Mathematical model of differential reflectometry for use in the investigation of thin film corrosion products

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Title:
Mathematical model of differential reflectometry for use in the investigation of thin film corrosion products
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Differential reflectometry for use in the investigation of thin film corrosion products
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vi, 165 leaves : ill. ; 28 cm.
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Urban, Frank Karl, 1947-
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Thin films   ( lcsh )
Corrosion and anti-corrosives   ( lcsh )
Reflectance spectroscopy   ( lcsh )
Materials Science and Engineering thesis Ph. D
Dissertations, Academic -- Materials Science and Engineering -- UF
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bibliography   ( marcgt )
non-fiction   ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 1981.
Bibliography:
Bibliography: leaves 161-164.
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Typescript.
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Vita.
Statement of Responsibility:
by Frank Karl Urban III.

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Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
    Table of Contents
        Page iii
        Page iv
    Abstract
        Page v
        Page vi
    Chapter 1. Introduction
        Page 1
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    Chapter 2. Literature review
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    Chapter 3. Experimental procedure
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    Chapter 4. Results and discussion
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    Chapter 5. Conclusions and suggestions for future work
        Page 123
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    Appendix A. Computer model derivations
        Page 126
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    Appendix B. Multiple internal reflection equation
        Page 139
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        Page 144
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    Appendix C. Difference in thickness
        Page 146
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        Page 152
        Page 153
    Appendix D. Single internal reflection program
        Page 154
        Page 155
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    Appendix E. Program function
        Page 157
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    Appendix F. Thickness from peak position
        Page 159
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    References
        Page 161
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    Biographical sketch
        Page 165
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        Page 167
        Page 168
Full Text












MATHEMATICAL MODEL OF DIFFERENTIAL REFLECTOMETRY
FOR USE IN THE INVESTIGATION OF
THIN FILM CORROSION PRODUCTS







By

FRANK KARL URBAN III








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 1981














ACKNOWLEDG-EMENTS



The author wishes to express his most sincere

appreciation to his advisor, Dr. R. E. Hummel, for his long hours of assistance, patience and reverence

for high ideals. Thanks especially to Dr. IE. D. Verink, Jr.

for support, encouragement and faith. Special thanks are

U ~also due Dr. R. T. DeHoff for encouragement and help

above the call of duty. Dr. J. R. Ambrose, Dr. P. H.

Holloway and Dr. G.. Schmid were of much help in the

z performance of this work.

Thanks also to the many people who supported me

during this effort including my family and friends. C) Also the support of the National Science Foundation

rJ~ is gratefully acknowledged.
M


9z














TABLE OF CONTENTS

PAGE

ACKNOLDGEMENTS ..................

ABSTRACT . . . . . . . . . .. v

SECTION

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

TI LITERATURE REVIEW............................... 9
II- Introduction.................... ... 9
11-2 General Theory of Optical Properties
of Materials............................ 9
11-3 Techniques for Bulk Optical
Properties.............................. 11
11-4 Techniques for Absorbing Films ..........13
11-5 Techniques for Non-absorbing Films 17
11-6 Non-uniform Films...................... 18
11-7 Metal Oxides................... ........ 20
11-8 Optical Properties..................... 21
11-9 Phase Diagrams......................... 32
II-10 Electronic Structure.................. 3

III EXPERIME~NTAL PROCEDURE......................... 39
III-1 Introduction................... .... 39
111-2 Sample Preparation..................... 39
111-3 Corrosi~on.............................. 42
111-4 Half-polishing......................... 45
111-5 The Differential Reflectometer ..........47
111-6 Summary................................. 50

IV RESULTS AND DISCUSSION......................... 51
TV-i Introduction........................... 51
IV-2 Model Geometry....................... 51
IV-3 Interaction Between Light and Matter. 53
IV-4 Computer Program....................... 62
TV-5 Simp~lification of Equations .............64
IV-6 Conceptualization of the Simplified
Equation................................ 65
IV-7 Results of the Calculations .............69
IV-8 Calculated Differential Reflectograms,
Thickness Effects...................... 70
IV-9 Effect of n1 Variation ..................77
TV-10 Effect of kj Variation ..................80

111











IV-11 Spectral Range Without Interference
Peaks ................................ 83
IV-12 Interference Peak Shapes ............. 87
IV-13 Theoretical Reflectogram. Summary ..... 90 IV-14 Tungsten Oxide ....................... 90
IV-15 Interband Transition Peaks ........... 95
IV-16 Obtaining k and d .................... 103
IV-17 Effect of Film Thickness on Signal
Strength, R/R ...................... 10
!V-18 Effect of External Medium Variation.. 117 IV-19 Practical Aspects .................... 119

V CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK.. 123
V-1 Conclusions .............. ........... 123
V-2 Suggestions for Future Work .......... 124

APPENDICES

A COMPUTER MODEL DERIVATIONS ................... 126
A-1 Transmission and Reflection
Coefficients ......................... 126
A-2 Impedance in Terms of Optical
Parameters ........................... 128
A-3 Attenuation Coefficients ............. 130
A-4 Simplification of Master Equation .... 131

B MULTIPLE INTERNAL REFLECTION EQUATION ........ 139
B-1 Derivation of Equation ............... 139
B-2 Comparison of Multiple and Single
Reflection Models ...... ........... 144
B-3 Multiple Internal Reflection
Computer Program ..................... 144

C DIFFERENCE IN THICKNESS ...................... 146

D SINGLE INTERNAL REFLECTION PROGRAM ........... 154

E PROGRAM FUNCTION ............................. 157

F THICKNESS FROM PEAK POSITION ................. 159

REFERENCES ....................................... 161

BIOGRAPHICAL SKETCH .............................. 165






i77














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



MATHEMATICAL MODEL OF DIFFERENTIAL REFLECTOIVETRY
FOR USE IN THE INVESTIGATION OF
THIN FILM CORROSION PRODUCTS By

Frank Karl Urban, III

March, 1981

Chairman: Rolf E. Hummel

Major Department: Materials Science and Engineering


A mathematical model dealing with the interaction of

light with thin corrosion product films on metal substrates was developed. This model is different from previous models because it is based on Maxwell's equations only, utilizing no simplifying assumptions.

The model was developed to produce computer calculated differential reflectograms (difference in reflectivity as a function of photon energy) which were compared with experimentally obtained reflectograms. The calculated differential reflectograms are distinguished by a sequence of interference peaks if the product of film thickness and index of refraction of the corrosion product film is greater than 40 nanometers. For an oxide film thickness below this critical value, no interference peaks are


v











expected. The peaks obtained experimentally for these thin films are therefore caused by electron interband transitions. They can be used to identify specific metal oxides on metal substrates.

The computer model was tested by comparing the

calculated differential reflectograms with experimental reflectograms of a number of oxides of various thicknesses grown on metal substrates by heating in dry oxygen. Specimens were prepared as follows. The oxide on half of the specimen surface was removed by mechanical polishing. The other half of the specimen was unaltered. The resulting reflectograms provide information based on the difference in reflectance between pure metal and corrosion product on metal substrates. High purity tungsten was chosen for the primary experimental thrust. Chromium, molybdenum, zinc, magnesium and copper also were investigated. Matching the calculated and experimental reflectograms leads to the identification of the source of each peak, the establishment of the interband peak energies and the determination of optical constants and thicknesses of the film. The Differential Reflectometer is becoming an important surface analytical instrument comparable with x-ray or Auger analytical techniques.




vi














SECTION I
INTRODUCTION



One of the most serious materials problems today is corrosion. Corrosion costs over eighty billion dollars annually in the United States alone. 1 Corrosion is consuming capital investments including buildings, roads, vehicles and machinery. Costly mineral, energy and human resources are used up to replace the losses. Clearly impQrovements in corrosion control are of major importance.

To obtain this goal one must have a thorough understanding of corrosion processes. Eight basic forms of corrosion have been identified. 2 ost observed corrosion may be categorized into one or more of these forms. The composition of the corrosion product, its thickness and growth rate are some of the key information needed to understand corrosion mechanisms and facilitate corrosion management.

Corrosion of bulk material commonly involves the formation of a thin film of corrosion product on the outside surface. This thin film is a compound of the base material and some substance(s) from the material environment. For example, Al.0 3can form on aluminum exposed to air or oxygen.3, CuO may form on copper and copper alloys exposed to air-(.





2





The qualities of the film have a strong influence on the further corrosion behavior of the material.5 Some films protect the bulk from further corrosion, for example A12 0 3 protects Al. Other films may not provide protection. The films are initially extremely thin and therefore require special techniques to characterize.

Analysis of the composition and behavior of the thin

film is essential in the understanding of the total corrosion response of the material. Knowledge of film composition allows one to observe changes in composition produced by environmental changes and alloying additions. Computation of film growth rates is made possible by film thickness measurements. Thus thin film characterization is a necessary step in understanding corrosion.

A number of techniques are suitable for characterizing thin films. Each has advantages and limitations. There are chemical methods and techniques which probe the electronic structure of the film by sending in energy, particles or electromagnetic waves, and measuring energy absorbed or given off.

Among the most common thin film techniques are Auger Electron Spectroscopy (AES) and Electron Spectroscopy for Chemical Analysis (ESCA). In the case of the AES technique, electrons bombard the sample surface and ejected Auger electrons are detected. 6In the case of ESCA, x-rays impinge on the sample and eject ed electrons are sensed.7





3




Both methods probe the first few atomic layers (about 0.3 nanometers) and are truly surface techniques.

AES and ESCA both suffer from some limitations. Both employ expensive equipment and require skilled operators. But most important, both require that the sample be placed under vacuum. The vacuum may have an effect on the thin, sometimes delicate corrosion films. Both methods are relatively slow, partly due to the vacuum required.

The traditional optical technique for characterizing thin films is ellipsometry. Other optical methods are also in use. They are usually variations of ellipsometry or techniques based on reflection and transmission of
8-20
light. Ellipsometry is the most common method; it

measures the extent to which plane polarized incident light is converted to elliptically polarized reflected light. From one set of measurements, two of the three film parameters (n, k, and d) can be calculated if the third is known. The parameter n is the index of refraction, k is the index of absorption and d is the thickness of the film. By performing additional measurements all three parameters may be calculated.17 In traditional ellipsometry the measurements have to be taken at each desired wavelength. Scanning ellipsometers have overcome this problem.

The new method of Differential Reflectometry (DR)8

employs a monochromatic beam of light which scans between two samples having slightly different reflectivities (R).





4





The difference in reflectivity between two samples (6R= R 1 R 2 is electronically divided by the average reflectivity ( F= (R 1 + R 2)/2). The A R/ P. is plotted as a function of wavelength of the light to produce the DR spectrum. This differential reflectogram yields peaks at characteristic energies which can be interpreted by using the electronic band structure of the material under investigation, such as a corrosion filmn.8,21,22 Thus Differential Reflectometry provides optical information in a different form than ellipsometry or other transmission and/or reflection techniques.

in previous corrosion studies, 8DR measurements were made by scanning the light beam between one sample half which consisted of bare metal and the other half which was the same metal covered with a corrosion film. See Figure 1.1. Two types of peaks may appear in the differential reflectograms obtained this way: firstly peaks due to electronic interband transitions and secondly peaks due to interference effects of the thin film on a substrate. In addition to the information obtainable involving only interband peaks, information is also obtainable from the interference peaks. This is possible by using the sample configuration described above in which one sample side is bare metal or by using a configuration in which both sample sides may be oxide covered to different thicknesses. See Figure 1.2. Using the latter technique, n and d may be found for films of



























OXIDE














METAL






Figure 1.1 Schematic drawing of the sample configuration
of a metal oxide on a metal substrate,

























OXIDE













METAL





Figure 1.2 Schematic drawing of the sample configuration
of metal oxide of two different thicknesses on
a metal substrate.






7



low absorption. In addition the value for k may be obtainable.

The primary advantages of Differential Reflectometry are that measurements can be pQerformed in air as well as in an electrolyte, i. e. in situ, and that extremely small differences in refllectivity may be measured allowing the identification ofL very thin corrosion films. Measurements are fast; an entire spectrum is recorded within one to three minutes depending upon the scanning speed selected. Finally the measurements cover a continuous band of light wavelengths ranging for example from 200 to 800 nanometers.

In this work a model was developed which takes into account the interactions between the incoming light and an oxide film on a metal substrate. The model relys upon several basic assumptions. Firstly, the validity of the Maxwell equations is assumed. Further the conditi on that the electric and magnetic fields are continuous at a boundary must hold. Finally it is assumed that no substantial error is introduced by describing the corrosion film as having average optical constants and a uniform thickness even though there may be variations in n and k with depth. From these considerations an equation for the reflectivity may be written that relates the electric field of the incident light to the electric field of the reflected light.






8




Two reflection equations were written, one for each of the sample halves, see Figure 1.2. The difference of the two calculated reflectivities was divided by their average to obtain AR/ R. The result was plotted as a function of the wavelength of the light. In this way calculated differential reflectograms were generated. The computations were found to be quite elaborate. Therefore, a digital computer was employed to generate the AR/ R v. s. X( plots.

The normalized differential reflectivities of the corrosion products of various metals were also taken experimentally. The metals which were investigated included copper, magnesium, zinc, molybdenum, chromium and tungsten. The specimens were mechanically polished and heated in dry oxygen to obtain the desired corrosion product. Subsequently part of the corrosion product was removed from one half of the specimen as described in subsection 111-4 in order to obtain the step configuration shown in Figures 1.1 and 1.2.

The experimental differential reflectograms were

compared with the calculated ones. These studies led to new insights into the general structure of differential reflectograms and how these reflectograms are altered by the thickness of the corrosion films. In section IV the method for determining optical properties and oxide film thicknesses from the experimental curves is discussed.














SECTION II
LITERATURE REVIEW



TI-i Introduction

In the development of a line shape analysis of differential reflectograms of metal-oxide corrosion products, it is necessary to consider previous research in two areas, optical computations and studies of the specific corrosion films in question. The line shape analysis leads to the computation of the optical properties of the corrosion film as well as the identification of the composition of the film. Previous research in optical properties which is relevant to this work includes basic optical theory and various computational techniques. Experimental investigations have been performed on bulk as well as thin -film specimens. Previous studies on oxides include thermodynamics, kinetics of formation, crystal structure, transformations, optical and other properties. A review of this work forms a basis for the computer model developed in the next section.



TI-2 General Theory of the O-ptical Properties of Materials

number of authors have reviewed the theories of the optical properties of materials 23and the optical propQerties of metals.124J The theories of the



9






10




interaction of light with matter form the basis of the methods for calculating the optical parameters n and k. Each of the theories enjoys success within certain ranges of applicability. The Drude free electron theory established in 1899 models a metal as containing a "sea" of free electrons which are able to move in response to an applied electric field. As long as only free electrons are involved, the theory describes observed physical phenomena. Lorenz models insulators as containing bound electrons which are attached to their nuclei by electrostatic forces similar to springs. These electrons are assumed to oscillate in response to an applied ac electric field. The Lorenz model (1910) describes the behavior of insulators although the d~c conductivity is not accounted for. A combination of the Drude and Lorenz theories can account for the behavior of most materials over a wide frequency range of the applied electric field.

The quantum-mechanical theory considers electrons to exist in energy bands and to respond to externally applied fields by experiencing interband and intraband transitions. This quantum-mechanical band theory of solids appears to encompass observed light-material interactions.

The values for the index of refraction, n, and the

index of absorption, k, are obtainable from the theory in conjunction with experimental measurements. One author 18 notes that although many papers exist in the literature











which deal with optical measurements and computations, there is poor agreement in the values of n and k for the materials studied. Also different values are commonly reDorted for bulk and for thin film specimens,

In addition to establishing values for n and k,

optical property measurements have been shown to provide insight into the electronic band structure of a solid. 8 Reflectivity peaks correspond to interband transition energies and serve as an experimental test of band calculations. With established band diagrams, the peaks may be used to identify an unknown material.



11-3 Techniques for BulkOptical Properties

Theoretically the interaction of light with a solid is simplest for bulk solids because there are on1y two media, the specimen and the surrounding medium. Transmission measurements are possible only if the material has a very low absorption constant, i. e. if it is essentially transparent. Reflection measurements on bulk metals have been performed to 0.1% accuracy 26,27 in the ultraviolet, visible and infrared regions of the electromagnetic spectrum. The results are shown to be more accurate when the light impinges upon the material at near-normal incidence. 26

A number of different approaches have been used to calculate optical properties. A traditional method of





12




optical measurements is ellipsometry.2 Ellipsometry measures the extent to which plane polarized light, incident on a material at an angle cc is converted upon reflection to elliptically polarized reflected light. The elliptically polarized light may be viewed as a superposition of two normal plane polarized rays with an amplitude ratio Y- and phase difference A~ The values of n and k are calculated f rom Y and A.

The most popular method in the past twenty years for obtaining the optical properties of metals has been by the measurement of the reflectivity over a large frequency range and obtaining n and k by the Kramers-Kronig analysis. This analysis applies a dispersion relation which relates phase shift at a given frequency to an integral involving the reflectivity at all frequencies. 230,31 Unfortunately
considerable error may be introduced because the reflectivity is not measured over all frequencies and instead extrapolations must be substituted. Another approach is to use an iterative computer program such as the "NOTS Multilayer Film Program"'. 10This program calculates reflection and transmission values from assumed n and k values. The program automatically increments n and k until the resultant calculated reflection and transmission values most closely agree with the experimental values.
Yet another approach is a grap~hical determination of the optical constants first using light polarized normal






13




then parallel to the plane of incidence.32 The value of n and k may be found by varying the angle of incidence, and plotting the reflection ratio Rjj/R. as a function of q n and k. The surfaces must be very clean for this technique.33 The measurements are very sensitive to adsorbed species on the surface of the specimen. This effect is the basis for "Surface Reflectance Spectroscopy", a technique which measures adsorbed species on bulk metal.34



11-4 Techniques for Absorbing Films

A different approach is required when n and k for thin absorbing films need to be obtained. In a review article, Abeles discusses reflectance, transmittance and ellipsometric techniques which are used to determine n and k for homogenous, isotropic as well as for anisotropic metallic films.

The theory linking these measurements to optical parameters is more complicated than in the bulk case. Equations which relate n and k with reflectivity (R) and transmissivity (T) have been put forth by Heavens.36 The reflection equation is:

(2 +h2 e2l+(g+h 2 )e-2dj1Ac2+Bi2

R =(2-1)
e2ml 2+h~
e 01+ (g2+h 2)(g2 +h 2)e- X+Cc os2e!+]Dsin2S1





14




where



A = 2(glg2+hlh 2) ,(2-2)


B 2(glh2-gh) (2-3




C = 2(g1g2-hlh2) ,(2-4) D = 2(glh2+g2 hl) ,(2-5)


9 -1 y (2-6)
1


n2 n2+ 2 -2
1 21 2 (2-7)
tt2 (n1-in2) +(k1+k2)2



h -2 k 2 2 (2-8)
(n 0+n1 +k1



h 2(n ik2-nk) (29
2 7 j 2 (2-9)(k k


21k d1
C( 11 (2-10)


and
27'n dl (2-11)

A






15




The transmission equation is:


n ((l+g1l)2+h ((+g 2+h 2 (2-12)
T n2 1)(+2 2
e 2M )(g2+h 2)e +Cc os2S +Dsin2
0 1 1


These equations apply to a thin absorbing film on an absorbing substrate. The indices of refraction are as follows:



n. = index of refraction of surrounding, nonabsorbing medium



ni = index of refraction of the absorbing film



k1 = index of absorption of absorbing film



n2 = index of refraction of absorbing substrate



k2 = index of absorption of absorbing substrate. The complex index of the film is


A
n! = n ikI (2-13)



and that of the substrate is



n2 = n2 ik2 (2-14)





16




These equations may be simplified, with some loss in accuracy. The approximations may be solved graphically or numerically for n and k of the films.!9 The graphical technique using the reflection ratio,6R / mentioned in subsection 11-3 is also possible for thin films.16

The application of digital computers to calculate n and k has allowed more accurate and faster methods. An iterative "hill-climbing" technique was described by Ward et al.7 utilizing data from reflection and transmission measurements using incident light of parallel and normal polarization. To accomplish this an objective function, y, is formed:



y = (Rp(eXpt.)-Rp(n,k,d))2 + (Rs(expt.)-R(n,k,d))2 +(Tp(eXpt)-T (n,k,d))2+(T (expt.)-Ts(nk,d))2 ,(2-15)

(Tp (ep)p s d) (


where

R (expt.) = experimentally measured reflection of
P parallel polarized incident light,

T (n,k,d) = calculated value of transmission of
normally polarized incident light.


The values of n, k and d are varied until y in equation (2-15) reaches a preselected small value. The associated values of n, k and d are then taken to be those of the film.






17




The Heavens equations may be directly used to obtain computer calculated solutions for the optical constants through another method. Reflection and transmission measurements at near-normal incidence allow the calculation of the optical constants as long as an independent measure of the thickness is known. 17,38-41 The computer methods are in general iterative, solving for the values of n and k which produce a close match between the measured and calculated values of reflection and transmission.

The optical constants of a thin film deposited on a non-absorbing substrate may be calculated from transmittance using a Kramers-Kronig analysis. 42The equations relate the real and imaginary parts of the transmittance. The real part is the change in amplitude and the imaginary part is the change in phase of the light.



11-5 Techniques for Non-absorbing Films

Techniques have been developed for non-absorbing

thin films on absorbing substrates. These are reviewed by Bennett and BennetUt.9 Methods for calculating optical properties and thickness of non-absorbing thin films using ellipsometry have been reviewed by Neal and Fane. 28 In the case of thin films, the same type of measurements are performed as for bulk mat erials. 43,44 Two of the values of n, k and d may be calculated from a single ellipsometric





18




measurement. All three parameters may be found from two ellipsometric measurements at differing angles of incidence. It is possible to calculate the refractive index, n, and thickness, d, from one ellipsometric measurement taken at a single angle of incidence by neglecting the absorption, k. The error in the calculated n and d has been shown to be more significant for low index films. 45

Each ellipsometric measurement is taken at a particular light wavelength. Thus, many measurements must be made to obtain the spectral dependence of the optical properties.

Differential Reflectometry has been shown to correlate signal intensity to thickness for low absorbing films on metal substrates. 82



11-6 Non-uniform Films

All the methods previously discussed assume homogeneous, isotropic films which are perfectly flat and are deposited on a flat substrate. The problems introduced by the conditions of non-uniformity, anisotropy, inhomogeneity and surface roughness have been the subject of inquiry by a number of investigators. An early study explains how the apparent color of a glass can be produced by the presence of microscopic particles of metal. 46it has been shown 47 that average values of the optical properties can be used instead of a continuous variation in optical properties.





19




It also has been shown that it is possible to represent a real surface film having a rough surface with an equivalent flat film having a perfectly smooth surface. 48 The notion of an equivalent flat film is further supported by results originating from ellipsometric measurements on absorbing films in which the calculated film thickness was in agreement with two independent thickness measuring techniques. 49This result is particularly important for the development of the model in the following section.

Polishing techniques affect the surface roughness and hence the measured reflectance. 50 This fact throws all absolute measurements into question since none of the authors reported or controlled surface roughness. The Differential Reflectometry technique circumvents this problem by comparing two surfaces which are equally prepared and hence surface roughness effects are subtracted out instrumentally,

A method for ellipsometric measurements and calculations for films containing foreign particles has been developed.5 The foreign material was treated as though it were an equivalent film, in addition to the film in which the particles are located. The equivalent film was assumed to have the same volume as the total volume of the foreign particles.

Computations for multilayer thin films have also been investigated. Weinstein52 discusses various approaches





20




to c computing thin film optical constants with particular attention to multilayer films. He describes graphical and explicit formula methods for obtaining reflection magnitude by keeping track of the E and H fields of the incident and subsequently multiply reflected and transmitted electromagnetic waves. The transmission and reflection factors are found by assuming continuity of the E and H fields at a boundary. He concludes that for the purpose of multilayer film design, the four terminal network model is insufficient in that it cannot account for the polarization and obliqueness of incidence.



11-7 Metal Oxides

The oxidation of metals is by no means an untouched field. A complete review of the literature is beyond the scope of this work. An excellent early review is given by McAdam and Geil.53 There are a number of texts in the field including one by Fromhold.54 The following focuses upon three specific areas involving metal oxides. These are optical properties, phase diagrams and electronic structure of the metal oxides of particular interest. Since the primary thrust of this work is in the tungstenoxygen system, most of the previous work in this area is shown here. Other metal-oxygen systems were investigated primarily to support the findings for tungsten oxide.





21





11-8 Optical Properties

The optical properties of two tungsten oxides, W02 and WO3 have been reported by several authors.
55
Dissanayake and Chase recorded the reflectivity of W02 as a function of energy. By using the Kramers-Kronig analysis described in subsection 11-3, the real and imaginary parts of the dielectric constant, P of W02 were found, Figure 2.1. In Figure 2.2 the spectral reflectivities of WO2 and MoO2 are shown. The following equations relate the optical constants to the dielectric constants:


^ 2
E n = (n ik) (2-16)



where



= 2 k2 (2-17)



and



2= 2nk (2-18)



where


A
E -1 + (2-19)





22






20



16


12l













12 1 2' 345 E E 2J


Fiur 2.Ieladmgnr F-2 prso
the dilcrccntn /fW '(eeec 5





23











0.8


0.7


0.6
Mo 02
1- 2
0.5


0.4
o.4o
2
~ 0.3
1,W 2



0.2


0.1

0
0 I I I I
01 2 3 4 5 6

E [eV]




Figure 2.2 Unpolarized reflectivity spectra of W02
and MoO2 at 300 OK. (reference 55)





24




The index of refraction of WO was reported by
3
Sawada and Danielson56 to be n = 2.45. They measured an average value over crystallographic direction using white light from a tungsten source. Deb reported57 the spectral dependence of n for WO3 from 450 to 1799 nanometers, Figure

2.3. The average value reported by Deb is in good agreement with the value obtained by Sawada and Danielson. Deb also reported optical absorption, k, of WO3 as shown in Figure 2.4 He reports absorption peaks at 4.39 eV (282 nm) and at 5.25 eV (236 nm) for thin films. These peaks moved to lower energies (longer wavelengths) with increased annealing time. Deb observed that the thin films are commonly amorphous and suggests that this fact may explain the lack of certain absorption peaks predicted by band calculations based upon the assumption of crystalline material. He reported an absorption peak for MoO3 at 2.85 eV (434 nm) for thin films.58'59 The index of refraction of MoO3 is shown in Figure 2.5 as measured by Deb. Chase6o has determined the dielectric constants of Cr02 and Mo02 and these are presented in Figure 2.6 and Figure 2.7.

Ellipsometric and transmission investigations of the optical properties of ZnO show an interband feature at

3.3 eV (375 nm). In an investigation using ellipsometric spectroscopy on the ZnO nonpolar (1100) surface, Matz and Luth61 report the values of n and k of bulk ZnO. See Figure 62
2.8 and Figure 2.9 Aranovich et al. report the





25





















C


4.0 3.0


2.0




500 1000 1500

X(nm] Figure 2.3 The refractive index, n, of amorphous WO 3
films as a function of 14Lght wavelength.
(reference 57)





26










200 107



"7
v E

CU 6
CDC
10

100 0.
00
C. .0
-<
0
105






200 300 400 200 300 400

[kinm] \ [nm]


Figure 2.4 The absorption factor of amorphous WO3 films
is reported in the right graph of the figure.
The data is replotted in the left graph to conform to the definition of the absorption
coefficient, k, where absorption factor =
41T(k/A (reference 57)





27











2.7







c 2.6







2.5 2.4




700 800 900 1000

(nm] Figure 2.5 The refractive index, n, of M'oO thin films
as a function of wavelength. (reference 58)






28

















10 5 0

IEI




-10

0 1 2 3 4 5 6

E [eV]




Figure 2.6 Real part of the dielectric constant El of
CrO2 as a function of photon energy.
(reference 60)







29




40






30






20






o 10



oooo

0





0 1 \E ,1

Figure~~~~ ~ ~ ~ ~ ~ 2. ilcrccntn o o2a ucino

phtneeg. (eeec 0







30









2.6



2.4



2.2 n.
2.0 II

n

1.8



1.6 I I
1.5 2.0 2.5 3.0 3.5 4.0

photon energy [eV]




Figure 2.8 Optical index of refraction of ZnO at room
temperature as measured by ellipsometry.
(reference 61)







31









1.0



0.8


CK
0.6K


0 K
+ I
') 0.4
0.

X

0.2



0.0 I !
1.5 2.0 2.5 3.0 3.5 4.0


photon energy [eV]




Figure 2.9 Optical extinction coefficient, k, of ZnO at
room temperature as measured by ellipsometry.
(reference 61)






32



transmission of Zn0 as a function of wavelength from 300 to 900 nanometers, see Figure 2.10.

The reflectivity of Cu20 is reported by Brahms and

Nikitine63 as shown in Figure 2.11. The electroreflectance of Cu20 is reported by Daunois et a6 4 as shown in Figure

2.12. Ladeife et al5 present the optical constants of Cu20. These are shown in Table 2.1.



11-9 Phase Diagrams

A number of phase diagrams of the tungsten-oxygen system have been published.6668 The phase diagram of St. Pierre et al.68 is presented in Figure 2.13. It can be seen that below 484 0C the only phases thermodynamically expected are W30, W02 and WO 3 Between 484 0C and 585 C W20058 becomes thermodynamically stable. Above 585 0C W18049 also becomes stable.


11-10 Electronic Structure

The electronic structures of cubic NaWO3 and of NaWO3

with a vacancy on the Na site have been calculated. Figure

2.14 presents the band diagram of OW 3.69 It can be seen that four symmetry point interband electron transitions are indicated. The P point transition is generally believed not to occur. It has the same energy as the X point transition, 1.9 eV (651 nm). The M point transition is at 4.6 eV (269 nm) and the R point transition is at 6.12 eV (202 nm).







33










100
90 80 70

60
a
50
*; 50E
p 40

30

20 10

400 500 600 700 800 900

A[nm]




Figure 2.10 Optical1 transmission of a ZnO film.
(reference 62)







34

















S40




30


20




500 400 350 300 250

X (nm] Figure 2.11 Reflectivity of' cuprous oxide (Cu2O0)
single crystal at 77 OK. referencee 64)






35





ARR 54

R

+2




+1








-1l




-2


I I I I I
500 400 300 250

A [nm] Figure 2.12 Electroreflectance of Cu20 at 85 oK in a
field, E = 60 kV/cm. (reference 64)






36





TABLE 2.1


The wavelength dependence of n and k for Cu0 films.




wavelength (nm) n k


450 2.45+0.08 0.744+0.011o
500 2.590.09 0.650+0.017
550 2.57+0.10 0.539+0.019
600 2.60+0.11 0.449+0.025
650 2.65+0.12 0.345+0.028

700 2.64+0.13 0.236+0.030
750 2.65+0.10 0.170+0.024
800 2.62+0.09 0.123+0.022




(reference 65)






3?












W20058+WO3



800 W+W02 co
O
725 WO2+W18049 '
725
+
700 -00
co


600 W W30 + WO 585 W03 + 02
+
W30
WO2 + 20058

500 484



400 WO2 + WO3


0 .2 .4 .6 .64 .66 .68 .70 .72 .74 .76 .8 1.0

Oxygen Content [at. fraction]


Figure 2.13 Phase diagram of tungsten-oxygen at
atmospheric pressure. (reference 68)







38












1.4 1.3

1.2 1.1 1.0 S0.9


0.8 0.7

0.6 0.5

0.4 0.3

r A x Z M I r A R S X R T MV






Figure 2.14 Energy bands of NaWO3 with a vacancy on
the Na site along the symmetry axes.














SECTION III
EXPERIMENTAL PROCEDURE



III-1 introduction

The basic steps of the experimental procedure are

sample preparation, corrosion, preparation for measurement and recording the differential reflectogram.



111-2 Sample Preparation

Discs, approximately 25 mm in diameter and 12 mm

high were cut from high purity rods. Table 3.1 lists the sources and purities. One part of the disc was cut away to produce a "D" shaped sample, Figure 3.1. The flat side of the sample serves as a guide for the half polishing technique (see below).

Rough polishing was performed using 180, 320 and finally 600 grit silicon carbide paper using soap as a lubricant, followed by Microcut Paper Sheets* to remove the silicon carbide grit from the sample. The samples were then polished on polishing cloth to a 6~4 diamond and finally to a iP. diamond finish using Metadi Fluid** as a lubricant, This procedure worked well for most metals.


*M~icrocut (registered) Paper Sheets, Buehler Ltd., Evanston, Illinois
** Metadi Fluid (registered), Buehler Ltd., Evanston Illinois

39






4o






TABLE 3.1

Sources of metals.



metal j source

tungsten A. D. MacKay, Rare Metals and Chemicals,
10 Center St. Daren, CT 06820 molybdenum same

magnesium same

zinc [ same

copper same

chromium Materials Research Corp., Orangeburg,
New York 10962






41









































Figure 3.1 Sample configuration for corrosion experiments.
The hole is to allow a thermocouple to be
inserted into the sample.






42



In the case of zinc however, Emery Paper Sheets* grades I through 0000, were used. The papers were lubricated by rubbing them with wax and applying a light coating of mineral spirits. The samples were rinsed in mineral spirits before changing to a finer grade of paper. Samples were rotated 90 0 between each grade.

After polishing, the specimens were rinsed with a soapwater mixture, followed by a rinse in methanol and drying in a stream of filtered air. Finally the samples were warmed by hot air to reduce the possibility of water condensation on the surface.

It was observed that one of the corrosion products

of molybdenum dissolved in water. Therefore these samples were rinsed in methanol only.



111-3 Corrosion

The metal discs were positioned in a quartz tube and heated in a radiant furnace, Figure 3.2. During heating, dried** oxygen is passed over the specimens, The temperature of the sample was measured using a thermocouple inserted into a hole in the metal disc.

Specimen temperature was controlled by a variable

transformer. Figure 3.3 shows a typical temperature curve.



*Emery Pa Qer Sheets, Buehler Ltd., Evanston, Illinois

**W. A. Hammond Drierite Co., Xenia, Ohio








43









E
0



CL

x 0


CL
0
u
0
E
0







C"4
CL 0) Cd cz ;11 cz

4-q


0











4-0 CD




q E--i


C)
H cz Q)

(1)







44 500





400 300





200 100







10 20

t[min] Figure 3.3 Temperature of a tungsten disc as a function
of time as measured by a thermocouple mounted
inside the disc. Power was applied at zero
time and removed at twenty minutes.






45



Most specimens were heated for twenty minutes.

Heating less than twenty minutes resulted in temperatures which did not approach a constant level, Figure 3.3. At longer heating times the corrosion product became too thick and opaque for useful optical measurements. At the end of each heating the sample was allowed to cool in an oxygen atmosphere.



111-4 Half-polishin,

Corrosion product on half of the specimen was polished partly or completely away using a polishing gauge, polishing cloth, 1,i4 diamond polishing compound* and Metadi Fluid** as a lubricant. Figure 3.4 shows how the flat side of the sample serves as a guide. To remove a small amount of corrosion product, very light pressure was applied and the sample was moved only a few cm along the guide. As before the sample was rinsed with alcohol, dried with dessicated air and gently warmed with hot air. After this procedure the sample had the configuration of Figure 1.1 or 1.2. The sample was then immediately transferred into the instrument and a differential reflectogram was taken. Since any further oxidation caused by the environment is believed to affect both


*Metadi Diamond Polishing Compound (registered), Buehler Ltd. Evanston, Illinois
**Metadi Fluid (registered), Buehler Ltd., Evanston Illinois






46



































diamond paste





Figure 3.4 This schematic drawing of' the half-polishing
guide shows how the sample may be accurately
positioned to remove corrosion product from
part of the reacted surface.






47



sides of the sample equally, the formation of further corrosion products can essentially be neglected.



1T1-5 The Differential Reflectometer

The Differential Reflectometer has been described in detail elsewhere. 8,1Thus only a brief description of the instrument and its operation is given here.

Absolute reflectivity measurements are extremely

difficult to make with a high degree of accuracy except for vapor deposited films in high vacuum. Surface contamination, surface preparation, variation in light source output, line voltage changes and sample alignment all contribute to error. The problems are exaggerated when an attempt is made to measure a small reflectivity difference between slightly different alloys or metals covered with thin films. A differential technique has been developed especially to make these kinds of measurements. The Differential Reflectometer is capable of detecting differences in reflectivity as small as 0.001%.

Briefly the DR technique produces a plot of the difference in reflectivity between two sample areas, divided by the average reflectivity, versus light wavelength. A high pressure xenon light source provides a broad band output to the scanning monochromator, Figure 3.5. The very narrow bandwidth light beam emerging from the monochromator is focused by mirrors onto an








48













C


X,

L

IL 4-1 CL CH









4H










c cz
a I; I ILA
x







49



approximately 2 mm diameter spot on the sample. One of the mirrors vibrates at 60 Hz and scans the spot up and down about 6 mm. The scan dimensions are adjustable by regulating the voltage to the coil of the vibrating mirror.

The operator can manually shift the sample position to cause the beam to spend half of the scan time on each part of the sample. An oscilloscope trace is used to monitor the reflected intensities of each specimen half, thus allowing very accurate positioning of the sample. The reflected beam is directed onto a frosted quartz glass plate immediately in front of the photomultiplier tube to diffuse the beam evenly over the photosensitive surface.

The photomultiplier output consists of a direct current component modulated by a 60 Hz square wave whose amplitude is proportional to a R. By means of a lock-in amplifier tuned to 60 Hz, the square wave is detected and a direct current signal proportional to the difference in reflectivity (A R) is produced. A low pass filter smooths the 60 Hz square wave modulation in the original signal to produce a signal proportional to the average reflectivity (T). These two signals are passed to a divider where the ratio, A R/17, is formed.

The monochromator produces a direct current voltage proportional to wavelength. This signal is applied to







50



the X axis of the X-Y recorder. The AR/Rf signal is applied to the Y axis. A DR spectrum is obtained by scanning the wavelengths from 200 to 800 nanometers. A typical scan takes approximately one and one-half minutes.



I11-6_Summary

Samples were prepared from high purity metals. The samples were polished on one surface to an optical quality, IA diamond finish.

Each sample was individually heated in a furnace

in a dried oxygen atmosphere. The corroded sample was half-polished to produce a step in the corrosion product film and the differential reflectivity between the corroded and polished areas was recorded as a function of wavelength.














SECTION IV
RESULTS AND DISCUSSION



TV-i Introduction

This section presents model calculations for the

normalized difference in reflectivity (LAR/ff) versus light wavelength A for a metal/metal-oxide couple as would be expected by measurements using a differential reflectometer. Much information exists in these reflectograms. A set of equations or an algorithm was developed to extract certain data from the experimental curves. From these the index of refraction, n, the index of absorption, k, and the thickness, d, of the oxide film may be calculated. In other words a bridge between the DR spectrum and n, k and d of the film was developed,



IV-2 Model Geometry

For the model used here, an average complex index

of refraction, n, is assumed for each medium. This complex index of refraction contains the real index of refraction, n, and the real index of absorption, kz



n = n ik (4-1)



In Figure 4.1 let medium "CO" be air or an electrolyte,

51






52
















MEDIUM '0'



._.I
dl2
T

MEDIUM '1"












MEDIUM '2'






Figure 4.1 Geometric configuration of the model used
in this work.







53



medium "1" be the oxide and medium "2" be the metal substrate. The light beam of the differential reflectometer is thought to scan between the two parts of this sample and thus produce the A R/N curve as described in section Ill.



IV-3 Interaction Between Light and Matter

A light ray which impinges upon a sample from medium "0" (r 0 ) is partly reflected (r and partly transmitted (r 2) into medium "1", Figure 4.2. The transmitted wave (r 2 ) is attenuated as it passes through medium "1" having a thickness, d. This attenuated wave is partly reflected (r4 and partly transmitted into medium "2" (r 3 ). Ray (r 4) is again attenuated by medium "1". Finally this wave reaches the 0-1 interface and is partly reflected (r6 ) and partially transmitted (r 5 Rays r 1 and r 5 are assumed to add by superposition. The internal reflection process in medium "1" continues with ever diminishing contributions from each subsequent reflection. Appendix B shows how these considerations may be taken into account. In this work however, the additional contributions are considered to be negligibly small and are therefore omitted.

The Maxwell equations,


A A
curl H = E 8/c + E 4'rf (3"/c (4-2)






.54










r5






Medium '0' \








d




Medium '1' 1
Medium '2'
r3


Figure 4.2 Interactions between a light beam, ro, and a
solid consisting of a semi-transparent medium
"1" (e. g. oxide film) and an absorbing medium
"2" (e. g. metal). For clarity the incident, reflected and transmitted rays are shown at
an angle. The calculation is carried out
assuming normal incidence.







55



and



cur"l E = -H A/c ,(4-3)



can be solved by assuming a plane polarized wave propogating in the z direction having an electric field strength E Xgiven by



=EeiW (t nz/c) (4



where

Ex = complex value of the magnitude and phase of
the x component of the electric field,
E 0=the real value of the peak of the electric field,

e = the natural base,

i = the square root of -1,

C... = the radian frequency of the electromagnetic wave,

t =time,
A
n = complex index of the medium in which the wave
propagates,

z =z axis position, axis of propogation,

c =speed of light,

6= permittivity of the medium, C= conductivity of the medium,

H =magnetic field vector,

A. = permeability of the medium and

E = the electric field vector.






56




The electric field strengths of the incident,

reflected and transmitted waves at a given boundary are

related by the materials properties. The following

equations hold, see figure 4.3.


A A A A A A
T ab = Et /2i = 2 Zb/(Zb + Za) (4-5)



and


A AA A )(
P ab= Er/E. (Zb Za Z )(+ z) .(4-6)



They are derived in Appendix A. In these equations


A
T ab = the ratio of the transmitted electric field
strength in medium "b" to the incident field
in "la"l

P ab = ratio of the reflected electric field'in medium
"a" to the incident field in "a", reflection
occurs at the a-b interface,

E. = the complex electric field strength of the
incident electromagnetic wave,

Et= the complex electric field strength of the
transmitted electromagnetic wave,
A
Er= the complex electric field strength of the
rreflected electromagnetic wave,

Za = the complex impedance of medium "a" (note that
aby definition impedance is the ratio of the
x component of the E field to the y component
of the H field in medium "a". These vectors
are normal to each other and to the direction
of propagation. The material may be anisotropic)
A
Z b =the complex impedance of medium "b"







57













MEDIUM 'b'
Et




.00 Ht





Ei
a-bINTERFACE Hi

MEDIUM 'a'


Er



Hr




Figure 4.3 Reference directions for the incident,
transmitted and reflected E and H components
of an electromagnetic wave. The rays are
shown at an angle for clarity only.







58



Equations (4-5) and (4-6) result from assuming

continuity of E and H fields at an interface. The notation of media "a" and "b" has been chosen in order to emphasize the general nature of the equations.

It is possible to express Z in terms of rl in order to rewrite equations (4-5) and (4-6) in terms of the variables of the model, n 0 n 1 and n**' 2' See the derivation in Appendix A, subsection A-2. Thus we write



Z A
n (4-7)



In the following, magnetic influences are not considered. Therefore k is set to unity. This yields for medium

"a" and medium "b",


A
Z a = 1/ n^ a (4-8)

and

Z b = 1/ n b (4-9)



Substituting (4-8) and (4-9) into (4-5) and (4-6) yields



T ab 2 n b (na + n b) (4-10)


and


P (n (4-11)
ab a b (^a b







59



Now it is possible to write the specific transmission and reflection ratios which appear in the model at the 0-1 and 1-2 interfaces. Figure 4.3 defines the incident field direction relative to the locations of the two media composing the interface. The reflection ratio for an incident field in medium "0" falling on an interface with medium "1" is



( n n n h-2
01 0 1' 0+


The transmission ratio for an incident field in medium "0" passing into medium "1" is







The reflection ratio for an incident field in medium "1" reflecting from an interface with medium "2" is







The transmission ratio for an incident field in medium "1" passing into medium "0" is


AA
T 10= 2 A ( + n0(4-15)






6o



The equation for the attenuation of the electric field as the wave penetrates perpendicularly into a medium can be written according to (4-4). See Appendix A, subsection A-3.



A= e (4-16)



Applying this to thicknesses d 1 and d 2 of Figure 4.1 yields



A dl= e ~fiic(4-17) and
AA
A d2=e 2/ (4-18)



It is now possible to write an equation in accordance with the reflection model of Figure 4.4. Let the incident
A
plane wave contain the electric field, E. Consider the

series of interactions in Figure 4.4.


A A
dl i ~01 01 adl 12 dl1


Similarly for thickness d 2



E 2 E.i ( P1+ T 1A d 12A ) T1 (4-20)



Because the photomultiplier is sensitive to intensity i. e. to fBI12 we obtain






61






MEDIUM '0'








... A A A 2 A ... [E 01l Adl P12 1T1





AA A
011


A /, A AA]
MEIU '1 AAi A o [E* P12 Ad.IP








MEDIUM '2'




Figure 4.4 Schematic representation of the reflection of
an electromagnetic wave from a thin film covering a substrate. The incident wave
begins at the upper left. At each significant point along the way, the E field is indicated.
For clarity the incident, reflected and
transmitted rays are shown at an angle. The
calculations are carried out assuming
normal incidence.





62




~R/ F = 22 1E2)/(d 2 ~\dI2),(4-21)



Combining (4-19) through (4-21) yields

l "" "" l~2TloI Po+o~ 2PiTi 2
~A A2 ^ ^1012_~~ 1,0+ A2 1
A R P T0 A p P2T P TAP
2P01 01 dl 12 z0 -P T d2^12^T10 (4-22)
01+ 0Ad!Pl2T 12+P o+ 12P2
IP +T P T P T A-P 01P0 01dl121 d2 12 i


Equation (4-22) expresses the normalized difference in reflectivity in terms of the physical parameters of the model, n, n1, n2, d1 and d2 which are contained in the A's, T's and P's.


IV-4 ComPuter Program

In order to obtain a differential reflectogram from equation (4-22) it is necessary to plot the value of A R/R as a function of wavelength A Recall that for light waves


c = =

or

Lj = 2'tlc/$. (4-23)


A
The value of A appears in Adl and Ad2 of equations (4-17) and (4-18). The values of P01, T01, P12 and T10 will also be functions of wavelength if the values of 0, 1
and n2 are functions of wavelength. Because the n's are







63



not all known in advance, a difference approach is used in which the values of the unknown n s are assumed to be constant with wavelength. As solutions are achieved, the unknown dependence may be found by interpreting the difference in the calculated and the experimental reflectogram to be due to the variation of n with wavelength.

Literature studies have shown that in a number of instances the complex index of refraction for a metal is nearly constant with wavelength over the range from 200 to 800 anaometers. In the case of air, 'no = 1 + i0; therefore n 0 is constant with wavelength.

A digital computer was used to calculate and plot theoretical reflectograms (A R/p-R versus A ) using equations (4-12), (4-13), (4-14), (4-15), (4-17), (4-18) and (4-22). The program is listed in Appendix D. The computer was programmed to calculate A R/iR for 800 individual wavelengths between 200 and 1000 nanometers. The differential reflectograms obtained for various values of d,1L, d 2 and n1show a wide variety of curve shapes (see subsections IV-8 through IV-12). Substantial structure is observed in these graphs. As will be shown herein, the structure contains the desired information about thickness and optical properties of the film.






64



IV-5 Simplification of Equations

Equations (4-12) through (4-15), (4-17), (4-18) and (4-22) lack easy visualization of how the spectrum varies as a function of wavelength or frequency. It is possible however, to alter equation (4-22) into a pure real expression. This is accomplished by assuming that k is small. The derivation listed in Appendix A, subsection A-4, yields the following expression


6R 2 sin(211nl (dl-d2)/( ) sin(9 +2T1nl(dl+d2)/A )
= (4-24)
R g+c os (2f nl (d1-d2)/X ) cos( q +2TYnl(dl+d2)/A) where
A
n1 = real part of n1, the index of medium "1"

= the light wavelength
dl = the thickness of medium "1" on one sample side
(see Figure 4.1)
d2 = the thickness of medium "1" on the other sample
side (see Figure 4.1)

tan- Imag(P01 -1 Imag(T012T10 (4-25)
q)=tan -tan (AIA(4-25)
Real(Po1) Real(T01P12T10)

g = (a2 + f2)/2af 2 +1T 0112 1(4-26)
-= A= (4-26)
2 P01) T01P12T101

a = P011o (4-27)


S01 12T101 (4-28)







65



IV-6 Conceptualization of the Simplified Equation

Equation (4-24) is seen to be a pure real expression consisting of a ratio of trigonometric functions of two different arguments. The first argument contains the difference in thickness and the second contains the sum of the thicknesses plus an angle P In addition the denominator contains the factor g. It is possible to understand the general behavior of equation (4-24) by analyzing the numerator and denominator and observing their individual effects upon A R/f.

The denominator of (4-24) consists of the sum of a factor "g" and a product of two cosine terms with different periods. The maximum range of the cosine product is between plus and minus one 7-P "g" would assume the value one, the denominator could reach zero, causing an infinite value in equation (4-24). Rearranging (4-26) yields



g = (a 2 + f2 )/2af a/2f + f/2a (4-26a)



Let

f/a = x (4-29)



Then

g = 1/2x x/2 (4-30)







66



Inserting g = 1 yields x = 1. The value of g cannot be negative because it is composed of absolute values, (4-26). A plot of (4-30) shows that g is greater than or equal to unity for ail allowable values of x (positive numbers), see figure 4.5. For x = 1, it follows that, from (4-29),



f/a = 1

or

f = a (4-31)



This yields from equations (4-27) and (4-28)


A011 = 1^01A
P T P 12 T '10 (4-32)



When this condition exists and the wavelength is such that the cosine product term equals minus one, AR/ 'goes to infinity. Inspection of the cosine product term shows that it may reach minus one only if 0 and either

d 1 = 0 or d 2 = 0.

The value of CP, defined by equation (4-25), will not be zero for the cases under consideration. Therefore the cosine product will not equal minus one. For the product to take on values close to minus one, the periods must be very different such that one cosine term is nearly one (or minus one) while the other periodically goes from one to minus one. This condition occurs when d is small.
1






67








9







2 1 *








I I
1 2 X





Figure 4.5 Variation of g = (a2 + f2)/2af as a function
of x = f/a.






68



This causes the following:


cos(2'1n1 (dI d2)/A ) is close to +1 or -1 (4-33) and additionally


cos(T +2Tr'n1(d+d2)// ) rapidly varies between (4-34) +1 and -1 .


ThLsthe value of fR/R may reach large but not infinite values under the conditions of (4-32) and


d close to d2 (4-35)


The numerator of (4-24) forces the value of AR/R
to zero whenever either of the two sine terms is equal to zero. This occurs when

27(n1(d-d 2)// mlV m = integer (4-36) The numerator is also zero when


+ 2T1 n(d1+d2)/A i2 = m2l' m 2 = integer. (4-37) Solving (4-36) and (4-37) for the wavelengths at which the numerator is zero and hence AR/R is zero yields







69





il= 2n1(d1 d2)/ml m, = integer (4-38)



Xi2 2n,(d1 + d2)/(m2 M/'Y) m2 = integer.(4-39)


Equation (4-24), when programmed, generates data that closely match the results of the original equation, (4-22) as long as the conditions of equation (A-31 of appendix A are maintained. It should be mentioned however that the exact equation (4-22) was used for all calculations and plots. Equation (4-24) is less accurate but it is much easier to conceptualize. Hence (4-24) is used primarily in rationalizing the results and in pointing the way to solutions.



IV-7 Results of the Calculations

Line-shape analysis of the calculated differential reflectograms was employed to gain new insights into experimental differential reflectograms of the tungsten/ tungsten-oxygen system. By comparing the experimental and theoretical results it is possible to identify interband as well as interference peaks and to quantify the differential reflectometry analysis. Additionally, a number of further metal/metal-oxygen systems were investigated to demonstrate that the method is generally applicable.







70



IV-8 Calculated Differential Reflectograms, Thickness Effects

The possible wavelength range of the calculated differential reflectograms is from zero to infinity. The experimental range is,however, only from 200 to 800 nm. Plotting the calculated DR spectra at both a wide range and a range corresponding to the experimental reflectograms gives a more comprehensive insight into the problem at hand.

Figure 4.6 shows calculated differential reflectograms (AR/f as a function of A ) for transparent dielectric films on a metal substrate. The thicknesses, d and d2f of the two transparent adjacent layers are assumed to be different, Figure 4.1. The thickness d1 varies in the three curves of Figure 4.6, whereas d2 remains small and constant. The optical constants are noted in the figure caption. The wavelength range was chosen to be from 0 to 5000 nm. The calculations were performed using equations (4-12)through (4-15) and (4-17), (4-18) and (4-22). The computer program is contained in Appendix D.

Inspection of Figure 4.6 reveals that at low photon energies (long wavelengths) the differential reflectivity is relatively constant and structureless. At high photon energies (shorter wavelengths) pronounced maxima and minima develop which have the appearance of interference type oscillations. The first maximum (moving along the horizontal axis toward higher energies) shifts to lower energies with increasing thickness, d, of the dielectric film.







































Figure 4.6 Calculated differential reflectograms of a
metal oxide on a metal substrate. The sample
configuration is shown in Figure 4.1. The
value of d increases from curve (a) through
(c) and d2 is 0.1 nm. The values of the
A
optical constants were taken to be, n0 = 1,
n= 1.6 -i 2 x 10- and A = 3-i 3.







72





AR
R


100


50


0 [a] di=10 nm







100


50

0 [b] dl= 50 nm






100


50
dl= 100 nrm [c]


5000 4000 3000 2000 1000 0

A [nm]







73



This can also be seen from equation (4-24) by setting d2 = 0.


A R 2 sin(2nidi/A ) sin( +2Vnldl// )
11 (4-4o)
g +cos(2Vn 1nd/A ) cos( +2Trn1d nI1/


In Figure 4.7 the same specimen configuration as before was assumed; however, the wavelength range for which the differential reflectograms were calculated was taken to be between 200 and 1000 nanometers, which more closely resembles the range for the experimental reflectograms shown later in this section. Similar structure, as seen in Figure 4.6, can be observed. Again the structure is more pronounced the thicker the dielectric film, dip assumed.

Inspection of equation (4-24) reveals that the first peak comes mainly from the term



sin(q + 211Yn1(d1 + d2)/A ) (4-41)



This is true because at large wavelengths the cosine terms are nearly unity and the sine terms are nearly zero. As A becomes smaller, the value of (4-41) grows the fastest. It approaches values significantly greater than zero as:



= 2tn1(d 1 + d2)/A (4-42)








































Figure 4.7 Calculated differential reflectograms of a
metal oxide on a metal substrate. The sample
configuration is shown in Figure 4.1. The
value of d Iincreases from curve (a) through
(e) and d 2 is 0.1 nm. The values of the
AA
n1 = i2x1-6and n 2 =3 -i 3.







75






100


50 dl= 25 nm


0
[a]


50

5dl= 75 nm


0







I
dl= 90 nm S[d][
AR


50 0 *
*[e]
dl= 125 nm
100


50


0

4 I--4 1000 800 600 400 200

A Inml







76



However it has been observed that the peak position is primarily a function of the greater of the two thicknesses, Considering this fact yields



T j2 V n 1dm/ A



Solving f or A Pyields



A 21Yn 1 d/ ( (4-43)


Referring to (4-25) the value of is dependent upon the particular optical constants. A typical value is 21Y/5. This yields



X p _5 n 1d m(-4 where



A is the approximate position of the lowest energy
P(longest wavelength) interference peak, and

d mis the larger of the two thicknesses, d 1 and d 2.



Equation (4-44) gives a value of A in the vicinity of the first peak. As n 1 increases and as d increases, the position of the first peak moves to lower energies, longer wavelengths. Thus the condition under which







77



geometric peaks will nor appear in the differential reflectogram is for a low n 1 x d product.

The shortest wavelength in the experimental differential reflectograms is 200 nm. If all interference peaks are required to be at a shorter wavelength than 200 nm. then



n 1d mK<40nm '(4-45)



For WO03, n 1 is 2.45 which yields



d < 16 nrn (4-45a)



So as long as the WO film is less than 16 nm thick, no interference peaks will occur in the differential reflectogram. Figure 4.7 shows the expected peak position as calculated by equation (4-44).



IV-9 Effect of n 1 Variation

The calculated differential reflectograms presented in Figure 4.8 assume a sample configuration as shown in Figure

4.1. The value of n 1 varies from curve (a) through (f). Other parameters are indicated in the caption of the figure. It can be seen that the effect of increasing n 1 is similar to that of increasing the thickness, see Figure 4.7. This also can be deduced by inspecting equation (4-24)






































Figure 4.8 Calculated differential reflectograms of a
metal oxide on a metal substrate. The
sample configuration is shown in Figure 4.1.
The value of nI is varied from curve (a)
through (f) as indicated in the figure. The value.of the optical constants were taken to
A -6
be n0 = 1, n2 = 3 i 3 and k1 = 2 x 10-6.
The thicknesses were dI = 20 nm and d2 = 10 nm.







79








( a] n =1.1







50 bE] nl1.5

0







AR










50=

0







[0 1



1000 800 600 400 200

A [nm]







80



in which thickness d and n 1 are products in the arguments of the sine and cosine terms.

Refer also to the results presented in subsection

IV-12 in which changes in peak shape can be seen when n 1 is varied and the difference in thickness (d I d 2) is small.



IV-10 Effect of k I Variation

The calculated differential reflectograms of Figure

4.9 result from a sample as shown in Figure 4.1. The value of k I increases from (a) through (f). Other parameters are indicated in the caption of the figure. The curve is virtually unaffected by increasing k until the value of k 1 has changed five orders of magnitude, curve (d). Inspection of equation (A-25), along with (A-31) of Appendix A, provides an explanation of this effect. As long as the inequality of equation (A-31) is maintained, the value of k 1 can be expected to have a negligible effect on the differential reflectogram. When the inequality is no longer met, equation (A-25) can be seen to approach the following as k I increases further:


E dl 2
A a (4-46)
E.
I

Similarly





































Figure 4.9 Calculated differential reflectograms of a
metal oxide on a metal substrate. The
sample configuration is shown in Figure 4.1.
A
The optical constants were taken to be no = 1,
n 2.5 and n2 3 -i 3. The value of the
1 2 A
imaginary part of n1 was varied as shown on
the figure, k 1. The thicknesses were
di = 20nm and d 2 l=10nm.







82





AR
R






50. [a] k1=2xl106




10







[d] ki = 0.2







1000 800 600 400 200




X~nml







83




A
fEd2 a .Q-47)




Substitution into (4-21) yields



0 (4-48)



Consequently as k 1 increases, initially there is no significant effect upon the differential reflectogram. As ki1 increases, the differential reflectogram amplitudes are reduced toward zero.



IV-11 Spectral Range Without Interference Peaks

It now has been demonstrated that interferencetype oscillations observed in the differential reflectograms presented in Figures 4.6 4.9 are entirely caused by the geometry of the specimens as shown in Figure 4.1. The observed peaks are neither due to a change in optical properties with wavelength (dispersion) nor interband transitions. Therefore it is important to identify the conditions for which these geometric peaks can be eliminated or at least minimized in the spectral range used for experimental investigations, i. e. between 200 and 800 nm.

There are two possible ways to limit the influence of the interference peaks. One is to prepare specimens







84



such that the interference peaks all occur at wavelengths below 200 nm. The other is to prepare specimens so that the amplitude of the interference peaks in the spectral range is minimized.

Equation (4-45) contains the conditions under which

the interference peaks will all fall to shorter wavelengths than the minimum of the spectral renge, namely 200 nm.



n 1 d m < 40 rim (4-45)



If equation (4-45) cannot be satisfied because both

n 1 and d m are large, then the amplitude of the interference peaks which will occur in the spectral range can be minimized by minimizing the difference in thickness between the two specimen halves. Figure 4.10 shows calculated differential reflectograms for samples represented by Figure 4.1. The value of d I is constant at 500 nm while d 2 increases steadily in curves (a) through

(d). It can be seen that the peak heights are reduced as the difference in thickness (d 1 d 2 ) becomes smaller.

It can be shown that the greatest peak amplitude may be restricted to a maximum value, D, under the following conditions; see Appendix C,


nj(d, d 2)1A F (4-49)








































Figure 4.10 Calculated differential reflectograms of a
metal oxide on a metal substrate. The sample
configuration is shown in Figure 4.1. The
thickness of the two sample sides is indicated
on the curves. The values of the optical
constants were taken to be 0 = 1 n1= 1.6
i 2 x 10-6 and n = 3 i 3.







86








dl=500 rim d=450 rim
50 0



AR [a]
R





50.- d=50n




d2 4900nm







-50- dl=500 nm


0




1000 800 600 400 200

[~nmI







87



where



F (1/21W) sin- (D (4-50)

and

1.2 .(4-51)



IV-12 Interference Peak Shapes

Figure 4.11 shows calculated differential reflectograms assuming the difference in thickness between two transparent films to be one nanometer (d, = 50 nm and d 2 = 49 nm). The The value of n 1 increases from curve (a) through (d). The other optical parameters are listed in the figure caption. The value of "g" was calculated from equation (4-26) and is listed on the figure. Recall that "g" is a strict function of ntn1and n 2' Thus the variation in "g" results from the variation in n 1. Figure 4.11 reveals that the interference peaks become more pronounced when "g" approaches unity. This confirms the result already discussed in subsection iv-6 stating that the peak shape and size may change when "g" is close to unity and the difference in thickness between the two oxide layers is small. The result of subsection IV-9 which shows that the peak position is dependent upon n 1 is also confirmed.

The significance of this result is that it may permit an estimate of n 1 from experimental differential reflectoA A
grams if both n 0 and n2are known and the peaks are








































Figure 4.11 Calculated differential reflectograms of a
metal oxide on a metal substrate. The sample
configuration is shown in Figure 4.1. The
value of the real part of n 1 (n 1) is varied
as shown from curve (a) through (d). The
optical constants were taken to be no = 1,
k= 2 x 10-6 and ^n 3 i 3. The
thicknesses were d 1 =50 nm and d 2 = 49 nm.






89







[a] nl=1-8 g= 1.19
50





0 0

AR


120 nl= 2.

(c] g= 1.0000

0






50 [d] n= 2.6 g= 1.011

0


"I"' II .. .

1000 800 600 400 200

Inml







90



pronounced as shown i n Figure 4.11, curve (c). The estimate is made by comparing calculated differential reflectograms for various values of n 1 with the experimental differential reflectograms. The value of n1which yields a close match is taken to be ani estimate of n 1 in the experimental case. An example of this is presented in subsection IV-14.



IV-13 Theoretical Reflectogram Summary

Table 4.1 summarizes the effects of the basic

parameters upon calculated differential reflectograms. Equations are shown in the table.



IV-14 Tungsten Oxide

Figure 4.12 shows experimental differential

reflectograms of tungsten oxide films on a tungsten substrate. The specimens were produced by the method described in Section III and have the configuration of Figure 4.1, i. e. the thickness of the oxide on the two sample halves is different. It was observed that both sample halves of each sample show visible coloration resulting from interference effects. This verifies that some oxide is present on each half after polishing. The sample was polished to bare metal after each corrosion treatment.









91











ct


C\2 Cd

M
(D Cd --q
4-;' E Q)
a) CH P-4 CD
Q) Ld -r-i 40 b.0
4
Cd cz CH
4-, 0 4--) (D
-H 0
0 (D r- :
m F=i > E Fi 'Ci "
0) m Cd m Cd Cd -P E
4 r:; -4

4-' 0 r-i M 4 -H
PLI 4-1
C\l


Cd -r-i 4-;'

Cd m _:t -H
0 4 -4 4
CH o (D CD (D
u q 0 3 40 -4 40 -P C\j
rc (D H pr; r I
Q) i 4-3 0 > 1 rr r-i a) -i --I -H .4
S-41 -H -H C- f Q) 0) co r4
Cd 4- M C\l 0 -H 0 -H Z m 5. Cd N",
4f -4 ro +-", +3 uro 0 m r--l
ID $: Cd 4 r-i I
0 -H pq 4-1 0 a) a) a) r-A 4--l
'Z 0 > > 5. r -H 0)
M 4-D r- 0 0 0 0) m
4-:' Cd m F= -H 'd N ::;
1--i 4H -P 4-:' Pj cd rc I
m CH c- a)
114 : 4Cd 0 P Cd 0 Cd Cd Id 1C. 10
Q) Q) r- a) r- a) -i -P Q)
m m -4 PA- Pq :Z rS4 0 4 4-3 Z
4
0 u Q)
0 4-:)
>5 4-D CZ 5-A 0 ; Cd (D 4-:'
r--l m




C
Q)

0 Q) Q) C\2
-4 4 ro
0 C) m
0 (D
m
4-l
rc:i

Cd CH CH CH CH
> . r-i H H H 'C





































Figure 4.12 Experimental differential reflectograms of
tungsten oxides on tungsten substrates.
each curve has been obtained by heating
a tungsten sample in dry oxygen for 20
minutes to the final temperature indicated
on the curve. Part of the oxide is removed
from half of the specimen as described in
Section III to produce a sample configuration
shown in Figure 4.1







93






4-
AR 100%
--4%]a
R

[a] 4000C

[b]
4150C



4541C
a

[d
460 oC
a

[e]

8 477 OC


4890C




5040C





5200C
[h]








800 700 600 500 400 300 200
X[nm] IIII I I '
1.6 2 2.5 3 4 5 6
E leV]







94




Curve (a) of Figure 4.12 was obtained by annealing tungsten metal for twenty minutes at 400 00 in a pure, dry oxygen atmosphere. Starting from low photon energies curve (a) shows a gradual rise in 6R/ff with a maximum, termed peak d.., around 400 nm. A minimum exists at 385 nm and a broad flat maximum around 242 nm. The differential reflectogram represented in (b) of Figure

4.12 was obtained by annealing tungsten at 415 00 similarly as above. A shift of the position of peak oC. to smaller photon energies is evident. With increasing annealing time the peaks are shifted further toward lower photon energies and new peaks emerge. For example, in Figure 4.12 curve (c), a pQeak appears at 325 nm and is termed peak The peak structure observed in Figure 4.12 is interpreted to be caused by interference effects resulting from the slightly absorbing film on the absorbing substrate as discussed in subsection IV-3. See Figure 4.2.

The curves obtained by annealing below 520 00, (a) through (g), exhibit a sharpened peak shape as discussed in subsection IV-6. This peak shape indicates a value of nin the vicinity of 2.5, see Figure 4.11.' This is in agreement with the value of Sawada and Danielson56 who found n1 = 2.45 for tungsten trioxide, see subsection 11-8. Starting with the curve obtained by annealing tungsten at 520 00, Figure 4.12 (h), the interference pattern changes in sequence and shape. More peaks