Reference

NS5

Publi - cations

NATL INST. OF STAND & TECH

A111DS 17»42bE

Q NBS TECHNICAL NOTE

U.S. DEPARTMENT OF COMMERCE/National Bureau of Standards

Measurement Assurance for

Dimensional Measurements on

Integrated-Circuit Photomasks

100

. U5753

no, 1164

1982
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards' was established by an act of Congress on March 3, 1901. The 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 per- formed by the National Measurement Laboratory, the National Engineering Laboratory, and

the Institute for Computer Sciences and Technology.

THE NATIONAL MEASUREMENT LABORATORY provides the national system of

physical and chemical and materials measurement; coordinates the system with measurement

systems of other nations and furnishes essential services leading to accurate and uniform

physical and chemical measurement throughout the Nation's scientific community, industry,

and commerce; conducts materials research leading to improved methods of measurement,

standards, and data on the properties of materials needed by industry, commerce, educational

institutions, and Government; provides advisory and research services to other Government

agencies; develops, produces, and distributes Standard Reference Materials; and provides

calibration services. The Laboratory consists of the following centers:

Absolute Physical Quantities2 — Radiation Research — Chemical Physics —

Analytical Chemistry — Materials Science

THE NATIONAL ENGINEERING LABORATORY provides technology and technical ser- vices 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 Engineering2 — Manufacturing

Engineering — Building Technology — Fire Research — Chemical Engineering2

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.

'Headquarters and Laboratories at Gaithersburg, MD, unless otherwise noted; mailing address Washington, DC 20234. 'Some divisions within the center are located at Boulder, CO 80303.
BATIONAL BUREAU

OF STANDARDS

LIBRART SEP 1 5 1982

Otf Measurement Assurance for

Ho

Dimensional Measurements on

Integrated-Circuit Photomasks n°

Nte

Carroll Croarkin

Ruth N. Varner

Statistical Engineering Division

Center tor Applied Mathematics

National Bureau of Standards

Washington, DC 20234

,^T °

F

'o.

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued August 1982
National Bureau of Standards Technical Note 1164

Natl. Bur. Stand. (U.S.), Tech. Note 1 164, 50 pages (Aug. 1982)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1982

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

Price (Add 25 percent for other than U.S. mailing.)
FOREWORD

This document has evolved from material presented as part of an on-going series

of NBS training seminars on linewidth measurements for integrated-circuit

photomasks and wafers sponsored by the Semiconductor Materials and Processes

Division of the Center for Electronics and Electrical Engineering in

cooperation with the Statistical Engineering Division of the Center for Applied

Mathematics. The seminars are conducted under the NBS Semiconductor Technology

Program which serves to focus NBS research on improved measurement technology

for the semiconductor device community in specifying materials, equipment, and

devices in national and international commerce, and in monitoring and

controlling device fabrication and assembly. This research leads to carefully

evaluated, well-documented test procedures and associated technology which,

when applied by the industry, are expected to contribute to higher yields,

lower cost, and higher reliability of semiconductor devices and to provide a basis for controlled improvements in fabrication processes and device

performance.

The document is intended both as an instructional aid for future NBS training

seminars on linewidth measurements and as a procedural guide for the

calibration of an optical-microscope linewidth-measurement system using an NBS

Optical Microscope Linewidth-Measurement Standard (SRM-474 or SRM-475) or any

properly calibrated linewidth standard. It also serves as the basis for a standard method for calibration of an optical microscope for linewidth or linespacing measurement which is currently being prepared under the auspices of the

American Society for Testing and Materials (ASTM) Subcommittee F-l on Electronics.

Special acknowledgment is gratefully made to the following NBS staff:

Dr. Diana Nyyssonen and Mr. John Jerke for their helpful suggestions and

discussions; Dr. James Filliben for modifications to the software package

DATAPLOT which produced the graphs and tables; Dr. Clifford Spiegelman for

advice and consultation particularly in regard to the derivation of

uncertainties; Dr. Lee Kieffer for suggestions for clarification on many

points; and Mrs. Janet Couch for her expertise and patience in the preparation

of the manuscript. Special acknowledgement is also made to the Office of

Measurement Services of the National Bureau of Standards which supported this

work.

ill
TABLE OF CONTENTS

Page

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

2. What exactly is a calibration curve? . 4

3. Is the optical system in need of calibration and, if so, how is

this accomplished? . 5

4. What can go wrong with the calibration procedure? 16

5. What is the precision of the optical system and how is it

estimated? 18

6. How long will the derived calibration curve be adequate for

correcting dimensional measurements? ................ 20

7. Can the control data be used to update the calibration curve?. ... 26

8. What is the uncertainty of the values corrected by the calibration

curve? ..................... 30

References 41

Appendix 42

LIST OF WORKSHEETS

A. Calculations for a line-spacing calibration curve. ......... 10

B. Calculations for a linewidth calibration curve . 14

C. Calculations for a pooled standard deviation 19

D. Calculations for control of measurement process. .......... 23

E. Calculations for updating the calibration curve. . 28

F. Calculation of limits to error and uncertainty ........... 33

IV
LIST OF FIGURES

Page

1. Schematic diagram of calibration curve 4

2. Differences between line-spacing measurements and

NBS values plotted against NBS values 9

3. Deviations of line-spacing measurements from a linear

calibration curve plotted against NBS values 9

4. Differences between linewidth measurements and

corresponding NBS values plotted against NBS value 13

5. Deviations of observed linewidths from linear calibration curve plotted against NBS values 13

6. Comparison of repetitions of linewidth measurements 17

7. Control chart for opaque linewidths for data in Worksheet D 22

8. Differences of observed linewidths from NBS values plotted

against NBS values 24

9. Control values for the lower, mid and upper endpoints of data

in Figure 7 24

LIST OF TABLES

1. Critical values of Student's t distribution for a two-sided test . 34

2.1. Critical values of t distribution for controlling the output

of a calibration curve at m points at significance level a = 0.05 35

2.2. Critical values of t distribution for controlling the output

of a calibration curve at m points at significance level a = 0.01 36

3.1. Percentage points of the x distribution-lower 1% points 37

3.2. Percentage points of the x distribution-lower 5% points 38

4.1. Percentage points of the F distribution-upper 1% points 39

4.2. Percentage points of the F distribution-upper 5% points 40

v

Measurement Assurance for Dimensional

Measurements on Integrated-Circuit Photomasks

Carroll Croarkin

Ruth N. Varner

Statistical Engineering Division

Center for Applied Mathematics

National Bureau of Standards

Washington, DC 20234

Optical Microscope Linewidth-Measurement Standards, SRM-474 and

SRM-475, have been developed by NBS for optical imaging systems

capable of making line-spacing and linewidth measurements in the

0.5)jm -12pm regime on IC photomasks. Each artifact affords a means

of reducing systematic errors via a calibration curve and keeping the

optical system in statistical control. Procedures are given for

accomplishing these goals along with a discussion of the uncertainty

of the calibrated values.

Key Words: IC photomask; linear calibration curve; line-spacing; linewidth;

measurement assurance; photomask; SRM; statistical control of

measurement process; tests for systematic error; uncertainty.

1. Introduction

Optical imaging systems which use an optical microscope fitted with some type

of measurement attachment are commonly used for line-spacing and linewidth

measurements in the 0.5um -12um regime on integrated circuit (IC) photomasks.

In an effort to provide the microelectronics industry with calibration

standards for determining and reducing systematic errors in line-spacing and

linewidth measurements , the National Bureau of Standards undertook to develop a dimensional artifact and procedures for calibrating a variety of measurement

systems. References [1], [2], and [3] describe this effort.

Past attempts to improve the accuracy of linewidth measurements have been

hampered by the lack of a suitable linewidth artifact and misconceptions about

the nature of line-spacing and linewidth measurements. Linewidth determination

is a more difficult measurement than line-spacing determination. Line-spacing

measures the displacement between two objects; it can be a left-edge to

left-edge, right-edge to right-edge, or center to center measurement in which

the errors in detecting the line edges cancel each other. Linewidth measures

the width of a physical object; it is basically a left-edge to right-edge

measurement in which the errors in detecting the edges are additive.

Therefore, systematic errors in linewidth measurements are fundamentally

related to edge detection [4] and to a lesser extent may also depend upon the

inherent metric or line-spacing calibration of the system. Theory of the

optical microscope in relation to linewidth measurements is discussed in

references [5] and [6].
In the absence of a linewidth standard, the IC industry has sometimes relied on line-spacing calibration to resolve the linewidth measurement problem. This

relationship was investigated in an NBS sponsored interlaboratory study

involving ten IC companies [2]. Of the twenty systems analyzed in the study,

only seven had substantial systematic line-spacing errors while seventeen of

the systems had substantial systematic linewidth errors. A line-spacing

calibration curve created for each of the seven systems with line-spacing

offsets from NBS was used to correct linewidth data for those systems. In no case did the line-spacing correction eliminate the linewidth errors although it

did reduce some substantially. In at least one case the offset was worse after

correction. As far as systems in the study are representative of optical

systems available in the industry at this time, it does not seem reasonable to

expect a correction for metric to solve the linewidth measurement problem.

A solution to the linewidth measurement problem is made possible by the

development of SRM-474 and SRM-475. The two artifacts are essentially the same except that the former contains four rows of lines, alternately opaque and

clear, which are calibrated for linewidth; the latter contains only two rows of

lines. Each row contains ten randomly arranged lines in the regime from 0.5um

to 12um. In addition, both SRMs contain a row of ten randomly arranged

spacings in the regime 0.5um to 12um.

The procedures in this document serve as a guide for establishing a measurement

assurance program for linewidth or line-spacing measurements. The measurement

assurance concept as set forth in reference [7] requires that a measurement

system be tied to a defined unit of measurement or to a single measurement

system such as the NBS system. The line-scale of an optical system is tied

to the defined unit of length (wavelength of radiation of krypton 86) [8] by a calibration curve derived from measurements made on the calibrated spacings on

the SRM. The linewidth capability of an optical system is tied to the unit of

length via an NBS photometric system in which the image profile of the

calibrated lines on the SRM are made to agree with theoretical profiles within

the reported uncertainty [9]. This is accomplished by adherence to NBS

recommended procedures for adjusting the microscope [10] and by a calibrated

curve derived from measurements made on the calibrated lines on the SRM.

The measurement assurance concept also requires that the measurement system be

maintained in a state of control, thus assuring that the uncertainty ascribed

to the output of the system is valid at all times. Methods suggested for

maintaining control on the values reported by the calibrated system and for

computing the associated total uncertainty are applicable to either a linewidth

or line-spacing discipline.

Although the procedures outlined in this document specify the use of SRM-474 or SRM-475, they are valid for any properly calibrated dimensional artifact as long as it is understood that several substantial issues such as the proper

form of the calibration curve and possible sources of error should be addressed

before attempting to adapt this analysis to a particular situation. Attention

should be given to the discussions in section 4 concerning between-day

differences in the calibration curve, the linearity assumption, and operator

differences, keeping in mind that any problems in these areas must be resolved

for a valid application of the calibration curve.
All analyses are illustrated using measurements made on an artifact similar to

SRM-474 by a participant in the interlaboratory study. The raw data from the

interlaboratory study are reported to two decimal places. Calculations were

made via a computer, and the results are rounded to four decimal places in most

of the tables. Consequently, values calculated from these rounded values may

vary slightly from those reported in this document.

As a final note, the term system as used in this publication should not be

thought of as only the microscope and measurement attachment per se but as the

total configuration of operator, environment and apparatus that go into

producing a measurement. It is recommended that a minimum of four sets of

measurements be made of the lines or spacings on the SRM in order to establish

a calibration curve for an optical system. The calibration curve will be

valid for the system only insofar as the initial measurements that produce the

calibration curve are representative of operating conditions and only as long

as the system response remains in a state of control.
2. What exactly is a calibration curve?

Calibration is a process of intercomparing an unknown with a standard and

assigning a value to the unknown based on the accepted value of the standard.

The intercomparison can also be accomplished by treating the standard as if it

were an unknown in the user's measurement process, thus eliminating any offset

that may exist between the user's measurement system and the measurement system

used by the laboratory such as NBS, which measured the standard. When the

intercomparisons are extended over a measurement regime of interest and a functional relationship is shown to exist between the user's measurements and

the accepted values for the standard, the assignment of values to the user's

system is based on this functional relationship.

For the present exercise, the NBS dimensional artifact covering the 0.5um-12ym

regime is the standard, and measurements made on the individual lines on the

artifact with an optical imaging system form the basis for establishing a functional relationship between the two systems. A least-squares technique is

employed to derive a best fitting straight line to the measured linewidths as a function of the NBS values. This empirical fit is called the calibration

curve.

In Figure 1, each optical measurement is plotted against the corresponding NBS

value, and the calibration curve fitted to all the measurements is shown by the

solid line. The offset between the user's system and the NBS system is

reduced by relating any future linewidth value back to the NBS value.

Schematically, for a future linewidth value y(T) as shown on the y-axis, a dotted line is drawn through y(T) parallel to the x-axis. At the point where it

intersects the calibration curve another dotted line is drawn parallel to the

y-axis, and its point of intersection on the x-axis, x(T), is the corresponding

calibrated value relative to NBS.

Y-AXIS

X-AXIS

NBS VALUES

Figure 1. Schematic Diagram of Calibration Curve

4
3. Is the optical system in need of calibration and, if so, how is this

accomplished?

The question of whether or not a calibration is needed for an optical imaging

system can be resolved by applying the procedures in this section to

measurements made on the NBS artifact. If a calibration is indicated, the same measurements are used to construct a calibration curve which in turn is used to

correct all future measurements on the optical systems.

The determination of systematic error is made independently for line-spacing

and linewidth measurements. A line-spacing calibration curve for reducing

systematic line-spacing errors is derived from measurements on the ten

line-spacings on the SRM. Linewidth calibration curves for reducing systematic

linewidth errors for opaque and clear lines are derived from measurements on

the ten opaque lines and the ten clear lines on the SRM. Separate calibration

curves are needed for each polarity because it is expected that offsets for

clear lines will be in the opposite direction from offsets for opaque lines.

A minimum of four repetitions of the measurements on the ten lines or spacings

on the SRM are needed in order to establish a calibration curve. These

repetitions should be spaced over roughly a two-week interval to ensure independence among the measurements, and it is reemphasized that the efficacy

of the calibration curve as a device for reducing systematic errors is

dependent upon the repetitions being truly representative of operating

conditions in the laboratory. The user is further cautioned that different

operators may produce measurements with systematic errors that are substantially different in sign or in magnitude, thereby necessitating separate

calibration curves for each operator.

The procedures and analyses that are set forth in this publication are intended

for either line-spacing or linewidth measurements. Because it is anticipated

that the majority of users will be primarily concerned with linewidth

determination, the text and examples are in terms of linewidth, but the total

content including equations and tests are equally applicable to line-spacing

measurements. Examples of the derivation of both a line-spacing calibration

curve and a linewidth calibration curve are included at the end of this

section.

To return to the calibration question, it is noted that an optical imaging

system with negligible linewidth errors will exhibit the following

characteristics. The observed linewidths will be a linear function of the

nominal linewidth values assigned to the artifact by NBS. This linear function

will have a slope b=l and an intercept a=0, and the scatter about the fitted

line will be sufficiently small for the system.

The thrust of the remainder of this section is to examine the relationship

between the NBS values for the linewidths and the user's observations on the

lines. This is accomplished by fitting the observed linewidths to a linear

function of the NBS linewidth values and testing the resulting estimates of the

slope and intercept for the characteristics cited above. If these tests show

that linewidth errors are not negligible, thus indicating a need for correction

to the optical system, the fitted curve becomes the calibration curve for the

system.
Before undertaking this analysis on the data, it is suggested that the

differences between the observed linewidths and the corresponding NBS values be

plotted against the NBS values. Any truly aberrant or oscillatory behavior of

the system will show up on the plot, and such behavior obviously precludes

using such data to calibrate the system.

A plot of the differences will also make it easier to identify any outliers

that are present in the data set. For the purpose of this exercise an outlier

is defined as any measurement that is obviously inconsistent with the other

measurements in the data set whether it be an individual point or a group of

points representing a single repetition. Any such discordant points should be

deleted from the data set because even a single isolated outlier can seriously

perturb the calibration curve. An outlier can be particularly disruptive if it

happens to lie near one of the endpoints of the calibration interval.

Formal tests for outliers are not given in this publication, but they are discussed in detail in reference [11]. A careful study of the difference plot

should be adequate for spotting outliers. When one is satisfied that the data

set is free of any discordant points and that it consists of measurements that are representative of the diversity of conditions affecting the measurement

system, then the following analysis is performed.

Given linewidth measurements z^j and corresponding NBS linewidth values Wji

(1=1,..., k; j=l,..., 10) where i denotes the repetition and j denotes the

linewidth, least-squares estimates are obtained for a linear function of the

form

(3.1) z-m = a + b w-M 1=1,... ,k

j-l,...,10

The equations for estimating the parameters are as follows:

I I(Wij ~ w)(zij - z)

b = U

I I ( WjLj - w)2

i J

a = z - b w

I I wij _ I I z ij

where w = i j and z = i j

The deviations from the linear fit are

6 ij = Zij - a - b Wij

and the residual standard deviation of the linear fit is

a <\
where n is the total number of observations, n = 10k.

Before proceeding with the tests for the slope and intercept, one should be

satisfied that a linear function is indeed a good characterization of the

system response as a function of the assigned values of the lines. A formal

statistical test for deciding the appropriateness of a linear function is given

in the appendix. A plot of the deviations from the linear fit provides

considerable insight into this question.

The deviations are perhaps the best diagnostic tool for deciding whether or not

the linear fit is a proper representation of the relationship between the NBS

values and the user's system. The deviations, when plotted against the NBS

values, should scatter randomly about a line drawn horizontal to the x-axis at zero with approximately an equal number of points falling above and below the

line. Any apparent cyclic behavior or obvious clustering of the deviations is

evidence that a linear fit is not appropriate. The NBS artifact is designed

only for systems with linear responses and probably cannot be used if a higher

order curve is indicated because of the limited number of lines on the

artifact.

Test statistics, tj and t2, for testing if the intercept a=0 and if the slope

b=l , respectively, are as follows [12]:

t

l = £ t 2 = l~t> s a

s b

where

/

I lyu* Y s b

= s_

= s | i j

1 and [ n IXO^j " wj2 I (HCwij " w) : / V

j

A value of |t]J > t a/2(n-2) indicates that the intercept a is significantly

different from zero, and a value of 1 1 2

1 > ^-a/2^ n~^ indicates that the slope b

is significantly different from one.* The parameters a and b are tested

separately so that the nature of the systematic errors can be ascertained. If

only the intercept is significantly different than assumed, the errors are constant, and a constant correction would be sufficient to reduce this source

of error. If the slope is significantly different than assumed, the errors are related to width of the lines, and a calibration curve is used to correct for

For probability level 1-a the critical value t a/2( v ) can be found in a table

of Student's t distribution where v is the number of degrees of freedom in

the residual standard deviation of the fit. Critical values for a = 0.05 are tabulated in Table 1 for v = 30 (2) 120.
systematic errors. A significant value for either of these parameters implies

that there are systematic linewidth errors and, as a practical matter, any

future linewidth measurement z should be corrected by

(3.2)

z** = z - a

Calculations for a line-spacing calibration curve based on measurements that

were made by a participant in the interlaboratory study on an artifact similar

to SRM-474 are illustrated in Worksheet A. The calculations utilize four

repetitions on the spacings, and the total number of observations is n=40.

Worksheet A lists the NBS assigned values of the line-spacings, the

participant's observed values, and the differences between the observed and NBS

assigned values. The differences are plotted against the NBS values in Figure

2, and for the system to be properly adjusted for scale, these differences

should be scattered randomly about the zero line drawn horizontal to the

x-axis. It is obvious from Figure 2 that the line-spacing scale of the system

is offset from the NBS system and that the magnitude of this offset is related

to the size of the spacing. This judgment is confirmed by the t-tests for the

slope and intercept of the linear function that has been fit to the data. Both

t-tests are significant.

The deviations from the linear fit are plotted in Figure 3. They are sufficiently random to conclude that a linear function is appropriate for the

data. Because the t-tests for the intercept and slope are significant, all

future line-spacing measurements should be corrected by the calibration curve

as defined in equation 3.2.

Calculations for a linewidth calibration curve based on measurements by the same participant in the interlaboratory study on the ten opaque lines on the same artifact are illustrated in Worksheet B. The calculation utilizes four

repetitions on the opaque lines, and the total number of observations is n=40.

Worksheet B lists the NBS assigned values of the lines, the participant's

observed values, and the differences between the participant's values and the

NBS assigned values. The differences are plotted against the NBS values in

Figure 4. It is obvious from Figure 4 that the linewidth calibration of the

system is also offset from the NBS system and that the magnitude of this offset

is related to the width of the lines. This is confirmed by the t-tests for the

intercept and slope as shown in the worksheet.

The deviations from the linear fit are plotted in Figure 5. In this case, the

deviations are randomly distributed about the zero line indicating that the

linear fit is a good characterization of the data and that the calibration

curve will adequately correct the system. Because the t-tests for the

intercept and the slope of this linear function are significant, all future

linewidth measurements should be corrected by equation 3.2.
D

IFF

E

R

E

N

C

E

S

F

R

M

N

B

S

0.3

0.2 -

0. 1 -

0.0 -

-0. 1

*

* **

i

2 2 *

*

3

*

2

*

*

* *

1

*

* 1 *

*

**

*

*

1 -j—,- 1 i

1

I

1

*

-i—,

2 4 6 8

NBS VALUES (MICROMETERS)

10 12

Figure 2: Differences between line-spacing measurements and NBS values plotted

against NBS values.

D

E

V

IA

T

I

N

S

0.2

0. 1 —

0.0 —

-0.1 —

-0.2

* —

* *

* *

* 2

*

* * 1

i 2

* *

* * 3 2

*

*

*

*

* 1

*

*

*

1 -j—,- 1i

1 i

*

1

i

1

*

i 4 6 8

NBS VALUES (MICROMETERS)

10 12

Figure 3: Deviations of line-spacing measurements from a linear calibration

curve plotted against NBS values.

9
Worksheet A

Calculations for Line-spacing Calibration Curve

Line-Spacing Measurements

Values in Micrometers

NBS Observed Differences Fitted Deviations

Values Values From NBS Value From Fit w z d z' 6

6.19 6.31 0.12 6.3455 -0.0355

9.17 9.27 0.10 9.2869 -0.0169

1.99 2.21 0.22 2.2000 0.0100

7.77 8.00 0.23 7.9050 0.0950

4.00 4.27 0.27 4.1839 0.0861

10.77 10.93 0.16 10.8662 0.0638

4.78 4.95 0.17 4.9538 -0.0038

2.99 3.24 0.25 3.1870 0.0530

6.98 7.14 0.16 7.1253 0.0147

9.98 10.23 0.25 10.0864 0.1436

6.19 6.27 0.08 6.3455 -0.0755

9.17 9.21 0.04 9.2869 -0.0769

1.99 2.19 0.20 2.2000 -0.0100

7.77 7.81 0.04 7.9050 -0.0950

4.00 4.15 0.15 4.1839 -0.0339

10.77 10.73 -0.04 10.8662 -0.1362

4.78 4.87 0.09 4.9538 -0.0838

2.99 3.17 0.18 3.1870 -0.0170

6.98 7.07 0.09 7.1253 -0.0553

9.98 10.02 0.04 10.0864 -0.0664

6.19 6.31 0.12 6.3455 -0.0355

9.17 9.34 0.17 9.2869 0.0531

1.99 2.22 0.23 2.2000 0.0200

7.77 7.95 0.18 7.9050 0.0450

4.00 4.15 0.15 4.1839 -0.0339

10.77 10.92 0.15 10.8662 0.0538

4.78 5.00 0.22 4.9538 0.0462

2.99 3.21 0.22 3.1870 0.0230

6.98 7.18 0.20 7.1253 0.0547

9.98 10.07 0.09 10.0864 -0.0164

6.19 6.28 0.09 6.3455 -0.0655

9.17 9.23 0.06 9.2869 -0.0569

1.99 2.20 0.21 2.2000 0.0000

7.77 7.84 0.07 7.9050 -0.0650

4.00 4.15 0.15 4.1839 -0.0339

10.77 10.89 0.12 10.8662 0.0238

4.78 5.00 0.22 4.9538 0.0462

2.99 3.21 0.22 3.1870 0.0230

6.98 7.20 0.22 7.1253 0.0747

9.98 10.17 0.19 10.0864 0.0836

10
Worksheet A (continued)

Calculations for finding a spacing calibration curve such that z=a + bw for n measurements:

Differences from NBS

n = 10k = 40

d =z-w

w = ZZw/n = 6.462

z = ZZ z/n = 6.614

Estimate of slope b = ZZ(w-w)(z-z)/ZZ(w-w)

2 = 0.9870

Estimate of intercept a = z - b w = 0.2358

Deviations from linear fit

/\ ^ 6 = z - z' where z' = a + bw

Standard deviation from linear fit s = yjl6

Z /(n-2) = 0.06203

with degrees of freedom n-2 = 38

Student's t-test for intercept:

Standard deviation of a = s^l ZZw

^JnZZ(w - "w = 0.02430

T statist!c for intercept t = a/s

Critical value t # Q25 OS) = 2.0

= 9.7

c

i ^ t .025 implies a is significantly different

than zero Student's t-test for slope:

Standard deviation of b Sh s/ ^ZZ(w - w) 2 - 0.00344

T statistic for slope

Critical value t # Q25 ( 38 ) =2.0

t

2 = (l-b)/s b

= 3.8

fc 2 ^ t .025 implies b is significantly different

than one

11

D

IF

F

ER

EN

C

ES

FR

n

N

B

S

0.4

0.3 -

0.2

0.0

0. 1 6 8

NBS VALUES mi CROflETERS )

10 12

Figure 4: Differences between linewidth measurements and corresponding NBS

values plotted against NBS values.

D

E

V

IA

T

I

N

S

0. 15

0. 10

0.05

0.00

-0.05

-0. 10

-0. 15 1

* * *

*

1 T

'

1

'

1

' 1 2 4 6 8

NBS VALUES (MICROMETERS)

10 12

Figure 5: Deviations of observed linewidths from linear calibration curve plotted against NBS values.

13
Worksheet B

Calculations for Linewidth Calibration Curve

Linewidth Measurements on Opaque Lines

Values in Micrometers

NBS Observed Differences Deviations

Values Values from NBS From Fit w z d 6

2.50 2.62 0.12 -0.1036

1.94 2.17 0.23 -0.0066

.74 .99 0.25 -0.0145

4.25 4.39 0.14 -0.0429

10.56 10.63 0.07 0.0339

5.29 5.54 0.25 0.0913

3.67 3.94 0.27 0.0736

7.45 7.57 0.12 0.0116

1.30 1.43 0.13 -0.1215

6.14 6.26 0.12 -0.0189

2.50 2.76 0.26 0.0364

1.94 2.28 0.34 0.1034

.74 1.01 0.27 0.0055

4.25 4.54 0.29 0.1071

10.56 10.49 -0.07 -0.1061

5.29 5.48 0.19 0.0313

3.67 3.88 0.21 0.0136

7.45 7.48 0.03 -0.0784

1.30 1.52 0.22 -0.0315

6.14 6.20 0.06 -0.0789

2.50 2.64 0.14 -0.0836

1.94 2.09 0.15 -0.0866

.74 .91 0.17 -0.0945

4.25 4.41 0.16 -0.0229

10.56 10.59 0.03 -0.0061

5.29 5.51 0.22 0.0613

3.67 3.78 0.11 -0.0864

7.45 7.49 0.04 -0.0684

1.30 1.52 0.22 -0.0315

6.14 6.22 0.08 -0.0589

2.50 2.79 0.29 0.0664

1.94 2.21 0.27 0.0334

.74 1.08 0.34 0.0755

4.25 4.49 0.24 0.0571

10.56 10.62 0.06 0.0239

5.29 5.55 0.26 0.1013

3.67 3.91 0.24 0.0436

7.45 7.65 0.20 0.0916

1.30 1.62 0.32 0.0685

6.14 6.29 0.15 0.0111

14
Worksheet B (continued)

Calculations for finding a linewidth calibration curve such that z = a + bw

for n measurements:

n = 10k = 40

d = z - w

w = ZZw/n = 4.384

1 = EE z/n = 4.564

Estimate of slope b = E2(z -z)(w-w)/EZ(w-w) 2 - 0.9767

Estimate of intercept a = z - b w = 0.2817

Deviations from linear fit 6 = z - z" where z" = a + bw

Standard deviation of linear fit s = |ZE6

2 /(n-2) = 0.06826

with degrees of freedom n-2 = 38

Student's t-test for intercept:

1

2 Standard deviation of a s = s I ZEw_ 2 0.01955

n££(w-w)

t statistic for intercept t = a/s = 14.4

1 a Critical value t (38) = 2.0

.025

t > t.o25 implies that a is significantly

different than zero. Student's t-test for slope

I ITT

Standard deviation of b s = s/JZZ(w-w) = 0.00372

b

j2Z(w-w;

t statistic for slope t = (l-b)/s = 6.3

b

Critical value t (38) = 2.0

.025

t > t ^^ implies that b is significantly

different than one.

2 .025

15
4. What can go wrong with the calibration procedure?

Specifically what can go wrong with the calibration procedure so that values

corrected by the calibration curve do not have the desirable property of

negligible offset from NBS? Possible sources of error that could affect this

accuracy, some of which have already been discussed, are summarized in this

section. A thorough analysis of the initial calibration data should be

undertaken before the calibration program is put into operation, and if

significant sources of errors are identified by this analysis, such problems

should be rectified, and the initial calibration experiment should be repeated.

i) Lack of precision.

The inability to repeat measurements adequately with the system

operating over a range of diverse conditions affects the accuracy of the

results. Such lack of precision as indicated by a process standard deviation

that is large relative to the accuracy goal of the program, is a key statistic

alerting one that unidentified sources of error are afflicting the system.

There is nothing intrinsic to the statistical treatment that will improve

precision. The calibration curve is a device for reducing systematic error,

and improvements in precision will have to come from adherence to the NBS

recommended procedures for adjusting the microscope and from changes in

operating conditions in the laboratory. It is important to realize that

measurement assurance, while making abundant use of statistical methodology, is

not achieved by statistics alone, but by the totality of procedures that relate

the system to the NBS photometric system.

ii) Outliers in the initial calibration data.

Outliers in the initial calibration data can seriously distort the

calibration curve especially if they lie near one of the endpoints of the

calibration interval. Formal tests for the types of outliers encountered in

the interlaboratory study are not given in this document because they require

considerable computing capability, but they can be found in reference [2].

iii) Non-linearity of the calibration curve.

The assumption of linearity is crucial in justifying the validity of

the results for the entire calibration interval. The NBS artifact is designed

for systems with linear responses relative to the values assigned to the lines on the artifact. Modelling a process that is non-linear in a given region of

the calibration interval would require an artifact with lines closer together

over that region. A statistical test for linearity is included in the

appendix.

iv) Differences among operators.

It is possible for different operators to produce measurements that

have offsets that are different in sign and in magnitude. If the measurement

system includes several operators, one may accept this source of error as part

of the imprecision of the process by including several operators in the initial

calibration experiment. This would certainly require a total of more than four

repetitions. If the measurement system includes very few operators who

consistently produce significantly different offsets, it may be worthwhile to

establish separate calibration curves for each operator.

16
v) Lack of system control.

The calibration procedures outlined in this publication are dependent

upon a calibration curve which is assumed to be valid for the system in its

future operation. This assumption must be checked continuously because if the

system response drifts or occasionally takes unpredictable excursions, the

resulting values will not be properly corrected for offset. A procedure for

exercising control over the calibration curve is discussed in section 6.

vi) Differences among repetitions in the initial calibration data.

The calibration curve will not produce satisfactory results if there is

an essential difference in the response of the system from repetition to

repetition; i.e., if the slope and intercept of the calibration curve do not

correctly characterize the individual repetitions. This source of error is

checked by plotting the differences between the observed linewidths and the NBS

values for the linewidths against the NBS values. Connecting the points for a single repetition enables one to see the type of variation that exists among

repetitions. If the slopes and intercepts of linear functions fitted to the

data for each individual repetition are changing in sign or are significantly

different in magnitude, the calibration curve, which is based on the averages

of the repetitions, will not properly correct the system at any given time.

Comparison of slopes and intercepts can be done graphically as in Figure 6 in

which the responses for each repetititon are connected by solid lines with the

corresponding fitted linear functions shown by dashed lines. This example

illustrates that even where the overall cluster of calibration data appears to

indicate a linear response, examination of the finer structure of the individual

repetitions reveals very disparate responses including changes in the sign of

the slope. A formal statistical test is given in appendix II.

2

DAY 4

0.0 —

0. 1

N -0.2

B

S

-0.3

4 6 8

NBS VALUES (MICROMETERS!

10

Figure 6: Daily changes in calibration curve. Differences for one day are connected by solid lines, and the corresponding calibration curve is represented

by a dashed line.

17
5. What is the precision of the optical system and how is it estimated?

The term system precision refers to the ability to repeat measurements on the

optical system from day to day or from occasion to occasion. It is based on

the fact that repeated measurements on the same physical object such as an

opaque line will not be identical, and the variation or scatter inherent in

such measurements is referred to as the system precision. It is usually

estimated by a standard deviation.

The standard deviation is computed from repeated measurements of the linewidths, and the repetitions must include all conditions for which the precision is assumed to be valid; i.e., they should include all the normal

operating conditions in the laboratory. The repetitions should be

sufficiently separated in time so as to adequately characterize the system.

They should also span the range of linewidths of interest.

The standard deviation from the linear fit for the linewidth calibration curve

is the usual estimate of the system precision. It is a proper estimate as long as the mathematical model, in this case we have assumed a linear

relationship, is valid.

If there are a sufficient number of repeated measurements at each line over

the entire regime of 0.5 um -12 |jm, then the standard deviation of each line

is an estimate of the system precision, and the pooled standard deviation over all lines is also an estimate of this precision. For example, for linewidths

Zjlj (i=l,..,k; j=l,..,10) where i denotes the repetition and j denotes the

line, the standard deviation for the jth line is

(5.1) s

i = I i=l I where z •-. = i=l

The pooled standard deviation is

(5.2)

S P =

Each of the standard deviations s\ t S2,.». , s^o nas (k"l) degrees of freedom,

and the pooled standard deviation s p

has 10(k-l) degrees of freedom. These

calculations are illustrated for data on opaque lines from the interlaboratory

study in Worksheet C.

The question arises, "Should the standard deviation from the linear fit or the pooled standard deviation be quoted as the estimate of process precision?"

The preferred one is usually the one with the larger number of degrees of

freedom, i.e., the standard deviation from the linear fit, but there are occasions when the pooled standard deviation is the more useful as described

in section 7.

18
Worksheet C

Calculations for a Pooled Standard Deviation

Linewidth Measurements on Opaque Lines

Values in Micrometers

Repetitions

1st 2nd 3rd 4th Averages Std Devs

Lines z\ Z2 Z3 Z4 z s

Standard deviation for each line Sj =1 £(z-z)2/3 , j=l, ..., 10

Calculation of pooled standard deviation

;{\

.oTk^fj

Degrees of freedom = 10(k-l) = 30

= 1(1

V

i( Sp •': Is-; «

J

1 2.42 2.56 2.44 2.59 2.502 0.0850

2 1.96 2.07 1.88 2.00 1.978 0.0793

3 0.76 0.78 0.68 0.86 0.770 0.0739

4 4.21 4.36 4.23 4.31 4.278 0.0699

5 10.53 10.39 10.49 10.52 10.482 0.0640

6 5.37 5.31 5.38 5.38 5.360 0.0337

7 3.75 3.69 3.59 3.72 3.688 0.0695

8 7.43 7.34 7.75 7.51 7.408 0.0793

9 1.21 1.30 1.30 1.40 1.302 0.0776

10 6.10 6.04 6.06 6.13 6.082 0.0403

19
6. How long will the derived calibration curve be adequate for

correcting dimensional measurements?

Establishment of a calibration curve based on the recommended number of

repetitions (a minimum of four sets of measurements on ten lines) constitutes

the first step in what can become a full fledged measurement assurance program

for linewidth measurements. The measurement assurance concept requires

continual surveillance on the output of the calibrated system to ensure that

the system is behaving accordingly; otherwise, the applied corrections based on the established calibration curve may degrade instead of improve the

accuracy of the resulting value. That is, it must be demonstrated that the

system is responding as presumed by utilizing a control procedure that is

capable of detecting significant changes.

This can be accomplished by exercising quality control on the calibration

curve itself in much the same way that quality control is exercised on individual components in a process. A test built into the operational

procedure at given intervals can be used to see if the values corrected by the

calibration curve meet the requirement of having negligible offset from NBS.

The method for controlling the output of the system as corrected by the

calibration curve is in keeping with a control chart approach.

This approach requires making one set of measurements on the ten lines on the

artifact at least at the start-up of the measurement sequence or as often as experience dictates and testing for out-of-control conditions after each set

of measurements. If ten measurements impose too great a burden on the

workload, one should consider measuring only three of the lines—making sure

that two of the lines are near the two endpoints of the calibration interval

and that one is near the center. In fact, fewer control measurements at

frequent intervals may be preferable to all ten measurements at infrequent

intervals. One could alternate lines from occasion to occasion. The

application of the control test will be easier, however, if the same number of

lines are checked each time.

JLJL In order to conform to our previous notation, let z^ ' (i=l, .

.

, m) denote the

corrected linewidth measurements (see equation 3.2) and w^ denote the assigned

values^ for the respective lines where m < 10. If the optical system is,

perfectly calibrated by the calibration curve, the quantities v^, referred to as the control values, have expected values of zero where

it it vi TM z i ~ wi 1=1,...,

* The quantities w^ refer to the NBS assigned values if the calibrated

pattern on the SRM is used for control purposes. In order to minimize

contamination on the SRM, it is preferable to use a secondary standard for

control purposes, in which case the quantities w^ refer to the values

assigned to the secondary standard according to the procedures in

section 3.

20
The control values should be evenly distributed about zero and fall within the

appropriate upper and lower control limits. It is recommended that all

control values be plotted as a function of time as in Figure 7. This will

result in m values at each interval. If one or more values falls outside of

the control limits, the system is out-of-control at that time. Such a finding

precludes using the calibration curve to correct the system. The control

measurements and statistical test for control should be repeated at that

point, the workload being discontinued until the system is brought back into

control. Continued failures should trigger an investigation into the cause and may indicate a permanent change, in which case the initial calibration

measurements should be repeated and the calibration curve reestablished.

The calculation of the upper and lower control limits depends on s the

estimate of system precision from the linear fit, v the number of degrees of

freedom associated with s, and m the number of control values. For

probability level 1-a, the upper and lower limits should be chosen as follows:

* Upper control limit &i = s c

t a/2(W

* Lower control limit &2 = ~s c

t a/2^ v )

where the standard deviation of the control is s c = s

and the critical values of the t distribution with v degrees of freedom are tabulated for m = 10 and m = 3 for a =.05 in Table 2.1 and for a = 0.01 in

Table 2.2.

Statistical control in the context of a calibration curve infers not only that

measurements on a given line are repeatable within certain limits but also

that the system response remains linear. Because the test is sensitive to

departures from linearity, control limits based on a = 0.05 will produce

stringent control on the calibrated values; less stringent control can be

achieved with control limits based on a = 0.01.

One example of calculations for control purposes for m=3 and a = 0.05 is shown

in Worksheet D, and the control values are plotted in Figure 7. Note that the

lines defining the lower endpoint, mid-point, and upper endpoint of the

calibration curve are denoted by L, M and U respectively in the graph. This

convention enables one to identify consistently non-linear behavior.

21
0.4

C

N

T

R

L

V

A

L

U

E

S

0.2 -

0.0 —

-0.2 —

-0.4

DAYS

Figure 7: Control chart for opaque linewidths for data in Worksheet D« Control limits computed for significance level a = 0.05.

The graphical method of looking for control values that are outside of the

control limits is synonomous with testing each control value in the following

way:

If v^ > &i or if v^ < %i f°r any i— 1, • •

•

, m

the system is judged out-of-control. This test is easily implemented on a computer or even by hand, but the benefit derived from plotting the control

data periodically should not be overlooked. The test is capable of detecting

only large changes in the behavior of the system, and the plots provide

additional information alerting one to a system that is gradually changing

over time and providing information about the precision of the process.

22
Worksheet D

Calculations for Control of Measurement Process

at the a = 0.05 Significance Level

Linewidth Measurements on Opaque Lines at m = 3 Points

Values in Micrometers

Repetition NBS Observed Corrected Cont.rol Upp er Lower

Value Value Value Limit Limit w 2

f **

z V *1 *2

1 0. 76 1. 12 0. 86 0. 10 0. 17 -0. 17

3. 29 3. 49 3. 28 -0. 01 0. 17 -0. 17

8. 89 9. 11 9. 04 0. 15 0. 17 -0. 17

2 0. 76 0. 99 0. 73 -0. 03 0. 17 -0. 17

3. 29 3. 53 3. 33 0. 04 0. 17 -0. 17

8. 89 8. 89 8. 81 -0. 08 0. 17 -0. 17

3 0. 76 1. 05 0. 79 0. 03 0. 17 -0. 17

3, 29 3. 46 3. 25 -0. 04 0. 17 -0. 17

8. 89 9. 02 8. 95 0. 06 0. 17 -0, 17

4 0. 76 0. 76 0. 49 -0. 27/ 0. 17 -0. 17

3. 29 3. 75 3. 55 0. 26/ 0. 17 -0. 17

8. 89 9. 30 9. 23 0. 34/ 0. 17 -0. 17

5 0. 76 0, 96 0. 69 -0. 07 0. 17 -0. 17

3, 29 3, 53 3. 33 0. 04 0. 17 -0. 17

8. 89 9. 05 8. 98 0. 09 0. 17 -0. 17

6 0. 76 1. 03 0. 77 0. 01 0. 17 -0. 17

3. 29 3, 52 3. 32 0. 03 0. 17 -0. 17

8. 89 9.,02 8, 95 0, 06 0. 17 -0, 17

/ Indicates that control value is either > l\ or < %i*

Linewidth calibration from Worksheet B

a = 0.2817 b = 0.9767

s - 0.06826

Correction z** = 1 (z - a)

Control Values

Upper Control Limit

Lower Control Limit

**

v = z - w

% \ = s_ t* #0 25( 38 ) t*.025( 38 ) = 2.498 from Table 2.1

b

i2 - -H

23
D

IF

FE

R

E

N

C

E

S

F

R

M

N

B

S

0.600

0.433

0.267

0. 100

-0.067

-0.233

-0.400

2 4 6 8

NBS VALUES (MICROMETERS)

10

Figure 8: Differences of observed linewidths from NBS values plotted against

NBS values. Differences for days 1-4 are connected by solid lines,

and differences for days 5-8 are connected by dashed lines.

0.4

C

N

TR

L

V

A

LU

E

S

0.2 —

0.0

-0.2

1 2

*

* *

i:

:: * * 1

:: * *

*

*

*

*

::

*

1

—

4

r r-T i

*

i i i i

'

i i i i

*

i i i i 'i i i i

- -0.4 i i i i i i i i

1 2 3 4 5 6 7 8

DAYS

Figure 9: Control values for the lower, mid and upper endpoints of data in

Figure 7. Control limits are for significance level a = 0.05.

24
Another example is presented to clarify some of the ideas presented in this

section. A linear calibration curve was established for an optical system

using the NBS SRM, and the subsequent control measurements taken over eight

days are plotted in Figure 8; measurements for each day are connected in order

to demonstrate the strengths and weaknesses of a control procedure based on m=3 control points. Control values for the lower, mid, and upper endpoints

and associated control limits were calculated for the existing calibration

curve and are plotted in Figure 9.

The four data points that are outside of the control limits in Figure 9 are marked accordingly in Figure 8. Notice that the endpoints are the most

sensitive to change and that point #4 is unquestionably out-of-control. But

what about the other three data points that are outside of the control limits?

Is it as apparent that they represent a significant change in the system? One

fact is apparent from a perusal of Figure 9 that cannot be gleaned from

Figure 8; i.e., almost all of the control values are above the zero line

indicating that the system has drifted since the initial calibration curve was established.

Furthermore, the curves in Figure 8 raise a basic question about whether or not a linear calibration curve is adequate for this system. Significant

findings for points #1 and #2 reflect a combination of non-linearity in the

lower end of the system and overall drift. The question about the linearity

assumption cannot be answered directly by the control procedure, but the

out-of-control points flag a deficiency in the system; i.e., that the values

corrected by the linear calibration curve will not meet the requirement of

having negligible offset from NBS. That is precisely the question in which

the user should be interested.

Finally, the system drift and multiple out-of-control conditions that are documented by the control procedure provide sufficient evidence that the

derived calibration curve no longer is adequate for correcting linewidth

measurements. The calibration curve should be reestablished from more recent

measurements on the system, and this new calibration curve should be monitored

at frequent intervals until one is satisfied that the system is no longer

drifting and is operating consistently in control.

25
7. Can the control data be used to update the calibration curve?

If all ten lines are being measured as part of the control process, it is

advantageous to keep track of the individual measured values, and

periodically, at convenient intervals, to add the accumulated control data to

the calibration data. If a smaller number of control measurements are made

regularly, the entire calibration curve should be repeated occasionally so that the parameters can be updated.

The slope and intercept and associated estimate of random error should be

revised based on this updated data set. This can be done simply by using the

updated data as in Worksheet B to estimate the necessary parameters which in

turn become the accepted parameters for the system and are used to correct the

linewidth values until the next update. Assuming the system remains in

statistical control, this will result in a smaller uncertainty for the

corrected values as the calibration curve is tied down by a larger data base.

However, for some computer automated systems with limited storage capacities,

it may be more convenient to save the averages of k repetitions made on each

line during the calibration sequence rather than the individual measurements

and to update this data base after every K succeeding control sequences. In

this case the pooled standard deviations from each sequence should be combined

to form the estimate of process precision. Note that a calibration curve based on averages will yield the same estimates of the slope and intercept as one based on the raw data as long as each average contains the same number of

data points, but the residual standard deviation, in this case, is not an estimate of process precision.

For example, let Zj (j=l, •••, 10) be the averages of k repetitions on each

line and sp

the corresponding pooled standard deviation. Also let

Zj (j"l, •••, 10) be the averages of K succeeding repetitions on each line

with pooled standard deviation S

p

. (See equations (5.1) and (5.2) in the last

section for pooled standard deviations.)

Thus the averages

i = kzj + K Z-j j=l 10

3 & + K

J

form a data base for updating the calibration curve given the model

(7.1) z'j = a' + b'wj j-1, ..., 10

26
The estimates of the updated parameters (denoted by ") are:

(7.2) b' = J, (z-j - 1' ) (wj - w)

J n - o

2, (wi - w) z

j

(7.3) a' = Y ~ b' w

The residual standard deviation of the fit with 8 degrees of freedom is

, 2\ J, I (z'j - a' - b' wj

8

Although s' can be used to estimate the system precision s where

(7.4) s = s' v^HC

the pooled standard deviation

s P

10 (k-1) s p + 10 (K -1) S

p

10 (k + K - 2)

with 10 (k + K-2) degrees of freedom is a better estimate of process

precision. These calculations are illustrated in Worksheet E.

Obviously the data base can be updated after each control sequence; i.e., for

K = 1, but no additional information is available for computing s'

p

. In this

case, the only way to get an updated estimate of the process precision is via

equation (7.4).

There are several possible variations on the schemes outlined above for

updating the data base. The mechanics of updating the data base are usually

dependent on the degree of automation that exists in the laboratory and should

be carefully thought out before implementation.

27
Worksheet E

Calculations for Updating the Calibration Curve

Linewidth Measurements on Opaque Lines

Values in Micrometers

Averages of Linewidth Measurements

NBS From Calibration From Control Updated

Lines Values Runs Runs Values w z Z z'

1 2.50 2.502 2.498 2.499

2 1.94 1.978 1.983 1.981

3 0.74 0.770 0.775 0.773

4 4.25 4.278 4.277 4.277

5 10.56 10.482 10.478 10.479

6 5.29 5.360 5.360 5.360

7 3.67 3.688 3.688 3.688

8 7.45 7.408 7.409 7.409

9 1.30 1.302 1.298 1.299

10 6.14 6.082 6.077 6.079

28
Worksheet E (continued)

Values z"* are averages of k=4 calibrations from Worksheet C.

Values Z" are averages of K=8 control runs.

Pooled standard deviation for calibration data is

s p

= 0.0692 from Worksheet C

Pooled standard deviation for control data is

S

p = 0.0610

Updated data base:

z

' = kz" + KZ

k + K

7' = Zz*/10 = 4.3844 w = Zw/10 = 4.384

Updated calibration curve:

b' = Z(z'-z') (w-w)/E(w-w) 2 = 0.9893

a" = 7' - b* w = 0.0473

Updated pooled standard deviation

s' p = 10(k-l)sp + 10(K-l)Sp = 0.0636

10(k + K-2)

with Y

= 10(k + K-2) = 100 degrees of freedom.

29
8. What is the uncertainty of the values corrected by the calibration curve?

Because the functional relationship between the user's system and the NBS

system is not known exactly but is estimated by a series of repeated

measurements, the calibration curve itself has some imprecision associated with

it. Also if the measurement on a line is repeated, the resulting value will

not be exactly the same; i.e., the optical system has some imprecision

associated with it. This combination creates some difficulty in finding the

uncertainty appropriate for the calibrated values. The difficulty arises from

the fact that the initial set of measurements used to derive the calibration

curve is used over and over again, and limits to error must be computed for all

future corrected linewidth values based on this calibration curve. -*

Tracking system performance with control data and consistently revising the

linewidth calibration curve with the same control data leads to either

refinement of the calibration curve or its periodic reestablishment. In either

case, the control data is a connecting thread that characterizes the output of

the system.

The standard deviation of a calibrated value will be largest for lines near the

endpoints of the calibration interval. Therefore, it is suggested that a standard deviation be computed for the endpoints based on the control data and

that it be used as a miximum for all intervening lines. Assume that there have

been p control sequences each of which produced m acceptable control values,

v-n,...,v-jm

(j=l,...,p). Then the control values for the endpoints, namely v-n and Vjm

(j=l,...,p) can be used to cbmpute

1.711

2 + I 'J.2

! CAL =

\ i_

2p

with 2p degrees of freedom. For a given line with calibrated value z**, the

usual confidence interval statement that (1-a) percent of all calibrations for

that line are contained in the interval z** ± L where

L = SCAL t a/2(2p)

is approximately true for the endpoints and conservatively true for all other

lines.

When sufficient data does not exist for this analysis, a tolerance type

interval, called a Scheffe' interval, based on the initial calibration curve can be constructed. This calculation, although cumbersome, should be used at

least for the initial uncertainty statement and can be revised after several

updates of the calibration curve have been completed.

J In cases where the calibration curve is being utilized only once, the reader

is referred to a paper by C. Eisenhart (1939) "The interpretation of certain

regression methods and their use in biological and industrial research,"

Annals of Math Stat , 10 , pp. 162-202.

30
The approach developed by Scheffe -* [14] guarantees that a large percentage of

statements that are made about future calibrations are correct. The

statements refer to limits to error which are appropriate for the corrected

values. For example, these limits L can be constructed such that the

probability is > 0.99 that 95 percent of all intervals z + L so constructed

contain the linewidth value that is properly related to NBS where z is a corrected linewidth measurement.

The Scheffe procedure is conservative and will yield fairly wide intervals

depending on the choice of a and 6. The statement that the probability is 1-6

that (1-a) percent of the intervals so constructed are correct should be

defined according to the user's requirements. Scheffe suggests that 6 be

chosen smaller than a, and a combination of a = 0.10 and 6 = 0.05 may be

appropriate for most situations. More conservative limits are given by a choice of a = 0.05 and 6 = 0.01. Tables are provided to accommodate both sets

of statements.

The calculation of the limits produces upper and lower curves bounding the

calibration curve resulting in upper and lower limits for each nominal

linewidth. The upper and lower limits are not necessarily symmetrical, but were sufficiently close for the data in the interlaboratory study that only

the procedure for calculating the upper limits is given here. For example, the

following limits are calculated for the linewidth calibration curve derived in

Worksheet C for a = 0.05 and 6 = 0.01.

Values in Micrometers

Nominal Linewidth Limits to Error

12

3

4

5

6

78

9

10

The maximum of these limits to error occurs at one of the endpoints of the

calibration interval and is appropriate for use in the uncertainty statement.

The steps necessary for computing the maximum limit to error L based on reference [14] are given on the next page, and it is noted that the

calculation should not be extended to linewidths outside of the calibration

interval. The steps outlined assume that the calibration curve is linear and

that the estimate of precision has n-2 degrees of freedom.

pper L Lower

0.25 -0.26

0.24 -0.25

0.24 -0.24

0.23 -0.23

0.23 -0.23

0.24 -0.24

0.25 -0.24

0.26 -0.25

0.27 -0.26

0.28 -0.27

31
Choose probabilities a and 6.

Look up Z a/2 the upper a/ 2 percentage point of the normal

distribution. Za/2 = 1.96 for a = 0.05

z a/2 = 1-6449 for a = 0.10

2 X (v) The lower 6-percentage point of the chil -

° square distribution with v degrees of

freedom. Table 3.1 gives values of x^ ( v ) f°r

6 = .01 and v = 30(2)120. 1_

<5 and Table 3.2 gives values for 6 = 0.05.

Fg(2,v) is the upper 6-percentage point of the F

distribution with 2 degrees of freedom in

the numerator and v degrees of freedom in

denominator (v = n-2) . Table 4.1 gives values of

F 6 (2,v) for <5 = .01 and v = 30(2)120, and Table

4.2 gives values for 6 = 0.05.

Compute

The value of C3 depends on several parameters and ranges between 0.95 and 1.05

for 6=.01 and n=40. The exact value of C3 which is appropriate for any data

set can be found using methods given in reference [5]. The calculation of L

is not overly sensitive to C3. For the interlaboratory study C3 was taken to

be 1.05, and this value is suggested as a conservative estimate of C3 and is

used in the sample calculation in Worksheet G.

Find the least-squares estimates a, b and s for the calibration curve

z-ji = a + bw-ji i=l,..., k; j = l,...,10

and choose wmax to be the maximum linewidth of interest.

Compute

C = b - (C2

s b ) 2

D = wmax ~ a - bw + Ci s

where w is the average of the NBS values. The limit L is given by

L = w - w' +

The computations for the linewidth calibration curve derived in Worksheet C

for a = .05 and 6 = .01 are shown in Worksheet F.

Finally, the uncertainty that is appropriate for a corrected linewidth value

is + U where U = L + | ^NBS I an<* ^NBS is t *ie systematic error from the NBS

system.

32
Worksheet F Calculation of Limits to Error L for Opaque Linewidth Values

Corrected by a Linear Calibration Curve

Values in Micrometers

Intercept a 0.2817

Slope b

Standard deviation of intercept sa Standard deviation of slope s^

Standard deviation of linear fit s Degrees of freedom v

0.9767

0.01955

0.003717

0.06826

38

Mean of NBS values w 4.384

Choose ct=.05, 6-. 01

Look up in tables:

Z =1.96 the upper 2.5% point of the normal distribution

.025

-"

.99

(v) = 20.6914 the lower 1% point of the x^ distibution

with v degrees of freedom

F.01 (2, v) = 5.2112 the upper 1% point of the F distrubution with 2

and v degrees of freedom

Compute:

C X = 1.05 Z AT//x2 .99(v) = 2.795 C 2 = 1.05 /2F. l(2, v) = 3.3898

Choose wmax 10 and compute

C b

2 - (C 2

s b ) 2 = 0.9794

D = wmax " a - b w + Cj s = 5.8144

w w. - a = 10.1520 max

w I w-j/n = 4.384 _ b D + C 2 IC w - w

1 + jn

2 j. n2 2

s + D s b = 0.28

The maximum limit to error that is appropriate for any corrected opaque

linewidth in the l-10um interval is L = 0.28pm.

33
Table 1 Critical Values of Student's t Distribution

for a Two-Sided Test

Significance Level a = 0.05

t.025< v ) v t.025( v )

30 2.042 76 1.992

32 2.037 78 1.991

34 2.032 80 1.990

36 2.028 82 1.989

38 2.024 84 1.989

40 2.021 86 1.988

42 2.018 88 1.987

44 2.015 90 1.987

46 2.013 92 1.986

48 2.011 94 1.985

50 2.009 96 1.985

52 2.007 98 1.985

54 2.005 100 1.984

56 2.003 102 1.984

58 2.002 104 1.983

60 2.000 106 1.983

62 1.999 108 1.982

64 1.998 110 1.982

66 1.997 112 1.981

68 1.995 114 1.981

70 1.994 116 1.981

72 1.993 118 1.980

74 1.993 120 1.980

v is the number of degrees of freedom in the estimate of process precision.

34
Table 2.1

Critical Values of t distribution

for controlling the output of a calibration curve

at m points

Significance level a = 0.05

t

• 025< v > t • 025< v ) V m=3 m=10 T m=3 m=10

30 2.528 3.021 76 2.442 2.883

32 2.519 3.006 78 2.440 2.881

34 2.511 2.993 80 2.439 2.879

36 2.504 2.982 82 2.437 2.877

38 2.498 2.972 84 2.436 2.875

40 2.492 2.963 86 2.435 2.873

42 2.487 2.954 88 2.434 2.872

44 2.482 2.947 90 2.433 2.870

46 2.478 2.940 92 2.432 2.868

48 2.474 2.934 94 2.431 2.867

50 2.470 2.929 96 2.430 2.866

52 2.467 2.923 98 2.429 2.864

54 2.464 2.919 100 2.428 2.863

56 2.461 2.914 102 2.428 2.861

58 2.459 2.910 104 2.427 2.860

60 2.456 2.906 106 2.426 2.859

62 2.454 2.903 108 2.425 2.858

64 2.452 2.900 110 2.425 2.857

66 2.450 2.896 112 2.424 2.856

68 2.448 2.893 114 2.423 2.855

70 2.446 2.891 116 2.423 2.854

72 2.444 2.888 118 2.422 2.853

74 2.443 2.886 120 2.421 2.852

Critical values t a/2(v) correspond to the upper z, percent point of student's t

distribution where

C = 1/2 { 1 - exp I &n( 1-ct)

m

and v is the number of degrees of freedom in the estimate of process

precision.

35
Table 2.2

Critical values of t distribution

for controlling the output of a calibration curve

at m points

Significance level a = 0.01

t

• 005< v ) t • 005< v > V m=3 m=10 V m=3 m=10

30 3.187 3.644 76 3.030 3.422

32 3.171 3.620 78 3.027 3.418

34 3.156 3.599 80 3.025 3.415

36 3.143 3.581 82 3.023 3.412

38 3.131 3.564 84 3.020 3.409

40 3.121 3.549 86 3.018 3.406

42 3.111 3.536 88 3.016 3.403

44 3.103 3.524 90 3.014 3.400

46 3.095 3.513 92 3.013 3.398

48 3.088 3.504 94 3.011 3.396

50 3.082 3.495 96 3.009 3.393

52 3.076 3.486 98 3.008 3.391

54 3.070 3.479 100 3.006 3.389

56 3.065 3.471 102 3.005 3.387

58 3.060 3.465 104 3.003 3.385

60 3.056 3.459 106 3.002 3.383

62 3.052 3.453 108 3.001 3.382

64 3.048 3.448 110 3.000 3.380

66 3.045 3.443 112 2.998 3.378

68 3.041 3.438 114 2.997 3.377

70 3.038 3.434 116 2.996 3.375

72 3.035 3.429 118 2.995 3.373

74 3.032 3.426 120 2.994 3.372

Critical values t a/2(v) correspond to the upper £ percent point of student's t

distribution where

S = 1/2 Jl - exp (in(H0)

' \ m

and v is the number of degrees of freedom in the estimate of process

precision.

36
Table 3.1

Percentage points of the x^ distribution

Lower 1% points

V

2

.99

V

2

.99

30 14.953 76 50.286

32 16.362 78 51.910

34 17.789 80 53.540

36 19.233 82 55.174

38 20.691 84 56.813

40 22.164 86 58.456

42 23.650 88 §0.103

44 25.148 90 61.754

46 26.657 92 63.409

48 28.177 94 65.068

50 29.707 96 66.730

52 31.246 98 68.396

54 32.793 100 70.065

56 34.350 102 71.737

58 35.913 104 73.413

60 37.485 106 75.092

62 39.063 108 76.774

64 40.649 110 78.458

66 42.240 112 80.146

68 43.838 114 81.836

70 45.442 116 83.529

72 47.051 118 85.225

74 48.666 120 86.923

v is the number of degrees of freedom in the estimate of process precision,

37
Table 3.2

Percentage points of the x

2

Lower 5% points

2

V

*.,5

<U) V

30 18.493 76

32 20.072 78

34 21.664 80

36 23.269 82

38 24.884 84

40 26.509 86

42 28.144 88

44 29.787 90

46 31.439 92

48 33.098 94

50 34.764 96

52 36.437 98

54 38.116 100

56 39.801 102

58 41.492 104

60 43.188 106

62 44.889 108

64 46.595 110

66 48.305 112

68 50.020 114

70 51.739 116

72 53.462 118

74 55.189 120

distribution

\95

(V)

56.920

58.654

60.391

62.132

63.876

65.623

67.373

69.126

70.882

72.640

74.401

76.164

77.929

79.697

81.468

83.240

85.015

86.792

88.570

90.351

92.134

93.918

95.705

v is the number of degrees of freedom in the estimate of process precision,

38
Table 4.1

Percentage points of the F distribution

Upper 1% points

F. 01 (2,v) v F. 01 (2,v)

30 5.390 76 4.896

32 5.336 78 4.888

34 5.289 80 4.881

36 5.248 82 4.874

38 5.211 84 4.867

40 5.178 86 4.861

42 5.149 88 4.855

44 5.123 90 4.849

46 5.099 92 4.844

48 5.077 94 4.838

50 5.057 96 4.833

52 5.038 98 4.828

54 5.021 100 4.824

56 5.006 102 4.820

58 4.991 104 4.815

60 4.977 106 4.811

62 4.965 108 4.807

64 4.953 110 4.803

66 4.942 112 4.800

68 4.932 114 4.796

70 4.922 116 4.793

72 4.913 118 4.790

74 4.904 120 4.786

v is the number of degrees of freedom in the estimate of process precision,

39
Table 4.2

Percentage points of the F distribution

Upper 5% points

F. 05 (2,v) v F.o5(2>v)

30 3.316 76 3.117

32 3.294 78 3.114

34 3.276 80 3.111

36 3.259 82 3.108

38 3.245 84 3.105

40 3.232 86 3.103

42 3.220 88 3.100

44 3.209 90 3.098

46 3.200 92 3.095

48 3.191 94 3.093

50 3.183 96 3.091

52 3.175 98 3.089

54 3.168 100 3.087

56 3.162 102 3.086

58 3.156 104 3.084

60 3.150 106 3.082

62 3.145 108 3.080

64 3.140 110 3.079

66 3.136 112 3.077

68 3.132 114 3.076

70 3.128 116 3.074

72 3.124 118 3.073

74 3.120 120 3.072

v is the number of degrees of freedom in the estimate of process precision,

40
REFERENCES

[1] Jerke, J. M. , ed. (February 1980). Semiconductor Measurement Technology:

Accurate Linewidth Measurements on Integrated-Circuit Photomasks, NBS

Spec. Publ. 400-43.

[2] Jerke, J. M. , Croarkin, M. C. , and Varner, R. N. , Semiconductor

Measurement Technology: Interlaboratory Study on Linewidth Measurements

for Antiref lective Chromium Photomasks, to be published as an NBS Spec.

Publ.

[3] Swyt, Dennis A. (January 1978). Design of a Pattern on a Photomask-Like

Physical Standard for Evaluation and Calibration of Linewidth-Measuring

Systems, Solid State Tech. 21, 35-42.

[4] Reference [1] pp. 7-14.

[5] Nyyssonen, D. (August 1977). Linewidth Measurement with an Optical

Microscope: The Effect of Operating Conditions on the Image Profile,

Appl. Opt. 16, No. 8, 2223-2230.

[6] Nyyssonen, D. (1979), Application of Optical Coherence: The Key to

Accurate Optical Micrometrology, Application of Optical Coherence,

Vol. 194 (Society of Photo-optical Instrumentation Engineers, Bedingham,

Washington).

[7] Cameron, J. M. (April 1977). Measurement Assurance. NBSIR 77-1240.

[8] Reference [1] p. 43.

[9] Reference [1] p. 78.

[10] Procedures for Using SRM-474. Submitted to ASTM Committee F-l as proposed ASTM Standard.

[11] Reference [2] sec. 3.3.

[12] Neter, John and Wasserman, Wm. (1974). Applied Linear Statistical Models,

Richard D. Irwin, Inc., Homewood, IL, 57-62.

[13] Reference [2] sec. 3.8.

[14] Scheffe', H. , Rosenblatt, J. R. and Spiegelman, C. A Statistical Theory

of Calibration to be published.

[15] Draper, N. R. and Smith, H. (1966). Applied Regression Analysis, John

Wiley & Sons, Inc., New York, NY, 26-32.

41
APPENDIX

I. Test of Linearity Assumption*

The test of whether or not a linear function of the measurements constitutes

an appropriate calibration curve is based on the information in the

repetitions [15]. The fitted value for each line

Zj = a + bwj j = l» •••» 10

is compared to the average of the repetitions for each line; namely,

k

I z ii j=l, •••> 10

zj = 1=1

where k is the number of repetitions of each linewidth.

Close agreement between the fitted curve and these averages is evidence that

the hypothesis is correct. The test is based on an F statistic, and the

assumption of linearity is rejected if

F = 10(k-l) . SSL - SSR > F.oi (8,10(k-l))

8 SSR

where

10 k

SSL = I I ( Zii - Zj ) 2

j=l 1-1

10 k

SSR = I I ( Zl , - zj) 2

j=l 1=1

and F^qi (8,10(k-l)) is the upper one percent point of the F distribution with

eight degrees of freedom in the numerator and 10(k-l) degrees of freedom in

the denominator.

42
II. Test for Differences Among Repetitions .

The test of whether or not a system with linear response is changing from

repetition to repetition is based on the following model; namely,

z ij = ai + &i xij i=1 » •*•» k

j=l, ..., 10

where x^j = w-jm - w^

and _ I Wj[j wi = J

10

Estimates of the intercepts a^ and the slopes 3^ and corresponding standard

deviations are computed for each repetition.

I < xij " xiX z ij " z i)

! i =

j , y i=1 » •••• k

ai " 2 i " Bi x±

_ I xij I

z ij xi = j

z i 1

10

v 2

I xij

J

10 I ( Xlj - Xi) :

j

H±

I (Xij - x± ) 2

j

and ft (z ± j - Oi - Bi x± j)

2

s i = J

43
The test statistic is

F = 1 I (o± - a)2 I (3 ± - 3)2

2(k-l) l_ ~yn?

+

i—r^2- k i 3,

where a = I I «i 3 = 1 l3i

k i k i

It is concluded that the system is not consistent from repetition to

repetition if

F > F.oi (2(k-l), 8k)

where F # qi (2(k-l), 8k) is the upper one percent point of the F distribution

with 2(k-l) degrees of freedom in the numerator and 8k degrees of freedom in

the denominator.

4A
NBS-114A (REV. 2-8C) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBS TN 1164

2. Performing Organ. Report No. 3. Publication Date

August 1982

4. TITLE AND SUBTITLE

Measurement Assurance for Dimensional Measurements on Integrated-Circuit Photomasks

5. AUTHOR(S)

Carroll Croarkin, Ruth Varner

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

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

^\ Document describes a computer program; SF-185, FlPS Software Summary, is attached.

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

bibliography or literature survey, mention it here)

Optical Microscope Linewidth-Measurement Standards, SRM-474 and SRM-475, have been

developed by NBS for optical imaging systems capable of making line-spacing and

linewidth measurements in the 0.5um -12um regime on IC photomasks. Each artifact

affords a means of reducing systematic errors via a calibration curve and keeping

the optical system in statistical control. Procedures are given for accomplishing

these goals along with a discussion of the uncertainty of the calibrated values.

12. KE.Y WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

IC photomask; linear calibration curve; line-spacing; linewidth; measurement

assurance; photomask; SRM; statistical control of measurement process; tests for

systematic error; uncertainty.

13. AVAILABILITY

Q>§ Unlimited

| | For Official Distribution. Do Not Release to NTIS

[~7| Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

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

14. NO. OF

PRINTED PAGES

50

15. Price

USCOMM-DC 6043-P80

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