Reference

NBS

PUBLICATIONS

AlllDE MT17flb

NATL INST OF STANDARDS & |ECH R- -C

A1 11 02491 786

CX:100 .U57 NO.400-78 V1986 C.I NBS-PUB-M \

CO Q

NBS SPECIAL PUBLICATION 400-78

Analytic Analysis of

Ellipsometric Errors

pooooooooooooooooooooooooooooooooo pooooooooooooooooooooooooooooooooc

mr\nrtr,rir\r\rinrinrir<r\nrtr\r,nr,rtririr,r\r,rir\r\r\r~,r,r.

QC

100

.U57

#400-78

1986
M he National Bureau of Standards' was established by an act of Congress on March 3, 1901. The

M Bureau's overall goal is to strengthen and advance the nation's science and technology and facilitate their effective application for public benefit. To this end, the Bureau conducts research and provides: (1) a

basis for the nation's physical measurement system, (2) scientific and technological services for industry and

government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety.

The Bureau's technical work is performed by the National Measurement Laboratory, the National

Engineering Laboratory, the Institute for Computer Sciences and Technology, and the Institute for Materials

Science and Engineering

The National Measurement Laboratory

Provides the national system of physical and chemical measurement;

coordinates the system with measurement systems of other nations and

furnishes essentiaJ services leading to accurate and uniform physical and

chemical measurement throughout the Nation's scientific community, in- dustry, and commerce; provides advisory and research services to other Government agencies; conducts physical and chemical research; develops,

produces, and distributes Standard Reference Materials; and provides

calibration services. The Laboratory consists of the following centers:

• Basic Standards^

• Radiation Research

• Chemical Physics

• Analytical Chemistry

The National Engineering Laboratory

Provides technology and technical services to the public and private sectors to address national needs and to solve national problems; conducts research in engineering and applied science in support of these efforts; builds and maintains competence in the necessary disciplines required to carry out this research and technical service; develops engineering data and measurement

capabilities; provides engineering measurement traceability services; develops

test methods and proposes engineering standards and code changes; develops

and proposes new engineering practices; and develops and improves

mechanisms to transfer results of its research to the ultimate user. The

Laboratory consists of the following centers:

Applied Mathematics

Electronics and Electrical Engineering^

Manufacturing Engineering

Building Technology

Fire Research

Chemical Engineering^

The Institute for Computer Sciences and Technology

Conducts research and provides scientific and technical services to aid Federal agencies in the selection, acquisition, application, and use of computer technology to improve effectiveness and economy in Government

operations in accordance with Public Law 89-306 (40 U.S.C. 759), relevant Executive Orders, and other directives; carries out this mission by managing

the Federal Information Processing Standards Program, developing Federal ADP standards guidelines, and managing Federal participation in ADP

voluntary standardization activities; provides scientific and technological advisory services and assistance to Federal agencies; and provides the technical foundation for computer-related policies of the Federal Government. The Institute consists of the following centers:

Programming Science and

Technology

Computer Systems

Engineering

The Institute for Materials Science and Engineering

Conducts research and provides measurements, data, standards, reference

materials, quantitative understanding and other technical information fundamental to the processing, structure, properties and performance of materials; addresses the scientific basis for new advanced materials technologies; plans

research around cross-country scientific themes such as nondestructive

evaluation and phase diagram development; oversees Bureau-wide technical programs in nuclear reactor radiation research and nondestructive evaluation; and broadly disseminates generic technical information resulting from

its programs. The Institute consists of the following Divisions:

Ceramics

Fracture and Deformation

Polymers

Metallurgy

Reactor Radiation

'Headquarters and Laboratories at Gaithersburg, MD, unless otherwise noted; mailing address

Gaithersburg, MD 20899, 'Some divisions within the center are located at Boulder, CO 80303. 'Located at Boulder, CO, with some elements at Gaithersburg, MD.
Semiconductor Measurement Technology:

Analytic Analysis of

Ellipsometric Errors

Deane Chandler-Horowitz

Semiconductor Electronics Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

Gaithersburg, MD 20899

^

Tt:—7 «

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued May 1986

z CO Q
Library of Congress Catalog Card Number: 86-600541

National Bureau of Standards Special Publication 400-78

Natl. Bur. Stand. (U.S.), Spec. Publ. 400-78, 36 pages (May 1986)

CODEN: XNBSAV

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1986

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402
Table of Contents

Page

Abstract 1

Introduction 1

Theory ajid Calculations 2

Program Description 8

Example Program Outputs 10

Conclusions 10

Acknowledgments 11

References 11

Appendix 18

List of Figures

1. Program flow chart 12

2. Plot of percent error in thickness and error in refractive index for a silicon dioxide

film 100 nm thick on a silicon substrate 13

3. Plot of percent error in thickness and error in refractive index for a silicon dioxide

film 10 nm thick on a silicon substrate 14

4. Plot of percent error in thickness and error in refractive index for a silicon dioxide

film 280 nm thick on a silicon substrate 15

5. Plot of percent error in thickness and error in refractive index for a

silicon nitride film 80 nm thick on a silicon substrate 16

6. Plot of error in refractive index (real part) and error in refractive index

(imag part) for a bare silicon substrate 17

iii

SemiconductorMeasurement Technology:

Analytic Analysis of Ellipsometric Errors

Deane Chandler-Horowitz

Semiconductor Electronics Division

National Bureau of Standards

Gaithersburg, MD 20899

Abstract

A computer program is given that contains an explicit error analysis (EEA) for

ellipsometric measurements. The program can identify the ellipsometric inaccuracies for any ellipsometer, can be used to determine which parameters contribute

the most to the overall measurement inaccuracy, and can lead one to an optimum

measurement procedure. A FORTRAN program that performs the evaluation of

the pajtial derivative expressions needed to analyze ellipsometric measurement

uncertainties is listed. The program determines the uncertainty in the calculation of the refractive index of a bare isotropic substrate or the uncertainty in

the determination of the thickness and refractive index of a nonabsorbing film on

a substrate of known refractive index. These are the two most commonly used

surface models used in ellipsometry performed at single angle of incidence and

a single wavelength. The program input parameters include the wavelength of

light, the angle of incidence and its uncertainty, and the uncertainties in the ellipsometric parameters A and ip. They also include in the ambient-substrate model

an estimated value for the substrate's refractive index, and in the film-substrate

model the refractive index of the substrate and its uncertainty and estimated

values for the film's refractive index and thickness. The case of the conventional

null ellipsometer utilizing a quarter-wave plate is treated to find the uncertainties

in A and 4> from the uncertainties in the polarizer and analyzer null values and

the waveplate constajits.

Key words: bare substrate model; computer program; ellipsometric error analysis;

film refractive index; film-substrate model; film thickness; substrate refractive

index.

Introduction

A FORTRAN program was developed that calculates the ellipsometric measurement uncertainties for two models of a surface. The first is the simple bare isotropic substrate

model. The second is the isotropic nonabsorbing film-substrate model. It is assumed that

the sample to be measured ellipsometrically can be best described by one of these two

models. From the ellipsometer, one obtains values for the two angular quantities A and ip.

These allow a determination of at least two of the unknown optical properties of the sur1
face. This corresponds to a determination of the real and imaginary parts of the substrate

refractive index for the bare substrate model and a determination of the film's refractive

index and thickness for the film-substrate model. In each calculation involving

the inversion of the ellipsometric equations is performed to solve for the unknown optical

properties [l].

For an ideal ellipsometer, A and xf) are functions of up to six independent parameters:

the ambient refractive index, the substrate refractive index, the film refractive index and

thickness, the angle of incidence, and the wavelength. The measured or assumed values of

these parameters may have systematic errors in addition to random errors. In an actual

ellipsometer, the measured values of A and ijj are susceptible to other sources of errors. These can include azimuthal-angle errors, component and cell-window optical imperfections, beam deviation errors, parasitic beams, source-beam polarization and coUimation

errors, polarization-dependent detector sensitivity, and residual mechanical imperfections.

Therefore, the ultimate accuracy for a nonideal ellipsometer can be limited by these other

errors, and, for a further error investigation, one should assess the above errors. In the case

of the conventional null instrument, the method of measuring A and 0 using four-zone

averaging best ensures that A and t/^ are free from most above cited imperfections [2]. The

analytic functions for A and x}j can then be solved for the two unknown differentials that

lead to a calculation of the uncertainty in the ellipsometrically measured quantities [3,4].

Theory and Calculations

The ellipsometric parameters A and are defined by the complex reflectance ratio p as follows [5]:

p = ^ = taml> e Ra

iA

(1)

Here, Rp and iE« are the ratios of the reflected to the incident electric field amplitudes of

the light for polarization parallel, p, and perpendicular, s, to the plane of incidence. In the

case of light reflection from the ambient-substrate surface, the complex reflectances are:

Ua cos(f> — no\ 1 —

Rp —

no sin<t) n<

ria cos(f) + no\ /

no cos(f) — Ug

no sin4>

-I 2

n<

1 -

Ra —

no sin(l)

no cos4> na\ 1 —

no sin(f>

In the case of the ambient-film-substrate model, the reflectances are:

,-i2l3 „ _ Tpof + rpfa e itp —

1 + TpOf Tpfa e

-»2/3

(2)

(3)

(4)

2
where

R/t —

fsQf + rafs e

-i2/3

1 + TsOf Tsfs e-»2/3'

Uf coscf) — now 1

TpOf =

rif coscj) + no\ / 1 —

no sin(f)

^/

no sin<f)

(5)

(6)

no cos4> — ^/\/ 1 —

no CO50 + 1

no sin(f)

^/

no sin(j>

n.A/

no sin(f> - nf \/l -no sin<f> n. rpfa =

ns\ 1 +

no s^n<j)^ —J no sin<f) n.

/.-

no sin(f) -no sin(f)

2

. ^/ . ^«

n/^

no sin4>

+

no sin(j)

2

. ^/ . n«

(8)

(9)

Tpo/ and r,o/ are the real Fresnel reflection coefficients for the p and 5 polarizations of the

electric field for the air-film interface. Tpja and Tafs are the corresponding complex Fresnel

coefficients for the film-substrate interface. The refractive indices are no for the ambient,

rif for the film, and Ug for the substrate. The angle of incidence is 0 and the film phase

thickness /? is given in terms of the wavelength, A, and film thickness, t, as follows:

no sin(f>

rif

T 2 (10)

It should be noted that there is no explicit dependence on wavelength for the bare substrate

model case.

The measurement uncertainties are derived for the bare substrate model as follows. The

function p as given by eq (1) is a function of no, ng, and ^:

P = p{no, ria, <i>). (11)

The substrate's refractive index has, in general, a real and imaginary part:

ria = riar - i nai. (12)

3
rier is the real part of the index of refraction, and Ugi is the imaginary part, normally

expressed as the extinction coefficient k.

By taking the total differential of eq (11), dp, and assuming the differential quantities dA,

d'4), dno, d(f) correspond to experimentally known uncertainties, the following result for the

uncertainty of is obtained, neglecting the variation in no:

dp-^d<i>

drts = . (13)

dria

By eq (l), this expands to

i p dA-\- sec^t/j e*^ d^f - d(f> dn, = . (14)

op

Fromeqs (l), (2), and (3),

dp _ Ua [1 + p)^ cos^^

and

dria 2 (1 — p) no^ sin'^(j)

dp

dp dut

[us^ (1 + sec^(f)) + no^(l - sec^<i>)]

Therefore,

where

a2 = 2 I p —^ -5- riQ^ sin^<f) tan^<f> (19)

Ua (1 +py

and

(15)

(16)

d(l> ria tan<f>

dria = oi d(i) + a2 dA + as dxfj, (17)

^ _

[ria^ (1 + sec^(f>) + np^ (1 - sec^(f>)] ^^^^ Ha tan<l)

as = 2 —^—^—3 no^ sin^cf) tan^<l> sec^if) e*^ (20)

Ha (1 + p)

duar = real{dna) and dugi = i'mag{dna). (21)

4
Here, dugr and dusi are the differentials in the substrate's real and imaginary parts of

refractive index. This leads to a standard deviation of the ellipsometrically measured

quajitities Uar and Uai as follows:

(^nar = yJreaP [ui) crj + reaP{a2) + real^{a3) a^^- (22)

<7n8i = yjimag^{ai) + imag^[a2) (7^ + imag^{a3) a^^-, (23)

where cr^, aa, and Cpgi are the standard deviations of (j), A, and iJj when the meajsurement

errors axe rajidom. This present FORTRAN program calculates the maximum uncertainty

for treating systematic errors, which are more likely for an ellipsometer, and gives the

uncertainties in n«r and

Suar = \real[ai) <50| + \real[a2) SA\ + \real{a3) 6<p\ (24)

^i^ei = \imag[ai) Sxp] + \imag[a2) 6A\ + \imag{a3) 6<f>\ , (25)

where 6 represents systematic measurement uncertainties.

The film-substrate uncertainty equations are more complicated but are derived as follows.

A and from eq (1) have the following functional form:

A = A{(f>, A, no, n/, t, Ua) (26)

tp = 0((/>, A, no, n/, t, Ua). (27)

By taking the total differential of eqs (26) and (27) and then assuming all variations are known except the variations in film thickness, dt, and refractive index, dnj, the following

results of the solution for dt and drif are obtained, neglecting the variations in no and A:

dt= ^

[6i dilj + 62 rfA + 63 d(f) + 64 duar + 65 duai] (28)

ki

where

duf = [ci dip + C2 dA + C3 d4> + C4 dugr + C5 dugi], (29)

ki

h = l^ (30)

h = 1^ (31)

63 = £^it_£*^ (32)

d(f) drif d(f> duf

t,, = _ (33)

dUar dUf dUar dUf

5
b. = dA dtp dip dA

duai duf dUai duf

dA

'dt

dip

ci =

C2 =

dt

and

C3 =_ dA dip

d<f) dt

dip dA

d(p dt

C4 = dA dip

dusr dt

dip dA

dUar dt

C5 = dA dip

duai dt

dip dA

driei dt

=

dip dA dA dip

dnf dt dnf dt

The uncertainties for the two measured quantities f and n/ are then

1

(34)

(35)

(36)

(37)

(38)

(39)

(40)

St =

\k,

[\bi «VI + \h IA\ + I63 *<*| + \b4 6n„\ + \h 6n,i\]

= mll'l ^'f'\ + l<'2 + |C3 S<l>\ + |C4 6n,r\ + \cs «n„|]

(41)

(42)

The desired partial derivatives contained in eqs (30) through (40) are derived explicitly

from eq (1) by first taking the partial derivative of eq (1) with respect to either t, n/,

Han or Usi. For example, to find the quantities ^ and one gets [6]:

2 , dip . , dA

sec w ——I- 1 tanip -r— ^ dt ^ dt

,iA ^ 4^)\ V ^ ^^^^ gt-A (n, dt

where C is defined as

1 dRp 1 dR

O —

Rp dt R, dt

(43)

(44)

By separating eq (43) into its real and imaginary parts, one obtains for the solutions for

dip sin2ip

dt

real{C)

and

^ = imag[C).

(45)

(46)
The following calculation allows one to determine the uncertainties in A and when

performing the conventional null ellipsometric measurement in one zone with an ideal

ellipsometer having a simple quarter-wave plate. The uncertainties in the calculated values

for A and rp can be calculated from the uncertainties in the polarizer and analyzer null

positions, P and A, and the quarter-wave plate's azimuth angle, Q, transmittance ratio,

Tcj and relative phase retardation, Ac, uncertainties. From eq (l)

A = tan ^

and

imag[p)

real[p)

(47)

'tp — tan ^y/imag'^[p) + reaP[p). (48)

For this ellipsometer [7],

tanA[tanQ + pc tan{P — Q)] . . . "= p.tanQtan{P-Q)-l ="- + "'•• (^9)

where Pc for the waveplate is given by

Pc^TcC*^'. (50)

Pr ajid Pi are the real and imaginary parts of p. pc is a complex number characterizing

waveplate by the ratio, Tc, of the transmittance along its fast axis to the transmittance

along its slow ajcis and the relative phase retardation, Ac, between these two axes.

The differentials of A and tp are given by

dA = ^ [di dP + d2 dA + dz dQ + ^4 dTc + ds dAc] (51)

«2

ajid

where

= -L [ei dP + 62 dA + 63 dQ + 64 dTc + 65 dAc] , (52)

^ dp^_Pi_dpr. ^ dP PrdP ^

^

^

dpi^ _

pj^dpr_

^ dA PrdA ^

'

^ dpi__pi_dp^

(55) ^ dQ PrdQ ^

^

d, = ^-ei^ (56)

7
dpi _ Pi dpr

dAc pr 5A< (57)

^l=PiJp+PrJp (58)

dpi dpr , .

dpi dpr , ,

dpi dpr fX e4 = P.^+P.^ (61)

e,=p,^+p^^ (62)

and

2 k2 = Pr + — (63)

pr

A:3 = (l + Pr'+Pi')V(Pr2 + Pi2). (64)

This gives the uncertainties in A and tp for the conventional null ellipsometer as

6A = r^ [\di 6P\ + \d2 6A\ + \d3 6Q\ + \d4 6Tc\ + {ds 6Ac\] (65)

Srp= [|ei 6P\ + \€2 6A\ + |e3 SQ\ + |e4 (^rd + [eg 6Ac\\. (66)

The above equations are explicitly contained in the FORTRAN program which is described

in the next section. These calculated uncertainties can be listed or plotted as a function of the angle of incidence, usually the one most easily varied ellipsometric parameter.

Program Description

The FORTRAN program which appears in the appendix of this report is described by

the line numbers at the left-hand margin. In addition, a flow chart showing the program

input and output data is given in figure 1. The program contains the introduction and

declarations, line numbers 1 to 60; the places to enter in input data, line numbers 61 to

8
249; the theoretical calculations at every one degree of incident angle, line numbers 250

to 690; the program printout and graphics, lines numbers 691 to 757; and three short

trigonometric subroutines. Except for the graphics subroutines for a Digital Equipment

Corp. VAX-11 computer,* a FORTRAN IV or FORTRAN 77 compiler may be used for

the program. A brief listing of some of the program line numbers with further explanation

are as follows. The variables that need to be abbreviated or changed for the FORTRAN

prograjn are shown in parentheses.

Line Numbers Description

69 to 75 choose between the bare- or film-substrate models

101 to 105 if bare substrate, film thickness is zero and

refractive index equals that of air

153 to 161 choose between entering conventional null data

- or errors in A (DEL) and V (PSI)

248 defines substrate refractive index Ug (NS) in eq (12)

257 defines incident angle in radians, (f> (PHI)

266 calculates eq (10) in text

283 to 286 calculates eqs (6) , (7) , (8) , and (9)

303 to 304 calculates eqs (1), (2), (3), (4), and (5)

308 to 313 calculates A, xp , and p (RHO) in eqs

- (11) , (26) , and (27)

314 calculates Pc (RHOC) of the waveplate in eq (50)

333 to 340 calculate positions of the polarizer and analyzer at

null for zones one and three

410 to 424 <5A (DDEL) and Sip (DPSI) are calculated for the null ellipsometer

- case, eqs (63) and (64). In some variables

- pr (RHOR) and p, (RHOI) are shortened to (RI) and (RR) . 443 to 449 calculates ^ (DPSIDT) and ^ (DDELDT) needed

- in eqs (35) to (40)

506 to 513 calculates (DPSIDN) and #^ (DDELDN)

onf arif 565 to 573 calculates || (DPSIDO) and |^ (DDELDO)

596 to 603 calculates (DPSDNR) and (DDEDNR)

an,r on,r 626 to 633 calculates (DPSDNI) and (DDEDNI)

636 to 640 calculates ki (Kl) in eq (40)

644 to 661 calculates 6t (DT) in eq (41) and Snj (DNF) in eq (42)

665 to 691 calculates Srisr (DNSR) and (5n« (DNSI) in eqs (24) and (25)

* Certain commercial equipment, instruments, or materials are identified in this paper

in order to adequately specify the experimental procedure. Such identification does not

imply recommendation or endorsement by the National Bureau of Standards, nor does it imply that the materials or equipment identified are necessarily the best available for the

purpose.

9
See the program listing itself for the complete description.

Example Program Outputs

The following j&gures show how the uncertainties in refractive index and thickness vary with

the angle of incidence. Figure 2 shows the uncertainties in refractive index and thickness for a 100-nm silicon dioxide film on silicon. There is relatively little angular variation for angles

of incidence form 25 deg to 80 deg, and for uncertainties in the angle of incidence and A and

tjj of 0.01 deg, the thickness and refractive index of the 100-nm film can be measured quite

accurately. Figure 3 shows the variation of thickness and refractive index uncertainty for a

thin 10-nm silicon dioxide film on silicon. Here, for uncertainties in the angle of incidence

of 0.001 deg and A and tjj of 0.05 deg, the principal angle of incidence at 75.5 deg is the

preferred angle of incidence to make the measurement. Figure 4 plots the uncertainties

for a 280-nm silicon dioxide film thickness near the first order film-phase period occurring

at about 70 deg. Although an accurate measurement cannot be made near 70 deg, one

can still make a reasonably accurate measurement at an angle of incidence around 25 deg.

Figure 5 plots for a silicon nitride film on silicon the uncertainties of thickness and refractive

index. This differs from a silicon dioxide film in that its higher index, 1.98, gives rise to a more accurate measurement because of its index mismatch with air, and because the 80-

nm silicon nitride film has its principal angle near 15 deg which is near the condition for

maximum accuracy. Figure 6 shows how both the real and imaginary parts of the refractive

index uncertainty vary as a function of the angle of incidence for the bare silicon substrate.

Conclusions

Many ellipsometers can make measurements to high precision. However, the present program allows one to calculate the maximum accuracy obtainable from any ellipsometer

when performing an ellipsometric measurement using either one of two specific surface

models: the refractive index of a bare substrate or the thickness and refractive index of a nonabsorbing film on a substrate. By inputting the appropriate uncertainties in the

known measurement parameters, the maximum accuracy as a function of the angle of incidence can be displayed. By changing the magnitude of the input uncertainties, one can

ascertain which parameter uncertainties are contributing the most to overall measurement

uncertainty. When measuring an unknown bare substrate or a film of unknown thickness, one or two iterations of the input parameters, such as the incident angle, are necessary to

maximize the accuracy.

An ellipsometer is an instrument that can be used to measure either the absolute optical

properties of a surface or relative changes in them. Although the concern here is only with

the accuracy of an absolute ellipsometric determination of these properties, because of the

nonlinear nature of the electromagnetic theory that describes the measurement process, even relative changes in a surface's optical properties can be subject to large measurement

error. An example of this type of measurement would be the calculated change in film

thickness as a function of some growth parameter in terms of changes in A and ip.

10
Requests for this program on recorded medium should be addressed to the author.

Acknowledgments

The author wishes to thank George A. Candela for the many enlightening discussions on

ellipsometric instrumentation and theory.

References

1. F.L. McCrackin, A Fortran Program For Analysis Of Ellipsometric Measurements,

NBS Technical Note 479 (April 1969).

2. R.M.A. Azzajn and N.M. Bashara, Ellipsometry and Polarized Light (North-Holland,

New York, 1977), pp. 388-389.

3. D. Chandler-Horowitz, Ellipsometric Accuracy and the Principal Angle of Incidence,

SPIE Vol. 342, Integrated Circuit Metrology (1982), pp. 122-123.

4. D. Chandler-Horowitz and G.A. Candela, On the Accuracy of Ellipsometric Thickness

Determinations For Very Thin Films, Journal De Physique CoUoque CIO, Ellipsometry

'83 (Dec. 1983), pp. clO-24.

5. R.M.A. AzzEun and N.M. Bashara, Chapter 4.

6. J. Humlicek, Evaluation of Derivatives of Reflectance and Transmitteince by Stratified

Structures and Solution of the Reverse Problem of Ellipsometry, Optica Acta, 1983,

vol. 30, no. 1, p. 102.

7. F.L. McCrackin, p. 23.

11
Figure 1. Program flow chart.

12
X 100 .10000

E+01

to

111 Z

H

I

10. 20 . 30. 40. 50. 60. 70. 80. ANGLE OF INCIDENCE < deg

)

- .10000

= E+00

.10000 ^ E-01

M

z

o

m

X

m

7)

o

:dz

.10000

90. E-02

Figure 2. Plot of percent error of thickness (DT/T), solid line, and

error in refractive index, dashed line, for an oxide film 100 nm thick

on a silicon substrate. The input parameters are:

wavelength 632.8 nm

incident angle uncertainty 0.01 deg

film refractive index 1.46

film thickness 100 nm

substrate refractive index (real part) 3.865

uncertainty in real part 0.005

substrate refractive index (imag part) 0.018

uncertainty in imag part 0.002

uncertainty in DEL 0.05

uncertainty in PSI 0.05

13
I

I- ANGLE OF INCIDENCE < deg

)

Figure 3. Plot of percent error in thickness (DT/T), solid line, and

error in refractive index, dashed line, for an oxide film 10 nm thick

on a silicon substrate. The input parameters are:

wavelength 632.8 nm

incident angle uncertainty 0.001 deg

film refractive index 1.46

film thickness lo nm

substrate refractive index (real part) 3.865

uncertainty in real part 0.001

substrate refractive index (imag part) 0.018

uncertainty in imag part 0.001

uncertainty in DEL 0.02

uncertainty in PSI 0.02

14
X 100.

0)

(0

Ulz

u

H

I

H

$

10. 20. 30. 40. 50 . 60 . 70. 80. ANGLE OF INCIDENCE < degf

)

.10000

E+01

.10000

90. E-02

Figure 4. Plot of percent error of thickness (DT/T), solid line, and

error in refractive index, dashed line, for an oxide film 280 nm thick

on a silicon substrate. The input parameters are:

wavelength 632.8 nm

incident angle uncertainty 0.002 deg

film refractive index 1.46

film thickness 280 nm

substrate refractive index (real part) 3.865

uncertainty in real part 0.005

substrate refractive index (imag part) 0.018

uncertainty in imag part 0.002

uncertainty in DEL 0.02

uncertainty in PSI 0.02

15
H ANGLE OF INCIDENCE <clegf)

%

Figure 5. Plot of percent error of thickness (DT/T), solid line, and

error in refractive index, dashed line, for a silicon nitride film 80 nm

thick on a silicon substrate. The input pstrameters are:

wavelength 632.8 nm

incident angle uncertainty 0.01 deg

film refractive index 1.98

film thickness 80 nm

substrate refractive index (real part) 3.865

uncertainty in real part 0.005

substrate refractive index (imag part) 0.018

uncertainty in imag part 0.002

uncertainty in DEL 0.05

uncertainty in PSI 0.05

16
7 . 10000 ,=r. Q E*00

a:o

X .10000

liJ E-01

X

lli Z .10000 M E-02

111

<I 10000

E-03

n

.10000

E+01

.10000

E+00

^.10000

E-01

0

(0

H

:oH

m

H

z

m

X

m

n

n

o

n

10. 20. 30 . 40. 50. 60. 70. 80. ANGLE OF INCIDENCE < deg

O

.lOOOOZ

90. E-02 (0

M

Figure 6. Plot of error in the refractive index (real pjtrt), solid line,

and error in refractive index (imag part), dashed line, for a bare silicon

substrate. The input parameters are:

waveleni^th 632.8 nm

incident angle uncertainty 0.01 deg

substrate refractive index (real part) 3.865

uncertainty in real part 0.005

substrate refractive index (imag part) 0.018

imcertainty in imag part 0.002

uncertainty in DEL 0.05

imcertainty in PSI 0.05

17
APPENDIX

0001 c 0002 r 0003 r 0004 c 0005 c 0006 c 0007 c 0008 c 0009 c 0010 c 0011 c 0012 c 0013 c 0014 c 0015 c 0016 WX V c 0017 WX f c 0018 WX \J c 0019 WX V c 0020 \J\J c 0021 Wx. X c 0022 c 0023 c 0024 c 0025 c 0026 c 0027 W^1 c 0028 c 0029 \/\J ^ ^ c 0030 WvV c 0031 c 0032 c 0033 r 0034 r 0035 r\» 0036 V/VOO r 0037

003A r\> 003Q rv« no40

0041 r 0042

0043 1X9

0045 3

0046 4

0047 5

0048 6

0049 7

0050 C

0051

0052 C

PROGRAM ERROR. FOR

WRITTEN BY DEANE CHANDLER-HOROWITZ

NATIONAL BUREAU OF STANDARDS

SEMICONDUCTOR MATERIALS AND PROCESSING DIVISION 725

BLDG. 225, RM. A331

GAITHERSBURG, MD 20899

DATE: NOVEMBER 19, 1984

PURPOSE: TO CALCULATE THE UNCERTAINTY IN THE ELLIPSOMETRIC

DETERMINATION OF THE THICKNESS AND INDEX OF REFRACTION

OF A FILM ON A SUBSTRATE

OR

TO CALCULATE THE UNCERTAINTY IN THE ELLIPSOMETRIC

DETERMINATION OF THE INDEX OF REFRACTION OF A BARE

SUBSTRATE

AS A FUNCTION OF ANGLE OF INCIDENCE

ASSUME THE AMBIENT-FILM-SUBSTRATE MODEL OR THE BARE

SUBSTRATE MODEL BEST DESCRIBES THE ACTUAL SURFACE

GIVEN THE UNCERTAINTIES IN THE ANGLE OF INCIDENCE,

DEL AND PSI (OR IN THE CASE OF THE NULL ELLIPSOMETER,

NULL VALUES OF POLARIZER AND ANALYSER, WAVEPLATE ANGLE,

AND WAVEPLATE CONSTANTS) , AND FOR THE BARE SUBSTRATE

MODEL THE SUBSTRATE'S REFRACTIVE INDEX

ASSUME NO WAVELENGTH UNCERTAINTY

THIS PROGRAM VERSION CONTAINS EXPLICIT EXPRESSIONS FOR THE

PARTIAL DERIVATIVES OF DEL AND PSI.

MAKE NEEDED DECLARATIONS

REAL Q,NF,NSR,NSI,N0,X(91) ,YT(91) ,YN(91) ,K1

COMPLEX NS , CS , CRPFS , CRSFS , RP , RS , UP2 , VP2 , US2 , VS2 , EJ , AP , BP , AS , BS

,

DAPDT , DBPDT , DASDT , DBSDT , DRPDT , DRSDT , CT , QC , UP2DNF , VP2DNF , CPFSDN

,

CSFSDN , Zl , Z2 , DAPDNF , DBPDNF , DASDNF , DBSDNF , DRPDNF , DRSDNF , CNF , UP2DI

VP2D0 , CPFSDO , US2D0 , VS2D0 , CSFSDO , Yl , Y2 , Y3 , Y4 , DAPDO , DBPDO , DASDO

,

DBSDO , DRPDO , DRSDO , CO , Z , RHO , RHOC , US2DNF , VS2DNF , US2DNR , VS2DNR

,

LOGICAL Ql

18
0053 WRITE (6,10)

0054 10 F0RMAT(1X, 'THIS PROGRAM, "ERROR. FOR", CALCULATES THE

0055 1 UNCERTAINTY IN THE FILM')

0056 WRITE (6,20)

0057 20 FORMAT (IX, 'THICKNESS AND REFRACTIVE INDEX OR SUBSTRATE INDEX

0058 1 VERSUS INCIDENT ANGLE AS')

0059 WRITE (6,30)

0060 30 FORMAT (IX, 'FUNCTION OF THE EXPERIMENTAL ERRORS'/)

0061 C

0062 C

0063 C ASK THE OPERATOR FOR VALUES FOR PSI, DEL, WAVELENGTH, ANGLE OF

0064 C INCIDENCE, NO, NF,NS, AND THEIR UNCERTAINTIES

0065 C

0066 RD=57. 29578

0067 C CONVERTS DEGREES TO RADIANS

0068 C

0069 C CHOOSE BETWEEN THE BARE OR FILM-SUBSTRATE MODEL

0070 WRITE (6,40)

0071 40 FORMAT (IX, 'ENTER 1 FOR BARE SUBSTRATE')

0072 WRITE (6,50)

0073 50 FORMAT (IX, 'ENTER 2 FOR FILM-SUBSTRATE')

0074 READ (5,60) II

0075 60 FORMAT (II)

0076 C

0077 C ENTER INPUT DATA

0078 C

0079 IF (II .EQ. 1) WRITE (4,70)

0080 70 FORMAT (T30, 'SUBSTRATE REFRACTIVE INDEX UNCERTAINTY'/)

0081 IF (II .EQ. 2) WRITE (4,80)

0082 80 FORMAT (T30, 'FILM INDEX AND THICKNESS UNCERTAINTY'/)

0083 C

0084 WRITE (4, 90)

0085 WRITE (6, 90)

0086 90 FORMAT (IX, 'ENTER WAVELENGTH IN NANOMETERS')

0087 READ (5, 100) WL

0088 WRITE (4, 100) WL

0089 100 FORMAT (FIO. 2)

0090 C

0091 WRITE (4, 110)

0092 WRITE (6, 110)

0093 110 F0RMAT(/1X, 'ENTER UNCERTAINY IN ANGLE OF INCIDENCE IN DEGREES')

0094 READ (5, 120) DPHI

0095 WRITE (4, 120) DPHI

0096 120 FORMAT (FIO. 3)

0097 DPHI=DPHI/RD

0098 C

0099 N0=1. 00036

0100 C INDEX OF AIR

0101 C IF BARE SUBSTRATE, FILM THICKNESS IS ZERO AND

0102 C REFRACTIVE INDEX EQUALS THAT OF AIR

0103 C

0104 125 IF (II .EQ. 1) NF=NO

19
0105 IF (II .EQ. 1) T=0

0106 126 IF (II .Eq. 1) GOTO 170

0107 WRITE (4, 130)

0108 WRITE (6, 130)

0109 130 FORMAT (/IX, 'ENTER INDEX OF REFRACTION OF FILM (NON-ABSORBING)*)

0110 READ (5, 140) NF

0111 WRITE(4,140) NF

0112 140 FORMAT (F9. 4)

0113 C

0114 WRITE (4, 150)

0115 WRITE (6, 150)

0116 150 FORMAT (/IX /ENTER THICKNESS OF FILM IN NANOMETERS')

0117 READ (5, 160) T

0118 WRITE (4, 160) T

0119 160 FORMAT (FIO. 2)

0120 C

0121 170 WRITE (4, 180)

0122 WRITE (6, 180)

0123 180 F0RMAT(/1X, 'ENTER REAL PART OF SUBSTRATE REFRACT INDEX')

0124 READ (5, 190) NSR

0125 WRITE (4, 190) NSR

0126 190 FORMAT (F9. 4)

0127 C

0128 IF (II .EQ. 1) GOTO 220

0129 WRITE (4, 200)

0130 WRITE (6, 200)

0131 200 FORMAT (/IX, 'ENTER UNCERTAINTY IN REAL PART OF SUBSTRATE

0132 1 REFRACTIVE INDEX')

0133 READ (5, 210) DNSR

0134 WRITE (4, 210) DNSR

0135 210 FORMAT (F9. 4)

0136 C

0137 220 WRITE (4, 230)

0138 WRITE (6, 230)

0139 230 F0RMAT(/1X, 'ENTER IMAGINARY PART OF SUBSTRATE REFRACT INDEX')

0140 READ (5, 240) NSI

0141 WRITE (4, 240) NSI

0142 240 FORMAT (F9. 4)

0143 C

0144 IF (II .EQ. 1) GOTO 270

0145 WRITE (4, 250)

0146 WRITE (6, 250)

0147 250 FORMAT (/IX, 'ENTER UNCERTAINTY IN IMAGINARY PART OF SUBSTRATE

0148 1 REFRACTIVE INDEX')

0149 READ (5, 260) DNSI

0150 WRITE (4, 260) DNSI

0151 260 FORMAT (F9. 4)

0152 C

0153 C CHOOSE BETWEEN ENTERING CONVENTIONAL NULL DATA

0154 C OR ERRORS IN DEL AND PSI

0155 C

0156 270 WRITE (6, 280)

20
01 ^7 280 F0RMAT(/1X,'D0 YOU HAVE NULL DATA AVAILABLE? (T/F) ') KJLOO RFAn^<^ 9QoS ni

990 FORMAT fLl'i 0160 r

01 fil TF ^Tll FQ FAI ^ fiDTfl 460

0162 WRITE M 300^

01 fi^ 01 64 ^00 FORMATS /IX 'FNTFR UNCFRTATNY FOR POI ART7FR NIJI 1 IN OFORFF^^M

01 6'^ READ (5, 310) DP

01 WRTTP^4 ^10"\ OP

01 67 ?1

ox w FORMAT ^'FQ 3^

01 6R r)p_r)p /DP)

01 RQ r 0170 WRITE M 320"^

01 71 WRTTP^fi ^90^

01 79 ^90 FORMATS /IX 'FNTFR UNCFRTATNY FOR ANAI Y<^FR NLJI 1 IN DFfiRFF^iM

01 7^ READ (5, 330) DA

01 lA WRTTP/'4 ^^0"^ OA

01\Ji. 7^1 o ^^0 FORMAT /"FQ 3^

01 7ft OA-OA /RO

01 77 c 0178 WRITE f4 340"^

01 70

0180 340 FORMATS /IX 'ENTER THE WAVEPLATE RETARDATION IN DEGREES '"^ 0181 READ (5, 350) DELC

01 89 WRTTPM ^^0"^ OPI C

01 8? ^^0 PORMAT^PQ

01 84 DEI C-DFl C/RD

0185 r

01 86 WRITFM 360"^

0187 WRITE f 6 360^

01v/xoo 88 ?60 FRRMAT/'/lX 'FNTFR THF WAVFPI ATF RFTARDATION UNCERTAINTY IN

0189 1X DEGREES l/^\]l\L_^s/ ''i1 0190 VX V V READ (5, 370) DDELC

01V/l Q19X WRTTF^4 ^70'\ OOPI C

0192 vX 7^ 370 FORMAT /F9 3"^ 01V/X 93s70 DDFl C-DDFI C/RD

0194 wX c

01 Ql^ WRTTP^4 ^ftO^

0196 w X 9w WRITE (6, 380)

01 07 ^80 FQRMAT(/1X, 'ENTER THE WAVEPLATE TRANSMISSION CGEFF.')

01 Oft 01 QQ WRITFM ^QO"^ TC

0900 ^QO FORMAT ^FIO e;"^ 0901 r

\jz\jz

0203 WRITE (6, 400)

0204 400 FORMAT (/IX, 'ENTER THE WAVEPLATE TRANS. CGEFF. UNCERTAINTY')

0205 READ (5, 410) DTC

0206 WRITE (4, 410) DTC

0207 410 FORMAT (FIO. 5)

0208 C

21
0209

0210

0211 420

0212

0213

0214 430

0215

0216 C

0217

0218

0219 440

0220

0221

0222

0223 450

0224

0225 C

0226 c 0227

0228 c 0229 c 0230 460

0231

0232 470

0233

0234

0235 480

0236

0237 C

0238

0239

0240 490

0241

0242

0243 500

0244

0245 C

0246 C

0247 C

0248 1000

0249 C

0250 C

0251 C

0252 C

0253

0254 c 0255 c 0256 c 0257

0258

0259

0260

WRITE (4, 420)

WRITE (6, 420)

FORMAT (/IX, 'ENTER THE WAVEPLATE AZIMUTH ANGLE (ie +-45 deg) ')

READ (5, 430) Q

WRITE (4, 430) Q

FORMAT (F9. 3)

Q=Q/RD

WRITE (4, 440)

WRITE (6, 440)

F0RMAT(/1X/ENTER THE WAVEPLATE AZIMUTH ANGLE UNCERTAINTY

IN DEGREES')

READ (5, 450) DQ

WRITE (4, 450) D(l

FORMAT (F9. 3)

DQ=DQ/RD

GOTO 1000

ERRORS IN DEL AND PSI GIVEN

WRITE (4, 470)

WRITE (6, 470)

FORMAT (/IX, 'ENTER UNCERTAINTY IN DEL IN DEGREES')

READ (5,480) DDEL

WRITE (4, 480) DDEL

FORMAT (FIO. 3)

DDEL=DDEL/RD

WRITE (4, 490)

WRITE (6, 490)

F0RMAT(/1X, 'ENTER UNCERTAINTY IN PSI IN DEGREES')

READ (5, 500) DPSI

WRITE (4, 500) DPSI

FORMAT (FIO. 3)

DPSI=DPSI/RD

REFRACTIVE INDEX OF SUBSTRATE, NS AS IN EQUATION (12)

NS=CMPLX(NSR,-NSI)

BEGIN LOOP FOR ALL ANGLES OF INCIDENCE, EVERY DEGREE

DO 5000 1=1,89

INCIDENT ANGLE IN RADIANS, PHI

PHI=I/RD

SO=SIN(PHI)

CO=:C0S(PHI)

YY=NO*SO

22
0261 CF=SQRT(1.-(YY/NF)**2)

0262 CS=CSQRT(1.-(YY/NS)**2)

0263 C

0264 C DEFINE BETA,B AS IN EQUATION (10)

0265 C

0266 B=8.*ATAN(1.)*T/WL*NF*CF

0267 C

0268 C CALC FRESNEL EQUATIONS

0269 C BREAK UP INTO SMALLER PIECES

0270 C

0271 UP1=NF*C0-N0*CF

0272 VP1=NF*C0+N0*CF

0273 UP2=NS*CF-NF*CS

0274 VP2=NS*CF+NF*CS

0275 US1=N0*C0-NF*CF

0276 VS1=N0*C0+NF*CF

0277 US2=NF*CF-NS*CS

0278 VS2=NF*CF+NS*CS

0279 C

0280 C CALC THE FRESNEL COEFFICIENTS RPOF AND RSOF, EQS.

0281 C (6) AND (7) AND CRPFS AND CRSFS IN EQS. (8) AND (9)

0282 C

0283 RP0F=UP1/VP1

0284 CRPFS=:UP2/VP2

0285 RS0F=US1/VS1

0286 CRSFS=US2/VS2

0287 C

0288 C DEFINE EJ, IMAGINARY EXPONENT E TO THE -2*B

0289 C

0290 Z=CMPLX(0.,-2.*B)

0291 EJ=CEXP(Z)

0292 C

0293 C

0294 C

0295 AP=RPOF+CRPFS*EJ

0296 BP=1+RP0F*CRPFS*EJ

0297 AS=RSOF+CRSFS*EJ

0298 BS=1+RS0F*CRSFS*EJ

0299 C

0300 C CALC RP AND RS FOR THE BARE SUBSTRATE MODEL, EQS. (2) , (3)

0301 C AND FOR THE FILM-SUBSTRATE MODEL, EQS. (4) AND (5)

0302 C

0303 RP=AP/BP

0304 RS=AS/BS

0305 C

0306 C CALC DEL AND PSI FROM INPUT PARAMETERS

0307 C

0308 PSI=ATAN (CABS (RP/RS)

)

0309 DEL=ATAN2 (AIMAG (RP/RS) , REAL (RP/RS)

)

0310 C

0311 C DEFINITION OF RHO, RP/RS AS IN EQUATION (1)

0312 C

23
0313 RHO=RP/RS

0314 RHOC=TC*CEXP ( (0 . , -1 . ) *DELC)

0315 C

0316 C

0317 C IF ERRORS IN DEL AND PSI GIVEN JUMP AHEAD

0318 IF (Ql .EQ. .FALSE.) GOTO 1150

0319 C

0320 C FOR CONVENTIONAL NULL ELLISOMETER CALCULATE NULL CONDITIONS

0321 C FOR ZONE 1 ~ BEGIN CALC FOR DDEL AND DPSI

0322 C

0323 IF(ABS(DEL) .LT. l.OE-8) GOTO 1100

0324 R= (TAN (Q) SIN (DEL-DELC) -SIN (DEL+DELC) /TAN (Q) ) /2 . /SIN (DEL)

0325 C

0326 TP1= (-R+SQRT (R*R+1) ) /TC

0327 TP2= (-R-SQRT (R*R+1) ) /TC

0328 P1=ATAN(TP1)+Q

0329 P3=ATAN (TP2)+(i

0330 A1=ATAN (REAL (RHO* (TP1*TAN (Q) *RH0C-1 . ) / (TAN (Q) +TP1+RH0C) )

)

0331 A3=ATAN (REAL (RHO* (TP2*TAN (Q) *RH0C-1 . ) / (TAN (Q) +TP2*RH0C) )

)

0332 GOTO 1110

0333 1100 P1=Q

0334 P3=H-90./RD

0335 A1=ATAN (REAL (RHO/TAN (Q) )

)

0336 A3=ATAN (REAL (RHO*TAN (Q) )

)

0337 1110 P1=AM0D (Pl+360 . /RD , 180 . /RD)

0338 A1=AM0D (Al+360 . /RD , 180 . /RD)

0339 P3=AM0D (P3+360 . /RD , 180 . /RD)

0340 A3=AM0D (A3+360 . /RD , 180 . /RD)

0341 C

0342 C AND NOW FOR THE ERRORS IN DEL AND PSI

0343 C ZONE 1 0344 P=P1

0345 A=A1

0346 C

0347 C DEFINE QUARTER-WAVE PLATE AZIMUTH ANGULAR QUANTITIES

0348 C

0349 TQ=TAN(Q)

0350 TQ2=TAN(q) *TAN(Q)

0351 TPQ=TAN(P-Q)

0352 TPQ2=TAN (P-Q) TAN (P-Q)

0353 SQ=SEC(Q)

0354 SQ2=SEC(Q)*SEC(Q)

0355 SPQ=SEC(P-Q)

0356 SPQ2=:SEC(P-Q)^SEC(P-Q)

0357 C

0358 C CALC RHOR AND RHOI AS IN EQN. (31)

0359 C

0360 DD=TC^TC^TQ2^TPQ2+l+2 . TC^TQ+TPQ^COS (DELC)

0361 RHOR= (TAN (A) /DD) (-TQ+TC^TC^TQ^TPQ2+TQ2^TC^TPQ^C0S (DELC)

0362 1 -TC^TPQ^COS(DELC))

0363 RHOI= (TAN (A) /DD) (-TQ2^TC*TPQ^SIN (DELC) -TC^TPQ^SIN (DELC)

)

0364 C

24
0365 C CALC ALL PARTIAL DERIVATIVES NEEDED IN EQNS. (33) AND (34)

0366 C

0367 C DRRDA=dRHOR/dA,DRIDA=dRHOI/dA

0368 C

0369 DRRDA=2.*RH0R/SIN(2.*A)

0370 DRIDA=2.*RH0I/SIN(2.*A)

0371 C

0372 C DRRDP=dRHOR/dP,DRIDP=dRHOI/dP

0373 C

0374 DRRDP=TAN (A) *SPQ2*TC* (TC*TQ*2 . *TPQ+COS (DELC) * (TQ2-1 .)) /OD

0375 DRRDP=DRRDP-RH0R*2 . SPG12*TQ*TC* (TC*TQ*TPQ+COS (DELC) ) /DD

0376 DRIDP=TAN (A) *SIN (DELC) *TC*SPQ2* (-TQ-1 . ) /DD

0377 DRIDP=DRIDP-RH0I*2 . *SPQ2*TQ*TC* (TC*TQ*TPQ+COS (DELC) ) /DD

0378 C

0379 C DRRDQ=dRHOR/dQ,DRIDQ=dRHOI/dQ

0380 C

0381 DRRDQ=TAN (A) * (-SQ2+TC*TC* (SQ2*TPQ2-TQ*2 . *TPQ*SP(12) +TC

0382 1 *COS (DELC) * (2 . *TQ*SQ2*TPQ-TQ2*SPQ2) +TC*SPQ2*C0S (DELC) ) /DD

0383 DRRD(1=DRRDQ-RH0R* (TC*TC* (2 . TQ*SQ2*TPQ2-2 . *TQ2*TPQ*SPQ2) + 0384 1 2.*TC*C0S(DELC)*(SQ2*TPQ-TR*SPQ2))/DD

0385 C

0386 DRIDQ=TAN (A) SIN (DELC) *TC* (-2 . TQ*SQ2*TPQ+TQ2*SPQ2+SPQ2) /DD

0387 DRIDQ=DRIDQ-RHOI* (TC*TC* (2 . *TCl*SQ2*TPGl2-2 . *TQ2*TP(;i*SPQ2) +2 . *TC*

0388 1 COS (DELC) (SQ2*TPR-TQ*SPQ2))/DD

0389 C

0390 C DRRDTC=dRHOR/dTC,DRIDTC=dRHOI/dTC

0391 C

0392 DRRDTC=TAN (A) * (2 . *TC*TQ*TPQ2+TQ2*TPQ*C0S (DELC) -TPQ*COS (DELC)

)

0393 1 /DD-RR*(2.*TC*TQ2*TPQ2+2.*TQ*TPQ*C0S(DELC))/DD

0394 C

0395 DRIDTC=TAN (A) * (-TQ2*TPQ*SIN (DELC) -TPQ*SIN (DELC) ) /DD-RHOI* (2 .

0396 1 TC*TQ2*TPQ2+2.*TQ*TPQ*C0S(DELC))/DD

0397 C

0398 C DRRDDC=dRHOR/dDELC , DRIDDC=dRHOI/dDELC

0399 C

0400 DRRDDC=-TAN (A) *SIN (DELC) * (TQ2*TC*TPQ-TC*TPQ) /DD+RHOR*SIN (DELC)

0401 1 2.*TC*TQ*TPQ/DD

0402 C

0403 DRIDDC=-TAN (A) *COS (DELC) * (TQ2*TC*TP(l+TC*TPQ) /DD+RHOI*SIN (DELC)

0404 1 *2 . *TC*TQ*TPGl/DD

0405 C

0406 C DDEL=dDEL, DPSI=dPSI AS IN EQNS. (33) AND (34) DUE TO ERRORS

0407 C IN THE QUARTER-WAVE PLATE AND POLARIZER AND ANALYSER NULL

0408 C POSITIONS FOR THE CONVENTIONAL NULL ELLIPSOMETRIC METHOD.

0409 C

0410 DDEL=(DRIDP-RHOI*DRRDP/RHOR) **2*DP*DP

0411 DDEL=DDEL+ (DRIDA-RHOI*DRRDA/RHOR) **2*DA*DA

0412 DDEL=SQRT(DDEL)

0413 DDEL=DDEL+ABS (DRIDR-RHOI*DRRDQ/RHOR) *DQ

0414 DDEL=DDEL+ABS (DRIDTC-RHOI*DRRDTC/RHOR) *DTC

0415 DDEL=DDEL+ABS (DRIDDC-RHOI*DRRDDC/RHOR) *DDELC

0416 DDEL=1/ABS (RHOR) / (1+RH0I*RH0I/RH0R/RH0R) *DDEL

25
0417 c 0418

0419

0420

0421

0422

0423

0424

0425 c 0426 c 0427 1150

0428 c 0429

0430 c 0431 C

0432 C

0433 C

0434 C

0435 C

0436

0437

0438

0439

0440

0441 c 0442 c 0443 C

0444 c 0445 c 0446

0447

0448 c 0449

0450 C

0451 c 0452 c 0453 c 0454 c 0455

0456

0457 c 0458 c 0459 c 0460 c 0461

0462 c 0463 c 0464 c 0465 c 0466

0467

0468

DPSI=(RH0I*DRIDP+RH0R*DRRDP)**2*DP*DP

DPSI=DPSI+ (RHOI*DRIDA+RHOR*DRRDA) **2*DA*DA

DPSI=SQRT(DPSI)

DPSI=DPSI+ABS (RHOI*DRIDQ+RHOR*DRRDQ) *DR

DPSI=DPSI+ABS (RHOI*DRIDTC+RHOR*DRRDTC) *DTC

DPSI=DPSI+ABS (RHOI*DRIDDC+RHOR*DRRDDC) *DDELC

DPSI=1/ (1+RHOR*RHOR+RHOI*RHOI) /SRRT(RHOR*RHOR+RHOI*RHOI) *DPSI

CONTINUE

COME HERE FOR DEL,PSI ERROR INPUT

IF (II .EQ. 1) GOTO 1300

BEGIN UNCERTAINTY CALC HERE FOR FILM-SUBSTRATE MODEL

CALC PARTIAL DIFFERENTIALS OF B, AP,BP,AS,BS WITH RESPECT TO T

DBDT=dB/clT, DAPDT=dAP/dT, DBPDT=dBP/dT, DASDT=dAS/clT, DBSDT=dBS/dT

DBDT=8 . *ATAN (1 . ) /WL*NF*CF

DAPDT=-2 . * (0 . , 1 . ) *CRPFS*EJ*DBDT

DBPDT=-2 . * (0 . , 1 . ) *RPOF*CRPFS*EJ*DBDT

DASDT=-2 . (0 . , 1 . ) *CRSFS*EJ*DBDT

DBSDT=-2 . * (0 . , 1 . ) *RSOF*CRSFS*EJ*DBDT

CALC PARTIAL DIFFERENTIALS OF PSI AND DEL WITH RESPECT TO T

DPSIDT=dPSI/dT,DDELDT=dDEL/dT AS IN EQNS (22) AND (23)

DPSIDT=SIN (2 . *PSI) /2 . *REAL (DAPDT/AP+DBSDT/BSDASDT/AS-DBPDT/BP)

DDELDT=:AIMAG(DAPDT/AP+DBSDT/BS-DASDT/AS-DBPDT/BP)

CALC PARTIAL DIFFERENTIALS OF RP, RS AND B WITH RESPECT TO NF

FIRST DEFINE P AND QC TO HELP WITH CALCULATIONS

P=SQRT (NF*NF-NO*NO*SO*SO)

HC=CS(1RT (NS*NS-NO*NO*SO*SO)

CALC PARTIAL OF B WITH RESPECT TO NF

DBDNF=dB/dNF

DBDNF=B/NF/CF/CF

CALC PARTIAL DIFFERENTIALS OF RPOF, CRPFS, RSOF, CRSFS

UPlDNF=dUPl/dNF,VPlDNF=dVPl/dNF,DUP2DNF=dUP2/dNF,DVP2DNF=dVP2/dNF

UP1DNF=C0-N0**3*S0*S0/NF**2/P

VP1DNF=C0+N0**3*S0*S0/NF**2/P

UP2DNF=NS*N0*N0*S0*S0/NF**2/P-CS

26
0469

0470 C

0471 C

0472 C

0473 C

0474

0475

0476 C

0477 C

0478 C

0479

0480

0481

0482

0483 C

0484 C

0485 C

0486 C

0487 C

0488 C

0489

0490

0491 C

0492 C

0493 C

0494 C

0495

0496

0497

0498 C

0499 C

0500 C

0501

0502

0503

0504 C

0505 C

0506 c 0507 c 0508 c 0509 c 0510

0511

0512

0513

0514 c 0515 c 0516 c 0517 c 0518 c 0519 c 0520 c

VP2DNF=NS*N0*N0*S0*S0/NF**2/P+CS

PARTIAL DIFFERENTIAL OF RPOl AND CRPFS WITH RESPECT TO NF

RPOFDN=dRPOF/dNF,CPFSDN=dCRPFS/dNF

RP0FDN=UP1DNF/VP1-VP1DNF*UP1/VP1/VP1

CPFSDN=UP2DNF/VP2-VP2DNF*UP2/VP2/VP2

USlDNF=dUSl/dNF,VSlDNF=dVSl/dNF,US2DNF=dUS2/dNF,DVS2DNF=dVS2/dNF

US1DNF=-NF/P

VS1DNF=NF/P

US2DNF=NF/P

VS2DNF=NF/P

ALL THE ABOVE PARTIALS ARE ALMOST THE SAME

PARTIALS OF RSOF AND CRSFS

RSOFDN=dRSOF/dNF , CSFSDN=dCRSFS/dNF

RS0FDN=US1DNF/VS1* (1 . +US1/VS1)

CSFSDN=US2DNF/VS2* (1 . -US2/VS2)

NOW GET THE PARTIALS OF AP,BP,AS,BS

DAPDNF=dAP/dNF, DBPDNF=dBP/dNF

Zl=2 . * (0 . , 1 . ) *CRPFS*EJ*DBDNF

DAPDNF=RP0FDN+EJ*CPFSDN-Z1

DBPDNF=EJ*(RP0F*CPFSDN+CRPFS*RP0FDN-RP0F*Z1/EJ)

DASDNF=:dAS/dNF, DBSDNF=dBS/dNF

Z2=2 . * (0 . , 1 . ) *CRSFS*EJ*DBDNF

DASDNF=RS0FDN+EJ*CSFSDN-Z2

DBSDNF=EJ*(RS0F*CSFSDN+CRSFS*RS0FDN-RS0F*Z2/EJ)

PARTIAL DIFFERENTIALS OF PSI AND DEL WITH RESPECT TO NF

AS IN EQNS. (21) AND (23)

DPSIDN=dPSI/dNF , DDELDN=dDEL/dNF

DPSIDN=SIN (2 . *PSI) /2 . *REAL (DAPDNF/AP+DBSDNF/BS1 DASDNF/AS-DBPDNF/BP)

DDELDN=AIMAG(DAPDNF/AP+DBSDNF/BS-DASDNF/AS1 DBPDNF/BP)

DEFINE ALL PARTIAL DIFFERENTIALS WITH RESPECT TO PHI

PARTIAL DIFFERENTIAL OF B WITH RESPECT TO PHI

DBDO=dB/dPHI

27
0521 DBD0=-4 . *NO*NO*T*ATAN (1 . ) /WL*SIN (2 . *PHI) /P

0522 C

0523 C PARTIALS OF RPOF,CRPFS WITH RESPECT TO PHI

0524 C UPlD0=dUPl/dPHI,VPlD0=dVPl/dPHI,UP2D0=dUP2/dPHI,VP2D0=clVP2/dPHI

0525 C

0526 UP1DO=-NF*SO+NO**3*SIN (2 . *PHI) /2 . /NF/P

0527 VP1DO=-NF*SO-NO**3*SIN (2 . *PHI) /2 . /NF/P

0528 UP2D0=-NS*NO*NO*SIN (2 . *PHI) /2 . /NF/P+NF*NO*NO*SIN (2 . *PHI)

/

0529 1 2./NS/(lC

0530 VP2D0=-NS*NO*NO*SIN (2 . PHI) /2 . /NF/P-NF*NO*NO*SIN (2 . *PHI)

/

0531 1 2./NS/QC

0532 C

0533 C RPOFD0=dRPOF/dPHI,CPFSD0=dCPFS/dPHI

0534 C

0535 RPOFDO=UP1DO/VP1-VP1DO*UP1/VP1/VP1

0536 CPFSD0=UP2D0/VP2-VP2D0*UP2/VP2/VP2

0537 C

0538 C PARTIALS OF RSOF AND CRSFS

0539 C USlDO=dUSl/dPHI , VSlDO=dVSl/dPHI , US2D0=dUS2/dPHI , VS2D0=dVS2/dPHI

0540 C

0541 US1DO=-NO*SO+NO*NO*SIN (2 . *PHI) /2 . /P

0542 VS1DO=-NO*SO-NO*NO*SIN (2 . *PHI) /2 . /P

0543 US2D0=NO*NO*SIN (2 . *PHI) /2 . * (-1 . /P+1 . /QC)

0544 VS2D0=NO*NO*SIN (2 . *PHI) /2 . * (-1 . /P-1 . /QC)

0545 C

0546 C RSOFD0=dRSOF/dPHI , CSFSDO=dCSFS/dPHI

0547 C

0548 RSOFDO=US1DO/VS1-VS1DO*US1/VS1/VS1

0549 CSFSD0=US2D0/VS2-VS2D0*US2/VS2/VS2

0550 C

0551 C PARTIALS OF AP,BP,AS, AND BS WITH RESPECT OF PHI

0552 C DAPDO=dAP/dPHI , DBPDO=dBP/dPHI , DASDO=dAS/dPHI , DBSDO=dBS/dPHI

0553 C

0554 Y1=2.*(0.,1.)*CRPFS*EJ*DBDO

0555 DAPDO=RPOFDO+EJ*CPFSDO-Y1

0556 Y2=2 . * (0 . , 1 . ) CRPFS*RPOF*DBD0

0557 DBPDO=EJ* (CRPFS*RPOFD0+RPOF*CPFSD0-Y2)

0558 Y3=:2.*(0.,1.)*CRSFS*EJ*DBDO

0559 DASD0=RSOFD0+EJ*CSFSD0-Y3

0560 Y4=2 . * (0 . , 1 . ) *CRSFS*RSOF*DBD0

0561 DBSDO=EJ* (CRSFS*RSOFD0+RSOF*CSFSD0-Y4)

0562 C

0563 C (PARTIALS OF RP AND RS WITH RESPECT TO PHI)

0564 C

0565 C CALC PARTIAL DIFFERENTIALS OF PSI AND DELTA WITH RESPECT TO PHI

0566 C TRIG FUNCTIONS OF DEL AND PSI

0567 C AS IN EQNS. (21) AND (22)

0568 C DPSIDO=dPSI/dPHI,DDELDO=dDEL/dPHI

0569 C

0570 DPSIDO=SIN (2 . *PSI) /2 . *REAL (DAPDO/AP+DBSDO/BS0571 1 DASDO/AS-DBPDO/BP)

0572 DDELDOsAIMAG (DAPDO/AP+DBSDO/BS28
0573 1 DASDO/AS-DBPDO/BP)

0574 c 0575 c TAKE DERIVATIVES WITH RESPECT TO THE REAL PART OF THE

0576 c SUBSTRATE REFRACTIVE INDEX NSR

0577 UP2DNR-dUP2/dNSR VP2DNR-dVP2/WNSR US2DNR-HIJ*52/HN<5R V<i9nNR-H\/<;9 /HM<^R

0578 c 0579 WW 1 7 UP2DNR-CF+NF*YY*YY/ f'NS**3WCS

0580 VP2DNR=CF-NF*YY*YY/('NS**3") /CS

0581 US2DNR=-CS+YY*YY/ (NS*NS*CS)

0582 wwO^ VS2DNR=CS-YY*YY V wi»L/l «l\*ww 1 1*1 1 // rNS*NS*CS'i I llw*llw*Ww J 0583 c 0584 www~ cw CPFDNRsdCRPFS wl 1 l^lll^^wwlM 1 w//dNSR Wllwl\ ^ CSFDNR=dCRSFS wwl W 1 11 \"\J wl \wl w//dNSR Ullwl\

0585 wwUw cw 0586 wwWW CPFDNR=UP2DNR/VP2-VP2DNR*UP2/VP2/VP2 wl 1 will \~wl ^Wlll\/ VI ^ Vl ^Wlll\'^Wi ^ M Tl ^/ wl ^ 0587 CSFDNR=US2DNR/VS2* WW1 will \~ww^w 1 «l^/ Vww^lX* fl +US2/VS2') ^ WW^ # w w^ J 0588 wwww cw 0589 wwU 9 cw DAPDNR=dAP/dNSR l/ni l/lll>~wr\l ^MIlwl\^L/LJI DBPDNR=dBP/dNSR wIllN^ULJI ^VJIlwl\^ DASDNR=dAS/dNSR L/r^W Wl 11 >~U/«w # Ullwl \ | DBSDNR=dBS/dNSR l/L^wwlll\~ULyw / \Jllwl\

0590 WW 7w cw 0591 DAPDNR=EJ*CPFDNR

0592 DBPDNR=EJ*RPOF*CPFDNR

0593 WW 7w DASDNR-EJ*CSFDNR L/riwL/lll\-"Lw*\»wl L/lil\ 0594 DBSDNR=EJ*RSOF*CSFDNR W W 111 ^W ^ 1 \ WW1 ^ WW1 W 1 11 ^ 0595 c 0596 c CALCULATE Wfl^B W\^^4« 1 PARTIAL I fll \ I ^r^b_ DERIVATIVES W ^_l \ ablf1l Ih\^ OFW PSI 1 AND fll 1 W DEL W WITH II A III RESPECT 11 1 ^B^l k_ W I TO1 W NSR 1 IN^I 1 0597 c ASw\y^ IN^1 1 EQNS w1 1%^ • (21) I ^ Ay AND rill w (22) 4b *B f rw DPSDNR-dPST/dNSR DDFDNR-dDFl /dN*5R r DPSDNR-<5TNf2 *PSTW2 *RFAI rOAPONR/AP+DR^iDNR/RS0601 1A DASDNR/AS-DBPDNR/BPI w f»ww 1 1 1 \ / r»w w w 1 1 1 \ / wi # 0602 wwW^ DDFDNR-ATMAG TDAPDNR / AP+DBSDNR /BS-DASDNR /AS0603 1A DBPDNR/BP') Wk^l W 111*/ L^l f 0604 c 0605 c TAKE 1 rii \ DERIVATIVES w L_4 \ x *« 1 X V ^w WITH iixiii RESPECT i \ ^wi w TO1 w THE II Ik* IMAG Alli#i w PART 1 ni i i OFw i THE i i i^ 0606 c SUBSTRATE REFRACTIVE INDEX NSI

0607 c UP2DNI-dUP2/dNST wl ^L/llX~SJV^I ^/UllWX^VIVP2DNI-dVP2/dNSI ^L/liX~UYI ^ / UllwX ^ US2DNI=dUS2/dNSI Vw^wliX^VIV^^^ \JliwX ^ VS2DNI=dVS2/dNSI w >./^l/liX"\J vw^^ VJIis^X 0608 c 0609 WWW ^ UP2DNI=fCF+NF*YY*YY/rNS**3WCS'i*ro VI ^wliX^ -1 v^' "111 * 1 1 ~ 1 1 / 1 ll%,^**w J # WW J * I w • ^ X • J 0610 VP2DNI= f CF-NF*YY*YY/ f NS* *3) /CS) * f0 . . -1 .

)

0611 US2DNI= (-CS+YY*YY/ (NS*NS*CS) ) * (0 . , -1 .

)

0612 WW X ^ VS2DNI=:fCS-YY*YY/fNS*NS*CS'i'J*ro T w^wliX* 1 WW 1 1*1 1 / I llw*llw*ww -1 J J ^ • 1 J 0613 c 0614 WWX ~ c CPFDNIsdCRPFS wl 1 wliX^wwIXI 1 w//dNSI \JllwX ^ CSFDNI=dCRSFS wwi wilX~Uwl\wl w^/dNSI \JiiwX

0615 WW X w cw 0616 WWX w CPFDNI=UP2DNI/VP2-VP2DNI*UP2/VP2/VP2 wl 1 w 1 1 X *WI ^wllXf Tl ^ wl ^WllX^WI ! ***/ 0617 CSFDNI=US2DNI/VS2* wwl wllX~ww4bwllX / wwiL* flIX • +US2/VS2') "V/w^/ « w^ 1 0618 rV

0619 c DAPDNI=dAP/dNSI , DBPDNI=dBP/dNSI , DASDNI=dAS/dNSI , DBSDNI=dBS/dNSI

0620 c 0621 DAPDNI=EJ*CPFDNI

0622 DBPDNI=EJ*RPOF*CPFDNI

0623 DASDNI=EJ*CSFDNI

0624 DBSDNI=EJ*RSOF*CSFDNI
0625 C

0626 C PARTIAL DERIVATIVES OF PSI AND DEL WITH RESPECT TO NSI

0627 C AS IN EQNS. (21) AND (22)

0628 C DPSDNI=dPSI/dNSI , DDEDNI=dDEL/dNSI

0629 C

0630 DPSDNI=SIN (2 . *PSI) /2 . *REAL (DAPDNI/AP+DBSDNI/BS0631 1 DASDNI/AS-DBPDNI/BP)

0632 DDEDNI=AIMAG (DAPDNI/AP+DBSDNI/BS-DASDNI/AS0633 1 DBPDNI/BP)

0634

0635 C

0636 C FINALLY DEFINE PARTIAL OF NF, THICKNESS DENOMINATOR TERM

0637 C AS IN EQN. (23)

0638 C

0639 C

0640 K1=:ABS (DPSIDN*DDELDT-DDELDN*DPSIDT)

0641 C

0642 IF (Kl .EQ. 0) Kl=1.0E-35

0643 C

0644 C FINAL RESULT OF SUMS IN EQNS. (21) AND (22)

0645 C

0646 DET1=ABS (DDEL*DPSIDN) +ABS (DPSI*DDELDN)

0647 DET2=ABS (DDEL*DPSIDT) +ABS (DPSI+DDELDT)

0648 DET1=DET1+ABS (DPHI) *ABS (DDELDO*DPSIDN-DPSIDO*DDELDN)

0649 DET2=DET2+ABS (DPHI) ABS (DDELDO*DPSIDT-DPSIDO*DDELDT)

0650 DET1=DET1+ABS (DNSR) *ABS (DDEDNR*DPSIDN-DPSDNR*DDELDN)

0651 DET2=DET2+ABS (DNSR) *ABS (DDEDNR*DPSIDT-DPSDNR*DDELDT)

0652 DET1=DET1+ABS (DNSI) *ABS (DDEDNI*DPSIDN-DPSDNI*DDELDN)

0653 DET2=DET2+ABS (DNSI) ABS (DDEDNI*DPSIDT-DPSDNI*DDELDT)

0654 DET1=DET1/K1

0655 DET2=DET2/K1

0656 C

0657 C DEFINE UNCERTAINTY DATA ARRAY

0658 C X(I) ANGLE OF INCIDENCE

0659 X(I)=PHI*RD

0660 YT(I)=DET1*100./T

0661 YN(I)=DET2

0662 C YT(I) X THICKNESS UNCERTAINTY YN(I) FILM INDEX UNCERTAINTY

0663 GOTO 5000

0664 C

0665 C —

0666 C BEGIN CALC OF UNCERTAINTIES FOR BARE SUBSTRATE MODEL

0667 C CALCULATE DNSR AND DNSI AS DERIVED IN EQUATIONS

0668 C (13) THROUGH (18)

0669 C

0670 1300 ZA= (NS*NS* (1 . +SEC (PHI) *SEC (PHI) ) +N0+N0* (1 . -SEC (PHI) SEC (PHI) )

)

0671 1 /NS/TAN(PHI)

0672 C

0673 ZD=:2 . (RHO-1 . ) NO^NO^SIN (PHI) SIN (PHI) TAN (PHI) TAN (PHI) /

0674 1 NS/ (1 . RHO) / (1 . RHO) / (1 . +RHO)

0675 C

0676 ZB=ZD^RHO^(0,1.)

30
0677 C

0678 ZC=ZD*SEC (PSI) *SEC (PSI) *CEXP ((0,1.) *DEL)

0679 C

0680 DNSR=ABS (REAL (ZA) *DPHI) +ABS (REAL (ZB) *DDEL)

0681 1 +ABS(REAL(ZC)*DPSI)

0682 C

0683 DNSI=ABS (AIMAG (ZA) *DPHI) +ABS (AIMAG (ZB) *DDEL)

0684 1 +ABS (AIMAG (ZC)*DPSI)

0685 C

0686 C PUT RESULTS IN ANGLE OF INCIDENCE DATA ARRAY

0687 C

0688 X(I)=PHI*RD

0689 YT(I)=DNSR

0690 YN(I)=DNSI

0691 C

0692 C

0693 5000 CONTINUE

0694 C

0695 C DISPLAY RESULTS NEXT

0696 C

0697 C THIS PROGRAM CONTAINS A PRINTOUT OF RESULTS COMMENTED OUT

0698 C AND INCLUDES VAX-11 RGL GRAPHICS TO PLOT UNCERTAINTIES AS A

0699 C FUNCTION OF ANGLE OF INCIDENCE.

0700 C THE GRAPHICS CAN ONLY BE USED WHEN COMPILED PROGRAM IS

0701 C LINKED WITH RGL GRAPHICS SUBROUTINES

0702 C

0703 C INITIALIZE GRAPHICS, CLEAR SCREEN AND TEXT, DRAW GRAPH

0704 CALL INITGR(5)

0705 CALL CLRSCR

0706 CALL CLRTXT

0707 CALL SC0L0R('GRAY3',1)

0708 CALL DPAPER('LIN',9,10,'LOG',3,9,'GRAY3')

0709 C

0710 C HEADINGS FOR PRINTOUT RESULTS OF FILM-SUBSTRATE MODEL

0711 IF (II .ER. 1) GOTO 5250

0712 C WRITE (6, 5200)

0713 C WRITE (4, 5200)

0714 C F0RMAT(/1X, 'ANGLE OF INCIDENCE (deg) ' ,3X, 'PERCENT THICKNESS UNCER

0715 C 1TAINTY',3X, 'INDEX UNCERTAINTY'/)

0716 C

0717 C LABEL GRAPHS FOR FILM-SUBSTRATE MODEL

0718 CALL LNAXIS('XB', 'ANGLE OF INCIDENCE (cleg)',0,

0719 190.,. TRUE.)

0720 CALL SC0L0R('GRAY1',2)

0721 CALL LNAXIS('YL', 'THICKNESS ERROR DT/T (%)', .1,100.

,

0722 l.TRUE.)

0723 CALL SC0L0R('GRAY2' ,3)

0724 CALL LNAXIS('YR' , 'INDEX ERROR DNF' , .001,1 .TRUE.)

0725 GOTO 5300

0726 5250 CONTINUE

0727 C HEADING FOR PRINTOUT RESULTS OF BARE SUBSTRATE MODEL

0728 C WRITE (6, 5260)

31
0729 C WRITE (4, 5260)

0730 C F0RMAT(/1X, 'ANGLE OF INCIDENCE (deg) ' ,2X, 'INDEX (REAL) UNCERTAIN

0731 C 1TY',2X,' INDEX (IMAGINARY) UNCERTAINTY'/)

0732 C

0733 C LABEL GRAPHS FOR BARE SUBSTRATE MODEL

0734 CALL LNAXIS('XB' , 'ANGLE OF INCIDENCE (cleg)',0,

0735 190.,. TRUE.)

0736 CALL SCOLOR ( ' GRAYl '

, 2)

0737 CALL LNAXIS('YL' , 'SUBSTRATE INDEX ERROR DNSR' , .001,1.

,

0738 l.TRUE.)

0739 CALL SCOLOR ('GRAY2' ,3)

0740 CALL LNAXIS('YR', 'SUBSTRATE INDEX ERROR DNSI' , .001,1 .

,

0741 l.TRUE.)

0742 C

0743 C LIST DATA

0744 5300 CONTINUE

0745 C DO 6000 J=l,89

0746 C WRITE(4,5400) X(J) , YT(J) , YN(J)

0747 C WRITE (6, 5400) X(J) , YT(J) , YN(J)

0748 C FORMAT(5X,F10.3,20X,F12.5,20X,F9.5)

0749 C CONTINUE

0750 C

0751 C GRAPH DATA

0752 CALL PDATA(89,X,YT, 'L', 'GRAYl', ,1, .FALSE. , ,)

0753 CALL PDATA (89 , X , YN ,

' R

'

, ' GRAY2 ' ,

, 9 , . FALSE . , ,

)

0754 C

0755 CALL SCOLOR (' GRAY3 ' ,1)

0756 C

0757 C END OF MAIN PROGRAM

0758 END

32
FEDERAL INFORMATION PROCESSING STANDARD SOFTWARE SUMMARY

01. Summary date

Yr. Mo. Day

8 5 0 8 0 1

04. Software date

Yr. Mo. Day

8 0 6 10

02. Summary prepared by (Name and Phone)

Deane Chandler-Horowitz (301) 921-3625

05. Software title Semiconductor Measurement Technology:

A FORTRAN Program for analysis of the uncertainty ir

the ellipsometric determination of the thickness and

refractive index of a film on a substrate or the re- 07. internal Software ID 06. Short title ERROR , FOR fractive index of a bare substrate

03. Summary action New Replacement Deletion s

Previous Internal Software ID

08. Software type

Automated Data Q System

[X] Computer Program

I I Subroutine/Module

09. Processing mode

1^ Interactive Q Batch

I I Combination

10. GeneralApplication area

Computer Systems

|—I Support/Utility

["^ Scientific/Engineering

I I Bibliographic/Textual

Management/

Business

Process Control Other

Specific Analysis of ellipsometric accuracy

11. Submitting organization and address Semiconductor Electronics Division

National Bureau of Standards

Gaithersburg, MD 20899

12. Technical contact(s) and phone

Deane Chandler-Horowitz

(301) 921-3625

13. Narrative The program performs the following functions: user can choose either the bare substrate

or film substrate model; user inputs ellipsometric data (wavelength of light, angle of

incidence, etc.); computes from the known input uncertainties the ellipsometric output

uncertainty in either the film's thickness and refractive index or the substrate's re- fractive index; a graph of uncertainty as a function of angle of incidence is plotted.

14. Keywords

ellipsometry ; bare-substrate model; film-substrate model; thickness uncertainty; refractive index uncertainty

15. Computer manuf r and model DEC VAX 11/78016. Computer operating system

VMS 4,0

17. Programing language(s)

FORTRAN

18. Number of source program state- ments 766

19. Computer memory requirements

55 KB

20. Tape drives one for program

loadinq

21. Disk/Drum units none22. Terminals

1 video VT 125

23. Other operational requirements

24. Software availability 25. Documentation availability

Available Limited m

In-house only Available Inadequate In-house only

m .

A user's manual is available as NBS

Special Publication 400-78.

26. FOR SUBMITTING ORGANIZATION USE

185-101 standard Form 185

1974 July

U.S. Dept. of Commerce—NBS

(PIPS. Pub. 30)
NBS-lUA (REV. 2-ec) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO. NBS/SP-400/78

2. Performing Organ. Report No. 3. Publ ication Date May 1986

4. TITLE AND SUBTITLE

Semiaonduator Measurement Teohnology

:

Analytic Analysis of Ellipsometric Errors

5. AUTHOR(S)

Deane Chandler-Horowitz

6. PERFORMING ORGANIZATION (If jo/nt or other than NBS, see instructions)

National Bureau of Standards

Department of Commerce

Gaithersburg, MD 20899

7. Contract/Grant No.

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City, State. ZIP)

Same as item 6

10. SUPPLEMENTARY NOTES

Library of Congrss Catalog Card Number: 86-600541

[2J Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most si gnificant information. If document includes a si gnificant

bi bliography or literature survey, pient/on it here) A conputer program is given that contains an explicit ellipsometric error analysis.

The program can identify the ellipsometric inaccuracies for any ellipsometer, can be

used to determine which parameters contribute the most to the overall measurement

inaccuracy, and can lead one to an optimum measurement procedure. A FORTRAN program

that performs the evaluation of the partial derivative expressions needed to analyze

ellipsometric measurement uncertainties is listed. Ihe program determines the

uncertainty in the calculation of the refractive index of a bare isotropic sxibstrate or the uncertainty in the determination of the thickness and refractive index of a nonabsorbing film on a substrate of known refractive index. These are the two most

commonly used surface models used in ellipsometry performed at single angle of

incidence and a single wavelength. Ihe program input parameters include the

wavelength of the light, the angle of incidence and its uncertainty, and the

uncertainties in the ellipsometric parameters A and Ihey also include in the

ambient-substrate model an estimated value for the substrate's refractive index, and

in the f ilm- subs t rate model the refractive index of the substrate and its

uncertainty and estimated values for the film's refractive index and thickness. The

case of the conventional null ellipsometer utilizing a quarter-wave plate is treated

to find the uncertainties in A and \\) from the uncertainties in the polarizer and

analyzer null values and the waveplate constants.

12. KEY WORDS (S/x to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

bare substrate model; ellipsometry; film substrate model; FORTRAN program;

measurement uncertainties; systematic errors.

13. AVAILABILITY

[X] Unlimited

I I For Official Distribution. Do Not Release to NTIS

Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

36

15. Price

ir U.S. QOVEENMEWTPRINTLNU urriUE,: i?oo - usi.-u/u - 265/40091 USCOMM-DC 6043-P80


Technical Publications

Periodical

Journal of Research—The Journal of Research of the National Bureau of Standards reports NBS research and development in those disciplines of the physical and engineering sciences in which the Bureau is active. These include physics, chemistry, engineering, mathematics, and computer sciences. Papers cover a broad

range of subjeas, with major emphasis on measurement methodology and the basic technology underlying

standardization. Also included from time to time are survey articles on topics closely related to the Bureau's

technical and scientific programs. Issued six times a year.

Nonperiodicals

Monographs—Major contributions to the technical literature on various subjects related to the Bureau's scientific and technical aaivities. Handbooks—Recommended codes of engineering and industrial practice (including safety codes) developed in cooperation with interested industries, professional organizations, and regulatory bodies.

Special Publications—Include proceedings of conferences sponsored by NBS, NBS annual reports, and other

special publications appropriate to this grouping such as wall charts, pocket cards, and bibliographies.

Applied Mathematics Series—Mathematical tables, manuals, and studies of special interest to physicists,

engineers, chemists, biologists, mathematicians, computer programmers, and others engaged in scientific and

technical work.

National Standard Reference Data Series—Provides quantitative data on the physical and chemical properties

of materials, compiled from the world's literature and critically evaluated. Developed under a worldwide pro- gram coordinated by NBS under the authority of the National Standard Data Act (Public Law 90-396).

NOTE: The Journal of Physical and Chemical Reference Data (JPCRD) is published quarterly for NBS by

the American Chemical Society (ACS) and the American Institute of Physics (AIP). Subscriptions, reprints,

and supplements are available from ACS, 1155 Sixteenth St., NW, Washington, DC 20056.

Building Science Series—Disseminates technical information develofjed at the Bureau on building materials,

components, systems, and whole structures. The series presents research results, test methods, and perfor- mance criteria related to the structural and environmental functions and the durability and safety charaaeristics of building elements and systems.

Technical Notes—Studies or reports which are complete in themselves but restrictive in their treatment of a

subject. Analogous to monographs but not so comprehensive in scope or definitive in treatment of the subject

area. Often serve as a vehicle for final reports of work performed at NBS under the sponsorship of other government agencies. VolunfTr> Product Standards—Developed under procedures published by the Department of Commerce in Part 10, Title 15, of the Code of Federal Regulations. The standards establish nationadly recognized re- quirements for products, and provide all concerned interests with a basis for common understanding of the

characteristics of the products. NBS administers this program as a supplement to the activities of the private

sector standardizing organizations.

Consumer Information Series—Practical information, based on NBS research and experience, covering areas of interest to the consumer. Easily understandable language and illustrations provide useful background

knowledge for shopping in today's technological marketplace.

Order the above NBS publications from: Superintendent of Documents, Government Printing Office,

Washington, DC 20402. Order the following NBS publications—FIPS and NBSIR's—from the National Technical Information Service, Springfield, VA 22161.

Federal Information Processing Standards Publications (FIPS PUB)—Publications in this series collectively constitute the Federal Information Processing Standards Register. The Register serves as the official source of

information in the Federal Government regarding standards issued by NBS pursuant to the Federal Property

and Administrative Services Act of 1949 as amended, Public Law 89-306 (79 Stat. 1127), and as implemented

by Executive Order 1 1717 (38 FR 12315, dated May 11, 1973) and Part 6 of Title 15 CFR (Code of Federal

Regulations).

NBS Interagency Reports (NBSIR)—A special series of interim or final reports on work performed by NBS

for outside sponsors (both government and non-government). In general, initial distribution is handled by the sponsor; public distribution is by the National Technical Information Service, Springfield, VA 22161, in paper copy or microfiche form.
U.S. Department of Commerce

National Bureau of Standards

Gaithersburg, MD 20899

Official Business

Penalty for Private Use $300