by

Vogel and co-workers for germanium single crystals. A germanium crystal was grown from a

seeded melt along <100> direction. When the surface of this crystal was etched with a

suitable chemical (acid), the terminus of a dislocation at the surface become a nucleus of the

etching. action and a row of each pits was formed. It is shown diagrammatically in fig. 27. On

examining these boundaries under very high optical magnification they were found to

consist of regularly spaced conical pits. By counting the number of these etch pits, we are

able to find out the number of dislocations in the crystal grain boundaries. The distance D

between the pits is obtained by counting. The relative inclination angle was also measured

by means of X –ray diffraction experiments. From this value of and knowing the value of b

= 4.0 A0 in germanium, the value of D was calculated theoretically. This was found to be in

very good agreement with the experimental etch pit interval.

Defects in Crystal

Fig. 27 Diagram of optical micrograph of a low

angle grain boundary in Ge

If is less than 50, the value of D is quite large as compared with the interatomic distance

and so each dislocation can be considered as isolated. If is about 150 than D is only a few

interatomic distance and we get a collection of irregular, diffused and deformed vacancies.

At present the etch pit method is the most direct method of determining the dislocation

density. The density of dislocations is the number of dislocation lines which intersect a unit

area in the crystal. It ranges from 102 – 103 in the best germanium and silicon crystals to 1011

– 10-12 dislocations/cm2 in heavily deformed metal crystals. The density of dislocations can

be estimated in solids by the following methods:

(i) By plastic deformation of crystals, just like the bending of a pack of playing cards.

(ii) By X-ray transmission method.

(iii) By X-ray reflection.

(iv) By electron microscopy

(v) By measuring the increase in the electrical resistivity produced by the dislocations in

heavily cold – worked metals

(vi) By measurement on magnetic saturation of cold- worked fermagnetic materials.

(vii) .By decoration methods. Decorated helical dislocations can be produced in calcium

fluoride by decorating particles of CaO.

(viii) By etch pit methods.

4.13 ETCHING- TYPES OF ETCHING

In order to form a functional Micro-Electro-Mechanical Systems (MEMS) structure on a

substrate, it is necessary to etch the thin films previously deposited and/or the substrate

itself. In general, there are two classes of etching processes:

1. Wet etching where the material is dissolved when immersed in a chemical solution

Defects in crystals and Elements of Thin Films

2. Dry etching where the material is sputtered or dissolved using reactive ions or a vapor phase etchant

In the following, we will briefly discuss the most popular technologies for wet and dry etching.

Wet etching: This is the simplest etching technology. All it requires is a container with a

liquid solution that will dissolve the material in question. Unfortunately, there are

complications since usually a mask is desired to selectively etch the material. One must find

a mask that will not dissolve or at least etches much slower than the material to be

patterned. Secondly, some single crystal materials, such as silicon, exhibit anisotropic

etching in certain chemicals. Anisotropic etching in contrast to isotropic etching means

different etch rates in different directions in the material. The classic example of this is the

<111> crystal plane sidewalls that appear when etching a hole in a <100> silicon wafer in a

chemical such as potassium hydroxide (KOH). The result is a pyramid shaped hole instead of

a hole with rounded sidewalls with a isotropic etchant. The principle of anisotropic and

isotropic wet etching is illustrated in the figure below.

This is a simple technology, which will give good results if you can find the combination of

etchant and mask material to suit your application. Wet etching works very well for etching

thin films on substrates, and can also be used to etch the substrate itself. The problem with

substrate etching is that isotropic processes will cause undercutting of the mask layer by the

same distance as the etch depth. 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. If this is a limitation for you, you

should consider dry etching of the substrate instead. However, keep in mind that the cost

per wafer will be 1-2 orders of magnitude higher to perform the dry etching

If you are making very small features in thin films (comparable to the film thickness), you may

also encounter problems with isotropic wet etching, since the undercutting will be at least

equal to the film thickness. With dry etching it is possible etch almost straight down without

undercutting, which provides much higher resolution.

Defects in Crystal

Figure 28: Difference between anisotropic and isotropic wet etching.

Dry etching: The dry etching technology can split in three separate classes called reactive

ion etching (RIE), sputter etching, and vapor phase etching.

In RIE, the substrate is placed inside a reactor in which several gases are introduced. A

plasma is struck in the gas mixture using an RF power source, breaking the gas molecules

into ions. The ions are accelerated towards, and reacts 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. 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 etch processes

that balance chemical and physical etching, since there are many parameters to adjust. By

changing 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. A schematic of a typical reactive ion

etching system is shown in the figure below.

A special subclass of RIE which continues to grow rapidly in popularity is deep RIE (DRIE). In

this process, etch depths of hundreds of microns can be achieved with almost vertical

sidewalls. The primary technology is based on the so-called "Bosch process", named after

the German company Robert Bosch which filed the original patent, where two different gas

compositions are alternated in the reactor. The first gas composition creates a polymer on

the surface of the substrate, and the second gas composition etches the substrate. The

polymer is immediately sputtered away by the physical part of the etching, but only on the

horizontal surfaces and not the sidewalls. Since the polymer only dissolves very slowly in the

chemical part of the etching, it builds up on the sidewalls and protects them from etching.

As a result, etching aspect ratios of 50 to 1 can be achieved. The process can easily be used

to etch completely through a silicon substrate, and etch rates are 3-4 times higher than wet

etching.

Defects in crystals and Elements of Thin Films

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

subjected to the ion bombardment instead of the material target used in sputter deposition.

Vapor phase etching is another dry etching method, which can be done with simpler

equipment than what RIE requires. In this process the wafer to be etched is placed inside a

chamber, 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 diflouride (XeF2), 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.

The first thing you should note about this technology is that it is expensive to run compared

to wet etching. If you are concerned with feature resolution in thin film structures or you

need vertical sidewalls for deep etchings in the substrate, you have to consider dry etching.

If you are concerned about the price of your process and device, you may want to minimize

the use of dry etching. The IC industry has long since adopted dry etching to achieve small

features, but in many cases feature size is not as critical in MEMS. Dry etching is an enabling

technology, which comes at a sometimes high cost.

Figure 29: Typical parallel-plate reactive ion etching system.

Defects in Crystal

4.14 LET US SUM UP

Like anything else in this world, crystals inherently possess imperfections, or what we often

refer to as 'crystalline defects'. The presence of most of these crystalline defects is undesirable

in silicon wafers, although certain types of 'defects' are essential in semiconductor

manufacturing. Engineers in the semiconductor industry must be aware of, if not

knowledgeable on, the various types of silicon crystal defects, since these defects can affect

various aspects of semiconductor manufacturing - from production yields to product reliability.

Crystalline defects may be classified into four categories according to their geometry. These

categories are: 1) zero-dimensional or 'point' defects; 2) one-dimensional or 'line' defects; 3)

two-dimensional or 'area' defects; and 4) three-dimensional or 'volume' defects. Table 2

presents the commonly-encountered defects under each of these categories.

Table 2. Examples of Crystalline Defects

Defect Type Examples

Point or Zero-Dimensional

Defects

Vacancy Defects

Interstitial Defects

Frenkel Defects

Extrinsic Defects

Line or One-Dimensional

Defects

Straight Dislocations (edge or

screw)

Dislocation Loops

Area or Two-Dimensional

Defects

Stacking Faults

Twins

Grain Boundaries

Volume or Three-Dimensional

Defects Precipitates

Defects in crystals and Elements of Thin Films

Voids

There are many forms of crystal point defects. A defect wherein a silicon atom is missing

from one of these sites is known as a 'vacancy' defect. If an atom is located in a non-lattice

site within the crystal, then it is said to be an 'interstitial' defect. If the interstitial defect

involves a silicon atom at an interstitial site within a silicon crystal, then it is referred to as a

'self-interstitial' defect. Vacancies and self-interstitial defects are classified as intrinsic point

defects.

If an atom leaves its site in the lattice (thereby creating a vacancy) and then moves to the

surface of the crystal, then it becomes a 'Schottky' defect. On the other hand, an atom that

vacates its position in the lattice and transfers to an interstitial position in the crystal is

known as a 'Frenkel' defect. The formation of a Frenkel defect therefore produces two

defects within the lattice - a vacancy and the interstitial defect, while the formation of a

Schottky defect leaves only one defect within the lattice, i.e., a vacancy. Aside from the

formation of Schottky and Frenkel defects, there's a third mechanism by which an intrinsic

point defect may be formed, i.e., the movement of a surface atom into an interstitial site.

Extrinsic point defects, which are point defects involving foreign atoms, are even more

critical than intrinsic point defects. When a non-silicon atom moves into a lattice site

normally occupied by a silicon atom, then it becomes a 'substitutional impurity.' If a non-

silicon atom occupies a non-lattice site, then it is referred to as an 'interstitial impurity.'

Foreign atoms involved in the formation of extrinsic defects usually come from dopants,

oxygen, carbon, and metals.

The presence of point defects is important in the kinetics of diffusion and oxidation. The

rate at which diffusion of dopants occurs is dependent on the concentration of vacancies.

This is also true for oxidation of silicon.

Crystal line defects are also known as 'dislocations', which can be classified as one of the

following: 1) edge dislocation; 2) screw dislocation; or 3) mixed dislocation, which contains

both edge and screw dislocation components.

Defectsin Crystal

An edge dislocation may be described as an extra plane of atoms squeezed into a part of the

crystal lattice, resulting in that part of the lattice containing extra atoms and the rest of the

lattice containing the correct number of atoms. The part with extra atoms would therefore

be under compressive stresses, while the part with the correct number of atoms would be

under tensile stresses. The dislocation line of an edge dislocation is the line connecting all

the atoms at the end of the extra plane.

Fig. 28. An edge dislocation; note the insertion

of atoms in the upper part of the lattice

If the dislocation is such that a step or ramp is formed by the displacement of atoms in a

plane in the crystal, then it is referred to as a 'screw dislocation.' The screw basically forms

the boundary between the slipped and unslipped atoms in the crystal. Thus, if one were to

trace the periphery of a crystal with a screw dislocation, the end point would be displaced

from the starting point by one lattice space. The dislocation line of a screw dislocation is the

axis of the screw.

Figure 29. A screw dislocation; note the screw-like

'slip' of atoms in the upper part of the lattice

Defects in crystals and Elements of Thin Films

If the dislocation consists of an extra plane of atoms (or a missing plane of atoms) lying

entirely within the crystal, then the dislocation is known as a 'dislocation loop.' The

dislocation line of a dislocation loop forms a closed curve that is usually circular in shape,

since this shape results in the lowest dislocation energy.

Dislocations are generally undesirable in silicon wafers because they serve as sinks for

metallic impurities as well as disrupt diffusion profiles. However, the ability of dislocations

to sink impurities may be engineered into a wafer fabrication advantage. i.e., it may be used

in the removal of impurities from the wafer, a technique known as 'gettering.'

A grain boundary is the interface between two grains in a polycrystalline material. Grain

boundaries disrupt the motion of dislocations through a material, so reducing crystallite size

is a common way to improve strength, as described by the Hall-Petch relationship. Since

grain boundaries are defects in the crystal structure they tend to decrease the electrical and

thermal conductivity of the material. The high interfacial energy and relatively weak bonding

in most grain boundaries often makes them preferred sites for the onset of corrosion and for

the precipitation of new phases from the solid. They are also important to many of the

mechanisms of creep.

Luminescence: The term luminescence is used to describe a process by which light is

produced other than by heating. The production of light from heat, or incandescence,

is familiar to everyone. The Sun gives off both heat and light as a result of nuclear

reactions in its core. An incandescent lightbulb gives off light when a wire filament

inside the bulb is heated to white heat. One can read by the light of a candle flame

because burning wax gives off both heat and light. But light can also be produced by

other processes in which heat is not involved. For example, fireflies produce light by

means of chemical

4.15 CHECK YOUR PROGRESS : THE KEY

1. Point defects can be divided into Frenkel defects and Schottky defects, and these often

occur in ionic crystals. The former are due to misplacement of ions and vacancies. Charges

are balanced in the whole crystal despite the presence of interstitial or extra ions and

vacancies. On the other hand, when only vacancies of cation and anions are present with no

interstial or misplaced ions, the defects are called Schottky defects.

Defects in Crystal

2. Few, if any, crystals are perfect in that all unit cells consist of the ideal arrangement of atoms

or molecules and all cells line up in a three dimensional space with no distortion. Some cells

may have one or more atoms less whereas others may have one or more atoms than the

ideal unit cell. The imperfections of crystals are called crystal defects. The missing and

lacking of atoms or ions in an ideal or imaginary crystal structure or lattice and the

misalignment of unit cells in real crystals are called crystal defects or solid defects. The two

terms are interchangeable. Crystal defects occur as points, along lines, or in the form of a

surface, and they are called point, line, or plane defects respectively.

3. Diffusion is a process where material is transported by atomic motion. One of the

simplest forms of diffusion , "diffusion bonding", occurs when two materials come in contact

with each other. There are two basic mechanisms for diffusion: Vacancy diffusion and

Interstitial diffusion. The reason for the two types of diffusion stems from the relationship

between their relative atomic sizes. Vacancy diffusion occurs primarily when the diffusing

atoms are of a similar size, or substitutional atoms. The movement of a substitutional atom

requires a vacancy in the lattice for it to move into. Interstitial diffusion occurs when the

diffusing atom is small enough to move between the atoms in the lattice. This type of

diffusion requires no vacancy defects in order to operate. Diffusion depends on five main

variables. These variables are: initial concentration, surface concentration, diffusivity, time,

and temperature. The initial concentration of the material that is diffusing into the base

("solid") material is often referred to as the "solute" or impurity which can be but is not

always zero. The surface concentration is the amount, weight %, of solute near the surface

of the "solid", base, material. Diffusivity is defined as the rate at which the solute or impurity

penetrates into the solid, base, material. The time and temperature are fairly self-

explanatory, though it should be noted that the temperature increases the rate of diffusion

with increasing temperature

4. A point lattice defect, which produces optical absorption bands in an otherwise

transparent crystal.

Color centers are imperfections in crystals that cause color (defects that cause color by

absorption of light). Due to defects, metal oxides may also act as semiconductors, because

there are many different types of electron traps. Electrons in defect region only absorb light

at certain range of wavelength. The color seen are due to lights not absorbed. For example,

a diamond with C vacancies (missing carbon atoms) absorbs light, and these centers give

Defects in crystals and Elements of Thin Films

green color as shown here. Replacement of Al3+ for Si4+ in quartz give rise to the color of

smoky quartz.

A high temperature phase of ZnOx, (x < 1), has electrons in place of the O2-

vacancies.

These electrons are color centers, often referred to as F-centers (from the German

word farben meaning color). Similarly, heating of ZnS to 773 K causes a loss of

sulfur, and these material fluoresces strongly in ultraviolet light.

Some non-stoichiometric solids are engineered to be n-type or p-type semiconductors.

Nickel oxide NiO gain oxygen on heating in air, resulting in having Ni3+

sites acting

as electron trap, a p-type semiconductor. On the other hand, ZnO lose oxygen on

heating, and the excess Zn metal atoms in the sample are ready to give electrons. The

solid is an n-type semiconductor.

Defect in Crystal

List of References and Suggested Readings

Luminescent Materials And Applications- Adrian Kitai

Solid State Physics By A J Dekkar

Crystals, Defects and Microstructures Modeling Across Scales-ROB PHILLIPS

Solid-State Physics- Introduction to the Theory - James D. Patterson Bernard C. Bailey

Solid State Physics -by Neil W. Ashcroft N. David Mermin David Mermin

Modern Physics & Solid State Physics 3rd Edition by Pillai S O

UNIT-V ELEMENTS OF THIN FILMS

Structure

5.0 Introduction

5.1 Objectives

5.2 Concept of Thin Films

5.3 The Electrical Conduction in Thin Films

2.3.1 Generations of charge carriers

5.4 Deposition of Thin Films by Thermal Evaporations

5.5 Cathodic Sputtering

5.6 Evaporation at Reduced Pressure (Vaccum Evaporation)

5.7 Thickness Measurement

2.7.1 Multiple-beam Interferometry

2.7.2 Tolansky Technique

2.7.3 Four Probe Method

5.8 Size Effect

5.9 Fuchs- Sondheimer Model

5.10 Lets Us Sum Up

5.11 Check Your Progress: The Key

VII. 5.0 INTRODUCTION

The use of thin films for the construction of resistors goes back at least 50 years When used for the

fabrication of discrete resistors, thin films offer improved performance and reliability as compared

with resistors of the composition type-and lower cost for comparable performance when compared

with precision wire wound resistors. It is in the area of integrated circuitry, however, that thin film

resistors have really come into their own. For resistors having minimum dimensions of 5 to 10 mils,

fired glazes can compete very well with thin films; but where precision resistors are needed (with

dimensions of 5 mils or less), the use of thin films becomes mandatory. The application of thin films

to discrete resistors has been reviewed in a number of places. More recent work has been done on

thin film resistors in integrated circuit applications.

Choice of Materials

A. Film-resistor Requirements

Most film-resistor requirements can be met with films having Rs (sheet resistance) in the range 10 to

1,000 ohms/sq. Resistors below 10 ohms are rarely needed, whereas resistors with values in the

megohm range can be realized through use of very long path lengths. There remains, however, a

limited, but urgent, need for films with Rs greater than 1,000 ohms/sq, and much of current

research on thin film resistors is devoted to finding a solution to this problem. Besides a suitable

sheet resistance, films must possess a low temperature coefficient of resistance (generally less than

100 ppm/0 c). They must also be sufficiently stable so that any changes in resistance value that

Occur during their operating life may reliably be expected to fall below some pre specified value.

Finally, the process that is used to prepare the resistors must be such that the final product can be

made to meet its specifications at a reasonable cost.

B. Sources of Resistivity in Films

It can be inferred from the foregoing remarks that materials used for resistive thin films should have

resistivities in the range 100 to 2,000 µohm-cm. It will be recalled, however, that metals in bulk

cannot have resistivities much in excess of the lower limit of this range (as is summarized in Table 1).

Bulk semiconductors can readily satisfy these resistivity requirements, but this is invariably at the

price of a large negative temperature coefficient. Semi metals such as bismuth and antimony (and

their alloys) show about an order of magnitude increase in resistivity over the metals but their low

melting points and relatively large temperature coefficients make these materials unattractive for

resistor applications.

TABLE 1 Approximate Maximum Contribution to the Residual Resistivity by Various Types of Defect

Defects in crystals and Elements of thin Films

e

Type of defect Contribution, µohm-cm

Dislocations. 0.1

Vacancies. 0.5

Interstitials 1

Grain boundaries. 40

Impurities in equilibrium, 180

Fortunately, many materials, when deposited In film form, achieve resistivities that are significantly

higher than their bulk counterparts, without necessarily acquiring large temperature coefficients.

Some of the ways in which this comes about include:

1. There may be a significant amount of scattering of the conduction electrons at the film surface

(Fuchs-Sondheimer effect), leading to high resistivity as well as low temperature coefficient.

However, because of the very small thickness normally required to produce the effect, this increased

resistivity is extremely sensitive to any changes in the thickness. In addition, such films are liable to

agglomerate rather easily and therefore have very limited mechanical integrity. Practical thin film

resistors rarely rely directly on this phenomenon as a source of resistivity.

2. The material may contain impurities or imperfections in concentrations greatly in excess of

thermodynamic equilibrium. This, too (by Mathiesseri's rule), will lead to a low temperature

coefficient. However, drastic departures from equilibrium are liable to lead to precipitation later

(during the operating life of the component). Even if excessive defect concentrations are not

present, any change in the defect concentration (for whatever reason) will be reflected as a

resistance change during the life of the film. In practice, this problem is overcome either by

incorporating a stabilizing heat treatment into the resistor fabrication process or by employing only

very refractory materials, or both.

3. Two-phase systems (metal-insulator or cermet films). This type of system "dilutes" a conductive

film by dispersing it in an insulator matrix so that the physical thickness of the film is considerably

greater than its electrical thickness. The resistivity of such a film may, consequently, include a

significant contribution from the surface scattering of electrons. The film itself will be much more

robust than a film in which surface scattering results from a straight forward reduction in thickness.

Elements of Thin Films

e

A significant problem with such films is the control of composition which, if lost, may lead to large

negative temperature coefficients as well as to poor stability.

4. Low-density or porous films. These are similar to those of type 3 above, in that they have a

physical thickness considerably greater than their electrical thickness. An example of such a film is

low-density tantalum. One problem with this type of film is that it has a very large surface area and

is therefore very susceptible to oxidation effects. If suitably protected, however, such a film can have

high resistance with low temperature coefficient and adequate stability.

5. Semi continuous films. These are films that are still in the "island" stage of growth. The spacing

between islands is such that the positive temperature coefficient of the metal islands just balances

the negative temperature coefficient associated with electron transfer between islands . In such

films, there is always a danger of agglomeration. The films are also susceptible to oxidation effects,

as well as presenting a control problem during deposition. Successful resistors of this type have,

however, been reported for the case of rhenium.

6. Stratified films. A thin layer with a positive temperature coefficient and low resistivity may overlay

a thicker layer having negative temperature coefficient and high resistivity, giving a combination that

has high resistivity and low temperature coefficient. Such films are obtained as a natural result of

gettering during deposition . Many chromium and Nichrome films are in this category. Their

principal problem is control since the exact amount of contaminant taken up by such films varies

with the deposition conditions.

7. New crystal structures. Certain materials may assume, when in thin film form, a crystalline

structure, which does not exist in bulk. These structures often exhibit relatively high resistivity and

low TCR, probably as a result of having a low density of conduction electrons. The best-known

example is ß-tantalum.

The phenomena are summarized in Table 2.

Description

Mechanism for Effect on Example

Resistivity increase TCR

Ultrathin. ............. Fuchs-Sondheimer Effect ->0

Trapped gas. ........ . ... Impurity scattering ->0 T a. nitride Insulating phase. .... . ... In tergra.in barriers ->

Cermets

Defects in crystals and Elements of thin Films

e

TABLE

2

Mecha

nisms

Causin

g Metal Films to Have Resistivities Greater than the Bulk

Netlike. . . . . . . . . . . . . . . . Construction resistance ->0 Low-density Ta

Discontinuous. ...... Particle separation ->-

Rhenium

Double Iayer

.

- _ .. TCRs cancel ->0 I Cr

New structure. Fewer carriers ->0 Ta

Check Your Progress 1

Notes : (i) Write your answer in the space given below

(ii) Compare your answer with those given at the end of the unit

What is thin films deposition?

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5.1 OBJECTIVES

The Main aim of this unit is to study Thin flim.After going through the unit you will be able to

understand

Describe the Thin Film

Method for creating thin films

Method for measuring thickness of thin film

5.2 CONCEPT OF THIN FILMS

The thin films have got the surface only not thickness but this is ideal difference of films. All the films

have surface thickness and therefore when the thickness of the surface layer is of the order of a

fraction of the millimeter this called the thin films. The thin films may be of metallic it is called

metallic thin films. The thin films may be of polymers are isutullared materials called polymers thin

films.

The thin films can be deposited add by the following methods

(i) Blowing methods (ii) Photolytic methods (iii) Vaccum Evaporation (iv) Films formation from solution (v) Iso thermal immersion technics (vi) Costing from solution (vii) Thermal Evaporation

VIII. 5.3 THE ELECTRICAL CONDUCTION OF THIN FILMS Electrical conduction in polymeric dielectrics is mainly due to transport of free charge carriers

present in the bulk of the polymer and from a number of different conduction processes taking place

simultaneously depending upon the experimental conditions. The structures of these materials are

sensitive to their electrical, mechanical and thermal history so that the mode of conduction differs

from polymer to polymer and the sensitivity of measurement is different for different materials.

Elements of Thin Films

e

When a polymer is subjected to different conditions they often undergo structural transitions

making charge carrier generation and transport phenomenon more complicated. No universally

accepted theory has been propounded till date, which can explain the conduction phenomenon in all

the polymeric dielectrics. However, attempts have been made to explain the observed conduction

behavior on the basis of various existing theories. Many workers have tried to explain the dark

conduction in polymers in their own way, such as traps and their energy distribution, tunnelling of

charge carriers, Schottky emission avalanche breakdown etc. Still, despite inconsistencies in

understanding the conduction mechanism, one can conclude on the basis of various studies

reported in the literature that some of the phenomena Occurring during electrical conduction in

polymers are similar and have physical origin similar to those observed in solids of poor electrical

conductivity. In general, polymers are amorphous or semi crystalline substances. The transport

mechanism in amorphous bodies is more complicated than the crystalline materials, especially for

mono crystals where long range order exists. Thus, the charge transport mechanism in dielectric

solids can be better understood from modifications applied to the quantum mechanical band theory

of solids. Hence, band model of disordered materials has some of the gross features of that of

crystalline Structure, but with significant differences details. Electronic conduction may be due to

the motion of free electrons in the conduction band or holes in the valence band or alternately due

to the motion of quasi localized carriers. If the concepts of band model are applied directly to

organic solids, a very large energy gap between valence and conduction bands is expected, so that

thermal activation in the normal temperature range is too small to transfer an electron from the

valence band to the conduction band. In amorphous substances, there are many localized charge

carrier levels and carrier mobility is very low. The low lying states may be treated as trapping sites

(levels) but in comparison with crystalline substances they are not related to the discrete activation

energy values because they are situated in the broadened edges of conduction band and valence

band. Hence, it is difficult to consider the transport behavior of polymers in terms of a generalised

theory. It is, therefore, not surprising to see controversies on transport theories in the literature or

no single mechanism is able to explain the entire conduction in these materials. However, the

theories proposed for amorphous and polycrystalline inorganic solids are normally applied to

describe the conduction behavior of these Materials with a few limitations.

5.3.1 Generation Of Charge Carriers

Most of the materials reveal in dark an exponential temperature dependence of conductivity (σ) of

the form, σ = σ0 exp (- A/kT) Where A is the activation energy, k is the Boltzmann constant and T is

the temperature. This led the earlier workers to assume that carriers are intrinsic in nature and

Defects in crystals and Elements of thin Films

e

hence they equated the experimental activation energy to half the band gap. The resistivity of

polymers is high because both the mobility and carrier concentration are low. The concentration of

carriers produced intrinsically by thermal ionization is also very low since the band gaps are several

electron volts. Hence, it seems more likely that ionization of impurities is responsible for any

outstanding concentration of carriers Impurities may also provide carriers by internal field emission

in the presence of gross doping. The charge carrier generation through the injection of electrons and

holes from the electrodes has been widely accepted and this is probably the main source of carriers

in high polymers. It is important to note here that the carrier density within the material should be

much greater than the material being treated, i.e., the contact should act as the reservoir of carriers.

Carriers once injected have appreciable mobilities and life times. Several workers have studied the

injection of charge carriers in polymer. Hofman has shown that conduction in atactic polystyrene

(PS) depends on the injection of excess of electrons from metals. Davies studied injection in

polyvinyl fluoride and has shown that injection continues for a long time and for both polarities of

applied potentials, although some asymmetry is indicated. Despite a great deal of work done, there

are still plenty of unanswered questions about the origin of free charge carriers, which take part in

conduction under electrical stream. It is still not evident whether the measured current is by the

motion of charge carriers inherent to the polymers or those injected from the electrodes. Adamec

and Calderwood measured current in polymethyl methacrylate (PMMA) under two conditions; first

when the specimen was in direct contact of the electrode, and second when an insulating air gap

was present between the specimen and electrode. The finding that the conductivity determined by

the experiment with contact less electrodes is the same as that obtained with evaporated electrodes

supports the contention that the free charge carriers originate in the bulk of the polymeric dielectric.

Doping of polymers with donors and acceptors and blending of two or more polymers

increases/decreases the conductivity by several orders of magnitude and also modifies the charge

carriers response for conduction.

IX. 5.4 DEPOSITION OF THIN FILMS BY THERMAL EVAPORATIONS

Usually, the choice of deposition method is made after the material has been selected. In a limited

number of cases, however, a particular deposition technique may be preferred if it fits more easily

into a larger process. At any rate, before the final choice can be made, three questions must be

asked: Will the method work for this type of material? What degree of control will it allow? How

much will it cost?

Elements of Thin Films

e

Thin film are produced from polymers by thermal evaporation of bulk material here material to be

deposited is heated to a high temperature at very low pressure and in extremely clean conditions

where it vaporizes .The vapour is then allowed to condense on a substrate placed above the source

to form a film. Thermal evaporation was reported to lead to a wax like deposit on a substrate,

together with gaseous fractions and solid residue. Evaporated polymer films are contaminated due

to the vigorous boiling action of the molten polymer and due to the rapid evaluation of break down

products. However uncontaminated film can be obtain by choosing a low evaporation temperature

and thus a slow rate of deposition and by specially designed thermal evaporation method,

combination of internal baffles and flash evaporation and laser evaporation.

5.5 CATHODIC SPUTTERING

This is a preferred. method for very refractory metals such as tantalum) and for alloy systems (such

as nichrome) when a very high degree of control is required. During conventional sputtering, there is

a greater likelihood of contamination than during evaporation. However, the introduction of

techniques such as bias sputtering and getter sputtering has considerably improved this situation

resistance monitoring during sputtering is difficult because of interference from the discharge

plasma. However, control of thickness through deposition time alone is usually easier during

sputtering than during evaporation

One of the major limitations to sputtering is that the material to be deposited may not always be

available as a sheet large enough to form a cathode. Relatively little work has been done on the use

of very large cathodes in batch systems. However, sputtering is well suited for use in a continuous-

feed system, since there is no source replenishment problem. Masking however is difficult during

sputtering, unless in-contact masks are used. Substrate temperatures are comparable with those

required for evaporated films, but their control is much more difficult during sputtering.

5.6 EVAPORATION AT REDUCED PRESSURE (VACCUM EVAPORATION)

The most widely used method for resistor film- deposition is vacuum evaporation, since most

materials lend themselves to deposition through this technique. The commonest exceptions are the

refractory metals and materials such as tin oxide which may decompose on evaporation. Major

problems associated with vacuum evaporation arc the great sensitivity of the amount of

contamination to the deposition conditions, and the difficulty of obtaining uniform film thickness

Defects in crystals and Elements of thin Films

e

over large areas. This, in turn, is intimately related to cost, since if large number can be processed at

one time, the process will obviously be cheaper.

Resistance monitoring during vacuum evaporation is straightforward and easily implemented,

provided the rate of deposition is not too great. Considerable engineering work has already been

done in industry on large vacuum-evaporation systems so that much tooling and fix Turing are

already commercially available. To date. Most evaporation systems have been of the batch rather

than of the continuous-feed type because it is difficult to replenish the evaporant source constantly

without breaking the vacuum. In cases where the tolerances involved allow the use of masks to

define the resistor patterns, evaporation is the preferred method, since manipulation of mask

changers in vacuum does not present any serious problem.

X. 5.7 THICKNESS MEASUREMENT In this topic, the most useful techniques for determining film thickness and composition will be

discussed in sufficient detail for the reader to understand them, but references will be given for

further details. References will also be given for some film-thickness-measuring techniques, which

are of limited applicability for general laboratory use. In other cases, we may not go into great detail

Check Your Progress 2

Notes : (i) Write your answer in the space given below

(ii) Compare your answer with those given at the end of the unit

What are different deposition methods

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Elements of Thin Films

e

because excellent reviews, books, and monographs have already been written on these particular

techniques. Specifically the reader is referred to the books by Tolansky. Heavens. Vasicek. Mayer,

and Francon, as well as to recent reviews written by Heavens' and by the Bennetts. Some of the

advantages and disadvantages of many film-thickness-measuring techniques have been listed in a

recent article by Gillespie. "The reader may also find of interest a survey by Keinath which covers

very briefly Thickness-measuring techniques with emphasis on industrial applications up to 1955.

The best technique for a specific application' or process, depends upon the film type, the thickness

of the film, the accuracy desired, and the use of the film. These criteria include such properties as

film thickness, film transparency, film hardness, thickness uniformity, substrate smoothness,

substrate optical properties, and substrate size. In many cases there is no single best technique, and

the particular one chosen will be determined by the personal preferences of the investigator.

Since thin film thicknesses are generally of the order of a wavelength of light, various types of optical

interference phenomena have been found to be most useful for the measurement of film

thicknesses. We have thus tended to emphasize these. In addition to interference phenomena, there

are other optical techniques, which can be used to measure thickness. Examples are ellipsometry

and absorption spectroscopy.

In addition to the optical techniques, there are mechanical, electrical, and magnetic techniques,

which have been used for film-thickness measurements. Among these, the one that has found the

widest acceptance is the stylus technique

5.7.1 Multiple-beam Interferometry

The sharpness of the fringes increases markedly if interference occurs between many beams. This

can be accomplished if the reflectivities at the two interfaces are very high as indicated in Fig. 1,

where each of the two parallel glass plates has a thin partially transparent silver film indicated by

CDEF and IJLK

deposited on

it. Another

condition for

multiple-

beam inter-

ference is

small

absorptivity A

Defects in crystals and Elements of thin Films

e

of the silver film through which light must be transmitted. In the case of multiple beam reflection,

only the silver film CDEF need be fairly transparent (low absorptivity), whereas for multiple-beam

transmission both silver films must have low absorptivities. More details on the conditions necessary

for good multiple-beam interferometry may be found in the works of Tolansky or Flint

Fig.1 Schematic representation of multiple-beam interference between two silvered glass plates. In this figure, R and T denote the

amplitudes of reflection and transmission.

To understand multiple-beam interferometry better, the equations which describe the fringe

positions and relative intensities must be discussed. In Fig. 1, the medium between the two glass

plates has a refractive index n1 and thickness t, The angle of incidence of the radiation is . In the

case of transmission the intensity is given by the Airy formula.

Equation (2) does not rigorously consider the phase changes occurring on reflection however; in

practice this has little effect

Fabry called F the "co-efficient of finesse." In English texts it is generally called the coefficient of

fineness. This coefficient is a measure of how fine or sharp the interference fringes can become.

With properly evaporated silver films, a reflectivity of near 0.95 is possible, which gives a value of F

of over 1,200 (for details see Tolansky"). Thus, as long as sin (' /2) has any significant value, the

intensity of the transmitted light I is very small, as can be seen from Eq. (1). If sin ('/2) becomes

zero, I reaches the maximum of Imax, This occurs only if ' is an integral value of 2π and therefore,

very sharp interference fringes with intensity maxima for integral values of N are observed. In the

case of multiple-beam interference by reflection, the interference pattern formed-the so-called

interferogram-is just the opposite of that seen in transmission provided the absorptivity is small. In

other words, where there are sharp, bright fringes on a dark background in transmission for integral

values of N, observation of the reflected light gives sharp, dark fringes on a bright background.

With an increase in the absorption A, the intensity of the entire transmitted pattern is decreased by

[T/(T + A)]2. In the reflected-fringe system, however, an increase in the absorption A prevents the

fringe minimum from going to zero. Hence, with larger A, the reflected fringes tend to become

Elements of Thin Films

e

washed out under the same conditions where they would still be seen on transmission.

Consequently, to obtain a sharply delineated reflected-fringe system, the absorption A should be

kept to a minimum. In addition to the requirements of high reflectivity and small absorption for

good quality fringes, the interplate distance t should be as small as possible. Referring back to Eq.

(2), we can see that there are several factors which contribute to the formation of fringes. For

practical applications, fringe systems are identified according to the method of fringe formation, and

two cases are distinguished in multiple-beam interferometry. Fizeau fringes are generated by

monochromatic light and represent contours of equal thickness arising in an area of varying

thickness t between two glass plates similar to those shown in Fig. 1 This is accomplished by

contacting the two glass plates such that they form a slight wedge at an angle α so that t varies

between the two plates. The angle a is generally made very small so that consecutive fringes are

spaced as far apart as possible. The angle of incidence is typically kept near 00 and the medium is

air (n1 = 1.0). Hence, the spacing between fringes corresponds to a thickness difference of λ/2,

where λ is the wavelength of the monochromatic radiation being used.

Fig.2 Semitransparent optical flat (ABEF) with film sample (KLIM) including step (LM) for multiple –beem interferometric measurement

of film thickness

The second multiple-beam interferometry technique is referred to as fringes of equal chromatic

order, or FECO. In this case, white light is used at an angle of incidence of 00 and the reflected or

transmitted white light is dispersed by a spectrograph, thus offering a means of varying λ. According

to Eq. (2), fringes will form for certain values of t/ λ. Thus, FECO fringes can be obtained with the two

silvered surfaces parallel to each other, whereas the plates must be inclined relative to each other to

Defects in crystals and Elements of thin Films

e

produce Fizeau fringes. The spacing between FECO fringes on the interferogram (or spectrum) is

inversely proportional to the thickness.

Multiple-beam interferometry for the measurement of film thickness can be implemented by the

method of Donaldson and Kharnsavi, which is shown in Fig. 2. The arrangement differs from Fig. 1 in

that the substrate GHIJ supports a film KLM I whose thickness is to be measured. A highly reflective

opaque metal film NKJO is evaporated on top of the film KLMI. Silver films of about 1,000 A0

thicknesses are typically used for this purpose. The step LM in the original film may be formed either

by etching the film after deposition or by masking the MJ part of the substrate during deposition.

Evaporated silver replicates such steps accurately: the bottom surface of the reference plate ABCD

has a thin, highly reflective, semitransparent film as in Fig. 1. To produce Fizeau fringes, the

reference plate ABCD is inclined at an angle with respect to the substrate underneath it, as shown in

Fig. 3 In the case of FECO fringes, the film substrate and optical flat are parallel to each other. The

distance between the plates is adjusted according to the film thickness since the magnification, i.e.,

the spacing between the fringes on the interferogram, increases with decreasing separation

between the plates.

Note that with either of these techniques the film KLMI may be either opaque or transparent. The

requirements for the methods are that a step or channel can be made in the film down to the

substrate surface, that the substrate is fairly flat and especially smooth, that the film itself has a

smooth surface for fringe formation, and that the film should not be altered by the deposition of the

reflective coating. For example, some organic films are altered by the heat generated during the

evaporation of the reflective metal and should not be measured by this technique.

5.7.2 Measurement of 'Fizeau Fringes (Tolansky Technique)

The use of Fizeau fringes for thickness measurements is commonly called the Tolansky technique in

recognition of Tolansky's contributions to the field of multiple-beam interferometry. A schematic

representation of Fizeau fringes produced by multiple-beam interference is shown in Fig. 3 The

Elements of Thin Films

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sample, channel size, wedge angle, etc., are exaggerated for illustrative purposes and not drawn to

scale.

The film thickness is given by d = ΔN λ /2 where ΔN is the number of fringes or fraction

Fig. 3 Schematic view of apparatus for producing multiple –been Fizeau fringes (Tolansky technique).

thereof traversing the step. In the interferogram shown in Fig. 3, the depth

of the channel (the film thickness) is exactly one-half of the separation between fringes (ΔN = 0.5),

and therefore the film thickness is λ./4. If the wavelength were that of the green mercury line, the

film thickness would be 1,365 A0.

Commercial microscopes which utilize Fizeau multiple-beam interferometry are available. Examples

are the Sloan Angstrometer M-1OO·* and the Varian A0-Scope Interferometer. In addition,

conventional metallurgical microscopes can be easily equipped with Fizeau plate attachments for

interferometric measurements. These plate attachments are generally equipped with three

adjustable screws to determine the tilt of the plate relative to the specimen and thus control the

direction and spacing of the interference fringes. These adjustments can be very tedious. An

Defects in crystals and Elements of thin Films

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example of such an instrument has been described by Klute and Fajardo. Whose stage inter-

ferometer facilitates fairly convenient adjustments. Accurate thickness measurements require

careful evaluation of fringe fractions. These may be measured by a calibrated microscope eyepiece,

or more accurately and commonly on a photomicrograph of the fringe system. Either way, the

evaluation requires a linear measurement, the accuracy of which is strongly dependent on

Fig.4 Schematic of apparatus for producing multiple-beam fringes of equal chromatic order (FECO)

the definition and sharpness of the fringes. As previously mentioned, this would require the optical

flat (Fizeau plate) to have high reflectivity and low absorptivity.

Two other prerequisites for an accurate thickness measurement are (1) extremely flat, smooth film

surface and (2) very well collimated and narrowband monochromatic light. Thickness measurements

from 30 to 20,000 A0 can be made routinely to an accuracy of ± 30 A0. With care, film thicknesses can

be measured to an accuracy of ± 10 A0

(b) Fringes of Equal Chromatic Order (FECO). Fringes of equal chromatic order are more difficult to

obtain but yield greater accuracy than Fizeau fringes, especially if the films are very thin. A more

detailed discussion of the subject can be found in the works of the Bennetts' and Tolansky. The

principle will be understood from the schematic of the apparatus shown in Fig. 4.

Elements of Thin Films

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Collimated white light impinges at normal incidence on the two parallel plates. The reflected light is

then focused on the entrance slit of a spectrograph. The image of the channel in the film must be

perpendicular to the entrance slit of the spectrograph. Assuming an angle of incidence of = 0°,

then sharp, dark fringes occur for integral values of N = 2t/ λ, as shown schematically in Fig. 5 for two

different plate spacings t. The fringes are observed as the wavelength λ is varied by the spectrograph

and are recorded on a photographic plate corresponding to λ = 2t/N. The resulting interferograms

are shown schematically in Fig. 5 where the scale is assumed to be linear in wavelength. with a

linear-wavelength scale, the fringes are not equidistant in the interferogram. Note, too, that the

order of the fringe increases with decreasing wavelength of the fringe.

Fig. 5. (a) Interferogram for parallel plate spacing o 2 (b) Interferogram for the same parallel plate spacing of 2 but

with a channel 1,000 A0 deep corresponding tp a 1,000 –A

0 film on the lower plate (c) Interferogram for parallel – plate

spacing of 1 (d) Interfergram for the 1 but with a channel 1,000 Ao- deep corresponding to a 1,000 – A

0 film .

Defects in crystals and Elements of thin Films

e

It is sometimes preferable to give the order as N = 2tѵ, where ѵ is the reciprocal wavelength or the

frequency in wave numbers, as this relation shows N to be linear with wave number. However, this

linearity is not found to be exactly true if the phase changes at the two reflecting interfaces vary

with wavelength. Hence, there may be a slight dispersion with wavelength, but the effect of phase

change on the determined fringes is very small. This has been treated more extensively by Bennett.

In practice, the spacing between plates is not known a priori as assumed in the hypothetical case

shown in Fig. 5. It is therefore necessary to deduce t as well as N from the wavelengths of the

observed fringes. If N1 is the order of a fringe corresponding to wavelength λ1 , then N1 + 1 is the

order of the next fringe with a shorter wavelength λo on the interferogram. Neglecting the small

phase-change dispersion, we have

Solving for N1 we obtain )5(21 0111 tNN

)6(01

0

1

N

And can now express t solely in terms of measured wavelengths :

)(22 01

0111

Nt (7)

To determine the film thickness, consider that a channel of depth d causes the fringe of the order N1

to be displaced to a new wavelength 1 Consequently the film thickness d must satisfy the relation

2

'

11Ndt (8)

Thus, the film thickness d is given by

01

01

'

111

'

11

222

NNd (9)

In fig. 5 b and d, the fringe displacements due to the channel are clearly related to the order N of the

fringes outside the channel . However if the slop were so steep that the order in the channel could

not be related ti that outside the channel, the film thickness could still be derived from the equation

Elements of Thin Films

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)(2

111

'

1

'

1 NNd (10)

Whereby the unknown order N1' must be obtained by observation of a second displaced fringe of

order N1' + 1 as in Eq. (7).

The effect of interplate spacing is shown by comparison of Fig. 5 a and b with Fig. 5 c and d. It is clear

from these interferograms that the "magnification" increases with decreasing interplate spacing. The

Bennetts' have concluded that it is possible to measure film thicknesses with an accuracy of 1 or 2

A0, provided very smooth optical flats are used, the films are carefully evaporated, the plates are

carefully aligned, and the fringes are accurately measured.

5.7.3 Four Probe Method

Many conventional methods for measuring resistivity are unsatisfactory for semi-conductors

because metal-semiconductor contact are usually rectifying in nature. Also there is generally

minority carrier injection by one, of the current carrying contacts. An excess concentration minority

carrier will affects the potential of other contacts and modulate the resistance of the material •.

Described here overcomes the difficulties mentioned above and also offers several other,

advantages. It permits measurement 0f resistivity in sample’s having a wide variety of shapes,

including the resistivity of small volumes within bigger pieces of semi-conductors. In this manner the

resistivity on both, sides of a p-n junction can be determined with good accuracy before the material

is cut in bars making devices. This method of measurement is a1so, applicable to • silicon and other

semiconductor materials.

The basic model for all· these measurements is indicated in fig 6 four sharp probes are placed on a'

flat surface of the material to be measured, current is passed through the two outer electrodes and

the floating potential is measured across the inner pair. If the flat surface on which the probes rest is

adequately large and the crystal is big the semi conductor may be considered to be a semi infinite

volume. To prevent minority carrier injection and make good contact, the surface on which the

probes rest may be mechanically lapped. the experimental circuit used for measurement illustrated

schematically in fig 7. A nominal value of probe spacing which has been found satisfactory is an

equal distance of 1.25 mm between adjacent probes. This permits measurement with reasonable

current of n or p- type semi conductors from 0.001 to 50 ohms. cm.

Defects in crystals and Elements of thin Films

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Fig.6 Model for the four probe resistivity Measurements

Fig.7 Circuit Used for Resistivity Measurement

XI. 5.8 SIZE EFFECT OR FOWLER NORDHEIM EQUATION

The tunnel effect between metal electrodes was first studied, in elementary fashion, by Frenksl.

Sommerfeld and Bethel made the first comprehensive studies, in which they included image-force

effects but confined their calculations to very low

(V « o/e) and very high (V» o/e) voltages. Holm made the next notable investigations and extended

the calculation to intermediate voltages, although approximations he used have been found

questionable. Stratton and Simmons further extended the theory, and the results of these studies

are those currently most commonly used in the analysis of experimental data. These models appear

to be quite suitable for predicting the salient features of the tunneling I- V characteristics. There

have been several other studies of a more detailed nature. These have considered the effect of

space charge, traps and ions in the insulator, the effect of the shape of forbidden band,

representation of the insulator by a series of potential wells electric-field penetration of the

electrodes and diffuse reflection. We will discuss the theory of the tunnel effect using the notation

and type of approximation developed by the author. This formulation is readily applicable to

potential barriers of arbitrary shape and to all practical voltage ranges.

The generalized formula gives the relationship connecting the tunnel current density with the

applied voltage for a barrier of arbitrary shape (see Fig. 8) as

2

1

20

2

1

2

1

0

)2(4

)(2

])(exp[)()exp(

mh

sAand

sh

eIwhere

eVAeVAII

(11)

8 = width of the barrier at the Fermi level of the negatively biased electrode

= mean barrier height above the Fermi level of the negatively biased electrode

h = Planck’s constant

m = mass of the electrons

e = unit of electronic charge

Elements of Thin Films

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= a function of barrier shape and is usually approximately equal to unity, a condition we will

assume throughout

Expressed in conventional units, except for s, which is expressed in angstroms, becomes

Fig. 8 Energy diagram arbitrary barrier [see (14) illustrating the parameters and s.

Since J1 = J2, the I-V characteristic is symmetric with polarity of basis for the Voltage range 0 V

1/e.

(2) Voltage Range V > 1/e. From Fig. 10a we have for the reverse – biased condition ucing 1 =2 -

1.,

eV

ss

11

2 (12)

Defects in crystals and Elements of thin Films

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(a)

(b)

Fig. 9 Energy diagrams of potential barriers for V > 2/e (see Fig. 10 for explanation of dotted lines)

Which on substitution in (15) yields

)13()/21(169.0

exp2

1

69.0exp

)(1038.3

2

1

.12

1

1

2

1

1

2

1

210

1

V

VsV

V

sVj

From Fig 9b we have for the forward- biased condition

eV

ss 22

2 (14)

Elements of Thin Films

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Which on substitution in (15) yields

)15()/21(69.0

exp2

1

269.0exp

)(1038.3

2

1

.22

1

2

1

2

1

2

1

210

1

V

VsV

V

sVj

In this case, Eqs. (13) and 15) are not equivalent. It follows, then that the J-V characteristics is

Defects in crystals and Elements of thin Films

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Fig.10. (= J/V) – V tunnel characteristics for 1 = 1 eV , 2 = 2 eV, and s = 30, 30, and 40 A.

At high voltages , i.e. V >> 2/e, both (20) and (21) reduce to the familiar Fowler Nordheim34

form :

F

FJ

2

1

210 69.0exp

1038.3

(16)

asymmetric in this range. In actual fact not only is the J-V characteristic is asymmetric in this range.

In actual fact, not only is the J-V characteristic asymmetric with polarity of bias, but also the

direction of easy conductance reverse at some particular voltage , as shown in Fig. 10.

5.9 FUCHS- SONDHEIMER MODEL

Consider now a metallic thin film with an electric field x . The thickness of the film (in the z-

direction) is t.

When we lose the translational invariance in bulk conductors by using thin films, we have to take

this into account by using the full time derivative of the electron momentum distriburion in the

Boltzmann equation:

(17)

x

relaxationdu

df

m

e

t

g

dt

dz

dz

dg

dt

df 0

Elements of Thin Films

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the first term is called the convective and can be rewritten dz

dgu z . Clearly this term (and its x and y

counterparts) is zero in the bulk caso because g does not depend on any spatial variable when full

translational symmetry is present. This equation has the solution

zu

z

x

euFdu

df

m

eg )(1 (18)

where F (u) is determined by boundary conditions. Since the sign of uz changes at either boundary

(at z = 0 and z = t), we must break up the solution into two pieces: one for the distribution traveling

up and the traveling down:

)20(1

)19(1

z

z

u

zt

x

u

z

x

eFdu

df

m

eg

eFdu

df

m

eg

The F+ and F- coefficients are determined by the boundary conditions. We assume diffusive

scattering, meaning strong relaxation at the boundary imposes

)22(0)(

)21(0)0(

tzg

zg

This determines the coefficients, F- = F+ = -1.

Now we can determine the current flowing in the film, with the usual expression:

dkkuzkfe

j x )(),(4 3

(23)

where f(u,z) = [g+ (u,z) +g- (u,z)]. We again use spherical coordinates, and the dk

df o term reduces the

volume integral to one over the sphere of radius kF. We also remember that since we have broken

up our solution to uz > 0 and uz < 0 cases and )cos(m

ku F

z

in spherical coordinates, g+ is

integrated only over the =0../2 hemisphere, and g- is integrated over the remainder. After doing

the trivial integration of Cos2 (), we have

2/

0 2/

cos

)(

3cos33

2

2

1sin1sin4

)(

dedek

m

ezj

ztz

F (24)

where m

kF is the mean free path. we can combine these two integrals into one with the same

bounds and simplify using

2

2cosh2 2/)( zt

eee tztz (25)

giving

2/

0

cos232

2

3

2

2

cos2

2cosh1sin3

342)(

d

zte

k

m

ezj

t

F (26)

Now since this is a function of z (i.e. the current density is different near the boundaries) we need to

average over the cross section:

t

t

det

dzzjt

j0

cos232/

00 1cossin

2

31)(

1

(27)

Defects in crystals and Elements of thin Films

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5.10 LET US SUM UP

Thin films are thin material layers ranging from fractions of a nanometer to several micrometers in

thickness. Electronic semiconductor devices and optical coatings are the main applications

benefiting from thin film construction.

Work is being done with ferromagnetic thin films for use as computer memory. It is also being

applied to pharmaceuticals, via thin film drug delivery. Thin-films are used to produce thin-film

batteries.

The act of applying a thin film to a surface is known as thin-film deposition.

Thin-film deposition is any technique for depositing a thin film of material onto a substrate or onto

previously deposited layers. "Thin" is a relative term, but most deposition techniques allow layer

thickness to be controlled within a few tens of nanometers, and some (molecular beam epitaxy)

allow single layers of atoms to be deposited at a time.

Deposition techniques fall into two broad categories, depending on whether the process is

primarily chemical or physical.

Chemical deposition is further categorized by the phase of the precursor: Plating, Chemical

solution deposition, Chemical vapor deposition

Examples of physical deposition include:

A thermal evaporator uses an electric resistance heater to melt the material and raise its vapor

pressure to a useful range. This is done in a high vacuum, both to allow the vapor to reach the

substrate without reacting with or scattering against other gas-phase atoms in the chamber, and

reduce the incorporation of impurities from the residual gas in the vacuum chamber. Obviously,

only materials with a much higher vapor pressure than the heating element can be deposited

without contamination of the film. Molecular beam epitaxy is a particular sophisticated form of

thermal evaporation.

An electron beam evaporator fires a high-energy beam from an electron gun to boil a small spot

of material; since the heating is not uniform, lower vapor pressure materials can be deposited.

The beam is usually bent through an angle of 270° in order to ensure that the gun filament is not

directly exposed to the evaporant flux. Typical deposition rates for electron beam evaporation

range from 1 to 10 nanometers per second.

Elements of Thin Films

e

Sputtering relies on a plasma (usually a noble gas, such as argon) to knock material from a "target"

a few atoms at a time. The target can be kept at a relatively low temperature, since the process is

not one of evaporation, making this one of the most flexible deposition techniques. It is especially

useful for compounds or mixtures, where different components would otherwise tend to

evaporate at different rates. Note, sputtering's step coverage is more or less conformal.

Pulsed laser deposition systems work by an ablation process. Pulses of focused laser light vaporize

the surface of the target material and convert it to plasma; this plasma usually reverts to a gas

before it reaches the substrate.

Cathodic deposition are PVD which is a kind of ion beam deposition where an electrical arc is

created that literally blasts ions from the cathode. The arc has an extremely high power density

resulting in a high level of ionization (30-100%), multiply charged ions, neutral particles, clusters

and macro-particles (droplets). If a reactive gas is introduced during the evaporation process,

dissociation, ionization and excitation can occur during interaction with the ion flux and a

compound film will be deposited.

5.11 CHECK YOUR PROGRESS: THE KEY

1. The act of applying a thin film to a surface is known as thin-film deposition. Thin-film

deposition is any technique for depositing a thin film of material onto a substrate or onto

previously deposited layers. "Thin" is a relative term, but most deposition techniques

allow layer thickness to be controlled within a few tens of nanometers, and some

(molecular beam epitaxy) allow single layers of atoms to be deposited at a time.

It is useful in the manufacture of optics (for reflective or anti-reflective coatings, for

instance), electronics (layers of insulators, semiconductors, and conductors form

integrated circuits), packaging (i.e., aluminum-coated PET film), and in contemporary art

(see the work of Larry Bell). Similar processes are sometimes used where thickness is not

important: for instance, the purification of copper by electroplating, and the deposition of

silicon and enriched uranium by a CVD-like process after gas-phase processing.

Deposition techniques fall into two broad categories, depending on whether the process is

primarily chemical or physical

Defects in crystals and Elements of thin Films

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2 A thermal evaporator uses an electric resistance heater to melt the material and raise its vapor

pressure to a useful range. This is done in a high vacuum, both to allow the vapor to reach the

substrate without reacting with or scattering against other gas-phase atoms in the chamber, and

reduce the incorporation of impurities from the residual gas in the vacuum chamber. Obviously,

only materials with a much higher vapor pressure than the heating element can be deposited

without contamination of the film. Molecular beam epitaxy is a particular sophisticated form of

thermal evaporation.

An electron beam evaporator fires a high-energy beam from an electron gun to boil a small spot

of material; since the heating is not uniform, lower vapor pressure materials can be deposited.

The beam is usually bent through an angle of 270° in order to ensure that the gun filament is not

directly exposed to the evaporant flux. Typical deposition rates for electron beam evaporation

range from 1 to 10 nanometers per second.

Sputtering relies on a plasma (usually a noble gas, such as argon) to knock material from a

"target" a few atoms at a time. The target can be kept at a relatively low temperature, since the

process is not one of evaporation, making this one of the most flexible deposition techniques. It

is especially useful for compounds or mixtures, where different components would otherwise

tend to evaporate at different rates. Note, sputtering's step coverage is more or less conformal.

Pulsed laser deposition systems work by an ablation process. Pulses of focused laser light

vaporize the surface of the target material and convert it to plasma; this plasma usually reverts

to a gas before it reaches the substrate.

Cathodic arc deposition (arc-PVD) which is a kind of ion beam deposition where an electrical arc

is created that literally blasts ions from the cathode. The arc has an extremely high power

density resulting in a high level of ionization (30-100%), multiply charged ions, neutral particles,

clusters and macro-particles (droplets). If a reactive gas is introduced during the evaporation

process, dissociation, ionization and excitation can occur during interaction with the ion flux and

a compound film will be deposited.

Elements of Thin Films

e

List of References and Suggested Reading

Thin Film Materials- Stress, Defect Formation and Surface Evolution By L. B. Freund

An introduction to physics and technology of thin films By Alfred Wagendristel,

Yu-ming Wang

An introduction to thin films- By Leon I. Maissel, Maurice H. Francombe

Thin Film Device Applications-By K. L. Chopra, I. Kaur

Size effects in thin films-By Colette R. Tellier, André J. Tosser KL Chopra. Thin Film

Phenomena. McGraw-Hill

Handbook of Thin Film Technology-By Leon I. Maissel, Reinhard Glang

Handbook of thin-film deposition processes and techniques By Klaus K. Schuegraf