TECHNK

Nation:.! :

Library, E- APR 2 1 1969 479

A Fortran Program for

Analysis of Ellipsometer Measurements

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

UNITED STATES DEPARTMENT OF COMMERCE

Maurice H. Stans, Secretary

NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director

NBS TECHNICAL NOTE 479

ISSUED APRIL 1969

A Fortran Program for

Analysis of Ellipsometer Measurements

Frank L McCrackin

Polymers Division

Institute for Materials Research

National Bureau of Standards

Washington, D.C. 20234

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office

Washington, D.C, 20402 - Price 75 cents

Contents

Page

1 Introduction .................. . . . . 1

2. Data Input Required for Calculations ........... 3

2.1. Table of Values for A, \|/, P, A, and

Reflection Coefficients for a FilmCovered Surface - CTABLE 3

2.2. Optical Constants of a Film-Free

Substrate - CNS 8

2.3. Input of Ellipsometer Readings, P, A, and Q . . . . 9

2.4 Corrections to Ellipsometer Readings:

ALINE, AT, CAT, CATA. „ . 10

2.5. Film Thickness for a Film of Known

Refractive Index -CD.. ........... ..12

2.6. Film Thickness and Refractive Index - CND ..... 13

2 7. Confidence Limits of Film Thickness and

Refractive Index - CNDE and CNDC. ... o .... » 15

2.8

o

Calculations Involving Multiple Films - FILM. ... 16

2.9. Inhomogeneous Films - IFILM ....... 17

2.10. Multiple Reflections of the Light ....... ..19

2 oil. Films Composed of a Mixture of Materials, o .... 19

2.12. New Pages, Reset of Variables, and Comments .... 20

2.13. Optical Constants of an Adsorbing Film of

Given Thickness - CNK, FT 21

3. Theory and Calculations. ... 22

3.1. Computation of A and \|/ from Ellipsometer

Readings. ................. ... 22

3.2. Computation of Constants T and A of the

Wave Plate °. . . . C

24

in
Page

3.3. Computation of the Angle of Tilt of the

Reflecting Surface. ....... 25

3.4. Calculation of the Thickness of a Film

from A and if 27

3.5. Calculation of Both the Refractive Index

and Thickness of a Film From Values of

A and if 28

3.6. Confidence Limits for the Refractive Index

and Thickness of a Film..... ...31

3.7. Volume Fractions and Total Amount of Material

for a Film Composed of a Mixture of Two

Materials 32

3.8. Computation of Ellipsometer Readings from

A and ljr . . . 33

3.9. Computation of n and % of a Film of Given

Thickness ......... 33

4. The Fortran Program 37

4.1. Fortran IV Program .....38

4.2. Subprograms 56

4.3. Variables Used in Program 58

a. Variables in Common a 58

b. Complex Variables Not in Common 60

5. References 62

Appendix I. Use of the ALINE Instruction in Relation to

Alinement of the Ellipsometer 64

Appendix II. Derivation of Equation 3 67

IV
A Fortran Program for

Analysis of Ellipsometer Measurements

Frank L. McCrackin

A general Fortran program is given that performs

the calculations required to analyze ellipsometer

measurements. This program replaces the program given

in NBS Technical Note 242, entitled "A Fortran Program

for Analysis of Ellipsometer Measurements and Calculation of Reflection Coefficients From Thin Films." The

main changes from the previous program are: 1) the

new program is in Fortran IV and V rather than Fortran

II; 2) the relative transmission of the wave plate is

considered in analyzing ellipsometer readings; 3) an

improved method is used for calculating the refractive

index of a film; 4) the form of the input data is

improved; 5) a method for calculating and correcting

for tilt of the reflecting surface is given; 6) an

improved method of calculating confidence limits for

the calculated values of thickness and refractive

index of a film is used; and 7) a method for calculating the optical constants of an adsorbing film of

given thickness is included.

Key words: Computer programs; ellipsometry; Fortran

program; multiple films; optical constants; reflection

coefficients; thin films.

1. Introduction

The reflection of polarized light from surfaces and thin films

has been discussed by various authors [1-4].* The reflection is

expressed in terms of the reflection coefficients R^ for light

polarized parallel to the plane of incidence and R for light

polarized perpendicular to the plane of incidence. Since these

coefficients are complex quantities, they represent both the change

of amplitude and phase of the light on reflection.

*Figures in brackets indicate the literature references on page 62,
The ellipsometer is an optical instrument used to studysurfaces and surface films by reflection of light from the surface.

The ellipsometer measures the ratio of the reflection coefficients

p = lP/RS = tan J elA

. (1)

In the ellipsometer, as considered by this program, monochromatic light passes through a polarizer and a wave plate (linear

retarder). It is then reflected from the surface under investigation, and passes through another polarizing prism called an analyzer

and to a detector. The wave plate is generally set at a fixed

azimuth and the polarizer and analyzer rotated until the light is

extinguished as determined by the detector. The azimuths of the

polarizer, fast axis of the wave plate, and analyzer, (P, Q, and A),

measured from the plane of incidence are recorded. From the

reading P, Q, and A, A and i|f of eq (1) are first calculated, then

the optical constants of the surface or of films on the surface are

calculated from A and \|/.

In order to calculate A and ijf from the ellipsometer readings,

the constants of the wave plate must be known. Until recently, the

relative phase retardance, A , along the fast and slow axes was

believed to describe the optical properties of the wave plate.

However, it has been shown [5-8] that the transmissions along the

two axes are different. Therefore, the ratio T of the trans

' c mission along the fast axis to the transmission along the slow axis

of the wave plate must also be considered. In addition, Oldham [6]

has shown that a tilt in the mounting of the wave plate can cause

A and T to vary with the azimuth of the wave plate. This

program takes both A and T and their variation with Q into

c c consideration in computing A and i|j . The values of A and T

may also be computed from the ellipsometer readings.

Another possible source of error is a tilt of the reflecting

surface. The instrument is alined so that P, Q, and A are read

with respect to the reflecting surface. If the reflecting surface

is changed, and the new surface is tilted with respect to the
surface with which the ellipsometer was aligned, the readings of P,

Q, and A will be in error. This error could be eliminated by re- alinement, but this is time consuming. Therefore, the program has

been written to calculate the angle of tilt from the ellipsometer

readings and correct for the tilt.

2, Data Input Required for Calculations

The data and instructions are given to the computer by punched

cards. Each card requires an instruction starting in column 1.

The data required for the instruction start in columns 10, 20, 30,

40, 50, and 60. All numbers must contain a decimal point.

Instructions that start with C cause the computer to perform

calculations. These instructions are listed in table 1. The other

instructions are used to give data for future calculations. Table

2 gives the required data with each instruction.

2,1. Table of Values for A, i|r, P, A, and Reflection

Coefficients for a Film-Covered Surface - CTABLE

The instruction CTABLE will produce calculation of ellipsometer readings and reflection coefficients for reflection of

light from a film-covered surface. The thickness of the film is

given on the same card as the instruction. However, before the

CTABLE instruction is given, instructions NM, NF, NS or CNS, AI,

and WL (see table 2) must be used to give values of the refractive

index of the medium, film, and substrate and the angle of incidence

and wave length of the light, respectively. For example,

1 10 20 30 40

NM 1.

NF 1.4

NS 1.7

AI 65.

WL 4800.

CTABLE 40.
Table 1„ Computations

Instruction Computes

Preceding Instructions

Required If Standard

Values Are Not Used

CAT Angle of tilt of reflecting

surface

WP or CWP

CD Thickness of film

CND Refractive index and thickness of film

CNDC Refractive index and thickness of film with allowed

combination of values

CNDE Refractive index and thickness of film with confidence

limits

CNK Refractive index and % of

film of given thickness

CNS Refractive index of

substrate

CTABLE Table of A and \|/ and

reflection coefficients

CWP Constants T and A of

c c wave plate

NM, NF, NS or CNS, AI,

NREF, WL1

NM, NF, NS or CNS, AI,

NREF, WL, ELDP1

NM, NF, NS or CNS, AI,

NREF, WL, ELDP, EDP 1

NM, NF, NS or CNS, AI,

NREF, WL, ELDP, EDP 1

NM, NF, NS or CNS, AI,

NREF, WL, FT1

NM, AI, NREF1

NM, NF, NS or CNS, AI,

NREF, WL

None

1 Instructions WP or CWP required if ellipsometer readings

(P, A, and Q) are given.
Table 2. Data Given With Instructions

10

Card Column No. 20* 30 40 50* 60

AI I(«p) 5(cp) E(cp) ALINE AP AA AQ AT Angle

CAT P

1 Al P2 A

2 Q

CATA P A

l P

2 A2 Q

CDCND A t CNDC P A Q

CNDE

>4 P

l A

l P2 A

2 Q

CNK blank

CNS

CTABLE 1(d) 6(d) E(d) Q**

CWP P A

l P

2 A2 Q

EDP EA E^

ELDP ELA ELt

FILM No

.

d n H FT 1(d) 6(d) E(d) IFILM No, films d n^_top Ktop oottom xiottom MIXED p P

P

nP

NF I(n) 6(n) E(n) 1(H) &(h) E(H) NM I(n) 5(n) E(n)

NREP No. Reflections

NS I(n) 6(n) E(n) 1(h) 6(h) E(h)

PAGE See Section 2 12 RESET See Section 2 12 ST0P WL I(\) 6(\) E(X)

WP T

c

A c (other) Comment

* The symbol 6 refers to incremental values.

**The value of Q with this instruction is optional. See Section 2.1,
Table 3. Assigned Values

Symbol Data Instruction

Assigned

Values

N

n

N

<Pi

T ,A

c' c

Index of medium

Index of film

Index of substrate

Angle of incidence

Vacuum wavelength of light

Constants of wave plate

Number of reflections of light

Experimental errors of A and i|r

Error limits of A and \|r used

for computing n„

Angle of tilt

NM 1

NF 1

NS 1

AI 70°

WL 5461

WP 1, 90°

NREF 1

EDP .015, .015

ELDP .002, .002

AT
will compute A and i|j and reflection coefficients for the condition:

Refractive index of medium =1.

Wave length of light = 4800. A

Angle of incidence = 65.°

Thickness of film = 40. A

Refractive index of film = 1.4

Refractive index of substrate = 1.7

A range of values of the input parameters may also be specified by giving the initial value, increment, and final value to be

used. Thus, the input data

1 10 20 30 40

NM 1.

NF 1.4

NS 1.7

AI 65.

CTABLE 0.

0.1 1.6

20. 100.

will produce tables for films of refractive index of 1 4, 1.5, and

1.6. In each table, the ellipsometer readings and reflection

coefficients will be given for films 0, 20, 40, 60, 80, and 100 A

thick. The results are given in the upper portion of figure 1.

The other parameters may also be similarly varied.

In this figure the values of A and \(i for the sample, then the

complex reflection coefficients for light polarized parallel and

normal to the plane of incidence, are printed. For the complex

reflection coefficients, the ratios of reflected to incident amplitudes and the change in degrees of the phase of the light are given.

The intensity ratios for parallel and normally polarized light are

the squares of the corresponding amplitude ratios.

The refractive index and value of k for the film and substrate

are next printed. In this example, the film and substrate are nonabsorbing so the values of x are zero.
A setting, Q, for the wave plate starting in column 40, may

also be given with the CTABLE instructions. In this case, the two

sets of ellipsometer readings, P1,A1 and P2,A2, (see Section 2.3)

that will produce extinction of the light are also calculated and

printed, as shown in figure 1.

Listed in table 3 are values for certain of the input parameters. These assigned values are automatically used for following

calculations unless instructions are given to change them. Thus,

in the above examples, the NM instruction was not necessary as the

assigned value of 1, was used for the index of the medium.

The film or substrate may also be absorbing. In this case the

optical constants are n and % of the complex refractive index

n - nni. The values of n and k are given starting in columns 10 and

40. Thus, the card

Column No,

1 10 20 3(3 40_

NS 3.316 1.4

defines an absorbing substrate with n=3,316 and k=1.4 or of complex

refractive index of 3,316(1-1.41).

The value of n may be varied by giving the initial, incremental, and final values starting in columns 10, 20, and 30, and

n may be varied by giving the initial, incremental, and final

values starting in columns 40, 50, and 60,

2,2. Optical Constants of a Film-Free Substrate - CNS

The instruction CNS calculates the optical r constants n and

s x of a substrate from measurements of A and f for the substrate

without a film. The values of A and ijr start in columns 10 and 20

of the card containing CNS. The values of n and k are printed

and also retained by the computer so they will be used in following

calculations, such as CTABLE, for which they are required. Thus,

the optical constants of the substrate may be given by either an

NS or CNS instruction.

8
Ellipsometer readings of P, A, and Q, rather than values of

A and ty may be used with the CNS instruction, as explained in the

next section.

2.3, Input of Ellipsometer Readings P, A, and Q

As previously described, the ellipsometer readings must first

be analyzed to obtain the values of A and f that describe the

reflection of the light from the surface before the instruction

CNS, or the instructions CD, CND, CNDC, CNDE to be later described,

may be used. However, the ellipsometer readings may also be used

as input data with these instructions if the components of the

ellipsometer are arranged in the order of light source, polarizer,

wave plate, reflecting surface, analyzer, and detector.

Let the wave plate be set with its fast axis at an azimuth Q

and let extinction of the light be obtained with the polarizer and

analyzer at azimuths P and A (see Reference 4). All azimuthal

angles are measured positive counterclockwise from the plane of

incidence. Then these readings of P, A, and Q may be used with

the above instructions, as shown in table 2. The computer first

calculates the values of A and \|f and uses them in the following

calculations

.

Let P. and A, be settings of the polarizer and analyzer

that produce extinction of the light for a given value of Q. Then,

settings of for example P.. + 180°, A, ; P,, A, + 180° / and

P. + 180°, A.. + 180° produce the same effect on the polarized light

so will also give extinction. That is, P and A are indeterminate

to ±180°, This duplication of settings may be eliminated by, for

example, taking only P and A readings between and 180° . After

eliminating duplicate readings, two sets of readings P ,A. and P 9 ,A9

will be found to give extinction for a given setting Q of the wave

plate. Both sets of readings may be used with the instructions CD,

CND, CNDE, and CNS. They are given in the order P,, A,, P

2

, A2

, Q.
The computer calculates values of A and ij; from each set of readings

and uses the averages of them in the following calculations.

In order for the computer to calculate A and \|i from the

ellipsometer readings, the constants of the wave plate must have

been previously made available to the computer. The constants are

the ratio, T , of the transmittance of the wave plate along the

fast axis to the transmittance along the slow axis and the phase

retardance along the fast axis minus the phase retardance along the

slow axis, A . They may be given in the order T , A with the

instruction WP or may be calculated from a set of readings P , A.,

P„, A? and Q by the CWP instruction

More than one calculation may be performed with a single set

of ellipsometer readings. In this case, a set of ellipsometer

readings from a film-free metal surface is used to first calculate

the constants of the wave plate by the CWP instruction, and then

is used to calculate the optical constants of the metal by the CNS

instruction. The readings do not have to be repeated with the CNS

instruction.

2.4. Corrections to Ellipsometer Readings:

ALINE, AT, CAT, CATA

The ellipsometer angular readings of P, A, and Q must sometimes be corrected to obtain angles measured from the plane of

incidence of the reflection from the surface, as is required for

this program. For example, if a dial reads from the plane of the

surface, 90° must be added to the dial readings. As another

example, in alining the ellipsometer the dial readings when the

component is alined in the plane of incidence may be recorded and

used as a correction to future readings of the dial, rather than

adjusting the dial or the vernier on the dial to read correctly.

This method of alinement is described in Appendix I.

10
The corrections (AP, AA, and AQ) may be given by the ALINE

Instruction (see table 2). Then the corrections will be added to

values of P, A, and Q given with all following instructions

„

A correction to the ellipsometer readings may also be required

due to tilting of the reflecting surface. This may occur when the

ellipsometer is alined with one surface, and then the surface is

replaced by another surface that is to be measured. The second

surface may be tilted with respect to the orientation of the first

surface so the ellipsometer will no longer be properly alined but

all its readings must be corrected by the angle of tilt. This

correction may be given to the computer by the AT instruction. The

AT instruction differs from the ALINE instruction in that one

correction is given to be added to all values of P, A, and Q, while

different corrections for P, A, and Q are given with the ALINE

instruction.

The angle of tilt of the reflecting surface is generally

unknown o It may be computed from ellipsometer readings by the

instruction CAT. The instruction CAT requires the two sets of P

and A readings for a setting of Q and the constants T and A of

the wave plate. The calculated value of the angle of tilt will be

added to all ellipsometer readings given by following instructions

The theory of this calculation is given in Section 3.3. This

theory requires that the angle of tilt is small. The calculation

has been found to be accurate for angles of tilt up to one degree

The constants of the wave plate may not be calculated from the

same readings used to compute the angle of tilt, since readings , from an untilted surface are required to compute the wave plate

constants. The usual procedure is to aline the ellipsometer and

calculate the constants of the wave plate from ellipsometer readings.

Then, every time the surface is changed, its angle of tilt is

calculated using the previously determined constants of the wave

plate. The angle of tilt of the surface is then used to correct

all ellipsometer readings obtained using the surface

11
Often two sets of ellipsometer readings are taken for the same

surface. For example, readings may be taken for two readings of Q

or readings may be taken both before and after a film forms on the

surface. Values for the angle of tilt of the surface may be calculated by the instruction CAT for both sets of readings and should

be nearly equal. However, subsequent ellipsometer readings will be

corrected by the angle of tilt calculated by the last CAT instruction, while for greatest accuracy, the average of the angles

computed by the two CAT instructions should be used Q This is

accomplished by using the instruction CATA to compute the second

angle of tilt. The instruction CATA performs the same calculations

as CAT, then averages the computed and previously computed values

of the angle of tilt. Subsequent ellipsometer readings will then

be corrected by the average angle of tilt.

2.5. Film Thickness for a Film of

Known Refractive Index - CD

The instruction CD computes the thickness of a film from

values of A and \jj . The ellipsometer readings P, A, and Q may also

be used as described in the previous section. Before this instruction is used, the following data must have been previously given to

the computer or standard values will be used:

1) Wave Length of Light by instruction WL

2) Angle of Incidence by instruction AI

3) Refractive Index of Film by instruction NF

4) Refractive Index of Substrate by instruction NS or CNS

5) Refractive Index of Medium by instruction NM.

If ellipsometer readings are given, the wave plate constants

must also be given by instruction WP or CWP, or the standard values

will be used.

12
For example, suppose a film-free surface was measured under

water (refractive index of 1.33) with a perfect quarter wave plate

set with its fast axis at 45.° from the plane of incidence. Let

the values obtained be P = 86.35, A = 25.19 and P = 176.35,

A = 154.81 for the two zones. Then let an absorbing film of

refractive index 2.5(l-0.3i) be placed on the surface and measurement P = 94.30, A = 24.44 be obtained in one zone. Let the angle

of incidence be 65.° and the vacuum wavelength of the light be

o 5461. A. Then the following input data will calculate the refractive index of the substrate and the thickness of the film.

Column No.

1 10 20 30 40 50

0.3

154.81 45.

Notice that the wavelength is not given since its assigned value

is used. The computer output is shown in figure 2.

2.6. Film Thickness and Refractive Index - CND

The instruction CND calculates both the thickness and refractive index of a film from ellipsometer measurements A and ty or

from ellipsometer readings P, A, and Q. The data given to the

computer before this instruction is the same as for the CD instruction. However, values for the refractive index of the film given

by the NF instruction do not have their usual meaning. Consider

first the case that the film has a real refractive index (nonabsorbing film) . Then the CND instruction causes the computer to

search for the refractive index of the film between the limits of

the initial value and the sum of the initial and increment values

13

NM 1.33

NF 2.5

AI 65.

CNS 86.35 25.19 176.35

CD 94.30 24.44 45.
!

Then the search is repeated between the sum of the initial value

and increment and the sum of the initial value and twice the

increment. This is continued until the final value is reached.

For example, if the instruction

NF 1.4 .1 1.6

is followed by a CND instruction, the computer will search for

values of n,. between 1.4 and 1.5, and between 1.5 and 1.6. If

the increment is not given, the computer will perform a single

search between the initial and final values.

The search procedure is explained in section 3„5. The limits

of film refractive index given by the NF instruction must be chosen

with care. If the limits do not include the solution for the film

refractive index, the solution will not be found. On the other

hand, if the limits are too broad, a solution may be missed.

An example of instruction CND is given in figure 3. In this

example, the values of A and \j/ are given with the instruction CND.

The solution of a film of refractive index of 1.2 and thickness of

1000 A is found. This solution reduces the error terms ADEL, APSI,

and ERROR to be essentially zero. The terms labeled ADSORPTION and

WT FRACTION have no meaning since the instruction MIXED was not

given (see 2.11). For the meaning of K, see section 4.3.

Consider next the case that the film is absorbing but its

value of k is known. Therefore, only the real part of the refractive index and the thickness of the film are to be determined. In

this case the imaginary part of the refractive index of the film

is given, starting in column 40 of the NF instruction card and the

calculations are the same as the previous case.

In the most difficult case, the film is absorbing and both the

real and imaginary part of its refractive index as well as its

thickness are to be determined. For this case the range of values

of the imaginary part is given by the NF instruction and the real

14
part of the index and thickness of the film is calculated for each

value. Thus, the instructions

Column No.

1 10 20 30 40 50 60

NF 1.0 2.0 0. 0.1 1.

CND 152.30 29.00

will compute the real part of the refractive index and thickness of

the film for assumed values of the imaginary part of the refractive

index of 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8; 0.9, and 1.

The real part of the refractive index is assumed to lie between

1. and 2 OQ In order to determine which of the calculations correspond to the actual film, additional information is required.

Obviously, three values describing the film cannot be determined

from the two ellipsometer measurement A and i|i For example, if

the thickness of the film was known independently of the ellipsometer readings, the refractive index of the film can be read

from the calculations. Other possible methods of determining

the refractive index of adsorbing films have been given by

McCrackin and Col son [9].

2<>7. Confidence Limits of Film Thickness

and Refractive Index - CNDE and CNDC

The instruction CNDC performs the same calculations as

instruction CND discussed in the previous section and, in addition,

computes limits of error for the calculated refractive index corresponding to the experimental errors in the ellipsometer measurement

A and i|r . The experimental errors, EA and E\|i in A and ty must have

been previously given by an EDP instruction or their standard

values of 0.015° are used. This instruction is illustrated in

figure 4.

15
The 95% confidence limits of the experimental errors based on

a large number of degrees of freedom must be used for EA and Eijj.

That is, the true value of A lies in the range A-EA to A+EA with

95% probability, and likewise for ijj. An ellipse is then calculated

such that the true values of A and ilr will lie within it with 95%

probability. Figure 5 shows the ellipse for the conditions of figure 4.

(When the errors are equal, this ellipse becomes a circle.)

After calculating the best values for the refractive index

n^ and the thickness d,. of the film from the measured A and \jj readings, the computer chooses eight evenly-spaced pairs of values

from the probability ellipse and computes values of n,. and d^ for

each pair. An example is shown infigure4. The best way to represent these values is to plot n versus d,. as shown in figure 6. The

points define a curve (approximately an ellipse) surrounding the

best value. The true values of n and d lie within this curve

with 95% probability.

This error analysis only takes the errors in the ellipsometer

readings into account. Other errors, such as errors in the refractive indexes of the medium and substrate, errors in the angle of

incidence, etc., will, of course, increase the error in the calculated values of n- and d .

2 8. Calculations Involving Multiple Films - FILM

The calculations previously described may also be performed

for a number of films on the substrate Three films on a substrate

are represented infigure7. In order to compute the values of A and

\!/ for reflection from this surface, the optical constants and thickness of the three films and the optical constants of the medium and

substrate must be given. The films are numbered starting from the

surrounding medium. The optical constants of the surrounding

medium, film no. 1, and the substrate are given by NM, NF, and NS

or CNS instructions as previously described. The thickness of film

16
number 1 is treated in the same way as previously described for a

single film, i.e., it is given with the CTABLE instruction when the

CTABLE instruction is used, or is calculated by the CD, CND, CNDE,

or CNDC instructions. The optical constants and thicknesses of the

films are given by the FILM instruction. With this instruction:

FILM starts in column 1, the number of the film starts in column

10, the thickness of the film in column 20, and the optical constants of the film, n and x, start in columns 30 and 40. Up to 900

films may be specified.

2.9. Inhomogeneous Films - IFILM

All the previously described computations are for homogeneous

films, i.e., films for which the optical constants are the same at

all points in the film. However, inhomogeneous films for which the

optical constants vary along the thickness of the film may be

accurately approximated by multiple homogeneous films [9] by use

of the instruction FILM described in the previous section. To

o illustrate, consider a film 100 A thick whose optical constants

vary from n=l, yj=0 adjacent to the surrounding medium to n=3, x=1.4

adjacent to the substrate. Let the variation be linear with

distance. Then the film may be approximated by 10 homogeneous

films as shown in figure 8. Let the optical constants of the substrate be n=3 and x=1.4 and the other parameters of the system have

their standard values. Then the reflection coefficients and A and

\|r for reflection from the film are calculated by the instructions

given below.

17
Column No.

1 10 20 30 40

NS 3. 1.4

NF 1.1 .07

FILM 2. 10. 1.3 .21

FILM 3. 10. 1.5 .35

FILM 4. 10. 1.7 .49

FILM 5. 10. 1.9 .63

FILM 6. 10. 2.1 .77

FILM 7. 10. 2.3 .91

FILM 8. 10. 2.5 1.05

FILM 9. 10. 2.7 1.19

FILM 10. 10. 2.9 1.33

CTABLE 10.

The results are shown in figure 9. Approximating this inhomogeneous

film with 60 homogeneous films gives A=117.625° and \|/=30.852° o Thus, approximating this inhomogeneous film by 10 homogeneous films

gave results accurate to 0.1° in this case.

The amount of input data for this type of calculation may be

reduced by using the instruction IFILM. The data given with IFILM

are the number of homogeneous films to be used to represent the

inhomogeneous film, the thickness of the inhomogeneous film, the

optical constants of the surface and the inhomogeneous film in contact with the immersion medium, and the optical constants of the

surface of the film in contact with the substrate. However, film

number 1 (the film next to the surrounding medium) is not assigned

a thickness or optical constants by IFILM. Thus, the previous

calculation may be obtained from the data:

Column No.

1 10 20 30 40 5(3 60_

NS 3

.

1.4

NF 1.

IFILM 10. 100. 1. 0. 3. 1.4

CTABLE .

18
The inhomogeneous film is represented by films 2 to 11. Film

number 1 is given zero thickness so has no effect on the results.

The values given with IFILM are the number of homogeneous films

used to approximate the inhomogeneous film, the thickness of the

inhomogeneous film, the values of n and k at the top (next to

surrounding medium) of the inhomogeneous film, and the values of n and k at the bottom (next to substrate) of the inhomogeneous film,

respectively. (See table 2, page 5.) Note that all data, including

the number of films, must contain a decimal point.

2.10. Multiple Reflections of the Light

The sensitivity of ellipsometer readings to a thin film on

a reflecting surface may be increased by reflecting the light from

the surface a number of times. An example of this technique, as

well as a discussion of the increase of sensitivity, has been given

by Stromberg, Passaglia, and Tutas [10], To analyze such results

the number of reflections may be entered, starting in column 10,

on a NREP instruction. The number of reflections must have

a decimal point even though it is an integer. All following calculations will then be performed for multiple reflection of the light.

Notice that this instruction is not necessary for the multiple

reflections of the light that occur from the surfaces of multiple

films, but only for the case that the light emergent from reflection from the surface is again incident on and reflected from the

surface.

2.11. Films Composed of a Mixture of Materials

In some applications of ellipsometry, the film being studied

is a mixture of a material with the medium above the film. For ex- ample, polymer films adsorbed from a solvent on a solid surface [10]

19
consist of a mixture of the polymer and solvent. As another

example, a rough clean substrate may be approximated by the system

of the substrate, a film consisting of a mixture of the substrate

and the medium, and the medium. For these applications, after the

refractive index of the film, n_ , is determined by ellipsometry,

the fraction of the film composed of the material and the amount

of material per unit area of the surface is desired. These

quantities are computed by the instructions CD, CND, CNDC, and

CNDE. The amount of the material per unit area of the surface in

2 units of mg/cm is printed in the column labeled ADSORPTION and the

weight fraction of material in the film is printed in the column

labeled WT FRACTION*

In order for these calculations to be meaningful, the density

of the medium, p1

, the density of the material in the film, p , and the refractive index, n , of the material in the film must

P

have been previously given by the instruction MIXED (see table 2

on page 5). The refractive index of the material in the film of

the medium are assumed to be real.

The calculations are described in Section 3.

2„12. New Pages, Reset of Variables, and Comments

The instruction PAGE will start a new page of output. For the

Fortran V version of the program, comments may be punched in

columns 10 to 80 on the same card and will appear on the new page

of output o For the Fortran IV version, the comment must be punched

on a card following the card containing the instruction PAGE. For

this case, any location on the card is satisfactory.

The instruction RESET has the same function as the instruction

PAGE and in addition resets all the parameters to the values given

in table 3. This instruction placed between two sets of calculations insures that none of the data given for the first set will

be used in the second set of calculations.

20
In addition, for the Fortran V version of the program, any

card that does not contain one of the instructions listed in table

2 starting at the first column of the card will be printed as

a comment. Therefore, comments may be freely interpreted with the

data cards. For the Fortran IV version of the program, comments

can only be inserted following a PAGE or RESET instruction.

The instruction ST0P is placed at the end of the program.

2.13. Optical Constants of an Adsorbing Film of

Given Thickness - CNK, FT

The instruction CNK computes values of n and k of a film from

ellipsometer readings. Values of A or i|; or the ellipsometer

readings are given with the CNK instruction in the same way as

described in section 2.3 for instructions CD, CND, etc. The

instruction CNK must be preceded by instructions FT, NF, and NS

or CNS. Also, CWP or WP must be given if ellipsometer readings are

given with the CNK instruction and NM, AI, and WL instructions if

values other than the assumed values are to be used for the refractive index of the medium, the angle of incidence, and the wavelength of the light.

The instruction FT gives the thickness or thicknesses of the

film for which the optical constants are to be calculated. The

values given with the FT instruction are the same as for the CTABLE

instruction (table 2)

The values given with the NF instruction give the ranges in

which the computer will search for values of n and k that correspond to the given ellipsometer measurements. Multiple solution's

may be found, in which case, means other than the ellipsometer

measurements must be used to chose the correct solution. The

incremental values for n and k given with the NF instruction determine the fineness of the search for the values of n and h. If "two solutions for n and k exist, the n values or k values must differ

by more than the incremental values for both solutions to be found.

21
An example of these instructions is given in figure 10. The

assumed values are used for the index of the medium, angle of

incidence, and wavelength of light. For every given value of film

thickness, three solutions for n and % (NF and KAPPA) are found in

this example. The accuracy to which they are determined is given

in the ANF and AKAPPA columns.

Thus, for these data, the exact solutions of n and h, may differ

by up to .0002 and .00002, respectively, from the solutions given.

For each solution of n and % and each film thickness, values

of A and \|/ are calculated,, The differences between these calculated values and the values given with the CNK instruction are

given in the ADELTA and APSI columns. These values shall be less

than the inaccuracies of the given values of A and ij; for the

solution to be satisfactory. If A and ijj are correct to 0.01°,

some of the solutions, for example the first solution, are

not satisfactory. These data could be rerun with smaller incremental values for n and k on the NF instruction in order to locate

the solutions more accurately.

3. Theory and Calculations

The theory of the ellipsometer has been given by various

investigators [1-8] „ Therefore, only special points of the theory

necessary to understand the calculations performed by this program

will be given.

3.1. Computation of A and ty From Ellipsometer Readings

Let P, A, and Q be the azimuths of the polarizer, analyzer,

and wave plate, respectively, for extinction of the light by the

ellipsometer For the calculations described in this section, the

components of the ellipsometer must be arranged in the order of

light source, polarizer, wave plate, reflecting surface, analyser,

22
and light detector. All azimuths are measured counter-clockwise

from the plane of incidence looking into the light beam.

The wave plate is characterized by the ratio, T , of the

transmittance along its fast axis to the transmittance along its

slow axis* and the relative phase retardation, A , along the axes.

In the following equation it is characterized by the complex

number

n = T exp(-iA ) = T cosA - iT sinA (1) 'c c

v c' c c c c v/

that has been previously computed. The surface is characterized

by the complex value

p = R /R = tanf exp(iA) (2)

p s where R and R are the complex reflection coefficients for light

polarized parallel and normal to the plane of incidence of the

light beam. The value of p is to be calculated from readings P,

A, and Q, and p . This calculation is based on the formulas

derived by Holmes and Feucht [7], Modifying their equations 19

and 21 for the conventions used in this report for measurement of

P, A, and Q gives

tan A [tan Q + p tan (P-Q)]

P

= p tan Q tan (P-Q) -1

^

far extinction of the light. A derivation of eq (3) is also given

in Appendix II.

A and \|/ are computed from eqs 1 through 3. If two sets of

values of P and A are given, p is computed for each set and the

average of the two values used to compute A and \|/

.

* When the previous program was written, the importance of T in

ellipsometry was not recognized, thus no correction for it was

included in the calculations.

23
3.2. Computation of Constants T and A of the Wave Plate

For computation of the constants of the wave plate, two sets

of readings of P and A (P, and A, and P„ and A_), that produce

extinction of light for the same value of Q are required. Then,

by eq 3

tan A, [tan Q + p tan (P -Q)

]

P

p tan Q tan (P,-Q) -1

*c v 1

'

tan A

2 [tan Q + Pc tan (P -Q)

]

pc tan Q tan (PT^Q) -1 *

Solving for p gives

B ± j [-B2

- tan (P^Q) tan (P

2 -Q)] 3 Pc =

- tan (P^Q) tan (P

2 ~Q)

^

where

tan A, [tan (P 9 -Q) tan Q - tan (P -Q)/tan Q]

g _ ± £

J; 2 (tan A, - tan A

2

)

tan A

2

[tan (P

1 -Q) tan Q - tan (P2 ~Q)/tan Q]

2 (tan A

±

- tan A

2

)

'

Using eq 1 we find

c ±[-tan (Pj^-Q) tan (P

2 ~Q)] 2

(5)

T = i . (6)

Since T is real and positive, tan (P.. -Q) tan (P 9 -Q) must be

negative and the plus square root is used. Also,

±[-B2 - tan (P

1 -Q) tan (P

2 ~Q)]

tan A c

= 1 . (7 )

24
A is defined as the phase of light transmitted along the fast

c axis of the wave plate minus the phase of light transmitted along

its slow axis. Therefore, ^ A ^ 180°, so sinA is positive.

Then, by eq 1, the imaginary part of p is negative. This

requires the negative sign for the imaginary term in eq 4 to be

used since the denominator has been shown to be positive. Thus,

+ [-B2

- tan (P-^Q) tan (P

2 ~Q)]

tan A c = - . (8)

3.3. Computation of the Angle of

Tilt of the Reflecting Surface

For this calculation, the reflecting surface is assumed to be

rotated through a small angle 6 around an axis of the intersection

of the specimen and the plane of incidence. This rotation or

tilting of the specimen has many effects. The angle of incidence

to the surface will be changed, but by a negligible amount for

small angles of tilt. The direction of the reflected light beam

will be changed but is assumed to still pass through the aperatures

in the instrument. The main effect will be a reading error of the

azimuths of the polarizer, analyzer, and wave plate. Since these

elements are alined with respect to the reflecting surface, a tilt

of the surface by an angle 6 will change the readings of these

azimuths by 6 . That is

P

l = V+ 6 P

2 = V + 5

A

l = A

l'

+ 6 A

2 = A

2

7 + 6 ( 9 ) Q = Q

7 + 6

where P , A.. , P , A9

, and Q are readings of the azimuths and

J_ _L c* £i P, , A,, P_, A9

, and Q are the true azimuths that would be measured

-L X Zj a by a properly alined instrument.

25
By eq 3, for the first set of readings

tan (A

1

/+6)[tan (Q +6) + Pc tan (P^-Q')]

P

p tan (q'+8) tan (P '-q') -1

(10)

c

The tangents are expanded in 8 and only the term in 6 retained,

since 8 is small.

tan (A '+ 6) = tan A '+ 8/cos9 A

tan (q' + 8) = tan q' + 8/cos2 Q

7 (11)

Eqs 11 are substituted in eq 10 and the result simplified to give

(12)

i

where

a, = tan An

' [tan C}' + p tan (P ' - q')]

y n = O tan (P ' - q') tan Q

7 1 *c K 1

$ 1 = ^/(sin A

1

') +

[tan A

1

/ -a

1 Pc tan(P

1

/ -Q / ) / y ] / cos

2 Q

/

„ (13)

Similar formulas apply to the other set (P_ , A_ , and Q) so

a, + 6,6 a + 6 6 P

= y

1 = y

' <">

X lX 2

Solving for the angle of tilt gives

a

27 l

" a

l72

6 = -LJz LJl- . (15)

?172

" Wl K

When 8 is calculated, the azimuths are corrected by eq 9

and p is calculated by eq 3.

26
3.4 e Calculation of the Thickness of a

Film from A and ^

This calculation has been previously described [4], but since

the final formulas were not given, they are summarized here.

The total reflection coefficients Ir and R for light polarized parallel and perpendicular to the plane of incidence are given

by

T?P - 4 + 4 y

XV —

1 + 4 4 Y

R

n

=

n

r 12 + 4*

1 + nr12

n

r23

Y (16)

where r^„ , rl_ , r.. 9

, and r„_ are the reflection coefficients

between medium and film (indices 1 and 2) and between the film and

substrate (indices 2 and 3).

Y = exp[-4Tri n

2 coscp2

d

2

/X] (17)

where n9 and d are the index of refraction and thickness of the

film and cp_ is the angle for light in the film given by

i

coscp

2 = [1 -(n^in^/n^ 2

] 2

. (18)

The ellipsometer readings determine

p = tan\|/ exp(iA) = R^R*3

. (19)

From eqs 16 and 19,

EY2 + BY + C = , (20)

where

_ p p n 21 p n

E = 0r21

r32r32 - rnr32

r32

27
B = ^( r32

+ r 21

n

r21rP

32,)

C n

= <> r21 " rP

21 •

The two solutions of eq 2C1

Y1i = [-B +![B3 - 4EC)

X

S

] /

32 p n n

T -TV T

p 21 21 32

(21)

(22)

Y

2 = [-B -(B2

- 4EC) 2

] Ae (23)

are calculated. Then, from each solution a value for the film

thickness d is calculated. These values, called D2C1 and D2C2

are calculated by subroutine CALDX.

These values are complex numbers. Since the film thickness

is a real number, the imaginary part is due to error in either the

ellipsometer readings or the values of the refractive indices or

in the angle of incidence used for the calculation, so must be

small. Therefore, the value of D2C1 or D2C2 with the smallest

imaginary part is taken as the correct solution. The real part is

taken as the film thickness while the imaginary part is taken as

a measure of error D

3.5 Calculation of Both the Refractive Index

and Thickness of a Film From Values of A and \|/

The preceding section described the calculation of the thickness of a film for a given value of the refractive index of a film,

Two complex numbers, D2C1 and D2C2, were obtained for the film

thickness. The real parts of the numbers were taken as the film

thickness and the corresponding imaginary parts as measures of

error

The subroutine S0LN2 determines the refractive index of the

film that will minimize the absolute error of the error term.

28
For simplicity, we consider that the index of the film is

real, i.e , n=0. The calculations may also be performed for a

given value of k.

The method requires limits, n and n , for the refractive

index n of the film, such that n < n . First, the complex

film thickness D2C1 is computed with n_ and n used for the

index. Since the error term, the imaginary part of D2C1, is zero

for the correct value of n_, it is expected to change sign as n„

is varied from n~ to n . Therefore, if the error terms for

n9 and n_ have the same sign, it is assumed that no solution for

n exists using D2C1 for the film thickness. See figure 11a.

If the error terms have different signs for indices n„ and

n_ , the error term is evaluated for the intermediate index

n_ = (n + n9

7/ )/2. Then the interval containing the index n„ is

narrowed by consideration of signs of the error terms. For example,

suppose that the signs of the error terms for the indices n , n , n" are -, -, +, respectively Then the error term changes sign

from n'" to n

'' so the solution for n for which the error term is

zero must lie m this range,, That is n„ < n < n .

The value of n? is then tested to determine if it is sufficiently close to the solution for n . This is done by computing

values of A and \|/ using the real part of D2D1 for the film thickness. Of course, use of the complex value of D2C1 for the film

thickness would give the experimental values of A and \|/, but

dropping the imaginary part will change the computed values of A and \|/. The value of n?

' is then judged to be sufficiently close to

the solution for n? if the computed values of A and \|/ are both

within 0.005° of the experimental values. If the differences

between the computed and experimental values of A and \j/ are larger

than 0.005°, the range is further reduced until they agree to

within .005°. (This value of 0.005° can be changed by the ELDP

instruction.) The process is then repeated using D2C2.

29
Note that the values given by the ELDP instruction determine

the closeness of the search for the refractive index n of the film,

They should be less than the experimental errors of A and t

? but if

they are too small, the computation time will be excessive.

Figure 11a shows a possible relationship of the error term to

assumed values of the film index. For the limiting values of n? and n9 of the index, the signs of the error terms are different

(- and +) so the computer will search for a solution by computing

the error term for the intermediate index n9

. The signs of the

_ . i in a error terms corresponding to n9 , n„ , and n_ are -, -, and +,

respectively. The solution is then correctly determined to lie

between n9 and n9 and this range will be subdivided. The procedure is seen to converge to the solution n9

.

Cases in which the procedure fails are shown in figures lib

to lid. In figure lib the error terms are of the same sign

(positive) for both n? and n_ so the computer assumes that no

solution exists. However, there is a solution, n , outside the

assumed range. In this case, too narrow a range for n was used.

In figure lie the error terms corresponding to the limits n„

and n are also of the same sign (positive) so no solution will

be found although two solutions between n9 and n exist. In this

case, the assumed range was too broad. If the range has been subdivided and both sections searched for a solution, both ^solutions

for n would have been obtained.

In figure lid the error terms for n„ and n„ have different

signs so the computer will search for a solution by first computing

the error term for index n9

. The signs of the error corresponding

i in - a , .-,-,. i to n , n , and n are +, -, -, so the computer will continue the

search in the interval n„ to n and converge to the solution.

However, the two solutions between n and n

" will be missed.

Again, the range of the index must be subdivided to obtain all the

solutions.

30
3 6. Confidence Limits for the Refractive Index

and Thickness of a Film

The preceding section described the calculation of the refractive index and thickness of a film from measured values of A and \|f as performed by the instruction CND Instruction CNDC performs the

same calculations and, in addition, calculates the uncertainty in

the refractive index and thickness due to uncertainty in the values

of A and \|r. Let the measured values of A and \j/ be A n and \|f_ and let their

95 percent confidence limits be A_-6A to A n+6A and \|/ n -5^ to ^ +6^.

That is, the true value of A lies between A.-.-6A and A-.+ 6A with 95

percent probability, and similarly for \|/. (6A and 6ty equal 1.96

times the standard deviations of A and ^ when based on a large

number of degrees of freedom.) Then both A and ^ will lie within

the joint confidence ellipse (also called contour ellipse)

(A-A ) 2 /(6A) 2 + (\h^Q

) 2 / (8\|0 2 = 1.560 with 95 percent probability*

[11, 12]. An example is shown in figure 5. For each value of A and ty on the ellipse, values of n and d9 , the refractive index and

thickness of the film, may be calculated. Plotting these values of

n„ versus d , a curve, approximately an ellipse, is obtained as

illustrated in figure 6 This curve is the 95 percent joint confidence curve for n„ and d • i.e., the true values of n_ and d„ lie

within this curve with 95 percent probability.

The instruction CNDC determines eight points of the joint

confidence curve for n9 and d9

. The maximum and minimum values of

n? and d9 are also printed as their individual confidence limits*

The instruction CNDE performs the same calculations as CNDC but

prints only the individual confidence limits of n^ and d^.

*The equation of the joint confidence ellipse is generally written

in terms of standard deviations. For convenience, we have ex- pressed it in terms of 95 percent confidence limits.

31
The confidence limits for A and \|f, 6 A and 6\|/, are given by an

EDP instruction before the CNDC or CNDE instruction. If an EDP

instruction is not given, the value 0.015° is used for both 6A and

3.7. Volume Fractions and Total Amount of Material

for a Film Composed of a Mixture of Two Materials

Consider a film that is a mixture of the immersion medium of

refractive index n.. and density p.. and a material of index n and

density D . Let the refractive index of the film that has been

calculated be n... We desire to compute the weight fraction, f, of

the material in the film.

The specific refraction, S^, of the film is additive in the

specific refractions of the medium, S, , and the material, S . Thus

S f/Pf

= f S

p

/p

p

+ (1-f) S

1 /p]

_ , (24)

where p, is density of the film and S^ = (n ,2 -l)/(n ^+2) and

similarlyJ for S, and S . To determine the density of the film,

1 p

J ' we have the volume of a unit weight of the film as f/p + (1-f) p,

assuming additivity of the partial volumes of the components of

the film. Therefore, the density of the film is

pf

= l/[f/P

p

+ (l-f)/p

±

] . (25)

Substituting in the previous equation and solving for f gives

P

B_ (26) Pp + ( Sf -sp

)p1/(s

1 -sf

)

•

The amount of material per unit area of film is p d f where d

is the thickness of the film. If the densities are in g/cm3 and

o the film thickness in A, the amount of material per unit area of

film in mg/cm2 is given by 10~ 5 p^df

„

32
3.8. Computation of Ellipsometer

Readings From A and \|/

Substituting eqs 1 and 2 in eq 3 and rearranging gives

(cosA+i sinA) [T tanQ tan(P-Q) (cosA -i sinA )-l]

TCU1A _ C C C /07 \ tan\|/ " tanQ + T tan(P-Q) (cosA -i sinA )

* ^

'

This expression is reduced to the form of a real and imaginary

term. Then, the imaginary term must be zero since tanA/tan^ is

real Solving gives

tan(P-Q) = (-R ± >/r3 + 1 )/T

c

, (28)

where

tanQ sin (A -A ) - sin(A+A )/ tanQ

R =

°- °- (29)

2 sinA

or

P = arctan[(-R ± v/R2 + l) /T ] + Q . (30)

Two values of P are thus calculated. A value of A may be

calculated by eq 27 for each value of P

.

3.9. Computation of n and n of a Film of Given Thickness

This computation is best described by considering the

instructions

NS 2 5

FT 50.

NF 1.1 0.1

CNK 90.5 32.

1.3 0.2 0.2 0.2

33
The computer searches for a value of n between 1.1 and 1.3 and

a value of % between 0.2 and 0.6 for a film of 50 A thickness that

will give values of A and \|/ near 90.5° and 32°. First, the

computer increases the range of n and % by their increments. That

is, the range of n becomes 1.1-0.1 to 1.3+.1 and the range of x becomes 0.2-0.2 to 0.6+0.2. This is graphically illustrated below.

.8

.6

4

.2

0.0

X X X X X

X * * * X

X * * X

x * * * X

X X X X X

1.0 1.1 l e 2 1.3 1.4

n

Increasing the ranges of n and % prevents solutions near the

boundary of the range being missed in the searching procedure.

For every pair of values of n and k in this figure an error

term |p -p is computed. In this error term, p is the value of

p computed for the value of n, k and thickness of the film, and p

is the value of p corresponding to the values of A and \|/ ( eq 1)

given with the CNK instruction. This error term is zero only when

the values of n and k are a solution, and is small when n and k are

near a solution Q Therefore, the computation consists of searching

for values of n and % that minimize the error time.

This is accomplished by comparing each error term with the

error term at surrounding points. If the error term is less than

that at the surrounding points, a minimum is assumed to occur near

the value of n and % for the error term. For example, in the above

figure, if the error term for n=1.2 and h=0.4 (represented by 0) is

34
less than the surrounding error terms represented by *, a minimum

of the error term is tentatively assumed to occur in the range of

n=l.l to 1.3 and h=0„2 to 0.6. All interior points of the grid of

n and k values are thus tested.

The next step is to search the range of n and k to locate the

minimum. This procedure is illustrated by the following figure.

7 24 8 25 9

X X X

19 20 21 22 23

4 17 5 18 6

X X X

12 13 14 15 16

1 10 2 11 3

X X X

26 27 28

n

This figure represents the ranges of n and K for which a minimum is found to occur. That is, the error terms at points 1 to 9,

F, to F„ (represented by x on the figure) have been calculated and

the error term to point 5, F,-, has been found to be less than F.

,

Fo* **V ^4» ^6* ^7' ^R» an(^ ^Q' "I

"*le P rocedure is to calculate the

error term at point 13, F, , . If it is greater than F,-, F, . is

calculated. If F is also greater than F,-, F^ ^ is calculated,

and so forth. If F^, F^, F^, F

1Q , F^, F^ F

2Q , and F±7

are

all greater than F,-, they define a new reduced range for the

minimum and the process is repeated.

35
Consider the case that one of the terras, say F, 5

, is less than

F,.o Then F.. _ is also less than F

?

, F_, F

g

, and F. . since these terms

are greater than Fc. Terms F, , , F, 6

, and F,

g

are then calculated. If these terms are all greater than F, j-> a new reduced

range centered around F, ,- is found and the process is repeated.

Next, consider the case that one of these terms, say F, , , is

less than F, (-° In this case terms F9fi , F„7

, and F9fi are computed

If these terms are all greater than F, , , a range centered around F,

,

is found. However, if one of these terms, say F96 , is less than

F, , , another process to follow the minimum is used. All eight

terms surrounding F9fi are calculated and if any of them is less

than F9fi , its eight surrounding terms are calculated and compared

with its value. This secondary process is repeated five times.

If an error term less than its eight surrounding terms is obtained,

the process of reducing the range continues. If not, the process

is considered to fail and a diagnostic message printed.

The process of reducing the range is repeated ten times,

thereby reducing the original range by a factor of about 2000

„

These calculations are performed by the subroutine MIN2V and the

error terms are calculated by the function FUNC.

Oldham [14] has programed and tested another method for determining the optical properties of adsorbing films of given thickness

from ellipsometer readings. His method is a generalization for

complex variables of Newton's method of finding roots of a

function. His method probably finds solutions with less calculation than the method used in this program. However, it is more

likely to fail in finding solutions and does not ordinarily give

multiple solutions as does the method used in this program.

36
4. The Fortran Program

This computer program was originally written in Fortran V for

use with the Univac 1108 computer. Because some of the features

of this Fortran compiler do not apply to other Fortran compilers,

the program in Fortran IV is given in Section 4.1. The data input

preceding is identical for both programs except for the instructions discussed in Section 2.12. With the Fortran IV program the

first card of the data and the card following the PAGE and RESET

instructions must be a comment card, and no other comment cards

may be used in the program. Also, comments may not be placed on

the PAGE and RESET instruction cards. These restrictions are

required because the instruction of the form, READ (0,i) List,

in Fortran V is not generally available in Fortran IV.

The Fortran IV program used logical unit 5 for input and

logical unit 6 for output. These logical unit numbers may have to

be changed for some computers.

For both versions of the program, complex and real values are

combined in arithmetic statements with the result being complex

The Fortran V version of the program and both versions on

cards are available from the author.

37
4.1 FORTRAN IV PROGRAM

CALCULATION FOR ELLIPSOMETER

FOR iFlLMf THE NUMBER OF FILMS ADD. RESET IS NECESSARY TO ERASE FILMS

CHANGED TO USE KAPPA INSTEAD OF IMAGINARY PART OF REFRACTIVE INDEX

DIMENSION IDT (It) »FNSI(2) ,DFNS(2) »FNSE(2)*»FNSX (2) »TN<8) »TD(8)

DIMENSION INSTOH

COMMON FNlrSl»ClrN»FNX(900) »D(900) r FN2r CRP,CRN» CRP32> CRN32» WL»

1 RP21»RN21rDEL»PSIrC2»C3»FN2K2) »DFN2(2) >FN2E(2) >FN2X(2) »T1»T2»T3»

2 T4rD2»DELDX>PSIDX» ELDELfELPSI »DENP»FNP»DEN1»MX»K» IS

COMPLEX C2»C3#CI»CIP1»CN»CNM1»CRN»CRN21>CRN32>CRP»CRP21»CRP32

COMPLEX FN2tFNX»FNS» RH0>RN»RN21. RP»RP21» RCrSRC

COMPLEX Tl>T2»T3»T4»RH0C»RH01»RH02»CAT»ALPl»ALP2rBETl>BET2

COMPLEX GAM1»GAM2»CAT

INSTRUCTIONS TO BE COMPARED WITH INSTRUCTIONS GIVEN WITH

DATA

M= 1 DATA INST/6HPAGE

7 8

L 6HWP »6HNREF

If 15

> 6HCTABLE»6HCD

21 22

5 6HCDN r6HCDNE

28 29

f 6HALINE »6HST0P

3

»6HNF

10

4 »6HNS

11

5 »6HAI

12

6

»6HWL

13

6HELDP

17

»6HCNDE

»6HRESET

18

>6HCNDC

»6HFILM

19

»6HCNS

»6HMIXED

20

»6HCWP

24

»6HAT

25

»6HCAT

26

»6HCATA

27

»6HIFILM

2 fbHNM

9

»6HEDP

16

»6HCND

23

6HCDNC

30 31

»2HFT»3HCNK/

SRC IS SMALL REFLECTION COEFFICIENT (WITHOUT CONSIDERING

UNDERLYING LAYERS)

SRC(A.B)=(A-B)/(A+B)

SIN»C0S»ATAN»TAN FOR ANGLES IN DEGREES

SIND (X)=SIN(X/57. 29578)

COSD (X)=C0S(X/57. 29578)

ATAND (X)=ATAN(X)*57. 29578

TAND(X)=SIND(X)/COSD(X)

S IS USED FOR CALCULATING SPECIFIC REFRACTION

S ( XN) = ( XN*XN-1 . ) / (XN*XN+2 .

)

GIVE STANDARD VALUES TO VARIABLES

10 FN1I=1.

DFN1=0.

FN1E=0.

AT=0.

ALP=0.

ALA=0.

ALQ=0.

FN2I(1)=1.

FN2I(2)=0.

DFN2(1)=0.

DFN2(2)=0.

FN2E(1)=0.

FN2E(2)=0.

FNSK1)=1.

FNSI(2)=0.

DFNS(1)=0.

DFNS(2)=0.

FNSE(1)=0.

FNSE(2)=0.

AII=70.

AIE=0.

WLI=5461.

38
DWL=0.

WLE=0.

RHOC=(0.»-1.)

TC=1.

DELC=90.

DEN1=1.

DENP=1.

FNP=1.5

N=2

ELDEL=.002

ELPSI=.002

D2I=0.

DD2=0.

D2F=0.

FN=1.

£DEL=.015

EPSI=.015

C READ AND PRINT TITLE CARD

12 READ(5fl5) IDT 15 F0RMAT(13A6»A2)

WRITE(6»17) IDT

17 FORMAT UH1»1<+A6)

C READ AND PRINT CARD OF DATA

20 READ(5»25) L>Z1»Z2»Z3»Z4,Z5»Z6

25 FORMAT(A6»3X»6F10.0)

WRITE<6»27) LfZl»Z2»Z3»Zi+fZ5»Z6

27 FORMATQH t A6» 3X»6F15.5)

C COMPARE L WITH INSTRUCTIONS TO FIND M. DO 30 M=l»31

30 IF(L.EQ.INSTCM)) GO TO 35

GO TO 20

C M= 1 2 3 I 5 6 7 8 9 10 11 12 13 14 15 16

35 GO T0(12»50»70»80»90»100»110»120»130»1^0»10»150»160»305»170»170»

C 17 18 19 20 21 22 23 21 25 26 27 28 29 30 31

1 170 » 170 1 170 » 730 t 170 t 170 t 170 » 250 » 750 1 750 1 270 1 260 , 810 t 305 1 170) »M

C INSTRUCTION NM»REFRACTIVE INDEX OF MEDIUM

50 FN1I=Z1

DFN1=Z2

FN1E=1.U00001*Z3

GO TO 20

C INSTRUCTION NF» REFRACTIVE INDEX OF TOP FILM

70 FN2I(1)=Z1

DFN2(1)=Z2

FN2E(1)=1.000001*Z3

FN2I(2)=Z4

DFN2(2)=Z5

FN2E(2)=1.000001*Z6

C IF DFN2(D IS LEFT BLANK ON DATA CARD* SUBSTITUTE A VALUE OF

C DFN2(1) SO THAT ONLY ONE CALCULATION WILL BE PREFORMED.

IF ( ABS ( DFN2 ( 1 ) ) .EQ . . ) DFN2 ( 1 ) =1 . 000002*ABS ( FN2E ( 1

) -FN2I < 1 )

)

C INSTRUCTION NS» REFRACTIVE INDEX OF SUBSTRATE

80 FNSI(1)=Z1

DFNS(1)=Z2

FNSE(1)=1.000001*Z3

FNSK2)=Z4

DFNS(2)=Z5

FNSE(2)=1.000001*Z6

GO TO 20

C INSTRUCTION AI> ANGLE OF INCIDENCE

90 AII=Z1

DAI=Z2

AIE=l.O0000l*Z3

39
GO TO 20

C INSTRUCTION WLV"WAVE LENGTH OF LIGHT

100 WLI=Z1

DWL=Z2

WLE=1.000001*Z3

GO TO 20

C INSTRUCTION WPr CONSTANTS OF WAVE PLATE

110 TC=Z1

DELC=Z2

RHOC=TC*CEXP ( ( .

» -1 . ) *DELC/57 .29578)

GO TO 20

C INSTRUCTION NREFr NUMBER OF REFLECTIONS OF THE LIGHT

120 FN=Z1

GO TO 20

C INSTRUCTION EDPt CONFIDENCE FIMITS FOR DELTA AND PSI

130 EDEL=Z1

EPSI=Z2

GO TO 20

C INSTRUCTION ELDP, ERROR LIMITS FOR DELTA AND PSI

140 ELDEL=Z1

ELPSI=Z2

GO TO 20

C INSTRUCTION FILM #NUMBER» THICKNESS AND REFRACTIVE INDEX OF FILM

150 I=1.000001*Z1+1

DU)=Z2

FNX ( I ) =CMPLX ( Z3 » -Z3*ABS ( Z4 )

)

N=MAX0(N»I)

GO TO 20

C INSTRUCTION MIXED' DATA FOR MIXED FILMS

160 DEN1=Z1

DENP=Z2

FNP=Z3

GO TO 20

C VALUES OF DELTA AND PSI OR OF PrArAND Q HAVE BEEN GIVEN

C USE PREVIOUS VALUES OF DEL AND PSI IF NO DATA IS GIVEN* Z1=Z2=0

C IF FIRST VALUE(Zl) =0»USE PREVIOUSLY GIVEN VALUES OF

C DELTA AND PSI

170 IF(ABS(Z1).EQ.0.) GO TO 310

IF(ABS(Z5).EQ.O.) GO TO 180

C Z5 NOT =0» SO P1»A1»P2»A2»Q ARE GIVEN

C COMPUTE TWO VALUES OF RHO

Z1=Z1+AT+ALP

Z2=Z2+AT+ALA

Z3=Z3+AT+ALP

Z4=Z4+AT+ALA

Z5=Z5+AT+ALQ

WRITE(6»27) L»Z1»Z2»Z3»Z«WZ5

174 RH01=TAND(Z2)*(TAND(Z5)+TAND(Z1-Z5)*RH0C)/(TAND(Z1-Z5)*TAND(Z5)*

1 RHOC-1.)

RH02=TAND(Z<+)*(TAND(Z5)+TAND(Z3-Z5)*RH0C)/(TAND(Z3-Z5)*TAND(Z5)*

1 RHOC-1.)

C AVERAGE VALUES OF RHO » COMPUTE VALUES OF DELTA AND PSI

RHO=(RH01+RH02)/2.

DEL1=ATAN2(AIMAG(RH01) » REAL (RHOl) ) +57.29578

PSI1=ATAND(CABS(RH01)

)

DEL2=ATAN2(AIMAG(RH02) *REAL(RH02) ) +57.29578

PSI2=ATAND(CABS(RH02)

)

U/RITE (6*175) DELI , PSI 1»DEL2»PSI2

175 FORMAT(bH DEL1=»F12.6»6H PSI1=»F12.6» 6H DEL2=»F12.6»

1 6H PSI2=rF12.6)

IF(FN.EQ.L) GO TO 210

GO TO 200

40
C IF Z3=0» DELTA AND PSI ARE GIVENrOTHERWISE PrAfQ ARE GIVEN

180 IF(ABS(Z3).EQ.O.) GO TO 190

Z1=Z1+AT+ALP

Z2=Z2+AT+ALA

Z3=Z3+AT+ALQ

WRITE(6»27) L»ZW2»Z3

185 RH0=TANu(Z2)*(TAND(Z3)+TAND(Zl-Z3)*RH0C)/(TAND(Zl-Z3)*TAND(Z3)*

1 RHOC-l.)

IF(FN.EQ.L) GO TO 210

GO TO 200

190 DEL=Z1

PSI=Z2

RHO=TAND( PSI )*CEXP((0.»l.)*DEL/57. 29578)

IF(FN.EQ.L) GO TO 310

C CORRECT FOR NUMBER OF REFLECTIONS IF FN NOT =1

200 RHO=CEXP(l./FN*CLOG(RHO))

210 DEL=ATAIM2(AIMAG(RH0) rREAL (RHO) ) *57. 29578

PSI=ATAND(CABS(RHO)

)

IF(M.EQ.25 .OR. M.EQ.26) GO TO 755

215 WRITE(6»220) DELrPSI

220 F0RMAT(6H DEL=»F13.6»5H PSI=»F13.6)

GO TO 310

C INSTRUCTION ATr ANGLE OF TILT OF SAMPLE

250 AT=Z1

GO TO 20

C INSTRUCTION ALINEf ALlNEMENT CORRECTIONS

260 ALP=Z1

ALA=Z2

ALQ=Z3

GO TO 20

C INSTRUCTION IFILM* CALCULATES THICKNESSES AND REFRACTIVE

C INDICES OF A SERIES OF FILMS

270 XNF=Z1

TH=Z2

C REMOVE ALL PREVIOULSY DESIGNATED FILMS.

C WITHOUT NEXT CARD* FILMS WILL BE PLACED BETWEEN PREVIOUSLY

C DESIGNATED FILMS AND SUBSTRATE

N=2

C DN AND DK ARE INCREMENTS OF REFRACTIVE INDEX AND K DN=(Z5-Z3)/XNF

DK=(Z6-Z<+)/XNF

C COMPUTE INDEX AND K (KAPPA) OF TOP FILM

Z3=Z3+DN/2.

Z<+=Z4+DK/2.

C INDEX OF TOP FILM IS Nl» INDEX OF BOTTOM FILM IS N Nl=N+l

NF=1.000001*XNF

N=N+NF

DO 280 I=N1»N

D(I)=TH/NF

FNX(I)=CMPLX(Z3»-Z3*Z4)

Z3=Z3+DN

280 Z4=Z<++DK

GO TO 20

C THICKNESS OF TOP FILM AND t OPTIONALLLY» SETTING OF WAVE PLATE

305 D2I=Z1

DD2=Z2

D2E=l.000001*Z3

Q=ZU

IFCM.EQ.30) GO TO 20

C SET INITIAL VALUES

310 FN1=FN1I

41
311 AI=AII

312 S1=SIND(AI)

C1=C0SD(AI)

IFCM.EQ.19) GO TO 710

315 WL=WUI

WRITE (6»316) FN1»AI»WL»TC»DELC

316 FORMATQ7H0INDEX OF MEDlUM=»F10.5»2lH ANGLE OF INCIDENCE=»F10.3r

1 14H WAVE LENGTH=»F10.2f5H TC=»F10.5>7H DELC=»F10.3)

IFCM.EQ.14) GO TO 320

IF(M.EQ.31) GO TO 318

C PRINT TABLE HEADINGS EXCEPT FOR CTABLE OR CNK

WRITE(6»317)

317 FORMATU23H0 THICK NF REAL KAPPA ADSORPTION D DEL D P 1SI ERROR NF-NM NS-NF WT FRACTION NS REAL KAPPA

2 K)

GO TO 320

C PRINT HEADINGS FOR INSTRUCTION CNK

318 PRINT 319

319 FORMATClH0#8X»2HNFf 15X»5HKAPPA»9Xr4HD NF.11X»7HD KAPPA»9Xf

1 5HERR0Rr3X»7HD DELTA»5X»5HD PSI > 10X>5HTHICK)

320 NP1=N+1

C SET VALUES FOR REFRACTIVE INDEX OF SUBSTRATE

325 FNSX(1)=FNSI(1)

326 FNSX(2)=FNSI(2)

327 FNXCNP1)=CMPLXCFNSX(1) »-FNSX(2)*FNSXCl)

)

C COMPUTE REFLECTION COEF BETWEEN LAYERS 3 AND 4 IFCN.EQ.2) GO TO 335

X=SIND(AI)*FN1

CN=CSQRTC1.-(X/FNX(N+1))**2)

CNM1=CSQRT(1.-(X/FNX(N) )**2)

CRP=SRC(FNX(N+1)*CNM1»FNX(N)*CN)

CRN=SRC ( FNX ( N> *CNM1 t FNX (N+l ) *CN)

IF(N.EQ.3) GO TO 335

NX=N-3

I=N-1

DO 330 J=1»NX

CIP1=CSQRT(1.-(X/FNX(I+1) )**2)

Cl= CSQRTC1.-CX/FNXCI)) **2)

RP=SRCCFNXCI+1)*CI>FNXCI)*CIP1)

RN=SRC(FNX(I)*CI»FNX(I+1)*CIP1)

T1=FNX(I+1)*CIP1*D(I+1)/WL

CRP=RC(RP»CRPrTD

CRN=RC(RN»CRNrTl)

330 1=1-1

335 IFCM.EQ.31) GO TO 850

340 FN2X(2)=FN2I(2)

342 FN2X(1)=FN2I(D

343 FN2=CMPLX(FN2X(1) »-FN2X (2) *FN2X ( 1 )

)

C CALCULATE REFLECTION COEF BETWEEN LAYERS 2 AND 3 CALL CRC

IFCM.EQ.14) GO TO 350

K=l

C EXCEPT FOR CTABLErCALCULATE VALUESrTl AND T2# OF FILM

C THICKNESS

CALL CALDX (T1»T2)

IFCM.EQ.15) GO TO 455

C EXCEPT FOR CTABLE OR CD»CALCULATE COMPLEX VALUES-T3 AND

C T4 OF FILM THICKNESS FOR UPPER VALUE OF FILM

C REFRACTIVE INDEX

FN2=CMPLX C FN2X C 1 ) +DFN2 C 1 ) , -FN2X C 2 ) * C FN2X C 1 ) +DFN2 C 1 ) )

)

CALL CRC

CALL CALDXCT3fT4)

42
C IS=1 FOR NO SOLUTION FOUND IS=2 FOR SOLUTION FOUND

IS=1

IF(( AIMAG(Tl) . GT.O.. AND. AIMAG (T3) .LT. . ) .OR. < AIMAG(T1 ) .LT.0..AND.

1 AIMAG(T3).GT.O.)) GO TO 485

C NO SOLUTION FOR N2 FOR FIRST SOLUTION OF QUADRATIC

345 IF <( AIMAG (T2). GT.O.. AND. AIMAG (T4) .LT.O.). OR. (AIMAG(T2) .LT.O.AND.

1 AIMAG(T4).GT.O.)) GO TO 487

C NO SOLUTION FOR N2 FOR SECOND SOLUTION OF QUADRATIC

347 IFUS.EQ.2) GO TO 670

C NO SOLUTION FOUND» TRY OTHER QUADRANTS FOR SQRT IN CALDX

K=3

FN2=CMPLX(FN2X(1) ,-FN2X (2) *FN2X (1)

)

CALL CRC

CALL CALDX (T1*T2)

FN2=CMPLX ( FN2X ( 1 ) +DFN2 ( 1

)

t -FN2X ( 2 ) * ( FN2X ( 1 ) +DFN2 ( 1 ) )

)

CALL CRC

CALL CALDX(T3rT4)

IF(( AIMAG(Tl) .GT.0..AND.AIMAG(T3) .LT. 0. ) .OR. ( AIMAG (Tl ) .LT. . .AND.

1 AIMAG(T3).GT.O.) ) GO TO 490

348 IF (( AIMAG <T2). GT.O.. AND. AIMAG (T4) .LT.O. ) .OR. (AIMAG (T2) .LT.O.AND.

1 AIMAG(T4).GT.O.)) GO TO 489

GO TO 670

C TABLES OF DEL AND PSI FOR CTABLE

350 WRITE<6»360)

360 FORMAT (1H0 »3X»5HTHICK» 4X, 3HDEL» 5X# 3HPSI »4Xr 14HRF.CF. PARALLEL >6X»

1 12HRF.CF. NORMAL »4X»2HNF»4X»5HKAPPA»2X»2HNSr5X» 5HKAPPA,4HW

2 2HPl»5X'2HAl»5Xr2HP2r5X»2HA2)

390 D2=D2I

400 CRP21=RC(RP21»CRP32»FN2*C2*D2/WL)

CRN21=RC(RN21»CRN32»FN2*C2*D2/WL)

RH0=CRP21/CRN21

IFCFN.NE.l.) RHO=CEXP(l./FN*CLOG(RHO)

>

DEL=ATAN2(AIMAG(RHO) »REAL(RHO) )*57. 29578

IF(REAL(RHO).LT.O. .AND. ABS( AIMAG(RHO) ) .LT. 1.0E-6) DEL=180.

PSI=ATAND(CABS(RHO)

)

XM1=CABS(CRP21)

THETA1=ATAN2(AIMAG(CRP21) #REAL (CRP21) ) *57. 29578

IF(REAL(CRP21) .LT.O. .AND. ABS( AIMAG(CRP21 ) ) .LT. 1. 0E-6)THETA1=180.

XM2=CABS(CRN2D

THETA2=ATAN2(AIMAG(CRN21) > REAL (CRN21) ) *57. 29578

IF(REAL(CRN21> .LT.O. .AND. ABS( AIMAG(CRN21) ) .LT. 1 .0E-6) THETA2=180.

FN2R=REAL(FN2)

FNXR=REAL(FNX(N+D)

FK2=-AIMAG(FN2)/REAL(FN2)

FKX=-A I MAG ( FNX ( N+l ) ) /REAL C FNX ( N+l )

)

IF(ABS(Q).NE.O.) GO TO 420

WRITE (6 » 410) D2,DEL»PSI»XMl»THETAlrXM2rTHETA2»FN2R>FK2rFNXRrFKX

410 FORMAT (1H F8.1 »2F8.3»2(F9.5»F9.3) #4F7.4»4F7.2)

GO TO 450

C Q NOT 0> COMPUTE P AND A READINGS FOR VALUE OF Q

C CORRECT Q FOR TILT AND ALINEMENT

420 TQ=TAND(Q+AT+ALQ)

IF(ABS(DEL).LT.1.0E-6) GO TO 430

R=(TQ*SlND(DEL-DELC)-SIND(DEL+DELC)/TQ)/2./SIND(DEL)

TP1=(-R+SQRT (R*R+1 ) ) /TC

TP2= ( -R-SQRT ( R*R+1 ) ) /TC

P1=ATAND ( TP1 ) +Q-ALP+ALQ

P2=ATAND ( TP2 ) +Q-ALP+ALQ

Al=ATAND(REAL(RHO*(TPl*TQ*RHOC-l.)/(TQ+TPl*RHOC) ))-AT-ALA

A2=ATAND(REAL(RH0*(TP2*TQ*RH0C-1.)/(TQ+TP2*RH0C) ))-AT-ALA

GO TO 440

QT~ USE "LIMITING" "FORMS OF EQUATIONS FOR DEL NEARLY =0.

43
430 P1=Q-ALP+ALQ

P2=Q-ALP+ALQ-90.

Al=ATAND(REAL(RHO/TQ) )-AT-ALA

A2=ATAND(REAL(RH0*TQ) )-AT-ALA

C REDUCE ANGLES TO LIE BETWEEN AnD 180 DEGREES

440 P1=AMOD(P1+360.»180.)

A1=AMOD(A1+360.»180.)

P2=AM0D (P2+360 . » 180 .

)

A2=AM0D ( A2+360

.

t 180 .

)

WRITE (6 > 410) D2>DEL»PSI»XMl»THETAl>XM2>THETA2>FN2RrFK2»FNXR»

1 FKX»P1»A1»P2»A2

450 D2=D2+DD2

IF(D2.LT.D2E) GO TO 400

GO TO 670

C COMPUTE FILM THICKNESS M=15 CD

C PICK THICKNESS WITH SMALLEST IMAGINARY PART

455 IF(ABS(AIMAG(T2)).LT.ABS(AIMAG(T1))) GO TO 460

D2=REAL(T1)

ERR0R=AIMAG(T1)

K=l

GO TO 470

460 D2=REAL(T2)

ERR0R=AIMAG(T2)

K=2

C COMPUTE DEL AND PSI FOR CALCULATED FILM THICKNESS D2

470 CRP21=RC(RP21»CRP32>FN2*C2*D2/WL)

CRN21=RC(RN21»CRN32»FN2*C2*D2/WL)

RH01=CRP21/CRN21

PSIX=ATAND(CABS(RH01) )-PSI

DELX=57.29578*ATAN2(AIMAG(RH01) »REAL(RH01 )

) -DEL

DIFN21=REAL(FN2)-FN1

DIFN32=REAL(FNX(N+1) )-REAL(FN2)

C WEIGHT FRACTION AND WEIGHT OF MATERIAL IN UNIT AREA OF FILM

FRACTN=DENP/(DENP+(S(REAL(FN2) )-S(FNP) ) *DEN1/(S(FN1)-S(REAL(FN2).)

)

1 ) WTMT=1 . 0E-5*D2*FRACTN/ ( FRACTN/DENP+ ( 1

. -FRACTN ) /DEN1

)

FN2R=REAL(FN2)

FNXR=REAL(FNX(N+D)

FK2=-AIMAG(FN2)/REAL(FN2)

FKX=-AIMAG(FNX(N+1) )/REAL(FNX(N+l)

)

WRITE (6> 480) D2»FN2R»FK2» WTMTf DELX»PSIX t ERROR »DIFN21 »DIFN32r

1 FRACTNrFNXR»FKX»K

480 FORMAT (1H »F8. 1 »2F9.5» 1PE12.4* 0P2F9.3»F12.5»3F10 . 5»2F9.5» 16)

GO TO 670

C CALCULATE SOLUTION FOR N2 FOR FIRST SOLUTION OF QUADRATIC

485 K=l

GO TO 490

487 K=2

GO TO 490

489 K=4

C MX=16 SO THAT S0LN2 WILL PRINT

490 MX=16

CALL S0LN2

MX=M

IF(M.EQ.16 .OR. M.EQ.21 .OR. IS.EQ.l) GO TO (345» 347» 348»670) »K

C PRINT BLANK SPACE

WRITE(6»27)

C CALCULATE ERROR LIMITS FOR N2 AND D2

DEL0=DEL

PSI0=PSI

DEL=DEL0+1.249*EDEL

CALL S0LN2
TN(1)=REAL(FN2>

TD(1)=D2

DEL=DEL0-1.249*ED£L

CALL S0LN2

TN<2)=REAL(FN2)

TD(2)=D2

DEL=DELO

PSI=PSI0+1.249*EPSI

CALL S0LN2

TN(3)=REAL(FN2)

TD(3)=D2

PSI=PSI0-1.249*EPSI

CALL S0LN2

TN(4)=REAL(FN2)

TD(4)=D2

DEL=DEL0+0.883*EDEL

PSI=PSI0+0.883*EPSI

CALL S0LN2

TN(5)=REAL(FN2)

TD(5)=D2

DEL=DELO-0 . 883*EDEL

CALL S0LN2

TN(6)=REAL(FN2)

TD(6)=D2

DEL=DEL0+0.883*EDEL

PSI=PSI0-0.883*EPSI

CALL S0LN2

TN(7)=REAL(FN2)

TD(7)=D2

DEL=DELO-0 . 883*E0EL

CALL S0LN2

TN(8)=REAL(FN2)

TD(8)=D2

FN2MIN=AMIN1(TN(1) »TN(2) »TN(3) »TN(4) >TN(5) »TN(6) »TN(7) »TN(8)

)

FN2MAX=AMAXl(TN(l)rTN<2)»TN(3)rTN(4)»TN(5),TN(6)rTN(7)>TN(8))

D2MlN=AMINl(TD(l)»TD(2)»TD(3)»TD(4>»TD(5)rTD(6)»TD(7) rTD(8))

D2MAX=AMAX1(TD(1) >TD(2) »TD(3) »TD(4) »TD(5) r TD(6) »TD(7) r TD(8)

)

WRITE (6*495) D2MIN»D2MAX»FN2MIN»FN2MAX

495 FORMAT (39H0CONFIDENCE LIMITS OF FILM THICKNESS ISF8.1»3H T0F8.1/

1 69H CONFIDENCE LIMITS OF THE REAL PART OF INDEX OF REFRACTI

20N OF FILM ISF12.5»3H TO F12.5)

GO TO (345r347r348»670)»K

670 FN2X(1)=FN2X(1)+DFN2(1)

IFCFN2X(1).LT.FN2EU)> GO TO 343

700 FN2X(2)=FN2X(2)+DFN2(2)

IF(FN2X(2).LT.FN2E(2)) GO TO 342

FNSX(2)=FNSX(2)+DFNS (2) IF (FNSX(2).LT.FNSE(2)) GO TO 327

FNSX ( 1 ) =FNSX ( 1 ) +DFNS ( 1

)

IF(FNSX(1).LT.FNSE(D) GO TO 326

WL=fcL+DWL

IF(WL.LT.WLE) GO TO 320

705 AI=AI+DAI

IF(AI.LT.AIE) GO TO 312

FN1=FN1+0FN1

IFCFN1.LT. FN1E) GO TO 311

GO TO 20

CALCULATE REFRACTIVE INDEX OF SUBSTRATE CNS

710 FNS=FN1*S1/C1*CSQRT(1.-4.*RH0*S1**2/(RH0+1.)**2)

FNSI(1)=REAL(FNS)

FNSI(2)=-AIMAG(FNS)/FNSI(1)

FNSE(1T=0. '

45
FNSE(2)=0.

WRITE(6>720) FNSIrDEL»PSI

720 FORMAT(20H0INDEX OF SUBSTRATE=F9.4»F9.4»2H »2X>4HDEL=r

1 F9.3»2X»4HPSI=F9.3)

IF(FNSI(2).LT.O.) GO TO 725

GO TO 705

725 WRITE (6» 727)

727 FORMAT (62H0KAPPA IS SET EQUAL TO OrSINCE A NEGATIVE VALUE WAS CALC

1ULATED)

FNSI(2)=0.

GO TO 705

C COMPUTE CONSTANTS OF WAVE PLATE CWP

C ASSUMES ANGLE OF TILT AT IS ZERO

730 Z1=Z1+ALP

Z2=Z2+ALA

Z3=Z3+ALP

Z4=Z4+ALA

Z5=Z5+ALQ

WRITE(6»27) L»Z1>Z2»Z3»Z4.Z5

IF(ABS(Z3-Z5).LT.0.1 .OR. ABS(Z1-Z5) .LT.O.D GO TO 745

735 B=(TAND(Z2)*(TAND(Z3-Z5)*TAND(Z5)-TAND(Z1-Z5)/TAND(Z5))-TAND(Z4)*

1 (TAND(Zl-Z5)*TAND(Z5)-TAND(Z3-Z5)/TAND(Z5)))/2./

2 (TAND(Z2)-TAND(Z4))

DELC= ATAN2(SQRT(-B*B-TAND(Z1-Z5)*TAND(Z3-Z5) > »B)*57. 29578

TC=1./SQRT(-TAND(Z1-Z5)*TAND<Z3-Z5)

)

RHOC=TC*CEXP((0.»-l.)*DELC/57. 29578)

RHO =TAND(Z2)*(TAND(Z5)+TAND(Z1-Z5)*RH0C)/(TAND(Z1-Z5)*TAND(Z5)*

1 RHOC-1.)

DEL=57.29578*ATAN2(AIMAG(RH0) »REAL(RHO)

)

PSI=ATAND(CABS(RHO)

)

WRITE(6»740) TCrDELCrDEL»PSI

740 FORMATUH TC=»F10.5»9H DELTA C=rF10.5r7H DELTA=»F15.5»5H PSI=r

1 F15.5)

GO TO 20

745 WRITE (6»747)

747 FORMAT ( 73H0C0NSTANTS OF WAVE PLATE CAN NOT BE ACCURATELY CALCULAT

1ED FROM ABOVE DATA)

GO TO 20

C CALCULATE ANGLE OF TILT OF STAGE ON ELLIPSOMETER CAT IS COMP

C ANGLE WITH REAL PART ANGLE AND IMAGINARY PART ERROR

750 Z1=Z1+ALP

Z2=Z2+ALA

Z3=Z3+AlP

Z4=Z4+ALA

Z5=Z5+ALQ

WRITE(6»27 ) L»Z1»Z2»Z3»Z4»Z5

TP1=TAND(Z1-Z5)

TA1=TAND(Z2)

TP2=TAND(Z3-Z5)

TA2=TAND(Z4)

TQ =TAND(Z5)

ALP1=TA1*(TQ+TP1*RH0C)

ALP2=TA2* ( TQ+TP2*RH0C

)

GAM1=TP1*TQ*RH0C-1

.

GAM2=TP2*TQ*RH0C-1

.

BETl=ALPl/SIND(Z2)/C0SDtZ2)+(TAl-ALPl*TPl*RH0C/GAMl)/C0SD(Z5)**2

BET2=ALP2/SIND(Z4)/C0SD(Z4)+(TA2-ALP2*TP2*RH0C/GAM2)/C0SD(Z5)**2

CAT=ULP2*GAM1-ALP1*GAM2)/(BET1*GAM2-BET2*GAM1)*57. 29578

ATX=REAL(CAT)

Z1=Z1+ATX

Z2=Z2+ATX

Z3=Z3+ATX

46
Z4=Z*++ATX

Z5=Z5+ATX

C COMPUTE RHO»DEL»PSI

GO TO 174

755 PRINT 760fCAT»DEL»PSI

760 F0RMATU5H ANGLE OF TILT=»F13.5» 15H DEGREES>ERR0R=»F13.5»

1 7H DELTA=»F15.5»5H PSI=»F15.5)

IF(M.EQ.26) ATX=(ATX+AT)/2.

AT=ATX

GO TO 20

810 STOP

C INSTRUCTION CNK M=3l COMPUTES N AND KAPPA OF FILM

650 D2=D2I

852 PRINT 855rD2

855 FORMAT(lH0»90XfF15.1)

860 CALL MIN2V(FN2I(1) »DFN2(1) »FN2E(D »FN2I(2) .DFN2(2) »FN2E(2))

D2=D2+DD2

IF(D2.LT.D2E) GO TO 852

GO TO 20

END

SUBROUTINE S0LN2

C SOLVES FOR REAL PART OF REFRACTIVE INDEX AND THICKNESS OF

C A FILM FOR MEASUREMENTS OF DELTA AND PSI

C ER2 IS ERROR OF SOLUTIONrTHICK IS FILM THICKnESS> INDEX IN FN2

C IF K=l OR 3» FIRST(POSITIVE SIGN) SOLUTION IS USED

C IF K=2 OR 4» SECOND(NEGATIVE SIGN) SOLUTION IS USED

COMMON FNl»Sl»Cl»NrFNX(900)»D(900) rFN2> CRPrCRNf CRP32» CRN32* WLr

1 RP21>RN21»DEL»PSI»C2»C3,FN2I(2> »DFN2(2> >FN2E(2) >FN2X(2) »T1»T2»T3»

2 T4»D2»DELX »PSIX t ELDEL»ELPSI»DENPrFNPr DEN1»M»K» IS COMPLEX FNXrFN2»CRP»CRN»CRP32»CRN32»RP21»RN21»C2rC3»TlfT2»T3»T1»

1 T5»T6»RH0»RC

C S IS USED FOR CALCULATING SPECIFIC REFRACTION

S(XN)=(XN*XN-l.)/(XN*XN+2.)

ATAND(X)=ATAN(X)*57.29578

X1=FN2X(1)

X3=FN2X(1)+DFN2(1)

GO TO (10»20»10»20) »K 10 ER1=AIMAG(T1)

ER3=AIMAG(T3)

GO TO 30

20 ER1=AIMAG(T2)

ER3=AIMAG(T4)

30 DO 80 I=l#20

X2=(Xl+X3)/2.

FN2=CMPLX(X2»-FN2X(2)*X2)

CALL CRC

CALL CALDX(T5»T6)

GO TO (fO>50»40»50)»K

40 D2=REAL(T5)

ER2=AIMAG(T5)

GO TO 60

50 D2=REAL(T6)

ER2=AIMAG(T6)

C COMPUTE DEL AND PSI FOR FILM THICKNESS D2

60 RHO=RC ( RP21 » CRP32 » FN2*C2*D2/WL ) /RC ( RN21 1 CRN32 » FN2*C2*D2/WL)

DELX =ATAN2(AIMAG(RH0) »REAL(RHO) )*57.29578-DEL

PSIX=ATAND(CABS(RHO) )-PSI

IF(ABS(DELX ) .LT.ELDEL. AND. ABS(PSIX ).LT.ELPSD GO TO 100

C HAS NOT CONVERGED TO ELDEL AND ELPSI. CHANGE LIMITS X1»X3

C IF ER1 AND ER2 HAVE SAME SIGNfX1=X2 AND ER1=ER2»0THERWISE

C ER2 AND ER3 HAVE SAME SIGN»X3=X2 AND ER3=ER2

47
IFUER1.GT.0.0 .AND. ER2.GT.0.0) .OR. (ER1.LT.0.0 .AND.

1 ER2.LT. 0.0) ) GO TO 70

X3=X2

ER3=ER2

GO TO 80

70 X1=X2

ER1=ER2

80 CONTINUE

90 D2=1.0E30

FN2=1.0E30

RETURN

C IF FILM THICKNESS D2 IS NEGATIVE* TREAT AS NO SOLUTION

100 IF(D2.LT.O.) GO TO 90

C PRINT SOLUTION AND ERROR LIMITS

IF(M.EQ.17 .OR. M.EQ.22) GO TO 120

DIFN21=X2-FN1

DIFN32=REAL(FNX(N+1))-X2

FRACTN=UENP/(DENP+<S(X2)-S(FNP))*DEN1/(S(FN1)-S(X2)))

WTMT=1 . 0E-5*D2*FRACTiM/ ( FRACTN/DENP+ ( 1 . -FRACTN ) /DEN1

)

FN2R=REAL(FN2)

FNXR=REAL(FNX(N+D)

FK2=-AIMAG(FN2)/REAL(FN2)

FKX=-AIMAG(FNX(N+1) )/REAL(FNX(N+l)

)

PRINT 110»D2#FN2R»FK2fWTMT»DELX»PSlX»ER2rDIFN21rDIFN32r

1 FRACTN'FNXRrFKXrK

110 FORMAT (1H >F8. 1 »2F9.5r lPE12.fr 0P2F9.3»F12. 5> 3F10 .5r 2F9.

5

t 16)

IS=2

120 RETURN

END

SUBROUTINE CRC

C CALCULATES CRP32 AND CRN32 REFLECTION COEF OF MEDIUM 2 AND 3 COMMON FN1 1 SI t CI t N » FNX ( 900

)

> D ( 900

)

t FN2 > CRPf CRN t CRP32 > CRN32 1 WL

»

1 RP21»RN21rDEL'PSI»C2»C3.FN2I(2) »DFN2(2) »FN2E(2) »FN2X(2) »Tl»T2»T3r

2 Tf rU2»DELDXrPSIDXr ERDELr ERPSI t DENPrFNPrDENl

COMPLEX FNXrFN2rCRPrCRNrCRP32»CRN32rRP21»RN21rC2rC3»TlrT2rT3rTf

COMPLEX RC»RP32»RN32

SRC<ArB)=(A-B)/(A+B)

X=FN1*S1

C2=CSQRT(1.-(X/FN2)**2)

C3=CSQRT(1.-(X/FNX(3) )**2)

RP21=SRC(FN2*C1»FN1*C2)

RN21=SRC(FN1*C1»FN2*C2)

IF(N.EQ.2) GO TO 10

RP32=SRC(FNX(3)*C2»FN2*C3)

RN32=SRC(FN2*C2fFNX(3)*C3)

CRP32=RC(RP32»CRP»FNX(3)*C3*D(3)/WL)

CRN32=RC(RN32»CRN»FNX(3)*C3*D(3)/WL)

RETURN

10 CRP32=SRC(FNX(3)*C2»FN2*C3)

CRN32=SRC(FN2*C2»FNX(3)*C3)

RETURN

END

COMPLEX FUNCTION RC (R21 »CR32»Q ) C RC=REFLECTION COEFFICIENT INCLUDING EFFECT OF UNDERLYING LAYERS

C FOLLOWING CARD IS REQUIRED FOR FORTRAN V ONLY

COMPLEX CEXPfCLOGrCSQRT

COMPLEX R21rCR32»Q »T T=CEXP( (0.»-12.56637)*Q)

C Q=N2*C0S(PHI2)*D/WL (0. r-12.56637)=-f*PI*J

RC=(R21+CR32*T)/(1.+R21*CR32*T)

48
RETURN

END

SUBROUTINE CALDX (D2C1 >D2C2)

C FOLLOWING CARD IS REQUIRED FOR FORTRAN V ONLY

COMPLEX CEXP»CLOG»CSQRT

C CALCULATES FILM THICKNESS AND ERROR TERMS

COMMON FNl»SlfCl>N»FNX(900) »D(900) »FN2»CRP,CRN»CRP32» CRN32* WLr

1 RP21»RN21»0EL»PSI»C2»C3»FN2I<2) >DFN2(2) »FN2E(2) fFN2X(2) »T1»T2>T3»

2 T4»D2»DELDX»PSIDX» ERDEL»ERPSI »DENP»FNP» DEN1'M»K

COMPLEX FNX»FN2rCRP#CRN»CRP32»CRN32»RP21»RN21fC2»C3»Tl»T2»T3»T4

COMPLEX E»B»C»Yl»Y2»D2Cl»D2C2rRH0rT5»T6

RH0=SIN(PSI/57. 29578) /COS (PSI/57. 29578) *CEXP(<0.»l.)*DEL/57. 29578)

C Y=EXP(-2 J DEL2) LET E*Y**2+BY+C=0 Y1»Y2 ARE TWO SOLUTIONS

E=RH0*RP21*CRP32*CRN32-RN21*CRP32*CRN32

B=RHO*(CRN32+RP21*RN21*CRP32)-CRP32-RP21*RN21*CRN32

C=RH0*RN21-RP21

T5=CSQRT(B**2-<+.*E*C)

GO TO (20f20>10»10) »K

C MAKE IMAGINARY PART OF SQUARE ROOT POSITIVE

10 IF(AIMAG(T5).LT.O.) T5=~T5

20 Yl=(-B+T5)/2./E

Y2=(-B-T5)/2./E

T5=CL0G(Y1)

T6=CL0G(Y2)

D2Cl=WL*T5/(0. t -12. 56637) /FN2/C2

D2C2=WL*T6/(0. » -12. 56637 J/FN2/C2

RETURN

END

SUBROUTINE MIN2V ( X I » DX , XF > Y

1

1 DY t YF

)

C USES FUNCTION FUNC(X»Y) WHICH MUST BE INCLUDED. DETERMINES AND PRINTS

C VALUES OF X BETWEEN XI AND XF AND VALUES OF Y BETWEEN YI AND YF THAT

C MINIMIZE FUNC. WILL FIND MULTIPLE MINIMUMS IF THEY ARE SEPARATED

C BY MORE THAN DX AND DY DIMENSION F(50»50)

COMMON FNl»Sl»ClrN»FNX(900) rD(900) »FN2»CRP»CRNf CRP32f CRN32» WL»

1 RP21»RN21»DEL»PSI > C2»C3,FN2I(2)»DFN2(2)»FN2E(2)»FN2X(2)»T1»T2»T3»

2 T4,Q2»DELDX,PSIDX» ERDELf ERPSI >DENP»FNP,DEN1 »MX»K» ISr RHO

COMPLEX FNX»FN2»CRP»CRN»CRP32»CRN32»RP21rRN21»C2»C3»Tl»T2»T3»Ti+

COMPLEX RC»RHO

C COMPUTE NUMBER OF VALUES OF X AND Y FOR RANGES XI-DX TO XF+DX

C AND YI-DY TO YF+DY.

NX=(1.000001*XF-XI)/DX+3

NY=(1.000001*YF-YI)/DY+3

IF(NX.GT.50) PRINT 10'DX

10 FORMAT COINCREMENT OF X»'E12.5»» IS TOO SMALL')

IF(NY.GT.50) PRINT 20»DY

20 FORMAT (»0 INCREMENT OF Y»'E12.5r» IS TOO SMALL')

C COMPUTE INITIAL VALUE OF X X=XI-DX

C COMPUTE TABLE F OF FUNCTION FUNC

DO 40 I=1»NX

Y=YI-DY

DO 30 J=1»NY

F(IrJ)=FUNC(X>Y)

C********D I AGNOSTIC PRINTING**************************************'

C PRINT 35fI»J»F(I» J)

30 Y=Y+DY

35 F0RMAT(2I5»E20.7)

40 X=X+DX

NX1=NX-1

49
NY1=NY-1

DO 910 I=2mMX1

DO 900 J=2»NY1

C SEARCH FOR MINIMUMS OF F(VALUES LESS THAN ALL 8 SURROUNDING VALUES)

IF(F(If J).LT.F(I-lf J-l) .AND. F(

I

t J) .LT.F( 1-1 t J) .AND.

1 F(I»J) .LT.Fd-lrJ+1) .AND. F( I » J) .LT.F( I » J-l> .AND.

2 F(I»J) .LT.F(I»J+1) .AND. F(

I

t J) ,LT.F( I+l» J-l

)

3 .AND. F(I»J).LT.F(I+1#J) .AND.

4 F(If J).LT.F(I + l»J+l))GO TO 50

GO TO 900

50 XMIN=XI+(I-2)*DX

YMIN=YI+(J-2)*DY

FMIN=F(I»J)

C A MINIMUM FMIN OCCURS NEAR POINT I»J AT XMIN-DX TO XMIN+DX AND

C YMliM-DY TO YMIN+DYr

C SUBDIVIDE RANGE TO LOCATE MINIMUM.

XU=DX/2

YD=DY/2

C SUBDIVIDE RANGE 10 TIMES

DO 550 K=l»10

F13=FUNC(XMlN-XD»YMiN-YD)

F14=FUNC(XMINfYMIN-YD)

IF(F13.LT.FMIN) GO TO 110

IF(F14.LT.FMIN) GO TO 120

C F14 GT FMIN

105 F15=FUNC(XMIN+XDrYMIN-YD)

IF(F15.LT.FMIN) GO TO 130

C F15 GT FMIN

107 F18=FUNC(XMIN+XD>YMIN)

IF(FIS.LT.FMIN) GO TO 140

F22=FUNC(XMIN+XD»YMIN+YD)

IF(F22.LT.FMIN) GO TO 150

C F22 GT FMIN

F21=FUNC(XMIN»YMIN+YD)

IF(F21.LT.FMIN) GO TO 160

C F21 GT FMIN

F20=FUNC(XMIN-XD»YMIN+YD)

IF(F20.LT.FMIN) GO TO 170

F17=FUNC(XMIN-XD»YMIN)

IF(F17.LT.FMIN) GO TO 180

C F MIN LT SURROUNDINGS

GO TO 500

110 IF(F1<+.LT.F13) GO TO 120

C F13 LT Fit AND FMIN

F17=FUNC(XMIN-XDrYMIN)

C IF F17 LT F13» CONTINUE SCAN WITH Fl5

IFCF17.LT. F13) GO TO 105

C F13 LT F1<+»FMIN>F17

F12=FUNC(XMIN-2*XD»YMIN-YD)

IF(F12.LT.F13) GO TO 190

F10=FUNC(XMIN-XD»YMIN-2*YD)

IF(F10.LT.F13) GO TO 200

C F13 IS NEW MINIMUM

FMIN=F13

XMIN=XMIN-XD

YMIN=YMIN-YD

GO TO 500

C F14 LT FMIN»F13

120 F15=FUNC<XMIN+XDrYMlN-YD>

IF(F15.LT.F1<+) GO TO 130

F11=FUNC(XMIN+XD»YMIN-2*YD)

IF(Fll.LT.Flt) GO TO 210

50
F10=FUNC(XMIN-XDrYMlN-2*YD)

IFCF10.LT.F14) GO TO 200

F17=FUNC(XMIN-XDrYMlN)

IFCF17.LT.F14) GO TO 107

F18=FUNCCXMIN+XD»YMIN)

IFCF18.LT.F14) GO TO 140

C F14 IS NEW MINIMUM

FMIN=F14

YMIN=YMIN-YD

GO TO 500

C F15 LT FMIN »F14

130 F11=FUNC(XMIN+XD»YMIN-2*Y0)

IF(F11.LT.F15) GO TO 210

F18=FUNC(XMIN+XD>YMIN)

IF(F18.LT.F15) GO TO 140

Fl6=FUNC(XMIN+2*XDfYMIN-YD)

IF(F16.LT.F15) GO TO 220

C F15 IS NEW MINIMUM

FMIN=F15

XMIN=XMIN+XD

YMIN=YMIN-YD

GO TO 500

C F18 LT FMIN,F15»F14

140 F16=FUNC(XMIN+2*XD»YMIN-YD)

IFCF16.LT. F18) GO TO 220

F22=FUNC(XMIN+XD»YMIN+YD)

IF(F22.LT.F18) GO TO 150

F21=FUNC(XMIN»YMIN+YD)

IF(F21.LT.F18) GO TO 160

F23=FUNC(XMIN+2*XD»YMIN+YD)

IFCF23.LT.F18) GO TO 230

C F18 IS NEW MINIMUM

FMIN=F18

XMIN=XMIN+XD

GO TO 500

C F22 LT FMIN>F18

150 F21=FUNC(XMIN»YMIN+YD)

IF(F21.LT.F22) GO TO 160

F23=FUNC(XMIN+2*XD»YMIN+YD)

IF(F23.LT.F22) GO TO 230

F25=FUNC(XMIN+XDrYMIN+2*XD)

IF(F25.LT.F22) GO TO 210

C F22 IS NEW MINIMUM

FMIN=F22

XMIN=XMIN+XD

YMIN=YMIN+YD

GO TO 500

C F21 LT FMIN»F22»F18

160 F25=FUNC(XMIN+XD»YMIN+2*XD)

IF(F25.LT.F21) GO TO 240

F20=FUNC(XMIN-XD»YMIN+YD)

IF(F20.LT.F21) GO TO 170

F17=FUNC(XMIN-XD»YMIN)

IF(F17.LT.F2D GO TO 180

F24=FUNC(XMIN-XD»YMIN+2*YD)

IFCF24.LT. F21) GO TO 250

C F21 IS NEW MINUMUM

FMIN=F21

YMIN=YMIN+YD

GO TO 500

C F20 LT FMIN»F2l

170 F17=FUNC(XMIN-XDVYMIN1

51
IF(F17.LT.F20) GO TO 180

F24=FUNC(XMIN-XD»YMIN+2*YD)

IF(F24.LT.F20) GO TO 250

F19=FUNC(XMIN-2*XD»YMIN+YD)

IFCF19.LT. F20) GO TO 260

F20 IS NEW MINUMUM

FMIN=F20

XMIN=XMIN-XD

YMIN=YMIN+YD

GO TO 500

F17 LT FMIN F20f F21*F13f F14

180 F12=FUNC(XMIN-2*XD»YMIN-YD)

IF(F12.LT.F17) GO TO 190

F19=FUNC(XMIN-2*XD»YMIN+YD)

IF(F19.LT.F17) GO TO 260

F17 IS NEW MINUMUM

FMIN=F17

XMIN=XMIN-XD

GO TO 500

F12 LT F13#FMINfF17

190 F10=FUNC(XMIN-XD»YMIN-2*YD)

IF(F10.LT.F12) GO TO 200

FMIN=F12

XMIN=XMIN-2*XD

YMIN=YMIN-YD

192 FX=FUNC(XMIN-XD»YMIN-YD)

IF(FMIN.GT.FX) GO TO 194

FX=FUNC(XMIN-XD»YMIN)

IF(FMIN.GT.FX) GO TO 196

FX=FUNC(XMIN-XO»YMIN+YD)

IF(FMIN.GT.FX) GO TO 198

GO TO 500

194 YMIN=YMIN-YD

196 XMIN=XMIN-XD

FMIN=FX

GO TO 300

198 YMIN=YMIN+YD

GO TO 196

F10 LT F13»F14,F12»FMIN

200 FMIN=F10

CHECK TO SEE THAT F10 IS NEW MINIMUM

XMIN=XMIN-XD

YMIN=YMIN-2*YD

202 FX=FUNC(XMIN-XD»YMIN-YD)

IF(FMIN.GT.FX) GO TO 204

FX=FUNC(XMIN»YMIN-YD)

IF(FMIN.GT.FX) GO TO 206

FX=FUNC ( XMIN+XD» YMIN-YD

)

IF(FMIN.GT.FX) GO TO 208

GO TO 500

204 XMIN=XMIN-XD

206 YMIN=YMIN-YD

FMIN=FX

GO TO 300

208 XMIN=XMIN+XD

GO TO 206

Fll LT F14»F15

210 F18=FUNC(XMIN+XDrYMIN)

IF(Fia.LT.Fll) GO TO 140

F16=FUNC(XMIN+2*XD»YMIN-YD)

IF(F16.LT.F11) GO TO 220

FMIN=F11

52
XMIN=XMIN+XD

YMIN=YMIN-2*YD

GO TO 202

C F16 LT F15>F11»F18

220 FMIN=F16

XMIN=XMIN+2*XD

YMIN=YMIN-YD

222 FX=FUNC(XMIN+XD»YMIN-YD)

IF(FMIN.GT.FX) GO TO 224

FX=FUNC(XMIN+XD»YMIN)

IF(FMIN.GT.FX) GO TO 226

FX=FUNC(XMIN+XD»YMIN+YD)

IF(FMIN.GT.FX) GO TO 228

GO TO 500

221 YMIN=YMIN-YD

226 XMIN=XMIN+XD

FMIN=FX

GO TO 300

228 YMIN=YMIN+YD

GO TO 226

C F23 LT F18»F22rFMIN

230 F25=FUNC(XMIN+XD»YMIN+2*XD)

IF(F25.LT.F23) GO TO 240

FMIN=F23

XMIN=XMIN+2*XD

YMIN=YMIN+YD

GO TO 222

C F25 LT F21#F22»F23»FMIN

240 FMIN=F25

XMIN=XMIN+XD

YMIN=YMIN+2*Y0

242 FX=FUNC(XMIN-XD»YMIN+YD)

IF(FMIN.GT.FX) GO TO 244

FX=FUNC(XMlNrYMlN+YD)

IF(FMIN.GT.FX) GO TO 226

FX=FUNC(XMIN+XD'YMIN+YD)

IF(FMIN.GT.FX) GO TO 248

GO TO 500

244 XMIN=XMIN-XD

246 YMIN=YMIN+YD

FMIN=FX

GO TO 300

248 XMIN=XMIN+XD

GO TO 246

C F2H LT F21»F20>F17»FMIN

250 F19=FUNC(XMIN-2*XD>YMIN+YD)

IF(F19.LT.F24) GO TO 260

FMIN=F24

XMIN=XMIN-XD

YMIN=YMIN+2*YD

GO TO 242

C Fl9 LT F20rFl7»FMIN

260 F17=FUNC(XMIN-XD»YMIN)

IF(F17.LT.F19) GO TO 180

FMIN=F19

XMIN=XMIN-2*XD

YMIN=YMIN+YD

GO TO 192

C FOLLOW MINIMUM FOR 5 STEPS

300 DO 400 KK=1»5

FX=FUNC(XMIN-XD»YMIN-YD)

IF(FX.LT.FMIN) GO TO 310

53
FX=FUNC(XMIN-XD»YMIN)

IF(FX.LT.FMIN) GO TO 320

FX=FUNC(XMIN-XD»YMIN+YD>

IF(FX.LT.FMIN) GO TO 330

FX=FUNC(XMIN»YMIN-YD)

IF(FX.LT.FMIN) GO TO 340

FX=FUNC(XMIN»YMIN+YD)

IF(FX.LT.FMIN) GO TO 350

FX=FUNC(XMIN+XDrYMIN-YD)

IF(FX.LT.FMIN) GO TO 360

FX=FUNC(XMIN+XD»YMIN)

IF(FX.LT.FMIN) GO TO 370

FX=FUNC(XMIN+XD»YMIN+XD)

IF(FX.LT.FMIN) GO TO 380

GO TO 500

310 YMIN=YMIN-YD

320 XMIN=XMIN-XD

GO TO 400

330 YMIN=YMIN+YD

GO TO 320

340 YMIN=YMIN-YD

GO TO 400

350 YMIN=YMIN+YD

GO TO 400

360 YMIN=YMIN-YD

370 XMIN=XMIN+XD

GO TO 400

380 YMIN=YMIN+YD

GO TO 370

400 FMIN=FX

C A MINIMUM IS NOT FOUND IN 5 STEPS

GO TO 600

500 XD=XD/2

YD=YD/2

550 CONTINUE

C NEW MINIMUM IS FOUND

GO TO 700

560 F0RMAT(5E15.6»3I5/(7F15.6))

t>00 PRINT 610rKfFMIN»XMIN»YMIN»XD>YD

610 FORMATCO ITERATION NUMBER' I3»» FAILED FOR MINIMUM'E12.5» • AT X 1=»E12.5»' Y='E12.5»' RANGES = 2E12.5)

C*****************PRINT VALUES AT MINIMUM

700 FN2=CMPLX(XMIN»-XMIN*YMIN)

Tl=RC(RP21rCRP32>FN2*C2*D2/WL)/RC(RN21»CRN32»FN2*C2*D2/WL)

DDEL=ATAN2 ( AIMAG ( Tl

)

t REAL (Tl ) ) *57 . 29578-DEL

DPSI=ATAN(ABS(T1) )*57.29578-PSI

XD=2*XD

YD=2*YD

PRINT 800» XMIN»YMIN>XD»YD»FMIN»DDEL>DPSI

800 FORMATC »5F15.5»2F10 .3)

900 CONTINUE

910 CONTINUE

RETURN

END

FUNCTION FUNC(FRN,FK)

C COMPUTES VALUES OF DELTA AND PSI» DELC AND PSIC» FOR A FILM

C OF N=FRN.KAPPA=FK» AND THICKNESS=D2 AND ERROR TERM»FUNC

COMMON FNlrSl»Cl>N»FNX(900)»D(900)»FN2»CRP»CRN»CRP32>CRN32»WL»

1 RP21»RN21fDEL»PSI»C2»C3fFN2I(2) »DFN2(2) »FN2E (2) rFN2X(2) »TlrT2»T3»

2 T4»Q2»DELDX»PSIDX» ERDELrERPSI r DENP» FNPrDENl 'MX »K» IS> RHO

COMPLEX FNX»FN2rCRP»CRN»CRP32»CRN32»RP21»RN21fC2»C3.Tl»T2»T3»T4

54
COMPLEX RC#RHO

FN2=CMPLX(FRN»-FRN*FK)

CALL CRC

FUNC=AB3(RC(RP21»CRP32»FN2*C2*D2/WL)/RC(RN21»CRN32fFN2*C2*D2/WL)

1 -RHO)

RETURN

END

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60
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61
5. References

[I] P. Drude, Ann. Physik. 272 , 532 (1889); ibid 272 , 865 (1889);

ibid 275 , (1890).

[2] A. B. Winterbottom, Optical Studies of Metal Surfaces, Royal

Norwegian Scientific Society Report No. 1, 1955,

published by F. Bruns, Trondheim, Norway; Trans.

Faraday Soc. 42, 487 (1946).

[3] M. Born and E. Wolf, Principles of Optics (Pergamon Press,

1959).

[4] F. L. McCrackin, E. Passaglia, R. R. Stromberg, and

H. Steinberg, J. Res. Nat. Bur. Stds. 67A, 363 (1963).

[5] T. Yolken, R. Waxier, and J. Kruger, J. Opt. Soc. Am. 57^, 283

(1967).

[6] Wo G. Oldham, J. Opt. Soc. Am. 57, 617 (1967).

[7] D. A. Holmes and D. L. Feucht, J. Opt. Soc. Am. 57 , 466 (1967).

[8] R. J. Archer and C. V. Shank, J. Opt. Soc. Am a 57, 191 (1967).

[9] F. L. McCrackin and J. P. Colson, in "Ellipsometry in the

Measurement of Surfaces and Thin Films," edited by

E. Passaglia, R. Stromberg, and J. Kruger, NBS Misc.

Publ. 256, p. 61 (U. S. Government Printing Office,

Washington, D.C., 1961).

[10] R. R. Stromberg, E Passaglia, and D. J. Tutas, see Ref. 9,

p. 281.

[II] C. A. Bennett and H. L. Franklin, Statistical Analysis in

Chemistry and the Chemical Industry, p. 128 (J. Wiley

& Sons, Inc., New York, 1954).

[12] J. Mandel, The Statistical Analysis of Experimental Data,

p. 282 (j Wiley & Sons, Inc., New York, 1964).

62
[13] This formula follows from the relationship for the molar re- fraction of mixtures given by, for example,

S. Glasstone, Textbook of Physical Chemistry

(D. van Nostrand Co., New York, 1946).

[14] W. G. Oldham, 'A Numerical Technique for the Analysis of

Lossy Films' to be published in the Proceedings of

Symposium on Recent Developments in Ellipsometry

at the University of Nebraska, August 1968.

[15] W. A. Shurcliff, Polarized Light (Harvard Univ. Press,

Cambridge, 1962).

I wish to thank Carl Fenstermaker for contributing Appendix

A, John Mandel for help in the statistical analysis, and Robert

Stromberg for help and encouragement.

63
Appendix I

Use of the ALINE Instruction in Relation to

Alinement of the Ellipsometer

Carl Fenstermaker

This appendix describes a modification of the alinement procedure of McCrackin, et al [4] . In the procedure described below the

components are not adjusted in their holders to correct errors in

the readings of their associated scales. Instead, errors in the

scale readings are measured and used to correct future readings.

Physical adjustment to 0.01 degree of the components within their

scales is frequently very difficult, requiring tedious small

changes and rechecks to approach the accuracy of readability. This

modified procedure is much faster and the alinement can be done to

the accuracy of the scales on the first trial. Although all ftiture

readings must be corrected, this correction is easily made with the

ALINE instruction.

We define P, A, and Q as the actual azimuthal position of the

plane of transmission of the prisms of polarizer and analyzer and

plane of the fast axis of the wave plate respectively, and P , A , and Q as the corresponding scale readings. All azimuthal angles

are measured positive counter-clockwise from the plane of incidence

when looking into the light beam.

The corrections used on the ALINE card (AP, AA, and AQ) are

the values which, when added to the scale readings, give the actual

positions of the plane of transmission of the polarizer and

analyzer and the position of the fast axis of the wave plate, so

(31)

(32)

(33)

64

p - P

s + AP

A = A

s + AA

Q = Qs + AQ .
As in Section 2.3 "Alinement Procedure" of reference [4], the

wave plate is removed, the arms of the ellipsometer set in the

straight through position, the scale reading of the polarizer, P , set to zero and the azimuth of the analyzer is adjusted to a value

near 90° that gives a minimum in the transmitted light. At the

minimum the polarizer and analyzer are crossed so that

A = P + 90°,

or by eqs 31 and 32

A + AA = AP + 90°

s since P = 0.

s Defining e = A -90° gives

AA = AP -e . (34)

We record e instead of adjusting the analyzer scale to make it zero

as in reference 4.

For the alinement of the polarizer, a metal surface is used

with the arms set for reflection at an angle near the principal

angle. The analyzer is set with the plane of transmission approximately in the plane of the surface (A « 90°). As described in

reference 4, the polarizer then is adjusted for minimum transmission (P » 0°) and the intensity of the minimum noted. A is moved

by 0.1 degree and P is again adjusted for a minimum and the

intensity noted If the intensity has increased A is adjusted in

the opposite direction. By taking a series of readings, setting

the analyzer by increments, and adjusting the polarizer for a minimum, we can plot P versus A and determine the point on this curve

which intersects the line P = A -e -90° which is also the line

s s P = A -90° „

We may also approach this point directly by using smaller and

smaller increments of A in an attempt to find scale readings P

and A such that

s

65
A -P = 90° +e (35)

s s

v ' within 0.01 degree. Thus, the point at which both P = A -90° and

minimum transmission occur is obtained by setting A and adjusting

P . This method has been shown to be the only unambiguous way of

alining the ellipsometer [4] . At this point the plane of transmission of the polarizer is in the plane of incidence; therefore,

P = and, by eq 31, AP = -P and AP is determined. AA can then

be calculated by eq 34

„

The wave plate is alined by inserting it between crossed

polarizer and analyzer in the straight through position with the

polarizer set at zero (P = 0° , A = 90° or P = -AP, A = 90° -AA)

and adjusting it near the zero reading of the fast axis for minimum

transmission. At this minimum the fast axis of the wave plate will

be set to the azimuth of the polarizer which is zero. By eq 33,

Q + AQ = 0°

s AQ = -Q . (36)

The three alinement corrections, AP, AA, and AQ, have now

been obtained and may be used on the ALINE card as shown in table 2,

66
Appendix II

Derivation of Equation 3

The derivation uses Jones vector and matrices as described

in "Polarized Light" by W. Shurcliff [15],

The normalized Jones vector of the light from the polarizer,

expressed in x and y axes along the fast and slow axes of the wave

plate, is I . / p \

I where P and Q are azimuths of the polarizer

and fast axis of the wave plate measured from the plane of incidence of the reflection. The matrix for the wave plate is | n

,

The light is now expressed with respect to x and y axes along

and normal to the plane of incidence by use of the counter-rotator

matrix cos Q -sin Q

sin Q cos Q

The matrix for reflection from the surface is simply l^ -i •

Therefore, the light after reflection from the surface is given by

[p

0"| Icos Q -sin Q"| |~1 1 [cos (P-Q)1

lj [sin Q cos qJ p

c bin (P-Q)J

[p[cos Q cos (P-Q) -p sin Q sin (P-Q)

]

sin Q cos (P-Q) +p° cos Q sin (P-Q)

For this light to be extinguished by the analyzer, it must be

linearly polarized. In this case, its azimuth e is given by

sin Q cos (P-Q) + p^ cos Q sin (P-Q)

.[cos Q cos (P-Q) - p sin Q sin (P-Q)]

tan Q + p tan (P-Q)

p[l - p tan Q tan (P-Q)] *

67
The analyzer must be set 90° from e to extinguish the light so

tan A = -1/tan e / therefore,

p[pc tan Q tan (P-Q) -1]

tan A = tan Q + p tan (P-Q)

Solving for p gives eq 3.

68
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71
12.00

11.95

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pst

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16.80 16.90 17.00 17.10 DELTA

Figure 5. Confidence ellipse for measurements A and \|/ .

72
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X30NI 3AllDVdd3d Wild

73
Surrounding

Medium

Film 1 Dl

Film 2 D2

Film 3 D3

Substrate

Fig. 7. . - Schematic diagram of three films on a

Substrate.

74
1.5-

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8 10

20 40 60

DISTANCE FROM MEDIUM

80 100

Figure 8. Approximation of an inhomogeneous film by ten

homogeneous films.

75
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76
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0)

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77
Uo

u

uw

Figure 11a. Refractive index of film,

a)

Uo

u

uw

Figure lib. Refractive index of film,

78
*u

a) EhUo

1-1 w

Figure lie. Refractive index of film.

0)

uuw

Figure lid. Refractive index of film,

* U. S. GOVERNMENT PRINTING OFFICE: 1969—31(1-025/155 79

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