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UNITED STATES DEPARTMENT OF COMMERCE Luther H. Hodges, Secretary NATIONAL BUREAU OF STANDARDS A. V. Astin, Director

Ellipsometry in the Measurement

Of Surfaces and Thin Films

Symposium Proceedings

Washington 1963

Symposium held September 5-6, 1963, at the

National Bureau of Standards, Washington, D.C.

Edited by E. Passaglia, R. R. Stromberg, and J. Kruger

National Bureau of Standards Miscellaneous Publication 256

Issued September 15, 1964

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C, 20402 - Price 32.25
Abstract

A Symposium on the Ellipsometer and its Use in the Measurement of Sur- faces was held at the National Bureau of Standards in September 1903. This

volume contains 19 of the papers included in the. program, together with any dis- cussions which followed oral presentation. Topics covered include historical review, theory, computational techniques, measurement techniques, and the use

of ellipsometry in measuring metal surface oxide films and organic films.

Library of Congress Catalog Card Number : 64-60043

ii
Foreword

An important responsibility of the National Bureau of Standards

is the development and improvement of measurement techniques and

the dissemination of information on them For many years the

Bureau has supported work in the study and measurement of properties of surfaces and, as part of this activity, sponsored a "S^P«TM

on the Ellipsometer and its Use m the Measurement of Surfaces and

Thin Films'' The Symposium was held at NBS m Washington,

D C on September 5 and 6, 1963 and was attended by approximately

200 scientists. Partial financial support was provided by the Army

Research Office (Durham).

This volume contains most of the papers presented at the Symposium as well as the discussions that followed the presentations.

The papers represent contributions from a number of aboratories in

this country and abroad. Primary responsibility for their techmeal

content must rest, of course, with the individual authors and their

organizations. A y ^STIN

Director

in
Preface

The study of surfaces and films is important in many branches

of physics, physical chemistry, and engineering. The measurement

of 'the changes in polarization upon the reflection of light from a

surface is a very sensitive technique for studying the properties

of a surface and any films overlaying it. This technique is com- I monly called "ellipsometry" since it involves measurement of the

state of polarization of elliptically polarized light. While the theory

of the method is old and well-known, it is only relatively recently

that serious and concerted application of the method to various 1 surface problems has been made. This has been occasioned by recent developments in instrumentation and computational techniques,

and particularly by the general availability of electronic computing

facilities. The Symposium, which was held to summarize developments in ! this field of measurement, consisted of 20 technical papers. Eighteen i of the papers were presented orally and two others were submitted but

not presented because the authors were unable to attend. Most

of these papers are published together in these Proceedings to pro- vide m one volume a "state of the art" for this measurement technique. The papers include historical review, theory, computational

techniques, measurement techniques, and the use of ellipsometry in studying metal surfaces, oxide films, and organic films. As this

book is a group of papers rather than a series of integrated chapters,

each paper is written to be self-supporting. An introduction by the

Chairman provides scientific background and discusses the differences,

similarities, and interrelationships among the various contributions.

We, the members of the Symposium Committee, wish to express our appreciation to those who contributed to this conference, the

authors and the discussants. E. Passaglia, Chairman

R. R. Stkomberg

J. Kruger

Ellipsometer Symposium Committee

National Bureau of Standards

IV
Contents

Foreword

Preface

Introduction

1. Historical Review

Measurements of the thickness of thin films by optical means, from

Rayleigh and Drude to Langmuir, and the development of the present ellipsometer. Alexandre Rothen

2 Theory

Optical study of a thin absorbing film on a metal surface. Antonin Vasicek s Optical properties of inhomogeneous films. Florin Abeles

3. Techniques

Computation . Computational techniques for the use of the exact Drude equations

in reflection problems. Frank L. McCrackin and James P. COLSON r u ~tV

~ A Factors influencing the experimental sensitivity of the Drude

technique. Richard C. Smith and Michael Hacskaylo—

Instrumentation . Increased scope of ellipsometric studies of surface film formation.

Page

III IV

1

25

41

6183

97

A. B. WlNTERBOTTOM

Electronic polarimeter techniques. Jerome M. Weingart and ^

Alan R. Johnston "

On determining optical constants of metals in the infrared Shepard Roberts

119

131

4. Applications

Oxidation and Corrosion of Metals Use of ellipsometry for in situ studies of the oxidation of metal

surfaces immersed in aqueous solutions. J khomb Kbuger„

An assessment of the suitability of the Drude-Tronstad polarized

light method for the study of film growth on polycrystallme

metals. P. C. S. Hayfield and G. W. T. White — 157

Studies of thin oxide films on copper crystals with an ellipsometer. J V. Cathcart and G. F. Petersen.-

Optical study of the formation and stability of anodic films on aluminum. M. A. Barrett - Ellipsometry in electrochemical studies. A. K. N. Reddy and ^ J. O'M. Bockris

Application of ellipsometry to the study of phenomena on surfaces prepared in ultra-high vacuum. J. F. Dettorre, T. G. Knorr, and D. A. Vaughan___

v
Adsorption

Measurement of the physical adsorption of vapors and the chemisorption of oxygen and silicon by the method of ellipsometry. R. J. Archer

Application of ellipsometry to the study of"adsorption'from "solution. Robert R. Strombero, Elio Passaolia, and Daniel

J. TUTAS

Determination of thickness and refractive index of thin films as an

approach to the study of biological macromolecules. J B

Bateman * „„„ Other 297 Blood coagulation studies with the recording ellipsometer. L Vroman

Ellipsometry for frustrated total reflection. T. D. Young and J M. Fath

255

281

335

349

VI
Introduction

Elio Passaglia

National Bureau of Standards, Washington, D.C., 20234 The studv of surfaces and thin films overlaying them has been

carried out for many years, but recently has become increasingly

important in several fields of study. Notable among these are he

nhvsics and physical chemistry of adsorption, metallurgy (where the concern Is with

7

oxidation and corrosion processes), and the genera

fielcI of"solid-state electronics, where surface phenomena are of crucial

'Tithsuch interest, an old and very elegant optical method for. the

study of surfaces has been further developed and applied to various

problems The method has been called by several names, among

themTel ipsometrv" and "polarization spectrometry," and is based nnon the classic theory of Paul Drude [1]

1 concerning the change in Z state of polarization of light upon reflection from a bare surface or

a surface with a film on it. By measuring the change in the state of polar zation it is possible to measure the optical constants of a bare

surfaTe or f these are known and the surface is covered by a dielectric film it is possible to calculate the thickness and refractive index of he film' The method involves only the.measurement of polanzat on

and not Photometry, which makes it relatively simple m application

Although the theory and equations of Drude are exact the exact equlSare cumbersome in form. They cannot be solved in closed

tern for the desired quantities (refractive index and thickness of

fitaO n terms of the measured change in state of Polarization as represented by tan the ratio of the magnitude of the reflection

coeffictent for light polarized in the plane of incidence to that polarized

normal the |lane of incidence, and A, the relative phase difference

forThese two polarizations. Thus trial-and-error and iteration methods must be used, and before the availability of electronic computers

the routine application of the exact equations was almost^mpos Al^

Thus Drude and many subsequent workers used only^the hrst linear

terms in the expansion of the equations relating tan^and Ato rrfrac

live index and thickness, and it is these approximate equations which

are (unfortunately) often referred to as the "Drude equations

This limited the application of the Drude theory to verv thin in comparison to the wavelength of light, However with

modern electronic computers the use of the exact equations is routine,

Ze Ihe initia programming is complete. Most workers have some

kuid of computational facilities, and programs have been described by

Archer [2] and McCrackm et al [3]. . The collection of papers in this volume is intended both to serve as an inroduction to tUmethod of ellipsometry for the worker who is

i Figures in brackets indicate the literature references on p. 4
2 PASSAGLIA

embarking on a study of surfaces and thin films, and to present for the expert a summary of some of the latest theoretical[and exDeri

Theori

aC^Tements- The papers are collected into three categori

nrnriZ'H W^bT*'

^ APPhcati°ns- ««d a historical review is provided by Rothem who reviews the history of optical studies of surfaces and gives the. physical basis of the ^thodVempsomXy ft is the purpose of this introduction to give a brief summary of the

HWI I X

&A °{ the

f three categories so that the reader may go directly to the papers of interest. y S

1. Theory

filmrnTM/^ek gh'f

a Tiew of his theor-v of reflection from a film-covered surface and applies it to an absorbing film on a metal

ZnrT.f n°^abS°vT fflmn 2* theory differs TMhat from the theory of Drude, which is well described in several places [4, 5] and in

Pressor \LTn? t

the aUthOTS

' n0tably Hayfield and White.

w ^r,1118 paper §

lves a discussion of the very im- portant problem of ^homogeneous films, and gives the differential equations for the case in which the refractive index of a film varies only m the direction normal to the surface. For the case of expo- nential variation he presents an exact solution for A and * from these equations, and points out how numerical solutions may be obtained for other cases by considering the inhomogeneous film to be a stack ot very thin homogeneous films (a so-called stratified medium).

2. Techniques

Poknn fn^°

d ^ d

??cribed ha

f been aPPUed by McCrackin and Colson m computing the properties of films whose refractive index

laTtZ an fP°nent

! a > hnear, and Gaussian manner with distance

%Z ttbet

, surface> and

fthereby .elating average quantities of these films to the average refractive index and thickness which would be measured experimentally by ellipsometry. These authors also ZlTJn'T n?lh°ds for determining thickness and the real and imaginary parts of the complex refractive index of absoibing filmsr Akn Ing

t if cannot in general be done by a single measurement. (Also see the paper by Hayfield and White.) This paper illustrates the essential role of the electronic computer in modem studies ofthe optical properties of surfaces.

in The" iiS!^mm t^W*. of ellipsometry has been described n the literature [4] and in this volume bv Rothen. However ex- tensions of it are continuing. Thus Winterbottom describes certain ways in which automatic elhpsometers might be made. This is not IThth

a P m reduC

ing the labor of takm? measurements which is very small), but of increasing the speed of the method so

elfinsZiZ T

TeTMUT "light be followed - ( The use of eonvential elhpsometers: to study fast reactions is illustrated in the papers by Keddy and Bockris and by Vroman.)
INTRODUCTION 3

Weingart and Johnston present elegant electronic techniques for measuring very accurately the state of polarization of light. For

cSl values of the parameters of film, substrate and surrounding

medium, ellipsometry has a calculated semstivity of only a. fraction

of an Angstrom [4, p. 47] even for ordinary means of measuring state

of polarization. However, in other cases experimental sensitivity

is of essential importance' (see for example, the paper by SRomberg, Passaglia, and Tutas) and the application of some of the methods

discussed by Weingart and Johnston would be of very real value n

these problems. The whole question of sensitivity is very thoroughly

aiscussed by Smith and Hacskaylo for thin films where the thin

film approximation of the Drude equation is applicable. A less thorough discussion for thicker films has been given by McCrackm

6t An extension of ellipsometry to measuring the optical constants of

metals in the infrared is given by Roberts. This is very difficult £ compared to measurements in the visible range and involves

making photometric measurements at selected azimuths of incident

finearly polarized light and analyzing prisms in the reflected beam.

3. Applications

One of the most widely exploited applications of ellipsometry is the study of metal surfaces and oxidation and corrosion processes on then/ In all of this work the object is to identify the process

occurring and its rate by measuring the refractive index ol the film

formed and the change in its thickness with time

The papers by Kruger and by Hayfield and White present reviews

of this subject describing the type of information that can be obtained

from these studies. The latter authors represent these data by

making a polar plot of P (called Z by them) which is the (complex)

ratio of the reflection coefficient for light polarized m the plane ol incidence to that polarized normal to the plane of incidence, t his has the advantage that a closed curve indicates a dielectric film

whereas a nonclosing curve represents an absorbing film with a

finite value of the imaginary part of the refractive index the same

type of representation is used by Barrett in her study of the corrosion

^ThToxiZtion of copper is described in papers by Cathcart and

Petersen and by Dettorre, Knorr, and Vaughan. The former authors,

studying specific surfaces of single crystal specimens, show the

presense of epitaxially induced strains m the oxide film, resulting in optical anisotropy of the film. This illustrates very clearly the type

of detail which can be obtained by use of this method

Reddy and Bockris illustrate the application of ellipsometry m

electrochemical studies. In this study one of the parameters varied

is the electric potential of the surface, and electrochemica processes

at the surface may be studied as a function of this potential.

As previously mentioned, one of the common problems in all these

studies is that of rapidity of measurement. Often the processes
PASSAGLIA

involved m oxidation corrosion, or electrochemical processes are extremely rapid and the finite time necessary to take measurements is a distinct handicap This clearly calls for some type of automatic

bottom

11

' P PS °ne °

f the d6signS descr*ibed by WmterAnother widely exploited application is the study of adsorption

vJnnr^ t^TTM * YV thoT0^

h

^of the ^sorption of several vapors on silicon surfaces, taking specific account of the roughness

ot the surfaces and deriving the dependence of A and * on surface

coverage and thickness from two points of view: (1) considering the

£?«r nfrt£ %

refract

.1V

fe 1?

dex

T

hlch is the avera^e of the refractive index ol the film, substrate, and surrounding medium, and (2) con- sidering the film to be a two-dimensional collection of Hertzian

and Whrte

imP°rtant problem is also discussed by Hayfield

Adsorption ofmacromolecules is described in papers by Strombere Passagha, and Tutas, and by Bateman. In both cases the aim of the work is to determine the conformation of the adsorbed macro-

,mcolfCa, • 5

e

°n mer workers> working with synthetic polymers, used a strictly elhpsometric technique, while Bateman used the stepped interference reflector method described by Rothen in his

paper.n°T^ Y* °f ?j ellipsometer is described by Vroman, who used this method for studying the coagulation of blood. By calibration with barium stearate layers, Vroman is able to convert measurements

ol intensity of extinction to equivalent thickness of barium stearate. Ihis gave him a very fast instrument to study the rapid changes occurring m this process. s Young and Fath also describe an interesting application of ellip- sometry I hey studied the thickness and refractive index of wringing

films between metallic gage blocks and optically flat transparent

glass surfaces. The angle of the incident light in the glass was greater than the critical angle, resulting in frustrated total reflection from the glass-hlm surface.

It is hoped that the present collection of papers will fulfill its aim

ot being useful to both experts and new workers in ellipsometry. By illustrating the many different fields of application, the potentialities

ot this technique have been demonstrated. It is hoped that new ap- plications will be forthcoming, perhaps stimulated by this volume.

4. References

U] P

'4?r(

d

1890) nD

" PhySik

" 272i 532 (1889); ibid

- 272> 865 (1889>; ibid - 2TM>

Wi\ Ji ATM

h^

r

' J- 0pt- Soc- Am- 970 (1962). NB^67A!f63'a96

P

3r

aglia

' R' R" Str°mbeT^

and H" L" Steinberg, J. Res.

r41b] AF'RL'^^rt

Cki

D andi- P- Colson' NBS Tech- Note 242 (1964). wpJ^rb0U

+T' ;°PticalStudies of Metal Surfaces," The Royal Nor- T^^X^7 ReP°rt N°- X

> 1955' Published by

Bruim,

[5] M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1959).
I

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1. Historical Review

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Measurements of the Thickness of Thin Films

by Optical Means, From Rayleigh and Drude

to Langmuir, and the Development of the

Present Ellipsometer

Alexandre Rothen

The Rockefeller Institute, N.Y. The historical review begins at the end of the last century when

Rayleigh remarked that "having proved that the superficial viscosrty

of water was due to a greasy contamination whose thickness might be much less than one mulionth of a millimeter, I too hastily.concluded

that films of such extraordinary tenuity were unlikely to be of optical

imrortance " The fundamental work of Drude is then discussed. In

theThirtiS, Blodgett and Langmuir used a very srmple ^

^eautrful

interferometric method to measure the thickness of very thin films. The review ends by summarizing how in order to obviate definite drawbacks of this method, the ellipsometer was developed in 1944.

1. Early Studies of Elliptically Polarized Light

Before going over the historical development of this chapter of optics it might be well to define the term "eUipsometry." Elhpsometryfs he aft of measuring and analyzing the elliptical polarization of

light. This is a definition in the broadest possible sense, no mstriction

bfing placed on the cause of the elliptical polarization. Methods for analyzing elliptical polarization go way back to Senarmon in the

middle of the nineteenth century, and more recently, Skinner, from

the National Bureau of Standards, in 1925 published a comprehensive

artick describing an apparatus which he called a universal Perimeter

permitting the determination of the elhpticity and orientation of an

ellipse characterizing a state of polarization 11]. I shall limit the discussion on ellipsometry to the case where elhptical polarization is caused by reflection from the surface of^electee

or a metal coated with a film, and we shall be concerned chiefly

with

the measurement of the thickness of these films or transition layers.

Genuine ellipsometry does not involve any photometric measurements

One measures angles, ratio of amplitude vectors and P^se differences^

Without the fundamental work of the great Augustm Fresnel early

in The nineteenth century, there would be no ''^^ VTMTM

the Fresnel formulas are the foundations upon which fXTli^t

could be constructed. The Fresnel formulas indicate that when light

is reflected, let us say, at the air-water interface the two components

vibrating in and perpendicular to the plane of incidence, generally

rXd r and rs undergo a phase shift of 180° or 0°, and furthermore,

that r^becomes zero ag

t a certain angle called the Brewstenan angle.

i Figures in brackets indicate the literature references on page 21.
8 ROTHEN

I might mention parenthetically that the use of the subscript s to refer to the component vibrating perpendicularly to the plane of incidence

r°nZli

from

, ^e German word senkrecht, meaning "perpendicular."

tW t thl ? formulas expenmentaUy, Jamin [2] and others found that in the neighborhood of the Brewsterian angle, the reflection of hght from many liquids and solids deviates sensibly from Fresnel's law, and that an appreciable amount of ellipticity was present (It is interesting to notice that Jamin described the failure of the component r to vanish at the Brewsterian angle and the ellipticity in its neighbor- hood as two phenomena of different natures.) In other words if the

reflSnf^^R 13 '",6 POlariZe<

! at,45 ° t0 the PrinciPal ^after reflection at the Brewsterian angle, the ratio K of the reflected ampli- tudes in and perpendicular to the plane of incidence was called elfipnnofi Tk

r ms,tance > Jamin found in the case of water a K as large as ? °?u p l;T

he/

alM

S °

f F

W6re 6lther positive or negative, depending on the liquid. No explanation was available for this departure from i<resnel s law and from Jamin 's own words, it was "impossible to find the cause of the ellipticity m an abnormal molecular constitution "

1 hen ten years before the end of the last century, two men, Lord Rayleigh [3] in Eng and and Paul Drude [4] in Gerniany, found the correct explanation for the failure of Fresnel's law. It resulted from the presence of thin films deposited at the solid or liquid interface these two great physicists had opposite temperaments. In their respective papers dealing with this subject, Rayleigh emerges as an outstanding experimentalist. Rayleigh was experimenting^ at the time W1th surface viscosity of different liquids repeating some of ^T„fW°r

i HaJ

mg pr° Ved '" said he> "tha» tfa e superficial

r^ttf!

7 W

l T

wa

fu due t0 a ^asy contamination whose thickness 'h?* ® than one-milhonth of a millimetre, I too hastily concluded that films of such extraordinary tenuity were unlikely to be ot optical importance, until prompted by a remark of Sir G. Stokes

I made an actual estimate of the effect to be expected." And then

ffif his famous experiments on the state of polarization

ot light reflected at the Brewsterian angle from clean water and water ST greaSy co?

tai??ination. His apparatus, at first, was very

wfw IT T WaS

J"?

h?

ht source' a Porcelain dish contained the water, and he observed the image of the sun through a nicol "Thus encouraged, I returned to the attack and, on October 2, 1890 ex- tZl^ the TMage

T of sun as reflected from water at the polarizing angle, and "on June 26th, the dish was placed on a table below the

ThP

unda*ened room and the passage of the spot across the sun s disk was watched. The spot was central at about 4

h 0m and the instant of centrahty could be determined to within 10 s " the simple and precise description of his experiment reminds one of Newton s optics. For instance, one reads in Newton's Opticks, WVerj

, darkJ ch

. TM

aml?er, at a round hole about one-third part of an b

K

road> TMade »n the shut of a window, I placed a glass prism, whereby the beam of the sun's light which came in at the hole might be refracted upwards toward the opposite wall of the chamber"
HISTORICAL DEVELOPMENT 9

This could just as well have been written by Rayleigh. And thus,

the great Rayleigh was experimenting in a lucky period where the

scientists were small in number and the periodicals not too crowded.

One might imagine the reaction of today's editors of a scientific journal if they received an article with the dates and the hours at which the experiments were performed. The general conclusion

drawn by Rayleigh was that the ellipticity of the light reflected by

clean liquids was very much less than that of a contaminated surface,

and he made the final statement that "it is even possible that there

would be no sensible ellipticity for the surface of a chemically pure

body But the surfaces of bodies are the field of very powerful forces

of whose action we know but little." A very prophetic remark.

At about the time that Rayleigh was investigating the optical

properties of the light reflected from the surface of liquids, Drude,

in Germany, was testing the optical properties of the light reflected

from solids. In two fundamental articles published m 1889 and 1890,

he was able to correlate quantitatively the optical thickness of a film

and the optical constants of the material upon which the film was

deposited with two parameters characterizing the ellipse representing

P^ the reflected light. The two parameters were the ratio, tan

and A=5P — 5

s

; where p

p and p

s are the reflection coefficients of the

components in and perpendicular to the plane of incidence after

reflection; 5

P and 8

s

, the absolute phase shifts of the same two components, brought about the reflection. Drude started from the Maxwell

equations and the appropriate boundary conditions. He obtained a most general equation, valid for films of any thickness deposited on

metallic surfaces, which reads:

IP, s TP,s \_ r

P,s p -ix

0

P,s e nv<s =L_ e i^ s =

r f e (1 P e 7?

. s e l+rf 8 r% s elx where /*• s represents the amplitudes of the electric vector in (p) and

perpendicular to (s) the plane of incidence after reflection, and 8

P' S represents the corresponding absolute phase shifts. II s are the amplitudes of the components before reflection, rf s and r% s are the

Fresnel coefficients for reflection at the film and the metal surface,

respectively. The variable x is equal to Ijit cos <p-> I being the thickness

of the film; n, its index of refraction, and <p, the angle of incidence.

The superscripts, p and s, indicate that this formula holds for each

component separately. Writing the above formula for the two

components p and s separately and taking the ratio, Drude obtained:

tan ^ ""(l+rjrfce-**) (r}+r^

The values of i> and A can be obtained experimentally from ellipsometric measurements, that is, by measurements of orientation ot
ROTHEN

amplitude vectors. No photometric measurements are involved In order to test his theoretical results for the case of extremely thin films much thinner than the wavelength of light, Drude developed the

terms ^containing e~

ix in a serial development, using only the first term Thus, he obtained the two following linear equations, one

relating the change in the ratio tan and the other, the change in the difference of phase A = 8

P -8S brought about by the film. These equations are:

and

(3) 2^-2^' =B(1- C) (l-^) £

where the primes refer to the values obtained in the absence of films

In these equations, A and B are functions of the angle of incidence and the optical constants of the support and C=n2 cos <p, n being; the

index of refraction of the film. These equations are commonly called the Drude equations valid

lor very thin films only. Unfortunately, it is not always recognized nowadays that Drude also obtained the most general equation (2) given above. The name of Drude should be attached to this equation

as it was done by Hauschild [5] in 1920, who used a two-term serial development of Drude's formulation. It is to be regretted that in bis recent book on Optics of Thin Films [6] Vaslcek does not make it clear that the general equation can be found in Drude's work.

2. Early Studies of Thin Film Thicknesses

Tronstad [7] was the first one in 1935 to test Drude's first-term ex- pressions for films of known thickness. He measured the thickness

ol monomolecular films of fatty acids adsorbed on a mercury surface and found good agreement with the theoretical values.

In the late thirties, chemists' interest in surface films was suddenly

aroused by one of the most important single developments in the

technique of surface films since the introduction of the spreading trough by Langmmr. It was the discovery made in Langmuir's

laboratory by Katherine Blodgett [8] that coherent films formed on a water surface could easily be transferred to solid slides if they were maintained at a certain minimum pressure during the transfer. In many cases, a large number of monomolecular layers could be trans- ferred on top of each other, a process akin to the building up of a

crystal layer by layer. This opened up a vast field for experimentation which has not yet been completely covered. I still remember the

great enthusiasm created by this new technique. I believe that the
HISTORICAL DEVELOPMENT 11

scientific spirit of Langmuir's laboratory was close to that which must

have permeated Rayleigh's.2 With this new technique mainly developed by Blodgett [9], a large

amount of experimenting went on in many laboratories with all kinds

of materials, from proteins to viruses and bacteria, capable of being

spread and transferred. One of the main characteristics of a film

is, of course, its thickness. From the historical background which I have just reviewed, one would think that the next step would have

been to use the basic formulation of Drude to measure the thickness

of all the films now available. Nothing happened for the following

reason. Blodgett and Langmuir built up on glass slides multilayers

sufficiently thick to have shown interference colors. But there were

no interference colors. This puzzled Blodgett and Langmuir for a while, until they realized that the refractive index of their slides was

not what they expected it to be. When they deposited multilayers

of stearic acid on glass of high index of refraction or on chromiumplated slides, they observed indeed, the most brilliant colors.3 Being accustomed to build films thick enough to show interference

colors, Blodgett and Langmuir developed in a most ingenious way a method to measure the thickness of films whose thicknesses comprised

a very small fraction of the wavelength of light. It consisted in building on a slide multilayers of barium stearate in a series of steps; the

difference in thickness of two adjacent steps corresponded to a double

layer of stearate (~48A). The slide was coated with 35 up to 55

monolayers of stearate. Monochromatic plane polarized light vi- brating perpendicularly to the plane of incidence was reflected at such an angle that two adjacent steps appeared equally dark. If a very thin film of unknown thickness was then deposited on the slide,

the angle of incidence had to be increased to maintain equal darkness

of the same adjacent steps. From this change in the angle of inci- dence, it was possible to estimate the thickness of a film. If the

unknown film had the thickness of two monolayers of stearate, the

angle of incidence remained the same, but equal darkness occurred for the pair of steps directly below the one matched before the deposition

of the unknown film, for instance, between 47 and 45 layers instead of 49 and 47. The method, extremely simple, permitted an accuracy of

3 to 4 A at best, but very often differences as large as 10 A occurred,

when the matching had to be made between two different pairs of

steps, before and after the deposition of the film.

2 1 shall always remember the essentially practical laboratory of Langmuir in one of the old buildings of G.E. in Schenectady. Langmuir liked to work with simple apparatus. His genius more than compensated for his primitive tools. Rayleigh also used simple apparatus, and both men made important discoveries. I visualised Blodgett and Langmuir experimenting with a large trough covered with a film of stearic acid and noticing the sudden change in surface tension when a slide was dipped into the trough, as I visualize Rayleigh forty years before, investigating properties of surface films of greasy material on top of the water

contained in his porcelain dish, with the same enthusiasm and keen sense of observation displayed by his successors in Schenectady. , 3 1 wish to mention here a detail so typical of Langmuir's practical sense. When he and Blodgett startea using chromium-plated slides, they bought from the 5 and 10 cent store electrical appliances consisting of rectangular chromium-plated brass sheets with a hole in the middle used to cover ordinary house switches. They sawed pieces, roughly 1 in. X 3 in., with reflecting properties suitable for their optical studies onnim

transfers. This practical sense reminds one of Millikan's when he made his first apparatus to determine the charge of the electron. 728-101 0-64-2
12 ROTHEN

I Most of the experiments with films carried out in this country just

before ] 940 were made by chemists or biologists not especially familiar

with the field of optics. Blodgett and Langmuir had shown what could

be done with their simple method which could be mastered without

difficulty. This is why the work of Drude and his direct successors was ignored in this country, at least for the time being, and why most

of the experimenters used the elegant interference method developed

by the Schenectady scientists. At that time, I became very much interested in Langmuir's work

and had the good fortune to spend a few days in his laboratory. I wished to pursue some biochemical investigation with surface films,

mostly protein films. This involved many measurements of film

thickness, and for each measurement by the Langmuir technique it meant the deposition of some 50 odd layers of stearic acid. This was time-consuming. Also, it was desirable to have a method per- mitting a greater accuracy by one order of magnitude, if possible.

Finally, for certain experiments, it was very objectionable to have a

large number of stearate films on the slides. Thus, I had to turn my

back to the beautifully colored slides of Blodgett and Langmuir as I recalled the experiments of Drude, done at the end of the last century.

I had a little more experience in optics than most chemists, having

built one of the first polariscopes for the ultraviolet range, and I decided to develop an apparatus according to the fundamental

principles of Drude.

Let me summarize the situation with the help of figure 1, in which

the plane of the paper is perpendicular to the light beam. A light

I II III 5 5 5

5 5 5

Figure 1. Representation of polarized light before and after reflection.

la and lb represent the incident plane polarized light in the two positions vibrating at 45° to the plane of

incidence. II a and II 6 represent the ellipses obtained after reflection. Their orientation and elhpticity

are functions of the angle of incidence and the optical constants of the metal. The two ellipses, II a and

II b , are symmetrically oriented with respect to the plane of incidence and they have opposite sense of ro- tation. Ilia and Illb show the parameters, 0 and R, which are required to analyze the ellipse. (Repro- duced from [12].)
r

HISTORICAL DEVELOPMENT 13

! beam plane polarized at 45° to the plane of incidence impinges on a metalized slide. (It might be in order at this point to recall that, in

optics, following a tradition which goes back to the early part of the

nineteenth century, the plane of polarization is perpendicular to the

plane of vibration of the electric vector. The radio engineers call the plane of vibration "plane of polarization." Certain authors, in recent articles on ellipsometry have adopted this definition, thus

creating unnecessary confusion.) Before reflection, the two components, OP, OS, that is, in the plane of incidence and perpendicular

to it, are in phase and of equal magnitude. After reflection, each of them suffers a phase shift (5

P

, 8

s ) and the ratio \p of their amplitudes,

ij which is now equal to tan V/, is decreased. For most metals at about

70° incidence, the phase shift A=8P —8S is greater than 90°. The

light has become elliptically polarized. The ellipticity is defined,

j contrary to the custom of geometers, as the ratio of minor to major

axis. If a film is present on the slide, the orientation of the ellipse

and its ellipticity are both changed. Drude's equations correlate the

change in \p and the change in A with the film thickness. If the

orientation of the ellipse and its ellipticity are known, ^ and A are

easily calculated, as was shown a hundred years ago by Senarmont.

One should then analyze the ellipse. By placing, for instance,

a A/4 plate oriented with its principal directions along the axis of the

| ellipse, the light becomes again plane polarized. The position of the

X/4 plate determines the orientation of the ellipse. The reestablished

j plane polarized light can be extinguished with an analyzer whose

position with respect to the axis of the ellipse determines the ellipticity.

The only art is to determine accurately the position of the X/4 plate

and of the analyzer, if the method is to be sensitive. No photomultiplier tubes were available at the time I was investigating the

question. Hauschild, in 1920, had used a Savart plate which would

tell by the disappearance of interference fringes produced by the

plate whether the analyzer was properly oriented. The only difficulty

with this method is that one should observe the beam while focusing

at infinity. The presence of scratches on the slides disturbed the

measurements so much that, for easily scratched slides, the method

was useless.

3. Development of Early Ellipsometer

A visual method had to be used which would permit focusing on the

slide so that the eye would ignore the scratchy or defective areas of the surface in the same way as the eye focuses on the slide in the

I Blodgett-Langmuir method. It occurred to me that a halfshade

device analogous to the halfshade device of Lippich could be used. In the Lippich polarimeter, the polarizing unit is made up of two

nicols so oriented that the field is divided into two half fields, each

! half field transmitting plane polarized light whose direction of vibration makes a small angle of about 3° with the direction of vibration

! transmitted by the other half field. Both half fields appear of the
14 ROTHEN
HISTORICAL DEVELOPMENT 15

same intensity when the analyzer bisects the angle made by the

directions of vibration transmitted by the components of the polarizer.

The idea was to make the halfshade device on the slide itself by

depositing, let us say, one monolayer of stearate on the upper part

and three monolayers on the lower part of the slide. Then the X/4

plate would be oriented with its principal directions bisecting the

angle made by the axis of the ellipses characterizing the light after

reflection from the upper and lower parts respectively.

With the help of Drude's approximate formulas, one can calculate

that under such conditions the light, after passage through the X/4

plate, becomes nearly plane polarized, with an ellipticity of about

0.5°, and that the major axes of these two elongated ellipses, characterizing the light from the upper and lower parts of the slide, make

an angle of about 3°, the best angle for a visual halfshade device.

The analyzer is placed in the position for which both half fields appear

equally illuminated (see fig. 2). When an unknown film is deposited on the slide, the analyzer has to be turned by a certain angle which

can be estimated within ± 0.01 ° to reestablish the equality of intensity

of the two half fields. The angle of lotation is a measure of the

thickness. The simplest way is to correlate angles of rotation versus

thickness of known films. The apparatus did work indeed according

to these simple theoretical considerations, and the aim to better the

interferometric method of Blodgett and Langmuir by one order of magnitude had been achieved. My own original contribution was

limited to the halfshade device. The apparatus was described in the

Review of Scientific Instruments [10], and I had to give it a name.

"When a=a' Correct measurement

"When a<or>a' Incorrect measurement Figure 3. Measurements of film thickness with the half- shade device.

Material placed on top of barium stearate indicated by a and a'.
16 ROTHEN

Up to that time, methods for measurements of states of polarization were called polarimetric methods. I could not call the apparatus a polarimeter because it simply was not one according to the chemist's

vocabulary. So T hit upon the word "ellipsometer." Thus, in the

year 1944, the term " ellipsometer" was born. Apparently, it was a good choice, because the word was adopted and it was a very satisfying feeling for me to find out that the word "ellipsometer" was in the

title of this symposium.

There is one point I should like to discuss with the help of figure

3, in connection with the halfshade device. The diagram represents

the cross section of a slide upon which one and three monolayers

of barium stearate have been deposited stepwise. If the added

unknown film or layer formed on top of the two steps is of uniform

thickness, then the halfshade method gives a correct measurement.

But there are some cases in which this is not so, and whereby the

halfshade device yields incorrect results. I shall mention but one

example. A few layers of protein are deposited on both steps and

heated to 75° C [11]. A thin membrane of formvar is placed on top.

The slide is treated with a solution of trypsin for a few minutes, then

the membrane is dissolved, and finally a solution of antibodies homologous to the protein film is smeared on the slide. It is found that

the protein layers deposited on three layers have not been much

inactivated by trypsin and that they can adsorb a thick layer of antibodies, whereas those deposited on one layer of stearate have been

completely hydrolyzed and are unable to adsorb an appreciable

amount of antibodies, as illustrated in figure 4. This is a very nice

illustration of the influence of supposedly inert layers serving as anchorage on the interaction between protein layers and enzymes

across a protective membrane. It is one example among many others

which shows that chemical reactions taking place at interphases

cannot be predicted by our knowledge of chemical reactions takingplace in a homogeneous liquid phase. If the Blodgett-Langmuir

Figure 4. Action of tryspin

through a membrane of formvar 5 70 A thick on three double

layers of protein deposited on

1, 3, 5, 7, 9, 11, and 13 monolayers of barium stearate. 11

13

The protein layers deposited on 3 and 5 monolayers of barium stearate have been very

little acted upon by trypsin. (Reproduced

from [11], p. 155.)
HISTORICAL DEVELOPMENT 17

j technique had been used, this most interesting result could not have

I been observed.

After the end of the war, the photomultiplier tube became available. The tube was substituted for the eye, and the halfshade device

became obsolete. The advantage realized by the substitution of the tube for the eye was an increase in the sensitivity of the apparatus

by a factor of two. This is easily explained by the following considerations. Using the limiting laws of Drude, it can be shown that a 48 A thick barium stearate film on chrome brings about a change

of 3° in the ellipticity and a rotation of 2.3° in the orientation of the

ellipse characterizing the reflected light. With the half shade method,

the A/4 plate remains stationary, and the angle by which the analyzer

has to be turned in order to bring back equal intensity of the half

fields is about 3°. On the other hand, with a phototube in conjunction with a photometer used as a null instrument, both plate and

analyzer can be turned simultaneously to bring back the intensity

of the reflected light to a minimum, with the result that the analyzer

has to be rotated by 2.3 +3 = 5°.^ It means that the optical thickness can be measured within ±0.1 A, since, with a good instrument, the

position of the analyzer to obtain a minimum intensity of the light can be determined within ±0.01°. It is important to notice that

the compensation of the ellipse has to be done with the proper orientation of the X/4 plate, the fast direction alined with the retarded axis

of the ellipse. If the compensation is done with the slow direction

of the plate alined with the retarded axis, a 180° phase shift is in- troduced. The two angles, 2.3° and 3°, characterizing the change

in the ellipse of the reflected light become of opposite sense, and the

analyzer has to be turned by their difference, 3°— 2.3°=0.7°, instead

of their sum. The method under these conditions is several times

I less accurate.

4. Measurement

Shortly after the birth of the term "ellipsometer," Vasicek published

a long series of articles on the optical properties of thin films. It was

his merit to show that the Drude general equation could be developed

in a usable form without approximation. He did it in the case of glass reflection, and I did it afterwards for metallic reflection. Not

only could A be plotted for any film thickness (see fig. 5), but the

individual values of b

v and 5

s

, which cannot be determined experimentally, could be calculated as shown in figure 6 [12]. These curves gave an explanation of some observations made earlier by Blodgett

and Langmuir when they were experimenting with their interferometric method. In this method the light is polarized either in or perpendicular to the plane of incidence.

The component I

s is used preferably to I

p for two mam reasons. The first reason is that, in order to achieve a 180° phase difference

j between light reflected from the air-film interface and that reflected
18 ROTHEN

300°

260° w 220°

<DoI

180° * 140° a

100°

60°

20

^

-'80°

69.2^

\

50° . —7—

/50° cycle)

10 20 30 40 50 60 TO 80 90 100 110 560 510 580 590 600 610 620 630 640 650 660 Thickness of film m Bq stearate monolayers

Figure 5. Phase shift produced by films of barium stearate at 50°, 69.2°, and

80° incidence. The dotted curves are theoretical and the solid and dashed lines are experimental. (Reproduced from [12].)

6P o

: 140 Or

100 -40

60 -80

20 -120

- 20 -160

- 60 -200

-100 -240

-140 -280

-180 -320

-220 -360

\

\ o = 6P

\ \

\ \ \

\\

\

\\

\\ \

0 10 20 30 40 50 60 70 80 90 100 Thickness of film in £>a stearate monolayers

Figure 6. Absolute phase shift, 5

P

, 5

s of each component produced by films of barium stearate at 69.2 incidence.

(Reproduced from [12].
HISTORICAL DEVELOPMENT 19

from the metal, the necessary number of layers of barium stearate

is about 45 layers when I

s is used, in accordance with the relation

7&X45X24 cos r =\. This means that the phase shift for 5

s produced

by the bare metal is close to zero at about 75° incidence. On the

other hand, there is a phase shift close to 140° for the component 8

P

. This phase shift 8

P diminishes with increasing thickness of a film; when the thickness of the film is equal to 45 layers of stearate, 8

P equals

zero and reaches the desired 180° when 95 layers of stearate have been

deposited on the slide. Thus, if one wishes to measure an unknown

film using the Ip component, the optical gage must have twice as many

: layers as when I

s is utilized. The second reason for using I

s instead of I

p comes from the fact

1 that Blodgett and Langmuir had observed that measurements with I

s could be made between 68° and 85° incidence, whereas the range of

i angles of incidence was limited between 77 ° and 85 ° when the measure- ments were made with I

p

, on account, said the authors, of lower

i

] contrast. The curves of 8

P and 5

s versus thickness shown in figure 6 gave an explanation for this lower contrast. The tangent to the

: curve for a thicknes corresponding to 180° phase shift is a measure

!> of the sensitivity of the method; it measures the contrast mentioned

by Blodgett and Langmuir. One can see that the theoretical curves

j corresponding to 8

P and 8

s have not the same shape and that the

1.5 ""Nil

-—7

1.0

0.5 A/

-i Vi -0 5 () 0 5

1 V -0.5

Figure 7. Ellipticity of reflected light at 69.2° incidence, produced by films of barium stearate from OA up to 5 wavelengths of light.

Arrow indicates beginning of first circle. u+iv= cot^e^, u= cotycosA, v= cotysinA. (Reproduced from [12].)

<i

il
20 ROTHEN

| tangent at 180° phase shift is much steeper for the component r

s than for r

v

. These theoretical considerations explain nicely the

observations of Blodgett and Langmuir.

In one way, the method of Blodgett and Langmuir is theoretically

simpler than the ellipsometric method which uses both r

s and r

p for a single measurement. If the film is optically equivalent to a uniaxial

crystal with its optic axis perpendicular to the surface of the slide, as in the case of barium stearate layers which form a positively birefringent crystal, only one index of refraction no is involved for any

angle of incidence when the component r

s is used. In the case of ellipsometric measurements, two indices are involved, n

s0 and n

p

e , the

value of n

p

e varying between two extreme values, n0 and n u with n0 <C.ni. In the Drude formula, an average n is introduced. Thus,

one should not expect the Drude method to be very precise for these

films. A second periodicity is introduced due to the birefringence of the film. The birefringence of barium stearate films is ne —

Q = 1 .551 —

1.491=0.06. At 50° incidence, the value of ne is 1.528. One can

thus estimate this longer period to be iV~3800, N being the number

of layers of stearate in the longer period. A plot of the experimental

values of \p and A can be seen in figure 7 for films up to 765 monolayers.

One cycle corresponds obviously to an optical path of X. From the

rate of precession of each cycle, it can be estimated that the 46th

cycle should come close to the first one, as predicted by the preceding

formula.

Shortly after the incorporation of the photomultiplier tube in the

ellipsometer, Trurnit followed a rate of adsorption by using a recording

device. This involved a mixed method; it is partially ellipsometric

and partially photometric. At the beginning of the experiment, one compensates partially or completely the ellipse representing the

reflected light and measures the increase in the intensity of the reflected

light which occurs as the adsorption proceeds, since both the orientation of the ellipse and its ellipticity change. In order to estimate a

thickness, a calibration curve has to be prepared with films of known

thickness. With a light source that is powerful and stable enough,

which is a difficult thing to achieve, the method may become extremely

sensitive, and differences in optical thickness much smaller than

0.1 A can be detected.

I wish to end this review by mentioning two entirely different

applications of the ellipsometer. In 1949, Jackson and Burge [13] in England were able to measure

with an ellipsometer the thickness of the creeping film of helium n

which was found, roughly, 150 A thick.

The second application is the use of the ellipsometer to follow

immunological reactions, based on the fact that an antigen deposited on a slide is capable of adsorbing specifically a relatively thick layer

of antibodies from a homologous antiserum. This I have shown years

ago in the case of systems giving a precipitin reaction [14]. Quite

recently, we have taken up the matter again in collaboration with

Dr. J. Casals of The Rockefeller Foundation and Mr. Christian
HISTORICAL DEVELOPMENT 21

Mathot of our laboratory, for systems which do not give precipitin

reactions. The method looks very promising and would be very

much faster than either the hemagglutinin test or the _ complement

fixation test, which are commonly used in this field of immunology.

5. References

! [1] C. Skinner, J. Opt, Soc. Am. 10, 491 (1925).

I [2] J. Jamin, Ann. Chim. 31, 165 (1851).

[3] Ravleigh, Phil. Mag. 33, 1 (1892).

[4] P. Drude, Ann. Physik Chemie 39, 481 (1890).

[5] H. Hauschild, Ann. Phvsik Chemie 63, 816 (1920).

[6] A. Vasfcek, Optics of Thin Films, North Holland Publ. Co., Amsterdam

j (I960).

[7] L. Tronstad, Trans. Faraday Soc. 31, 1151 (1935).

[8] K. B. Blodgett, J. Am. Chem. Soc. 56, 495 (1934).

[9] K. B. Blodgett, J. Opt. Soc. Am. 24, 313 (1934).

1 [10] A. Rothen, Rev. Sci. Instr. 16, 26 (1945).

[11] A. Rothen, Physical Chemical Techniques in Biological Research, II, p. 194, Academic Press (1956).

[12] A. Rothen, Ann. N.Y. Acad. Sci. 53, 1054 (1951).

[13] L. C. Jackson and E. J. Burge, Nature, 164, 660 (1949).

[14] A. Rothen and K. Landsteiner, J. Exp. Med. 76, 437 (1942).

2. Theory
I

I
Optical Study of a Thin Absorbing Film on a

Metal Surface

Antonin Vasicek

Institute of Solid State Physics of J. E. Purkyne University, Brno,

Czechoslovakia

The classical theory of the reflection of light from a metal with a

thin superficial film was deduced by Drude for very thin films up to about 100 A. The author deduces from the interference of light in a thin absorbing film the formulae for the reflection of light from a metal with such an absorbing thin film. The formulae fulfill the energy conditions on both boundaries of the absorbing film. The

apparatus, the methods of measuring the elliptically polarized light, and some experimental results obtained by this polarimetric method

are mentioned.

1. Introduction

The optical studies of a thin superficial film on a metal have re- cently become a branch of physics, although such problems are not

new. As much as 75 years ago Drude examined the influence of a

thin superficial film produced on a metal on the measured optical

quantities [l].

1 These formulae of Drude 's were completed by

Tronstad. These formulae of Drude-Tronstad are very often applied,

e.g., in the literature about the corrosion of metal, in the following

form [2]:

4:ird (cos (p0 sin2 <pQ (nl— n$) |~, 2 2 \ /l \ i 2 ,2!

2\P— 2^=sin2 \f/ 4cird cos <p0 sin2 ^0 (^i — ^0) X (cos2 <p0 —nla) 2j\-nt>a'

2

^n\a f (J-*—

( cos2 <^o—^o«)^J'

(!')

where \p indicates the azimuth, A the phase difference measured on a

clean metal surface, ^ and A the same quantities measured on a metal with a thin, superficial, transparent (nonabsorbing) film. In

the previous formulae, d denotes the thickness of the film, ni its re- fractive index, n0 the refractive index of the first medium (usually the

air having refractive index n0 =l). These formulae also include the

angle of incidence <p0 of monochromatic light having the wavelength X. The expressions a, a' are determined by the formulae

1 Figures in brackets indicate the literature references on p. 39. 25
26 VASICEK

1-K2 n

2 (l+K2

) 2K

a —

n

2 (l+#2

)

(2)

(3)

where n, K are Drude's optical constants computed for the normal

incidence of light. The complex refractive index of a metal can then

be written in the form N=n(l—iK) = n—ik.

The formulae (1) and (!') may be written briefly in the form

A-A=-Cd, (4) 2t-2f=C'd. (4')

It is evident that with increasing thickness of the thin film the measured azimuth increases linearly, but the phase difference decreases

linearly. Kecently the Drude-Tronstad formulae have been considered as a special case of the more general formulae, being deduced

from the interference of light in a thin film. We show later that the

validity of Drude's formulae may be assumed up to a thickness of about 100 A, i.e., for the thinnest film. Drude-Tronstad's formulae

are simpler and more convenient for practical calculations of the

refractive index and of the thickness of a thin transparent film than more general formulae, which are true for greater film thicknesses,

because they are deduced supposing the interference of light in a thin

film. In this time of rapid computers this greater complexity of the

general formulae is no obstacle to the determination of the corresponding optical quantities.

2. Theory

2.1. The Determination of the Optical Constants of a Clean

Metal Surface

When light incident on a plane boundary between two dielectrics

(air and glass) is plane polarized, the reflected light is also plane polarized. If a thin superficial film is present on a dielectric, the reflected

light is elliptically polarized. In this case it is very easy to decide

whether or not a very thin superficial film is present on the surface of

dielectric. According to the author, it is possible to determine the

refractive index and the thickness of a thin film from the ellipticity of the reflected light by the polarimetric method [3, 4]. A new situation occurs at the reflection of light from a metallic

medium, because the reflected light is elliptically polarized. The

measure of this ellipticity is the angle 7, determined by the ratio of
OPTICS OF ABSORBING FILM ON METAL 27

the two semi-axes of the ellipse according to the relation tan y = b/a. For aluminium, at an angle of incidence of 60°, this ellipticity angle is about 15°, for silicon only about 30'. A thin superficial film on a metal surface changes this ellipticity in a certain way. Therefore the

determination of a clean surface of metals (without any superficial

film) is more difficult than in the case of dielectrics. We mention this problem again in the experimental part.

According to the author, the optical constants of a clean metal are determined"as follows: We start from the relation for the square of the complex refractive index of a metal [5] N2 =n2 —k2 —i2nk cos <?i=(n— ik)\ (5) where n, k are the optical constants of the metal for the angle of re_- fraction in the metal <pi, for the angle of incidence <^0 , respectively, n

and Tc are Drude's optical constants for the normal incidence of light

(^=0°).

Denoting by n0 the refractive index of the first medium, ^ the

azimuth defined by the relation tan and A=8P —8 S the phase dif- ference, then the refractive index of the metal is calculated from the

formula

tan

2 <p0 cos

2 2\p sin 2^ cos A)

nanism* vJl+j^ : L (1 and the absorption coefficient

^

rip tan (pQ sin <?0 sin 2\p sin A ^

1+sin 2\p cos A

The optical constants of the metal depend upon the angle of refrac- tion tpi and also on the angle of incidence <p0 , because the light waves

propagating in a metal are nonhomogeneous. Therefore it is necessary

to calculate these optical constants always for the normal incidence

of light according to the formulae ,_

(A2 +B2

) 1/2 +A

2 n*=^ —> (8)

A=n2 -k2 =n2 -k2

, (10)

B=2nk cos ^=2ri£ (11)

728-101 0-64-3
28 VASICEK

The angle of refraction in a metal is determined by the formula

2 _n2 —n2

0 sin2 (p0 _ tan2 <p0 cos

2 2\p , COS <p\ — 9 — — — _ • (lz) n

2 (l+sin2^cos A) 2 +tan2 ^0 cos

2 2^

2.2. The Calculation of the Amplitudes rp, rs and of phases 8P, 8S of Light Reflected on a Metal from n, k, <p0 For these calculations the starting points are the generalized

Fresnel relations

j 6p _tan ((pQ— <$i) _N2 cos (p0 —n0N cos $i

tan (^o+^i) N2 cos (po+rioN cos $i

AT"

2 cos <p0 —nQ (n cos <pi—ik)

(13) iV2 cos ^0 +^o(^ cos vi—ik)

id s _ sm (<Po —<^i)_% cos (Pq—N cos <£i_n0 cos <p0 —(n cos ^—ik)

sin Opo+^i) 7?o cos <^0 +^V cos <3>i n0 cos <p0 +(^ cos pi—ik)

(14)

where we substitute for the complex expression N cos <E>i the relation

N cos $i=n cos <pi—ik. (15)

Direct calculation from the complex amplitude rpe

idp is very diffi- cult. It is more convenient to calculate the ratio of the two complex

amplitudes first

re

u, =

rv

e

i(sr -8,) =

cos (go-H&i) =n cos ^pcos (pi—nQ sin2 go—ik cos^0 r s cos (<p0 —$i) n cos <p0 cos <^i+n0 sin2 (p0 —ik cos<p0 (16)

and to determine r=tan and the phase difference A = 5 P —

S

. Snell's law is also valid for the complex quantities in the form n0 sin (p0 =N sin <E>]

. (17)

After the calculations we obtain the following formulae: ^= (t*o cos <po—n cos <pi) 2jrk2

;

, s (no cos (p0 -{-n cos <^1 ) 2 +F

(n cos <po cos (pi—n0 sin2 <p0)

2 +&2 cos2 <p0 (rs cos <^0 cos (Pi +Uq sin2

<p0)

2 +&2 cos

2 <p0 (19)
OPTICS OF ABSORBING FILM ON METAL

rp =r-rs , 2n0k cos <p0 tan 8

tan A= 2n0k sin <p0 tan <p0 —[n2 -\-k2 —nl tan2 <pQ ]

29

(20)

(21)

(22)

8P =A+8S . (23)

It is evident that the author does not calculate with Drude's

] optical constants of a metal n, k, but with the optical constants

|] n,k for a certain angle of incidence <£0 (<Po^O°).

2.3. The Reflection of Light on a Metal With a Weak Absorbing

Film

Experience has shown that many oxide films on metals, e.g., oxide

films on steel, have weak absorption. Therefore it is necessary to present the theory of the reflection of light from a thin film on a metal

in the most general form, i.e., for a thin absorbing film. This theory

I was published by the author in 1961 [6]. In this case both amplitudes

1 on the boundaries of the absorbing film are complex quantities

1

j (%. i). For the reflection of light from a metal with a thin absorbing

film, the following formulae are valid:

r'p e

is '»+rp

' eKe-*' e- ix _ rp e

i5 *-\-r

p e~x

' e~i(~x-s

'p,) (24)

• if

r e

r e if

Nzn.ik

Figure 1. The reflection of light from a metal with a thin absorbing film.
30 VASI^EK

rs e

i5 S: r'a e

id: +r',' e-x

' e-*1-*")

(240

The phase difference of the light rays in an absorbing film is the

complex expression, which in the previous formulae is divided into

two parts:

the real phase difference independent of the absorption

x—— 2nx d cos <p\ A and the imaginary part

being used in the expression

A

_2j

e x

2k ^

(25)

(26)

(260

The last expression determines the damping of the amplitude owing

to the absorption.

Formulae (24) and (24 ') can be written in the form

rn e

l °pr e

*os

. Ct3 +i@3

(27)

(270

In the latter expression it is necessary to substitute

ai=r'p cos 8p +r'p'f-x ' cos (x—8p ), a2 =l+rprp e-x' cos (x+8'p —8p ), a3 =r's cos d's +r" e~x ' cos (x—5"),

oci=l+r'sr's ' e~x' cos (x +8'

s —8"),

Pi=r'p sin 8'p +rp e-x' sin (x—8p ), &2=r'prp e-x' sin (x+8'p -8p )

P3 =r's sin <5;+r'/e-x ' sin (z— 5"), sin (x+8',-%').

Multiplying the formulae (27) and (270 by the complex conjugates

r^e l5 p and rs e~ Us

, we get

(28)
OPTICS OF ABSORBING FILM ON METAL 31

2 o$+fi_ r'p

2 +r'p

' 2 e2*'+2rfp

r'

p'e-*' cos {x^8'p -8'

p

f )

? ( Tv ~*%+& l+r'p

2rp

' 2 e2x '+2r'p

r'p'e-*' cos (x+8'p -8'

p

')

2 _a|+M_ r's 2 +r's

' 2 e2x '+2rys 'e-*' cos (x+d'-d'/) _ f ^""oJ+jSS l+r;2 r;

,2 e2i '+2r>;, ex ' cos (z +^-C)

The azimuth \p, determined by optical measurements, is

tan ^=^- (30)

rs The amplitudes rp and rs , as the absolute values, are always positive.

Calculating the phase 8 P we introduce the auxiliary angles % p and

7i P according to the formulae

tan f,=^i tan Vp

=^ (31)

Similarly, calculating the phase h s we introduce the angles £s and

rj s by the relations

tan £s =

The phases are then

tan £,=&, tan „s =^- (32)

«3 a4

8 P =^P —Vp, 8s =^s —rjs , (33)

and the phase difference, as the quantity following from the optical

measurements, is finally

A=a,-*,=(fc-&) - Oh-*) • (34)

The amplitudes rp, rs and the phases 8 P , 8S completely determine

the optical characteristic of the system—a thin absorbing film on a metal—at the reflection of light.

The author points out that the formulae (24) and (24') differ from

Murmann's formulae for light reflected from a thin metallic film. The formulae following from (24) and (240 fulfill the corresponding

energy conditions on both boundaries of the absorbing film, but

Murmann's formulae fail in the energy conditions [7]. The preliminary calculations require the determination of the

phase difference according to the formula (25) and of the expression

(260 for the damping of the amplitude. The amplitudes r'P) rs and

the phases 8'

p , 8'

s are calculated from formulae (18) to (23); we

merely substitute the quantities nh kx for n, k. The amplitudes rp , r" and the phases dp, «" are calculated

separately from the supposition that for the thickness of the film

d->0, x->0, z'-»0, respectively, the reflection of the light is the same

as from a clean metal without any superficial film. Then the formulae
32 VASTER

(24) and (24 ') are simplified as follows:

<> r,V8-'+r,'V!-"

Because the calculations are the same for both components of light, we calculate the corresponding formulae for one component only; e.g., for the ^-component.

From the formula (35) we calculate the complex amplitude r'v

'

u», as the unknown quantity. We obtain

From the last formula we get the final relations

^. P +^2 —2r0 , p r; cos (<3 0 , p —8P ) l+r20tPr'p

2 -2r0 , p

r'

p cos (50 .„— O'

tan y/_ ^o. P sin3o, p -r;(l-fr^) sin g;-r0 ,

2 sin (80 , p -28'

p ) o. P rn . cos <5 0 , P -rp (l+r* p ) cos 5; +r0 , pr;

2 cos (50iJ,— 2«i)

(37)

(38)

Similar formulae are also written for the s-component.

The amplitudes of the light reflected from a clean metal r0 p , rQ s and their phases 50. ?

, 50 , s are given by the formulae (18) to (23).

' Then

all the quantities in the formulae for the reflection of light from a metal with a thin absorbing film are determined.

The reflection of light from metal with a thin absorbing film ex- presses the most general case, where k^O, k^O, and therefore zVO, 8'p ^o, ssvo, «;vo, «;vo.

For the reflection of light from metal with a thin dielectric film

kl = 0, k^O. Then we have x' =0, tan d'p =0, tan 8'

3 =0. The corre- sponding phases 8'p and 5S

' are 0 or w. For the phases 8" and 5," we

have s;V0, 5S'V0.

The simplest case is the reflection of light from a dielectric (glass)

with a thin dielectric (transparent) film, where ^=0, k =0. Then

x', =?' ^

n ^ —tan 5,' = tan 5p =tan 8

f/ =0, and the corresponding phases

8'p , 8'

s , 8P , 8" are 0 or ir. When calculating the Fresnel amplitudes on the boundary between

two dielectrics (these amplitudes are real quantities), it is more con- venient to start from tbe formulae for the first boundary

, tan fa-Pi),

(39) >_ sjnfa-jgQ
OPTICS OF ABSORBING FILM ON METAL 33

and for the second boundary

tan (<pi —<p) tan (<pi+<p)

(40)

sin (<pi—<p)

sin (<pi+<p)

(400

We obtain the amplitudes with positive or negative sign. The

positive amplitude means the phase 0 or 2tt; the negative amplitude

is actually the positive amplitude with phase tt according to the

relation re

±iir =—r.

3. Experiment

3.1. Apparatus for Measurement of Elliptically Polarized Light

When studying reflection in polarized light we use the polarization

spectrometer, which is also called an ellipsometer. Measurements

with this apparatus have been made by the author since 1937. It is a spectrometer furnished with polarization equipment—polarizer, analyzer, and compensators. One compensator is placed on the side of the polarizer, the other on the side of the analyzer. The schematic

arrangement of this apparatus is seen in figure 2. The third compensator is not necessary. Senarmont's compensators (quarter-wave

mica plates) are used. The author's measurements were made in

half-shade using Nakamura's double-plate placed in front of the

analyzer. The subjective measurements by this half-shade method

are more sensitive than the original arrangement in the dark field of a telescope. Using a detector photocell or photomultiplier tube, we come back to the original arrangement, because these measure- ments can be made with very high exactness without the half-shade

arrangement.

Figure 2. Schematic diagram of the polarization spectrometer: S—source of monochromatic light, Si—slit; C—collimator; P—polarizer; T—rotating table; Ku K2 , K%—

compensators; D—Nakamura's double-plate; A—analyzer; D—telescope, (a) R—eyepiece for measurements

in a dark field, (b)—telescope 0 with eyepiece E for measurements in half-shade.
34 VASICEK

M

Figure 3. Measurement of elliptically polarized light.

Figure 4. Measurement of the optical

quantities x and y. The axis x is the plane of incidence.

The measurements of the elliptically polarized light can be made

in two ways:

(a) From the polarizer the parallel rays of the plane polarized

light are incident on the metallic mirror. The polarizer is set at 45°

(fig. 3a). The light reflected from a metallic mirror is elliptically

polarized. We rotate the compensator circle on the side of the

analyzer and the analyzer, until we get the smallest intensity of the

reflected light. The vibration direction of the compensator determines the angle x between the great semi-axis of the ellipse and the

plane of incidence. The difference between the position of the

compensator and the analyzer gives the ellipticity angle y, where

tan y=- (fig. 4). This measurement can be made in other ways,

where the analyzer is set at 45° and we rotate the polarizer and the

compensator on the side of the analyzer, until the detector gives the

minimum of the reflected light (fig. 3b). The roles of the analyzer

and polarizer are changed.

Because the azimuth \p and the phase difference A are deduced from

the theoretical formulae, but with the optical methods the angles x
OPTICS OF ABSORBING FILM ON METAL 35

; and 7 are measured, we perform the calculations according to the

! formulae.

cos 2^= cos 27 cos 2x,

tan A= tan 27

and reciprocally

sin 2x

tan 2x=tan 2\f/ cos A,

sin 27=sin 2\p sin A.

(41)

(42)

(43)

(44)

(b) The second way of measuring elliptically polarized light is described by Archer [8, 9]. With this method the compensator (on

the side of the analyzer or on the side of the polarizer) is set so that

the direction of the fast vibration is at 45°. If we consider the

compensator placed on the side of the analyzer (fig. 5a), then we

rotate the polarizer and analyzer until we obtain the minimum of the reflected light. The corresponding position of the polarizer is indicated by P and the corresponding position of the analyzer is indicated by A. Then we set the compensator (the direction of the

fast vibration) at 135°. The corresponding position of the analyzer

at the minimum of the reflected light is A! . For the calculations

Archer and Winterbottom [10] give the following formulae

tan2 ^=tan A tan A'

,

tan ^=cot L tan A=tan L tan A

cos 2 L-—cost e cos 2 P,

tan A=sin e cot 2P,

(45)

(46)

(47)

(48)

M

A 45°

S

M

A 45c

Figure 5. Archer's method for the measurement of ellipti- cally polarized light.
36 VASICEK

where e indicates the relative retardation of the quarter-wave mica

plate. For the chosen wavelength, this relative retardation is exactly - or |- Archer's method not only enables one to check

exactness of this supposition, but also permits the application of the

quarter-wave plate for other wavelengths, because the previous

formulae allow a different relative retardation than ^ to be taken

into account. This fact is very important for Archer's method.

Similarly, Archer's method can be used in the second arrangement

(fig. 5b), where the compensator placed on the side of the polarizer

is set at 45° and 135°, respectively. The roles of the polarizer and

and analyzer are again changed.

3.2. Measurements and Results

In our institute this polarimetric method is used to study thin

transparent oxide films on aluminium and on the monocrystal plates

of germanium and silicon (Sladkova) [11] and thin absorbing oxide

films on steel (Ruzicka). The most difficult problem is the determination of the optical constants of the clean metal surface. This

problem is relatively simple when studying metallic mirrors evaporated in vacuo, where the measurements are made immediately

after the production of these metallic mirrors. In this case it is possible to neglect the thicknesses of the thinnest natural oxide

films. With the measured samples which are mechanically polished,

e.g., steel, the determination of the clean metal surface is more difficult. This problem is especially difficult with semiconductors,

where etching by various convenient agents is recommended. We

took the cleanest metal surface to be one giving minimum value of the measured azimuth \p and maximum value of the phase difference A. The difficult and lengthy calculations of the corresponding tables

(usually for angle of incidence 50° or 60°) are performed by rapid

computers. The refractive index of the first medium (air) is practically nQ =l. In these calculations it is necessary to know the optical constants of the clean metal surface N=n—ik. The optical constants

of the film Ni=rii—i kx are chosen according to the values of the bulk

material (e.g., for oxide films on steel according to hematite) or ac- cording to known values of the oxides. For example, when calculating

the oxide film on silicon we take the values 7ii = 1.40, 1.45, 1.50, 1.55. The tables are calculated as a function of the thickness d of the

measured film and the phase difference x of the light rays in the film. Figures 6 and 7 are examples of these calculations for the oxide film

A12 03 on aluminium having nx = 1.635, according to the author's

measurements.

Because the film is nonabsorbing, these curves are periodically

repeated, increasing the thickness d and the phase difference x, respectively. From the graphs in figures 6 and 7, it is evident that

the initial part of these curves may be taken as linear up to thicknesses
OPTICS OF ABSORBING FILM ON METAL 37

Figure 6. The dependence of the azimuth \p on the thick- ness d and on x, respectively, of thin AI2O3 film produced on aluminium.

Figure 7. The dependence of the phase difference A on

the thickness d and on x, respectively, of thin A12 03 film produced on aluminium.

of about 100 A in accordance with Drude's theory. According to Winterbottom, it is very useful to plot the polar curve for the film

in question. The polar curve combines the two graphs (figs. 6 and 7) into one. For the transparent film, the polar curve in figure 8 is closed. Archer plots these curves in the orthogonal system \p, A [9]. When calculating the optical constants of an absorbing film on a metal, the determination of the optical constants of such a film is very

complicated because the number of various possible combinations 01

Tii and ki increases. Owing to the damping of the amplitude of light

in the absorbing film, the curves of the dependence ^ and A on the

thickness d or on x are no longer periodical; they are damped to a

greater or lesser degree according to the value of the absorption

coefficient kx

. With increasing thickness of the film, the values

yp, A approach the constant values which correspond to the optical constants of the same material as the film. We can suppose that m

the case of the reflection of light from a metal with a thin absorbing

i

I
38 VAsfcEK

Figure 8. Polar curve accord- ing to Winterbottom for thin Al^O^ film produced on alu- minium.

Figure 9. Polar curve for thin absorbing oxide film with

111 =2.48, ki = 0.96S -produced on steel.

200

600 0

film we have few equations for the determination of the optical constants rii and ki of the film and its thickness d, and that it is necessary

to determine the thickness of the measured film by another method

(e.g., by the interferometric method). Because these measurements

are made at various thicknesses of the film (not for one case), it is possible to determine the third unknown quantity, e.g., from the

damping of the curves. In the case of the reflection of light from a metal with a thin absorbing film, Winterbottom 's curves are not closed,

but create spirals, which are closed the more rapidly, the greater is the absorption coefficient k\. An example of such a spiral is shown

in figure 9 according to Ruzicka for an oxide film with n1 =2.42,

&i=0.968 on steel.

Polarimetric measurements for the study of a thin oxide film on

metals haye many advantages, but they also have some disadvantages.

First, it is very difficult to obtain a polished clean metal surface

having the necessary optical qualities. The polarimetric measure- ments are also difficult, and it is also difficult to prepare the tables

and evaluate the measured results from these tables. The polarimetric method can be applied as long as the surface of the sample

at the production of the film reflects light. The polarimetric method

enables one to determine the optical constants and the thickness of the film with great accuracy, especially for the thinnest films, where

other methods fail. Recently the microbalance method was elaborated, enabling the study of the continuous increase of oxide films on metal by heating [12]. The polarimetric method will certainly

retain its importance in physical and technical practice.

4. Conclusion
OPTICS OF ABSORBING FILM ON METAL 39

5. References

[11 P. Drude, Wiedeman Ann. Phys. 36, 884, 886 (1889).

2 U R Evans, Korrosion, Passivitat und Obernachenschutz, p. 715, Berlin

(1939).

[3] A. Vasicek, Phys. Rev. 57, 925 (1940).

[4] A. Vasicek, J. Opt. Soc. Am. 37, 145 (1947)

5 A Vasicek, Optics of Thin Films, p. 288, Amsterdam (1960).

[6] A. Vasicek, Opt. Spectry. 11, 128 (1961).

7 A. Vasicek, Optik 19, 584 (1962); 20, 225 (1963).

[81 R. J. Archer, Phys. Rev. 110, 354 (1958).

9] R. J. Archer, J. Opt. Soc. Am. 52, 970 (1962)

[10] A. B. Winterbottom, Optical Studies of Metal Surfaces, pp. 44, 57, Trondheim (1955).

[11] J. Sladkova, Czech. J. Phys. B13, 452 (1963).

[12] E. A. Gulbransen and K. F. Andrew, J. Electrochem Soc. 109, 560 (1962).

Optical Properties of Inhomogeneous Films

F. Abeles

Faculte des Sciences and Institut d'Optique, Paris

The inhomogeneous films investigated in this article have a

refractive index n which is a function of the single coordinate z, the

axis Oz having the direction of the normal to the planes which limit the layer. The discussion is limited to non-absorbing films, although

mention is made of the possibilities of using the same methods for absorbing layers. Three different methods of computation are described: (a) use of the differential wave equation; (b) WKB ap- proximation; (c) replacement of an inhomogeneous layer by a pile of homogeneous films. A numerical example is discussed, which cor- responds to a refractive index of the inhomogeneous layer given by n=nl exp (az). Exact solutions are given in terms of Bessel functions and they are compared with approximate solutions. Drude's equations, valid for very thin films, are not discussed in this article.

1. Introduction

The aim of this review article is to give some information concerning

the computation methods which can be used when discussing the

optical properties of inhomogeneous films. Such films may be en- countered quite often in physico-chemical investigations, and it may

be useful to have some indications concerning their properties. We

are dealing mainly with nonabsorbing films, which are such that

their refractive index is a function of the coordinate z only, the

direction Oz being that of the normal to the planes which limit the

film. We do not discuss anisotropic films, because it would take us

too far and because the mathematical investigation of this problem

is only in its infancy. In fact, an inhomogeneous and isotropic film can be rather complicated in its optical properties and we do not haye

yet any means by which to make sure that such a film is anisotropic, at least if its anisotropy cannot already be detected by measurements

in normal incidence. , We shall discuss three different methods of computation. Two ol them, the use of the differential equation and the replacement of the

inhomogeneous layer by a number of homogeneous layers, are quite

general, which means that they can be employed in any circumstance.

The third method, which is an adaptation of the well-known WKB

method for the solution of second-order differential equations, can be

used only when the refractive index of the film is a slowly varying

function of z. We have omitted from our discussion the famous

Drude equations, which are valid only when dealing with films which

are very thin with respect to the wavelength of the incident light.

These generally have a very rapidly varying refractive index, so that

they represent the case which is the opposite of the WKB films. It should be pointed out that the different methods described

below are not specific to ellipsometry, because they enable the compu41
42 ABELES

tation of the reflected complex amplitudes separately for the s- and

the ^-polarizations. We think that they are convenient, as they

distinguish the quantities which concern the inhomogeneous film

from those which pertain to its substrate. A modification of the

substrate is easily taken into account. On the other hand, if photometric measurements are used in order to supplement the ellipsometric

measurements, their results could be readily interpreted.

2. Definitions and Notations

The thin film is bounded by the planes z=0 and z=d. It is comprised between two semi-infinite media, the refractive indices of which are n0 (for 2<0) and ns (for z>d). Light is incident on the

plane 2=0 and we shall generally have n0 =l, although it is possible

to examine thin inhomogeneous films embedded in other media (for

instance, fluids). The refractive index of the film under study,

n(z), is a complex function of z, which means that it can be absorbing!

In fact, we shall discuss mainly the properties of nonabsorbing films. But the considerations which lead us to the differential equations

satisfied by the function U(z) (see below, eqs (4) and (8)) are valid even if the inhomogeneous layer is absorbing and so are the expressions

for the reflected amplitudes (eqs (17) and (22)). It is convenient to use the dielectric constant of the film, e(z) = n

2 (z). The refractive

index of the substrate, ns , may be complex, a situation which occurs whenever it is metallic or, more generally, absorbing.

The law of refraction will be written nQ sin 4>Q =ns sin 4> S =S

and it is valid even if ns is a complex number. Here 0O is the angle

of incidence and 05 is the angle of refraction (a complex quantity, when ns is complex)

3. Fields in the Layer

In the layer, the fields verify a wave equation which is a direct

consequence of Maxwell's equations. Let us discuss first the s vibration. We shall choose the xoy coordinates in such a way as to have

E= (Ex , 0, 0) , which means that the plane yoz is the plane of incidence.

We are looking for solutions of Maxwell's equations which are Ex =U(z) exp (i^t-kSy)) (l) Hv =V{z) exp (i^t-kSy)). (2) We have used the notation ^=2tt/X, X being the wavelength of the

incident light in vacuo. If, for the sake of symmetry, we imagine
OPTICS OF INHOMOGENEOUS FILMS 43

that our layer has a magnetic permittivity /x which is a function of z, it may be shown that m v(s)=rf w (3)

k/j, dz

and U verifies the second-order differential equation (wave equation)

d

2U_dJog» dU+k2{^_S2)U= ^ (4) dz2 dz dz

The equations for the p vibration may be deduced from^qs (3) and

(4) by the simultaneous interchanges of m and e and of E and —H.

This signifies that if H= (Hx, 0, 0), then we may set Hx =U(z) exp (i(ut-kSy)) (5) Ey =-V(z) exp (i(cot-kSy)) (6) and now

y=f^ (7) ke dz

dJU_d_Jioge ^+F(€M_sW=(). (8) dz2 dz dz

We shall suppose throughout all the following that n=l. The point

in using a n{z) was only to better show the symmetry of our wave

equations for Z7((4) and (8)). When „=1, this symmetry is lost and (4) becomes

^+P(e-S2 )Z7=0. (9)

4. The Reflected and Transmitted Amplitudes

Let ZJiO) and Z7,(g) be two arbitrary solutions of the wave equation, and let us discuss first the s polarization, maintaining tor the

moment /jl{z)^0. The general integrals of (3) and (4) are U=aUl (z) +bU2 (z), (10)

V=r^(aU[(z) + bU'2 (z)), C 11 )

a and b being two arbitrary constants, which must be chosen in order

to satisfy the boundary conditions. On the plane 2=0, we must have

E0=aUl (0)+bU2 (0) } (12)

728-101 0-64-4
44 ABELES

Hr

"MO) (13)

EQ and ZZq being the tangential components of the fields in the first medium on the plane 2=0. The eqs (12) and (13) may be written

in matrix notation

.770_

77,(0) U2 (0) iU[(0) i(7M)

a~ a —ia»(o)j

_b_

(14)

MO) MO) J

In the same way, the boundary conditions on the plane 2=d are ~a~

_b_

(15)

E, and 7/, being the tangential components of the fields in the substrate on the plane z=d.

Combining (14) and (15), we may write

A.

=[2R(0)] W(d)}-

~E,-

=[M] (16)

This shows that an inhomogeneous film may be characterized by

a two-by-two matrix, like a homogeneous one. The advantage of

this representation is that the complex reflected and transmitted

(if it exists) amplitudes will have exactly the same expression as for a homogeneous film, namely

rs =Ps exp (^=^4^ (17)

2Y

t s =rs exp (^s)==— —= -5- , ( 18) where> [M] =[mv] (i, j=l, 2), Y0 =n0 cos 0O , F,=n5 cos 0,. For an

absorbing substrate, Ys is computed by using the following relation,

which is a consequence of the law of refraction

Y2

s =n2

s —S2

. The mtj can be expressed by using the two functions Ul and U2 is found that:

It

As m {UMU'M-UMU'M))
OPTICS OF INHOMOGENEOUS FILMS 45

As m12 =Ul (d)U2 (0)-U1 (0)U2 (d)

A,m21 =

*V(0)/z(d)

(uMum-umu'M)

with

kn(z)

(u1 (d)um-umu2 (d))

(Ul (z)U2 (z)-U[(z)U2 (z)) = constant.

(19)

The value of As may be obtained by replacing z either by zero or by d, as it is a constant, independent of z. In fact, eq (17) is a homogeneous

expression in the mih so that it is unnecessary to calculate As when

we are interested in the reflected (complex) amplitude only.

The value of the determinant of the matrix [M] is unity, which means that

mnm22 —mi2m21 =l.

If we use As mn , will be

instead of mll3 the value of the determinant

A

2

s (m nm22 —

l2 m2 i) =A2

S

.

All that has been said concerning s-polarization may be immediately

transposed to p-polarization. It should be borne in mind the fact that \M] is connecting now

"Ho" ~fff

to .Socontrarily to the previous situation. So, let us write

Then, it is easily seen that

"ffo" ~H,~ = [M]

_£o- -E,_

"£o" ~Et - = [N\

Ho. _H,_

(20)

(21)

where [N] = [n ij ](i, j= 1, 2) and n i5 =mn for i^j and nn =m22 , n22 =mn . Taking this into account, it can be shown that the reflected and

the transmitted (if it exists) amplitudes are given by the following

expressions
46 ABELES

r =o exn (is )- -Z°( n"+Z'n>2)+nu+Zsnn /„„,

' P' e P ^o(«n+Zs%2)+n21 +ZA2 (22)

t,= r, exp (ft„) = 2WcoS »s

_ Z0 (W11 +ZaW12)+7l2i +Z,W22

V y We have used the following notations: Z0 =n0/cos <f>0 , Zs =ns/cos <J> S

. Zs can be computed, when ns is a complex quantity, by using the

following relation: Zs =n2J(ns cos <f> s), as ns cos 0S has already been

computed when studying the s-vibration. It is assumed that, in our case, the n lj have been written in terms of U functions which are solutions of eq (8). Although not indispensable for ellipsometry, we may remind the

reader of the fact that the reflectance and transmittance (intensities) are given by

7? — o

2 /? — n

2 T —— t £l 2 T ^2

s —ps , Ilp —pp , l s —v Ts , J p ^ Tp

- 1 0 J- 0 The quantities which are measured by ellipsometry are ^ and A,

defined by

(tan ifd exp {iA)=f=^ exp i(8rp -8TS )- (24)

"s Ps

5. A Special Inhomogeneous Layer

We shall use now the general method outlined in the preceding

paragraph, in order to solve a particular problem. We shall suppose

that the refractive index of the inhomogeneous film has an exponential dependence on the z coordinate. More precisely, its real refractive

index can be written

n^n^\ (25)

rit and a being two parameters. It should be remembered that n t is the refractive index at the "entrance" of the layer, whereas a is related to its thickness d by the following relation

ad=\og^ (26)

Tii being the refractive index at the "exit" from the layer.

5.1. ^-Polarization

The use of the new variable

u=k ndz=knla (27)
OPTICS OF INHOMOGENEOUS FILMS 47

transforms the wave equation (9) into the classical equation for the

Bessel functions W l^+A^Wc, (28)

du2 u du \ cfu2 J

Thus, by using the optical thickness for normal incidence on the

layer as a new variable, we can write

U{z)=aJm {u) +bYm (u), (29)

Jm and Ym being Bessel functions of the first and second kind of the

variable u and of order m. We use the following notation:

m=kdS/\og (njrii) =kS/a, (30)

which shows that we are dealing with Bessel functions, the order m

of which is a function of the angle of incidence, whereas the variable u is independent of it. The matrix elements are mn =ni

-imu=Jm (u0)Ym {u1 ) -Jm(v>i)Ym(u0) -im21 =n<rh (J'm (uo) YmM- Y'm (u0) J'm M

)

m22 =m (Jm (Ui) Y'm (U0) —J'm (Uq) Ym (Ui) ) with u0 =knila and ux =knx \a.

5.2. p-Polarization

It is convenient now to use not only a change of variable but a change of function too. We shall set U=nZ, which is quite natural,

because U is now that part of the amplitude of the magnetic vector H

in the film which is a function of z, and we know that \H\==n\H\.

The new variable will be again u, which was defined by (27). It can

be shown that Z verifies the following differential equation

d

2Z ( ldZ.(, l + (kS/a) 2\ 7_() (31)

the solutions of which are again Bessel functions of the variable u, but of order g=(l +m2

) 1/2

. Then

U=n(aJq (u) +bYg (u)), (32)

where the order q of the Bessel functions is again variable with the
48 ABELES

angle of incidence, like m previously.

The matrix elements are mn=n^JQ (u0

) (I^+Y>M))-Yg(u0)

100

105

115

120 -

125

SA= + .02I

41.5 42.0 42.5 Figure 1 (a and b). A versus * for a thin inhomogeneous layer on a substrate with

refractive index hs =0.8—5.51

experimental errors. Angle of incidence: <£0 = 70°. *a-±u.j are estimated limits of
OPTICS OF INHOMOGENEOUS FILMS 49
50 ABELES

5.3. Numerical Results

The Bessel functions are available in the IBM library, and our problem was programmed on a 7094 IBM computer. We shall give

here some of the results which were obtained for a film having the

following characteristics: 7^=1.38, n1 =2.45, deposited on a substrate

which is characterized by ns =0.S—i5.5.

We examine a situation where the two extreme values of n are always the same, the thickness of the film being one of the variables.

In fact, it is more convenient to have a variable without dimensions

and, therefore, our variable will be rj =2ird/X. If X = 5460 A (green

mercury line), c/=870r? A. Figure 1 shows A versus \p, when <p=70°,

for very thin films (rj < 0.20, r/<174 A). If the experimental errors are ±0.2 ° for A and ±0.1° for \p, it may be seen that only A is measured with sufficient accuracy for most purposes. To a change of
OPTICS OF INHOMOGENEOTJS FILMS 51

215

210

A

205

200

195

190

I 8A = ±0.2 S^= + O.I

43.9 4 3.95 44.0

Figure 3. Same as figure t, but for different angle of incidence,

0.2° in \p corresponds a change of about 0.05 in rj, i.e., 45 A in thickness

if \=5460 A. For thicker films, the situation is much better, as it can be seen from figure 2. For instance, when 77=0.75 (c/=650 A

for \=5460 A), a change in ^ of 5^=0.2° corresponds to 5r7 =0.005

(5d=4.5 A for the same wavelength). The accuracy on A is always

higher. When o

r? =0.75, 5A =0.4° corresponds to 5r7 =0.0014 (5d=1.5

A for A=5460 A).

We have chosen a relatively large angle of incidence </> 0 , because the

accuracy is higher. Figure 3 shows the results obtained for 4>0 —45 . It may be seen that the variation in ^ is less than 0.1 for 0<T7^0.4.

This confirms the fact that small angles of incidence are less sensitive

to the influence of a surface layer than large ones. On the other hand, we have a variation of 3° in A when 77 changes by 0.05, at least tor
52 ABELES

41 42 43 44 45

l 1 ^A

290

/ r

A0 ^V50 280 ISA y /

380 H / 25/ / 270

2o/ (525 V60 A

A

.15/ /

/ /

15

260

370 0/ / 250 aL

p. 10 .65 *

.05 ?.05

.01

240

360 1

Figure 4. A versus ^ curves for the same inhomogeneous layer as previously de- posited on two different substrates.

For a film on metallic substrate (n«=0.8—5.5i); for a film on glass (rc 8 =1.52). The angle of incidence is <Ao=70°. Note the different scales for the two curves. small values of 77. Again we see here that A is a rapidly varying

function of 77, so that we find the generally well-known result stating

that the variation in thickness of a layer is reflected by a change of A. One might ask if it is better to have the thin layer deposited on

metal or on glass. In our case, and for an angle of incidence </> 0 =70°, we have d^/dr; =3.2

0 for very small 77 and a refractive index of the

substrate which is either ns =0.S —i5.5 or ns — 1.52. In the first instance, the substrate is an aluminum mirror, whereas n s =1.52

corresponds to a glass substrate. Concerning the variation of A, we

find dA/d7? =175 0 for n5 =0; 8-5.5i and dA/d77 =97° for n s ==1.52,

always for very small 77. This demonstrates the increased sensitivity, at least concerning A, for very thin inhomogeneous layers deposited on a highly reflecting mirror. For thicker films, the situation is
OPTICS OF INHOMOGENEOUS FILMS 53

somewhat different. Figure 4 shows A versus $ curves for films

deposited on both substrates (<£ s =70°).

6. Films With a Slowly Varying Refractive Index

This is the situation where the well-known WKB approximation

may be used. We shall discuss it briefly, as we feel that m some

instances it can give very easily and without too lengthy computations

approximate results which can serve at least as a first approximation.

6.1. s-Polarization

It is useful to make a change of variable as well as a change of function. The new variable will be

=kj(n2 -S2) l/2 dz. (33)

It should be pointed out that £ is the phase corresponding to the

optical thickness, for the given angle of incidence <fo, and, therefore,

! is not the same as the variable u which was chosen m the preceding

| example. The new function will be y(f). It is chosen m such a U=(KWy (34)

where K=W-8T^ (35)

K is the characteristic impedance of our medium, i.e., the ratio of the

tangential components of E and H. The change of function is a mathematical device useful for the supression of the term containing

the first derivatives in the differential equation. This can be written

now

|i+H(§)'+I(t))-

the derivatives of K being taken with respect to f

The WKB approximation consists in the neglect of the terms

different from unity in the bracket. It shows that, when

\Js\ we have

1 (K"\ 3 (K\

2\KJ Ak) i(ln K)"-\((lnK)') «1, (37)

y=ae i *+be-i * ( 3§) so that /onv U=K(aei !+be- i !)=aUl JrbU2 . (39)

L
54 ABELES I

! We may use now the method described previously, which is valid in any circumstance. The only approximation which will be made

consists in writing

Ul

dz d$ dz K d^~^K (4Uj

and

We have neglected the terms containing the derivatives of K with

respect to f, i.e. which are smaller than those which were retained by

a factor K'\K.

Making use now of eqs (3), (17), and (18), we can write

r =n Pvn (4* , ^^a -Ys Kb) COS {f +i(Y0 Ys KaKb -l) sin

. Ps exPWs; (Y0Ka+YsKb) cos tf+i(Y0YsKaKb +l) sm {/

s 3 P { ts) (Y0Ka+Ys Kb) cos tf+i(Y0YsKaKb +l) sin f,

(41)

Ka and i£6 are the values of K at the entrance and exit of the film,

i.e. on the planes which limit the film and $f

is the value of f corre- sponding to the thickness of the film.

6.2. />-Polarization

The discussion and the results are quite analogous to the preceding

case. It is only necessary to replace the previous definition of K by

a new one, viz

K=

(n2 -S2

) 1/2 ' (42)

This change is due to the fact that U is now that part of the magnetic

-> -» field which is ^-dependent, and we know that n\E\=\H\. (Throughout

all the discussion of the WKB approximation, it is assumed that p=l.)

Using the eqs (22) and (23), it can be shown that: r-n PYntt , _ (Zs Ka -

0 Kb) cos {f +i(Ka Kb -Z0 Zs ) sin f, r,_p, exp

{ZsKa +Z0 Kb) cos tf +i(KaKb +Z0Zs ) sin

(4' 3) t=r exp (id, )= grWggycos 0s

P r* exp

^ f44 {ZsKa+Z0Kb) cos sin jy

(44)

jp is given by the same expression as/s in terms of K, K'', If" (see eq

(37)), but K is now given by eq (42).
OPTICS OF INHOMOGENEOUS FILMS 55

A straightforward computation shows that

/•--sp&fcs? (

n{n!-si) ski oy

One should remember that the two approximations adopted for the

s- and ^-polarization are: |/|<<0 and |(di£/c^)/i£|<<l too.

It may be shown that

n(dn/dz)

\Kjs ~ k{n2 -S2f

12 and

(IC\ v

2 -2S2 \K)~ n

2 \Kj;

so that we always have

I (§),!<

7. An Inhomogeneous Film Replaced by a Pile of

Homogeneous Films

It is also possible to replace an inhomogeneous film by a superposition of homogeneous films of different refractive indices. In that

case, the problem pertaining to the solution of the differential field equations is eliminated. The only difficulty is the estimation of the

number of homogeneous layers which is necessary for the replacement of the inhomogeneous one. In practice, this is done by successive trials. The inhomogeneous film is replaced first by a unique

layer having an average refractive index

1 C

d

and the computation is done. Then the film is replaced by two,

three, etc., homogeneous films and the values of \f/ and A are computed.

The refractive index of each layer is an average of the refractive

indices of the portion of the inhomogeneous layer it replaces. The

satisfactory result will be obtained when \f/j} AJf and Aj+i do not

differ from each other to the desired accuracy (the index indicates

the number of layers which replace the inhomogeneous film)

There are no practical difficulties when using a large and rapid

computer. The computation may take some time when the variation

of the refractive index is rapid.
56 ABELES

Characteristics of multiple homogeneous layers on glass substrate

(na — 1.52), as a function of the number of layers, H

The total thickness of the layers is d=X/27r and the angle of incidence is 0O =7O° H 1 2 3 4 5 6 7 8 15 00 R,_ Rp *A

0. 5292

0. 0094

7°. 58

182°. 96

0. 5682

0. 0143

9°. 02

241°. 18

0. 5740

0. 0179

10°. 02

245°. 31

0. 5760

0. 0193

10°. 365

246°. 40

0. 5769

0. 0199

10°. 525

246°. 85

0. 5774

0. 0203

10°. 61

247°. 09

0. 5777

0. 0205

10°. 66

247°. 22

0. 5779

0. 0206

10°. 70

247°. 31

0. 5784

0. 0209

10°. 77

247°. 51

0. 57855

0. 02106

10°. 80

247°. 59

2. Same as table 1, but for different values of the angle of incidence

00=10° H 1 2 3 4 Rt rp :: A

0. 1496

0. 1412 44°10'17" -0°. 1204

0. 1783

0. 1695 44°16'29" -0°. 4905

0. 1827

0. 1738 44°17'02" -0°. 5447

0. 1842

0. 1753 44°17'12" -0°. 5624

0. 1862 0.1772 44°17'24" -0°. 5844

00 = 40° H 1 2 3 4 5 R.A

0. 2334

0. 0798

30°. 32 -1°. 7040

0. 2697

0. 1075

32°. 275 -8°. 7808

0. 2752 0.1115

32°. 478 -9°. 8064

0. 2772

0. 1128

32°. 54 -10°. 1417

0. 2781

0. 1134

32°. 567 -10°. 2928 H 6 7 8 9 15 00 R...- RP *A

0. 2786

0. 1138

32°. 583 -10°. 3739

0. 2788 0.1140

32°. 591 -10°. 4224

0. 2790

0. 1141

32°. 596 -10°. 4537

0. 2792

0. 1142

32°. 60 -10°. 4752

0. 2795

0. 1144

32°. 612 -10°. 5266

0. 27966

0. 11451

32°. 615 -10°. 555 00=80° H 1 2 3 4 5 6 R,RP *A

0. 7249

0. 1673

25°. 655 -179°. 7967

0. 7521

0. 1539

24°. 333 -165°. 9209

0. 7561

0. 1574

24°. 511 -163°. 7149

0. 7575

0. 1589

24°. 617 -163°. 0100

0. 7581

0. 1597

24°. 650 -162°. 6955

0. 7585

0. 1601

24°. 675 -162°. 5277 H 7 8 9 15 R,rp :: *A

0. 7587

0. 1603

24°. 688 -162°. 4276

0. 7588

0. 1605

24°. 697 -162°. 3631

0. 7589

0. 1606

24°. 702 -162°. 3190

0. 7591

0. 1609

24°. 717 -162°. 2134

0. 75923

0. 16104

24°. 72885 -162°. 15445

7.1. Numerical Example

Let us indicate here the results which are obtained for the films

which were investigated and which have an exponential variation of the refractive index. The homogeneous films had the following
OPTICS OF INHOMOGENEOUS FILMS 57

characteristics:

total number of layers: H

refractive index of layer p:

v+1 n(z)dz=HX

In thickness of layer p: dp =d/H.

The numerical values of the different quantities are the same as before.

It should be stressed that the average values of the rip's are independent of the angle of incidence. They are functions of H, the total number of layers, only.

The values of nx and rii are the same as in the previous example

(^=1.38, ni=2.45), and ns =1.52.

Table 1 gives the results obtained when the incidence is <£ 0 =70° and

for 77 = 1, i.e., for an inhomogeneous layer having a thickness d equal

to X/27T. (For visible light, that means a thickness of about 900 A).

It can be seen that \p approaches the true value more rapidly than A. But already for six layers the difference A (00)— A(6)=0?5 is quite

small. It can be even smaller for smaller angles of incidence. We

have chosen </> 0 =70°, because, in our opinion, it is a convenient angle

to use in ellipsometry. In order to get more insight into the rapidity

of convergence, we give in table 2 the results obtained for rj—1 and

00=10°, 40°, and 80°. It will be seen that, for large angles <f> 0 , if we

decide that the largest error admitted on A is 0.2°, it is necessary to replace the inhomogeneous film by at least nine homogeneous layers.

As it can be expected, for small values of 0O , the same accuracy is obtained with a smaller number of homogeneous layers.

It is impossible to discuss all the situations which may arise when

dealing with inhomogeneous layers. Our purpose was only to indi- cate some computational methods. We did not discuss the problem

of inhomogeneous absorbing films. In principle, it is not different

from those already treated, but the introduction of a complex di- electric constant leads to solutions of the differential wave equation,

which are complex functions.

The special inhomogeneous layer which was examined may serve as

a starting point for more complicated inhomogeneous layers. In

fact, we might approximate a rather complicated law of variation of e by a pile of thin films, each of them having an exponentially varying

refractive index. The question is still pending as to the easiest and more accurate way of computing the ellipsometric properties of in- homogeneous layers. The replacement of an inhomogeneous film by a

pile of homogeneous ones is a straight forward method, but it is im8. Concluding Remarks
58 ABELES

possible to know from the beginning the number of the latter which is necessary in order to know the result with a given accuracy. It would be very useful to examine the approximate methods which can be used, especially concerning the accuracy which can be expected in each case. But this is a more difficult problem than those we have

discussed here. This work has been sponsored in part by the Office of Scientific

Research, O.A.R., through its European Office, United States Air

Force, under Grant AF-EOAR 63-48.

Discussion

A.N. Saxena (Fairchild Semiconductor Co.):

I want to ask you whether your calculation is restricted to in- homogeneous but transparent film only because you say nothing

about k. I wonder whether you can extend this for inhomogeneous

but absorbing film? F. Abeles:

Yes. It can be done. As we saw on the slide one gets Bessel functions

of complex variable and complex arguments, but otherwise it is exactly the same and can be extended very easily. A. N. Saxena:

It is good that your calculations can be extended for inhomogeneous

films also ; however, your present paper is restricted only to transparent

films. F. Abeles:

Yes, what I have done was only on transparent films. A. N. Saxena:

The inhomogeneity in thick transparent films on an absorbing substrate can be studied by performing ellipsometer measurements to get

(ip, A). Knowing the film-substrate interface optical constants, one

can compute A) curves for various refractive indices nx and thick- ness d of the film. From these curves then, one can determine ni and

d. By etching the film successively to obtain smaller thicknesses

and making ellipsometer measurements each time, the variation in

rii due to inhomogeneity can be studied. We have made such measurements on Si02 films grown on Si,o and I find that they are oquite

homogenous all the way from 9000 A down to a few hundred A. I am attempting to assign a value to the optical constants of Si02 -Si interface which seem to be different from that of bulk Si. J. T. Bloxsom (Air Reduction Corporation)

Would you comment briefly on how you would modify your equations to incorporate scattering in the internal portion of your films. Would you have to use a two-dimensional function?

F. Abeles:

That is completely different. All that supposes that you have an inhomogeneous but isotropic film and with no scattering in the film,

but if you had to have scattering, that would bring you to a completely

different problem. It cannot be treated in this way.
J

!

3. Techniques

728-101 0-64-5

Computational Techniques for the Use of the

Exact Drude Equations in Reflection Problems

F. L. McCrackin and J. P. Colson

National Bureau of Standards, Washington, D.G. 20234 The Drude equations applied to a thin film give the ellipsometer

readings in terms of the refractive index and thickness of the film. A solution of these equations for the film thickness when the refractive index of the film is known, and methods of determining both the

thickness and refractive index of a film from ellipsometer readings, are presented for both dielectric and absorbing films. Ellipsometer readings for an inhomogeneous film are often analyzed as if the film were homogeneous, yielding an effective thickness and refractive index for the film. The relation of _ these effective quantities to various averages of thickness and refractive index of the

film is investigated. In particular, the effective thickness determined

by the ellipsometer is found to be greater than the root mean square average thickness of the film.

1. Introduction

When polarized light is reflected from a surface, the polarization

of the light is changed. This change of polarization may be repre- sented by the ratio of the reflection coefficient r\ for light polarized

with its electric vector parallel to the plane of incidence, to the

reflection coefficient r

s

, for light polarized with its electric vector perpendicular to the plane of incidence. These reflection coefficients are in general complex numbers so they express the change m both

the amplitude and phase of the light, and they are given by the

well-known Fresnel coefficients, n2 cos <£i— rii cos 02 yV } 12

rii cos0i+7i2 cos02

8 ui cos 0i— n2 cos <fr 2 12 —

rii cos 0i +^2 cos 02 The indexes of refraction of the medium above the surface and of the surface are nx and n2, respectively, and the latter may be complex. The angle of incidence is 0i, and the angle of refraction, 02 , may be computed from rii, n2 , and 0i. . The changes in the polarization of light by reflection from a surlace

are conveniently measured by an ellipsometer. The quantity measured by an ellipsometer is the ratio of the reflection coefficients, p:

p =rp/rs =tan \{/e iA

. (2) This ratio represents the relative attenuation, tan ^, and the difference

61
62 McCRACKIN AND COLSON

of phase shifts, A, for the components of the electric vector in the

plane of incidence and normal to it. The refractive index of the substrate may be computed from p. Solving the previous equations gives

I" 4psin2 0j"1 01 77-2=7*1 tan^

L

1 — WJ-——-y • (3)

If the substrate is absorbing, its refractive index will, of course, be

complex.

If a film is now placed on the surface, the reflection coefficients of the surface are changed. The appreciable changes in the reflection

coefficients for very thin films are the reason that the ellipsometer

is such a sensitive instrument. The exact Drude equations [1, 2, 3]

1 for the reflection coefficients for a film-covered surface are:

l+rf2rf3 expZ>

(4) Rs^_ 7l2 H-r23 exp£>

l+7l27-23 expD

where

D=—±irin2 cos<fo d2/\. (5) The subscripts 1, 2, and 3 are used for the medium, film, and substrate,

respectively. The Fresnel coefficients r\2 and r\ 2 refer to reflection

between the medium and film, and the coefficients r

v

2Z and r*2Z refer to reflection between the film and substrate. The ratio of the reflec- tion coefficients is, as before,

P =Rp/Rs =t2M fe**. (6) For a given angle of incidence, thickness of film, and indexes of refraction of surrounding medium, film, and substrate, the expected

ellipsometer values ^ and A may be computed. First the angles of refraction in the film and substrate, then the Fresnel coefficients rf2 and r\2 , and finally D are computed. These are substituted in the expressions for the reflection coefficients and finally the ratio of the reflection coefficients gives the ellipsometer values of \p and A. Typical results of such calculations are given in figure 1. Values of A and ^ are shown for films oof refractive index of 1.4, 1.5, and 1.6 and thicknesses of 0 to 1000 A on a substrate of the refractive index

of chromium. The index of refraction of the medium above the

film is 1.359. The thickness of films of known refractive index may be determined

from graphs such as figure 1, or from tables. From the measured

values of \p and A, the thickness may be read off the curve for the

1 Figures in brackets refer to the literature references on p. 80.
T COMPUTATIONAL TECHNIQUES

128

I 32

o

140

1 1 1 1 500 — y° ^0600

_

400 Cf >v / \ 700 30o/ \

/ 600 \

X800

o //

00 \8 °° —

// 30° \

// » 900

// \

IP200 \

l00» \ 1000

//cnn 700 1

// ^*V800 n^'1.50

6900

y^00%900 1/400 ^|Q00 fi»300 n

2 =|.40 #2 00

1100 0#

1 1 1

1

\ n 2 =i.60

1 \ 34

= 36 ~ 38 40 42

\j/,deq Figure 1. A -plot of A against \p for films of refractive index n2 as shown, in a medium with a refractive index of 1.359 on

a chrome substrate. The numbers along the curves are film thicknesses in Angstrom units.

! index of refraction of the film. However, a different curve must be

calculated for each set of values of the indexes of medium, film, and

substrate, and for each angle of incidence. Therefore, it is usually more

efficient to solve the equations for the thickness of the film in terms

of the ellipsometer readings. The ratio of the reflection coefficients,

p, is given in terms of the ellipsometry readings by eq (6) . Since the

thickness, d2 , of the film occurs only in the term exp D in eqs (4) and

(5), we may first solve for exp D. Substituting for Rp and Rs and

clearing fractions, an equation is obtained of the form:

C^exp D) 2 + <72 (exp D) + C3 =0, (7) where Gu C2 , and C3 are complex numbers depending on the refractive

indexes, angle of incidence, and measured values A and \p. Solving

this quadratic equation gives two solutions for exp D. From each

solution, a value of D is computed and a thickness, d2 , is determined

from D by eq (5) . The value of d2 that is correct must now be chosen.

Since the coefficients Ch C2 , and C3 are complex numbers, the

film thickness calculated from this equation would also be expected to be complex. However, the correct film thickness must be a real
64 McCRACKIN AND COLSON

number, as it represents a real quantity. Therefore, the calculated j thickness with a zero imaginary part is the correct solution. In

practice, experimental errors will result in both computed thicknesses

being complex. The thickness with the smallest imaginary component

is selected as the correct solution, the real part is taken as the thickness

and the magnitude of the imaginary part is arbitrarily taken as a measure of error. The real part of d is then used to compute ellipsometer values A

and } by eqs (4), (5), and (6). As the imaginary part of d has been

dropped, these computed values will differ from the measured values

of A and ^ by error terms 5A and 8f. In order for the measurements

and interpretation to be satisfactory, the error terms must be within

the limits of experimental error. Thus far, the refractive index of the film has been assumed to be

known. The film refractive index and thickness may both be determined from ellipsometry readings, even though the equations have

not been solved for the film refractive index in closed form. We

consider first the case of a dielectric non-absorbing film, so that its refractive index is a real number. To determine the refractive index,

several refractive indexes are assumed for the film and the film i thickness and error terms computed for each refractive index. An

example is shown in table 1 for a barium fluoride film on a chrome 1 Table 1. Thickness of evaporated barium fluoride film on chrome slide *- h A SA 8& ri2 d

deg deg deg deg A

133. 96 3. 64 36. 30 0. 60 1.420 753

2. 64 .40 1.430 726 1.58 .22 1.440 701 1.08 .14 1.445 688

0. 58 .07 1.450 676

.28 .03 1.453 668

.18 .02 1.454 665

. 10 .01 1.455 662 .00 .00 1.456 660

.10 .01 1.457 657 .20 .02 1.458 654

.28 .03 1.459 651 .38

. 70 .04 1.460 647

.38 1.475 453

aI?Q?|rfooo? acetone ' ni=1.359, at an angle of incidence of 60°. The refractive index of the slide was Til —o.olo—4.0S0Z.

b Film prepared by Frank E. Jones, NBS.

slide. Values from 1.42 to 1.475 were assumed for the refractive

index of the film and the thickness and error terms 8A and b\p computed

for each assumed refractive index. Choosing the row with zero

error terms gives a refractive index of 1.456 and a thickness of 660 A j for the barium fluoride film. All the previously mentioned calculations have been programmed

[4] in Fortran for the IBM 704, 7090, and 7094 computers and all calculations have been performed on these computers. The computer

has also been programmed to determine the refractive index of a •
COMPUTATIONAL TECHNIQUES 65

i film by calculating error terms for assumed values for the index and

! choosing the refractive index that gives the smallest error term, so

! the table of error terms does not need to be examined as in the previous

example. Also, the computer can use ellipsometer readings obtained

I by multiple reflections of the light or a compensator with arbitrary

phase retardation. Calculations have also been extended to reflection

j from multiple films and absorbing films.

2. Determination of Complex Index of Refraction of

Films

.We now consider the case of an absorbing film which has, of course,

s a complex index of refraction. If the imaginary part of the refractive

I index of the film is given, the real part of the index and the film

thickness can be determined as previously by searching for the real part of the refractive index that gives zero error terms. If the real part of the refractive index of the film is given, searching oyer the

imaginary part for zero error terms gives the solution. If the thickness

of the film is given, the real and imaginary parts of the refractive

j index of the film may be searched until a complex refractive index is found which yields the given thickness as well as zero error terms.

I However, the real and imaginary parts of the refractive index and

i the thickness of the film cannot all be determined from a single set of ellipsometer readings. Since a set of ellipsometer readings consists

of only two values, \p and A, it cannot determine the three quantities

for the film. However, by increasing the number of ellipsometer

measurements, both real and imaginary parts of an unknown complex

refractive index of a film may be determined.

We shall now investigate the following four means of increasing

the number of ellipsometer readings:

(1) A series of ellipsometer readings is obtained for films of different

but unknown thicknesses, but of the same refractive index.

(2) Ellipsometer readings are obtained on a single film for different

surrounding media of known refractive index.

(3) Ellipsometer readings are obtained for a single film (or a series

of films of the same refractive index) on various substrates of known

refractive index.

(4) Ellipsometer readings are obtained for a single film at various

angles of incidence.

The possibility of determining the complex refractive index and

thickness of a film from a series of ellipsometer readings obtained

by each of these methods was investigated. The series of ellipsometer data that would be obtained for an assumed complex refractive

index and film thicknesses was calculated for each method. Then

various schemes of determining the refractive index and thickness were tried, using these computed ellipsometer measurements, to see if they could determine the refractive index and thickness of the film

that was originally used to compute the ellipsometer measurements.
66 McCRACKIN AND COLSON

2.1. Variation of Film Thickness

One method of obtaining many ellipsometer measurements is to make measurements on a series of films with the same complex

refractive index but with different thicknesses. To obtain such a

series of measurements, a series of films of thicknesses 40 to 100 A,

all with a refractive index of 2.5 —0.5i, was assumed. The ellipsometer readings that would be obtained by measurements on these

films were calculated, as described, using eq (6). A series of measure- ments of this type could be obtained, for example, on a vacuumevaporated film. Readings taken during the deposition would all be for films of the same refractive index but different, unknown,

thicknesses if the structure of the film did not change during its deposition. From this series of readings, an effort was made to calculate the refractive index of the films. Values for the imaginary

part of the refractive index were assumed and a value for the real part of the refractive index was computed for each assumed value

of the imaginary part and for each film. The results are shown

in table 2. Table 2. Calculated values of the real part of the refractive index for films from

40 to 100 A for assumed values of the imaginary part, m, of refractive index

Thickness Assumed m

-0.2 -0.3 -0.4 -0.5 -0.6 A

40 2. 952 2. 812 2.668 2. 500

60 2. 960 2. 820 2. 668 2.500 2. 262 80 2. 974 2. 828 2. 673 2. 500 2.279 100 2.990 2. 839 2. 677 2. 500 2.279

If one assumes —0.4, for example, for the imaginary part of the

refractive index, the computed real part of the refractive index is seen to vary from 2.668 to 2.677 for the films from 40 to 100 A thickSince it is known that the refractive index is the same for all the

films, the value —0.4 for the imaginary part of the refractive index

is incompatible and is discarded. Examining the other columns of the table, we see that only for the value —0.5 for the imaginary part

of the refractive index are constant values for the real part obtained.

Therefore, the refractive index of the films is correctly determined

to be 2.5 —0.5i After the refractive index of the films is determined,

the thicknesses of the films may be calculated.

This calculation shows that it is possible in principle to determine

the complex index of refraction of a series of films of the same refractive index but varying thicknesses by this method, but it does not

show that the method is practical. This is because exact values of the ellipsometer readings A and ^ were used, while in practice the

readings would have experimental errors. In order to make the test
i

I COMPUTATIONAL TECHNIQUES 67

of the method more realistic, random errors were added to the computed

readings before they were analyzed. The errors were random normal

errors with a standard deviation of 0.04° for A and 0.02° for \p. The

I values computed for the real part, nR, of the film refractive index

from these readings with errors are shown graphically in figure 2.

i The computed real parts of the refractive indexes are plotted versus

' the film thickness for assumed values of the imaginary part, nt. For

; the method to work, the computed values for nR for ni= —0.5 would

have a constant value, so they would lie on a horizontal line, while

j the other values of ni would give more scatter about a horizontal line. Since the graphs for all values of n f show the same scatter about a

! horizontal line the method does not work for these conditions.

The method was also tried with thicker films, from 200 to 1000 A,

I with errors added to the computed ellipsometer readings and the other

| conditions the same. The results are shown in figure 3. The range

j 1.5 to 3.5 was searched for values of nR for each assumed value of rn and film thickness. For some cases, two values of nR were found in

this range. The method worked very well in this case, since nt = —0.5

gave much smaller scatter around a constant value of nR =2.5 than

did 7ii=— 0.54 or —0.48. The correct value of nR was obvious when

two values were obtained. Therefore, the imaginary part of the

2

1 1

—o

n. =-o.5 —° O o ^~

n ( =-o.4 o 0 o o—

o O o o °-

1

n,-=- 0.2

i i l

1 0 20 40 60 80 FILM THICKNESS, A

100

Figure 2. Calculated values of the real part, nR, of the re- fractive index of films from 20 to 100 A thick for assumed

values of the imaginary part, lii, of the refractive index of the film, random errors having been added to the ellipsometer

readings. The true index of the film is 2.5-0.5L
68 McCRACKIN AND COLSON

3 :

1 1 1 1 s

0

O

Q ° n, ---.54

2 -

o

o o o o o— n. = -.5o

2 -

2 - O

| I 1 I I I I | 200 400 600 eoo 1000 FILM THICKNESS, A Figure 3. Calculated values of the real part, nR, of the re- fractive index of films from 200 to 1000 A thick for assumed

values of the imaginary part, ni} of the refractive index of the film, random errors having been added to the ellipsometer

readings. The true index of the film is 2.5-0.5i.

refractive index may be determined to better than 0.02 for films up

to 1000 A thick for these conditions.

The computed values of nR were averaged for the film thickness

range 200 to 1000 A for each assumed value of ra,. The method was

applied two more times for the same conditions, except different sets of errors were added to the ellipsometer readings. This simulates

repeating an experiment. The average values for nR that were

obtained for each calculation are given table 3. The spurious values

of nR were omitted.

Table 3. Average values of calculated real part of film refractive indexes for assumed values of rij Assumed nt

-

-0. 46 -0. 48 -0. 50 -0. 52 -0. 54 -0. 56

2. 513 2. 508 2. 504 2. 492 2. 483 2. 470

2. 516 2. 510 2. 498 2.492 2. 491 2. 474

2. 510 2. 504 2. 502 2.489 2. 480 2. 463
COMPUTATIONAL TECHNIQUES 69

Table 4. Standard deviations of calculated real part of film refractive indexes Random errors added to ellipsometer readings. Assumed rii

-0. 45 -0. 48 -0. 50 -0. 52 -0. 54 -0. 56

0. 127 0. 055 0. 006 0. 066 0. 134 0.207

.129 .066 .004 .066 .130 .205

.123 .060 .012 .070 .146 .212

Table 4 gives the standard deviations of nR computed for the various

film thicknesses, choosing the values closest to the average value for each film thickness. The correct value of nt = —0.50 is easily chosen,

since it gives much smaller standard deviations for all three cases than the other values for Ui. Therefore, the complex index of refraction of a series of films of varying thickness but the same index of refraction may be determined

if the films are reasonably thick but not for very thin films. Of

course, the thickness of films required and the accuracy to which the

index is determined depend on the actual experimental conditions.

For the conditions of the calculations presented here, the film refractive index could be determined for a series of films between 100 A and

1000 A thick but not for a series of films up to 100 A thick.

2.2. Variation of Refractive Index of Surrounding Medium

A series of ellipsometer readings may also be obtained from a

single film by varying the index of refraction of the medium above

the film, the index of refraction of the substrate, or the angle of incidence of the incoming light. The possibility of determining the

complex refractive index of a film from a series of ellipsometer readings

obtained by varying each quantity was investigated.

First, a series of ellipsometer readings was calculated for a film

of 100 A and refractive index 2.5 —0.5i on a substrate with the refractive index of chromium for indexes of the surrounding medium from

1.0 to 1.6. A series of readings of this type could be experimentally

obtained for a single film by placing the film on the substrate in a

cell as described by Stromberg et al. [5]. By using liquids of various

refractive indexes in the cell, ellipsometer readings for the film could

be obtained for various refractive indexes of the surrounding medium.

However, it would have to be established that the refractive index

of the film did not change because of the change in the surrounding

liquid, due to the film absorbing the liquid or the liquid either dis- solving or depositing another film onto the film that was being

investigated.

The real part, nR, of the refractive index of the film was then

computed for each surrounding medium and for assumed values from —0.3 to —0.6 for the imaginary part, nu of the refractive index of
70 McCRACKIN AND COLSON

n, =-0.3

1.0 1.2 1.4 1.6 REFRACTIVE INDEX OF SURROUNDING MEDIUM Figure 4. Calculated values of nR, for a film 100 A thick for varying refractive index of surrounding medium and assumed

values for n ;

. The true refractive index of the film is 2.5-0.5i.

1.0 1.2 1.4 |.6 REFRACTIVE INDEX OF SURROUNDING MEDIUM Figure 5. Calculated values of the thickness of a film for varying refractive index of surrounding medium and assumed

values for rij. The refractive index of the film is 2.5-0.5? and thickness is 100 A.
COMPUTATIONAL TECHNIQUES 71

the film. The calculated values of nR are shown in figure 4 and the

corresponding film thicknesses are shown in figure 5. From figure 4, the only value of nt that gives constant values of nR independent

of the surrounding medium is the true value —0.5 for which the value nR =2.b is obtained. For 7ii=— 0.6 and for larger negative values,

there were no solutions for nR for these ellipsometer readings when

the refractive index of the surrounding medium is greater _ than 1; therefore, these values of nt may be discarded. The refractive index

of 2.5 —0.5^ used for calculating the series of ellipsometer readings,

has been determined from the ellipsometer readings.

The value of nt may also be determined from the condition that

calculated thickness of the film shall also be constant, independent

of the refractive index of the surrounding medium, as in figure 5. The correct value of —0.5 is again obtained.

2.3. Variation of Substrate

The effect of variation of the refractive index of the substrate was

investigated. The ellipsometric readings for films of refractive

index 2.5-0.5^ and a thickness of 100 A on chrome, gold, silver, and

steel substrates were computed, using refractive indexes of the substrates obtained from the literature. Then the real parts of the

refractive index of the films for various assumed nt were calculated

and are shown in figure 6. When the correct value of —0.5 was

assumed for nu the calculated value of nR was 2.5 for all substrates.

However, when —0.4 or —0.45 was assumed for n u the calculated

2.0

1.5

1 1 1

1

ASSUMED - ASSUMED 0-.4

-.4 Q

-.4 5 c\ / D-.45 - « J/m-.5 - -.5 «^ •

-. 55 O \ y 0-.55 _ // A-.66 _ -.60 A V

-.60 A \\r

-. 55 O \

-.5 • \ y

-.45 -.4

O

O \Y/

1

1

1 1 CHROME GOLD SILVER STEEL SUBSTRATE MATERIAL Figure 6. Calculated values of nR for a film 100 A thick for varying refractive index of the substrate for assumed values

of rii. The true refractive index of the film is 2.5-0.5?.
72 McCRACKIN AND COLSON

value^of nR varied greatly with the substrate material, and when

—0.55 or —0.60 was assumed for n u there was no solution for nR for gold and silver substrates. Therefore, since all values except —0.5

are eliminated for ni} the method works. Some spurious solutions

for n t are found for the chrome substrate, but they do not seriously

interfere in the determination of nt

. The effect of experimental error on this method was also investigated. Random normal errors with a standard deviation of 0.04° for A and

0.02° for \p were added to the ellipsometer readings and the values of nR shown in figure 7 were calculated. The correct solution of nt = —0.5

is still clearly indicated as giving the most nearly constant value for nR, even though the values of nR show some variation due to the

experimental error.

3.0

2.5

2.0

1 1 1 I

ASSUMED ASSUMED _ n

-.4 q -.45 CA - - .5 _ Ai:

- 55 0 \^ — / 0-.55-

- -6 A \ // A_6

-.6 A \\s -.550

-.45

- 5

°" \ y \ / y 1 1 CHROME GOLD SILVER STEEL SUBSTRATE MATERIAL Figure 7. Calculated values of nR for a film 100 A thick for varying refractive index of the substrate and for assumed

values 0/ rij. Errors have been added to ellipsometer readings. The true refractive index of the film is 2.5-0.5*'.

2.4. Variation of Angle of Incidence

A series of ellipsometer readings for a single film may also be

obtained by varying the angle of incidence of the light. The analysis

of such a series of ellipsometer readings has been considered by

Hayfield [8]. Such a series of readings is convenient to obtain

experimentally on a single film without the complications due to the

possibility of changing the index of refraction of the film. Such

complications occur when the substrate, or the surrounding medium,

or thickness of the film are varied.

Ellipsometer readings were computed for angles of incidence from

10 to 85° for a film of 100 A thickness and index of 2.5-0.5i on a
COMPUTATIONAL TECHNIQUES

chrome substrate in air. For each eilipsometer reading, the value

for the real part of the refractive index of the film was computed for various assumed values of the imaginary part, nt, of the refractive

index, as shown in table 5. In some cases two values were obtained.

It is desired to determine the true value of n< by choosing the column

of the table which gives constant values for nR independent of the

angle of incidence. However, the calculated values of nR are nearly constant with respect to the angle of incidence for all assumed

values of nif so the true value of 7i z

- cannot be determined from this information.

! Table 5. Calculated values of the real part of the refractive index of a 100 A film

for varying angles of incidence and assumed values for the imaginary part, m,

of the refractive index of the film

Angle of incidence Assumed n»

-0.2 -0.3 -0.4 -0.5 -0.6

deg

10 2. 994 2.853 2.684

1.569 2.500

1.672

2. 284

1.809 20 2.994 2.839 2. 677

1.577 2.498

1.689 2.264

1.827 30 2. 994 2.841 2. 681 1.573 2.503

1.683

2. 274

1.822

40 2.990 2. 839 2. 677

1. 572

2. 500 1.682

2. 277

1.820

50 2.990 2. 839 2.677

1.570 2.500

1.679 2.279

1.815 60 2.988 2. 837 2. 677

1.568

2. 500 1.677

2.281

1.812

70 2.988 2.837 2.677

1.567

2. 500

1. 675

2. 282

1.808 80 2. 988 2.836 2. 676

1.566 2.500

1.674 2.283

1.806

85 2.987 2.856 2. 676

1.566

2.500

1.673

2. 282

1.805

The method was then repeated for film thicknesses of 500 and 1000 A,

with the other parameters unchanged. The results are shown in tables 6 and 7. For a film thickness of 1000 A, some variation of the

calculated values of nR with the angle of incidence is observed for

values of n 2 different than the true value of —0.5, as may be seen from

table 7. However, this variation is still not sufficient for the method

to determine nt accurately when the eilipsometer readings contain

experimental error. For a film thickness of 500 A, the method gave the surprising results shown in table 6. Solutions for nR were found only when the correct value of —0.5 was used for n z ; when other values were assumed for 71* no solutions for nR in the range of 1.5 to 3.0 existed. Therefore, the

correct value of nt was determined. More exactly, the value of rii must be between —0.4 and —0.6. To determine how closely may

be determined for a 500 A film, values of from —0.4 to —0.6 in
74 McCRACKIN AND COLSON

steps of -0.02 were assumed and the results shown in table 8 were calculated. Ihe value of nt is between -0.40 and -0.54 by table 8

Therefore, ellipsometry readings at various angles of incidence of a film determine the complex refractive index of the film only for a

special thickness of the film (in this case 500 A).

Table 6. Calculated values of the real part of the refractive index of a 500 A film for varying angles of incidence and assumed values for the imaginary part n of the refractive index of the film

y p ' " 7

A T) crip nf incidence Assumed w,- -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -0. 10 deg

1020

30

2. 500 2.500

2.500

1.536

40 2.500

1.588 50 2.500

1.636 60 2.500

1.677

7080

2.500

1.707

2. 500 1.726 85 2.500

1 ion

Table 7. Calculated values for the real part of the refractive index of a WOO A film for varying angles of incidence and assumed values for the imaginary part n t

- of the refractive index of the film

Angle of incidence Assumed w< -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9

deg. 102030

40

50

1.550

1.621 1.688

1.746

1.990

2.012 2.033

2. 057

2.080

2. 284

2. 293

2.302

2. 311

2. 319

2. 498

2. 500 2.500

2. 500 2.500

2. 656 2.654

2.652

2. 648

2. 644

2.776

2. 775

2. 770

2. 766

2. 763

2. 871 2.870

2. 868

2. 866

2. 869

2. 944 2.948

2. 951 2.958 2.969

60

708085

1.793

1.826

1.846

1.852

2. 100 2.115

2. 123

2. 126

2. 325 2.330

2.333

2. 334

2. 500

2. 500 2.500 2.500

2.642

2. 640

2. 640

2. 640

2.763

2. 764

2. 765

2. 765

2. 874

2. 880

2. 886

2. 886

2.984

3. Inhomogeneous Films

Only homogeneous films have been discussed so far. However the index of refraction of many real films varies in the direction perpendicular to the plane of the films. In many cases, it is not possible to determine the manner in which the index of refraction of the film
COMPUTATIONAL TECHNIQUES

| Table 8. Calculated values for the real part of the refractive index of a 500 A film

j for varying angles of incidence and assumed values from —O.4O to —0.60 for the imaginary part, n,, of the refractive index of the film

Angle of Assumed

incidence -0. 40 -0. 42 -0.44 -0.46 -0.48 -0. 50 -0.52 -0.54

deg

10 2. 984 2. 880 2. 767 2. 642 2. 500 2.320

1.597 20 2. 983 2. 879 2. 767 2. 643 2. 500 2. 318 1.630 30 2.981 2. 878 2. 766 2. 643 2. 500

1.536

2.316

1.678 40 2. 979 2. 876 2.766 2. 643 2. 500

1.579

2. 312

1.733 50 2.976 2. 875 2. 765 2. 643

1.538

2. 500

1. 636 2.308

1.785 60 2. 973 2. 873 2. 764 1.505

2. 643 1.580

2. 500

1.677

2. 303

1.829 70 2.971 2. 872 2.764

1.537

2. 643

1.610

2. 500

1.707

2. 300

1.862 80 2. 970 2. 871 2. 763

1. 556

2.643 1.629 2.500

1.726 2.2981.882 85 2. 970 2. 871

1. 502

2. 763

1.561

2. 643

1.634

2. 500

1.730

2. 297

1.888

I varies. Therefore, the inhomogeneous film is analyzed as though the

film were homogeneous, i.e., the index is regarded as constant through1 out the film. Therefore, a thickness, 2

tav, and a refractive index,

??/ av, are obtained.

This refractive index is some kind of average of the variable re- fractive index of the film, and the thickness is an average of the

thickness of the film averaged over the refractive index. A homogeneous film with this average index of refraction and thickness is equivalent to the inhomogeneous film with respect to ellipsometer

measurements.

To determine the type of averages represented by t&y and nav , the

ellipsometer readings were calculated for assumed inhomogeneous

films, £av and n&v being computed from the ellipsometer readings.

Then, various averages over the thickness and refractive index of the

inhomogeneous films were computed and compared with the corre- sponding values of f av and nav . The ellipsometer readings were computed by replacing the inhomogeneous films by a series of homogeneous films and calculating the reflection coefficients by the method

of Rouard [2, 6, 7]. The three variations of the refractive index with distance from the

substrate shown in figure 8 were assumed. Only non-absorbing films were considered. In all three cases, the refractive index varied from

1.6 at the surface to 1.3, and the index of the surrounding medium

2 The symbol t rather than d is used for film thickness in this section to avoid confusion when derivatives

are discussed. 728-101 0-64-6
McCRACKIN AND COLSON

o

1.6

1 1 1

_

1.5 EXPONENTIAL

- 10" d

n = I. 3 t 0.3 e -

1.4 - -

1.3

1.6 -

1.5 \ GAUSSIAN \ -io- 4 \ d

n = 1.3 + o. 3e lu 0 -

1.4 - -

1.3

1.6 LINEAR -

1.5 n = 1.6- io _3

d

1.4

1 1 100 200 300 DISTANCE FROM SURFACE, A Figure 8. Assumed variations for refractive index of inhomogeneous films.

i-80

r

1 ,

1 1 1 1 1

1 |

| | | |

DISTANCE FROM SURFACE, A Figure 9. Approximation of an inhomogeneous film by many homogeneous films.
COMPUTATIONAL TECHNIQUES 77

was 1.3. Exponential, Gaussian, and a linear variation of the re- fractive index were assumed. These films are expected to be similar

to the films of adsorbed polymer discussed by Stromberg et al. [5]. Small differences between the refractive index of these films and the

surrounding medium were assumed, so the conclusions that were obtained are probably limited by this condition.

The expected ellipsometer readings that would be measured for these films were then computed as illustrated in figure 9 for the case

of the linear film. The inhomogeneous film was approximated by

about 15 homogeneous films and the reflection coefficients and ellipsometer readings for these multiple films were computed.

In order to determine if the inhomogeneous film was sufficiently well approximated by the homogeneous films, the inhomogeneous

film was more closely approximated by about 30 and about 60 homogeneous films and the ellipsometer readings recomputed. In all cases, the readings computed by using 30 and 60 films agreed within

0.02° of each other, so 30 homogeneous films were considered a

sufficiently good approximation for the inhomogeneous films. From the calculated ellipsometer values for each of these films, an average index of refraction nav and an average thickness fav were

computed. The variation of these averages with the angle of inci- dence of the light is shown in table 9 for the linear and Gaussian

films. Both ?i av and / av are seen to be almost independent of the

angle of incidence so the type of average is, for practical purposes,

independent of the angle of incidence. For this reason, the variation

of ellipsometer readings with angle of incidence cannot determine

the type of variation of refractive index in an inhomogeneous film, or even distinguish between a homogeneous and an inhomogeneous

film. Table 9. Variation of n av and t av with angle of incidence for linear and Gaussian

variation of refractive index of film (see fig. 8)

Angle of incidence

7ll

Linear Gaussian

tav

deg A A

30 1.3 1.514 213 1.519 123 50 1.3 1.514 213 1.518 123

70 1.3 1.513 215 1.518 123

85 1.3 1.513 215 1.518 123 Values of nav and / av were also computed for distributions with a

smaller range of index of refraction. The index, nu of the surrounding medium was 1.4268 and the index of the film varied from 1.4268

to 1.4768 at the substrate. The results are shown in table 10. The types of averages over the distribution of index of refraction

of the films that approximate these averages were developed by the

following plausible argument. A film of the index nx of the sur- rounding medium has, of course, no effect on the reflection of light or the ellipsometer reading. Therefore, the effect of a film will
78 McCRACKIN AND COLSON

Table 10. Average values of the refractive index, nav , and thickness, t av , and

total optical effect for inhomogeneous films calculated from ellipsometer readings

ni ta.v rim m.\Hf J \"i III )Ul

Linear

1.3-1.6 1.3 1.513 214 45. 69 45. 00

1.4268-1.4768 1. 4268 1.462 215 7. 49 7. 50 Exponential

1.3-1.6 1.3 1.442 207 29.5 30.0

1.4268-1.4768 1. 4268 1.451 206 4. 96 5.0 Gaussian

1.3-1.6 1.3 1.518 123 26.6 26.6

1.4268-1.4768 1.4268 1.466 112 4.44 4.43

Table 11. Averages for the refractive index and thickness of inhomogeneous films compared with the ellipsometer averages

Weal «av trma fcal tivltitne

Linear

1.3-1.6 1.513 1.500 214 122 225 1.75

1.4268-1.4768 1.462 1.460 215 123 227 1.75 Exponential

1.3-1.6 1.442 1.450 207 141 200 1.47

1.4268-1.4768 1. 451 1.452 206 141 200 1.46 Gaussian

1.3-1.6 1.518 1.512 123 71 126 1.73

1.4268-1.4768 1.466 1.462 112 71 126 1.58

depend on n—nx

. Since the effect will also depend on the thickness

of the film, the effect of a film of infinitesimal thickness, dt, may be

expected to be (n-njdt and the effect of a film of varying index of refraction is obtained by integrating this quantity over the film, i.e.,

j (n— ni )dt. This integral will be called the optical effect of the film. JoThe value of this integral for the equivalent homogeneous film of index nav and thickness t&v is (n^-n,)^. This integral for the inhomogeneous film and for the equivalent homogeneous film, shown m

table 10, are seen to agree closely with each other. Since the optical effect of a thickness dt of the film is {n—njdt,

the refractive index was weighted by this quantity to give the average

refractive index

/» oo

J n(n—ni)dt

J

(n—n^dt

This average is given in table 11 and is in good agreement with the

average index nav that would be determined by the ellipsometer.

Various types of average thicknesses for the film were tried. Ine

only average that gave reasonable agreement for these films with

t&v determined by the ellipsometer was obtained by dividing the
COMPUTATIONAL TECHNIQUES 79

total optical effect by ne& i—ni:

This average is compared with £av in table 11. An average for film thickness of special importance is the root mean

square thickness given by

As may be seen from table 11, the average, tav, by the ellipsometer

is from 1.4 to 1.8 times larger than the root mean square average,

The reflection of light from an inhomogeneous film may therefore

be calculated by replacing the inhomogeneous film by a series of homogeneous films. Measurements of an inhomogeneous film by

the ellipsometer give rise to averages of the refractive index and

thickness of the film. Plausibly defined averages over the refractive

index and thickness of the film were found that agree approximately

with the averages that would be obtained by ellipsometry for three

assumed variations of refractive index in the film.

The ellipsometer measures the change of polarization of light

produced by reflection from a surface. For a film-free surface, the

refractive index of the surface may be computed from ellipsometer

readings.

Most applications of the ellipsometer involve measurement of a thin film on a substrate. For a film of known refractive index on

a substrate of known refractive index, the thickness of the film may

be calculated from ellipsometric measurements. For a nonabsorbing

film (real refractive index), both the refractive index and thickness

may be calculated from ellipsometric measurements. However, for an absorbing film (complex index of refraction) the refractive index

and thickness of the film cannot be determined from a single set of

ellipsometric measurements.

Therefore, a method was presented for analyzing sets of ellipsometer

readings of absorbing films to determine their refractive indexes and

thicknesses. The methods consisted of assuming values for the

imaginary part, nu of the film refractive index and computing values

of the real part, nR , and the thickness of the film for each assumed

value of rii and each set of ellipsometer readings. Then the value of

rii is chosen that gives a constant value of nR for the various sets of ellipsometer readings.

rms

4. Conclusions
80 McCRACKIN AND COLSON

This method was tested for computed sets of ellipsometer readings

obtained by varying (1) the thickness of the film, (2) the surrounding

medium, (3) the substrate, and (4) the angle of incidence of the light.

The method worked for sets of readings obtained by varying the

film thickness when the film thicknesses were not too thin, but failed, for the assumed conditions, for film thicknesses up to only 100 A. The method worked for sets of readings obtained by varying the

surrounding medium or varying the substrate. However, the method

worked only for a particular thickness of film for sets of readings

obtained by varying the angle of incidence of the light. Therefore,

the refractive index of an absorbing film may be determined from sets of ellipsometer readings obtained in various ways. The method of obtaining the sets of ellipsometer readings will depend on the particular

experimental conditions.

The effect of variation of the refractive index in the direction

perpendicular to the plane of the film was investigated. Such films are often measured by the ellipsometer and analyzed as though the

refractive index of the film was constant. Therefore, average values

for the refractive index and thickness of the film are obtained. The

type of averaging that is obtained was determined. For this purpose,

several films with varying refractive index were assumed and different

kinds of averages for the refractive index and thickness of the films were calculated. The ellipsometer readings that would be obtained

for these films were calculated and a refractive index and thickness were calculated from the ellipsometer readings. By a comparison of the averages determined by the ellipsometer readings with the directly

computed averages, the kind of averaging obtained from the analysis

of ellipsometric measurements was determined. Simple averages for refractive index and thickness of a film that agreed closely with those

obtained from the ellipsometer readings were found.

5. References

[1] P. Drude, Ann. Physik 212, 532 (1889); ibid. 212, 865 (1889); ibid. 215, 481 (1890).

[2] A. B. Winterbottom, Optical Studies of Metal Surfaces, The Royal Norwegian

Scientific Society Report No. 1, 1955, published by F. Bruns, Trondheim,

Norway; Trans. Faraday Soc. 42, 487 (1946).

[3] F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. NBS 67A (Phys. and Chem), 363 (1963).

[4] F. L. McCrackin and J. Colson, A Fortran program for analysis of ellipsometer measurements and calculation of reflection coefficients from thin films, NBS Tech. Note 242 (1964).

[5] R. Stromberg, E. Passaglia, and D. Tutas, this book, p. 281.

[6] P. Rouard, Ann. Phys. 1, 291 (1937).

[7] 0. S. Heavens, Optical Properties of Thin Solid Films, Butterworths Scientific Publications, London (1955).

[8] P. C. S. Hayfield, The Drude-Tronstad polarimeter and its use in corrosion

studies, First Intern. Congr. on Metallic Corrosion, Butterworths, London

(1962).
COMPUTATIONAL TECHNIQUES 81

Discussion

H. H. Soonpaa (Minneapolis-Honeywell Regulator Co.):

You assumed the index of refraction was constant for films of varying thickness. In single crystal films, the index is known in at least two cases to vary with the thickness.

F. L. McCrackin:

Yes, I quite agree. None of the results I presented were real data,

except for the barium fluoride data. I assumed constant index. For

actual films, the index may vary with the thickness so the method

could not be applied. Maybe the index would be constant with

thickness for film thicker than 100 A so the method could be applied

for this case. A. N. Saxena (Fairchild Semiconductor Co.): In your presentation, you do not refer to Archer (J. Opt. Soc. Am.

52, 970, 1962), who has already used a program of Miss R. E. Cox,

Bell Telephone Labs, and given \f/, A values for transparent films of various refractive indices on silicon. Actually it can be used for any

film on any opaque substrate having known optical constants. I pre- sume your program is either similar to the one used by Archer or else,

is it an advanced version of this program?

F. L. McCrackin:

The program was developed independently from Dr. Archer's

program.

A. N. Saxena:

Are your 8A and 8tp the differences between the experimentally

measured and calculated values for A and \p for a given index for the

substrate and for the film?

F. L. McCrackin:

Yes.

A. N. Saxena:

The differences 8\f/ and 8A are rather large. Does that signify

changes in the film itself or the error in the theoretical calculation due

to inaccurate values of the optical constants of the film-substrate

interface?

F. L. McCrackin:

For the barium-fluoride film, various values of the index of refraction were assumed in order to find the true index. For some values of the

refractive index, large values of 8\f/ and 8A were calculated. This

indicated the calculation was inconsistent so these values of the re- fractive index were eliminated.
82 McCRACKIN AND COLSON

A. N. Saxena:

What was the experimental accuracy of A and \p? F. L. McCrackin:

Several hundredths of a degree. A. J. Mabis (Procter and Gamble Company):

I would like to ask for which computer and which computer language

the program is written and are copies available?

F. L. McCrackin:

The program I discussed is written in Fortran and can be used on

an IBM 704, 7090, or 7094 computer, and probably others. A some- what simplified version of the program has been run on an IBM 1410

computer. I will be happy to give copies of the program to anyone

who wants them. The program is available as an NBS technical

note. (NBS Tech. Note 242 (May 27, 1964), available from Superintendent of Documents, U. S. Government Printing Office, Washington,

D.C. 20402. Price, 30 cents.) A. Rothen (Rockefeller Institute):

What wavelengths did you use and have you tried different wave- , lengths?

F. L. McCrackin:

All the results were for the mercury green line. We have not used

different wavelengths.

A. Rothen:

When you used different substrates, how thick was the film on the

substrate?

F. L. McCrackin:

The films were 100 A and had an index of 2.5-0.5i.
Factors Influencing the Experimental

Sensitivity of the Drude Technique

Richard C. Smith and Michael Hacskaylo

When measuring film thickness with a polarizing spectrometer,

the lowest measurable film thickness and the accuracy of the meas- urement are governed by the limitations of the optical apparatus. In

addition, the film thickness calculated using -the Drude approximate

equations will be in error due to the inherent error in the approximation.

The application of the Drude equations requires the measurement of the phase difference A and A' and the amplitude ratios \p and \p' of the uncoated and film-coated surfaces, respectively. The film thickness is then given in terms of the differences in these quantities, A — A' and

^ — The sensitivity, S, of a film of thickness t is defined as (A — A') /t and —. 9

—, 1 1- The remaining experimental parameters or combination

of parameters in the Drude equation are chosen so that the sensitivity

is maximized, i.e., the maximum difference between the optical properties of the coated and uncoated surface per unit thickness will be

realized. Curves and equations are presented to illustrate the de- pendence of the sensitivity on the experimental parameters and the technique is illustrated by practical cases where the experimental

conditions are specified and the corresponding accuracy and limits of thickness determination are discussed.

The state of polarization of light reflected from a metallic surface

is altered when a thin film is introduced on the surface. Drude [1, 2]

1 derived exact equations relating the thickness and refractive index of the film in terms of the angle of incidence, the optical constants of the

metal surface, and the state of polarization of the reflected light. The

exact equations, however, cannot be solved explicitly for the index of refraction and the thickness of the film. Drude therefore expanded

the equations and retained only the first-order terms in thickness.

The resulting equations are the well-known approximate equations,

which are valid if the film thickness is small compared with the wavelength oi the incident light. Although the validity of the Drude

approximate equations has been confirmed many times [3, 4, 5, 6], their value in determining the refractive index of the film has been

the subject of much conjecture [7, 8, 9]. The ellipsometer measures changes in the state of polarization of

light reflected from the film-coated and uncoated surfaces, and the

Drude equations relate these measurements to the thickness of the

film. The purpose of this paper is to give the experimental conditions

derived from the Drude approximate equations which will make the

ellipsometric measurements most sensitive to film thicknesses. It

will then be shown that both the minimum measurable film thickness

Melpar Inc., Falls Church, Va.

1. Introduction

1 Figures in brackets indicate the literature references on p. 93. 83
84 SMITH AND HACSKAYLO

and the greatest accuracy are achieved under the conditions for which

the sensitivity is a maximum.

2. The Drude Technique and the Sensitivity Equations

The "Drude technique" involves the use of an ellipsometer to determine the state of polarization of the light reflected from a bare

metal surface and from the same metal surface coated with a thin

transparent film. The state of polarization of light reflected from

the metal surface is completely described by the two measured parameters, 2 A' and \f/ f

, while the state of polarization of the light reflected from the film-coated surface is described by A and \p. To avoid sys- tematic error, the metal surface is only partially coated with the

films so that A, A', \p, and \p

f can be measured simultaneously. The

optical constants can then be calculated from A' and \J/' using the

equations of Ditchburn [10]. These values are then substituted into

the Drude approximate equations from which the thickness and re- fractive index are calculated. The Drude approximate equations are given in the following form:

. Mn2 —! f , . 2jL cos

2 <£—ax \ A—A = 5— -s cos 4> sin 2 0 r2 >

X n

2 \ ^ a

2

2 + (cos2 (jy-a,) 2 J

_,, Mn2 —l f , . 2

, l —n

2 cos2

<j) \ . 2^—2^'=— 0 a2 cos 4> sm2 4> 3— V2 >-sin2f

X n

2 \ a2+(cos2 </>— diY J

(1)

(2) where:

k

2 _ 2k

V

2 (l+K?

)

2

',a2 ~V2 (l+K2) di= 2( ?

. 2 ; a2 = 2( , 2

, 2

; n* =v(\—%K)

The terms in the Drude equations can be divided into three groups

the measured parameters, A, A', \p, and xp' ; the unknowns, n and t; and the constants of measurement n*, X, and <£. The constants of measurement can be chosen such that the difference in the measured

parameters per unit thickness is a maximum. This difference per

unit thickness is defined as the sensitivity.

The Drude approximate equations can be rearranged to yield the

following sensitivity equations:

&-A-A'/t-|-^|; St =H=££ /t = \A,t \, (3) where

, . 2 , cos

2 0—(Li , A

, <rA =cos </> smJ 2 , , 0— r-2 (4) a2+(cos2 0—a0

2

,

. - 1—n

2 cos2

(f> , . <j^=a2 cos <f> sim0 9 , , ^— ^ {£>) w a

2 + (cos2 4>— di)2

See Glossary of Symbols in table 1.
EXPERIMENTAL SENSITIVITY

| and

A V n

2 J

I The ratios SA and S+, called the A- and ^-sensitivities respectively,

are expressed in units of degrees per Angstrom (when the wavelength

i is in Angstroms). Thus the sensitivity of the Drude technique is not uniquely determined by either SA or S$, but by some combination

| of the two.

j 3. Constructing the Nomographs

Because of the form of the sensitivity equations, the method of

| nomographical representation was chosen to display the interdependence of the maximum sensitivity, and the conditions for which

the maximum sensitivity is derived. The optical constants, the

maximum sensitivity, and the angle of incidence for maximum sensitivity are all functions of ax and a2

. For this reason ax and a2 are

j chosen as coordinates for the nomographs. If the sensitivity is plotted

j as a function of the angle of incidence for fixed values of ax and a2 ,

| there will be a maximum in the sensitivity at some characteristic

| angle of incidence. The nomographs are constructed by plotting the

maximum sensitivity and the characteristic angle of incidence cor- responding to that sensitivity for every combination of ax and a2 . Each of the aA, f equations has terms which will vanish under certain

I measuring conditions. The factor (cos2 4>—aY ) in the numerator of eq

(4) will vanish where ax assumes the value of cos

2 0. However, if k assumes only values which are greater than unity, then ax will be

always negative, thereby precluding the possibility of a vanishing

sensitivity. This restriction does not limit the choice of metals

appreciably, since most metals which are satisfactory for use as substrates have extinction coefficients larger than unity [11]. The

Table 1. Glossary of symbols A = 8 P —8 S ; the difference in the phase changes of the p-component, S p , and the s-component, «5 S , on reflection from the film-coated substrate.

A' = A for the uncoated metal substrate ^ = arctan the arctangent of the factor by which the ratio of the

p- and s-component reflection coefficients change on reflection from the

film-coated substrate.

4>'

' = \f/ for the uncoated metal substrate. Z=thickness of the film in Angstroms.

n = index of refraction of the transparent film. X = wavelength of the incident light, n* = complex index of refraction of the metallic substrate. v=resd part of the complex index of refraction.

k = extinction coefficient.

<^ = angle of incidence. # a = angle of incidence for which the A-sensitivity is a maximum.

<t>l = angle of incidence for which the ^-sensitivity is a maximum.
86 SMITH AND HACSKAYLO

factor (1— n

2 cos

2

cj>) in the numerator of eq (5) cannot be handled in

this way, since there is an infinity of combinations of indices and angles

of incidence for which the product is unity. Thus, this term must

be carried along with no restrictions on the values of the index or angle of incidence. The wavelength appears in the factor A explicitly. At first thought, the sensitivity would be expected to in- crease for small wavelengths of incident light. But there is a higher

order dependence of aA, t on the optical constants which in turn exhibit

a nonlinear dependence on wavelength, namely dispersion. Thus

the wavelength of the incident light should be chosen such that the

values of the optical constants at that wavelength will offer the

greatest sensitivity. This leaves the angle of incidence as the major

experimentally independent variable affecting the A- and ^-sensitivities. For particular values of ax and a2 , plots of sensitivity versus

angle of incidence result in a peak for each sensitivity equation at some angle of incidence. This angle of incidence is defined as 0j, +. Curves of the A- and ^-sensitivities as functions of angle of in- cidence are shown in figure 1. The sensitivities are calculated for the optical constants of chromium which are v=2.97 and k=1.63

(see table 2). Under these conditions the A-sensitivity reaches a maximum at an angle of incidence, <j>%, of 76.50. The ^-sensitivity curves are plotted for three film indices. The maximum values all occur at about the same angle, i.e., near 82.50° with only a slight

increase in the characteristic angle of incidence and decrease in

sensitivity with increasing film index. These effects are not appreciable. In general each combination of ax and a2 has its characteristic angle of incidence for which maximum sensitivities are achieved.

2.5&

2.0A r\ -SA /

VAn=l.5

n=2.5

0° 10° 20° 30° 40° 50° 60° 70° 80° 90' ANGLE OF INCIDENCE Figube 1. A-and \p- sensitivity as a function of the angle

of incidence.

These curves are plotted from the era- and o>-equations (eqs. 4 and 5) using

the values for ai and ai corresponding to a chromium substrate. The values

of Sa and are given in units of A, The curves for are given for three

indices to = 1.1, 1.5, and 2.5.
EXPERIMENTAL SENSITIVITY 87

.3 oC

Qj^ noon 111

So ^

o |TM

O C © ^ HH

03-2 O

> S G

j> 03 03 -2d

« «3

MM

31

CUD H C '3 w 03

o3 03 >> d H —I fl O

03 03 pL| ^3

ft ft..(S

nun . •

| 03X3
SMITH AND HACSKAYLO

°2 0.1

0.001 mmsm

'A 0.0001

-0.001

THE OPTICAL CONSTANTS

ARE GIVEN BY THE COMPLEX

INDEX OF REFRACTION

n*= v(l-iK) THE VALUES FOR THE ABSCISSA AND ORDINATE ARE OBTAINED FROM

V 2 (1+K 2

} 2 V 2

(1 + K 2)2

•1.0 10.0 -100.0

Figure 2. Nomograph giving ai and a2 as a function of

the real part of the index of refraction, v, and the extinction

coefficient k.

o.oi

0.001

0.0001

Figure 3. Nomograph giving al and a2 as a function of

the peak A sensitivity
EXPERIMENTAL SENSITIVITY

i -J- 4U L 45° - 50° 54

1° 54°- i i-

- h

f —39° ~

!-

-

- r- —1 1

40° /

45°

50° \M 55° 60°

65° 70°

_ 75°

80° s

85°

86° 87° R \

88 o

g 7

o 86° 85° 80° 75° 70° 6 5° ( 0° 56 c 55°

1 1 1 1 1 1 1 1 II 1

°1 Figure 4. Nomograph giving ax and a2 as a function of

the angle of incidence for maximum ^-sensitivity

100.0

-0-001 -0.01 -0.1 -1.0 -10.0 -100.0 Figure 5. Nomograph giving ax and a2 as a function of peak \p- sensitivity
90 SMITH AND HACSKAYLO

For the condition where the two sensitivity maxima are not at the same angle of incidence, the optimum angle of incidence lies somewhere between the two maxima. Since both equations are needed to determine the index and thickness of the film, both sensitivities should

be as high as possible at the selected angle of incidence. The sensitivity of the Drude technique will be the smaller of the two sensitivities at the chosen angle of incidence. For the case of chromium where the

separation in the characteristic angles of incidence is about 6 deg,

the optimum angle can be approximately at the value of </>! At this angle of incidence the ^-sensitivity is a maximum and the A-sensitivity is very nearly equal to this value.

The nomograph shown in figure 2 gives the variation of a{ and a2 for combinations of the optical constants v and k. The practical

range of values of v was chosen to be from 0.1 to 5.0 and of k from 1.1 to 20.0. The corresponding values for ax and a2 can be found on the

abscissa and ordinate, respectively. • , . A plot of the maximum values of <rA as a function of ax and a2 is plotted in figure 3. The values for ax and a2 were selected to encompass the useful and practical range of values of v and k given above. , The range of a, is from -0.001 to -100, and the range of a2 is from

0.0001 to 100. The highest values of aA , and consequently the I maximum sensitivity, occur at the lower magnitude of ax and a2

. The magnitude of the sensitivity is obtained by multiplying the values

of (ta of figure 3 by the factor A. From the nomograph m figure 4

2

"1 Figure 6. Nomograph giving a x and a2 as a function of

the angle of incidence for maximum ^-sensitivity.
EXPERIMENTAL SENSITIVITY 91

one can determine the angle of incidence <j>% for which aA is a maximum.

A combination of values of ax and a2 which falls, for example, on the curve marked 70° has a characteristic angle of incidence for maximum

sensitivity of 70 deg. Clearly, all combinations of ax and a2 within

the limits of the nomograph, which have peak sensitivities at 70°,

are included in this curve. The nomograph giving o> as a function of

di and a2 is shown in figure 5. The values of ax and a2 cover the same

ranges as before. The maximum values of o> are less by an order of magnitude than the aA values for the same ax and a2 ranges. A plot

of the angles of incidence for which maximum values of o> are obtained

at given values of ax and a2 are shown in figure 6. To obtain the

magnitude of the ^-sensitivity the values of o> in figure 5 must be

multiplied by the factor A.

4. Discussion

The metals most suitable for accurate thickness measurements by

this technique must afford maximum sensitivities for both sensitivity

equations at the same angle of incidence. It is clear that such metals

must have optical constants which give values for ax and a2 located

in the lower left-hand corner of the nomograph in figure 2. For combinations of lower values of ax and a2 the characteristic angles for maximum A- and ^-sensitivities correspond more closely than for

combinations of higher magnitudes. As the magnitudes of ax and a2 increase, the separation of characteristic angles <£j and </>J also in- creases. In general, the larger the real index and the extinction coefficient the closer the values of <f>t and <£j. However, the choice

cannot be strictly limited by coincidence of the angles of incidence.

In table 2 are listed various metals, their optical constants, calculated characteristic angles and sensitivities, and principal angles of incidence. In the visible region of the spectrum, silver has one of the

highest extinction coefficients [11] and as a consequence a2 becomes

quite small. Since the ^-sensitivity is directly proportional to a2 , this sensitivity becomes much smaller than the A-sensitivity. Thus, even though the A-sensitivity is one of the highest for metals, its relatively

low ^-sensitivity does not make silver particularly useful as a substrate. The four metals, chromium, mercury, aluminum, and stainless steel,

have A- and ^-sensitivities which are of the same order of magnitude.

In each case the A-sensitivity peaks and the ^-sensitivity peaks occur

at angles of incidence which are separated by about 5° to 7°. Each

metal may be treated in the same manner as was previously employed

for chromium (fig. 1), and the optimum angle of incidence will be that near <£j. Thus, for the cases where the sensitivities are comparable

in magnitude and the characteristic angles are within 7 °, the optimum

angles of incidence are those near the ^-characteristic angle. It has

been suggested that an angle of incidence near the principle angle

would afford good sensitivity [9]. Note that for the cases where the

principal angle of incidence is given, the A-characteristic angle is less

728-101 0-64-7
92 SMITH AND HACSKAYLO

I than the principal angle by about one degree while the ^-characteristic

j angle is greater by about 6 deg.

The sensitivity is of interest in extending the minimum measurable

film thickness as well as increasing the accuracy of the measurement

technique. The Drude technique involves the measurement of the

difference in the values of A and A' and \p and xj/''. Since in general

there is a fixed experimental error involved in the determination of each of these parameters, the error in determining the difference in parameters is also fixed. Thus, the percent error decreases as the i difference in the parameters increases.

Consider an ellipsometer which can read the orientation angle of plane polarized light to P± dP=0.01 ±0.005 deg. Thus the dif- ference in two angles, P and P' which can be experimentally determined can be represented as (P-P') ± (8P+5P') =0.01 ± 0.01 deg. (6) This gives an error of 100 percent at the minimum measurable dif- ference in angles. In an effort to determine the minimum measurable

thickness one must specify what percent error is permissible. Considering 10 percent as the minimum permissible error, and if the error

in determining the difference in these angles is ±0.01 deg the dif- ference in the angles must not be less than 0.1 deg for the 10 percent

maximum error. Thus for 10 percent error the minimum determinable

difference in parameters can represented as (P-P')±(5P+5P')min=0.1±0.01 deg. (7)

Now from the sensitivity equation, the minimum thickness is given

in terms of the minimum in the differences in the parameters. For

the A-equation, _ (A-A')±(3A+5A') mlD , rmln \-AaA \

' W

If the A-sensitivity, | —^4.crA | , under optimum experimental conditions is given as 0.1 deg/A, then

*min=l±0.1 A. When the sensitivity increases for films of the same thickness the

percent error is decreased by the same factor. This does not hold for

films at the minimum measurable thickness. The percent error will always remain the same for a film of "minimum thickness" regardless

of the sensitivity.

As a practical example consider chromium. At a wavelength of 5790 A, v=2.97, k=1.63, and the angle of incidence for maximum

sensitivity is at 76.5°, aA =2.30, and for a barium stearate film of index 1.5, the value of A is 0.07. This gives the minimum measurable
EXPERIMENTAL SENSITIVITY 93

film thickness with 10 percent error as ftnm=0.625±0.0625 A.

This, of course, does not include the inherent error of this technique

due to the theoretical Drude approximations cited before. This

error in calculations due to the approximate nature of the Drude

equations has been reported to be about 5 percent up to 100 A in

thickness [12]. It is clear that one must choose the optimum experimental conditions, i.e., those conditions for which maximum sensitivity will be

afforded, to achieve both high accuracy and minimum measurable

thickness. Thus, two methods of choice of conditions are open to the experimenter. He can use the nomographs to determine the

optical constants and the corresponding angle of incidence which

afford maximum sensitivities and select the metals or form alloys

which exhibit these optical constants. Or he may use the nomographs

to choose the particular metal substrate of those he has available to him which will afford the highest sensitivity.

The authors acknowledge the assistance of J. C. DeBrie, J. H. Hoskinson, and R. J. Murphy in the preparation of the nomographs.

5. References

[1] P. Drude, Ann. Physik Chemie 36, 532 (1889).

[2] P. Drude, Ann. Physik Chemie 39, 481 (1890).

[3] A. Rothen, Rev. Sci. Instr. 16, 26 (1945).

[4] A. Rothen, Rev. Sci. Instr. 28, 283 (1957).

[5] J. A. Faucher, G. M. McManus, and H. J. Trurnit, J. Opt. Soc. Am. 48, 51

(1950).

[6] I. N. Shkylarevskii and N. A. Nosulenko, Opt. i Spektroskopiya 12, 435

(1962).

[7] A. B. Winterbottom, Kgl. Norske Videnskab. Selskab. 45, 128 ff (1955)

[8] R. D. Mattuck, Rev. Sci. Instr. 23, 844 (1957).

[9] F. P. Mertens, P. Theroux, and R. C. Plumb, J. Opt. Soc. Am. 53, 788 (1963).

[10] R. W. Ditchburn, J. Opt. Soc. Am. 45, 743 (1955).

[11] See, e.g., Am. Inst, of Physics Handb., sec. 6K (McGraw-Hill Book Co.,

Inc., New York, 1957).

[12] O. S. Heavens, Optical Properties of Thin Solid Films, p. 143 (Buttersworth's

Scientific Publications, London, 1955). Discussion

A. Baidins (E. I. duPont deNemours Company):

I noticed that in this symposium practically all the papers deal

with ellipsometry on metal substrates and I know in principle that

it is possible to determine the thickness of coatings on dielectrics

what are the sensitivities in that case?
94 SMITH AND HACSKAYLO

R. C. Smith:

The work which I have done here has only been in terms of metal

substrates. This was part of a feasibility study in the use of the

ellipsometer to measure the thickness of evaporated films on metal

substrates and we did not consider anything on dielectric substrates.

One would have to make a study under much the same conditions as

this. E. Passaglia (National Bureau of Standards):

I can comment on that to some extent. We have never done work

in our laboratory with direct reflection from a dielectric substrate, nor

to my knowledge has anyone else in the Bureau. But we have done

work with total reflection. Here, of course, one does not have the

problem of low sensitivity because of the low reflection coefficient. On the other hand, one has only one parameter to work with, for tan

\j/ is always unity. A theoretical calculation shows that for a film

with refractive index 1.6 on glass with refractive index 1.838 the whole

being immersed in water, ao film 250 A thick causes a change in A of about 4°, this is about 60 A per degree, so there is no problem with

sensitivity, provided the refractive index of the film is not too close to that of the substrate or surrounding medium.

A. C. Hall (Socony Mobil Oil Company):

Antonm Vasicek, among others, has used the reflected polarized

light method in studying thin films on dielectric substrates. He has

contributed two concise analyses of experimental limits; they appear

in the Journal of the Optical Society of America 37, 979 (1947), and

47, 565 (1957).

D. J. Tutas (National Bureau of Standards):

I would like to know how you prepared these films? We prepared some films of some of these metals you discussed and also made some

tables calculating the sensitivity and find the angles different from

what you gave.

R. C. Smith:

The angles for maximum sensitivity reported in the table are given

for optical constants reported by previous workers. The preparation

of metallic surfaces and subsequent measurement were not in the

scope of this study. The Drude approximate equations were here

used to calculate the sensitivity of the parameters by the Drude

technique, i.e. A, A', \p, and The sensitivities and consequently

the optimum angle of incidence vary very rapidly with the optical constants and with the wavelength of light. Thus if your metals had

the same optical constants as reported here and your definition of " sensitivity" was similar, then I would expect you to obtain the same optimum angle of incidence. If a different set of equations is used to calculate a sensitivity, as for instance the exact Drude equations, one would not expect to obtain exactly the same angle of incidence—even
EXPERIMENTAL SENSITIVITY 95

for the same optical constants—because of a different definition of the " sensitivity." A. K. N. Reddy (University of Pennsylvania)

I am sure that Dr. Archer will agree with me that in connection

with photoelectric ellipsometry it would be useful to have tables of

maximum A sensitivity and minimum i> sensitivity so that one can throw the whole burden of change of intensity on one of the parameters.

Increased Scope of Ellipsometric Studies of

Surface Film Formation

A. B. Winterbottom

Norges Tekniske Hogskole (The Norwegian Institute of Technology),

Trondheim, Norway

Recent developments in ellipsometric instruments and methods

should increase the scope of profitable applications of this technique even though the principles of such applications remain unchanged.

The more sensitive photoelectric methods now available will un- doubtedly replace classical visual procedures. The development of systems capable of following and registering rapidly changing phenomena would not be expected to involve any specially sophisticated

electronics. An experimental photoelectric ellipsometer to explore

such possibilities is described.

1. Historical Introduction

Paul Drude [l], 1 with an amazing combination of theoretical insight

and experimental skill, derived the basic theory of film optics and

demonstrated its validity elegantly by measuring the progressive

changes in reflection on cleavage faces of stibnite and tetradymite.

Sporadic applications of ellipsometry to the study of film formation were made subsequently by various students of such phenomena.

The instruments used were usually adapted spectrometers or so- called crystal ellipsometers, perhaps best suited for simple classroom

demonstration but which required the addition of many appurtenances

to enable experiments to be conducted under controlled conditions

of environment and temperature and even for the reading of scales [2]. Nevertheless, useful results were obtained and these were improved

in quality and precision by improvement in the details of the measure- ment technique and particularly by the preparation of simple, well- defined surfaces free from surface damage and contamination, by

the use of single crystal specimens and of electrolytic polishing, etc. [3]. In addition to the desire for higher precision in the quantitative

results obtained and a finer discrimination as regards the nature of film substance formed, there was also a need for an automatic recording

instrument to make studies of rapid surface changes. The late Professor Bonhoeffer was keenly interested in exploring this aspect

and initiated work at Gottingen over ten years ago. A servo- operated ellipsometer was actually built there and this has since

been presented to Professor K. Vetter, after Professor Bonhoeffer's

death. However, this instrument was apparently much too sluggish

to be used to follow rapid changes [4]. Professor J. O'M. Bockris,

who is participating in this conference, was also one of those who

has long been interested in new and more thorough studies of electrode

phenomena with improved ellipsometer technique.

1 Figures in brackets indicate literature references on page 110. 97
98 WINTERBOTTOM

At Norges Tekniske Hogskole (NTH), ellipsometric studies of film formation in both liquid and gaseous environments have been

made from the late 20's onward by Tronstad [6] and various collab- orators, later by the present author [7], and most recently by Marjorie

A. Barrett [8]. All these studies have been made with a visual

instrument that has previously been described in some detail in various papers [9]. Quite recently a request was made by Norsk Blikkvalseverk for the development of simple ellipsometer equipment to monitor oil film thickness on tin plate. It was decided to undertake this task as, although automatic recording equipment for the purpose could then

be purchased in the United States [10], it was felt that it would

provide an opportunity to gain experience in photoelectric ellipsometry and at the same time possibly lead to che development of a

simpler, cheaper photoelectric ellipsometer suited to this kind of application. This led to studies of the literature on the subject

and to the discussion of the problems involved in polarization modulation for simultaneous independent feedback compensation of the

two magnitudes involved in ellipsometry with electronic engineer

colleagues, both at Norges Tekniske Hogskole and at the National

Physical Laboratory (NPL)

As a result of these studies and discussions, an experimental ellipsometer with facilities for trial of various modulation and compensation systems is now being built. The design of this instrument will be briefly discussed in the following.

2. Basic Instrument Design

As previously noted, the majority of the ellipsometric studies so

far made have been carried out with adapted optical instruments of the classic type to which many auxiliary services have been added.

The space for experimental cells or chambers has often been cramped

and access to it severely limited. There is also the question whether

the plane of incidence shall be horizontal or vertical, or whether the

instrument should be adapted to both arrangements. The visual

instrument in use at Trondheim can be operated in both ways, but

it has been used almost exclusively with the plane of incidence vertical

and specimen surface horizontal (figs. 1 and 2). We know of other workers who have exclusively used the alternative

arrangement with plane of incidence horizontal and the experimental

surface vertical [11] (fig. 3). Evidently each arrangement has its advantages and disadvantages.

We have chosen to design the new photoelectric instrument for use with plane of incidence primarily horizontal but with facility for adapting to the other arrangement with the plane of incidence vertical. In view of the desire to experiment with modulation and compensation based on magnetic rotation (Faraday effect), the new

instrument construction aims at the minimum use of metal in the
INCREASED SCOPE OF ELLIPSOMETRY 99

Figure 2. Optical system of Leiss ellipsometer
100 WINTERBOTTOM

Figure 3. Ellipsometer designed and used by P. C. S. Hayfield with quarter-wave

plate compensator in rejected system and applicable to studies of vertical surfaces

(ca. 1950). [11]

Figure 4. General arrangement of experimental photoelectric ellipsometer under

development at Trondheim (1963). vicinity of the optical components and avoidance of magnetic

materials entirely. It seems appropriate to enclose the optical

systems of the incident beam and the reflected beam, as well as the

experimental space, in light-tight chambers so that the instrument can normally be used in a lighted laboratory.

Although it is expected that photoelectric operation will ultimately

supersede visual operation entirely, the present experimental arrangement does permit alternative visual operation with halfshades. The

instrument table also provides accommodation for the various elec- tronic units and auxiliary service units under the table plate and in

the dead quadrant (fig. 4). 3. Basic Ellipsometer Optics

Discussion of the changes in the state of polarization produced on

passing the various optical units in an instrument system is of course
INCREASED SCOPE OF ELLIPSOMETRY 101

facilitated by the use of appropriate terminology and mathematical

manipulation methods. A specially clear exposition of the analytical

; treatment of such problems has been given by the late Dr. L. B.

I Tuckerman of the National Bureau of Standards in a classic paper [12].

! The Poincare method of representation of a general state of polariza-

! tion as a point on the surface of a sphere has much to commend it for rapid examination of the action of optical systems containing doubly

I refracting and rotating elements and also reflectors. It has been

described in Poincare's monograph, Theorie Mathematique de la Lumiere, and in a series of papers concerned with its utility in various

practical cases [13]. A recent monograph on polarized light also dis-

! cusses and describes this and other methods of treatment [14]. Turning now to the basic optics of the ellipsometer, these aim at determining the two magnitudes characterizing the reflection of a

, polarized wave, viz, the relative amplitude reduction and the relative

! phase retardation of the components vibrating parallel and perpendicular to the plane of incidence. It turns out that these magnitudes

can be determined for any surface by allowing a monochromatic

I polarized coherent beam to pass through a system composed of a

polarizer, experimental reflecting surface, and an analyser with a

1 quarter-wave plate fixed in a diagonal azimuth either before or after

| the reflector. It is not necessary that the quarter-wave plate be exactly

quarter wave. If the polarizer and analyser settings are found which

' give perfect extinction, then the two required magnitudes can be

I computed from these settings and the constants of the quarter-wave

plate compensator. The two alternative arrangements with the

Poincare representations of their action are given in figure 5. These

i n

P Pol a riser A Analyser

K Compensator in diagonal position

P Polariser azimuth ZZ' Elliptic polarisations KFKS Compensator axis LU Linear polarisations

l s

l p Mirror axis A Crossed analyser azimuth NB. Primed polarisations result from relative amplitude reduction Figure 5. Alternative arrangements of ellipsometer optical systems with Poincare' representations of their respective modes of action.
102 WINTERBOTTOM

alternative arrangements each have their advantages and disadvan- ! tages, and the choice of one or the other will depend on the relative

importance attached to these differences. Since the quarter-wave

plate does not occupy much space, it was deemed wise in the present

case to provide for easy insertion and removal of plates in both the

alternative diagonal positions, together with facilities for fine adjustment of their settings and for changing to alternative diagonal settings

quickly with click action as in some types of microscope polarizers

The polarizer and analyser will initially be of the usual polaroid ! type mounted in circles with manual setting and scales with verniers,

but servo-operation may also be tried. The polarizer and analyser | may of course be used with fixed settings when some other means is used for varying the polarizer or analyser azimuth, such as magneticrotation units.

Elementary considerations of error in such systems reveal that

multiple reflection can give rise to systematic effects that are nevertheless difficult to predict and correct [9,13b]. On these grounds

there should be the minimum of air/glass (or other transparent solid)

surfaces in the system and those surfaces that cannot be avoided

should be provided with antireflection coatings or nondeviating prismatic cover glasses. The random error involved in setting the

instrument is of the greatest importance for the precision of the

measurements and as it is really the essential property of the setting

device it is treated in the next section.

4. Ellipsometer Setting Devices: Halfshades or Modulators

Inevitably the precision with which an ellipsometer can measure

depends on the precision with which the settings of the adjustable

optical units can be made and read. In visual work halfshades are used for increasing the accuracy of setting as in conventional polarimetry. One set of halfshades enables the analyser to be set precisely

crossed to the plane of polarization of the final beam and thus to determine the azimuth of the final polarization while the other set must enable another of the optical units, the polarizer, to be set so that the final polarization is precisely plane polarized. The sensitivity with which settings can be made with halfshades depends on the ability of the eye to distinguish intensity differences at various

intensity levels, the intensity of the source and the halfshadow angle.

This has been treated exhaustively in many papers, monographs,

and handbooks [13b, 15]. The Poincare representations in figure 6 show how the amplitudes

and accordingly the intensities in the two fields of halfshade sj^stems

may be studied. For a symmetrical system it is evident that equality

of intensity for the two half fields would be obtained when the points

characterizing the polarizations for the two half fields are equidistant

from the point representing the polarization crossed by the analyser.

To obtain this, the incident polarization or the analyser setting
INCREASED SCOPE OF ELLIPSOMETRY 103

MODULATOR

SOLENOID CURRENT

OPTICAL SETTING: EXTINCTION

PHOTOMULTIPLIER + VVWWAA

DISPLACED FROM EXTINCTION 1 OPPOSITE DISPLACEMENT FROM EXTINCTION /VAAAAiA/UA CURRENT 0

1 ^NOTE PHASE DIFFERENCE

MODULATION COMPONENT

FREQUENCY

WITH

0 O/wWxXAAAy

Figure 7. Effect of modulation on photomultiplier output with extinction setting and slight deviations from this in both senses. must be altered. In the case of a biplate with clockwise and anticlockwise rotations of equal magnitude for the two half fields, evi- dently the system is sensitive solely to azimuth. Deviation from

the truly crossed position leads to differences in intensity for the

half fields. Similarly a symmetrical doubly refracting biplate system

would be sensitive only to ellipticity.

Similar Poincare representations may be used to discuss the effects of periodic modulation of a polarization with suitable periodic alter- ations in either azimuth or ellipticity. However, in this case the

extreme points represent the extreme alterations produced in the

polarization arriving at the analyser at various times. If the polarization that is modulated would be precisely crossed by the analyser

setting, then successive extreme positions of the modulated polarization would be equidistant from the extinguished polarization in the Poincare representation and the heights of successive peaks m

a photomultiplier output would be identical. It is thus apparent

that the output from a photomultiplier analyser combination receiving

a polarized beam modulated symmetrically about the truly crossed

polarization would only contain a double modulation-frequency component superposed on the dark current. On the other hand, un-
104 WINTERBOTTOM

symmetrical modulation or deviation from the true crossed polarization, whether it be in azimuth or in ellipticity, would evidently cause successive peaks in the photomultiplier output to alternate

in magnitude. That is, this output would contain a modulationfrequency component in addition to the double-modulation frequency.

Furthermore, it turns out that as the deviation alters sense when

the polarization passes through the truly crossed azimuth polarization, then the modulation-frequency component of the photomultiplier output would change phase by a half wave (see fig. 7). Evidently these features of modulation bear in themselves the

possibilities not only for sensitive setting of polarizers and the like but also for automatic compensation [16]. This is precisely what is done in the ETL-NPL automatic saccharimeter which is shown in

figure 8 and diagrammatically in figure 9 with a block diagram in figure 10. A diagram showing the way in which compensatory feedback is obtained is given in figure 11. This polarimeter is based

Figure 9. ETL-NPL automatic saccharimeter {diagrammatic)
INCREASED SCOPE OF ELLIPSOMETRY 105

on the utilisation of Faraday magnetic rotation of the plane of polarization with the output feedback. In a commercial version it can measure rotations of up to 0.5° with an accuracy of ±0.0001°.

It is of course feasible to modulate symmetrically by oscillating

the polarizer or analyser equally about their settings [17]. This is also done in one type of commercial polarimeter and a sensitive

galvanometer used directly in the photomultiplier output. Here

the precise crossed position is set with an accuracy of about 0.001°

by simply adjusting until the galvanometer spot does not move on

on flipping the polarizer. The Faraday rotation is given by the

expression:

a=vlH and H= 1,257 ni,

where

a is the instantaneous rotation in minutes of arc,

i is the instantaneous current in amperes,

H intensity of magnetization, n the number of turns per centimeter length of solenoid,

I the length of solenoid in centimeters,

2J0v. A C. MAINS 50c/s

LAMP

SUPPLY PHASE

SENSITIVE DETECTOR

1STABILIZED

E.H.T. SUPPLY [1FIRST ! HARMONIC

REJECTOR

PHOTOMULTIPLIER MODULATOR C0MPENSFigure 10. ETL-NPL automatic saccharimeter-diagrammatic section of optics of the saccharimeter and block diagram of the electronic circuit.

SAMPLE ROTATION UNDERCORRECTED BALANCED 0VERC0RRECTED

TIMING SIGNAL A: ^Y/TYA /i\/i\/r

PHOTOMULTIPLIER 'a SIGNAL

DFUNDAMENTAL COMPONENT vL ^

DETECTOR A x C ,/|\/\/N/\/N

FEEDBACK D |

| |~

I I

I Figure 11. Derivation of feedback signal giving automatic compensation in a

photoelectric polarimeter.
106 WINTERBOTTOM

I and

v the Verdet constant in minutes/gauss/centimeter for the core 1 substance.

For a long core extending to negligible fringe field strengths on each

side of the solenoid it can be shown that <*= 1,257 vNi, where N is now the total number of turns. The refractive index and Verdet

constant are given in table 1 for a number of transparent materials.

Table 1. Values of Verdet constant for a few transparent solids and liquids at room temperature and for sodium light

Substance

Refractive index for NaD

Verdet constant minutes of arc/gauss/cm Carbon disulfide. _ . . ... ... 1.6994

1. 6578

1. 70035

1. 74842 1.80120

1. 92707

0. 04255

.0519

.052

.062

.074 .106

Monobromnaphthalene

Chance Pilkington glasses EDF 700303 DEDF 748278 . DEDF 801255 DEDF 927210

Table 2. Specifications of some solenoids used in magnetic rotation experiments No 1 2 3 4 5 Name,. . Perkin Dowry Gates Gillham and King

Gillham and King

Literature references [length mm..

18 146

76 476

2000 3X2.7 mm

100

19 16 115

1436

1657

10

16

57

10 Dimension-^IdiameterM 1 {ext. mm..

Winding turns 7620 65000 428WG

8

6 mm bore copper tube

Conductor dimensions Weight (approx) kg. Resistance . w.. 504

200 800

11

11

12500

0, 007

87.5 0.6

Current A 12, 550 625

1000 water

±4. 5° ironshroud and end

plates

v.. W-.

Max. field COS

170 Core substance CSj±20°

ironshroud and end

plates

glass DEDF

930, 181 Max. rotation deg.. ±3° ±0. 1° ±1.8° Remarks

A group of workers at NPL, England, developing photoelectric

polarimeter equipment, have used fused quartz glass or a special

double, extra-dense flint glass from Chance Pilkington for both

modulators and compensators. The double, extra-dense flint glasses

have high refractive indices, as the name implies, and reflected parasite beams, approaching 10 percent per surface, might be ex- pected to be embarrassing. It would therefore be desirable to
INCREASED SCOPE OF ELLIPSOMETRY 107

provide end-cover glasses coated for reduced reflection and possibly

to arrange for continuous core rods for each modulator-compensator

set if such long rods could be obtained. Alternatively, it might be

simpler to use long jacketed polarimeter tubes with a liquid of high

refractive index and Verdet constant, such as monobromnaphthalene. The compensators and probably also the modulators should

be provided with thermostat-controlled water circulation in order to define the core temperature and hence the Verdet constants. In table 2 are tabulated the specifications of some Faraday

modulator and compensator coils and their performances with specified core material. The coils used by Perkin [18] and Lowry [19] in

their classical work on magnetic rotation are also included. Since

these were used with steady currents, the materials and design

would have to be modified appropriately if modulators or compensators were to be made on similar lines. 4.1. Electro-Optic Effect, Pockels Effect

Other optical effects that might be utilised in automatic ellipsometry are the electro-optic effects: the Kerr effect in liquids and

the Pockels effect in crystals. The latter is the converse of the piezoelectric effect, viz., the setting up of anisotropy as a result of an

applied electric field. One kind of crystal in which this effect is pronounced is ammonium dihydrogen phosphate (ADP), which is tetragonal, i.e. uniaxial. The properties of this crystal have been

studied and reviewed by Billings [20]. A so-called Z-cut slice of ADP

is generally mounted between conducting transparent cover glasses

(e.g., glass coated with SnO such as NESA) for electro-optic work.

On applying a field in the Z direction, the crystal slice becomes

TEST

Figure 12. Photoelectric polarimeter after Takasaki (diagrammatic)

728-101 O- -64 8
108 WINTERBOTTOM

biaxial with induced axes at 45° to the X and Y axes with orientation

independent of the field strength. The retardation of such a slice is! a linear function of the potential applied and independent of the i thickness. A retardation of a halfwave requires a potential of about

9 kv which is near the breakdown voltage for thicknesses less than s about 0.2 mm.

It is evidently possible to use ADP to modulate polarizations except

linear ones with azimuths identical with those of the induced axes. The modulation will naturally be greatest for polarizations with!

azimuths diagonal to the induced axes. The modulation will follow 1 small circles having the induced axes as centre in the Poincare rep- resentation.

Takasaki has described a number of applications of such modulators including a photoelectric polarimeter [21[. This utilizes a

polarizer and Soleil-Babinet compensator system driven by servo- motors (see fig. 12). The elliptic polarization coming from the

polarizer quarter-wave plate in zero azimuth is modulated in two

directions at right angles on the Poincare sphere by ADP modulators

with induced axes in 45° and 0° azimuths. Thus one modulation is i effective only in the azimuth of the ultimate polarization arriving at the analyser, while the other is effective only in the ellipticity. If

| the final polarization is not exactly crossed to the analyser fixed at I 45° in both aximuth and ellipticity, modulation-frequency components appear in the photomultiplier signal. The servos acting as ; phase-sensitive detectors react only to modulation-frequency components supplied to their control phases derived from modulation

by their respective reference phases.

I. 5. Experimental Ellipsometer for Trial of Alternative

Modulation-Compensation Systems

It has been seen that automatic compensation can in general be I derived from suitable modulation of the polarization in the optical I system followed by photoelectric detection, amplification, and feedback. Furthermore it is apparent that various methods of modulation and compensation are feasible, ranging from mechanical move- ! ment of components to magnetic or electrical effects. Since the

sensitivity and precision of automatic polarimeters based on the use

of magnetic rotation for both modulation and compensation have

been amply demonstrated, it would be appropriate to try to adapt ] the same methods to ellipsometry. Evidently, servo-operation with

either ADP modulation in the manner of Takasaki or magnetic rota- 1 tion modulation, could be applied to measurements where the reflecting , surface only changes slowly.

The experimental photoelectric ellipsometer now being built at Trondheim provides for manual operation with either visual halfshade

systems or with photomultiplier-detector-galvanometer or with van- . ous modulator-compensator system combinations (see fig. 13).
INCREASED SCOPE OF ELLIPSOMETRY 109

Alternative quarter wave

plates Q 1 & Q, in diagonal positions

Figure 13. Optical system of experimental photoelectric ellipsometer with arrangements for operation in various ways (diagrammatic).

Figure 14. Combined diagram of optical system and block diagram of the electronic

circuit for a trial arrangement of the experimental photoelectric ellipsometer.

One system that seems particularly promising is indicated diagrammatically in figure 14. With this experimental instrument it is hoped

that it will be possible to determine the feasibility of a purely elec- tronic ellipsometer with fixed optical units, providing indications and

records of the two reflection parameters A and \p, even when changing

rapidly. Such an instrument it is felt might find wide applications

in both the technological and pure research fields.
110 WINTERBOTTOM

This work has been supported with grants from The Royal Norwegian Council for Scientific and Industrial Research over a number

of years, which support is gratefully acknowledged. A special grant , awarded by Norges Tekniske Hogskoles Fond has made the design

and construction of the new ellipsometer possible, for which the

author expresses his sincere thanks. He is also indebted to the Industrial Research Division (SINTEF) and to many friends and

colleagues at the Division of Automatic Control for assistance with

electronic problems and to Mr. E. J. Gillham and Dr. King of NPL,

Light Division, for stimulating discussions and advice.

6. References

[1] P. Drude, Uber Oberflachenschichten I and II, Wiedeman Ann. Phys. 36,

532, 865 (1899).

[2] A. B. Winterbottom, The study of immersed surfaces based on the reflection

of polarized light. Principles and possibilities, Z. Electrochemie 62, 811 (1958).

[3] F. W. Young, Jr., J. V. Cathcart, and A. T. Gwathmey, Acta Met. 41, 145

(1956).

[4] K. H. Bonhoeffer, private communication (1957); M. A. Barrett, Norges

Tekniske Hogskole, Trondheim, Norway, private communication (1961).

[5] J. O'M. Bockris, Univ. of Pennsylvania, Philadelphia, Pa., private communication (1960).

[6] L. Tronstad, The validity of Drude's optical method of investigating trans- parent films on metals, Trans. Faraday Soc. 31, 1151 (1935).

[7] A. B. Winterbottom.

[8] M. A. Barrett, B 173 Report 10, Royal Norwegian Council for Scientific and

Industrial Research; M. A. Barrett and A. B. Winterbottom, 1st Intern. Congr. Metallic Corrosion, p. 657, Butterworths, London (1961).

[9] A. B. Winterbottom, (a) A polarization spectrometer for investigation of surface films on metals, J. Sci. Instr. 14, 203 (1937) ; (b) Optical studies

of metal surfaces, Kgl. Norske Videnskab. Selskabs, Skrifter No. 1, 52 et

seq. (1955).

[10] T. P. Murray, Oil film thickness gage, Proc. ISA (Iron and Steel Conf., 1959).

[11] P. C. S. Hayfield, Imperial Metal Industries (Kynoch) Ltd., Birmingham,

England, private communication (1960); J. Kruger and W. T. Ambs,

J. Opt. Soc. Am. 49, 1005 (1959).

[12] L. B. Tuckerman, Univ. Nebraska Stud. IX, No. 2, 1909.

[13] (a) H. Poincare, Theorie Mathematique de la Lumiere 2, 275 (1892).

(b) L. Chaumont, Ann. Phys. 4, 103 (1915).

(c) B. H. Zocher and F. C. Jacoby, Kolloid Chem. Beih. 24, 395 (1927).

(d) A. F. Turner, J. R. Benford, and W. J. McLean, Econ. Geol. 40, 18 (1945).

[14] William A. Shurcliff, Polarized Light: Production and Use, London, Oxford

Univ. Press (1962).

[15] G. Bruhat, Traite de Polarimetrie, Paris Dunoyer.

[16] J. W. Gates, An automatic recording saccharimeter, Chem. Ind. (1958), p. 190; E. J. Gillham and R. King, New design of spectropolarimeter, J. Sci. Instr. 38, 21 (1961).

[17] B. R. Malcolm and A. Elliot, J. Sci. Instr. 34, 48 (1957).

[18] W. H. Perkin, J. Chem. Soc. 89, No. I, 608 (1906).

[19] T. M. Lowry, J. Chem. Soc. 103, No. II, 1322 (1913).

[20] Bruce H. Billings, J. Opt. Soc. Am. 39, 797, 802 (1949).

[21] H. Takasaki, Photoelectric measurement of polarized light by means of an ADP modulator. II. Photoelectric elliptic polarimeter, J. Opt. Soc. Am.

51, 463 (1961).
INCREASED SCOPE OF ELLIPSOMETRY 111

Discussion

i A. C. Hall (Socony Mobil Oil Company)

I should like to know how rapidly a servo-controlled ellipsometer

might be expected to respond. What time lapse would occur between

j two readings? A. B. Winterbottom:

I feel that this is an electronic problem. The commercial photoelectric polarimeter reacts in a matter of seconds, but it is very

heavily decoupled. The type of instrument with servo-operation

! would also respond in a matter of seconds. However, ADP crystals or Faraday rotators would respond very rapidly similar to Kerr cell which is used for rapid photo shutters, but to realise such possible

high speeds, the time constants of the circuits would obviously have

| to be equally small so far as this is consistent with other requirements,

such as noise.

J. Kruger (National Bureau of Standards)

You mentioned that there are advantages and disadvantages to the two different ways one can arrange the components on the ellip-

| someter. Would you care to point out the advantages and disad-

| vantages of each of those two arrangements?

A. B. Winterbottom:

I had thought that the second arrangement enabled Nakamura

biplates to be used in different positions in the system as azimuth and

ellipticity half shades respectively. However, on further consideration it seems that this applies to both arrangements and we are left with the convenience of placing more components in the incident

section or in the reflected section of the instrument as being the

decisive factors when choosing one or the other arrangement. Thus

if one arm is fixed and the other moved to alter the angle of incidence,

it might be convenient to have fewer components on the mobile arm. M. A. Barrett (Institutt for Fysikalsk Metallurgi, Norway)

In the case of a technological control instrument in which only

variations in A were followed, it was convenient to use the second

arrangement as then the Nakamura biplate could be used in setting

the analyser and determining A. A. N. Saxena (Fairchild Semiconductor)

Another comment that could be made on the two methods of using

the ellipsometer is that in the first case where one uses the quarter- wave plate before reflection, the computations of

\f/ and A from the

measured polarizer and analyzer readings is much simpler than in the

second case in which one uses the quarter-wave plate after reflection. Also, the disadvantage you have in the second case is that if the
112 WINTERBOTTOM

reflecting surface introduces a phase retardation such that you get

circularly polarized light then you can't analyze with the second

arrangement.

A. B. Winterbottom:

No. With either arrangement the parameters sought are found by N solving spherical triangles which incidentally are easily discerned in the Poincare representations. The error situation can be examined

by writing down the partial derivative expressions in the usual way. | M. A. Barrett: i Although circularly polarized light (or tan ^=1 and A=90°) can | be analyzed by both arrangements, when tan \p approaches 0 or oo

; the measurement of A becomes increasingly more difficult and of course at these extreme values, A is meaningless. Here again the

situation appears to be roughly equivalent for the two possible

arrangements of the ellipsometer.

R. R. Stromberg (National Bureau of Standards)

How accurately can A and \p be determined using conventional

methods and light sources? A. B. Winterbottom:

The accuracy of setting the various circles with half shades depends on the light source, the half shades and the observer. This random

error of setting naturally also varies with the quality of the mirror,

especially the degree of depolarisation. The further effect of random

errors of setting on A and \p will be given by the partial derivative

expressions obtained from the relevant formulae for A and ^/ or from

the tables relating instrument readings to these parameters. With

the instrument I have used under favorable conditions, polarizer

readings were made with settings accuracy of ±0.1° corresponding

to about ±0.2° in A and about the same accuracy of setting for I analyser corresponding to ±0.1° in A. However, the commercial

photoelectric polarimeter to which reference has been made, measures

rotations down to 0.001° and is thus able to work with very small

sample thicknesses and samples with high absorption with sufficient accuracy.

M. A. Barrett:

I would estimate that ideally with half shades you can get to 0.02

centesimal degrees. However, when an actual specimen is involved,

the accuracy is somewhat lower.
Electronic Polarimeter Techniques

Jerome M. Weingart and Alan R. Johnston

Jet Propulsion Laboratory, California Institute of Technology, Pasadena,

Calif. A technique for measuring changes in the orientation of linearly

polarized light with a precision of a few seconds of arc is described. A

Faraday cell and appropriate electronics were added to a standard

photoelectric polarimeter. The cell provided an a-c signal at the

photomultiplier which was accurately phase-detected to produce a sharp null. A similar technique was used to measure the orientation and ellipticity of elliptically polarized light. Accurate measurement

of a-c birefringence into the megacycle region can be made by additional application of an electronic sampling technique.

The instrument described below (fig. 1) was developed to measure

electrically induced birefringence in crystals. However, it can be

applied whenever precise measurement of optical retardation is desired, and can measure periodically varying retardation into the

radio frequency region with a precision approaching that of the static

measurements. The collimated, monochromatic light source consists

of a PEK 109 high-pressure, mercury-arc lamp in a suitable housing,

a collimating lens, and a narrow-band filter centered on one of the

prominent lines in the Hg discharge spectrum. With this particular

lamp, a collimated beam with ± 8 min of arc divergence is obtained

with a collimating lens of 4-in. focal length. Both polarizing and

analyzing elements are Glan-Thompson prisms set in rotatable mounts.

The analyzer is set in a modified Gurley Unisec divided circle, which

will measure rotations to within 2 sec of arc. The polarizer setting can be read to 1 min of arc. The quarter-wave plates are mounted

in a holder designed to support them in as strain-free a manner as

possible. The holder, in turn, fits in a rotatable mount identical to the polarizer mount.

Previous to 1950, the addition of electronics to a conventional

polarimeter generally consisted of substituting a photocell for the eye

ELECTRO -OPTIC MONOCHROMATIC

FILTER

MERCURY COLLIMATING GLAN-THOMPSON

ARC LENS PRISM LAMP (POLARIZER)

QUARTER- WAVE

PLATE

FARADAY

CELL

GLAN - THOMPSON T PRISM

(ANALYZER) ,,

DC TO 30 -Mc

HIGH -VOLTAGE

SOURCES LOWFREQUENCY

PHASE SHIFTER DEMODULATOR

Figure 1. Optical bench and electronics for measurement of optical retardation. 113
114 WEINGART AND JOHNSTON

Figure 2. Parameters describing

j elliptically polarized light.

a - AZIMUTH ANGLE

9 SPECIFIES DEGREE OF ELLIPTICITY and adding the appropriate electronics for measuring a minimum in d-c voltage level from the photocell. In 1950, A. C. Hardy [1]

1 suggested inserting a Faraday cell (a solenoid containing a transparent

material with a large magneto-optical response) in the light path

between the analyzer and the rest of the optical elements. A lowfrequency current in the coils of the Faraday cell results in a sinusoidal

rotation or nutation of the polarized light traversing the cell, thus

providing an a-c signal at the photomultiplier which can then be

accurately phase-detected using standard demodulator techniques.

The Faraday cell contains pure water with a 10-in. pathlength. The

polarimeter described here operates on this phase-detection principle

and is essentially a refined version of an instrument described by

Weingart et al. [2] which is based on the original version by Trageser,

Fopiano, and Hardy [3]. A null occurs when the azimuth (fig. 2) of linearly or elliptically

polarized light entering the Faraday cell is perpendicular to the

transmission axis of the analyzer. The waveform of the intensity

If) transmitted by the analyzer and detected by the photomultiplier

depends on the orientation of the analyzer. In the null position, I(t) contains only even harmonics of the 100-c/s fundamental in the

Faraday cell. For positions near null, 1(f) contains the fundamental

component as well. In passing through the null, the amplitude of the

fundamental component passes through zero and changes phase by

180 deg. This behavior around the null position is shown in figure 3. Phase detection of I(t) is accomplished using the circuits shown in

figure 1 providing extremely precise setting of the analyzer to the null

position. With linearly polarized light the analyzer setting to null was repeatable to within ± 1 sec of arc. The precision with which

the analyzer can be crossed with the major axis of elliptically polarized

light depends on the degree of ellipticity of the light entering the

Faraday cell. The precision for various degrees of ellipticity is shown in table 1.

1 Figures in brackets indicate the literature references on p. 117.
ELECTRONIC POLARIMETER TECHNIQUES 115

NULL POSITION

+ 3 mm OF ARC FROM NULL

14 mm OF ARC FROM NULL Figure 3. Photomultiplier current I(t) for different analyzer settings near the null position.

Measurements of static birefringence are quite straightforward.

One method is based on the Senarmont method [4] and another, which

avoids using quarter-wave plates, is described by Robinson [5]. The

measurements described below are based on the Senarmont method.
116 WEINGART AND JOHNSTON

Table 1. Precision of null setting for varying degrees of ellipticity of light entering

the Faraday cell

6, deg Precision of null setting

0 ± 1 sec

2 db 2 sec 3 ± 5 sec

5 ±10 sec

10 ±15 sec

15 ±30 sec 20 ± 3 min

25 ± 5 min

30 ± 6 min

40 ±40 min

The birefringent sample is placed in the optical bench between the

crossed analyzer and polarizer (fig. 1) and is then rotated until the

principal directions of the sample are alined with the polarizer and

analyzer. The sample is properly alined when the polarimeter again

indicates a null. Since birefringence and retardation will depend on

wavelength, all measurements described here are made in monochromatic light, although alinement of the sample can be made in

white light. After rotary alinement, the sample is removed from the

optical path by means of a crossfeed attached to the optical bench,

which does not disturb the alinement. The analyzer is rotated

through an angle of 45 deg, and the polarizer is then recrossed with the

analyzer. A quarter-wave plate then inserted in the optical path

between the sample and the Faraday cell is alined with the polarizer

and analyzer using exactly the same method described for the birefringent sample. The sample is then placed back into position.

Linearly polarized light emerging from the polarizer enters the birefringent sample with the plane of polarization making an angle of 45

deg with the principal directions of the sample. The emerging

elliptically polarized light enters the quarter-wave plate and emerges

linearly polarized, with the plane of polarization now making an angle

of 45 + 6 deg with the principal directions. The retardation 8 of the sample along a given direction is the optical phase shift induced between the

ordinary and extraordinary rays as they traverse the sample in that

direction. The retardation is related to the birefringence of the

sample by 8=2irAnd/\, where An is the difference in indices for the ordinary and extraordinary rays, d is the path length through the material,

and X is the free-space wavelength of the monochromatic light. With

this alinement of quarter-wave plate and birefringent crystal, 6=8/2.

With the sample in place, the analyzer now has to be rotated through an angle 6 to reestablish the null condition described above. This

readout, therefore, is a direct measure of the birefringence of the

sample. It has been shown [2, 6] that first-order errors in either the

alinement or the angular retardation of the quarter-wave plate result

in second-order errors in measuring 6. The alinement error can be

completely eliminated as a result of the sensitivity of the instrument.
ELECTRONIC POLARIMETER TECHNIQUES 117

INPUT FROM

PHOTOMULTIPLIER SAMPLE

AND HOLD OUTPUTTO AMPLIFIER

AND FILTER

PHASE

SHIFTER SAMPLING

PULSE GENERATOR

STEP-UP

; Eq sin oi/ TRANSFORMERAC VOLTAGE

APPLIED TO CRYSTAL

SAMPLE Figure 4. Polarimeter a-c sampling electronics.

The actual retardation of the quarter-wave plates available can be

accurately measured using the methods described in references [2] and

[5], and a correction factor introduced into the measurements if necessary. Measurements of periodically varying birefringence [7] were made

by inserting an electronic sampling network (fig. 4) between the out- put of the photomultiplier and the input to the phase-detection net- work used in the steady-state measurements. The sampling network

provides a signal which is phase detected and permits using all the

techniques available in the steady-state measurements to make

measurements into the rf region. Measurements on samples of ADP

and KDP indicate that precise, reliable measurements can be made to 120 kc/s with the particular circuitry available. Using the head of a sampling oscilloscope in place of the sampling network, precise measurements to 30 Mc/s are foreseen without much difficulty.

References

[1] A. C. Hardy, private communication.

[2] S. J. Williamson, J. M. Weingart, and R. D. Andrews, Jr., J. Opt. Soc. Am.

54, 337 (1964).

[3] A. C. Hardy, P. J. Fopiano, and M. B. Trageser, U.S. Patent No. 2 974 561

(1961).

[4] J. R. Partington, An Advanced Treatise on Physical Chemistry, IV, p. 156, Longman's Green and Co., New York (1953).

[5] C. C. Robinson, J. Opt. Soc. Am. 53, 681 (1963).

[6] R. C. Plumb, J. Opt. Soc. Am. 50, 892 (1960).

[7] Further details are available in the Bimonthly Space Programs Summary

No. 37-22, Vol. IV, p. 33, Jet Propulsion Laboratory, California Institute

of Technology, Pasadena, Calif.

On Determining Optical Constants of Metals

in the Infrared

Shepard Roberts

General Electric Research Laboratory, Schenectady, N.Y. A review is given of several methods for measuring optical properties of metals in the infrared, starting with simple measurements

of reflectivity and leading to a description of a system employing

three polarizing prisms. The discussion is limited to those methods

which are based on intensity ratios observed at predetermined instru- mental settings. The three-prism method is reported to be of comparable precision to the instruments used by Drude in the region of

visible light. 1. Introduction

Optical constants of metals in the infrared are of interest for

several reasons, one of which is that they give information about

the behavior of free electrons, which generally predominate over

other electronic processes in this wavelength range. At any rate,

it is of some importance to know the optical constants in a broad

range of wavelength extending beyond the limits of sensitivity of photographic film or of the human eye, or even of the photomultiplier. In this extended range special measuring techniques are required

The purpose of this paper is to describe several photometric techniques which have been used in the infrared and to point out some

principles according to which high precision may be obtained.

The most effective photometric techniques are those which are based on Drude's method of determining optical constants from

the relative phase and amplitude of the two reflection coefficients,

Rs and Rp , for perpendicular and parallel components, respectively,

of a beam incident at an angle to the metal surface. In other words,

the quantity to be measured is the complex ratio: RJRs =t8in ^eiA

, where tan \p is the ratio of the amplitudes of Rp and Rs and A is their

relative phase difference. . . In order to measure yp and A, Drude used incident light polarized

at an azimuth of 45°. The elliptically polarized light reflected from

the metal was then analyzed by passing it through a compensator

and a second polarizing prism, the latter being called the analyzer.

The values of ^ and A could then be related in a simple manner to the settings of the compensator and analyzer required for extinction.

For infrared work a different method has to be used for determining

the phase and amplitude ratio in the reflected beam. The reason

for this is that the signal intensity from infrared sources is not great

enough, in relation to the minimum level which can be detected, to

119
120 ROBERTS

permit an accurate null setting. For example, if the analyzer is displaced 0.1° from the null position, it will transmit only 3 parts

in a million of the incident intensity. This would be several orders

of magnitude smaller than the noise level in the detector. However, 1 if the compensator is omitted, one can measure relative intensities for different azimuthal settings of the analyzer. In principle, the

shape and orientation of the polarization ellipse may be determined

from the measured intensity ratios. The question to be considered

is how to do this most expeditiously and accurately.

In Drude's method, the final settings of the compensator and

analyzer are arrived at after a certain amount of manipulation

involving successive trials and approximations. It is these final settings, then, which make up the raw data from which other information is calculated. In contrast to this, the work to be described

here is based on an entirely different philosophy of measurement.

It is centered on the idea that the instrumental settings should be

entirely prescribed at the beginning of the experiment and that only

the indicated intensity ratios should appear in the raw data from

which other things are calculated. This principle is emphasized

because it is believed to be a prerequisite for the greatest speed and

accuracy, though it has not been consistently followed by all workers

j in this field. The development of appropriate measuring techniques, J based on intensity ratio measurements, may be traced through a i series of stages in which a number of individuals have made important contributions. The earliest of these did not measure either ; \p or A, nor make use of Drude's formulas.

2. Earlier Experimental Procedures

The method used by Collins and Bock [1]

1 in 1943 is an example

of the application of the principle outlined above. Thtir method

involved a direct measurement of reflectivity of linearly polarized

light at various angles of incidence. To do this a mechanical linkage was employed which moved the test surface out of the way of the

collimated beam and at the same time moved the receiver into a

position to receive the direct beam. The plan of their equipment is shown in figure 1. The polarizing prism which they used is of the

square-ended type and operated in a wavelength range from 0.48 to

2.4 fi. Keflectivities for light polarized parallel to the plane of incidence were measured to an accuracy somewhat better than 1 percent. Optical constants of the metal were obtained from the reflectivities at different angles of incidence by a graphical process. The results were said to be comparable in sensitivity and accuracy with other

methods for certain limited ranges of the values of the constants. The method is better adapted to materials having a relatively low

reflectivity, since reflectivities close to unity are hard to measure to the required accuracy.

1 Figures in brackets indicate the literature references on p. 126.
OPTICAL CONSTANTS IN THE INFRARED 121

Figure 1. Optical system used by Collins and Bock.

Exit slit of monochromator—Si; Polarizer—N; Specimen—MM; Thermopile receiver— T. (Reproduced from [7].) In 1952 Avery [2] described a modification of the method of Collins and Bock which avoided the mechanical motions of the test surface and the receiver during a measurement. Avery simply

noted the ratio of received intensity when the polarizer was rotated

through an angle of 90°. This measurement in effect gave \RP/RS \ 2 =

tan2 but it gave no information whatever about A. Avery made

use of data at different angles of incidence in a manner similar to that used by Collins and Bock and subject to much the same limitations. Avery noticed that it was necessary to make a correction for the

incidental polarization produced by the monochromator and other

optical components. For careful work, this incidental polarization

is not a trivial matter, and it is not so easy to make a correction for

it, because the polarization ratio depends in a complicated manner

both on the wavelength and the slit settings of the monochromator.

In 1955 Beattie and Conn [3] pointed out that the limitation on the

range of optical constants that can be handled by the foregoing

methods is really very serious. A good many metals have high

reflectivity, very close to unity, in the infrared, and cannot be measured adequately by these techniques at convenient angles of incidence.

They concluded that the most suitable parameter to measure, in addition to the ratio of reflectivities, tan 2 ^, was the phase difference A. These are the parameters which Drude measured in the range of

visible light, so that the conclusion of Beattie and Conn really amounts

to a reaffirmation of the effectiveness of Drude's technique.

Beattie [4] worked out the experimental procedure for measuring A. This fact alone is not significant because A had been measured in the

infrared by numerous investigators prior to Beattie. However,
122 ROBERTS

Figure 2. Optical system used by Beattie. Source—N; Chopper—C, Polarizer—P, Specimen—S, Analyzer—.4, Entrance slit of monochromator—St. (Reproduced from [4].) Beattie did make a significant advance in that he showed how to determine both yp and A solely from intensity ratio measurements at predetermined instrumental settings and he showed which settings were best. . . „ To do this Beattie used two polarizing units as indicated m figure

2. The polarizing units shown schematically in the figure are selenium film polarizers of the type described by Elliott, Ambrose,

and Temple [5] in 1948. These units produce only a very small

displacement of the beam and have a high polarizing efficiency in a wavelength range from 2 to 14 /x. With the polarizer set at an azimuth

of 45°, Beattie showed that it was only necessary, in principle, to observe the relative transmitted intensity for three positions of the

analyzer: 0°, 45°, and 90°. He also added a fourth position, 315°, not

because it was necessary but because it served as a useful check.

The equations for transmitted intensity are perfectly symmetrical

with respect to the azimuths of polarizer and analyzer. Hence it does

not matter, in principle, which polarizing unit is left fixed at the

azimuth of 45° and which one is rotated to different positions m

determining intensity ratios. In practice it does matter, however,

because there is more incidental polarization connected with the

monochromator than there is in the source. For this reason Beattie

preferred to rotate the polarizing unit nearest the source, leaving

the other unit fixed. 3. The Three-Prism Method

Any partial polarization of the source will result in errors in Beattie's

system of measurement. For the highest precision it appears preferable to use a perfectly polarized source, since this is easier to control

than a perfectly unpolarized source. For this reason the author [6] introduced a third polarizing prism in work reported in 1959 and 1960.

This addition has a slight effect on the method of taking data and

calculating results as will be shown.
OPTICAL CONSTANTS IN THE INFRARED

PLANE MIRRORS- M|,M3 ,M6 POLARIZING PRISMS - P, ,P2; P

3 SPHERICAL MIRRORS

2,M4l M5 COLLIMATING SLITS - S,, S

3 , S

4 IMAGE SLITS- S 2 , S 5 GLASS WINDOWS - W, ,W 2 Figure 3. Optical system with three prisms.
ROBERTS

The experimental layout is given in figure 3. This equipment was | used in a wavelength range from 0.365 to 2. 65^ and employed prismtype polarizing units. The main polarizing units, Pi and P2 , are

j identical and are arranged symmetrically around the sample. Since

the beam travels in opposite directions in the two units, the calibration is right-handed for one unit and left-handed for the other, relative

to the beam. The third prism, P3 , has just two positions and selects 1 either the parallel or perpendicular polarized component of the source, i One of the main prism rotators is shown in figure 4. The rotator has no graduated scale, but instead eight equally spaced holes, which

are used in conjunction with a locating pin to set the azimuth of the

prism. This construction permits a precision and reproducibility of settings which exceed those of the usual vernier scale and at the same

time permit more rapid shifting of settings. The azimuth calibrations indicate the angle of the electric vector in the transmitted beam

relative to the plane of the instrument.

The measurement consists in observing the ratio of two intensities. These intensities are designated by appropriate symbols in table 1, which also shows the azimuth settings of the polarizers corresponding

to each intensity value. It should be noted that there are 16 possible

combinations when Px and P2 are each set to one of the four positions

45°, 135°, 225°, or 315°. These combinations may be classified as odd or even depending on whether P] and P2 are parallel or at right

angles. All of the odd settings are equivalent insofar as the trans- mitted intensity is concerned and so are the even combinations. The

odd settings result in readings of IZs or I3p , for example. However

equal readings will not be observed for all equivalent positions if the

polarizing prisms are incorrectly oriented in the rotators or if they

allow a small displacement or deviation of the beam. A careful

study of these effects may be applied to working out a routine which

minimizes the systematic errors. Table 1. Polarizer azimuths for intensity readings

Pi Pi P3

Iu 45° 90° 90°

73. 45° 315° 90° "odd"

Ii. - 45° 45° 90° "even"

ii" 45° 0 0

J3p 45° 315° 0 "odd"

u*. 45° 45° 0 "even" Two prerequisites for accurate intensity measurements are a highly

stable source and a stable detector connected to an output indicator

whose readings are accurately proportional to the intensity being

measured. The former is met by a tungsten ribbon-filament lamp

with a stable regulated power supply. The detector is a lead sulfide

cell connected to a stable a-c amplifier of adjustable gain. The

output of the amplifier is converted to d-c by a synchronous rectifier coupled to the light chopper. After filtering, this output is displayed
OPTICAL CONSTANTS IN THE INFRARED 125

by means of a self-balancing potentiometer indicator. This system

gives results which are reproducible to about 0.1 percent of the full scale reading.

The gain of the amplifier may be adjusted so that a reference

intensity, say Ils, will read exactly full scale. Then, to read the

ratio lis/lis directly, it is necessary only to move the polarizer P2 from azimuth 90° to 45°. This ratio is called es

. A complete set of measurements includes the following intensity ratios:

lis „ Izs y- =e s j-=as

J- Is J- Is ^4p_ lip / 1 \ T~— e p T~— (Jv U;

J-2p J-2p

If the beam falling on P3 were perfectly unpolarized, 73s and 74s would be the same as I3p and I4p. In any case these intensities should

be in the same proportion so that:

Consequently,

!f=^ (2)

J- is J-4P

(3) There must be one more identity between the four measured

quantities, because two parameters are sufficient to describe the

reflection ratio. Either of the following identities may be chosen for

this purpose, since they are equivalent.

1 1 =2 (4)

1. (5) (2«,-l)(2g,-l)

(2<r,-l)(2cr.-l)

These identities form a convenient means of checking the self-con- sistency of the data. When some of the ratios are close to 1/2, the

latter identity is a very sensitive check on the linearity of the detector

and output indicator.

The final equations for ^ and A may be written as follows: tanV=—> or tani£=4 (J-+J-) ( 6)

(T P e p 2 \ \ ep V (jpj

and

cos

2 A = ((Ts —es) (<jp—e p). (7) The sign of cos A is the same as that of as —

s , but this method does

not determine the sign of A. This is not a severe handicap, however,
126 ROBERTS

because it is always clear from the experiment what the sign of A

ought to be. In addition, this method cannot detect a partial depolarization of light reflected from the sample. Except for these

drawbacks, which are not generally very serious, the method and

apparatus described here, covering a wide range of wavelength in the

infrared, are believed to be comparable in sensitivity and accuracy to the instruments used by Drude in the range of visible light. For ex- ample, a change in A as small as that attributable to a single molecular

layer of water on a copper surface can be detected easily at 1 n wavelength.

4. References

[1] J. R. Collins and R. O. Bock, Rev. Sci. Instr. 14, 135 (1943).

[2] D. G. Avery, Proe. Phys. Soc. B65, 425 (1952).

[3] J. R. Beattie and G. K. T. Conn, Phil. Mag. 46, 222 (1955).

[4] J. R. Beattie, Phil. Mag. 46, 235 (1955).

[5] A. Elliott, E. J. Ambrose, and R. Temple, J. Opt. Soc. Am. 38, 212 (1948).

[6] S. Roberts, Phys. Rev. 114, 104 (1959); ibid. 118, 1509 (1960).

[7] R. O. Bock, Phys. Rev. 68, 210 (1945). Discussion

C. E. Jones, Jr. (Texas Instruments)

What type of prisms were you using in your polarizer that allowed

you to make measurements at 2.5 ju?

S. Roberts:

The conventional Glan-Thompson prism will work up to and

beyond this limit ; however, our prisms were somewhat unconventional.

If I may, I will describe the prisms used in our work and, also, a second design which we have tested and found to be superior. yr- FLUORITE

VISUAL

AXIS

CALCITE- ^CRYSTAL AXIS Figure 1. Beam-splitting prism (air spaced).

VISUAL AXIS

CRYSTAL AXIS -CALCITE Figure 2. Double Glan prism (air spaced).
OPTICAL CONSTANTS IN THE INFRARED 127

We used beam splitting prisms having a central element of calcite between two fhiorite elements as shown in figure 1. These are air spaced. The optic axis of the calcite is perpendicular to the visual

axis as shown. The ordinary ray is refracted about 8° in this prism

and misses the apertures which follow. The E'-ray goes almost

straight through, being deviated by less than ±2}{ min in a wavelength range from 0.5 to 2.15 m- Even this small deviation is a nuisance, however, in that it makes it difficult to reproduce readings

when the polarizer is rotated 180°. The second design, developed jointly with D. T. F. Marple early

in 1960, might be called a double Glan prism and is shown in figure 2. This consists of three air-spaced calcite elements with optic axis

perpendicular to the visual axis as shown. The operation of this

prism is similar to the Glan prism except that it has about twice the

angular field of view and it has better transmission characteristics.

However, it requires only about half as much calcite as a GlanThompson prism. The choice of the angle 6 depends on the wavelength range to be covered. Our prisms were designed for operation

to a maximum of 0.8 n with 0=37°25'. These prisms actually gave

good results up to 2.2 n, thereby surpassing the calculated limit. In a prism 25 mm square the transmission of the E-tblj was 89 percent

at 7000 A and 38 percent at 2200 A. These prisms were used in work

reported by D. T. F. Marple and H. Ehrenreich in Phys. Rev. Letters 8,87 (1962).

I

4. Applications

Use of Ellipsometry for in Situ Studies of the

Oxidation of Metal Surfaces Immersed in

Aqueous Solutions

Jerome Kruger

National Bureau of Standards, Washington, D.C. 20234

Ellipsometry enables one to study in situ the formation, dissolution,

and optical properties of oxide films on metal surfaces immersed in aqueous solutions while simultaneously measuring or controlling other system parameters, e.g., electrochemical ones. Experimental aspects such as ultra-high-vacuum or electrochemical techniques of surface preparation, effect of changes in the optical properties of the solution and the film, the effect of surface roughness, and the effect of light are

discussed. The application of ellipsometry to studies at the metalsolution interface is illustrated by examples from studies of the passivity

of iron and the oxidation of copper in water.

1. Introduction

Virtually all processes occurring on metal surfaces immersed in solutions involve at some stage the formation of a very thin film. This

film, in the majority of cases an oxide or hydroxide, usually has a

vital influence on the course of the process studies. Its existence

may be fleeting or its stability tenuous except under the very special

conditions existing in solution at the time it is being studied.

Thus in order to study phenomena occurring on metal surfaces

immersed in solution one needs to be able to measure these thin films

in situ while they are forming or disappearing without disturbing the

system. Ellipsometry answers these requirements beautifully. This was appreciated in a most effective manner by Tronstad who over 30

years ago carried out a number of pioneering studies on the films that

form on metal surfaces immersed in inhibitor solutions or while anodically or cathodically polarized [1,2].

1 It was Tronstad's application

of the polarimetric technique of ellipsometry that first yielded meaningful results and pointed the way for further exploitation of this powerful

tool. Yet in spite of Tronstad's signposts showing the way and m

spite of the great need for a tool to study the initial stages m corrosion

processes and electrode kinetics, almost a third of a century passed

before ellipsometry was applied to immersed surface studies. A few

studies are now appearing in the literature, e.g., the work of Barrett

and Winterbottom [3], Kruger [4,5], Hayfield [6], and Andreeva [7] on oxide film formation, of Stromberg et al. [8] on polymer adsorption,

and of Reddy et al. [9] on electrochemical studies.

Since the theoretical principles underlying the method are well

established, and the ellipsometric technique itself has also been admirably covered elsewhere by Winterbottom [10], the scope of this

1 Figures in brackets indicate the literature references on p. 150. 131
132 KRUGER

I presentation will be limited to a discussion of the experimental aspects

of studying oxide film formation and dissolution with ellipsometry.

j Further, this discussion will be concerned entirely with the films that

form on metal surfaces immersed in aqueous solutions, with most of

its attention directed at films thinner than 100 A, the films that play

a very decisive role in corrosion phenomena. For such systems the

difficulties associated in achieving meaningful results lie not so much

in some of the errors introduced by shortcomings inherent in ellipsometric technique as they do in the problems associated with all surface

studies—those of working with a clean, well-characterized surface

and solution. In other words, when working with oxide films of less 1 than 100 A thickness, as will be shown in what follows, the errors j introduced by not knowing the optical constants of the film or metal

substrate precisely, or by indeed not being able to measure A, the

relative phase retardation, or \p, relative amplitude reduction to 0.01°,

have much less importance than the introduction of a part per million

of impurity. For thicker oxide films, however, where the effects of strains begin to affect A and ^ appreciably [11], or for solvent-swollen

polymeric films whose optical constants are very close to that of the

surrounding solution [8], purely ellipsometric considerations become

all-important.

Hence the experimental questions of how to carry out an ellipsometric study of oxide films on immersed metal surfaces will be discussed,

mentioning some of the pitfalls, giving techniques we have found

useful, and describing examples from our own work which will show

some of the information one obtains using ellipsometry. It is probably

these experimental considerations that have inhibited others for over

30 years from following Tronstad's brilliant beginnings.

2. Method

This section will be concerned with the apparatus, experimental and

computational techniques, and complicating factors. The latter,

although they generally limit ellipsometric studies, can sometimes

provide valuable information. Four kinds of information can be

obtained from studies of metal surfaces immersed in aqueous solutions:

1. Growth of oxide films, i.e., kinetics of growth from thickness

measurements.

2. Film breakdown or dissolution.

3. Film formation or dissolution while electrochemically polarizing

the surface.

4. Film properties.

The section following this one will give examples of experiments that were concerned with obtaining one or more of these.

2.1. Ellipsometer

No discussion will be given here of the ellipsometer itself, as this is amply and ably covered elsewhere [10]. It will suffice to mention
OXIDATION IN AQUEOUS SOLUTIONS 133

that, in studying processes involving rapid film formation, the photometer method of measuring extinction point offers the possibility of using fast recording techniques. This can prove greatly advantageous,

as the work of Reddy et al. [9] has shown. It should also be mentioned

that an ellipsometer that can conveniently use a monochromator as

its light source may be of considerable help in studying the optical

properties of the thin films formed on a metal. When a number of different wavelengths of light are employed, it is not necessary to have

a different quarter-wave plate compensator for each wavelength.

This was pointed out by Archer [12], who also showed how to take

this into account in calculating A and \f/ values from polarizer and

analyzer readings.

2.2. Experimental Cells and Surface Preparation

The types of experimental cells that one can use with an ellipsometer

to study immersed surfaces are, of course, limited only by the system

under study and the imagination of the experimenter. There are, however, a few general requirements for carrying out ellipsometric

measurements on immersed metals that, though they may be obvious,

should be listed as follows

1. Entrance and exit windows set at the proper angle of incidence.

If a second angle of incidence is desired, the advantages of which are pointed out by Hayfield [6], a third window is necessary.

2. Means of adjusting specimen surface position. This is not

always necessary if the experimental cell is so constructed that the

specimen is always positioned accurately. Also, if one is concerned

with relative changes that occur on the specimen surface rather than

absolute values, as is the case frequently in kinetic studies, precise

positioning becomes less important. Sometimes alinement can be

achieved by adjusting the position of the ellipsometer rather than that

of the specimen.

3. A smoothly polished, strain-free specimen surface.

4. Means for making measurements on a bare specimen surface

prior to the introduction of the solution, or a way of producing a filmfree surface while the specimen is in the solution. This requirement

is especially important since in many studies it is the change in A from

the bare value A that is of prime interest.

5. Means for introducing purified solution.

6. Means, if desired, for making non-ellipsometric measurements,

e.g., electrochemical ones, while simultaneously using the ellipsometer.

Three examples of experimental cells and the techniques in using

them that more or less conform to the requirements listed above will now be given. They each handle the requirements in a different manner. The first is shown in figure 1. This apparatus can be used to study film formation on four different crystal orientations simultaneously, illustrating a very important advantage that ellipsometry

has in following the kinetics of film growth on single crystal surfaces.
134 KRUGER

PLAN VIEW

AT SECTION A—

Figure 1. Apparatus for studying by ellipsometry the for- mation of films on a single crystal immersed in solution. (Reproduced from [4].) The ellipsometer allows one to select areas such as different crystal

planes on a specimen to study whereas in using, for example, a microbalance one has to study the whole specimen.

In the study in which this apparatus was used [4], the copper

single crystal was machined as a cylinder with its axis in the <110>

direction. Using a crystal of this orientation, it was then possible

to cut flat surfaces parallel to the {ill}, {110}, {100}, and {311} crys- tallographic planes, these planes being parallel also to the cylinder's

axis as shown in figure 1. With such a crystal sealed into this apparatus, the various crystallographic planes can be rotated into

position by means of a magnet so that measurements can be made

on them with the ellipsometer at various times during the oxidation

process. The method of preparation to be used to render the crystal surface

smooth and strain-free and the manner in which high-purity water

or solution can be introduced into the apparatus containing no

lubricated joints are described elsewhere [13]. The optical parameters of the film-free copper surface are obtained by measurements

on the surface after it has been annealed in purified hydrogen at 500

°C. The values for the film-free copper surface in water can be
OXIDATION IN AQUEOUS SOLUTIONS 135

Figure 2. Ultra-high-vacuum apparatus for studying the optical properties of and the electrochemical potential of a

metal surface immersed in a solution. (Reproduced from [5].) 25 | I

i

20

15

8.0XI0

5 T0RR f 2.5 XIO

5

,_^aox io6

10 ^^f^.Q)* io6

5

0 —o ^5X1^

i i i 0 10 20 30 40 50 60 70 80 90

TlME.min Figure 3. Change of oxide film thickness with time at different oxygen pressures at 300 °K.

(Reproduced from [14]) calculated from these values. Once these values are obtained, the

solution is introduced and the kinetics of film growth are followed.

Such an experimental arrangement as that just described is quite

adequate unless one is concerned with the very beginning stages

of film formation, where it is necessary to start with a surface that

is as bare as is possible using modern ultra-high-vacuum techniques.

The apparatus shown in figure 2 was constructed to meet such a requirement in a study [5] whose aim was to determine whether an

adsorbed film of oxygen or an oxide film was responsible for the

passivity of iron. Figure 3, taken from some recent work of Kruger

and Yolken [14], an ellipsometric study of the early stages on the
136 KRUGER

oxidation of iron, shows the film growth that occurs on surfaces pre- pared in an ultra-high vacuo when they are placed in ordinary vacua. This then emphasizes the necessity for going to ultra-high-vacuum

techniques to prepare a metal surface prior to immersion when one

1 wishes to see if a monolayer or two of oxgyen is responsible for a given

phenomenon. The apparatus shown in figure 2 is a typical ultra- high-vacuum apparatus capable of attaining pressures of less than j 10"9 torr. The reaction cell is provided with optically flat 3-mm-thick

quartz windows set at a chosen angle of incidence, usually between 60°

and 70°. During bakeout the ellipsometer has to be moved, but by

putting guide marks on the table upon which it rests it is possible to move it back exactly into its previous position. The specimen is screwed into place in the reaction cell and the bottom of the cell sealed

with a hydrogen-oxygen torch. After outgassing in vacuo by induction heating at 800 °C, 85X10-3 torr, hydrogen is introduced at the same temperature. At this pressure a glow discharge is obtained in

the field of the induction coil. The glow discharge is maintained for a 3-min duration, followed by a pumping-out of the hydrogen. After

the system has been baked at 430 °C for 6 to 8 hr, this treatment

is repeated a number of times, introducing fresh hydrogen after each

3-min glow discharge. Such a procedure, which bombards the

surface with hydrogen atoms and ions, was found to be very effective i in reducing the initially present oxide film, as judged by measure- I ments made with the ellipsometer. These are made after each

exposure to hydrogen ; the values of A and yp will change in a direction that indicates that the surface is becoming cleaner. When

further hydrogen treatment produces no changes in the optical

parameters, another bakeout is carried out with no change in parameters noted. This will indicate a very low partial pressure of oxygen in the residual gases present during this second bakeout.

These readings are taken as the values for the bare surface or, more

probably, a surface having a monolayer or less of adsorbed gas. Evaporated metal surfaces can also be used in such an ultra-high-vacuo

apparatus if the limitations on their cleanliness mentioned by Hickmott and Ehrlich [15] are borne in mind.

The solutions are prepared so that their dissolved gas content can

be controlled. The solution in the bulb shown above break-seal C 1 in figure 2 is degassed in another vacuum system by alternate freezing

and thawing a number of times in vacuo and then introducing other

assayed reagent grade gases of interest. This bulb, in which the

solution is in equilibrium with about 0.5 atm of gas, is then sealed off from the solution preparation system and sealed onto the ultra-high- vacuum system above break-seal C. The space between A and C is then pumped out by a zeolite sorption pump at B, the pump then

being removed by sealing off at B. A run is started by breaking

break-seal A and then C, the solution covering the specimen in less than 15 sec. Readings can then be made with the ellipsometer.

Measurements of the potential between the specimen surface and a platinum electrode provided in the apparatus can also be made if desired.
OXIDATION IN AQUEOUS SOLUTIONS 137

SCHEMATIC OF APPARATUS FOR OPTICAL STUDIES OF ANODIC FILMS

Figure 4. Cell for studying an electrochemically polarized

surface by ellipsometry.

Another approach to starting with a bare metal surface, which

allows one also to control electrochemical parameters while measuring

film optical properties and growth, can be carried out by using apparatus similar to that shown in figure 4. This apparatus consists of a means for observing the specimen surface with the ellipsometer, a

reference electrode for measuring its potential, and a counter electrode

for polarizing the surface at selected potentials by the potentiostat on

the right or at selected currents shown on the left in simplified form.

With this apparatus one can cathodically or anodically polarize the

specimen surface being studied ellipsometrically and control the

potential with a potentiostat or control the current galvanostatically.

A description of these electrochemical techniques is given by Schuldi- ner [16]. It is possible to remove any existing oxide films on a number

of metals by first making the surface cathodic and thus reducing this

oxide film. When the film is so reduced, optical readings can be

taken and the surface can then be rapidly brought by the potentiostat

to an anodic potential where anodic oxide films start to form. There

are a number of difficulties associated with this technique. First

when the surface oxide has been cathodically reduced, in a number of

cases, hydrogen evolution takes place and the gas interferes with the

optical measurements. Secondly, ions adsorbed from solution,

cation concentration gradients, and the electrical double layer may

influence the values obtained for bare metal. Finally, metal dissolution may occur in some low pH solutions before the surface is polarized

to the proper value. If this occurs while film growth is going on, as it does in some cases, errors in film-thickness measurements are introduced. Besides selecting systems where these effects are minimized,

it is possible to obtain a good indication of the bare metal value by

cathodically reducing the existing oxide films slowly while con-
138 KRUGER

I

l tinuously taking ellipsometer readings before hydrogen evolution and

extrapolating these optical readings to the potential where the metal

would be bare. Such a procedure is possible for some metals, e.g., iron in solutions of neutral or alkaline pH.

Obviously, measurements of the optical constants of the bare

surface made, if possible, under the ultra-high-vacuum conditions

described previously would be the most satisfactory approach. As

will be shown in the applications section, very useful information can

be obtained, in spite of the limitations mentioned, just from the I changes in A and \p observed when electrochemical changes are effected.

2.3. Computation

A number of excellent descriptions of computational methods exist

[10, 17, 18]. (See also the paper by McCrackin and Colson in this

volume.) This section will be concerned briefly with the aspects of computation associated with the thin films formed on metal surfaces

in solution. If one can obtain experimentally a number of A and \p values for different thicknesses of the film on the metal surface and

these values are sensitive to changes in the film's complex refractive

index n2 (n2 =N2 (l —iK2 )), then the conventional technique of comparing the experimental curve with one calculated for different values

of n2 by a computer until a good fit is obtained is the recommended

approach. There are difficulties in this procedure when K2 5*0, as is the case for most oxides (A1203 is an example of an exception), because

there is no unique set of N2 and K2 values that can be obtained by this procedure. In other words, good fits may be obtained for a number

of n2 values. Hayfield [6] suggests that one can determine a unique

Nt by measuring A and \p values at two different angles of incidence

33

31 30

142 140 A 138 136 134 132 130

10 20 30 40 ^0

THICKNESS, A Figure 5. Variation of A and \p with thickness for an n2 whose N varies from 2 to 3 and whose NK varies from 0.1

to 0.9 for iron immersed in a solution whose refractive index is 1.33.

(0—60°, X—5461A, m—3.06-3.67i)
OXIDATION IN AQUEOUS SOLUTIONS 139

thereby overcoming this difficulty. Once this is done K2 can be

determined from the curve fitting technique.

The main factor that should be pointed out here is in connection

with the determination of the film thickness of films less than 50 A

thick. Figure 5 shows the sensitivity of the thickness measurement

to refractive index. This can be diminished as long as one has a rough idea of the value of n 2 . The bulk value provides a very useful

starting point. In this range of film thicknesses A is the sensitive

parameter so that in studying a system that is changing rapidly only

the polarizer readings (from which A is obtained) need be followed.

Further, the calculation using the exact theory need not be used but

instead the simple Drude approximation [1] becomes adequate. For

many studies of the phenomenon occurring on immersed metal surfaces, such thin films « 50 A) play an all-important role in controlling

the process.

2.4. Complicating Factors

A description has been given of the technique of the use of elliptically polarized light in the study of film growth. Some of the complicating factors that must be borne in mind by anyone using the

technique should now be pointed out. This is done for two reasons:

(1) To show some of the limitations of the method and (2) to point

out that some of these complications actually give information that

adds to its power. The first group of items are of lesser importance

and are the difficulties connected with the technique.

a. Rough Metal Surface

This can arise from difficulty in properly polishing a surface or from dissolution taking place in the solution in which the metal is immersed. Drude and later Tronstad [1] have found as long as the

roughness is small as compared to the wavelength of the light its effect is negligible, especially on A. If it is larger it can have a marked effect on the readings.

b. Strains in Viewing Windows

These can affect the readings drastically and hence precautions to minimize thermal strains such as using quartz windows or avoiding uneven heating during bakeout are necessary. Strain introduced

when readings are taken in vacuo is best eliminated by using thick

windows or obtaining bare metal readings in a H2 or inert gas

atmosphere.

c. Change of the Refractive Index of the Solution

This can occur if the process under study introduces ions continuously into the solution or some sort of decomposition of the

solution is taking place. One good way to take care of this is to introduce a gold mirror into the solution from which readings can be made

from time to time. These readings then allow one to compute the

728-101 0—64 10
140 KRUGER

changes in n

2 , if they occur, assuming that nothing is happening to the gold.

d. Difficulty in Using Constant-Temperature Baths

Two difficulties exist here: (1) There is not enough room in commercially available ellipsometers to provide for the use of a constant temperature bath. (2) Windows have to be provided in the

bath and stirring introduces alinement problems and variations in

light intensity. This latter difficulty can perhaps be overcome by

using an air thermostat.

e. Absorption by the Solution In this case the refractive index of the solution is complex complicating the calculations. This is best overcome by choosing aodifferent

wavelength for the light source than the conventional 5461 A Hg line when it is absorbed by the solution.

f . Influence of Light on Process Being Studied

Figure 6, from some work by Kruger and Calvert [19], shows how

light can affect the oxidation of copper in water. The light used was

0

I04 X 400

1 1 1

290 ^xw/cm 2 (100) O 190 " DARK

1

(a)

I 1

A 370 /xw/cm 2 O 290 " • 190 » » A DARK

I — a—o

(in) —zr (b)

i 100 200 300

TIME, MINUTES Figure 6. The effect of light intensity on rate of film growth on copper immersed in water in equilibrium with an inert atmosphere containing 1% O2.

(Reproduced from [19].)
OXIDATION IN AQUEOUS SOLUTIONS 141

white radiation of different intensities. Because of this effect of

light on some oxidation processes in solution, Cu, Fe, Zn being some

of the metals affected, one must bear this effect in mind in studying

oxidation using light. Thus it is necessary to expose the specimen

! to the light from the ellipsometer as little as possible (shielding it when a measurement is not being made). If this is done the effect

is negligible.

The next group of items arises because of the oxidation process

itself. In this case the ellipsometric technique becomes less precise in that one cannot always determine film thickness or optical constants precisely, but on the other hand it reveals events happening

at the metal surface and in the film just as they occur—an extremely

valuable aspect of the technique.

g. Two Phase Films

When a fiJm consisting of two components with different refractive

indices forms on a metal surface in solution, the situation becomes

considerably more complicated than that for the one-component case. This is so because three new unknown parameters are introduced:

the thickness of the new phase, its refractive index, and absorption

coefficient. However, the very fact that the ellipsometer reveals

the formation of a second phase at the time it happens is of great value,

as will be shown in the section on applications.

h. Nonuniformity of the Films

By this is meant variations in oxide thickness over the metal surface. Winterbottom [10] points out that what one actually measures

according to theory is the average amount of matter in the film

equivalent to that of a thin uniform film of the measured thickness.

However, the refractive index measured is probably made up of an

average index that includes empty spaces between oxide nuclei.

i. Anisotropy of Film and/or Substrate

Here there is a variation in the refractive index with direction in the plane of the surface studied. This serves to introduce another

variable, the orientation of the specimen with respect to the light beam,

and the different refractive indices for the different crystal directions.

Winterbottom [10] has treated this theoretically, but all of the extra parameters make the task of computation formidable indeed. Cathcart [11] has recently shown that a normally optically isotropic

oxide can become optically anisotropic when strain is present in the

oxide for epitaxial reasons. Thus a study of the anisotropy of the

oxide film may throw some light on the important question of stram

in the oxide film.
142 KRUGER

3. Applications

In this section examples are given from four different studies which

illustrate in a specific way the use of the techniques described earlier to obtain varied types of information about the films that form on immersed surfaces.

This is the most frequent and obvious application of ellipsometry.

Here the ellipsometer is used to measure the change of film thickness

with time. It can, as is shown in figure 7, relate the kinetics of growth

to another simultaneously measured parameter (e.g., potential). The

results in figure 7 obtained for iron immersed in 0.1 N NaN02 solution [5] are plotted logarithmically showing that the film growth

kinetics follow a logarithmic law. Similarly, film dissolution could

be followed.

Using the apparatus shown in figure 1, it was found in a study of the oxidation of copper single crystal surfaces [4] that when the copper was oxidized in pure water in equilibrium with 1 atm of oxygen that

very rapid oxidation took place initially, followed by a leveling off at 65 to 95 A (fig. 8). The {111} face exhibited a smaller limiting film

thickness than the other three faces studied. During the formation

of these initial films the value of A decreased with_time until the limi ing film thickness was reached. The fact that A-A (with an increase

in film thickness) was negative is what one would expect when cuprous

oxide forms on copper immersed in water.

3.1. Kinetics of Film Growth or Dissolution

3.2. The Formation of a Second Phase

V)

- 0.4| o

30

[ - 0.2 I| <

(110) RON SURFACE

IN 0.1 M NAN02 S o

a. Q_ 20

x

=! io -

10 100 1000

(t+ l) L MINUTES

10000 Figtjee 7. Growth of a passive film on a (110) iron surface

in an air-saturated 0.1 N NaN02 solution.

Logarithmic plot . (Reproduced from [5].)
OXIDATION IN AQUEOUS SOLUTIONS 143

However, the value of A reached a minimum after from 90 to 120

min and then began to increase. This is shown in figure 9, where A

is plotted versus time. It can be seen that the greatest change in

A occurred_for the {100} and the {111} faces. This change in the

i sign of A-A was probably associated with the formation of a new

component or layer in the oxide film. X-ray diffraction studies made

I after removal of the crystal from the solution bear this out by showing

the presence of CuO. Also if one uses a reasonable approximation

TIME -MINUTES Figure 8. Growth of oxide film on copper in unstirred water in equilibrium with 1 aim of 02 . (Reproduced from [4].)

1

i

"50 iOO 150 200 25(3 300 TIME IN MINUTES Figure 9. Change in relative phase retardation for film growing on copper in unstirred water in equilibrium with

1 atm of 02. (Reproduced from [4].)
144 KRUGER

for the refractive index for CuO (2.7-9$ based on the few bulk values

available in the literature [20] such a behavior can be reproduced by

computer calculations using a program for double films. The reasonableness of this assumption, i.e., that a new component starts to form,

is also made clear by considering figure 10. In this figure are plotted

the two optical parameters, A and tan \f/, determined by the ellipsometer for films of different thicknesses both for the all-Cu20 film, and

for the two-layer film found experimentally. If the Cu20 film had

continued to thicken, the optical parameters would have moved to the right on the solid curve. Instead, after a film of 65 to 95 A had

formed on the copper surface after 90 to 120 min, the parameters

suddenly followed the dashed curve. It is also known from previous

work on the system that the introduction of C02 will convert the

CuO to Cu20. When this was done, after the parameters measured

were those found on the two component curve, subsequent measure- ments of A and tan \J/ always lay on the solid curve that was calculated for a film made up of Cu20 only. If the sudden change in optical parameters was associated with changes in structure or the intro- duction of strain, as is sometimes the case, it is highly unlikely that

introduction of C02 would have brought the values of the optical

parameters back to the curve calculated for Cu20. _Hence it seems

reasonable to say that the change in the sign of A-A from negative

to positive was associated with the formation of CuO.

Figure 10. A comparison between the polar plot of relative phase retardation, A, and relative amplitude reduction, tan \//, calculated for a CU2O film growing on copper and

obtained experimentally for a film starting out as C112O and

then starting to form a second component.

Cu2 0 FILM

(Reproduced from [4].)
T OXIDATION IN AQUEOUS SOLUTIONS 145

One observation made when a crystal was removed after being

] immersed for 18 hr should be mentioned. The CuO formed was

j loosely held and could be partially wiped off. Beneath it a film was

; observed showing interference colors, presumably Cu2 0. These

interference colors indicated thicknesses greater than the limiting

I thickness of the Cu2 0, 65 to 95 A, observed before the formation of

1 CuO. Thus it appeared that, as the CuO formed, the Cu20

layer underneath was also increasing in thickness. Therefore, since

both CuO and Cu20 were growing simultaneously and at different

j rates, the values of A plotted in figure 9 were related in a complicated manner to the thickness of the combined film. Hence one cannot say

I from looking at this plot that the {100} and {111} planes have thicker

films on them than the {311} and {110}.

The value of the observations is that for the particular situation

described here ellipsometry could detect the formation of second phase

j in the oxide film forming on the copper—a valuable piece of information even though this complication of a second phase makes determina-

| tion of film thickness most difficult.

3.3. Distinguishing Between Adsorbed Oxygen and an Oxide

Film

As pointed out earlier in describing the ultra-high-vacuum apparatus

in figure 2 there exist two theories of the origin of the passivity of iron. Experiments using this apparatus and the technique described earlier

I [5] were aimed at fulfilling the need for a positive way for distinguishing between the adsorption and solid film theories of passivation.

By using ellipsometry it is possible to determine whether the passivation process involves the adsorption of an oxygen film, the growth of a three-dimensional oxide, or a combination of both, a transformation

of an adsorbed film into an oxide. The theoretically derived variation

| of A with thickness for each of these three models is shown in figure 1 1

When the iron surface was immersed in an inhibitor solution

(0.1 NaN02 ) saturated with air the A at the beginning of the

passivation process increased rather than decreased with time. Such

initial increases in A were found occasionally by Tronstad and

Borgmann [21] for films formed in concentrated HN03 . This is shown in figure 12, along with a simultaneous plot of the potential of the iron surface versus platinum for an air-saturated 0.1 A^ NaN02 solution.

Because the growth of iron oxide films is accompanied by a decrease

in A, the increase found indicates that at the beginning of the passivation process something else was occurring. Figure 11(a) points to one

possible reason for the increases. It implies that a rise in A could

indicate that a film is forming with a refractive index, n2 , less than

1.3349, the refractive index of 0.1 N NaN02 solution, nu for the

wavelength of the light used in all of the studies reported here, 5461 A. None of the oxides of iron whose refractive indices are known or indeed

any of the possible solids whose values are known, which could

constitute the film, have a refractive index lower than that of the
146 KRUGER

Figure 11. Theoretically derived curves for the variation of

the phase retardation with film thickness for: (A) a film whose refractive index is less than the solution (n2 <C^i); (B) a film whos refractive index is greater than the solution

(n2 <^ni) (C) a film where initially n2 <^i, but later n2 becomes greater than n\. (Reproduced from [5].)

1 0.5 I1301 | i | l_

0 100 200 300 400 TlME.min Figure 12. Change in phase retardation with time for an

iron surface immersed in air-saturated 0.1 N NaN02 solution. (Reproduced from [5].) nitrite solution. On the other hand, an oxygen film [22] or an adsorbed

nitrite film would probably have a refractive index lower than that

of the solution.

This interpretation does not apply in this case, however, because

although a rise in A-values was initially observed, A was always less than A, the value of the relative phase retardation for the film-free surface. This quantity (shown in fig. 12) was obtained from a

calculation of its value in nitrite solution based on the all-important

measurements made in ultra-high vacuo. During the time when

the solution was introduced into the cell containing the specimen, and
OXIDATION IN AQUEOUS SOLUTIONS

a reading was made (in some cases less than 0.5 min), A had decreased

4°. This corresponds roughly to the formation of a 20 A film,2 whose

refractive index was higher than that of the solution, during the

! time of solution introduction. Thus the rise in A-values was due

1 either to the adsorption of a low index film (n2 <nx ) on top of the

film formed almost instantaneously at the beginning of the process,

or to the dissolution of this initial film, where decreasing film thickness

values cause an increase in A-values. The fact that Amax always

I approached A closely, but never exceeded it, argues for the film dis- solution interpretation. This interpretation is also bolstered by

thermodynamic considerations [23] for an air-saturated solution

I (pH 5.8).

In order to see if initial film dissolution was occurring because of the low pH of the air-saturated solution due to the C02 present in

it, outgassed solutions in equilibrium with 0.5 atm of high-purity

oxygen were used. As can be seen from figure 13, the initial rise in A

is completely absent, the instantaneously formed film instead continuing to increase (A decreasing) with time. Thus it appears that the

initial rise in A for air-saturated solutions was due to film dissolution

rather than to the adsorption of a low-index film on top of the in- stantaneously formed film. The condition for the latter possibility

still existed when oxygen was the only gas present. This experiment

2 These were based on the approximation, used by Tronstad and Borgmann [21], for films less than 100 A

that the mean thickness isproportional to the difference in the phase

o retardation values of the " film-free" and film-covered surface (A-A), 1° corresponding approximately to 5 A.

130 1

I

1

1 0 IOO 200 300

TIME, MINUTES Figure 13. Change in phase retardation with time for an

iron surface immersed in 0.1 N NaN02 solution in equi- librium with 0.5 atm of pure oxygen.

(Reproduced from [5].)
148 KRTJGER

0..5

0.4 -

0.3

0

-

<3 140

0 100 200

TIME, MINUTES Figure 14. Change in phase retardation with time for an

iron surface immersed in a deaerated 0.1 N NaNC>2

solution. (Reproduced from [5].) with the pure oxygen also casts doubt on the very remote possibility

that the initial rise in A was due to the formation of a second film with

a very high adsorption coefficient (a situation analogous to that

described earlier in the oxidation of copper). This is so because it is not likely that raising the pH by using pure oxygen would have done

anything but enhance the formation of the second phase layer, and if this had the proper optical constants, a rise in A would have again

been noted.

To ascertain whether the instantaneously formed film was produced

during the solution introduction interval by interaction with oxygen

and water vapor, experiments were carried out using de-aerated solutions and high-purity argon as the gas for pushing the solution in rapidly. As figure 14 shows, an extremely thin film (<5 A) formed

in the almost completely deoxygenated solution. An initial adsorption on top of or dissolution of this film appeared to take place; it then started to grow.

Thus by the use of ellipsometry, ultra-high-vacuum techniques, and

a control of the atmosphere in equilibrium with the solution covering

the metal surface studied, it is possible in some cases to distinguish

between adsorbed and three dimensional oxide films.

3.4. Study of Electrochemically Polarized Surfaces

The same arguments used to distinguish between an adsorbed

oxygen film and a three-dimensional film can be used in studies

carried out on polarized surfaces using the apparatus shown in figure 4
OXIDATION IN AQUEOUS SOLUTIONS 149

COMPARISON BETWEEN OPTICAL AND ELECTROCHEMICAL FILM THICKNESS MEASUREMENTS IN PASSIVE REGION OF ANODIC POLARIZATION CURVE

POTENTIAL (S.C.E.), VOLTS Figure 15. Anodic polarization curve for iron in 0.15 N sodium borate-0.15 N boric acid solution with a comparison between film thicknesses obtained by coulometric measurements {from Nagayama and Cohen [22]) and those obtained by ellipsometry.

and described in section 2, Method. It is very important wherever

possible to supplement ellipsometric results with those obtained from

other techniques. By combining ellipsometry with an apparatus

capable of polarizing a metal surface it is possible to intercompare the

optical with electrochemical measurements of film thickness. This is done in figure 15 on a typical anodic polarization curve obtained

from the work of Nagayama and Cohen [24] for iron in a boric acidsodium borate solution. In the passive region of this curve are shown

values of film thickness measured at three different potentials by the

ellipsometer and by measurement of total coulombs passed to form

the film made by Nagayama and Cohen. The first value listed for the ellipsometer was taken as zero thickness, since good A and yp values

could not be obtained from cathodically reduced surfaces, as mentioned

in the previous section, because of the presence of hydrogen bubbles.

At this point on the polarization curve this assumption of a very thin

film (thickness «0) is reasonable. The electrochemical measure- ments indicate a thickness of <10 A. Using such an assumption, the

values obtained optically and electrochemically compare quite well.

If the film thickness was not zero at —0.6 V, the optical values would

then be somewhat higher than those obtained by coulombic

measurement.

Such an optical-electrochemical approach was applied to a system

where the nature of the film formed in the passive region of the anodic

polarization curve is a subject of controversy. This system is iron in

1 M H2S04 . When the iron is polarized to a potential of around 1 V and kept at this potential by a potentiostat, the flow of current is markedly reduced and the iron is said to be passive even though dis-
150 KRUGER

112

CO

UJ

LU

eno

LUQ III zfo

<tQ

<Z<

LU

q:LU HO

CO<

X

CLJ

LU

or 109 0 I 1.2 1.4 1.6 POTENTIAL ( S.C.E.), VOLTS Figure 16. Variation of A with potential in the passive

region of the anodic polarization curve of iron in 1 M

H2 S04 . solution continues, albeit at a diminished rate. Figure 16 shows how

the argument applied in the case of inhibitor formed films may be

applied to this system. It is known that film thickness increases

with increasing potential in the passive region of the polarization curve

[25]. Figure 16 shows that with this increase in potential and film thickness a decrease in A is obtained. On the basis of what was said

before this indicates the formation of a film of greater refractive index

than the solution, thus arguing against the formation of an adsorbed

oxygen film. Further work has to be done in this difficult system

(surface roughening occurs if the experiment is carried out for times

greater than a few minutes) to establish the optical constant of the

film and its thickness. Similar electrochemical-ellipsometry experiments have been carried out by Hayfield [6]. These examples of applications of ellipsometry to the study of immersed metal surfaces are just a small sampling of the sort of studies that wait to be carried out in the future. Although the

difficulties of preparing clean, well-characterized surfaces and solutions,

of determining accurate film optical constants, and understanding the

effects of the double layer and cation concentration gradients near

the metal surface remain, the prospect of obtaining many valuable

insights into processes at the metal solution interface is, as Tronstad

believed 30 years ago, promising.

4. References

[1] L. Tronstad, Optische Untersuchungen zur Frage der Passivitat des Eisens und Stahls, Kgl. Norske Videnskab. Selskabs Skrifter 1931, 1, F. Bruns

Bokhandel, Trondheim (1931).

[2] L. Tronstad, Trans. Faraday Soc. 29, 502 (1933).

[3] M. A. Barrett and A. B. Winterbottom, First Intern. Congr. on Metallic Corrosion, London, Apr. 1961, p. 657 (Butterworths, London, 1962).

PASSIVE REGION OF

ANODIC POLARIZATION CURVE

(INCREASING FILM THICKNESSI I
OXIDATION IN AQUEOUS SOLUTIONS 151

[4] J. Kruger, J. Electrochem. Soc. 108, 504 (1961).

[5] J. Kruger, ibid. 110, 654 (1963). . [6] P. C. S. Havfield, First Intern. Congr. on Metallic Corrosion, London, Apr.

1961, p. 663 (Butterworths, London, 1962).

[7] V. V. Andreeva, Corrosion 20, 35t (1964).

[8] R. R. Stromberg, E. Passaglia, and D. J. Tutas, J. Res. NBS 67A, 431 (1963) . [9] A. K. N. Reddy, M. A. V. Devanathan, and J. O'M. Bockris, J. Electroanai. Chem. 6, 61 (1963). , e [101 A. B. Winterbottom, Optical studies of metal surfaces, Kgl. Norske Videnskab. Selskabs Skrifter 1955, 1, F. Bruns Bokhandel, Trondheim (1955).

[Ill J. V. Cathcart, J. E. Epperson, and G. F. Peterson, Acta Met. 10, 699 (1962).

[12] R. J. Archer, Phys. Rev. 110, 354 (1958).

[131 J. Kruger, J. Electrochem. Soc. 106, 847 (1959).

[141 J. Kruger and H. T. Yolken, Corrosion 20, 29t (1964).

[15] T. W. Hickmott and G. Ehrlich, J. Phys. Chem. Solids 5, 57 (1958).

16] S. Schuldiner, Naval Research Laboratory Report 5775 (1962).

[17] F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg,

J. Res. NBS. 67A, 363 (1963).

[18] R. J. Archer, J. Opt. Soc. Am. 52, 970 (1962).

[19] J. Kruger and J. P. Calvert, accepted for publication in J. Electrochem. Soc.

[20] Gmelins Handbuch der Anorganischen Chemie, System-Nummer 60B1, p. 76, Verlag Chemie, Weinheim (1958). ,* MA< [21] L. Tronstad and C. W. Borgmann, Trans. Faraday Soc. 30, 349 (1934).

[22] J. Kruger and W. J. Ambs, J. Opt. Soc. Am. 49, 1195 (1959).

[23] M. Pourbaix, Z. Elektrochem. 62, 670 (1958). 24 M. Nagayama and M. Cohen, J. Electrochem. Soc. 109, 781 (1962).

[25] K. G. Weil, Z. Elektrochem. 62, 638 (1958).

Discussion

A. N. Saxena (Fairchild Semiconductor):

Whenever you quote the thickness of a film in your results what

value of the optical constants do you use for the substrate? Are

these for the clean bulk surface?

J. Kruger:

Yes.

A. N. Saxena:

Would you agree or disagree that the biggest source

< of error in interpreting the ellipsometer measurements to give thickness and

refractive index of thin films on metallic (or opaque) substrates comes

from the fact that the optical constants of the film-substrate interface

are not really that of the true surface? The true bulk surface could

be obtained when we cleave the specimen in ultra-high vacuum.

J. Kruger:

I agree with you very strongly. In fact, that is the whole point

in using an ultra-high-vacuum system. I am not saying that we

have achieved this completely but we have reasons for believing

that we have come fairly close. The whole point in that experiment was to determine carefully the optical constants of bare surfaces.
152 KRTJGER

A. N. Saxena:

You may start with a bare surface whose optical constants you

know. However, as soon as you start growing a film or when you

have grown a film (for example an oxide layer) on top of the original

"bare" surface, the film-substrate interface has no longer exactly the same optical constants as the original "bare" surface. The difference

between the two sets of optical constants depends on a number of factors, e.g., ambient temperature, type of film growth, etc. The

optical properties of the film-substrate interface can be studied by

using non-ellipsometric techniques to measure film thickness (like

multiple beam interferometry which does not depend on the optical constants of the interface) and combine this with the ellipsometer

measurement which does depend on the optical constants of the interface. This procedure is quite tedious but I am not aware of other

methods by which one can study the optical properties of the interface. The agreement between the multiple beam interferometer and

ellipsometer measurements of thickness of a film will be good if we

use the correct optical constants of the interface in the ellipsometer

calculations.

J. Kruger:

Perhaps we are not talking about the same thing. I never refer to the optical constants of the interface itself. I refer to the optical constants of the substrate, the bare metal, and the optical constants

of the film. The combination will from ellipsometric measurements

give a value for A and tan \p which depends on the thickness of the

film, its optical constants and optical constants of the substrate.

One needs to know accurately the optical constants of both the

substrate and the film to determine the thickness. I don't know

what you mean by "the optical constants of the interface" itself unless you mean the A and tan \f/ of the film-substrate combination.

A. N. Saxena:

By optical properties of the interface I mean its optical constants. Let me give you an example to explain this. Archer (J. Op. Soc. Am. 52, 970, 1962) quotes for the optical constants of Si, n-ik, as 4.050-0.028 i. He estimates k from the transmission measurements

of the absorption coefficient of Si and using the standard formulas

for n and k and measured \p, he estimates n. To relate the A)

(which you measure with an ellipsometer), for a specimen of a trans- parent film of refractive index rii on Si, with the thickness d of the

film and^!, the exact ellipsometer equation is used in which the optical constants of the substrate used are that of Si, viz, 4.050-0.028 i. If Ui <1.49 or >3.5, a family of closed contours is obtained for \p, A each corresponding to a particular nx and the thickness of the film

is marked off in terms of path difference in the film a. In a set of experiments, we obtained Si02 films on a batch of identical Si wafers

in which nx was the same, 1.46, but d was different. The measured

(^, A) points for each sample did not coincide with the theoretical
OXIDATION IN AQUEOUS SOLUTIONS 153

curve for 7^= 1.46 which was obtained using 4.050-0.028 i for the

Si substrate. I am attempting to analyze this experimental curve

to assign more accurately the optical constants for Si02-Si interface

obtained during thermal oxidation in steam at 1200 °C. I am also trying to investigate the Si02-Si interface for the case of anodic

oxidation where I feel the optical constants should be closer to the

bulk value.

I feel that the changes that you measure are due to the changes

in the optical properties of the film and not changes in the mterface. A. N. Saxena:

The changes in the optical properties of the film will show up m the

refractive index measurement as you etch the film gradually to smaller

thicknesses. In the refractive index measurements on our films

performed by Dr. A. E. Lewis of our laboratory, we found that the

refractive index nx was

o 1.45 ±0.003 as we etched the Si02 films from

9000 A down to 600 A. So the discrepancy between experimental

A) and theoretical (ft A) is not due to changes m the optical

properties of the film but due to the difference in the optical constants

of the Si02-Si interface. Regarding what Dr. Plumb said about

charge transfer at the interface which changes the potential barrier seen by electrons on either side, I think it will be very interesting to compare the measured tunneling currents across a thin insulator film

with a theoretical calculation of the tunneling current in which the

thickness of film used is that determined by ellipsometer.

J RTJGER*

One thing I agree very strongly with is that one should not rely on

one technique to get things like film thickness. If possible an in- dependent and different technique should be employed. For instance,

interferometry is still optical, one might use electrical techniques or something of this sort. A. N. Saxena: * I must emphasize that when the film thickness gets <100 A, other

techniques like interferometry fail to retain any useful accuracy.

This thin film region, however, is of interest to those who want to look at tunneling currents. Jean Bennett (Michelson Laboratory):

There has been some discussion about the use of multiple-beam

interferometry for film thickness measurements. I may be able to

clarify this point somewhat. In the method we use at Michelson

Laboratory, we evaporate the film whose thickness is desired on part

of the surface of an optical flat. We then overcoat the whole surface

of the flat with an opaque material, such as silver, and form a FabryPerot interferometer using this overcoated flat as one of the plates.

I
154 KRUGER

The other plate would be coated with a semi-transparent material,

such as a 90 percent reflectivity silver film. We then photograph the

fringes of equal chromatic order which are formed in reflection in white light. From the measured wavelengths of these interference

fringes the desired film thicknesses may be obtained. Using proper

care, film thicknesses may be determined with an accuracy of ±2 A. Although we normally measure film thicknesses in the range 100 A to 10,000 A this method should be able to be used for thicknesses from i 25 A up to several microns. The limitation of the method is that part

of the surface has to be free of the film, so that there is a sharp step

at the edge of the film. I don't know of any way to remove part of an oxide film from a metal surface. I think this is a major limitation

of the interferometric method.

A. N. Saxena:

In reference to Mrs. Bennett's statement, I do not disagree that

theoretically you could measure film thicknesses from 10 A up to several thousand A, and if one is careful, one may achieve an accu- racy of ±10 A with multiple-beam (Fizeau) interferometry (monochromatic light, measure shift of fringes at the step), and an accuracy

of ±5 A or maybe even ±2 A with multiple-beam (fringes of equal chromatic order) interferometry (white light, measure wavelength of fringes). To achieve this in practice is not very easy. Problems are encountered due to incomplete removal of the film at the step (typically for the case of Si02 on Si, after etching and rinsing,

the thickness of the film which grows again on the so-called clean Si

is about 50 A), uneven metallization of the step and surface roughness

of the substrate. I must point out that this method is a measurement

of the physical height of the film above the substrate at the step, and

this method is independent of the optical constants of the interface

and the film. William Primak (Argonne National Laboratory) (communicated):

Measurements of birefringence of vitreous silica have been in progress in our laboratory for a number of years under the auspices of the U.S. Atomic Energy Commission. Some of the technical information may be of value to those seeking strain-free vitreous silica windows.

Several commercial grades of vitreous silica were tested about 1960,

mostly optical blank stock purchased about that time; some from

purchases dating back perhaps up to a decade are labeled "old."

Since manufacturing procedures change from time to time, present

stock may be different. Some of our birefringence retardation

results obtained in % to 1 in. thick sections follow. General Electric: grade 101, even in thin sections, patches about

50 mn

grade 103, patches about 6 m^u

grade 104, fine patchwork about 5 m.fj..
OXIDATION IN AQUEOUS SOLUTIONS 155

Corning High Purity, optical and schlieren grades, <K mM; all Corning blanks we have examined showed small bubbles observed as stars in our polarimeter and our schlieren apparatus.

Engelhard Industries, Amersil: one Amersil grade (old) spiral

patterns ~50 m/*; Optosil very fine patchwork about 1% m^; Homosil

<K m/x, no refractive index variations; Suprasil <K m/z, variation in refractive index in sixth decimal from center to edge of a 2-in. section

found in one block. Faint patterns are seen in Homosil and Suprasil

as described in literature from Engelhard Industries and taken from

work published by Mohne. These are more distinct in our polarimeter

than in our schlieren equipment. They are more distinct in one

direction in the stock, and if they are of concern, the stock should be

cut accordingly. A specimen of selected Dynasil supplied for test showed some bubbles, edge strain from sawing, and a rectangular

pattern birefringent patchwork much like that of GE 104, but with

lower birefringence, about % m/i. In our past work we have preferred

Homosil or Suprasil, depending on whether radiation-induced coloration (occurs in Homosil) is of concern. We have heard a rumor there

has been some alteration in Suprasil manufacture to reduce its rela- tively large OH- content but we have not yet seen this stock.

Cutting stock is a severe problem in working strain-free silica. Our diamond saws readily introduce 15 mn retardation along edges,

30 to 50 at corners. In our most careful work, a rubber-bonded

carborundum wheel with copious coolant was used, and 1 -mil-deep

cuts were taken in a surface grinder so that many hours were spent

cutting a %-in. -thick slab of Herasil I. The birefringence near the

edge was. doubled by this operation to about 2 m/*. We have been

told mud saws (copper disk dipping in carborundum slurry) make

relatively strain free cuts but have not had the opportunity to examine

such work. We have found that wet grinding and polishing, as practiced by opticians, if done on all faces of a piece, will remove most

of the birefringence introduced by cutting if the work had not been

pushed through the saw. If the piece is sawed after a pair of faces

have been finished, it is necessary to refinish these as well as finish the

other faces. The ordinary edging machines introduce birefringence,

but if small cuts are taken and care is taken not to chip the corner at the edge of the work, the birefringence introduced is rather less than

by a saw cut. We have been told that etching will remove birefringence introduced by cutting, but we have not tested this ourselves.

Waxing or cementing plates will usually introduce considerable

birefringence. Even with the soft wax, Cenco Tackiwax, it is easy to introduce a retardation of 15 m^/cm in a small piece. It is virtually

impossible to cement silica to another material with a cement which

requires melting or thermal curing and obtain a strain-free piece.

The problem for other glasses has become severe in optical manufacture as thermally cured polymers have supplanted balsam as a cement.

If silica pieces are waxed to other silica pieces, a reasonably strain-free

joint can be obtained by placing the waxed piece into the oven,

turning off the heater, and letting the work cool slowly overnight to anneal the wax.

728-101 0—64 11
156 KRUGER

Our polarimeter for the above tests consists of a lantern with a 100-W projection bulb focussed at a distance, polarizer and analyzer

j of Polaroid sheet crossed at 45° azimuth, plastic sheet quarter-wave

material parallel to the analyzer between it and the specimen which

is mounted at 0° or 90° azimuth. A 15 to 30 cm lens is used either as a magnifier or the objective of a telescope for viewing. It is important

to use a large aperture (3 to 4 in. recommended) so that a large field ! outside of the specimen is viewed. The angular rotation of the analyzer to secure extinction is proportional to the retardation, about 1 3 mju/deg. At high light intensities it is easy to detect 0.3 m/x retarda-

( tion.
An Assessment of the Suitability of the DrudeTronstad Polarised Light Method for the

Study of Film Growth on Polycrystalline

Metals

P. C. S. Hayfield and G. W. T. White

Imperial Metal Industries (Kynoch) Limited, Birmingham, England

Theoretically derived changes in ellipticity resulting from reflec- tion of polarised light from different film-covered metals are considered. The examples illustrated include single layer and two layer

films, and films varying in thickness laterally across the surface under

examination. On the basis of this information, interpretation is made

of Drude-Tronstad experimental data recorded during a study of several gaseous and aqueous oxidation reactions on polycrystalline metals,

as well as aqueous corrosion reactions. The significance of changes

both in film thickness and film absorptivity is briefly discussed.

1. Introduction

The reflection and transmission of polarised radiation at interfaces

has been understood mathematically in terms of Maxwell's electro- magnetic theory for many years. One of the practical implications

of this theory in the field of optics is the measurement of the optical

properties of metal surfaces, in which determination is made of the

ellipticity, after reflection at a flat surface, of incident plane polarised

light. As shown by Drude [1]

1 and later Tronstad [2], the reflected component of light is greatly changed by the presence of contaminating film. Indeed, the change in ellipticity produced by films 10 to 1000 A is so great that the phenomenon forms the physical basis of one of the most powerful techniques presently available for measure- ment of thickness of thin tarnish films. The title Drude-Tronstad was chosen for the technique in order to distinguish it from sac- charimeters and similar instruments, but the name ellipsometer seems

to have been chosen very largely by other contributors to this symposium when describing the same technique.

In this paper emphasis is given to the suitability of the technique

to measurement of the thickness of tarnish films formed on polycrystalline metals during corrosion. Because films formed under these

conditions are frequently non-uniform in thickness and optically in- homogeneous, section 2 of the paper is devoted to development of equations describing reflection of polarised light by various types of non-uniform film. The equations are used in the remaining sections

of the paper as a basis for the interpretation of experimentally determined data.

1 Figures in brackets indicate the literature references on p. 198. 157
158 HAYFIELD AND WHITE

I 2. Theoretical Principles

The object of this section is to present the physical and mathematical principles underlying the theoretical calculations needed for the interpretation of the experimental work presented in the following

sections of this paper. For this purpose two models will be considered

in detail, namely,

(i) a surface covered by two or more homogeneous films of uniform

thickness, and

(ii) a surface covered by a single homogeneous film of non-uniform

thickness.

As a necessary introduction to the theoretical treatment of these

models, an outline is first given of the traditional treatment of the

single, uniform layer.

2.1. The Optical Properties of a Plane Boundary

In order to understand the properties of a thin film, it is first neces- sary to evaluate the reflection and transmission coefficients of a plane

boundary between two media, simply because the thin film comprises

two such boundaries in close proximity. The problem is to determine

the amplitude and phase of the electric vector of the reflected and

transmitted beams when a plane polarised beam having an electric vector E1+ is incident at an angle 4>\ on a plane boundary between

media 1 and 2 of refractive index nx and n2 (see fig. 1). The solution

to this problem is most conveniently expressed by representing amplitude and phase as a single complex number denoted by a boldface

symbol, say E, such that E equals (a+ib), where a and b are real numbers, and i is the square root of —1. In this notation the amplitude is taken as the modulus of E, viz, (a2 +62)^; the phase of E

relative to some arbitrary standard, such as the phase of the incident

beam at the point of incidence, is taken as the argument of E, viz,

Figure 1. Reflection and transmission at a plane interface.
FILMS ON POLYCRYSTALLINE METALS 159

arg (E)= tan-1 (b/a). With this notation the reflected and trans- mitted beams can be formally related directly to the incident beam

by the equations

E!_=rE1+ (1) E2+ =ZE1+ (2) where r and t are the complex reflection and transmission coefficients.

It is shown in standard textbooks (Heavens [3] and Vasicek [4]) that

when the incident beam is polarised parallel to the plane of incidence,

r and t have the values _7ii cos 4>2 —n2 cos <j>i v ~ ^ rii cos <f>2 -\-n2 cos 0i

t =

2ni CQS ^ (4) P

Tbi COS 02+^2 COS 0i and when the incident beam is polarised normally to the plane of incidence, r and t assume the values

^

S _rii cos 4>\—n2 cos 02 ^

Tbi COS 0i+7i2 C0S 02

t =s 2nx cos 0i _ ^

% cos 0i+ri2 cos 02 These reflection and transmission coefficients are known as the Fresnel

coefficients. If either one or both of the media are absorbing, the

corresponding refractive indices n are complex, and the Fresnel

coefficients are similarly complex: in the following work the assumption is made that this is so. In all cases 02 is obtained from Snell's law

rii sin <t>i=n2 sin <f>2 . (7) It should be noted that if the direction of incidence is reversed,

the transmission or reflection coefficients are changed. By interchanging rii and n2 in eqs (3) to (6) it can be seen that the Fresnel

coefficients r' and t for the reflection and transmission of light travelling in the reverse direction to that shown in figure 1, are related to

r and t by the equations

r' =-r, (8)

tt' = \-r\ (9) These equations are required in the following subsections.
160 HAYFIELD AND WHITE

2.2. The Optical Properties of Thin Film

At this stage the properties of a uniform thin film can be discussed.

Suppose the film is of thickness du and that a plane polarised beam

of unit amplitude is incident upon the 0-1 boundary, as shown in figure 2. The light will suffer multiple reflection within the film and

upon each reflection at the 0-1 interface, a fraction t[ will be trans- mitted and a fraction r[ (equal to —rr) will be reflected. Likewise,

at the 1-2 boundary, a fraction t2 will be transmitted and r2 reflected. Thus, the amplitudes of the successive beams from C, E, G, etc., decline in geometric progression. However, in order to calculate the

total amplitude reflected into medium 0, it is not sufficient to sum

these intensities, for in travelling through the film from A to B to C,

the beam acquires a phase lag of 25= (2ir/X)nid1 cos <\> with respect to the incident beam at the point A, and on each successive double reflection an additional lag of 25 is introduced. Consequently, the amplitudes at C, E, G, etc., must be multiplied by the

factors exp (— 2i8), exp (— 4*5), exp (—6*5), etc. Thus, the phase

and amplitude of the total reflected beam are represented by the sum

B=n+tit[r2 exp (-2*5) {l-nr2 exp (—2id)+r2r

2

2 exp (—4*5)— . . .} _ tit[r2 exp (— 2i8) ~n

\+nr2 exp (-2*5)'

Finally, eliminating tit[ by use of eq (9)

n+r2 exp (—2*5) , . U

l+rir2 exp (-2*5)' V ;

Figure 2. Multiple reflections and transmissions of a light beam incident on a thin

film.
FILMS ON POLYCRYSTALLINE METALS 161

The Fresnel reflection coefficients, rx and r2 in this equation, must

be those appropriate to the direction of polarisation. Thus the

i reflection coefficient, Rp, defining reflection when the incident beam

i is polarised parallel to the plane of incidence, will be obtained by

substituting into eq (10) values of rx and r2 derived by use of equations

of the type (3). Similarly, for polarisation in the normal direction,

the reflection coefficient Rs is obtained by substituting into (10) the

Fresnel reflection coefficients obtained from equations of_ type (5)

At this point, in preparation for discussion in subsection 2.4, it

is worth paying attention to the following aspects of eq (10). Firstly,

I the derivation of (10) by use of figure 2 makes no assumption conl cerning the coherence of the incident light over the region of film

illuminated. The phases of the reflected beams are all related to the

' phase of the light arriving at A. Reflection of light incident at some other point can be described by the same reflection coefficient

j provided, in this case, phases are all referred to the new point of incidence. If these two light beams are not coherent, then the

i resultant reflection cannot be described, a priori, by an expression

such as (10), because it is then not possible to define uniquely the

| phases of the incident and of the reflected beams.

In the Drude-Tronstad technique as normally practised, this diffi- culty is overcome by illuminating the sample with light polarised

at approximately 45° to the plane of incidence. In this situation

the p and s waves incident at any point must be coherent with each

other because they are resolved components of the same incident

i wave. Thus, the ratio Z=RP/RS , which involves only the difference

in phase between the reflected p and s components, is uniquely deterI mined. Moreover, this phase difference is independent of the absolute

phase of the incident light, and thus it is unaffected by coherence of the incident light over the region of film illuminated. The DrudeTronstad polarimeter is therefore designed to measure solely this

1 phase difference—a feature which is extremely important to the

discussion in section 2 describing reflection from non-uniform films.

2.3. A Matrix Approach to Multilayer Films

This next theoretical section is concerned with a detailed consideration of reflection from multilayer films. Unfortunately, the method of summing the contributions from individual multiple reflections, as given in section 2.2, is not suitable for calculating the properties

of multilayer films. A more practical approach is provided by the

use of matrix algebra more commonly associated with the theory

of four-pole electrical networks or with the theory of periodic heat

flow [5]: this approach has been used for calculating the properties

of optical filters [6] and its pertinency to calculations for the DrudeTronstad polarimeter has been mentioned by Winterbottom [7]. As a starting point consider an optical system involving, perhaps,

many layers of various materials ; in general the light at any point in the system can be resolved into two components—a forward travelling
162 HAYFIELD AND WHITE

wave, Ei+, say, and a backward travelling wave, Ei_, say. It is worth emphasising, at this point, that Ei+ is not the complex amplitude

of a single light beam subject to multiple reflection as in the previous

section, but is the resultant total beam, observed, as it were, after all the multiple reflections have taken place; likewise, Ex _ represents the

whole resultant wave. At some other point in the system the light

may again be resolved into forward and backward travelling waves,

say E2+ and E2 _ (see fig. 3). Because the equations governing the

propagation of light are linear, the relationships between Ei+, E^

and E2+ , E2 _ must be of the form

or, in the matrix form,

E1 _ =C

rE2+ +DE2 _

IA B\ /E2+\

\Ei-/~ \0 D/ \E2 _/

(11)

(12)

but the relationship between E1+ , Ei_ and E2+ , E2 _ must necessarily

be governed by the properties of that part of the optical system which

lies between the two points at which Ei+, Ei_, etc., are defined: These

properties enter eq (12) in the form of the matrix of coefficients A, B,

C, D—indeed, the matrix ABCD may be completely identified with

E.4. PART OF

OPTICAL

SYSTEM

2 +

V-l

( E 2 +

Figure 3. Illustration of relationship between an optical system and the matrices used to describe it.
FILMS ON POLYCRYSTALLINE METALS 163

the optical properties of the system, as is shown diagrammatically in

figure 3. This representation of optical properties by a matrix is made clearer

by study of the situation shown in figure 4 where two optical systems

are placed in series. The matrix equation relating the amplitudes of the light entering and leaving the first system may be taken as that

given by eq (12) and the corresponding equations for the second sys- tem may be written

\E2 _/ \G H/\E3 J

The intermediate amplitudes E2+, E2 _ may now be eliminated between

eqs (12) and (13) to give

AAA. * /« r\ /E,A

\b,_/ \c dJ\g h/\e3 J

but, since the two juxtaposed optical systems may be regarded as a

single system, E1+ , Ex _ may be related to E3+, E3 _ by an equation

analogous to (12), namely

\ti \*i3-/

where the matrix PQRS represents the properties of the composite

system. Comparison of (14) and (15) shows that

1ST OPTICAL SYSTEM

E 2 + 2ND OPTICAL SYSTEM

E 3-

E 3-

Figure 4. Illustration of matrix representation of duplex optical system.
164 HAYFIELD AND WHITE

(P Q\ (A B\ (E F\ /(AE+BG) (AF+BH)

\R s)\C D/ \G H/ \(CE+DG) (CF+DH)

That is to say the matrix representing the properties of a composite

system is obtained simply by multiplying together the matrices repre- senting the individual components of the system taken in order. This

rule is clearly applicable to any number of components, and herein

lies the great usefulness of this approach in the study of multilayer

problems, for matrix multiplication is a simple, systematic operation.

At this stage it is informative to evaluate the precise values taken

by the coefficients ABCD, etc., when the optical system is particularly

simple. As a first case, the matrix elements for a plane interface will be determined. To do this, E1+ , Ex _ are defined as the amplitudes in medium 1 immediately adjacent to the interface, and E2+ , E2 _ as the

corresponding amplitudes in medium 2. In the special case of a beam

incident from medium 1, as shown in figure 1, E2+ =flEi+ ; E!_ =rE1+ and E2 _ =0, where r and t are the Fresnel coefficients appropriate to the direction of polarization. Substitution into (12) now gives

E1+=^E1+ E1 _ = aE1+ =rE1+ (16) or A=l/t; C=r/t

Similarly, when light is incident from medium 2 only,

E2+ =r'E2_ and

E1+ =0

which, in conjunction with (12) leads to the equations

B=r'/t

D=(tt'-rr')ll (17)

The reverse Fresnel coefficients r' and V can be eliminated by use of the eqs (8) and (9) leading to the simpler relations

B=r/t

D=l/t (18)

Thus, the matrix representing the properties of a plane interface is
FILMS ON POLYCRYSTALLINE METALS 165

/l/t r/t\ (I r\ =1/4 V (19)

\r/t l/t/ \r I

)

The second elementary optical system that it is necessary to consider is a simple homogeneous slab. The objective is to determine

the matrix relating the amplitudes of the forward and reverse travelling light, Ei+, Ei_ at one point in a homogeneous medium, say at x=xu to the corresponding quantities E2+ , E2 _ at another point,

Say x=x2

. In a medium of refractive index n the electric vector E+ (x) of a plane wave of wavelength X propagating in the forward

x direction is given by the equation

E+ (z)=E0 exp (iut—ikx)

where E0 is the amplitude at the point x=0 and

K=-r- COS </>, A whence it follows that

E1+ =E0+ exp (iut—ikx^ (20)

and

E2+ =E0+ exp (iwt—ikx2). (21)

Now for the special case of a forward travelling wave in a homogeneous medium, E2 _ =0, and there is no reflected wave, i.e., Ex_==0;

taking this in conjunction with (20) and*(21) and substituting into

(12) at once reveals that

^4=E1+/E2+ =exp (i8) and C=0

where

8=k(x2 —Xi) 2irnd cos <f>. X

Similarly, consideration of a backward travelling wave shows that

B={) and D=exp (—id).

In other words, the matrix of a homogeneous slab is

e u 0 \ II 0

0 On the basis of principles already discussed, the matrix representing

the properties of a thin film is simply the product of the matrices
166 HAYFIELD AND WHITE

representing the properties of the 0-1 interface, the homogeneous

film of thickness d, and the 1-2 interface, taken in order, viz,

Equations (16) show that the reflection coefficient for a plane interface

is given by the ratio C/A; by analogy with this result it follows that

the reflection coefficient of a thin film is the ratio of the second to the first elements of the first column of the matrix (22), i.e.,

This result agrees with that previously obtained by detailed analysis

of multiple reflections. The advantage of the matrix approach is the ease with which it can be extended to deal with multilayer cases. In order to calculate

the properties of two-layer films, it is necessary only to multiply

(22) by the matrices

and compute the ratio of elements of the first column of the resulting

matrix product. Because the reflection coefficient is calculated as a

ratio, the calculation may be further simplified by ignoring the common

terms eu/t which are factored out of the individual matrices. Moreover, this approach is well-suited for programming on a digital

computer, because it provides a routine for calculating the reflection

coefficient of a composite system comprising any number of films, as opposed to providing an explicit expression which, of course,

varies according to the number of films in the multilayer.

The calculations of the reflection coefficients for light polarised

parallel and normal to the plane of incidence proceed along exactly

similar lines, except that in the first case, the Fresnel reflection coefficients, rp , are computed from eq (3), whereas in the second case the coefficients, rs , are calculated from eq (5). Z is then calculated

from the ratio Rp/Rs in the usual way.

For the present work this matrix procedure was programmed for a Ferranti Mercury computer which enabled Z for approximately

100 double layer cases to be computed in about 1 min. The use of these Z curves is discussed in subsequent sections.

R

ri+r2 e l+7y2 e

-2is'
FILMS ON POLYCRYSTALLINE METALS 167

2.4. Films of Non-uniform Thickness

For practical work with specimens which are optically imperfect

by reason of roughness or uneven film growth, it is essential to be able

I to assess, at least in a qualitative way, how far such departures from

perfection will affect Drude-Tronstad measurements. Unfortunately,

I an accurate theoretical treatment of non-uniform films is difficult to devise, and the following brief account describing one of the more

I plausible lines of approach serves only to illustrate some of the difficulties of constructing a satisfactory theory.

The oversimplified model chosen to represent a non-uniform film

sample is one in which the film is laid down on a plane substrate in such a way that it comprises finely divided and intermixed areas of

! distinct thicknesses dx and d2 (see fig. 11). The aggregate area of

! thickness di, represented as a fraction of the whole area illuminated,

| is a, and the corresponding fractional area of thickness d2 is (1—a).

Apart from this discrete difference in thickness the film is assumed to

I have uniform properties.

The reflection coefficients of such a uniform film will be given as usual by eq (10) wherein the value of 5 is that appropriate to the

thickness dl or d2 . These reflection coefficients can be denoted by

Ri P) Rls for that part of the film which is of thickness du and by

| R2p and R2s for that part of the film which is of thickness d2

: the

j corresponding ellipticities can be Zx and Z2

. The assumption is now made that in the experimental determina-

| tions of ellipticity the relative importance of reflections from the two

areas is proportional to the aggregate relative areas, a and (1 — a);

in the expression for the ellipticity of the whole film the reflection

coefficients must therefore be weighed accordingly. Because this assumption is independent of the orientation of the plane of polarisation, it follows that

In general this equation will not describe experimental observations

because the derivation just given tacitly assumes total coherence of the incident light, in addition to optical perfection of the film and

substrate—conditions which are not satisfied in the conventional

Drude-Tronstad setup. In an attempt to understand the effects

of incoherence it is necessary to recall the discussion given at the end

of section 2.2, where it was suggested that provided the DrudeTronstad apparatus was arranged to detect solely the phase difference

between the reflected p and s components of the same incident plane

polarised beam, coherence was not essential to the validity of eq (10).

Thus, if there exists an arbitrarily changing phase difference 4> between reflections from different points on the sample given by the

reflection coefficient Rlp and R2p the same phase difference must exist

Z=aRlp -\-(l—a)R2p aRis +(l—a)R2s (23)
158 HAYFIELD AND WHITE

between components polarised normally to the plane of incidence,

e.g., between Rls and R2s . Accordingly, eq (23) is modified to aRlp +(l-a)R2 pe

i + aRls +(l-cdR2 se

i * The observed value of Z will be the average Z given by

z=do Z^J0 \aRM-«)R^r*' (24)

| Direct integration of this expression shows that if aEisXl—a)R2s then _Z=R\v \R\ s z=l Z\

and if aRls <(l-a)R2s then _ _

2=i?2p/^2s =22. As intimated at the beginning of this section this result is not

entirely satisfactory: 'it indicates that the presence of small areas of a different thickness in a film which is otherwise uniform, has no

influence on the measured ellipticity, and implies that Z changes

abruptly from Zx to Z2 and vice versa according to the relative magnitudes of aRu and (1 —<*)&.. In section 3 it is pointed out that these

conclusions are not verified by experiments, and the assumptions

leading to eqs (23) and (24) are therefore suspect.

Accordingly, at present we cannot satisfactorily account lor the

phenomenological conclusions given in the following section, that the

ellipticity of a non-uniform film is the same as that of a film having

a uniform thickness equal to the mean thickness of the non-uniform

film, nor can we justify the empirical equation

Z=aZ1 +(l-a)Z2) (25)

which is briefly discussed in the next section, and which appears to describe the formation of oxide on uranium (see section 4.2a) and the

corrosion of aluminium in demineralized water (see section 4.2f),but

does not describe non-uniform films of calcium fluoride on aluminium

see section 3).

3. Experimental Verification of Formulae Derived in Section 2

Before making use of formulae derived in section 2 to interpret the

thickness of films formed during corrosion processes, experimental

evidence for their validity is essential. In the past, strong circum-
FILMS ON POLYCRYSTALLINE METALS 169

stantial evidence has existed for eq (10), in the close correspondence

between ellipticity changes predicted on the basis of this formula,

and experimental results relating to the growth of film of constant

chemical composition. This is most conveniently seen in the polar

plots of ellipticity with increasing thickness of film, as practised by

Winterbottom [8]. These curves trace out distinctive shapes, but it transpires that mere correspondence of shapes is not in itself a suffi- cient criterion to define absolutely the parameters in eq 10 [9]. Other

evidence takes the form of comparison between film thicknesses

determined optically and by other techniques. Thus Winterbottom

has shown close correspondence of data obtained optically and by

electrometric reduction for oxidised copper [10]. However, in order

to be able to obtain more direct proof of eq (10), and other equations

relating to more complicated film configurations, some attempt has

been made to reproduce films of known thickness and uniformity.

The deposition of barium stearate monolayers, as practised by Rothen

[11] and Mertens et al. [12], was rejected on the grounds of optical

anisotropy within each layer deposited, and the experimental difficulty

of putting down thick films. As an alternative, experiments were

carried out using controlled vacuum evaporations, the evaporated

material being most usually calcium fluoride, and the substrate electro- polished aluminium. This condition was chosen because the polar

plot of the reflection characteristics of aluminium covered with trans- parent film (frequently referred to as the Z curve) is a very distinctive

crescent-shaped curve. This shape of curve is known to be very

sensitive to change in film absorptivity and thickness. Also, calcium

fluoride is readily evaporated without decomposing, and as a conse- quence the deposited film has a composition close to stoichiometric.

Greater sensitivity would have been obtained using a film material

of higher refractive index, such as zinc sulfide, but against this advantage it is considerably more difficult to deposit films of stoichiometric composition.

The amount of calcium fluoride deposited was estimated entirely

from the weight of sample evaporated, using a V-shaped molybdenum

boat, and assuming uniform evaporation over a solid cone of angle 60°.

The accuracy of measurement can be considerably improved by multiple beam interferometry on evaporated deposits, but this is timeconsuming and it was not adopted for the investigation now reported.

A polar graph illustrating the experimentally determined ellipticity

values is shown in figure 5, and these data are to be compared with a corresponding theoretically derived curve using eq (10) (heavy line in

fig. 5). There are differences in detail in the shape of the two curves.

First, the width of the crescent formed by experiment is smaller than

that of the theoretical curve, and also the azimuth at which the end

point of the crescent turns is approximately 310°, whereas that for the theoretical curve is 272°. Contained in figures 6 and 7 are additional theoretically computed curves in which different optical constants have been employed for the aluminium substrate as well as changes in angle of incidence. The former changes do not provide

a reason for the changed angle of the turning point of the curve.
170 HAYFIELD AND WHITE

Figure 6. Theoretically derived Z curves indicating the influence of substrate

optical constants upon ellipticity of light reflected from calcium fluoride films.
FILMS ON POLYCRYSTALLINE METALS 171

i Figure 7. Theoretically derived Z curves indicating the influence of angle of incidence upon ellipticity of light reflected from aluminium covered with calcium fluoride film.

I whereas the latter do. Examination of the surface for possible

faceting that might have resulted in a component of light reflected at a smaller angle of incidence than that measured using the plane

of the specimen revealed there was none, and this could not therefore be the explanation. Similarly, calculations on films of different

refractive index revealed that a fall in refractive index (as might be

produced by a non-stoichiometric, possibly partially hydrated film)

would not account for the observed shape of the Z curve. This discrepancy between theory and practice is still unexplained by us. Comparison between estimated thickness of evaporated material, and

values computed from optical data on the basis of a film refractive

index of 1.434 a^e given in table 1. Bearing in mind the possible

large error in evaporation, the agreement to a first order is satisfactory.

3.1. Two-Layer Film

To test the applicability of a two-layer model, a condition was re- quired that would give some change in shape of the Z curve in order to compensate for the relatively poor accuracy available in ascertaining

the thickness of evaporated deposits. Merely to change the refractive

index of transparent film would not appreciably change the shape of

728-101 0—64 12
172 HAYFIELD AND WHITE

Table 1. Comparison between apparent computed film thickness of nonhomogeneous

calcium fluoride on aluminium and mean estimated thickness from vacuum evaporation data

9 percent coverage of surface

Estimated thickness evaporated through mask

Estimated mean thickness of overall film Apparent film thickness computed from comparison with

theoretical Z curve for uniform film growth

Film thickness, angstroms

250 1250 2250 2500 3000

22 112 180 225 270

20 90 120 180 180

25 percent coverage of surface Estimated thickness evaporated through mask

Estimated mean thickness of overall film Apparent film thickness computed from comparison with

theoretical Z curve for uniform film growth

250 1250 1750 2000 2500 3500

62 312 437 500 625 875

50 325 420 380 690 820

percent coverage of surface Estimated thickness evaporated through mask

Estimated mean thickness of overall film Apparent film thickness computed from comparison with

theoretical Z curve for uniform film growth

250 750 1250 1750 2000

98 293 488 682 780

75 400 380 950 980

2500

970 1010

100 percent coverage of surface Estimated thickness evaporated through mask

Estimated mean thickness of overall film Apparent film thickness computed from comparison with

theoretical Z curve for uniform film growth

250 750 1250 1750 2000 3000 3500

250 750 1250 1750 2000 3000 3500

160 640 1140 2100 2260 3000 3500

the crescent-shaped curve; calculations for the systems air/ZnS/CaF2 / aluminium and air/CaF2 /ZnS/aluminium confirmed this point. Some

advantage would be obtained by using water as an environment in

place of air, but although this would produce change in curve shape more readily, the film thickness dispersion along the curves is considerably reduced and in any case it was feared it would create problems of film dissolution, hydration, etc. The condition finally

selected was to deposit vacuum evaporated calcium fluoride upon

absorbing metallic oxide films. For this purpose copper samples were oxidised to thickness levels of 209, 350, 490, and 625 A by heating them to 120 °C in 30 mm 02 for different periods; the cuprous

oxide thus produced had an absorptivity lying between 0.2 and 0.25. Onto this was evaporated calcium fluoride in successive layers up to 2500 A, optical readings being taken at each stage. Comparison

between theoretical and experimental data is shown in figures 8 to 10. Results for the cuprous oxide film 490 A thick are very similar to those of the 625 A film shown in figure 10, and are therefore not in- cluded. To a first order the expected shape of Z curve is reproduced

experimentally, and this is true also for the expected thickness of
FILMS ON POLYCRYSTALLINE METALS 173

FAINT CONTINUOUS BACKGROUND

THEORETICAL

Z CURVES

FOR SINGLE

MODEL

N| 1.0 AIR

n^-2J5-(v«R,oU s) 1 [j48L[ «f

rij = 0.98G-2.G47i COPPER

0 O 0

THEORETICAL

ELLIPTIC ITY VALUES

FOR TWO LATER MODEL

N, - 1.0 AIR

H^=l.« JVARIABLE fftS"

n'

2'-2.75-0.24^09S TMPTMS

n 3

- 0.986-2. 647 i COPPER

o o o

EXPERIMENTAL ELLIPTICITY VALUES FOR CALCIUM FLUORIDE FILM ON CUPROUS OXIDE

CflVFRFO COPPFR

Figtjee 8. Comparison of experimental

and theoretical ellipticity values for the formation of calcium fluoride film on a 209 A thick layer of cuprous oxide on

copper.

Figure 9. Comparison of experimental

and theoretical Z values for formation of calcium fluoride on a 850 A thick layer of cuprous oxide on copper.

THEORETICAL Z CURVE FOR SINGLE LAYER MODEL

N, =1.0 AIR +-•-+

n"i 2.75 — 0.24i d

VARIABLE CUPROUS OXIDE n

3

- 0.986 -2.647i

(COINCIDENT WITH EXPERIMENTAL CURVE

COPPER

FOR Cu 2 0 ON CU)

THEORETICAL Z CURVE FOR TWO LAYER MODEL

N| 1.0 AIR d

N'

2

= 1.434 VARIABLE CALCIUM FLUORIDE

n'

2

' - 2.75 - 0.24i 350? CUPROUS OXIDE n3

= 0.986 -2.6471 COPPER

EXPERIMENTAL Z CURVE FOR CALCIUM FLUORIDE 000

FILM ON CUPROUS OXIDE COVERED COPPER

300 200 100 ~o
174 HAYFIELD AND WHITE

Figure 10. Comparison of experimental and theoretical ellipticity values for the formation of calcium fluoride on a 625 A thick layer of cuprous oxide on copper. calcium fluoride on the thicker deposits of cuprous oxide. The significance of the large discrepancy in film thickness values for the case

of cuprous oxide 209 A is unknown, more especially because the calcium

fluoride evaporations were always made onto the four specimens at the same time. The conclusion is reached, however, that the formula

relating to the two-layer model can be used to predict experimental

results.

3.2. Non-Uniform Thickness Films

To study the influence of non-uniform films of the type illustrated

in figure 11, calcium fluoride was vacuum evaporated in the form of small islands or cells upon electropolished aluminium surfaces. Three

variables were studied:

(i) the relative area of surface covered by the cells—-9.5, 25, 39,

and 100 percent;

(ii) the relative thickness of film present over each cell compared

with surrounding surface;

(iii) the influence of such a cellular structure as provided by (i) and (ii) upon surfaces already covered with uniform thickness

film.
FILMS ON POLYCRYSTALLINE METALS 175

SUCCESSIVE VACUUM - EVAPORATIONS OF CALCIUM FLUORIDE

( SMALL CELL

\ AREA

POLYCRYSTALLINE ELECTROPOLISHED

' ALUMINIUM

/INTERMEDIATEn

- SIZE CELL

\ AREA J

/UNIFORM \ VERTICAL MAGNIFICATION

~ - —[ THICKNESS EXAGGERATED FOR PURPOSE

7 ^ \ FILM / OFDIAGRAM CLARITY. Figure 11. Diagram illustrating sections through various types of non-uniform

thickness calcium fluoride films deposited upon electropolished aluminium.

Table 2. Details of masks employed for the preparation of non-uniform thickness

films

Description of grid Shape of cells Distance between

cells

No. of

cells per cm2

Transmission through cells as determined

spectrophotometrically

Square cm

0. 034 500 percent

9.5 Image of grid system reduced photographically onto sensitised copper sheet. Holes prepared by etching.

Circular. . ... .018 1600 25

Chemically thinned stainless steel mesh Square ... .013 330 39

Area of sample illuminated when using the Drude-Tronstad technique 0.56 cm2 ± 0.02.

The cellular structure was obtained by evaporating through grids

having dimensions as given in table 2. The grids were affixed horizontally to a framework such that the aluminium samples could be

rested on top of them during the period of vacuum evaporation from a source located vertically underneath. The distance between specimen

and source was 20 cm in order to ensure that the edge of the film

transmitted through the mask was quite sharp. This point was con- firmed microscopically when the calcium fluoride deposits were

sufficiently thick to show temper colours, but no multiple beam interferometric traces were taken to confirm the actual profile.

Prepared samples were examined by the Drude-Tronstad method,

data being recorded in figures 12 and 13. The first figure shows results from samples representing all the different types of film thickness

configurations tested, and all the points follow fairly closely to the

Z curve for uniform thickness film. Figure 13 provides rather more

detail of the example where film over only 9.5 percent of the surface was increased, while the remainder was zero or at some fixed level
176 HAYFIELD AND WHITE

Figure 12. Comparison of experimental ellipticity values from aluminium covered with non-uniform thickness calcium fluoride films, with experimental and theoretical Z curves for uniform thickness films.

THEORETICAL Z CURVE FOR THE

CONDITION 1.0-4. Oi 5461?

N0N- UNIFORM THICKNESS CALCIUM FLUORIDE FILM 9.5%

TRANSMISSION

GRID

THICKNESS OF FILM EVAPORATED THROUGH

THE GRID

THICKNESS OF FILM OVER THE REMAINDER

OF THE SURFACE

o

VARIABLE

(AS INDICATED ALONGSIDE EXPERIMENTAL POINTS) oX

500 1

ioooS 0 1500?

2000? AREA OF SAMPLE ILLUMINATED 0.565 cm 2 NUMBER OF CELLS ILLUMINATED 500 Figure 13. Experimental ellipticity values from non-uniform thickness calcium

fluoride films on aluminium.
FILMS ON POLYCRYSTALLINE METALS 177

such as 500, 1000, and 1500 A. For the thinner films sufficient results were obtained to show that the Z curve traced was in the same

direction as the one for uniform film growth, and the variation ot

ellipticity with film change appeared to follow the average thickness

of film existing over the whole surface. Interpolation of apparent

film thicknesses for all three nonhomogeneous film cases studied, i.e., cell areas covering 9.5, 25, and 39 percent of the surface, are given m

table 1 ; and to a first approximation the apparent thickness levels recorded are an average for the surfaces examined, in spite of widely

ranging configurations.

In section 2 consideration was given to the expected ellipticity ot light reflected from non-uniform thickness films. Data contained

in figure 14a represent ellipticity values calculated on the basis ot eq (23), derived on the assumption of a coherent light source, while

the curves in figure 14b are computed from eq (25). Data m the

first of these figures take the form of closed loops, with increasing

thickness of film producing movement in a clockwise direction.

Further, the increase in thickness to produce a complete rotation is considerably less than that necessary to give a complete cycle ot the

uniform film example (see fig. 5) and is completely contrary to observed experimental results. The method adopted for calculating

data given in figure 14b produces miniaturised forms of the Z curve

for uniform film growth, and as the area of surface being covered

with film increases, so the resultant size of Z curve increases and

finally for 100 percent becomes identical with that for a uniform thick- ness film. Although there is some correspondence with observed

experimental results in that the movement around the Z curve with

increase in film thickness is the same, and also the basic shape ot the

Z curve is the same, there is no correspondence m the positioning ot the curve on the polar plot, and also the increase in film thickness to give a rotation of the miniaturised Z curve is the same as necessary

to give a complete rotation from a uniform thickness film. One ot the

results of the experimental work carried out with non-uniform thick- ness films was that movement along the Z curve not only follows the curve for a uniform film, but moved at a rate equal to the average

thickness of film, i.e., if 10 percent of the surface only was being

covered, then these areas would need to increase 25,000 A in thick- ness before a complete rotation of the Z curve was made. It is clear,

therefore, that the empirical eq (25) does not represent the conditions

encountered experimentally.

Experiments with absorbing films are less easy to contrive on

account of the difficulty in depositing films of controlled absorptivity,

but a naturally occurring example exists which goes some way towards

providing an illustration. Previously [13] it has been reported that

prior vacuum annealing of copper changes the subsequent oxidation

rate at temperatures up to 120 °C very considerably (see the lower

part of fig. 15), this effect being produced by impurity atoms spreading

from grain boundary sites which effectively prevents the surface

covered from reacting with oxygen at temperatures up to 120 U.
178 HAYFIELD AND WHITE

Figure 14b. Theoretical Z curves for non-uniform thickness films of calcium

fluoride on aluminium, calculated using eq {25).
FILMS ON POLYCRYSTALLINE METALS 179

EXPERIMENT

NO. AND SYMBOL

VACUUM I HEAT I TREATMENT

PRIOR TO I OXIDATION

SKETCH OF SECTION THROUGH SAMPLES AFTER

OXIDATION AT 120 °C AND 30m m 0

2

FILM THICKNESS VERSUS TIME CURVES

FOR THE OXIDATION AT I20°C AND 30m m 0

2

) MOMENTARILY (cu

L

2 Q) }at 400°C SUBSTRATE GRAIN BOUNDARIES

I hr. AT 400°C

1 2 hr. AT 400 °C 3hr. AT 400°C

7 hr. AT 400°C

10 20 30 40 50 60 TIME IN MINUTES

EXPERIMENTAL Z CURVES FOR OXIDATION

AT I20°C AND 30 m m 0

2

270° Figure 15. Formation of non-uniform thickness optically absorbing films of cuprous

oxide on copper and its effect upon experimentally determined Z curves.

Such samples provide some naturally occurring examples of extreme

non-uniformity of oxide film thickness, and it is interesting to note

that the polar plots of optical readings all follow the same course, in confirmation of the data already described in this section for trans- parent non-uniform films on aluminium. Furthermore, although

the observed optical changes for film growth on samples vacuum

annealed prior to oxidation represent only short sections along the

polar plot (see fig. 15) the islands of oxide exhibit first- and secondorder temper colours, indicating that the apparent optical readings

displayed in the graph are average film thickness values only.

On the basis of the experimental work carried out on non-uniform

thickness films, the conclusion is reached that even under extreme

conditions of non-uniformity, the shape of the recorded Z curve is not appreciably altered, and the interpolated apparent film thicknesses

correspond to the mean thickness of film.

3.3. Anisotropy of Film and Substrate Optical Constants

The problems implicit in the study of systems involving optically

anisotropic media have been outlined in the work of Mallemann

and Suhner [14] and Blodgett and Langmuir [15]. The number of variables to be considered becomes so great that it is doubtful whether

much progress can be made in interpreting results, save those from

single crystals where precise orientation details are available. For

polycrystalline materials the exact interpretation of waves reflected

from film and metal which are made anisotropic by strain, or which are inherently anisotropic, would seem to pose insuperable difficulties. There is some possibility, however, that provided both films and

substrate are fine grained and randomly oriented, the effects of anisotropy are likely to be cancelled out. A practical test for the
180 HAYFIELD AND WHITE

I presence of anisotropy consists of taking optical readings at different

orientations of the sample, any anisotropy being detected by a

periodic change every 180° in analyser reading A, where Z=peiA

. As an illustration of the absence of overall anisotropy on samples

where substrate and film are known to be optically anisotropic, listed in table 3 are readings taken from film-covered titanium surfaces. Table 3. Drude-Tronstad optical readings from chemically polished and anodised

polycrystalline titanium samples

Method of preparation Angle of rotation

of speci- men

Annealed 76 percent cold rolled

Polished with alumina + 10% oxalic acid, bright dipped 20 sees in warm mixture of 40% HN03 , 30% H2SO4, and 30% HF.

deg0

4590

135

0. 466

.462

.462

.466

.477

.483

.490

471

deg min

93 56

93 44

94 00

93 44

93 44

94 08

93 20

92 44

93 12 93 08

0.490

.490

494

494

.492

deg min

94 48 94 08

95 36

95 08

94 12 93 24

94 44

94 32

93 24

Prepared as above, anodised in 0.005% wt/wt H3 P04/water at 100 V for 5 min.

04590

135

180

0.445

.433 .420

.424 .464

.462

.422

.431

.431 .433

-99

-99 104

102

105 105 -99

100

101 -99

0. 262

.261 .259

.253

.300

.294

.264

.287

.264

.261

117

118

•114 112

•118 -116

-113

-112 116

-116

Under a polarising microscope, film overlying individual substrate

crystals of such samples can readily be shown to have optical anisotropy.

The influence of oxide nuclei upon reflected ellipticity has been

briefly mentioned elsewhere [9]. Nuclei which contribute largely to

diffuse reflectivity are undetected by the Drude-Tronstad method.

This means that measurements taken during early stages of some

reactions, where the nuclei form an appreciable fraction of the total

reaction product, may be very misleading. Such conditions exist

during the early stages of thermal oxidation when metals are subjected

to high temperature and low partial pressure of oxygen, e.g., during

bright annealing processes. Under these conditions the electrometric

or manometric techniques provide a more reliable result than the

optical technique. In order to assess how accurate the interpretation

of Drude-Tronstad measurements is likely to be, there must clearly

be electron microscope inspection, or direct comparison from other

techniques.

For nuclei shapes which take part in determining specular reflectivity, it can be assumed that the contents will be included in the average

film thickness computed, provided the dimensions are such as to allow

the requisite number of multiple reflections within the film as neces-
FILMS ON POLYCRYSTALLINE METALS 181

sary for the derivation of eq (10). To a first approximation the

j width of nuclei must be the same order as the height, and known

| examples of nuclei which contribute to specular reflection are covered

by this criterion.

4. Application of the Drude-Tronstad Method to the

Study of Film Growth on Metals During Oxidation

and Corrosion

In section 3 of this paper it has been established that ellipticity

changes produced during reflection of polarised light from film-

! covered surface are a function of average film thickness and of the

optical constants of the media concerned. It follows that in applications of the method to oxidation and corrosion studies, interpretation

' will be most straightforward for those conditions where optically

homogeneous films of uniform thickness are formed. In general,

these conditions of film formation are most frequently encountered on polycrystalline metals during anodic aqueous oxidation: the film

profile becomes progressively more complicated in conditions promoting gaseous oxidation and corrosion respectively. Each reaction

! condition, however, presents its own special characteristics, and

because of this it is now proposed to describe a number of specific

1 examples.

I 4.1. Aqueous Anodic Oxidation

a. Aluminium

Commercial purity polycrystalline aluminium, having surfaces

I prepared either by mechanical or chemical processing methods, was

anodised in ammonium borate solution at a range of voltages up to 120 V, the samples being periodically removed, washed and dried,

and then examined optically. The shape of the Z curve produced by

plotting ellipticity values on a polar plot can be matched very closely

by theoretical data, on the assumption of a single uniform thickness

film of transparent and isotropic oxide. The results are not reported

here in detail because of the similarity of the data to that reported

by Barrett for anodic oxide film growth on single crystals of high

purity aluminium in a similar electrolyte [16, 17]. It is noteworthy

that the shape of the curve is very similar to the example illustrated

in figure 5, and in particular that the lower turning point of the

crescent-shaped curve occurs at the theoretically determined value

between 270 and 280°. Film thickness data has been studied up to one complete cycle of the Z curve, i.e., up to a thickness of about

2500 A. Metallographic examination confirms qualitatively the

uniformity of film thickness, even although temper colours are only

weakly developed.
182 HAYFIELD AND WHITE

b. Uranium

Data for the anodic oxidation of polycrystalline uranium in ethylene

glycol/ammonia/water mixtures at constant current density are shown

in figures 16 a and b. The temper colours exhibited are uniform over

the surface, and up to film thickness of 1800 A there is a close similarity between theory and experiment.

c. Titanium

A wide range of titanium samples, varying in purity and metallurgical condition, have been examined during anodisation in both

40 percent sulfuric acid and 0.005 percent phosphoric acid electrolytes.

Temper colour formation was found to be uniform only on substrates

produced by vacuum evaporation; in all other instances the colour,

and therefore anodic film thickness, showed a marked dependency

upon substrate orientation. The wide differences in shape of observed

Z curves can best be attributed to differences in film composition, a

fact already briefly reported [9].

Figure 16a. Z curves recorded for the anodization of uranium in ethylene glycol/ ammonia/water mixture at constant current.
FILMS ON" POLYCRYSTALLINE METALS 183

4.2. Gaseous Oxidation

a. Uranium in Carbon Dioxide

Gaseous oxidation of single crystal and polycrystalline uranium in carbon dioxide of various purities has been studied up to 550 °C.

Data for reaction in very pure gas [<0.1 ppm H2 0; <1.0 ppm 02 ] at 300 °C is represented in the Z curves of figures 17 a andb. Uniform

film formation, as judged metallographically, is observed on single

crystal material whether the surface preparation is made by chemical

or mechanical methods, but large differences are observed with polycrystalline material; with the latter, marked differential oxidation

j rates with substrate crystal orientation are observed for electropolished

surfaces, but not for those prepared by mechanical polishing on

selvyt impregnated with ){ \x diamond dust paste. The single crystal

specimens (of considerably better purity than the polycrystalline

| material), both electropolished and mechanically polished, exhibited

identically shaped Z curve plots (fig. 17a) even though the oxidation

rates derived from the results were markedly different (see fig. 18a).

Figure 16b. Theoretical curves for growth of oxide on uranium during anodization

in ethylene glycolIammoniaI water mixture at 13 °C.
184 HAYFIELD AND WHITE

y SINGLE CRYSTAL URANIUM OOO ELECTROPOLISHED. OXIDISED IN PURE CO2

• • • MECHANICALLY POLISHED. OXIDISED IN PURE C02

(a)

270°

POLYCRYSTALLINE URANIUM

HI O O O ELECTROPOLISHED. OXIDISED IN PURE C0 2 HZ • • • MECHANICALLY POLISHED. OXIDISED IN PURE C02

Figure 17 a and b. Experimental Z curves obtained from the oxidation of single

and polycrystalline uranium at 300 °C in carbon dioxide.
FILMS ON POLYCRYSTALLINE METALS 185

POLYCRYSTALLINE URANIUM AAA MECHANICALLY POLISHED. OXIDISED IN C0 2 + 150 wt.p.p. m.H 2 0 MECHANICALLY POLISHED. 0X1 DISED IN C02 + 130 wt. p. p.m.0 2

180° / [ 0°

a** y 1

<J^-t—z—

1.5

(0

270° Figure 17c. Experimental Z curves obtained from the oxidation of polycrystalline

uranium at 300 °C in carbon dioxide.

By comparison, although a similar difference in oxidation rate is exhibited by polycrystalline samples (fig. 18b), the Z curve for the

samples displaying differential oxidation rates is now drawn in towards

the origin (fig. 17b). However, although theoretical curves of the Z

curve shape resulting from oxidation of electropolished polycrystalline

material can be derived on the basis of eq 25, the conclusions reached

in section 3 suggest this cannot be a valid explanation, and make it necessary to attribute the change in shape to a true increase in oxide

film absorptivity. Similar relatively high film absorptivity is detected on electropolished surfaces oxidised over a range of temperatures, and

also for dilute uranium-molybdenum alloys. The tentative explanation must be advanced that such films differ either in structure or composition from film built up on mechanically polished surfaces of the same material, but the nature of the differences is not known.

Additional information yielded during optical examination of polycrystalline uranium is illustrated in figures 18b and 19. Data in the

first of these diagrams show the influence of gaseous impurities in

carbon dioxide upon the rate of growth of uniform thickness film on

mechanically polished samples, leaving no doubt that separate additions of oxygen and water vapour are responsible for considerable
186 HAYFIELD AND WHITE

SINGLE CRYSTAL URANIUM

ELECTROPOLISHED, OXIDISED IN PURE C0 2 MECHANICALLY POLISHED, OXIDI SED IN PURE CO?

1000

Figure 18 a and b. Oxidation of single and polycrystalline uranium at 300 °C in carbon dioxide.
FILMS ON POLYCRYSTALLINE METALS

20 30 40 50 60 70 80 90 100 TIME.min Figure 19. Weight uptake of polycrystalline uranium heated at 400 °C in carbon

dioxide of various impurity contents.

(Published by permission of the Institute of Metals.)

increases in growth of film over the surface. Identical conclusions

j are drawn from examination of single crystal material.

In figure 19 comparison is made between optical and gravimetric results taken on samples subjected to the same reaction

conditions. The fact that the two sets of results are quite different is attributable to the growth of irregularly shaped mounds of reaction

product not contributing significantly to specular reflection, and yet

containing the greatest percentage of oxygen. The point in time when

the two sets of results begin to differ defines the stage at which the

overall film is weakening sufficiently to allow a "breakaway" type of oxidation to begin, and the subsequent divergence between the results becomes a measure of the severity of the localised, as opposed to

general, oxidation.

b. Beryllium

Oxidation of polycrystalline beryllium at temperatures up to 700 °C

in carbon dioxide of varying pressure and purity has been examined.

A comparison between optical and gravimetric results is instanced in

figure 20, the overall result showing similarities to the example already

quoted for uranium. The divergence between the two sets of results when the film reached 1000 A in thickness indicates that at this point,

localised oxidation, which with beryllium takes the form of intergranular oxidation, is becoming the dominant cause of oxygen uptake.

Qualitative confirmation of this point is provided by metallographic

examination of samples so severely internally oxidised to be grossly

swollen and distorted, whilst the outer grain surfaces still remain

locally flat and covered with second order temper colours. Figures 21a and b illustrate Z curves obtained from oxidation under

different conditions of humidity and pressure of the carbon dioxide.

Three observations of particular interest are :

728-101 0^64 13
188 HAYFIELD AND WHITE

DRUDE-TRON STAD OPTICAL RESULTS

GRAVIMETRIC

RESULTS (THERMOBALANCE ) *-a-*-a

OXIDATION AT 700°C

,. 650 °C

.. 600 °C

» 700 °C

» 650 °C

.0 10 TIME IN HOURS Figure 20. Oxidation of beryllium over the temperature range 600-700 °C

carbon dioxide. in wet

(i1 the "hump" in the Z curve produced by very thm films formed in

wet carbon Sde (fig. 21a) . This change in shape of the Z curve at ful an early stage must result from a marked change m optical

nrouerties of the film Under these conditions the only variable capaEkoSroducing the desired change in a theoretical model is

, a sharp

fall inSm refrfctive index, suggesting the formation of an hydrated

°XigThKrad

se

e in optical absorptivity of film formed in wet as 0P^^^iyTncrease°Fn"absorptivity of oxide formed at 300 psi

compared with carbon dioxide at atmospheric pressure

The differences in film absorptivities noted m W^W

thought to arise from defects in the oxide, producing absorption peaks
FILMS ON POLYCRYSTALLINE METALS 189

Wjm OXIDISED 600- 700°C IN WET

C02 AT ATMOSPHERIC PRESSURE

Fv^l OXIDISED 600-700°C IN DRY

t^ C0 2 AT ATMOSPHERIC PRESSURE

(b)

270°

Figure 2 1 a and b . Experimental Z curves for oxidation of polycrystalline beryllium.
190 HAYFIELD AND WHITE i

in the visible spectrum. Some metallographic confirmation of this

point is provided by the appearance of oxide overlying gram boundaries. Examination of specimens in the very early stages oi reaction

in wet gas, before localised oxidation of gram boundaries has been

initiated, shows a discolouration effect over grain boundaries which

serves to highlight the grain structure. Covering such surfaces with vacuum evaporated aluminium causes this network to disappear

suggesting that the observed appearance of dark lines is associated

with a colouration effect in the oxide as opposed to a topography variation The evidence is consistent with the film covering gram boundaries becoming highly non-stoichiometric prior to breakdown. On
FILMS ON POLYCRYSTALLINE METALS 191

DRY

ATMOSPHERE

C0 2 7GRAIN BOUNDARY

BeO FILM BERYLLIUM

SUBSTRATE

WET 7

300 p.S.i. C0 2

Figure 22. Diagram illustrating the probable degree of disorder existing in beryl- lium oxide formed under different conditions of oxidation in carbon dioxide.

Density of spots is a measure of disorder in the beryllium oxide film as deduced from micro-discolouration

effects and Drude-Tronstad optical measurements. the basis of this and Drude-Tronstad measurements of average film

absorptivities, the lattice disorder in beryllium oxide formed under

different conditions can be represented by the model shown in figure 22.

c. Copper

Optical data obtained during the oxidation of copper over the

temperature range room temperature to 200 °C in pure oxygen has

been reported elsewhere [18]. The observed changes in film absorptivity with conditions of formation are attributed to variations in

cupric ion content of cuprous oxide. These results are confirmed

by Wieder and Czanderna [19], who have reported, on the basis of additional techniques, including electrometric reduction of oxide and

microchemical analysis, that film forming at the temperature under

consideration has an overall composition of CuO0 67 which changes

to CuO at temperatures above 300 °C.

4.3. Aqueous Corrosion

a. Brass in Seawater

Polycrystalline aluminium brass, 76/22/2, polished on % » diamond

dust, has been exposed to circulating seawater at room temperature
192 HAYFIELD AND WHITE

WEIGHT LOSS 3. 5 m g TORBAY SEAWATER FOR 94 h rs WEIGHT LOSS 5.3 m g

Figure 23b. Theoretical Z curves describing growth of cuprous oxide and. cupric

chloride hydroxide films on aluminium brass immersed in sea water.
FILMS ON POLYCRYSTALLINE METALS 193

for periods of several weeks, and subjected to intermittent optical

; observations without removal from the solution. Film formed during

the first 24 hr on test, i.e., the portion ab of the Z curves illus- trated in figure 23a, corresponds to the growth of a highly absorbing

form of cuprous oxide. This film attains a limiting value even though

| the metal oxide interface is receding at a rate considerably faster than

film growth. After a further period of immersion, the length of time

varying with batch of seawater and other variables, a second film, this one of cupric chloride hydroxide, begins to form rapidly on top

of the first film layer, producing a range of Z curve shapes as illus- trated. The direction of this second portion of curve is variable,

I and some possible explanations for the courses taken can be gaged

from the theoretically derived Z curves in figure 23b, derived on the

1 basis of a two-film model. On this interpretation the overlying film

varies widely in absorptivity and therefore composition, and may

| reach thicknesses considerably in excess of the initial cuprous oxide.

b. 70/30 Cupro-Nickel in Seawater

In a similar manner to that just described for aluminium brass,

j samples of 70/30 cupro-nickel were prepared by mechanical polish-

| ing and then exposed to natural seawater for periods of several weeks,

j Z curves describing optical observations recorded during the first 30 hr

of immersion are illustrated in figure 24a. Electron diffraction examination revealed the presence of a layer with crystal size too small

for identification, but some deductions about its properties can be

made from trying to prepare theoretical models capable of reproducing

the Z curves obtained (fig. 24 b and c). To produce crescent-shaped

curves of the type illustrated, rather than the spiral-type growth

observed for brass, it is necessary to postulate a film which is not only

of low refractive index but which is also transparent. These findings

preclude the possibility of cuprous oxide being formed, and suggest

a corrosion product containing principally salts precipitated from the

seawater. X-ray microanalysis of reaction products indicates the

existence of magnesium and sulfur, suggesting that the reaction

product contains magnesium sulfate.

c. Aluminium in Demineralised Water at 80 Q C

Specimens of polycrystalline commercial purity aluminium were

polished on % \x diamond dust and exposed to demineralised water

at 80 °C for periods of a few days. At frequent intervals samples were removed from the solution for optical examination; the surfaces

dried off quickly and naturally with no visible staining. Examples

of the types of Z curve produced are shown in figure 25, other variable

loop-type shapes being observed. These curves show a habit that is very different from the theoretical curves illustrated in figure 5, or the Z curves produced by anodisation of the same material in am- monium borate solution. Similar differences have been reported by

Tronstad [20] and Barrett [16, 17] under slightly different conditions.
194 HAYFIELD AND WHITE

Figure 24a. Experimental Z curves describing the corrosion of 70/80 copper/nickel

(_|_ 1% iron and 1% manganese) in various natural sea waters.

Figure 24b. Theoretical curves for growth, on 70/30 copper/nickel, of film having

refractive index 2.75 and various optical absorbtivity.
FILMS ON POLYCRYSTALLINE METALS 195

270'

Figure 25. Experimental Z curves for the formation of Bohmite film on mechanically

polished aluminium exposed to deionized water at 80 °C.
19Q HAYFIELD AND WHITE

Examination of samples corroded in the present experiments revealed

Bohmite formation (A1203H20) only, but metallographically the

film was found to be very non-uniform in thickness. This latter

effect arises from the presence of pits which are each surrounded by

an annulus of thick reaction product.

In order to interpret the observed Z curve, the ellipticity to be

derived from a wide range of theoretical models has been computed.

Uniform thickness single layer films of a wide range of refractive

index produce crescent-shaped curves in which the lower turning point

is always near 270°. Other models, based upon a high refractive

index "barrier" layer adjacent to the film, and with an outer lower

refractive index film, also produce similar shaped Z curves, which , differ only in the dispersion of film thickness values around the curve and in the total thickness necessary to give a complete rotation.

Neither do models, based on a large number of films in which the

refractive index is made to decrease from metal to the air interface m a , more uniform manner, produce any major change in Z curve shape.

Introduction of absorbing film at either of the interfaces, using either

two or three layer models, fails to reproduce in any way the shape

of the experimental curves illustrated in figure 25. Theoretical curves derived on the basis of eq (25) can be made to reproduce the

desired curve but, as described in sections 2 and 3, there is no theoretical justification for doing so. Interpretation of optical data m this instance remains unsolved.

5. Discussion

Examples illustrated in the foregoing section show that where re- action conditions give rise to formation of uniformly thick optically

homogeneous film, whether the substrate is a single crystal or polycrystalline, there is close correspondence between experimental DrudeTronstad observations and theoretical data derived from eq (10).

Under these conditions optical observations may be interpreted with

the full accuracy inherent in the technique. Although this will vary

widely with the optical constants of the particular media concerned,

with film thickness levels, etc., it generally means that absolute film

thickness measurements can be made accurate to a few angstrom units.

There are, however, few conditions encountered during oxidation

and corrosion of metals where such ideal filming conditions exist, the

principal reason being non-uniformity of composition caused by

concentration gradients, these being responsible for influencing optical properties. The closest approach to simple optical properties

results from such conditions as are met in aqueous anodic oxidation, more especially on single crystal as opposed to polycrystallme material.

However, even under apparently uniform film thickness conditions.

Heavens and Kelly [21] have shown that, for the anodisation oi tantalum in N/10 sodium sulfate, the average film refractive index

varies linearly from 1.95 to 2.15 with increasing thickness. Although

the advent of computers has made possible calculations which take

account of constantly changing refractive index, such changes are not
FILMS ON POLYCRYSTALLINE METALS 197

readily detected experimentally except by the special means employed

by Heavens and Kelly, a procedure not always possible to apply during

i corrosion studies.

As the conditions of film formation become more complicated, e.g. by the formation of non-uniform thickness oxide, layered oxide structures, mixed oxides, stressed oxides, oxide nucleaton etc., the expectancy of providing a perfect interpretation of observed optical changes

becomes smaller. Interpretation cannot be made without recourse

! to examination by a wide range of additional techniques. For data

reported in section 4 it was found mandatory to obtain reflection

electron diffraction examination of samples both before and at the

I end of a reaction, and also to assess qualitatively the homogeneity

of film by reference to optical and electron microscopy. By this means an interpretation may be attached to optical observations which

is sufficiently unequivocal to be of scientific value. The failure to

j achieve maximum sensitivity of absolute film thickness determinations does not mean that the method becomes generally inapplicable.

Indeed, because of the considerable sensitivity still achievable, it remains under many conditions the most sensitive technique still available for measurements of film thickness in the 10 to 1000 A range.

6. Future Developments

In the U.K. there are no instruments available with electronic

recording, and a method utilised by Hodgson [22] would appear to

offer considerable advantages. With such an improved method of recording, the logical extension of the technique to the measurement

of optical changes over a wide range of wavelengths would become

possible. Up to this stage in its development the technique has been

employed almost exclusively for measurement of film thickness, but

in future the detection of absorption peaks corresponding to various

lattice imperfections may be of equal or greater value. For most

semiconducting materials the spectrum of the visible and ultraviolet

wavelength radiations is dominated by the fundamental lattice absorption, and consequently is a field not greatly investigated be- cause crystals cannot be thinned sufficiently for conventional examination by transmission. Using the optical reflection technique

described in this paper, and by utilisation of computing aids now

available, the derivation of absorption data in this wavelength range

is entirely practical, and it should now be possible to study absorption

peaks arising from lattice imperfections in this wavelength range, a subject only occasionally mentioned in the literature. On the high

wavelength side of the fundamental absorption band for semiconductors, and more especially in the infrared, the correlation between

lattice imperfections and associated optical transitions has been

widely reported for germanium and silicon, and to a considerably

smaller extent for metallic oxides. Drude-Tronstad observations m

this wavelength range represent a method of detecting the presence

of absorption peaks in thin films, and offers the means of classifying
198 HAYFIELD AND WHITE

the type and average concentration of impurities present, necessary

data for the confirmation of theoretical models describing thin film

growth of the type proposed by Uhlig [23]. General experience on corrosion of metals and alloys in service

promotes the idea that microfailures in thin films are more important

than macrofilm properties of the type described in the present paper.

The development of a micro-optical technique would seem to merit

consideration, but it is unlikely that the minimum area illuminated

could be made sufficiently small to be of practical value. This is because the area for illumination must be a few microns, i.e., close

to the resolving power of microscopes using visible wavelength radiation, and manifestly too small for measurements with infrared i radiation.

7. Conclusions

Equations have been derived describing reflection of plane polarised

light from surfaces covered with various film structures, and some

experiments designed to check their validity have shown that al- though the theory of two layer films is in broad agreement with

experimental observations, the theory of films of non-uniform thick- ness is still inadequate. Subsequently, a survey has been given of the applicability of the Drude-Tronstad polarised light method to the

study of film growth on polycrystalline materials under different

corrosion conditions, and it is concluded that results may be obtained

which are helpful for understanding the mechanism of the processes under examination.

The authors gratefully acknowledge the sanction of the Directors

of Imperial Metal Industries (Kynoch) Limited to publish this paper. In addition, sections 4.3(a) and 4.3(b) are included by

permission of Yorkshire Imperial Metals Limited, sections 4.2(a)

and 4.2(b) by permission of U.K.A.E.A., and figure 19 by permission

of The Institute of Metals.

8. References

[1] P. Drude, Ann. Phys. 32, 584 (1887); 34, 489 (1888); 36, 532, 865 (1889);

39, 481 (1890).

[2] L. Tronstad, Trans. Faraday Soc. 29, 502 (1933).

[3] O. S. Heavens, Optical Properties of Thin Solid Films, Butterworths, London

(1955).

[4] A. Vasicek, Optics of Thin Films, North Holland Publ. Co., Amsterdam

(1960).

[5] G. W. T. White, J. Appl. Sci. Res. 6A, 433 (1957).

[6] F. Abeles, Ann. d. Physique 3, 504 (1948).

[7] A. B. Winterbottom, Norske Videnskabers Selskab 45, 1 (1955).

[8] A. B. Winterbottom, J. Opt. Soc. Am. 38, 1074 (1948).

[9] P. C. S. Hayfield, First Intern. Congr. Metallic Corrosion, p. 663, Butterworths, London (1962).
FILMS ON POLYCRYSTALLINE METALS 199

[10] A. B. Winterbottom, J. Electrochem. Soc. 76, 327 (1939).

[11] A. Rothen, Ann. N.Y. Acad. Sci. 53, 1054 (1951).

[12] F. P. Mertens, P. Theroux, and R. C. Plumb, J. Opt. Soc. Am. 53, 788

(1963).

[13] E. C. Williams and P. C. S. Hayfield, Institute of Metals Monograph No.

23, p. 131 (1957).

[14] R. Mallemann and F. Suhner, Rev. D'Opt. 23, 20 (1940).

[15] K. B. Blodgett and I. Langmuir, Phys. Rev. 51, 964 (1937).

[16] M. A. Barrett and A. B. Winterbottom, First Intern. Congr. Metallic Corrosion, p. 657, Butterworths, London (1962).

[17] M. A. Barrett, p. 213 in this volume.

[18] P. C. S. Hayfield, First Intern. Congr. Metallic Corrosion, p. 670, Butterworths, London (1962).

[19] H. Wieder and A. W. Czanderna, J. Phys. Chem. 66, 816 (1962).

[20]' L. Tronstad and T. H0verstad, Trans. Faraday Soc. 30, 362 (1934).

[21] O. S. Heavens and J. C. Kelly, Proc. Phys. Soc. 122, 906 (1958).

[22] J. N. Hodgson, Phil. Mag. 4, 183 (1959).

[23] H. H. Uhlig, Acta Met. 4, 541 (1956).

Studies of Thin Oxide Films on Copper

Crystals With an Ellipsometer

J. V. Cathcart and G. F. Petersen

Metals and Ceramics Division, Oak Ridge National Laboratory 1 The problems encountered in the use of an ellipsometer to study

the oxidation of copper single crystals are described. Thin > Cu? 0

films are crystalline, highly oriented, and subject to large epitaxial

stresses. The resulting strains, investigated by both optical and

x-ray techniques, cause the films generally to be optically anisotropic. In addition the average index of refraction varies with film thickness,

and a gradient in the refractive index exists across the films. Modifications of the classical equations for reflection from a film- covered interface are suggested in an effort to provide a more realistic optical model for the films. A description is given of the construction

of a thickness calibration curve for the ellipsometer using these equations in conjunction with a series of thickness values obtained from

x-ray measurements. One of the uses for which the ellipsometer is in principle admirably

suited is the investigation of the kinetics of the early stages of oxidation of metal single crystals. Various advantages afforded by the

ellipsometer, such as (1) the nondestructive nature of the measurements, (2) the possibility of making essentially continuous measure- ments at a distance, and (3) the extreme sensitivity of

# the method,

have all been mentioned in previous papers. In practice, however,

serious complications may arise from the fact that oxide formed on

such metal specimens is frequently a highly oriented, crystalline material [l].

a The existence of epitaxial forces between oxide and metal

may lead to an inhomogeneous distortion of the oxide and possibly

of the metal substrate as well. In general, films will not only be

optically anisotropic, but a gradient in their index of refraction will This paper reports on the problems encountered in the use of an

ellipsometer in the study of the oxidation of copper single crystals m

which both ellipsometry and x-ray measurements were used to characterize the oxide films.

The initial indication of the existence of anomalous optical effects in Cu20 films came in the course of a study of oxide films formed on

the (100), (111), (110), and (311) faces of copper [2]. At that time an

ellipsometer, which had been calibrated in terms of a literature value

for the index of refraction of Cu2 0 on the assumption that the Cu2 0

1. Optical Anisotropy in Cu2 0 Films

1 Operated for the TJ.S Atomic Energy Commission by Union Carbide Corporation.

2 Figures in brackets indicate the literature references on page 210. 201
202 CATHCART AND PETERSEN

440

430

420

410 400

405 P•- 395 |- 205

195 260

\

: 3

1 u H

\

» V

/ L

-/

J L ^(100)_

240 230 220

210 200

190

180

--

1 4= u

r\— ±t\

- f

-

/

-1

- \ - X-JFigure 1. Variation of the apparent oxide film thickness on four crystallographic planes

of copper as a function of the angle of rotation about the film normal.

After Cathcart, Epperson, and

Petersen [2].

60 120 180 240 300

ANGLE OF ROTATION ABOUT SPECIMEN NORMAL (deg

films were optically isotropic, was being used to measure the thick- nesses of the oxide films. It was discovered, however, that the ro- tation of the (110) and (311) specimens about their surface normals

led to periodic changes in the ellipsometer readings. Such optical

anisotropy obviously invalidated the previously used calibration scale

for the ellipsometer, but it was felt that the anisotropy of the films

might still be usefully characterized qualitatively in terms of changes

in "apparent film thicknesses" obtained from this calibration curve. Typical sets of data for the four crystallographic planes of copper

investigated are shown in figure 1. For the (311) and (110) specimens, maxima and minima in the thickness values occurred at 90°

intervals during rotation. No such variation in apparent film thick- ness was observed for oxide films formed on the (111) and (100) faces

of copper, nor did these films exhibit optical anisotropy when examined with a polarizing microscope.

These effects may be rationalized through a consideration of the

epitaxial relationships between the oxide and metal and of the result- ing strains in the oxide. As is well known, thin oxide films on copper

single crystals are highly oriented and may be regarded as pseudo

single crystals ; the epitaxial relationships are summarized in table 1

When a cuprous oxide film is viewed symmetrically with respect

to the film normal, as is the case for ellipsometer measurements, the

observation of optical anisotropy requires that the crystal lattice in the plane of the oxide film be strained in such a way as to have lost the elements of cubic symmetry. It is to be expected that this condition is fulfilled for the oxide formed on both the (110) and (311)
OXIDE FILMS ON COPPER CRYSTALS 203

Table 1. Epitaxial relationship between C112O and copper after Lawless and

Gwathmey [1]

Direction in sur- Plane of Plane of oxide face of oxide copper parallel to parallel to [110] copper surface in surface of metal

(100) (111) 110

(111) (111) 110

(110) (110) 110

(311) (110) 110

faces of copper, either because the ratio of the elastic constants for the principal directions in the oxide is different from that for the metal,

or because of differences in orientation of oxide and metal which lead

to a variation in misfit as a function of direction in the plane of the

film. The situation for the (111) faces of copper is different. Not only

are the lattices of the oxide and metal completely parallel to one another, but the threefold symmetry exhibited by both requires circular

symmetry for the stresses developed in the (111) plane. Thus a

distortion of the oxide lattice should be circularly symmetrical about

the surface normal, and the film should exhibit no optical anisotropy.

A similar argument, based also on the high degree of symmetry of both oxide and metal, may be used to explain the lack of anisotropy

of the (100) films. In general, however, it would appear that epitaxially induced strains in oriented oxide films will lead to an anisotropic

distortion of the films except in special cases. An attempt was made to determine the change in the degree of anisotropy of the oxide as a function of average apparent film thickness.

The "degree of anisotropy" for a given film was defined in terms of

its maximum and minimum apparent thickness values:

Degree of anisotropy =(

C?ma^~^mln) 100. A plot of the degree of anisotropy versus the average apparent film

thickness is shown in figure 2 for oxide films on the (110) face of copper. A similar curve was obtained for films on the (311) face. The most

striking feature of the figure is the large peak in the degree of anisotropy which occurs at an average apparent thickness of about 200 to 300 A. Given the association of optical anistropy with anisotropic strains

in the oxide film, it is logical to relate a change in the degree of optical

anisotropy with a corresponding change in the degree of strain anisotropy. It appears that the existence of the pronounced peak in the

degree of anisotropy curve reflects a more rapid change in the strain

in one principal direction of the oxide than another, although the

nature of such a change is still not fully understood. In any event,

728-101 0—64 14
204 CATHCART AND PETERSEN

1 I OXIDATION TEMPERATURE

a 150°C 200°C

o 250°C

\ —1-

1

11

0 300"J 0

-0—

Figuee 2. Variation of degree

of anisotropy of oxide on the (110) plane of copper as a

function of film thickness.

After Cathcart, Epperson, and

Petersen [2].

200 400 600 800 1000 AVERAGE APPARENT FILM THICKNESS (A)

the data illustrate a sensitive, although qualitative, method for studying structural changes in growing oxide films with an ellipsometer.

2. Strain Model for Oxide Films Deduced From X-Ray

Diffraction Data

The optical studies just described illustrate the dangers of considering the oxide as a uniform, homogeneous film and emphasize the need

for the development of a more realistic optical model for the tilms it the full potential of the ellipsometer is to be realized. Fortunately,

the existence of an x-ray technique, recently developed at the Oak

Ridge National Laboratory by B. S. Bone and C. J. Sparks L3, 4J, has permitted them to make useful measurements of lattice strain,

strain gradients, and oxide film thicknesses for Cu20 films with aver- age thicknesses as small as 20 A, and the data thus obtained provide

considerable insight into the detailed structure of the oxide films. Lattice parameter measurements for films on the (110) lace ot copper [5] showed the existence of a lattice expansion of 1 to 2 percent

in a direction normal to the surface of the oxide, and preliminary

results for films on the (111) [6] and (311) [7] faces suggest the pres- ence of comparable strains in these films as well. The strain appears

to be of epitaxial origin, and oxide on the (110) is compressed m the

plane of the film. These lateral strains are themselves anisotropic,

the oxide being compressed to a maximum degree in the direction

parallel to the [001] direction in the surface of the metal. In the [110J

direction, on the other hand, the average lattice parameter of the oxide

is equal to that for bulk oxide. The average unit cell volume ot the

oxide is also that of the bulk material. # An analysis of the line shapes of the diffraction maxima lor the

films indicates the existence of a strain gradient m the oxide on the

(110) face of copper. At the oxide/gas interface the oxide has bulk

lattice parameters, and the strain varies approximately linearly as a function of position in the film, reaching a maximum at the oxide/

metal interface. It has been suggested that this strain gradient m

the oxide may be described in terms of an array of edge dislocations

in the oxide [5].
OXIDE FILMS ON COPPER CRYSTALS

^ 2.480

J

§X

-t

t

200 300 400

OXIDE THICKNESS (A) Figure 3. Variation of the average lattice parameter measured along film normal as a function of film

thickness [6]. Finally the average lattice parameter of the oxide changes as a function of film thickness. In figure 3 the average lattice parameter,

measured normal to the surface of the film on a (111) of copper, is plotted against film thickness. For extremely thin films, the oxide

has bulk lattice parameters, but din then increases rapidly and passes

through a maximum. Similar measurements on oxide films on the

(110) face of copper also demonstrated the existence of variations of duo as a function of film thickness.

The observation that a 25-A-thick Cu20 film has bulk lattice parameters was a surprising one, since epitaxially induced strains

would be expected to be at a maximum in the thinnest oxide films. This result may be rationalized in terms of a discontinuous oxide

film and lattice distortions in the substrate metal [6], but the possible

consequence for the optical properties of the system occasioned by

changes in the index of refraction of the substrate with film thickness

has not been investigated.

The physical picture of the oxide as indicated by the x-ray data

may be summarized as follows: In general the oxide film is subject to

epitaxial forces in the plane of the surface, causing the oxide to be

compressed in a lateral direction; normal to the surface there is a cor- responding expansion of the lattice. A strain gradient exists in the

film, which implies the existence of a corresponding gradient in the

index of refraction of the oxide. Likewise, the average lattice parameter of the film varies with film thickness, implying that the average

index of refraction of the oxide is also a function of film thickness.

3. The "Gradient" Optical Model for the Oxide

An attempt was made to formulate an optical model (hereafter

referred to as the "gradient model") for the oxide films which takes
206 CATHCART AND PETERSEN

i

! into account these findings. The film was considered as consisting of ! ra layers, each of which was treated as a homogeneous, isotropic I medium. Reflection from this "composite film" was described in terms of the standard Drude equations [8], making use of the recursions

formulas developed by Wolter [9] for reflection from multiple films. The only modification made in the equations was that suggested by

Vasicek [10] to take into account the effect of the optical absorbancy

of the Cu20 films.

Figure 4. Schematic repre- sentation of an m-layered

oxide film with refractive in- dices ui, 112, . . . nm on a

substrate with refractive index

no immersed in a transparent

medium with refractive index nm+i.

Reflection from a layered system consisting of m+1 interfaces may

be described in terms of the ratio of the Fresnel reflection coefficients

^|±^=tan^exp iA

where Rm+l and R

sm+i are the reflection coefficients for respectively,

the parallel and the perpendicular components of the electric vector of the incident light. The experimentally determined quantities, tan \f/ and A, represent the relative amplitude reduction and the phase

retardation, respectively, suffered by the two components on reflec- tion. Values of R^+i are given by „a _ C+i+C exp (—2pmdm) m+1

1 + MUi) *K exp (-2Pmdm) where a=p or s, W+i)* represents the complex conjugate of r^+1 , and r« and r%+1 , the reflection coefficients for the rath and (ra+l)th

interfaces, are obtained from the recursion formulas given below.
OXIDE FILMS ON COPPER CRYSTALS 207

(92—gi)Nl exp (Pid1 ) + (g2 +g1 )Zl exp (- Pld1 ) _Z2 (92+giWi exp (pid1 )-jr (g2 —g1 )Z1 exp (—Pldi) ~N2

(gm+i—gm)Nm exp (pA) + (gm+i+gw)Zw exp (—Pmdm) (gm+i Jrgm)Nm exp (pmdm) + (gm+1 +gm)Zm exp (—pmdm) For a=s, gm =nm cos <j> m, and for a=p, gm =cos <j>Jnm where nm is the index of refraction of the mth layer and $m is the angle of incidence in the mth layer. The thickness of the mth layer is

j designated by dm, and pm =2irinm cos <j)J\ where A is the vacuum

I wavelength of the incident radiation. In accord with the results from the x-ray study, it was assumed that

! a gradient in the refractive index exists across the oxide and that the

average refractive index of the oxide is a function of film thickness.

The functions chosen to express these variations are shown below for the refractive index of the jth layer, n^rij—ikj.

where nh kj} nm, and km are the real and imaginary parts of the

refractive indexes of the jth and mth layers, m is the total number of

layers, a, a', b, b'', c, and c' are constants, and d is the oxide thickness.

The three unknown quantities in the classical Drude equation for

I reflection are the thickness, d, and the index of refraction of the oxide,

fh=n\—ikx . The usual procedure for determining nx involves

varying nx and kx until tan ^ and A values computed for a series of thicknesses agree with a corresponding set of experimentally determined

values of tan xf/ and A. Equations (1) and (2) introduce an additional

set of adjustable constants (a, a', b, c, and c') into the basic Drude

equation, and values for these constants must be chosen in order to calculate tan \p and A for a given oxide thickness.

These equations were programmed for an IBM 7090 computer, and

values of tan \p and A, expressed in terms of the corresponding analyzer

(4) and polarizer (P) readings, were computed for a series of film

thicknesses for specified ranges of both the real and imaginary parts

of the index of refraction of the oxide.

The object of these calculations was to find those values for nm, km, and the other adjustable constants which produce a fit between plots

of calculated and observed A and P values. To this end the differ- ences between Acalc and ^lob8 were determined for each value of Pob8 and the squares of these differences summed. The combination of

(1)

(2)
208 CATHCART AND PETERSEN

n and k for the oxide yielding a minimum for S(^.cai c —

obs) was

taken as the value of the refractive index providing an optimum lit between calculated and observed data.

4. Results

The equations described above do not take into account the existence

of optical anisotropy in the oxide films, and, as might be expected,

considerably greater success was achieved m applying the equations

to data for oxide films on the (111) face of copper than for films on

the (110) face where, as already pointed out, the oxide is distorted

anisotropically. Only the results for the (111) films will be discussed

m

l

hlcomparison of calculated and observed data for an oxide film

formed at 200 °C and 1 atm 02 pressure on the (111) face of copper

is shown in figure 5. Analyzer readings are plotted against polarizer

readings As may be seen, the two curves match reasonably well,

especially for small thickness values. On the basis of curve httmg

alone, however, it was very difficult to select a unique value lor the

index of refraction of the oxide ; the nature of the problem is illustrated

in figure 6, which shows a plot of km (the imaginary part of the index

of refraction, the plotted value being that at the oxide/gas interlace) versus S(Acalc -Aohs) 2

. Each curve represents tests of the fit between observed and calculated data for a fixed value of the real part

of the index of refraction of the oxide while the imaginary part is varied Clearly, for each real value of the index, there exists a value

of k which minimizes 2 (Ac&lc -

ohB)

2

, and while it is possible to identity an ''optimum curve," the improvement m fit, measured m terms ol the average deviation per experimental data point, lies within ex- perimental error for a modest range of values for the index ol retraction

100 120 140 160 POLARIZER READING (deg) Figure 5. Plot of polarizer versus analyzer reading for oxide film grown on the (111) face of copper at 200 C

and ~1 atm oxygen pressure.
OXIDE FILMS ON COPPER CRYSTALS 209

80 120 160 2<ACALC -A OBS ) 2 Figuee 6. Curves measuring the "fit" between

calculated and observed ellipsometer data for various values of the real and imaginary parts

of the index of refraction of the oxide.

40 80 120 160 200 240

OXIDE THICKNESS (A)

2.455 320 360 400 Figure 7. Plot showing the variation of dm and k, the imaginary part of refractive index of the oxide, as a

function of oxide film thickness. of the oxide. This problem is especially serious for films <250 A

where uncertainties in thickness values corresponding to uncertainties

m the index of refraction of the oxide may run as high as ± 50 percent.

For these reasons it was concluded that the construction of a

thickness calibration curve for the ellipsometer by curve-fitting

procedures was not satisfactory. Instead a purely empirical approach

to the problem was adopted, and a comparison was made between a

series of thicknesses obtained by x-ray measurements and film

thmknesses calculated for the ellipsometer. The best agreement was obtained using the gradient model for the oxide film with nm =3.00,

a =o=c=0, km =0.190, a'=c' = l, and with the value of V chosen so that the variation of the average value of k for the oxide and its value at the oxide/metal interface were those shown in figure 7. As may be

seen, the change in the average index of refraction for the oxide

corrpsponds roughly to the change in dm as a function of film thickness.
210 CATHCART AND PETERSEN

A comparison of thickness values from x-ray measurements with

those from ellipsometer data is shown in table 2. The two sets of ellipsometer thickness values correspond to thicknesses determined

from the "gradient" model for the oxide and from the classical equations in which the oxide is assumed to be homogeneous and isotropic.

Comparison of thickness values obtained by x-ray and ellipsometer measurements Table 2.

x-ray Ellipsometer

Gradient model Isotropic model A A A

76 92 126 130 125 169

148 141 197 205 205 296 The data for the isotropic oxide model were obtained with the index

of refraction for the oxide of 2.50 to 0.265 i, which was the index

producing the "optimum" fit for that oxide model. The average

deviation per experimental data point between the calculated and

observed curves over the thickness range tested was 0.25° for the

isotropic model and 0.11° for the gradient model.

Compared to the isotropic model, the gradient model tor the oxide

not only produced a better fit between calculated and experimental

data but also gave thickness values in much better agreement with the

x-ray data Thus, it seems reasonable to assume that the thickness

calibration curve obtained with the gradient model is accurate to thicknesses of at least ~250 A. It should be noted, however that

the changes made in the optical equations take the form of the addition

of adjustable constants in the equations. A better fit of experimental

and calculated data is, hence, to be expected, but it is questionable

that real significance should be attached to the index of retraction

values obtained from the equations.

5. Summary

Optical and x-ray studies of thin Cu20 films on copper crystals

demonstrated that the classical equations for reflection from a tilmcovered surface are not adequate when the films are crystalline and

subject to epitaxial forces. A more realistic optical model lor the

oxide was proposed on the basis of these studies. This new model

permitted an empirical thickness calibration curve to be constructed

for the ellipsometer, but it was felt that little physical significance

could be attached to the refractive index values for the oxide indicated

by the model.

6. References

[1] K. R. Lawless, and A. T. Gwathmey, Acta Met. 4, 153 (1956)

[2] J. V. Cathcart, J. E. Epperson, and G. F. Petersen, Acta Met. 10, 699-703

(1962).
OXIDE FILMS ON COPPER CRYSTALS 211

[3] B. S. Borie, Acta Cryst. 13, 542 (1960).

[4] B. S. Borie and C. J. Sparks, Acta Cryst. 14, 569 (1961).

[5] B. S. Borie, C. J. Sparks, and J. V. Cathcart, Acta Met. 10, 691-697 (1962).

[6] J. V. Cathcart, G. F. Petersen, and C. J. Sparks, Lattice disregistry in very

thin oxide films on copper, Proc. Symp. on Nucleation Processes in Gas- Metal Reactions, Paris (1963).

[7] C. J. Sparks, unpublished research.

[8] A. B. Winter-bottom, J. Opt. Soc. Am. 38, 1074-82 (1948) ; K. Danske vidnska- bernes selskab Skrifter, 1955 NR 1, p. 1.

[9] H. Wolter, Optik diinner Schichten, Handbuch der Physik, Vol. XXIV, pp. 461-544, Springer-Verlag, Berlin-Gotingen-Heidelberg (1956).

[10] A Vasi'cek, Opt. and Spectr. 11, 128-130 (1961). Discussion

H. E. Bennett (Michelson Laboratory)

I have two questions. First, could you tell us about the surface

preparation of the copper? Was it electropolished, and did you look

at surface damage using electron diffraction or a similar technique?

Second, could you give a fuller description of the gradient model

used? Was the variation linear with thickness or exponential?

J. V. Cathcart:

To answer the first question, we used electropolished copper

crystals which were nominally 99.999 percent copper. The surfaces

of these crystals, as far as we could tell from replica studies with

the electron microscope, were essentially featureless after electro- polishing. We used the standard phosphoric acid polishing solution

for the specimens.

With regard to the form of the gradient of the index of refraction

in the oxide, we assumed the film to consist of m homogeneous layers,

the index of refraction of the jth. layer then being given by

rij and nm are the refractive indices of the jth and mth layers, re- spectively, d is the oxide thickness, and a, b, and c are constants. In most of our calculations we set a==c=l.

K. H. Zaininger (Radio Corporation of America) : How sensitive is this x-ray method that you mentioned?

J. V. Cathcart:

t We think it is very sensitive. The method is based on the assumption that all the oxide film present is very highly oriented; in other

words, all of the oxide film that the x-ray beam intercepts is in position to diffract when the specimen is rocked through a modest

angle. Within the limits of this assumption, which I think is a

fairly good one based on electron diffraction evidence, films approximately 20 A thick can easily be studied. This technique, which was
212 CATHCART AND PETERSEN

1 developed by B. S. Borie and C. J. Sparks at the Oak Ridge Nationa

Laboratory, is described in two papers that appeared in Acta

Crystallographica 13, 542 (1960) and 14, 569 (1961). Two additional

papers describing the application of the technique to thin Cu20

films on copper crystals may be found in Acta Metallurgica 10,

691-97 and 699-703 (1962). F. L. McCrackin (National Bureau of Standards)

What is the modification of the Drude equation by Vasicek and

where is it published?

J. V. Cathcart:

Vasicek's paper appeared in the Russian Journal of Optics and

Spectroscopy 11, 128-30 (1961). The basic change he makes comes

from applying the Stokes reversibility principle in deriving the

reflection equations. Such a consideration leads to the substitution

in the denominator of the general equation for reflection from a thin

film of the complex conjugate of the reflection coefficient for the

first interface of the system.
Optical Study of the Formation and Stability

of Anodic Films on Aluminium

M. A. Barrett

Norges Tekniske Hbgskole, Trondheim, Norway

The ellipsometer has been employed in a study of anodic films on aluminium primarily with the aim of understanding the effect of this type of film on the corrosion resistance of aluminium in hot water. Measurement of the thicknesses of the two layers resulting either from

pore formation during anodising, or hydration by the action of hot water was accomplished by a combination of total thickness measure- ments and determinations of the anodic polarization necessary for the recommencement of the film growth, a quantity dependent only on the

thickness of the inner compact layer.

1. Introduction

The ellipsometer built by Tronstad [1]

1 has been used in a study

of anodic films formed on aluminium, and on the corrosion of alu- minium in hot water [2] . Visual half shades have been used in making

most of the measurements, rather than a photomultiplier, enabling

observations to be made which relate to specific small areas when the

reaction rate varies markedly over the surface. In all cases, measure- ments were made in situ, and it has been found profitable to combine data from the ellipsometer with electrical measurements, since

the combination provides a means of studying the porous outer layer

and the compact inner layer independently.

The two types of film substance involved, the anhydrous and the

mononydrated aluminium oxide, are both completely transparent,

but they vary markedly in the ease with which the results can be interpreted. To obtain a growth curve for the anodic film, the electrolyte

3 percent ammonium tartrate was chosen. This produces an essentially nonporous film. Films produced with this electrolyte yielded

a closed curve when the delta and tan psi data from the ellipsometer were plotted on a polar graph for a series of thicknesses (fig. 1), corre- sponding to the growth of a homogeneous, transparent film. The

extent to which the curve is retraced on the second cycle can be seen on the figure, where the circles represent readings taken in the first cycle and the triangles those in the second cycle. The dashed line represents the theoretical curve for growth of film in the electrolyte

with those optical constants that have been chosen as best fitting the

data. Agreement between the experimental and theoretical curves

is not perfect, and the small discrepancies have not yet been explained.

1 Figures in brackets indicate the literature references on p. 226. 213
214 BARRETT

Figure 1. The reflection coefficients are plotted for a growth curve for aluminium

anodized in 3 percent ammonium tartrate electrolyte. The closed circles represent observations in the first cycle, triangles are for the second cycle. The dashed

curve is one computed for a substrate with optical constants 1.0 — 6.7 j for a transparent film with refractive index «2 = 1.66 immersed in a transparent medium with refractive index ?zi = 1.346.

2. Determination of the Refractive Index of

Aluminium

The complex refractive index of aluminium would be determined

by a single observation on a film-free surface, but it is extremely diffi- cult to obtain massive aluminium in this state, and it is doubtful

whether evaporation under the best vacuum conditions would produce

deposits strictly comparable optically to massive aluminium. However, the following conclusions can be reached from the experimental

observations thus far described. The reading of a clean surface lies along the experimental line of figure 1 in the vicinity of the estimated

zero point, and certainly to the left of all observations made on freshly

polished surfaces. This effectively determines the minimum possible

value of the absorption coefficient as well as the relation between this and the real part of the refractive index. A shift of the zero point results in only a minor shift of the growth curve, and making observations at other angles of incidence does not remove the uncertainty of the zero point.

In an attempt to remove these difficulties, another method [2] was

employed. This method, which involved varying the refractive index

of the surrounding medium, is theoretically a more effective approach,
T ANODIC FILMS ON ALUMINIUM 215

and was the method adopted although it presented experimental

, difficulties. This technique was based on the principle that if a filmed surface is

i immersed in a medium of the same refractive index as the film sub-

! stance, the reflection at the interface between the film and the imj mersion medium is eliminated. The refractive index of the substrate can then be determined directly as though the surface were film free. This method requires a knowledge of the refractive index of the film.

! It was here assumed to be the same as that of anodic film. CompariI son of the observed individual reflection parameters for the various

! immersion media with the corresponding computed ones furnishes an

i alternative approach for interpreting the data, one that has the ad-

! vantage of treating equally well the case where no medium is found

1 matching identically the refractive index of the film. It is advantageous to use aluminium with the least possible film in

i order to minimize error due to misjudgment of the film substance.

The observations were made on aluminium freshly electropolished in

i a solution consisting of 700 ml alcohol, 200 ml perchloric acid and 100

! ml 2-butoxy-ethanol. The immersion liquids used were benzene,

! o-dichlorobenzene, monochloronaphthalene and alpha-bromonaphi thalene. The constants of aluminium at the wavelength 5461 A were

i estimated as 1.0 — 6.7j by this method.

3. Determination of the Refractive Index of Anodic

Oxide Films on Aluminium

An estimate of the refractive index of the anodic film can be made

directly from the growth curve shown in figure 1 with reasonable

accuracy, since a higher film refractive index results in an increase

in the length of the crescent-shaped curve. The estimate obtained

this manner for the anodic film was 1.66 The immersion method described above was also used. In this case there was an advantage in using relatively thick films. Two

polished aluminium specimens were anodised to produce films of the same character but with such different thicknesses that the reflection

coefficients in ammonium tartrate were widely different. The

reflection coeficients were then determined in a series of immersion

liquids with a range of refractive indices. On inspection of the

observations, it was then possible to interpolate a refractive index

for that immersion medium in which the two specimens would have

identical values for the reflection coefficients and which therefore

would have the same refractive index as the film substance.

Here again, a comparison of the individual reflection parameters

with calculated ones provides a check. An example is taken from

the observations of a single filmed surface immersed in the above

mentioned liquids. The observations are plotted as crosses in

figure 2. The match of computed points with observations naturally

involves the assignment of values for the film thickness and the

refractive index of the substrate. To obtain the corresponding com-
216 BARRETT

1.517

1.667

o n 2

= 1.65

. n 2 = 1 .66

+ experiment

1.553

•1.639 Figure 2. Computed and experimental reflection

coefficients for anodized specimen in various

liquids, assuming n2 = 1.65 and 1.66. The refractive indices of the immersion media are marked at the points.

puted points the substrate was assumed to have the refractive index

1.0 — 6.7j as determined in the experiment mentioned above. The

film thickness was determined first assuming a film refractive index

1.65 from the observation taken in ammonium tartrate. Using this thickness, the points to be expected with immersion in the various

media were computed and are shown as open circles in the plot.

The procedure was repeated with a value of 1.66 for the film refractive index, and the results are shown as filled circles. Agreement was best assuming the value of 1.66 for the anodic film. A serious uncertainty when using the immersion liquids on filmed

samples, not present using the growth curve to estimate directly the

refractive index of the film, arises from the assumptions that no

adsorbed water is left on the surface, but that any water incorporated

in the film during formation (approximately 15 percent for compact

anodic films) remains in the film. The treatment adopted was

similar to that used in botany and zoology; the specimens were immersed in absolute alcohol for several hours after formation.

The method depends also on the various immersion liquids replacing one another in the pores if the film is porous.

Alterations in the film as a result of reaction in the course of a

series of immersions would be expected to give rise to differences in the ellipsometer readings on repeating observations in the earlier immersion liquids. Alteration in water adsorbed or absorbed in porous films would give similar indications. In the immersion

experiments made, the films appeared reasonably stable except that

there were indications of some action with alphabromonaphthalene.

4. Refractive Index of the Corrosion Film

Regardless of which method is used, direct estimation from the

growth curve or the immersion method, the problem of interpreting

results obtained during the corrosion of aluminium in hot circulating

water is much more complex than that for the anodised aluminium.

The difficulties associated with the growth curve method are evident

from a typical curve for corroding freshly electropolished aluminium,
ANODIC FILMS ON ALUMINIUM 217

2 \\ \

n

2 =1.55- x

\>\

\ \M

Figure 3. Typical curve for changes in reflection coefficients during corrosion of aluminium in hot water compared to theoretical curves (dashed lines) for films with refractive indices 1.50 and 1.55.

as shown in figure 3. It does not repeat itself, but follows a spiral

track whereas computations of curves for growth of homogeneous

transparent films with various refractive indices give a series of closed curves with the characteristic crescent shape. The lower the

| film refractive index, the smaller the loop, while the apparent loop

center approaches the zero point or point for a clean metal substrate

; reading. The spiral curve in figure 3 has diminishing convolutions

that might be associated with a lower film index, but the location of the center is displaced from the expected location. Electron diffrac- tion studies indicate that the corrosion product in these conditions

is a monohydrate whose refractive index is reported to be nD =\.§b.

The position of the curve suggested that each reading might be

the average of a series of points on the growth curve for a uniform

homogeneous film. Such an averaging process might be conceived

as arising from a surface consisting of a patchwork of small areas with different film thicknesses or different refractive indices. In

figure 4 a curve has been computed for film growth in which the

parameters delta and tan psi have been averaged for 10 thicknesses

ranging from 60 to 140 percent of the nominal value, in which no

interference effects are assumed to arise between adjacent areas. The computed curve shows some resemblance to the observed curve

in figure 3. Similar curves could presumably be derived on the
Figure 4. Computed curves where each reflection coefficient is averaged over a range

of thicknesses from 60 to 140 percent of the nominal thicknesses. The dashed curve is the normal growth curve for the same constants. The optical constants employed

are n\= 1.326 (water at 70 °C), 7*2 = 1.54, 713 = 1.66 (inner barrier layer with constant thickness of 20 A), m=

1.0-6.7 i.

I

1 = 500 K

i 2

= iooo£

Figure 5. Computed curve assuming a double film in which the outer layer (n =

1.40) is half the thickness of the inner layer {n—1.54)-
ANODIC FILMS ON ALUMINIUM 219

assumption of the film refractive index varying over small areas with

the same film thickness, i.e., variations in the density of the film. Computations have also been made for the case of a double-layered

film in which the layers grow in constant proportions. The results

of some computations of this type are presented in figure 5, and it

is apparent that it is possible to obtain a curve similar to that observed on these assumptions.

It seems reasonable to assume that a variety of models could lead

to curves for progressive changes in the optical parameters that would

resemble the observed ones. In other words the optical observations

alone cannot be expected to yield an unambiguous interpretation

and it would therefore be reasonable to apply several independent

techniques to distinguish between such alternative interpretations.

Immersion in a series of media of various refractive indices would

be expected to be fraught with extra difficulties on account of the

probability that these heterogeneous films would change on immersion

and replacement of liquids in pores by subsequent immersion in

different media might be rather incomplete. Nevertheless the esti- mate made by this method must be considered the most reliable at

present. The observations were compared with theoretical points

calculated on the basis of homogeneous films only, and the best match

was obtained by taking the refractive index to be 1.54 for the corrosion

product.

5. Stability of Anodic Films in Hot Water

The changes occurring in anodic films in hot water have been

studied quite extensively in connection with the process of sealing

of porous anodised films. It is accepted that this process involves

the formation of the hydrated oxide, boehmite, which plugs or seals the pores. There is, however, some doubt as to whether the boehmite

arises from conversion of existing film substance or from the reaction

of further metal ions transmitted through the film. It seems that anodising is not effective in increasing protection

above the boiling point for any length of time, and the implication is that the anodic film has been destroyed under these conditions [4]. In the present studies the ellipsometer was used to follow the change

in anodised surfaces exposed to doubly distilled water at 70 and 90 °C.

Observations for the initial experiment with rapidly circulating water

are plotted in figure 6. The first changes in the optical parameter

were a retracing of the forming curve from point R to E in the figure, at an approximate rate of 5 A/min. o After reaching E, corresponding

to a thickness of only about 100 A, the film growth recommenced,

and the further behaviour was similar to that observed in the direct

corrosion of aluminium already discussed.

Further experiments, in which the water conditions were varied,

produced the whole spectrum of results ranging from the type shown

in figure 6, with vigorous circulation to those made with stagnant

water in which the reflection coefficients remained stationary for

728-101 0—64 15
220 BARRETT

Figure 6. Experimental curve representing an aluminium surface previously anodized in ammonium tartarate, then exposed to circulating hot water.

Observations started at S. There was first a slight increase of film to i? then steady changes following

closely the computed growth curve of fig. 1 (shown dashed), in the reverse direction. At E reversal again took place and the opln circles represent this final phase of formation of corrosion product. (Reproduced from [2].)

/ / / /

/ /

/ /

/ / M\ \ \ \ \ \ \ \ \ \

Figure 7. Experimental curve representing changes of a

pre-anodized aluminium specimen exposed to still water at 70° C. The initial point is at S. The dashed curve is the theoretical growth curve for anodic film.
ANODIC FILMS ON ALUMINIUM 221

Figure 8. Theoretical curve for the two-layer film model in which the inner layer (anodic film, n=1.66) is progressively

replaced at the outer interface by hydrated film (n = 1.54). The numbers represent the thickness ratio of the loss of the inner film to the gain of the outer film. The dashed curve is the theoretical growth curve for a

single anodic film. some minutes, and then spiraled in anticlockwise. A typical curve

of the intermediate type is shown in figure 7. It is tempting to interpret the initial section of all curves of this type as representing dissolution, possibly accompanied by hydration, and the second portion as representing corrosion film growth. However, some computations

were made for a model process assuming homogeneous films, in which

a film 1080 A thick with a refractive index 1.66 was progressively

converted to a film with refractive index 1.54 accompanied by some

loss by dissolution. The ratio of the decrease of the inner compact

layer to the increase of the outer layer was varied in different series with the values .3, 1, 1.2, and 1.4 to represent varying degrees of dissolution. It will be noted (see fig. 8) that in all these hypothetical

processes the curves change direction during the progress of the

assumed reaction. Consequently an alteration in the sign of the

change in delta is not necessarily indicative of a change in process;

it is often merely the result of the periodic character of the optical

phenomena for uniform growth or conversion of films. Conversely,

the change of process to pure corrosion film growth when the entire

anodic film has been consumed is not accompanied by any appreciable

change in the direction of the curve, except for examples similar to

figure 6. In other words, the commencement of the last stage is generally not detectable by a plot of the reflection coefficients alone.
222 BARRETT

On the other hand, the location of the turning point might be useful

in estimating the ratio characterising a conversion process. Un- | fortunately, this requires a knowledge of the refractive index of the i film formed. It is not to be expected that the final corrosion product on corroding previously anodised aluminium will necessarily be the same as for the corrosion of freshly electropolished aluminium, and

the corrosion behaviour over a longer time indicated that the two

products must differ in some respects. Characteristic of the curves ; of the type shown in figure 6 was that the last section, representing

corrosion film growth, proceeded at a rate an order of magnitude

slower than that for corrosion of freshly polished aluminium under 1 similar conditions. Curves of the type of figure 7, or those indicating

higher ratios of anodic film lost to corrosion film gained, if continued

for a longer time, were impossible to interpret as they did not resemble

any of the theoretical curves thus far computed. Taking the value

of 3.2 for the specific gravity of anodic film and 2.8 for the specific

gravity of corrosion film [4], the ratio should be 1.3 if the anodic

film is simply hydrated without loss by dissolution or gain by further

reaction with the metal. Even allowing for the uncertainty of the

film refractive index it can be stated that values considerably above

and below this were obtained in practice. The amount dissolved was clearly dependent on the speed of circulation. In still water the

value of the ratio did not fall below about 1.2. Some dependence was

also found on the amount of aluminium already present in the water,

dissolution occurring more readily if the water had previously been

boiled with aluminium shavings. This seemingly anomalous result

is in agreement with the findings of the Institutt for Atomenergi [4], and Draley [5].

6. Determination of Barrier Layer Thickness

An independent method of measuring the amount of compact anodic

type film in the presence of an overlying porous film is based on the

difference in electrical properties of the two oxide types. The method

as first devised by Hunter and Fowle [6] involves the study of current

flow when anodically polarising the surface under study in a suitable

electrolyte, e.g., ammonium tartrate, at gradually increasing voltages

until the occurence of a sudden increase of current. This increase is characteristic of the resumption of film formation. The voltage at which this occurs may then be taken as a measure of the barrier layer

film thickness, since the anodic film formation is dependent on ionic

conduction which only occurs when the field strength is above about

107 V/cm. It is assumed that the entire potential drop lies across the

barrier layer.

However, by using the ellipsometer to detect the onset ot film

growth, this technique can be applied to less perfect films, where

leakage currents are substantial and interfere with the detection of the onset of formation by current measurements. An example of the

application of the ellipsometer to the determination of the barrier
ANODIC FILMS ON ALUMINIUM 223

1160

140

00 Figure 9. Thickness curve for the w

determination of the voltage for re- ^

1120 anodisation for aluminium anodized ^

-previously at 80 V. U

1100

10800 50 100 POTENTIAL ( V)

layer thickness is given in figure 9. Each increment in voltage was

maintained for 5 min before the optical reading was taken. The

slight dip in the thickness curve that occurs before the rise indicating

resu nption of film formation is thought to be due to the dissolution

of the film being stimulated by the polarisation.

It would not be possible to employ optical measurements for this purpose of growth if the barrier layer occurred solely at the expense

of the outer layer since the optical effect of such conversk n would

be only minor. This mode of barrier layer growth would immediately

be evident in that all barrier layer determinations would then simply

give the total film thickness as the barrier layer thickness. Conversely,

the fact that this is not so gives an indication that barrier layer

growth occurs partially within the layer itself or at the metal oxide

interface. The slope of the curve such as that in figure 9 when

anodising has recommenced should give an indication of the extent

to which further anodic oxidation takes place without the conversion

of the porous layer, a steep slope indicating relatively independent

growth of the barrier layer. The maximum possible slope would be

13.7 A/V as found when the entire film is of the barrier type.

The slope found in an experiment with a porous anodic film was

10.3 A/V. If it is assumed that the formation of the inner barrier

layer film in all cases corresponds to the factor found in the formation

of compact film, then for each volt applied, an increase of 13.7 A in the

barrier layer thickness follows.

o If, however, the optical indication

is that an increase of only 10.3 A has occurred in total film, then the

difference must represent either that part of the potential drop lies m the outer layer, or that there is a reduction in the fraction of the

film that is porous as shown diagrammatically in figure 10. The

film substance of both layers is supposedly the same and therefore
224

POROUS

LAYER

BARRIER LAYER

Figube 10. Suggested model of film growth.

(a) film as formed after several hours at constant voltage, (b) structure

after further anodisation at increased voltage. conversion of porous to barrier type film would result from further

anodic oxidation occurring at the base of the pores filling them up. This is the reverse of the process of porous anodic film formation

which is believed to take place by dissolution at the base of pores, consuming the outside of the barrier layer, which is replenished else- where within the barrier layer by further oxidation to maintain the

equilibrium thickness approximate to the applied voltage. A porous

layer is formed, although slowly, even in ammonium tartrate [7] which is normally considered to produce compact film. The above

results relate to films formed in this electrolyte at constant voltage

for many hours.

From the evidence that oxide formation may take place in part at the solution interface at the base of the pores as well as at the metal

film interface or within the barrier layer, it can be argued that both

anions and cations diffuse through the film during the formation

process. Mott and Hoar have proposed that both Al+++ and O

or OH~ diffuse, cation migration predominating at high field strengths

and anion migration at low field strengths [8] . The mobility of both

ions has been demonstrated experimentally by Bernard [9]. It would be interesting to apply a similar analysis to the case where the barrier layer is covered with boehmite as in the hydration

process described above. The optical inhomogeneity of the hydrated

layer is again a hindrance to interpretation, and no certain conclusion

has yet been reached. A value for the voltage for re-anodisation can

be found, but the slope of the curve above this point, requiring

thickness evaluation, is in doubt.

7. Influence of Field Strength on Film Dissolution

Another aspect of the anodising process studied with the ellipsometer was the influence of field strength on the solution rate at the
ANODIC FILMS ON ALUMINIUM 225

14O0

0 100 200 300 TIME (MIN.) Figure 11. Film growth during anodizing. The initial applied voltage is 70.25 V. At point A

the voltage is lowered to 69.25 V and at C to 61.25 V. base of the pores. The first indication that this could be of importance came from the invariable small decrease in thickness just prior to the sharp bend upwards in curves such as that in figure 9. For this study ellipsometer readings were made during the formation of a two-layer film in the ammonium tartrate electrolyte. During

the first % hr of anodising with constant voltage a linear rate of growth of porous film was established (see fig. 11). After a period

of such growth the applied voltage was reduced by 2 V at A, when

film growth stopped immediately and was resumed after 4 min at B.

After a further period the applied voltage was reduced by 5 V at C

when growth again stopped and resumed after 15 min at D.

If it is assumed that when uniform growth of porous film is occurring,

there is also established an equilibrium barrier layer thickness cor- responding to the applied voltage, then this thickness has decreased

during the pause by an amount corresponding to the voltage reduction

of two V for the first time and five V for the second time. If the total equivalent thickness remains constant, then this reduction in barrier

layer thickness must be balanced by a corresponding increase in the

porous layer thickness, presumably by dissolution at the base of the

pores just as during linear anodic growth. From the time taken to
226 BARRETT

resume growth the average rate of thinning of

othe barrier layer I

at the base of the pores would be about 6.5 and 4.5 A/min respectively

in the two experiments. This rate exceeds that during the linear I growth of porous anodic film, which is known to be 2.5 A/min in the

j example shown in figure 11, because the dissolution rate must equal I the film growth rate to maintain the barrier layer at the characteristic I thickness for the forming voltage. Similar results were obtained

using electrolyte of pH 5.5. However, in a further experiment, a

film was anodised at 85 V, after which the applied voltage was re- duced to 60 V, a reduction of 25 V, or, expressed in terms of the

relative field in the barrier layer, this was reduced to 0.7 of the value

maintained during linear growth. Growth of the film was not resumed | in this case after 26 hr at the reduced voltage, i.e., the rate of thinning

of the barrier layer in this case was less than 0.2 A/min.

These experiments tend to suggest that dissolution at the base of pores is strongly dependent on the applied field in a similar way to the barrier layer formation process itself, but that at a field strength

just below that required for barrier film growth, the dissolution reaction

is appreciable.

8. Concluding Remarks tfi The foregoing are some examples of applications of ellipsometry

to the study of rather complex phenomena for which it is difficult to obtain reliable optical constants and where the film structure is probably rather complicated. In such cases it is profitable to make ! collateral studies with ellipsometry and other techniques applicable

to films in situ to give independent information regarding film I formation.

It is also apparent that with the increased availability of computer I facilities many aspects of measurements and interpretation can now i be effectively studied and elucidated by computation surveys.

The present work was initiated at the request of Institutt for Atomenergi (Kjeller Research Laboratories) and of AB Atomenergi,

Stockholm, in the framework of the program of optical studies of metal surfaces at Norges Tekniske H0gskole, supported by grants I from the Royal Norwegian Council for Scientific and Industrial Research. The author thanks the above-mentioned organizations for r financial support and Prof. A. B. Winterbottom and Messrs. K. I Videm and B. Forsen for stimulating discussions and encouragement. :

I 9. References

[1] A. B. Winterbottom, this volume p. 91.

[2] M. A. Barrett and A. B. Winterbottom, First Intern. Congr. on Metallic Corrosion, London, 1961, p. 657, Butterworths, London.
ANODIC FILMS ON ALUMINIUM 227

j [3] Alcoa Monograph, Tech. Paper No. 10.

j [4] K. Videm, private communication, 1963.

[5] J. E. Draley, J. Electrochem. Soc. 110, 622-27 (1963).

[6] M. S. Hunter and P. Fowle, ibid. 103, 481 (1954).

j [7] M. S. Hunter and P. Fowle, ibid. 108, 139 (1961).

[8] T. P. Hoar and N. F. Mott, Chem. Solids 9, 97 (1959).

; [9] W. J. Bernard, J. Electrochem. Soc. 109, 1082 (1962). Discussion

A. Rothen (Rockefeller Institute)

There is one point which was not quite clear to me. What kind of formulation did you use to get your thickness from your ellipsometry

j data?

i M. A. Barrett:

We used the complete Drude formula, with no approximations.

| H. E. Bennett (Michelson Laboratory)

Could you tell us more about the electropolishing technique that

you mentioned you used to get the optical constants of the aluminum?

! Also, how pure was the aluminum and how thick were the oxide films

| you had in that case? M. A. Barrett:

The aluminium used was electrolytic aluminium of purity 99.993

percent. The analysis was 0.003 percent Cu, 0.0012 percent Fe, 0.003

percent Si. The ingots were subjected to a homogenizing anneal for 24 hr at 500 °C. Some of the later anodising experiments were done

with zone refined aluminium, but no difference was observed.

The electropolishing solution was 70 percent alcohol, 20 percent

perchloric acid, and 10 percent 2-butoxy-ethanol. The solution was

circulated during electropolishing, and around 35 V was applied.

According to our best estimates the thickness after electropolishing

varied from 20 to 40 A. Above about 40 A the sample was discarded.

These thicknesses are the most difficult to determine absolutely because of the difficulty of determining the zero point on the calibration

curve. R. C. Plumb (Worcester Polytechnic Institute):

I would like to ask whether you have observed any effects with

ammonium tartrate electrolyte which indicated departure in surface

topography of the oxide-solution electrolyte interface from a simple

plane surface. M. A. Barrett:

There are deviations from the theoretical curve that we cannot explain, but there is another complication not yet mentioned. This is that with ammonium tartrate solution pore formation does occur. In fact, all the results I have given for porous anodic films were obtained with ammonium tartrate solution but anodising for a long
228 BARRETT

time at constant voltage. The pore formation is slow, 1 to 2 A/min,

but a small amount of pore formation would be expected to occur in

all anodising, so we must always expect some deviation from the

theoretical curves derived for uniform compact film growth.

R. C. Plumb:

The reason I bring it up is in connection with your atomistic model

for the decrease in thickness as you re-anodize, which you have at- tributed to dissolution. I wonder if it would be appropriate to do the same experiments with an electrolyte having a higher buffer capacity.

Apparently the dissolution during anodization in an ammonium

tartrate electrolyte is associated with the buffer capacity. Ammonium

tartrate solutions have a rather weak buffer capacity. Perhaps some

other electrolytes with a greater buffer capacity would not show that

decrease and perhaps substantiate that mechanism.

M. A. Barrett:

I think that is a good point to make. We would have liked to make

the same studies using a borate solution, but oddly enough the curves deviated from the closed curve that would correspond to growth of a uniform transparent film. They did not follow the first convolution

in the second time round and did not pass through the starting point.

Ammonium tartrate appeared to be the best electrolyte of those we

have tried so far, but we hope to examine other electrolytes with

different anions and differing buffer capacities later. K. H. Zaininger (Radio Corporation of America)

How large is your illuminated sample area? How uniform do you

think the films are over this area? And how is the accuracy dependent on sample area? Have you ever tried to correlate your

thickness measurements determined by ellipsometry with measure- ments made by multiple beam interferometer?

M. A. Barrett:

No, multiple beam interferometry has not been used in this work.

The illuminated area of the specimen in the ellipsometer is 5 mm wide

and about 12 mm long. Using half shades, an uneven film shows up

as an uneven field in the telescope, but the analyser and polariser

adjustments can be made for a small portion of this field, say 1x2 mm,

lying on the line of demarcation of the half shade. This involves no loss of sensitivity since with the eye the rest of the field of view

has no effect, in marked contrast to measurements made with the

aid of the photomultiplier.
Ellipsometry in Electrochemical Studies

A. K. N. Reddy and J. O'M. Bockris

The Electrochemistry Laboratory, The University of Pennsylvania,

Philadelphia, Pa. 19104 This communication presents techniques for utilizing ellipsometry for in situ electrochemical studies of film formation and growth on mirror

electrodes immersed in aqueous solutions. In such studies, two major

problems arise: (a) the presence of electrolyte may be incompatible with

the establishment of a film-free state of the metal surface, required as an optical reference state; (b) film growth may occur much faster than

can be matched by methods normally used for analyzing elliptically polarized light. In this laboratory, these problems have been circumvented by the use of an electronic potentiostat which has permitted

the development of new types of techniques.

1. Introduction

This Laboratory has been involved in investigations which range

over most of the basic aspects of electrochemistry, such as the struc- ture of the electrical double layer, the kinetics of metal deposition,

! the mechanism of hydrogen and oxygen evolution, the electrochemistry

| of fused salts, etc. In 1961, work was commenced on the formation

and growth of anodic films, and ellipsometry recommended itself as

I an eminently suitable technique, particularly in view of the classical studies of Tronstadt and Winterbottom [1, 2].

1

I Ellipsometry is the optical examination of surfaces, based on the

laws of reflection of polarized light. The term ellipsometry is generally defined in a more restricted sense as an optical technique for the

determination of the thickness and optical constants of surface films

I on metals. The technique is based on the fact that when linearly

polarized light is incident on film-covered metals, the polarization

state (i.e., the azimuth, X, and ellipticity, «j of the reflected light is different from that observed (X°, co°) in the case of reflection from a

film-free surface. The magnitude of the changes in the polarization

state, introduced by surface films, depends on the thickness and optical

constants of the film material. Ellipsometry is noted particularly

for its sensitivity in detecting and measuring extremely thin surface

films (-1 A) [3]. There are two main requirements for experimental ellipsometry:

(1) a film-free state of the metal surface must be produced as an

optical reference state; (2) the thickness of the surface film must

either be constant, or must change at a much smaller rate than the

rapidity with which elliptic polarization states are determined.

In utilizing ellipsometry for in situ electrochemical studies of film

formation and growth on mirror electrodes immersed in aqueous

solutions, two major problems arise: (1) interaction between the metal

mirror and the electrolyte may lead to spontaneous film formation

1 Figures in brackets indicate the literature references on p. 244. 229
230 REDDY AND BOCKRIS

and to the necessity of reestablishing the film-free, optical reference

state ; (2) anodic film growth generally occurs much faster than can be

matched by methods normally used for analysing elliptically polarized

light. . Notwithstanding these problems, some progress has been achieved

in this laboratory towards the goal of adapting ellipsometry for electro- chemistry. This communication is intended to describe the new

techniques which have been developed, and then to discuss future

possibilities. It will be seen that a distinctive and novel feature of the techniques

described here is the use of the electronic potentiostat as an integral

part of an ellipsometric study of electrochemical processes. The basic

viewpoint implicit in this approach is that, under steady-state conditions, the state of the metal-solution interface (particularly with

reference to the existence, composition and thickness of surface films)

is determined by the potential difference across the interface. Thus,

the potentiostat, by electronically regulating the potential of a mirror

electrode with respect to the solution, also determines the state of the

metal surface with respect to surface films. For example, if it is desired to obtain a film-free, optical reference state of a metal M in a

solution S, then a knowledge of the electrochemistry of the metal M,

often conveniently summarized in potential-pH diagrams [4], enables one to choose the potential range in which the mirror electrode is thermodynamically immune to film formation.

There are many roles which a potentiostat can play in an ellipsometric experiment, and each method of utilizing the potentiostat

yields a particular type of technique.

2. Qualitative Ellipsometry of Growing Films

The potentiostat in this technique serves to maintain an initial,

film-free optical reference state of the electrode surface. Cessation of potentiostatic control of the electrode potential makes film formation

possible (either spontaneously or with the passage of anodic current or by establishing a new electrode potential with a potentiostat) and

ellipsometrically detectable.

For this type of investigation, however, it is necessary that the

electrolytic cell be designed to permit electrochemical and ellipsometric experiments to be conducted simultaneously. Requirements of current density distribution and minimization of natural convection

restrict the choice of cell and electrode geometry. In this laboratory,

rectangular, fused-quartz cells were used with the metal under study

covering the cell b3ttom. A pair of opposite walls of the cell serve as entry and exit windows for the light beam. The cell cover is designed

to permit introduction into the cell of a reference electrode and a

counter-electrode. . The assembled cell is mounted on the specimen carrier of an ellipsometer having a vertical plane of incidence (fig. 1). The horizontal

metal functions as an electrode for the electrochemical experiment and

as a mirror for an optical experiment. Thus, the state of the metal
ELECTROCHEMICAL STUDIES 231

MONOCHROMATIC

LIGHT SOURCE

Figure 1. Fused quartz cell mounted on specimen carrier of ellipsometer.

(Reproduced from [7].)

2.04

0.241 1 I I

I I I I

I I i 0 0.2 0.4 0.6 0.8 LO 1.2 1.4 1.6 1.8 2.0 TIME (seconds) Figure 2. Potential-time and ellipsometer intensity-time

transients obtained from platinum in 0.1 N H2 S04 at a

current density of 1 mA/cm2

. surface is continuously monitored in situ both electrochemically and

optically.

The experimental technique consists of the following steps:

(1) the potentiostat is switched on and a suitable potential difference

is maintained across the electrode-solution interface to ensure that the

mirror surface is in a film-free state;

(2) when the mirror is in the film-free, optical reference state, the

optical components of the ellipsometer (polarizer, analyzer, and compensator) are adjusted for "extinction" at first visually and finally by

using a photomultiplier. These settings are not altered during the

course of an electrolysis experiment;

(3) the extinction intensity is fed via a preamplifier to an oscillo- scope;
232 REDDY AND BOCKRIS

(4) a fast switch is used to switch off the potentiostat and to impose a constant anodic current across the mirror electrode-solution interface;

| (5) the formation of a surface film on the mirror anode degrades the i fixed settings of the components into "off-extinction" settings and the

off-extinction is manifested as an intensity increase. 2 With increasing

film thickness (for thin films) the fixed settings of the optical components depart more and more from the true extinction settings at any

particular instant, and, therefore, the intensity of light incident on the

j photomultiplier also increases;

(6) if a dual-trace oscilloscope is used, a simultaneous record of 1 potential versus time can be made. In this manner the information

from the ellipsometer intensity-time transient can be used to augment

the information gained from the potential-time transient.

Such a study has been carried out on the anodic oxidation of platinum in 1 N H2 S04 solutions [5]. It suffices to point out here that

the transients (fig. 2) show that an "oxide" is formed only at potentials

greater than about 1.0 V (versus normal hydrogen electrode). 3. Chronoellipsometry

In the case of the anodic formation of calomel, the potential-time

and ellipsometer intensity-time transients (fig. 3) (obtained by the

technique described in section 2) presented an interesting problem [6]. The galvanostatic charging transients showed that the calomel

2 Film formation can be made to reveal itself more distinctly when the optical components are set slightly

off-extinction when the mirror is in its initial, film-free state. To the extent that the settings of the optical components are not used to calculate thickness and optical constants, the actual intensity emerging from the analyzer serves only as an arbitrary "zero."

POTENT IAL /

^4 ZERO

EL

INJPSOMElTENSITY

rER\ -

-—Icm—

Figure 3. Anodic formation of calomel films.

Potential-time and ellipsometer intensity-time transients recorded with a 2 sec/cm Time Base on a Tektronix RM35A Oscilloscope, with a Type CA

Dual-Trace Plug-in. Potential versus time transient with 200 mv/cm

amplification. Initial potential held at -440 mv versus SCE with Wenking potentiostat. Zero of potential is indicated in figure. (Reproduced

from [6].)
ELECTROCHEMICAL STUDIES 233

10 15 20 25

TIME (IN SECONDS) Figure 4. Ellipsometer intensity versus time transient to illustrate induction time n.

[Reproduced from [7].) potential was attained in a few hundred milliseconds (the integral

capacity corresponding to this process being of the order of 120

^farads/cm2). In contrast, an induction time (r t ) of several seconds was required for the formation of a calomel film (signalled by an

intensity jump in the ellipsometer intensity transient) (fig. 4).

The induction time, r h was found to be dependent on mass trans- port factors. Thus, the induction time increased considerably when

the solution was stirred during the passage of anodic current. The mechanism of growth of the calomel film was elucidated by

switching off the galvanostatic current during the growth of the film. It was observed that the film continued to grow for some time even

after the cessation of the current. The anodic formation of calomel films on mercury electrodes was,

therefore, interpreted in terms of a dissolution-precipitation model,

a brief version of which can be written thus:

1. Specific adsorption of chloride ions:

2 Hg+Cl" >2 Hg cr

2. a. Dissolution of mercury with formation of chloromercurous ions

b. Disproportionation of chloromercurous ions2Hg Cl-J±*(Hg,Cl)-

3. Precipitation of calomel

Mercuric entities

(Hg2 Cl)++Cl- >Hg2Cl2 j. The solution of the diffusion problem [7] showed that the induction

time Tf should increase with a decrease of current density, and that

ivi 12 should increase with current density, and above a critical current
234 REDDY AND BOCKRIS

200

180 ~I60

cI

140

cC'20

Ld2

hIOO z° 80

=3Q?60-

40-

20-

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 CURRENT DENSITY (in mA/cm 2

) Figure 5. The induction time (n) required for calomel

film formation observed at various current densities. (Reproduced from [6].) 8r

\T 4

Figure 6.

Ql 0.15 0.2 CURRENT DENSITY The function vtp plotted against current

density (i). (Reproduced from [6].)
ELECTROCHEMICAL STUDIES 235

density, should become constant with current density. The experimental results on the induction time for the formation of calomel

are shown in figures 5 and 6. We see, therefore, that the dissolutionprecipitation model is consistent with the pattern of experimental

results, as is argued in detail elsewhere [6]. A preliminary study has also been carried out on the mechanism

of film formation during the anodic passivation of nickel in acid solutions. The ellipsometer intensity-time and potential-time transients

are shown in figure 7. An induction time was observed for the formation of a film on a nickel mirror electrode. The general behavior of the induction time was similar to that in the case of the anodic formation of calomel. It increased with stirring, and varied with current

density as shown in figure 8. These results will be analyzed in greater

detail in a forthcoming publication [8]. It will only be mentioned

here that the surface film (not necessarily the passivating film itself)

is apparently formed by a dissolution-precipitation mechanism.

The experimental determination and theoretical analysis of induction times was termed "chronoellipsometry" in view of its close analogy with "chronopotentiometry." In the latter technique, the

"transition time" is determined from the potential "jump" in the

potential-time plot obtained at constant current density, and is theoretically significant as the time required for the concentration

of the reducible species to attain the value CX=Q =0. In chronoellipsometry, the "induction time" is determined from the intensity

"jump" in the ellipsometer intensity-time plot obtained at constant

current. The induction time is theoretically analyzed as the time

required for the product of anodic dissolution to attain a concentra-

+ 1.84

4 8 12 16 20 24 28 32 36 40 44 48

TIME (sec ) Figure 7. Potential-time and ellipsometer intensity-time transients obtained from nickel in 0.5 M K2 SO4 + 0.01 M

H2 S04 (pH = 3.1) at a current density of 1.5 mA/cm2

. 728-101 O—64 16
236 REDDY AND BOCKRIS

_J I I I I L

0.5 I 1.5 2 2.5 , 3 CURRENT DENSITY (mA/cm2

) Figure 8. The induction time (n) required for a formation of a surface film on nickel as a function of the constant (anodic ) current density.

tion C* near the electrode, O* being determined by the solubility

product of the insoluble salt which forms the anodic film by precipitation.

In the qualitative ellipsometry of growing films, the record of

ellipsometry intensity versus time curves serves to provide ellipsometric monitoring of the surface and yields the following information : (a) whether a surface film is formed or not; (b) the time, after the

commencement of anodic polarization, at which a film is formed on

the surface; and (c) whether the film thickness increases with time

or remains constant. It is to be noted, however, that though chronoellipsometry provides only qualitative information on the characteristics of surface films, it yields quantitative data on induction times.

4. Quantitative Ellipsometry of Growing Films

The ellipsometric analysis of growing films is bound to rest on a

qualitative level until one is able to determine the time variation of the thickness and optical constants of a film. This advance, however, can be achieved only if the elliptic polarization states of the reflected
ELECTROCHEMICAL STUDIES 237

POI

Linearly

Polarized

Light

Rotating Analyser

(b)

POI

Rotating

JJ Analyser

Figure 9. An automatic ellipse-meter using a rotating analyzer.

(a) Schematic of ellipsometer.

(b) Sinusoidal intensity output in the case of linearly polarized light.

(c) Sinusoidal intensity output in the case of elliptically polarized light. light can be continuously determined by methods other than the

classical method of analysis. The latter method requires the manual

alteration of the azimuths of the compensator and analyzer by means

of independent mechanical controls and demands that the film be

stable over several minutes. There is a vital need therefore for an

automatic recording ellipsometer.

An attempt has been made in this laboratory to use a rotating

analyzer. 3 If one provides a "phase marker" to register the time

when the transmission direction of the analyzer is parallel to the plane

of incidence, the azimuth and ellipticity of the reflected elliptically

polarized light can be determined from the phase of the maxima, and

the ratio of the maxima to minima respectively of the sinusoidally

varying intensity emerging from the rotating analyzer (fig. 9). For

time-independent elliptic polarization states, the method has proved

successful [9]. For growing films, at least one complete rotation of the analyzer should be completed before there is any significant change

in film thickness. The problems arising from the mechanical

"wobble" of the rotating analyzer and from accurate (to 0.01°)

3 We are grateful to Mr. V. Radhakrishnan (Department of Radio Astronomy, California Institute of Technology, Pasadena, Calif.) for this suggestion.
238 REDDY AND BOCKRIS

determination of the phase and amplitude ratio of sine waves of in- creasing frequency (the faster the film growth, the higher should be

the rpm of the rotating analyzer) have not yet been solved in detail

but the method seems sufficiently promising to merit further attention.

At the present juncture, therefore, the best that can be achieved is to resort to approximation methods and to recover a thickness-time

plot from the ellipsometer intensity-time curves. A significant step

in this direction has been taken by Archer [10], who developed a method (which he called "photoelectric ellipsometry") in the course

of a study of the adsorption of water on silicon substrates.

A derivation of his equations is presented here in terms of the

Poincare sphere representation [11], and an ellipsometer geometry in which the quarter-wave plate is situated in the incident beam, its fast axis being set at a 45° azimuth with respect to the plane of incidence.

The present treatment is based on a well-known property of the

Poincare sphere [11] viz, given an analyzer which transmits completely

light of polarization state A(\A , coA), the fraction (I) of the intensity

of light of polarization state Z(XZ , o>z) emerging from the analyzer A

is given by

1= cos >

where ZA is the arc on the Poincare sphere joining the points Z and A.

If the azimuth (a) of the analyzer is known, then the problem is to express the polarization state Z of the reflected light in terms of the

polarizer azimuth, p, the relative phase retardation (A) and the rela- tive amplitude diminution ty)

introduced by the reflection. The

transformations of polarization state as the light passes through

various optical components is shown in figure 10. From the Naperian

triangle \\Z°KF, we have

5=2^-90°,

where 5 is the phase difference introduced by the quarter-wave plate.

Expressions for sin 2XZ and cos 2XZ are obtained from the Naperian

triangle | \ZX. Then cos ZA is determined from a well-known formula

giving the distance on a sphere between two points whose latitude and

longitude are known. In this manner we obtain

sin2 (\p+a) n ,

. „ 2 /A+2p-90°\

sin 2yp sin 2a sim ( ^ )

which is Archer's expression. If the analyzer is set in the extinction

azimuth \p=a, then

/ocsin2 (|+p-45°)-

It is obvious, therefore, that changes of intensity can result either from
ELECTROCHEMICAL STUDIES 239

Figure 10. (a) Sequence of optical components.

1. Mercury light source; 2. collimator and monochromator; 3. polarizer; 4. quarter-wave plate; 5. mirror;

6. analyzer; 7. photomultiplier. Polarization states at various stages indicated by 0->P->Z°-*Z.

(b) Poincare sphere representation of transformation of polarization states 0->P-+Z°^Z.

|| and J_ represent linearly polarized light vibrating parallel and perpendicular to plane of incidence

respectively, Kp and Ks , the fast and slow axes of quarter-wave plate.

changes in A due to film formation or from changes in the polarizer

azimuth. The following procedure is therefore adopted:

(a) the mirror surface is maintained in a film-free state in which the

relative phase retardation is A°;

(b) starting from an initial polarizer azimuth of p° (yielding an

extinction intensity 1°), the polarizer azimuth is decreased in steps

of say 0.1° and at each value the deflection on the instrument which

records the intensity is noted
240 REDDY AND BOCKRIS

(c) the polarizer is returned to the p° azimuth;

(d) the film formation process is allowed to proceed revealing itself as an intensity-time plot;

(e) from the initial calibration, the intensity-time curve can be reduced to a A-time curve. A basic assumption in this whole approach is that film growth

results only m changes of A, leaving ^ constant. The conditions under which this approximation is justified can be defined by taking the

Drude-Tronstad thin-film equations from which [12] dA

and

dL=~a

dL P' Expressions 4 for the coefficients a and 0 are available in the literature

[12] and it is found that &l<*=j£ depends essentially on the magnitude

of the extinction coefficient (K) of the reflecting substrate and the

angle of incidence, 6. If K->0 and 6=6B, where 0B is the Brewster

angle, then -->0. In Archer's experiments on silicon [10], - =Q ' QQ3

. mi i a

. . . , a 0.256

I he above approximation is less suited for metal substrates, but even

here its validity increases as film thickness decreases.

Once a is calculated on the basis of a known (or assumed) value

lor the refractive index of the film, the A-time plot yields a thicknesstime plot from which the kinetics of film formation can be interpreted.

[II, during the course of an ellipsometric experiment on growing

films, the intensity becomes constant with time (even for a matter of minutes) indicating that the film is stable, then the stable thickness

ot the film can be determined by rapidly carrying out an extinction

experiment to analyze the reflected elliptically polarized light. From

the ellipse parameters the thickness can be calculated either by using approximate formulae for thin nonabsorbing films or by solving exact equations by computer techniques. In experiments on the anodic

formation of calomel, it was observed that on switching off the

electrolysis current during film growth, the film continued growing

lor some time and then was stable over several minutes. Under these

circumstances, its thickness was calculated [6].] 5. Quantitative Ellipsometry of Steady- State Films

The functional relationship between the potential of a metal-solution

interface and the state of the metal surface can be used not only to maintain a film

:free metal surface but also a steady-state thickness of anodic

him. This can be achieved once again with the aid of a potentiostat

set to a suitable anodic potential. Under these conditions of constant

nW Zpt o?rSoSS °/ thA

a?

gIe 0f in

2idence fo). the tadex of refraction of the surface film nF , the com- plex index of refraction of reflecting substance n*=n 0—iK) and the wavelength (X).
ELECTROCHEMICAL STUDIES 241

potential and therefore steady-state thickness, the parameters of the

elliptically polarized light reflected from the film-covered metal surface can be determined by setting the optical components to extinction.

From the ellipse azimuth and ellipticity, the relative phase retardation (A) and relative amplitude diminution of the parallel and

perpendicular components of the reflected light, and thus the thickness

and optical constants of the surface film can be quantitatively determined. By analyzing the reflected light at various potentials, a thickness versus potential curve can be plotted.

It has been hitherto customary to use cells with optical windows

normal to the light beam. In electrochemical studies, it is often convenient to use rectangular cells made of optical-grade, synthetic

quartz with highly reflecting metal tape held between the bottomless

cell and a bottom glass plate. The tape projects outside the cell and

electrical contact can be made to the projecting part. Such an

arrangement corresponds to the ideal current distribution of a rec- tangular conductor.

It is important, however, to ensure that the cell is alined with its sides parallel and perpendicular to the plane of incidence. A very

sensitive method of carrying out this alinement has been described in a communication which deals with the theory and practice of the use of rectangular cells in ellipsometry [12]. Figure 11. Sequence of optical components and optical path through the ellipsometer.

1. Hg vapor lamp; 2. 5461 A filter; 3. collimator;

4. polarizer; 5. quarter-wave plate; 6. analyzer;

7. photo multiplier. The polarization states at the

various stages in the optical path: 0, P", P', P, Z, Z\ Z", and L are represented on the Poincare sphere

(see fig. 12) . di" is the angle of incidence read on the

ellipsometer, and 0< is the true angle of incidence at the mirror. (Reproduced from [6].) Apart from this experimental precaution of cell alinement, the

use of a rectangular cell requires that careful Fresnel corrections be

made at four interfaces (Air-Glass I, Glass-Electrolyte I, ElectrolyteGlass II, and Glass-Air II) . The optical path through the ellipsometer

is shown in figure 11 and the Poincare sphere representation of the

polarization state of the light beam at various stages in the optical path

is shown in figure 12. The changes in polarization state P"—>P'

are due to the entry of the light beam into the electrolyte after tra- versing two interfaces, Air-Glass I and Glass-Electrolyte I. These

azimuthal changes are given by

tan «i=(tan oT)/{cos (0<— cos (6 t —6'/)}.

When the plane polarized light is reflected from the metal, a relative

phase retardation A and a relative amplitude diminution tan 4*=

^ °^ results in elliptically polarized light 5. On exit from the cell and
242 REDDY AND BOCKRIS

Figure 12. P&incare sphere representation of polarization

states at the various stages in the optical path shown in figure 11. (Reproduced from [6].) traversing Electrolyte-Glass II and Glass-Air II interfaces, two more

changes (5->6->7) in polarization state occur which are once again

given by equation:

tan ar =tan a^' cos —

t") cos (0*— 0*)- Thus, it is seen that the Fresnel corrections P"^Pf -^>P and

Z^Z'^Z" can be quantitatively determined knowing the angle of incidence, 0 f , and the refractive indices of the glass and the electrolyte.

The results of a quantitative study of steady-state oxide films on platinum in IN H2 S04 solutions are shown in figure 13. Prior to a computer calculation of the thickness and optical constants of the

oxide films, preliminary calculations were made. Assuming a value

of nF =2 for the refractive index of the oxide, the thickness of the

oxide at various potentials has been calculated (fig. 14). The

significance of these results has-been discussed elsewhere [5].

6. Adsorption of Organic Molecules on Solid Electrodes

A knowledge of the adsorption behavior of organic materials is of crucial importance to electrochemistry from both a theoretical and

an applied point of view [14]. It is our intention here to follow a suggestion made by Devanathan [15], and to point out the method of using ellipsometry in such studies.
ELECTROCHEMICAL STUDIES 243

36.20°

3600°

35.80

35.60°

16.60

16.4(5

1620° b

1 16.00 <

1580

1560°

0.34 0.54 0.74 0.94 1.14 1.34 1.54 174 POTENTIAL (volts vs. NHE) Figure 13. (a) Change of relative amplitude diminution (\p) with potential;

(b) Change of relative phase retardation (A) with potential

Consider the adsorption of an organic molecule L A long and let us assume that the molecule adsorbs "head-on." A potentiostat is used to hold the solid electrode at various potentials and at each value

of potential the extinction settings of the optical components of the

ellipsometer are determined. The instrument readings would cor- respond to a hypothetical film of thickness L and refractive index n

which would depend on the percentage coverage of the solid electrode

with the adsorbent. The observed film refractive index (nF) can be

related to the coverage by referring to calibration curve giving the

refractive index of bulk mixtures of the organic substance and the

electrolyte as a function of the percentage of the organic ranging

from zero (pure electrolyte) to 100 (pure organic). In this manner

the variation of the adsorption with potential can be determined.

The authors acknowledge financial support from the Aeronautical

Systems Command, Air Force Systems Command, U.S. Air Force

under Contract No. AF 33(616)-8150, and from the National

Aeronautics and Space Administration under Contract No. NsG-325.
244 REDDY AND BOCKRIS

POTENTIAL (volts vs.NHE) Figure 14. Approximate thickness of platinum "oxide" at various potentials.

They thank E. Passaglia, R. Stromberg, and J. Kruger of the National

Bureau of Standards for helpful discussion, and Dr. R. J. Archer

of Bell Telephone Laboratories for a preprint of his paper on

"Photoelectric EUipsometry." One of us (A. K. N. R.) thanks the

Council of Scientific and Industrial Research, India, for the grant

of Leave of Absence.

5. References

[1] L. Tronstadt, Det Kgl. Norske Videnskabers Selskabs Skrifter, No. 1 (1931).

[2] A. B. Winterbottom, Det. Kg. Norske Videnskabers Skrifters, No. 1 (1955).

[3] D. S., Heavens, Optical Properties of thin Solid Films, Butterworths Scientific Publications, London (1955).

[4] M. Pourbaix, Atlas d'Equilibres Electrochimiques a 25° C, Gauthier-Villars & Cie

, Paris (1963).

[5] A. K. N. Reddy, J. O'M. Bockris, and M. Genshaw, in preparation.

[6] J. O'M. Bockris, M. A. V. Devanathan, and A. K. N. Reddy, Proc. Roy.

Soc. 379, No. A1378 (1964).

[7] A. K. N. Reddy, M. A. V. Devanathan, and J. O'M. Bockris, J. Electroanalytical Soc. 6, 61 (1963).

[8] A. K. N. Reddy, and J. O'M. Bockris, in preparation.

[9] W. Budde, Appl. Optics. 1, 201 (1962).

[10] R. J. Archer, HP Associates, Palo Alto, Calif., private communication.

[11] G. N. Ramachandran, S. Ramaseshan, Handbuch der Physik, Vol. XXV/1

(1961).

[12] R. J. Archer, J. Electrochem. Soc. 104, 619 (1957).

[13] A. K. N. Reddy, in preparation.

[14] J. O'M. Bockris, and D. A. J. Swinkels, in preparation.

[15] M. A. V. Devanathan, private communication.
Application of Ellipsometry to the Study of

Phenomena on Surfaces Prepared in UltraHigh Vacuum

J. F. Dettorre, T. G. Knorr, and D. A. Vaughan

Battelle Memorial Institute, Columbus 1, Ohio

A bakeable vacuum system has been built which allows ellipso- metric observations of evaporated films as prepared and held at pressures of 10~9 torr or lower. Valving permits the introduction of controlled atmospheres. The dispersion of evaporated and bulk

copper specimen show significant differences between samples oxidized

in dry oxygen and samples corroded in air. Changes in optical properties immediately after deposition indicate the films are losing surface material at room temperature. This can possibly be explained as desorption of gases collected during film formation. In the case of aluminum the changes reverse direction, and after about 4 hr indicate the growth of a surface film approximately 10 A thick.

1. Introduction

There have been no reports, as yet, of the in situ study of the surface

of metal films evaporated in ultra-high vacuum. This paper contains

such observations and indicates some of the potential of a system

specifically designed to make measurements of this type [l].

1 The

observations reported are somewhat preliminary inasmuch as they were used to both evaluate the vacuum system and to establish

possible directions for fundamental investigations.

1.1. Apparatus and Techniques

The vacuum system developed for these studies has been discussed

in detail elsewhere [1]. A brief description of the system and its capabilities will be listed here for completeness. Space limitations were imposed upon the system design, since it was intended for use on

an unmodified commercial ellipsometer. Construction was of 304

stainless steel with connections made through standard copper shear

seals. Ion pumping was used exclusively to maintain the highest

degree of cleanliness. The system can be isolated for bakeout to assure operating pressures in the range of 10~9 torr or below.

Samples were mounted in a furnace made from a copper block,

treated to prevent oxidation at high temperatures in air. Heating

of the samples was achieved by means of coiled, platinum-wire

heaters contained in ceramic tubes passing through the block. Substrates for evaporated film samples were clamped in the furnace with

chamfered stainless steel jaws which also define reproducible edges.

The furnace was mounted such that the sample substrate could be

1 Figures in brackets indicate the literature references on p. 252. 245
246 DETTORRE, KNORR, AND VAUGHAN

alined with respect to the windows before sealing the vacuum system. j Leads for simultaneous electrical measurements made contact to the ! films through spring clips and baked-on platinum probes. The i sample and furnace were contained in a chamber which also contained

a coil of copper tubing used with coolants for supplemental cryogenic

j pumping.

The windows used for the ellipsometric observations were 1-in. circles of ^-in.-thick quartz or Pyrex. These windows were sealed to the main chamber with indium gaskets held in stainless steel containing rings. The use of indium limits bakeout temperature to 150 °C

which nevertheless has been found to be sufficient to insure operating

pressures in the range 10~9 torr or below. The windows were fixed in | position and the sample was alined such that the angle of incidence ; was 66 deg.

An auxiliary chamber containing an evaporation source was located on a direct line in front of the sample. The distance between source and substrate was about 6 in. Pumping, gaging, and valving connections for the introduction of controlled atmospheres were contained on a manifold at the opposite side of the main chamber.

Copper was chosen for most of the investigations since comparisons

could then be made with other studies already reported for this ma- | terial. Such comparisons provide a most effective method for evalj uating the usefulness of the technique and apparatus discussed above.

Data were taken at points of equal intensity around the minimum for both the analyzer and polarizer. Averages of these readings were

used as the input to a computer program which calculated values for

\f/ and A. These values were ultimately used to determine optical constants of either the base material or the combination of base material plus added film. Kinetic data for the changes yp and A with

increase in oxygen partial pressure were obtained by following these

parameters as a function of time. The data were typical of those ex- pected and observed elsewhere on samples cleaned by various techniques. Analysis of the kinetics permitted evaluation of activation

energies for oxide development which, however, are not included in the present discussion. The oxide overgrowth studied on the copper

films reported here had an index of refraction of 2.80-i0.165 and grew

at room temparature, to a thickness of about 40 A as determined from

the ellipsometric data. Some of the most interesting data were obtained from the dispersion relations and will be discussed below.

1.2. Experimental Results

Figure 1 shows the dispersion of the reapportion of the complex

dielectric constant for a copper film 1500 A thick evaporated and

measured on a substrate held at room temperature. Copper films of

this thickness do not possess bulk electrical properties but are almost

optically opaque. Effects of the glass substrate on the observed

values of \}/ and A were assumed negligible in the range of film thickness
SURFACES IN ULTRA-HIGH VACUUM 247

100

Ev

c aporated

3 In vqcul

< Anneale

~> Oxidize

Copper

)

< m

d

j in 02

—7"X

X

_ B

cr

4000 5000 e 6000

X, A Figure 1. The real portion of the complex dielectric constant for a copper film 1400 A thick. Three conditions are shown; as evaporated in vacuum, after anneal at 150 °C for 2 hr, and after oxidation in dry oxygen. All measurements were taken at room temperature.

studied here. Three dispersion curves are shown; the curve labeled

"in vacuum" was measured as soon as the sample stabilized. In the

case of copper, these data were usually taken about an hour after formation. The second dispersion curve, "annealed," was measured

at room temperature after the copper film had been held at 150 °C

for 2 hr. Treatment of this type was sufficient to significantly

anneal lattice defects in the evaporated film. Measurements on other

samples of similar thickness indicated that the change of electrical

resistivity was less than one percent during the last half hour of anneal.

On this basis, the anneal was arbitrarily taken as complete. The

third dispersion curve, "oxidized," was measured after the film had

stabilized at atmospheric pressure, usually at least 24 hr after formation. The three curves in figure 1 indicate no significant changes in
248 DETTORRE, KNORR, AND VAUGHAN

10.0

c 1.0

0.1

X B-

\ x

X

Evaporatec o In vc Copper jcuum

x Ann< o Oxid?aled ized in 02

4000 5000 6000

X,A

Figure 2. T%e imaginary portion of the complex dielectric constant for the copper

film shown in figure 1 . dispersion after annealing, whereas the oxidation treatment appears

to have resulted only in a vertical downward displacement of the

curve. The values shown in these curves are in fair agreement with

Roberts' [2] values when the difference in thickness of the samples is considered. The dispersion curves after oxidation are made for two

reasons: (1) to determine the effect of oxide layers on the dispersion

curve, and (2) to determine any window effects. Corrections for the

effects of strain birefringence are made in the following manner:

with the specimen inside the system the dispersion curve is measured

after the sample is allowed to stabilize at atmospheric conditions.

The specimen was removed from the vacuum system and the dispersion curve measured again. The difference in readings represents

the effects of strain birefringence and is the correction that must be

applied to measurements taken when the specimen was inside the vacuum chamber. Corrections measured in this way were of the

order of two degrees.
SURFACES IN ULTRA-HIGH VACUUM 249

90

<

70

Copper in Vacuum

+ Cu 7 x Cu 13 o Cu 14

+

o

• o

•—o

Cu 15 • Cu 17 Bulk

X

+ .

X

+

X

o o 8

•

• D

O•

X #

m

X X i

+ .

o

•

° °. X

+

a

X X

+

"

+ 4000 5000 6000

X, A Figure 3. Analyzer readings for the six copper samples studied. The data are corrected for the strain birefringence of the windows in the vacuum system. All measure- ments were taken at room temperature. Figure 2 shows the dispersion of the imaginary portion of the complex dielectric constant for the same film. The general shape of the curves is similar to that shown by Roberts [2]; however, the minima

are not as deep. Likewise, the changes in magnitude of the values

shown for — (n2 —

2), see figure 1, are not as great as those reported

by Roberts. Some preliminary data, other than that shown, suggest

that the dispersion observed in both parts of the complex dielectric constant are related to the thickness of the metal film. Relative changes among specimens can be compared more easily

by direct inspection of the data after correction for the strain birefringence of the windows. Corrected analyzer readings for five evaporated copper specimens of different thickness, and a bulk

specimen, are shown in figure 3. These data are measurements of the dispersion in vacuum. The curves indicate a relationship between

the dispersion and film thickness for four of the films and the bulk

specimen. The exception is Cu 17. There is no experimental evi- dence to suggest exclusion of the Cu 17 data. The thicknesses of
250 DETTORRE, KNORR, AND VATJGHAN

i

4000 5000 6000

. o X, A Figure 4. Corrected analyzer readings for the bulk copper specimen.

Three conditions are shown: as prepared in atmospheric conditions (labeled "oxidized in air"), after hydrogen reduction in vacuo, and after oxidation in dry oxygen. All measurements were taken at room

temperature.

the samples measured were Cu 14, o 650 A; Cu 15, 1500 A; Cu 13,

1700 A; Cu 17, 1800 A; Cu 7, 3300 A. Polarizer data for all samples

are essentially the same and typical measurements will be shown in figure 5. Data obtained for a polished bulk specimen are shown in figure 4. This specimen was mounted in a small glass vacuum system, hydrogen

reduced and sealed in high purity dry nitrogen. The system was then

baked to 400 °C for 18 hr which resulted in a pressure of 10~9 torr. The corrected analyzer data of the bulk specimen are shown in the

upper curve in figure 4, which is labeled "oxidation in air." The curve measured in vacuo is again shown for comparison. The effect of oxidation at 1 atm of pure dry oxygen at ambient temperature can be

seen from the middle curve. The shapes of these three curves are

also characteristic of those observed for evaporated films treated similarly. There is a significant difference between the dispersion charac-
SURFACES IN ULTRA-HIGH VACUUM 251

80

40

Bulk o Oxidi

In va a Oxid rCopper

zed in air cuum

zed in 02

A -\a M^-

A 4000 5000 6000

X, A Figure 5. Corrected polarizer readings for the bulk copper specimen shown in

figure 4.

teristics of specimens oxdized in oxygen and those corroded in air. In the case of the bulk specimen, the air oxidized curve, which represents the "as prepared condition," probably contains the effects of cold work introduced during the polishing procedure which possibly was modified or removed during the high temperature cleaning.

Corrected polarizer data for the bulk specimen are shown in figure 5. Two of the curves are drawn and indicate the typical spread in data

observed for all samples studied. In addition, the two curves drawn

in figure 5 define a band of values which contains all of the curves measured for the evaporated copper films. Observations of oxidation rates at reduced pressures have also been

made. In the case of the bulk copper specimen just described, no

changes in analyzer and polarizer readings were observed at oxygen

pressures equal to or less than 100 n. At this point the oxygen pres- sure was increased to atmospheric and the oxidation rate followed.

The observed absence of change at pressures below 100 n is in agree- ment with an extrapolation to room temperature of Yoda and Siegel's

728-101 0—64 17
252 DETTORRE, KNORR, AND VAUGHAN

[3] electron diffraction studies of the nucleation of copper oxide.

Similar observations were made in the case of evaporated copper

studied here, except that immediately after evaporation there was an

initial period of change indicating the loss of surface film material.

Although not completely understood, this change is probably due to the desorption of gases pumped by the evaporant stream during deposition. After about 20 min the surface stabilized and again no changes were observed at oxygen pressures up to 100 m- Observations of the behavior of aluminum evaporated and maintained at 10-9 torr and room temperature exhibit a behavior similar

to that discussed above. However, after the initial period of observations, indicating the loss of surface film material which could be

desorption or possibly annealing in the case of aluminum, the changes

in the measurements reversed direction indicating the formation of a surface film. The growth, at 10~9 torr, is completed in 3 to 4 hr. It is of interest to calculate the time required for a monolayer to form

at this pressure. If one assumes the residual gas in the system is 20 percent oxygen and the sticking factor is unity, it would require

approximately 4 hr for the surface to be covered by a monolayer of oxygen. The ellipsometric data indicate the thickness of oxide formed

at 10~9 torr is approximately 10 A. This value agrees with the limiting thickness proposed by Lanyon and Trapnell [4] based on a place

exchange mechanism of oxidation. This mechanism has also been

employed by Eley and Wilkinson [5] to explain the oxidation of aluminum at 10"^ torr. A limited amount of electron microscopy has been done on surface

replicas to investigate the possibility of observing nucleation phe- nomena in the oxide overgrowth. There is evidence to suggest

nucleation in the oxides grown in dry oxygen, however the density of

sites varies by orders of magnitude from sample to sample. No

correlation with preparation conditions or film surface have been

investigated as yet. The overgrowth produced by corrosion in air is observed to be much rougher and does not contain obvious nucleation

sites. A group of experiments has been started in another ultra- high vacuum system where the availability of multiple evaporations

will allow us to compare simultaneously deposited specimens immediately after evaporation and after oxidation

One of the most interesting questions still to be investigated is the

interpretation of the data immediately after evaporation. In addition, the effects of evaporation parameters, annealing, desorption,

and oxidation must be isolated and characterized. Also, the meaning

of 1 'index of refraction" and "thickness" must be established when the

film consists of isolated nuclei or small patches. Future investigations

are being directed toward a fundamental understanding of the processes involved.

2. References

[1] J. F. Dettorre, T. G. Knorr, N. F. Hartman, and G. G. Cocks, Rev. Sci. Instr. 35, 503 (1964).
SURFACES IN ULTRA-HIGH VACUUM 253

[2] S. Roberts, Phys. Rev. 118, 1509 (1960).

[3] E. Yoda, B. M. Siegel, J. Appl. Phys. 34, 1512 (1963).

[4] M. A. H. Lanyon and B. M. W. Trapnell, Proc. Roy. Soc. A. 234, 405 (1956).

[5] D. D. Eley and P. R. Wilkinson, Structure and Properties of Thin Film,

ed. C. A. Neugebauer et al., John Wiley & Sons Inc., New York (1959)

H. T. Yolken (National Bureau of Standards) : I wonder if (in the

aluminum studies) you made any effort to get back to the beginning

point before your 20 minutes were up by using bombardment, heating, or some other method; or do you think this is an annealing

process relative to the film growth?

J. F. Dettorre: No, we have not. These were deposited and

observed at room temperature with no special treatments. H. T. Yolken: Could the difference between the oxidized in air specimens and the oxidized in dry oxygen specimens be due to surface preparation? The oxidized in air specimens were never heated

or reduced with hydrogen prior to oxidation while the oxidized in dry

oxygen specimens were given a hydrogen anneal and bake at 400 °C.

J. F. Dettorre: You are referring to the bulk specimen. Yes, the

differences could be due to surface preparation in this case. For the

bulk specimen air oxidation data were taken after the specimen was

prepared, that is, after the specimen was placed in the vacuum sys- tem, but had not been evacuated. Measurements were made at that

time, then the specimen was hydrogen reduced, sealed off in nitrogen

and baked at 400 °C. Surface preparation, however, can not explain

differences, similar to those observed in the bulk specimen, between

thin film samples prepared under equivalent conditions but oxidized

either in air or dry oxygen.

C. Del Carlo (Chrysler Corporation) : This change in A which you

described with respect to time, we observed also with aluminum films,

and at a pressure of 5X10-8 torr; we found the effect is largely

reversible with the subsequent heating of the film after evaporation.

The process can be repeated a number of times.

H. T. Yolken: I wondered if you were able to determine any dif- ferences in your equilibrium film at the end for different pressures?

J. F. Dettorre: From 10-9 torr to less than 100 microns we reproduced our instrument settings to the nearest 0.02 degree at each

pressure, so we essentially observed no changes until greater than

100 microns. At 100 microns we stopped our pressure variation, and

increased the pressure to atmospheric.

H. T. Yolken: How do you account for no film growth up to the

high pressure of 100 microns?

page 508. Discussion
254 DETTORRE, KNORR, AND VAUGHAN

J. F. Dettorre: I can't exactly, We observed this in the case of copper. A possible explanation is that if there is an adsorbed film

already on the copper, it must be replaced by another gas before

changes could be observed. The work of Yoda and Sigel extrapolates

to this case, if the extrapolation is valid and indicates that you must

have an oxygen pressure greater than 2,000 microns to observe changes

in copper.

H. T. Yolken: We have done work with iron where the sticking

probability appeared to be 5X 10~ 3 after the first monolayer is formed.

We did oxidations from 1X10-7 torr oxygen to 1X10-3 torr oxygen;

the rate changes, but the equilibrium film thickness is almost independent of pressure.

J. F. Dettorre: At any given pressure the maximum time we used was around 10 to 12 hours and the minimum time was about 8 hours.

Any changes that might have occurred for times much longer than

these would not have been detected. All changes observed occurred

in the first few minutes at any given pressure. A. K. N. Reddy (University of Pa.): The statement you made

about the nodules raises a very interesting problem of the optics of such discontinuous films. I wonder whether there could be some

discussion on this subject? Perhaps Dr. Passaglia would like to comment on this. E. Passaglia (National Bureau of Standards): The problem we

have been concerned with is scattering from inhomogeneous films, in which the refractive index varies in directions parallel to the surface

as well as normal to the surface. The films were polymer films and

are discussed in "Application of Ellipsometry to the Study of Adsorption from Solution" by R. R. Stromberg, E. Passaglia and D. J. Tutas,

this book, p. 281. In a very crude approach, I assumed the films

would scatter much like a polymer solution of the same concentration.

Then, assuming that the scattering would do the worst possible thing,

namely subtract from the p component, and not the s component,

etc., one can get a rough estimate of the error in \p due to this scattering. This was less than 0.01° in this case. In the case of nodules, I have always thought, rather naively, that one takes the horizontal projection of the area, if the surface is rough,

for everything else reflects off in other directions and does not enter the analyzer. However, if such a rough surface is covered with a

film, the simplest thing one can do is assume the refractive index to be a linear combination of the index of the substrate and of the film,

weighted by volume fraction or some other means. Dr. Archer

discusses this in his paper, "The Measurement of the Physical Adsorption of Vapors and the Chemisorption of Oxygen on Silicon by the

Method of Ellipsometry", this book p. 255.
Measurement of the Physical Adsorption of

Vapors and the Chemisorption of Oxygen

on Silicon by the Method of Ellipsometry

R. J. Archer 1

Bell Telephone Laboratories, Incorporated, Murray Hill, N.J.

Physical adsorption isotherms for water, carbon tetrachloride, and

acetone have ellipsometric thickness scales which agree with the dimensions of the adsorbed molecules for substrate surfaces prepared

by etching at the maximum rate in an HNO3-HF solution. All other methods of preparing the substrate surface result in isotherms which

have ellipsometric thicknesses at monolayer coverage which are smaller than the diameters of the adsorbed molecules by factors as small as one-half to one-third. The effect is attributed to the degree of irregularity of the substrate surface. A calculation for a specific model

of an irregular surface predicts effects of the same magnitude as observed experimentally. The adsorption of water is found to be

sensitive to the chemical structure of the silicon surface. The chemisorption of oxygen on atomically clean, cleaved silicon surfaces gives ellipsometric thicknesses for the monolayer of 2.5 to

2.8 A. Sticking coefficient as a function of coverage for four pressures

of oxygen from 7X 10~8 to 1.6 X 10~6 mm of Hg shows that the sticking

coefficient is independent of pressure, and that the average value,

6X 10-3

, is independent of coverage up to coverages of 0.6 to 0.7. The

magnitude of the sticking coefficient agrees, within a factor of two, with most previous measurements.

1. Introduction

The use of the exact equations of reflection theory to obtain the

optical constants and thicknesses of films on silicon by the method of ellipsometry has been recently reported [l].

2 The emphasis was on

relatively thick films in the range from 10~6 to 10" 4 cm. This paper is concerned with the use of the method for measuring very thin films. The phenomena studied are the physical adsorption of water vapor

and some organic compound vapors on practical silicon surfaces and

the chemisorption of oxygen on atomically clean silicon surfaces. The thickness range of concern is from submonolayer coverage up to several layers, with an upper limit of the order of 10 A. This paper is divided into three main sections plus a summary.

The remainder of the present section deals with the principles and

theory of interpreting ellipsometric measurements on very thin films. The presentation and discussion of experimental results will occupy

the following two sections. The first will be concerned with physical

adsorption, the second with chemisorption.

The effect of such very thin films on the state of polarization of reflected light requires some discussion. The experimentally measured

1 Present address: HP Associates, Palo Alto, Calif.

2 Figures in brackets indicate the literature references on p. 277. 255
256 ARCHER

quantities are A and \f/, the angles which define the ratio of the Fresnel

coefficients for the p and s component waves, i.e., ^=tan*e<A

. (1) The formation of a surface film and changes in its thickness are detected as changes in A and \p. The dependence of A and \p on

the properties of very thin films can be derived from two points of view. The film may be considered to be a three-dimensional phase

characterized by a thickness and an index of refraction, or, for the

specific case of monolayer films, a two-dimensional model may be

used in which the film is treated as a surface distribution of Hertzian

oscillators. The results of both derivations will be discussed.

1.1. Three-Dimensional Theory

The general, exact relationship between A and \p and the properties

of the surface film when the film is characterized by a thickness and

an index of refraction is given, for example, as eq (3) in ref. [1]. By

the use of electronic computer techniques it is easy to determine

the dependence of A and \f/ on the film properties from this equation.

However, first-order approximate equations can be derived which

are accurate for very thin films and which allow easier illustration

of the properties of the relationships.

For the ratio of the Fresnel coefficients, first-order theory [2, 3] gives

t&n\pe iA =teai\pe'

Hit cos <p sin2

<p \~{}__±\ (7) 2 -in/H

(2) (See the Glossary of Symbols for definitions.) From eq (2) are derived

A=A—ad (3) and

*=*+/3d (4) where

4-7T cos (p sin2 <p(jii —1) "^^^2 —&^[cos2 <p —& +sin2 ^(a2 -a,2)]+a,2 (l-2asin2 <^) ^

X { [cos2 ^-a+sin2 ^(a2 -a/2

)

2 +a,2 (l -2a sin2 <p)

2

} (5)
ADSORPTION ON SILICON 257

and

27rsin 2\p cos <p sin2 <p(nl —l)a f ^(~2~a)

(1— 2a sin2 ^) —[cos2 <p— a-\-sm2 <p(a

2 —

a' 2)] ^

X { [cos2 <p-a+sin2 *>(a

2 -a/2)]2 +a'2 (l-2a sin2 *>)

s (6) Equations (5) and (6) contain more terms than the expressions

usually quoted [4, 5], and have been given before [6]. They are repeated because of typographical errors in the indicated reference. In all of the work described here, the following conditions have

been constant: <?=70.00°, A=5461 A, and n(Si) =4.050— 0.028i[l].

Evaluation of a and j8 for these parameters using %= 1.5 gives

a-0.3 deg/A (7) 0-0.001 deg/A (8) so that for films in the thickness range under consideration, \J/ is effectively constant, and changes in film thickness are detected only

as changes in A. In order to calculate the change in thickness cor- responding to a measured change in A it is necessary to assume a value

for the index of refraction of the film in order to evaluate a. The limitation of having to guess a value for nx is not too severe. This can be shown with reference to figure 1 which plots a as a function

Figure 1. Calculated dependence of the thickness coefficient, a, on the index of

refraction, ni, of a surface film.
258 ARCHER

of n1 . Assuming a likely lower limit for % of 1.2 for any film leads

to the condition 0.188 <a<0.380 for all films in the range 1.2 <ni< 3.4. An intermediate value, a, can be selected for evaluating

all films. It follows that if2(0.188)(0.380) a " 0.188+0.380 -a2iH W

the maximum error is ±34 percent of the thickness calculated from

*=M- (10) a The question of the effect of multiple films on A arises in the physical

adsorption work. The surfaces used for physical adsorption have

permanent films 10 to 20 A thick upon which the adsorbed layer

forms. Also, the interpretation of some of the data uses a model of the surface which pictures adsorption as resulting in multiple films. For multiple, very thin films, each film affects A independently and

the change in A due to any one film is the same as that which would

occur if the film were present as a single layer. That is, A=A-SaA, (11)

subject to the condition 2<ii<<X where the a/s are calculated for each film from eq (5). Equation (11) may be derived by expanding

the exact expressions for the Fresnel coefficients of a multiple film

system and retaining only terms in thickness to the first power.

The use of the three-dimensional theory to deduce the properties

of very thin films must be questioned. A reasonable criterion for

its validity, and for the validity of the concept of index of refraction,

is that the thickness of the film be large compared to the dimensions

and separations of the constituent atoms or molecules. For monolayers, for example, it is a priori unreasonable to expect the quantity

Ui in eq (2) to be of the same magnitude as the index of refraction

of the film material in a bulk phase.

For films which consist of only partial monolayer coverage, the

interpretation of measurements with the three-dimensional theory, even assuming its validity for such films, is complicated by the fact that for whatever value for the coverage, 0, the thickness, d, of the

film is constant, at least for the case of spherical atoms or molecules,

or for any other case if the orientation of the atoms or molecules

relative to the surface is independent of coverage. Only the density

and index of refraction of the film vary with 6, and to deduce 6 from

changes in A requires a knowledge of the functional dependence of % on 6. One might, for example, assume that nx is the sum of the

indices of refraction of occupied and unoccupied adsorption sites,

each weighted by relative coverage. Then, n1 = (l-6)+dn0 ,
ADSORPTION ON SILICON 259

where nQ is the index of refraction of a complete monolayer. Again,

one might assume a surface corollary of the Clausius-Mosotti rela- tionship so that

7&1+2

Neither model is convincing, and both lead to complicated relationships between A and 6. Fortunately a very simple relationship is predicted by the derivation of the Fresnel coefficients based on a two-dimensional, monolayer model of the film.

1.2. Two-Dimensional Theory

Strachan [7] and more recently Sivukhin [8] have derived the

Fresnel coefficients on the basis of a monolayer model with identical

results except for differences in terminology and symbolism.

Strachan's results will be given.

Strachan postulates that a monolayer is equivalent to a twodimensional distribution of scattering centers and assumes that an

element of surface dS when influenced by a light wave whose electric vector has components with amplitudes Au A2 , and A3 becomes

a Hertzian oscillator of strengths

dS{al Al + a2A2 + <i3A3 } (12)

parallel to the axes xu x2 , and xz . xx and x2 are mutually perpendicular

directions in the plane of the surface. xs is normal to the surface.

The possibility of molecular orientation in the film is allowed for by

the use of the three scattering indices a1} a2 , and tr3 . Generally,

ffl From this point of view, Strachan derives the following first-order

expression for the ratio of the Fresnel coefficients

tan^z

' A =tani/^M 1 i^T cos <p sin2

<p

(13)

This equation is identical to the corresponding three-dimensional rela- tionship, eq (2), except for the quantities in brackets which contain

the terms that characterize the film, scattering indices in the present

case, and thickness and index of refraction in the former case. Corresponding to eqs (3) and (4), eq (13) gives

=*+S«** (15)
260 ARCHER

Since the scattering indices define the oscillator strengths per unit

area, it is reasonable to assume that they are proportional to coverage.

Writing o-,=0o-J, it follows that

The instrument and the measuring technique have been described

before [1, 6, 9]. To recapitulate briefly, the ellipsometer is a standard

polarizing spectrometer with divided circles which can be read to 0.01°. The light source is a mercury arc with an interference filter for the 5461

A line. The quarter-wave plate transmits the incident beam and

is fixed in orientation with the fast axis inclined at 45° to the plane

of incidence. A and \p are given by

8 is the retardation of the compensator, A and P are the orientations

of the analyzer and polarizer which extinguish the beam. The detector is a photomultiplier tube at the exit of the telescope. A and P

are measured by determiniug settings of equal intensity on each side

of the extinction minimum, using the procedure described in ref. [1]. Extinction settings can be made rapidly, e.g., 20 sec, and with a reproducibility equal to the precision with which the divided circles can be

read. The relative retardation of the wave plate is approximately

90°, therefore A~90° —2P, and since P can be measured to ±0.01°,

A can be determined to at least ±0.02°. If A is changing rapidly with

time, the precision will be less good, depending on the rate of change.

It follows, that since o;^0.37A, changes in thickness of ±0.07 A

can be measured.

tan A=sin 8 tan (90°— 2P) (18)

where

tan ^=tan (—A) cot L (19)

j (20) cos 2L= —cos 8 cos 2P.

2. Physical Adsorption

2.1. Technique

Physical adsorption was measured on surfaces which were pre- pared and cleaned chemically in air. Such surfaces are relatively

complex, compared, for example, to atomically clean surfaces, so that
ADSORPTION ON SILICON 261

it is necessary to give some of the details of their preparation and

chemical properties.

The substrate material was single-crystal, 10 ohm-cm, p-type silicon with, in most cases, surfaces parallel to the (110) crystal plane.

Plates 2X1X0.1 cm were mechanically polished to optical flatness. After polishing, the surfaces were cleaned in hot organic solvents, hot

nitric acid, and hot quartz-distilled water. In addition to the polished

surfaces, measurements were also made on etched surfaces, on

heavily oxidized surfaces, and on cleaved surfaces. In all cases the

substrate was covered with a permanent film 10 to 20 A thick.

Depending upon the chemical treatment, the permanent film may be

either silicon dioxide or silicon hydride, and the surface may be

either hydrophobic or hydrophilic.

Etched surfaces were prepared by agitating a polished silicon plate

in a solution composed of concentrated nitric acid (70 percent) and

concentrated hydrofluoric acid (48 percent). Because the adsorption results depend on the composition of the etchant, some details

must be specified. Three different solutions were used; designated

by the volume proportions of nitric acid and hydrofluoric acid, they

are: 10:1 (10 parts HN03 , 1 part HF), 3:1, and 3:7. The approximate etching rates for the three etches, in mils per mmute, are, re- spectively, 1, 3.4, and 65 [10]. The last rate is the maximum for HF-HNO3 solutions.

The etched surfaces without subsequent treatment are hydrophobic

and have permanent films 15 to 20 A thick which are probably mixtures of silicon hydrides and oxides [ll]. 3 The surfaces are rendered

hydrophilic and have permanent films 10 to 15 A thick of silicon di- oxide, presumably vitreous silica, if they are treated as follows: (1) hot nitric acid; (2) rinse in hydrofluoric acid; (3) hot nitric acid; (4) hot quartz-distilled water. This treatment will be referred to below

as the hydrating treatment. From the fact that these surfaces are hydrophilic, from the water adsorption results, and by analogy to the

surface structure of hydrophilic silica gel [12], it is concluded that

the oxide films are terminated by a two-dimensional array of hydroxyl

groups bonded to silicon atoms in the permanent film. This is the

so-called silanol structure. The cleanest chemically prepared surfaces, on the criterion of the

thickness of the permanent film, result from rinsing the hydrated surfaces in hydrofluoric acid. For this case, ellipsometric measure- ments indicate permanent film thicknesses of 5 to 10 A. Such

surfaces are hydrophobic.

The cleaved surfaces (111) used for physical adsorption are obtained by cleaving a specially machined silicon bar in air. Details are given below in describing the chemisorption work.

The following are some details of the vacuum apparatus and

technique. The silicon substrate was mounted at the axis of the

3 Unpublished measurements by the author, G. W. Gobeli, and G. E. Becker of the infrared absorption

spectra of stain films on silicon substantiate the speculation in ref. [11] that these films are silicon hydride.
262 ARCHER

I

Figure 2. Physical adsorption vacuum cell. spectrometer in an all-Pyrex vacuum cell, figure 2, with fused-in

optically finished windows 1.5 in. in diameter and 0.25 in. thick.

The windows were oriented for an angle of incidence of 70°, and the

light beam was normal to the windows within 2 to 3°. Normally

the beam was reflected from an area about 0.5 cm in diameter at the

center of the sample and was transmitted by the center portion of the windows. The windows were strain-free before evacuation, but

under the pressure differential caused by evacuation, they introduced

a few tenths of a degree change in A. In no case was it necessary to measure the absolute magnitude of A, so that this small strain

birefringence was unimportant.

The vacuum system consisted of an oil mechanical pump in series with a three-stage mercury diffusion pump separated by two liquid

nitrogen traps from the vacuum cell. There were no greased joints or

valves. The gas delivery system and manometers were isolated with

all metal valves. The vacuum cell could be evacuated to 10~7 mm of Hg. Water vapor pressures were measured with a Dubrovin mercury

gage accurate to ±0.02 mm of Hg. Organic compound vapor pres- sures were measured with a Wallace and Tiernon aneroid manometer

accurate to ±0.2 mm of Hg. The liquids were multiply distilled and

outgassed under vacuum. The silicon plate was mounted in the vacuum cell in only approximate alinement relative to the spectrometer. Precise alinement was

effected by adjusting the position of the spectrometer which, for this purpose, was mounted on a milling machine head on an adjustable

tripod base.
ADSORPTION ON SILICON 263

After mounting and alining the sample, the cell was evacuated to

1 0~6 to 1 0~7 mmof Hg, and A and $were followed until they were constant

Most of the small change observed following evacuation was due to

strain birefringence in the windows. To measure the adsorption

isotherm, the vapor was admitted to the vacuum cell to the desired pressure and A was measured as a function of time until both the

pressure and A were constant. Usually this condition was obtained

within 10 min. This time lag is caused by the relatively slow equilibration of the vapor with the walls of the vacuum system. In some

cases subsequent points for the isotherm were obtained by changing

the pressure in small increments up to near saturation values. In

other cases, each point on the isotherm was obtained by completely

evacuating the system before and after each change m pressure.

2.2. Results

Figure 3 presents data which show the influence ot the chemical

nature of the substrate surface on water adsorption at room temperature <5A is plotted as a function of relative humidity. pa is the

equilibrium vapor pressure of liquid water. Since A decreases with

increasing thickness, 5A is the negative of the change in A relative to the value before adsorption. The thickness scale is calculated using

Figure 3. Water adsorption on silicon 25 °C.

(1) normally etched (3: 1) surface; (2) etched surface after hydrating treatment; (3) hydrated surface after heating to 940 °C for 30 minutes; (4) 3 days in water vapor at 70 percent relative humidity (25 C) after heat treatment; (5) hydrated surface after immersion in HF.
264 ARCHER

a=0.260/A from figure 1 assuming %= 1.33, the value for liquid

water at room temperature. Curve 1 is for a normally etched (3:1)

surface with no subsequent treatment. The extremely meager adsorption is consistent with the hydrophobic nature of etched surfaces. Following the hydrating treatment, curve 2 is obtained. This type II isotherm, according to the BET classification [13], indicates strong

interaction between the adsorbed film and the substrate surface. If the hydrated surface is heated in vacuo, curve 3 results. For this case the silicon was heated, by passing a current, to 940 °C for 30 min

at 10~6 mm of Hg. Like curve 1, this type III isotherm shows little interaction between adsorbed film and substrate. By analogy to the

effect of heat treatment on the structure of a silanol surface on silica

[12], it is concluded that the heat treatment causes a condensation

of the hydroxyl groups to evolve water and to form a hydrophobic

terminating layer of Si-O-Si groups, the so-called siloxane structure, on the permanent silica film. Long exposure to water vapor causes

a partial rehydration of the surface, as shown by curve 4, which is obtained after the heated surface is exposed to water vapor at 70

percent saturation (25 °C) for 3 days. If the heated surface is immersed in liquid water it is completely rehydrated and curve 2 is obtained again. Following a rinse in hydrofluoric acid, the initially

| hydrated surface shows a partial hydrophobic character, curve 5. The hydrophobicity is attributed to the replacement of hydroxyl

groups in the terminating layer by fluoride ions. These data exemplify chemical effects on water adsorption. The

following results show what may be characterized as a physical effect on ellipsometric thickness measurements and bring into question the

interpretation of measurements on very thin films. Figure 4 gives isotherms for water adsorption on silicon surfaces

with identical chemical properties but with different physical structures. The five isotherms are for substrates prepared by different

methods except that the final treatment was in all cases, the hydrating treatment described above that produces hydrophilic silanol surfaces. The substrates were prepared as follows: (1) mechanically

polished; (2) normal etch (3:1); (3) very rapid etch (3:7); (4) following

a 3:1 etch, this surface was ooxidized in steam (1 atm) at 1000 °C to produce a film about 5000 A thick, the adsorbed film thickness was

calculated from the exact Fresnel coefficient relationship for a two

film system using an electronic computer (the A scale does not apply

for this curve) ; (5) cleaved in air. It is interesting to compare these isotherms with one obtained on

silica by conventional manometric measurements. The broken

curve 6 is such an isotherm reported by Livingston [14] for water on

crushed quartz. His data (micromoles per square meter, 77) are con- verted to the thickness scale using the assumption that the density,

p, of the adsorbed film is the same as that of liquid water, i.e., rf=„^X 10- 2 A. (21)
ADSORPTION ON SILICON 265

Figure 4. Water adsorption on hydrated silicon surfaces, 25 °C.

(1) mechanically polished; (2) normal etch (3: 1); (3) fast etch (3:7); (4) 5000 A silica film (8 A scale does not apply U; Ssother

P

m); (5) cleaved surface; (6) Livingston's data for water on crushed quartz.

Figure 5 Water adsorption isotherms plotted according to the BET equation using

the data for the rapidly etched (3: 7), hydrated surface and the data of Livingston for water on crushed quartz.
266 ARCHER

M is the molecular weight of water. It is seen that the thicknesses

for all of the isotherms are considerably smaller than Livingston's

result, except for the case of the very rapidly etched substrate, which

agrees closely. All of the data of figure 4 have been analyzed by the BET method

[13] to determine the film thicknesses at monolayer coverage. According to the BET theory,

d0 is the thickness of the film at monolayer coverage. The symbol c

is a measure of the heat of absorption of the first layer. Figure 5 plots Livingston's data and the 3:7 etched isotherm according to eq

(22). The deviation from the predicted linearity at high pressures is typical of the fit of all adsorption data to BET theory. From the

linear parts of the curves at low relative saturations and eq (22), are calculated the values 3.2 A for the monolayer thickness of Livingston's

isotherm and 3.7 A for the rapidly etched substrate.

The value for d0 obtained for Livingston's data gives credence to the BET analysis for the water adsorption isotherms because it agrees

nearly exactly with the diameter of a water molecule calculated from

the density of liquid water,

NA is Avogardro's number. The reproducibility of the 3:7 isotherm

and the uncertainty in assessing the best straight line on the BET

plot lead to a variation of about 20 percent in the values for d0

. Values for d0 for curves 1, 2, 4, and 5 of figure 4 are, respectively,

0.8 A, 1.1 A, 1.6 A, and 1.0 A, all of which are anomalously small

in view of the diameter of a water molecule.

The same kind of dependence of the isotherms on the method of preparation of the surface is found for the adsorption of the vapors

of carbon tetrachloride and acetone. Curves 1, 2, and 3 of figure 6

for carbon tetrachloride show the dependence of the isotherm on the

composition of the etch. The compositions are, respectively, 10:1,

3:1, and 3:7. The dependence is not as strong as for the water case, but qualitatively the effect is the same—the more rapid the etching

rate, the larger the calculated thickness (again, thicknesses are cal- culated using the index of refraction of the liquid, 1.46). The cleaved

surface, curve 4, and the mechanically polished surface, curve 2 (the same isotherm as for the 3:1 etched surface) have isotherms that are intermediate in thickness. BET analysis of the data gives d0 values

for curves 1, 2, 3, and 4, respectively, of 2.0 A, 3.1 A, 3.6 A, and 2.7 A.

(22)
ADSORPTION ON SILICON 267

P/Ps Figure 6. Carbon tetrachloride adsorption on silicon, 25 °C.

(1) slow etch (10: 1) ; (2) normal etch (3: 1) or mechanically polished; (3) fast etch (3: 7) ; (4) cleaved surface. The largest value, 3.6 A, for the very rapidly etched case, is 66 percent

of the molecular diameter, 5.5 A, calculated from eq (23). Consistent with the inertness of carbon tetrachloride toward silica at room temperature, the isotherms are not affected by the chemical

nature of the substrate surface. The same results are obtained for

either hydrophobic or hydrophilic surfaces. Two isotherms for acetone on silicon are given in figure 7. The

cleaved surface has a dQ value of 2.1 A (% =1.36). The surface etched

in the 3:7 etchant has a value for d0 of 3.9 A which is 78 percent of the value 5.0 A calculated from eq (23). Palmer's data [15] for acetone on vitreous silica powder obtained by the conventional

manometric method are also plotted (the broken curve). d0 for this isotherm is 6.1 A.

It is concluded that for all of the vapors—water, carbon tetrachloride, and acetone—the monolayer thicknesses from BET theory

are comparable to the diameters of the adsorbed molecules for adsorption on the very rapidly etched surfaces (3:7) but that for other

methods of preparing the substrate, the monolayer thicknesses are smaller than the molecular dimension by factors as large as 2 or 3. 728-101 O—64 18
268 ARCHER

i

P/Ps Figure 7. Acetone adsorption on silicon, 25 °C.

Results for a cleaved surface and a rapidly etched surface are compared with the results of Palmer on powdered silica.

2.3. Discussion

r This section will be concerned with a proposed explanation of the

effect of the method of preparation of the substrate on the ellipsometric

thicknesses of adsorbed films. The basis for the explanation of the r differences in the isotherms is the following hypothesis. For the data ,. of figures 4, 6, and 7 the amount of adsorbed vapor, for a given pressure

and vapor, is the same for all of the isotherms, and the differences in

j the ellipsometric thicknesses are owing to differences in the smoothness

or regularity of the substrate surface on a microscopic scale. A model

will be proposed to explain the influence of surface regularity on the ! effect of the thickness of the adsorbed film on A. First, a qualitative picture of the effect of surface irregularities on ellipsometric thicknesses. Consider a chemically clean surface

which is irregular on a scale small compared with the wavelength

of light. The region of the irregularities, e.g., hills and valleys,

constitutes a surface film with an index of refraction smaller than

i

i
ADSORPTION ON SILICON 269

that of the substrate. If the valleys of the irregular region are

filled by an adsorbed film, the average index of the surface region

increases, and the surface film becomes optically more like the substrate. Depending upon the exact structure of the irregular region,

A could decrease less following adsorption on such a surface than for a film of the same thickness on a perfectly smooth surface, or even,

for some cases, A could increase following adsorption. That is, the

surface could ellipsometrically appear to be cleaner after adsorption

than before.

SUBSTRATE

Figure 8. Model for irregularities of substrate surface.

CROSS-SECTION A quantitative treatment will be given from this point of view

for the specific model of the irregular surface shown in figure 8. This model is arbitrary and is selected for ease of computation.

The irregularities are taken to be a periodic distribution of cubes

with the dimension D for the cube edge. It is assumed that there

is no permanent film on the surface. If an adsorbed film of thickness

d forms, the surface region consists of three optically homogeneous

regions (D is taken as small compared to the wavelength of light)

That is, the surface region is composed of the three films a, b, and c, as sketched. The index of refraction of each of the films is some

averaged value of the indices of the media constituting the film. The simple assumption will be made that the index of each film is the sum of the indices of its constituent regions each weighted by

its volume fraction. From this point of view, one obtains the fol- lowing relationships for the indices of refraction and the thicknesses

of the three films. n.= (n+wi)/2 (24)

da =d (25) nb = [nD2 + (Wd-4d2) %+ (D +2d) 2 }/2D2 (26)
270 ARCHER

db =D-d, (D>2d) (27) ^= [(D2 +4M-4^)^+ (Z>-2^) 2 ]/2Z)2 (28)

dc =d (29)

where nx is the index of refraction of the adsorbed film material and n is the index of refraction of the substrate (the substrate is assumed

to be nonabsorbing, a reasonable assumption in view of the small

absorption coefficient of silicon). In the absence of an adsorbed

film, the surface region is a film of thickness D and index of refraction

(ft + l)/2.

Using the above expressions for the indices of refraction, a can be

calculated for each of the three films as a function of the index of refraction and thickness of an adsorbed film and also for the single

film of the chemically clean condition. Designating the latter quantity a0, the change in A that results from the formation of an

adsorbed film is given from eq (11) by

A0 —A=aA+-o:6d6 +acdc —aj), (30)

where A0 is the value for the chemically clean surface.

It is desired to compare the ellipsometric thickness, d', of the film | with the true thickness, d, where the former quantity is: d'=^=± (31) a and a is calculated from the index of refraction of the absorbed

film material. Effects comparable to the experimental findings are obtained if D=50 A. Using this value and 7^ ==1.46, n=4.05, it is found that for d=5 A, d'=2.0 A and for d=10 A, d'=5A A. Thus,

the model predicts differences between true thicknesses and ellipsometric thicknesses of the same magnitude as the differences between

the isotherms of figures 4,6, and 7. The implication is that the very

rapidly etched (3 : 7) surfaces are smooth on a scale of tens of atomic

dimensions and that the other methods for preparing the substrate

yield surfaces that are rough on this scale. There is at present no

independent evidence concerning the relative smoothnesses of the

different surfaces, so that the proposed theory as an explanation of the data is speculative. However, it is clear that surface irregularities can lead to calculated thicknesses that are smaller than the

true values.

3. Ghemisorption

3.1. Technique

The formation of the first layer of chemisorbed oxygen was measured

on atomically clean, cleaved silicon surfaces in ultra-high vacua by
ADSORPTION ON SILICON 271

following the change of A with time when the clean surface was ex- posed to oxygen, gas. The kinetics of the formation of the second

and subsequent layers were also measured. Those results, as yet

incomplete, will be reported elsewhere.

The cleavage technique was developed by Gobeli and Allen [16], to which reference the reader is directed for details. The surfaces

obtained by this technique consist of two regions. Part of the

surface fractures irregularly; the other part, generally the smaller

of the two, is a flat, nearly ideal (111) cleavage plane. Electron

micrographs show that the better examples of the flat region consist

of plateaus several thousand Angstroms in width with steps between

plateaus ranging from 50 to 300 A. The plateaus are featureless to ±30 A, the resolution of the microscope. The ellipsometric measure- ments were made by reflecting light from the flat region by masking

the incident beam with an iris diaphragm in front of the polarizer.

This condition often entailed using surface areas as small as 10-2 cm2

. Figure 9 is a photograph of our optical cell and associated vacuum

apparatus. The stainless steel cell contains a vise which holds a

silicon plate several inches long, 1 cm wide, and 0.2 cm thick with an

L-shaped cross section as described in reference [16]. The rod ex- tending from the top of the cell manipulates a diamond scribing

point and the cleaving hammer by means of a sylphon bellows. Pressure is measured with the Baird-Alpert ionization gage. The

pump is an 8-liter-per-minute Vac-Ion pump. Oxygen is admitted

to the system by means of a silver leak. This is contained in the

long tube above the pump and consists of a silver cylinder wrapped

Figure 9. Chemisorption vacuum cell and associated apparatus.
272 ARCHER

with a heating element. The rate at which oxygen enters the system

depends on the temperature of the leak. The oxygen transmitted

by the leak is spectroscopically pure. Our procedure will be illustrated by describing a typical experiment.

The vacuum tube is loaded with the silicon bar and is sealed to a conventional high-vacuum station which is pumped and baked at 300 °C for 12 hr. The tube is then sealed off from the baking station

and after several hours the Vac-Ion pump exhausts the vessel to a pressure of about 5X 10~ 10 mm of Hg. The vacuum tube is mounted

so that the axis of the cell coincides with the axis of the spectrometer

and the silicon bar is cleaved. Immediately the spectrometer is alined relative to the cleaved face and measurements of A are begun.

The time after cleavage before the first measurement is made is generally not more than 10 to 15 min. A in the ultra-high vacuum

is measured continuously for 10 to 15 min, during which time it is always constant to the precision of the measurements. Then, with

the pump still on, the silver leak is heated by controlling the current

in its heating element. Any desired pressure can be obtained within

a few minutes and maintained indefinitely with occasional adjustments of the heating current. A is measured continuously during

this process and as frequently after the establishment of the desired

pressure as the rate of adsorption warrants. At the termination of an experiment the leak current is turned off and within from 1 to 12

hr, depending upon the pressure during the experiment, the vessel

again pumps down into the 10~ 10 range. This process has been

repeated as many as five times before a new silicon cleavage bar was

inserted and the vessel was again baked out. 3.2 Results

A typical result is shown in figure 10. The points plot A, relative

to the average value before introducing oxygen, as a function of time.

The broken curve is the pressure of oxygen versus time. For times

greater than 8 min, the pressure was constant at 3.1X10

mm of Hg. These and all other measurements were made at room

temperature (about 24 °C). Typical of chemisorption kinetics, the

rate is relatively rapid at first and becomes very slow near the completion of a monolayer.

Chemisorption kinetics are conveniently described m terms ot the

dependence of the sticking coefficient, s, on the coverage, 0. The

sticking coefficient is the fraction of the molecules colliding with the

surface which is adsorbed. The number of molecules impinging on tne

surface per unit time and area is vp; p is the pressure and v is a kinetic

factor given by nc/4, where n is the number of molecules per unit

volume per unit pressure and c is the average velocity of the molecules.

For oxygen at 300 °K, *>=3.48X1020 cm"2 sec" 1 mm"1

. Defining 7V0 as the total number of oxygen molecules required to form a complete

monolayer, it follows that

dt N0
ADSORPTION ON SILICON 273

° i

i i i i i i i i i i i i i i i i i i i i i i i i 0 4 8 12 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42

TIME MINUTES Figure 10. Chemisorption of oxygen on cleaved silicon at room temperature.

6A and oxygen pressure as a function of time. Using the two-dimensional formulation resulting in eq (16), and defining 8A0 as the change in A corresponding to complete monolayer

coverage, one obtains

Combining eqs (32) and (33), the sticking coefficient is given by

S=\M^)pW (34)

A point-by-point graphical differentiation of the kinetic curves, such as

figure 10, yields the plots of i versus <5A given in figure 11 for the

four indicated oxygen pressures. The plotted quantities are proportional, respectively, to sticking coefficient and coverage. These data

appear to fall into two sets of similar curves, and although the characteristics of the curves are different, the average behavior indicates an

approximately constant sticking coefficient up to 8A equal to about

0.45° followed by a rapid decrease. In order to arrive at values for

s and 0, values for 5A0 must be deduced from these curves. For the

lowest pressure curve, 7.2 X10~8 mm, A stopped changing after about

100 min and remained constant at 0.63° for another 300 min. The
274 ARCHER

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 8A° Figure 11. Chemisorption kinetics at four pressures of oxygen.

_L is proportional to the sticking coefficient and 5A is proportional to coverage. p dt curve for a pressure of 1.35 X10~7 mm was terminated at 5A=0.60°

while A was still changing slowly with time. From the similarity of these two curves it seems reasonable to take the value <5A0 =0.63° for both. In the region of the rapid decrease of s with 8A the curves for the two higher pressures are displaced about +0.08° on the 8 A scale relative to the lower pressure curves. We take therefore the value

5Ao=0.71° for the two higher pressures. We have no explanation

for this 12 percent difference in the values for 5A0 for the two sets of curves. Perhaps differences in the nature of the surface from cleavage

to cleavage account for the effect. From the measurements of Law [17], Law and Eisinger [18], and

Green and Maxwell [19] for the total amount of oxygen taken up m

forming the first layer of chemisorbed oxygen, we conclude that

approximately one oxygen atom per surface silicon atom is the

monolayer coverage. The number of surface silicon atoms on the

(111) surface is 7.8X1014 cm"2

; therefore N0 of eq (34) has the value

3.9 X1014 cm-2 oxygen molecules.
ADSORPTION ON SILICON 275

1- 3.1 x icr 7 mm of h< 2- 1.6 X I0~ 6 3- 1.35 X I0 -7 4- 7.2 X I0"8 EIS1NGER AND LAW

Figure 12. Sticking coefficient as a function of coverage at four pressures of oxygen.

It is assumed that at monolayer coverage there is one oxygen atom per surface silicon atom, and <5Ao is taken

to be 0.63° for the two lower pressure curves and 0.71° for the two higher pressure curves. The broken curve

is the average of the results of Eisinger and Law obtained by flash filament manometry. With these values for 5A0 and N0 we arrive at the plots of s versus

6 of figure 12. The average characteristics of these curves indicate

that the sticking coefficient is nearly constant at about 6X10-3 up

to values for coverage between 0.6 and 0.7, after which values of s decrease at a rate that increases with B. From the fact that the data

represent a range of pressures of a factor of 20, it is concluded that

there is no dependence of s on pressure.

3.3. Discussion

Included in figure 12 as the broken curve is the result of Eisinger

and Law [18] obtained by flash-filament manometry. Their values

for s vary by about a factor of 2 from run to run. We plot the

average of their curves. Our values for s agree with theirs within

a factor of 2 for all values of coverage, but their finding that s decreases exponentially with coverage throughout most of the range is in disagreement with the characteristics of our curves. This differ-
276 ARCHER

erence is probably attributable to differences in surface structures. I Their flashed filaments are considerably roughened by heat treatment ji and have roughness factors of the order of 3. Hagstrum [20] from measurements of the Auger effect, and Allen U [21] from field emission microscopy, deduce values for s at low cover- \ ages of 1X10-2 and 1.7 X10-2

, respectively, in order of magnitude i agreement with our results. Schlier and Farnsworth [22], using slow | electron diffraction, obtained values for s that range from a factor of 5 to 20 larger than all other results, depending upon crystal orienta- I tion and the method of interpreting the measurements.

The value for the thickness of the monolayer, 2.5 to 2.8 A, obtained

from 5A0 and a of eq (9), is reasonable for a layer of oxygen atoms b and suggests that there is no anomaly in thickness of the kind found

in physical adsorption and attributed to surface irregularity.

It is possible that our results were affected by the use of the ion

gage to measure pressures. CO is known to be formed [23] when a tungsten filament is heated in oxygen, and the possibility exists that

CO could be produced by the tungsten filament of the ion gage. To eliminate, or at least minimize, CO formation, the gage was

operated in oxygen at 10"4 mm for many days for the purpose oi removing carbon contamination from the tungsten filament. Because

of the CO problem and because other species might be produced by

the ion gage which could contaminate the oxygen ambient at low

pressures, we are repeating the measurements without using the ion

gage. Initial results indicate that the magnitude of s and the variation of s with 6 are essentially the same with or without the ion gage. There is an indication, however, that the rate of growth of subsequent layers of oxide is strongly affected by the use of the ion gage. When completed, these results will be published elsewhere.

4. Summary

Measurements of physical adsorption of vapors on silicon are interpreted using the standard theory of ellipsometry. The ellipsometric thicknesses at monolayer coverage obtained from a BET

analysis of the isotherms agree with the dimensions of the adsorbed

molecules for water, carbon tetrachloride, and acetone if the substrate

surface is prepared by etching in the HN03 -HF solution which gives

the maximum etching rate. For all other surfaces, however prepared,

monolayer thicknesses are smaller than molecular diameters by

factors as small as }{ to ){. This effect is attributed to the influence

of irregularities in the structure of the substrate surface on the scale of tens of atomic dimensions. Such irregularities can cause the

change in A resulting from the formation of a surface film to be smaller

than that which a film of the same thickness would cause on a perfectly

smooth, flat surface. On the basis of the theory proposed, a calculation is made for a specific model of the irregular surface region.

Calculated effects agree with experimental ones if the dimension of the irregularities is taken to be 50 A.
ADSORPTION ON SILICON 277

Water adsorption depends on the chemical structure of the silicon surface. Surfaces which are macroscopically hydrophobic have

very thin absorbed films even at high relative humidities and the

isotherms are type III, according to the BET classification. Macroscopically hydrophilic surfaces have type II isotherms and cor- respondingly thicker adsorbed films. The kinetics of the formation of the first layer of chemisorbed

oxygen were measured on cleaved silicon surfaces produced in ultra- high vacua. On the basis of the standard, three-dimensional, theory

of ellipsometry the thickness of the monolayer is 2.5 to 2.8 A, in reasonable agreement with the assumption that the layer consists

of one oxygen atom per surface silicon atom. Interpretation of the

kinetics using the two-dimensional theory, which allows the assumption that changes in A are proportional to coverage, gives curves of sticking coefficient versus coverage which have, on the average, constant values for the sticking coefficient of about 6X10 3 up to coverages of about 0.6 to 0.7, after which the sticking coefficient

decreases rapidly. Sticking coefficient does not depend on pressure

in the range from 7X10-8 to 1.6 XIO"6 mm of Hg. The magnitude

of the sticking coefficient agrees, within a factor of two, with previous

measurements.

The author acknowledges the active cooperation of G. W. Gobeh

in the study ot the chemisorption of oxygen on cleaved silicon. A

joint paper covering in more detail this aspect of the present paper

will be published elsewhere.

5. Glossary of Symbols, Equations (2) through (6)

a—(l-/c2 )/n2 (l + /c2

)

2

. a'—2/c/n2 (l + «

2

) 2 . d—thickness of film, m—index of refraction of film. n —index of refraction of substrate = n—ik = n(l — iic). A—A for a film-free surface. ^

\J/ for a film-free surface. X—vacuum wavelength of light.

4>—angle of incidence. 6. References

[1] R. J. Archer, J. Opt. Soc. Am. 53, 970 (1962).

[2] C. E. Leberknight and B. Lustman, J. Opt. Soc. Am. 29, 59 (1939),

[3] M. Born, Optik, p. 36, Verlag, Julius Springer, Berlin (1933).

[4] P. Drude, Wied. Ann. 36, 532, 865 (1889) ; ibid., 39, 481 (1890).

[5] L. Tronstad, Trans. Faraday Soc. 31, 1151 (1935).

[6] R. J. Archer, J. Electrochem. Soc. 104, 619 (1957).

[7] C. S. Strachan, Proc. Cambridge Phil. Soc. 29, 116 (1933).

[8] D. V. Sivukhin, Soviet Phys. JETP 3, 269 (1956).

[9] R. J. Archer, Phys. Rev. 110, 354 (1958).

[10] H. Robbins and B. Schwartz, J. Electrochem. Soc. 106, 505 (1959).

[11] R. J. Archer, J. Phys. Chem. Solids 14, 104 (1960).
278 ARCHER

[12] R. K. Her, The Colloid Chemistrv of Silica and Silicates, Cornell Univ. Press, Ithica N.Y. (1955).

[13] S. Brunauer, The Adsorption of Gases and Vapors, Princeton Univ. Press, Princeton, N.J. (1945).

[14] H. K. Livingston, Adsorption and Free Surface Energy of Solids, Ph. D.

dissertation, Univ. of Chicago (1941).

[15] W. G. Palmer, Proc. Rov. Soc. 160, 254 (1937).

[16] G. W. Gobeli and F. G. Allen, J. Phys. Chem. Solids 14, 23 (1960).

[17] J. T. Law, J. Phys. Chem. Solids 4, 91 (1958).

[18] J. Eisinger and J. T. Law, J. Chem. Phys. 30, 410 (1959).

[19] M. Green and K. H. Maxwell, J. Phys. Chem. Solids 13, 145 (1960).

[20] H. D. Hagstrum, J. Appl. Phys. 32, 1020(1961).

[21] F. G. Allen, J. Phys. Chem. Solids 8, 119 (1959).

[22] R. E. Schlier and H. F. Farnsworth, J. Chem. Phvs. 30, 917 (1959).

[23] R. E. Schlier, J. Appl. Phys. 29, 1162 (1958). Discussion

A. C. Hall (Socony Mobil Oil Company):

I should like to know at what relative pressure of water vapor the

water monolayer was complete, and also whether j^our measurements

go to high enough relative pressures to confirm Deryagin's observation

that isotherms of polar substances intercept the ordinate P/P0 =l,

while isotherms of nonpolar substances become asymptotic to it? R. J. Archer:

In most cases the monolayer is complete at relative saturations of about 20 percent. There are no definite indications in the data of the

behavior of the isotherms at 100 percent saturation. A. C. Hall:

Is the phase difference, A, a function of surface coverage below a monolayer? It seems to me that the physical models used in deriving

the relationship of A and film thickness may be invalid below a monolayer. In Strachan's theory it is possible that whereas a complete

monolayer would give rise to coherent scattering, an incomplete one would scatter incoherently.

R. J. Archer:

For submonomolecular coverages, changes in coverage should

produce proportional changes in A according to my interpretation of Strachan's theory. Only for separations not small compared to the

wavelength of light would incoherent scattering be expected. A. N. Saxena (Fairchild Semiconductor):

Would you care to comment how you arrived at the value of &=0.007?

R. J. Archer:

It is in the literature [1]. On the basis of accumulated experience

in measuring silicon surfaces we are able to deduce fairly well the
ADSORPTION ON SILICON 279

thickness of the permanent film on the surface. One can prepare

surfaces so that this film is only about 10 A thick. This thickness is taken into account in calculating optical constants. There is an

independent measurement of this 0.007 by Dash and Newman by

transmission through silicon plates. Their curve must be extrapolated

a little but the agreement is clear. A. N. Saxena:

I thought you used 0.028 in the paper you wrote? Or perhaps it is k=n £ =4.050X0.007^0.028.

R. J. Archer: p The complex index of refraction of silicon at 5461 A is 4.05 —0.028i—

4.05(l-0.007i).

M. Kopp (University of Pittsburgh):

It is remarkable that, unlike transparent media and though the

absorption coefficient is so small, a cleaned silicon surface does ellip- ticize light at an incidence so far from Brewster's. The effect must

however be very weak for the ideal interface. Haven't you observed

the better marked ellipticization near or at principal incidence (maximum) on reflection by a bare new surface protected from any con- tamination (e.g., cleaved in vacuo), and subsequently followed the

building up of the transitional film?

!R J A.RCHER'

The angle of incidence, 70°, is not far removed from the principal

angle, 76°. Most of the ellipticity is caused by surface films. Identical results would be obtained on transparent substrates. It is a fact

that a is a maximum near the principal angle. Our measurements

were limited to 70°.

Application of Ellipsometry to the Study of

Adsorption From Solution 1

Robert R. Stromberg, Elio Passaglia, and Daniel J. Tutas

National Bureau of Standards, Washington, D.C., 20234

Ellipsometry has proved to be a valuable tool in the study of the adsorption of polymers on solid surfaces from solution. Polystyrene

with molecular weights of 76,000 and 537,000 was adsorbed from cyclohexane on chrome surfaces, and the thickness and refractive index of the adsorbed film were measured in situ. The swollen film thicknesses measured were 200 A for the lower molecular weight polymer at 24 °C

for most of the concentration range studied and 500 A for the higher molecular weight polymer at 34 °C for a solution concentration of 0.23 mg/ml. The concentration of polymer in the film and the amount

adsorbed are also determined from ellipsometry measurements. The

application of ellipsometry to this type of problem, including the use

of multiple reflection techniques, as well as the results obtained are

discussed.

1. Introduction

Ellipsometry has proved to be a valuable tool in the study of adsorption from solution on solid surfaces. Using this technique,

measurements can be made in situ and both the thickness and refractive index of the adsorbed film, as well as the amount of material

adsorbed, can be simultaneously determined.

The determination of these parameters for a film under liquid is of particular importance in the case of polymer adsorption. _ One of the

principal objectives in polymer adsorption studies is information

concerning the conformation of the adsorbed molecule on the surface.

Much is known about the shape of polymer molecules in a solution,

coil dimensions, interaction between polymer and solvent, etc.; however, only little is known about the molecular dimensions of a molecule adsorbed on a surface. Experimental results have been interpreted as evidence for both very thin and very thick adsorbed films. (See ref. 1 for discussion.) Different theoretical treatments have also predicted both thin and thick layers. (See ref. 1.) It is, therefore,

very desirable to study the thickness of these films, exposed to the

solution and still in a swollen state. The simultaneous measurement

of the refractive index also permits determination of the concentration

of polymer in this film. This paper reports the type of calculations employed, the techniques used, and some of the results obtained from a study of the

adsorption of polystyrene from dilute solutions on metal surfaces.

1 This work was supported, in part, by the Army Research Office (Durham)

281
282 STROMBERG, PASSAGLIA, AND TUTAS

2. Calculations

The programming of the original exact Drude equations [2],

2 for high-speed computers has permitted their use without any limiting

assumptions [3, 4]. For the purpose of analysis, the electric vector

of the light is considered to be resolved into two components, one

in the plane of incidence and one in the plane of the surface. The

ellipsometer measures the ratio of the reflection coefficients of these

two components. For single reflection this can be represented by

A=£=tan*e" (1)

in which p is the ratio of the reflection coefficients, r

p is the reflection

coefficient for the component in the plane of incidence, r

s for the

component in the plane of the surface, tan \p is the relative amplitude

reduction, and A is the relative phase change.

For a bare surface, r

v and r

s are given by the Fresnel equations,

v _n3 cos 0i —7bi cos 03 n3 cos 0i+% cos 03

(2) r

s _nl CQS &~^3 COS 03

rii cos 0i +n3 cos 03 where nx is the refractive index of the surrounding medium, n3 (which

may be complex) is the refractive index of the surface, and 0 is the

angle of incidence. For a film-covered surface with a film of refractive

index n2 and thickness d2 on a substrate of refractive index n3 , the

whole in a medium of refractive index nu the total reflection coeffi- cients are given by

p== yf2 +yf3 exp D

l+r?2rf3 expZ>

(3) rs_ rI2 +r|3 expP

l+^VIs expZ?

where r12 and r23 are the reflection coefficients at the film-medium

and film-substrate interfaces respectively, and D is given by

T)——^kWj2 cos 4>2d2 /\ (4) in which 02 is the angle of incidence on the substrate, and X is the

wavelength of the light in vacuo. The optical constants for a bare metal surface can be determined

by measurements of A and \p on that surface. If a film has a real refractive index, a single measurement of A and \p for the film-covered

2 Figures in brackets indicate the literature references on p. 295.
POLYMER ADSORPTION 283

surface will permit determination of the refractive index and thickness

of that film if the optical constants of the substrate are known [3]. The ellipsometer measurements become more and more insensitive

to the thickness and refractive index of a film as the refractive index

of the film approaches that of the substrate or surrounding medium.

For the cases to be discussed here, the refractive index of the metallic

substrate is very different from that of the films and immersion

medium. However, the adsorbed layer on the metal surface consists

of a highly swollen polymer, resulting in a film with a refractive

index very close to that of the solvent. It is, therefore, essential

to increase the sensitivity of the measurements. This can be accomplished by means of multiple reflection. The relationships between the measured values, p and a, obtained

from the polarizer and analyzer, respectively, to A and \p have been

described [3, 5]. Extinction of the reflected light from a surface can be accomplished by a number of arrangements of the polarizer,

analyzer, and compensator scale settings. The arrangement of the

possible values of p and a into four zones, and the calculations required

to determine the zone of a specific combination of p, a, and compensator

values have also been .described for single reflection [3]. The basic

criterion applied is that the values of A and \p are within the limits

0<A<2tt (5a)

0<i£<tt/2. (5b)

However, for multiple reflections of the light, the value of the total relative phase change, A«, may be greater than 2x and this must be

taken into account ; thus eq (5a) will no longer apply. For an unknown

surface it may therefore be desirable first to determine A and ip for a

single reflection. The zone corresponding to a given reading of p

and a may then be determined quite easily and the values of A and \p determined accordingly. The limits for ^ are not affected by multiple

reflections. The ratio of the reflection coefficients for n reflections is

Pn =P

7t (6) and, therefore, from eq (1),

p„ = (tan i/0VnA =tan \M'A

» (?) where \p and A are the values to be expected for a single reflection,

and \pn and An are the overall values determined for n reflections. We can now examine the benefits and limitations that result from

multiple reflections. The measurement error may change slightly

as a result of multiple reflections. In our case we have determined

the standard error in the measurement of A to be 0.042 for a single

reflection and 0.052 for three reflections from a chromium surface.

728-101 0—64 19
284 STROMBERG, PASSAGLIA, AND TUTAS

I The standard error in the measurement of \f/ was 0.018 for single

reflection and 0.009 for triple reflection. The value of A required for eq (1) is related to An by

A=^- (8) n

If the error in the measurement of A is 8A, then the error in the value

of A required for eq (1) for a measured An is 5A=^- (9)

Therefore, increasing the number of reflections decreases the error

in A without limit, assuming that no additional errors, e.g., the two

surfaces not being completely parallel, are introduced. For meas- urement of A it is desirable to increase the number of reflections

until the measurement error, 8An , is increased as a result of additional

reflections. For reflection from a metal surface using a conventional

light source, this would eventually occur as a result of decreased

light intensity. For total internal reflection when no loss in intensity

results from a reflection, the number of such reflections can be made

very large [6]. The relation between the measured value of \f/ n and the value of

\f/ required for eq (1) is tan ^=(tan +n) 1/n

. (10)

If the error in \p is 8\f/, then the error in the value that is required for use in eq (1) for a measured \J/n is

1 [(tan^) 2-+l] (tan jj^ nn

The difference between the values of dxf/ and 8\f/n which results from

multiple reflections is not, as in the case of A, related only to the

number of reflections. It also varies with the value of tan which

is dependent on the surface. Further, 8\p is not necessarily smaller

than 8\pn . If eq (11) is represented as b+=mn (12)

where k is dependent upon the value of \p and n, it is clear that 8\p will be smaller or greater than 8xl/n as k is less or greater than unity.

For a gold or a silver surface the value of \f/ is about 40°. The

value of k decreases from approximately 0.6 for two reflections to approximately 0.2 for seven reflections. It then increases to a value

of approximately 0.3 for 10 reflections and is approximately unity

at about 20 reflections. It is possible, therefore, for gold and silver
POLYMER ADSORPTION 285

surfaces to increase the precision in yp with multiple reflections, and

have as many as 20 reflections without losing any precision. As the

value of n increases beyond 20, the value of k increases to values

greater than 1, decreasing the precision.

If the value of tan \f/ is 1/2, which is the case when \p is approximately

26.5°, then eq (11) becomes

Hs^*-- (13)

Here an increase in the number of reflections is not nearly as favorable

for reducing the value of 8\p as it was for values of <p near 40°. The

value of k is approximately unity for three reflections and becomes

larger as n increases. Equation (13) approximates the relationship

between 8\p and b\p n for steel and chrome surfaces which have a value

of \p near 30°. For such surfaces there is a slight advantage in three

reflections, as k is approximately 0.8. Multiple reflections, therefore, are of considerable aid in improving

the precision. The precision of A is improved by a factor directly

proportional to the number of reflections. The precision of can be

either improved or at least not changed, depending on the surface

and the number of reflections. All of the measurements on chrome

116

1

114 %V n 2 =l.44

112 \\\ \ "v<n2=l.46

110 200K\\\ \ \

108 - „ n 2 =l.50- )ibooS

106 400&^^ \ \ / ^A-—

-^800 }i 104 1 |600S 1 29 30 31 32 33 34

* , DEGREES Figure 1. A plot of A versus 4> of several possible refractive indexes for an adsorbed

polystyrene film on a chrome surface immersed in cyclohexane. The solid lines are constant refractive index and the dashed lines are constant film thickness. The

refractive indexes and the film thicknesses in angstrom units are given on the figure. «i = 1.4268; 7i 8 =3.3006 -4.2456i; X=5461 A; 0=70°.
286 STROMBERG, PASSAGLIA, AND TUTAS

surfaces reported here were carried out with three reflections. The

precision of A, therefore, was improved by a factor of 3, although

that of \p was increased by only a small amount.

The need to decrease the values of 8A and 8\f/ for the investigation

of adsorbed polymer films, measured in situ, can be demonstrated

by figure 1. Here A is plotted against \p using the optical constants

of a typical chrome surface for the substrate and the refractive index

of cyclohexane for the immersion medium. The film is an unknown

mixture of polystyrene and cyclohexane, requiring consideration of

all refractive indexes, ranging from that of the solvent, 1.4268, to that of polystyrene, 1.59. Curves for four possible refractive indexes

included in this range are given.

On this plot, curves of constant refractive index and curves of constant thickness form a network, or mesh. The thickness and

refractive index represented by the mesh point with a value of A and

\(/ equal to the experimental values are the desired thickness

and refractive index. However, because of the experimental errors, any

mesh point which falls in the area of the rectangle defined by 8A

and 8\p also represents allowed values of thickness and refractive index.

It can be seen from figure 1 that the number of mesh points per

unit area is much greater for thin films and films whose refractive

index is close to that of the medium. Hence, the thinner the film

and the closer its refractive index is to the index of solution, the

greater the range of values of film refractive index and thickness per- missible within the experimental error. In this work we are unfortunately working just in the region of thin films with refractive index near that of the solution. The use of triple reflection and the conse- quent lessening of 8 A will clearly increase the experimental precision.

Although the graphical method demonstrated here will permit the

determination of the best fit values of index and thickness as well

as the range of values falling within experimental error, it is more

convenient and rapid to determine these quantities by computation

[1, 3]. A number of refractive indexes, n2 , are assumed for the adsorbed film and the corresponding thicknesses, d2 , calculated for the

experimental A and \J/. Because of the method of calculation [3], calculated values of d2 will, in general, consist of a real and an imaginary part. The true thickness of the films must of course be real. The size of the imaginary component is taken as an indication of the

error in the assumed value of n2

. This error can be related more directly to the experimentally determinable values 8A and 8\p. The imaginary portion of d2 is dis- carded and A and \J/ calculated by means of eq (1) combined with eq

(3) , using the real portion only. These calculated values of A and \f/ are compared with the experimental values. There will generally be

a difference, 8'A and 8

f

\f/, between these calculated values and the

observed values. The best fit value of the refractive index and thick- ness is then that value of n2 and d2 for which the differences between

the calculated and measured values of A and \f/ are zero. However,
POLYMER ADSORPTION 287

a range of values for n2 and d2 , determined by the experimental error, are also permissible. These are determined for the system studied

here by finding the range of n2 and d2 for which d'A= ±0.052/3 and

b'6= ±0.009.

3. Experimental Details

3.1. Experimental Procedure

A diagram of the cells used to study adsorption under liquid is shown in figure 2. The cells were fabricated from optical glass.

The optically flat windows were attached either by fusion or by means

of epoxy resin. The epoxy seals generally were found to be more

successful, as there was less distortion of the window resulting from

the attachment. The principal difficulty encountered was to obtain

windows with no birefringence. The windows were arranged so that

light entered and left at normal incidence to the window.

For studies at 24 °C, the measurements were carried out in a con- stant-temperature room. Two cells were usually placed side by side on the ellipsometer stage, thus permitting two sets of measurements

to be made at the same time. For measurements above room temperature, the cells were placed in a hollow-walled chamber through

which water at constant temperature was circulated. This chamber,

which was sufficiently large to accommodate two cells, was placed on

the ellipsometer stage.

The arrangement of the metal surfaces for three reflections is also shown in figure 2. The upper slide was placed on two gage blocks

which provided a level, constant height from the lower reflecting

surface. By moving the ellipsometer stage it was possible to make

measurements at two locations on each set of slides. The work reported here was carried out on the chrome surfaces of 1X2 cm slides sheared from commercial chrome ferrotype plate.

They were cleaned by immersion in warm sulfuric acid-chromic

acid cleaning solution, followed by thorough washing in hot distilled water. The slides had a hydrophilic surface after this procedure, but

upon standing in the laboratory air for a short period of time, gradFigure 2. Cell and metal surfaces arranged in cell for triple reflection.
288 STROMBERG, PASSAGLIA, AND TUTAS

ually became hydrophobic. It was possible to keep the surfaces

hydrophilic for longer periods of time by storing in a dessicator under

clean nitrogen. It was also possible to restore the hydrophilic character of the surface by passing the slide through an oxygen-gas flame.

The slides were flamed and placed, while still warm, under solvent

in the cells. The optical constants of the metal were obtained under the solvent.

These optical constants are, of course, an average of the constants

at the three reflection points. The solvent was then removed from

the cell with a hypodermic syringe and polymer solution placed in the cell. Measurements at the same location were made of the film- covered surface as a function of time.

Polystyrene was chosen for study because of its availability in relatively narrow molecular weight fractions and because of the

wealth of knowledge about this polymer. Two samples of different

molecular weights, kindly provided by Dr. H. W. McCormick of the

Dow Chemical Company, were used in this study. Both had been

prepared by anionic polymerization. One sample, of relatively low

molecular weight (Dow Sample No. S-102), had a number average

molecular weight, as determined by the supplier, of 78,500 with a Mw/Mn ratio of 1.05. For some of the experiments, this sample was

used as received (unfractionated) . For most of the experiments, a fraction with a viscosity average molecular weight of 76,000 was

used. This fraction was prepared by conventional precipitation methods. The other sample was of much higher molecular weight (Dow

Sample No. S-1195, Mw/Mn=1.09). From this a fraction with a

viscosity average molecular weight of 537,000 was prepared and used

for the experiments reported here.

The measured thickness of the layer of the 76,000 molecular weight

polystyrene sample, adsorbed on a chrome surface from a cyclohexane

solution with a concentration of 0.18 mg/ml, is given in figure 3 as a

Figure 3. Thickness of adsorbed film of polystyrene, M.W. = 76,000, versus time,

for solution concentration of 0.18 mg/ml at 24 °C.

3.2. Materials

4. Results

0 i 1 1

1 1 0 50 100 150 200 250 TIME, min

Each set of points was obtained on a different set of slides. O. Fractionated polymer

, Unfractionated polymer.
POLYMER ADSORPTION 289

function of time. The film was measured in contact with the solution and consisted of a mixture of polymer and solvent. The points

represent the best-fit values for each individual measurement. The

vertical lines represent the range of thicknesses that fit within experimental error, determined as described earlier. In some cases, the

lower thickness value was determined by physical considerations, i.e., the upper value of the refractive index of the film was limited to the

index of the bulk polymer.

Three runs are shown in figure 3, one using fractionated polymer and

the other two using unfractionated. There are no differences among

the three runs. The average film thickness, shown by the curve drawn

through the average of the best-fit values, is approximately 80 A. There was no measurable change in thickness during the time interval

studied. The range of film thicknesses that fit within experimental

error for each point is relatively large. This is a result of two factors. The films are highly swollen, with a refractive index of about 1.465,

which is fairly close to that of the solvent, 1.4268, and they are also relatively thin. As previously discussed, and as can be seen from

figure 1, a relatively large range of refractive indexes and thicknesses

of the adsorbed film are allowable within the experimental error of the measurements.

The curves shown in figure 4 represent the measured thickness

values obtained for the 76,000 molecular weight polymer at a concentration of 5.0 mg/ml. Curves A and B were obtained using the

fractionated sample and curves C, D, and E using the unfractionated

material.

100 150 TIME, min 200

Figure 4. Thickness of adsorbed film of polystyrene, M.W. = 76,000, versus time, for solution concentration of 5.0 mg/ml at 24 °C.

Curves A and B were obtained using fractionated polymer, curves C, D, and E, using unfractionated. Curves A-l and A-2 refer to measurements made on different portions of the same set of slides, curves B-l

and B-2 to measurements on different portions of another set of slides. , Curve A-l , Curve C

A, Curve A-2 Curve D

Curve B-l fi, Curve E

O. Curve B-2
290 STROMBERG, PASSAGLIA, AND TUTAS

Curves A-l and A-2 represent two different locations on one set of

slides; curves B-l and B-2, two different locations on a second set of

slides. The differences among these four sets of data are most probably

caused by differences in amount adsorbed at the different locations on

the surfaces. The differences in measured thicknesses can be larger

between two locations on the same set of slides than between different

sets of slides. The average thickness obtained from the four curves

for the fractionated polymer is approximately 200 A. Although the range of thicknesses within experimental error for each point on curves A and B are somewhat smaller than those in figure 3, they are still relatively large. The films represented in figure

4 are thicker; however their refractive index of 1.45 is closer to that of the solvent than was the index of the films obtained from the 0.18

mg/ml solution. As can be seen from figure 1, the benefit derived

from the increased film thickness is almost lost by the decrease in film refractive index. The experimental values of A and $ are again

located in a region where a large range of film thicknesses fall within

the values of 8A and 5\p. The range of possible thicknesses for the points on curves C, D,

and E is much smaller than those for curves A and B. The film thick- nesses represented by curve C are approximately the same as those

given by curves A and B, although curves A and B are flat and curve C continues to rise for the entire time period. The refractive index of the film shown by curve C was approximately 1.468, which was near

that for the films represented in figure 3. The effect of the increased

refractive index in reducing the uncertainty in the calculation of the

thickness can be seen, therefore, by comparing the spread of the data on curve C with the spread in figure 3. The refractive index of the films represented by curves D and E

was approximately 1.442. Although this was closer to the index of the

solvent than was the index of the films represented by curves A and B,

the range of possible thicknesses for each experimental point was

reduced. It can be seen from figure 1 that as the film thickness in- creases, the curves of constant refractive index diverge, resulting in a

smaller range of permissible thickness values within the experimental

error. The differences between the curves for the fractionated sample and

those for the unfractionated probably are caused by precipitation of higher molecular weight components in the unfractionated sample

even though such components made up only a very small portion of this sample. The fact that precipitation occurred was immediately ap- parent from curves D and E. Although the thickness of the film

represented by curve C continued to rise, while those of curves A and

B remained flat for most of the time interval studied, it is not completely evident from the thickness curves alone that precipitation

occurred, but does become evident when the refractive index of the

film, and hence the concentration of polymer in it, is considered.

The concentration of polymer in the film can be readily determined if the refractive index increment, dn/dc, is known. The rela-
POLYMER ADSORPTION 291

tionship between the concentration of polymer and refractive index- was determined experimentally to be 0.168 ml/g and was found to be

linear for the concentration range studied. Assuming additivity of

specific volumes, a linear relation for this concentration range was

also obtained using the molar refraction relationships of Lorentz

and Lorenz. Therefore, the concentration of polymer in the film can

be determined directly from the refractive index. The product of this

concentration and the film thickness gives the amount of polymer

adsorbed or deposited per unit area. The computer program designed

to determine the refractive index and thickness of the adsorbed films

from the experimental ellipsometer quantities can also compute the

amount deposited, using the Lorentz-Lorenz relation [4]. The curves in figure 5 represent the same films whose thicknesses

are given by curves A, B, and C in figure 4. In figure 5 the adsorbance

(amount adsorbed per unit area) in mg/cm2 is plotted versus time of

immersion. The symbols for the individual points and the lettering of the curves are identical to those used in figure 4. The average adsorbance of the fractionated sample was approximately 2X10 4 mg/cm2

. Although the thickness values of curve C (fig. 4) were near

those of curves A and B, the larger refractive index, and consequently

the more dense nature of the film resulted in an "adsorbance" curve markedly different from those of the fractionated sample. The range

of uncertainty for each point on all the curves in figure 5 is relatively

small and within the dimensions of the symbols used.

The thickness of the adsorbed layer of the fractionated 76,000

molecular weight polystyrene sample is shown in figure 6 for the entire

concentration range studied, 0.18 to 9.7 mg/ml. Each point represents the average of several runs at that concentration. Differences

Figure 5. Adsorbance of polystyrene, M.W. = 76,000, for solution concentration

of 5.0 mg/ml at 24 °C.

A, Curve A-l O, Curve B-2 A, Curve A-2 , Curve C

Curve B-l
292 STROMBERG, PASSAGLIA, AND TUTAS

<300

CO^200 o

^100 2

U_ o 2 4 6 8 10 SOLUTION CONCENTRATION, mg/ml

Figure 6. Average swollen-film-thickness of polystyrene, MM.= 76,000, as a

function of solution concentration at 24 °C.

o

2 3 4 5 6 7 SOLUTION CONCENTRATION , mg/ml

Figure 7. Concentration of polystyrene, M.W. = 76,000, in adsorbed layer as a

function of solution concentration at 24 0 C.

among the individual runs, such as is illustrated in figure 4 by curves A-l, A-2, B-l, and B-2, were attributed to differences in the surface

from one location or set of slides to another. The average of the

individual best-fit point averages results in a thickness for the adsorbed

polymer-solvent layer of approximately 200 A for most of the

o concentration range studied, with a thickness of approximately 80 A for the

lowest concentration.

The concentration of polymer in the film is directly determined from

a knowledge of dn/dc, as described earlier. This concentration is given in figure 7 for the solution concentration range studied. For

most of the solution concentration range, the concentration of the

polymer-solvent mixture is approximately 12 g of polymer per 100

ml of the mixture. This concentration of polymer in the film was

higher at both low and high solution concentrations.

The measured thickness of the adsorbed layer of the 537,000 molecular weight sample of polystyrene on the chrome surfaces is shown in

figure 8. These measurements were carried out at 34 °C for a solution

concentration 0.23 mg/ml. A maximum thickness for the adsorbed

film was reached much more slowly for the higher molecular weight

material than for the fractionated sample of the lower. The decrease

in the range of possible thicknesses for each experimental point in figure 8 at increasing times results from the increase in film thickness.

The refractive index of these films was approximately 1.44 and that

of the solvent 1.4200 at this temperature. The film refractive index

did not change significantly with time. This difference in refractive
POLYMER ADSORPTION 293

2 00 3 00 TIME , min OVER NIGHT

Figure 8. Thickness of adsorbed film of polystyrene, M.W. =537,000, versus time, for solution concentration of 0.23 mg/ml, at 34 °C.

Figure 9. Adsorbance of polystyrene, M.W. =537,000, for solution concentration

of 0.23 mg/ml, at 34 °C.

Curves A and B each represent measurements made on separate sets of slides. The open and closed points on each curve represent separate measurements made on that set of slides.

index between the film and solvent is approximately the same as was

obtained for the lower molecular weight sample.

The adsorbance curves for these films are shown in figure 9. Although the thicknesses were approximately the same for all four films

measured, the adsorbance curves were somewhat different for each set

of slides, resulting from differences in the refractive index of the films. The thickness and adsorbance for this molecular weight sample

were both somewhat higher at higher solution concentrations.
294 STROMBERG, PASSAGLIA, AND TUTAS

I

I

! 5. Discussion

The ability to measure the adsorbed layer in contact with the solu- tion is especially valuable for polymers because of the highly swollen

nature of the film. Theories of polymer adsorption have led to predictions of both relatively thick [7] and relatively thin [8] films. The thicknesses reported here are based on a simple model of a single

homogeneous film on a surface in contact with a liquid medium.

However, the polymer segment density in the adsorbed film is probably not uniform, but rather decreases at increasing distances from

j the surface. The actual distribution is not known.

McCrackin and Colson [9] have theoretically studied films with

| refractive indexes varying between that of the solvent and that of

either the pure polymer or some intermediate value between solvent

and polymer, using in particular, the values for the cyclohexanepolystyrene system. Three different types of distributions which are reasonable for a polymer film adsorbed on a surface—linear, ex- ponential, and Gaussian—were studied. They found that the thick- ness value calculated using the simple homogeneous single film model was approximately 1.7 times larger than the root-mean-square aver- age of film (thickness) calculated for all of these distributions.

The values of the thicknesses shown in figures 3, 4, 6, and 8, therefore, must be divided by a factor of 1.7 to obtain the root-meansquare average thickness. This results in a film thickness of approximately 115A for the 76,000 molecular weight sample in the plateau

region of the solution concentration curve, and approximately 290A

for the 537,000 molecular weight sample reported here. These re- duced values can be compared to the dimensions of the polymer in solution at the theta temperature (35 °C). The radii of

Q gyration of the high and low molecular weights are 80 A and 210 A, while the

root-mean-square end-to-end distances are 196 A and 514 A. The

observed thickness is in each case somewhat larger than the radius of

gyration but smaller than the end-to-end distance. Therefore, the

configuration of the polymer on the surface is probably close to a random coil. The simultaneous determination of the concentration of polymer

in the adsorbed layer provides another indication of the conformation

of the polymer molecule at the surface. The higher concentration of polymer in the film adsorbed from the 0.18 mg/ml solution, shown in

figure 7, may indicate that the polymer molecule exists on the surface

in a more uncoiled state than at higher solution concentrations, re- sulting in denser films. This is consistent with the lower thickness

values obtained at this solution concentration. The high value of the concentration of polymer in the film for the 9.8 mg/ml solution

concentration may indicate that there is entanglement of non adsorbed polymer molecules in the adsorbed layer, increasing its density,

but not its thickness.

There are very few techniques by which adsorbed material can be

measured on a surface directly. The determination of adsorbance by
POLYMER ADSORPTION 295

measuring changes in the concentration of a solution is a very satis- factory method for many materials. However, in the case of poly- mers the rate of desorption is usually many orders of magnitude

slower than that of adsorption, and it is advantageous to adsorb

from a system with no significant change in solution concentration

resulting from adsorption. Ellipsometry provides such a system.

6. References

[1] R. R. Stromberg, E. Passaglia, and D. J. Tutas, J. Res. NBS 67A (Phys. and

Chem), 431 (1963).

[2] P. Drude, Ann. Physik. 272, 532 (1889); ibid. 272, 865 (1889); ibid. 275, 481 (1890).

[3] F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. NBS 67A (Phys. and Chem.) 363 (1963).

[4] F. L. McCrackin and J. Colson, NBS Tech. Note 242 (1964).

[5] A. B. Winterbottom, Optical Studies of Metal Surfaces, The Royal Norwegian

Scientific Society, Report No. I, 1955, published by F. Bruns, Trondheim,

Norway.

[6] E. Passaglia and R. R. Stromberg, Optical properties of thin films on trans- parent surfaces by ellipsometry; internal reflection for film covered surfaces near the critical angle, J. Res. NBS 68A (Phys. and Chem.), No. 6 (Nov.-

Dec. 1964) (to be published).

[7] R. Simha, H. L. Frisch, and F. R. Eirich, J. Phys. Chem. 57, 584 (1953); H. L. Frisch and R. Simha, ibid. 58, 507 (1954); H. L. Frisch, ibid. 59, 633 (1955); H. L. Frisch and R. Simha, J. Chem. Phys. 24, 652 (1956).

[8] A. Silberberg, J. Phys. Chem. 66, 1872 (1962) ; A. Silberberg, ibid. 66, 1884

(1962).

[9] F. L. McCrackin and J. P. Colson, this book, p. 61. (See table 11). Discussion

R. Ullman (Ford Motor Co.)

If you were to take pure solvent, cyclohexane, let it sit on the

surface, and then make measurements as a function of time, would

you get any changes in the readings at all? R. R. Stromberg:

Changes in A and \p that occur as a result of exposure of the

surface to solvent for many hours are negligible in relation to the

changes occurring as a result of adsorption. The surface is usually

exposed to solvent for at least 2 hours before solution is added.

R. Ullman:

How does the radius of gyration of the polymer in solution compare

with the measured thicknesses either corrected or uncorrected?

R. R. Stromberg:

The radii of gyration are approximately 80 A for the 76,000 molecular weight polymer and 210 A for the 537,000 molecular weight

sample. The root-mean-square average measured thickness values of the adsorbed polymer are 115 A and 290 A respectively.

R. Ullman:

What is interesting about this is that the result is in fair agreement

(making appropriate corrections for molecular weight and solvent
296 STROMBERG, PASSAGLIA, AND TUTAS

power) with that obtained by measuring the alleged polymer thickness

in terms of the flow through fine capillaries, where Ohrn found that

the average thickness associated with the blocking of the fluid flow near the edge of the capillaries is of the order of 2 to 3 times the radius

of gyration in solution. Now, of course, it may be that both of these

methods of measurement yield awkward and different average values,

but it is interesting that they are not too different from each other. R. R. Stromberg:

Ohrn reported thicknesses that are larger than those we measured.

This was also true of the results of the other workers who have

applied this capillary technique. Rowland (Polytechnic Institute of Brooklyn) reported a thickness of about 200 A for polystyrene with a molecular weight of 110,000 adsorbed from cyclohexane. Our value

of 115 A is less than that found by Rowland.

R. Ullman:

Rowland's method was carried out using a sintered glass powder,

not a capillary, and only a nominal average capillary dimension was

calculated. This number must be corrected by some unknown

factor. C. L. Sutula (Marathon Oil Company)

I understand you are performing radioactive isotope measurements

along with your ellipsometer work. Can you comment on the agree- ment in the adsorption isotherms determined by both methods?

R. Stromberg:

Using the radiotracer technique one cannot, of course, measure

thicknesses; however, one can quite accurately measure the amount

of polymer adsorbed. We have used the same molecular weight

polymer and the same chrome surfaces with both techniques and the

agreement is very good as far as we can tell. C. L. Sutula:

Do you find, as did Bartell and Betts, that the ellipsometer readings

are linearly related to surface coverage if coverage is inferred from

radiotracer measurements?

R. R. Stromberg:

In the case of polymer adsorption we think it highly probable that

at all the concentrations reported here the surface is completely

covered. Differences resulting from different concentrations may

be caused by differences in polymer conformation on the surface. Our studies, therefore, cannot be directly compared to those of Bartell.
Determination of Thickness and Refractive

Index of Thin Films as an Approach to the

Study of Biological Macromolecules

J. B. Bateman

U.S. Army Biological Laboratories, Fort Detrick, Frederick, Md.

Some years ago a stepped interference reflector, consisting of barium

stearate multilayers on chromium, was devised for determining the

thickness and refractive index of films deposited upon it. Tests of the method were encouraging. Thin films thus measured yielded values

of refractive index in good agreement with values obtained on much

thicker films by the Abeles method. Values of film thickness were

often compatible with known molecular dimensions. Preliminary data contrasting the properties of "intact" and "unfolded" protein molecules will be shown in illustration. Areas for future study and

possible modifications of the method will be indicated, with the warning that limitations arising from dispersion, anisotropy and film hetero- geneity also require further examination.

1. Introduction

It is perhaps trite to say that phenomena in two dimensions represent a sort of halfway house between one dimension and three, but

1 this is nevertheless the chief reason for the interest and fascination

of thin films in biology. We now realize that information is most

effectively stored in one dimension, on a thread or wire; but between

the abstract instructions and their three-dimensional realization we

must have boundary layers, films, membranes, and septa to implement the creation of structure. Since these boundary elements are

far from being mere partitions, but possess activities, specificities,

structures, information-bearing capabilities of their own, their study

is of the greatest importance. They are, moreover, amenable in many of their aspects to in vitro study. Such study has both intrinsic

significance and significance in the investigation of the fragile threads

that store genetic information, since these can be more readily pre- served and manipulated by being incorporated in films. As an ultimate accomplishment one may imagine the message of a

biological linear polymer being translated in a sort of molecular wire

recorder, or a minute area of interfacial film being scanned for the

distribution of specific reaction foci. In the absence of any immediate

prospect of such an achievement, aside from the use of the living cell

itself as translator, we are left with the necessity of drawing what

inferences we can from examination of macroscopic regions of the

objects of interest. The value of studies using visible light in this

connection rests upon the fact that, while the cross section of the

optical probe is necessarily very large, viewed on an atomic scale,

the short dimensions of biological polymers are of the order of one

percent of a wavelength, so that polarimetric or related methods can

297
298 BATEMAN

provide significant information concerning these dimensions. This

is not all, for by the same token, these methods can detect, and

measure, processes involving deposition of biological macromolecules

at active interfaces, or involving their release, degradation, or change

of configuration.

Such considerations as these, for which we are indebted above all to the writings of Bideal [27, 28] 1 and Langmuir [19], have inspired our interest, over a number of years, in the optical study of thin films. The remainder of this paper will be devoted to the description of a method invented by Richard Mattuck, which permits the "optical

thickness", or mass per unit area, of a film to be resolved into metrical

thickness and refractive index. The existence of such a method carries us a step forward in the study of biological interaction at interfaces, for there is a very clear advantage in being able to distinguish, for example, between attachment of large compact molecules to a rela- tively small number of reaction sites, and the formation of a thinner,

but denser, film covering the entire surface. 2. Principles of Determination of Thickness and

Refractive Index of Thin Transparent Films

2.1. Outline

The description of the method and of the preliminary results obtained with it is spread through several papers published from 1956

to 1961 [23, 24, 6, 7, 8]. Our hope in the present paper is to consolidate this material and to present it in easily assimilable form for the interested biological scientist. The initial description will be more or less qualitative, followed by statement, without proof, of the simple approximate equations. Experimental procedures and

results will then be given, while the derivation of the equations, and

various other matters such as limitations on validity, sources of error, and future developments, will be dealt with later.

2.2. Qualitative Sketch of Optical Principles

In attempting a simple approach, consider a semi-infinite flat sheet of some material illuminated at some specified angle of incidence

by a parallel beam of plane-polarized light. In general the light

suffers complicated changes of amplitude (and therefore of intensity)

and of phase upon reflection, which can however easily be understood

by considering two simple special cases. In the first, the plane of polarization is adjusted so that the electric vector oscillates in the

plane of incidence (p-, or £>ara^-vibration) 5 m tne second, the

electric vector oscillates perpendicular to the plane of incidence

0-, or senkrecht-vibration) . The changes of amplitude and phase

upon reflection are in general different in the two cases; they are

1 Figures in brackets indicate the literature references on p. 332.
BIOLOGICAL MACROMOLECULES 299

described by the well-known Fresnel equations, which show a differ- ent functional dependence of the reflected amplitude and phase upon

the relative refractive index of the reflector. Thus, in principle,

absolute measurements of the two reflection coefficients (amplitude

and phase) will describe the properties of the reflector in terms of the

two components of the refractive index (refraction and absorption)

if the reflector is a transparent dielectric, a single measurement will

suffice. If now the plane surface is coated with a thin film, we shall have to deal in general with reflections from the two surfaces of the film, and

the amplitude and phase of the reflected ray will be the result of compounding that arising from the upper surface of the film with

those emerging after successive pairs of reflections within the film. Since the appropriate Fresnel coefficients will again be involved,

depending as they do upon different functions of the film refractive

index, it would seem that if the substrate has been adequately characterized, further measurements of the new absolute reflection coefficients

and phase changes for the s- and ^-vibrations will serve to characterize

the film in terms of thickness and refractive index—remembering

that the optical thickness of the film also contributes to the phase

change and to the resultant amplitude.

Unfortunately, this type of procedure, so transparent in principle,

suffers from severe practical drawbacks. It is quite usual in optics

to try to avoid absolute measurements, and at the same time to seek

conditions that will provide either qualitative distinctions, as between

light and dark, or at least easily recognizable extrema. In the field of thin film measurements, a useful stratagem first exploited by Drude

has had a dominating influence. In illuminating the reflector we

have just been discussing, the plane of polarization of the incident

light can be shifted (for example) through 45°, so that instead of using

the s- and ^-vibrations singly, we are in effect applying them simultaneously with equal amplitudes and exactly in phase with each other.

After reflection they no longer agree either in phase or in amplitude,

but if the reflected beam is passed through suitable optical devices

suitably adjusted, the agreement can be restored, and the settings of the instruments that accomplish this are simply related to the ratio of the reflected amplitudes for the s- and p-rays and to the difference

between the two changes of phase. When the reflector is covered

with a thin film, the amplitude and phase changes are modified in a manner that is dependent upon the optical parameters of the film. The sensitivity of these polarimetric quantities to changes in the film

parameters can be increased in various ways. For example, it is not

necessary for the reflector to be a simple semi-infinite solid slab of the

kind postulated above; more typically it is laminated, a metal base

coated with a dielectric layer of optimum thickness for the purpose at hand. It is this use of a compound substrate that is partly responsible

for the power and versatility of Rothen's famous "ellipsometer."

One might hope that the ellipsometric technique itself would provide

a means of resolving film thickness and refractive index. Analytical

728-101 0—64 20
300 BATEMAN

approximate expressions for the effects of thin films on the reflection of polarized light, such as those of Drude [14] and Lucy [21], provide some encouragement, but the experimentally observable quantities

usually turn out to be dependent essentially upon optical thickness

with very slight second order sensitivity to variations in refractive in- dex per se.

2 Attempts to get around this limitation, or to supplement

the polarimetric measurement in some suitable manner [16] have been

discouragingly unsuccessful [22]. 2.3. The Stepped Interference Reflector Method

In the method now to be described, the traditional polarimetric

measurements have been abandoned, and replaced by observations of reflected intensity minima that occur when the s- and #>-rays are in- cident upon suitable compound reflectors and then upon the same

reflectors coated with the "unknown" thin film. In this respect the

technique has more in common with a simple method of measuring

small changes in film thickness devised years ago by Blodgett [9, 10], than with that of Drude. The basic observation is illustrated in figure 1, adapted from Blodgett's paper. If a plate of polished metal

is coated with a layer of a dielectric such as barium stearate, which can be deposited quite conveniently in successive bimolecular sheets, each

adding a constant increment of thickness, the reflection coefficient for the s-ray of monochromatic light at a definite angle of incidence will pass through a minimum when the thickness of dielectric is such that

the optical path difference between the rays reflected at the two interfaces of the reflector is approximately 180°. The word "approxi2 From the recent work of McCrackin et al. [25] It appears that this statement is not always correct, and

several papers in this symposium contain evidence to the same effect. Our conclusion, right or wrong,

provided the impetus for developing the method described here.

Angle of Incidence 8 0°

Wavelength 5 89 3

50 100

Double Layers of Barium Stearate Figure 1. Illustrating the conditions for formation of s- and p- fringes when

different numbers of barium stearate double layers are deposited upon chromium.

Modified from Blodgett [9]
r BIOLOGICAL MACROMOLECULES 301

mately" is used because the condition for minimum reflectance is determined not only by the thickness of the dielectric but by any

contribution that may result from differences between the phase

changes at the two interfaces. These differences are rather small

with the s-ray. In the case of the p-ray, on the other hand, at large

angles of incidence, they approach 180°, so that in order to produce

minimum reflectance it is necessary for the optical path difference

within the dielectric layer to approach 360°: thus with increasing

dielectric thickness the reflection minima will alternate in the manner

shown in figure 1. The underlying phase relationships that produce

180'

Figure 2. Phase changes at air-dielectric (13) and barium stearate-chromium (34)

interfaces.

Abscissa: angle of incidence in air. Ordinate, phase

change. Letters s and p prefer to plane of polarization of incident light, parallel (p) or perpendicular (s) to the plane of incidence.

e

9Qf

2.

13s 13c lip 34s

13p

1 34p

l 1 3 30 60 90°

The existence of a minimum in the reflected intensity depends

not only upon the establishment of suitable phase relationships. It

is also necessary that there should not be too great a disparity between

the reflected amplitudes from the two interfaces. Obviously, if one

amplitude were to be very much greater than the other, the minimum

would be reduced to a mere ripple very hard to locate and to measure.

Conversely, the simple equations for reflection from a dielectric

coated metal reflector show that, in principle, the minimum can be

converted to an extinction if, together with the correct phase relationships, the reflected amplitudes are exactly equal. Calculations using

these equations and the known optical constants of metals and barium

stearate show that the necessary conditions should readily be attainable in practice. There is always an angle of incidence at which

the two reflection coefficient curves intersect (see examples in fig. 3). Here, then, we see the origin of the stepped interference reflector. We have, for example, a chromium-plated glass slide, one half of which

has about 20, the other half about 40, double layers of barium stearate

(fig. 4). At the correct wavelength and angle of incidence, the thinner

step will give the s-ray extinction; at a slightly different angle of incidence the thicker step will give the p-ray extinction. Alternatively,

thicker dielectric films can be used to give the corresponding first- order extinctions. If instead of using monochromatic light the slide
302 BATEMAN

0.:

0.'

P ____AiL 1 Au -

_ ^

J—

^Cr

Ta__Z

^[a / ^^BaSt

"i 1 II

BaSt ^/

1 1 1 40° 80° 40° 80° 4>

Figure 3. Illustrating first condition

for formation of s- and p- extinction fringes by compound reflector of barium

stearate on metal.

Curves labeled BaSt give reflected intensity at interface between air and a semi-infinite layer of barium stearate. Curves labeled Au, Cr, Ta give the corresponding reflected intensities at a barium

stearate-metal interface. Abscissa, angle of inci- dence in air; ordinate, square of reflected amplitude, i Wavelength about 5890 A throughout.

thin added film

stearate double layers chromium Figure 4. Chromium plated glass slide with stepped multilayer of barium stearate-stearic acid. (Reproduced from [6].)

Figure 5. Calculated extinction fringes pro- duced when stepped reflector of barium stearate on chromium is examined at the correct angle

of incidence.

Fringes are calculated to coincide at 5400 A. Dispersion

of metal is ignored. Abscissa: wavelength. Ordinate:

reflected intensity. Curve 1: zero-order s-fringe. Curve 2: zero-order p-fringe. Curves 4 and 3 are the corresponding

first-order fringes respectively.
BIOLOGICAL MACROMOLECULES 303

is illuminated with white light and the reflected light dispersed in a spectrometer, a black band will be seen at the extinction wavelength.

Figure 5 shows curves calculated for zero and first order s- and pextinction fringes produced by barium stearate on a metal plate

having optical constants resembling those of chromium, but assumed

to be independent of wavelength.

What happens if a thin transparent film of arbitrary refractive

index and thickness is now deposited upon both steps of the interference reflector? Clearly the reflection coefficient at the original

upper surface of the dielectric substrate will be decreased; the rays

reflected from this interface, now representing the boundary between

the "unknown" film and the substrate (fig. 6), will continue to interfere with those reflected from the metal, with their optical path

difference unchanged, and will thus produce a minimum reflected

intensity at the wavelength of the original extinction. However,

upon this will be imposed a second minimum produced by combining

the ray reflected from the upper surface of the added film with that

coming from the metal; this second minimum will occur at a new,

longer, wavelength corresponding to the total optical thickness of

dielectric substrate and test film (regions 2 and 3 of fig. 6). These

two minima will not actually be resolved; they are introduced for purposes of exposition only. The result of combining the two hypothetical fringes will be a single rather broad minimum at an intermediate wavelength. The same process of combination will occur both on the s- and p-steps, but since the profiles of the two minima

in each case will depend upon the values of the respective Fresnel

reflection coefficients, and since these in turn will involve different

functions of the refractive index of the unknown film, the wavelengths

of the s- and ^-minima will depend in distinctive ways upon film

refractive index. The theory of the fringe shifts justifies this prediction and shows, furthermore, that the difference referable to film

refractive index is a readily measurable quantity. After drastic

approximation, the following simple equations are obtained: n2 -1.5(6LAs/5^Ap ) 1/2 =1.5(€s /6 p) 1/2 (1)

Figure 6. Simplified scheme, ignoring

multiple reflections, illustrating displace- ment of extinction fringe when thin test film is formed upon the dielectric film 8. Observed fringe is compounded from fringe formed

(a) by addition of rays 23 and 3c, at the wavelength \ b of the original extinction fringe, and (b) by addition of 12 and 3c, at some longer wavelength. Resultant minimum is at an intermediate wavelength X 0 .

4
304 BATEMAN

4~0.0885;A,/(n3-l) (2)

(3) These equations refer to the case of a film (fig. 6) deposited upon

barium stearate of refractive index 1.5 and observed at an angle

of incidence 81°. Film refractive index and thickness are denoted

by n2 , d2 , respectively; fringe shifts are A; the 8'

3 values are effective

optical path differences for the barium stearate substrate, and are

calibration constants measured as described later.

Glass microscope slides coated with a thick film of chromium by

vapor condensation in vamp can be prepared in the laboratory as needed, or can be obtained from commercial sources.

3 Sample slides from each batch should be examined for suitability since the optical

properties of the chromium films are quite variable and occasionally

will fail to give extinction fringes when coated with the usual thick- nesses of stearate and examined at the usual angle of incidence. A

check by this means is more satisfactory than a measurement of the

optical constants of the metal film since the tolerable range of variation has not been established.

Procedures for transfer of stearate multilayers vary considerably

from laboratory to laboratory and have undergone many modifications

in our own laboratory. Although studied by many workers, some of the factors involved are so subtle that all-inclusive specifications can- not be given. The composition of the water upon which the stearic

acid is spread, for example, is certainly of major importance, but the

usual electrical criterion of purity is irrelevant, and more importance

must be attached to such factors as presence of impurities picked up

from ion exchange materials during the course of preparation.

Our most recent procedures have been described in some detail [8]. The aqueous subphase is prepared from laboratory distilled water of

specific resistance around 80,000 ohm/cm, and containing spectrographically detectable traces of B, Si, Al, Fe, Mg, Ca, and Cu; the

salts added to this are barium chloride, 10~4 M; potassium bicarbonate, 2X10"4 M; cupric chloride, 10-7 M. The final pB. is 6.8 and

the temperature must not exceed 20 °C. This solution is placed in a waxed Plexiglas film tray (fig. 7). The surface is repeatedly "swept"

clean with a Plexiglas barrier, and on the clean surface is placed a polyethylene float in position 2 (fig. 7). A solution of stearic acid

(Eastman, M. p. 69.5 to 70 °C), 3X 10~ 2 M in reagent grade benzene,

is then dropped from a micropipet upon the surface at the center of the ring 3; the ring retains the solution while permitting outward

spreading of the monolayer formed as the benzene evaporates. Spread3 Evaporated Metal Films Corp., 436 W. State St., Ithaca, N.Y.

3. Apparatus and Procedures

3.1. Fabrication of Stepped Reflectors
BIOLOGICAL MACROMOLECULE

305

ing is continued, as rapidly as possible, until the float has been propelled to position 2'. A residual lens of solution is aspirated off the

surface, and after a few minutes the film is compressed to 29 dynes/cm

by applying a drop of purified oleic acid to the surface on the left

(in fig. 7) of the floating barrier. Several clean chromium slides,

made hydrophobic by rubbing with cotton wool, are then mounted

on an automatic dipping device above the trough (position 4 in fig. 7)

and dipped vertically at about 5 cm/min. The result of repeated

dipping is a very stable, uniform multilayer of barium-copper-hydrogen

stearate. For many purposes zero-order reflectors are satisfactory, with about

20 stearate double layers on one portion and about 40 on the other.

Sharper extinction fringes are obtained with first-order reflectors (61

and 80 double layers) (cf. fig. 5), with reduction of fringe shift for a given unknown film, but with an overall gam m precision and

convenience.

. Figure 7. Plexiglas trough for monolayer formation and transfer.

See text and ref. [8] for further details. (Repro- duced from [8].) L

50cm «-i

3.2. Optical Apparatus and Procedure

The various optical elements are shown in figure 8 and identified

in the accompanying legend. The arrangement is such that at an

angle of incidence of about 80° the area of the stepped reflector viewed

by the photomultiplier is 2 mm high X0.6 mm. The spectral bandwidth received by the photomultiplier is estimated to be about 10 A,

representing a compromise between desirable fringe resolution and

available intensity.

Assuming the necessary alinement and other adjustments to have

been checked, the first step is to locate the s- and ^-fringe minima on

the reflector. Formerly this was done at the angles of incidence for true extinction fringes, 79°10' and 82°30' respectively for s- and

p-rays. A convenient practical compromise is to work at a constant

angle of incidence of 81°, with negligible loss of accuracy. A manual

null-searching method has proved adequate for locating the fringe

minima, although scale readings taken on either side of the minimum

may also be useful and are convenient indicators of any unusual

change in symmetry. In general, observations are made at ten

places on each step of the reflector, placed in two vertical rows on

either side of the axis. The uniformity of the multilayer plates is such that discrepancies between readings exceeding 2 A are quite
306 BATEMAN

Figure 8. Apparatus used for measurement of film thickness and refractive index

by the s-p stepped reflector method.

a is zirconium arc; 6, collimating lens; c, first collimating slit; d, polarizer; e, second collimating slit;/, stepped reflector with test film; g, analyzer; h, and i are focusing lenses; j is mirror; k, constant deviation spectrometer mounted on turntable; I, Hilger slit; m, photomultiplier; n, movable mirror; o, collimating

lens for mercury vapor lamp used for calibration. (Reproduced from [24].) unusual, and within a series of established uniformity the number of observation areas may be reduced. The test films added subsequently are generally less uniform.

After locating the minima at Xos and \op , calibration is done by

adding a known small number of double layers of BaCuH-stearate to the entire reflector and repeating the fringe measurements, which now

yield the wavelength values X6s and \bp . From the four wavelength

values the calibration constants 8'

3 of eqs (1) to (3), can be calculated by eq (69).

The reflector is now ready for the test film, deposition of which

produces a further displacement of the fringe minima to Xas and Xap . These displacements (Xas — \bs ) and (Xap —\bp), are the values A used

in eqs (1) to (3). If more than a rough estimate of the test film

parameters is required, more involved calculations can be carried out

as indicated later.

4. Methods of Checking Validity of Film Thickness and

Refractive Index Data

4.1. Test Criteria

Independent methods are not available for determining the optical

constants, or alternatively the mass per unit area and density, of the

test films deposited upon the reflector. Indirect tests must be applied,

recognizing that each is open to some objection. The following

criteria have been used, with results that will be presented briefly in

the next section:

a. Successive transfers of monolayers of test substance: constant

apparent refractive index combined with constant apparent thickness

increment.
BIOLOGICAL MACROMOLECULES 307

b. Agreement of film refractive index and thickness per layer with

values found using multilayers built up to several higher orders (see

section 4.2).

c. Variation in apparent thickness of monolayers prepared from

members of homologous series correlated with molecular dimensions;

apparent refractive indices compared with values for organic crystals.

d. Apparent refractive index of thin films formed by condensation

of vapor compared with values for thicker films, similarly prepared,

measured by the Abeles method (see section 4.3).

e. Apparent refractive index of test film formed either by vapor

condensation or by monolayer transfer shown to be independent of film thickness, and correlated with refractive index of substance in

bulk. . . f. Demonstration that the apparent refractive index of certain

porous films of very low apparent index can be increased by ' 'filling in."

4.2. Independent Determination of Thickness and Refractive

Index of Thick Multilayers

When successive monolayers can be transferred to a chromium

slide in sufficient numbers to give s-ray extinction fringes of several

orders, the average refractive index and thickness per monolayer can

be determined by measuring the fringe wavelengths X for various

angles of incidence above and below the extinction angle (17). The

fringe condition is

(7>

2 -sin2 0)1/2 +0=(l+2m)7r (4) A where the first term corresponds to the optical path difference in the

added film. The additional phase term 0 incorporates (a) phase

changes on reflection at the two interfaces, and (b) the retardation

within any "rubbed-down" monolayer present on the metal surface. The fringe order, m, is 0, 1, 2 ... . If 0 is not wavelength dependent,

the plot of N/\ against m at any angle of incidence will be a straight

line: m=2d(n2 -sm2 0)

1/2 (A7X)-(l-0/7r)/2.

=b(N/\)+a. (5) Equation (5) can then be solved for d and n by plotting b

2 against sin2

<f>: b

2 =-4d2 sin2 0+4n25

2

. (6) Some dependence of 0 upon wavelength would be expected. A basal

layer, d0 , n0 , would contribute a term 0/ in X

-1

, and <f>, while there

would also be a dispersion term 0r at the metal-dielectric interface

dependent mainly on X: e=0f+0r =4irdo(n2

o -sm2

<t>) 1/2/\+0T . (7)
308 BATEMAN

In practice, measurements cannot be made at exactly constant wavelength; there is some variation of fringe wavelength within each set of orders at any angle of incidence, and a much larger systematic variation with changing angle of incidence, resulting in apparent dej pendence of the intercept a upon angle of incidence. If this is attributed to 9f, varying according to eq (7), it is found experimentally

that d0 has a large negative value, while the observed small overall

values of 8 (around 20°) have to be maintained by an unreasonably

large compensating value of 0r

. Thus no reliable estimate of 6(<f>) can be obtained, and it is preferable to consider it as exclusively wavelength-dependent. With this assumption, the values of a obtained

at each angle of incidence can be plotted against average fringe wavelength and revised values of a can be obtained by interpolation for the

fringe wavelength measured at each value of N. Each measurement

then yields a new value of the slope b by the equation:

b=(m-a)l(N/\) (8) and the averaged values of b plotted against sin2

<f> lead to the corrected

estimates of n and d. If consideration is being given to the nonisotropic character of the multilayers, measurements can be made at each angle of incidence

with the s-ray at right angles to the direction of film deposition (the

usual position of the slide) or parallel to it. In this case it is assumed

that the metrical thickness of the films, Nd, is independent of the

direction from which they are viewed [8]. Using one prime and two

primes, respectively, to denote values referable to the two mutually

perpendicular directions of incidence, one obtains

(by+(b'y=4d2[(ny]-8d2 sin2

cf> (9) (b'y-(b"y=±d2 [(n') 2 -(n")2

]. (10)

4,3. Determination of Refractive Index by the Abeles Method

In the Abeles method [1, 2], a portion of the surface of a plane

reflector of any type is coated with the test film. The boundary

between the covered and uncovered portions is illuminated with

collimated monochromatic light plane polarized in the plane of incidence

(p-vibration) and the angle of incidence is varied until the images of the areas on either side of the boundary appear equally bright.

The reflection coefficient of the film-covered system is obtained

simply by adding the multiple reflections (see, for example, ref. [11], pp. 61, 323; ref. [18], p. 55), while that of the clean reflector can be ex- pressed in terms of the film parameters by bringing the film thickness

to zero: ri3+r3c exp (—^3) 33 =0 ) ^i3+^_

rib l+r

r ^ 13r3e exp (-£53) l+n3r3c 10
BIOLOGICAL MACROMOLECULES 309

Figure 9. The two Brewster angle methods of Abeles [1, 2] for determining index of refraction of thin films. Top: multiple reflections between surfaces b and c of a

dielectric film on an arbitrary base. Middle: Multiple

reflections abolished when light (p-vibration) is incident at Brewster angle for interface 6. Bottom: Film on dielectric substrate. Multiple reflections abolished when incidence corresponds to Brewster angle for interface c.

where the various reflection coefficients, which may be complex, are denoted by r. Matched fields can be obtained either by eliminating

reflection at the upper film surface (r13 =0) or at the lower surface

(r3c =0), as shown in figure 9. The second condition cannot readily

be satisfied on metallic substrates. The first condition is generally

applicable; it shows that if, by eliminating r13 , the multiple reflections are reduced to a single reflection r3c plus complete transmission at 13

and 31, the reflected amplitude r3c at an internal angle of incidence of

sin-1 (sin 4>ijnz ) will be equal to that at the clean substrate surface at an

angle of incidence <P. This is not generally the case; in a system

governed by Fresnel reflection coefficients it is satisfied only when tan

01= ^3. Thus the matching angle of incidence, the Brewster angle,

at once gives the refractive index of the film, no matter what the

optical properties of the substrate.

Abeles [1] remarks that there may be an optimum film thickness

for the detection of the matching angle, and it has been noted [24] that when a metallic base is used, the condition for maximum contrast

in the vicinity of the matching angle is periodic in film thickness, with

53 , the optical path difference in the film, equal to ir, Sir, 5tt; this is, remembering the additional phase difference close to tt between r13 and r34 , the condition for maximum rlb . The contrast is much less in the vicinity of the p-ray extinction. The upshot of this is that the

minimum film thickness for satisfactory measurements is about 1000

A. The method, in this form, is not suitable for direct check of results obtained by the stepped reflector method, but furnishes an

interesting comparison of thin and thick films deposited under

similar conditions. 4 The apparatus used consisted of a sodium vapor source, collimatmg

lens, two collimating irises, Nicol prism, Spencer student spectrometer

table with added slide holder, and a viewing telescope focused on a

dust particle at the center of the slide. The matching condition was

« More recent work on the method is referred to in the review by Abeles [3].
310 BATEMAN

established visually by simultaneous manual rotation of the slide holder and telescope arm, 10 to 20 such readings being taken.

5. Some Experimental Results

5.1. Measurements on Thick Fatty Acid Multilayers

a. Illustration of Procedure

The method outlined in section 4.2 has been used for measurements

of double-layer spacing and refractive index of barium stearate, barium copper stearate, and barium copper nonadecanoate mixed in unknown proportions with the free acids. An illustration of the

primary fringe wavelength measurements has been given in an earlier paper [17], while the data used for the initial calculations from equations (5) and (6) are shown graphically for a typical case in figure 10.

Figure 10. Experimental data illustrating higher order s-fringe method of deter- mining film thickness per monolayer and refractive index. Ba Cu H multilayers on chromium plated glass. Open circles: Nf\ as function of fringe order, to (text eq (5)) for angles of incidence 70° and 80°. Closed circles: W (sin2 0), text eq (6), for angles of incidence

70.0°, 72.5°, 75.0°, 77.5° and 80.0°. (Reproduced from [8].) Examples of the graphs used to obtain corrected values of a(X), for application of eq (8), are given in figure 11. The resulting individual

values of b(m,<f>) are then averaged for each <f> and eq (6]_ again applied

for calculation, by least squares, of final values of n and d. The values

of b(m) prove to be compatible with a straight line passing through the

origin, but they nevertheless show some nonrandom variation that seems to depend upon the nature of the film substance. Illustrations

of this slight systematic variation, which suggests a variation of film

properties with thickness, are given in table 1.
BIOLOGICAL MACROMOLECTJLE

311

0,43

-0.41

-0.39 4900

Figure 11. Apparent s-vibration phase change at surface of "clean" chromium

reflector, obtained by extrapolation of straight line relating fringe order, m, to iV/\

(eq (5)).

Abscissa- average fringe wavelength for 1st- to 5th-order fringes. Ordinate: a, equal to -0.5(1 -dh), w4?e fi pSe chaSft "clean" "surface. If 0=0, .--JUL Curve 1: for Ba> Cu H stearate slide III; 2: Ba Cu H stearate checkerboard slide V; 3: Ba Cu H nonadecanoate shde II. See table 2. Table 1. Average variations of b(m)

^^^^^Fih^^m^^^

b/bi No. of

slides examined

1 2 3 4 5 BaCuH stearate.. 1l

1

0.99583

. 99915

. 99974

0. 99464

. 99980

1. 00061

0. 99667

. 99963

. 99974

0. 99834 3

12 BaCuH stearate, BaCuH nonadecanoate. .99850

Values averaged over five angles of incidence.b. Results

The measurements are summarized in table 2. The variation from

slide to slide is rather greater than might be hoped, apparently because

of variations in the nature of the multilayer rather than poor reproducibility of the optical measurements themselves. The average spacmgs

for stearate and nonadecanoate are compatible with a vertical orientation of the extended chains, although the extended chain lengths themselves are open to uncertainty as to the effect of the barium and copper

atoms and the percentage of free acid in the films. The values of the

refractive index of stearate are compatible with the assumption that

the index that is effective for the s-vibration is the ordinary index n0 if the hydrocarbon chains are normal to the slide. In the case of the

nonadecanoate films, it is apparent that the chains are either more

loosely packed than is the case for stearate, or that there is a certain

proportion of vacancies throughout the film. An indication of this can be obtained from the values of (n2 — l)/(n2 +2) since n

2 -l=

l #

7i

2 +2 p'M

(12)
312 BATEMAN

Table 2. Spacing and refractive index of thick fatty acid multilayers by measurement

at several orders of interference

Slide Refer- ence Substance Fringe type

a to to n-n

*15]

*[4

[8] BaH stearate s 45.7 1.508 BaH stearate s 47. 575 1. 498

1 BaCuH stearate s 51. 326 1. 492 BaCuH stearate P

s 1.493

II [8

[81 [8 BaCuH stearate .. 49. 485 1. 510 1.508 0. 00200

III BaCuH stearate s 49. 256 1.515 V BaCuH stearate checkerboard __ s 48. 535 1. 52250 1. 52269 -0. 00019

I' [8[8

[8 BaCuH nonadecanoate s 51. 409 1. 5108 1. 5125 -0. 00170

II' BaCuH nonadecanoate s 54. 225 1. 4661 1. 4649 0. 00120

TT' BaCuH nonadecanoate, 2 53. 908 1. 4679 1. 4667 0. 00120 months later. Averages. BaH stearate s 46. 638 1.503 BaCuH stearate+ s 49. 762 1.508 BaCuH stearate checkerboard.

BaCuH nonadecanoate § s 48. 535 1.522

s 52. 738 1.488 jy=number of double molecular layers (range) d=thickness per double layer, A

n'=refractive index for light incident at right angles to monolayer transfer direction n"= value for incidence parallel to transfer direction

* = Values as given in reference. Remaining values have been revised as described in text section 4.2. Differences from values given in reference [7] are small except for slide II' remeasured after two mouths;

values given there were d=52.53, »'= to" =1.491. +=Weighted average, with double values assigned to II and III, since edge effects between orders were

suspected in slide I. §Weighted average, with double values to I" and single values to each measurement of II

where R is molecular refractivity, M is molecular weight, and p is density of the solid-air mixture 5 having refractive index n. The

highest value found was 3.07 for the BaCuH stearate checkerboard

slide (q.v.) ; for stearate transferred in the usual manner, 2.90 to 3.02;

for nonadecanoate, 2.76 to 3.00. A test for vertical orientation was done by examining some of the

film-coated slides after turning them through 90° in the slide holder.

In the usual measurements the vertical axis of the slide coincides

with the direction of dipping during film transfer, and is perpendicular

to the plane of incidence; in this case, if the molecular chains are not

exactly perpendicular to the dipping axis, the refractive index for the

s-vibration will be a little larger than the ordinary index, n0 . Rotating

the slide through 90° will then bring the refractive index back to n0 . The expected difference, n' —n", would be about 0.0010 for a tilt of

5°, 0.0015 for 10°, 0.0040 for 15°. Thus the observed differences,

0.0019 for stearate and 0.0012 for nonadecanoate (table 2), suggest a

tilt of about 6 to 10°. A further test was made by building a stepped multilayer in the

usual way and then adding to this a second series of steps at right

angles, producing a checkerboard effect for which the refractive

indices n' and n" should be equal and, in presence of molecular

tilting, slightly larger than n0

. This slide proved to be superior to all others in that the intercept a was independent of wavelength, the

s Although RIM is not exactly constant for the different fatty acids, the difference is probably less than

1 percent—e.g., 0.3054 and 0.3079 for palmitic and stearic acids, respectively, at 80 °C [12].
BIOLOGICAL MACROMOLECULES 313

values of b showed no dependence on fringe order (table 1), and the

I difference between n' and n" was reduced by a factor of 10 (table 2).

! At the same time, the film density was higher than usual, suggesting

that transfer in two directions may be advantageous in the production

of uniformly packed multilayers.

5.2. Measurements with s-p Stepped Reflector

a. Transferred Monolayers of Aliphatic Acids

Since stearate is used in determining the calibration constants

8'

3s and 5'3p , eq (1), calculation of apparent refractive index and thick- ness of subsequently added stearate amounts to a further calibration

and should yield the same values as those initially assumed. The

fairly satisfactory check obtained is shown in figure 12. The results

for stearic acid transferred from a distilled water substrate show

equally satisfactory constancy of double-layer spacing and refractive

index, with values that are significantly different from the calibration

values for barium stearate.

200

d AU

100

0

1.6

1.5

1.4 0 12 3 4 DOUBLE LAYERS Figure 12. Thickness (above) and refractive index (below) as a function of number

of double layers of stearate transferred to s-p stepped reflector.

Vertical bars indicate standard deviations of each value. Overall thicknesses per double layer, d, and re- fractive index, nit are given at right, together with standard deviation for each series. Ba St indicates barium hydrogen stearate. H St is free stearic acid. Measurements with a few other long-chain saturated acids are shown in Table 3; the values of d are compared in table 5 with the

actual molecular dimensions obtained in x-ray diffraction studies.

It is of interest that while the experimental values of d-shown the correct overall trend, there are considerable irregularities and even a

reversal of the expected order. This reversal is avoided if the

amounts of material per monolayer are estimated in terms of a "Lorentz-Lorenz" function, (n2 — l)d/(n2 +2) : the values of this quantity, with increasing chain length, are 11.934, 13.705, 14.817, 16.045,
314 BATEMAN

; Table 3. Spacing and refractive index of thin aliphatic acid films on s-p stepped

reflector

Ref. Sn V*qf c\ n oP C

atoms F N d n d-d(cal) 7i-n (cal)

[24]

[8]

[8] [24]

1 Q Dynes/ cm

29

29

16

929

29

929

2929

1-5

2

64

4

1-2.5

2

4

64

24

1. 73 -.099 -.319

—0. 012 .002

.001 BaCuH stearate BaCuH stearate 47. 89

45. 16 41.9 41.0

39.7 46.6

56.1 53.9

53.7

1.518 1.528

1. 537 1.535

1.513 1.500

1. 487

1.529

1.531

BaCuH stearate. Stearic acid 18

1516

1920

BaCuH pentadecanoate BaCuH palmitate - [8] BaCuH nonadecanoate. BaCuH arachidate

_F=film surface pressure at time of transfer to stepped reflector iV=number of double molecular layers used in measurement

d(cal), to (cal) refer to values for stearic acid used in stepped reflector calibration In ref. [24]: d(cal) = 47.575 Au, 7i(cal) = 1.498 In ref. [8]: 49.499 1.510 16.619. Some irregularity remains, but this may be due to differences ; in structure of films formed from the odd and even members of the

aliphatic series. The improvement resulting from calculation of the

Lorentz-Lorenz function is probably an example of the n—d

j "coupling" discussed in the next paragraph.

b. Transferred Monolayers and Adsorbed Films of Protein

Two distinct procedures in the study of protein films, and their results [6, 7] are relevant to the present display of the stepped reflector method.

Direct adsorption. The stepped reflector is immersed m a solution [ of protein, the free remaining protein washed away in a stream of solvent, the solvent in turn displaced by water, and the reflector with

adsorbed protein then dried. The entire process may be repeated.

Monolayer transfer. A monolayer of protein is formed upon an aqueous surface and the monolayer transferred by dipping, in the manner used in preparation of stearate films. According to the

conditions of transfer the films produced may be "AB", double

layers produced by attachment of the monolayer during both downward and upward passage of the slide through the interfaces, or "B", with attachment only during emergence.

A complete protocol of an experiment on direct adsorption of protein has been published [6]. Figure 13 shows that with sucessive

transfers of bovine plasma albumin monolayers, the relative s- and

^-fringe displacements remain constant, as does the corresponding

increment in thickness per added monolayer. Other data on the

thickness and refractive index of adsorbed and transferred films are ; given in table 4. In instances where several successive monolayer

or double-layer transfers were made to the same reflector or pair of

i

i
BIOLOGICAL MACROMOLECULES 315

0 400 800 2 4 6

Figure 13. Results of measuring s- and p-fringe shifts upon transfer of monolayers of bovine plasma albumin to stepped reflector from distilled water surfaces.

Film pressure, 9.5 dynes/cm. Left side: Abscissa, e„. Straight line is regression line that gives rough

estimate of film refractive index by text eqs (89) or (90). Right side: apparent film thickness d and number

of transferred monolayers N. Slope of regression line gives average thickness per monolayer, ~d. (Diagram

reproduced from [7].)Table 4. Thin protein films on s-p stepped reflector

Solvent Concentration X n

d d A

Adsorbed Films of Bovine Plasma Albumin

Water

percent

0. 67

.67

. 0067-10. 00

. 05-5. 00

.1

.1

.1

1

169

2

2

1

1. 600±0. 032 1.588±0. 042

1. 598±0. 018

1. 571±0. 058

1.595 1.27

1.48

1.589

29. 2±1.6

27. 8±2.0

20. 2±10.6

30. 6±5.8

28.4

14.5 26.6

29.0

Water

Acetate buffer, protein re- moved with water _ 4.7

5. 5-6.0 5.0 3.0 Water HC1

(a) HC1

(b) Water, 2d exposure Averages for water and HC1 pH 5 Adsorbed Films of Botulinum Toxins Type A in 0.05 Macetate 3.38 0. 357 3 1.593, 1.574, 1.598 24.5, 26.1, 24.2

avg. 1.588 avg. 24.9 Type B in 0.2 Msuccinate-.. 5.5 0. 06 3 1.530, 1.536, 1.513 31.1, 29.3, 29.9

avg. 1 M6 avg. 50.1 Transferred Monolayers of Bovine Plasma Albumin

Liquid Substrate Type

Water 9 AB 1 1.526 20.9 10.45 AB2 29.2 9. 73 AB5 57.6 9. 60

Dilute zinc chloride 9 AB 2 1.583 19. 97, 19. 07 9. 99, 9. 54 (AB) 2 36. 10, 35. 93 9. 03, 8. 98 Dilute zinc chloride. 16 AB 2 1.600 17. 67, 24. 25 8. 84, 12. 13 (AB) 2 36. 22, 36. 76 9. 06, 9. 19 (AB) 3 54.35, — 9. 06 Averages for ZnCh.. 1.592 9.6S

rc= Film refractive index. d= Total thickness. d= Average thickness per monolayer. .X=Number of slides. ^Surface pressure at time of transfer, dynes/cm. A indicates film transferred during downward motion of slide. B indicates transfer during emergence of slide from film tray. 728-101 O—64 21
316 BATEMAN

reflectors, a single value of refractive index was obtained by calculating

the linear regression lines of es on ep

. Individual values of d were'

then calculated for each experimental point.

A notable feature of the measurements on protein films is a marked

coupling between film thickness and refractive index, of such a nature

that, under similar conditions of deposition, rather wide variations of n and d are associated with highly reproducible values of the surface

density, y mg/m2

. For a protein film regarded as a mixture of solid protein and air, y is given by the Lorentz-Lorenz equation in the form n

2—1 7 1 rij—\ , 13y

7i

2 +2 d'p'nl+2

where p~1.3 [20] and n,~1.60 [26, 13] for solid protein, giving

1=0.38 (14)

d n

2 +2

The n—d coupling has been confirmed by using 25 sets of data in I which duplicate slides were coated with protein under various condi- I tions. For each pair of slides, with protein films nu dh yi} and nHf d2 , 72, the angle 6 given by

tan 0=5 (o.38 jjj=|) /8(yjd) (15)

is a useful measure of the degree of coupling; in absence of couplingthe average value will be zero, and with perfect coupling, 45°. The

value obtained for the 25 pairs of slides was 38.3°. ml

c. Other Measurements

Several substances were deposited upon stepped reflectors by con- densation in vacuo, and monolayers of one organic substance of high

bulk index of refraction were successfully transferred. The results of, these miscellaneous measurements are included in table 5. d. Summary and Discussions

I The evidence for the validity of the stepped reflector method rests! upon the accumulation of data that can be considered in connection

with auxiliary information that is not strictly comparable. For,, example (table 5): (a) Using calibration constants that are fairly close to values obtained by the s-fringe method, measurements of

the thickness per monolayer of several other aliphatic soap films are

in fair agreement with values anticipated on stereochemical grounds.,

(b) The corresponding indices of refraction are reasonable since (i)U low values for the odd carbon acid, nonadecanoic, are found by both

stepped reflector and s-fringe methods; (ii) some uncertainty as to the actual values arises from the presence of Ba and Cu, and from the
BIOLOGICAL MACROMOLECTJLES 317

Table 5. Summary of data on validity of s-p stepped reflector methods

Substance

Stepped s-p reflector Higher order

s-fringes Abeles method

Other values

n,p fit a rip d n bulk n film

Stearic acid BaH stearate

1. 537±0. 014

1. OoO±u. UZo 41.9±L3

41. 0±1. 4 1. 549±0. 006 1.508±0. 014

b 48. 84 (-41?)

1 1.51 1.503 46. 638 d 1. 4917

1. 001 BaCuH soaps pentadecanoate 1. 513 39. 7 • (41. 22) palmitate

stearate..

1 500 46 6 (43. 76)

(48. 84)

(41. 38)

* 1. 5225 1.489

f 48. 53 52.74

8(1.559) nonadecanoate

arachidate-.. Bovine

plasma albumin

Adsorbed h „

Monolayers k Botulinum

toxins Adsorbed,

type A

Adsorbed,

type B_-_

Fused ring pom nnnnH 0

\s\JLLl \J\J Li-Livl Monolayers

CaF2

1. 487 56.1

1. 530 53.8. (53. 92)

^>22 m 10. 0

» 25.5

1.589 29.0 i 1.6

1. 592 9 63

1. 588 24. 9

1. 526 30. 1

1. 676±0. 028

1. 362±0. 015

1. 393±0. 009 1.411±0. 014

1. 563±0. 020 1.51±0. 06 1.50±(0. 04)

1. 728±0. 014

1. 250±0. 022

1. 405±0. 006

1. 412±0. 006

1. 596±0. 006 1.98±0.02

p 1.7

« 1. 434

1. 364

» 1. 384

v 1. 572 "2.37

258 (±10) 106-197..

63, 246

' 1. 23-1.28

* 1. 30-1. 31 1.400±0.002

Cryolite, Na3 AlF6 - MgF2 CaCOs 150 ZnS 93~100

Symbols:

n,p =refractive index determined by stepped reflector method. Calibration with BaCuH stearate for all BaCuH compounds using w3 ==1.510, d3 =49.499 A. For remaining experiments, calibration with BaH stearate using w3 = L498, d3 = 47.575 A.

ra s =refractive index by s-fringe method

np =refractive index by Abeles method with directions of incidence and reflection parallel to long axis of film slide. jd=film thickness (evaporated films) d=thickness per double layer (fatty acids) or monolayer (proteins and ring compounds) Notes

» cf. ref [8], fig. 7.

t> Stearic acid unit cell: c=48. 84 A, c sin 0=43. 75 or 39. 75 [30].

0 [10]. d n0 and ne , respectively [10].

e Values in this column obtained by adding 2.54 A per carbon atom to extended chain length of stearic

acid. No allowance made for metal atoms. anglJ

alU6S obtained with cneckerDoard reflector, with equal numbers of double layers deposited at right

e n» for BaPbH stearate [10].

h Adsorbed from HC1 solution at pH 5. ^Molecule 22X49X106 A [4] if deposited in orientation similar to that in unit cell would form film about

22 A thick [6].

J Calculated for solid protein from refractive index increment of solutions, using Lorentz-Lorenz equation

[26, 13]

k Protein spread on dilute ZnCl2 . m Side chain spacing in extended /8-form [20]. n From sedimentation and diffusion measurements, axial ratio ~18, molecular weight ~106

, consistent with ellipsoid of revolution 460X25.5 A [32]. °8, 9, 10, 11—tetrahydro-benzanthracene; 8—malonic acid obtained from the laboratory of Dr. W. E. Bachmann through the Chemical Biological Coordination Center of the National Research Council. See

[5].P A|sumed from structural resemblance to other fused ring compounds of refractive index around 1-7.

1 [15].

r Range of values given for very thin films [18]

8 Value for 0 index [15].

* np determined by Abeles [2]. u Average of nQ , 1.378 and rh, 1.390 [15].

v Average of values given for calcite: 1.6583, /3, 1.4864. Value for CaO =1.838 [15]. w Average of 2.36 and 2.38 [15].
318 BATEMAN

fact that the ordinary and extraordinary indices are appreciably dif- ferent, (c) Protein adsorbed films and monolayers prepared under

optimum conditions appear to have refractive indices close to 1.60, i the value for dry protein in bulk, (d) The thicknesses of adsorbed

films of two different proteins are consistent with known molecular

dimensions, while transferred monolayers have the thickness of ex- tended polypeptide chains, (e) The refractive indices of several

substances deposited in vacuo, and of a fused ring organic compound

transferred in monolayer form, were in satisfactory agreement with i values obtained on much thicker films prepared by similar methods

and examined by the Abeles method. Agreement to be expected with

bulk values is limited by the fact that the substances are not always

isotropic, and by a possible decrease of refractive index in very thin

layers.

Two discrepancies exist.

(a) Zinc sulfide, of bulk index ~2, appeared to be much less dense i (n~1.5) in very thin layers, while the Abeles method gave an intermediate value. Zinc sulfide is used routinely as a highly refractive

material in preparing interference filters [31] and we know of no prece- dent for these low values.

(b) The apparent refractive index of stearic acid is unexpectedly

high (1.536) by the stepped reflector method, and still higher (1.549) by

the Abeles method. The first result has been explained [8] by calculating the s- and ^-indices as a function of angle of molecular tilt and showing that the value 1.537 is compatible with the values of np and ns expected when the molecules are tilted at an angle 30.3° to the slide normal. The still higher value obtained by the Abeles

method becomes reasonable when one remembers that the slide was

viewed parallel to the monolayer dipping direction at an angle of incidence tan-1 1.549 or 56.45°. The incident beam is nearly parallel

to the molecular chains and a value approaching the extraordinary

index, 1.55, is to be expected.

A final point of relevance to these tests is the coupling between n and d, which suggests, under carefully duplicated conditions, the

formation of films having the same mass but different thicknesses.

A result of this tendency, which has been noted with the aliphatic

acids as well as the proteins, is that the reproducibility is less than

that calculated for the known accuracy with which the fringe wavelengths can be measured. Mattuck [23] gives the following standard

deviations upon the latter basiso

: for films 12 A thick, <r n = ±0.054,

<rd =±lA A; for films 142.5 A thick, trn= ±0.010, <rtf =±3.2 A.

Larger standard deviations in n and d in replicate experiments, with

small deviations in y, apparently show that with extremely thin

films the orientation is far from perfect, although on ^ the average

the same number of molecules is always needed to

1 'saturate" a

receptor surface. This question needs further study, but for the

moment we shall assume that it does not point to any shortcomings

of the optical observations.
BIOLOGICAL MACROMOLECULES 319

6. Theory of Stepped Interference Reflector

6.1. Derivation of Fringe Shift Equations

Having presented evidence of the usefulness of the s-p stepped

reflector method, we now sketch the derivation of the fringe shift equations for those who are interested; the original derivations are spread over three papers [16, 22, 23] and a brief self-contained state- ment may be of value.

a. Stepped Reflector Without Test Film

We refer to the diagram of the stepped reflector given in figure 14,

with the interfaces represented by lower case letters and the substances forming them by numbers. The letters s and p indicate the

portion of the reflector under consideration, and the plane of polarization of the light impinging upon it.

Figure 14. Illustrating designation of films and interfaces in s-p stepped reflector. Medium 1 is air; 2, the test film; 3s, barium copper stearate of thickness suitable for production of s-extinc- tion fringe in absence of film 2; 3p, barium stearate of thickness suitable for production of p-extinction fringe

in absence of film 2; 4 is chromium or chromium-plated glass. tf> 8 and <f> P are angles of incidence appropriate

to the extinction conditions; in the theoretical development given here, <j> s = <t> P = <t> while in ref. [23J the

distinction is maintained. Reflection at a single interface separating two semi-infinite media is designated with a numerical suffix. Reflection at an interface within the actual system shown in the diagram is desig- nated by the number of the medium containing the incident beam, followed by the letter designating the

interface—e.g. ri a , the reflection coefficient for the whole reflector; ne , the reflection coefficient for a clean metal-coated glass slide in air. Considering first the case of a dielectric film, 3, between a metal

plate, 4, and air, 1, we repeat eq (11):

where

r13 -f r3c exp (—i83) ~l-\-rlzr3c exp (—id3) cos 0-W-sin2

<t>) 1/2 cos0+«~sin2 0)

1/2

(16)

(17)
320 BATEMAN _n\ cos cf>— (n2

3 —sin2

(f>) l/2

, , Tuv ~nl cos 0+ (Til-sin2

<*>) 1/2 K *}

53 =(47r/X)^(n23-sin2 0)

1/2 (19)

while rZc is given by equations similar to (17) and (18) using n4 =n4 —iki. (20)

For present purposes it is sufficient to write

r'z c =Pzc exp (i6Zc) (21)

giving

riB+Ps* exp *(*3C -y (22) l+n3 P3c exp i(03c —

At a wavelength X6 and a film thickness d'3 giving a phase change

5a at a suitable angle of incidence, the reflected amplitude will be

zero (extinction fringe) if rn=Pn exp ^180° =p3c (23)

and

ezc -h' =2irm (m=0, 1, 2 . . . )• (24)

The condition for (23) has already been illustrated in figure 3; it is found empirically to be satisfied when <f> is about 78° for the s-, and

81° for the ^-vibration in the case of barium stearate films on chromium-plated glass. The second condition is satisfied when the

s-layer has about 20 or 60 barium stearate double layers, while about

40 or 80 are required for the p-layer, because of the different value of

03c illustrated in figure 2.

b. Stepped Reflector With Test Film

If a film of substance 2, thickness d2 , is placed on the step reflector,

the reflected amplitudes will now be given by

rTla _

l+r^t

ri2+ r26 exp

exp

(—

(—

i8i8

2

2

)

) ^

where _

r23 +r3c exp (~^3) (26)

26 l+r23r3c exp (-ifi8) and 5 2? <5 3 are values appropriate to the wavelength of observation,

X, by equations similar to (19). By suitable manipulation the new \
BIOLOGICAL MACROMOLECTJLE

321

fringe condition can now be calculated by differentiating (25) with

| respect to wavelength.

The original fringe condition can be used to define r3c by equations

(16), (23), and (24), and applying this to any wavelength X by assum- ing no dispersion of the reflection coefficients: r3c =—

3 exp id'z (27)

r23~ exp i

(8'

3 —83) r26— ^8) 1—

23ri3 exp i(5'z —53 ) Substitution in (25) and introducing some abbreviations then leads

to ria_

a—b exp (—i3)+c[exp (—i£)— afr exp (—iip] ^

l-a6 exp (—i5)+c[a exp (—#) —& exp (—iij)] where

a=r12

c=r23 e=52 +53 —53 =£ +??

£ =52

ry =53 —

3

. Multiplying ri a by its complex conjugate we then obtain

» (a— 6 cos 6+c cos g—a6c cos t?)

2 +(6 sin e—c sin g+fl&c sin rj) 2 Tla ' Vla ~ (l—ab cos e+oc cos £— 6c cos r?) 2 +(a& sin e—oc sin £+6c sin t?) 2

(30)

(31)

where

A=a2 +62 +c2 +aW (32)

5=l+a26

2 +a2c

2 +62c

2 (33)

/=/(€) = —2a6[cos €+c2 JJ cos (£— 77)] +2c[a(l+62

) cos£-6(l+a2) cos n]. (34)

The variables £ and e are readily expressed in terms of e by introducing the phase shift 8'

2 due to the new film at the original fringe
322 BATEMAN

wavelength A6 and noting that

83/8'

z =82l8'2 . (35)

One finds

,

S'2 (€ +5'3 ) (36)

" =^+sT (37)

The fringe condition is now given by

rie =

(5+7?*^-° (38)

remembering that

de/dX^O.

Hence

dflde=0. (39)

Differentiation of (34), using (36) and (37), and applying the argument used in deriving eq (11) by placing

c =(6-a)/(l-a6) (40)

then gives

b-a fl+b2 8'2

. t ,l+a2 8'

3 sin e- . r —^— • 77-77 sin H —• ^7-77 sin y l—ab\_ b 83 -\-8 2 a o z -\-b 2 b-a 8'

s -8'2 _: _ j

=0. (41) 1—a& 62+5:

sin

It is now convenient to eliminate e, £, and 77, using the definitions; eq (29) , together with

X6/Aa =63/^=62/5; (42)

Ji where Xa is the new fringe wavelength. It is also convenient to desig- : nate the two unknowns x and y:

i x(n2 , d2 )^8'2 (43)

[I y(n2 ) =rl2 =a (44) I and to define the fringe shift A:A=Aa -X& . (45)
BIOLOGICAL MACROMOLECULES 323

Equation (41) then becomes

(8'3 +x) sin (ax-e)+f1 (5'3 -x) sin (-coz-0)

+f2x sin o>x+f38'

3 sin (-0)=O (46)

where

£0 =X&/Xa =l-A/Xo (47)

d=8'3 Al\a (48)

fi=[(b-y)Kl-by)}2 (49)

f2=-(b-y)(l +b

2)/b(l-by) (50)

f*=(b-y)(i+y2)ly(i-by). (51) 6.2. Approximations

a. Simple Approximate Solution to the Fringe Shift Equation

An exact analytical solution to eq (46) has not been obtained, but

expansion of the trigonometric functions and neglect of square and

higher powers of ux and 6 and their products leads to a quadratic in x that can be solved. The result is

3 ~^'l-b2 '~y~ (52)

and the approximation is equivalent to specifying that x must be

less than 6° and the film thickness less than 150 A. Also, by an

equation similar to (14),

x=47rd2/32/X6 (53)

where

/3!=7i|-sin2 0. (54)

Combining (52) and (53) one finds

where, it will be recalled, /3 2 and y are both functions of the refractive

index of the test substance, n2 ; A is measured experimentally; 5s is determined by calibration ; b can be calculated from the refractive index

of substance 3 and the angle of incidence.

Equation (55) applies to both s- and ^-vibrations, and it must now

be shown that after inserting the appropriate reflection coefficients

d2 can be eliminated from the two resulting equations and an explicit

solution for n2 obtained.
324 BATEMAN

Substituting (eqs (17) and (18);,

y,= (cos /3 2 )/(cos 4>+/32) (56)

yv =(n2 cos2 <j>— j8 2)/(ni cos <f>+/32) (57) one obtains

1 cos0 6, ,

1 7i 2 COS 0 6j

(58)

2 *,',A,. (59)

7r w| cos

2 0—wl+sin2 4> 1—bl

Neglecting n\ cos

2

<f> and placing sin2 <£~1,

n2 1-61' bp 8; p

'

p which gives, on substitution for b s and 6 P , eqs (17) and (18),

(60)

712 l + (l+w5) cos

2

</> «P AP ^ }

If n3 is 1.5 and 4> is 81° (values used in the most recent work),

j we find, calculating the term in n3 or placing it nearly equal to n\,

respectively,

r*2 ~1 AU(5'3AMp±p)1,2 ~1 .50(5LAS/^AP) 1/2 (62) | while, from (58)

4-0.088«A/(wJ-l). (63)

b. Calibration for Determination of 8Z The values of 8'

3 cannot be calculated reliably because the thickness

of the dielectric film 3 is not determined solely by the number of double

layers transferred; calibration, however, is a simple matter. This is done by measuring the fringe shifts produced by the last one or more

substrate double layers prior to deposition of the test film. Preparation of the stepped reflector is stopped when fringes can be

observed at some suitable wavelengths Xos , A op . By eq (19), and

abbreviating with

^(ni-sin^)1 '2 (64) we find

aJ=4*Wa/Xo- (65)

Upon transfer of TV additional double layers of thickness Nd to both

steps of the reflector, the fringe wavelength shifts to the higher value
BIOLOGICAL MACROMOLECULES 325

\b in order to keep 8'

3 constant

8'

z =Mdzo+Nd)faf\b (66)

=±wd3 p3/\b . (67)

Eliminating c40 and placing X6 -X0 =Ao (68)

H=4arNdpjAo. (69)

— O Typically, if n3 is 1.5 and d 50 A,

53 =70W/AQ (70)

where the fringe shift is in A and 83 in radians; 8'

3s is somewhat less than 180° and 8'3p somewhat less than 360° (again, compare fig. 2).

c. A More Refined Approximation

For a given value of the fringe shift A the exact eq (46) and the

simple analytical approximation (52) both represent curves x(y)

corresponding to pairs of values of n2 and d2 . For the special case when the test film has the same refractive index, 1.5, as the substrate

(i.e., y=b), the exact slope of this curve can be written in simple

form and compared with the approximate slope obtained from eq

(52) . The latter is then modified to conform to the equation for the

exact slope, and the modified equation (assumed also to be valid

for other values of n2) is integrated. The resulting improved approximation gives a curve x(y) which (a) coincides with the exact curve when n,2=1.5; (b) has the same slope as the exact curve when 7i 2 =1.5;

and (c) has the same form as eq (52) in the limit as x approaches zero. Differentiating (46) and setting y=b, one finds

/i=/s=/i=0 (71)

sin (cox- 0)~O (72) cos 0)~1 (73)

df3/dn2 = —df2/dn2 (74)

giving finally

dj2/dn2 = —df3/dn2 =^ • j-ip • (75)

'dx\ 1 1 +6

2 1 ,

. Q. (dy\
326 BATEMAN

The result of a similar calculation with eq (52) is the same as (76)

except for the replacement of sin (0) by its argument. Assuming (76) to be approximately correct for any n2 and integrating, we get

l>l=£.*ki+K (77) 1— b

2 y co and, finally, evaluating K by placing n2 equal to 1.5, X~- • ^-r2 sin 0+0-sin 6. (78)

co y 1 —tr As before, separate expressions for d2 are now obtained by substituting

for ys and yp

\ d2 =^~ (ft*

1 sin d+d-sin 0) (79)

where for the s-step we use Xas , 6„ and Fs , and for the p-step the

corresponding p-terms

*-AriS? (80)

w __K

^cos <f> (S1)

p l-b2p 'ntcos2 <f>-nt+sm2

<l> Eliminating d2 and proceeding without further approximations,

H-Ei+B • E*—E* (82) F~Bj* Ds where , v Z?=Afl sin0 (83)

#=Xa (0-sin 0) (84)

B=l/faFP

. (85)

Equation (82) is quite suitable for calculation of n2 from the meas-

| ured fringe shifts, since an approximate value can be obtained by i equating Dp/Ds to Fs/Fp , using this to calculate B (approximately) | and then applying the complete equation to obtain a second, more [ accurate value of n2

. The film thickness can then be obtained by | eq (79), which may be abbreviated:

4Trd2 =DF+E/(32

. (86)

6.3. Methods of Computation; Numerical Examples

The fact that the s- and ^-fringe shifts are sufficiently different to permit resolution of film thickness and refractive index is evident

j
I

BIOLOGICAL MACROMOLECULES 327

Figure 15. Plot of d2 versus n2 for % added double layer of barium stearate.

; Letters s and p indicate curves obtained from s-fringe and p-fringe equations. Each curve represents the

| solutions (22(712) that are compatible with a constant fringe shift, 52.5A for the s-step and 26.25 A for the pj step. (Adapted from ref. 23, fig. 2.)

I from the first crude approximation, eq (62), and is illustrated graphi-

| cally in figure 15. That the magnitudes of the fringe shifts are sufficient

I for experimental purposes is evident from the data already presented;

I some typical sets of values for quick reference are collected in table 6. The choice of the approximation to use will depend upon the

! accuracy desired and the time available for computation. For

! convenience we restate the several approximate equations for the

I refractive index and present examples of results obtained by the use

of each:

FJFP = (Dp/Ds) +B(EP -ES)/DS (87)

nl=nm/Fp)(es/ep) (88)

nl=bs (l-bp) es/bp (l-tf)ep (89)

nt=2.25es/eP . (90) In eqs (87) to (89),

Fs =[bJ(l-b*m cos */(l-nJ)] (91)

Fp = [bP/(l-b2p)][±n2

2 cos <j>/(ni cos

2 </>-n!+sin2 0)] (92)

Table 6. Approximate fringe shifts as a function of film thickness and refractive index

Calculated by eqs (62) and («3) using A, =420 and Ap =220 for calibration.

1.50 d(A)

23.7

95.0

A,59 236 AP 40.7

163

A.

105

(420) AP55

(220)

A.257

1026 AP77 300
328 BATEMAN

D=\a sin 0 (93)

E=\a (e—sm d) (94)

B=l/p2Fp (95)

e — o3Z\ — o3 {Aa —A6; (9o)

d=8sA/\a=e/\a (97)

b=rlz (eqs (17) and (18)).

The use of successive approximations with eq (87) has been outlined already. Equations (88) and (89) can be used similarly; (88)

is simply, in other notation, the result of combining (58) and (59)

without omitting n\ cos

2 0 and sin2

<£, and can be used after a preliminary value of n2 has been obtained by (89). Equation (90) is of

little value since the numerical factor in eq (89) can always be used.

Examples of calculated values of n2 for several values of As and Ap are collected in table 7. The most reliable values are those in the

final column, since Mattuck [22] has shown by direct comparison with

the exact eq (46) that eq (87) is reliable to 3 decimal places in n2

. If permanent tables of FJFP are available, some time is saved by using

(89) and (88) ; (90) or (89) alone give only a crude estimate.

Table 7. Film refractive indices calculated from fringe shifts by approximate

equations

nj A8 /A„ Eq (90) Eq (89) Eq (88) Eq(87)

1 2

59.0

105 420

257

40.7

55 220

77

1.4496

1. 0991 1.9091

3. 3377

5059

5105 5420 5257

5048

5055 5220 5077

1.2770

1. 4655

1. 4655

1. 9378

1. 3405

1. 5383

1. 5383

2. 0340

1. 2956

1. 4759

1. 4759

1. 9055

1. 2970

1. 4773

1.4785

1. 9120

1. 2967

1. 4772

1.4768

1. 9159

rca = 1.510 and <£=81° throughout.

68 /(l-6j) = -1.791102.

bPl(X-bl) = -0.722482.

7. Some Possible Sources of Error

7.1. Dispersion Effects

The dispersion of the metal base has been neglected in the development of the s-p stepped reflector. In this respect the choice of chromium appears to have been fortunate; moreover, it can be assumed
BIOLOGICAL MACROMOLECULES 329

Figure 16. Functions used in calculation of film thickness and refractive index from s-p stepped reflector data, using eq (87), defined in eqs (91), {92), and {95).

Calculated for films of refractive index m on dielectric of refractive index m equal to 1.51. Angle of in- cidence, 81°.

I 1 I

- /

III. 1 1

1 0.96 1 1.04 5400/X Figure 17. Extinction fringe amplitude profiles for p-vibration incident upon

metal slides coated with barium stearate.

Calculated from ea (23); values for complex Fresnel reflection coefficients r3e calculated from optical constants' bfeauatSns g ven by Born and Wolf [11], p. 624 ff. In absence of dispersion effects, fringes are S pLXucTvidStyo^ the extinction position^ giving a straight line ^jtott^m oppo«te qu^-

rants as in diagram. This is true for Cr and Ta which give fairly symmetrical frmges, but not for Au which

gives grossly distorted fringes. All extinctions normalized to occur at 5400 A. Abscissa, 5400/X. Ordinate,

pibp.

that if the optical constants of the metal vary in a regular manner

over the range of wavelengths used, the calibration procedure will minimize dispersion errors. When less regular variations occur, the

fringes may become quite unsymmetrical, as shown in the case ot gold in figure 17; the effect of this on the fringe shifts has not yet been

calculated. Ideally, the angle of incidence for extinction fringes

should be independent of wavelength; the extent of actual deviatioan

from this is illustrated in figure 18.
330 BATEMAN

4000 5000 6000 AU A

Figure 18. Reflection coefficients as a function of wavelength. The diagrams show that as a result of dispersion the first extinction fringe condition (ri3=p3c, eq (23)) may be dependent upon wavelength. The assumption that this is not so, used in the theory of the stepped

reflector, is very nearly true for tantalum over the range 5000 to 6000 A, and for chromium between 5000 and

5400, but not for gold at any wavelength between 4000 and 5800 A. Horizontal lines give ru2 (=p\32

) for the air-barium stearate interface; closed circles give fac assumed equal to phi, for the stearai e-metal interface. Optical constants of the metals from International Critical Tables and Landolt-Boernstein "Tabellen."

d n'=f[n,<p]

Figure 19. Film sections illustrating departures from uniformity of thickness and

refractive index.

Second from top, homogeneous film with irregular surface. Third, film with random pores. Fourth,

film density decreasing with distance from substrate. (Reproduced from [6].)

7.2. Anisotropy of Dielectric Substrate and Test Film

The theory has been developed for isotropic films, but it is certain

that the refractive index of barium stearate, at an angle of incidence

81°, for the ^-vibration is about 0.18 higher than for the .s-vibration. Mattuck [23] has given approximate fringe shift equations appropriate

to this difference and has concluded that errors arising from birefringence will usually be 1 percent or less. The effects are nevertheless

detectable, and some observations on aliphatic acid films already

discussed in section 5 have been analyzed in terms of molecular

orientation [8].
BIOLOGICAL MACROMOLECULES 331

7.3. Film Heterogeneity

It has been pointed out [6] that the theory envisages plane-parallel

homogeneous films with sharp boundaries. However, the existence

of (a) skeleton films of stearate [10], (b) of incomplete or mosaic-like

films of protein [6], changes in optical properties with thickness [18],

| and the n-d coupling already discussed, all suggest that various types

1 of irregularity may be encountered (fig. 19) and that it may be diffi- cult in practice to detect these or to characterize them. Chemical

non-uniformity may exist in the stepped reflector itself; Schaefer [29]

has observed periodic variations in the surface properties of BaH

stearate multilayers, and it has been noted [8] that with respect to

I certain polycylic organic compounds the s- and p-surfaces behave

quite differently, so that optical measurements on these substances were impossible.

8. Conclusion

The outcome of these investigations has been a method that, within

certain clearly defined limits, should be very serviceable for the study

of dielectric films of thickness greater than a few angstroms, and of index of refraction between 1.1 and 2.0. Certain technical improvements and certain other developments are of course to be expected.

The former should include development of stepped reflectors with

isotropic dielectric layers, preferably with zero and first-order steps on the same slide as an aid to the detection of differences of surface

reactivity with changing thickness.

For the biologist two matters will be of special interest. The first

is the need, in many types of investigation, to characterize thin test films placed upon the stepped reflector and then to observe their

ability to take up additional material by adsorption or chemical

attachment. The determination of the optical constants of a second

or a third test film requires a modification of the fringe equations

that has not yet been attempted, although the basic theory of multiple layers is available [18]. The second matter of biological interest is the possibility of characterizing films at the liquid-solid interface. This presents special

difficulties because the extinction fringe condition (r13 =r3c ) occurs at rather higher angles of incidence than in air: <£s ~88°, 0P ~83° for the system water-barium stearate-chromium. A solution to the problem exists in principle; if the reflector is composed of a dielectric

material (medium 4 in fig. 15) of refractive index equal to that of water, ru is equal to r34 at all angles of incidence, so that there should

be no difficulty in obtaining extinction fringes with suitable thick- nesses of barium stearate. In practice, the only solid substance said

to have a sufficiently low refractive index, polytetrafluoroethylene,

also possesses undesirable optical and chemical properties. In all these matters, it need hardly be said, the biologist depends

upon the work of the physicists, and we have made this contribution

to the Symposium on the Ellipsometer and Its Use in the Measurement

728-101 0—64 22
332 BATEMAN

of Surfaces and Thin Films in the hope that the solutions to some of our problems may be present, or implicit, in the work that others will be describing on this interesting occasion.

In preparing this paper, the author has made use of the published

work and the notebooks of his former colleagues R. E. Hartman and

R. D. Mattuck. He also acknowledges with thanks the cooperation

of E. D. Adams, T. T. Bannister, A. Calio, E. J. Covington, Roberta S. Hartman, K. Larson, R. L. Parker, R. D. Petti, and A. Rosen, who

in varying degrees have been responsible for much of the experimental

work, and that of H. R. Peplow in some of the calculations. The

standard tables for the stepped reflector were calculated by the Biomathematics Division, Fort Detrick; the friendly help of Mr. V. S. Palmer and Mi. G. I. Eccles is acknowledged.

9. References

[11 F Abeles, Sur la determination des indices et des epaisseurs des couches

minces, Compt. rend. Acad. Sci. (Paris) 228, 553-555 (1949^.

[21 F Abeles, La determination de l'indice et de l'epaisseur des couches minces

transparentes, J. Phvs. Rad. 11, 310-314 (1950).

[3] F. Abeles, Methods for determining optical parameters of thin films, Progr.

in Optics, 2, 249-288 (1963).

[4] J. W. Anderegg, W. W. Beeman, S. Shulman, and P. Kaesberg, An investi- gation of the size, shape and hydration of serum albumin by small-angle X-rav scattering, J. Am. Chem. Soc. 77, 2927-2937 (1955).

[5] W.E.Bachmann, The synthesis of compounds related to 1,2-benzantnracene

and cholanthrene, J. Org. Chem. 3, 434-447 (1938).

[61 J. B. Bateman, E. D. Adams, The thickness and refractive index of plasma

albumin films on stearate, J. Phys. Chem. 61, 1039-1049 (1957) . [71 J. B. Bateman, Continuous and disperse f lms of bovine plasma albumin on

stearate, Proc. first nat. Biophysics Conf., 298-313 Yale Univ. Press, New Haven, Conn. (1959).

[8] J. B. Bateman, E. J. Covington, Molecular tilt in fatty acid multilayers,

J. Coll. Sci. 16, 531-548 (1961).

[9] K. B. Blodgett, Properties of built-up films of barium stearate, J. Fnys,

Chem. 41, 975-984 (1937). . , . [10] K B. Blodgett, I. Langmuir, Built-up films of barium stearate and their

optical properties, Phys. Rev. 51, 964-982 (1937)

[11] M. Born, E. Wolf, Principles of Optics, Pergamon Press, New York, JN.Y., 1959

[12] H. J. Deuel, The Lipids. Vol. I. Chemistry. Interscience Publishers,

Inc., New York, p. 103 ff. (1961). m ^ ^ . . [13] P. Doty, and E. P. Geiduschek, Optical Properties of Proteins, m the

Proteins, ed. H. Neurath and K. Bailey, Vol. I, Part A. Academic Press, New York (1953). _

[14] P. Drude, Ueber Oberflaechenschichten. II. Theil. Ann. Physik u. Chem.,

Neue Folge 36, 865-897 (1889). . . [15] Handbook of Chemistry and Physics, 1961-2, 43d ed., Chemical Rubber

Publishing Co., Cleveland, Ohio. . [16] R. E. Hartman, The fringes of equal reflection coefficient ratio and their application to the determination of the thickness and refractive index oi monomolecular films. I. Theory. J. Opt. Soc. Arm 44, 192-196 (1954).

[17] R. E. Hartman, R. S. Hartman, K. Larson, and J. B. Bateman, lhe fringes

of equal reflection coefficient ratio and their application to the determma-
BIOLOGICAL MACROMOLECULES 333

ation of the thickness and refractive index of monomolecular films. II. The determination of the thickness and refractive index of barium stearate double layers. J. Opt. Soc. Am. 44, 197-198 (1954) . . [18] O S. Heavens, Optical Properties of Thin Solid Films, Academic Press, New York (1955). . Tr , _ a

- [19] I. Langmuir, The Collected Works of Irving Langmuir, Vol. 9: Surface Phenomena. Pergamon Press, New York, N.Y. (1961).

[201 B W. Low, The Structure and Configuration of Amino Acids Peptides and

Proteins, in The Proteins, ed. H. Neurath and K. Bailey, Vol. 1, Part A. Academic Press, New York (1953). . [21] F \. Lucy, Studies of surface film by reflection of polarized light. L. A

more rigorous algebraic method for computing results, J. Chem. Phys. lb,

167-174, (1948). ,

, , ,. [221 R D. Mattuck, Recent attempts to determine the thickness and refractive index of unimolecular films, J. Opt. Soc. Am. 46, 615-620 (1956).

[23] R D. Mattuck, A stepped interference reflector for determining the optical constants of non-absorbing unimolecular films. I. Theory, J. Opt. hoc. Am. 46, 621-628 (1956). . [241 R D Mattuck, R. D. Petti, J. B. Bateman, A stepped interference reflector for determining the optical constants of non-absorbing unimolecular films.

films. II. Experimental, J. Opt. Soc. Am. 46, 782-789 (1956)

[251 F. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg Measurement of the thickness and refractive index of yery thin films and

the optical properties of surfaces by ellipsometry, J. Res. JNtfb t>7A, 363-377 (1963). , , o„n

[261 P Putzeys, and J. Brosteaux, L'indice de refraction des proteides, Bull. Soc. Chim. Biol. 18, 1681-1703 (1936).

[27] E K Rideal, Discussion on surface phenomena, Proc. Roy. boc, .London A155, 684-690 (1936). ^n fi71 fi79nQ„,

[281 E. K. Rideal, Surface action m biology, Nature 140, 671-672 (1937)

[29] V J. Schaefer, Studies of surface properties by the light scattering of deposited

liquid films, J. Phys. Chem. 45, 681-701 (1941).

[30] T. Schoon, Polvmorphe Formen kristalliner Kohlenstoffverbindungen mit

langen gestreckten Ketten (Nach Strukturuntersuchungen durch Elektron- enbeugung), Z. phys. Chem. 39B, 385- 410 (1938); Strukturber. 6, 238-

240 (1938). .. , fll t [31] A F Turner, Some current developments in multilayer optical nims. Phys. Rad. 11, 444—460 (1950). . . [32] J Wagman, and J. B. Bateman, The behavior of the botulmus toxins m the

ultracentrifuge. Arch. Biochem. Biophys. 31, 424-430 (1951). Discussion

A. Baidins (E. I. du Pont de Nemours and Co.):

Generally it is assumed that large molecules are adsorbed vertically

and weakly on surfaces of low energy and horizontally on surfaces ol high energy. As barium stearate has a very low surface energy, I wonder why all the protein molecules are adsorbed horizontally.

J. B. Bateman:

The protein molecule can be considered to be an assembly of polar

and nonpolar groups covalently linked. In an aqueous environment,

the molecule assumes a configuration in which the ambient water has

the greatest possible opportunity, consistent with stereochemical

limitations, to interact with the polar groups, while the nonpolar

groups participate in internal hydrophobic interactions. In presence

of a hydrocarbon-water interface, however, it is thermodynamically

advantageous for the nonpolar groups to form hydrophobic associa-
334 BATEMAN

tions with the hydrocarbon since this decreases the area of the inter- face occupied by water molecules. This explanation, of course, detracts somewhat from the significance of the apparent correlation

between film thickness and the small molecular dimension, since

emergence of the previously buried nonpolar groups is likely to be

accompanied by a certain amount of distortion of the "native"

molecule.
Blood Coagulation Studies With the Recording

Ellipsometer

L. Vroman

Veterans Administration Hospital, Brooklyn, N.Y. An ellipsometer was used which records the difference between outputs

of one phototube receiving light passed by the polarizer only and one

at the end of the light path. Polished silicon crystal slices and

anodized tantalum sputtered glass slides were used as reflecting surfaces, with and without treatment to affect wettability. Out of solution, blood plasma was shown to be adsorbed, and then, only if containing both intact clotting factors XI and XII, partially desorbed

again, in agreement with current coagulation theories. Other series demonstrated the adsorption and polymerization of fibrinogen by

films of thrombin. Though macroscopically visible fibrin formed on

the slides, instrument recordings and minimum settings revealed that only a film with a thickness equivalent to less than 400 A of barium

stearate was adsorbed, indicating that polymerization took place only

at a fraction of the interface.

1. Introduction

The coagulation of blood is generally regarded as a chain of reactions among several proteins; it is thought to be initiated at

glass or other wettable solid surfaces by adsorption of one specific

protein, named factor XII. This adsorbed factor in turn apparently

adsorbs factor XI, and the resulting ''activation product' 7 of the

two proteins may stay at the interface [1]

1 or be released [2]. The

problem appears to cry out for ellipsometric observations, but no

studies have thus far been reported in which physical measurement

of the adsorbed layer and its release was attempted.

The ellipsometrist should also be attracted by the last phase of- coagulation, in which the enzyme thrombin removes peptide groups

from fibrinogen molecules, thus transforming them into fibrinogen monomers ; these will then polymerize to form macroscopically visible

fibrin. Like other proteins, those involved in coagulation must contain

amino acid residues with varying degree of polarity ; after adsorption

of a protein from bulk solution onto predominantly polar or apolar

solid surfaces, and its more apolar or polar groups are exposed to reactants in solution, further reactivity of the film may indicate

which type of residues is most involved [3]. Specific coagulation

tests using hydrophobic and hydrophilic powders as adsorbents, as

well as other approaches including ellipsometry to some extent, have

suggested that certain coagulation factors are more active at hydrophilic interfaces, and that exposure of hydrophobic bonds may be

involved in coagulation [4, 5, 6]. We therefore decided to obtain

1 Figures in brackets indicate the literature references on p. 347. 335
336 VROMAN

ellipsometric recordings of surface "activation" and of the thrombinfibrinogen reaction at both hydrophilic and hydrophobic interfaces.

2. Technique

2.1. Instrument

The elements of our recording ellipsometer (Rudolph & Sons,

Inc., Caldwell, N.J. ; model #436-200E) are arranged as follows:

Mercury light source; filter passing the 5461 A line; polarizer (reading

to 0.1°); beamsplitter passing adjustable amount of light to photomultiplier tube a (IP 21) ; specimen holder (see below) ; quarter-wave

plate; analyzer (both reading to 0.01°); plate allowing selection

between a telescope and photomultiplier tube b. The specimen

holder consists of a clamp allowing a slide to be moved vertically

along a scale and into a strainfree glass cuvette; a 300-rpm motor

for stirring; and a base on which the assembly can rotate or slide in

the horizontal plane for adjustment.

The signals from tubes a and b are amplified and their difference serves as input for a voltmeter (Photovolt #520SP) and a recorder

(Honeywell #KSCl53xl7; recording speed % in. per minute).

2.2. Reflecting Surfaces

(a) Glass slides, vacuum sputtered with tantalum (kindly pro- vided by Dr. B. H. Vromen, Bell Tel. Labs., Inc., Murray Hill, N.J.) were anodized in 0.01 percent nitric acid at room temperature until

appearing violet (zero order) when viewed in air with daylight at about 45° incidence. The slides were washed, air-dried, and returned

to the anodizing bath with only about %& in. submerged wnile the potential was increased less than 1 V, to create an oxide "step" near the

end of the slide. This step served for calibration of the recorder (described below). Before use, the slides were cleaned with a detergent

(Sparkleen, Fisher Sc., N.Y.), rinsed with tap and distilled water, air- dried, and passed five times over the blue section of a Bunsen flame.

They were quite wettable and are indicated in this text as TaW. To

render them hydrophobic, they were covered with a solution of lignoceric acid in benzene, dried, and then polished vigorously, or dipped

in and out of a compressed film of barium lignocerate on a Langmuir

tray. These nonwettable slides are indicated as TaN.

(b) Silicon crystal slices with one surface polished optically flat (kindly provided by Dr. R. J. Archer, Bell Tel. Labs., Murray Hill,

N.J.) were cleaned as follows: washed three times in boiling benzene,

placed in concentrated nitric acid which was then brought to near

boiling, rinsed with large amounts of distilled water, dipped for 15 to 20

sees in concentrated HF, and again rinsed in distilled water. These

hydrophobic surfaces are indicated below as SiN. They could be

rendered wettable by again heating in concentrated nitric acid,

followed by rinsing with large amounts of distilled water and air drying. Such slides are indicated as SiW.
BLOOD COAGULATION STUDIES 337

(c) Other surfaces, which yielded curves of desired sensitivity but

are not reported here, were titanium-coated glass slides, with and

without oxide film, and anodized tantalum slides treated with ferric or barium stearate. The latter was avoided as a substrate in this

study because of its possible removal by plasma albumin under our

conditions [7]. 2.3. Optimal Conditions

With the polarizer at 135° and our slides submerged in a mixture

of 1 part veronal buffer (pH 7.4) to 4 parts of 0.85 percent NaCl at room temperature (22 to 26 C C), greatest sensitivity was obtained at 75° incidence. All observations reported below were made under

these conditions. Optimal settings for recording were sought by

plotting quarter-wave plate and analyzer positions against recorder

readings using first a blank surface and then one on which various

proteins had been adsorbed (figs. 1 and 2). As a compromise between

138 136 134 132 130 ANALYZER

Figube 1. Analyzer and quarter wave plate readings plotted against recorder output; values of the latter beyond 100 obtained by zero adjustment of instru- ment.

Blank «hape- blank TaW slide. Flowered shape: same surface after completing adsorption from 0.067% Thrombin Topical 2 crosses on X/Y plane indicate minimum light settings for blank and coated slide.
338 VROMAN

maximum sensitivity as well as linearity, and ability of the beamsplitter to balance a large light output, a position of the analyzer 6°

| above or below minimum setting was decided upon; the quarter-wave i plate was maintained at minimum position, and a 1-mra aperture was

used to collimate the reflected light.

ANALYZER Figure 2. Construction as of figure 1. Checkered shape: blank SiW surface. Flowered shape: same surface after completing adsorption from

1.3% serum. 2.4. Calibration

Barium stearate layers, compressed by 23.5 dynes, were prepared on a modified Langmuir trough [8], and deposited as 1, 3, and 5 monomolecular layers on clean TaW and SiW, and as 2 and 4 layers

onto TaN and SiN slides. Ta slides were dipped at right angles to their extra oxide steps, so that the effects of stearate and oxide steps,

upon the minimum transmitted light position of the analyzer could

be compared. The stearate layers were then removed and the oxide

step could serve as a permanent calibrated property of the slide,
BLOOD COAGULATION STUDIES 339

allowing adjustment of the recorder span at the onset of each experiment as described below. Since the Si slices could not be provided

with permanent steps, the relationship between blank surface and

stearate layer was noted by its effect on minimum transmitted light

position of the analyzer as well as by the recorded change in light

output when the slide was shifted from stearate to blank under ex- perimental conditions. Both the specific anodized Ta slide used in

the present experiments, and the Si slideo

s, happened to demand a change of 0.1° in analyzer position per 3 A of barium stearate, when

observed under the conditions given above. However, we would be

disregarding the large differences between refractive indices of tanta- lum oxide, barium stearate, and protein if we listed the recorded light

and analyzer changes as changes in actual film thickness. Instead, we will use the oterm basta to indicate Ba Stearate Angstrom, being

the number of Angstroms of compressed barium stearate film which

would cause the same change in A and \p being the analyzer azi- muth), and the same recorder pen travel under the same recording

conditions.

2.5. Experiments

Actual tests were run as follows: the slide was lowered in buffer

mixture as described and positioned for light incident at 75°; minimum position of analyzer and quarter-wave plate were determined;

the analyzer was turned 6° up or down; the beamsplitter filter was

adjusted to bring the meter back to 0; the recorder was actuated;

when Ta oxide surface was used, the slide was moved up briefly,

causing the recorder to deflect so that its span could be set to the

desired value (80% of total scale for an oxide step of 24 basta) ; if an

Si slice was used, the recorder span was adjusted to give a certain

deviation when the analyzer was moved from 6° to 7° off minimal

position (1° giving 80% of total scale). The recorder was run for several minutes with the blank slide in buffer mixture; then the

sample was added. When the recording indicated that adsorption

or desorption had ceased, the instrument was briefly switched from

recorder to meter, and minimum positions of quarter-wave plate and

analyzer were determined. The values obtained, as well as our

ability to record more or less symmetrical recordings when repeating

experiments with the analyzer at symmetrical deviations from minimum, indicated that the direction of recorder deviations represented

that of actual optical thickness changes at the film and not turbidity

or other phenomena that would affect light output.

3. Results

3.1. Surface Activation

If certain presumptions quoted in our introduction [1, 2] are correct,

it must be possible to show a difference between adsorption of intact

and of glass-activated plasma. For example, if intact factor XII,
340 VROMAN

Table 1. Change in compensating analyzer position, and its barium stearate equivalent in angstroms, after adsorption and desorption of intact plasma

Degree Equivalent

Slide Sample change ± Std. of com- N*

in ana- dev. pressed

lyzer BaStea A

TaW Intact normal plasma. . - . . . _ 1.90 ± 0. 25 57 13 TaW Spontaneous desorption of intact normal plasma. __ .44 ± .27 13 13 TaW Intact Ba stearate treated plasma.

_

1.91 ± .18 57 11 SiW 1. 13 ± .07 34 4 SiW Spontaneous desorption of intact normal plasma

.16 ± .04 5 4

> jV~=number of experiments.

MINUTES

Figure 3. Superimposed photocopies of recordings. Curve A: adsorption and spontaneous desorption of intact plasma. B: adsorption of activated plasma. C : adsorption of barium stearate treated plasma. once adsorbed, will in turn adsorb intact factor XI, after which the

XI-XII product is released, the adsorption of intact plasma at the

proper concentration will come to an optimum, and then show a decrease, while no decrease will be shown by plasma which has been

activated. This was indeed found (table 1 ; fig. 3, curves A & B)

when 1 ml of citrated, intact plasma was added to the 60 ml buffer

mixture in the cuvette, adsorption onto TaW and onto Si slides started at a rate of much more than 10 basta per minute but decrrased rapidly and appeared to cease after about 2 to 6 min, when

about 40 to 60 basta had been adsorbed; this was followed by slow
BLOOD COAGULATION STUDIES 341

desorption of about 5 to 15 basta (fig. 3, curve A). Adsorption of plasma which had been shaken gently with 1 part of quartz powder

(3 to 5 n, Engelhard, Newark, N.J.) or Speedex (Great Lakes Carbon

Corp Los Angeles, Calif.) for 10 min and then centnfuged, was not

followed by desorption (fig. 3, curve B). Specific coagulation tests, to be published elsewhere, indicated that the quartz or Speedex

treatment had indeed activated our plasma. Presumably, the "activation product" of factors XII and XI had formed and was at least partially released by the powder; during subsequent ellipsometry,

either no intact factor XII was left available for adsorption, or no

intact factor XI was left to remove adsorbed factor XII.

It appeared likely that desorption began before adsorption ol the

intact plasma was completed; separation of the two processes in time

to reveal the full extent of each would be possible if plasma could be

prepared which contained intact factor XII but no factor XI activity.

Treatment of normal intact plasma with barium stearate powder

(neutral, Witco Chem. Co., N.Y.) did yield such a preparation [9] Fifty milligrams of the powder per milliliter of plasma was rubbed

into suspension at room temperature, and removed by high-speed

centrifugation after 20 min. Such plasma showed adsorption onto

our slides; no spontaneous desorption followed (fig. 3, curve C).

About 10 min after adsorption had ceased, 1 ml untreated intact

plasma (therefore containing intact factor XI as well as XII), was

m

h i

f - -

1* 1

20 3.0 MINUTES

Figure 4. Superimposed photocopies of recordings.

Effect of the following upon films of barium stearate treated intact plasma ?

UT^rP^^t^lSlf

by whole intact plasma. E: removal of film by "activated" congemtally factor XII deficient plasma.

F: no removal by whole activated plasma.
342 VROMAN

added to the cuvette; removal of some of the substrate followed (fig. 4, curve D). A substrate which lacked intact factor XII, such as a

film deposited by barium stearate-treated serum instead of plasma,

or by barium stearate-treated activated plasma, could not be removed 1 by untreated intact plasma. This would indicate that intact factor

XI does not remove substrate unless it contains intact factor XII.

A substrate which presumably did contain intact factor XII (being

deposited by barium stearate treated intact plasma) could not be

removed by plasma which had been activated with Speedex or quartz

(fig. 4, curve F) . All factor XI in this plasma may have been changed

during its reaction with factor XII by the activating Speedex treat- ment. However, when congenitally factor XII deficient plasma was

treated with Speedex, it was still able to remove adsorbate left by i barium stearate-treated intact plasma (fig. 4, curve E). This finding

fits the theory that in absence of factor XII, factor XI cannot be

activated [1, 2]. On TaN slides, adsorption of intact plasma was only

rarely followed by measurable desorption, but results were erratic

and better temperature control for this substrate may be required.

rOO

Dip

MINUTES

Figure 5. Adsorption of Thrombin Topical; recorder tracings. Curve 1: TaW slide, 1 ml 0.4% thrombin added to cuvette at 0 minutes. Dipping causes second adsorption curve. Curve 2: TaN sliae, same treatment. Dipping causes instantaneous increment. Curve 3: as curve 2, but liquid in cuvette covered with plastic film. Instantaneous increment prevented. Curve

4: 0.2 ml amounts of thrombin added at zero, 14 and 18 to 20 minutes. Upon dipping, new adsorption slope

reflects total thrombin concentration. (Reproduced from [6].)
BLOOD COAGULATION STUDIES 343

3.2. Thrombin- Fibrinogen Reaction

(a) Adsorption of Thrombin

Two preparations of thrombin were used: Thrombin topical

(bovine, approximately 5,000 N.I.H. units per 200 mg, Parke Davis &

Co., Detroit, Mich.), a rather crude preparation, and resin thrombin

(7,100 U/ml, 25,400 U/mg tyrosine), kindly supplied by Dr. W. H.

Seegers, Detroit, Mich. One milliliter of a 0.4 percent solution of the

former in saline or 0.1 ml of the latter were added per 60 ml buffer

mixture. The crude preparation yielded rather convex adsorption

curves (fig. 5, curves 1-3), while the purified (resin thrombin) sample

showed quite linear adsorption rates (fig. 6). Into both preparations,

TaW and TaN slides could be dipped deeper after adsorption at the

observed site appeared complete; thus, whereas the first observation was made on the slide placed into buffer mixture before protein solution was added, the second observation showed adsorption upon a

section of the slide submerged more than 20 min after protein solution

had been admixed.

20

MINUTES

Figure 6. Adsorption of Resin thrombin; recorder tracings.

Curves 1 and 2: adsorption onto TaW; curves 3 and 4: adsorption onto TaN. At "observe", instrument was switched temporarily to observe by telescope, or to read analyzer and quarter wave plate angles at minimum light transmission. (Reproduced from [6].)
344 VROMAN

This technique revealed that adsorption upon sections of TaW

slides submerged after adding the thrombin proceeded quite similarly

to adsorption upon sections which had been submerged into the buffer

mixture before thrombin was added ; but when TaN slides were dipped

deeper in the protein solution, they appeared to have become coated

immediately with a layer of about 20 to 35 basta (fig. 5, curve 2 ; fig. 6, curves 3 and 4), upon which only a small additional amount was

adsorbed. By floating a fitted piece of polyethylene film on the thrombin solution in the cell and dipping the TaN slide through a narrow

slot in this plastic film, "instantaneous adsorption" could be pre- vented, and the initial adsorption curve repeated itself completely

after dipping (fig. 5, curve 3). These findings suggest that a film, i presumably of protein, had formed at the air/liquid interface; it was | dragged down by the hydrophobic slides but not by the hydrophilic

ones. The progressive dipping technique also allowed a comparison between adsorption rates at various concentrations in the same cuvette. For example, repeated addition of 0.2 ml amounts of thrombin yielded

a lower rate of adsorption onto a TaW slide; dipping the slide deeper

showed an adsorption rate corresponding to the total concentration of added thrombin (fig. 5, curve 4). Under these conditions 45 to 70 basta of thrombin were adsorbed on TaW and TaN slides in 30 min, and 30 to 40 basta onto SiW and

SiN slides. In one experiment, a TaW slide was dipped into 400

mg/100 ml instead of 6.7 mg/100 ml crude thrombin, and a total of 54 basta was adsorbed rapidly.

(b) Adsorption of Fibrinogen Onto Thrombin

As a source of fibrinogen, either a fresh, centrifuged, and decanted

0.3 percent solution of human fraction I (Mann Res. Labs., N.Y.)

in 0.85 percent NaCl, or normal ACD plasma was used as follows. Slides onto which thrombin had been adsorbed to equilibrium, were

rinsed with buffer mixture and, while wet, submerged into a clean cuvette of buffer mixture; the stirrer too was cleaned before reuse. Recording and minimum quarter-wave plate and analyzer readings

rarely revealed significant thickness changes of thrombin as a result of

this treatment; if these were found, the experiment was repeated.

When a thrombin coated slide had shown no change for several minutes, 1 ml of fraction I or 0.2 ml of plasma was added to the cuvette. In a few experiments, adsorption of concentrated fibrinogen onto

thrombin was observed by dipping thrombin coated slides into undiluted plasma or into a 300 mg/100 ml solution of fraction I. From

the dilute solutions, all types of thrombin-coated slides except TaN

adsorbed about 50 basta in 50 min (fig. 7), and from the concentrated

samples, about 150 to 250 basta. In all instances however, after about 40 min of contact between thrombin film and fibrinogen or plasma solution, the TaW, SiW, and SiN slides began to grow grossly

visible beards of fibrin, waving in the stirred solution and causing

the light output to oscillate and to decrease; yet A and yp did not con-
BLOOD COAGULATION STUDIES 345

rO

10 20

_j i 1 1 1 1— MINUTES

Figure 7. Adsorption of plasma onto Resin thrombin; recorder tracings.

Curves 1 to 4 are continuations of the correspondingly numbered curves in Fig. 6. Curves 1, 2 and 5: pl?sma

on thrombin on TaW slides. Curves 3 and 4: plasma on thrombin on TaN slides. Cl ae.ac: nonochloroacetic acid V/% 1 ml added at arrow. Note absence of oscillations from then on. (Reproduced from [6].)

tinue changing beyond the indication that the equivalent of a mono

molecular layer of fibrinogen had been deposited upon the thrombin [6]. When thrombin had been deposited upon TaN slides, it appeared

able to attract the usual amount of fibrinogen in a fibrinogen solution

and cause it to form fibrin, but out of whole plasma such thrombin

films could only adsorb about 20 basta and did not form visible fibrin.

4. Evaluation of Results

4.1. Optical Versus Actual Thickness of Films

Some of the .preceding papers in this volume have dealt thoroughly

with the problem of obtaining both thickness and refractive index by

ellipsometer measurements. By merely noting A and \p values on

our recordings where an obtained equilibrium allowed time for such

observations, we have delayed our encounter with this problem until

computer programs such as described here by McCrackin and Colson

[11], Archer [12], and others may be available to us; we are not re-
346 VROMAN

I

i porting these values now. We did gain the impression that the most \ delicate change recorded, namely the "desorption" of intact plasma, j) was accompanied by analyzer (see table 1) and quarter-wave plate

setting changes which were too small to show statistically significant mutual relationships; from these we must predict that our [ measurements could not have been accurate enough to tell whether

thickness or refractive index was affected most during desorption. ? In our use of barium stearate steps for the calibration of slides,

too, we confined ourselves to the values of compensating analyzer

j position and recorded light output, without referring to quarter-wave

plate settings. We may therefore not be allowed to equate basta

j units on Ta with those on Si slides.

4.2. Validity and Meaning of Results

In spite of the restrictions given above, correlation between re- s cording and analyzer values indicates that true optical thickness \ changes were observed though not accurately measured. If so, the

finding that only a substrate containing intact factor XII can be

removed in part only by a solution containing intact factor XI, , agrees alarmingly well with the current opinion of coagulationists,

that intact factor XI must form a complex with adsorbed intact

j factor XII. Release of the complex (activation product) then may

leave holes in the film that can be refilled by more intact factor XII,

and the process may continue—as some preliminary ellipsometry

studies are indicating—until the supply of intact factors XI and/or

XII is exhausted.

Plasma or serum treated with barium stearate powder was found ( to have solubilized a significant amount of fatty acid. It was con- | ceivable that such plasma would carry this contaminant to the inter- I face, and that subsequently added intact plasma would remove this stearic acid. The various control tests, including use of stearate

| treated serum instead of plasma as a substrate, or of albumin instead

of intact plasma to remove the adsorbate with, and of various states

g of activation, all yielded results pointing away from implication of such a nonspecific interaction. Yet, further studies, preferably

with purified intact factor XII, may yield further information, such

as its need for orientation at the interface. Our intact plasma

behaved similarly on SiW and on SiN; this may have been due to

| its rather high dilution, allowing factor XII to spread upon the sur-

[j face with all its amino acid residues available, whether resting prone ! or supine. Studies with higher plasma concentrations, as well as

j with metal interfaces (allowing application of electrostatic charges)

are in progress. We are hoping that others will develop some rapid 1 technique for actual thickness measurement; we shall be eager to see then if our spontaneous desorption of intact plasma is actually the

| production of small holes in our film, accompanied by a drop of re-

| fractive index only.

The thrombin-fibrinogen reaction, though perhaps not interfacial

by nature, was a rewarding subject for ellipsometry because of the
BLOOD COAGULATION STUDIES 347

apparent discrepancy between progressive macroscopic fibrin formation and unchanging optical thickness. The most likely explanation

for this finding may be that once a monomolecular layer of fibrinogen

is deposited upon the thrombin film, rapid polymerization of the

! now forming fibrinogen monomers takes place only at certain points on the film; thus, the isolated strands of fibrin growing to relatively

very great and random lengths will merely be recorded as a loss of

total light, but they will not affect the A and \p values of undisturbed

I film. The complex of events leading from surface activation to thrombin

| formation have not been discussed here, because measurements m

| this area were not yet attempted. Some involve a phospholipid.

! Its physiological orientation in suspension, as well as that of proteins

| reacting with it during clotting, will be tested in future by observing

their interactions at hydrophilic and hydrophobic interfaces in the

i ellipsometer, and compared with "breath pattern" [10, 4] and other

The technical assistance of Miss Ann Lukosevicius is gratefully

acknowledged.

! [1] B. A. Waaler, Scand. J. Clin. & Lab. Invest 11 (Suppl. 37), 1-130 (1959).

[2] P. Ollendorff, Scand. J. Clin. & Lab. Invest 14 641-647 (1962).

[3] J. B. Bateman, H. E. Calkins, and L. A. Chambers, J. Immunol. 41 321-341

(1941).

[4] L. Vroman, Nature 196 476 (1962).

[5] L. Vroman, A possible role of hvdrophobic bonds in coagulation proteins, 11th Annual Symp. on Blood, Wayne State Univ. Coll. of Medicine,

Detroit (1963).

[6] L. Vroman, Effects of hydrophobic surfaces upon blood coagulation, Thromb.

Diath. Haemorrhag. 10 455-493 (1964).

I [7] H. Sobotka, R. Santamaria, and M. Demeny, Interactions of molecular

layers of fatty acids with urea treated serum albumin, Proc. 2d Intern. Congr. of Surface activity, 307-309 (1957).

[8] I. H. Sher, and J. D. Chanley, Rev. Sci. Instr. 26 266-268 (1955).

[9] L. Vroman, Surface Contact and Thromboplastin Formation, Thesis, Utrecht,

The Netherlands (1958).

[10] A. Rothen, Surface Film Techniques, in Physical Techniques in Biological

Research, ed. G. Oster and A. W. Pollister, Academic Press, New York

(1956). Vol. II, 193-194.

[11] F. L. McCrackin, and J. P. Colson, this publication, p. 61.

[12] R. J. Archer, J. Opt. Soc. Am. 52 970-977 (1962). 728-101 O—64 23

studies.

5. References

Ellipsometry for Frustrated Total Reflection

T. R. Young and J. M. Fath

National Bureau of Standards

Washington, D.G. 20234

In the field of precise length measurement there exists the need

to measure precisely the proximity of two surfaces, nominally "in contact." Often at least one of these surfaces transmits light. For

instance, in the interferometric determinations of the lengths of metallic end standards, called gage blocks, one of the end surfaces may be contacted to a transparent plate and the optical length between

the plate and the other end surface of the gage block is determined.

By correcting for the phase-change difference of the two dissimilar

surfaces, the practical length may be determined from the optical

length. Therefore the practical length of the gage block is dependent

upon the thickness of the film between the two contacting surfaces. The relationship between this wringing film thickness and its effect upon incident parallel and perpendicular polarization has been

theoretically determined for the case where the light is incident upon

the first surface (transmitting surface) at an angle greater than the

critical angle and where the second surface is metallic (complex index)

The theoretical results indicate that when the optical constants of the two contacting surfaces and the wringing film are known, a polari- zation analysis of the reflected light by the methods of ellipsometry

offers an extremely sensitive and unique determination of the wringing

film thickness. Attempts to verify the theory experimentally are

discussed. 1. Theory

In the field of precise length measurement there exists the need

to measure precisely the proximity of two surfaces that are " n contact." Frequently at least one of the surfaces transmits light. For

instance, in the interferometric determinations of the lengths of metallic end standards, called gage blocks, one of the end surfaces

may be contacted to a transparent plate and the optical length between

the plate and the other end surface of the gage block is determined.

After correction of the phase-change difference between the two

dissimilar surfaces, the practical length is obtained from the optical

length. This practical length is therefore dependent upon the thickness of the film between the two contacting surfaces. Measured values of wringing film thicknesses range from —2.4 to +2.8 yidn. for lapped metallic surfaces [l].

1 The negative values

are explained as being caused by an intermeshing of the high points

of the surfaces. Some of this inconsistency in results might be due

to the use of questionable assumptions; however, there is a distinct

possibility that wringing film thickness is dependent upon the material

forming the surface, the planeness of the surfaces involved, and

the surface finish. There is agreement among many investigators

1 Figures in brackets indicate the literature references on p. 358. 349
350 YOUNG AND FATH

that the thickness of a wringing film between two given surfaces is repeatable to a fraction of a microinch [1, 2, 3]. Thus it is obvious that in the measurement of the practical length

of gage blocks to an accuracy of 0.1m in., stringent control and meas- urement of wringing film thicknesses are required. What seems

to be required is a measurement facility that will directly determine

the wringing film thickness between two surfaces to an accuracy

better than 0.01 interference fringe (0.1 pin.). Extension of fundamental physical theory led to the method of frustrated total reflection

to measure these quantities to a high degree of accuracy.

As light travels from a dense medium of index of refraction mi to a rarer medium of index m2 , making an incident angle d1 with the

normal, the light is refracted away from the normal by an angle 02 . This phenomena is governed by Snell's Law: mi sin #i =M2 sin 02 . As

the angle of incidence 61 is increased the critical angle is approached

beyond which all the light is totally reflected back into the incident

medium. The energy of the light is not reflected exactly at the

boundary surface; it is carried in an attenuated manner into the

rarer medium [4]. The magnitude of energy rapidly damps off exponentially with increasing d, d being the normal distance measured

from the boundary into the rarer medium.

COLLIMATOR

L FILAR EYEPIECE

L MICROSCOPE STEEL BALL Figure 1. Schematic outline of apparatus used in experimental verification of

theory. The basic equipment consists of a spectrometer with the vertical axis of rotation being perpendicular to the plane of the paper. The prism, end caps, and ball rests on the spectrometer table with the ball held

against the prism by a 0.1 g force due to a 1 deg tilt in the table. By bringing up a third medium of index Ms, e.g., the steel ball m

figure 1, some of this energy is frustrated in the sense of prevention of

its return to the first medium. The third medium will absorb or transmit part of the frustrated energy, depending on whether Ms is real or complex. This will naturally affect the light reflected back

into the first medium. For an all dielectric system, such as glassair-glass, the relative amount of energy intercepted is known to depend upon the polarization of the incident light, the thickness ot the wringing film, the three refractive indices, m, p2 , Ms, the wavelength of the incident light, and the angle of incidence 0i [5]. The ef- fect of conducting surfaces (complex Ms) was investigated recently 16J.
FRUSTRATED TOTAL REFLECTION 351

For light polarized perpendicular to the plane of incidence, no

unexpected phenomena occur when conductors are used as the third

medium. However, for parallel polarized light a minimum in re-

! flectivity occurs when n? has a complex index, figure 2. These curves,

! plotted from an equation [6] developed in an analysis of electro- magnetic wave theory, indicate a sensitive method of determining d

by measuring the reflectivity of the light. This presumes a knowledge of the indices of refraction of all the media, the wavelength, and

I the angle of incidence 0r. Each one of these values can be determined

by measurement with the possible exception of /x 2 , the index of the

wringing film. By methods of ellipsometry m3 , the index of the frus-

| trating medium, may be measured even if it is complex. The pertinent questions are the value of the index of refraction and the

! thickness of the wringing film. From the form of the curves in figure

I 2, it is apparent that for a very thin film increasing from zero m

FILM THICKNESS, in. Figure 2. Variation of reflectance with film thickness of air at various angles

of incidence of parallel polarized light for the case where M3 is of complex index. thickness a pattern of minimum reflectance should be obtained

resembling an interference fringe of equal thickness. Unlike an interference fringe pattern, only one minimum can be obtained, corresponding to a thickness of film dependent upon the angle of incidence.

Therefore no ambiguity concerning the order of interference will

exist. Other important differences are the increased sharpness of the

reflectance dip obtained for larger angles of incidence, and, for an

increase in the index ratio mi/m*, the minimum occurs at smaller d and

the system becomes very sensitive since there is a rapid change in

reflectivity for small changes in d. Starting with Maxwell's equations and modifying them for the

total reflection case one obtains for the perpendicular polarization,
352 YOUNG AND FATH

sinh u^sin2 6,-n^-p1 '2 cos 6, cos^jp

1/2 (sin2 ^-/i2,)

1/2 sin » cosh u -i |V12 cos ex sin|sinh u- (sin2 6t -n\2)

1 12 (cos 0X +p

1 12 cos |) cosh uJ

sinh u|j$in2 ^-w?a +p

1/2 cos Bl cos |J-p

1/2 (sin2 61 -ni2)

1/2 sin|cosh u +^P1/2 cos^sin|sinh^-(sin2 ^-7t2

2) 1/2 (cos ^-p1/2 cos0 cosh^J

and for the parallel polarization,

[?&/4p

1/2 cos 0l cos \~n2 {l -a2

) (sin2 ^-ri?2)J

sinh u

+|jufpl/2 sin|-2n22n

2a cos (sin2 B.-n2^)

112 cosh u -H^sinh u ^nW2 P

112 cos 0, sin ^4-2^0: (sin2 0i—*J0

J

-cosh u [m!p1/2 cos |+ri22ri

2 (l-a2

) cos [sin2 ^-r^]172 }

(1)

r?. 22M

2 p

1/2 cos 0! cos |+7i2 (l -a2

) (sin2 0 -rm!p

1/2 sin

i—^12) J

sinh u ^+2n212n

2a cos flij [sin2 61 —n2

12] 1/2 cosh u +£^sinh u [n212n

2

2 p

112 cos 6, sin^-2TM2a(sin2 J

-fcosh u [pJp1/2 cos ^-ri2

2n

2 (l-a2

) cos^J [sin2 O.-nU112 }

where

(2)

M2

Mi n(l—ia)

ra= refractive index

a—absorption index
I FRUSTRATED TOTAL REFLECTION 353

—i2irfjL2d cos 62 u= \>

\0 =vacuum wavelength

2n\2n

2a

sin <j>= 2—-

M2P

n\2n

2 {l-a2)-&sm2 Bl cos <f>= 2 M2P Equations (1) and (2) may be used to determine the magnitude

and phase of the reflected electric vectors. These theoretical equations have been qualitatively and quantitatively verified by placing a steel ball against the boundary surface as

in figure 1. The minimum indicated by theory has been observed

and its formation verified with respect to film thickness. This minimum is formed because as the surface of the ball recedes from the

:.S* /,.T'<

B \

Figure 3. Newton's rings obtained by interference between glass boundary surface and steel ball of figure 1.

(A) angle of incidence less than the critical angle. (B , C, & D) reflectance patterns obtained by frustrated total reflection at angles of incidence 45, 65, and 75 .
354 YOUNG AND FATH

! total reflecting surface, d increases and the reflectivity decreases, ; goes through a minimum, and then increases as predicted by theory. | Figure 3 shows a series of patterns formed between a glass surface

| and a steel ball. Preliminary investigation and results indicate the method of frustrated total reflection applicable to the precise measurement of thin films such as those existing between a dielectric and a conductor. I There are two areas of investigation of thin films in which this method

of frustrated total reflection can be significantly employed. One is a

1 reflectometer to measure the intensity of the reflected light, and the

j other is an ellipsometer to analyze the polarization characteristics of | the reflected light. As can be seen in figure 4, if the optical constants

j of the metallic surface (these determine the terminal point of the

family of curves) and the index of the wringing film are known, and

a polarization analysis of the reflected light is made, an extremely

sensitive and single-valued determination of the wringing film thick- ness can be obtained. This sensitivity can be shown by assuming

a hypothetical case of the wringing film index being 1.3 and the

thickness going from 0 to 1.0 ii in. Over this range of film thickness,

for this particular index, the relative phase change of the reflected

light changes by 70 deg and the azimuth angle by 19 deg. Whereas

if the intensity of the reflected light is measured with a reflectometer,

figure 2, the result obtained is not single-valued, since it can not be ' determined on which side of the minimum the measured reflectivity

value falls. Thus there will be two film thicknesses for each reflectivity

value that is experimentally determined. The problem of knowing

the index of the wringing film still exists if a reflectometer is used.

This, and the fact that with an ellipsometer the optical constants of a conducting material can be measured, led to the selection of an

ellipsometer for investigating the thickness of a film existing between

two surfaces that are nominally "in contact."

2. Procedures

Various techniques have been used for the cleaning and preparation

of the end surfaces of the gage blocks employed in this investigation.

Initially the surface of the gage block was cleaned with trichloroethylene, to remove any organic contaminates, and then with ethyl

alcohol to remove any residue from the trichloroethylene. The

polarization characteristics of the reflected light were then measured

for this cleaned surface. Although the angle of incidence was not

changed, and the surface was not disturbed, the measurements were found to vary considerably throughout the day. From the characteristics of the variations in the measurements it seemed likely that

a transparent film was building up on the surface throughout the day.

By directing a beam of microwave radiation at normal incidence upon

the surface while it was being measured this film was " boiled off" the surface without any heating effects to the surface. The measure- ments of the light reflected from the surface, at the same angle of
FRUSTRATED TOTAL REFLECTION 355

incidence, were then found to be highly reproducible, showing random

character and having variations limited to about 0.05°. Another

cleaning solution that was tried was spectral grade acetone. The

; experimental results of surfaces cleaned with acetone showed that a

film was being deposited upon the surface. When the cleaning was

repeated the thickness of the film increased. A third method that was

tried was flaming. The surface was cleaned with trichloroethylene

and alcohol and then rapidly passed through a flame several times.

¥ , DEGREES Figure 4. Relative phase change vs. azimuth angle for various values of /jl 2 . Numbers on curves represent index of refraction of wringing film. Each successive point, beginning at

P, represents an increase in d of 0.5 ^ in. 01=65°, m= l. 576, w=2.41 (1—3. 326i ), X=21. 50 microinches.
356 YOUNG AND FATH

This made the surface hydrophilic for about an hour, at the end of which it again became hydrophobic. This method produced a clean

surface, but it was found that after flaming a surface could not be

wrung. In the following experimental results the method mentioned

above of cleaning a surface with trichloroethylene and alcohol and

then placing it in a beam of microwave radiation was employed.

By using the following formula, the measured polarization characteristics of the reflected light can be utilized to determine the complex

index of refraction of the metallic surface under study.

f a n 4/3 sin2 fl^172 M3 =Mi tan 0i ^1 (£+1)2 J

where,

M3 =^(l

ia)

i8=tan ype^

01 = angle of incidence

^= azimuth angle

A =relative phase shift. As stated above the measurements of optical constants of the

surfaces prepared with this cleaning procedure were found to be

highly reproducible at the same angle of incidence. However, when

the angle of incidence was changed the optical constants of the

surface were found to vary slightly. The possibility that this variation

could be due to the formation of an oxide film on the surface was

investigated. This investigation consisted of a thorough study of a theoretical case of a surface with a transparent oxide film of complex index on it. Upon the completion of this study it was found that

this variation in the index of refraction of the metallic surface could

not be explained by an oxide film, regardless of the thickness of the

film. At the present time this variation of index with angle of in- cidence is still unexplained.

If the complex index of the metallic surface is measured and the

index of the glass to which the surface is to be wrung is known, the

index of the wringing film must still be determined in order to measure

the thickness of the film. In figure 4 the starting point, P, of the curves is the point at which the film thickness is zero. It is dependent

upon the wavelength of the light, the angle of incidence, the index

of the glass, and the index of the metallic surface. Starting at this point of zero thickness the curves then undergo a separation dependent upon the index of the film. If the index of the film is known

the thickness of the film can then be computed from the polarization

analysis of the reflected light.
FRUSTRATED TOTAL REFLECTION 357

3. Results

The first approach to this problem was to measure the bulk index

of the fluid (e.g. kerosene) used to deliberately contaminate the

surfaces in order to wring them together. Using this measured index,

approximately 1.4, and the polarization characteristics of the reflected

light from the wrung combination of glass and metallic surfaces a

film thickness for this index was calculated, using an electronic

computer program [7]. The measured polarization for different

angles of incidence for this combination of glass and gage block in- dicated a rather large variation in the derived film thickness even though the complex^ index of the metallic surface was corrected for the angle of incidence at which the film was being measured. Both

! the film thicknesses and refractive indices were then calculated from

' the experimental data for each angle of incidence. The film index

j which gave the minimum spread of film thicknesses for various

' angles of incidence was in the range 1.00 to 1.02. The index of the film being equivalent to the index of air instead

of approximately 1.4, the index of the contaminate used to wring the

combination together, seems reasonable when two facts are considered.

One, the surface of a gage block is not a completely smooth finish,

but a series of irregularities. The arithmetical average surface

! finish is defined as the average value of the departure of the surface

from a centerline, whether above or below it, over the length of the

1 block. It is the sum. of all the areas of the peaks and valleys divided

I by the length. The average height of the irregularities with respect

to the centerline, for the type of surface used in this investigation was about 0.07 /x in. Since the metallic surface was contaminated

by putting a drop of the fluid on it and then rubbing the surface with

a lens tissue until the contimate was no longer visible, the possibility

existed that the contaminating fluid had flowed into the valleys of

I the surface. If this were the case the material immediately adjacent

to the glass surface would be a thin layer of air. Second, since the

real Poynting vector, or the real energy flow, is parallel to the surfaces of the glass and the gage block, the real energy would be radiating in the thin layer of air and would not encounter the contaminating

fluid in the valleys. However, when the film index was taken to be

1.00 to 1.02, the deviations of the measured polarization characteristics, the azimuth angle and the relative phase change, from theoretical curves similar to those of figure 4 were too large to be attributed

to experimental errors. The next phase of the investigation of the film index was to use a "dry" wring to contact the surfaces. That is, the surfaces were

cleaned with trichloroethylene and alcohol, wiped dry with a lens

tissue and then wrung together. A series of measurements was then

taken of this wringing film at an angle of 65 deg. Film indices and

corresponding film thicknesses which fell within experimental errors were then calculated. For a wringing combination that had been

wrung for a period of 18 hr the film index was in the range 1.35 to 1.36,

and the film thickness was 0.615 to 0.654 m in. After a total wringing
358 YOUNG AND FATH

time of 41 hr another measurement of this film was made. The index

had now increased to the range 1.43 to 1.44 and the thickness to the

range 1.217 to 1.340 n in. After a lapse of another 96 hr the index was

still about 1 .43 to 1 .44 but the thicknesses had increased to the range

1.378 to 1.519 n in. Since the last cleaning fluid used on the surfaces was alcohol and

alcohol has an affinity for water the initial index value of about 1.35 seems realistic. A visual observation of the metallic surface after it had been pulled free from the glass revealed that some unknown

substance had seeped in between the metallic surface and the glass.

This seepage appeared to have originated at the edge of the metallic

surface. Further investigation of this phenomenon revealed that the

amount of seepage increased with time. Experimental attempts to identify the index of this substance proved unsuccessful, but they did

reveal that the index was rather high, approximately 1.5. This high

index value would explain the increase of the film index observed

experimentally. The observed continual seepage of this substance

would explain the index value appearing to reach a plateau of 1.43 to

1.44, while the thickness of the film continued to increase. This

specimen was a 405 nitrided stainless steel gage block. The four nongaging surfaces of the block had not been subjected to any finishing process. The unfinished sides of the specimen probably served as a

reservoir for the unknown substance that seeped into the wringing

area existing between the glass and the gage block.

Th3 reported results are of a preliminary nature. The investigation of the index of refraction and the thickness of the thin films

existing between two surfaces nominally "in contact" with each other

win b/ continued along the same lines of investigation indicated in

this paper.

4. References

[1] C. F. Bruce and B. S. Thornton, J. Appl. Phys. 27, 853 (1956).

[21 A. M. Dexter, ASTE 59, Tech. Paper 195 (1959).

[31 F. H. Rolt and H. Barrell, Proc. Roy. Soc. A116, 401 (1927).

[4] R. W. Ditchburn, Light, Interscience Publishers, Inc., N.Y., p. 434 (1955).

[5] E. E. Hall, Phvs. Rev. 15, 73 (1902). _x 6 T. R. Young and B. D. Rothrock, J. Res. NBS 67 A, 115 (1962).

[7] F. L. McCrackin and J. Colson, NBS Tech. Note 242 (1964). Discussion

C. Del Carlo (Chrysler Corp.):

If I understand you correctly, the variation of the length ot the

gage block can be as much as a 10th of a wavelength whereas the

thickness of the wringing film is about an order of magnitude less than this. In view of this difference, I can't understand your concern about an exact measurement for the thickness of the film. T. R. Young:

The thickness of the film must be measured at one point. If
FRUSTRATED TOTAL REFLECTION 359

there is a convexity of the surface the forces of attraction working

| on the surfaces tend to draw them together. R. J. Archer (Bell Telephone Laboratories)

Recognizing that the surface of the metal gage block is not perfectly

flat, it must be considered that an irregular surface region can constiI tute a film on the gage block. The thickness of this film is the height

of the irregularities, and its optical constants are a combination of

; the optical constants of the metal and of the medium that fills the

| "valleys" of the irregular region. Thus, the net change in A and \p following the formation of a wringing film results from the combined

effect of the wringing film plus the change in the optical constants of

I the film constituted by the irregular surface region due to the change

in the index of refraction of the medium that fills the irregularities.

! It follows that the properties of the wringing film calculated from

changes in A and ^ would be incorrect if the irregular surface region were not taken into account. The likely effect would be for the

calculated thickness and index of refraction to be smaller than the

| true values. T. R. Young:

This irregular surface region of the gage block we feel is taken into

account since the optical constants of the "clean" block are measured

and the block is then wrung to the glass without the use of any wringing lubricant or agent. Therefore, the effective surface of the gage

block in the wrung combination is essentially the same surface for which the optical constants were determined.

A. B. Winterbottom (Norges Tekniska Hogskole)

Experiments have indicated that all surfaces have a contaminated

film therefore the actual model that would closely resemble the

physical model would be a three-film case. This may explain the

discrepancy in your results. T. R. Young:

In the measurement of the optical constants of the surfaces in- volved we have undoubtedly obtained some resultant effects, hopefully

the optical constants of an effective surface.

U.S. GOVERNMENT PRINTING OFFICE ; 1964—O-728-l 01