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MIST PUBLICATIONS

'-SS^ NIST SPECIAL PUBLICATION 260-131 ,1999 Ed.

U.S. DEPARTMENT OF COMMERCE/Technology Administration

National Institute of Standards and Technology

Standard Reference Materials:

The Certification of 100 mm Diameter

Silicon Resistivity SRMs 2541 Through 2547

Using Dual-Configuration

Four-Point Probe Measurements, 1999 Edition

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James R. Ehrstein and M. Carroll Croarkin
rhe National Institute of Standards and Technology was established in 1988 by Congress to "assist industry in the development of technology . . . needed to improve product quality, to modernize manufacturing processes,

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NIST Special Publication 260-131, 1999 Ed.

Standard Reference Materials:

The Certification of 100 mm Diameter

Silicon Resistivity SRMs 2541 Through 2547

Using Dual-Configuration

Four-Point Probe Measurements, 1999 Edition

James R. Ehrstein

Semiconductor Electronics Division

Electronics and Electrical Engineering Laboratory and M. Carroll Croarkin

Statistical Engineering Division Information Technology Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899-0001

(Supersedes NIST Special Publication 260-131, August 1997)

U.S. DEPARTMENT OF COMMERCE, William M. Daley, Secretary

TECHNOLOGY ADMINISTRATION, Gary R. Bachula, Acting Under Secretary for Technology NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, Raymond G. Kammer, Director Issued June 1999
National Institute of Standards and Technology Special Publication 260-131, 1999 Edition

Natl. Inst. Stand. Technol. Spec. Publ. 260-131, 1999 Ed., 117 pages (June 1999)

CODEN: NSPUE2

U.S. GOVERNMENT PRINTING OFFICE WASHINGTON: 1999

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402-9325
FOREWORD

Standard Reference Materials (SRMs®) as defined by the National Institute of Standards and Technology (NIST) are well-characterized materials, produced in quantity and

certified for one or more physical or chemical properties. They are used to assure the accuracy and compatibility of measurements throughout the Nation. SRMs are widely used as primary standards in many diverse fields in science, industry, and technology, both within the United States and throughout the world. They are also used extensively

in the fields of environmental and clinical analysis. In many applications, traceability of

quality control and measurement processes to the national measurement system is carried out through the mechanism and use of SRMs. For many of the Nation's scientists and

technologists, it is therefore of more than passing interest to know the details of the procedures, modes, and philosophy used at NIST to use, produce, and certify SRMs and RMs. The NIST Special Publication 260 Series is a series of papers reserved for this purpose and can be accessed via internet: http://ts.nist.gov/srm.

This 260 publication is dedicated to the dissemination of information on different phases of the preparation, measurement, certification, and use of NIST SRMs. In general, much

more detail will be found in these papers than in generally allowed, or desirable, in

scientific journal articles. This enables the user to assess the validity and accuracy of the measurement processes employed to judge the statistical analysis, and to learn details of

techniques and methods utilized for work entailing greatest care and accuracy. These

papers also should provide sufficient additional information so SRMs can be utilized in new applications in diverse fields not foreseen at the time the SRM was originally issued.

Inquiries concerning the technical content of this paper should be directed to the

author(s). Other questions concerned with the availability, delivery, price, and so forth, will receive prompt attention from:

Standard Reference Materials Program

Bldg 202, Room 204

National Institute of Standards and Technology

Gaithersburg, MD 20899-2320

Telephone: (301)975-6776 FAX: (301)948-3730

e-mail: srm in foffi.rrist. go

v

, or www:http://ts. nist.gov/srm

Thomas E. Gills, Chief

Standard Reference Materials Program

iii
Table of Contents Page

Abstract 1

Introduction 2 Previous Resistivity SRMs 2 Improved Resistivity SRMs 2

1 . Certification of Improved Resistivity SRMs 3

1.1 General Comments 3

1 .2 Resistivity Standards vs. Sheet Resistance Standards 5

1.3 Traditional Description of Uncertainty and the ISO Formulation 6

1 .4 Acquisition and Characteristics of Silicon Wafers for the SRMs 7

1.5 Measurement Concerns and Control of the Certification Procedure 7

2. Certification Procedure 9

2.1 Wafer Thickness Screening and Thickness Measurement 9

2.2 Four-Probe Measurements of Sheet Resistance 11 2.3 Reporting of Data from 18 Measurements on Each Wafer 13

3. Control Procedures for the Certification Process 14

3.1 Control Procedure for Probe Effects 14

3.2 Control Procedure for Day-to-Day (Environmental) Effects 15

3.3 Control Procedure for Other Longer Term Effects and Drift of Wafer or Probe 15

4. Equations Used for Calculating Sheet Resistance and Resistivity Values 16

4.1 Rewriting the Equation to Relate to Evaluation of Uncertainty 17

5. Sources of Measurement Uncertainty — Detailed Discussion 19

5.1 Type A Evaluations of Components of Uncertainty 19

5.1.1 Short-Term Precision; Repeatability 19

5.1.2 Intermediate and Longer Term Precision; Reproducibility of Wafer-Center Average Value 20

5.1.3 Uncertainty Due to the Selection of a Particular Probe 21

5.2 Type B Evaluations of Components of Uncertainty 21

5.2.1 Discussion of Components Related to Electrical Measurements 22

5.2.2 Evaluation of Uncertainty in Electrical Measurement Scale 25

5.2.3 Evaluation of Uncertainty Components Related to Temperature Measurements 28

5.2.4 Evaluation of Uncertainty Components Related to Geometry

Measurements 30

5.2.5 Evaluation of Uncertainty Due to Thickness Measurement Scale 33

5.2.6 Evaluation of Uncertainty Due to Thickness/Probe Separation Scaling Factor 34

iv
Table of Contents (cont'd.)

Page

6. Unanticipated Effects 35

6.1 Resistivity Shift with Repeated Probing 36

6.2 Photosensitivity of Resistivity Value 36

7. Compilation of Uncertainty Components 38

7. 1 Summary of Statistical Analysis Parameters from the Appendices 39

7.2 Type A and Type B Variance Terms 40

7.3 Combined Standard Uncertainty and Expanded Uncertainty 42

7.4 Corrections Applied to Measured Values 43

7.5 Estimated Degrees of freedom for the Uncertainty Values of the SRMs 43

8. Conclusion 44

9. Acknowledgements 44

References 44

Appendix 1. Summary of Important SRM Wafer Material and Measurement Condition Parameters 46 Appendix 2. Analysis of Certification Data and Control Experiments for SRM 2547 47

Appendix 3. Analysis of Certification Data and Control Experiments for SRM 2541 66 Appendix 4. Analysis of Certification Data and Control Experiments for SRM 2542 70 Appendix 5. Analysis of Certification Data and Control Experiments for SRM 2545 74 Appendix 6. Analysis of Certification Data and Control Experiments for SRM 2546 78 Appendix 7. Analysis of Certification Data and Control Experiments for SRM 2543 82 Appendix 8. Analysis of Certification Data and Control Experiments for SRM 2544 98 Appendix 9. SRM Values after an Extended Period of Time 104

List of Figures

Page

1 . Scaled drawing of a 100 mm diameter wafer showing locations of thickness measurements and the locations of the four-point probe measurements in the 2X magnification at the bottom 10

2. Schematic of probe wiring for dual-configuration measurements 12

3. Conceptual drawing of wafer with two textured surfaces during thickness measurement by electromechanical gauge 32

v
List of Tables

1. Manufacturers' Specifications for the Current Supplies and DVM Used for Certification 24

2. Standard Resistor Values, and Typical Measurement Voltages for Each of the SRM Levels 27

3. Temperature Coefficients of Resistivity of Silicon for the Nominal Values of the SRMs 30

4. Variance in Resistivity Value Due to Temperature Error 30

5. Specifications for Haidenhain Certo 60 Thickness Measurement Instrument 33

6. Components Identified in Statistical Analyses of Certification and Control Experiment Data 39

7. Values of the Components Identified in Statistical Analyses for the Various SRMs in Appendices 2 through 8, mQ-cm 40

8. Type A Standard Uncertainty Values, w/ , Taken from appendices 2 through 8 40

9. Type A Variance Values, u

2

t , Obtained by Squaring the Entries in Table 8 41

10. Type B variance Values, u

2

^

, Calculated from Summation of Terms in Section 5.2 ... 41

1 1 . Combined Variance Values, u~c

, from Addition of Terms from Tables 9 and 10 42

12. Combined Standard Uncertainty Values, uc 42

13. Expanded Uncertainty Values, U 43

14. Bias Corrections Applied to Measured Values 43

1 5 . Estimate of Degrees of Freedom from the Welch-Satterthwaite Formula 43

vi
Standard Reference Materials: The Certification of 100 mm Diameter Silicon Resistivity SRMs

2541 through 2547 Using Dual-Configuration Four-Point Probe Measurements,

1999 Edition by James R. Ehrstein Semiconductor Electronics Division, EEEL

and M. Carroll Croarkin

Statistical Engineering Division, ITL

ABSTRACT

This report documents the selection of material, the certification procedure and its control, and

the analysis of measurement uncertainty for a family of new and improved Standard Reference Materials (SRMs) for sheet resistance and resistivity of silicon wafers, SRMs 245

1

through 2547, covering the resistivity range 0.01 Q cm through 200 Q cm. These SRMs,

made from 100 mm diameter silicon, replace previous SRM sets 1521 through 1523, which

used 50.8 mm (2 in) diameter silicon at the same nominal resistivity levels. This revised report replaces the original report issued August 1997 and contains new appendices 7 and 8 that

detail the data analyses of SRMs 2543 and 2544; these appendices were not available at the time the original report was issued. It also contains updated Tables 6 through 14, new Table

15, and a revised Section 8 on Conclusions to reflect the incorporation of material on SRMs

2543 and 2544. The certification of the new SRMs uses a dual-configuration four-point probe procedure

rather than the single-configuration procedure of ASTM F84 [1], as used for previous SRMs.

The new SRMs offer better handling compatibility with current user instrumentation, better uniformity of wafer thickness and of resistivity, more extensive spatial characterization of the near-center wafer resisivity, and reduced measurement uncertainty. The general procedures for the certification measurements, the control of the certification process, and the analysis of the results are based on experience gained from numerous

preliminary experiments that allowed evaluation of the importance and relative magnitude of many possible measurement effects. The validity and effectiveness of the resulting

certification and control procedures were tested during the analysis of results from the first of

the SRMs to be certified, that at 200 Q cm. The body of this report details the background

and principles of the certification process and the approach to analyzing the experimental

data needed to calculate the uncertainty of the certified values. This report details the evaluation of underlying Type B components of uncertainty that apply to all SRM levels.

1
Additional Type A components, derived from statistical analyses of the actual certification data, are done individually for each SRM level and are reported in separate appendices for each of the SRMs. Key Words: four-point probe; resisitivity; semiconductor; silicon; SRM; standards

INTRODUCTION

This Special Publication summarizes the certification procedure for a new generation of

silicon resistivity Standard Reference Materials (SRMs) 2541 through 2547. It includes,

in individual appendices, the analysis of the associated uncertainty levels calculated from

the certification data for each of the resistivity levels.

Previous Resistivity SRMs

For a number of years, the Semiconductor Electronics Division of the National Institute of

Standards and Technology has issued three sets of silicon resistivity SRMs. These sets, designated 1521, 1522, and 1523, contained two, three, and two wafers, respectively, of

50.8 mm (2 in) diameter silicon with the resistivity values in each set having been chosen to serve a particular application need in the silicon semiconductor industry. The generic purpose of each ofthese sets was to allow a user to verify the performance of a four-point

probe* test instrument, or to calibrate the output of a noncontact eddy current conductance- measuring instrument. The resistivity values of these SRMs ranged from about 0.01 Q cm to about 200 Q cm. More than 1300 sets of these SRMs have been certified and sold worldwide. Each wafer in each set was certified for resistivity using a four-point probe

following the measurement procedure of ASTM Method F84 [1]. This procedure is also

referred to as a "single-configuration" four-probe procedure in the remainder of this report. Improved Resistivity SRMs

After several years of exploratory work, the Semiconductor Electronics Division is issuing

improved SRMs at the same resistivity levels as in the previous sets, but having four salient upgraded features:

1 . The new SRMs are wafers of 100 mm diameter silicon which enables better compatibility with present generation user instrumentation.

* In the remainder of this report, the term "four-point probe" is used when referring to the probe itself. The term "four-probe" is used when referring to the measurement process.

2
2. They are fabricated from silicon with improved uniformity of resistivity and thickness. This will reduce ambiguities of interpretation related to measurement sampling volume, and it will improve transferability of the certified value to the end user.

3. They are certified with a modification of the original certification procedure. This modified procedure is referred to as "dual-configuration" or "configuration-switched"

four-probe measurements and is implemented on most commercial four-point probe

instruments that are automated for thin-film sheet resistance mapping. Tests at NIST have shown there is also a significant reduction of uncertainty when using this procedure for measuring bulk silicon wafers.

4. Measurements and analysis are provided at the wafer center, as was done with the original SRMs, and also around two small circles with sizes related to the requirements of commercial resistivity-measuring instrumentation. These additional data serve to characterize the small nonuniformities in resistivity that are present even in these wafers. While these improved SRMs are issued singly, rather than in sets, it is strongly recommended

that for all purposes of calibration or testing of instrument linearity, SRMs at two or more

resistivity levels be used, with the values being chosen according to the user's application needs. A major goal of these SRMs has been to meet or exceed the requirements set forth at the SEMATECH Workshop on Silicon Materials for Mega-IC Applications [2]: "That layer

resistivity measurements be improved to an accuracy of 1 % and repeatability of 0.5 %

and that NIST provide the SRMs required for such measurements."

1. CERTIFICATION OF IMPROVED RESISTIVITY SRMs

1.1 General Comments

While it may seem trivial to generate silicon resistivity standards that far exceed the SEMATECH Mega-IC Workshop requirements for precision and accuracy, this is not the

case. Silicon is a semiconductor, nonuniform in resistivity in both lateral and vertical directions, unpassivated for use as an SRM, that can be measured with a four-point probe and,

therefore, subject to possible surface effects due to storage and handling environments that can modify the near-surface resistivity. A lapped surface is used on SRM wafers to increase

surface recombination velocity, to improve the quality of contact with the spring-loaded probe

tips, and to improve the long-term stability of measured resistivity by reducing the

susceptibility to changing surface conditions. This, in turn, introduces compromises in terms of near-surface damage, and of the definition and measurement of wafer thickness. Fourpoint probes are used for certification measurements, and the probes are subject to wear and

to changes in contact quality and performance that may be either gradual or rather sudden.

Despite a number of efforts, no simple characteristic of a probe pin has been identified that is

3
a clear indicator of how that pin will contribute to the quantitative performance of a given

four-point probe. It has been found that measurement precision with a single probe head, as well as measurement variability among probe heads, are functions (among other things) of

resistivity, conductivity type, specimen surface preparation, environmental conditions, and

present condition of the probe pins themselves.

It is possible, using a technique such as the van der Pauw procedure [3] with contacts bonded

to the perimeter of a polished wafer and with measurements done in an ambient capable of

controlling wafer surface charge, to eliminate many of the concerns related to measurements

with mechanical probe contacts. It might be possible in this way to eliminate or reduce

noticeably a number of sources of measurement variability. Such measurements would then have a lower uncertainty than those made by four-point probe on a lapped wafer and might

well provide the best estimate of overall volume average resistivity for an entire silicon wafer. However, this would probably not be particularly useful for calibrating or verifying the performance of instruments used in production environments if the measurements required a

special ambient for measurement or if the full-wafer average resistivity did not bear a clear relationship to the localized (small area) value measured by the production test instruments. An important distinction needs to be made. The principal objective of these SRMs is not to provide the best value of the volume resistivity of the silicon wafer itself, but to use the SRM

wafer to help define and transfer a functional resistivity/sheet resistance measurement scale to users of common instrumentation in various parts of the semiconductor industry. Currently, most such equipment is based on four-probe dc resistance or on eddy current measurements

and has spatial sampling volumes on the centimeter scale. There is no known analytic expression for the exact volume weighting of measurements by a four-point probe or by an eddy current tester with a ferrite core. As a result, it is not possible to guarantee perfect equivalence between four-probe and eddy current instruments for specimens with various and

arbitrary patterns of resistivity nonuniformity. Nevertheless, resistivity SRMs based on lapped

silicon wafers with certification measurements by four-point probe, particularly when done

with a well-controlled measurement system used in the dual-configuration mode, and with measurements in well-specified locations on the SRM wafer, offer the user community several

significant benefits. These are: stable SRM artifacts, measurement sampling volume

generally comparable to that of the user's instrumentation, and certified measurement

precision and resolution that more than meets the requirements of the semiconductor industry. Thus, in developing these SRMs, the interest is not so much in what the true bulk resistivity of

each silicon wafer is, but rather in how the measurement values on these wafers behave as a

function of measurement conditions, and how the SRMs transfer between NIST and the user community. There are two principal reasons for preferring the dual-configuration implementation of

four-probe measurements for the SRM certification. First, the probe-to-probe differences are reduced noticeably compared to those that exist when using the single-configuration (ASTM)

procedure; such differences are generally only several tenths of a percent, but make it difficult to reach or exceed the accuracy goals in the SEMATECH Mega-IC Workshop report.

4
Second, the scatter, or random error, is reduced in a set of measurements taken with any

given probe. Both improvements are interpreted as being due to the ability of

dual-configuration measurements to correct more exactly for the true electrical probe

separations than can be done with the auxiliary optical and mechanical separation measurements required by the ASTM Method, in combination with single-configuration

electrical data. For some time, it was common among users of the technique to speak of configuration- switched rather than dual-configuration four-probe measurements. The term dual

configuration is used in this report when the term is written out in order to reinforce the operational difference from the ASTM, or single-configuration, procedure.

The following sections discuss the details of the procedures used for certification, and its control, as well as the manner of analyzing, and reporting the results. They also give a brief description of the components of measurement uncertainty in relation to the equation used to calculate the reported results from the raw data. Section 5 discusses the evaluation of

uncertainty in more detail, but organizes the discussion according to whether the various

contributions are evaluated by ISO Type A or Type B evaluation procedures (see 1.3).

1.2 Resistivity Standards vs. Sheet Resistance Standards

This SRM is called a "resistivity" standard, and much of this report and the SRM certificate focus on describing it and analyzing the measurements in terms of a resistivity value. This is done primarily as a .concession to customary terminology and conceptualization in the semiconductor industry wherein "sheet resistance" is a property associated almost exclusively with a thin film of conducting material rather than with a substrate wafer. However, these SRM wafers do have sheet resistance values associated with them (resistivity divided by

thickness), and moreover, the functional need of most user instrumentation is actually for calibration or verification of a (sheet) resistance scale, and not of a resistivity scale.

This distinction is not simply one of semantics. There is an actual benefit to the user from

treating the SRMs as sheet resistance reference artifacts. To obtain resistivity values for a

silicon wafer, it is necessary to know the wafer's thickness. But when silicon wafers are lapped in order to improve their stability as electrical reference standards, the surface texture compromises the possibility of a wafer having a single, unique wafer thickness. The

determination of the wafer's resistivity value is therefore poorer than that of its sheet

resistance value because of the added uncertainty due to thickness. Thus, each of the SRM

wafers has a somewhat larger relative uncertainty of resistivity than it does for sheet

resistance.

Further, if the user employs these SRMs to establish a scale for resistivity but uses an

independent measurement of thickness such as from a capacitive- or sonic-gauge, then the user must add yet another component of uncertainty to the transfer process. The reason is that these other instrument types are different in operating principle from that of the

5
contacting electronic-micrometer which is used at NIST for the determination of SRM wafer

thicknesses. Therefore, they are not likely to give the same functional value of wafer

thickness that is reported on the SRM certificate, and an additional measurement error is incurred in establishing a resistivity scale. However, sheet resistance values do not depend on

measured thickness value, and transfer of SRM (sheet) resistance values are unaffected by this consideration. It is therefore recommended: 1) that these SRMs be used as sheet resistance standards whenever possible and 2) that the thickness value given on the certificate be used whenever a resistivity value is needed.

[Note: Thickness values for lapped surface wafers typical of those being used for these SRMs have been found to be about 0.5 % smaller when measured with a

capacitance gauge than when measured with an electronic-micrometer. These

capacitance-gauge thickness values are probably closer to the actual thickness of the

electrically conducting portion of the wafer (beneath the lapped texture) than are those from the electronic-micrometer. However, for the purposes of SRM certification, it is easier to establish traceability of thickness scale to dimensional standards when using an electronic-micrometer.]

1.3 Traditional Description of Uncertainty and the ISO Formulation Measurement uncertainty for these SRMs is reported in conformance with guidelines formulated by the International Standards Organization, ISO [4, 5]. Sources of uncertainty

are classified as Type A or Type B according to whether their values are estimated from

repeated measurements (Type A), or are inferred in another manner (Type B). A variance is calculated, or estimated, for each contribution to the uncertainty of the measured value; a sum

of variances is then done separately for Type A and Type B evaluations. The square root of

the sum of the Type A and Type B variances is calculated and is called the combined

standard uncertainty, uc. A quantity called the expanded uncertainty, U, is calculated by

multiplying the standard uncertainty by a coverage factor, k. This factor can often be taken from the Student t tables to give a stated coverage, say 95 %, if the degrees of freedom can

be calculated. The effective degrees of freedom in the analyses of each of the SRM levels are

sufficiently large, typically 60 or more, that a factor of k = 2 gives a coverage of 95 %. Where sources of uncertainty for this SRM are estimated from other than repeated measurements, it is generally assumed that the affected measurements come from a

rectangular distribution, the limits of which are the values that would have been assumed as the maximum systematic error for that quantity. For a rectangular distribution, the variance is the half-width divided by a/3. There is not always a one-to-one correspondence between the categorization of traditional sources of measurement error as being random or systematic and

the uncertainty components determined by Type A or Type B evaluation procedures.

6
1 .4 Acquisition and Characteristics of Silicon Wafers for the SRMs

Wafers at all SRM resistivity levels were bought, having been already cut, etched, and lapped

by the supplier. The supplier for each of the resistivity levels is identified on the SRM

certificate. All wafers are nominally 625 urn thick. The perimeters of all wafers were

contoured to reduce breakage; a single primary orientation flat was ground onto all crystals prior to slicing. The supplier of the wafers for the three lowest resistivity levels, 0.01 Q-cm,

0. 1 Q-cm, and 1 Q-cm, used a laser marking technique to engrave a unique wafer

identification into each wafer just above this flat; the suppliers of the four highest levels did

not offer such a marking process. Wafers at the three lowest resistivity levels are from (100) boron-doped Czochralski-process

(Cz) silicon crystals, while wafers at the four highest levels are from (111) crystals phosphorus-doped by the neutron-transmutation doping (NTD) process. These combinations have been found to be appropriate for meeting the goal of high uniformity of resistivity across a resulting wafer. The suppliers** selected (Recticon Inc. for the three lowest resistivity levels, Wacker

Siltronic for the middle level, and Topsil Semiconductor Materials A/S for the three highest

levels) specialize in the types of growth processes listed. Preliminary batches of wafers from

each supplier were evaluated for thickness and resistivity uniformity. These evaluations

indicated a high degree of likelihood of total thickness variation being less than 1 \xm over the wafer surface and of resistivity uniformity being 1 % or better within the central 50 mm

diameter of the wafers. These levels of uniformity are not guaranteed, however.

1.5 Measurement Concerns and Control of the Certification Procedure

Extensive preliminary testing was done to reach a reasonable optimization of the wafer

preparation and test conditions, to minimize or eliminate effects that would degrade the

certification uncertainty, and to estimate the relative importance of the various known

remaining effects. These tests then led to the design of several experimental control procedures to monitor and evaluate possible changes in probes, wafers, or instrumentation during the certification. The following sources of experimental variability and possible error were identified and are

listed along with the procedure that was developed to minimize their effect and to estimate

their value.

**Certain commercial equipment, instruments, or materials are identified in this report to specify adequately the experimental procedure. Such identification does not imply

recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.

7
1 . Short-term imprecision (repeatability of measurements taken within a period of

minutes in a small, uniform region of material) is believed to be controlled by

probe contact fluctuations and electronics noise; it is minimized by using the dual- configuration procedure and choosing a probe with low noise. Short-term

imprecision is evaluated from data at the centers of the certified wafers, as well as from all wafers used in the control procedures.

2. Longer-term imprecision (the ability to reproduce an average value at a fixed point on a wafer over a period of days or weeks) is related to changes in the measurement environment, e.g., power-line conditions, electromagnetic

interference, or humidity. No measurements are taken at a relative humidity above

50 %, and residual long-term imprecision can be evaluated from the control experiments.

3. Probe-to-probe differences in measured value have been seen to exist. Although

small (0.1 %, or less), it is necessary to identify and select for certification a probe

with low bias. This is done through the design of one of the control experiments. Residual offset for the selected probe is estimated through analysis of this control experiment data, and a correction applied to the measured results if the offset is statistically significant.

4. Possible drift of the measurement process with time, whether due to changes in the probe used for certification, the wafers being tested, or to strong changes in the measurement environment. Drift can be estimated from the design of one of the

control experiments.

5. Possible dependence of the measured resistivity value on the current value is controlled by a very tight procedure for selecting the current level for each SRM

wafer.

6. Wafer nonuniformity effects on the certified values of resistivity are minimized by

using very high uniformity wafers and by using a tightly controlled procedure for selecting the measurement locations.

7. Error related to the temperature dependence of resistivity value is controlled by

measuring the temperature of the wafer stage and applying a correction for the difference between ambient and a reference temperature of 23 °C for each line of measured data. An estimate is made of the uncertainty of the temperature

correction, and this estimate is part of the Type B standard uncertainty.

8. Possible error related to the accuracy of the measurement current supply and the

digital voltmeter (DVM) are minimized by using standard resistors to measure the current value and by using the same scale of the DVM for measurement of both

8
wafer voltage and standard resistor voltage drops. Residual uncertainty related to the voltage measurements is estimated by Type B procedures.

9. Possible error related to the accuracy of the thickness measurement tool is minimized by instrument checks, several times a day, on NIST-traceable gauge

blocks of thicknesses very close to those of the wafers. All wafers with a total indicated runout in excess of 1 um over a nine-point thickness measurement pattern

are rejected. Residual thickness measurement uncertainty is estimated by Type B

procedures. Accumulated probe damage in the wafers should not be detectable within the duration of the

tests being performed. Previous tests on approximately a dozen wafers similar to these SRMs

showed no effect out to 3000 probings for most wafers. However, a few wafers in those tests did show noticeable shifts (about 5 %) in average resistivity and greatly reduced measurement

precision after about 1500 probings.

2. CERTIFICATION PROCEDURE

To minimize the effects of test instrument performance on measurement accuracy, a high

degree of reliance is placed on ratioing techniques for both wafer thickness and electrical measurements, with the instruments being checked against precision calibration standards. Thus, the instrument used for wafer thickness measurement is regularly verified against gauge

blocks having thicknesses very close to those of the SRM wafers, and measurements of the voltage drops across the silicon wafer and the standard resistor are read on the same scale of

the same digital voltmeter. The standard resistors employed for monitoring the current serve

as the primary reference point for all electrical measurement values.

2.1 Wafer Thickness Screening and Thickness Measurement

Preliminary screening with a capacitance-type thickness instrument of a small random

selection of wafers from each of the actual SRM batches showed typical within-wafer

thickness variation to be 0.2 um, or less, for the central region where four-probe measurements are taken. This is noticeably better than the uniformity requirement of 1 %

(which would be about 6.2 um for the SRM wafers), as required by ASTM F84 for referee

resistivity measurements. For the certification procedure, thickness measurements of each wafer are taken on a three- row by three-column grid with a distance of 1 9 mm between the wafer center and the corners

(Fig. 1). (The locations are approximate since the wafers are positioned manually.) This

nine-site sampling gives a reasonable measure of the thickness and its variation in the area used for electrical measurements. Because the small contact area of the electronic-micrometer

is more sensitive to local fluctuations due to variations in lapped surface texture, there is more

9
. Electrical Measurement Locations:

; q ; Six Probe Orientations at Center

Six Locations on Each of Two

Small Diameter Circles

Figure 1 . Scaled drawing of a 100 mm diameter wafer (top) showing locations of thickness measurements (x) and locations of the four-point probe measurements in the 2X magnification

at the bottom.

10
variation in thickness values obtained by this instrument than those obtained by the capacitance gauge. Nevertheless, the range of thicknesses from this nine-point sampling plan

is less than 0.5 urn for most wafers, and wafers are excluded from use as SRMs if there is an

indicated variation of more than 1 um among the nine sites. The average of all nine

thickness values is used for conversion from sheet resistance to resistivity values on the SRM

certificates. (ASTM Method F84 requires the use of only the thickness measured at the wafer

center for this conversion.) The use of a nine-point average thickness reduces small errors due to local fluctuations in surface texture, is more representative of the area over which

electrical measurements are taken, and improves the consistency among all wafers certified at a given SRM level. The standard deviation of these nine measurements is reported on each

certificate for each wafer. Specifications for the electronic micrometer used for the thickness measurements can be found in Section 5.2.4.

2.2 Four-Probe Measurements of Sheet Resistance

Certification measurements are taken using a single four-point probe head, selected from five available (see 3.1). The specific probe used may differ from one resistivity level to another according to results of preliminary tests. All probe heads are constructed with in-line mounted tungsten-carbide probe pins, with a nominal separation of 1.59 mm between adjacent

pins, with a spring-loaded force of about 1.5 N per pin and a nominal 40 urn (0.0016 in) tip radius. Eighteen sites are measured on each wafer, and wafers are allowed to equilibrate with

the environment of the lab module for at least 24 h and with the temperature of the heat sink on the probe station for at least 1 min before being measured. Basic equipment requirements

for all measurements follow ASTM Method F84; manufacturers' specifications for the equipment used can be found in Section 5.2.1. The measurement procedure for the first wiring configuration at each site follows ASTM F84, and that for the second configuration follows ASTM F 1529 [6]. At each of the 1 8 sites for electrical measurement, the probe is connected first to the dc

current source and DVM as in ASTM F84 (current through the outer probes and potential drop across the wafer measured with the inner probes). The current supply is set to give a specimen voltage drop between 9.95 mV and 10.05 mV for the forward current polarity at the

first wafer-center measurement site. The current-supply controls remain set at this position

for all remaining measurements on the wafer. The standard resistor for measurement of

current value is chosen so that the voltage across the standard resistor is larger than that across the wafer. Applied current and specimen voltages are measured for both current

polarities, and the average voltage-to-current ratio is calculated from these "forward" and

"reverse" readings (to eliminate Seebeck voltages) [7]. Standard resistor and wafer voltages

are recorded to a resolution of 0.1 uV. (More detail on the voltage measurements is given in Sec. 5.2.1.) While still in contact with the wafer, the probe head is connected to the current supply and DVM in the second wiring configuration, with the current passing between one

outer probe pin and the nonadjacent interior pin, and the specimen voltages being measured

with the two remaining pins. Again, forward and reverse direction current and wafer voltage values are measured, and an average voltage-to-current ratio is calculated. (See Fig. 2 for

11
Standard

Probe

Configuration

(ASTM F-84)

Second Configuration, bl

(There are two symmetric choices

for the second configuration)

I

® ©n ® @ ® © ® ®

Figure 2. Schematic of probe wiring for dual-configuration measurements.

12
schematic of probe wiring.) For the wafers being certified, only one of the two nominally

symmetric choices for this second wiring configuration is used, although both choices are used

for measurement of the "control" wafers. Using a theoretically derived relation between the voltage-to-current ratios from these two

configurations, a scaling factor for lateral geometry effects is calculated [6]. This scaling

factor, Ka

, in eq (1) of Section 4, is multiplied by the voltage-to-current ratio from the first configuration to give a value for sheet resistance that has been corrected for wafer-edge

boundary condition effects, and for variations of probe separation (at least to first order); an

additional scaling factor is required if the wafer thickness is more than about 0.4 times the probe spacing (the largest ratio of wafer-thickness-to-probe-spacing for any of the SRM

wafers is 0.4008). Multiplication of the sheet resistance values by wafer thickness produces

values of wafer resistivity. The measurement results are corrected to 23 °C using the temperature of the heat sink at time of measurement and empirical temperature coefficients of

resistivity [1]. A complete control and certification procedure is applied to a batch (approximately 125

wafers) at one of the seven resistivity levels; all data are analyzed for that batch and any

necessary auxiliary measurements taken before proceeding to another resistivity level. This is to assure that any wear-induced drift, or other change that may be experienced by any of the four-point probes being used, is contained in and analyzed as part of the certification of a

single batch. Two levels of control detailed in Sections 3.1 and 3.2 are used. In the first, and

simpler part of certification control, a monitor-, or check-wafer, selected at random from the batch being certified, is measured at random times at least twice a day, during actual

certification, to check for time-of-measurement effects due to factors other than changes in the probe. In the second part, a formal control experiment is conducted just prior to and just following the certification measurements. The entire cycle for initial control-wafer measurements, certification data for a batch of wafers, and final control-wafer measurements

takes approximately 5 to 6 weeks.

2.3 Reporting of Data from 18 Measurements on Each Wafer The 18 electrical measurement sites are distributed as follows: 1) six are located at the wafer

center, with the wafer being rotated 30° between them, 2) six are spaced 60° apart around a

circle of 5 mm (0.2 in) radius, and 3) six are spaced 60° apart around a circle of 10 mm

(0.39 in) radius. (See Fig. 1.) Average values of both sheet resistance and resistivity are reported for the center of the wafer, where wafer nonuniformity effects should be negligible. For measurements taken around the

5 mm and 10 mm radius circles, where additional variability due to material nonuniformity

can be detected, individual site values are reported. To reduce clutter on the certificate, these

individual values are given only for sheet resistance. A procedure for converting them to

resistivity values follows eq (1), and is outlined on the certificate using values that are

specific to each individual wafer. The data entries on the certificate are generally only

13
significant to several counts in the last digit. This digit is retained, however, to avoid

additional error due to truncation. The values reported for the two circles give the user a measure of the radial variation of

resistivity for the wafer, although some azimuthal variation can also be detected on many

wafers. This radial variation information is important for improving measurement transfer to instrument types having different integration volumes from that of the four-point probe used

for certification. However, it is left to the user to determine how to weight the resistivities from the three regions of the wafer for the particular application of interest.

In cases where the user does not specifically need the resistance information from the 5 mm

and 10 mm circles, it is strongly suggested that only the certified values from the wafer center be considered and that all user measurements be restricted to the wafer center.

3. CONTROL PROCEDURES FOR THE CERTIFICATION PROCESS

3.1 Control Procedure for Probe Effects Immediately prior to certification of a batch of wafers and again at the end of certification, a

"control" experiment is performed as follows. Each of five wafers, referred to as "control- wafers" and randomly selected from the batch to be certified, is measured for six orientations of the probe at the wafer center, using each of five probe heads in turn and the dual-configuration procedure. For the reason discussed in the next paragraph (criterion 4), both choices for the second wiring configuration are used for this test. This sequence is repeated until six rounds of measurements have been obtained on each wafer with each probe;

the order of probes used and of the wafers measured is randomized for each round. Experience has shown that there is no reason to extend these measurements over a protracted period of time; this test is completed in about 7 days. The results are analyzed to give

baseline values so the performance of the probe to be used for certification can be checked

later if needed, to provide both short-term and longer-term estimates of measurement precision

at that resistivity level, and to estimate the contribution to measurement uncertainty of the choice of measurement probe. For this latter purpose, the five available probes are assumed

to represent a random sampling of all possible probes meeting reasonable operating

requirements; they are not brought to like-new conditions prior to the tests. Probe heads

are, however, prechecked for a number of operating characteristics, and individual pins

replaced, if necessary.

After the results of the initial control experiment are analyzed, one of the probes is selected

for certification measurements of all wafers in the batch. The following criteria are used when reviewing the initial control-wafer test data: 1) preference is given to probes with low

average within-run standard deviations for six replicate runs on the five wafers; 2) preference

is given to probes having high reproducibility of average value from the six rounds for each of the wafers; 3) preference is given to a probe that gives resistivity values in the middle of

14
the distribution for the six-round, five-probe, five-wafer data set which should ensure minimum bias to the ensuing certification data; and 4) preference is given to probes having

good consistency of measured values between the two choices for wiring the second measurement configuration. Experience has shown that most probes do not give exactly the same measurement results on bulk substrate wafers for the two choices of second

configuration. The small differences that are generally seen are believed due to the inability of dual-configuration measurements on bulk silicon to completely account for variations in probe spacing. Therefore, it is important to identify, and use for certification, probes that behave according to the theory for dual-configuration measurements where the theory does not admit to a distinction between the two choices for the second measurement configuration. There is no a priori formula or weighting factor used for these preference criteria; the goal is simply to identify and use the probe that has the lowest short-term and longer-term "noise" or imprecision and the least bias in measurement results.

This control experiment is repeated upon completion of the certification measurements for a wafer batch. This repetition is used to test for a change in response of the probe used for

certification (which would indicate wear or contamination during certification). This should be distinguished from possible drift that might show up for most, or all, of the probes and which would more likely be due to changes in the measurement environment or to changes in the control-wafers themselves. Small changes in the response of only the certification probe would need to be accounted for by use of an additional contribution to the uncertainty statement for the SRM value. Larger changes in the response of the certification probe, if they occur, might require the probe to be rebuilt, and the entire sequence comprising initial control experiment, probe selection, batch certification, and final control experiment to be

repeated. Changes in the response of all probes, if observed, would be analyzed for consistency or randomness of behavior and appropriate components estimated for the uncertainty statement. Data from this multi-wafer control experiment also serve to estimate short-, intermediate-, and

longer-term random variations in the certification process; see Appendix 2.

3.2 Control Procedure for Day-to-Day (Environmental) Effects

Acquisition of all the certification data on a batch of about 125 wafers takes approximately

10 to 12 days. Humidity is monitored, and no wafers are measured when relative humidity readings are in excess of 50 %. To monitor for possible effects due to changes in humidity, power-line fluctuations, or similar environmental problems, one wafer from the

batch, referred to as a "monitor-wafer" or "check-wafer," is measured at random times approximately twice a day for the duration of certification. This results in 20 to 30 sets of measurements on the check-wafer (six wafer-center measurement sites each); this number is well below any level that has been found to cause significant change of value due

to accumulated probing damage. These check-wafer data are analyzed for possible day-to-day

(or time-of-day) variations in value, either random or systematic, that need to be incorporated

into the uncertainty statement. The check-wafer data can also be used to give another

15
estimate of the short-term precision of the measurement process, and may serve to corroborate wear or contamination in the certifying probe.

3.3 Control Procedure for Other Longer Term Effects and Drift of Wafer or Probe The sets of control-wafer measurements that are taken both before and after the certification data can be used to determine whether average resistivity has increased or decreased between

the two series of measurements. If they have changed by a statistically significant amount,

the multiplicity of probes involved can be used to determine whether the changes are likely due to wear of the certification probe (only that probe should show significant change) or whether the same changes are detected by most or all of the probes being used. The

latter condition would indicate likely changes in the control wafers themselves or in the measurement equipment. Appropriate follow-up tests would then need to be made or suitable additional terms added to the uncertainty statement.

4. EQUATIONS USED FOR CALCULATING SHEET RESISTANCE AND

RESISTIVITY VALUES

The following equations are used for calculating sheet resistance and resistivity values from

dual-configuration four-probe measurements. Rs = -Ka FT F(t/S) =XKa FT F(t/S) (1)

p = -K FT

t F(t/S) = XKa FT

t F(t/S) (2)

where: Rs is the sheet resistance of the wafer, in ohms;

p is the volume resistivity of the wafer, in ohm centimeters;

(V/I) is the first-configuration (ASTM F84) voltage-to-current ratio (also called Ra

), in ohms;

t is the wafer thickness, in centimeters; S is the average probe separation, in centimeters;

F(t/S) is a thickness-related scaling factor (near unity for t < 0.4 S); FT

is a correction from the temperature of measurement to a reference temperature (23 °C); Ka

is a geometric scaling factor that is calculated from electrical data in the two

configurations; and X is a shorthand for the voltage-to-current ratio in the first configuration.

16
The equations are applied at each measurement site to the average of the voltage-to-current

ratio for the forward and reverse currents. From the theoretical development of the dual-configuration measurement, the scaling factor, Ka

, is determined from a transcendental equation from reference [8], but a simplified

calculation that is a highly accurate representation is given by the following quadratic

equation, also from reference [8]

where Rais the voltage-to-current ratio in the first electrical configuration, and Rb

is the voltage-to-current ratio in the second electrical configuration.

4. 1 Rewriting the Equation to Relate to Evaluation of Uncertainty

In the ISO formulation of uncertainty, the standard uncertainty is the square-root of the sum

of variances of the components evaluated by Type A procedures and of those evaluated by

Type B procedures. Those components (e.g., short-, intermediate-, and longer-term measurement system imprecision) that enter through the measurement data are evaluated by

statistical analysis of the actual measurements, in units of resistivity, and give a Type A

standard uncertainty "directly in units of resistivity. Those that enter through one of the

scaling or correction factors in eq (1) or eq (2) must be multiplied by an appropriate prefactor

to give a Type B standard uncertainty in the same units. The development of these prefactors

is most readily done through a propagation of variance formulation for the variance of

resistivity, a

2(p), in terms of the variances of the quantities in eq (2). The variance of

resistivity can then be expressed as:

where % is the product X Ka

.

All certification and control experiment data that are supplied for statistical analysis are in units of resistivity, corrected to a temperature of 23 °C, with dimensions of ohm centimeters. The statistical variations in these data are principally manifestations of variations in the measured electrical quantities: the first term in eq (4); and to a lesser extent, variations in the temperature of measurement and the associated temperature correction: the second term in eq (4). Since each of the wafers being analyzed has a fixed assigned thickness value, there is no statistical variation due to thickness: the last two terms in eq (4). The statistical analyses

(R \ (R \2 K = -14.696 + 25.173 — - 7.872 — , (3)

17
look at total change in resistivity from all sources and are not partitioned into variability of

voltage and current or temperature correction. The results of the statistical analyses give

values of uncertainty, in ohm centimeters, from which a Type A variance, in ohm

centimeters squared, is calculated and then summed with the Type B variance.

All terms in eq (4) need to be considered in ISO Type B analyses of uncertainty related to measurement scale calibration errors. All terms, as written, have dimensions of ohm

centimeters squared, but it is convenient to rearrange the first square-bracketed term of the equation so that it shows the same explicit dependence on p

2 that can be seen for the second

square-bracketed term. By multiplying numerator and denominator of the first term by

yjIFj, the equation can be rewritten as:

a

2 (p) = \F

2T o

2 (x)+X

2 o2 (Ft )]+ P

2 [F\tlS) a\t)H2o

2 (F(t/S))} . (5) X

2F

2

t 2F

2(t/S)

It is useful to summarize the nominal values of the various terms that appear as part of

prefactors in eq (5). All such terms are sufficiently constant from wafer to wafer that use of nominal values will suffice. Nominal wafer thickness is 0.628 cm; the thickness-related

scaling term F(t/S) is dimensionless and is taken as unity for all wafers because of the values of t/S for the wafers being certified. The temperature correction factor, FT

, is dimensionless and is very close to unity, being no smaller than 0.985, nor larger than 1.005 for any SRM

wafer. The term Ka

is dimensionless and has a slightly different value for each line of data

for each wafer, but the value is always close to 4.50.

In order to facilitate Type B evaluation of measurement uncertainties, it is helpful to split the preceding equation into separate variance terms that can be related to the background

discussions of Type B standard uncertainty evaluations in Section 3. These terms deal with:

electrical measurements:

X

2F

2T [F

2T o

2 (x)}

o\Vll)

+ °

2(*)

iVlI)

2 K2 (5a)

temperature measurements:

X

2F

2 P

2 o

2(FT) (5b)

18
and thickness measurements:

t

2 F\tlS)

[F2(t/S) o\i) + t 2 a

2 (F(t/S))] = p

: V_(0

+ o

2(F(t/S)y

t 2 F

2(t/S) . (5c)

Thus, following this rearrangement, each of the contributions reduces to the relative variance of a variable times the square of the resistivity.

5. SOURCES OF MEASUREMENT UNCERTAINTY - DETAILED DISCUSSION

5.1 Type A Evaluations of Components of Uncertainty

The contributions to uncertainty from sources discussed in this section are evaluated solely from certification and control experiment data taken at the time of certification for each of the

resistivity levels. A variance is calculated for each of the Type A contributions. These

variances are then combined in a root-sum-of-squares fashion to give a standard deviation from the combination of effects; this standard deviation is the Type A standard uncertainty. Data from SRM 2547, at 200 Q-cm, are used in Appendix 2 to illustrate the analysis

procedures used. Abbreviated summaries from SRMs 2541, 2542, 2545, and 2546, which

follow the same procedures, are given in Appendices 3 through 6 (some of these latter appendices also contain analysis details of a specific additional term which was not pertinent

to the data analyzed in Appendix 2). The statistical reports in the appendices state the standard uncertainties for the resistivities at the wafer-center and for the 5 mm and 10 mm

measurement circles. In Section 7, those values from Appendices 2 through 6 are summarized, the variances from Type A and Type B analyses are tabulated, and the combined

variances, combined standard uncertainties, and expanded uncertainties are given for both

sheet resistance and resistivity for each of the SRMs.

5.1.1 Short-term precision; repeatability There is expected to be negligible effect from wafer nonuniformity on the six measurements

at the wafer center; ideally, these measurements would all have the same value. The standard

deviation of the six values is a measure of the repeatability, or short-term precision, under

tightly controlled conditions. The repeatability is evaluated from data taken over periods so

short that there should be no changes in measurement environment, or wear or damage effects on the wafers or the probe. The variability among the data being analyzed is a combination of two effects, both of which cause fluctuations in the voltage-to-current ratios, and as a

result, in the calculated Ka scaling factor that is based on those ratios. (See eq (1).) The first of these effects is the scatter in the electrical data due to pure electrical or electronic sources such as variations in probe contact quality, power supply noise, or DVM noise; the second is scatter in electrical data due to small fluctuations in probe separation, from one site to the

19
next, that is not fully corrected for by use of the dual-configuration technique. These two

effects are the primary mechanisms causing short-term data scatter; they cannot be separated

functionally, and there is no need for doing so. They are accounted for in calculations of

short-term standard deviation of resistivity. Typical values for the standard deviation of a set of six measurements at the center of a wafer have been found to range from 0.03 % to about

0.30 % for single-configuration data and about 0.02 % to 0.12 % for dual-configuration data. The actual values for standard deviation depend somewhat on the probe used and on the wafer

resistivity level. One of the causes of the spread in the observed values is the small sample

size (six measurements in the NIST certification procedure) for calculating the standard

deviation. The short-term precision for the certification process at each resistivity level is estimated from a pooling of variances of the wafer-center data from the wafers being certified and similar data from the wafers in both types of control experiments. There are typically 1000 or more degrees of freedom to this pooled estimate, depending on the number of wafers

in the batch being certified.

5.1.2 Intermediate and longer term precision; reproducibility of wafer-center average

value Assuming there is no significant change in the probe used for certification measurements and

no change in the resistivity of any of the wafers due to the probing process, it should be

possible to remeasure any of the wafers and obtain average values that fall within limits based on the short-term precision value. In fact, this generally is not found. The small excess

variation is believed due to changes in the measurement environment, such as power-line

variations and changes in the radiated noise in the laboratory environment, humidity changes,

etc., that are not readily identified over the short time spans used to measure individual wafers. Data from the replicate measurements on the check-wafer and also from the initial and final control experiment wafers are analyzed for a day-to-day (run-to-run) random

variation in the response of the certification probe that is in excess of the pooled short-term standard deviation. In addition, comparisons of preliminary and final control experiment data

for each of the probes on each of the control wafers are used to estimate any additional longer-term variations that are characteristic of the entire measurement system and changes in the environment, not just of the certification probe or the check-wafer. Such contributions to uncertainty are termed "long-term variations" in the statistical analysis reports. There are

typically 50 degrees of freedom in the determination of the day-to-day variability in the

analysis of the control-wafer data and 20 or more degrees of freedom in such a determination from the check-wafer data. There are 5 degrees of freedom for the calculation of long-term

variability in the comparison of initial and final control experiment data. The same sets of

data are also analyzed for possible systematic data trends in the measurement process or specimens and corrections terms applied or additional uncertainty components evaluated, as necessary. Such a systematic trend was identified for the 200 Q-cm wafers. It is discussed

separately in Section 6, and the analysis of a resulting asymmetric modification of the uncertainty interval is given in Appendix 2.

20
5.1.3 Uncertainty due to the selection of a particular probe

It has been found, based on the analysis of many experiments, that resistivity measurement

values obtained by four-point probe on bulk wafers have a small dependence on the probe

being used. Experience at NIST shows this to be the largest residual error when ASTM

Method F84 is used for measurement; measuring the geometric separation of probe

impressions made on a polished wafer, as required by ASTM F84, does not adequately

describe their functional electrical separation. This dependence is significantly reduced, but

not eliminated, by use of the dual-configuration procedure. This may be thought of as an

issue of the accuracy of the basic model of dual-configuration four-probe measurements

applied to real probes having finite size contact areas and contacts that are not purely ohmic,

but affected by metal-semiconductor interface effects. The result is a probe-dependent bias in the measured wafer resistance value that might normally be considered a systematic effect, the value of which could be evaluated, or estimated for any given probe. Because there is no model of the physics that causes the offset for a given probe, an estimate of the probable

distribution of probe offset values cannot be done on a theoretical basis. A numerical

evaluation of such a distribution could be done if given a sufficiently large number of probes, and the bias of a given probe could then be determined using the many-probe average as a

point of reference. However, there are only five probe heads available for use in the

certification procedure, thus making it impossible to obtain data from a sufficient variety of

probe heads to generate a distribution of probe-dependence values.

Instead, the five available probe heads are treated as a random sample from the universe of

probe heads, and sufficient replication data are taken with each probe head during the initial and final control experiments (Sec. 1 .4) that a statistical estimate can be made for a variance term due to probes as a variable. Thus, while the choice-of-probe effect is most simply

conceptualized as a systematic error, it is actually evaluated from statistical analysis of these

replicate measurements as a Type A contributor to the standard uncertainty of certification. The initial and final control experiments incorporate data from both choices for wiring the second probe configuration, while the certification measurements use only one of those two

choices. The initial and final control experiments are also analyzed for possible contribution

to certification uncertainty due to small differences between the two choices for second

configuration wiring.

5.2 Type B Evaluations of Components of Uncertainty

No corrections were applied to the SRM certification measurements for possible errors in voltage, current, or thickness values. However, a correction was applied for the difference between the temperature scale of the thermistor used to monitor measurement temperature and

that of a precision mercury bulb thermometer which is the customary reference to a NISTtraceable temperature scale following the procedure of ASTM F84.

In this section, with one exception, a single value is calculated for uncertainty in electrical and

thickness scales which is applicable to all resistivity levels. That exception is at 0.01 Q-cm,

for which the value related to electrical measurements is almost twice as large as that for the

21
worst case from any of the higher resistivity levels. A separate value is given for 0.01 Q-cm.

For the temperature correction term, it is necessary to calculate a separate value for each

resistivity level.

In the remainder of Section 5.2, individual effects are considered significant, and are retained,

if they are at least 0.01 % (one part in ten thousand) of the measured value. Values smaller than that are considered negligible.

5.2.1 Discussion of components related to electrical measurements

Measurement of Specimen Current — Four separate precision-current supplies are available, each calibrated and tested annually for ripple and noise. Measurement accuracy does not rely on this calibration, however. Instead, the measurement current is fed through a precision standard resistor in series with the wafer, and the voltage drop across the resistor is measured

with the same 6-1/2 digit DVM (Hewlett-Packard model 3456) used for the silicon wafer

measurements. Voltage measurements are taken with a resolution of 0.1 uV. Five precision

resistors from 0.01 Q to 1000 Q are available. Each is calibrated periodically at NIST. The

resistors have calibration uncertainties of 3 \xQ/Q to 5 \iQJQ. There is no meaningful change

of value of these resistors due to temperature variations for the temperature excursions encountered in the lab. Standard resistor and wafer voltages are measured on the same range

setting of the DVM. In typical practice, a standard resistor is selected for use so that it gives

a voltage drop that is a factor of 1 to 10 times that of the specimen being measured; e.g., a

1 0 Q standard resistor is used for the measurement of a 1 Q-cm wafer. Voltages measured

across the standard resistor are typically 25 mV, generally stable to 1 \xV and read to 0. 1 (J.V. One of the two available solid-state power supplies is preferred for measurement because of

the convenience of six-digit current selection; however, the regulation specifications for these current supplies (as a percent of full-scale) become marginal for the low currents used when

measuring 100 Q-cm and 200 Q-cm SRMs, and it has been found preferable to switch to a vacuum-tube supply to maximize measurement current stability for these resistivities. Specifications for the current supplies and for the digital voltmeter are given in Table 1

.

Measurement of Specimen Voltages — ASTM Method F84 requires that the measurement

current be set to give a specimen voltage drop, between the two inner probes, of 10 mV to 20 mV. NIST measurements for SRMs 1521 to 1523 were taken in the restricted range of

10 mV to 12 mV. For the 100 mm SRMs, 2541 to 2547, reported here, measurements are taken in the still more restrictive range of 9.95 mV to 10.05 mV. (See Sec. 2.2.) (With a

1.59 mm probe point separation, this gives a maximum field of less than 7 mV/mm across the wafer.) Once the current is adjusted to give a specimen voltage in this range for the ASTM wiring configuration of the very first measurement at the wafer center, the power

supply is left at this setting for all other measurements on that wafer. This specimen voltage range results in an acceptable number of digits of measurement resolution with minimal risk of Joule heating or minority carrier injection. The only exception to this procedure occurs for wafers with low resistivity (below about 0.05 Q-cm for a nominal 625 urn thickness) where

use of a current supply having a typical 100 mA maximum output will result in a maximum

obtainable specimen voltage that is below the range stated above. For the lowest resistivity

22
SRM, 0.01 Q-cm, the specimen voltage at 100 mA is about 3.1 mV; this causes a somewhat

larger relative uncertainty in the scale of the electrical measurements at this SRM level.

Typical stability of wafer voltage readings, as seen from the DVM display, ranges from

±1 jliV to ±3 uV (depending upon resistivity, probe, and environmental conditions). In

practice, after setting the switches for each desired voltage to be measured, the operator

verifies that there is no drift in the DVM display for that setting by observing five to ten readings, and then causes the next DVM reading to be stored in the computer with the expectation that the scatter noted above represents a random error in the stored value. Although the single DVM reading that is stored for each voltage or current measurement can

be said to be in error as long as there is any scatter in the DVM displays observed by the

operator, it is not necessary to do a first-principles propagation of error based on typical voltage scatter and eq (5a) in order to determine the random uncertainty in the voltage-to- current ratio. The standard deviation of a set of measurements taken in a fixed region of the wafer (e.g., the wafer center where material nonuniformity effects are negligible) encompasses

the uncertainty due to digital voltmeter noise just described, as well as that due to variations

in probe separation and probe contact quality. Thus, these sources of error are part of the short-term Type A uncertainty of measurement discussed in Section 5.1.1. It is not necessary

to do any other analysis for these factors. Accuracy, or systematic error, of the digital voltmeter is limited by the 24 count, or 2.4 \xV specification. However, relative accuracy of

the ratio measurement is better than 2.4 fiV and is essentially controlled by the accuracy of

the standard resistor values. The effect of digital voltmeter accuracy on measurement

uncertainty is given in Section 5.2.2. General Integrity of the Electronic Instrumentation — This is basically a problem of

elimination/rejection of noise, whether from electronic or thermal sources. When the current

supplies are sent for calibration, they are also checked to verify that they are within the manufacturers' specifications for ripple and noise; see Table 1 . The primary switch-matrix

in the instrumentation utilizes heavy copper contact posts and twin seven-wiper blade

construction designed to be thermal-voltage free. The common-mode and normal-mode noise

rejection specifications for the DVM are stated for the case of a 1000 Q measurement load;

this value is exceeded, however, for all SRMs above 1 Q-cm. To test the effectiveness of

noise rejection, as well as possible leakage currents, analog boxes with very large series

resistors (that represent probe contact resistance, see ASTM F84) are measured with, and

without, the series resistors in the circuit. This is done as a part of the preparation for

certification of each SRM level. Worst-case experience shows that analog boxes simulating

10 000 Q-cm silicon experience a measurement difference (error) of about 0.20 % between

these two setups. This decreases to about 0.02 % when simulating 1000 Q-cm silicon and is negligible for the simulation of 200 Q -cm and lower resistivity silicon.

23
Table 1. Manufacturers' Specifications for the Current Supplies and DVM Used for Certification ELECTRONIC MEASUREMENTS Inc. Model C612 Constant-Current Supply*

OUTPUT RANGES:

STABILITY:

CURRENT REGULATION:

RIPPLE and NOISE:

OUTPUT IMPEDANCE:

1 uA, 2.2 uA, 5 uA, multiplier xl, xlO, xlOO etc. to 100 mA max. (0 to 100 % vernier each range)

0.3 % of range setting (fixed line, load, and temp.)

0.1 % for 100 V step in compliance voltage

0.04 % rms of range setting + 0.5 uA (negative ground)

0.04 % rms of range setting + 0.1 uA (positive ground)

(floating output is used for certification) 30 000 MQ @ 1 uA to 500 kQ @ 100 mA

This current supply is operated at 50 %, or greater, of range setting.

ELECTRONIC DEVELOPMENT CORPORATION Model CR103 Constant-Current Supply

OUTPUT RANGES:

STABILITY (non-additive):

RIPPLE and NOISE: OUTPUT CONDUCTANCE:

10 mA and 100 mA full scale; 6 digit setability

1 h 0.001 % of range

8 h 0.005 %

"

1 Yr. 0.01 %

(0.1 Hz to 100 kHz) <0.5 uA

0.1 nS TEMPERATURE COEFFICIENT: 0.0005 %/K

24
Table 1. (cont'd.) HEWLETT-PACKARD #3456 DVM

(All values are stated for the 100 mV range)

RESOLUTION (Least Count): INPUT IMPEDANCE:0.1 juV

>10 10 Q

MEASUREMENT ACCURACY: For auto-zero on, filter off and >10 power cycle cycles): 24 h @ (23 ± 1) °C: ±(0.0022 % rdg. + 24 counts) 90 day @ (23 ± 5) °C: ±(0.0034 % rdg. + 24 counts) TEMPERATURE COEFFICIENT: ±(0.0002 % rdg. + 0.2 counts/°C)

NOISE REJECTION: Normal mode, ac: 60 dB

(1 kQ max. Unbalance in low)

Common mode, ac: 150 dB

Common mode, dc: 140 dB

5.2.2 Evaluation of uncertainty in electrical measurement scale

Electrical measurement scale contributions to the variance of resistivity value are found from

examining the right-hand side of eq (5a)

o

2(K/7)2 + °\Ka

) (V/I)2 + (Kf

(6)

Ignoring temporarily the term in Ka

, and replacing the current, /, with the ratio of standard

resistor voltage to standard resistor value, V/Rs

, in the first term, results in

o\VRjVs

) = P

2 o\VlVs)

^ o\R)

(V/Vf R

2S

(7)

25
Rather than expanding the term in (V/Vs

) to get separate terms in a

2(V) and o

2(Vs

), it is preferable to look at the way in which electrical measurement error affects the ratio, V/Vs

, as a whole. It is assumed that the 2.4 uV error (due to voltmeter accuracy limit statement)

affects the measurements of V and Vs equally (both voltages are measured on the same meter, and in quick succession). Then the worst-case error in their ratio occurs when Vs

is the

largest multiple of V. It can be seen from Table 2 that this occurs when Vs approximately

equals 3 x V. For the resistivities above 0.01 Q-cm, the wafer voltage-drop is 10 mV, the standard resistor voltage-drop is 30 mV, and the worst-case ratio V/Vs

, with no error in voltage values, is 0.333 333 3. A 2.4 uV error in both V and Vs for these wafers causes a change in the ratio to 0.333 386 7. The difference of the two ratios is 0.000 053 4 and will be taken as a limit of error in the voltage ratios due to DVM least-count error. Squaring this value, and dividing

by 3 (assuming a rectangular error distribution) gives a variance of 9.50 x 10" 10

. The

denominator, (V/Vs)2

, equals 0.111, so the contribution to variance from the first term above

is: (9.50 x 10" 10/0.111) p

2

, or 8.56 x 10'9 p

2

. At 0.01 Q-cm, because of smaller measurement voltage levels, the contribution to uncertainty from electrical measurements is actually larger than the worst-case value for the SRMs above

0.01 Q-cm. For this SRM level, the ratio without voltage measurement error, is 3.1 mV/10 mV, or 0.310 000; and with a 2.4 uV error, it is 0.310 166. The resulting error in the V/Vs ratio is 0.000 166. Squaring this, and dividing by 3, as above, gives 9.18 x 10"9

. The denominator, (V/Vs)2

, at 0.01 Q-cm, is 0.096. The resulting contribution to variance of

resistivity, at 0.01 Q-cm, is 9.56 x 10"8 p

2

. The second term in eq (7),

P

2 o2(Rs

)

can be shown to be negligible. The calibration uncertainty of all standard resistors used for the SRMs is <5 x 1 0"6 times the value of the resistor. Assuming a rectangular distribution for standard resistor calibration error, a

2(/y/(/y2 = (2.5 x lO'

11 ^) = 8.3 x 10" 12

. The

contribution to variance related to standard resistor calibration error is 8.3 x 10~ 12 p

2 and is negligible. Likewise, possible contributions due to drift, or to temperature dependence of

standard resistor values are negligible compared to the one part in ten-thousand criterion noted

above.

26
Table 2. Standard Resistor Values, and Typical Measurement Voltages

for Each of the SRM Levels Nominal Standard SRM Wafer Std. Res. SRM Value Resistor Voltage, V Voltage, Vs (Q-cm) (Q) (mV) (mV)

0.01 0.1 3.1 10

0.1 1 10 25

1 10 10 25

10 100 10 25

25 100 10 11.5 100 1000 10 30 200 1000 10 14

The other contribution to uncertainty due to electrical measurement scale error comes from

the second term in eq (5a)

Ka has the following characteristics. It is the solution to a transcendental equation based on

two configurations of electrical data taken at each measurement site. The solution has been

approximated by a quadratic equation in the argument RJR^ where Rais the ratio of V/I in the first (ASTM) wiring configurations and Xb

is the V/I ratio in one of the two choices for the second configuration. Specifically, the quadratic equation is

14.696 + 25.173 - 7.872

The accuracy of the fit over the range 1.20 <R£/Rb < 1.32 is reported to be better than 0.05 %

[9]. For the wafer diameter, measurement locations, and probe size used in this SRM

certification, the ratio, RJRj,, is approximately 1.255. There are small variations, from about

1 .25 to 1 .26, which encompass both the effects of electrical measurement noise and small

fluctuations in the separation of adjacent pairs of probe pins from one measurement position

to the next. A ratio of R/Rj, of 1.255 results in a Ka value of about 4.50. Over this restricted range, the accuracy of fit of the quadratic, is actually about 0.01 %.

27
There are two independent considerations in evaluating a (Ka

). The first is the relative inaccuracy, 0.01 %, of the quadratic representation of the transcendental equation. With the assumption of a uniform probability distribution, it results in a contribution to the variance of

[{(0.000 1 Ka)2/3}/Ka

2

] x p

2

, or 3.33 x 10"9 p

2

. The second is the error in Ka

that would occur because of an error in measured voltages. For

a nominal value of R^R^ - 1.255, and any of the SRMs above 0.01 Q-cm, a voltage measurement error of 2.4 uY would cause an error in RJRh of no more than 0.000 07. This

causes a change (error) in Ka of about 0.000 38. Again assuming a rectangular distribution of

error, this means that the voltage error contribution to variance is [{(0.000 38)

2/3}/(4.50)2

] x p

2 or about 2.39 x 10"9 p

2

. For the 0.01 Q-cm SRM, and under the same assumptions, a

voltage error of 2.4 uV causes an error in the ratio, RJR^, of 0.000 24. This, in turn, results in an error in Ka of 0.001 29 and a contribution to the variance of p

2 at 0.01 Q-cm of

[{(0.001 29)

2/3}/(4.50)2

] x p

2

, or about 2.74 x 10"8 p

2

. Adding these terms to that for possible error due to the quadratic representation of the transcendental equation, the variance in p

2 due to possible error in the factor, Ka

, is 3.07 x 10"8 p

2 at 0.01 Q-cm, and is 5.72 x 10'9 p2 for SRMs above 0.01 Q-cm.

No specific additional systematic error terms due to instrumentation integrity have been

identified in the resistivity range of these SRMs other than the 0.02 % offset that has been

seen with the 1000 Q analog box. Noise, due to poor contact quality, radiated signal pickup,

or other sources, may be present. It is believed to contribute scatter, in the low microvolt

level, to the data, and show up as a component of the standard deviation of the data. It is possible, but has not proven necessary, to integrate measurements on the DVM for 100

power-line cycles, instead of the customary 10 cycles, to suppress the effects of ac pickup.

Therefore, the total contribution to variance of resistivity due to electrical measurement

considerations discussed above is 1.263 x 10"7 p

2 at 0.01 Q-cm and 1.428 x 10"8 p2 at all higher resistivities.

5.2.3 Evaluation of uncertainty components related to temperature measurements

The variance of resistivity value due to temperature measurement errors arises as follows: During resistivity measurement, each wafer is placed on a copper block which is both

massive, to maintain temperature stability, and made of a good thermal conductor, to enhance

the speed of equilibration of temperature between the surface where the wafer is located and

the block's interior where the thermistor temperature sensor is located. A thin mica film provides electrical insulation between the wafer and the copper block. The measured

temperature (maintained in the range 22 °C to 24 °C for all SRM wafer measurements and

observed to be stable to 0. 1 °C, or better, for any given SRM wafer) is used in conjunction with an empirically evaluated temperature coefficient of resistivity for silicon to correct the measured resistivity to the standard value of 23 °C. The temperature coefficient of resistivity for silicon, which is a function of both resistivity and conductivity type, was evaluated at NBS in the mid- 1 960' s. This temperature coefficient is used internationally and is part of a

28
standard measurement procedure (ASTM F84) for silicon resistivity near room temperature.

It is expected that all users of these SRMs for application to silicon technology will use the same temperature coefficients for interpretation of their "unknown" or "test" wafers. No

evaluation of uncertainty of the coefficient itself is made here. The thermistor was calibrated against a precision mercury bulb thermometer over the range

15 °C to 35 °C. The mercury bulb thermometer itself was calibrated by NIST with a stated uncertainty of ±0.03 °C, or better. Thermistor resolution is better than 0.01 °C. Transfer

uncertainty between glass bulb and thermistor is estimated to be no worse than 0.02 °C; a

value of 0.02 °C will be used. The largest potential error is that the copper block temperature may not be the same as that of the wafer. This could be due to warm or cool air currents from the room ventilation system affecting the wafer and block exterior. Tests of consistency of resistivity measurement with controlled temperature increase and decrease indicate that potential error between sensor and wafer is less than ±0.08 °C. The calibrations of the glass bulb thermometer and that of the thermistor are added to give a worst-case temperature

calibration error of ±0.05 °C. This is added linearly to the possible wafer-sensor offset of

0.08 °C to give a worst case total temperature error of 0.13 °C. Because all possible temperature errors were added linearly to calculate worst-case error, above, it is overly conservative to assume a uniform distribution of error to calculate a

variance, and a triangular distribution for the temperature error is assumed instead. Thus, the variance of the distribution of possible temperature error is (0.13 °C)2/6 = 0.002 82 (°C)2

. To minimize possible temperature error in practice, wafers are kept in the vicinity of the measurement station for at least 24 h prior to measurement, and have at least 1 min to

stabilize on the copper block before taking measurements. Possible errors in resistivity values due to temperature enter through the term from the right-hand side of eq (5b)

where the temperature correction of resistivity, FT

, has the form, FT

= 1 - CT (T - 23 °C) ,

where CTis the temperature coefficient of resistivity, in degree Celsius , and T is the temperature at which measurements are made, in degree Celsius. The variance in FT

is really the variance in temperature (given above) times the square of the temperature coefficient, (CT)2

. Since the coefficient, CT

, varies noticeably as a function of

29
resistivity value, Table 3 summarizes the values of the temperature coefficient used in the calculation of uncertainty.

Table 3. Temperature Coefficients of Resistivity of Silicon

for the Nominal Values of the SRMs

Norn. Res. (Q-cm) 0.01 0.1 1 10 25 100 200 Temp. Coeff. (°Cy l 0.0031 0.0041 0.0071 0.0082 0.0083 0.0083 0.0083 Exact values are given on the certificate for each SRM wafer.

The variance of resistivity value due to temperature error is 0.002 82 (CT) p . Because of

the difference in the values of CT

for the various SRM levels, the variance in resistivity value due to temperature error is given in Table 4.

Table 4. Variance in Resistivity Value Due to Temperature Error SRM Resistivity o

2(FT) Contribution to Variance Q -cm of Resistivity

0.01 2.71 x 10"8 2.71 x 10"8 p

2 0.1 4.51 x 10"8 4.51 x lO

-8 p

2

1 1.42 x 10"7 1.42 x 10

-7 p

2 10 1.90 x 10"7 1.90 x 10"7 p

2 25 1.94 x 10"7 1.94 x 10"7 p

2 100 1.94 x 10'7 1.94 x 10"7 p

2 200 1.94 x 10"7 1.94 x 10'7 p

2

5.2.4 Evaluation of the uncertainty components related to geometry measurements

In single configuration (ASTM F84) measurements by four-point probe, it is necessary to measure accurately the wafer diameter, the wafer thickness, the average separation between the probe pins, and variability thereof, in order to calculate geometry-related scaling

factors that convert measured voltage/current ratios to sheet resistance and resistivity values. In dual-configuration measurements, only the measurement of wafer thickness and the average probe separation (for thicker wafers) enters into the calculation of sheet resistance and

30
resistivity. The following discussion deals with errors in geometry measurements as they

relate to possible uncertainties in the certification values. Comments on Nonideality of a Lapped Surface — A lapped surface texture is used to optimize

electrical stability of the SRMs and to improve contact quality between the probe and wafer. The wafer thickness, in centimeters, is used to multiply sheet resistance values to convert them to resistivity values. Fractional errors in thickness values are reflected 1 : 1 as fractional errors in calculated resistivity values. The lapped wafer surface has a peak-and-valley texture

that is related to, but generally smaller in size than, the abrasive used to do the lapping. Even

though the lapping process used to prepare the SRM wafers is known to give total (macroscopic) thickness uniformity better than obtainable on as-cut or polished wafers,

the existence of the surface texture precludes there being a unique thickness value at any location on the wafer. (The 100 mm SRM wafers for SRMs 2541 to 2547 were lapped

with a simultaneous two-side lapping process. An abrasive grit size of about 12 um was used

for the four lowest resistivities and a 7 urn grit size for the three highest resistivities. The

earlier 50.8 mm (2 in) diameter NBS silicon resistivity SRMs utilized a one-side-at-a-time process and a 5 um abrasive. As a result, the 100 mm wafers have much improved

macroscopic thickness uniformity, but a somewhat coarser surface texture compared

with earlier SRMs.)

Measured thickness values are somewhat dependent on the method of measurement.

Electromechanical-, capacitive-, acoustic-, or air-gauges are not expected to respond the same

to the hills and valleys of a textured surface or to average over the same surface area. A

mechanical method that measures front-surface-to-back-surface peak-to-peak thickness is the most idealized conceptually when dealing with these circumstances, and was used for thickness measurements of the SRM wafers. However, the peak heights on a lapped surface

are somewhat variable on both wafer faces (resulting in small local fluctuations in peak-to-peak thickness and some sensitivity to the location where the thickness measurements

are made). Figure 3 attempts to illustrate the situation of defining and measuring thickness on

a textured surface using an electromechanical gauge.

Calibration and Control of the Electronic-Micrometer — Wafer thicknesses of the SRMs were measured with an electronic-micrometer having a resolution of 0.05 um and a short-term

repeatability of about 0.1 um. The instrument's specifications state that its accuracy is ±0. 1 um if the ambient temperature is kept at 20 °C ± 1 °C. The requirement of a temperature of 20 °C is based on the temperature at which the instrument was calibrated by

the manufacturer. While the laboratory at NIST in which the instrument is used maintains

the required 1 °C temperature stability, the nominal working temperature is typically 23 °C. To maintain the calibration accuracy of the micrometer, standard practice is to calibrate, and

to recheck, the instrument a number of times a day against precision gauge blocks traceable to NIST and having thicknesses that are comparable to the SRM wafers. The gauge readout was

reset, as necessary, to match the gauge block value. Thus, the thickness measurement for the SRMs was a process of transfer of thickness value from a gauge block through the thickness

31
Section of Wafer in Position for Measurement

(Texture Exaggerated)

Figure 3. Conceptual drawing of wafer with two textured surfaces during thickness measurement by electromechanical gauge.

32
gauge to the wafer. Several error components can be identified that will affect the accuracy of this transfer. The thickness gauge specifications are given in Table 5.

[Note: These resistivity SRMs are not intended to serve as thickness calibration standards, and that it goes beyond the scope of this work to be able to relate the performances of electromechanical-, capacitive-, acoustic-, air-gauge, and other thickness methodologies on lapped surface wafers.]

Table 5. Specifications for Haidenhain Certo 60 Thickness Measurement Instrument Measurement Resolution 0.05 um

Measurement Accuracy ±0.1 urn (At an Operating Temp of 19 °C to 21 °C) Wafer Backside-Reference Pedestal 2 mm x 2 mm

Probe Tip Radius 1 mm

Probe Working Force 1 N

5.2.5 Evaluation of uncertainty due to thickness measurement scale Possible errors in the thickness measurement scale contribute to uncertainty of resistivity directly through the first term in eq (5c),

Based on the average SRM wafer thickness of 0.0628 cm, the denominator has a value of

0.003 94 cm2. Two error mechanisms contribute to a Type B estimate of variance of thickness values. The

first relates to the calibration of the thickness measurement tool with precision gauge blocks. Three blocks with thicknesses of 0.024 in, 0.025 in, and 0.026 in (0.060 96 cm, 0.063 50 cm,

and 0.066 04 cm), i.e., just spanning all expected values of wafer thickness, and having NISTtraceable thicknesses known to better than 0.000 004 in (0.000 01 cm) are used. In the tool calibration procedure, the tool is adjusted to read the known thickness of the 0.025 in (0.06350 cm) block and required to read the other two within 0.15 um (0.000 015 cm) of

their stated calibration values. This is 50 % larger than the uncertainty of individual gauge

block calibration values. The value 0.000 015 cm is taken as the half-width of the rectangular

33
distribution of possible error assignable to the calibration of the thickness measurement tool. As a result, there is a contribution to variance of resistivity of 1.90 x 10"8 p

2 from possible

tool calibration error. The second error mechanism relates to the transfer of the thickness measurement scale to the SRM silicon wafers. Various tests of consistency of wafer thickness values suggests that a rectangular distribution with a 0. 1 urn half-width should be sufficient to account for the thickness transfer error term. This results in a contribution to variance of resistivity of

8.37 x 10"9 p

2

. Combining these two terms gives a variance of resistivity value directly due

to variance of thickness of 2.74 x 10"8 p

2

.

5.2.6 Evaluation of uncertainty due to thickness/probe separation scaling factor A scaling factor, F(t/S), is used to correct the calculated sheet resistance values for layers of

finite thickness (greater than about 0.4 times the average probe spacing). Error in either the wafer thickness or in average probe separation value contributes to the variance of sheet

resistance or resistivity through the second term in eq (5c),

p2 o

2 (Fjt/S)) F\tlS)

where the denominator is effectively unity. The scaling factor, F(t/S), for dual-configuration measurements is similar to that, F(w/S), for single-configuration measurements in that they both asymptotically approach unity for values of w/S just below 0.4. These scaling factors are virtually identical for values of the ratio, w/S, below about 0.45, but diverge noticeably in value for wafer thicknesses that are a large

fraction, or a multiple, of the probe separation. ASTM F84 recommends the simplification that this factor be set to unity when the ratio of

wafer thickness to average probe separation is 0.4 or less. For all larger values of the ratio, the scaling factor is then computed from summation of a specified series and takes on values decreasing from unity as the ratio increases above 0.4. The ratio, 0.4, exactly corresponds to a wafer thickness of 635 urn (0.025 in) and a probe separation of 1587 urn (0.0625 in). This

is the nominal separation of the probes being used for SRM certification, and the SRM wafers

were, in fact, purchased with a target thickness of 625 um. Some fraction of the wafers in a

given SRM batch will exceed the ratio, 0.4, if only by a small amount, simply due to fabrication process tolerances. For the seven SRM levels, a total of 34 wafers (out of

approximately 800) had thicknesses such that the t/S ratio exceeded 0.4; the worst-case value of the ratio was 0.4008. Examination of the scaling factor shows that it actually has a value of 0.9995, not unity, for thickness-to-probe-spacing ratios that are infinitesimally above 0.4. When the procedure for

34
certification of these SRMs was devised, it was not known exactly how much variation in wafer-to-wafer thickness would be encountered. To avoid the inconsistency in scaling

factor that would result from using a default value of unity for ratios up to 0.4, and then a

calculated value of the scaling factor for all higher values of the ratio t/S, a decision was made to calculate and apply a correction term for all values of the thickness/probe separation

ratio. The result is to improve the SRM wafer-to-wafer consistency for resistivity value as a function of thickness, but to introduce an offset for most SRM wafers that makes their stated

resistivity 0.04 % to 0.05 % smaller than if the asymptotic value of unity had been used for

this scaling factor. The exact amount of the offset for a given SRM wafer can be found, if needed, by comparing to unity the value of this scaling factor as printed on the certificate for

that wafer. This offset is incorporated in both sheet resistance and resistivity values. There is no error, or uncertainty, term developed to relate to this change from the procedure of ASTM

F84. To calculate the variance in the scaling factor due to uncertainty in the measurements of

thickness and probe separation, typical results for probe separation measurements and

thickness data from one of the SRM levels are used. Following the procedures of ASTM

F84, probe separations can be measured to a resolution of about 1 um and with a typical precision for 10 readings of about 0.06 % (1 um). Wafer thickness and probe-spacing values for the 25 Q-cm SRM level are used to calculate the variance of the wafer-thickness probe-spacing scaling factor. For this SRM level, the

slightly larger upper end wafer thicknesses relative to the spacing of the probe used make the

sensitivity of this term a little larger than for the other SRM levels. For this SRM, assuming no error in thickness or probe separation value, the ratio, t/S, ranges from 0.387 26

(F(t/S) = 0.999 632), to 0.400 83 (F(t/S) = 0.999 506). A worst-case combination of probe

separation error (0.0002 cm assumed) and wafer thickness error (0.000 025 cm assumed)

causes a change in the scaling factor value of about 0.000 03 (a relative change of 0.003 %).

Using this value as a half-width (the error could also be the same amount in the opposite

direction), and assuming a rectangular distribution, the variance of F(t/s) is 5.07 x 10" 10

. Thus, this contribution is negligible.

6. UNANTICIPATED EFFECTS

During the course of certification of the seven SRMs, two effects were encountered that had

not been experienced previously and which were thus partially, or wholly, outside the design of the control experiments. The first of these was a shift, or drift, in measured resistivity during the first few rounds of probe measurements on the 200 Q-cm SRMs. The second was

a sensitivity of the measured resistivity to background illumination level for the 1 Q-cm

(SRM 2543) and the first batch of 10 Q-cm (SRM 2544) wafers.

35
6. 1 Resistivity Shift with Repeated Probing

The phenomenon of resistivity shift with repeated probing is documented in Sections 2.1 and

3.3 of the analysis of SRM 2547 which is given in Appendix 2. It shows up as a decrease of measured resistivity with successive sets of probe measurements made within a period of days

or weeks. It was found to occur for some but not all wafers tested, and where it exists, it is stronger for some probes than for others. The shift is not totally cumulative, but appears to

saturate.

Additional measurements of the original control wafers more than a year after the acquisition of the certification data showed nearly the same effect as shown in Appendix 2. The first of

these additional measurements started at almost the same value as originally (i.e., an upward

recovery of value had occurred in the interim), followed by a gradual decrease in resistivity by about the same amount as previously, then reached an asymptotic value. As previously

noted, some of the control wafers suffered the effect; others did not. The observed shift did

not accumulate beyond a few tenths of a percent. The mechanism for this shift is unknown. It is not believed to be experienced with repeated

eddy current measurements, but this hypothesis was not tested. It is expected that when

wafers from SRM 2547 are first measured by the user, they will manifest the resistivity (sheet

resistance) values listed on the certificate, and if measured by four-point probe, some of them

will show small decreases of resistivity if replicate probe measurements are made within a

period of days, or perhaps weeks. The additional term added to the estimated uncertainty

interval due to analysis of this effect on the control wafers is believed to fully cover any

manifestation of this effect to the user.

6.2 Photosensitivity of Resistivity Value Measurements being made separately from the certification of these SRMs, and after the time when most of the SRMs had been measured for certification, showed that certain types of

silicon had a resistivity value that was dependent on the level of background illumination. Extensive previous experience with four-probe measurements of many silicon specimens,

particularly the types used for previous SRMs, had shown that normal laboratory-level fluorescent illumination had no observable effect on the measurement value. It had been seen

that high resistivity silicon (perhaps 1000 Q-cm, and higher) must be measured in the dark.

It was also seen that bright incandescent illumination, with a significant component of

penetrating infrared radiation, would inject hole-electron pairs that would decrease the measured resistivity with a very rapid recovery (because of short minority-carrier lifetimes) to higher values when that illumination was turned off.

Previous resistivity SRMs up to 200 Q-cm, fabricated from float-zone or neutron- transmutation-doped silicon, showed no sensitivity of resistivity to normal laboratory levels of

fluorescent lighting, and certification measurements were taken on them without a dark-box

enclosure. The current 100 mm diameter SRM wafers were fabricated from boron-doped

36
Czochralski silicon from 0.01 Q-cm to 10 Q-cm and from float-zone grown, NTD-doped

silicon for the highest three resistivity levels. These choices of silicon types were made

specifically to optimize within-wafer uniformity of resistivity for the various SRM levels. The photosensitivity of resistivity that was detected subsequent to SRM certification occurred on boron-doped Cz silicon wafers that were not related to the SRM wafers. Subsequent

testing showed that the effect was measurable from a few tenths of an ohm centimeter to the highest resistivity boron-doped Cz silicon obtainable, approximately 80 Q-cm. The

magnitude of the shift in resistivity was found: 1) to be as high as 2.5 %, 2) to depend

roughly on resistivity level, and 3) to be present on all boron-doped Cz silicon wafers

available, independent of supplier and wafer surface type. Auxiliary tests, on wafers from the

1 Q-cm and 10 Q-cm SRM crystals, also showed a correlation between the magnitude of the

effect and the interstitial oxygen level. The effect could not be detected at all on boron-doped

float-zone silicon or on any phosphorus-doped silicon. Tests were then made of the existence and magnitude of this effect on wafers from the four boron-doped Cz silicon crystals that had

already been certified for SRMs. No effect could be detected for the 0.01 Q-cm or

0.1 Q-cm resistivity levels. A photoeffect as large as 0.4 % and decreasing to about 0.15 %,

as a function of wafer position in the starting crystal, was detected for wafers from SRM

2543, at 1 Q-cm. The effect ranged from 0.6 % to 1.2 % for wafers from the crystal initially used for SRM 2544, at 10 Q-cm.

The photosensitivity is unusual in its very long decay time from lower resistivity in normal room illumination to higher resistivity in the dark. Typical times for decay to the asymptotic

value typical of the new illumination state ranged from about 2 min to more than 20 min. Wafers used for SRMs 2543 and 2544 were at the lower end of this time scale. Because of the significantly large value of the photoeffect for wafers from the original 10 Q-cm boron-doped Cz crystal, these wafers were invalidated for use as SRMs. It was

possible to purchase a sufficient quantity of 10 Q-cm wafers grown by the float-zone process and phosphorus-doped by the NTD technique to be able to retain the 10 Q-cm SRM level using these replacement wafers. The NTD wafers are nearly as uniform as the boron-doped Cz silicon wafers they replaced and are suitable for use as SRMs since they show no evidence of a photosensitivity. The complete set of certification and control measurements have been

completed on the NTD wafers. At the time of publication, analysis of those data is not complete.

The case for the 1 Q -cm SRM level was not so straightforward. It was not possible to get float-zone grown, NTD-doped wafers that are irradiated heavily enough to produce 1 Q-cm

silicon. Possible replacement Cz silicon wafers doped with phosphorus were expected to be

free of photosensitivity, but to have sufficiently large nonuniformity of resistivity as to be

unacceptable for use as standards. No other alternative could be identified, and a choice had

to be made between voiding the 1 Q-cm SRM level altogether and a judicious use of the

1 Q-cm wafers already measured. The decision was made to retain only the best of the

original 1 Q-cm SRM wafers, i.e., those wafers having the lowest amount of photosensitivity,

37
about 0.25% and below. This will allow the retention of just over half of the originally measured batch of 125 wafers. The task of selection was made easy because the supplier

for those wafers laser-engraved a unique serial number on each wafer in the sequence the wafers were taken from the saw. The magnitude of the photoeffect had been found to decrease monotonically from the low numbered toward the high numbered wafers. During the analysis of the certification data for the 1 Q cm SRM, an estimate was made

of a new component of uncertainty, due to the level of illumination. This estimate was

based on measurements at normal operating illumination levels in the laboratory (ceiling fluorescent lights), measurements in the dark, and measurements at noticeably higher- than-normal levels of illumination. This latter condition served to evaluate shifts to lower

values of resistivity that might occur in a user facility having a higher illumination level than was present in the NIST laboratory module during certification. A variety of

sources of additional illumination were evaluated, and a two-cell flashlight with a krypton bulb flooding about a 5 cm. diameter an area, where the probe contacted the wafer, was chosen for the tests. This additional illumination caused saturation in the reduction of resistivity, but did not cause wafer heating. The results of the analysis of the

photosensitivity effect, which were not available for the original issue of this report, are contained in Appendix 7 of this revision of the report. Two notes of caution are in order regarding the use of moderate to lightly boron-doped Cz silicon wafers, regardless of source, for resistivity standards. Both are based on the assumption that photosensitivity, of the type described here, is a universal characteristic of boron-doped Cz silicon. First, it is not sufficient, in general, simply to take the

certifying data in darkened surroundings. Any user of such a standard who is not able to take measurements in similarly darkened surroundings will experience a different

resistivity value, and the difference between the dark-level and illuminated-level values may not be characterized adequately. Second, because the decay time for the

photosensitivity is so long, it is relatively easy, using most commercial, automated

instrumentation, to be fooled about whether a photosensitivity exists for a given wafer. Only a series of measurements over a period of minutes is likely to reveal the drift that is caused by this photosensitivity. There is a related consideration, for a wafer certified in the dark, that will be measured in a darkened, or shrouded, user-instrument, but which

has been stored in illuminated surroundings. Such a wafer will have to be allowed to equilibrate with the darkened interior of the instrument for a number of minutes before

valid readings can be taken.

7 COMPILATION OF UNCERTAINTY COMPONENTS

This section summarizes the Type A standard uncertainty terms for resistivity from

Appendices 2 through 8 and the Type B variance terms from Section 5. It uses these

inputs to obtain the combined variance, u] , the combined standard uncertainty uc

, and

the expanded uncertainty, U, for several parameters. The expanded uncertainty is stated on the SRM certificates for: 1) average resistivity at the wafer center; 2) average sheet

resistance at the wafer center; and 3) individual sheet resistance measurements at locations on the 5 mm and 10 mm circles.

38
The values of Type A standard uncertainty in the appendices are given only for resistivity values. To convert these to values appropriate to sheet resistance, it is necessary only to divide them by

the average SRM wafer thickness, 0.0628 cm. Separate values are needed for average sheet

resistance at the wafer center and for individual measurements on the 5 mm and 10 mm radius

circles. There are three considerations for converting the values of Type B variance of

resistivity, given in Section 5, to values of variance of sheet resistance. First, since sheet

resistance values do not depend on wafer thickness, only the terms in Section 5 from the variance of the electrical and temperature measurements contribute to Type B variance of sheet

resistance. Second, Sections 5 .2.2 and 5.2.3 give the contributions of electrical and temperature measurement variations to the variance of resistivity; it is necessary to divide those variance-ofresistivity terms by the square of the average SRM wafer thickness, ie. by (0.0628 cm)2

, to scale to the variance of sheet resistance. Third, the Type B variance terms are estimates of measurement scale error, and are the same for average measurements at the wafer-centers and

for individual measurements on the two small circles.

7. 1 Summary of Statistical Analysis Parameters from the Appendices

This section summarizes the information given in Appendices 2 through 8. It gives the symbols

used in the statistical analyses, the components of Type A standard uncertainty that they

represent, and a table of values obtained for these components for five of the SRM levels. It also gives a summary of the Type A standard uncertainty values for wafer-center averages and for individual values on the 5 mm and 10 mm radius circles for both resistivity and sheet resistance. The nominal resistivity for each of the SRMs is given in Table 1 of Appendix 1

.

Table 6. Components Identified in Statistical Analyses of Certification and Control Experiment Data The general form for the Type A standard uncertainty for the certification of these resistivity SRMs

'

tt

t = (s\l n +j,

2 + 5

r

2 + 5c2 + 5a2 + ^+ a

2 /3+b2 /3+c2 /3)'

/2

where n = 1 for individual measurements on the circles and n = 6 for the average value at the center, and

se Short-term imprecision of certification probe

ss Run-to-run measurement variability s Y Longer-term measurement variability se Uncertainty of non-zero correction for bias of certification probe

sa Uncertainty of non-zero correction for probing induced drift (wafer-probe interaction) scf,g Uncertainty of non-zero correction for probe-wiring configuration difference a/V3 Uncertainty of correction for probe wiring configuration (where the best correction = 0) [Type A estimate, but based on limit of error] b/^3 Uncertainty of correction for bias of certification probe (where the best correction = 0) [Type A estimate, but based on limit of error] c/V3 Uncertainty of correction for effect of illumination level (where the best correction = 0) [Type A estimate, but based on limit of error].

39
Table 7. Values of the Components Identified in Statistical Analyses for the various SRMs

in Appendices 2 through 8, mH cm* SRM sE s8 Sy Sc Sa Scfig a//3 b//3 c//3 2541 0.001 83 0.001 04 0.004 00 0 0 0 0 0.000 47 0 2542 0.062 0.032 0.004 0 0.011 0 0 0.016 0 2543 0.714 0.192 0.154 0 0 0.058 0 0.038 1.682 2544 4.662 1.198 5.646 0 0 0.287 0 0.204 0 2545 14.14 3.31 3.01 0 0 0 2.89 0 0 2546 72.0 13.4 14.6 5.1 0 0 0 0 0 2547 138. 64. 129. 5. 10. 0 0 0 0

*For ease of reading, this table is expressed in terms of milliohm centimeters.

Table 8. Type A Standard Uncertainty Values, u„ Taken from Reports in Appendices 2 to 8 SHEET SHEET

RESISTIVITY RESISTIVITY RESISTANCE RESISTANCE

SRM at center on circles at center on circles (mQ cm) (mQ cm) (mQ) (mQ)

2541 ±0.004 23 ±0.0045 ±0.067 3 ±0.0716 2542 ±0.045 ±0.072 ±0.725 ±1.16 2543 ±1.72 ±1.85 ±27.5 ±29.6 2544 ±6.09 ±7.43 ±96.9 ±118.3 2545 ±7.8 ±15.1 ±125. ±241. 2546 ±35.8 ±74.8 ±570. ±1190. 2547 ±155. ±199. ±2470. ±3180.

7.2 Type A and Type B Variance Terms

In this section, a table of Type A variance terms is constructed from the squares of the Type A standard

uncertainty values given in Table 8. A table of Type B variance values, obtained from the analyses in Section 5, is also given. Finally, a table of the combined variance, u

2

c , obtained by adding the Type A and Type B variances, is then given.

40
Table 9. Type A Variance Values, u

2„ Obtained by Squaring the Entries in Table 8 SHEET SHEET

RESISTIVITY RESISTIVITY RESISTANCE RESISTANCE oKM at center on circles at center on circles (Q-cm)2 (Q-cm)2 (Q2

) (Q2) 2541 1.79 x 10-" 2.02 x lO'

11 4.53 x lO'9 5.13 > 10"9 2542 2.02 x lCr

9 5.18 x 10"9 5.26 x 10"7 1.35 x 10"6 2543 2.96 x 10-6 3.42 x 10"6 7.57 x \04

8.76 x lO"

4 2544 3.71 x 10"3 5.52 x lO'

3 9.40 x 10"3 1.40 x 10"2 2545 6.08 x 10-5 2.28 x W4

1.56 x 10"2 5.81 x 10"2 2546 1.28 x 10

3 5.59 x 10"3 3.25 x 10"1 1.42 2547 2.40 x 10"2 3 .96 x 10'2 6.10 10.1

Table 10. Type B Variance Values, u

2h

Calculated from Summation of Terms in Section 5 .2 RESISTIVITY SHEET RESISTANCE

SRM at center & on circles at center & on circles (Q-cm) 2 (Q

2

) 2541 2.51 x lO"" 5.68 > 10"9 2542 8.68 xlO'10 1.51 x 10

7 2543 1.84 x lO'

7 3.96 x 10"3 2544 2.32 xlO'5 5.18 x 10

3 2545 1.47 x 10^* 3.30 x 10"2 2546 2.36 x 10"3 5.28 x 10"1 2547 9.43 xlO'3 2.11

Table 11. Combined Variance Values, u

2

c, from Addition of Terms in Tables 9 and 10 SHEET SHEET

RESISTIVITY RESISTIVITY RESISTANCE RESISTANCE SRM at center on circles at center on circles (Q-cm) 2 (Q-cm) 2 (Q

2) (Q

2

) 2541 4.30 x 10

-"

4.53 x

10'" 1.02 x lO"

8 1.08 x lO'

8 2542 2.89 x lO"

9 6.05 x lO'9 6.77 x 10"7 1.50 x lO

-6 2543 3 .14 x 10"6 3.61 x 10

-6 7.97 x 10^ 9.16 x W4 2544 6.03 x 10"3 7.84 x lO'

3 1.46 x 10'2 1.92 x 10'2 2545 2.08 x IQi4 3.75 x 10"4 4.86 x 10

2 9.11 > 10

-2 2546 3.64 x 10"3 7.95 x lO'

3 8.53 x 10

_1 1.95 2547 3.35 x lO'

2 4.90 x 10"2 8.21 12.2

41
7.3 Standard Uncertainty and Expanded Uncertainty

This section gives values of the combined standard uncertainty, ua and the expanded uncertainty, U, based on a coverage factor k = 2 where U= k uc. The combined standard uncertainty values are the square roots of

the entries in Table 1 1 for the combined variance.

Table 12. Combined Standard Uncertainty Values, uc SHEET SHEET

RESISTIVITY RESISTIVITY RESISTANCE RESISTANCE

SRM at center on circles at center on circles (Q-cm) (Q-cm) (H) (Q) 2541 ±0.000 006 56 ±0.000 006 73 ±0.000 101 ±0.000 104 2542 ±0.000 0538 ±0.000 0778 ±0.000 823 ±0.001 22 2543 ±0.001 77 ±0.001 90 ±0.0282 ±0.0303 2544 ±0.007 76 ±0.008 85 ±0.1208 ±0.1385 2545 ±0.0144 ±0.0194 ±0.220 ±0.302 2546 ±0.0603 ±0.0892 ±0.924 ±1.39 2547 ±0.183 ±0.221 ±2.86 ±3.49

Table 13. Expanded Uncertainty Values, U (Coverage Factor k = 2)

SRM

RESISTIVITY

at center (Q-cm)

RESISTIVITY

on circles (Q-cm)

SHEET

RESISTANCE

at center (Q)

SHEET

RESISTANCE

on circles (H) 2541 2542

2543 2544

2545 2546

254711

0.000 013 1 0.000 108

0.003 54

0.015 53

0.0288

0.121

-0.498, +0.366

0.000 013 5 0.000 156

0.003 80

0.017 70

0.0387

0.178

-0.575, +0.443

0.000 202

0.001 65

0.0564

0.241

0.441

1.85

-7.83, +5.73

0.000 208

0.002 45

0.0605

0.277 0.604

2.78

-9.08, +6.98

* Asymmetry is due to a contribution of 1.32 Q-cm, 2. 10 Q, from wafer-probe interaction, i.e., to a drift in value.

42
7.4 Corrections Applied to Measured Values

This section summarizes bias corrections that must be made to measurement results, as acquired, because of

effects that were identified during statistical analyses of the SRM certification experiment data. These

corrections are explained in the appropriate appendices. They are given for resistivity and sheet resistance

values. All resistivity and sheet resistance values shown on the SRM certificates have already been

corrected for these bias terms.

SRM

Table 14. Probe Bias Corrections Applied to Measured Values BIAS CORRECTION SOURCE

2541 2542

2543 2544

2545 2546

2547

Subtract 0.000 000 472 Q-cm (0.000 007 52 Q)

[This is a Negligible Amount]

Subtract 0.000 037 5 Q-cm

Subtract 0.000 131 Q-cm

Subtract 0.0011 Q-cm

None Add 0.0393 Q-cm

Subtract 0.0490 Q-cm

(0.000 597 Q)

(0.002 10 Q)

(0.017 Q)

(0.626 Q)

(0.78 Q)

Wiring Configuration Bias

Wiring Configuration Bias Wiring Configuration Bias Wiring Configuration Bias

Probe Bias Probe Bias

[wafer drift term of 0. 132 Q-cm, 2. 10 Q is built into expanded uncertainty]

7.5 Estimated Degrees of Freedom for Uncertainty Values of the SRMs

Table 15 summarizes the estimated degrees of freedom for the stated uncertainty values of each of the SRMs. These estimates are made using the Welch-Satterthwaite formula [5]. They can be used to estimate

the confidence interval covered by the expanded uncertainties with coverage factor k= 2. The wide variation

in the degrees of freedom listed arises from the difference in the number of degrees of freedom for the dominant contribution to the uncertainty of each of the SRMs.

Table 15. Estimate of Degrees of Freedom from the Welch-Satterthwaite Formula

Center Average Individual Measurement SRM Degrees of Freedom Degrees of Freedom

2541 6 8 2542 88 486

2543 16800 15415 2544 6 14 2545 52 528 2546 41 543 2547 26 66

43
8 CONCLUSION

When the certification procedure for the SRMs was being developed, the quantitative design

objective was to support the goal for layer resistivity stated in the SEMATECH Mega-IC

Workshop, i.e., measurements with a 1% accuracy and a 0.5% repeatability. NIST and ISO

practice is to state the total uncertainty of measurement values, rather than to state accuracy and

precision values separately. However, values of expanded uncertainty, listed in Table 13, can be

used to give a reasonable assessment of how well the design objectives for this SRM series were

met. Values of expanded uncertainty for resistivity averages at the center of an SRM wafer range from 0. 1 1% to 0.25%, of the nominal resistivity for all SRMs except 2543 at 1 Q cm; for this SRM the expanded uncertainty is 0.35%. The relative uncertainty of individual resistivity values on the two circles increases slightly for several resistivity levels where probe imprecision was one

of the larger contributors to uncertainty, but it is still less than 0.2% for five of the SRM levels and is 0.38% at 1 Q cm and 0.28% at 200 Cl em. Expanded uncertainties for sheet resistance values are slightly smaller since there is no uncertainty due to thickness scale in the sheet

resistance values. Thus, these SRMs should serve quite well to support the original design goals.

9. ACKNOWLEDGMENTS

It is a happy privilege to acknowledge the dedication and persistence of Donnie Ricks throughout

all the preliminary testing, the certification, and follow-up testing that was necessary for the

certification of these SRMs. Without her conscientious attention to detail, the uncertainty levels noted in the report could not have been achieved. Thanks also to Mike Thomas for his able assistance throughout the certification. A special thank you goes to Jane Walters for her diligent and meticulous work in preparing this report from a number of disparate-format pieces of source

material. REFERENCES

[ 1 ] ASTM Method F84-93, "Standard Method for Measuring Resistivity of Silicon Wafers

with an In-Line Four-Point Probe, " Annual Book of ASTM Standards Vol. 10.05, West

Conshohocken, PA 19428.

[2] Recommendations for Specific Standards Actions, Proceedings of the SEMATECH

Workshop on Silicon Materials for Mega-IC Applications, New Orleans, Louisiana, January 30-31, 1991, p.9.

[3] van der Pauw, L.J., A Method of Measuring Specific Resistivity and Hall Effect of Discs of Arbitrary Shape, Phil. Res. Rep. 13, 1-9 (1958).

[4] Guide to the Expression of Uncertainty in Measurement, ISBN 92-67- 1 0 1 88-9, 1 st Ed. ISO, Geneva, Switzerland, (1993).

44
Taylor, B.N. and Kuyatt, C. E., Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, NIST Technical Note 1297, U.S. Government Printing Office, Washington DC (1994). ASTM Method F 1529-96, "Standard Method for Sheet Resistance Uniformity

Evaluation by In-Line Four-Point Probe with the Dual-Configuration Procedure," Annual Book of ASTM Standards Vol. 10.05, West Conshohocken, PA 19428.

Harris, F. K., Electrical Measurements (John Wiley & Sons, New York, 1952), p. 431.

Perloff, D. S., Gan, J. N. and Wahl, F. E., Dose Accuracy and Doping Uniformity of

Ion Implant Equipment, Solid State Technol. 24 (2), 112-120 (1981).

Perloff, D. S., Four-Point Probe Correction Factors for Use in Measuring Large Diameter Doped Semiconductor Wafers, J. Electrochem Soc. 123 (11), 1745-1750

(1977).

45
Appendix 1 . Summary of Important SRM Wafer Material and Measurement Condition Parameters

This appendix consists of two tables that summarize important useful silicon wafer

characteristics and electrical measurement conditions that apply to the various resistivity levels of SRMs 2541 to 2547. Table 1 lists the nominal resistivity, crystallographic orientation of

the SRM wafer surfaces, crystal growth type, dopant species, and the commercial supplier for the wafers for each of the SRMs. Table 2 lists the four-point probe identification, the serial number and nominal resistance of the standard resistor used, as well as the nominal value of

the measurement current used for certification of these same SRMs.

Table 1 . Silicon Wafer Characteristics That Apply to Various Resistivity Levels of the SRMs 2541 to 2547 SRM SRM level Crystal Orient/Growth/Dopant Supplier

(in Q-cm)

2541 0.01 91905 (100) Cz Boron Recticon Corp. 2542 0.1 91904 (100) Cz Boron Recticon Corp. 2543 1 91907 (100) Cz Boron Recticon Corp. 2544 10 29473 (111) FZ-NTD Phos. Wacker Siltronic 2545 25 21565 (Ill) FZ-NTD Phos. Topsil Semi. A/S

2546 100 51939 (Ill) FZ-NTD Phos. Topsil Semi. A/S

2547 200 21566 (Ill) FZ-NTD Phos. Topsil Semi. A/S

Table 2. Electrical Measurement Conditions That Apply to Various Resistivity Levels of the SRMs 2541 to 2547

Standard Resistor Nominal

SRM Probe Nominal Value/ Measurement

Serial Number Current 2541 283 0.1 Q / 1771494 100 mA

2542 281 1 Q / 1594503 28 mA

2543 283 10 Q / 1593079 2.8 mA

2544 283 100 Q. 1 1598893 260 uA

2545 2062 100 Q / 1598893 110 uA

2546 2362 1000 Q 1 1592167 29 uA

2547 SRM1 1000 Q / 1592167 14 uA

46
Appendix 2. Analysis of Certification Data and Control Experiments for SRM 2547

1 . GENERAL COMMENTS AND SUMMARY OF TYPE A STANDARD

UNCERTAINTY COMPONENTS

1.1 Introduction

This appendix documents the statistical analysis leading to the certification of wafers from

crystal 21566 for SRM 2547 and outlines a general procedure for analysis of other SRMs in the series 2541 through 2547. The results of the analyses of the remaining SRMs are

briefly summarized in the following appendices. In addition to the three random

components and the first three systematic components listed below which are common to

all SRMs, this report also treats a small drift effect that was not found with any of the other SRMs.

The 137 wafers in this issue have nominal resistivities of 200 Qcm, and the wafers are assumed to be identical with regard to wafer face. Certification measurements are made

with a single probe, identified as SRM1. Data consist of measurements at six locations on

each of three circles located at 0 mm, 5 mm, and 10 mm from the center of each wafer, with

the wafer face chosen at random with respect to the crystal growth direction. Sources of

error which could contribute to the uncertainties of the certified values and which are examined in this appendix are: probe imprecision, run-to-run variability, long-term

variability, differences between wiring configurations, differences between wafer faces, differences among probes (probe SRM1 bias), and wafer drift with probing, which was an

unanticipated effect.

Only the standard deviation associated with probe spacing and electronic imprecision can

be estimated from the certification data for the SRM wafers. A series of control experiments was carried out to identify and estimate error components which cannot be

addressed by the certification measurements.

Measurements on a check-standard, chosen at random from the wafers in the issue, were made routinely during the certification procedure to: identify any anomalous behavior, document the stability of the process, and estimate a day-to-day component of measurement error. For this issue, the check standard is wafer #150; it was measured only with the certification probe SRM1.

Pre- and post-certification control experiments with five probes on five wafers with both

second-configurations, bl and b2, were repeated on 6 days. These measurements are

47
intended to estimate both the random and systematic components of the measurement

process. The next section summarizes the Type A standard uncertainty for SRM 2547. It also gives a statement of how the uncorrected term due to wafer drift contributes to the expanded uncertainty. Tables 1 and 2 in Section 1 .2 give an executive summary of the terms that contribute to the Type A standard uncertainty. The details of the calculation of

the component terms are given in subsequent sections.

1 .2 Certified Resistivities and Uncertainties

The average of six measurements on the 0 mm circle of each wafer, corrected for the

effect of probe SRM1, is reported as the certified resistivity value. The Type A standard

uncertainty associated with the certified value for the wafer center is

= 0.155 Q-cm .

The expanded uncertainty (coverage factor k = 2) allows for an uncorrected systematic

error of -A. See Section 3.3 for details. Because the uncorrected systematic error is always in one direction, the expanded uncertainty interval is nonsymmetric and is expressed as Certified value - (2 «/ + A), Certified value + 2 u\

,

where 2 u\ = 0.310 Q-cm, and A = 0.132 Q-cm .

Individual measurements on the 5 mm and 10 mm circles for each wafer, corrected for the effect of probe SRM1, are reported as certified values on the certificates. The Type A

standard uncertainty associated with each of these individual certified values is

«i =

[Sc + S * + 56

2 + + *aT

2 = 0- 20 &'Cm

The expanded uncertainty interval for individual measurements is then expressed by

Certified value - (2 u\ + A), Certified value + 2 u\

t where 2 u\ for individual measurements = 0.40 Q-cm.

48
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50
2. RANDOM COMPONENTS

2.1 Pre- and Post-Certification Control Experiments

A nested experiment was performed with five probes on five wafers. Six measurements

were made at the center position of each wafer with each probe; this sequence was

repeated on 6 days; and the entire experiment was conducted twice, i.e., prior to and at the conclusion of the certification experiment. The temporal error model for one probe and one wafer is Yijk^ + Yi +§ij i = 1,2; j = 1, 6; k= 1, ... ,6 (1)

where [J is the average value, Yj is a component for long-term error; 6^ is a component of

run-to-run measurement error; and eijk represents short-term measurement imprecision

error associated with the probe and electronics.

For this SRM, the pre- and post-certification measurements were made on opposite faces of the same wafers. Thus, there is a question as to whether the differences (see Fig. 1) between the pre- and post-certification measurements are caused by: (1) biases between

faces; (2) drift on the wafer surfaces; or (3) long-term error in the measurement process.

1) Because the faces for the pre-certification experiment were chosen at random,

it is unlikely that the differences, which are consistently in one direction, are caused by a front-to-back bias on the wafers. See Figure 1 where resistivity measurements on the five wafers are plotted versus the month/day of measurement. Also, measurements made 2 to 3 months after the conclusion of the certification process on additional wafers called #901, #902, #903, and #904 show differences which are consistently in the opposite direction.

2) There are not sufficient data from these experiments to judge inherent wafer

drift.

3) The behavior of the pre- and post-certification data, which show strong

correlations across wafers with time, is consistent with a components of

variance model such as eq (1).

Sources of error and root-mean-square error terms (RMSE) for this model are in Table 2. For analysis of the initial and final control experiments, the first day's measurements were

omitted. Estimates are made for each wafer individually and then pooled over wafers. The last column of the table shows the relationships between the results of the various experiments and the terms in the temporal error model above.

51
Crystal 21566

Wafer 20 Wafer 40

192.8-

Is0

•

192.7" a

192.6"

>i

4J •H>

192.5"

rl

+> 192.4'

ra H

CO0

192.3"

fas 192.2

Month/Date

Wafer 60

Month/Date

Wafer 100

V

CJ 193.9

193.8'

193.7'

193.&

193 o

• 192.9"

>i 192.

8

- -H>H 192.7"

•H

ra 192.6"

192.5

7.5 8 8.5 Month/Date

Wafer 80

6 6.5 7 7.5 8 Month/Date8.5

6.5 7 7.5 8 8.5 9 Month/Date

Figure 1. Resistivity (Q cm) on five control-wafers from crystal 21566 with probe

SRM1 plotted versus the month/day of measurement, showing change between

and within pre- and post-certification experiments.

52
2.1.1 Precision of probes The standard deviation, se , is directly computed from six measurements at the center and

estimates the precision for each probe. These standard deviations are shown in Table 3; the pooled values are also shown in Table 2.

Table 3. Within-Run Standard Deviations, se, Pooled over Five Wafers

and Six Runs, Qcm

Probe

Std. dev, se Config bl DF

Std. dev, se Config b2 DF

SRM1

281

Pre 283 2062 2362

0.1586

0.2235 0.2139

0.1645 0.1520

150

150

150 150

150

0.1907

0.2468 0.2389

0.2043 0.1635

150

150

150 150

150 SRM1

281

Post 283 2062 2362

0.1134

0.2102

0.1687

0.1568 0.1269

150

150

150

150 150

0.1280

0.2217

0.2115 0.1770

0.1374

150

150

150

150

150

Pooled

Value: SRM1 0.1379 300 0.1624 300

2. 1 .2 Run-to-run measurement variability from pre- and post-certification control experiments

Standard deviations and averages computed from the six repetitions with each probe on

each wafer are shown in Table 4. For this purpose, the first run with each probe on each wafer has been discarded. Each standard deviation is then estimated with four degrees of

freedom. The pooled standard deviation for SRM1 of 0.089 20 Qcm with 40 degrees of freedom incorporates both probe imprecision and day-to-day measurement error as shown

in the relationship column of Table 2.

53
Table 4. Run-to-Run Component of Error, Crystal 21566

Averages and Standard Deviations for Last Five Runs

on Each Control-Wafer, Q-cm

Wafer# Probe

Pre-certification Average Std dev

Post-certification Average Std dev

20 SRM1 196 .2431 0 .0875 196 .0078 0 .1574

40 SRM1 193 . 8663 o . 0615 193 .7433 o . 0605

60 SRMl 193 . 5259 o . 0795 193 .4869 o . 0693

80 SRM1 192 . 8096 o . 1127 192 . 6597 o . 0777

100 SRMl 192 . 5503 o . 0835 192 . 3768 o . 0596

20 281 196 .2443 0 . 1380 196 . 0423 0 .2617

40 281 193 . 8445 0 . 1105 193 . 7325 o .0948

60 281 193 . 5903 0 . 0935 193 . 4285 o . 1399

80 281 192 . 7595 0 . 0826 192 . 6768 o .1939 100 281 192 . 5428 0 , . 1189 192 .3991 0 . 0930

20 283 196 . 1670 0.. 0937 195 . 9598 0 . 1525

40 283 193 .7223 0 , . 0499 193 .6426 o .0951

60 283 193 .4253 0 . ,0536 193 .3253 0 . 0992

80 283 192 .7630 0 , 0396 192 .5120 0 .0945 100 283 192 .4705 0 . ,0545 192 .3259 0 .0824

20 2062 196 . 1481 0 . 1042 195 , 9211 0 . ,2248

4 0 2 062 193 .8217 0 0957 193 74 94 0 0711

60 2062 193 , . 4647 0 . 0723 193 , , 4411 0 . 0355

80 2062 192 , , 7436 0 . 0727 192 . , 6205 0. 1538

100 2062 192 . ,4263 0 . 0412 192 . , 3818 0 . 0644

20 2362 196 . , 1432 0 . 0884 195 . 8630 0. 2282

40 2362 193 . , 7696 0 . 0681 193 . 7181 0 . 0667

60 2362 193 . .4426 0. 0581 193 . 3722 0 . 0775

80 2362 192 . 7206 0. 0920 192 . 5816 0. 0810

100 2362 192 . 4557 0 . 1279 192 . 2694 0. 1589

Pooled Value

SRMl 0 . 085 73 0 . 092 5

54
2. 1 .3 Long-term measurement variability from the control-wafers Averages for each wafer from the pre- and post-certification experiments are shown in Table 5. The differences are assumed to be the result of a long-term component of measurement error. The standard deviations as estimated from the pre- and postcertification averages represent probe imprecision, day-to-day error, and long-term measurement error as shown in the relationships column of Table 2.

Table 5. Pre- and Post-Certification Averages with Probe SRMl, Q-cm

Wafer# Pre-Certification Post-Certification Difference Std dev DF

20 196.2431 196.0078 -0.2353 0.1664 40 193.8663 193.7433 -0.1230 0.0870 60 193.5259 193.4869 -0.0390 0.0276 80 192.8096 192.6597 -0.1499 0.1060 100 192.5503 192.3768 -0.1735 0.1227

Pooled

Value: . 0.1116 5

2.1.4 Long-term measurement error from #900 series wafers Averages of six center measurements made 2 to 3 months after the certification procedure

are shown in Table 6. The measurements were made on a random selection of additional wafers numbered #901, #902, #903, and #904. The differences are assumed to be the

result of a long-term component of measurement error. The standard deviations as estimated from the September and October averages represent probe imprecision, day-to- day error, and long-term measurement error as shown in the relationships column of

Table 2. The fact that the differences shown in Table 5 for the control-wafers are always

negative, whereas the differences observed for the #900 series of wafers are nearly always

positive, is taken to indicate that this is not an inherent systematic effect. Therefore, no

systematic correction term is applied.

55
Table 6. Long-Term Changes in Measurement Process with SRMl, Q-cm

Wafer Face September October Difference Std dev DF

901 1 192.748 192.839 +0.091 0.0643 902 1 195.654 195.593 -0.061 0.0431 903 1 200.866 201.096 +0.230 0.1626 904 1 190.982 191.381 +0.399 0.2821

901 2 192.704 192.974 +0.270 0.1909 902 2 195.478 195.533 +0.055 0.0389 903 2 200.955 201.098 +0.143 0.1011 904 2 191.198 191.367 +0.169 0.1195

Pooled

Value: 0.1477

2.2 Check-Standard Measurements

Twenty-three measurements (averages of six center measurements each) with probe

SRMl on wafer #150 were made over the course of the certification experiment. The

initial drop in resistivity after the first day, which can be seen in Figure 2, is assumed to be the result of wafer-probing damage. The first day's measurements are omitted from

the analysis. The slope of a straight line fit to the remaining 21 measurements as a

function of time is not significant, indicating that the measurement process is not drifting. Therefore, only run-to-run variations in the measurement process and probe imprecision

are reflected in the standard deviation which is shown in Table 2.

56
Crystal 21566

Date In August

Figure 2. Resistivity measurements (Q'cm) on check-wafer #150, with probe SRM1 as function of time (date in August)

57
3. SYSTEMATIC COMPONENTS

3.1 Systematic Differences between Probe-Wiring Configurations bl and b2

In the pre- and post-certification experiments, six measurements at the center with the probe in configuration bl were immediately followed by six measurements with the probe in configuration b2. The differences between configurations bl and b2 for the pre- and post-certification measurements are shown in Figure 3. Averages and standard

deviations for each probe over 6 days and five wafers are shown in Table 7. The

t-statistic,

shows no evidence of a significant difference between configurations bl and b2 for the

pre-certification measurements and some evidence of a difference for the postcertification measurements. These differences for the post-certification measurements

appear to be caused by the measurements on the first two wafers. No uncertainty from

this source is assigned.

t = y[30 Average/Std dev

,

Table 7. Average Differences over All Control-Wafers between Configurations bl and b2 for Probe SRM1, Q-cm

Probe

Pre-certification Average Std dev DFPost-certification Average Std dev DF

SRM1 -0.002 44 0.044 49 29 0.020 56 0.031 38 29

58
Crystal 21566

0 . 2

s

a

g 0.05-

to 0*0

0 o

0

* 0

o 0 0 o

0

0 0.1

1 0

"1 '

1 1 1 1—

15 20 25 30 Measurement Number for 5 Wafers over 6 Days Each

Figure 3. Differences between wiring configurations bl and b2. Five wafers in random

order over each of 6 days, with probe SRM1, Q-cm.

Legend: O = pre-certification; * = post-certification

59
3.2 Differences among probes

The probes in the SRM certification are assumed to be a random sample of similar probes. However, certification using a single probe can have a systematic effect on the measurements. For this SRM, the measurements with SRM1 are found to be high

relative to measurements with the other probes. Figures 4 and 5 show differences from

the mean for each wafer plotted by probe. The systematic nature of these differences argues that the measurements made with SRM1 (identified by the number 1 in the plots) should be corrected to the average of the five probes based on the pre- and postcertification control measurements.

The estimated correction is calculated as the average of the differences in the table below

to be C = -0.049 Qcm. The standard deviation of the differences is divided by y[TQ to obtain the standard deviation of the correction, sc = 0.0050 Qcm. The correction, C , is applied to all certified resistivity values, and its standard deviation is taken as a Type A

component of uncertainty.

Table 8. Differences between Multi-Probe Average and Probe SRM1 for Each of the Control-Wafers, Q cm

Wafer Pre-certification Post-certification 2040

60

80

100

-0.054

-0.061 -0.036

-0.050

-0.061

-0.049

-0.026

-0.076

-0.050

-0.026 Mean difference = -0.049 Q cm

Standard deviation, s = 0.0159 Q cm

Correction to be applied = -0.049 Q cm

Standard deviation of correction, s/vTO = 0.005 Q cm

60
Crystal 21566

0 . 2

0.15

S o.igg

0 . 05

-0.05 -

-

1 1 I

1 2

2 1

4 I

3 4

4 3

5 5

5 5

3 4 3

|

. | . | . | . | . 2 0 6 0 8 0 Control Wafer Number

Figure 4. Differences between individual probe responses and multiprobe average, for pre-certification measurements on each of the control-wafers. Plot symbol code: 1 = SRM1; 2 = 281; 3 = 283; 4 = 2062; 5 = 2362

61
Crystal 21566

o . 1

0.05

-0.05

0 . 1

b n r

20 4 0 6 0 8 0 Control Wafer Number

10 0 12 0

Figure 5. Differences between individual probe responses and multiprobe average, for

post-certification measurements on each of the control-wafers.

Plot symbols: 1 = SRM1; 2 = 281; 3 = 283; 4 = 2062; 5 = 2362

3.3 Initial wafer damage

There is evidence from previous as well as present experiments that initial probing may

change the surface characteristics of the 200 Qcm wafers. The phenomenon is not totally understood nor always consistent, but has displayed itself as an initial drop in resistivity. The resistivity on check-wafer #150 dropped 0.3 Q-cm after the first day's measurements

and then leveled off. For the pre- and post-certification measurements with SRM1, the

resistivities always dropped after the first measurement for probe SRJVI1 (see Fig. 1). The

62
average drop is A = 0.132 Q-cm, and the standard deviation is s = 0.0427 Qcm. A

correction of - A/2 would assume an equal probability of initial damage between 0 and

-A Qcm. However, we choose to apply a correction for this asymmetry not to the data, but

rather to the calculation of the uncertainty in Section 1 .2. The term A is added to the lower

limit of the expanded uncertainty. The standard deviation associated with A is s/V20 Q-cm

or 0.010 Q-cm, and is treated as a Type A component of uncertainty in the analysis.

Table 9. Drop in Resistivity between First and Second

Measurements of Control-Wafers with Probe SRM1, Q cm

Wafer Pre-certification Post-certification 20 0.086 0.185 40 0.122 0.072 60 0.174 0.122 80 0.177 0.093 100 0.113 0.176

The resistivity dropped after the first measurement (a measurement is the average of six readings at the wafer center) for all five control-wafers in both the pre- and postcertification experiments with probe SRM1, i.e., ten times out of ten possibilities. However, for the other probes, the number of times there was a drop after the first measurement in the same experiments was as follows: 2062 - seven out of ten

possibilities; 281 - four out of ten; 2362 - four out of ten; and 283 - six out of ten

possibilities. Thus, the effect is stronger for probe SRJV11 than for any of the other

probes. Plots of the complete data from the pre- and post-certification experiments are given in Figures 6 and 7.

63
Wafer 20

90 9 5 100 105 110 115 12 0 Wafer 100

12 0 125 130 135 140 145 150

Cumulative Measurement Number

Figure 6. Resistivity (Q-cm) from pre-certification measurements for five control-wafers from crystal 21566 vs. cumulative measurement run number.

Plot symbol code: 281 = 0; 2062 = 1; 2362 = 2; 283 = +, SRM1 = *

64
Wafer 20

12 0

12 0 125 13 0 135 140 145 150

Cumulative Measurement Number

Figure 7. Resistivity (Q-cm) from post-certification measurements for five control- wafers from crystal 21566 vs. cumulative measurement run number.

Plot symbol code: 281 = 0; 2062 = 1; 2362 = 2; 283 = +, SRM1 = *

Reference:

Graybill, F. A., An Introduction to Linear Statistical Models, Vol. 1 (McGraw-Hill, New

York, 1961), pp. 349-351.

65
Appendix 3. Analysis of Certification Data and Control Experiments

for SRM 2541

1 . GENERAL COMMENTS AND SUMMARY OF TYPE A STANDARD

UNCERTAINTY COMPONENTS

1.1 Introduction

This appendix documents the statistical analysis leading to the certification of wafers from

crystal 91905 for SRM 2541. The 130 wafers in this issue have nominal resistivities of

0.01 Q cm, and the wafers are assumed to be identical with regard to face. For this issue, the pre- and post-certification measurements were made on opposite faces of each wafer.

Certification measurements were made with probe 283.

This appendix includes a summary of the temporal and other components of uncertainty in Section 1 .2, and details of the analysis for a systematic bias for probe 283 in Section 2.1. Such a probe-bias calculation was not illustrated in Appendix 2. The analyses of all other

effects follow the procedures detailed in Appendix 2. The details are not included here.

1 .2 Certified Resistivities and Uncertainties

The averages of six measurements on the 0 mm circle, and individual measurements on the

5 mm and 1 0 mm circles of each wafer are reported as certified values. No correction is applied for probe or wiring effects.

Only Type A uncertainty evaluation procedures are treated in this appendix, and estimates of

all uncertainty components are shown in Table 1 . The Type A standard uncertainty for the average resistivity at the wafer center is

ui = y-j- + s

y

2 + s2& +-s] = 0.000 004 2 Qcm (0.0042 mQ-cm)

.

66
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The Type A standard uncertainty for individual resistivity values on the 5 mm and 1 0 mm

radius circles is —+ s* + + s; = 0.000 004 5 Q-cm (0.004 5 mO-cm)

.

2. SYSTEMATIC EFFECTS

2.1 Bias Effect of Probe 283

There is a small systematic bias for this probe (relative to the average over all probes); the average bias is -0.000 000 68 Q-cm with a standard deviation of the average of

0.000 000 21 Q-cm. This bias can be seen in the measurements on the control-wafers, but does not affect the values of the SRMs which are only reported to six places beyond the decimal point. Therefore, the correction is taken to be zero. A conservative assumption

is that during the certification the bias could fall somewhere within the limits ±b where

b = 0.000 000 82 Q-cm, and a contribution of b/V"3 = 0.000 000 47 Q-cm is included

as a systematic component of the Type A standard uncertainty.

Table 3. Bias of Probe 283 Relative to the Average of All Probes, Q-cm

Wafer Pre-certification Post-certification 24344

53 144

-0.000 000 16 -0.000 000 58 0.000 000 48

-0.000 001 44

-0.000 000 94

-0.000 000 32

-0.000 000 58

-0.000 001 10

-0.000 000 86

-0.000 000 26 Mean -0.000 000 53 -0.000 000 82

69
Appendix 4. Analysis of Certification Data and Control Experiments

for SRM 2542

1 . GENERAL COMMENTS AND SUMMARY OF TYPE A STANDARD

UNCERTAINTY COMPONENTS

1.1 Introduction

This appendix documents the statistical analysis leading to the certification of wafers from

crystal 91904 for SRM 2542. The 129 wafers in this issue have nominal resistivities of

0.1 Q- cm, and the wafers are assumed to be identical with regard to face. For this issue, the pre- and post-certification measurements were made on opposite faces of each wafer.

Certification measurements were made with probe 281.

This appendix includes a summary of the temporal and other components of uncertainty in Section 1 .2, as well as the details of analysis of a term due to differences in probe-wiring

configurations in Section 2.1. Such a probe-wiring calculation was not illustrated in Appendices 2 or 3. All other analyses follow procedures detailed in preceding appendices.

1 .2 Certified Resistivities and Uncertainties

The averages of six measurements on the 0 mm circle, and individual measurements on the

5 mm and 10 mm circles of each wafer, corrected for wiring-configuration bias, are reported

as certified values. No correction is applied for probe effect.

Only Type A uncertainty evaluation procedures are treated in this appendix, and estimates of

all uncertainty components are shown in Table 1 . The Type A standard uncertainty for the average resistivity at the wafer center is

The Type A standard uncertainty for the individual resistivity values on the 5 mm and

1 0 mm radius circles is

u\ = J— + s]

flg + + si + -si = 0.000 045 Q-cm .

70
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72
2. SYSTEMATIC EFFECTS

2.1 Differences between Wiring Configurations bl and b2

Differences are found between measurements in configurations bl and b2. Averages and

standard deviations (for the first four days of measurements on each wafer) are shown in Table 3. Rounds 5 and 6 of the pre-certification measurements were found to have been

adversely affected by a faulty power supply that was discovered and repaired shortly after the start of wafer certification. Rounds 5 and 6 are omitted from the analysis of the probe-wiring effect for both pre- and post-certification control-wafer data. The t-statistic for testing for a significant difference between wiring configurations bl and b2is

t = -sfZO Avg/SD. The values of the t-statistic suggest a slight difference between wiring

configurations for this SRM. The average difference between the pre- and postcertification measurements is 0.000 075 Q*cm. A correction of minus one-half this difference, or -0.000 037 5 Q'cm, is applied to all certification measurements to obtain an

average over the two configurations. The standard deviation of the correction,

Srf. = ~ — A2 W = 0.000 011 Q-cm , Cfi8 2

V

where s x is the standard deviation from the pre-certification and s2 is the standard

deviation from post-certification measurement, is taken as a component of the Type A

standard uncertainty for the process.

Table 3. Average Differences and Standard Deviations between Wiring Configurations bl and b2, Q-cm

Pre-certification Post-certification Probe Avg SD (s : ) DF t Avg SD (s2 ) DF t 281 0.000 085 0.000 064 19 5.9 0.000 065 0.000 072 19 4.0

73
Appendix 5. Analysis of Certification Data and Control Experiments

for SRM 2545

1 . GENERAL COMMENTS AND SUMMARY OF TYPE A STANDARD

UNCERTAINTY COMPONENTS

1.1 Introduction

This appendix documents the statistical analysis leading to the certification of wafers from

crystal 21565, SRM 2545. The 133 wafers in this issue have nominal resistivities of

25 Q-cm, and the wafers are assumed to be identical with regard to wafer face. For this SRM, the pre- and post-certification measurements were made on opposite faces of each

wafer. Certification measurements were made with probe 2062.

This appendix includes a summary of the temporal and other components of uncertainty in Section 1 .2, as well as details of an analysis for wiring-configuration differences of a form

not contained in any of the previous appendices. All other analyses follow procedures

detailed in Appendix 2.

1 .2 Certified Resistivities and Uncertainties

The averages of six measurements on the 0 mm circle, and individual measurements on the

5 mm and 10 mm circles of each wafer, are reported as certified values. There is no

correction for probe effect.

Only Type A uncertainty evaluation procedures are treated in this appendix, and estimates of

all uncertainty components are shown in Table 1 . The Type A standard uncertainty for the average resistivity at the wafer center is

The Type A standard uncertainty for the individual resistivity values on the 5 mm and

10 mm radius circles is

= 0.015 Qcm.

74
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76
2. SYSTEMATIC EFFECTS

2.1 Differences between Wiring Configurations bl and b2

Differences are found between measurements in configurations bl and b2. An obvious

outlier in the pre-certification measurements on wafer 39 was deleted from the database

for the purpose of the analysis. Averages and standard deviations are shown in Table 3. The t-statistic for testing for a significant difference between wiring configurations bl and b2 is t = -y/29 Avg/SD. The t-statistics suggest a slight difference among wiring

configurations for this issue, although the differences are in opposite directions for the

pre- and post-certification measurements. With no other information at hand, it is reasonable to assume that during the certification procedure, the difference between

wiring configurations could fall somewhere within the limits ±a, where a = 0.005 Qcm is based on the post-certification average value. It is also reasonable to assume that the best

correction is zero, and that the standard uncertainty for the underlying uniform

distribution is a/ V3 , or 0.002 89 Q-cm.

Table 3. Average Differences and Standard Deviations between Wiring Configurations bl and b2, Q-cm

Pre-certification Post-certification Probe Avg SD DF t Avg SD DF t 2062 -0.003 83 0.005 14 28 -4.0 +0.004 89 0.004 00 28 6.6

77
Appendix 6. Analysis of Certification Data and Control Experiments

for SRM 2546

1 . GENERAL COMMENTS AND SUMMARY OF TYPE A STANDARD

UNCERTAINTY COMPONENTS

1.1 Introduction

This appendix documents the statistical analysis leading to the certification of wafers from

crystal 51939 for SRM 2546. The 130 wafers in this issue have nominal resistivities of

100 Qcm, and the wafers are assumed to be identical with regard to face; all measurements

were made on the same face of each wafer. All certification measurements were made with probe 2362.

This appendix contains a summary of the temporal and other components of uncertainty in Section 1 .2. All analyses of the temporal components of uncertainty for this SRM follow procedures detailed in Appendix 2 for analysis of SRM 2547. Section 2.1 summarizes an

analysis of a probe bias correction that follows the procedures used in Appendix 2.

1 .2 Certified Resistivities and Uncertainties

The averages of six measurements on the 0 mm circle, and individual measurements on the

5 mm and 10 mm circles of each wafer, corrected for probe #2362 are reported as certified values.

Only Type A uncertainty evaluation procedures are treated in this appendix, and estimates of

all uncertainty components are shown in Table 1 . The Type A standard uncertainty for the average resistivity at the wafer center is

The Type A standard uncertainty for individual resistivity values on the 5 mm and 1 0 mm

radius circles is

1/2 = 0.075 cm .

78
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2. SYSTEMATIC EFFECTS

2.1 Bias Effect of Probe 2362

Differences from the multi-probe mean were found for probe 2362 for each wafer, and

are given in Table 3. The estimated correction for this probe over five wafers is C = + 0.0393 Q-cm; the standard deviation of this average correction is sc = 0.0051 Q-cm.

The correction, C, is applied to all certified values, and its standard deviation is taken as a component of the Type A standard uncertainty.

Table 3. Bias of Probe 2362 Relative to the Average

for All Probes, Q-cm

Wafer#

138

139 140

141 142

Pre-certification 0.0372 0.0094

0.0261 0.0252

0.0383

Post-certification 0.0507 0.0657

0.0398 0.0534

0.0469 Mean Bias, C 0.0393 Q-cm

Standard Deviation of Mean 0.005 1 Q-cm

81
Appendix 7. Analysis of Certification Data and Control Experiments

for SRM 2543

1 . GENERAL COMMENTS AND SUMMARY OF TYPE A STANDARD

UNCERTAINTY COMPONENTS

1.1 Introduction

This appendix documents the statistical analysis leading to the certification of wafers from

crystal 91907 for SRM 2543 at 1 Q-cm. It follows the general procedures outlined in Appendix 2, which documents general certification uncertainty analysis procedures and the

results for the first SRM, at 200 Q-cm, to be certified in this series. In particular, however,

the current appendix develops a component of uncertainty for a sensitivity of measured

resistivity value to ambient illumination level. This photosensitivity appears to exist in all boron-doped, Czochralski-grown silicon crystals, but was of a sufficiently low level to be

negligible in the previously issued resistivity SRMs in this series that used silicon of this type

(see Appendix 1). The photosensitivity effect was originally discovered through experiments

unrelated to SRM certification, and only after certification measurements had been taken for

this SRM (see section 6.2).

The photosensitivity in the crystal used for this SRM has a magnitude that decreases monotonically with increasing wafer serial number. Only wafers with serial numbers greater than 100 (76 wafers from about 125 measured in the initial round of certification measurements) are being issued as SRMs. Control wafers with serial numbers below 100,

i.e. #1 1, #26 and #42, as well as check-standard wafer #35, were used for various aspects of

temporal and measurement-condition control experiments and are retained for the uncertainty

analysis because they were measured under conditions of constant illumination level. Therefore, photosensitivity had no bearing on the function they served or on the validity of the

analysis results derived from their use.

The wafers issued as SRMs are assumed to be identical with regard to the two wafer faces, and the wafer face used for certification measurements was chosen at random with respect to the growth direction of the crystal. Certification measurements were made with a single probe

having serial number 283.

Section 1.2 summarizes the Type A standard uncertainty for SRM 2543. Tables 1 and 2

give an executive summary of the terms that contribute to the Type A standard uncertainty. The details of the calculation of the component terms are given in subsequent sections. Analysis of measurements for possible correction terms is covered in section 3 . No correction

to measurement values for choice of probe used, or for illumination level was required.

82
However, a correction for choice of probe wiring configuration was necessary and was

applied to all measurements on certified SRM wafers.

1.2 Certified Resistivities and Uncertainties

The average of six measurements at the center of each wafer, corrected for bias of the probe

wiring configuration used for the certification measurements, is reported as a certified

resistivity value. The Type A standard uncertainty associated with the certified value at the wafer center is:

+ S

cfig +Sr 0.001 72 Q-cm.

Individual measurements on the 5 mm and 10 mm circles for each wafer, corrected for bias of

the probe wiring configuration used for the certification measurements, are reported as

certified values on the certificates. The Type A standard uncertainty associated with each of

these individual certified values is:

Y

+ S

lfig +Sr +Si +5' = 0 001 85 Q' Cm83
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85
2 RANDOM COMPONENTS OF UNCERTAINTY

Probe imprecision, represented by the standard deviation, se, is obtained from a combination of the results of three different experiments: 1) from a pooling of the standard deviations of

the six measurements at the wafer center for each of the certified SRM wafers (this value is given as the first entry in Table 2); 2) from a pooling, across control wafers and measurement replications, of the measurements with the certification probe, #283, taken

during the pre- and post-certification control wafer measurements (section 2.1); and 3) from the pooled standard deviations of the check-standard measurements that were taken

concurrently with SRM certification (section 2.4). Run-to-run measurement imprecision is estimated both from the pre- and post-certification control wafer measurements (section

2.2), and from the measurements on the check-standard wafer (section 2.4). Long-term

imprecision is estimated from the control wafer measurements (section 2.3).

2. 1 Probe Imprecision from the Pre- and Post-Certification Control Wafer Measurements

The standard deviation, se, from six measurements at the center of the control wafers gives an

estimate, with five degrees of freedom, of the precision for each probe. The pooled values of

such standard deviations, over the six runs on each of five control wafers, are shown in Table 3. The pre- and post-certification standard deviations for probe #283 with configuration bl appear

as the probe imprecision RMSE entries in Table 2.

Table 3. Probe Imprecision Standard Deviations, Each with 150 Degrees of Freedom, after Pooling over Six Runs on Five Control Wafers of Crystal 91907, Q cm

Pre-certification Post-certification Probe Config. bl Config. b2 Config. bl Config. b2 SRM1 0.000 934 0.001 619 0.000 763 0.001 079

281 0.000 767 0.000 878 0.000 748 0.000 872 283 0.000 897 0.000 981 0.000 866 0.000 746 2062 0.000 908 0.001 793 0.000 799 0.001 497 2362 0.001 129 0.000 788 0.001 183 0.000 816

86
2.2 Run-to-Run Variability from Pre-and Post-Certification Control Wafer Measurements

Run-to-run variability for probe 283 is shown in Figure 1 where pre- and post-certification measurements are plotted for each control wafer. There is no evidence of change or drift in the process. Standard deviations and averages computed from the six repetitions with each probe on each wafer are shown in Table 4 for all probes used in the control experiments. The

pooled standard deviations, 0.000 3123 Q cm and 0.000 3363 Q cm, for probe #283 appear as the run-to-run RMSE entries in Table 2. They incorporate both inherent probe imprecision and run-to-run measurement error as shown in the relationship column of Table 2.

Table 4. Run-to-Run Variability for Crystal 91907 Control Wafers Averages and Standard

Deviations for Six Runs, Q cm

Pre-certification Post-certification Difference Wafer Probe Resistivity Standard Dev. Resistivity Standard Dev. (Pre - Post.)

11 SRM1 1.073 76 0.000 243 6 1.073 94 0.000 355 0 -0.000 18 26 SRM1 1.060 98 0.000 628 3 1.060 74 0.000 230 9 0.000 24

42 SRM1 1 f\ A /l A A1.046 49 A AAA 1/1 1 C 0.000 341 5 1 .046 63 A AAA Irtl C 0.000 302 5 A AAA 1 A

- 0.000 14 131 SRM1 0.991 54 0.000 274 6 0.991 62 0.000 173 6 - 0.000 08 208 SRM1 0.962 49 0.000 212 2 0.962 34 0.000 259 0 0.000 15

11 281 1.073 44 0.000 405 5 1.073 50 0.000 265 1 - 0.000 06

26 281 1.060 73 0.000 266 8 1.060 63 0.000 174 1 0.000 10 42 281 1.046 06 0.000 457 3 1.046 22 0.000 315 9 - 0.000 16

131 281 0.991 46 0.000 312 3 0.991 34 0.000 497 8 0.000 12 208 281 0.962 07 0.000 414 6 0.962 36 0.000 194 2 - 0.000 29

11 283 1.073 28 0.000 294 0 1.073 26 0.000 636 0 0.000 02

26 283 1.060 48 0.000 244 1 1.060 70 0.000 251 3 - 0.000 22 42 283 1.046 09 0.000 369 9 1.046 27 0.000 148 8 - 0.000 17 131 283 0.991 12 0.000 328 3 0.991 60 0.000 140 4 - 0.000 48 208 283 0.961 96 0.000 314 1 0.962 32 0.000 236 8 - 0.000 36

11 2062 1.072 90 0.000 477 0 1.073 28 0.000 326 2 -0.000 38 26 2062 1.060 63 0.000 305 4 1.060 49 0.000 509 5 0.000 14 42 2062 1.045 84 0.000 399 2 1.045 85 0.000 166 3 - 0.000 02

131 2062 0.991 02 0.000 447 2 0.991 33 0.000 405 8 -0.000 31 208 2062 0.961 65 0.000 406 5 0.961 77 0.000 751 0 - 0.000 12

11 2362 1.072 85 0.000 349 4 1.073 24 0.000 482 3 - 0.000 39

26 2362 1.060 08 0.000 352 7 1.060 51 0.000 518 5 - 0.000 43 42 2362 1.045 88 0.000 650 7 1.045 99 0.000 263 8 -0.000 11 131 2362 0.990 81 0.000 621 5 0.991 32 0.000 455 7 - 0.000 50 208 2362 0.961 75 0.000 180 8 0.961 87 0.000 493 8 -0.000 11

Probe 283 (pooled across wafers) 0.000 312 3 0.000 336 3

87
Wafer 1

Wafer 26

1 . 074 1.062

1 0466

1 0456

0.963

7.5 8 8 5 Month/Date Wafer 42

Month/Date Wafer 208

0.961- 7.5 8 8 5 Month/Date

£

• 1.0616: a

1.0612^ "> 1.0608 :

t CO 1.0604:

1.06

0.992-

Month/Date Wafer 131

7.5 8 8 5 9 Month/Date

Figure 1 . Resistivity (Qxm), with probe #283, for five control wafers from crystal 91907 plotted

vs. month/date of measurement, and showing consistency obtained both within and between and

the pre- and post-certification, (*) and (o) plot symbols, control measurements.
2.3 Long-term Measurement Variability of Control-Wafer Measurements with Probe #283

Table 5 shows averages and standard deviations computed from six runs (replications) on

each control wafer with probe #283. The differences listed are assumed to be the result of a long-term component of measurement error. The standard deviations of the differences incorporate probe imprecision, run-to-run variation, and long-term variability as shown in the relationship column of Table 2. The standard deviation resulting from pooling across

control wafers is the value shown as the long-term RMSE entry in Table 2.

Table 5. Long-Term Component of Uncertainty for Crystal 91907

Control Wafers with Probe 283, Q cm

Pre-cert. Post-cert. Wafer Average Average Difference Stand. Dev

11 1.073 28 1.073 26 0.000 02 0.000 014

26 1.060 48 1.060 70 -0.000 22 0.000 156 42 1.046 09 1.046 27 -0.000 17 0.000 127

131 0.9-91 12 0.991 60 -0.000 48 0.000 339

208 0.961 96 0.962 32 -0.000 36 0.000 255

DF

Standard deviation after pooling across control wafers 0.000 210

2.4 Check-Standard Measurements

Twenty-five measurement runs (of six wafer-center measurements each) were made with probe #283, on check-standard wafer #035, over the course of the SRM wafer certification, a period of five weeks. Run-to-run variations in the measurement process are shown in Figure 2. The standard deviations of the 25 individual measurement runs were pooled to give a value of 0.000 724 O-cm with 125 degrees of freedom. This pooled estimate is given

in Table 2 as the RMSE of the probe imprecision from check standard measurements. Also

given in Table 2 is the run-to-run standard deviation, 0.000 349 Q-cm, of the average values from the 25 measurement runs.

89
0.0011

0.0010

0.0009

0.0008-

0.0007

0.0006-

0.0005"

0.0004 I r

I

1

I 1

I 1I r 5 10 15 20 25 30 Run Sequence

Figure 2. Standard deviation (Q»cm), for resistivity measurement runs made with probe #283 on

check standard wafer #035 during the course of the certification measurements.

90
3. SYSTEMATIC EFFECT COMPONENTS OF UNCERTAINTY

3 .1 Effect of Using Probe #283 for Certification Probe #283 is shown as the symbol "3" in Figure 3 which plots the offset, or bias, of each probe compared to the multiprobe average. These data are also summarized in Table 6. There is a shift in the direction of the bias of probe #283 from the pre-certification measurements to the post-certification measurements. Therefore, any correction for bias of

probe #283 is taken to be zero. A conservative assumption is that during SRM certification the functional bias could have fallen somewhere within the limits ± b where

b = 0.000 0652 Q cm; a standard uncertainty of bj3 = 0.000 038 Q cm is included as a systematic component of the uncertainty.

Table 6. Difference (Bias) from Multi-probe Grand Mean for Each

Probe and Each Control Wafer, Qcm.

Wafer Probe Pre-certification Post-certification SRM1 r\ r\/-\/-\ c i a r\ 0.000 514 0 0.000 496 0

281 0.000 194 0 0.000 056 0 H 283 0.000 034 0 -0.000 184 1 11 2062 -0.000 345 9

r\ r\r\r\ i /z A f\ -0.000 164 0

. z36z c\ c\f\f\ "jot n -U.UUU ZU4 u 26 SRM 1 0.000 399 9 0.000 126 0 26 281 0.000 150 0 0.000 016 0 26 283 -0.000 100 0 0.000 086 1 26 2062 0.000 049 9 -0.000 124 0 26 2362 -0.000 500 0 -0.000 104 0 42 SRM1 0.000 418 1 0.000 438 1 42 281 -0.000 011 9 0.000 028 0 42 283 0.000 018 1 0.000 078 1 42 2062 -0.000 231 9 -0.000 341 9 42 2362 -0.000 191 9 -0.000 201 9

131 SRM1 0.000 349 9 0.000 178 0

131 281 0.000 269 9 -0.000 102 0

131 283 - 0.000 070 1 0.000 158 0

131 2062 -0.000 170 1 -0.000 112 0

131 2362 - 0.000 380 1 -0.000 122 0 208 SRM1 0.000 506 0 0.000 207 9 208 281 0.000 085 9 0.000 227 9 208 283 - 0.000 024 0 0.000 187 9 208 2062 - 0.000 334 0 -0.000 362 1 208 2362 - 0.000 234 1 -0.000 262 1 Probe 283 (average) - 0.000 028 4 +0.000 065 2

91
Pre-Certification Measurements

T

100 150 200 Control Wafer Number

Post-Certification Measurements

Control Wafer Number

Figure 3. Bias, in Q-cm, of individual probes, relative to the multiprobe average value for each of five control wafers during the pre- and post-certification control measurements; probe # 283 is symbol 3.

92
3.2 Difference Between Wiring Configurations bl & b2

Differences between measurements in wiring configurations b 1 & b2 are shown in Figure 4. Averages and standard deviations are shown in Table 7 and summarized in Table 8. The t-statistic for testing for a signficant difference between wiring configurations bl & b2 is t = V30 Avg/SD. The plots and t-statistics suggest a slight difference among

wiring configurations for this batch of SRM wafers. The average difference between the

pre- and post-measurements in configurations bl and b2 is 0.000 262 Q-cm. A correction of

minus one-half this difference, or - 0.000 131 Q-cm, is applied to all certification measurements to obtain an average over the two configurations. The standard deviation of

the correction is:

Table 7. Differences Between Wiring Configurations bl & b2 for Six Days of

Control-Wafer Measurements with Probe #283, Q-cm

Wafer Pre-certification Post-certification

0.000 28

0.000 37

-0.000 13 0.000 12 0.000 38

0.001 55

0.000 52 -0.000 10

- 0.000 09

-0.000 17 0.000 35 0.000 16 26

26

26

26

26

26

0.000 57

-0.000 03 0.000 03

-0.000 41

-0.000 60

0.000 35

0.000 29

0.000 11 0.001 40

-0.000 11 0.000 35 0.000 28 42

4242

4242

42

0.000 32

0.000 65

-0.000 39

0.000 86

-0.000 47

0.000 88

0.000 48

0.000 71 0.000 66

0.000 18

- 0.000 42

0.000 14

93
131 0.000 49 0.000 66

131 -0.000 03 0.000 61

131 0.000 02 0.001 26

131 0.000 42 0.000 03

131 0.000 23 0.000 33

131 -0.000 92 0.000 06 208 -0.000 29 0.000 30 208 0.000 30 0.000 56 208 0.000 25 0.000 73 208 0.000 18 -0.000 19 208 0.000 26 0.000 30 208 0.000 74 0.000 36 Average 0.000 199 0.000 326

Standard Dev. 0.000 498 0.000 402 DF 29 29

Table 8. Average Differences between Wiring Configurations b 1 & b2

for Probe #283, Q cm

Pre-certification Post-certification Probe Average Stand. Dev. DF t Average Stand. Dev. DF t 283 0.000 199 0.000 498 29 2.2 0.000 326 0.000 402 29 4.4

94
Pre-Certification Measurements

0.0005

Measurement Run

Post-Certification Measurements

0.0005- A C

E

B

B C c DEE

D D

o.ooil " 1

1 1

"

1

' 1

" 1

"

0 5 10 15 20 25 30 Measurement Run

Figure 4. Differences, Q-cm, between wiring configurations bl and b2 for six measurements on each of five control wafers, A to E.

95
3 .4 Differences Caused by Lighting Conditions SRM 2543 is unlike the other SRMs in the series 2541-2547 in that the measured resistivity values are photo-sensitive. The full change in resistivity between light and darkened room

conditions takes several minutes to occur. The certification measurements were made in, what is termed "standard conditions", i.e. the same ambient fluoroescent room light that was

used with the other SRMs. An experiment was performed on 7 wafers from crystal 91907

to evaluate the effect of darkness and brightness on the results. For those wafers, resistivity measurements were made under conditions of: standard illumination, dark-room, and verybright illumination (bright enough for photosensitivity to saturate). The data are summarized in Tables 9 and 10. The resulting differences can be seen to be asymmetric: there is a much larger change between standard and dark conditions than there is between standard and very-bright

conditions. For wafers with serial numbers higher than those in the table, the

photosensitivity decreases somewhat. It is not meaningful to attempt to apply a correction

to the as-measured SRM resistivity values to correct the values to either dark or very-bright

illumination conditions. Instead, a component of uncertainty is evaluated to account for other possible illumination conditions that might be experienced by the user of SRM 2543. To do this, the larger of the two differences, i.e. that between standard and dark conditions,

is used to evaluate the component of uncertainty. Use of data from test wafers in the low

end of the SRM wafer range ensures a conservative estimate of the uncertainty to

photosensitivity. Under these conditions, the expectation is that under lighting conditions other than standard, the difference of measured resistivity would be within the limits ± c where c = 0.002 914 Q-cm, and a component of c/ V3 = 0.001 682 Q cm is included as a

systematic component of the uncertainty.

Table 9. Resistivity Values under Standard Illumination and in Dark, and their Difference, Q-cm

Wafer

Standard

Illumination Dark Difference

102 104

106

107

108

123 124

0.9981

1.0095

1.0233 1.0047

0.9991 1.0057 0.9985

1.0013 1.0125

1.0261 1.0076

1.0018

1.0086

1.0014

-0.0032 -0.0030

-0.0028 -0.0029

-0.0027

-0.0029

-0.0029 Mean Difference Standard Deviation

-0.002 91 0.000 157

96
Table 10. Resistivity Values under Standard Illumination and in Very Bright Ambient, and their Difference, Q-cm

Standard Very

Wafpr vv aici lllUlllllidllUIl i-/iricicncc 102 0.9983 0.9976 0.0007

104 1.0094 1.0087 0.0007 106 1.0233 1.0225 0.0008

107 1.0047 1.0042 0.0005

108 0.9991 0.9984 0.0007

123 1.0057 1.0050 0.0007

124 0.9985 0.9978 0.0007 Mean Difference 0.000 686

Standard Deviation 0.000 090

97
Appendix 8. Analysis of Certification Data and Control Experiments For SRM 2544

1 GENERAL COMMENTS AND SUMMARY OF TYPE A STANDARD UNCERTAINTY

COMPONENTS

1.1 Introduction.

This appendix documents the statistical analysis leading to the certification of wafers from crystal 29473 for SRM 2544 at 10 Q cm. It follows the general procedures outlined in Appendix 2 of this

publication. The 144 wafers in this SRM have nominal resistivities of 10 Q cm; they are assumed

to be identical with regard to wafer face. For this issue, the pre- and post-certification control measurements were made on opposite faces of the control wafers. Certification measurements were made with probe #283.

1.2 Certified Resistivities and Uncertainties. The averages of six measurements at the wafer centers, and individual measurements on the 5 mm and

10 mm radius circles of each wafer are reported as certified values on the certificates. No correction

to measured values needed to be applied due to the choice of probe used for certification, but a

correction based on the difference between two probe-wiring configurations is applied to all certified values as discussed in 2.2, and listed in Table 1

Only Type A evaluations of uncertainty components are treated in this appendix. Estimates of all such uncertainty components are shown in Table 1 . The data from the check-standard show

degradation in the standard deviation over time. It is assumed that this finding is not symptomatic of degradation of the probe, but rather of debris collecting on the surface of the wafers with measurements over time. However, these data are not used to estimate either probe precision or

run-to-run variability.

98
The Type A standard uncertainty for the average resistivity at the center of each SRM wafer is:

b

2

1 - + s^

g + s

2 + s] +-s2e = 0.00609 Q cm.

The Type A standard uncertainty for individual resistivity values on the 5 mm and 10 mm radius

circles is:

4, + s

2 +s25 +s)= 0.00743 Q cm.

99
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101
2. SYSTEMMATIC EFFECTS

2 .1 Bias of Probe 283

There is a small bias for the certification probe, #283, (relative to the average over all probes) as shown in Table 3 for each of the control wafers. The differences from the average for all probes

are small; they are of both algebraic signs, and sometimes change signs between pre- and postcertification measurements. Therefore, the best value for a correction due to bias is taken to be

zero. A conservative assumption is that during certification the bias was within the limits ± b where b = 0.000 353 Q cm, the worst case mean bias below, and a standard uncertainty of bV3 = 0.000 204 Q cm is included as a systematic component of the uncertainty.

Table 3. Bias of Probe #283 Relative to the Average for All Probes, Q cm

Wafer Pre-certification Post-certification

1632

75

108 120

0.000 6742

-0.000 1497

-0.000 4139

-0.000 2584

0.000 3242

0.000 4902

-0.001 7977

0.000 5331

-0.000 2708

-0.000 4511 Mean Bias 0.000 3528 -0.000 2993

102
2.2 Difference Between Wiring Configurations bi & b2

Differences are found between measurements in wiring configurations bi & b2 . Averages and

standard deviations are shown in Table 4. The t-statistic for testing for a significant difference between wiring configurations is t = -/30 Avg/SD. The t-statistics show that the difference between

wiring configurations for this probe and resistivity level are significant. The average difference between the pre- and post-certification control wafer measurements in configurations bl and b2 is 0.002 1605 Q cm. Certification measurements were taken using configuration bi only. A correction of minus one-half this difference, or -0.001 085 Q cm, is applied to all certification measurements to report an effective average value over the two wiring configurations. The

standard deviation of the correction is:

Table 4. Average Differences and Standard Deviations Between Wiring Configurations

bi & D2, Q cm

Pre-certification Post-certification

Probe Average Standard Dev. DF t 283 0.001 358 0.001 914 29 3.9 Average Standard Dev. DF t 0.002 963 0.002 495 29 6.5

103
Appendix 9. SRM Values after an Extended Period of Time

It is useful to evaluate how closely the original SRM measurements can be reproduced after extended periods of time since the original certification and control measurements were taken for each of the SRM levels. To test this, a single set of six measurements was taken in February 1997

at the center of each of the original control wafers used for each of the SRM levels. This was

done with the same probe that was used for the certification measurements of each given SRM

level. None of the probes had been rebuilt or modified since the earliest of the certification measurements, namely those for SRM 2547, at 200 Q-cm. However, in the ensuing time, each probe was used for the control measurements of all subsequent SRM levels, and, in the case of

probe 283, was used as the certification probe for three different SRM levels. As a result, some

wear can be expected on all probes between the time they were first used and the time of the February 1997 follow-up measurements.

1. Summary of Results The results are summarized in Table 1 where the entries in the second and fourth columns are the grand averages from six runs using the probe noted during the pre- or post-certification control experiments. The entries in the third and fifth columns are the standard deviations, 2, of the average values from each of those six runs. The value in column six is the average of six measurements at the wafer center during a single run in February 1997. The entry in the seventh column is the standard deviation, a, of those six individual measurements. Finally, the value in the last column is the relative difference between the single average value from February 1997

and the grand average, or base-line value, of all 12 runs from the pre- and post-measurement

experiments. Data from pre- and -post measurement experiments are stated as actually acquired.

No correction for probe bias or for configuration bias, such as are identified in some of the

statistical analysis reports as being necessary for the SRM wafers to be issued, was applied.

2. Comments on the Results

For measurements on wafers at 0.01 Q-cm, 0.1 Q-cm, 10 Q-cm, 25 Q-cm, and 100 Q-cm,

the latest measurements appear to be randomly above and below the base-line values from the

control experiments. With the exception of wafer #141 at 25 Q-cm, recent measurements at those resistivities are all within 0. 10% of the base-line values. Measurements on wafers at

1 Q-cm show a consistent high-side bias of recent values over the base-line results. Because

of the known residual sensitivity of the 1 Q-cm material to illumination levels, it must be

considered that a difference in illumination levels between that during the latest measurement

and that at the time of the original measurements could be responsible for causing all measurement differences to be of the same sign. Measurements at 200 Q-cm also show a

systematic difference between recent and base-line values. In this case, present values are below the base-line values, in the direction of the shift with remeasurement previously noted. While this effect may be the dominant cause of the observed shift, average relative humidity was approximately 45% at the time of the base-line measurements and was approximately 32%

during the latest measurements. In order to put the latest values in perspective, an

additional column has been added to the table for the 200 Q-cm wafers. This column gives

104
the two standard deviation (2cr) lower limit value for each wafer, which is calculated from the base-line average value and the lower 2a uncertainty value given in Appendix 2. It can be

seen that the February 1997 values are clearly within the lower 2ct limit for the 200 Q-cm

SRM level.

Table 1. Summary of the Six-Round Grand Averages and Standard Deviations from Pre- and Post-Certification Measurements, Single Round Averages and Standard Deviations from

Recent Measurements, and the Percent Changes in Measurement Values CRYSTAL 91905 Probe 283 Elapsed Time 38 Months

Control Pre-certification Wafer# p.avg (Q-cm) (£,%)

Post-certification Pavg (Q-cm) (Z,%)

Feb. 1997

Pavg (Q-cm) (o,%)

Feb 1997 Minus

Pre/Post Avg

(Difference, %)

002 0.011 286 0.021 0.011 275 0.015 0.011 281 0.046 +0.004

043 0.010 974 0.013 0.010 972 0.014 0.010 970 0.024 -0.027 044 0.010 955 0.010 0.010 949 0.004 0.010 954 0.030 +0.018 053 0.010 923 0.015 0.010 926 0.006 0.010 923 0.023 -0.014 144 0.010 350 0.014 0.010 352 0.012 0.010 352 0.015 +0.010

CRYSTAL 91904 Probe 281 Elapsed Time 19 Months

003 0.114 51 0.137 0.114 59 0.039 0.114 59 0.091 +0.035 066 0.113 88 0.123 0.113 82 0.063 0.113 80 0.110 -0.044 097 0.112 61 0.143 0.112 52 0.016 0.112 56 0.062 -0.004

161 0.104 35 0.146 0.104 33 0.012 0.104 32 0.076 -0.019 287 0.099 61 0.129 0.099 60 0.035 0.099 63 0.047 +0.025

CRYSTAL 91907

011 1.0733 0.028 1.0733 026 1.0605 0.023 1.0607 042 1.0461 0.035 1.0463

131 0.9911 0.033 0.9916 208 0.9619 0.033 0.9623

Probe 283 Elapsed Time 31 Months

0.059 1.0745 0.068 +0.110

0.024 1.0613 0.022 +0.066

0.014 1.0470 0.078 +0.076

0.014 0.9916 0.077 +0.025

0.025 0.9630 0.125 +0.094

105
Table 1 (cont'd.) CRYSTAL 29473 Probe 283 Elapsed Time 3 Months

Feb 1997

Control Pre-certification Post-certification reb 199/ Minus

Waier# Pavg Pavg Pavg Pre/Post Avg

(Q-cm) (2,%) (fl-cm) (Q-crn) (ct,%) (Difference, %)

016 10.085 0.014 10.080 0.033 10.085 0.079 +0.025 032 10.105 0.025 10.096 0.021 10.109 0.049 +0.084 075 10.316 0.029 10.308 0.024 10.309 0.085 -0.029 108 10.186 0.017 10.177 0.020 10.181 0.041 -0.005 120 10.082 0.017 10.073 0.014 10.077 0.056 -0.005

CRYSTAL 21565 Probe 2062 Elapsed Time 29 Months

017 24.050 0.032 24.046 0.015 24.046 0.104 -0.008 039 24.695 0.029 24.699 0.022 24.701 0.061 +0.016

063 24.509 0.016 24.517 0.011 24.495 0.058 -0.073 103 24.135 0.031 24.142 0.025 24.124 0.044 -0.060 125 24.052 0.032 24.056 0.019 24.054 0.068 +0.001

CRYSTAL 51939 Probe 2362 Elapsed Time 34 Months

138 95.093 0.038 95.124 0.048 95.131 0.125 +0.024

139 99.306 0.048 99.310 0.022 99.252 0.076 -0.056 140 96.036 0.028 96.077 0.029 96.103 0.072 +0.048

141 101.060 0.023 101.079 0.053 101.277 0.097 +0.205

142 94.215 0.029 94.244 0.039 94.309 0.080 +0.084

CRYSTAL 21566 Probe

Control Pre-certification Post-certification Wafer# Pavg Pavg (Q-cm) (2,%) (Q-cm) (Z,%) 020 196.27 0.050 196.07 0.104 040 193.88 0.034 193.76 0.032 060 193.57 0.072 193.50 0.036 080 192.82 0.054 192.69 0.050 100 192.59 0.065 192.42 0.057

SRM1 Elapsed Time 54 Months

Feb 1997 Feb 1997 Minus 2c

pavg Pre/Post Avg Lower

(Q-cm) (a,%) (Difference, %) Limit

196.05 0.081 -0.061 195.60 193.59 0.102 -0.119 193.39 193.24 0.039 -0.152 193.10

192.35 0.097 -0.210 192.32 192.26 0.074 -0.127 192.07 106

AU.S. GOVERNMENT PRINTING OFFICE:1999-45<I-531/17108
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