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
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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 DataReducing Circuit .............. 60
3.3 Sample Preparation . . . . . . . ... .68
IV RESULTS AND DISCUSSION . . . . . . .... 70
4.1 General Remarks . . . . . . . . .. 70
4.2 CopperZinc Alloys . . . . . . . ... 72
4.3 CopperAluminum Alloys .... . . . . .... 76
4.4 CopperNickel 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 facecentered 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 e1 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 Cu30 at. % Zn with
those of copper . . . . . . . .... . 73
4.3 AR/R vs wavelength for various copperzinc 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
coppernickel alloys, showing the behavior expected
from rigidband and virtualboundstate models at
15 at. % Ni in Cu . . . . . . . . . 80
4.7 AR/R vs wavelength for various coppernickel 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 copperbased alloys at room temperature in the spectral
range between 1.8 and 6.2 eV.
For coppernickel alloys, a decrease in reflectivity in the
lowenergy 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
boundstate 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
nlphacopper zinc alloys. The agreement between theory and experiment
was satisfactory.
Results for alphacopperaluminum alloys were similar to those
for the copperzinc 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 socalled 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 threedimensional 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
virtualbound state and rigidband 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 coppernickel
alloys it is found quite conclusively that a virtualbound model applies
to copperrich alloys, in agreement with photoemission studies of these
alloys. Measurements on the copperzinc 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 copperaluminum 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 rigidband 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(nl)/(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
4Tcr 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 onedimensional form. With the direction of vibra
tion of the electric field along the xaxis and the direction of propa
gation along the zaxis, the equation takes the form
82E aE a2E
x 4Tra x x
+=  (9)
z2 2  2 t2 (9)
az c c at
A planewave 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
(nik) = 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 righthand 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 righthand 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 righthand 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 frequencydependent 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 socalled 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 positiely 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 socalled
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 Braggreflected 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 freeelectron
bands shown in the figure by the dashed curves. The freeelectron
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 freeelectron
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 facecentered cubic structure show
ing various points and lines with symmetry properties.
this figure is the fundamental unit cell of the facecentered 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 freeelectron 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 twoquantumlevel 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
freeelectron 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 socalled "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 fluidlike 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 frequencydependent 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 KramersKronig [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 freeelectron 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 freeelectron 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
socalled "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 randomphase 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 freeelectron 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 coworkers [1215] 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 freeelectron metal. The reflectivity for an ideal Drude
freeelectron 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 .EWYLO6 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 KramersKronig 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 freeelectron 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 socalled 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 C1 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 freeelectron
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 1015 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 freeelectronlike
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)
KramersKronig 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 lowfrequency range when freeelectron effects predominate.
In the far infrared the dielectric constant it determined by the
freeelectron 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 selfconsistent 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 oscillatorlike 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 freeelectron 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 freeelectron 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 freeelectron 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)
KramersKronig 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 welllocalized plasma resonance and the
sharp peak in the energy loss function at this energy. This is not
a freeelectron 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
KramersKronig 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 [1820]
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 errgies 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 freeelectron 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 freeelectron
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 welllocalized 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 coworkers
(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 KramersKronig 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 freeelectron mass. Figure 2.15 shows that n essentially
saturates at the freeelectron 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 highpressure 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 338607. 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 S20 "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 450500 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 DataReducing Circuit
The purpose of datareducing 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:140)
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 lockin 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 yaxis 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 lockin 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 "online" 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 lockin 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
lockin 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 lockin 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 lockin
amplifier is 50 mV, whereas the PMT output is typically a few volts.
The output from the lockin could be fed directly into the numerator
of the divider. It is, however, convenient to reduce the level of the
lockin 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 yaxis of an xy 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 xaxis 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 introduces a time constant to this signal. On the other hand,
the lockin 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
twotime 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 lockin 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 105 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 copperbased
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 cphase 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 CopperZinc Alloys
Reflectivity measurements were carried out in acopperzinc
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 ubrass 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 copperzinc 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 CuZn 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 Cu30
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 cbrass
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.
CuZn
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 copperzinc 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 righthand 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 CopperAluminum Alloys
Another interesting system to study changes of the band struc
ture by alloying is represented by the copperaluminum alloys. The
band structure of aluminum is well understood and CuAl alloys can be
considered to be a prototype of a noble metal alloyed with a metal of
nearly freeelectronlike 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 aCuZn 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/CuAl1/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 CopperNickel Alloys
Often regarded as characteristic of the many alloy systems
between a noble metal and a transition metal are the CuNi alloys.
They are ferromagnetic over more than onehalf 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 CuNi system. Early measurements on CuNi alloys led
Mott to propose the rigidband model. This model assumes that there
is one electronic densityofstates function which is the same for
copper, nickel, and CuNi 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
rigidband model for CuNi alloys and alternative models have been
suggested. In particular, the virtualboundstate model, developed by
Friedel [29] and Anderson [30] appears to be applicable to the Curich
CuNi alloys.
These two models predict quite different behavior for the
density of states in CuNi 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
RigidBand Model
.VirtualBoundState
Model
Figure 4.6 Illustration of the density of states in copper and
coppernicke? alloys, showing the behavior expected
from rigidband and virtualboundstate 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 CuNi alloys assumes that when nickel is added to
copper, electrons from the s and pderived 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 virtualboundstate 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
freeelectronlike 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 coppernickel
alloys in the composition range from 0.5 at. % Ni to about 15 at. % Ni.
In Figure 4.7 six differential spectrograms for various coppernickel
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 rigidband model.
The virtualboundstate 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 coworkers [31] on the CuNi system and photo
emission studies of this alloy made by Spicer [32] give further sup
port to the applicability of the virtualboundstate model to copper
rich CuNi 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 CuNi alloys using the microprobe. Two specimens
of the same average composition, namely, Cu15 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 Cu15 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 Cu15 at. % Ni alloys.
Figure 4.10
BIBLIOGRAPHY
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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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