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 sin22 . 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

000

33

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 3.3310 3 coulomb.

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

sin0

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

202

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:

tEtD

i0

"'

(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

tDtDdt

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 200EW (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

2/100

v (8.11)

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

21

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( /toss 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 ePPP00

** )(*

(9.8)

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

00/*

1

)(

iCP rs

t

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

)(***

iPD rs ie

(9.11)

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

irs

1

)(* 000

(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.

http://en.wikipedia.org/wiki/Planck%27s_law_of_black_body_radiationhttp://en.wikipedia.org/wiki/Electromagnetic_radiationhttp://en.wikipedia.org/wiki/Gas_in_a_box

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)

http://en.wikipedia.org/wiki/Electrical_resistivityhttp://en.wikipedia.org/wiki/Optical_absorptionhttp://en.wikipedia.org/wiki/Molecular_orbital

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

pneJa

J

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)()( 1221212 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

aik

e 11 and aik

e 22

(9.11)

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

)()( 11 xexuxk

)()( 22 xexuxk

(9.12)

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

)()()( 1)()(

111 xeaxeaxu

axkaxk

)()()( 1)( 111 xuxexee

xkaikaxk

(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

rkikk 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 kNaikx

k

(9.16)

)(xuee kikxikNa [From equation (9.13)]

)(xe kikNa [From equation (9.16)]

Similarly

)()( ** xeNax kikNa

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 kkikNa

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(.022

(6.8)

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

Tk

JJHNM

B

Bg

3

)1(.022

(6.9)

The paramagnetic susceptibility will be

Tk

JJN

H

M

B

Bg

3

)1(.022

(6.10)

or T

C

Tk

N

B

B 3

.202

(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

JxxJJeB eeedx

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

).......1(log)( 2JxxJxeB 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

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