Title: Optical reflectivity measurements on alloys by compositional modulation
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Title: Optical reflectivity measurements on alloys by compositional modulation
Physical Description: viii, 89 leaves. : illus. ; 28 cm.
Language: English
Creator: Holbrook, Juan Alfaro, 1930-
Publication Date: 1972
Copyright Date: 1972
 Subjects
Subject: Reflection (Optics)   ( lcsh )
Reflectometer   ( lcsh )
Copper-nickel alloys   ( lcsh )
Metallurgical and Materials Engineering thesis Ph. D
Dissertations, Academic -- Metallurgical and Materials Engineering -- UF
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non-fiction   ( marcgt )
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Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaves 87-88.
General Note: Typescript.
General Note: Vita.
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Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: alephbibnum - 000577371
oclc - 13959734
notis - ADA5066

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OPTICAL REFLECTIVITY MEASUREMENTS
ON ALLOYS BY COMPOSITIONAL MODULATION













By

JUAN ALFARO HOLBROOK




















A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA
1972
































PLEASE NOTE:


Some pages may have

indistinct print.

Filmed as received.


University Microfilms, A Xerox Education Company

















ACKNOWLEDGMENTS


The author would like to express sincere appreciation to his

advisor, Dr. R. E. Hummel, who suggested the topic for this disserta-

tion and provided constant support and encouragement during the course

of this work. He would also like to thank Dr. K. R. Allen, Dr, J, J.

Hren and Dr. R. Pepinsky for serving on his supervisory committee.

The valuable help from Dr. D. D. Dove in the initial stages of this

project is gratefully acknowledged.

Special gratitude is expressed to Dr. A. G. Guy for his

unselfish professional and moral support while the author was pursuing

graduate studies at the University of Florida.

Many thanks are also extended to Mr. Darry Andrews for

assistance in the design and construction of the equipment for this

project.

This project was partially financed by the Department of

Metallurgical and Materials Engineering of the University of Florida,

for which the author is very grateful.

















TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS . . . . . . . . .. . . . . ii

LIST OF FIGURES . . . . . . . . .. . . . iv

ABSTRACT . . . . . . . .... .......... vii

CHAPTER
I INTRODUCTION . .. . . . . . . . . 1

II OPTICAL PROPERTIES OF METALS . . . . . . . 8

2.1 Introduction . . . . . . . . . . 8
2.2 Relation Between the Optical
and Electric Parameters . . . . ..... . 10
2.3 Optical Excitation Processes . . . . ... 14

2.3.1 Classical Oscillators . . . . . 15
2.3.2 Energy Bands in Solids . . . . .. 18
2.3.3 Excitation Processes . . . . ... 23

2.4 The Interpretation of the Optical Properties . 27

III EXPERIMENTAL PROCEDURE . . . . . . . . 54

3.11 The Apparatus . . ... . . . . . . 54
3.2 Data-Reducing Circuit .............. 60
3.3 Sample Preparation . . . . . . . ... .68

IV RESULTS AND DISCUSSION . . . . . . .... 70

4.1 General Remarks . . . . . . . . .. 70
4.2 Copper-Zinc Alloys . . . . . . . ... 72
4.3 Copper-Aluminum Alloys .... . . . . .... 76
4.4 Copper-Nickel Alloys . . . . . . . .. 79
4.5 Effect of Inhomogeneities . . . . . ... 85

BIBLIOGRAPHY . . . . . . . . ... . . ... .87

BIOGRAPHICAL SKETCH . . . . . . . . ... . 89

















LIST OF FIGURES


Figure Page

2.1 Behavior of the function described in Equation (21)
where 1 and E2 represent the real and imaginary
parts, respectively . . . . . . . . . 19

2.2 Energy bands along two symmetry directions for
aluminum . . . . . . . . . . . 22

2.3 Brillouin zone for the face-centered cubic structure
showing various points and lines with symmetry
properties . . . . . . . . . . . 22

2.4 The spectral dependence of the reflectivity of
aluminum . . . . . . . . . . . 32

2.5 The spectral dependence of the real and imaginary
parts of the dielectric constant, the confuctivity a,
and the energy loss function Im e-1 for aluminum . 34

2.6 The real and imaginary parts of (b)(w) for aluminum 36

2.7 The calculated energy bands of aluminum along the
symmetry axes . . . . . . . . .. .. . 38

2.8 Spectral dependence of the reflectivity of silver . 43

2.9 Spectral dependence of the real and imaginary parts of
the dielectric constant and the energy loss function
62/(1 + e2) for silver . .... ......... 43
2 1 2

2.10 Decomposition of the experimental value of EC
for silver into free and bound contributions
f) and (b) . . . . . . . . . . 44
1 1

2.11 Spectral dependence of the reflectivity of copper . 46

2.12 Spectral dependence of the real and imaginary parts of
the dielectric constant and the energy loss function
2 2
E2/(C1 + C2) for copper . . . . . . . . 46
2 1 2













LIST OF FIGURES (CONTINUED)


Figure Page

2.13 Decomposition of the experimental value of
1 for copper into free and bound contributions e
(b)
and . . . . . . . . . . ... 48


2.14 Calculated band structure of copper along symmetry
axes . . . . . . . . ... .. . 50

2.15 Effective number of electrons per atom as a func-
tion of energy obtained from numerical integration
of e2 for silver, copper and gold . . . . .. 53

3.1 Schematic diagram of the optical part of the apparatus 55

3.2 Percentage of stray light from the monochromator as
a function of wavelength as measured by the voltage
output from the photomultiplier tube . . . ... 59

3.3 Spectral distribution of the xenon lamp multiplied
by the sensitivity of the photomultiplier tube . .. 61

3.4 Block diagram of the normalization circuit . . .. 64

3.5 Wiring diagrams for the dc amplifiers and divider . 66

4.1 Schematic representation of the spectral reflec-
tivities of two alloys . . . . . . .... . 71

4.2 Comparison of energy bands of Cu-30 at. % Zn with
those of copper . . . . . . . .... . 73

4.3 AR/R vs wavelength for various copper-zinc alloys . 75

4.4 Spectral c'-ange in reflectivity upon adding 1 at. % Al
to Cu . . . . . . . . ... .. . . 77

4.5 Shifts in the peaks marked A, B, and C in Figure 4.4
as a function of concentration of aluminum . . .. 78

4.6 Illustration of the density of states in copper and
copper-nickel alloys, showing the behavior expected
from rigid-band and virtual-bound-state models at
15 at. % Ni in Cu . . . . . . . . . 80

4.7 AR/R vs wavelength for various copper-nickel alloys . 83













LIST OF FIGURES (CONTINUED)


Figure Page

4.8 Compositional variation of as cast Cu -15 at. % Ni
alloy as revealed by microprobe examination . . .. 86

4.9 Compositional variation of homogenized Cu- 15 at. % Ni
alloy as revealed by microprobe examination . . 86

4.10 Spectral change in reflectivity between as cast and
homogenized Cu -15 at. % Ni alloys . . . ... 86













Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in Partial
Fulfillment of the Requirements for the Degree of Doctor of Philosophy


OPTICAL REFLECTIVITY MEASUREMENTS
ON ALLOYS BY COMPOSITIONAL MODULATION


By

Juan Alfaro Holbrook

August, 1972


Chairman: Dr. R. E. Hummel
Major Department: Metallurgical and Materials Engineering


A differential reflectometer, which is capable of measuring

small differences in reflectivity between two samples, is described.

Among the possible applications of this device are: to reveal the

structure in the spectral reflectivity of metals and alloys; to study

changes in reflectivity caused by surface reactions and environmental

changes, and to investigate metallurgical reactions.

The instrument has been used in the study of compositional

modulation of copper-based alloys at room temperature in the spectral

range between 1.8 and 6.2 eV.

For copper-nickel alloys, a decrease in reflectivity in the

low-energy region was observed with increasing concentrations of nickel.

No significant shift of the fundamental edge at 2.1 eV in pure copper

was detected when nickel was added to copper in concentrations up to

15 at. % nickel. These observations provide evidence that a virtual-

bound-state model applies to these alloyc, in agreement with results

obtained by photoemission studies.


vii












When zinc is alloyed to copper, the fundamental edge at 2.1 eV

of copper was shifted to higher energies; on the other hand, the struc-

ture at about 4.0 and 4.4 eV was shifted to lower energies. This behav-

ior was interpreted using existing band structure calculations for

nlpha-copper zinc alloys. The agreement between theory and experiment

was satisfactory.

Results for alpha-copper-aluminum alloys were similar to those

for the copper-zinc system.



































viii


L

















CHAPTER I


INTRODUCTION



In optical studies of metals and alloys the material is probed

with electromagnetic radiation in the spectral region from the infrared

to tile ultraviolet. Light reflected from a metal surface results from

interaction of electrons in the solid with the incident radiation.

The reflected radiation provides information on the energy distribu-

tion of electrons within the bands, and particularly on the optical

excitations involved inl these interactions.

Knowledge of the bnnd structure of metals has improved consider-

ably in recent years. The energy distribution of electrons within tho

bands is in general understood and in many instances can be calculated.

Measurements leading to the knowledge of the bond structure were at

first possible only for electrons near the Fermi surface. However with

optical techniques, which excite optical transitions over a region

covering a few electron volts above and below the Fermi surface, the

possibilities have been greatly extended.

The increasing awareness of the importance of understanding

the electron structure of solids in metals research can perhaps be

emphasized by quoting the words of J. J. Gilman and W. A. Tiller [1].

The progress made in the knowledge of atomic and electronic structure

of metals














will have an important influence on understanding and predict-
ing metallurgical reactions and processes, and is now in a form
that allows it to be incorporated into the general education of
metallurgists . as critical materials become more special-
ized, their invention and characterization will depend as much
on the control of their electronic structure as on the tradi-
tional control of chemical composition and microstructure (1, p. v].


Optical properties of solids received considerable attention

during the 1950's. This was in part due to the advances in materials

preparation technology and to theoretical work leading to detailed

knowledge of the band structure of solids.

Conventional techniques to measure the optical constants of

solids use the method known as ellipsometry. For an absorbing medium,

such as metal, the determination of the optical constants involves the

measurements of two parameters for each frequency for a given incident

radiation. If linearly polarized light impinges obliquely on a metal,

the reflected light will in general be elliptically polarized. It is

possible to determine the optical constants at a given wavelength by

measuring the state of polarization of the reflected light, that is,

the orientation and axes ratio of th.e polarization ellipse. A number

of techniques have been devised to perform this type of measurements

but they are restricted to the spectral region for which polarizing

equipment and wave plates are available. Moreover, they are limited

in resolution and the measurements are often tedious and time consum-

ing, particularly when the optical constants are to be obtained over

a range of wavelengths.

More recently optical studies have been carried out at normal

or near normal incidence, in conjunction with relations by which it is













possible to calculate one optical constant if the other is measured

over a sufficiently wide spectral range. The common practice is to

measure the reflectivity and to determine the optical constants (usu-

ally the real and imaginary parts of the dielectric constant) from

these reflectivity data. With normal incidence reflection techniques,

studies were extended in the late 1950's and early 1960's to photon

energies covering the infrared, visible and ultraviolet. By the mid

1960's the electronic structure and the optical constants of a large

number of materials had been investigated and tentatively interpreted.

From these studies the existence of the so-called critical points asso-

ciated with relatively narrow regions in frequency were discovered, which

produce pronounced structure in the optical constants. The identifica-

tion of this structure with particular optical transitions today consti-

tutes part of the main effort in the area of optical properties of

solids.

Optical properties regained interest in .965 when Seraphin and

Hess [2J introduced a modulation technique which prompted a series of

new experiments.

The optical structure associated with critical points can be

greatly enhanced by means of modulation or derivative techniques.

The general idea behind this method is as follows. The dielectric

constants near a three-dimensional critical point can be expressed as



e(J) = b(w -w ) + constant, (a)


where wg represents the critical frequency, a is the frequency of the

incident radiation, and b is a parameter. The constant background













in Relation (a) may be much larger than the singular part, and there-

fore the singularity may be lost in the noise of the background. It

is therefore convenient to measure, instead of e (or some other optical

parameter), the derivative of E with respect to some parameter p.

The constant background is thus eliminated. The derivative becomes


de b d(W db(W
d 2 )I .A( w 0_ ) + (! )_ (b)
dp (w- dp g dp g) (b)



It can be seen that the first term in Relation (b) exhibits n singular-

ity at the critical frequency wa and may therefore be easily detected.

The second term containing (w -wg ) on the other hand can be neglected

near w .

Several parameters p can be used to modulate the band struc-

ture. Seraphin and Hess [2] imposed an alternating electric field on

a semiconductor to induce changes in the reflectivity (eluctroreflect-

nnce). Engeler et al. [3] varied an external uniaxial stress in var-

ious crystallographic directions during reflectivity measurements

(piezoreflectance). Berglund [4] and Hanus, Feinleib and Scouler [5]

modulated the temperature of a specimen on optical absorption measure-

ments. It is also possible to obtain the first derivative of the

spectral reflectivity by imposing a small periodic variation on the

frequency of the incident radiation (frequency modulation). Each of

these modulation methods has its specific advantages and disadvantages.

Electroreflectance has been used successfully in the study of

the band structure of semiconductors; but it does not find application

in metals because of the difficulty of maintaining an electric field














within a conductor. Temperature modulation is used to study the influ-

ence of phonons on interband transitions. Modulation of uniaxial

stress upon a sample changes the symmetry of the crystal and this is

found useful in determining the symmetry of the critical points.

Frequency modulation enhances the structure in the reflectivity but it

has the disadvantage that it also reveals any structure present in the

light source.

The investigations herein reported use another type of modula-

tion: the variation of the composition of alloys. The addition of an

alloying element to a metal as a rule changes its band structure.

Compositional modulation provides therefore a means to study the evolu-

tion of the band structure of the alloy as its composition is changed.

The differential reflectometer which was developed in this work

has proved to be potentially a powerful tool for the investigation of

the optical properties of solids and the study of surface reactions.

Generally speaking the instrument is capable of detecting minute reflec-

tivity differences (a fraction of 1 percent at any given wavelength).

The change in reflectivity might be due to alloying, a surface reaction,

a difference in doping in a semiconductor, a change from crystalline to

amorphous structure, a transformation (short range order, allotropic,

etc.), or any phenomenon which brings about a change in the electronic

structure of the solid with which a change in optical parameters is

associated.

Because of computer elements incorporated into the instrument,

the data are processed instantaneously. This permits automatic














recording of a normalized change in reflectivity AR/R as a function

of wavelength. This normalization is extremely useful, for it makes

the recording of the data independent of source instability, spectral

sensitivity of detector, and intensity of light source.

Several models have been suggested to explain the effect of

alloying on the electronic structure. Notably among these are the

virtual-bound state and rigid-band models. In general different models

lend to different predictions about the effects of alloying on the band

structure. Therefore a study of the changes in reflectivity upon adding

solute atoms to a metal can provide a critical test for the validity of

current models.

Chapter II presents an outline of the theory from which inter-

pretation of the optical properties of metals is made. The optical

constants are defined, and their relation to electrical properties of

the solid is established. A brief discussion of early models and their

relevance to modern developments is also given. The chapter proceeds

with a description of the optical excitation processes involved in the

interaction of light in solids. The remainder of that chapter is

devoted to a review of some of the more relevant achievements in the

area of optical properties of metals.

The description of a new apparatus developed for optical

measurements by compositional modulation, including the circuit used

to process the data, is given in Chapter III.

In Chapter IV the results obtained by this technique on copper

based alloys, with additions of nickel, zinc and aluminum, are pre-

sented and discussed. The main effort is directed towards the










7



correlation of the experimental findings with other studies and to

current theoretical models of alloying. In the case of copper-nickel

alloys it is found quite conclusively that a virtual-bound model applies

to copper-rich alloys, in agreement with photoemission studies of these

alloys. Measurements on the copper-zinc system are compared with band

structure calculations on these alloys. The correlations are found to

be satisfactory in some respects and not so satisfactory in others.

Results obtained on copper-aluminum alloys are discussed in terms of

the evolution of its structure as a function of composition. It is

suggested that this system probably conforms to a rigid-band model.


















CHAPTER II


OPTICAL PROPERTIES OF METALS



2.1 Introduction

From the point of view of the propagation of electromagnetic

waves in an absorbing medium, the optical properties of the medium are

given in terms of two parameters: the index of refraction, n, and the

absorption coefficient, k. The two parameters may be defined by the

following equations. First,


C
na (1)



where c is the velocity of light in vacuum and v is the phase velocity

in the medium. This is the definition usually given for the refractive

index of a dielectric substance. The absorption coefficient k is

defined by the rate of decrease of amplitude E of the electromagnetic

wave as it progresses through the medium


2n
E = E exp (- k z) (2)



Here z is the coordinate in the direction of propagation, E is the

amplitude at z=0, and X is the wavelength of the light in vacuum.

The name "absorption coefficient" for k is perhaps not the best choice,

since it is not related to the absorption of radiation inside matter in

a very direct or intuitive way. Extinction coefficient would be a














better name but unfortunately this designation is reserved for a dif-

ferent parameter defined as k/n. The term absorption coefficient will

be used here since it has become the standard custom.

Together these two parameters form the complex index i of the

medium. This is defined by the equation


Sn ik (3)


The term "complex index" for the quantity defined by

Equation (3) is an appropri te one. If the index in the equation



E =E exp ilw(t -" z)J (4)
o c



for transparent media, is replaced by the complex index n, the result-

ing equation,

2n n
E = E exp (- -- kz) oxp iw(t z) (5)



has the properties required by the defining Equations (1) and (2).

This concept is convenient because it enables some of the formulas

of dielectric optics to be converted easily to the metallic case.

For example, the reflectivity of a dielectric is l(n-l)/(n+l1)1

It can be shown [6] that the reflectivity II of a metal is found by

inserting the complex index into the corresponding expression for the

reflectivity of a dielectric, namely,


(n -1) + k
S= + (6)
(n+ 1) + k














The conversion of optical formulas to the metallic case in this

way provides little insight into the physics involved and so gives no

aid in understanding or interpreting the equations. It is preferable,

therefore, to treat the absorbing case directly. Transparent media can

then be considered as n special case in which k =0.


2.2 Relation Between the Optical
and Electric Parameters

The classical theory of the optical properties of solids is

based upon Maxwell's electromagnetic equations. When the solid is a

conductor it is characterized by a conductivity a and a ditlectric

constant C. The permeability of the medium is assumed unity. For an

uncharged medium the equations take the form


4T-cr e E
div l H = 0 curl H= E +-
c c
I l ~(7)
div e E = 0 curl E = c



where the Gaussiun system of units is used. The equations as written

are supplemented by Ohm's law, J = a E.

For a homogeneous, isotropic (or cubic) medium the dielectric

constant and the conductivity are scalars instead of tensors. The

magnetic field H can be eliminated between the two curl equations and,

using the fact that div E=0, the wave equation can be derived, namely,


4- 4Ta E C a2
T cE 2 2 (8)
c c














This is the differential equation that governs the propagation uf

electromagnetic waves in a conducting medium. It can be seen that the

term containing the first derivative of E is a dissipative term. It can

be shown [7] that in a steady state situation no longitudinal electric

or magnetic field (other than uniform fields) can exist in a conductor

which satisfies Equation (8). It is possible therefore to write

Equation (8) in a one-dimensional form. With the direction of vibra-

tion of the electric field along the x-axis and the direction of propa-

gation along the z-axis, the equation takes the form



82E aE a2E
x 4Tra x x
+= -- (9)
z2 -2 -- 2 t2 (9)
az c c at



A plane-wave solution of (9) of angular frequency w will then have

the form


E = E exp [i(t -2 z) (10)



where the index of refraction is complex. Substitution of solution (10)

into the wave equation (9) yields



n2 = c 1 ,

or

2 4rra
(n-ik) = i -- (11)



which establishes a relation between the optical constants (n,k) with

the electric properties of the solid (e,o) for the particular frequency.














It will be observed from Equation (11) that it is possible to

introduce a complex dielectric constant


4Tra
= E i _. (12)
w



Here the real part of E is the conventional (actual) dielectric constant,

while the imaginary part includes the dissipative effects associated with

the conductivity. This is a convenient generalization of the dielectric

constant, since it allows the relation

2
n = e (13)


for transparent substances to be extended to the metallic case with

n and e substituted by the complex quantities introduced above.

It appears convenient to look at the complex nature of the

dielectric constant by considering the curl H equation in (7). It can

be seen that the contribution to the magnetic field comes from conduc-

tion currents through the conductivity and also from displacement cur-

rents arising from the time derivative of the displacement vector D.

For a particular angular frequency w the time dependency of the fields

will be of the form


E = E exp (iwt)



and the time derivative will be


= i E














With this last expression it is therefore possible to formally trans-

form the right-hand side of the curl H equation into terms containing

only the time derivative of the electric field, that is,



curl H = (14)



The two terms in parentheses represent the same complex dielectric

constant introduced in Equation (11).

In a similar way it is also possible to generalize the conduc-

tivity by converting the right-hand side of Equation (14) into terms

containing only the field E. The resultant relation is



curl HE= ( + i )E (15)



where the generalized conductivity can be defined as


we
a= + 1 (16)



It should be emphasized that this is just a convenient general-

ization of the parameters involved in the interaction of electromagnetic

waves with matter. Whichever formalism is used, the relation between

the optical and electrical parameters is the one given by Equation (11).

The aim of a theory of the optical properties is to obtain relations

for n and k, or equivalently for C and a based on a microscopic model

of the conductor. If a particular model of the conductor leads to an

equation relating the dielectric constant to fundamental parameters

such as the mass and charge of the electron and if the dielectric














constant is a complex number, then the imaginary part can be associated

with conduction currents, that is, with the conductivity. On the other

hand, if one is dealing with a complex conductivity, it can be asserted

that its imaginary part is taking care of the polarization current.


2.3 Optical Excitation Processes

The distinction between metals and insulators is made on the

basis of dc electrical properties. Upon the application of a steady

electric field, a metal conducts, whereas an insulator does not. This

distinction becomes increasingly vague when the applied field is fre-

quency dependent, particularly when the frequency falls into the optical

range, that is, into the visible and ultraviolet. In this range the

electrical, or, more appropriately, the optical properties of metals and

insulators have much in common. It seems appropriate therefore to

start this section with a discussion of some elementary properties which

apply to both metals and dielectrica.

The earliest useful model for the electrical and optical prop-

erties of metals was proposed by Drude early this century. In this

model it is supposed that the electrons form a gas of free particles

which can respond to the application of external fields by producing

currents. The ionic cores present in the solid are completely neglected.

The only statistics known at that time was that of Boltzmann. It was

later discovered that through the substitution of Fermi statistics for

Boltzmann statistics, Drude's picture could be transformed into a very

realistic and useful model for a metal. Even in its original form,

Drude's model is already adequate for the discussion of many aspects

of the optical properties of metals.














Another simple picture of an insulator was proposed by Lorentz.

Here the electrons are envisioned as tied to certain equilibrium posi-

tions by electrical forces arising from the ionic cores. Since the

electrons are not free to wander around, the crystal can have no dc

conductivity. On the other hand, polarization effects in the presence

of an ac electric field are quite possible.

In the light of sophisticated modern developments in solid

state physics, it is perhaps remarkable that such simple ideas still

have relevance in the interpretation of the optical properties of

solids. If one can understand qualitatively how electrons described

by these models respond to light, or equivalently to an ac electric

field, then a synthesis and generalization of these ideas will come

close to a true description of the complex nature of the interaction

of light with the electrons in a real solid.


2.3.1 Classical Oscillators

The response of the electrons in a Lorentz insulator to

a uniform oscillating electric field, E(t) = E(w) exp (-int), having

a single Fourier component corresponding to frequency w, is described

by the equation of motion


d2+ dI 2 =x eE(w) exp (-iut) (17)
dt


2 2
Here d x/dt is the acceleration term, 1/T(dx/dt) is a damping term

proportional to the velocity and characterized by the relaxation time T ,
2
and uwx is associated with the restoring force of the oscillator whose
s














natural frequency is ws. The right-hand side of the equation repre-

sents the driving term due to the electric field with e and m being

the electric charge and the mass of the electron, respectively. Since

the equation is linear in the displacement x, it can be assumed that

x(t), and all electromagnetic field quantities, have the same time

dependence as E(t); that is, x(t) = x(w) exp (-iwt). The substitution

of x(t) transforms Equation (17) into a linear algebraic equation for

x(w) that can be solved readily.

The vibrations of all the electrons in a unit volume of the

solid result in an induced polarization given by


P(t) = P(w) exp (-iwt) (18)


whose amplitude, P(w) =enx (w), is proportional to the displacement

x(w) and the concentration n of electrons. Using the value x(w) that

comes from the solution of Equation (17), P(w) is found to be


ne 2E(w)/m
P(w) = 2 2 (19)
w -ws + iw/T



The frequency-dependent dielectric constant describing the response

of the system to an external field E(w) exp (-iwt) having frequency w

is related to the electric field and the polarization by the expression


E(w) E(w) = E(u) + 4TP(w) (20)


using cgs units. Combining: (19) and (20) the complex dielectric

constant is obtained












2
P
e(W) = 1 2 2 (21)
w -w + iW/T
s

2 2
where up = (4nne /m) is the so-called plasma frequency associated

with electrons.

It should be emphasized that the dielectric constant is here

a complex quantity. The real part is C as it appears in Maxwell equa-

tions and the imaginary part includes the dissipative effects asso-

ciated with the conductivity. In fact, the model leading to Equa-

tion (21) contains both the Drude model and the Lorentz oscillators.

Equation (21) does not contain the conductivity explicitly, but if the

imaginary part is separated out, the conductivity can be shown readily

to be

1
U = o 1)2 (22)
1 + (UWT)

2
where o = ne T/m is the dc conductivity. It can be observed that,

by introducing a relaxation time 7 $ 0 into the equation of motion (17),

a dissipative contribution comes in, that is, the conductivity of the

solid. Furthermore, the real and imaginary part of the dielectric

constant (21) can be related to the index of refraction n and the

absorption coefficient k through the formula (11) derived in Section 2.2.

Thus, the complex dielectric constant and the parameters n and k contain

the same physical information. For purposes of theoretical descrip-

tions, however, the dielectric constant formulation is more convenient

because it can be related in a more direct way to a microscopic model

of the solid.














The behavior of the function described in Equation (21) is shown

in Figure 2.1 where s1 and e2 represent the real and imaginary parts,

respectively. For an insulator, in which the restoring force is finite,

the dielectric constant is that of an oscillator, Figurp 2.1A. On the

other hand, for a metal, the Drude model can be recovered simply by

equating the natural frequency ws to zero, amunilng that the electrons

are free to move through the crystal. The characteristic behavior for

the real and imaginary parts of the dielectric constant is now shown in

Figure 2.1B; 2 and e1 become positi-ely and negatively large, respec-

tively, with diminishing frequency. Thus, it is possible to character-

ize the electrons in a Drude metal as oscillators of zero natural

frequency.

2.3.2 Energy Bands in Solids

As mentioned earlier, Drude's model of a metal completely

ignores the presence of the ionic cores. On the other hand, the Lorentz

model of an insulator takes another extreme position: the electrons are

bound to the positive ions with elastic forces and can only oscillate

about their equilibrium positions but cannot move around inside the

crystal. The situation in a real metal has features of both the free-

electron model and the Lorentz insulator. Some electrons are tightly

bound to the positive nuclei and are little affected by the fact that

the atoms are part of a solid, while other electrons move almost freely

inside the crystal.

The model that emerges when the interaction between the elec-

trons and the periodic potential of the crystal is taken into account

is considerably more complex than the simple models described above



















S

LIA
A


o ^----- -o



B

Figure 2.1 Behavior of the function described in Equation (21) where
e1 and 62 represent the real and imaginary parts, respec-
tively. A Insulator (ws 1 0). B Metal (ws = 0).














and its development requires sophisticated techniques of quantum mechan-

ics. The description of the details leading to the band theory of

solids is beyond the scope of this dissertation. However, it seems

appropriate to present here some of the underlying principles of the

theory as well as some of its prominent features.

The electron in an isolated atom can only have certain pre-

scribed energy states and as a result the electron can change its

energy only by discrete amounts. When the atoms are put together to

form a crystalline solid, the levels of the individual atoms spread out

into bands. This phenomenon can be regarded as a consequence of the

Pauli exclusion principle which comes into effect when the wave func-

tions of the various atoms overlap in the solid. The amount of spread

is related to the extent of overlapping of the wave functions. The

energy states inside the crystal can thus be envisioned as a series of

bands with a quasicontinuous distribution of states within each band:

narrow bands for the more tightly bound electrons with the width becom-

ing progressively larger for higher energies and eventually overlapping

with each other at sufficiently high energies. The energy difference

between electronic states of an isolated atom become the "energy gaps"

in a crystal. There are, therefore, energy regions not allowed or

forbidden gaps.

In a solid, energy levels are generally described in terms of

a wave vector k (|kf= 2n/X) and a band index. In band calculations

one of the tasks of quantum mechanics is to find a relation between the

energy E and the vector k for each band. For free electrons this is

a simple relation, namely,













h2 2
E = k (23)
2m


where h = h/2z (h = Planck's constant) and m is the electronic mass.

When the electron moves in the periodic potential of the lattice, how-

ever, Equation (23) is not valid. In dealing with the motion of elec-

trons in a crystalline solid it is convenient to work with the so-called

reciprocal lattice or reciprocal space. A consequence of the periodic-

ity of the lattice in ordinary space is that the reciprocal space is

also periodic. When the wave vector, which is a vector in reciprocal

space, approaches one of the vectors of the reciprocal lattice, the

condition for Bragg reflection is satisfied. Bragg reflection occurs

also for electron waves in crystals and leads to energy gaps. When

the Bragg condition is satisfied for electrons of wave vector k, waves

travelling in one direction are soon Bragg-reflected and then travel

in the opposite direction, leading to standing waves. In other words,

no travelling waves of the wave equation can exist and therefore devia-

tions from the simple parabolic behavior of Equation (23) are expected.

As an example for a real metal, the bands along certain symmetry direc-

tions are shown in Figure 2.2 for aluminum. As it happens, aluminum

has a band structure which deviates very little from the free-electron

bands shown in the figure by the dashed curves. The free-electron

bands are simply obtained by reducing the interaction of the electrons

with the periodic potential to a negligible amount while keeping the

symmetry properties of the lattice. It is customary to present the

bands along certain directions of particular symmetry in k space. These

symmetry directions and points are shown in Figure 2.3. Also shown in










2


I







0)
L. 1.0

"0


.5






L
Figure 2.2 Energy bands along two
Solid curves represent
show the free-electron
sition. b Interband


F K X
symmetry directions for aluminum.
calculated energies; dashed curves
energy bands, a Intraband tran-
transition.


Figure 2.3 Brillouin zone for the face-centered cubic structure show-
ing various points and lines with symmetry properties.














this figure is the fundamental unit cell of the face-centered cubic

reciprocal lattice also called the Brillouin zone. One practical

importance of this cell is that the wave vectors corresponding to

independent solutions of the wave equation are all contained in this

unit cell. That is, the wave equation needs only be solved for k

vectors contained within the Brillouin zone. Although Figure 2.2

represents the energy levels for a single electron only, the electrons

in a solid can be regarded, to a good approximation, as independent and

the diagram can be applied to such systems. In this independent par-

ticle model the electrons know of one another's existence only through

the exclusion principle, which provides that no more than two electrons,

one of each spin, can occupy a given state. This means that at low

temperatures all levels below the Fermi level, indicated in Figure 2.2

by the horizontal line, are filled and all others are empty.


2.3.3 Excitation Processes

In this section the three relevant optical excitation

processes in metals will be described. Intraband transitions are

analogous to the optical excitations in a free-electron metal. As indi-

cated in Figure 2.2, intraband transitions involve the excitation of

an electron by a photon within the same band. Because of the exclu-

sion principle, the electron must go from an initial filled state to

a final empty state. These transitions therefore can only occur in

metals, but not in insulators, in which all bands are either completely

full or completely empty. By contrast, the second kind of excitation,

the interband transition, which involves two bands, is common to both














insulators and metals. This process has its analog in the optical

excitation of a Lorentz insulator and, in fact, reduces almost precisely

to that for the very artificial case of a solid consisting of a widely

separated array of two-quantum-level atoms whose wave functions do not

overlap appreciably. In the absence of any other interactions involv-

ing the electrons, the interband transitions can be shown to be essen-

tially vertical in k space, as a result of the fact that the total

crystal momentum of electron plus photon must be conserved during the

transition. For wavelengths in the visible and ultraviolet regions,

the momentum of the light wave is very much smaller than the diameter

of a Brillouin zone, and therefore the wave vector of the electron is

essentially unchanged in the transition.

Comparison of the results of a quantum mechanical derivation

of the dielectric constant of a solid with the elementary results out-

lined above shows that it is legitimate to regard the dielectric con-

stant as a sum of contributions from both intraband and interband

transitions. These transitions can be described as in the Drude

free-electron metal plus a contribution of oscillators at each k con-

necting vertically filled and unfilled states.

Since interband transitions are possible everywhere in the

Brillouin zone and for all energies beyond a certain threshold, these

transitions might be expected to present relatively structureless

contributions to the dielectric constant. The first experimental

evidence that this is not the case came in connection with semiconduc-

tor ..tudies by J. C. Phillips (8] where very large contributions from














certain small regions in k space were found that result in pronounced

structure in the optical parameters. This was very important, since

it led to a new branch of an old discipline, which can be called "band-

structure spectroscopy." These regions in k space associated with

strong interband transitions contain the so-called "critical points."

They are characterized by very rapidly varying, often large density of

states for optical transitions. For the most part, these occur at

points of high symmetry, points at which the hand structure of a given

material is most likely to be known. Thus, given a rough idea of the

band structure, it is possible to use optical techniques to obtain some

important band gaps, thereby providing a basis for better calculations.

The third kind of elementary excitation in a solid is the so-

called plasma or collective oscillation, which results from the high

density of electrons that are present in a metal and the fact that they

can act cooperatively due to the Coulomb interaction among them. This

cooperative motion may be regarded as a fluid-like vibration of the

electrons as a whole through the solid, which produces fluctuations

in the electron density. Under certain conditions, plasma oscillations

can represent normal modes of the entire system. This means that, in

the absence of dissipative effects, once such oscillations are excited,

they do not decay in time. Since they maintain themselves, an internal

electric field E(wp) is expected at the plasma frequency W due to the

oscillations in the electron density, without the presence of an exter-

nal field D(&u). Since


D( p) = c(wp) E(w ) = 0














and E( u) g 0, the condition for plasma oscillations is therefore


C(WU) = e (Wp) + i2 (p) = 0 (24)


Thus, strictly speaking, plasma oscillations can exist as normal modes

of the system only if both the real and imaginary parts of the dielec-

tric constant vanish at wp. In practice, however, the plasma frequency

will be well defined if e2( P) < 1 when eCl(W) = 0. Since C2 repre-

sents the damping of the plasma resonance, the condition e (wp) < 1

implies that the damping should be small.

The excitation of plasma oscillations can be described by

consideration of a uniform displacement of the electron gas in a thin

metallic slab. The electron gas is moved as a whole with respect to

the positive ion background. A displacement of amplitude x creates an

electric field E = 4nnex which acts as n restoring force. Therefore

the equation of motion of a unit volume of the electron gas is


d2x 22
nm -- = neE = 4nn ex (25)
dt


Equation (25) is recognized as the equation of motion of a simple

harmonic oscillation of frequency


2
[Op = 4ene (26)


This expression for the plasma frequency is identical with that appear-

ing in Equation (21) which arose in quite a different connection.














2.4 The Interpretation of the Optical Properties

The linear response of a conducting nonmagnetic medium to an

electromagnetic wave of angular frequency a can be characterized by

a frequency-dependent complex dielectric constant or equivalently by

a complex index of refraction. In Section 2.2, it has been shown that

through Maxwell's equations a relation exists between these two complex

quantities. From the point of view of the interpretation of the optical

parameters in terms of a theoretical model, the complex dielectric

constant plays a more important role because it can be related directly

to optical excitation processes. Experimentally, however, the dielec-

tric constant cannot be measured directly at optical frequencies and

therefore reliance has to be made on reflectivity measurements. These

measurements are based on Fresnel formulas for the reflection and

refraction at the boundary of a conducting medium. It is also possible

to introduce a complex reflectance. This is defined as the ratio of

the complex amplitude of the reflected to that of the incident electric

field. The complex reflectance r= r exp (iG) can be shown [9] to be

related to the complex index of refraction by the expression


n ik 1 iB
k 1 = r e (27)


The reflectivity R introduced in Section '.1 is obtained immediately

by R = r Measurements of either (n,k) or (r,6) permit therefore the

calculation of (6le2) and the complete description of the optical prop-

erties in the appropriate frequency range can be obtained. Usually the

real and imaginary parts of the complex index of refraction are measured














independently for each frequency of light. This procedure, however,

is long and tedious, since it involves measurements at different

angles of incidence for each frequency.

Fortunately, there are relations between the real and imaginary

parts of any of the complex quantities describing the optical properties.

These are very general equations, known as Kramers-Kronig [10] rela-

tions, that allow the calculation of one parameter at any given fre-

quency when the other is known over a wide range of frequencies.

This relationship is imposed by the causal nature of the response to

an electromagnetic field. That is, no response to an applied field

can appear before the field is applied. For the dielectric constant,

this relationship is



2 f 2 ) d
6(w) 1 = P -(2 2d (28)
a -a
o

and its inverse:

-'2 e1(w')
C() = 2P (29)
o


where Pf designates the Cauchy principal value of the integral. For

the complex reflectance the expression is:




e jw) = p J 2 d' (30)
oW/2 -2














The use of the integral (30) requires the knowledge of the reflectiv-

ity for all frequencies, in principle. Since in practice its measure-
I-

ment is not possible for all frequencies, an explicit assumption for

extrapolation to high frequencies is necessary. The results of the

method are more reliable if the measurements of R are extended to

higher frequencies. A great experimental advantage of the usage of

relation (30) lies in the fact that only the reflectivity at or near

normal incidence has to be measured. Therefore, the method using

reflectivity measurements at normal incidence has been widely used in

recent years.

As was pointed out in Section 2.3, the interpretation of the

optical properties of metals can be made on the basis of a dielectric

constant which includes contributions from free electrons plus a term

associated with bound electrons. These two contributions can be

thought of as generalizations of the simple models introduced in

Section 2.3; the free-electron term having the form of Drude equation,

whereas the bound part is the quantum mechanical counterpart of the

Lorentz equation for classical oscillators. Consequently, the

dielectric constant for a metal is composed additively of the contri-

butions of free electrons e (f)(), and interband transitions or

bound electrons (b)():


e(u) = e(f)() + e(b)() .


Both are, in general, complex.














The free-electron contribution is


2
(f)( ___ Pa
e ) 1 + I/- T (31)
W(+ i/TT
c

2 2
Here wPa = 4e n /m is the square of the conduction band plasma fre-
Pa c a
quency, nc is the conduction electron density,



-= 32n d3 Vk Ec(k) (32)
M a 4TT3h2n hn3 k c
c

is the average inverse effective mass of the conduction electrons, the

so-called "optical" mass defined by Cohen [11]. The two parameters

WPa and the relaxation time T7 characterize the conduction or free

electrons in the Drude theory.

An expression for the interband processes, and valid within

the framework of the random-phase approximation is


(10 1 ieN' f2(k) f(' L u
e (b) (e) f d2 2 (33)



Here e is the electron charge, m the free-electron mass, f(k) is the

Fermi distribution function for state k in band A,


2


is the oscillator strength,















PI t = u (k,x) pp u (k,x) d x
a


is the momentum matrix element for direction p integrated over the unit

cell of volume va, and


hw1,1 ='E. (k)- E, (k)



is the energy difference between bands A and A' at the point k.

Ta / is an interband relaxation time.

The above formalism has been developed largely by Ehrenreich

and his co-workers [12-15] and has been applied with a great deal of

success to the interpretation of the optical properties of the noble

metals as well as the metals aluminum and nickel. The rest of this

section is devoted to a review of this work with some emphasis on the

results for copper.

Aluminum. Figure 2.4 shows the reflectivity R uf aluminum as

a function of the energy of the incident radiation. It is character-

ized by its very simple optical properties that make this metal a good

example of a free-electron metal. The reflectivity for an ideal Drude

free-electron metal stays at 100 percent as long as C1 is negative.

When el passes through zero at the plasma frequency, the reflectivity

falls off very rapidly. It can be seen that Al appears to conform

qualitatively with this description except for the small dip near 1 eV,

which is due to interband transitions and therefore represents a

distinct departure from this model.













90.
AULMINUM



SE.BENNETT a R.P. MADDEN
E.J.ASHLEY 6. HASS
6M.SILVER -- -- LR.C EL

z ,IG.HASS





40 __ 0 .E-WYLO6 IS .

0 2 4 6 8 10 12 14 16 18 20 22
E(eV)

ALUMINUM


95 -DATA OF HE.BENNETT,
.. M. SILVER 8 EJ. ASHLEY







85 0.2 0.4 0.6 0 1 1 L8 1 0 22


E(eV)


Figure 2.4 The spectral dependence of the reflectivity of aluminum.
(From Reference 14.)















The results of a Kramers-Kronig analysis of the reflectivity

are shown in Figure 2.5. It can be observed that the real and imagi-

nary parts of the dielectric constant near zero frequency have the

behavior characteristic of a Drude free-electron metal. Around 1 eV

the effect of additive superposed structures can be seen, an effect

caused by interband transitions. As expected from the reflectivity,

the real part of the dielectric constant remains negative until 15 eV,

when the reflectivity begins to decrease. The imaginary part is seen

to be very small when eC passes through zero which is the condition for

strong plasma resonance. Figure 2.5 also shows the conductivity

a = a e2/4Tr and the so-called energy loss function Im 1. The latter

function represents the energy lost by fast electrons traversing a

thin film of a metal and can be shown to be very sharply peaked about

the plasma frequency when the conditions for plasma oscillations are

fulfilled. The peak value in the curve for Im C-1 in Figure 2.5 is

calculated at 15.2 eV and is in good agreement with characteristic

energy loss experiments.

The fact that in aluminum interband transitions do not set in

until about 1 eV makes possible the separation of the free-electron

contribution from the interband transitions. It is therefore possible

to fit the experimental results at low energies to Equation (31).

The results of this fitting to the experimental values of e1 and e2

allow the calculation of the plasma frequency and relaxation time

associated with the conduction band. At 0.4 eV the calculated values

are: hwPa = 12.7 eV and T = 5.12 X 10-15 sec.
Pa c













































2 -6

2- -4

1- -2

026 12 J I
E(eV)


Figure 2.5 The spectral dependence of the real and imaginary parts
of the dielectric constant the conductivity a, and the
energy loss function Inm C for aluminum. (From Ref-
erence 14.)














It is also possible to compute the dc conductivity. For the

Drude model the de conductivity is shown to be


2
Wpa T
Pa c
4 4 (34)


17 -1
Substitution into this equation yields a = 1.55 x 10 sec in com-
17 -1
prison to the experimental value 3.18 X 10 sec for bulk aluminum.

This rather large difference, a factor of about 2, is attributed to

the fact that reflectivity measurements pertain to thin films, or

alternatively that T decreases rapidly from its dc value as the fre-

quency is increased.

The value of the optical mass is also determined from w
pa
assuming there are three free electrons per atom. The value is

ma = 1.5. Since the calculated energy bands are so free-electron-like

it is clear that they are expected to lead to a smaller m The esti-
a
mated optical mass, mac, predicted by band theory yield a value close

to 1.15. The discrepancy between the two values is thought to arise

from the interactions neglected in the individual electron model.
(b)
Since 2 = 0 in the region near zero frequency where e2 is

largest, the separation into intra- and interband parts is most easily

achieved by calculating eb) from e2 (exp) e( and then using
(b)
Kramers-Kronig analysis to calculate 1 b)

The results for the interband contribution to the dielectric

constant are shown in Figure 2.6. These curves show quite clearly

that the important interband transitions, giving rise to structure in


















































E(eV)


Figure 2.6 The real and imaginary parts of e(b) () for aluminum.
(From Reference 14.)














the reflectivity, are confined to a small energy region surrounding

1.5 eV. The rapid decrease in magnitude of both dielectric constants

at higher energies indicates that any further interband transitions

can make only a small contribution.

In order to understand the interband effects in detail, it is

convenient to utilize the results of the energy band calculations for

this metal. The energy bands for the various symmetry axes are shown

in Figure 2.7. It can be shown [14] from Equation (33) that the con-

tribution to eCb) for a given W is proportional to the joint density
2
of states, times an average of lP /I2 over the surface w (k)=.

Structure ir. E2 is then expected to occur at critical points, i.e.,

where VP'km = 0. The critical points, in turn, generally occur at
k U
symmetry points where the gradients of E (k) and E ,(k) may vanish

separately due to symmetry.

The band calculations of Figure 2.6 show that the only inter-

band transitions below 8 eV at symmetry points are those between the

two occupied states W2,and W3 and the unoccupied state W1. The calcu-

lated transition energies are 1.4 and 2.0 eV, respectively. It can

also be noted that interband transitions in the vicinity of E also

set in at about 1.4 eV. A detailed study of the E(k) in this region

shows that a critical point exists on the [110] axis by virtue of the

fact that E and 1 bands below and above the Fermi level, respectively,

are parallel there. It is clear then that the regions around W and E

must contribute significantly to the large peak in e2.

























































Figure 2.7 The calculated energy bands of aluminum along the
symmetry axes. (From Reference 14.)















Band calculations predict that allowed higher energy transi-

tions at symmetry points set in at about 8 eV and above, whereas the

experimental 2 exhibits no structure in this range. It is argued

that, for the case of aluminum, the peak value of the contribution
-3/2
from a critical point is proportional to Eg where Eg is the energy

gap. This fret would account for the fact that no strong interband

contributions at higher energies are observed for this metal.

The outstanding feature at higher energies is the strong and

sharp peak in the energy loss function Im e- at 15.2 eV. This energy

is generally associated with the plasma resonance. It must be shown

that this value is consistent with the result hw = 12.7 eV obtained
pa

in the low-frequency range when free-electron effects predominate.

In the far infrared the dielectric constant it determined by the

free-electron expression (31), the contribution from interband processes

being relatively unimportant. Near the plasma resonance, w at about

15 eV, C2 is very small so that ap is determined quite accurately by

the condition that l(mp) = 0. This, together with the fact that

u T >> 1 at this frequency, implies that

2
pa (b)
1 + e (Wp) = 0 (35)
p


and hence that

w
pa
w [1 + C(b)














(b)
The contribution of Ce whose value is about -0.3, is therefore quite

important in determining the magnitude of w It is possible to deter-
p
mine a self-consistent solution of Equation (35), using the known value

of w and (b) as given in Figure 2.6. It is found that h w = 15.2 eV,
pa 1 p
in agreement with the location of the peak in the energy loss function.

The fact that w is greater than pa results from the negative value of
p pa
(b) which is a remnant of the oscillator-like dispersion associated

with the interband transitions near 1.5 eV.

It is interesting to note that tho dielectric constant can also
2 2
be approximated by its general asymptotic form C(m) = 1 4Tne2/mI2

where m is the free-electron mass and n corresponds to 3 valence elec-

trons per atom. Thus, the plasma frequency can be written in the form

previously used in connection with Equation (21), namely,
1
al = (4nne2 /m). Direct substitution into the preceding expression
P
yields h w = 15.8 eV, in good agreement with the earlier result of

15.2 eV. The negative value of (b) is therefore seen to raise the

value of the plasma frequency from its lower value pa as determined
2 '
by the optical effective mass, nearly to the larger value (4nne /m)"

required by the asymptotic behavior.

The Noble Metals. Figure 2.8 shows the reflectivity spectrum

of silver to energies up to about 22 eV. It can be observed that at

low frequencies the reflectivity is near 100 percent, already pointed

out for aluminum, which is characteristic of the free-electron range.

However, near 4 eV there is an extremely sharp dip, with the reflec-

tivity dropping to less than 1 percent in a very small energy range.















In fact, this dip is so narrow and the absorptivity so low at the

bottom of the dip that it is possible to construct a narrow bandpass
0
filter, using a thin film of silver (about 500 A) that will transmit

ultraviolet light in the vicinity of 315 nm.

At energies beyond the dip the structure is characteristic

of interband processes. Here the transitions are not confined to a

small energy region, as in the case of aluminum, but are rather widely

dispersed, as occurs with most metals. The reason for this is that in

the case of silver, and the noble metals in general, the d bands are

located close to the Fermi level and have oscillator strengths that

extend over a large frequency range.

It has already been pointed out that a sharp decrease in the

reflectivity is usually associated with a plasma resonance. Figure 2.9,

which shows the real and imaginary parts of the dielectric constant

and the energy loss function for silver, also exhibits a sharp peak

in the energy loss function near 4 eV. In this range el goes through

zero and C2 is very small as required for the existence of plasma

oscillations. The data shown in Figure 2.9 were obtained by Kramers-

Kronig analysis of the reflectivity data. For this analysis the reflec-

tivity curve was extrapolated beyond 22 eV so as to make the calculated

values of c1 and 2 agree with independent measurements of Schulz [16]

and Roberts [17] in the region near 1 eV.

It can also be observed in Figure 2.9 that e2 falls off rapidly

in the free-electron region, as expected from Equation (31), and

approaches a value very close to zero before the rise due to the onset














of interband transitions follows. This fact permits a relatively

unambiguous separation of e2 into its "free" and "bound" contributions.

That is, to a good approximation, it is assumed that below the onset
(b)
of interband processes 2 = 0, and that for energies above the crit-

ical energy e2 = 0. This fact may be used in conjunction with
(b)
Kramers-Kronig analysis to calculate C Once this contribution is
(f) (f) (b)
separated, (f)1 is obtained from e = 1 (exp) C The results

of this analysis is shown in Figure 2.10. The curves for f (W) are

found to fit accurately the description of the Drude region given by

Equation (31). The results also agree quite well with the measurements

of Schulz [16] and Roberts [17], giving a value for the optical mass

m = 1.03 and a relaxation time equal to that obtained from dc conduc-
a
tivity measurements.

In connection with the identification of plasma oscillations,

it is observed in Figure 2.9 that in the absence of interband transi-

tions, e1 would be given by (f) and e2 would vanish substantially for

frequencies greater than the critical frequency ni. The plasma reson-

ance would occur where s1 = 0, i.e., w = Wpa, the value calculated for

the density of free electrons having mass m given above. Figure 2.10
a

indicates that ha = 9.2 for silver. When interband transitions
pa
contribute, this simple picture is no longer valid, since the contribu-

tion of e(b) is such as to cause e1 to be zero in a region where 2 is

also small. That is, the increase in e, due to the onset of transitions

from the d band to the Fermi surface, causes e1 to vanish at an energy

just below i the threshold for these transitions. As shown in































Figure 2.8 Spectral dependence of the reflectivity of silver.
(From Reference 12.)


E l)


Figure 2.9 Spectral dependence of the real and imaginary parts of
the dielectric constant and the energy loss function
2 2
e /(e + e ) for silver. (From Reference 12.)
2 1 2

























































Figure 2.10 Decomposition of the experimental value of e1 for
silver into free and bound contributions
s) and (b) (From Reference 12.)
1 1














Figure 2.9, this occurs when e2 happens to be quite small and is thus

responsible for the sharp and well-localized plasma resonance and the

sharp peak in the energy loss function at this energy. This is not

a free-electron resonance but rather a hybrid resonance resulting from

the cooperative behavior of both the d and conduction electrons.

The investigation of the optical properties of copper follows

very much along the lines described for the case of silver. In partic-

ular, the separation of the dielectric constant into "free" and "bound"

contributions is carried out, using the same technique which employs

Kramers-Kronig analysis and the experimentally determined values of

l and e2 of Schulz [16] and Roberts [17] as a check to obtain a con-

sistent analysis of the data. On the other hand, the electronic band

structure of copper has been worked out by several investigators [18-20]

along the symmetry directions in k space. This allows the interpreta-

tion of the structure in the optical parameters to be analyzed in more

detail, particularly in regard to the assignment of structure to partic-

ular transitions.

The reflectivity spectrum of copper is presented in Figure 2.11

for er-rgies up to 25 eV. The calculated dielectric constant as well

as the energy loss function obtained from the reflectivity is shown

in Figure 2 12. Among the outstanding characteristics of the data for

copper, it will be noted that interband transitions in this metal set

in at about 2 eV, as can be seen from the sudden rise in e2 shown in

Figure 2.12. This in turn causes the reflectivity of copper to drop

from about 100 percent typical of the free-electron range, producing



































Figure 2.11 Spectral dependence of the reflectivity of copper.
(From Reference 12.)


Cu




.2




,4




-2









0 10 1p 20 2:


Figure 2.12 Spectral dependence of the real and imaginary parts of
the dielectric constant and the energy loss function
2 2
e2/(E + 2) for copper. (From Reference 12.)
2eeec 12.)














a sharp shoulder at this energy. In contrast to silver, the d band

in copper and gold lies closer to the Fermi level by about 2 eV.

In the latter metals the d bands are located about 2 to 2.5 eV below

the Fermi level. Therefore, the onset of interband processes occur

approximately in the middle of the visible spectrum, producing a drop

in the reflectivity which is responsible for the yellowish color of

copper and gold.

Another feature of the optical data of copper, associated with

the position of the d band, is the fact that the onset of interband

transitions takesplace when both e1 and 2 are relatively large in

magnitude and the strength of these transitions is not large enough

to force Cl through zero at a smaller energy. This point is illus-

trated in Figure 2.13, where the free and bound contributions to the

real part of the dielectric constant are shown separated. The free-

electron part corresponds to m = 1.42 and a relaxation time equal to
a
that obtained from dc conductivity measurements, namely,
-14
T = 3.5 X 10 sec. It can be seen that with the free-electron

contribution only, e passes through zero at h a = 9.3 eV. When
pa
the interband part is taken into account, this point moves to about

7.5 eV. However, the conditions for strong plasma resonance are never

fulfilled in copper and the energy loss function, shown in Figure 2.12,

does not show a sharp and well-localized plasma resonance like the one

observed in silver.

While the above interpretation of the optical properties of

the noble metals appears to be essentially correct, the original


























































Decomposition of the experimental value of e for copper
into free and bound contributions (f) and (b)
(From Reference 12.)
(From Reference 12.)


Figure 2.13















assignment of the optical transitions made by Ehrenreich and co-workers

(based on Segall's band calculations) is at variance with results of

later calculations. Benglehole [21] has made careful measurements on

the optical parameters of copper over a range up to 28 eV. His results

are in good agreement with those of Ehrenreich, but Benglehole suggests

that a definite assignment to particular transitions should wait for

better understanding of the electronic structure of copper. On the

other hand, Mueller and Phillips [22] made theoretical calculations of

the optical spectrum of copper. Their results appear to agree well

with Burdick's band calculations [19] for this metal and furthermore

predict optical transitions in agreement with suggestions made by

Benglehole. It should also be pointed out that the energy gaps at

symmetry points calculated by Segall differ from those calculated by

other authors [19,20] by as much as 20 percent. It would appear that

Burdick's band calculations on copper and the optical density of states

derived from his work fit experimental optical data a great deal better

than do the calculations of Segall. More recently, Fong et al. [23]

have made optical studies of copper, using wavelength modulation.

They have also made band calculations [20] that agree closely with

those of Burdick [19] and the interpretation of their experimental data

is consistent with the band structure.

The assignment of transitions at low and moderate energies in

copper appears to be well established at the present time. Figure 2.14

shows the electronic band structure for this metal as obtained by Burdick.

With reference to the structure shown in Figures 2.11 and 2.12, the


















































Figure 2.14 Calculated band structure of copper along symmetry
axes. Arrows indicate interband transitions produc-
ing structure in the optical constants. (From
Reference 19.)














sudden shoulder at 2.1 eV is associated with transitions from the top

of the d band t) the Fermi surface near X and L, whereas the structure

at about 4 eV is identified with the transitions X to X4 and L 2 to

L1. All these transitions are indicated by arrows in Figure 2.14.

The identification of optical transitions for higher energies

is more difficult, since the number of possible transitions is greatly

proliferated. In contrast to the case of aluminum and most semiconduc-

tors, the oscillator strengths in the noble metals appear to be widely

distributed in energy. Some appreciation of this fact can be obtained

from consideration of the sum rule yielding neff, the effective number

of electrons per atom contributing to the optical properties over

a given frequency range. This sum rule [10] is a general expression

obtained from the Kramers-Kronig relations which can be written as




J w C2(w) dW = 22 n e2(36)
Om



where n is the total density of electrons. If the core states in the

noble metals can be neglected over the entire far vacuum untraviolet

range, then a numerical evaluation of n, using the experimental values

of E2 in Equation (36), should saturate at a value corresponding to 11

electrons per atom (Is plus 10 d electrons) and the sum rule is said to

be exhausted. An empirical modification of Equation (36) can be made

if the integration is carried to a frequency rather than to infinity.

The calculated value of n gives therefore a measure of the electrons

taking part in the optical processes to the frequency wo and the
0














rapidity with which neff approaches its saturation value as a function

of energy gives some information concerning the distribution of oscil-

lator strengths. Figure 2.15 shows nef as a function of energy

obtained from numerical integration of the experimental 2 by Ehrenreich

et al. for the noble metals [12,15]. The singularity of e2 at zero

frequency causes neff to rise very rapidly to a value near unity. The

deviation from unity is due to departures of the effective mass value

from the free-electron mass. Figure 2.15 shows that n essentially

saturates at the free-electron value and is nearly constant until the

threshold for interband transitions is reached. The fact that neff

increases very slowly and is for copper and silver still far removed

from the expected saturation value of 11 at 25 eV indicates that the

oscillator strengths for transitions involving d and s electrons are

distributed over a rather wide energy range.



































'i 3 4 5 v6
1 .. -, --.



0 I 8 12 16 20 eV 214




Figure 2.15 Effective number ,of electrons per atom as a function
of energy obtained from numerical integration of 2
for silver, copper and gold. (From Reference 9.)


















CHAPTER III


EXPERIMENTAL PROCEDURE



3.1 The Apparatus

The differential reflectometer developed in this work is

described in this chapter. The instrument fulfills the requirements

set forth in the design; namely, the measurement of small differences

in reflectivity between two specimens at room temperature in the

spectrnl range from 200 nm to 700 nm. The discussion can be sepa-

rated conveniently in two parts: the first part contains the modu-

lating unit and various optical components; the second part is formed

by circuits and instruments arranged for the reduction of the data

and the recording of the changes in reflectivity as a function of the

incident light. The technique of compositional modulation and a

first version of the apparatus for differential reflectivity mea-

surements has been reported [24] in a preliminary paper.

Figure 3.1 shows a schematic diagram of the optical part of

the apparatus. Light that comes from a monochromator passes through

lens 1 and falls onto the mirror M which performs small oscillations

at the line frequency, i.e., 60 Hz. As a consequence the light re-

flected from this mirror illuminates alternately two specimens mounted

side yv side and which have a different composition. After reflection






























o

at O
'-0E

UK


w


E .0
oF-
0
0.


Ii





'4g














from the specimens the light goes through lens 2 and then into a photo-

multiplier tube (PMT).

If R1 and R2 represent the reflectivities of each of the speci-

mens and if I. is the intensity of light at n particular wavelength \,

then the light reflected from the specimens will have intensities I R1

and I R2. The reflectivities of the sample being in general different,

the output voltage from the PMT exhibits small steps corresponding to

the light being reflected from each of the specimens. This is indi-

cated by el and c2 in Figure 3.1.

The geometry of the optics is such that lens 1 images the exit

slit of the monochromator onto the samples so that the light spot that

scans over the specimens is a fine line parallel to the boundary sepa-

rating the alloys. Lens 2, on the other hand, is placed so as to give

a magnified image of the vibrating mirror on a ground plate of fused

quartz a short distance in front of the face of the photomultiplier

tube. This plate is necessary because the sensitivity of the photo-

cathode is not uniform over its entire surface. The light passing

through the ground plate is diffused in all directions, thereby min-

imizing the effect of the slight motion of the light spot that results

from the motion of the oscillating mirror.

It was shown in preliminary experiments that the response of

the PMT to the incident radiation is linear over a wide range of

intensities. In addition, the PMT has a very small dark current.

Therefore it is possible to express its output voltage e as


e = f(V,A) A IX R,















where A is the effective area of the exit slit and f(V,k) is some

unknown function of the wavelength X, which represents the spectral

sensitivity of the detector, and the voltage V applied to the detector.

It therefore follows that the step (e e2) is proportional to the
1 2
difference in reflectivities (R R2) between the two alloys and can

be written as


e e2 = f(V,X) A I (R R2) (38)


A high-pressure xenon lamp was used as a light source, which

covers the spectrum from the infrared (1000 nm) to the near ultra-

violet (200 nm) with an output intensity at 200 nm of about 1 percent

of the maximum output of the lamp.

The light coming from the light source is focused into the
**
entrance slit of a plane grating monocheomator that covers the range

from 200 nm to 700 am. The specifications given by the manufacturer of

this grating are a dispersion of 7.4 nm/mm, a blaze at 210 nm, and

a typical scattered light of 0.2 percent. With the slits opened at

about 0.3 mm the resulting dispersion at the exit slit is approx-

imately 2 nm which is quite adequate considering that the structure

to be resolved by the instrument is a great deal larger than the spread

in wavelengths. The scattered light, however, can be a problem at the

ultraviolet end of the useful spectrum, because here the output from

the light source is very small. This may be appreciated by writing


Model 613. McPherson Instrument Corporation.
**
Model 33-86-07. Bausch & Lomb, Inc.


_ _














for the light intensity from the exit slit


I' = Io + I (39)



that is, the output is the sum of a "pure" monochromatic component I

plus a "white" background I For most of the useful range the light

output from the lamp is large and I is negligible. Near the short-

wavelength end, however, I is quite small and the stray light becomes

increasingly disturbing. Since the output voltage from the PMT results

from both terms in Equation (39), an estimate of the relative magnitudes

of these two terms can be obtained by removing one of these components

from the light beam by an appropriate filter. In the ultraviolet,

where the background light is important, this can conveniently be done

by inserting n glass plate into the light beam. This plate blocks off

the ultraviolet, allowing most of the background to pass on to the PMr.

Therefore, by noting the readings from the PMT with and without the

glass plate for various wavelengths, it is possible to measure the

fraction of the PMT output which is due to strny light. Figure 3.2

shows the results of measurements carried out in this way. It can be

seen that at 200 nm the background light is responsible for about

15 percent of the PMT output and it follows, therefore, that results at

this end of the spectral range must be interpreted with caution.

The photomultiplier is of the S-20 "extended" type with fused

silica entrance window. The spectral sensitivity of the tube covers


EMI 9659.











e./e' (%)




60-




50-





40-




30-




20-





10





280 260 240 220 200 180
-A (rim)

Figure 3.2 Percentage of stray light from the monochromator as a
function of wavelength as measured by the voltage output
from the photomultiplier tube. e' is the output without
filter, e is the output with a glass plate across
the beam.














the range between about 900 nm to about 160 nm which matches well with

the monochromator range.

Figure 3.3 shows the spectral distribution of the xenon lamp

multiplied by the sensitivity of the photomultiplier tube as measured

by the PMT voltage output. It can be seen that the source contains

a few peaks around 450-500 nm which is characteristic of the xenon

emission lines, but otherwise it is a fairly continuous light source.

The vibrating mirror was constructed from a dc moving coil

galvanometer by dismounting the face and bending the pointer upward

into a vertical position. A small front face aluminum mirror was then

attached to the pointer. When a small ac voltage is applied to the

galvanometer, it vibrates at the line frequency. The amplitude of the

vibrations can be adjusted by changing the applied voltage. This

amplitude is normally set so as to produce a light spot on the speci-

mens that scans over a span of approximately 3 mm. In addition, the

center of oscillations can be varied by a small dc bias to the moving

coil which allows the scanning beam to be centered on the optical axis.

The specimens are placed on a moving vertical plate that can be

displaced on a plane parallel to the reflecting surfaces by means of

two micrometer screws. This is done in order to select the areas of

the specimens that are to be examined and also to center the boundary

between the specimens on the scanning beam.


3.2 Data-Reducing Circuit

The purpose of data-reducing circuit is to process the signal

from the PMT in order to obtain a graph of the changes in reflectivity

as a function of wavelength.

















































. .. ... a i i .
6bd:1-40)-




Spectral distribution of the xenon lamp multiplied by
the sensitivity of the photomultiplier tube.














The output e from the PUT can be split into two channels:

a dc channel with a signal proportional to R (where R is the average

value of R1 and R2) and an ac channel proportional to AR. The ac

channel, which contains the modulation in the reflectivity, is fed

into a phase sensitive amplifier (also referred to as lock-in amplifier).

This amplifier delivers an output which is proportional to the RMS value

of the fundamental frequency component of the input signal. That is,

its output is a dc signal proportional to Ae, or equivalently to AR.

This signal could be fed directly to the y-axis of a recorder.

However, as can be seen from Equation (38), this graph would contain,

in addition to the signal AR of interest, the structure in the light

source through the factor I and also the particular spectral sensitiv-

ity of the detector. Moreover, fluctuations in the light source,

which are always present, show up in the graph as background noise.

It is very desirable, therefore, to normalize the signal by

dividing the output from the lock-in amplifier by the average value

e = (el + e2)/2. This way the factor f(V,X) A Ix cancels out.

It follows that


el e2 R1 -R2

e R


Problems associated with the instability of the source are avoided if

the normalization is performed with an "on-line" computer element.

However, it is not easy to construct a simple electronic analog circuit

to carry out this division. The large dynamic range required and over-

loading of the PMT are two of the problems that arise.














One possible analog system [25] is obtained by feeding the sig-

nals from the lock-in amplifier and the PMT into two logarithmic con-

verters and the outputs of the converters to a differential amplifier

so as to obtain log AR/R.

A better solution to the normalization has been developed,

using operational amplifiers and a divider component. A block diagram

of the circuit is shown in Figure 3.4. The PMT output is fed into the

lock-in which picks up and amplifies the modulated part and becomes

the numerator channel. The PUT output is also fed into a dc amplifier

which includes a capacitor in its feedback loop. The signal is there-

fore filtered to a dc signal proportional to R = (R1 + R2)/2 becoming

the denominator channel. This denominator amplifier is wired so as to

satisfy several other requirements: it presents a very high impedance

to the PMT, drawing very little current from it; it changes the sign

of the PMT signal, converting it into a positive value as required by

the divider circuit; it also permits the adjustment of the voltage gain

in order to operate the divider in its optimum range.

The attenuator in front of the lock-in amplifier (see Fig-

ure 3.4) is necessary because of the relatively high level signal from

the PMT. The maximum input level that can be fed into the lock-in

amplifier is 50 mV, whereas the PMT output is typically a few volts.

The output from the lock-in could be fed directly into the numerator

of the divider. It is, however, convenient to reduce the level of the

lock-in output to match the input requirements of the divider. The dc

amplifier shown in the numerator channel acts therefore as an attenua-

tor, the gain of which can be adjusted by a potentiometer in the feed-

back loop.




































0
14








S..
4D
a







*0



0
'-4

w
0


o U














Figure 3.5 shows the circuit diagrams for the dc amplifiers

and the divider circuit. The function performed by each amplifier

is also indicated next to each circuit.

The divider output is proportional to the ratio of the two

input signals,

RI -R2





and this is connected to the y-axis of an x-y recorder. On the other

hand, a potentiometer is attached to the monochromator in a way that

when the monochromator scans over its entire wavelength range, a small

dc voltage proportional to X is produced to drive the x-axis of the

recorder. The recording operation is therefore performed in an auto-

matic way.

In order to monitor the PMT signal, an oscilloscope is con-

nected directly across the PMT output. This permits readings of the

voltage outputs and a direct determination of the percentage varia-

tion of the reflectivity of the samples.

Since the output from the divider is the ratio of two

processed signals, attention must be paid to the time constants intro-

duced into each signal. The filtering capacitor in the denominator

channel introdu-ces a time constant to this signal. On the other hand,

the lock-in amplifier in the numerator channel has an adjustable time

constant which can be varied in steps from 0.1 m sec to 30 sec. For

a proper performance of the instrument it is convenient to have these

two-time constants equal. An indication of any mismatch in time


















Numerator


e ---Z/R, elm










Denominator

---- +
e, -- R ,/R e,,











Divider


e =-10 e,/e,


Figure 3.5 Wiring diagrams for the dc amplifiers and divider.
The function performed by each element is indicated
next to each diagram.














constants can be obtained by suddenly increasing the applied voltage

to the PMT and noting the motion of the recording pen. If the time

delay introduced into each channel is equal for both, then no motion

of the recording pen is observed. On the other hand, if, say, the numer-

ator channel has a larger time constant, the pen will move to small

values upon increasing the applied voltage. Therefore, the mismatch

can be eliminated by adjusting the lock-in time constant until no

motion in the recording pen occurs when the applied voltage is suddenly

changed.

Perhaps the most outstanding advantage of the divider is that

the applied voltage to the PMT can be adjusted during recording. This

is very convenient because the light intensity from the source is very

small in the very important ultraviolet region of the spectrum.

In fact, the output from the monochromator tails off rapidly near

200 nm (Figure 3.3). By increasing the applied voltage in the ultra-

violet region it is possible to keep a nearly constant output from the

PMT, thereby lifting the signal from the background noise. Moreover,

the divider is kept within a small dynamical range where its perfor-

mance is optimum. This permits the extension of tha useful range of

the instrument to about 200 nm which corresponds to almost the limit

that can be achieved without vacuum.

If the applied voltage to the PMT were to be varied so as to

keep the output precisely constant, using a servomechanism, then the

denominator would be constant at all times during recording. This

means that the divider circuit would not be necessary. This is in fact














another way to obtain AR/R [26]. This is not, however, a very simple

method and its design requires close attention to transient effects

within the power supply.


3.3 Sample Preparation

The alloys are prepared from high purity metals, encapsulated
-5
in Vycor glass in a vacuum of approximately 10-5 Torr and melted in an

induction furnace. After cooling the specimens are cold worked and

homogenized for about two days at a temperature somewhat below the

solidus line. In order to make reflectivity measurements with the

instrument, two specimens having different compositions are cold

mounted side by side with no gap in between. The specimens are then

mechanically polished with standard metallographic techniques down to

1 diamond compound. Reflectivity measurements are carried out within

a few minutes after final polishing.

Since mechanical polishing leaves a layer of cold worked

material on the surface of the specimens, preliminary measurements

were conducted in order to get some appreciation of the effect of

mechanical polishing on the reflectivity. Two specimens of high pur-

ity copper were mounted side by side; one was heavily cold worked,

while the other was annealed. The composite was mechanically polished

in the way described above and subsequently electropolished to remove

a few microns of material, thereby leaving the two specimens with the

same surface treatment but one cold worked and the other annealed.

Examination of these specimens with the differential reflectometer

revealed no significant difference in reflectivity over the range of










69



the instrument. Therefore, mechanical polishing appears to be a satis-

factory means of surface preparation at least in the range from 200 nm

to 700 nm. There are indications, however, that in the vacuum ultra-

violet surface preparation becomes very critical [21].

It should be mentioned 'at this point that the differential

reflectivity measurements are made on two samples which have similar

composition. Since these samples receive the same polishing, the

error, if any, due to surface distortion (and also due to oxidation)

is approximately the same. Even if the absolute values of the reflec-

tivities are changed by surface effects, the difference in their

reflectivities is probably not affected by significant amounts.


















CHAPTER IV


RESULTS AND DISCUSSION



4.1 General Remarks

The results obtained by compositional modulation of copper-based

alloys containing additions of nickel, aluminum and zinc are presented

and discussed here. The alloys of copper with aluminum and zinc have

been kept within the c-phase region. At first a general description

on the main features of a "differential spectrogram" of two alloys

containing different amounts of a second phase X will be given. In

Figure 4.la the reflectivities of two alloys are schematically plotted

versus the photon energy of the impinging light. The spectral reflec-

tivities of the alloys show a marked decrease in the low energy region,

as copper does. In this schematic diagram it is assumed that the onset

of this decrease (marked A and C, respectively) is shifted in alloy 2

towards higher energies.

The differential spectrogram (AR/R versus photon energy,

Figure 4.1b) shows maxima and minima at energies where the difference

between the two reflectivities is largest, i.e., at places where

"structure" in the spectral reflectivity is found. For example,

a maximum in AR/R can be seen at that energy, where the slope of the

spectral reflectivity is largest. In the following, actual results

of differential spectrograms are given and discussed.


Here R stands for the average of R1 and '2.

70
















Alloy


Alloy 2:N--


Photon energy -


Figure 4.1 (a) Schematic representation of the spectral reflectivities
of two alloys (marked Alloy 1 and Alloy 2). (b) Differ-
ential spectrogram for the two alloys referred to in (a).














4.2 Copper-Zinc Alloys

Reflectivity measurements were carried out in a-copper-zinc

alloys with the aim to investigate the changes in its band structure

as a function of composition, in particular the behavior of the main

absorption edge around 2.1 eV and absorptions near 4.0 and 4.4 eV.

This alloy system is of particular interest because parts of the band

structure for u-brass are known [27] (Figure 4.2) which provides a basis

for comparison with our experimental data. For calculation, the Kohn-

Rostoker method in combination with a "pseudoperiodic" potential was

used. The bands of a copper-zinc alloy containing 30 at. % Zn have

been calculated [27] at high symmetry points of the Brillouin zone with

the assumption that the copperd bands are unchanged by alloying. If

one compares the bands of pure copper as calculated by Segall [18] with

the alloys bands (Figure 4.2), one can observe a similar shape. The

Fermi level., however, has changed relative to the d bands. This can

be understood by considering the fact that zinc has essentially one

extra electron compared to copper. This has the effect of raising the

Fermi level by filling empty states (rigid band behavior). A predom-

inant effect due to alloying is therefore an increase of the energy gap

between the top of the d bands and the Fermi surface. Of significance

also is the relatively large displacements of the L1 and X4/ levels

because of the shift of the s and d bands to lower energies. This

results in reduced L21 -L1 and X5 -X4 energy gaps.

Figure 4.3 shows four differential spectrograms (AR/R vs photon

energy) for Cu-Zn alloys varying from a medium composition of 2.25 at. %


































AA.





LI
Li /



Lr rA s A




Le Xi





L A r A X


Figure 4.2 Comparison of energy bands of Cu-30
cuirves) with those of copper (solid
Reference 27.)


at. % Zn (dashed
curves). (From


0.6-


0.4-


0.2 -

O.-



*0.2-


-0.4 -


-0.6 -


-0. -














Zn to 11 at. % Zn. The energy where AR/R starts to increase (marked

T in Figure 4.3) is 2.1 eV. We interpret this energy as the threshold

energy for optical interband transitions,

The maximum marked A in the differential spectrograms is shifted

to larger energies with increasing concentrations of zinc. We ascribe

this peak to the 5 A1 transition (Figure 4.2). According to the

considerations mentioned above, this energy gap increases with increas-

ing zinc concentration due to a raise of the Fermi level. For c-brass

with 25 at. % Zn a total displacement of point A of 0.5 eV has been

measured. The calculated increase of this energy gap was found to be

0.48 oV [27] which is in good agreement with our optical measurements.

The minimum B and the maximum C in the differential spectro-

grams of Figure 4.3 are shifted to lower energies with increasing zinc

concentration as the ,L2, L and the X5 X4' energy gaps (see above).

We associate therefore peaks B and C with these critical interband

transitions.

It should be emphasized at this point that there is no way of

knowing exactly that a certain maximum or minimum in a differential

spectrogram is caused by a very specific interband transition. It is

known, however, that optical interband transitions cause structure

in the function e2 = f(E) and therefore most probably cause structure

in R=f(E) (see Chapter II). Any structure in the spectral reflectiv-

ity curve shows up in a differential spectrogram AR/R = f(E) as an

extremum point. This may be seen by considering the equation


And all the transitions from the 5th band to tte Fermi surface.








Cu-Zn
75
A
2.25 ev
AR C
R 4.38 ev
(%)
2.27ev 4.25ev

10 +10




0 +0

3.80 ev
B
10 3.70ev

0/5 2.5/5
700 500 300 700 500 300


,R 3.83ev
'2.30ev
Q,) 4.13ev
10 +10 2.40ev




0



3.65ev -10
0 3.30ev


7.5/10 10/12
700 500 300 700 500 300
A (nm) A (nm)

Figure 4.3 JR/R vs wavelength for various copper-zinc alloys. The
first number on the lower left corner of each graph gives
the zinc concentration of specimen 1 in at. %, while the
second number gives the corresponding information for
specimen 2. Positive values of MR/R indicate that speci-
men 2 has higher reflectivity.










-76


de 2(w) dEc de2(w)
2 dC dC d40)
cr


where 2 is the imaginary part of the complex dielectric constant

(which is directly related to R) and Ecr is the critical energy for

optical interband transitions. The compositional difference between

the two samples, AC, is assumed to be small and constant. The first

factor on the right-hand side of Equation (40) is independent of w and

is a constant for a particular material and transition. With increas-

ing photon energy hw the second term is expected to peak at E cr

since C2 usually exhibits a singularity at this energy [28].


4.3 Copper-Aluminum Alloys

Another interesting system to study changes of the band struc-

ture by alloying is represented by the copper-aluminum alloys. The

band structure of aluminum is well understood and Cu-Al alloys can be

considered to be a prototype of a noble metal alloyed with a metal of

nearly free-electron-like band structure.

Figure 4.4 shows .data for the change in reflectivity upon

adding 1 at. % Al to Cu. As in the previous case, one can distinguish

the threshold energy T and various peaks designated A through C.

These peaks broaden as aluminum is alloyed to copper.

In Figure 4.5 the shifts of the aforementioned peaks are pre-

sented as a function of composition. It can be seen that similar to

the case of a-Cu-Zn alloys, peak A (which we ascribed to the A 1

transition) is shifted to higher energies when aluminum is added to












AR (o/.) For Cu/Cu-Al1/o
R


Wavelength A nm


Figure 4.4 Spectral change in reflectivity upon adding 1 at. % Al
to Cu. Positive values of AR/R indicate that the alloy
has higher reflectivity than pure copper.


















ENERGY (or)
A


C, ~A


0 2 4 6 I S 2 '14

at~. Al in Cu



Figure 4.5 Shifts in the peaks marked A, B and C in Figure 4.4 as
a function of concentration of aluminum.


`c"














copper, whereas peaks B and C are shifted to lower energies. The

interpretations of these shifts have to follow a similar line as

described in the previous section.


4.4 Copper-Nickel Alloys

Often regarded as characteristic of the many alloy systems

between a noble metal and a transition metal are the Cu-Ni alloys.

They are ferromagnetic over more than one-half of the composition

range and are substitutional solid solutions over the entire range.

These considerations, and the fact that copper and nickel. are among

the best understood of the noble and transition metals, has stimulated

research of the Cu-Ni system. Early measurements on Cu-Ni alloys led

Mott to propose the rigid-band model. This model assumes that there

is one electronic density-of-states function which is the same for

copper, nickel, and Cu-Ni alloys, with this density of states filled to

an energy level determined by the ratio of electrons to atoms. Many

subsequent measurements, however, have questioned the validity of the

rigid-band model for Cu-Ni alloys and alternative models have been

suggested. In particular, the virtual-bound-state model, developed by

Friedel [29] and Anderson [30] appears to be applicable to the Cu-rich

Cu-Ni alloys.

These two models predict quite different behavior for the

density of states in Cu-Ni alloys. This is illustrated in Figure 4.6.

The drawing on the left illustrates schematically the density of states

of pure copper which is characterized by a high density of d states for

energies greater than 2 eV below the Fermi level (E ), and a relatively





















- EF

2 ev


Rigid-Band Model


.Virtual-Bound-State
Model


Figure 4.6 Illustration of the density of states in copper and
copper-nicke? alloys, showing the behavior expected
from rigid-band and virtual-bound-state models at
15 at. % Ni in Cu.


;1- EF
1.4 ev


Pure Co














lower density of s and p states between the d bend and EF. The rigid-

band model for Cu-Ni alloys assumes that when nickel is added to

copper, electrons from the s- and p-derived states fill the nickel d

shell, thereby decreasing the separation between the Fermi level and

d band. For 15 at. % Ni in copper, the alloy density of states would

be as shown schematically in the central drawing of Figure 4.6, the

separation between EF and the upper edge of the d bands would be

decreased to about 1.4 eV, the total density of states remaining essen-

tially unchanged.

The virtual-bound-state model, on the other hand, assumes that

the impurity 3d orbitals of nickel form levels, highly localized around

the impurity atoms, that are broadened in energy around Ed, the center

of the state, through a resonant scattering interaction with the nearly

free-electron-like band of the host metal copper. The resultant density

of states expected is shown schematically in the right of Figure 4.6 for

15 at. % Ni in copper. Copper and nickel d electrons would form essen-

tially independent levels, the energy separation between copper d states

and EF would be unchanged, and the nickel d electrons would cause an

increase in the alloy density of states between the copper d band and

Fermi surface.

To test the validity of one or the other of these two models,

the following experiments were undertaken. Reflectivity measurements

with the differential reflectometer were carried out in copper-nickel

alloys in the composition range from 0.5 at. % Ni to about 15 at. % Ni.

In Figure 4.7 six differential spectrograms for various copper-nickel































9r3 r4
0 W *4


E rl 1
a) OH



0 m 4
o 0 0 H

r C a *H
*P T

1 0 U
0 H 0. #

O ( "'H P


0 *r 0.4


0 4. c *r
0. 0 H'



-0 4J0 *r
4H O 0
44a o -
Mr r
$4 0. 0 0O
> 4')H U*





0 M A 0
d 4.) E d











fk 0













83













i

2


U,




0

+ I 4






z C

0 0 r





U


ZN


U,



4 H '4















alloys are shown. It can be seen that the edge at 2.1 eV (590 nm)

in pure copper does not move to lower energies when nickel is added

to copper as would be expected on the basis of the rigid-band model.

The virtual-bound-state model, on the other hand, predicts that

this edge is unaffected by alloying. Moreover, this model predicts

a drop in the reflectivity at the red end of the spectrum from the

near 100 percent value in pure copper. The reflectivity measurements

(see Figure 4.7) show that this di ip does take place; from the graph

marked 0/10 it can be seen that the reflectivity at low energies

decreases by about 20 percent with the addition of 10 at. % nickel

to copper. This is expected because the d states from nickel, local-

ized in the gap between the copper d band and the Fermi surface, provide

a higher density of states spread in the energy gap, allowing transi-

tions to take place between these bands and the Fermi surface, thereby

absorbing energy from the Incident radiation. Theoretical calculations

made by Ehrenreich and co-workers [31] on the Cu-Ni system and photo-

emission studies of this alloy made by Spicer [32] give further sup-

port to the applicability of the virtual-bound-state model to copper-

rich Cu-Ni alloys.










*
The wavelength can be converted to energy of the radiation
with the expression E = 1240/X. With X in nm, the formula gives
E in electron volts (eV).














4.5 Effect of Inhomogeneities

In order to determine the amount of compositional inhomogeneity

of the samples and assess its effect on the optical properties, tests

were conducted on Cu-Ni alloys using the microprobe. Two specimens

of the same average composition, namely, Cu-15 at. % Ni, were mounted

side by side in the manner used for differential reflectivity measure-

ments. One specimen was as cast and the other cold worked and homogen-

ized at about 10000C for 24 hours. The specimens were then carefully

polished. Microprobe analysis (wit a spatial resolution of about 2p)

showed that there was a measurable difference in composition across the

surface of the as cast sample. The homogenized sample, on the other

hand, showed a very uniform composition. The chart recording from the

as cast sample is presented in Figure 4.8, showing a compositional

variation between 79 at. % Cu and 89 at. %Cu. Figure 4.9 shows the

corresponding data for the homogenized sample.

Of particular interest was the determination of any correla-

tion between the microprobe analysis and reflectivity measurements.

The same specimens were therefore examined with the differential

reflectometer. The measurements showed a difference in reflectivity

between 1 and 2 percent. The results are shown in Figure 4.10 where the

higher reflectivity corresponds to the as cast specimen.

Having these results in mind, particular care was taken in the

aforementioned experiments to properly homogenize the various alloys

before measurements.















89
79--- -- -- *


AS CAST




75p


Figure 4.8 Compositional variation of as cast Cu 15 at. % Ni
alloy as revealed by microprobe examination.






83 .


Figure 4.9 Compositional variation of homogenized Cu-15 at. % Ni
alloy as revealed by microprobe examination.



2






12 I -_-
1 -- ---- -


700 600 500


400 300


Spectral change in reflectivity between as cast and
homogenized Cu-15 at. % Ni alloys.


Figure 4.10
















BIBLIOGRAPHY


1. J. J. Gilman and W. A. Tiller, Atomic and Electronic Structure
of Metals, American Society for Metals, 1967, p. v.

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BIOGRAPHICAL SKETCH


Juan Alfaro Holbrook was born September 22, 1930, in Talcahuano,

Chile. He attended the State Technical University in Santiago, Chile,

where he graduated in 1954, with the degree of Electrical Engineer.

From 1958 to 1961, the author studied at the University of London,

England, where he graduated with the degree of Bachelor of Science

(Special) in Physics. From 1963 to 1965, the author attended the Grad-

uate School of Cornell University and graduated with the degree of

Master of Science with a major in Physics. Mr. Holbrook entered the

Graduate School of the University of Florida in September, 1968, and

has pursued the degree of Doctor of Philosophy since that date.

The author has served on the faculty of the Engineering School

of the State Technical University of Chile as a Teaching Assistant

since 1955, as Assistant Professor of Physics since 1961, and ac

Associate Professor of Physics since 1965.

Juan Alfaro Holbrook is married to the former Giovanna Ravetti

of Alessandria, Italy. He is the father of three children, David,

Maria and Andrea. He is a member of the American Physical Society,

the American Society for Metals, the American Institute of Mining,

Metallurgical and Petroleum Engineers, and Alpha Sigma Mu.














I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.




Rolf E. Hummel, Chairman
Associate Professor of
Metallurgical Engineering



I certify that I have read this study and that in my opinion it
,conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.




Kenneth R. Alien
Assistant Profefsor of Physics



I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.




Joh J. Hren
fessor of
Metallurgical Engineering



I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.




Rayoeid Pepinsky
Professor of Physics














This dissertation was submitted to the Dean of the College of Engineering
and to the Graduate Council, and was accepted as partial fulfillment of
the requirements for the degree of Doctor of Philosophy.


August, 1972


ea llege of Cngineering




Dean, Gradefite School


UNIVERSITY OF FLORIDA


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