2014

BY

Thesis

DEPARTMENT OF PHYSICS

UNIVERSITY OF LUCKNOW

LUCKNOW-226 007

INDIA

UNDER THE SUPERVISION OF

Satya Prakash Singh

Dr. Daya Shanker

SUBMITTED TO THE

UNIVERSITY OF LUCKNOW

FOR THE DEGREE OF

Doctor of PhilosophyIN

PHYSICS

Dedicated to

My Parents

CERTIFICATE

This is certify that all the regulations necessary for the submission

of Ph.D. thesis of Mr. Satya Prakash Singh have been fully observed.

Dr. U.D. Mishra

(Head)

CERTIFICATE

This is to certify that all the regulations necessary for the

submission of the Ph.D. thesis of Mr. Satya Prakash Singh have

been fully observed. The content of this thesis are original and have

not been presented anywhere else for the award of Ph.D. degree.

(Dr. Daya Shanker)

Supervisor

DECLARATION

I (Satya Prakash Singh) hereby, declare that the thesis entitled “Study

of szigetti effective charge on the surface of condensed materials at

nano scale parameter” being submitted to Lucknow university in

fulfillment of the requirements for the degree of Philosophy in Physics is

the original work of mine and has not previously formed the basis for the

award of any degree, diploma or any other similar title or recognition.

(Satya Prakash Singh)

ACKNOWLEDGEMENT

I (Satya prakash singh) am highly indebted to Dr. Daya Shanker

who supervised my research work. Without his valuable guidance, keen

interest and constructive criticism throughout the course of the

investigations, this thesis would probably never have appeared.

I would like to express my heartiest gratitude to Prof. U.D.

Mishra and Prof. Kirti Sinha, Head, Physics Department, Lucknow

University, Prof M. Husain (J.M.I, NDLS) for encouraging me in

carrying out this research thesis. I am also grateful to Dr. N.V.C. Shukla,

Deptt. of Maths, L.U. for valuable help to solve complicated

mathematical problems in this thesis.

Thanks are due to research fellows Dr. Hari Narayan , Dr.

Abhishek Tiwari and Mr.Sanjay Agrwal, Mr.Pradeep Sharma for their

assistance during writing and typing the present work.

Last but not least, I would like to mention the encouragement and

cooperation given to me by my family members specially my father,

mother& wife throughout the research work.

Satya Prakash Singh

Contents

Preface i-ii

Chapters Page No

1. Introduction 1-18

2. Theory of phonon,plasmons and polaritons 19-36

2.1 Surface phonon excitations

2.2 Modifed plasmon –matter interaction

2.3 Locally excited plasmons on the material surface

2.4 Theoritical & experimental study of plasmons

2.5 Excitation of surface polaritons on the surface of

Materials

3 Dielectric functions due to lattice vibrations 37-55

3.1 Dielectric constant and relative permittivity

3.2 Dielectric functions for materials

3.3 Polarization of dielectric materials

3.4 Dielectric function for polar semiconductor

3.5 Dielectric function for materials

4. Szigetti’s dielectric theory 56-75

4.1 Szigetti’s first relation

4.2 Anharmonic correction

4.3 Spectroscopic implication of Szigeti relation

4.4 Aspects of the szigetti charge

5 Filtering properties of materials 76-149

5.1 Modified Bloch’s hydrodynamical model

5.2 Special dispersion relations for two mode coupling

5.3Study of three mode coupling under special conditions

5.4 Local theory approximation for k>>1

5.5 Dielectric function on the surface of materials

5.6 Expression of szigetti effective charge for k>>1

5.7 Band attenuation properties of materials without

magnetic Field & with magnetic field

5.8 Comparitive study of filtering properties

5.9 Variation of the width of allowed band of polarized

materials with szigetti effective charge for k>>1

5.10 Advatages of szigetti effective charge study and Its

applications

6 Summary and conclusions 150-171

7. References 172-183

8. Publications 184

i

PREFACE

The study of surface polariton waves at the interface of isotropic, homogenous dielectric

media has recently recived considerable attention surface polaritons are defined as the coupled modes

of photons and elementary surface excitations of solids, so surface polariton waves propagates as

electromagnetic waves along the interface of two media but decay in non-oscillatory exponential

manner in a direction perpendicular to the interface. The surface properties of these modes depend

upon the type of materials that form interface .therefore these modes provide a sensitive probe for the

study of solid surface. the fact that the surface polaritons may be excited at some place on the surface

of a solid and detected at some other place on the surface so it makes it surface polaritons an extremely

sensitive tool for determining surface properties [v.m agranovich(1978)]

The frequency of the surface polariton waves depends upon the type of elementary surface

excitations to which the EM wave (photons) couples .The important elementary excitations in dielectric

media are plasnons, phonons, excitons etc. a study of these surface polariton mode give an insight into

the type of elementary excitations sustained by the medium .the presence of polariton waves at the

surface affect the optical properties of the medium , the study of dispersion of these surface waves

considering reflection and refraction at surface and interfaces ,which may be refraction at surfaces and

interfaces , which may be referred to as crystal optics of surfaces , plays a fundamental role in the

phyics of surfaces.

In view of the above discussion, in the present thesis, the Bloch’s Hydrodynamical model is

utilized to study the coupled surface polariton waves sustained by spatially –dispersive polar semi-

conductor medium for k≠0 bounded by a non-dispersive dielectric medium. Since the properties of

surface polariton waves are strongly dependent on the geometry of the interface and for spherical

geometry of the interface are derived and the effect of spatial dispersion and interface are divided and

the effect of spatial dispersion and interface geometry on surface polariton is studied. The dispersion

relation for both these cases is obtained by applying certain suitable boundary condition at the

interface. Also since the dielectric function plays important role in determining the dispersion constant

(wave vector) dependence of the dielectric function of the dispersive polar semiconductor medium is

also studied . Since the optical properties of the medium are affected by the presence of surface waves,

the refractive index of the polar semiconductor medium is also derived and its frequency &wave vector

dependence are studied . The inferences obtained from the study of the dielectric function and the

refractive index , when analyzed along with the dispersion curves obtained for surface polariton waves

leads to important conclusion regarding the effect of surface polariton waves on the optical properties

the of the polar semiconductor medium. Also the dispersion curves provide information regarding the

wave vector region for which the surface waves mixed photon-surface excitation character.

It is observed that the inclusion of spatial dispersion plays an important role in the study of

surface polariton waves at polar semiconductor dielectric interfaces. The surface polariton waves at the

ii

spherical interfaces become radius independent when spatial dispersion is neglected and as a result

becomes independent of the curvature of the interface. The exclusion of spatial dispersion , therefore

,will not give correct result for small spheres . For the plane interfaces too, the inclusion of spatial

dispersion effects is important, and leads to the infinite life-time of surface polaritons.

The effect of D.C Magnetic field on two mode coupling (SP-SOP) cylindrical polar

semiconductor for k=0 and k≠0 have been discussed by Dr. K.S. Srivastava and D.K Singh (1997). The

effect of D.C Magnetic field on spatial dispersion relation for three mode coupling in spherical polar

semiconductor for k≠0 gives an important result in the study of surface polariton waves at

semiconductor dielectric interface.

The theoretical study of polar semiconductor surface is essentially of importance because of

the great practical applicability of power semiconductor in solid state plasma devices. The foremost

characteristic property of the polar semiconductors which makes them suitable for experimental studies

and for their use in physical instrument, compared to metals and other non-polar semiconductor

compounds. The simultaneous existence of SP and SOP modes makes these materials stand apart from

the other and thus they are ideal for the study of coupling of these modes and the effects of these

coupling of these modes on the surface properties. Finally, by adequate doping, the electron

concentration can be adjusted leading to a wide range of frequency for which these materials can be

utilized.

Szigeti published four seminal papers on the dielectric behaviour of crystals during the

period 1949-1961. Szigeti’s theory is applicable to isotropic and anisotropic, ionic and covalent crystals

with different structures. Szigeti’s theory connects dielectric, spectroscopic and elastic properties. An

important outcome of Szigeti’s theory is the concept of the effective ionic charge (s). It is pointed out

that s correlates with a number of physical properties and is a measure of ionicity of the interatomic

bond. Since Szigeti’s work, several theoretical models have been proposed to account for the fact that s

< 1. These models provide an insight into the complex polarization mechanisms in solids. This review

summarizes Szigeti’s work and the work that followed; the implications and applications of Szigeti’s

theory are discussed.

1

CHAPTERCHAPTERCHAPTERCHAPTER----1111

INTRODUCTIONINTRODUCTIONINTRODUCTIONINTRODUCTION

The study of the properties of surface plasmons [SP], surface

polaritons and surface optical phonon [SOP] has recently received

considerable attention. Surface plasmons are sustained by media having

appreciable free electron concentration i.e. metals, semiconductors and

polar semiconductors. Surface polaritons are the coupled photon and

elementary excitation modes localized at the surface or interface of two

media such that they propogate in a wave like manner along the interface

but decay to zero exponentially in a perpendicular direction to the

interface surfaces. Polaritons are sensitive probes for the study of solid

state surfaces as the intensity of the associate electromagnetic field is

maximum just on the surface.

If electromagnetic radiation of appropriate frequency is incident on

the surface of metal or semiconductor then surface plasmons may couple

with photons giving rise to surface plasmons- polariton modes. In the

case of polar semiconductor (i.e. compounds like InAs, InSb,MgO , InP

etc) the dielectric function is frequency dependent and for certain

frequency ranges they can sustain surface optical phonons along with

plasmons.

Surface polariton waves are electromagnetic waves that remain

localized within the thin surface layer or bound along the interface of two

media [1, 2]. Thus these waves can be which it turn of great scientific and

practical importance [3, 4] not only in the field of Physics but also in the

field of Chemistry and Bio-chemistry. Important physical phenomenon

like quantum hall effect [5] surface enhanced Roman effect [6, 7] etc and

applied fields like micro electronics and integrated optics are all directly

2

related to study of solids. Maximum information regarding the surface

properties of a medium may obviously be obtained by studying the

propagation, not the bulk, but of surface electromagnetic energy just on

the surface.

There is great possibility of interaction between surface plasmon

and surface optical phonon leading to coupled plasmon – optical phonon

modes. The bound surface polariton are non radiative and the associated

electromagnetic field are irradiative and it away from the interface. The

phase velocity (vp) of these modes is less than the velocity of light (vp<c)

in the bounding medium and their surface wave vector (K) is greater then

that of light. As a result, surface polaritons can not be excited by direct

irradiation of smooth surface technique like the attenuated total reflection

(ATR) method and the periodic grating method have to be employed for

the excitation of these modes. The properties of surface polaritons are

strongly dependent upon the geometry of guiding surface.

When the dipole moment density and also the displacement field

depends not only on the value of electric field at a particular point but

also on the value assumed by the electric field in the near vicinity of that

point, then material is said to be exhibit “spatial dispersion”. The

inclusion of spatial dispersion in the study of surface polaritons leads to

important conclusion like the finite life time of the surface modes even

collision less plasma state..

When an EM wave is incident on a medium, then oscillating

electric and magnetic fields induce a polarization. In the medium leading

to the excitation of elementary dipoles excitation of medium like

plasmons [10], phonons [11], excitons [12], magnons [13] etc. The

incident EM field thus interacts with the medium via the polarization, it

3

induces and the resulting wave that propogates in the medium consists of

EM field (photons) coupled with elementary excitations of the medium.

This coupling is stronger when the frequency of the incident waves is

comparable with the frequency of the elementary modes of the medium.

The EM waves that arise as a result of this coupling are termed as

polariton waves [14, 15].

The incident electromagnetic radiation can couple to more than one

type of elementary excitations simultaneously if their frequencies are

comparable. These waves arise as a result of coupling of the EM radiation

and those elementary surface excitations of the medium that may couple

on linear manner to the incident EM field by virtue of their electric of

magnetic character [16]. The predominant surface excitations in the case

of dielectric media are surface plasmons, surface phonon, surface

excitons etc. The fact that surface EM waves at a metal surface involves

the coupling of EM radiation .with surface plasmons was first explain by

Stern [17] . Also surface of optical lattice vibrations that exist at the

surface of ionic and polar materials have a great tendency to couple with

electromagnetic radiation. These vibrations lead to surface phonon,

polariton modes [18]. In case of polar semiconductors, surface Plasmon

modes[19] and surface Phonon modes [20] have comparable frequencies

so that there is a possibility of interaction between the two leading modes

i.e. surface plasmon & surface polariton modes [21,22]. Special

techniques like attenuated total reflection method (ATR) [26.27] and

periodic grating method [28.29] are employed to study these modes

experimentally. The above mentioned techniques have been successfully

employed by several workers [30-36] to study surface polariton modes in

metal, semiconductors and polar semiconductors for different interface

geometries. The frequency dependent dielectric function for metals and

4

semiconductors are well known and have been recently employed by

several workers [24, 37-39] to study these waves. The most widely used

effective methods that have been employed for obtaining the dispersion

relation, are the Maxwell’s equation method [39-41] and the quantum

mechanical random phase approximation [RPA] [42].

The Maxwell’s equations method has been used to study Surface

Polariton modes in metals as well as Semiconductors but this method is

effective and suitable within the local limit theory[43-44] where the

response of the medium to the incident EM waves is taken to be a

continuum, so the spatial variation of electric and magnetic field induced

in the medium is neglected. Where as in the non-local limit [8] the

dielectric function is dependent on the frequency as well as wave vector

[45].

Several worker have also studied the coupling between Phonon-

polaritons [135-147] phonons- polaritons damping [143, 144] scattering

of surface plasmons [135] non-local exchange effect on the bulk

plasmons dispersion relation [104]. Coupled EM modes coherent

structure in cylindrically bounded magnetised density plasma [106, 111,

112, 125, 145] have been studied by several workers. The effect of

electric and magnetic field on dispersion relation on cylindrical and

spherical surfaces [121-123, 126, 130-133] energy loss in thin film [124],

fluid dynamical mode [127] normal mode double branch dispersion

relation [128-129], collective excitation [133] are utilized to find the

spatial dispersion relation for three mode coupling. Optical properties of

interaction between plasmons, phonons and polaritons [140] and its

coupled dispersion relation [141], have been discussed. The dispersion

relation between phonons, polariton and plasmons in non local limit for

5

polar semiconductors at plane surface [143] and cylindrical surface are

studied [144-145].

First principle of calculation of the plasmon resonance [146] and

optimization of surface plasmon enhanced magneto optical effect [147]

was investigated in surface plasmons in condensed materials. Infrared

surface plasmons [148], theoretical and experiment study of localized

plasmon [149] Szigeti dielectric theory [150] are recent study in this field.

Channel plasmon [151], single photon transistor [152], a hybrid

plasmonic wave guide [153], surface polariton of small coated cylinder

[154], have been investigated by several workers in condensed materials.

Light driven plasmonic switches [155], magneto-plasmons [156-157,

158] are recent research in surface plasmons. Non-local screening of

plasmons in grapheens [159-160].

Non relativistic local dielectric response theory has been used

with success for the interpretation on isotropic nanometer size particles

for different geometries such as slabs [47], spheres [48], layered spheres

[49-50], and sphere half way embedded in a supporting medium [51] and

cylindrical channels [52].

The matter is classified in three states- Solid, liquid and gas. The

distinction lies in difference in bond strength or the strength of

intermolecular forces which decreases considerably as substances change

from solid to liquid and the bonds break or intermolecular forces

practically disappear as liquid changes to gaseous state. Bonds can be

broken by increasing the kinetic energy of atoms and molecules, i.e. by

increasing the temperature of the substance. Sir William Crooks, in 1879,

studied electrical discharge of gases and observed that when the electrical

discharge was passed through the gas, the gas was ionized and the

6

temperature was increased in the process and called this collection of

positive and negative ions as the fourth state of matter. Langmuir [53]

called this as “plasma” and defined it in a more comprehensive way as a

system containing high and approximately equal concentration of

positively and negatively charged particle which are relatively mobile and

interact with each other collectively via long range coulomb forces.

These conditions may be satisfied in other forms of matter (Solid

and liquid) also. The conduction electrons moving under the influence of

positive ion background in a metal and similarly free electrons (or holes)

of an extrinsic semiconductor or those of a doped semimetal, all are

examples of solid state plasma system. The solid state plasma differ from

gaseous plasma in that, they are characterized by high density of charge

carriers ( 2010 per≈ cc) and low temperature. Therefore the solid state

plasma is degenerated and obeys quantum ‘Fermi-Dirac’ statistics. The

gaseous plasma, on the other hand, has low carrier density ( 1210 per≈ igh

temperature. Hence the system is nondegenerate and obeys the classical

‘Maxwell-Boltzmann’ statistics.

Bloch introduced a comparatively simple classical form known as

Bloch’s hydrodynamical model [54] which is applicable for small wave

vectors. Recently the hydrodynamical model has been applied to two

dimensional electron gases [55] of small spheres [56] to one dimensional

quantum wires [57] and in the deviation of additional boundary

conditions at the interface between two conductors [58].

Bloch introduced a comparatively simple classical form known as

Bloch’s hydrodynamical model [120] which is applicable for small wave

vectors. Recently the Hydrodynamical model has been applied to two

dimensional electron gases [121] of small spheres [122] to one

7

dimensional quantum wires [123] and in the deviation of additional

boundary conditions at the interface between two conductors [124].

The Bloch’s model has been used by Ritchie [78-81] to study

surface plasma excitations in metal film and later it was developed to

study the surface plasma oscillations in plane single surface [125]

interface between two metals [126] and in spherical geometry [127]etc.

A survey of literature shows that all these studies [ 81- 83] have

been made for metal, in which surface phonon waves are absent and

independent plasmons exist. Srivastava and Tondon [128] for the first

time modified the Bloch’s Hydrodynamical equations for metals [125]

to study the interaction between these two modes in semiconductors. The

modified Bloch’s equation for the semiconductor [128] may be written

as-

_

_'_ _ _ _ _

( , )

0 '

1 ( )n r tD v d nm e E B mv

Dt c n

ρ = − + ∇× − ∇− ∇ ∫

(1.1)

__ _ _4 1 D

B Jc c t

π ∂∇× = +

∂ (1.2)

_ _

.( )n

nvt

∂= −∇

∂ (1.3)

_ _ _ _4. ( ) ( , )

eE N r n r t

E

π+

∇ = −

(1.4)

The surface of polar semiconductor particle supported T.M.

surface polariton waves.

Dispersion relation given by the author by using Blochs-

Hydrodynamical model is the most general dispersion for phonon,

8

polariton and Plasmon for polar semiconductor of cylindrical interface

for k≠0.

Y[Y2-(az+b)Y+az] = 0 (1.5)

.Now the author study the surface of different materials(polar

semiconductors)for different parameters by using the relation given by

equation(1.5).

For lower value of k nearly linear variation with wsp/wt for radius

1A. & 2A

. .

When radius increases then variation of wsp/wt with respect to K

shown lesser linear variation.

Thus for higher value K it shows non linear variation. Hence for

higher value of K, it shows non linear agreement.

But at radius 8A. It shows again perfect linear variation for high

value of K.

Again linear variation between wsp/wt versus K disturb for radius

16A. for high value of K.

Variation of wsp/wt versus K for different radius observe that

variation is either perfectly linear or not perfectly linear.

When radius increases then it’s linear variation changes with

wsp/wt versus K, thus for high radius linear variation changes.

When radius increases then ratio of wsp/wt decreases. wsp/wt

inceases with increment in value of K. It mean for larger value of wave

vector K, more prominent wave passes through the substance KF in

epoxy resine medium.

Thus we conclude that with K Ratio of wsp/wt increases. It shows

linear variation for low radius but it deviates for high value of radius.

9

Author observed that:

In this graph we use different medium in 3D having constant radius

0.5 A. for KF substance.

In this we take wsp/wt along one axis, wave vector K along another

axis and variation of different medium along different axis.

In epoxy resine wsp/wt increases linearly with K but for transoil it is

not linear variation.

All medium except transformer oil shows linear variation for KF

substance.

wsp/wt increases linearly with increase in value of K.

For same radius neoprene use a medium variation of wsp/wt initially

increases but as K=5, it changes abnormally again at K=7 it fallow

simillar path as in epoxy resine medium follow.

When we increase K more than 9 (K > 9) than it increases rapidly.

Except trans oil medium other medium like mica, quartz, Bee wax,

transformer oil and vaccum shows linear increment of wsp/wt with

K.

Thus wsp/wt increases with increment in value of K

For KF substance at .5A.radius. It is observed that trans oil

medium show different variation otherwise other seven medium

shows same kind of linear variation.

Ratio of wsp/wt increases for same value of K for when we consider

different medium [Epoxy resine, Neoprene, Mica, quarz, lice, Bee wax,

Transformer Oil & vaccum] i.e. minimum value for Epoxy resine and

maximum value for vaccum.

Thus author obverse that all medium shows same type of variation

only transoil shows different type of variation.

10

Now Author observed that that

In this graph we use different medium in 3D having constant radius

.5A. for Mgo substance.

In this we take wsp/wt along one axis, wave vector K along another

axis and variation of different medium along different axis.

In this graph we take different medium like Epoxy Resine,

Neoprene, Mica, Quartz, Lice, Bee Wax, Transformer Oil &

Vaccum.

Ratio of wsp/wt increases when we move left to right for different

medium with constant value of K.

For same medium wsp/wt increases when K increases.

All medium like Epoxy Resine, Neoprene, Mica, Quartz, Lice, Bee

Wax, Transformer Oil & Vaccum share linear variation between

wsp/wt versus K. Thus for different medium variation are same.

For lice medium Ratio of wsp/wt increases with increase in K but

when K>8 then decreases rapidly.

Ratio of wsp/wt becomes constant at K = 9 or greater value for lice

medium.

Even at wave vector K = 0 all medium have same value of wsp/wt.

But when value of wave vector is higher (i.e. K>8) then for lice

medium its linear variation vanishes.

Thus for low value of K all medium shows linear variation while

for high value of K only lice shows different variation when we plot 3D

graph between wsp/wt versus K for different medium.

So author observe that all medium shows linear variation while lice

medium not shows linear variation.

11

Again author observe for different radius that

We plot graph between wsp/wt with propagation constant K for

Mgo substance in 3D for Epoxy resine medium. Graph plotted for

different radius of .5A. to 16 A

. .

In this 3D graph we take wsp/wt along one axis K, along other axis

while variation of different radius taken along new axis.

We plot graph between wsp/wt versus K is exactly linear for

different radius for Mgo substances in Epoxy resine medium.

When value of K increases then wsp/wt increases Radius of cylinder

increases then wsp/wt also increases.

Ratio of wsp/wt is lowest at 5A. radius and wsp/wt is maximum are

in cylindrical shape increases then more space to pass waves. Thus

wsp/wt increases with increases in radius.

For Mgo substance in epoxy resine medium there is constent

variation, It is free from radius.

It has only single effect when radius is more then large amount of

wave pass trough it easily otherwise less amount of wave pass

trough it.

When K = 0 then there is no propagation of waves value of K

increases then wave propagation also increases.

wsp/wt has max value at 16A. means at this radius waves passes

easily through the cylindrical geometry. It cappers for highest value

of K = 10.

Thus for low value of K, wave propagation is not significant but

when radius and K both increases then significant wave passes through it.

In this graph at all radius wsp/wt verses K shows linear variation.

12

Author observe at constant radius variation of InP:

We plot graph between wsp/wt with propagation constant k for InP

in different medium having radius of cylinder is 0.5 A..

In 3D graph we take wsp/wt along one axis K along other axis while

different medium along third dimensional.

For fixed value of K the ratio of the frequency of surface plasmon

to transverse medium is minimum in epoxy resin and very high in

vaccum medium.

wsp/wt versus K is almost linear increment in all medium except

lice medium.

Value of wsp/wt is high in vaccum medium and low in epoxy resine

but the range is nearly equal for same value of K.

When we plot graph wsp/wt versus K. Then InP substance for

cylindrical surface of 0.5 A. radius of different medium, then all

medium shows linear variation but lice shows different variation.

In lice medium wsp/wt increases but its value remains constant at

5.040028. So for lice, wsp/wt increases with increment in K but for

K = 7 or more then this, wsp/wt deceases rapidly.

Thus for InP at lone value of K all medium including lice shows

linear variation but for higher value of K, only lice shows different

variation also for high value of K.

Author observe at different radius in Epoxy resine,It shows

We plot graph for InP at different radius of 1 A. & 2 A

. in epoxy

resine medium for cylindrical surface.

We plot graph between wsp/wt with propagation constant K for

InP in epoxy resine medium having different radius.

13

In 3D graph we take wsp/wt along one axis. Wave vector K along

other axis while different radius consider along third dimension.

The wsp/wt varies linearly with respect to K (1 A. - 10 A

.) for

cylindrical surface of InP at different radius.

At different radius InP medium shows same linear variation.

wsp/wt increases vertically when we move from top to bottom,

while this ratio is decreases when we move Horizontally from left

to right.

We observe that for InP in epoxy resine medium variation of

wsp/wt is linear with K at different radius such that 1 A. & 2 A

., Both

shows linear variation in the epoxy resine medium as in fig(5.6).

Coupled SP-SOP modes arise on the surface of a polar

semiconductor as a result of frequency and wave vector dependence of

the dielectric surface function of the lattice dielectric surface function of

polar semiconductor. These coupled SP-SOP modes on coupling with the

incident EM radiation of comparable frequency lead to the coupled

surface plasmon, polariton phonon modes on the surface. The dispersion

relation for these modes can be obtained from equation (1.6) by

substituting the frequency and wave vector dependent form of 1( )kε ω we

get

12.3W6- (14.9+13.3K1

2) W

4+(34.33K1

2+18.4)W

2 - 18.4K1

2 = 0 (1.6)

Now equation (1.6) gives three values of w1 ,w2 and w3 for

different value of k1.

The comparative study for cylindrical surface of different substances like

KF, InP & MgO:

14

In this graph we are comparing variation of wsp/wt with wave rector K

for different substances KF, InP & Mgo.

wsp/wt in all substances increases from top to bottom when K

increases.

wsp/wt is lower for KF but in other two substances InP & Mgo wsp/wt

increases rapidly [slightly greater value].

When we plot comparative graph for cylindrical surface of different

substance like KF, InP & Mgo with respect to wave vector K at

same radius 1 A., then all substances have same linear variation.

This dispersion curves shows that the frequency of the lower mode

changes slowly and frequency of the upper mode varies rapidly

with the wave vector K.

Thus wsp/wt verses K is almost linear increment of substances KF,

InP & Mgo at same radius cylindrical surface (1 A.) these behaves like

same nature.

Now comparative study of filtering property of different polar

semiconducting substances discussed as-

Author observed that

In this graph we are comparing variation of wsp/wt versus K for

different substances KF, InP & Mgo.

When we plot comparative graph for cylindrical surface of different

substances at equal radius 2A.. then all substances have same linear

variation.

wsp/wt increases with increment in value of K.

15

For KF variation of wsp/wt with K varies with lesser effect where as

for other two substances InP Mgo show more effective linear

variation. At this radius (2 A.) both InP & Mgo having nearly same

value of wsp/wt in comparison to radius (1 A.).

Again author observed that

In this graph we are comparing variation of wsp/wt with wave vector K

for same substances KF, InP & Mgo but his time at higher radius

(16 A.) of cylindrical surface.

At this radius KF substances has lower value of wsp/wt even starts

from Zero. At this radius cylindrical surface has lower value.

For KF substance it has lower effect, it means lesser wave propagate

through the surface.

At this radius (16 A.) both InP & Mgo have same value of wsp/wt so

both have same type (kind) and equal linear variation.

Thus for these comparative study we observe that when radius of

cylinder increases then KF has lower linear variation and InP & Mgo

have greater linear variation.

The frequency of surface plasmon has very small variation in KF

compared to InP & MgO with respect to radius and propagation constant

(K). The reason is that KF is not a good conductor and it behaves like

polar semiconductor. Thus study is important in nanotechnology and

electronic communication.

But at higher electronic concentrations the frequency of this mode

become almost constant. It is clear that at the higher concentrations, the

lower mode become like pure SP mode and upper mode becomes like

pure sop mode. The coupling between SP and sop mode is stronger when

16

wsp = wsop i.e. where two uncoupled modes intersect. The stronger

coupling is observed at wsp/wt = 1.

Thus wsp and wt both are equal then coupling is stronger.

Thus the surface acts as band pass filter [BPF]. Agian their is no

propagation when ε(W) is negative.

for

, the surface become transparent again act as High pass filter

(HPF). from graph we observe band width (∆) of band pass filter is given

by

∆ 1 WW

Thus the allowed band ∆ will also be differed for different

compounds. Similarly

∆ 1 WW

Gives values at which the surface will act as high pass filter. from

study of Szigeti effective charge e∗ versus ∆ with the help of data in

table. from this the best fit of data we find the following relationship

between e∗ and ∆ as

∆=0.074+0.099∗ (1.7)

The values of ∆ can be estimated for different compounds with the

help of graphs. from these values ∗ can be calculated by using equation.

from table our calculated values agree well with the experimental values

of Hass and Henvis.

17

We can conclude that polar semiconductor surface acts as a band

pass filter for ω- <ω<ωt and as high Pass Filter for ωω+. The es2 is

measure of iconicity of polar semiconductors. thus width of band (∆)

increases with increase in ionicity. If there is more ionic character, band

of allowed frequency is wider and surface becomes high pass filter at

higher value of frequency ω.

There are some new features which are not seen when we observe

general effective charge in place of Szigetti effective charge so this

charge provides to be very useful tool in theoretical and experimental

study of Surfaces in the present work. The next chapter i.e. chapter-2

deals with elementary ideas of theory of phonon, Plasmon & polariton. In

this we study the existence of surface Plasmon and surface polariton in

polar semiconductor. Their interaction discussed with the help of its

dispersion relation.

Dielectric function and its effect due to lattice vibrations are

studied in chapter-3.The base of the thesis is szigetti effective charge

studied in chapter-4 i.e. Szigetti’s dielectric theory. Szigetti published

four papers on the dielectric behavior of crystals during the period 1949-

1961.This theory is applicable to isotropic and anisotropic, ionic and

covalent crystals with different structures. The method to solve the

problem is Bloch’s hydrodynamical model, which is explained in chapter-

5.The author solve his problem chapter-5,and investigated new recent

research work. The author also investigates expression of szigetti

effective charge for polar semiconductor also band attenuation properties

of polar semiconducting material in presence and absence of magnetic

field. In this chapter the author also study variation of allowed band

width of polarized materials with szigetti effective Charge. This is new

recent theory of Science. This study is new and can be utilized in

18

electronic communication system and is a measure of iconicity of polar

semiconductors. Thus width of band increases with increase in ionicity.

Author also explains his recent research problem in short as Summary

and conclusion in chapter-6.

19

CHAPTERCHAPTERCHAPTERCHAPTER----2222

THEORY OF PHONON, PLASMONS & POLARITONSTHEORY OF PHONON, PLASMONS & POLARITONSTHEORY OF PHONON, PLASMONS & POLARITONSTHEORY OF PHONON, PLASMONS & POLARITONS

In this chapter we study the existence of surface Plasmon and

surface polariton in polar semiconductor. Since both SP and SOP modes

exist in polar semiconductor.The possibility of their interaction will be

discussed with the help of its dispersion relation and also study the

plasma dielectric function, surface Plasmon, surface polariton and surface

phonon excitation.

Sir Willam Crooks, In 1879, studied electrical discharge passed

through the gas, the gas then gas ionized and the temperature increased in

the process. So the collection of positive and negative ions as the fourth

state of matter. Langmuir[53] called this as Plasma and defined it in a

more comprehensive way as a system containing high and approximately

equal concentration of positively and negatively charged particle, which

are relatively mobile and interact with each other collectively long range

Coulomb forces.

In the present work I shall study only the solid state plasma and

electron moving in a lattice of ions will be charge carriers. Due to the

motion of free electrons oscillations are set up in the system which are

known as plasma oscillation.

2.1 Surface phonon excitation:

Recently local probe techniques such as field emission microcopy

(FEM) [107, 108] scanning tunneling microcopy (STM) [109,110,111]

and atomic force microcopy (AFM) have given some insight into the

intriguing properties of those novel carbon based materials. Using FEM it

was possible to show that the tips of nanotubes are characterized by well

20

localized electron states form which the field emission takes place

preferentially [107,108] STM measurements allowed for the first time a

direct comparision of the electronic properties of a single wall carban

nanotubes with the calculated Density of states (Dos). Another

experimental set-up using STM principally for high resolution

characterization allowed multipurpose transport measurements to be

carried out on Single wall carban nanotubes [109,111] Another technique

giving complementary information about the electronic properties of

nanometer size particle is electron-energy loss Spectroscopy (EELS) in a

high resolution transmission electron microscope

Non-relativistic local dielectric response theory has been used with

success for the interpretation on isotropic nanometer size particular of

different geometrics such as thin Slabs, spheres [112, 113, 114, 115],

layered spheres [115], spheres halfway embedded in a supporting

medium and cylindrical channel [116]. Since a preliminary comparison of

experimental data with the simultaneous of the an isotropic spheres

(nested concentric-shell fullerenes) shows excellent qualitative agreement

between theory and measurement.

We consider the following equation to study the surface & volume

excitation.

(2.1)

(2.2)

(2.3)

(2.4)

( ) ( )

( )

( ) ( )

( ) ( )[ ] ( )ωρωωεε

ωρεε

ω

ωω

ω

ωωωπ

δω

ω

,,.

,1

,

.,exp2

2

0

rrV

rrV

dd

dp

dxdxEtie

o

o

E

trajectory

E

−=∇∇

−=∇

=∇

−=−=∇

∫

∫ ∫∫∞

∞

∞−

h

21

Basically equations (2.1) and (2.2) allow the total phonon excitation

probability to be determined using the solution of equations (2.3) and

(2.4). However, it is convenient to treat the surface and phonon

excitations separately for this purpose it can be noted that the general

solution of equation (2.3) and (2.4) is the sum of the homogeneous and

the particular solution of the problem. The two terms represent the

induced potential (Surface Phonon) and the direct potential (Volume

Phonon) respectively.

( ) ( ) ( )ωωω ,,, rVrVrV pind += (2.5)

Since the induced potential responsible for the surface losses can

be calculated separately. It is useful to introduce the notation of surface

Phonon excitation Probability dpsur

(w)/dw. Starting from equations (2.1)

& (2.2) it can be shown that the surface excitation probability for the

geometry shown in figure 2.1 is given by the following expression.

( ) ( )( )( )0,,

0,',,

'exp'

2 yxr

yxrrrV

V

YYilmdYdY

V

e

d

dP

o

oo

o

indsur

=

=

−= ∫ ∫

∞

∞−

∞

∞−

ω

πω h (2.6)

It is important to note that the time dependence of the problem has

been eliminated. ( )o

indrrV , is the induced potential at position r caused by

a stationary electron located at position ro. It is homogenous part of the

solution of

( )( )

( )o

o

o rre

rrV −=∇ δωεε

,2 (2.7)

If the electron is traveling in an isotropic medium and of

( ) ( )[ ] ( )o

o

o rre

rrV −=∇∇ δε

ωε ,. (2.8)

22

If it is an isotropic medium. The potential distribution therefore

is quasi-electrostatic and frequency dependent for each point along the

trajectory of the incident electron.

The volume phonon excitation in a uniaxial crystal such as

graphite has been treated theoretically by different workers [117,118].

For a uniaxial crystal with its axis inclined by an angle α with respect

to the optical axes of the microscopy (fig2.1) the volume phonon

excitation probability per unit path length is given by

( )( ) ( )∫ ∫

+

−=

⊥

c

IIcp

o

o

volume

qq

qdd

V

e

dYd

Pdθ π

ωεωεφθθ

επω

ω

0

2

0

22

2

23

22

4 h (2.9)

Where pq and cq are the projection of the transferred momentum

α on the coordinate system in which the dielectric tensor is a diagonal,

which is one of the unit vector parallel to the axis c of graphite and on the

plane perpendicular to the axis c respectively (fig 2.1)

Fig.2.1

The azimuthal angle φ and scattering angle θ and α angle between

the c axis of the crystal and the optical axis (2.1 fig)

23

(2.10)

(2.11)

where Eθ is given by AVqo

Eπ

ωθ

2= ( 2.12)

The integration over the angles θ and φ takes into account all

electrons scattered with in an angle smaller then the cut-off angle cθ [119].

2.2 Modified Plasmon Matter interaction

The electron density fluctuations i.e. the plasma oscillations which

occur at the surface of a material are known as surface plasma

oscillations and the quantum of their energy is called “surface plasmon”

As we know the dielectric function must be negative for surface

waves to exist, the frequency of this mode is always lesser than the bulk

plasma frequency.

Frequency of the surface plasmon can easily be deduced by

applying the boundary conditions i.e. normal components of Dielectric

displacement D and magnetic induction B and the tangential

components of electric field E and Magnetic field H of an

electromagnetic wave must be continuous at the interface two media

[76.77].

We shall calculate the frequency for a plane semi-infinite isotropic

plasma medium of background dielectric function Lε bounded by a non

dispersive medium of dielectric constant 2ε as shown in fig (2.2)

The total dielectric function of the conducting medium is thus

( ) ( )[ ][ ]222

2222

sincoscos

sincoscossin

αφθαθ

φθαφθαθ

−=

+−=

Eoc

Eop

qq

qq

24

( )2

2

1ω

ωεωε p

L −= (2.13)

Since the normal components of electric displacement vector D must be

continuous at z=0. we have

21 zz DD = at z=0 (2.14)

But we know that ED ε= , therefore

( ) ( ) 021011 == = zzzz EE ωεωε (2.15)

In absence of any external field, the electric field arises only due to

polarization charges and by symmetry it has the same magnitude but

opposite direction at the two sides of the interface, i.e.

21 zz EE −= at z=0 (2.16)

For eqn (2.15) & eq

n (2.16) we get

( ) 21 εωε −=

Substituting ( )w1ε from (2.13) we obtain surface mode frequency.

The coordinate system ( )ωε1 and ( )w2ε are the dielectric functions of

conducting and non- dispersive media respectively.

2

2

2

22

2

22

2

εε

ωω

εεω

ω

εω

ωε

+=

+=

−=−

L

p

s

L

s

p

s

p

L

25

Fig 2.2

or [ ] 2

1

2εε

ωω

+=

L

p

s (2.17)

If we take the conducting medium as metal 1=Lε and non-

dispersive medium as vaccum 12 =ε .the above eqn(2.17) reduces to

ω2

p

s

ω=

In both media are metals havings plasma of frequency 1pω and 2pω ,

the surface plasma frequency at the interface is given by

+=

2

21 pp

spwωω

(2.18)

The surface plasma excitation is a characterstic property of surface

and its frequency and dispersion depends on the nature and the geometry

of the surface and the properties of the bounding media.

26

R.H Rithcie [78] was the first to predict, theoretically, the

existance of surface plasmon in thin metallic foils and at the single plane

surface bounded by vaccum.

Later it was confirmed by stern and Ferrell. Both pointed out that a

part of plasmons excited in a thin metallic film by energetic electrons,

should be able to decay into photons of frequency equal to bulk plasma

frequency pω .

Although the retardation effect was ignored in his derivation, his

result were found to be correct by more rigorous theories. Ferrel

recognized that the radiative mode must be of antisymmetric (normal)

type, from which it follws that the radiation is plane polarised.

2.3 Locally excited plasmans on the material surface

Suppose we consider a medium in which there occur a surface

vibrational excitation with which we can associate wavelength λ and an

angular frequency ω . Then for d>>λ , where’d’ is the mean inter atomic

spacing, the response of the medium to the excitation is essentially that of

continuum, since within theλ there exist a large numbers of atoms. This

response is called ‘local’ due to the following reasons.

For simplicity, we consider a bulk homogenous isotropic system.

The relation between the averaged electric field ( )trE , and elastic

displacement ( )trD , can be written as

( ) ( ) ( )∫ ∫ −−∈=t

RA trEttrrradttrD0

3 ,''',''', (2.19)

where ( )',' ttrrRA −−∈ is the real space, real time response function.

The displacement at a particular position r and at particular time t

27

depends upon electric field over a region of space about r (i.e. the

response of medium is non local or spatial dispersion has been included)

and at earlier time 't . Thus

( ) ( )→−−=−− ','',' ttrrttrr RR εε for tt <' (2.20)

=0 for tt >'

Taking Fourier transforms of ED, and Rε from ( )tr, to ( )ω,K space

( ) ( )( ) ( )( ) ( ) rKit

R

rKit

rKit

eetrdtrdK

eetrEdtrdKE

eetrDdtrdKD

×∈=∈

×=

×=

−

−

−

∫ ∫

∫ ∫

∫ ∫

ω

ω

ω

ω

ω

ω

,,

,,

,,

3

3

3

multiplying eqn

(2.22) and (2.23) we find with the help of (2.19)

( ) ( ) ( )ωωω ,,, KEKKD =∈ (2.24)

Hence for a non local system the response function (which is the

dielectric function) is a function of wave vector and frequency, In general

a tensor. However a local response can be imposed by requiring

( ) ( ) ( )''',' ttrrttrr RR −∈−=−−∈ δ (2.25)

Where δ is Dirac delta function

( ) 1' =− rrδ for 'rr = (2.26)

0= for 'rr ≠

Which implies that the displacement D at a particular position

depends on the electric field E at the same position or in other words the

(2.21)

(2.22)

(2.23)

28

wavelength λ is much larger (or K is very small) then depends on the

electric field. Since in this limit

( ) ( )ωω =∈∈→ ,lim 0 KK (2.27)

Equation (2.24) modified to

( ) ( ) ( )ωωω ,, KEKD =∈ (2.28)

Local dielectric function is therefore scalar and depends only on

the frequency. So locally excited plasmon on the material surface explain

by frequency. Now we shall derive the dielectric function of material

(ionic and partially ionic compounds e.g. Polar semi conductors) having

cubic structure of Nacl, CsCl or ZnS type.

2.4 Theoretical & Experimental study of plasmons:-

The diffraction limit of light is a major obstacle on the way of

achieving high degree of miniaturization and integration of optical

devices and circuits. The main approach to overcome this problem is

related to use of surface Nanostructures Such as rectangular metallic

nanostructures; nanostructures, nano strips, nano rods , nano chains .

parabolic metal wedges could also be used for the design of

subwavelength waneguides, though such a possibility has not been

specified , Recently a new type of strongly localized plasmons called

channel plasmon–polariton [cpp] has been investigated theoretically in

metallic groovs. The major features of CCPs in V grooves include a

unique combination of strong localization and relatively low dissipation ,

single- mode operation, possibility of nearly 100% transmission through

sharp bends , and high tolerance to structural imperfections.

29

Only very few experimental attempts to investigate different types

of plasmonic waveguides with subwavelengh locolization have been

under taken so far .

It has been shown that nanochain waveguides have very strong

dissipation and short propagation distance ( 200≤ nm). The numerical

analysis is carried out by means of the three dimensional finite difference

time domain (FDTD) algorithm. This has been extended to plasmanic

wave guides with negative permittivity by using the local Drude model.

Generation of surface plasmons and wedge plasmons (wp) occurs

at the point of the end fire excitation. Beyond this point, surface

plasmons experience significant diffractional divergence. Therefore, if

we place a small aperture into the wedge at some distance from the point

of end-fire excitation, so that it blocks the diverged surface plasmons and

lets only wp through, this should significantly reduce the beats. The

fourier analysis of the field at the tip behind the aperture shows that the

maximum due to surface plasmons disappear from the spectrum.

Fig 2.3 Fig 2.4

30

Fig (2.3) shows the effect of dissipation on wp. It can be determind

that the intensity of the wp drops e times within the propagation distance

Ltheory mµ25.2≈ .Which is larger than the wavelength of the plasman and

thus is sufficient for a range of nano-optics applications.

The analysis of the end-fire excitation can only be conducted using

3D FDTD algorithm. At the same time the compact 2D FDTD provides

more accurate results for the field distribution and dispersion of localized

plasmons. The dependencies of the wp wave number on θ for a silver

wedge in vaccum, obtained in the compact-2D and 3D FDTD

formulation, demonstrate significant differences at o30<θ [ ]afig 5.2 . This

is due to low accuracy and efficiency of 3D FDTD at smallθ .

If cθθ = , then 17

01 10025.1 −×≈= mqq and the wp has infinite

penetration depth (zero localization) along the sides of the wedge. If

cθθ > wp does not exist as a structural eigen modes, since it looks into

surface plasmons. This is similar to the existence of the upper critical

angle for CPP modes in the metallic grooves.

The field distribution in the

fundamental wp mode in a cross

section of the o40 silver wedge is

presented in fig [ ]b5.2 demonstrating

strong subwavelength localization

within the region of`~50nm.

Plasmon parameters and field

structure were determined by means

of two different FDTD formulation. In particular, it has been shown that

wp modes do not exist. If the wedge angle is larger than critical angle.

Fig 2.5

31

Strong subwave length localization of wp modes has been

demonstrated and reasonable propagation distance have been predicted,

which makes wp modes a good candidate for the design of subwave

length plasmanic waveguides.

The first direct excitation of a nano-waveguide by bulk waves was

successfully counted. The predicted and experimentally observe

propagation distance are sufficient for design of nano-scale inter

connectors between nano optical devices.

2.5 Excitation of surface polaritons on the surface of materials

When an electromagnetic wave is incident an a solid, the electric

and magnetic fields associated with the waves tend to perturb the electron

density equilibrium and the lattice arrangement of the solid. This

perturbation lead to collective oscillation of electron density or lattice

vibrations, leading to excitation of plasmon modes or phonon modes

[79,80] in the crystal. other types of modes like exitons [81] magnons [82]

may also be excited by the incident radiation, but the types of modes

generated will depend upon the material and on the frequency of the

excited modes is comparable with the frequency of the incident radiation.

There is a possibility of interaction between the two leading to couple

modes that are termed as polariton modes [83, 84]. The generation of the

elementary excitations in the medium by the Incident electromagnetic

radiation is actually due to the polarization [85] induced in the medium

by the incident electromagnetic field. As a result, the coupled

polarization mode propagates in a crystal as an electromagnetic wave

consisting of incident oscillating electric and magnetic field

superimposed by the polarization induced in the medium by deriving

fields. The electromagnetic waves propagating in a medium thus a

32

complex entity. The energy stored in the wave is shared among the

incident field and the excitation induced in medium. Thus a polarization

wave is actually a composite wave or a coupled mode of the incident

electromagnetic field with the elementary excitations of the medium that

may coupled to electromagnetic field by virtue of their electric and

magnetic characteristics [86, 87]. Any elementary excitation of a medium

which gives rise to an electric and magnetic fields of the incident

Electromagnetic wave. The incident electromagnetic radiation can couple

simultaneously more then one predominant elementary excitation modes

of the medium if they have comparable frequencies.

Thus the term polariton is broad and general term for coupled

electromagnetic (EM) modes in solids [88]. However when a specific

designation is required for a particular case, For example, in the

frequency region where EM waves in a medium couple with optical

phonon or plasmon excitations of a dielectric medium, the coupled

modes are designated respectively as phonon- polariton or plasmon

polaritons as the case may be. Similarly in the case of magnetic materials,

the polarisation modes are termed as magnon polaritons. In the present

work only the polariton mode is non-magnetic, isotropic dielectric media

have been studied.

The properties of an isotropic, homogenous, dispersive dielectric

medium can be studied with the help of its frequency dependent

dielectric faction [89]. The nature of dielectric function gives a clear

insight into the type of elementary excitations that can be sustained by

the medium. The properties of Polariton waves can be studied with the

help of dispersion relation for these modes which depends upon the

dielectric function of the medium.

33

The coupled EM modes or polariton modes in a medium can be

both bulk modes [86,87] or surface modes [81] layer or are bounded

along the interface of two media such that they propagate without decay

in a wave like manner along the interface between two media having

different electrical or magnetic properties but decrease exponentially with

distance in the direction perpendicular to the interface.

These Surface modes are transverse magnetic in character and are

characterized by a negative value of the frequency dependent dialectic

function. The properties of surface waves are strongly dependent upon

the boundary medium and on the nature of the boundary at the interface

of two media along which they propagate. Thus surface polarizations can

participate in a number of physical processes that takes place near the

surfaces and interfaces and are an effective tool for the study of surfaces

and interface [90, 91]. On the other hand, bulk polarization modes are the

coupled modes of the elementary excitations of the medium and the

electromagnetic wave that are sustained in the whole volume of the solid

instead of being restricted to a particular surface or boundary between

two media. The incident electromagnetic wave in this case couples with

the bulk modes of the elementary excitations of the medium.

As discussed earlier the incident electromagnetic wave may excite

the bulk elementary modes in a medium. the surface excitations are

localized in a thin layer at the interface of the two dielectric media and do

not ordinarily link with the bulk elementary excitations of either medium,

but decay rapidly in directions perpendicular to the interface, when

frequency of incident electromagnetic radiation becomes comparable

with the frequency of the excited surface modes. Surface polarization

waves are obtained that propagates as electromagnetic waves along the

interface, but decay in a non oscillatory exponentially manner, in the

34

directions perpendicular to the interface [92]. As such, the surface

polarization waves represent pairs of plane waves, one on each side of

the boundary between two media having different electrical properties.

the phase velocity of these surface polarization modes is less than the

velocity of light in vaccum and thus these modes can not be excited

directly by striking light at the surface of solid, and therefore, do not

optically couple directly to the surrounding medium.

Similarly phase velocity of these surface polarization modes is also

less than that of bulk polariton modes and that the surface polarization

modes can not be couple to the bulk modes [82]. thus the surface

polariton mode remains localized at the surface. The properties of these

surface waves in a medium are depending upon the nature of the

boundary and the bounding medium [91, 93]. The properties of these

waves can be studies with the help of dispersion relation for these waves.

The dispersion relation for surface polariton waves can be obtained by

using the Maxwell’s equation [94] and applying suitable boundary

conditions [76, 77] This method has been used effectively by several

workers [94, 95] for metals as well as for semiconductors. Another

method, the quantum mechanical random phase approximation (R.P.A)

method [96] has been used to study the properties of surface polaritons.

Maxwell’s equations method has been found to be suitable and effective

for the study of surface polariton waves within the local theory limit [97,

98] only, where the dielectric functions of the dispersive medium that

sustain these modes is taken to be dependent only on frequency, so that

the form of the dielectric function is well known or can be easily derived

and can be used to obtain the dispersion relation. Difficulty arises when

the study of these surface polariton waves has to be extended to the non

local limit, where the dielectric function of the medium is dependent on

35

wave vector as well frequency. To understand this non-local limit [85], a

dispersive dielectric medium is considered in which there occurs a

surface vibrational excitation of frequency ‘ ω ’ due to an incident

wavelength ‘λ ’ then for λ >>d.

Where ‘d’ is the inter atomic spacing. This type of response of the

medium to the incident radiation is called the local response of the

medium, under such circumstances the propagating polariton waves is to

be within the local theory limits. This condition is easily satisfied by the

long wavelength surface optical Phonon waves so that the surface phonon

polariton modes can be studied accordingly within the local theory limits.

In general the relation between the displacement field and the electric

field for homogenous dielectric medium is given by [85]

( ) ( ) ( )KEKkD ,,, ωωεω = (2.29)

i.e. dielectric function of the medium is dependent both on

frequency w and on wave vector K. however ,if the wavelength of the

surface waves is larger, the wave vectors

=

λ

π2K tends to zero, so that

( ) ( )ωεω =∈= ,lim 0 KK (2.30)

i.e. the dielectric function in the local limit is dependent only on

frequency ( )K,ωε is called the non-local dielectric function .However

if the condition d>>λ is not satisfied i.e. the wavelength of the

incident radiation is not very large as compared to the inter atomic

spacing .

It is thus important to include the spatial dispersion effects in

the study of surface Plasmon polariton wave. There fore in order to

36

obtain the dispersion relation for surface Plasmon polariton waves

taking spatial dispersion into account the non-local form of the

dielectric function is required.

This non-local form of the dielectric function may be calculated

using the classical or the quantum mechanical method. However this

method requires complicated mathematical analysis. An alternative,

simple and effective approach is provided by the Bloch’s hydro-

dynamical method (101) for the inclusion of spatial dispersion in the

study of plasma system .This model will be discussed in detail in next

(chep-5) chapter.

The Bloch’s hydrodynamical model has been effectively used by

several workers (102,103) to study surface Plasmon polarition wave in

the case of metals, taking spatial dispersion into account .In the present

work, therefore, this problem is taken up for polar semiconductors

considering cylindrical geometry. There fore, deriving the dispersion

relations for surface polaritons, it is worth while to study first the plasma

oscillations and the phonon waves of the dispersive, dielectric medium

and to obtain the dielectric function of the medium.

37

CHAPTERCHAPTERCHAPTERCHAPTER----3333

DIELECTRIC FUDIELECTRIC FUDIELECTRIC FUDIELECTRIC FUNNNNCTIONS DUE TO LATTICE CTIONS DUE TO LATTICE CTIONS DUE TO LATTICE CTIONS DUE TO LATTICE

VIBRATIONSVIBRATIONSVIBRATIONSVIBRATIONS

Dielectrics are insulating materials which are capable of storing

electrical energy. The dielectric materials do not have free electrons and

hence are good insulators. When they are placed in electric fields,

intertial fields are set up in the dielectric materials which oppose the

externally applied field, there by reducing the net electric field and hence

the potential difference. If these dielectrics are placed between plates of

capacitor,the potential difference will be reduced without affecting the

charge on the plates, the capacitance of capacitor )(v

qc = increases

Dielectric materials are essentially insulating materials. However,

the dielectric material store electric energy while, insulating material

obstruct the flow of current. This difference between dielectric material

and insulating material explains that the material to be used as dielectric

must have properties some of which may not be answered by a material

ordinarily use as an insulating material. The concept of dielectric was first

introduced by Faraday. He discovered that when a space between the

plates of a parallel place capacitor is filled with a dielectric material, Its

capacitance increases. It is due to the fact that when a dielectric material

is placed in an electric field its electric properties get modified. The

increase in capacitance is due to the reduction of electric field between

the plates. The external electric field has an ability to polarize the

dielectric material to create dipoles.The atoms and moleclues of

dielectrics are influenced by an external electric field and hence, the

positive charges are pushed in the direction of the field while the negative

charges in the opposite directions from their equilbrium position.

38

Hence ,dipoles are developed and they produce a field of their own in a

direction opposite to the direction of an external field. Consequently, the

resultant electric field between the capacitor plates is reduced.

According to band theory of solids a dielectric is a material in

which the energy gap between valence band and conduction band is more

than 3ev and normally,no electron is able to cross this band gap.

Dielectric material can be solids, liquids or gases, hign vaccum can

also be useful,lossless dielectric even though its relative dielectric

constant is only unity. Solid dielectrics are perhaps the most commonly

used dielectrics.material is used extensively in electrical engineering.

Thus dielectric helps in the following three ways

It helps in maintaining two large metal plates at very small

separation.

It increases the potential difference which a capacitor can with

stand without breakdown.

It increases the capacitance of a capacitor.

3.1 Dielectric constant OR Relative permittivity

Dielectric constant define the following ways-

When a dielectric is placed between the plates of a capacitor its

capacity is increased. The ratio of the capacitance of a capacitor with

dielectric to the capacitance of the same capacitor without dielectric is

defined as dielectric constant. Thus

39

0C

Ck =

C = Capacitance of the condenser with dielectric

Co =Capacitance of the condenser without dielectric

It has been observed that the potential difference dV between the

plates of the capacitor filled with dielectric is smaller than the potential

difference oV without dielectric. Thus ratio of Potential difference without

dielectric oV to the potential difference with dielectric dV is defined as

dielectric constant.

hence d

o

V

Vk =

The ratio of permittivity of medium (ε ) to the permittivity of free

space ( 0ε ) is known as the dielectric constant (K) or relative permittivity

(ε r ) of the medium

ε r = 0ε

ε=k

3.2 Dielectric function due to lattice vibration

The dielectric function ( )k,ωε of electron gas, with its strong

dependence on frequency and wave vector, has significant consequence

for physical properties of solids. ( )o,ωε describes the collective

excitations of the volume and surface plasmons when k →0. ( )ko,ε

describes the electrostatic screening of the electron-electron, electron-

lattice and electron-impurity interactions in crystals when ( )o→ω .

Dielectric function of the ionic crystal has been used to derive the

40

polariton spectrum. Here the electron gas in metals has been taken the

dielectric constant ∈of electrostatic is defined in terms of the electric field

E and the polarization P, the dipole moment density

D = oε E+P (3.1)

The introduction of the displacement D is motivated by the

usefulness of this vector related to the external applied charge density extρ

in the same way as E is related to the total charge density

=ρndiext ρρ + (3.2)

Where indρ is the charge density included in the system by extρ .

Thus divergence relation of the electric field is

extoEdivdivD ρεε == (3.3)

( )

o

indext

o

divEε

ρρ

ε

ρ +== (3.4)

The long wavelength dielectric response ( )o,ω∈ or ( )ω∈ of an

electron gas is obtained from the equation of motion of a free electron in

an electric field.

eEdt

xdm −=

2

2

If x and E have the time dependence, then

2ωm

eEx =

The dipole moment of one electron is 2

2

ωm

Eeex −=−

41

The polarization defined as the dipole moment per unit volume, it

is

Em

nenexP

2

2

ω−=−= (3.5)

Where n is the electron concentration.

The dielectric function at frequency ω is

( ) ( )( )

( )( )ωε

ω

ωε

ωωε

E

P

E

D

oo

+== 1 (3.6)

Now from eqn (3.5) & (3.6)

( )2

2

1ωε

ωm

ne

o

−=∈ (3.7)

Now plasma frequency pω is defined as by the relation

m

ne

o

pε

ω2

= (3.8)

A plasma is medium with equal concentration of positive &

negative charges. In solid the negative charges of the conduction

electron are balanced by equal concentration of positive charge of the

cores. Now from eqn (3.7) & (3.8)

( )2

2

1ω

ωω p

−=∈ (3.9)

By above eqn (3.9) Dielectric function at frequency w calculated

when plasma frequency known for the substance.

42

3.3 Polarization of Dielectric materials

In atoms, because of their spherical symmetry, the centre of mass

of the electrons coincides with the nucleus. Therefore atoms do not have

permanent electric dipole moments. However when atoms are placed in

an electric field, they accuire an induced electric dipole moment in the

direction of the field. This process is called the electric polarization and

atoms are said to be polarized.

There are four different mechanisms by which electrical

polarization can occur in dielectric materials when they are subjected to

an external electric field. They are

Figure 3.1

Electronic polarization :- The electronic polarization occurs due to the

displacement of positive and negative charges in a dielectric material

when an external electric field is applied. This process occurs throughout

the material and the material as a whole is polarized. The electronic

polarization is independent of Temperature.

Ionic polarization :- Ionic polarization occurs in ionic materials when an

electric field is applied to an ionic material, cations and anions get

displaced in opposite directions, which gives rise to a net dipole moment.

The displacement causes an increases or decrease in the distance

43

separation between the atoms depending upon the location of ion pair.

This polarization is also independent of Temperature.

Figure 3.2

Orientational Polarization :- This type of polarization is found only in

substances that process permanent dipole moment e.g CH3Cl. When an

electric field is applied on such a molecule, then the dipoles tend to align

themselves in the direction of applied field. This polarization is

dependent on temperature. Polarization decreases with increasing

temperature.

SPACE-CHARGE Polarization :- Space charge polarization occurs due

to the accumulation of charges at the electrodes or at the interfaces in a

multiphase material. The ions diffuse over appreciable distance in

response to the applied field. This gives rise to redistribution of charges

in the dielectric medium. The Space-charge polarization is not an

important factor in most common dielectrics. This type is also known as

interfacial Polarization. This polarization is insensitive to temperature

charges.

44

Figure 3.3

Figure 3.4

The total polarization P of a multiphase material is equal to sum of

all types of polarization

soie ppppp +++=

For a single phase dielectric

oie pppp ++=

3.4 Dielectric function for Polar Semiconductor :-

In long optical vibrations, atoms of one type move as a body

against the atom of the other type by action of the field. For such a wave

the effective inertial mass per unit cell is reduced mass of +ve and –ve

ions.

−+

+=MMM

111so

−+

−+

+=

MM

MMM (3.10)

45

When external electric field is applied to a substance. It gets

polarized giving rise to a polarization field which always opposes to the

applied field. The macroscopic field in the insulator is

pe EEE −= (3.11)

and is always lesser than applied field.

But the field which is actually responsible for polarizing individual

ion is the macroscopic field, also known as effective electric field

PEEeff3

4π+= (3.12)

For a crystal with cubic symmetry. This field is always greater than

the macroscopic field. ‘P’ is known as Polarization vector.

To get an expression for the dielectric function we will first

determine the polarizability which characterizes the polarization of an

individual atom or molecule. Polarizability is defined as the dipole

moment an a molecule per unit effective electric field. In partially ionic

materials the dipole moment is due partly to the displacement of the ionic

charges of molecule and partly due to induced electric moments on the

ions.

When an ionic charge Ze is displaced by u , the net effect is as

through a charge - Ze has been placed at the undisplaced position to

annihilate the original charge and a fresh charge Ze has been created at

displaced position. The displacement is thus equivalent to the addition of

a dipole with the moment Ze . The induced electric moment on the other

hand, is produced due to the deformation of an atom or ion (electron are

displaced opposite to the field and no longer remain symmetrically

46

distributed about the nucleus). The dipole moment is proportional to the

effective field and is given by effE+α and effE−α on positive and negative

ions respectively. +α and −α are electronic Polarizability. The total dipole

moments on the two types of ions are thus

effe EuZ ++ + α

effe EuZ −− +− α

The dipole moment on a molecules is

( ) ( ) effe EuuZP −+−+ ++−= αα

in terms of polarization vector P , which can be defined as dipole

moment per unit volume, as

( )[ ]effe ExZNP −+ ++= αα (3.13)

where N is the numbers of molecules (pairs of ions) per unit

volume and −+ −= uux is the relative displacement.

The eqn of motion for both type of ions are

( )( ) effe

effe

EZuuuM

EZuuuM

−−=

+−−=

−+−

−+++

−µ

µ (3.14)

Multiplying equations respectively by +M and −M subtracting and

dividing the result by ( )−+ + MM we get

( ) ( )effeEZuuuuM +−−=− −+−+ µ (3.15)

effeEZXXM +−= µ (3.16)

47

The frequency dependence of the dielectric function is considered

by taking A.C field as

ti

oeff eEEω−= (3.17)

ti

oeXXω−= (3.18)

Substituting eqn (3.17) & (3.18) into eqn (3.16) we have

effeEZXXM +−=− µω2

effeEZXMXM +−=− 22 ωω (3.19)

where m

µω = is optical mode frequency which the lattice would

have in the absence of Coulamb forces between the ionic charges eZ± .

Therefore

( ) effeEZM

X22

1

ωω −= (3.20)

putting value of x in Eq (3.13), we have

( )

( )( ) eff

e E

M

ZNP

−+

−= −+ αα

ωω 22

2

(3.21)

Hence the polarizability is

( )

( ) ( )

−+

−= −+ αα

ωωα

22

2

M

Ze (3.22)

Here the first term in the atomic or ionic polarizability and the

second term is electronic polarizability.

48

The dielectric function due to lattice ( )ωε l or the background

dielectric function is related to polarizability α by Clausius Mossotti

relation

( )( )

απ

ω

ωN

L

L

3

4

2

1=

+∈

−∈ (3.23)

using eqn (22) in above relation

( )( )

( )( ) ( )

−+

−=

+∈

−∈−+ αα

ωω

π

ω

ω22

2

3

4

2

1

M

ZeN

L

L (3.24)

On applying an A.C field frequency ω , the system tries to respond to the

charge due to the field. Electrons are light so able to follow the field

instantly to high frequencies while the ions, because of their large mass

(104 times the mass of electron) are unable to follow high frequencies and

their polarizability tends to zero for ωω >> where ω is in infrared

region. It is convenient to write the dielectric function ( )ωL∈ in terms of

low )( ωω << and high ( ωω >> ) frequency dielectric constants, o∈ and

∞∈ , which are easily measurable.

o∈ , the static dielectric constant is obtained by setting ωω << in

eqn

1)(

1)(

−∈

−∈

ω

ω

L

L =

−Μ

Ζ++Ν −+

)(

)(

3

422

2

ωωαα

π e (3.25)

and similarly ∞∈ , the high frequency dielectric constant by taking

ωω >> , 2

10

+∈

−∈

∞

= [ ]−+ +Ν ααπ

3

4 (3.26)

49

High frequency means the frequency high enough to render lattice

polarizability zero. This range comes out to be in optical region and

therefore is related to the index of refraction ‘n’. Thus 0∈ is greater than

∞∈ .

Form eqn (3.16) & (3.17) we have

2

1

0

0

+∈

−∈=

2

1

+∈

−∈

∞

∞ +

Ν

2

2

3

4

ϖ

π

M

Ze (3.27)

)(3

)2)(2()(

34

0

0

22

∞

∞

∈−∈

+∈+∈=

M

ZeN

πω (3.28)

Thus eq (3.24) may be modified as

( )( )

−

++=+∈

−∈−+

2

22

2

13

4][

3

4

2

1

ϖ

ωϖ

παα

π

ω

ω

M

ZeN

L

L N (3.29)

by eqn (3.26)_& (3.27) the eqn reduces to

−

+∈

−∈−

+∈

−∈+

+∈

−∈=

+∈

−∈

∞

∞

∞

∞

)1(

1

2

1

2

1

2

1

2)(

1)(

2

2

0

0

ϖ

ωω

ω

L

L

After simplification it gives

+∈

+∈−

+∈

+∈∈−∈

=∈∞

∞∞

2

0

2

2

0

0

2

2

2

2

2

)(

ϖω

ϖω

ωL (3.30)

50

2

0

2

2

2ϖω

+∈

+∈= ∞

t (3.31)

tω is the transverse optical Phonon frequency. The distinction

between transverse optical and transverse acoustical waves in diatomic

lattice are illustrated by particles displacement for two modes.

3.5 Dielectric function for materials :-

In long wavelength (K=0) optical mode, the oppositely chargesed

ions on each primitive cell undergo oppositely directed displacement,

giving rise to a non vanishing polarization density. Thus by a

macroscopic electric field Ε and an electric displacement D , related by

PEED πω 4)( +==∈ (3.32)

Since there is no externally induced charge, we have

0=•∇ D

The macroscopic electric field is electrostatic hence it may be

expressed as the gradient of a scalar potential function

E= ψ∇− (3.33)

or curl of E vanishes thus 0=×∇ E

we have to choose P and )(ω∈ such that these two equations

are satisfied simultaneously. This can be achived in three different ways

0=Pdiv ∞=∈ )(ω

curl 0=P 0)( =∈ ω (3.34)

div 0=P 0=Pcurl

51

again in cubic crystal D is parallel to E (i.e D are parallel to

p ). So considering the spatial dependence

rkie

P

E

D

P

E

D

=

0

0

0

Re (3.35)

eqn (3.33) reduced to 0=⋅ DK

0=D or KPandED ⊥, (3.36)

Where as (3.36) reduced to

k ,0=× E i.e 0=E or PandDE, parallel to k.

So in order to satisfy both of these equtions simultaneously we must have

(a) 0=E KPandED ⊥, , (3.37 i)

(b) ) 0=D IIKPandED,

for above conditions (a) & (b) in eqn (3.37(1) are combining, we

get

(a) 0=E KPandED ⊥, , ∞=)(ωε

(b) ) 0=D IIKPandED, 0)( =ωε

(a) In this case the polarization density is ⊥ to the Propagation

vector therefore these are the transverse modes. The macroscopic electric

field E corresponding to these modes vanishes everywhere.

Therefore the effective electric field can be obtained by substuting

macroscopic electric field ( E ) equal to zero into eqn (12)

(3.37ii)

52

PEPE effeff

3

4

3

40

ππ=⇒+= (3.38)

Polarization density P (Eq 3.13) may be written with the help of

(Eq 3.26) equation of motion may be written as with the help of earlier

equations

effEZeNxP

+

−+=

∞

∞

2

1

4

3

ε

ε

π (3.39)

PZeNxP3

4

2

1

4

3 π

ε

ε

π

+

−+=

∞

∞ (3.40)

then ZenxP3

2+= ∞ε

(3.41)

The equation of motion (Eq 3.20) may be written as with the help

of earlier equations

PZexM t3

4)(

22 πωω =− (3.42)

xZeNZexM t

+

=− ∞

3

2

3

4)(

22 επωω

222 )(49

2)( ZeNM t π

εωω

+=− ∞

or 22

2

2ω

ε

εω

+

+=

°

∞t (3.43)

with the help of Eq(3.28) this eq gives the transverse optical

phonon frequency which is exactly the value defined in equation ( 3.31)

(b) In the case the polarization density is parallel to the propagation

vector therefore these longitudinal modes the electric displacement

53

D vanishes every where. Hence the electric field for such modes is given

by

PE π4−=

their frequencies are gives by the zeros of the dielectric

function .They are called longitudinal bulk modes since they are

analogous to the longitudinal plane waves which can propagate in an

infinite medium of dielectric constant . ).(ω∈

for such modes the effective electric field is

PPE eff

3

44

ππ +−=

PE eff

3

8π−= (3.44)

And polarization density (3.39) is

+= ∞

3

2εxZeNP (3.45)

The eqn of motion (3.20) for these modes is written as

( ) efft EZexM −=−22

ωω (3.46)

38

3

8 PZePZe ππ

=

−−= , ( ) 222

)(3

2

3

8ZeNM l

+=−

∞

∞

ε

επωω (3.47)

Which with the help of equation (3.21) gives the longitudinal

optical phonon frequency as

54

22

2

2ωω

+∈

+∈

∈

∈= ∞

∞ o

ol (3.48)

Equation (3.44 and 3.48) give the relation between the frequencies

of two modes as

22

to

l ωω∞∈

∈= (3.49)

Which is known as Lyddance-sachs-Teller (LST) equation or

relation . The acting of long range coulomb interaction between ions

causes splitting in mode frequencies otherwise they were degenerated at

W. the transverse mode frequency is lesser and longitudinal mode

frequency is greater than ω for long wave length in polar semiconductors.

The frequency dependence of dielectric function (3.30) may also be

written as-

( )22

ωωω

−

∞∈−∈+∞=∈∈

t

oL (3.50)

In order to express separately the contributions due to electronic

and ionic polarizabilities .

For ionic compounds the static dielectric contant is greater than the

high frequency dielectric constant which also confirms with the help of

(3.49).

In case of metals, the valence electrons remain free and those of

inner shell from an intert gas configuration with the nucleus. They do not

respond at all to the lattice is zero giving the dielectric constant as unity

(from eq 3.24) which is equal to that of vacuum. Thus we see that the

dielectric function of a polar material depends on the field and the phonon

55

frequency but for a material or non – polar substance .It is a constant thus

the lattice is equivalent to uniform background for this case .

The frequency-dependent dielectric function due to phonons for a

diatomic polar crystal. This is valid for cubic crystal having a single long

wave length infrared active, transverse optical phonon frequency ω t.

56

CHAPTER 4CHAPTER 4CHAPTER 4CHAPTER 4

SZIGETI’S DIELECTRIC THEORYSZIGETI’S DIELECTRIC THEORYSZIGETI’S DIELECTRIC THEORYSZIGETI’S DIELECTRIC THEORY

Szigeti published four seminal papers on the dielectric behaviour of

crystals during the period 1949-1961. Szigeti’s theory is applicable to

isotropic and anisotropic, ionic and covalent crystals with different

structures. Szigeti’s theory connects dielectric, spectroscopic and elastic

properties. An important outcome of Szigeti’s theory is the concept of the

effective ionic charge (s). It is pointed out that s correlates with a number

of physical properties and is a measure of ionicity of the interatomic

bond. Since Szigeti’s work, several theoretical models have been

proposed to account for the fact that s < 1. These models provide an

insight into the complex polarization mechanisms in solids. This review

summarizes Szigeti’s work and the work that followed; the implications

and applications of Szigeti’s theory are discussed. Some new results are

also included.

The electric polarization P in a crystal is related to the applied electric

field E by the relation

P = αE, (1)

where α is the polarizability. The dielectric constant ε is related to

the polarizability through the relation

ε − 1 = 4πα. (2)

This relation does not take into account of the internal field. When

the internal field is included, the Clausius–Mosotti relation

57

(ε − 1)/(ε + 2) = (4π/3)α. (3)

The Clausius–Mosotti equation is valid when the environment

around every ion has tetrahedral symmetry as in the alkali halides.

Further, it applies only to cubic crystals. The polarizability α is made up

of two components, the atomic polarizability αa arising out of the ionic

displacements and the electronic polarizability αe arising out of the

displacement of the electron cloud relative to the nucleus. With proper

substitution for αa and αe we get

[(εs − 1)/(εs + 2)] − [(ε∞ − 1)/(ε∞ + 2)] = (4π/3)αa, (4)

Where εs and ε∞ are the static and high frequency dielectric

constants. But this extended Clausius–Mosotti equation also does not

work well and alternate theories have been proposed to account for the

dielectric behaviour of solids. These theories are dependent on the details

of the polarization mechanisms considered in each theory. Thus,

assuming that the ions are rigid (non-deformable) and non-overlapping,

Born and Mayer [1] obtained

εs = ε∞ + (z2e

2N)/π µt ν

2 (5)

Where z is the valency, e the electron charge, N the number of

molecules in unit volume and νt the transverse optical frequency. µ is the

reduced mass given by

µ-1

=m1-1

+m2-1

(6)

m1 and m2 being the masses of the positive and negative ions.

Equation (5) is valid for an ionic crystal with two different ions having a

58

single transverse optic infrared active mode. Agreement of eq. (5) with

experiment is not satisfactory.

4.1 Szigeti’s first relation:

Szigeti developed a theory which resulted in a relation between the

dielectric constant and the lattice frequency. The derivation of the relation

is to be found in Szigeti’s paper [2] and also in Born and Huang [6] and

Brown [7]. The main features of Szigeti’s theory are:

(i) The ions are deformable and they overlap.

(ii) The polarization in dielectric crystals has two parts: the infrared

polarization and the ultraviolet polarization.

(iii) While the ultraviolet polarization is the same as the electronic

polarization, the infrared polarization has contributions from both

electronic and atomic polarizations.

(iv) The atomic and electronic polarizations are not independent but

there is an interaction between them. This interaction is separated

into short-range and long-range effects.

(v) The total dipole moment M is split into two parts:

M = Mi +Mu, (7)

where Mu is the contribution of the electronic oscillators with

ultraviolet frequencies and Mi is the infrared part. Mi is written as

(8)

where the Qj ’s are normal coordinates. The summation is over

normal modes of infinite wavelength as only such waves contribute to

macroscopic dielectric properties. It is to be noted that Szigeti included

59

only first-order terms in the summation. Szigeti [2] obtained the

following relation:

εs = ε∞ + [(ε∞ + 2)/3]2[s

2(ze)

2N/πµνt

2 ]. (9)

Equation (8) is known as Szigeti’s first relation. The first term on

the right-hand side (ε∞) is the contribution of the ultraviolet polarization

and the second is due to infrared polarization. Szigeti’s relation (eq. (9))

contains two factors not present in Born’s relation (eq. (5)); both arise out

of the electronic contribution to the infrared polarization. The factor [(ε∞

+ 2)/3]2 is due to the fact that the long range interaction does not vanish

for transverse waves. The other new factor is s which represents the

short-range interaction of electronic and atomic displacements. The term

sze is equivalent to replacing ze by an effective charge ze*; then sze = ze*

and s = (ze*/ze). In subsequent literature, s has come to be known as the

‘effective ionic charge’ or the ‘Szigeti charge’. As Szigeti’s theory was

developed mainly for the alkali halides. It is convenient to rewrite eq. (9)

as

s = (9πµ/N)1/2

(εs − ε∞)1/2

(ε∞ + 2)−1

(νt/ze). (10)

The value of s for some alkali halides calculated by Szigeti [2] are

given in table 1. The input data (εs, ε∞, νt) used by Szigeti was from

earlier sources [8,9]. Values calculated by Lowndes and Martin [10] from

their own more recent and accurate data are also included in table 1. The

values in both sets are mutually consistent.

60

A common feature in the s values is that they are significantly less

than unity. Szigeti [2] himself pointed out that the deviation of s from

unity could be due to a partial homopolar (partial ionic) character or due

to interpenetration of the ions. In deriving eq. (9), a harmonic crystal was

assumed as implied by eq. (8). Szigeti [3] qualitatively surmised that this

may not affect the s values. However, he examined this aspect rigorously

in 1959 [4]. These and other aspects will be considered in later sections.

4.2 Anharmonic correction to S

Szigeti [4] considered the effect of anharmonicity on s. For this, he

expressed the dipole moment M as

61

(11)

where the Q’s are the normal coordinates and α0, βij and γijk

are constants. Q0 is the transverse optic infrared active mode. Here the

first term contains the effect of displacement of ions and also the

electronic distortions while the second and third terms are due entirely to

electronic deformation. Similarly, he expressed the potential energy W as

W = Wh +W’ (12)

Wh = (1/2)Σi ωi2Qi

2 (13)

and

(14)

Here Wh represents the harmonic and W’ the anharmonic terms.

(ωi/2π) is the frequency νi of the ith mode and bijk and cijkl are constants.

While macroscopic polarization arises only out of long wavelength

modes, the summation in eqs (13) and (14) involves normal modes of all

wavevectors. W is the potential energy with respect to all forces arising

from lattice displacements. Therefore,

W = potential energy of lattice + EextM. (15)

where M and W are given by eqs (11) and (12) and Eext is the

external field. Writing the Hamiltonian and applying perturbation theory,

Szigeti [4] showed that

εs − ε∞ = η + G, (16)

62

where η is the harmonic and G the anharmonic contributions. η is

the second term in eq. (9) whereas G is a function of the two dielectric

constants and the constants in eqs (11) and (14). where α is the linear

coefficient of thermal expansion, ψ the compressibility and T the

temperature. s, the Szigeti charge corrected for anharmonicity is given by

s’ = s(1 − δ)1/2

, (17)

where δ is the an harmonicity correction G/(εs − ε∞). The

importance of Szigeti’s work lies in relating the anharmonic correction to

experimentally measured quantities like thermal expansion coefficient

and compressibility without assuming any detailed model for the

anharmonic forces. Szigeti [5] estimated the anharmonic correction G/(εs

−ε∞) only for three alkali halides (NaCl, KCl and LiF) as the required

experimental data was available only for these crystals. From the results,

Szigeti [5] concluded that the anharmonic correction (i) is very small, of

the order of 10−2, (ii) affects the s value by about 2% and (iii) enhances,

though very slightly, the deviation of s from unity. [11] extended these

calculations to crystals with CsCl structure. His results showed that the

anharmonic correction G/(εs- ε∞) is negative for these crystals and it

reduces the difference between s and unity. As mentioned, due to the

availability of limited data, Szigeti [5] could estimate the anharmonic

correction only for three alkalihalides. Further, he did not report the value

of s’. Recently, data on all the quantities in eq. (16) for several alkali

halides have been compiled by Sirdeshmukh et al [12]. Using these data,

the present authors have estimated the anharmonic correction G/(εs − ε∞)

and the corrected Szigeti charge s_ from eqs (16) and (17) for a number

of alkali halides. These results are included in table 1. It is seen that the

anharmonic correction G/(εs − ε∞) has a small value and the s_ values are

only slightly different from the s values. Thus, the neglect of

63

anharmonicity in the derivation of Szigeti’s first relation (eq. (10)) is not

the cause of the difference between s and unity. It can also be seen that

the anharmonic correction has different signs for different crystals. Thus,

it is negative for LiBr and the cesium halides and positive for the other

alkali halides. The negative value is of significance. It is observed [11,13]

that crystals with a negative anharmonic correction have a negative value

for the temperature derivative of static dielectric constant.

4.3 Spectroscopic implication of the Szigeti relation

Lyddane et al [14] derived the following relation:

(εs/ε∞) = (ωl2 /ωt

2 ), (18)

where ωl and ωt are the longitudinal and transverse optical phonon

frequencies respectively. This relation is referred to as the Lyddane–

Sachs–Teller (LST) relation. Combining the LST relation with Szigeti’s

relation (eq. (10)), we get

s2 = (9µ/4πNz

2e

2)[ε∞/(ε∞ + 2)

2][ωl

2 − ωt

2 ]. (19)

This relation relates s with the splitting (ωl2−ωt

2 ) of the optical

mode. This splitting is an essential consequence of Szigeti’s relation.

Mitra and Namjoshi [15] comment that ‘no matter how one defines

ionicity of a crystal, its only manifestation is in the splitting of the

longitudinal and transverse optic phonon frequencies at k ~ 0’.

Although the term s in eq. (9) was attributed by Szigeti to the

short-range interaction of electronic and atomic displacements, the fact

remains that the term was introduced by Szigeti almost arbitrarily to bring

about equality between the two sides of eq. (9). The value of s was not

evaluated by Szigeti independently, but rather from eq. (9). An evaluation

64

of s independently of Szigeti’s relation was certainly desirable. Several

models were developed to account for the s values and, more particularly,

to account for the deviation of s from unity. These models are discussed

in this section. Yamashita [16] and Yamashita and Kurosawa [17] treated

the problem by a quantum mechanical variation method. They calculated

the ionic charge for LiF. Narasimhan [18] extended Yamashita’s method

to MgO. Yamashita’s method has not found further application as it

involves more parameters than can be evaluated from experimental data.

Szigeti [3] himself suggested that the mutual distortions of the ions due to

their overlap could be a cause of the s values being <1.

Born and Huang [6] pointed out that such distortions would result

in a ‘distortion dipole moment’ m(r) and that this dipole moment would

cause s to be <1. However, the treatment of Born and Huang was

qualitative. Lundquist [19] used the Heitler–London approach to evaluate

the overlap distortion moment for NaCl. Dick and Overhauser [20]

reexamined Szigeti’s theory. They pointed out that apart from the

electronic polarization and polarization due to displacement of ions

included in Szigeti’s theory, it is necessary to include ‘some further

polarization’. Thus, they introduced the concepts of short-range

interaction polarization and exchange charge polarization into the theory.

These arise from charge redistributions which occur when ions move with

resulting changes in electron overlap. Formulating their theory in terms of

the shell model, Dick and Overhauser calculated the integrals Sij

necessary to estimate the exchange charge. Here e_, A, µ and λ are

parameters occurring in the theory; these can be theoretically estimated

for each crystal. The values of s calculated from eq. (17) are given in

table 2. These values follow the same general trends as the original

Szigeti (or Lowndes and Martin) values but are seen to be systematically

65

larger and hence closer to unity. Dick and Overhauser concluded that the

short-range interaction polarization and the exchange charge polarization

are responsible for the deviation of s from unity. Hanlon and Lawson [21]

observed that the values of the Szigeti charge s of alkali halides lie on a

smooth curve when plotted against the difference in the polarizabilities of

the negative and positive ions. In order to explain this empirical

observation, Hanlon and Lawson modified Szigeti’s theory by including

the contribution of a mechanical polarization. When an electric field is

applied, each ion will be polarized in the direction of the field. But the

ions will also move relatively closer. Their electron shells will

mechanically repel one another, resulting in a relative shift of each

electron cloud with respect to its nucleus. This shift corresponds to an

additional polarization. In the positive ion, this mechanical polarization

enhances the electrical polarization, but in the negative ion it detracts.

Hanlon and Lawson showed that this mechanical polarization contributes

to the value of s.

66

The concept of the deformation dipole moment originally

introduced to explain the Szigeti charge values was applied by Hardy to

the problem of lattice dynamics of alkali halide crystals. Hardy [24]

included the interactions due to the deformation dipole moment in the

dynamical matrix for KCl to obtain the dispersion curves. This method

was applied to study the lattice dynamics of several other alkali halide

crystals with NaCl and CsCl structures [25,26].

Goyal et al [28] examined the close dependence of the effective

ionic charge upon the lattice vibration model. They derived expressions

for s corresponding to the Lundquist (LM), simple shell (SM) and three-

body shell (TSM) models. The values are given in table 2. They are of the

same order and differ only slightly from those calculated by Lowndes and

Martin [10] from Szigeti’s relation (eq. (10)). In recent years, Mahan [29]

67

has revived the deformation dipole model originally introduced by Hardy

in the 1960s. Mahan [29] made ab initio calculations of the

polarizabilities of alkali ions and used them to calculate the refractive

index and the effective ionic charge by employing improved

‘calculational technology’. Mahan derived the perturbation equations for

the deformation dipole tensor by the combined use of self-consistent

treatment of local density approximation and the spherical solid model.

The effective ionic charge s is calculated from

s = (1 − τ) (20)

and

τ = 2(γl + 2γt). (21)

Here γl and γt are the longitudinal and transverse components of the

tensor. Mahan [29] calculated the values of s for alkali halides excluding

the iodides. Recently, Michihiro et al [30] extended Mahan’s method to

the alkali iodides. The resulting s values are included in table 2. Mahan

concluded that the deformation dipole model can explain the origins of

the deviation of s from unity. The attempts described above have

introduced some sophistication into Szigeti’s theory and have provided a

deeper insight into the polarization mechanisms. However, as far as the

effective charge is concerned, the values given by the simple Szigeti

relation have remained essentially unaltered.

4.4. Aspects of the Szigeti charge

4.4.1 Values of s for various crystals

The most important aspect of Szigeti’s theory is the concept of the

effective ionic charge s. This is also called the Szigeti charge and is often

68

denoted by q*. Though Szigeti [2] had mainly the alkali halides in view,

he evaluated the value of s for

Several other cubic diatomic crystals like TlCl, MgO and ZnS,

triatomic crystals like CaF2 and anisotropic crystals like TiO2. In the same

spirit, the Szigeti charge has been evaluated by several workers for

crystals belonging to different structures. Mitra and Marshall [31] quoted

values for the fully covalent diamond-type crystals and to several

partially covalent crystals like ZnS. s values for several crystals are given

in table 3.

4.4.2 Szigeti charge and ionicity

It can be seen from tables 1 and 3 that, in general, s ≤ 1. Some

views regarding the factors that contribute to the difference (1 − s) have

been discussed in §2.4. Szigeti [2] himself suggested that one of the

69

factors could be that ‘the bond has a partial homopolar character’ but he

did not pursue this possibility in any of his later papers. However, from

time to time, the view has been expressed that the Szigeti charge is a

measure of the ionicity of the bond in the crystal [15,36,41,43]. There are

several measures of the ionicity. A commonly employed measure is the

Phillips ionicity fi [44]. Phillips plotted fi against an effective charge S

and obtained a linear plot. However, the effective charge S used by

Phillips has a different definition and, further, he considered only crystals

with the zinc blende structure. Gervais [39] examined the correlation of fi

and the Szigeti charge s by plotting s against exp(fi) for over 60 crystals

belonging to several structures. The plot (figure 3) was linear with data

points evenly scattered around the straight line. Thus, the Szigeti charge s

can be considered as a measure of ionicity of the bond.

4.4.3 Empirical relation between the Szigeti charge and physical

properties of crystals

Apart from being an ionicity parameter, the Szigeti charge s is now

considered an independent physical entity and correlations have been

proposed between s and several physical properties. Some of them are

discussed in this section. 3.3.1 s and interionic distance r: Ram Niwas et

al [45] plotted log s against log r and obtained linear plots (figure 4) for

each ion. They proposed the relation

s = krm, (22)

70

Where m is 0.5 for all plots and the constant k has a different value

for each ion. From the plots it is seen that k increases with the size of the

anion. Lowndes and Martin [10] found that a plot of s and the radius ratio

(r+/r−) for the alkali halides is a smooth curve (figure 5) with a separate

curve for each halogen (or alkali ion). Koh [43] obtained linear s vs.

(r−/r) plots (figure 6). Assuming equations of the type

s = a(r−/r) + b, (23)

Koh obtained values of a and b for each cation. s and

polarizabilities: Hanlon and Lawson [21] plotted s against the

polarizability difference (α+

0− α−

0) for the alkali halides and obtained a

smooth curve (figure 7) in the form of a hyperbola. Sirdeshmukh [46]

used this curve to estimate values of s for the lithium halides which were

not otherwise known at that time. On the other hand, Mitra and Marshall

[31] found that the s vs. (α+

0− α−

0) plot is linear (figure 8) for the II–VI

and III–V compounds with ZnS structure.3.3.3 s and thermal expansion:

Sirdeshmukh [46] proposed the following relationbetween s and the

linear coefficient of thermal expansion α:

71

αs2 = constant. (24)

This relation holds well in the case of the alkali halides with NaCl

structure. Earlier, Megaw [47] proposed the relation

αq2 = constant. (25)

Here q is the ‘electrostatic share’ given by the ratio of ionic charge

ze to the coordination number C of an ion. Thus eq. (25) becomes

(αz2e

2/C) = constant. (26)

72

This relation explains the difference in the thermal expansion of

crystals with different valencies. Sirdeshmukh [46] pointed out that, in

view of Szigeti’s theory, eq. (26) reduces to eq. (24).

s and the hyperfine coupling constant A: The hyperfine coupling

constant (hfc) A of a crystal doped with ions can be theoretically

estimated. It has been established that A is related to the degree of

covalency in the bonding in the host lattice. Motida [51] calculated the

hfc for a number of divalent crystals doped with Mn++

ions and found that

73

the plot of A and (1−s) (figure 11) was linear. A linear plot was also

obtained for (1 − s) and A of alkali halides doped with Cr+ ions.

4.4.4 Temperature variation of s

By differentiating s in eq. (10) with respect to the temperature, Kim

et al [54] obtained, z is assumed to be 1, i.e., the equation applies to alkali

halides. The results of the calculations for four alkali halides are shown in

figure 13. The characteristic features are seen in the case of LiF. At low

temperature (1/s)(ds/dT) is negative. With increasing temperature, the

value increases to become positive, passes a peak and becomes negative.

The derivative, (1/s)(ds/dT) may be written as

(1/s)(ds/dT) = (1/s)(∂s/∂T)V + (1/s)(∂s/∂V )T(3αV ). (27)

The first term is due only to change in temperature and the second

represents change due to thermal expansion. The net value of (1/s)(ds/dT)

is due to the result of the two competing contributions at different

temperatures.

4.4.5 Pressure variation of s

Attempts to estimate the pressure (or volume) variation of s were

made by Jones [55], Barron and Batana [56], Mitra and Namjoshi [15],

and Batana and Faour [57]. The starting point in all these works is the

differential form of eq. (10):

(V/s)(ds/dV ) = 0.5 − γt − V (∂ε∞/∂V )[(ε∞ + 2)−1

+ (1/2)(εs − ε∞)−1

]

+(1/2)V (εs − ε∞)−1

(∂εs/∂V ). (28)

74

The results obtained by them were provisional as they used

theoretical values for the mode Gruneisen parameter γt. Shanker et al

[58], on the other hand, calculated γt from the experimental data of

Lowndes and Rastogi [59] on temperature variation of the transverse

optical phonon frequency. Using these experimental γt values, they

calculated (V/s)(ds/dV ) from eq. (43). These values of (V/s)(ds/dV ) are

given in table 4. Shanker et al [58] also calculated (V/s)(ds/dV ) from

various dielectric models. These values are also given in table 4. The data

given in table 4 reveal the following features:

(i) The experimental as well as model-based values of (V/s)(ds/dV )

are positive without exception. This implies that the decrease in

crystal volume due to pressure increases overlap and distortion of

ions, thereby causing a decrease in the effective ionic charge.

(ii) The values of (V/s)(ds/dV) obtained from the shell model and the

Lawaetz model [53] are systematically smaller than experimental

values, almost by a factor of 1/2.

(iii) The values obtained from the exchange charge model and the

deformation dipole model are comparable with each other and with

the experimental values.

4.4.6 Szigeti charge of mixed crystals

The composition dependence of physical properties of mixed

crystals differs from property to property and from system to system.

Using experimental data on εs, ε∞, N and νt, the Szigeti charge s has been

calculated as a function of composition for a few mixed crystal systems

having NaCl structure. The conclusions from these studies the

composition dependence is linear in all cases. It may be noted that in the

KCl–KBr system, the composition dependence is linear when the

75

frequency data of Ferraro et al [65] is used while it is non-linear with the

data of Fertel and Perry [66]. Thus, the frequency data plays an important

role.

76

CHAPTER 5CHAPTER 5CHAPTER 5CHAPTER 5

FILTERING PROPERTIES OF MATERIALSFILTERING PROPERTIES OF MATERIALSFILTERING PROPERTIES OF MATERIALSFILTERING PROPERTIES OF MATERIALS

Surface plasmons and Surface Optical phonons may interact with

each other in polar semiconductors if their frequencies are of the same

order. Surface properties of different geometrical surface of materials can

be studied with the help of the dispersion relation which can be obtained

by various methods. The hydrodynamical model is one of the various

methods to study the behavior of polariton, phonon and plasmons on the

geometrical surface of materials for two mode coupling or three mode

coupling. A Filtring property of matrials is the most important property of

surfaces of materials.

5.1 Modified Bloch’s Hydrodynamical Model

The modified Bloch’s equation for the semiconductor [128] may be

written as-

_

_'_ _ _ _ _

( , )

0 '

1 ( )n r tD v d nm e E B m v

D t c n

ρ = − + ∇ × − ∇ − ∇ ∫

(5.1)

__ _ _4 1 D

B Jc c t

π ∂∇× = +

∂ (5.2)

_ _

.( )n

n vt

∂= −∇

∂ (5.3)

_ _ _ _4. ( ) ( , )

eE N r n r t

E

π+

∇ = −

(5.4)

Equation (5.1) is the Euler’s equation of motion. The operator D

Dt is co-

moving time derivative given by-

77

x y z

DV V V

Dt t x y z

∂ ∂ ∂ ∂= + + +

∂ ∂ ∂ ∂ (5.5)

Where yx VV , and zV are the components of velocity _

( , )V r t in x, y

and z directions respectively. Dv

Dt gives the acceleration of an electron in

the fluid and may be written with the help of equation (4.5) as-

_

( . )Dv v

v vDt t

∂= + ∇

∂ (5.6)

The first term on the right hand side of equations (5.1) is force

exerted on a free electron by the electromagnetic field which is produced

due to the density of fluctuations of the electron gas. The electric and

magnetic fields may be written in terms of scalar and vector potential as-

__ _ 1 AE

c t

∂= −∇Φ −

∂ (5.7)

_ _ _

B A= ∇× (5.8)

The second term of equation (5.1) is the restoring force due to the

short range interactions with the background i.e. the interaction of the

electrons with lattice vibrations. υ represents the collision frequency and

is assumed to be constant. The last term represents the force due to

pressure ‘P’ and an electron in the mass of the fluid. The pressure is taken

to be of Fermi type assuming that the electrons are totally free i.e.

52 2 2 3 3 5

3(3 )

( )5

nP n n

m

πξ= =

h (5.9)

And m5

)3( 3/222 πξ

h= (5.10)

78

With these substitutions eqn. (5.1) reduces to

_

( , )

0

1 ( ')( ) ( . )

'

n r t

vm

t

A e d ne v A m v V v m v V

c t c n

ρφ

∂

∂

∂= + ∇ − × ∇ × − − − ∇

∂ ∫

(5.11)

Where ‘m’ and ‘e’ represent the mass and charge of a free electron

and ),( trn is the instantaneous electronic concentration at a position r .

Equation (5.2) is the Maxwell’s equation for displacement current

for non-magnetic materials considered in the present study the magnetic

permeability ‘υ ’ is equal to unity i.e. BH = . Now substituting electric

and magnetic fields with the help of equation (5.7),(5.8) in equations

(5.2) we obtain.

22

2 2

4 eA nv

c t c t c

ε ε φ π ∂ ∂ ∇ − = ∇ +

∂ ∂ (5.12)

Whereε is the dielectric function of the medium.

Equation (5.3) is the continuity equation giving the current flow

due to the oscillations of free electrons. Equation (5.4) is the Poisson’s

equation for free electrons moving in the background of positive charges.

( ) ( )ii

i

N r z r rδ+ = −∑ . Where ir is the effective position of vector and iz is

the effective charge of the thi ion. Since this equation describes the

motion of free electrons only or independent plasma oscillations, the

dielectric equation used is approximated to the frequency independent

value ε with the help of equation (5.7) and choosing the gauge 0. =∇ A

equation (5.4) reduces to –

79

2 4( , ) ( )

en r t N r

πφ

ε+

∇ = − (5.13)

Now equation (5.3), (5.11) and (5.13) may be simplified by

introducing a scalar velocity potential according to –

eV A

mcψ−∇ = − (5.14)

Then equation of motion (5.11) becomes-

2

0

1 1 ( ')

2 '

n

t

e e e dP nA v A

mc m mc m n

ψ

ψ φ ψ

∂∇

∂

= ∇ ∇ − − ∇ + − ∇ + ∇

∫ (5.15)

By using the identity

( )2

1.

2

e eV V V

mc mcψ

∇ + ×∇× Α = ∇ ∇ − (5.16)

The high frequency solid state plasma is characterized by collision

less region vω >> , therefore, author can neglect the collision term, and

then equation (5.15) reduces to-

2

0

1 1 ( ')

2 '

ne e dp n

At mc m m n

ψψ φ

∂ = ∇ − − + ∂

∫ (5.17)

And equation (5.12), (5.3) and (5.13) respectively change to-

22

2 2

4 e eA n A

c t c t c mc

ε ε φ πψ

∂ ∂ ∇ − = ∇ + −∇ +

∂ ∂ (5.18)

. ( )n e

n At mc

ψ∂

= ∇ ∇ − ∂ (5.19)

80

and 2 4( , ) ( )

en r t N r

πψ

ε+

∇ = − (5.20)

Here author shall give only non relativistic treatment that is the

speed of electromagnetic propagation c has been taken to be much larger

than phase velocity v of surface wave. Therefore, for c>> v equations

(5.14) and (5.17) to (5.20) reduce to

v ψ= −∇ (5.21)

( )2

0

1 1 ( ')

2 '

ne dP n

t M m n

ψ φψ

∂= ∇ − +

∂ ∫ (5.22)

[ ]n

nt

ψ∂

= ∇ ∇∂

(5.23)

( )2 4 en N

πφ

ε+∇ = − (5.24)

Hence ( , )r tφ represents the self consistent electrostatic potential

due to the system of ions plus the electron gas, ( , )n r t is the instantaneous

electronic concentration and is a function of position and time.

There is no externally induced charge present in the system and the

free electron gas is homogeneous and capable of maintaining self

sustained oscillations then one can use the standard method for linearising

equations (5.22) to (5.24) by expanding φ,n and ψ as

2

0 1 2( , ) ( ) ( , ) ( , )n r t n r n r t n r tλ λ= + + + -----------+----------

2

0 1 2( , ) ( ) ( , ) ( , )r t r r t r tφ φ λφ λ φ= + + + -----------+----------- (5.25)

2

0 1 2( , ) ( ) ( , ) ( , )r t r r t r tψ ψ λψ λ ψ= + + + -----------+------

81

Where 0 1 2n n n>> >> so that only first perturbation is sufficient to

describe the system,λ is a parameter and is a measure of perturbation.

Substituting the expansions in equations (5.25) into equation (5.22)-

(5.24) and equating separately the coefficients of various powers of ‘λ ’

one finds- In zero order

5 / 3

00

1 ( ')

'

ne dP n

m m nφ ξ= ∫ (5.26)

After substituting P(n) from equation (5.18), then

2/3

0

5

2

en

m mφ ξ= (5.27)

and 2

0

4[ ]

en N

πφ

ε+∇ = − (5.28)

In first order

5/3

1

0

1 ( ')

'

ne dP n

t m m n

ψφ ξ

∂= +

∂ ∫ or 1/311 0

5

3

en n

t m m

ψφ ξ

∂= +

∂ (5.29)

10( )

T

nn

tψ

∂= ∇ ∇

∂ (5.30)

and 2

1 1

4 en

πφ

ε∇ = (5.31)

0n is the equilibrium electronic density and 1( , )n r t is a small

perturbation term and is responsible for charge density fluctuations i.e.

inside the semiconductor

0 0( )n r n= (5.32)

82

Equations (5.29) to (5.31) then modify for semiconductor to give-

1/311 1

5

3

en n

t m m

ψφ ξ

∂= − +

∂ (5.33)

)( 1

2

01 ψ∇=

∂

∂n

t

n (5.34)

11

2 4n

e

ε

πφ =∇ (5.35)

Now eliminating ‘ψ ’, equations (5.33) and (5.34) and with the

help of equation (5.35), The eq.(5.33)can be written as

22 2

120

3

Fp

vn

tω

∂+ − ∇ =

∂ (5.36)

Where pω is the bulk plasma frequency for polar semiconductor

and Fv is the Fermi velocity given by

1/ 2

2/3

0

5F

v nm

ξ =

(5.37)

Equation (5.36) is the differential equation for volume plasma

oscillations in a semiconductor. Time dependence of the density

fluctuations ),(1 trn is taken as–

.

1( , ) ik r i tn r t Ne

ω−= (5.38)

Where k is the wave vector; ω is the wave frequency and N is a constant.

Substituting this equation (5.35), it is obtained

2 22 2

1( , ) 03

Fp

v kn r tω ω

− + + =

or 2

2 2 2

3

Fp

vkω ω= + (5.39)

83

This is the dispersion relation for bulk plasma. It shows that the

plasma energies are shifted by a term 2k . If one takes into account the

finite speed of propagation of hydrodynamic disturbance in the electron

gas. For a metal as 1=ε , one has only to modify pω from the value

1/ 22

04 n e

m

π

ε

to

1/ 22

04 n e

m

π

The relation (5.39) differs from the bulk plasma dispersion relation

of Pines [129], obtained by using quantum mechanical treatment of

random phase approximations viz.

2 2 22 2 2

23 4

Fp

v h kk

mω ω= + + + − − − − − − − (5.40)

It has no higher order terms of wave vector k other than square

which confirm the point made out earlier that the hydrodynamic treatment

is only for small wave vector limit, so that higher order terms of k are

negligible.

Secondly, the hydrodynamic theory gives a factor 2

Fv , in the second

term of R.H.S. instead of (3/5)F

v obtained by the microscopic RPA

theory. These terms are represented by β which signify the average

speed of free electrons of a solid in acoustic and in optical region. This

shows that the hydrodynamic theory is valid only for two plasma

frequencies that is vω << in the characteristics of hydrodynamic

equations.

Now the difficulty arises when the author want to apply the

hydrodynamic equations to solid state electron gas for relatively high

plasma frequencies. Futter [130] has considered this point in his study of

84

layered plasmas and suggested that for a Fermi gas of D dimensions, the

factor 3 in the value of / 3F

vβ = should be replaced by ‘D’ and to use β

at high frequencies, it must be multiplied by [ ]3 /( 2)D D + . Since in our

case D=3, therefore the low frequencies value (3 / 5)F

vβ = .Moreover,

Klienman [131] has shown that the surface plasma dispersion equation

contain an effective 2β equal to 20.53 Fv , which is sufficiently close to

2 2(3 / 5) .6F Fv v= .

The difficulties regarding the applicability of hydrodynamics when

Fvω << have been discussed by Ginzburg [103]. It is pointed out that in

spite of accepted colloquial references to a hydrodynamic model as (5.2)

is a quasi hydrodynamics description with a form that is similar to a

hydrodynamic equation. It is permissible to use it either Fvω << or when

Fvω >> . If F

vω << ,then β is the familiar hydrodynamic speed of

acoustic (sound) wave, but when Fvω >> ,then β becomes the speed of

optical wave. Therefore in this region the term due to restoring force

equation (5.15) may be neglected safely.

Considering these factors Boardman et.al. [132] have described the

properties of plane metal interface, by using (3 / 5)F

vβ = and their values

agree well with those measured by electron loss spectroscopy [133, 134],

making the above modification our equation (5.39) and (5.36)

respectively become

2 2 2 2

pkω ω β= + (5.41)

And 2

2 2 2

12( , ) 0p n r t

tω β

∂+ − ∇ =

∂ (5.42)

85

Where

(3 / 5)F

vβ = Or 1/ 2

2

0

33

5n

mβ π

=

h

This is a function of electronic concentration.

The equation (5.42) is a differential equation for volume plasma

oscillations and may be used to derive the dispersion relation for volume

plasmons in a polar semiconductor. The dispersion relation for surface

plasma oscillations at the interface between the two media is obtained by

applying the boundary conditions which should be satisfied in the

interface.

5.2 Special Disperssion relation for two mode coupling for k ≠ 0

The surface of polar semiconductor particle supported T.M.

surface polariton waves. The Bloch’s Hydrodynamical model be

employed in the case of cylindrical interface by obtaining their solution in

cylindrical co-ordinates ( , , )r θ φ . These equations are as follows:-

( )2 2 2 2 . 0p

Eβ ω ω∇ + − ∇ = ( 5.43)

And ( )2 2 2 20L pc Eε ω εω∇ + − ∇× = (5.44)

The time and position dependence of the electromagnetic field

may be written as exp ( ).ik r i tω− .

Therefore

( )2 2 . 0Eγ∇ − ∇ = (5.45)

86

( )2 2 0Eγ∇ − ∇× = (5.46)

Where

2 2

2

pω ωγ

β

−= (5.47)

2

2

( ) ( )p L

k k

c

ε ω ω ε ωα

−= (5.48)

the solution of eqn (5.45) and eqn (5.46) can be written in terms of

scalar potential function γψ and αψ [96]

Eγ

γψ= ∇ (5.49)

1 ˆ( )E rrα

αψ= ∇ × (5.50)

( )2

1( )E E

α α

α= ∇× (5.51)

Where r is radius of cylinder and r the unit vector along rr

. The

scalar potential functions γψ and αψ can be expanded in terms of

cylindrical harmonics, as

.

,

( ) ( ) ( ).m ik r

l l

l m

r r eγ γψ ψ γ θ=∑ (5.52)

And .

,

( ) ( ).m ik r

l

l m

r Y eα αψ ψ θ=∑ (5.53)

where l, m are integer

2∇ the Laplacian operator for cylindrical harmonics is

87

2 2 22

2 2 2 2

1 1

r r z r rθ

∂ ∂ ∂ ∂∇ = + + +

∂ ∂ ∂ ∂ (5.54)

For the cylindrical polar semiconductor the equilibrium

electronic polar semiconductor function ( )rγψ must satisfy the following

condition

0 0( )rγψ ψ= for r<R (5.55)

0= for r>R (5.56)

Now using equation (5.49) and equation (5.54)

2. .( )E γ γψ ψ∇ = ∇ ∇ = ∇

2 2 2

2 2 2 2

1 1

r r r r Zγψ

θ

∂ ∂ ∂ ∂= + + +

∂ ∂ ∂ ∂ (5.57)

The derivative with respect to r is

( ) ( )m ikr

l llm

r Y er r

γ γψ ψ θ∂ ∂

= ∑∂ ∂

( ) ( ) ( ) ( ) ( ).

,

m ik r m ikr

l l rl llm l m

r Y ik e r Y eγψ θ ψ θ= ∑ + ∑

The second derivative with respect to r is

( ) ( ) ( ) ( ) ( )2

. '

2,

m ik r m ikr

l l rl llm l m

r Y ik e r Y er r

γ γψ ψ θ ψ θ∂ ∂ = ∑ +∑

∂ ∂

88

= ( ) ( )( ) ( ) ( ) ( )( )2 . '

,

m ik r m ikr

l l rl llm l m

r Y ik e r Y ik eγψ θ ψ θ∑ +∑

( ) ( )( ) ( ) ( )( )2' . ''

,

m ik r m ikr

l l rl llm l m

r Y ik e r Y ik eγψ θ ψ θ+∑ + ∑

But

2.E γψ∇ = ∇

( ) ( )2 2 2

2 2 2 2

1 1 m ikr

rl llm

r Y er r r r Z

ψ θθ

∂ ∂ ∂ ∂= + + + ∑

∂ ∂ ∂ ∂

( ) ( )( ) ( ) ( )( )2

.

2

1m ikr m ik r

l l l llm lm

r Y e r Y er r r

γ γψ θ ψ θ∂ ∂

= ∑ + ∑∂ ∂

+ ( ) ( )( ) ( ) ( )2 2

.

2 2 2

1 m ikr m ik r

l l l llm lm

r Y e r Y er z

γ γψ θ ψ θθ

∂ ∂∑ + ∑

∂ ∂ (5.58)

Since z is constant the last term becomes zero

' 2

' ''

'

. ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1( ) ( )( ) ( ) ( )

m m ikr

l l l l

lm lm

m m ikr

l l l l

lm lm

m ikr m ikr

l l l l

lm lm

E r y r y ik e

r y r y ik e

r y ik e r y er

γ γ

γ γ

γ γ

ψ θ ψ θ

ψ θ ψ θ

ψ θ ψ θ

∇ = +

+ +

+ +

∑ ∑

∑ ∑

∑ ∑

''

2

1( ) ( )( ) ikrm

l l

lm

r y ik er

γψ θ+ ∑ (5.59)

Neglecting the derivative of ( )rγψ and ( )m

lrγψ because they have very

small values Thus eqn. (5.59) can be written as

2 1. ( ) ( )( ) ( ) ( )

m ikr m ikr

l l l l

lm lm

E r Y k e r Y er

γ γψ θ ψ θ∇ = − +∑ ∑

89

2( ) ( )( )m ikr

l l

lm

r Y ik k eγψ θ= −∑ (5.60)

2 *( )ik k a− = then eqn.(5.60) becomes

* .. ( ) ( )m ik r

l l

lm

E a r Y eγψ θ∇ =∑ (5.61)

Now from eqn. (5.45), we get

( )2 2 .Eγ∇ − ∇ = ( )2 2 * .( ) ( ) 0m ik r

l l

lm

a r Y eγγ ψ θ∇ − =∑ (5.62)

Further for cylindrical geometry, the Laplacian operator is

2 2 22

2 2 2 2

1 1

r r r r Zθ

∂ ∂ ∂ ∂∇ = + + +

∂ ∂ ∂ ∂

( )2 2 * .

2 2 22 * .

2 2 2 2

( ) ( )

1 1. ( ) ( ) 0

m ik r

l l

lm

m ik r

l l

lm

a r Y e

a r Y er r r r Z

γ

γ

γ ψ θ

γ ψ θθ

∇ −

∂ ∂ ∂ ∂= + + + − =

∂ ∂ ∂ ∂

∑

∑

Taking z as constant for cylindrical geometry

( )2

2 2 * . * .

2

2* . 2 * .

2 2

1( ) ( ) . ( ) ( )

1( ) ( ) ( ) ( ) ( ) 0

m ik r m ik r

l l l l

lm lm

m ik r m ik r

l l l l

lm lm

a r Y e a r Y er r r

a r Y e a r Y er

γ γ

γ γ

γ ψ θ ψ θ

ψ θ γ ψ θθ

∂ ∂∇ − = +

∂ ∂

∂+ − =

∂

∑ ∑

∑ ∑

2 2* .

2 2

2 * .

1 1( ) ( )

( ) ( ) ( )

m ik r

l l

lm

m ik r

l l

lm

a r Y er r r r r r

a r Y e

γ

γ

ψ θ

γ ψ θ

∂ ∂ ∂ ∂+ +

∂ ∂ ∂ ∂

−

∑

∑

90

= -

2* .

2 2

1( ) ( )m ik r

l l

lm

a r Y er

γψ θθ

∂

∂∑ (5.63)

Let us suppose that

2

* .

2* . 2 * . 2

2

( ) ( )

1( ) ( ) ( ) ( ) ( )

m ik r

l l

lm

m ik r m ik r

l l l l

lm lm

r

a r Y e

a r Y e a r Y e lr r r

γ

γ γ

ψ θ

ψ θ γ ψ θ

×

∂ ∂+ − =

∂ ∂

∑

∑ ∑

Then

2* .

2

22 * .

2

2* .

2

1( ) ( )

( ) ( )

( ) ( )

m ik r

l l

lm

m ik r

l l

lm

m ik r

l l

lm

a r Y er r r

la r Y e

r

la r Y e

r

γ

γ

γ

ψ θ

γ ψ θ

ψ θ

∂ ∂+

∂ ∂

− +

=

∑

∑

∑

2* .

2

22 * .

2

1( ) ( )

( ) ( ) 0

m ik r

l l

lm

m ik r

l l

lm

a r Y er r r

la r Y e

r

γ

γ

ψ θ

γ ψ θ

∂ ∂+

∂ ∂

− + =

∑

∑ (5.64)

Now

* .

* . * ' .

( ) ( )

( ) ( )( ) ( ) ( )( )

m ik r

l l

lm

m ik r m ik r

l l l l

lm lm

a r Y er

a r Y ik e a r Y ik e

γ

γ γ

ψ θ

ψ θ ψ θ

∂ ∂

= +

∑

∑ ∑ (a)

And

91

2* . * 2 .

2

* ' . * ' .

* '' .

( ) ( ) ( ) ( )( )

( ) ( )( ) ( ) ( )

( ) ( )

m ik r m ik r

l l l l

lm lm

m ik r m ik r

l l l l

lm lm

m ik r

l l

lm

a r Y e a r Y ik er

a r Y ik e a r Y e

a r Y e

γ γ

γ γ

γ

ψ θ ψ θ

ψ θ ψ θ

ψ θ

∂ = ∂

+ +

+

∑ ∑

∑ ∑

∑

2* 2 .

22( ) ( )m ik r

l l l l

lm

a ik k Y er r

γ γ γψ ψ ψ θ ∂ ∂

= + − ∂ ∂

∑ (b)

Substituting the values of r

∂

∂ and

2

2r

∂

∂from (a) and (b) in eqn (5.64)

2* .

2

22 * .

2

1( ) ( )

( ) ( ) 0

m ik r

l l

lm

m ik r

l l

lm

a r Y er r r

la r Y e

r

γ

γ

ψ θ

γ ψ θ

∂ ∂+

∂ ∂

− + =

∑

∑

22 .

2

* .

22 * .

2

( ) 2( ) ( ) ( ) ( )

1( )( ) ( ) ( ) 0

( ) ( )

m ik r

l l l l

m ik r

l l l

lm

m ik r

l l

lm

r ik r k r Y er r

a r ik r Y er r

la r Y e

r

γ γ γ

γ γ

γ

ψ ψ ψ θ

ψ ψ θ

γ ψ θ

∂ ∂+ −

∂ ∂ ∂ + + =

∂ − +

∑

∑

(5.65)

For l mode only

92

22

2

.

22 *

2

( ) 2( ) ( ) ( )

1( )( ) ( ) ( ) 0

( )

l l l

m ik r t

l l

l

lm

r ik r k rr r

r ik r Y er r

la r

r

γ γ γ

ωγ γ

γ

ψ ψ ψ

ψ ψ θ

γ ψ

−

∂ ∂ + −

∂ ∂ ∂

+ + = ∂

− +

∑

(5.66)

Equating real and imaginary parts separately

2 22 2

2 2

1( ) ( ) ( ) 0l l

lr r k r

r r r rγ γ γψ ψ γ ψ

∂ ∂+ − + + =

∂ ∂ (5.67)

12( ) ( ) ( ) 0ik r r

r rγ γψ ψ

∂ + =

∂ (5.68)

Let us assume that 2 2 2k qγ + =

2 22

2 2

1( ) ( ) ( ) 0

lr r q r

r r r rγ γ γψ ψ ψ

∂ ∂+ − + =

∂ ∂ (5.69)

Eqn (5.43) is a modified form of Bessel’s differential equation. Its

solution is

( ) ( )l

r qrγψ = ΑΙ

where 222 kq += γ (5.70)

)(qrI γ is modified Bessel’s function for integer γ and depend on

)(qr Hence solution be

93

( )

,

( , ) ( ) ( )m i kr t

l l l

l m

r t qr Y e ωγψ θ −= ΑΙ∑ r<R

=0 r>R (5.71)

The solution for electrostatic potential function

),(4

),(2

tre

tr rll ψε

πφ

=∇ (5.72)

Let us assume that

)(

,

)()(),( tkri

ml

m

lll eYrtrωθφφ −∑= (5.73)

)(

,

)()(),( tkri

ml

m

lll eYqrtrωθψ −∑ΑΙ= (5.74)

Now

ΑΙ=∇ −− ∑∑ )(

,

)(

,

2)()(

4)()(

tkri

ml

m

ll

tkri

ml

m

ll eYqre

eYrωω θ

ε

πθφ

)(

,2

2

2

2

22

2

)()(11 tkri

ml

m

ll eYrzrrrr

ωθφθ

−∑

∂

∂+

∂

∂+

∂

∂+

∂

∂

( )

,

4( ) ( )m i kr t

l l

l m

eqr Y e

ωπθ

ε−

= ΑΙ ∑

After eqn. (5.63)

)(

2

2

22

2

)()(11 tkri

lm

m

ll eYrrrrr

ωθφθ

−∑

∂

∂+

∂

∂+

∂

∂

94

( )

1

,

( ) ( )m i kr t

l l

l m

qr Y e ωθ −= Α Ι∑ (5.75)

Taking z as constant

)(

,

1

)(

2

2

)()()()(1 tkri

ml

m

ll

tkri

lm

m

ll eYqreYrrrr

ωω θθφ −− ∑∑ ΙΑ−

∂

∂+

∂

∂

=)(

2

2

2)()(

1 tkri

lm

m

ll eYrrr

ωθφ −∑

∂

∂ (5.76)

2 2( )

1( ) 2

1( ) ( )

( )

i kr t

l li kr t

l

lm

rr qr e

r e r r r

ω

ωφ

φ−

−

∂ ∂+ − Α Ι

∂ ∂ ∑

2

2

1( )

( )

m

lm

l

YY r

θθ

∂=

∂ (5.77)

Again L.H.S. of eqn (5.77)) is a function of r only and R.H.S. is

function of θ only. Therefore, each side must be equal to the same

constant 2l

2

2( ) 22

( )

1

1( )

( )( )

l i kr t

i kr t

l

lm l

rre lr r r

r eqr

ω

ω

φ

φ−

−

∂ ∂+

=∂ ∂ −Α Ι

∑ (5.78)

)(

2

)(2

12

2)(

)()(1 tkri

tkri

lm

l

ll er

erl

qrrrrr

ω

ωφ

φ −

−∑=ΙΑ−

∂

∂+

∂

∂

(5.79)

or

)(

12

2

2

2

)()(1 tkri

ll eqrrr

l

rrr

ωφ −ΙΑ=

−

∂

∂+

∂

∂ (5.80)

95

On Simplifying, we get

2 22

2 2

( )

2( ) ( ) ( )

12( ) ( ) ( ) ( )

l l l

i kr t

l l l

lr r k r

r r r r

ik r r qr er r

ω

φ φ φ

φ φ −

∂ ∂+ − +

∂ ∂

∂ + + = ΑΙ

∂

(5.81)

Equating real and imaginary parts of the equation, we get

2 22

12 2

2( ) ( ) ( ) ( )

l l l l

lr r k r I qr

r r r rφ φ φ

∂ ∂+ − + = Α

∂ ∂ (5.82)

0)(1

)( =+∂

∂r

rr

rll φφ (5.83)

The solution of the equation (5.83) may be taken as

r

Crl =)(φ (5.84)

The LHS of equation (5.84) is independent of l. in other words the

solution of equation (5.82) can be written as

IPFCrl ..)( +=φ (5.85)

By solving the eqn, the C.F is obtained as

)()( qrCKqrBICF ll += (5.86)

Where )(qrI l and )(qrK l are modified cylindrical Bessel function of first

and second kind respectively. The value of these modified cylindrical

Bessel function may be calculated in terms of the following relation

96

2

12

)(+

=

ll I

xxI

π (5.87)

2

12

)(+

=

ll K

xxK

π (5.88)

Where )(xI l and )(xK l are cylindrical Bessel functions of first

and second kind respectively.

The P.I of the eqn (5.83) is given by

22

1 )(.

kq

qrIIP l

−

Α= (5.89)

Substituting the eqn (5.89) and (5.86) in eqn (5.85), we get the

general solution as

22

1 )()()()(

kq

qrIqrCKqrBIr l

lll−

Α++=φ (5.90)

Where B and C are constant and ε

π Α=Α

e41 and

222 kq += γ

Thus )(rlγψ represent the scalar potential function )(rlφ , and then we can

write

)()()(

)(22

1 qrCKqrBIkq

qrIr ll

l

l ++−

Α=γψ (5.91)

The boundary conditions are

RrleRrli === φφ (5.92)

97

And

Rr

le

Rr

lirr ==

∂

∂=

∂

∂φεφε 21 (5.93)

Α= 11 MB (5.94)

Α= 11 NC (5.95)

Where

−

−=

)()()()(

)()()()(1''

'

21kRKkrkIkrKkrI

kRKqrkIkrKqrIp

kM

llllB

llBll

εε

εε

γ (5.96)

+= )(

1)(

)(

1211 qrI

rkRIM

kRKN ll

l

(5.97)

This with the help of eqn (5.94) and eqn (5.95), equation (5.91) can be

written as

)()()()( krCKkrBIqrIr llll ++Α=γψ (5.98)

)()()()( 11 krKNkrIMqrIr llll Α+Α+Α=γψ

[ ])()()( 11 krKNkrIMqrI lll ++Α=

)( krX l γΑ= (5.99)

Where )()()()( 11 krKNkrIMqrIkrX llll ++=γ

Now from eqn (5.53), we get

98

ikrm

l

lm lm

l

ikrm

ll eYkrXeY )()()( θγθψψ γγ ∑ ∑Α== (5.100)

Now electric and magnetic field components may be obtained from

eqn (5.50), (5.51) and (5.52)

( ) ( )m

rl l lE X kr Y

r

γγψ γ θ

∂ = ∇ = Α ∂

or ( ) ( )m

rl l lE X kr Y

r

γ γ θ∂ = Α ∂

(5.101)

. ( ) ( )m

l l lE X kr Yr

γθ γ θ

θ

∂ ∂ = Α ∂ ∂

(5.102)

Also the magnetic components are related to the electric field as

1( )B E

iω= ∇ × (5.103)

From eqn (5.50) and eqn (5.101) and (5.102) it is clear that

0rlBγ = (5.104)

And 0l

Bγθ = (5.105)

Similarly eqn (5.52) along with eqn (5.51) yields the electric and

magnetic field components as

2

( ) 2

2 ( ) ( )m

rl l lE l Y kr Yα α θ= Α (5.106)

2

( ) 2 ( )( )ml

l l

r Y krE Y

i r

αθ

αθ

α θ

Α ∂=

∂ (5.107)

99

and

2

( ) 0rlBα = (5.108)

2

( ) 2 ( )

sin

ll

Y krB α

θ

α α

ω θ

Α= (5.109)

The function

[ ]2 2 2( ) ( ) ( ) ( )l l l l

r I r N K kr M I krαψ α= Α + + (5.110)

[ ]3 3 3( ) ( ) ( ) ( )l l l l

r h r N K kr M I krδψ δ= Α + + (5.111)

where

' '

2 ' '

( ) ( ) ( ) ( )1

( ) ( ) ( ) ( )

Bl l l l

B l l l l

p I pR K kR I pR K kRM

k I kR K kR I kR K kR

ε ε

ε ε

− = −

(5.112)

[ ]2 1 1

1( ) ( )

( )l l

l

N M I kR I p Rk kR

= + (5.113)

' '

2

2 ' ' '

( ) ( ) ( ) ( )1

( ) ( ) ( ) ( )

Bp l l l

B l l l l

p h R K kR h R K kRM

k I kR K kR I kR K kR

ε δ ε δ

ε ε

− = −

(5.114)

Where p kR= (5.115)

22

2Bc

ωδ ε= (5.116)

100

Where 2 2

1p kα= + and 2 2

2p kδ= + ( )lh Rδ is the

cylindrical Hanke function and given in equation (5.111) and Bε is the

dielectric of the bounding medium.

The field component in the region outside and inside will be

(5.117)

2( ) ( )m

l l

l

r r YE

i r

δδθ

ψ δ θ

δ θ=

∂ (5.118)

0rl

Bδ = (5.119)

2

( ) ( )

sin( )

m

l ll

r YB

δ δθ

ψ θ

ω θ θ

∂ ∂=

∂ (5.120)

Now the field component in polar semiconductor and the

bounding medium are known, the dispersion relation may be obtained by

applying the boundary conditions which are

P BE Eθ θ= at r=R (5.121)

r P BD D= at r=R (5.122)

P B

B Bθ θ= at r=R (5.123)

Equation (5.122) may also be written as

( ) ( )L r B rP B

k E kr Eε ω ε= at r=R (5.124)

2 ( ) ( )m

l lrl

l r YE

i

δ δψ θ

δ=

101

Where the subscript ‘P’ and ‘B’ denotes the polar semiconductor

and bounding medium respectively. ‘R’ represents the boundary of the

polar semiconductor cylinder and the bounding medium such that

r<R represents polar semiconducting medium

r>R represents the bounding medium

and r=R represents the interface of the media.

Now let us assume that from equation (5.110) and (5.111), we get

2( ) ( )rl lr y krψ α= Α (5.125)

And 3( ) ( )

l lr z krδψ α= Α (5.126)

Where 3 3( ) ( ) ( ) ( )l l l lz kR h R M K kR N I kRδ δ= + + (5.127)

And 2 2( ) ( ) ( ) ( )

l l l ly kR I R M K kR N I kRα α= + + (5.128)

Now applying boundary condition (5.121), (5.122), (5.123) and

(5.124) at interface r=R using eqn (5.101) to eqn (5.117) to eqn (5.120)

with the help of eqn (5.98) and eqn (5.126), we get

[ ] [ ]

2 ' 2

1 2 1 1 3

' '

2 31

2 2

2 3

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( )0

( ) ( ) ( ) ( )0

sin sin

m m m

l l l l B l l

m m ml ll l l l

m m

l l l

k l Y kRY R k X kRY k l z kRY

R Y kR R z kRY X kR Y Y

i R R R

Y kR Y i z kRY

ε ω α θ ε ω γ θ ε ω δ θ

α δθ γ θ θ

α θ θ δ θ

α α θ δ δ θ

ω θ ω θ

Α Α Α

Α Α∂ Α ∂ ∂=

∂ ∂ ∂

Α ∂ Α−

(5.129)

Reduced form will be

102

[ ] [ ]

2 ' 2

1 1

' '

2 2 2

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) 0

( ) 0 ( )

l l B l

l l l

l

k l Y kR R k X kR k l z kR

RY kR X kR Rz kR

i Y kR i z kR

ε ω α ε ω γ ε ω δ

α γ δ

α α δ δ

=

−

(5.130)

On solving the 3×3 determinant, we get

( )

[ ]( )

[ ]

2 2 2

1

2 2

'

1 ' 2

( ) ( ) ( ) ( ) 0

( ) ( )( ) ( )

( ) ( )

l l

l

l

l l

k l Y kR X kR i z kR

RY kR i z kRk X kR

Rz kR i Y kR

ε ω α γ δ δ

α δ δε ω γ

δ α α

− −

− − −

2 2( ) ( ) 0 ( ) ( ) 0

B l l lk l z kR X kR i Y kRε ω δ γ α α − − × = (5.131)

Where the prime (‘) denotes differentiation w.r.to θ, and , ,α γ δ are in

units of /P

c ω and given by

2 2

1( ) ( )k kα ε ω ε ω = − − Ω (5.132)

22

2

1γ

ξ

− Ω= (5.133)

And 2 2( )

Bkδ ε ω= − Ω (5.134)

Now the two mode coupling eqn

of SP-SOP electro state surface

modes by taking the limit ∝R→O, R → O.

103

So that eqn becomes ε εl Ω ω

ω ϵ∞Ωε ω

ω l εl 1 Ω ωωΩ !"#$γ%!"&$γ% 'εl

εl 1(Ω ωω )ε∞Ω ε ω

ω l εl 1 Ω ωω*Ω + 0(5.135)

Where use has again been made of recurrence relations given by

equation

I1(x) = - [Il-1(x) - Il+1(x) ] (5.136)

and I' (x) = [Il-1(x) + (l+1) Il+1(x)] (5.137)

The equation (5.135) become

.'εl 1 ϵ∞l('1 !"#$/%!"&$/%(0ωω1 .'εl 1ϵ∞I

εl ωω 0'1 !"#$/%!"&$/%(0 ω

ω .εl'1 !"#$/%!"&$/%( .'εl 1'1 !"#$/%!"&$/%(0 ω

ω + 0 (5.138)

The equation (5.138) is the dispersion relation for the SP-SoP

modes of the spastically dispersive medium [54]. optical phonon

frequency has easily be obtained for the neglecting geometry of the

interface. This is obtained by neglecting the contribution due to plasma in

eqn (5.138) i.e. by taking ωP=0, So that eq

n becomes-

ω2

SOP = ε3ε4ε∞ε4 ωt

2 (5.139)

104

The uncoupled surface plasma frequency may also be obtained

from eqn (5.101) by neglecting the contribution due to surface optical

phonon modes i.e. setting ωt=0 and ε∞, ε6666 so eqn (5.138) becomes

7 !"#$/%!"&$/% 18ω9ω 1 + 1 ε4ε

εε4 !"#$/%!"&$/% (5.140)

The dispersion relation (5.140) for coupled modes at the spherical

polar semiconductor surface matches exactly with the relation obtained

by Srivastava et. All [(59], plotted the reduced surface mode frequency

ωω . The SP modes decreased with increase in radius, whereas the SOP

mode remains constant, i.e. independent of frequency. For both the

coupled branches also the surface mode frequency decreases as radius

increases.

Now again the equation (5.138) has been considered after

simplifying and arranging the terms, we get

!"&$/%!"#$/%!"&$/% =

ε4ω.ε;ε40ω#."<ε4&ε=ω

or !"&$/%!"&$/%!"#$/% =

.ε;ε4ω0.ε>?ε4ωε4ω

since ω>>ωp, we put ε@=ε, then we get

!"&$/%!"&$/%!"#$/% = = .ε;ε4ω0.ε>ωε4ωε4ω

subtracting 1 on both sides, we get

.!"&$/%!"#$/%0!"&$/%!"#$/% = [AB$BCAABC ]

ωω -

B6BC

105

(l+1)B = [AB$ABCABC ]

DDE - B6BC

l(l+1) FGH + .IFG I 1F0 DDE - F [-F A) - FGH] + BCA B$A ω2

=0 (5.141 )

If KJ0, then from the Bessel's modified equation, we get

- A =

KL&$MNKL&$MNAKL&$MNAKL#$MN (5.142a)

and A =

OL&$KNOL#$KNAOL&$KNAOL#$KN (5.142b)

B= OL&$PNOL#$PNAOL&$PNOL#$KNAOL#$KN (5.143)

The equation (5.139) becomes

.F KL&$KNKL#$KNAKL&$KNAKL#$KN - FG

OL&$PNOL#$PNAOL&$PNOL#$KNAOL#$KN0QR

+ .FG OL&$KNOL#$KNAOL&$KNAOL#$KN- F

KL&$KNKL#$KNAKL&$KNAKL#$KN] Q=0 (5.144)

This equation (5.144) is required dispersion relation for

surface phonon and plasma coupled mode. thus it is clear that the

dispersion relation given by the author is the most general dispersion for

phonon, polar ton and Plasmon for polar semiconductor of cylindrical

interface for KJ 0. The frequencies of coupled SP-SOP modes by

substituting the dielectric function of semiconducting medium F in

eqn(5.144), we get F as

106

F = FS - FDED (5.145)

where FS = B∞TBUT

Y = DDV & Z = DEDV (5.146)

then from equation (5.144), we get

(FW FGH)QX + [FGY-B∞TBUT FZTA] ω

2 =0

or (FW FGH)DEDV + [FGY-B∞TBUT FZTA] DDV= 0

Then on simplifying by taking FG=1 as vacuum we get

Y[Y2-(az+b)Y+az] = 0 (5.147)

Where a = GB6[BB∞[ , b=

BB∞[\B∞[ (5.148)

only Y=0 , does not give any meaningful solution, then we get

y2 - (az+b)Y+az =0 (5.149)

The above equation is quadratic in Y and for each mode l gives two

roots for the given value of the constant a,b for Z.

l=1 mode l=2 mode

A = KUKNKKNKUKNKKN A =

K$KNK]KNK$KNK]KN B =

OU^NOKNOU^NOKN B = O$^NO]^NO$^NO]^N

107

C = OUKNOKNOUKNOKN C =

O$KNO]KNO$^NO]KN

where P = √` a

and ` is given by[68] equation(5.48)

` + DVb [(DEDV -BcBB∞B0/ (5.150)

QR + 1efcgB6h and i= jkl Vf = jkl m 3opq/k

the values of cylindrical Bessel function In (x) and Kn (x) for n = 0, 1, 2,

3 are

rqs = tufvww aqs =

tufvww

rs = xytvww -

tufvww as = xytvww -

tufvww

rs = rys - kwI1(x) as = ays -

kwK1(x)

rks = rs - lwI1(x) aks = as -

lwK1(x)

for the Insb cylindrical bounding by vaccum, we have

Fy= 17.70, F∞= 15.60, F = Fy F∞ = 16.65

ωt = 1.39x1012

sec-1

py = 2.0x1017/

CC(at room temp 300o K) and ωp = 6.18x10

12

then Z = [(DEDV0= (4.46)2 = 19.89, ` = 3.7 x 10

-2/A°

108

from eqn(5.149) We get

Y2=(az+b)Y-az (5.151)

Using the above data and the dispersion relation, we have

calculated the frequency of two coupled mode SP-SOP modes arise on

the surface of polar semiconductor as a result of frequency and wave -

vector dependence of the lattice dielectric surface function of polar

semiconductor. These coupled SP-SOP modes, a coupling with the

incident EM radiation of comparable frequency lead to the coupled

surface plasmon, polariton - phonon modes on the surface.Now the author

study the surface of different materials(polar semiconductors)for different

parameters by using the relation given by equation(5.149).

We plot graph between wsp/wt with propagation constant k for KF

substance in 3D.

Graph plotted for different radius ranges from 1A. to 16A

. as

shown in fig (5.1)

In this graph along one axis wsp/wt taken along other axis k taken

while variation of different radius taken along third axis.

We observe that graph between wsp/wt versus k is nearly linear.

When value of k increases from o to 9 than its linear variation

changes.

For lower value of k nearly linear variation with wsp/wt for radius

1A. & 2A

. .

When radius increases then variation of wsp/wt with respect to K

shown lesser linear variation.

Thus for higher value K it shows non linear variation. Hence for

higher value of K, it shows non linear agreement.

109

But at radius 8A. It shows again perfect linear variation for high

value of K.

Again linear variation between wsp/wt versus K disturb for radius

16A. for high value of K.

Variation of wsp/wt versus K for different radius observe that

variation is either perfectly linear or not perfectly linear.

When radius increases then it’s linear variation changes with

wsp/wt versus K, thus for high radius linear variation changes.

When radius increases then ratio of wsp/wt decreases. wsp/wt

inceases with increment in value of K. It mean for larger value of wave

vector K, more prominent wave passes through the substance KF in

epoxy resine medium.

Thus we conclude that with K Ratio of wsp/wt increases. It shows

linear variation for low radius but it deviates for high value of radius.

Conclusion of the graph as shown in fig (5.2) is given as :-

In this graph we use different medium in 3D having constant radius

0.5 A. for KF substance.

In this we take wsp/wt along one axis, wave vector K along another

axis and variation of different medium along different axis.

In epoxy resine wsp/wt increases linearly with K but for transoil it is

not linear variation.

All medium except transformer oil shows linear variation for KF

substance.

wsp/wt increases linearly with increase in value of K.

110

For same radius neoprene use a medium variation of wsp/wt initially

increases but as K=5, it changes abnormally again at K=7 it fallow

simillar path as in epoxy resine medium follow.

When we increase K more than 9 (K > 9) than it increases rapidly.

Except trans oil medium other medium like mica, quartz, Bee wax,

transformer oil and vaccum shows linear increment of wsp/wt with

K.

Thus wsp/wt increases with increment in value of K

For KF substance at .5A.radius. It is observed that trans oil

medium show different variation otherwise other seven medium

shows same kind of linear variation.

Ratio of wsp/wt increases for same value of K for when we consider

different medium [Epoxy resine, Neoprene, Mica, quarz, lice, Bee wax,

Transformer Oil & vaccum] i.e. minimum value for Epoxy resine and

maximum value for vaccum.

Thus we obverse that all medium shows same type of variation

only transoil shows different type of variation.

Conclusion of the graph as shows in fig (5.3) is given as :-

In this graph we use different medium in 3D having constant radius

.5A. for Mgo substance.

In this we take wsp/wt along one axis, wave vector K along another

axis and variation of different medium along different axis.

In this graph we take different medium like Epoxy Resine,

Neoprene, Mica, Quartz, Lice, Bee Wax, Transformer Oil &

Vaccum.

111

Ratio of wsp/wt increases when we move left to right for different

medium with constant value of K.

For same medium wsp/wt increases when K increases.

All medium like Epoxy Resine, Neoprene, Mica, Quartz, Lice, Bee

Wax, Transformer Oil & Vaccum share linear variation between

wsp/wt versus K. Thus for different medium variation are same.

For lice medium Ratio of wsp/wt increases with increase in K but

when K>8 then decreases rapidly.

Ratio of wsp/wt becomes constant at K = 9 or greater value for lice

medium.

Even at wave vector K = 0 all medium have same value of wsp/wt.

But when value of wave vector is higher (i.e. K>8) then for lice

medium its linear variation vanishes.

Thus for low value of K all medium shows linear variation while

for high value of K only lice shows different variation when we plot 3D

graph between wsp/wt versus K for different medium.

So we observe that all medium shows linear variation while lice

medium not shows linear variation.

Conclusion of the graph shows in fig (5.4) is given as below :-

We plot graph between wsp/wt with propagation constant K for

Mgo substance in 3D for Epoxy resine medium. Graph plotted for

different radius of .5A. to 16 A

. .

In this 3D graph we take wsp/wt along one axis K, along other axis

while variation of different radius taken along new axis.

We plot graph between wsp/wt versus K is exactly linear for

different radius for Mgo substances in Epoxy resine medium.

When value of K increases then wsp/wt increases Radius of cylinder

increases then wsp/wt also increases.

112

Ratio of wsp/wt is lowest at 5A. radius and wsp/wt is maximum are

in cylindrical shape increases then more space to pass waves. Thus

wsp/wt increases with increases in radius.

For Mgo substance in epoxy resine medium there is constent

variation, It is free from radius.

It has only single effect when radius is more then large amount of

wave pass trough it easily otherwise less amount of wave pass

trough it.

When K = 0 then there is no propagation of waves value of K

increases then wave propagation also increases.

wsp/wt has max value at 16A. means at this radius waves passes

easily through the cylindrical geometry. It cappers for highest value

of K = 10.

Thus for low value of K, wave propagation is not significant but

when radius and K both increases then significant wave passes through it.

In this graph at all radius wsp/wt verses K shows linear variation.

Conclusion of the graph as shown in fig(5. 5) is given as

We plot graph between wsp/wt with propagation constant k for InP

in different medium having radius of cylinder is 0.5 A..

In 3D graph we take wsp/wt along one axis K along other axis while

different medium along third dimensional.

For fixed value of K the ratio of the frequency of surface plasmon

to transverse medium is minimum in epoxy resin and very high in

vaccum medium.

wsp/wt versus K is almost linear increment in all medium except

lice medium.

Value of wsp/wt is high in vaccum medium and low in epoxy resine

but the range is nearly equal for same value of K.

113

When we plot graph wsp/wt versus K. Then InP substance for

cylindrical surface of 0.5 A. radius of different medium, then all

medium shows linear variation but lice shows different variation.

In lice medium wsp/wt increases but its value remains constant at

5.040028. So for lice, wsp/wt increases with increment in K but for

K = 7 or more then this, wsp/wt deceases rapidly.

Thus for InP at lone value of K all medium including lice shows

linear variation but for higher value of K, only lice shows different

variation also for high value of K.

Conclusion of the graph as shows in fig (5.6) is given as

We plot graph for InP at different radius of 1 A. & 2 A

. in epoxy

resine medium for cylindrical surface.

We plot graph between wsp/wt with propagation constant K for

InP in epoxy resine medium having different radius.

In 3D graph we take wsp/wt along one axis. Wave vector K along

other axis while different radius consider along third dimension.

The wsp/wt varies linearly with respect to K (1 A. - 10 A

.) for

cylindrical surface of InP at different radius.

At different radius InP medium shows same linear variation.

wsp/wt increases vertically when we move from top to bottom,

while this ratio is decreases when we move Horizontally from left

to right.

We observe that for InP in epoxy resine medium variation of wsp/wt is

linear with K at different radius such that 1 A. & 2 A

., Both shows linear

variation in the epoxy resine medium as in fig. no. 6.

Calculation

0.576112 0.458197

0.945957 0.814658

1.344748 1.268568

1.768659 1.733666

2.210058 2.195285

2.660146

3.112725 3.109517

3.565661 3.563975

4.018341 4.017393

4.470037 4.470037

0

1

2

3

4

5

1

Wsp

/Wt-

----

----

Graph between Wsp/Wt vrs K in 3D of different radius

0.458197

0.814658 0.758192 0.753698

1.268568 1.257262 1.256526

1.733666 1.730612 1.73042

2.195285 2.194229 2.194164

2.65354 2.653105 2.653078

3.109517 3.109312 3.109299

3.563975 3.563869 3.563863

4.017393 4.017334 4.01733

4.470037 4.470038 4.470036

Fig. 5.1

3 5 7 9

Graph between Wsp/Wt vrs K in 3D of different radius

of KF in Epoxy resine

114

0.753698 0.753417

1.256526 1.25648

1.73042 1.730408

2.194164 2.19416

2.653078 2.653076

3.109299 3.109299

3.563863 3.563862

4.01733 4.01733

4.470036 4.470036

Graph between Wsp/Wt vrs K in 3D of different radius

1A

2A

4A

8A

16A

Calculation

0.578501 0.581682

0.970722 0.972601

1.395954 1.397331

1.829093 1.830354

2.264339 2.265705

2.701287 2.702849

3.141099 3.142831

3.584298 3.586117

4.030378 4.032206

4.47844 4.480221

0

2

4

6

1 2

Wsp

/Wt-

----

--

Graph between Wsp/Wt vrs K in 3D of .5A radius of KF

0.581682 0.607543 0.614589 0.623532 0.626

0.972601 0.988118 0.99242 0.998 0.999

1.397331 1.408752 1.411933 1.416 1.417

1.830354 1.840826 1.843748 1.847 1.848

2.265705 2.277062 2.280231 2.284 2.285

2.702849 2.71583 2.719454 2.724 2.725

3.142831 3.157223 3.161241 3.166 3.166

3.586117 3.601244 3.605468 3.611 3.612

4.032206 4.047402 4.051647 4.057 4.058

4.480221 4.495039 4.49917 4.481 4.506

Fig. 5.2

3 45

67

89

1011

K----------

Graph between Wsp/Wt vrs K in 3D of .5A radius of KF

in different medium

115

0.626 0.63 0.64

0.999 1 1

1.417 1.42 1.425

1.848 1.85. 1.86

2.285 2.285 2.293

2.725 2.725 2.734

3.166 3.168 3.178

3.612 3.612 3.622

4.058 4.58 4.069

4.506 4.5.5 4.515

Graph between Wsp/Wt vrs K in 3D of .5A radius of KF

epoxy resine

Neoprene

Mica

Quartz

Lice

Bee Wax

Transformer

Vacuum

Calculation

0.74 0.7373 0.7636

1.2145 1.2164 1.2324

1.7398 1.7413

2.2769 2.2782 2.2891

2.8178 2.8192 2.8309

3.3612 3.3628 3.3762

3.908 3.9098 3.9247

4.4586 4.4605 4.4761

5.0124 5.0143

5.5684 5.5703 5.5856

0

5

10

1 2 3

Wsp

/Wt-

----

-

Graph between Wsp/Wt vrs K in 3D of .5A radius of

0.7636 0.7708 0.7799 0.7819 0.7869

1.2324 1.2368 1.2425 1.2437 1.2468

1.753 1.7163 1.7605 1.7615 1.7638

2.2891 2.2921 2.2959 2.2968 2.2989

2.8309 2.8342 2.8384 2.8393 2.8416

3.3762 3.3799 3.3848 3.3858 3.3885

3.9247 3.9288 3.9341 3.9353 3.9383

4.4761 4.4805 4.4861 4.4874 4.4905

5.03 5.0343 5.04 5.0413 5.0444

5.5856 5.5898 5.5966 5.5996

Fig. 5.3

4 5 6 78

910K----------

Graph between Wsp/Wt vrs K in 3D of .5A radius of

MgO in different medium

116

0.7869 0.7989

1.2468 1.2544

1.7638 1.7694

2.2989 2.3041

2.8416 2.8473

3.3885 3.3949

3.9383 3.9454

4.4905 4.4979

5.0444 5.0519

5.5996 5.7069

Graph between Wsp/Wt vrs K in 3D of .5A radius of

Epoxy resen

Neoprene

Mica

Quartz

Lice

Wax

Trasformer oil

Vacuume

Calculation

0.798947 0.731606

1.254347 1.192797

1.769384 1.690038

2.304098 2.214776

2.847247 2.761879

3.394872 3.318774

3.94537 3.878741

4.497908 4.439395

5.051902 4.999995

5.606916 5.560347

0

5

10

12

Wsp

/Wt-

----

---

Graph between Wsp/Wt vrs K in 3D of different radius

0.731606 0.615327 0.320792 0.248437

1.192797 1.056934 1.001527 0.997161

1.690038 1.609697 1.598266 1.597522

2.214776 2.178956 2.175835 2.17564

2.761879 2.746706 2.745623 2.745556

3.318774 3.311976 3.311528 3.3115

3.878741 3.875435 3.875224 3.875211

4.439395 4.437656 4.437547 4.43754

4.999995 4.999017 4.998956 4.998953

5.560347 5.559765 5.559729 5.559727

Fig. 5.4

23

45

67

89

10K----------

Graph between Wsp/Wt vrs K in 3D of different radius

of MgO in Epoxy resine

117

0.248437

0.997161 0.996888

1.597522 1.597475

2.17564 2.175628

2.745556 2.745552

3.3115 3.311498

3.875211 3.87521

4.43754 4.437539

4.998953 4.998952

5.559727 5.559727

Graph between Wsp/Wt vrs K in 3D of different radius

.5A

1A

2A

4A

8A

16A

Calculation

0.734022 0.737243 0.763559

1.214465 1.216394 1.23235

1.739829 1.741248 1.753026

2.276939 2.278239 2.289049

2.817777 2.819187 2.830913

3.361179 3.362791 3.376195

3.908005 3.909793 3.924656

4.458626 4.460505 4.476127

5.012422 5.014309 5.030007

5.568406 5.570246 5.585554

1 23

4

Wsp

/Wt-

----

----

Graph between Wsp/Wt vrs K in 3D of .5A radius of InP

0.763559 0.770767 0.779936 0.781959

1.23235 1.23678 1.242454 1.243711

1.753026 1.75631 1.760523 1.761458

2.289049 2.292067 2.295941 2.296801

2.830913 2.834187 2.838392 2.839325

3.376195 3.379938 3.384746 3.385813

3.924656 3.928807 3.934139 3.935322

4.476127 4.480492 4.486097 4.487342

5.030007 5.034394 5.040028 5.041279

5.585554 5.589833 5.596551

Fig. 5.5

56

78

9

10

K----------

Graph between Wsp/Wt vrs K in 3D of .5A radius of InP

in different medium

118

0.781959 0.786993 0.798947

1.243711 1.246849 1.254347

1.761458 1.763793 1.769384

2.296801 2.298949 2.304098

2.839325 2.841657 2.847247

3.385813 3.38848 3.394872

3.935322 3.93828 3.94537

4.487342 4.490452 4.497908

5.041279 5.044406 5.051902

5.596551 5.599601 5.606916

Graph between Wsp/Wt vrs K in 3D of .5A radius of InP

epoxy

neopren

mica

quartz

lice

beewax

trance oil

vacuume

Calculation

0.731606 0.615327

1.192797 1.056934

1.690038 1.609697

2.214776 2.178956

2.761879 2.746706

3.318774 3.311976

3.878741 3.875435

4.439395 4.437656

4.999995 4.999017

5.560347 5.559765

0

5

10

1 2

Wsp

/Wt-

----

----

Graph between Wsp/Wt vrs K in 3D of different radius

0.615327 0.320792 0.248437

1.056934 1.001527 0.997161

1.609697 1.598266 1.597522

2.178956 2.175835 2.17564

2.746706 2.745623 2.745556

3.311976 3.311528 3.3115

3.875435 3.875224 3.875211

4.437656 4.437547 4.43754

4.999017 4.998956 4.998953

5.559765 5.559729 5.559727

Fig. 5.6

34

56

78

910

K------------------

Graph between Wsp/Wt vrs K in 3D of different radius

of InP in Epoxy resine

119

0.248437 0.242957

0.997161 0.996888

1.597522 1.597475

2.17564 2.175628

2.745556 2.745552

3.3115 3.311498

3.875211 3.87521

4.43754 4.437539

4.998953 4.998952

5.559727 5.559727

Graph between Wsp/Wt vrs K in 3D of different radius

1A

2A

4A

120

5.3 Special dispersion relation three for mode coupling

The dispersion relation for three mode coupling can be obtained

from (5.131) by substituting the frequency and wave vector dependent

form of FSQz given by the equation (5.48) for polar semiconductor. this

substitution leads to

R' (γkR) |B∞KDΩBUKDVEΩVE ~

FzQ F∞zQΩ FqzQ QQRΩ QQR

Ω

.z′`z FGzQΩ`z′z0 I 1szz.FzQFGzQ + 0 (5.152)

So RX1 (zFGzQΩ FyzQ DDRFzQ Ω DVDE F∞zQΩ FzQ DVDEΩ `z. z′ Ω DVDEΩFGzQ`z′z0 Ω DVDE II 1FzQFGzQz`z`z. `z + 0

(5.153)

The equation (5.153) is the required dispersion relation for the

surface Plasmon phonon - polaritan coupled modes in the case of spatial

dispersion relation polar semi-conductor sphere for KJ 0 embeded in

abounding. Non dispersive dielectric medium for dielectric constant FG

(kω). None we will take the case when k→o then from eqn(5.39) become

as

121

Yrl ( + WArA(pr)

=A A(

Because Ml =0

Nl = 0

Then z → I1 () (5.154)

from eqn (5.81) Y1 (`z → rαR (5.155)

from eq (5.156)we get k → 0

Z1(a →h1() (5.156)

And F (kω) → F (W) = F (5.157)

FG (kω) → FG (W) = F (5.158)

and M2 = M3 = N2 = N3 = 0 (5.159)

After putting the values of z, Y1(`zpZ1(zfrom

equation (5.154), (5.155), and (5.156) respectively in eq (5.153) we get

RI’()(F∞Ω Fy DVDEF Ω DDR F∞Ω Fy DVDEΩ m′ FGΩ Ω DVDEr` II 1FFG Ω DVDE rh + 0

(5.160)

n (5.160) is the dispersion relation for spherical surface Plasmon and

polariton coupled models for polar semiconductors in the case of K = 0

[150] for K → ∞.

122

The dispersion relation for surface plasma, polaritons and phonons

including spatial dispersion effect as given.

RX1(γkr)|B∞MDΩBcMDVEΩVE ~ .'FzQ |B∞MDΩB∞MDVE

ΩVE ~

ΩNMN( z`a FGzQΩ<z′z II 1z`zzFGzQFzQ + 0

(5.161)

Further study of the dispersion relation for surface Plasmon,

phonon, polaritan for non-spatially dispersion case, from eq (5.161) by

taking limit → ∞ and F → 0, then eqn (5.161) become in new form

Ω DEDV _-F∞[Fy 'F 1 F∞a(DEDV0Ω1 '1 Fy F DEDV* z F0Ω Fz + 0 (5.162)

For InAs F + 1.49, F∞ + 12.3, F + 13.6Q + 4.1 10k, ,QR + 4.77 10k,

Then eqn(5.162) becomes into new form

16.65Ω 33.3 18zΩ1 34.3z 13.6Ω

13.6z + 0 (5.163)

The eqn (5.163) gives three values of ΩΩpΩk for different

values of K.

then eqn(5.162) can also be written as

w = DDV and a> =

\ V

123

Then F∞W6-[F(

E V) - Fy+(1+F∞K12] W

4 + [(1+Fy K1

2+

F E V0W2- F E V K1

2 = 0 (5.164)

Now putting values we get

12.3W6- (14.9+13.3K1

2) W

4+(34.33K1

2+18.4)W

2 - 18.4K1

2 = 0 (5.165)

Now equation (5.165) gives three values of Q, Q, &Qk, for

different value of a.

Explanation of graph between w and k for Cscl & InP

We observe from the graph that the frequencies w1, w2 and w3 vary

with respect to propagation constant K. At high value of K, w1 & w2 are

always constant but the third coupling frequency w3 always increases with

respect to K.

Thus we observe that coupling frequency varies with K for cylindrical

surface of Cscl. When coupling frequency is low than with K. It shows

contant variation [as shows in fig. 5.7] but for high value of coupling

frequency. It always increases with K.

If we take InP substance to study the surface behavior. It is

observed that as propagation constant K increases there is no variation in

w1 and w2 but w3 varies very slowly till K = 20 but sharply above K

=20(as shown in fig,5.8).

This shows that frequency of polariton depends upon the

propagation constant K but phonon and plasmon frequency does not vary

against propagation constant K in the case of cylindrical surface of cscl.

Calculation

10

20

30

40

50

0

50

100

w--

----

--

10

w1 0.6501

w2 1.1607

w3 10.6681

0.6501 1.1607

0.6486 1.1575

0.6483 1.03818

0.6482 1.03798

0.8481 1.03789

Fig. 5.7

w1

w2

w3

1020

3040

50

20 30 40

0.6486 0.6483 0.6482

1.1575 1.03818 1.03798 1.03789

10.6681 23.3565 33.034 45.932 57.6918

Graph between W and K of CsCl

124

10.6681

23.3565

33.034

45.932

57.6918

w1

w2

w3

50

0.8481

1.03789

57.6918

Calculation

10

20

30

40

50

0

100

w--

----

-

10

w1 0.90419

w2 1.06

w3 10.1384

0.90419 1.06

2.858 1.05215

0.903706 1.05092

0.9036 1.0505

0.903667 1.0503

Fig. 5.8

w1

1020

3040

50

20 30 40

0.90419 2.858 0.903706 0.9036

1.05215 1.05092 1.0505

10.1384 20.7643 31.5085 38.2408

Graph between K and w for InP

125

10.1384

20.7643

31.5085

38.2408

50.8616

w2w3

50

0.903667

1.0503

50.8616

126

5.4 Local Theory approximation

The system of Coordinates such that the polar semiconductor -

dielectric Interface is situated at z=0. The Z¢0 space is filled by polar

semiconductor (medium-1) and z<0 space is filled by the dielectric

medium (medium-2) which is taken as vacuum.

Considering the local approximation i.e. kl<<1, where K is wave

vector and l is the electronic mean free path, the dielectric function for

the polar semiconductor (medium-1) is given by FQ

The dielectric function at frequency ω is

F£ + ¤DBc¥ =1+¦ωBc¥ P= -

§¨©ωE

So that FQ + 1 §¨Bc©ª

Let plasma frequency ωp is defined as

ωp2 =

§¨Bch

Thus FQ=1-ωω (5.166)

By consideration of local theory approximation. It is

FSQ=1 for metal

FSQ + F∞ For Non polar crystal

Now Dielectric function for polar semiconductor

FQ=FSQ - ωω (5.167)

127

FSQis dielectric constant of Lattice vibration. The coupling of

the electric field E of the photon with the dielectric polarisation P of the

transverse optical phonon is described by Electromagnetic wave eqn

C2k

2E=ω

2 (E+4πP)

or (ω2-C

2k

2) E + 4πω

2P = 0 (5.168)

At Small wave vectors the transverse optical phonon frequency ωt

is independent of k. the polarisation is proportional to the displacement of

the positive ions relative to the negative ions. So eqn of motion of the

polarization is like of an oscillator and may be written as

«¬­ E + (ω2-ωt

2) P = 0 (5.169)

When we neglect the electric contribution to the polarization for

simplicity, then from (5.168) & (5.169) eqn we get

®®ω ck4πω«¬© ω ω

®® = 0 (5.170)

At k=0 there are two roots ω=0 for photon and ω2 = ωt

2 +

1𫬭

for polarization the dielectric function obtained from eqn (5.168)

FQ= F∞ + 1π«¬/­ωω (5.171)

By definition of F∞ as the optical dielectric constant obtained as the

square of the optical refractive index.

At w=0 the static dielectric function

128

Fy = F∞ + 1𫬭ω

Which is Combined with (5.171) to obtain in terms of

FSQ + F∞+ .FS F∞0 ωωω

By LST Relation B∞BU =

ωω;

So F∞ω@ = Fyω

Then FS(ω) = F∞+ B∞

ωω .ω@ ω0 = F∞ 71 ω;ω

ωω 8 FS(ω)=

B∞D±B∞Dωω

or FS(ω)= B∞ωBcωωω (5.172)

Fq² Static dielectric function

F∞ + High limit dielectric function

Which gives the Background dielectric function of the lattice. Fy and F∞ correspond to low frequency and high frequency dielectric

constant respectively. In case of metals FSω + 1 and in the case of non

polar crystals FSω + F∞. This shows that the lattice dielectric function

is constant for metals and non polar semi-conductors but is frequency

dependent for polar semiconductors.

129

5.5 Dielectric function on the surface materials

A plasma is a medium with equal concentration of positive charges

and negative charges, in a negative charges of the conduction electrons

are balanced by an equal concentration of positive of the ion cores. If the

positive ion core background has a dielectric constant.

FQ= F (∞) - 1𫨩ω = F (∞)71 ω

ω8 (5.173)

Where QR= 1π«¨ε∞©

So ε =0 at ω=ωp

In a nonmagnetic isotropic medium the electromagnetic wave eqn is

³´³ = c2µ2

E

Now solution E∝exp(-iwt) exp(i.k6.r)

and D> = F E(ω, k) E> , then we have the dispersion relation for

electromagnetic waves

εω, k= ¸¬ω

This relation gives conclusion

¹º»¼½¼¾¿ ¢ À: for ω+

real, K is real and transverse EMW

propagate with the phase velocity Vp= Â√ε

¹º»¼½¼¾¿ à À: for ω- real, K is imaginary and the wave is

damped with a characteristic length |Å|

130

¹Complex :- for ω real, K is Complex and the waves are

damped in space.

¹ +O :- only longitudinally polarized waves are

possible only at the Zeros of ε.

¹ + ∞ ∶ System has infinite response in the absence of

an applied force, thus the poles of εω, K

define the frequency of the free oscillations of

the medium.

The dielectric function εω, Kof the electron gas, with is strong

dependence on frequency and wave vectors, has signifies consequences

for the physical properties of solids. In first limit εω, O describes the

collective excitations of the Fermi electron gas for surface Plasmon's and

phonons. In second limit εO, K describes the electrostatic screening of

the electron-electron, electron-phonon and electron impurity interaction

in crystals.

So εω, O=1- ωÉω and εO, K = 1+Ê9Å

Where VS is screening potential but k→ O does not approach the

same limit as Q → O. Thus great care must be taken with the dielectric

function near the origin of the ω-k plane.

The Total dielectric function, lattice Plus electrons, but without the

electronic polarizability of the ion cores, is

¹Ë,Ì=1- ËÍÎËÎ + ÏÐÎÌÎ (5.174)

131

Above eqn

is valid when we consider both frequency and

propagation constant k.

5.6 Expression for szigetti effective charge for kl¢¢1

ε@(ω) is given as from equation (5.172)

ε@(ω) = ε∞ω

εUωωω

Dielectric function is frequency depended for polar semiconductors-

ε(W)= ε∞ 71 Ωωω8 - ωÉ

ω (5.175)

Where Ω = 1π«¨9∗­ε∞ 7ε∞k 8 (5.176)

N= number of ions per unit cell

M= reduced mass of the ion

e∗Ó = szigeti effective charges

Let eqn (5.172) can also be written ass

ε@(ω) = ε∞ ωωεÔ ω

ω (5.177)

Using eqn (5.175) & (5.77) in eq

4(5.167) we get

ε∞ 71 Ωωω8=

ε∞ ωωεÔ ωω

using eqn(5.176) in above

132

ε∞ 1 ωω 1Ω«¨9∗­ε∞ ε∞k =

ε∞ ωωεÔ ωω

on simplifying

1 ωω ε∞+

1Ω«ε∞¨9∗kω­ =7ε ε∞ ωω8 (5.178)

eqn (5.178)

given relation between eÓ∗and ω

ω.

So ε∞+1Ω«ε∞¨9∗kω­ = ε

ε - ε∞= 1Ω«ε∞¨9∗kω­

eÓ∗ = 7εÔε∞1Ω 8 7­«8 7 kωε∞8

Or eÓ∗= 7εÔε∞1Ω 8 Õ .MVØ0 Õ 7 kωε∞8 (5.179)

Va = « , Va is volume of one unit cell. By eq

n(5.179) Szigeti

effective charge calculated.

5.7 Band attenuation properties of materials without magnetic field

and with magnetic field

(a) Filtering properties without magnetics field [Ù»ÚÛ=o]

Let dielectric function for metal is given by equations (5.166) and

equation (5.167)

ε(ω) = 1- ωÉω

133

None dielectric for polar semiconductor is as

ε(ω) = ε@(ω) - ωÉω

where ε@(ω) is given by equation (5.172)

ε@(ω) = ε∞ ωωε3

ωω

So eqn (5.167) become FQ =

ε∞ ωωεÔ ωω ω

ω ωω

Where εÜ, ε∞ are low frequency dielectric constants and high

frequency dielectric constant respectively. the values of ωÉω chosen

corresponds to strongest coupling between surface plasman [SP] and

surface optical phonon [SOP]. We can plot graph of ε(ω) versus ωÉω . Form

graph we observed that there exit two SP-SOP models. These modes do

not propagate when ε(ω) is negative.

i.e ωω <

ωω

but propagate when ε(ω) is positive.

i.e ω&ω Ã ω

ω <1.

(b) Effect of Dc magnetic field

In presence of dc magnetic field, the dielectric function not remains

scalar but becomes a tensor with non-Zero off diagonal element given by

134

F= ε@ δij - ε ωÉ

ωωωÝ .ω2δij - ωciωuj + iδijkωck] (5.180)

where ω¦= 1π§3¨ε©∗

and ωc = ¨>©∗

ωp and ωc are plasma frequency and cyclotron frequency. m* is

effective mass of electric in solids. ε@ is background dielectric constant of

the polar semiconducting medium.

δij =Kronecker delta function

it's value δij = 1, i = j

= 0, iJj

and δijk = third rank antisymmetric tensor

Its value δijk = +1 If i, j,k is an even permutation

= - 1 If i, j,k is an odd permutation

= o otherwise

Due to magnetic effect the crystal becomes anisotropic i.e.

dielectric function is different for different directions. Therefore, the

filtering properties of the substance vary with the direction of incident

beam with respect to the magnetic field. If we take the magnetic field in a

particulars direction out to be qualitatively similar to eqn (5.180). Hence

the filtering properties are also similar.

Calculation

0.5

1

1.5

2

2.5

3

0

10

20

0.5

w/w

t---

----

0.5

w1/wt 0.244323

w2/wt 1.2659

w3/wt 2.307404

Graph between K and w/wt of NaF tube in DC Magnetic field

0.244323 1.2659

0.854315 1.41424

0.999795 1.602071

0.999788 1.609127

0.834796 1.6123

0.834796 1.414214

Fig. 5.9

11.5

22.5

3

0.5 1 1.5 2 2.5 3

0.244323 0.854315 0.999795 0.999788 0.834796 0.834796

1.2659 1.41424 1.6020712771.609127 1.6123 1.414214

2.307404 4.182 5.0349 5.308 6.4668 10.48595

Graph between K and w/wt of NaF tube in DC Magnetic field

135

2.307404

4.182

5.0349

5.308

6.4668

10.48595

3

0.834796

1.414214

10.48595

136

Calculation

0.1 0.070065 0.9 37.81063

0.15 0.23472 0.95 79.73133

0.2 0.470575 1 00.0000

0.25 0.783538 1.1 -43.5334

0.3 1.182149 1.2 -21.9022

0.35 1.678527 1.3 -14.0337

0.4 2.289935 1.4 -9.54597

0.45 3.041383 1.5 -6.36034

0.5 3.970033 1.6 -3.78135

0.55 5.131269 1.7 -1.49663

0.6 6.57654 1.8 0.428607

0.65 6.443071 1.9 2.537843

0.7 -2.89487 2 4.527956

0.75 10.54477 2.1 6.509414

0.8 15.96352 2.2 8.50802

0.85 23.50687 2.3 10.53935

Fig. 5.10

-60

-40

-20

0

20

40

60

80

100

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.2 1.4 1.6 1.8 2 2.2

ev

X2

----

---

X----------

Graph between evX2and x for InP

evX2

137

The variation of w/wt versus propagation constant K in presence of

DC magnetic filed for NaF tube shown in fig (5.9). It shows the

dispersion curves for the coupled SP and SoP waves in NaF cylindrical

tube radius 100A.. Thus dispersion curves shows that the frequency of the

upper mode (Ώ3) varies slowly with the wave rector K. when K is more

than 2 it increases rapidly.

Lower mode (Ώ1) varies very slowly and remains almost constant

when value of wave vector is equal to or greater than unit. But Ώ2

mediator mode lies in between upper and lower mode is exactly constant

for all value of wave vector (K). So there exist a band gap between the

two modes showing that the coupled modes with frequencies lying in this

region can not be excited.

Theoretical study of InP from graph between εvX2 and as X shown

in fig (5.10).

The author observes that there are four range of EM wave for

which cylindrical surface behaves as filtering property of InP cylinder.

These ranges are :-

i. For w/wt = X = 0.1 to 0.69 the surface of InP behaves as filter of

EM waves because εvX2 because positive and increases slowly and

becomes maximum at X = 0.65, εvX2 =6.443071 now after X =

0.65, εvX2 decrease and thus the filtering property of InP decreases.

ii. For X = 0.69 to 0.72 the surface of InP stops to pass EM waves and

holds penetrating property.

iii. For X = 0.72 to 1.00 value of εvX2 increases rapidly so that surface

of InP allows EM waves to pass with high frequency range because

at this range εvX2becomes positive. The frequency X = 0.95, high

EM wave passes as at this frequency range εvX2 = 79.73133. This is

138

third order filtering property of InP for EM waves. For 0.95 to 1.00

EM wave decreases.

iv. For range X = 1.00 to 1.79 the surfaces of InP does not allow to

pass EM waves because at this range X, εvX2 becomes negative and

at X = 1.1, εvX2

= - 43.5334.

For X = 1.79 and above it becomes transparent for EM waves

because εvX2

becomes positive. Thus it allows to pass EM waves. The

cylindrical surface of InP used as switch on and switch off property for

electronic signal in electronic communication in science & technology

5.8 comparative study of filtering properties

The comparative study of cylindrical surface of substances like KF,

InP &Mgo Polar semiconductors at different radius. When radius of

cylinder increases then Propagates more wave through it. Only KF,

substance show non linear variation, at Lower value of k and at higher

value of radius it becomes zero. This comparative Study of different polar

semiconducting compounds is useful in communication, Travelling wave

tubes(TWT) backward wave oscillator(BWO).

Conclusion of the graph as shown fig (5.11)

In this graph we are comparing variation of wsp/wt with wave rector

K for different substances KF, InP & Mgo.

wsp/wt in all substances increases from top to bottom when K

increases.

wsp/wt is lower for KF but in other two substances InP & Mgo

wsp/wt increases rapidly [slightly greater value].

When we plot comparative graph for cylindrical surface of

different substance like KF, InP & Mgo with respect to wave

139

vector K at same radius 1 A., then all substances have same linear

variation.

This dispersion curves shows that the frequency of the lower mode

changes slowly and frequency of the upper mode varies rapidly

with the wave vector K.

Thus wsp/wt verses K is almost linear increment of substances KF, InP &

Mgo at same radius cylindrical surface (1 A.) these behaves like same

nature.

Conclusion of the graph as shown in fig. no. (5.12)

In this graph we are comparing variation of wsp/wt versus K for

different substances KF, InP & Mgo.

When we plot comparative graph for cylindrical surface of

different substances at equal radius 2A.. then all substances have

same linear variation.

wsp/wt increases with increment in value of K.

For KF variation of wsp/wt with K varies with lesser effect where

as for other two substances InP Mgo show more effective linear

variation. At this radius (2 A.) both InP & Mgo having nearly same

value of wsp/wt in comparison to radius (1 A.).

Conclusion of the graph as shows fig No. (5.13)

In this graph we are comparing variation of wsp/wt with wave

vector K for same substances KF, InP & Mgo but his time at higher

radius (16 A.) of cylindrical surface.

At this radius KF substances has lower value of wsp/wt even starts

from Zero. At this radius cylindrical surface has lower value.

For KF substance it has lower effect, it means lesser wave

propagate through the surface.

140

At this radius (16 A.) both InP & Mgo have same value of wsp/wt so

both have same type (kind) and equal linear variation.

Thus for these comparative study we observe that when radius of

cylinder increases then KF has lower linear variation and InP & Mgo

have greater liner variation.

The frequency of surface plasmon has very small variation in KF

compared to InP & Mgo with respect to radius and propagation constant

(K). The reason is that KF is not a good conductor and it behaves like

polar semiconductor. Thus study is important in nanotechnology and

electronic communication.

But at higher electronic concentrations the frequency of this mode

become almost constant. It is clear that at the higher concentrations, the

lower mode become like pure SP mode and upper mode becomes like

pure sop mode. The coupling between SP and sop mode is stronger when

wsp = wsop i.e. where two uncoupled modes intersect. The stronger

coupling is observed at wsp/wt = 1.

Thus wsp and wt both are equal then coupling is stronger.

Calculation

0.576112 0.731606 0.798947

0.945957 1.192797 1.254347

1.344748 1.690038 1.769384

1.768659 2.214776 2.304098

2.210058 2.761879 2.847247

2.660146 3.318774 3.394872

3.112725 3.878741 3.94537

3.565661 4.439395 4.497908

4.018341 4.999995 5.051902

4.470037 5.560347 5.606916

KF Inp MgO

0

5

10

1Wsp

/Wt-

----

--

1 2

KF 0.5761 0.9459

InP 0.7316 1.1928

MgO 0.7989 1.2543

Comparative study of 1A radius cylidrical substances

0

5

10

1 2

Ax

is T

itle

1 2

Series1 0.5761 0.9459

Series2 0.7989 1.2543

Series3 0.7316 1.1928

Comparative study of 2A radius cylidrical substances

Fig. 5.11

Fig. 5.12

2 3 4 5 6 7 8 9 10

3 4 5 6 7 8

0.9459 1.3447 1.7686 2.2100 2.6601 3.1127 3.5656

1.1928 1.6900 2.2147 2.7618 3.3187 3.8787 4.4393

1.2543 1.7693 2.3041 2.8472 3.3948 3.9453 4.4979

Comparative study of 1A radius cylidrical substances

2 3 4 5 6 7 8 9 10

2 3 4 5 6 7 8

0.9459 1.3447 1.7686 2.2100 2.6601 3.1127 3.5656

1.2543 1.7693 2.3041 2.8472 3.3948 3.9453 4.4979

1.1928 1.6900 2.2147 2.7618 3.3187 3.8787 4.4393

Comparative study of 2A radius cylidrical substances

141

KF

9 10

3.5656 4.0183 4.4700

4.4393 5 5.5603

4.4979 5.0519 5.6069

Comparative study of 1A radius cylidrical substances

Series1

9 10

3.5656 4.0183 4.4700

4.4979 5.0519 5.6069

4.4393 5 5.5603

Comparative study of 2A radius cylidrical substances

Calculation

0.753417

1.25648

1.730408

2.19416

2.653076

3.109299

3.563862

4.01733

4.470036

KF

0

2

4

6

1 2

Wsp

/wt-

----

-

1 2

KF 0 0.7534

InP 0.2429 0.9968

MgO 0.2429 0.9968

Comparative study of 16A radius cylidrical substances

0.242957

0.753417 0.996888

1.25648 1.597475

1.730408 2.175628

2.19416 2.745552

2.653076 3.311498

3.109299 3.87521

3.563862 4.437539

4.01733 4.998952

4.470036 5.559727

Inp MgO

Fig. 5.13

2 3 4 5 6 7 8 9 10

3 4 5 6 7 8

0.7534 1.2564 1.7304 2.1941 2.6530 3.1093 3.5638

0.9968 1.5974 2.1756 2.7455 3.3115 3.8752 4.4375

0.9968 1.5974 2.1756 2.7455 3.3115 3.8752 4.4375

Comparative study of 16A radius cylidrical substances

142

0.242957

0.996888

1.597475

2.175628

2.745552

3.311498

3.87521

4.437539

4.998952

5.559727

KF

10

9 10

3.5638 4.0173 4.4700

4.4375 4.9989 5.5597

4.4375 4.9989 5.5597

Comparative study of 16A radius cylidrical substances

143

5.9 Variation of the width of allowed band of polarized

materials with Szigeti effective charge

The dielectric function can be written as by eqn (5.180)

ε(ω) = ε∞ Þ ωωεÔ ωω ß

ωÉωà

ωωÕ (5.181)

After putting values of constant for different polar semiconductors

in above eqn. We observe that there exist two coupling modes (SP&SOP).

Here ωω and ω

ω are roots of equation ε(ω) = 0

for this we get

ωäω

= ε3ε∞

ε∞

ωω ä 1 ε3

ε∞

ε∞ωÉ

ω ε∞

ωÉω /

(5.182)

Thus the surface acts as band pass filter [BPF]. Agian their is no

propagation when ε(ω) is negative.

for ω

ω ¢ ω#ω , the surface become transparent again act as High pass

filter (HPF). from graph we observe band width (∆) of band pass filter is

given by

∆¦æ+ 1 ω&ω (5.183)

thus the allowed band ∆ will also be differed for different

conounds.

Similarly ∆ç¦æ+ 1 ω#ω (5.184)

144

gives values at which the surface will act as high pass filter. from

study of Szigeti effective charge eÓ∗ versus ∆ with the help of data in

table. from this the best fit of data we find the following relationship

between eÓ∗ and ∆ as

∆ =0.074+0.099¨9∗¨ (5.185)

The values of ∆ can be estimated for different compounds with the

help of graphs. from these values ¨9∗¨ can be calculated by using equation.

from table our calculated values agree well with the experimental values

of Hass and Henvis.

We can conclude that polar semiconductor surface acts as a band

pass filter for ω- <ω<ωt and as high Pass Filter for Qw¢ω+. The es2 is

measure of iconicity of polar semiconductors. thus width of band (∆)

increases with increase in ionicity. If there is more ionic character, band

of allowed frequency is wider and surface becomes high pass filter at

higher value of frequency ω.

When we plot graph between band gap (∆) and ¨9∗¨ , we obtain we

that various polar semiconductor shows linear variation(as shown in fig

5.14) . different polar semiconductors like Mgo , Agcl , Kcl , Cscl , KF ,

NaF , & RbF value of szigetti effective charge increases from mgo to cscl

respectively . Let value of band gap (∆) also increases from Mgo to Cscl.

From the graph we observe that the band width (∆) of the band

pass filter is given by 1-è&è depends on єy, єê, QR and Q. thus, the width

of allowed band (∆) will be different for different compounds. similarly

145

è#è , which gives the value at which the surface will act as high on the

szigelti effective charge.

The simultaneous existence of surface plasmons [SP] and surface

optical phonons[sop] leads to a strong coupling between them . These

coupled modes do not propagate when the dielectric function is negative

so surface waves exist. When the dielectric function is positive .The

surface waves do not exist and wave can propagate through the matter

.when dielectric function of positive values the surface acts as a band pass

filter[BPF] as well a high pass filter[HPF].The band width has been found

to be different for different compounds. In the presence of a dc magnetic

field the number of allowed band increases .It gives useful information

about the application of semiconductor in filter circuits as the range of

allowed frequencies by variation in the compounds.

146

Calculation

Compound e*/e Band Gap ∆

MgO 0.6 0.1334

AgCl 0.69 0.1423

KCl 0.83 0.1562

CsCl 0.84 0.1572

KF 0.91 0.1641

NaF 0.93 0.1661

RbF 0.95 0.1681

Fig. 5.14

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 0.2 0.4 0.6 0.8 1

Ba

nd

ga

p∆

----

--

e*/e---------

Graph between Bandgap∆ and e*/e for different compounds

147

5.10 Advantage of szigetti effective charge study and its application

Crystals with the triatomic and cubic structure, these properties

indicate that they are essentially ionic. Szigetti calculated the effective

ionic charge es for the alkaline earth materials using eqn (10). In this

equation the reduce mass M for this system as

(1/M) = 1/m1 + 1 / 2 m2

Where m1 is the mass of ion with higher valency and m2 is the

mass of ion with lesser valency. The value of t∗ obtained by szigetti [2]

for CaF2 type crystals. These value of es are of the same order as but

systematically lower than those for the highly ionic alkali halides.

The long wave lattice dynamics of the fluorite lattice was studies

by Axe [73]. For he obtained the same relation [(eqn) as obtained by

szigeti [3] except that the definition of the reduced mass used by Axe

[73]. The value of es obtained by Axe and his coworkers for several

crystals with caF2 structure. The es values for tow oxides with CaF2

structure are distinctly low and indicate ‘Increased covalency’ [75].

Szegeti’s [2,3] theory is by far the most comprehensive theory of

dielectric. It is applicable to diatomic as well as polyatomic crystals to

ionic as well as covalent crystal. Though the original theory assumed a

harmonic cystal szigetti [4,5] modified it to make it applicable to

anharmonic crystal. The most important contribution of szigeti’s theory is

the concept of the eefiective ionic charge, it has found applications, the

most important being its utility as an ionicity parameters.

Several theoretical models have been proposed to account for the

deviation of the effective ionic change from unity. These models have

148

provided an insight in the complex polarization mechanisms in solids.

Some new results have been discussed. There are : -

i. Szigeti’s second relation relating the compressibility to the

absorption frequency. It is useful as a consistency check on

compressibility date.

ii. Axe’s modification [75] of szigett’s theory theory has lead to a

method to estimate the mode Gruneisen parameter of fluorite type

crystals.

iii. Systematic calculation of the anharmonic correction for the alkali

halides with Nacl structure has revealed that only LiBr has a

negative correction.

iv. Two equations for the effective ionic charge by Havinga have been

systematically applied to a number of alkali halides.

In order to study the effect of szigeti effective charge on optical

parameters of coherent Raman scattered mode it shows behavior of

absorption coefficient α with excitation intensity Io. We have plotted

excitation intensity in absence and presence of szigeti effective charge es.

It may notice that increase in excitation intensity (I) decreases and

saturates at high excitation intensity. This is quite obvious because

increase in the transmitted power should decreases absorption.

In the absence of szigeti effective charge the results are well in

agreement with Dubey & Ghosh [24]. It infers that weakly polar

semiconductor plasma can be a potential candidate nonlinear medium for

the fabrication of cubic non linear optical devices as compared to highly

polar semiconductor plasmas.

149

it shows behavior of refractive index with excitation intensity Io. In

this figure it shows variation of excitation intensity in absence and

presence of szigetti effective charge.

An analytical investigation of non linear absorption coefficient and

refractive index of Raman Scattered stokes mode resulting from the non

linear interaction of an intense pumping light beam with molecular

vibrations of polar semiconductor [like InSb].

The szigeti effective charge contributes that Raman susceptibility

at moderate excitation intensity. At high excitation intensity the

contribution of szigetti effective charge is wiped off and non linearity in

the medium is only due to differential polarizability.

In the absence of szigetti effective charge the magnitude of Raman

susceptibility is found to agree with the other theoretical models. The

analysis establishes that a small absorption coefficient and large

refractive index can easily obtained under moderate excitation intensity in

weakly polar semiconductors, crystals which proves its potential as

candidate material for fabrication of cubic non linear devices.

150

CHAPTER 6CHAPTER 6CHAPTER 6CHAPTER 6

SUMMARY AND CONCLUSIONSSUMMARY AND CONCLUSIONSSUMMARY AND CONCLUSIONSSUMMARY AND CONCLUSIONS

Surface Polariton waves are the electromagnetic couple modes of

the Surface Elementary excitations and Photons. These Coupled modes

propagate in a wave like manner along the interface of two medial, but

decay in a non-oscillatory exponential manner in a direction

perpendicular to the interface. In order to study the Polariton waves

sustained by Polar Semiconductor Surfaces, the interface is considered to

be formed by Polar Semiconductor bounded by a non dispersive dielectric

medium can be taken to vaccum. The dielectric medium of Polar

semiconductor sustains Surface Elementary excitation is termed as the

surface active medium, whereas the bounding medium is termed as the

surface inactive medium. The Properties of the Surface Polariton waves

are thus determined by the type of elementary excitations that are

sustained by the Surface active medium.

The predominent elementary excitations sustained by polar

semiconductor surfaces are the Surface Plasmons and Surface Phonons.

In this regard, Polar semiconductor differ from metals and non polar

compounds which do not sustain Surface Phonon modes.

Surface Phonon arises in the case of Polar Semiconductor as result

of frequency dependence of lattice dielectric function ε(w) which is

constant in the case of metals and non-polar Compounds. The Surface

Optical Phonon (SOP) waves in Polar Semiconductors have frequencies

so that their Simultaneous existance at the Surface may lead to the

interaction between SP-SOP coupled modes. The Surface Polariton

waves that may exist at the Surface of Polar Semiconductor are Surface

Plasmon-Polariton, Surface-Phonon-Polariton and Surface Plasmon-

151

Phonon-Polariton for k≠o. The effect of Spatial dispersion on the

Properties of Surface Polariton wave have also been studied by

comparing the result obtained when Spatial dispersion is taken into

account, with those obtained in the local limit i.e. when spatial dispersion

is neglected.

If electromagnetic radiation of appropriate frequency is incident on

the surface of metal or semiconductor then surface plasmons may couple

with photons giving rise to surface plasmons- polariton modes. In the

case of polar semiconductor (i.e. compounds like Inas, Insb, Mgo, Inp

etc) the dielectric function is frequency dependent and for certain

frequency ranges they can sustain surface optical phonons along with

plasmons.

When the dipole moment density and also the displacement field

depends not only on the value of electric field at a particular point but

also on the value assumed by the electric field in the near vicinity of that

point, then material is said to be exhibit “spatial dispersion”. The

inclusion of spatial dispersion in the study of surface polaritons leads to

important conclusion like the finite life time of the surface modes even

collision less plasma system. The properties of surface polaritons can be

studied by the help of their dispersion relation. The non local dielectric

function is both frequency and wave vector dependent although in local

limit it becomes wave vector independent and depends only on

frequency. The non local dielectric function can be calculated either by

using the quantum mechanical or classical approach. The dispersion

relation for surface plasmon, surface optical phonon & surface polariton,

all modes has been obtained and an attempt to study the effect of spatial

dispersion on surface plasmon surface optical phonon modes of polar

semiconductor.

152

These modes indicate the type of elementary excitations that give

rise to the polariton waves. The incident electromagnetic radiation can

couple to more than one type of elementary excitations simultaneously if

their frequencies are comparable. These waves arise as a result of

coupling of the EM radiation and those elementary surface excitations of

the medium that may couple on linear manner to the incident EM field by

virtue of their electric of magnetic character. The predominant surface

excitations in the case of dielectric media are surface plasmons, surface

phonon, surface excitons etc. The fact that surface EM waves at a metal

surface involves the coupling of EM radiation .with surface plasmons was

first explain by Stern . Also surface of optical lattice vibrations that exist

at the surface of ionic and polar materials have a great tendency to couple

with electromagnetic radiation. These vibrations lead to surface phonon,

polariton modes . In case of polar semiconductors, surface Plasmon

modes and surface Phonon modes have comparable frequencies so that

there is a possibility of interaction between the two leading modes i.e.

surface plasmon & surface polariton modes.

The properties of surface polariton waves are dependent strongly

on the nature and geometry of the interface and the bounding medium.

The phase velocity of these waves is less than the velocity of light in

vaccum and the parallel component of wave vector for these waves is

greater than that of low wave vector, so that for given frequency of the

incident radiation, the condition for the conservation of energy and

momentum cannot be simultaneously satisfied at the interface. As a

result, it is not possible to excite surface polariton modes by directly

irradiating smooth surfaces. Special techniques like attenuated total

reflection method (ATR) and periodic grating method are employed to

study these modes experimentally.

153

The above mentioned techniques have been successfully employed

by several workers to study surface polariton modes in metal,

semiconductors and polar semiconductors for different interface

geometries. The theoretical study of these modes has also recently

received great attention. The properties of the surface modes can be

studied by deriving their effect on the reflection and refraction of incident

EM waves plays a fundamental role in the Physics of surfaces.

The basic requirement for derivation of the dispersion relation is

the dielectric function of the dispersion relation is the dielectric function

of the dispersive dielectric media that sustain these modes. The frequency

dependent dielectric function for metals and semiconductors are well

known and have been recently employed by several workers to study

these waves. The most widely used effective methods that have been

employed for obtaining the dispersion relation, are the Maxwell’s

equation method and the quantum mechanical Random Phase

Approximation [RPA]. Spatial dispersion effects play an important role in

the study of surface plasmon, polariton modes. Although its effect on

long wavelength has not been observed. The inclusion of spatial

dispersion relation in the study leads to important conclusions like finite

life time, time of surface polaritons even in collision less system, and

radius dependence of surface modes in the case of finite geometries of

interface.

In order to study the interaction of surface plasmons with surface

phonons, one has to select a polar semiconducting material, since in these

materials the surface plasmons and surface phonons exist simultaneously

at the surface. Srivastava and Tondon for the first time modified the

Bloch’s hydrodynamical equations for metals to study the interaction

154

between these two modes in se miconductor. The dispersion for surface

Plasmon and polariton is given as below

( )

[ ]( )

[ ]

2 2 2

1

2 2

'

1 ' 2

( ) ( ) ( ) ( ) 0

( ) ( )( ) ( )

( ) ( )

l l

l

l

l l

k l Y kR X kR i z kR

RY kR i z kRk X kR

Rz kR i Y kR

ε ω α γ δ δ

α δ δε ω γ

δ α α

− −

− − −

2 2( ) ( ) 0 ( ) ( ) 0B l l l

k l z kR X kR i Y kRε ω δ γ α α − − × = (6.1)

The solution of the dispersion relation will be complicated even

when the dielectric function of the medium is real as the solution will

contain the modified Bessel’s function terms. This implies that the

surface polariton modes are radiative in the case of cylindrical surface.

The case of cylindrical surface differs markedly from the plane interface

case where both radiative and non-radiative modes exist. The surface

polariton in the cylindrical geometry thus have finite lifetime due to

radioactive decay even in a collision less system. The surface polariton

modes can therefore couple to the electromagnetic waves in the

surrounding medium and as a consequence the surface modes of

cylindrical particles can easily be studied experimentally by absorption

and scattering of electromagnetic radiation.

For the Insb cylindrical bounding by vaccum, we have

= 17.70, ∞= 15.60, = ∞ = 16.65

ωt = 1.39x1012

sec-1

n= 2.0x1017/

CC(at room temp 300o K) and Wp = 6.18x10

12

155

Z = (= (4.46)

2 = 19.89, = 3.7 x 10

-2/A

from eqn(6.1) We get

Y2 = (az+b)Y-az (6.2)

Using the above data and the dispersion relation, we have

calculated the frequency of two coupled mode SP-SOP modes arise on

the surface of polar semiconductor as a result of frequency and wave -

vector dependence of the lattice dielectric surface function of polar

semiconductor. These coupled SP-SOP modes, a coupling with the

incident EM radiation of comparable frequency lead to the coupled

surface plasmon, polariton - phonon modes on the surface. Now the

author study the surface of different materials(polar semiconductors)for

different parameters by using the relation given by equation(6.2).

Author plot graph between wsp/wt with propagation constant k for

KFsubstance in 3D.

Graph plotted for different radius ranges from 1A. to 16A

. as

shown in fig(5.1)

In this graph along one axis wsp/wt taken along other axis k taken

while variation of different radius taken along third axis.

We observe that graph between wsp/wt versus k is nearly linear.

When value of k increases from o to 9 than its linear variation

changes.

For lower value of k nearly linear variation with wsp/wt for radius

1A. & 2A

.

When radius increases then variation of wsp/wt with respect to K

shown lesser linear variation.

156

Thus for higher value K it shows non linear variation. Hence for

higher value of K, it shows non linear agreement.

Again linear variation between wsp/wt versus K disturb for radius

16A. for high value of K.

Variation of wsp/wt versus K for different radius observe that

variation is either perfectly linear or not perfectly linear.

When radius increases then it’s linear variation changes with

wsp/wt versus K, thus for high radius linear variation changes.

When radius increases,ratio of wsp/wt decreases. wsp/wt inceases

with increment in value of K. It mean for larger value of wave vector K,

more prominent wave passes through the substance KF in epoxy resine

medium.

Thus we conclude that with K Ratio of wsp/wt increases. It shows

linear variation for low radius but it deviates for high value of radius.

The author observed that

In this graph we use different medium in 3D having constant radius

0.5 A. for KF substance.

In this we take wsp/wt along one axis, wave vector K along another

axis and variation of different medium along different axis.

In epoxy resine wsp/wt increases linearly with K but for transoil it is

not linear variation.

All medium except transformer oil shows linear variation for KF

substance.

wsp/wt increases linearly with increase in value of K.

157

For same radius neoprene use a medium variation of wsp/wt initially

increases but as K=5, it changes abnormally again at K=7 it fallow

similar path as in epoxy resine medium follow.

When we increase K more than 9 (K > 9) than it increases rapidly.

Except trans oil medium other medium like mica, quartz, Bee wax,

transformer oil and vacuum shows linear increment of wsp/wt with

K.

Thus wsp/wt increases with increment in value of K

For KF substance at .5A.radius. It is observed that trans oil

medium show different variation otherwise other seven medium

shows same kind of linear variation.

Ratio of wsp/wt increases for same value of K for when we consider

different medium [Epoxy resine, Neoprene, Mica, quarz, lice, Bee wax,

Transformer Oil & vaccum] i.e. minimum value for Epoxy resine and

maximum value for vaccum.

Thus we obverse that all medium shows same type of variation

only transoil shows different type of variation.

The author again observed that

In this graph we use different medium in 3D having constant radius

.5A. for Mgo substance.

In this we take wsp/wt along one axis, wave vector K along another

axis and variation of different medium along different axis.

In this graph we take different medium like Epoxy Resine,

Neoprene, Mica, Quartz, Lice, Bee Wax, Transformer Oil &

Vaccum.

158

Ratio of wsp/wt increases when we move left to right for different

medium with constant value of K.

For same medium wsp/wt increases when K increases.

All medium like Epoxy Resine, Neoprene, Mica, Quartz, Lice, Bee

Wax, Transformer Oil & Vaccum share linear variation between

wsp/wt versus K. Thus for different medium variation are same.

For lice medium Ratio of wsp/wt increases with increase in K but

when K>8 then decreases rapidly.

Ratio of wsp/wt becomes constant at K = 9 or greater value for lice

medium.

Even at wave vector K = 0 all medium have same value of wsp/wt.

But when value of wave vector is higher (i.e. K>8) then for lice

medium its linear variation vanishes.

Thus for low value of K all medium shows linear variation while

for high value of K only lice shows different variation when we plot 3D

graph between wsp/wt versus K for different medium.

So we observe that all medium shows linear variation while lice

medium not shows linear variation.

The other observe for epoxy resine that

We plot graph between wsp/wt with propagation constant K for

Mgo substance in 3D for Epoxy resine medium. Graph plotted for

different radius of .5A. to 16 A

. .

In this 3D graph we take wsp/wt along one axis K, along other axis

while variation of different radius taken along new axis.

We plot graph between wsp/wt versus K is exactly linear for

different radius for Mgo substances in Epoxy resine medium.

159

When value of K increases then wsp/wt increases Radius of cylinder

increases then wsp/wt also increases.

Ratio of wsp/wt is lowest at 5A. radius and wsp/wt is maximum are

in cylindrical shape increases then more space to pass waves. Thus

wsp/wt increases with increases in radius.

For Mgo substance in epoxy resine medium there is constent

variation, It is free from radius.

It has only single effect when radius is more then large amount of

wave pass trough it easily otherwise less amount of wave pass

trough it.

When K = 0 then there is no propagation of waves value of K

increases then wave propagation also increases.

wsp/wt has max value at 16A. means at this radius waves passes

easily through the cylindrical geometry. It cappers for highest value

of K = 10.

Thus for low value of K, wave propagation is not significant but

when radius and K both increases then significant wave passes through it.

In this graph at all radius wsp/wt verses K shows linear variation.

The author observe for different medium but at constant radius as

The author plot graph between wsp/wt with propagation constant k

for InP in different medium having radius of cylinder is 0.5 A..

In 3D graph we take wsp/wt along one axis K along other axis while

different medium along third dimensional.

For fixed value of K the ratio of the frequency of surface plasmon

to transverse medium is minimum in epoxy resin and very high in

vaccum medium.

160

wsp/wt versus K is almost linear increment in all medium except

lice medium.

Value of wsp/wt is high in vaccum medium and low in epoxy resine

but the range is nearly equal for same value of K.

When we plot graph wsp/wt versus K. Then InP substance for

cylindrical surface of 0.5 A. radius of different medium, then all

medium shows linear variation but lice shows different variation.

In lice medium wsp/wt increases but its value remains constant at

5.040028. So for lice, wsp/wt increases with increment in K but for

K = 7 or more then this, wsp/wt deceases rapidly.

Thus for InP at lone value of K all medium including lice shows

linear variation but for higher value of K, only lice shows different

variation also for high value of K.

Author observe at different radius of InP in epoxy resine as

We plot graph for InP at different radius of 1 A. & 2 A

. in epoxy

resine medium for cylindrical surface.

We plot graph between wsp/wt with propagation constant K for

InP in epoxy resine medium having different radius.

In 3D graph we take wsp/wt along one axis. Wave vector K along

other axis while different radius consider along third dimension.

The wsp/wt varies linearly with respect to K (1 A. - 10 A

.) for

cylindrical surface of InP at different radius.

At different radius InP medium shows same linear variation.

wsp/wt increases vertically when we move from top to bottom,

while this ratio is decreases when we move Horizontally from left

to right.

161

Author observe that for InP in epoxy resine medium variation of

wsp/wt is linear with K at different radius such that 1 A. & 2 A

., Both

shows linear variation in the epoxy resine medium as in fig. no. 6.

Coupled SP-SOP modes arise on the surface of a polar

semiconductor as a result of frequency and wave vector dependence of

the dielectric surface function of the lattice dielectric surface function of

polar semiconductor. These coupled SP-SOP modes on coupling with the

incident EM radiation of comparable frequency lead to the coupled

surface plasmon, polariton phonon modes on the surface. The dispersion

relation for these modes can be obtained from equation (6.2) by

substituting the frequency and wave vector dependent form of 1( )kε ω we

get three mode coupling

12.3W6- (14.9+13.3K1

2) W

4+(34.33K1

2+18.4)W

2 - 18.4K1

2 = 0 (6.3)

Now equation (6.3)gives three values of w1 ,w2 and w3 for different

value of k1.

Explanation of graph between w and k for cscl & InP

The author observe from the graph that the frequencies w1, w2 and

w3 vary with respect to propagation constant K. At high value of K, w1 &

w2 are always constant but the third coupling frequency w3 always

increases with respect to K.

Thus author observe that coupling frequency varies with K for

cylindrical surface of cscl. When coupling frequency is low than with K.

It shows constant variation [as shows in fig 5.7] but for high value of

coupling frequency, it always increases with K.

162

If author take InP substance to study the surface behavior. It is

observed that as propagation constant K increases there is no variation in

w1 and w2 but w3 varies very slowly till K = 20 but sharply above K

=20[as shown in fig 5.8].

This shows that frequency of polariton depends upon the

propagation constant K but phonon and plasmon frequency does not vary

against propagation constant K in the case of cylindrical surface of cscl.

In a nonmagnetic isotropic medium the electromagnetic wave eqn is

= c

2 2E

Now solution E exp(-iwt) exp(i.k.r)

and D=є(ω,k)E, then we have the dispersion relation for

electromagnetic εw, k=

This relation gives conclusion

! " #:% for ω+

real, K is real and transverse EMW

propagate with the phase velocity Vp= &√ε

! " #:% for ω- real, K is imaginary and the wave is

damped with a characteristic length (|*|

Complex :- for ω real, K is Complex and the waves are

damped in space.

+O :- only longitudinally polarized waves are

possible only at the Zeros of .

163

+ ∞ ∶ % System has infinite response in the absence of

an applied force, thus the poles of εW, K define the frequency of the free oscillations of

the medium.

The dielectric function εW, Kof the electron gas, with is strong

dependence on frequency and wave vectors, has signifies consequences

for the physical properties of solids. In first limit εW, O describes the

collective excitations of the Fermi electron gas for surface Plasmon's and

phonons. In second limit εO, K describes the electrostatic screening of

the electron-electron, electron-phonon and electron impurity interaction

in crystals.

So εW,O=1- 01 and εO, K = 1+23 *

Where VS is screening potential but K O does not approach the

same limit as W O. Thus great care must be taken with the dielectric

function near the origin of the W-K plane.

The Total dielectric function, lattice Plus electrons, but without the

electronic polarizability of the ion cores, is

εW, K=1- 01 + 23 * (6.4)

Above eqn

is valid when we consider both frequency and

propagation constant K.

So eqn become 5 =

ε∞67789 :ε;

67789 :(

% <7=78>

67789 (6.5)

164

Where ε?, ε∞ are low frequency dielectric constants and high

frequency dielectric constant respectively. the values of 0108

chosen

corresponds to strongest coupling between surface plasman [SP] and

surface optical phonon [SOP]. we can plot graph of (W) versus 0108

.

Form graph we observed that there exit two SP-SOP models. These

modes do not propagate when (W) is negative.

i.e 008

< 0@08

but propagate when (W) is positive.

i.e 0@08

A 008

<1

In presence of dc magnetic field, the dielectric function not remains

scalar but becomes a tensor with non-Zero off diagonal element given by

= εB ij - ε 01

0 0 :0C DW2

ij - WciWuj + i ijkWck] (6.6)

where WE= FπGHI

εJ∗

and Wc = ILJ∗

Wp and Wc are plasma frequency and cyclotron frequency. m* is

effective mass of electric in solids. εB is background dielectric constant of

the polar semiconducting medium.

ij =Kronecker delta function

165

it's value ij = 1, i = j

= 0, i j

and ijk = third rank antsymmetric tensor

Its value ijk = +1 If i, j,k is an even permutation

= - 1 If If i, j,k is an odd permutation

= o otherwise

Due to magnetic effect the crystal becomes anisotropic i.e.

dielectric function is different for different directions. Therefore, the

filtering properties of the substance vary with the direction of incident

beam with respect to the magnetic field. If we take the magnetic field in a

particulars direction out to be qualitatively similar to eqn (5.180). Hence

the filtering properties are also similar.

The variation of w/wt versus propagation constant K in

presence of DC magnetic filed for NaF tube shown in fig (5.9). It shows

the dispersion curves for the coupled SP and SoP waves in NaF

cylindrical tube radius 100A.. Thus dispersion curves shows that the

frequency of the upper mode (Ώ3) varies slowly with the wave rector K.

when K is more than 2 it increases rapidly.

Lower mode (Ώ1) varies very slowly and remains almost constant

when value of wave vector is equal to or greater than unit. But Ώ2

mediator mode lies in between upper and lower mode is exactly constant

for all value of wave vector (K). So there exist a band gap between the

166

two modes showing that the coupled modes with frequencies lying in this

region can not be excited.

Theoretical study of InP from graph between εvX2 and as X shown

in fig (5.10).

The author observes that there are four range of EM wave for

which cylindrical surface behaves as filtering property of InP cylinder.

These ranges are :-

i. For w/wt = X = 0.1 to 0.69 the surface of InP behaves as filter of

EM waves because εvX2 because positive and increases slowly and

becomes maximum at X = 0.65, εvX2 =6.443071 now after X =

0.65, εvX2 decrease and thus the filtering property of InP decreases.

ii. For X = 0.69 to 0.72 the surface of InP stops to pass EM waves and

holds penetrating property.

iii. For X = 0.72 to 1.00 value of εvX2 increases rapidly so that surface

of InP allows EM waves to pass with high frequency range because

at this range εvX2becomes positive. The frequency X = 0.95, high

EM wave passes as at this frequency range εvX2 = 79.73133. This is

third order filtering property of InP for EM waves. For 0.95 to 1.00

EM wave decreases.

iv. For range X = 1.00 to 1.79 the surfaces of InP does not allow to

pass EM waves because at this range X, εvX2 becomes negative and

at X = 1.1, εvX2

= - 43.5334.

For X = 1.79 and above it becomes transparent for EM waves

because εvX2

becomes positive. Thus it allows to pass EM waves. The

cylindrical surface of InP used as switch on and switch off property for

electronic signal in electronic communication in science & technology.

167

Now comparing the filtering properties of different substances as

The author observe that comparative study of 1A0

radius of cylindrical

substances as

In this graph we are comparing variation of wsp/wt with wave rector

K for different substances KF, InP & Mgo.

wsp/wt in all substances increases from top to bottom when K

increases.

wsp/wt is lower for KF but in other two substances InP & Mgo

wsp/wt increases rapidly [slightly greater value].

When we plot comparative graph for cylindrical surface of

different substance like KF, InP & Mgo with respect to wave

vector K at same radius 1 A., then all substances have same linear

variation.

This dispersion curves shows that the frequency of the lower mode

changes slowly and frequency of the upper mode varies rapidly

with the wave vector K.

Thus wsp/wt verses K is almost linear increment of substances KF,

InP & Mgo at same radius cylindrical surface (1 A.) these behaves like

same nature.

The author observe that comparative study of 2A0

radius of cylindrical

substances as

In this graph we are comparing variation of wsp/wt versus K for

different substances KF, InP & Mgo.

When we plot comparative graph for cylindrical surface of

different substances at equal radius 2A.. then all substances have

same linear variation.

168

wsp/wt increases with increment in value of K.

For KF variation of wsp/wt with K varies with lesser effect where

as for other two substances InP Mgo show more effective linear

variation. At this radius (2 A.) both InP & Mgo having nearly same

value of wsp/wt in comparison to radius (1 A.).

The author observe that comparative study of 16A0 radius of

cylindrical substances as

In this graph we are comparing variation of wsp/wt with wave vector K

for same substances KF, InP & Mgo but his time at higher radius (16 A.)

of cylindrical surface.

At this radius KF substances has lower value of wsp/wt even starts

from Zero. At this radius cylindrical surface has lower value.

For KF substance it has lower effect, it means lesser wave

propagate through the surface.

At this radius (16 A.) both InP & Mgo have same value of wsp/wt so

both have same type (kind) and equal linear variation.

Thus for these comparative study we observe that when radius of

cylinder increases then KF has lower linear variation and InP & Mgo

have greater liner variation.

The frequency of surface plasmon has very small variation in KF

compared to InP & Mgo with respect to radius and propagation constant

(K). The reason is that KF is not a good conductor and it behaves like

polar semiconductor. Thus study is important in nanotechnology and

electronic communication.

But at higher electronic concentrations the frequency of this mode

become almost constant. It is clear that at the higher concentrations, the

169

lower mode become like pure SP mode and upper mode becomes like

pure sop mode. The coupling between SP and sop mode is stronger when

wsp = wsop i.e. where two uncoupled modes intersect. The stronger

coupling is observed at wsp/wt = 1.Thus wsp and wt both are equal then

coupling is stronger.

Thus the surface acts as Band Pass Filter [BPF]. Agian their is no

propagation when (W) is negative for 008

" 0M08

, the surface become

transparent again act as High pass filter (HPF). from graph we observe

band width ( ) of band pass filter is given by

∆LEO+ 1 %0@08

(6.7)

Thus the allowed band will also be differed for different

compounds.

Similarly ∆QEO+ 1 %0M08

(6.8)

Gives values at which the surface will act as high pass filter. from

study of Szigeti effective charge eS∗ versus with the help of data in

table. from this the best fit of data we find the following relationship

between eS∗ and as

=0.074+0.099<I3∗I > (6.9)

The values of can be estimated for different compounds with the

help of graphs. from these values I3∗I can be calculated by using equation.

170

from table our calculated values agree well with the experimental values

of Hass and Henvis.

We can conclude that polar semiconductor surface acts as a band

pass filter for ω- <ω<ωt and as high Pass Filter for ω ω+. The es* is

measure of iconicity of polar semiconductors. thus width of band ( )

increases with increase in ionicity. If there is more ionic character, band

of allowed frequency is wider and surface becomes high pass filter at

higher value of frequency ω.

Crystals with the triatomic and cubic structure, these properties

indicate that they are essentially ionic. Szigetti calculated the effective

ionic charge es for the alkaline earth materials using eqn (6.10). In this

equation the reduce mass M for this system as

(1/M) = 1/m1 + 1 / 2 m2 (6.10)

Where m1 is the mass of ion with higher valency and m2 is the

mass of ion with lesser valency. The value of es obtained by szigetti for

CaF2 type crystals. These value of es are of the same order as but

systematically lower than those for the highly ionic alkali halides.

The long wave lattice dynamics of the fluorite lattice was studies

by Axe. For es he obtained the same relation [(eqn) as obtained by szigeti

except that the definition of the reduced mass used by Axe . The value of

es obtained by Axe and his coworkers for several crystals with caF2

structure. The es values for tow oxides with CaF2 structure are distinctly

low and indicate ‘Increased covalency’. Szegeti’s theory is by far the

most comprehensive theory of dielectric. It is applicable to diatomic as

well as polyatomic crystals to ionic as well as covalent crystal. Though

171

the original theory assumed a harmonic cystal szigetti modified it to make

it applicable to anharmonic crystal. The most important contribution of

szigeti’s theory is the concept of the eefiective ionic charge, it has found

applications, the most important being its utility as an ionicity parameters.

Several theoretical models have been proposed to account for the

deviation of the effective ionic change from unity. These models have

provided an insight in the complex polarization mechanisms in solids.

Some new results have been discussed. There are: -

i. Szigeti’s second relation relating the compressibility to the

absorption frequency. It is useful as a consistency check on

compressibility date.

ii. Axe’s modification of szigetti’s theory theory has lead to a method

to estimate the mode Gruneisen parameter of fluorite type crystals.

iii. Systematic calculation of the anharmonic correction for the alkali

halides with Nacl structure has revealed that only LiBr has a

negative correction.

Two equations for the effective ionic charge by Havinga have been

systematically applied to a number of alkali halides.

This is new recent theory of condensed material Science. This

theory can be utilized In electronic communication system and is a

measure of iconicity of polar semi conductors. Thus width of band

increases with increase in iconicity.

172

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PUBLICATIONS

1. Study of frequency of plasmons and polaritons on the cylindrical

surface of carbon nano tubes, Indian science congress-2011

2. Nanotechnology: Benefits and Applications, CVPI-NTAMST,

November-2011, Kanpur,

3. Effect of Szigeti effective charge on filtering properties of nano

polar semiconductor, N.C.M.S, Jalandhar, Punjab, March-2012

4. Study of electronic properties of CNT in the electric field,

N.C.M.S, Jalandhar, Punjab,March-2012

5. Surface polariton interaction on surface of condensed nano

materials, ISCA-ISC-2012-11 Mat S-22

6. Radiative and non Radiative Properties of semiconducting surface,

ISABNM, November-2013, LUCKNOW

7. Surface study of Agcl in presence of Dielectric medium, March-

2014, Gorakhpur, U.P

8. Szigeti effective charge interaction on the surface of condensed

nano materials, ISABMS-2014, Lucknow, University (Under

consideration)