Standard Reference Material®

 The Certification of 100 mm Diameter Silicon

 Resistivity SRMs 2541 Through 2547 Using Dual-

 Configuration Four-Point Probe Measurements, 

2006 Edition

 James R. Ehrstein, M. Carroll Croarkin, and Hung Kung Liu

 NIST Special Publication 260-131, 2006 Ed. NIST Special Publication 260-131, 2006 Ed.

 NIST Special Publication 260-131, 2006 Ed.

 Standard Reference Materials®

 The Certification of 100 mm Diameter Silicon 

 Resistivity SRMs 2541 Through 2547 Using 

 Dual-Configuration Four-Point Probe 

 Measurements, 2006 Edition

 James R. Ehrstein* 

 Semiconductor Electronics Division 

 Electronics and Electrical Engineering Laboratory

 M. Carroll Croarkin* 

 Hung Kung Liu 

 Statistical Engineering Division 

 Information Technology Laboratory 

 * Retired 

 National Institute of Standards and Technology 

 Gaithersburg, MD 20899 

 (Supercedes NIST Special Publication 260-131, 1999 Ed., June 1999) 

February 2006

 U.S. Department of Commerce 

 Carlos M. Gutierrez, Secretary 

 Technology Administration 

 Under Secretary of Commerce for Technology, William Jeffrey, (acting as)

 National Institute of Standards and Technology 

 William Jeffrey, Director 
Certain commercial entities, equipment, or materials may be identified in this

document in order to describe an experimental procedure or concept adequately. Such

identification is not intended to imply recommendation or endorsement by the

National Institute of Standards and Technology, nor is it intended to imply that the

entities, materials, or equipment are necessarily the best available for the purpose.

National Institute of Standards and Technology Special Publication 260-131, 2006 Ed.

Natl. Inst. Stand. Technol. Spec. Publ. 260-131, 2006 Ed., 134 pages (February 2006)

CODEN: NSPUE2
iii
Table of Contents

Page

Abstract .........................................................

Introduction........................................................

PreviousResistivitySRMs "

.

'" 2

ImprovedResistivitySRMs ........................................

1. Certification of Improved Resistivity SRMs . . . . . . . 

1 . " . . . . . . . . . . . . . . . . . 3GeneralComments.........................................

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................................... ...........

1 Wafer Thickness Screening and Thickness Measurement . . . . . . . . . . . . .. . . 9

2 Four-Probe Measurements of Sheet Resistance

. . . . . . . . . . . . . . . . . . . . .. 

2.3 Reporting of Data from 18 Measurements on Each Wafer

. . . . . . . . . . . . .. 

3. Control Procedures for the Certification Process. . . . . . . . . 

. . . . . . . . . . . . . . . . .. 

3.1 Control Procedure for Probe Effects

. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 

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

. . . . . . . . . . . .. 

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

4. Equations Used for Calculating Sheet Resistance and Resistivity Values

. . . . . . . . .. 

1 Rewriting the Equation to Relate to Evaluation of Uncertainty

. . . . . . . . . .. 

5. Sources of Measurement Uncertainty - Detailed Discussion. . . . . . . . . . . 

. . . . . . . . 

1 Type A Evaluations of Components of Uncertainty. . . 

 .  . . . . . . . . . . . .. 

1.1 Short-Term Precision; Repeatability. . 

. . . . . . . . . . . . . . . . . . . . . . . .. 

Intermediate and Longer Term Precision; Reproducibility of Wafer-Center

AverageValue ...........................................

1.3 Uncertainty Due to the Selection of aParticular Probe

. . . . . . . . . . . . . . 

2 Type B Evaluations of Components of Uncertainty

. . . . . . . . . . . . . . . . . . . 

1 Discussion of Components Related to Electrical Measurements

. . . . . . . . 

Evaluation of Uncertainty in Electrical.Measurement Scale

. . . . . . . . . . . 

2.3 Evaluation of Uncertainty Components Related to Temperature Measurements.......................................... 

2.4 Evaluation of Uncertainty Components Related to Geometry

Measurements ..........................................

Evaluation of Uncertainty Due to Thickness Measurement Scale

. . . . . . . 

Evaluation of Uncertainty Due to ThicknesslProbe Separation Scaling

Factor-

............................................... 
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 Apendices...............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

Appendix 10. Analysis of Certification Data and Control Experiments for SRM 2541, 

batch 2 ................................................................................................................ 106

Appendix 11. Analysis of Certification Data and Control Experiments for SRM 2543, 

batch 2 ................................................................................................................ 113

Appendix 12. Analysis of Certification Data and Control Experiments for SRM 2544, 

batch 2................................................................................................................. 120

Appendix 13. SRM Unit Serial Numbers included in the second certification batches of 

SRMs 2541, 2543, and 2544............................................................................... 126

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

Manufacturers' Specifications for the Current Supplies and DVM Used for

Certification ,"

......

Standard Resistor Values

, .

and Typical Measurement Voltages for Each of the

SRMLevels .............

...........................,......................

Temperature Coefficients of Resistivity of Silicon for the Nominal Values of the

SRMs "

...........

Variance in Resistivity Value Due to Temperature Error . . . . . . . . . . . . . . . . . . . . . . . . 30

Specifications for Haidenhain Certo 60 Thickness Measurement Instrument. . . . . . . . . 33

Components Identified in Statistical Analyses of Certification and Control Experiment

Data .................................................................

Values of the Components Identified in Statistical Analyses for the Various SRMs in

Appendices 2 through 8, mQ'cm ..........................................

Type A Standard Uncertainty Values U;, Taken from appendices 2 through 8 .......

Type A Variance Values

;, 

Obtained by Squaring the Entries in Table 8 .........

10. Type B variance Values

j

Calculated from Summation of Terms in Section 5.2 ...

11. Combined Variance Values c, 

from Addition of Terms from Tables 9 and 10 . . .,. . .

12. CombinedStandardUncertaintyValues, c ..................................

13. ExpandedUncertaintyValues U ..........................................

14. Bias Corrections Applied to Measured Values

. . . . . .. 

. . . . . . . . . . . . . . . . . . . . . . . . . . 43

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

. . . . . . .. . . .
1

Standard Reference Materials:

The Certification of 100mm Resistivity SRMs 

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

2006 Edition 

by 

James R. Ehrstein 

Semiconductor Electronics Division, EEEL 

and 

M. Carroll Croarkin and Hung Kung Liu 

Statistical Engineering Division, ITL 

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

the analysis of measurement uncertainty for a family of improved Standard Reference Materials 

(SRMs) for sheet resistance and resistivity of silicon wafers, SRMs 2541 through 2547, 

covering the resistivity range 0.01 S.cm through 200 S.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 both the 

original 1997 report and a 1999 update. It contains three new appendices: 10, 11, and 12 which 

summarize H.-K. Liu’s statistical reports of analysis of second batch certifications for SRMs 

2541, 2543 and 2544 as well as an appendix 13 that details the SRM serial numbers that were 

part of those second batch certifications. This revision also includes updates to tables 10 

through 15 to include summaries of the uncertainty component results from those “secondbatch” certifications. 

The certification of the improved 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 S.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. 
Additional Type A components, derived ITom 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 ITom

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 of these 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 conductancemeasuring 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 SRMsare wafers of 100 mm diameter silicon which enables better compatibility

with present generation user instrumentation.

* In the remainder ofthis 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. 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 needs. , with the values being chosen according to the user s application

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 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
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) 

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 waftr 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.
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 configurationswitched 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).

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 

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
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, 

c' 

A quantity called the expanded uncertainty, 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 v3. 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.
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 /-lm 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-em

1 Q -em, and 1 Q -em, 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 AlS 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 /-lID 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.

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

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 dualconfiguration 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. , 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 DC 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
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 f-lm 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.

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 f-lm, 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 f-lm 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 threerow by three-column grid with a distance of 19 rom 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
(!jj) 

xl 

~l 

. . . . ... .

Electrical Measurement Locations:

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.
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 5/lm for most wafers, and wafers are excluded from use as SRMs if there is an

indicated variation of more than l/lm 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 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 rom between adjacent

pins, with a spring-loaded force of about 1.5 N per pin and a nominal 40 /lm (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. 1. The measurement procedure for the first

wiring configuration at each site follows ASTM F84, and that for the second configuration

follows ASTM F1529 (6).

At each of the 18 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 mVand 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) I7). Standard resistor and wafer voltages

are recorded to a resolution of 0. 1 /lV. (More detail on the voltage measurements is given in

Sec. 5. ) 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
Standard

Probe

Configuration

(ASTM F-84)

Second Configuration, 

(There are two symmetric choices

for the second configuration)

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

a' 

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 (th~. I3!gest 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 oC 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 con,trol-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 300 between them, 2) six are spaced 600 apart around a

circle of 5 mm (0. 2 in) radius, and 3) six are spaced 600 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. F or 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
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 rom

and 10 rom 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

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 "controlwafers" 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
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.

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
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.

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.

V K

a FT F(tfS) 

= X a FT F(tls) (1)

p 

V K

a FT t F(tls) 

= X a FT t F(tlS)

(2)

where:

is the sheet resistance of the wafer, in ohms;

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

(VII) is the first-configuration (ASTM F84) voltage-to-current ratio (also called 

)

ohms;

is the wafer thickness, in centimeters;

S is the average probe separation, in centimeters;

F(tIS) isa thickness-related scaling factor (near unity for t ..::;: 0.4 g);

F T is a correction from the temperature of measurement to a reference temperature (23 OC);

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.
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

a' 

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)

= -14.696 + 25.173 (

:)

- 7.872 (

~:r

(3)

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

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

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

resistivity can then be expressed as:

(p) 

(t/S)) (F;a (x)+x

))

(F2(t/S) a (t)+t (F(t/S))J, (4)

tF(t/S)

where X is the product X 

All certification and control experiment data that are supplied for statistical analysis are in

units of resistivity, corrected to a temperature of 23 o , 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
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 p2 that can be seen for the second

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

IF 

l, 

the equation can be rewritten as:

(p) 

F;(J (xJ+X,

2 (F IF2(t/S) (J (t)+t 2 (F(tls))) F; t . (5)

(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 

nominal values will suffice. Nominal wafer thickness is 0.628 cm; the thickness-related

scaling term F(tIS) is dimensionless and is taken as unity for all wafers because of the values

of tiS for the wafers being certified. The temperature correction factor l' 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 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')

p

(V/l)

2 2 

X, 

(5a)

temperature measurements:

; 2

(X,

2 (J (F T

)) 

p 2 (J

(F 

X, 

(5b)
and thickness measurements:

(F2(t/S) . (t) (F(t/S))) 

p

2 .. (t) (F(t/S))

2 F2(tls) 2 F2(tls)

(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

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 rom and 10 rom

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.

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 K 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
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

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.

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 -em 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.
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

MethodF84 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 varianc.

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.

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

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.

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 6-112 digit DVM (Hewlett-Packard model 3456) used for the silicon wafer

measurements. Voltage measurements are taken with a resolution of 0. 1 flY. 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 flQ/Q to 5 flQ/O. 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. , a

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

across the standard resistor are typically 25 mY, generally stable tol flY and read to 0.1 flY.

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 -em and 200 Q -em 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 10m V to

20 mY. NIST measurements for SRMs 1521 to 1523 were taken in the restricted range of

10 mV to 12 mY. 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 mY. (See Sec. 2. ) (With a

1.59mm probe point separation, this gives a maximum field of less than 7 mV fmm 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-em for a nominal 625flm 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
SRM, 0. 01 Q' , 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

:tl J..I.V to :t3 J..I.V (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-tocurrent ratio. The standard deviation of a set of measurements taken in a fixed region of the

wafer (e. , 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 J..I.V specification. However, relative accuracy of

the ratio measurement is better than 2.4 J..I.V 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.

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

Certification

ELECTRONIC MEASUREMENTS Inc. Model C612 Constant-Current Supply

OUTPUT RANGES: 1 !lA, 2. 2 !lA, 5 !lA, multiplier xl , xlO, xlOO etc.

to 100 mA max. (0 to 100 % vernier each range)

STABILITY: 0.3 % of range setting.(fixed line, lo&d, and temp.

CURRENT REGULATION: 1 % for 100 . V step in compliance voltage

RIPPLE and NOISE: 04 % rms of range setting + 0.5 IlA (negative ground)

04 % rms of range setting + 0. 1 !lA (Positive ground)

(floating output is used for certification)

OUTPUT IMPEDANCE: 30 000 MQ (gJ 1 !lA to 500 kO (gJ 100 mA

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

ELECTRONIC DEVELOPMENT CORPORATION Model CRI03 Constant-Current Supply

OUTPUT RANGES: 10 mA and 100 mA full scale; 6 digit setability

STABILITY (non-additive): 1 h 0.001 % of range

8 h 0.005 % 

1 Yr. 0.01 % 

RIPPLE and NOISE: (0. 1 Hz to 100 kHz) ..:::0.5!lA

OUTPUT CONDUCTANCE: 1 !lS

TEMPERATURE COEFFICIENT: 0.0005 %/K
Table 1. (cont'

HEWLETT-PACKARD #3456 DVM

(All values are stated for the 100 mV range)

RESOLUTION (Least Count): 1 J.1V

INPUT IMPEDANCE: ).1010 Q

MEASUREMENT ACCURACY: For auto-zero on, filter off and ;:::10 power cycle cycles):

24 h (23 :t 1) oC: :t(0.0022 % rdg. + 24 counts)

90 day (23 :t 5) oC: :t(0.0034 % rdg. + 24 counts)

TEMPERATURE COEFFICIENT: :t(0.0002 % rdg. + 0.2 counts/o

NOISE REJECTION: Normal mode, ac: 60 

(1 kQ max. Unbalance in low)

Common mode, ac: 150 dB

Common mode, dc: 140 dB

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 (Sa)

(VIIl 

(VI/)2 (K

(6)

Ignoring temporarily the term in 

a' and replacing the current, with the ratio of standard

resistor voltage to standard resistor value, V IRs' 

in the first term, results in

aWR/V)

p

2 J (VIV.) CR.)

(VR/Vl (VIV )2 (7)
Rather than expanding the term in (V/V to get separate terms in (V) and 

)

it is

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

a whole. It is assumed that the 2.4 J..l V error (due to voltmeter accuracy limit statement)

affects the measurements of and equally (both voltages are measured on the same meter

and in quick succession). Then the worst-case error in their ratio occurs when is the

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

equals 3 x 

For the resistivities above 0.01 Q-cm, the wafer voltage-drop is 10 mY, the standard resistor

voltage-drop is 30 mY, and the worst-case ratio V/Vs' with no error in voltage values, is

333 333 3. A 2.4J..lV error in both and 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- . The

denominator (V/V equals 0.111 , so the contribution to variance from the first term above

is: (9.50 x 10- /0.111) p , or 8.56 x 10-

9 p

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

01 Q-cm. For this SRM level, the ratio without voltage measurement error, is

1 mV/I0 mY, or 0.310 000; and with a 2.4 J..lV error, it is 0. 310 166. The resulting error in

the V/V ratio is 0. 000 166. Squaring this, and dividing by 3, as above, gives 9. 18 x 10-

The denominator (V/V at 0.01 Q-cm, is 0. 096. The resulting contribution to variance of

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

The second term in eq (7),

can be shown to be negligible. The calibration uncertainty of atl standard resistors used for

the SRMs is .:;:5 x 10-6 times the value of the resistor. Assuming a rectangular distribution for

standard ~esistor calibration error )/(R )2 

== (2. 5 x 10- /3) = 8. 3 x 10- . The

contribution to variance related to standard resistor calibration error is 8.3 x 10-

12 p2 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.
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 Voltage

(Q'Cm) (0) (mV) (mV)

100

100 11.5

100 1000

200 1000

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

the second term in eq (5a)

t~?J .

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 bya quadratic equation in the argument RjRb' 

where is the ratio of VII 

the first (ASTM) wiring configurations and is the VII ratio in one of the two choices for

the second configuration. Specifically, the quadratic equation is

 

14.6% + 2$. 173 ( ) - 7.872 (

The accuracy of the fit over the range 1. 20 ~jR .::: 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, RjRb' 

is approximately 1.255. There are small variations, from about

25 to 1. , 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 RjR of 1. 255 results in a value of about 4.50. Over this

restricted range, the accuracy of fit of the quadratic, is actually about 0.01 %.
There are two independent considerations in evaluating cr

)

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

(t(0. 000 1 /3VK/) , or 3.33 x 10-

9 p

The second is the error in that would occur because of an error in measured voltages. For

a nominal value of RjR = 1.255, and any of the SRMs above 0.01 Q'cm, a voltage

measurement error of 2.4J.1.V would cause an error in RjR of no more than 0.00007. This

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

error, this means that the voltage error contribution to variance is H(O. OOO 38)2/3V(4.50)

2 or about 2.39 x 10-

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

voltage error of 2.4 J.l.V causes an error in the ratio RjRb' 

of 0. 000 24. This, in turn, results

in an error in of 0.001 29 and a contribution to the variance of p2 at 0. 01 Q-cm of

H(O. OOI 29) /3V(4. 50)2 x p , or about 2.74 x 10-

8 p

Adding these terms to that for possible error due to the quadratic representation of the

transcendental equation, the variance in p2 due to fossible error in the factor

a' 

07 x 10-8 p

at 0. 01 Q-cm, and is 5.72 x 10-9 p 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 p2 at 0.01 Q-cm and 1.428 x 10-8 p2 at all

higher resistivities.

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 oC to 24 oc for all SRM wafer measurements and

observed to be stable to 0. 1 oC, 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 OC. The temperature coefficient of resistivity

for silicon, which is a function of both resistivity and conductivity type, was evaluated at

NBS in the mid- 1960' s. This temperature coefficient is used internationally and is part of a
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 oC to 35 oC. The mercury bulb thermometer itself was calibrated by NIST with a stated

uncertainty of :to.03 o , or better. Thermistor resolution is better than 0.01 oC. Transfer

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

value of 0.02 oC 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 :to.08 OC. The calibrations of the glass

bulb thermometer and that of the thermistor are added to give a worst-case temperature

calibration error of :to.05OC. This is added linearly to the possible wafer-sensor offset of

08OC to give a worst case total temperature error of 0. 13 o

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 OC)2/6 == 0. 00282 eC)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)

(F T

where the temperature correction of resistivity, To has the form

= 1 ~ 

T (T 

~ 23 o c) ,

where is the temperature coefficient of resistivity, in degree Celsius

, .

and is the

temperature .at which measurements are made, in degree Celsius.

The variance in F r is really the variance in temperature (given above) times the square of the

temperature coefficient Since the coefficient, CTo varies noticeably as a function of
resistivity value, Table 3 summarizes the values of the temperature coefficient used in the

calculation of uncertainty.

Table 3. Temperature Coefficients of Resistivit,( of Silicon

for the Nominal Values of the SRMs

Nom. Res. (Q-cm) 0.1 100 200

Temp. Coeff. cocr1 0031 0041 0071 0082 0083 0083 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 

)2 . Because of

the difference in the values of 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 c:J (F r

Q-cm

Contribution to Variance

of Resistivity

100

200

71 x 10-

51 x 10-

1.42 x 10-

1.90 x 10-

1.94 x 10-

1.94 x 10-

1.94 x 10-

71 x 10-8 p

4.51 x 10-

8 p

1.42 x 10-7 p

1.90 x 10-7 p

1.94 x 10-

7 p

1.94 x 10-

7 p

1.94 x 10-

7 p

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
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 rom SRM wafers for SRMs 2541 to 2547 were lapped

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

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

earlier 50. 8 rom (2 in) diameter NBS silicon resistivity SRMs utilized a one-side-at-a-time

process and .a 5 p.m abrasive. As a result, the 100 rom 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. 

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 p.m and a short-term

repeatability of about 0. 1 p.m. The instrument's specifications state that its accuracy is

:1:0. 1 p.m if the .ambient temperature is kept at 20 oC :!: 1 oC. The requirement of a

temperature of 20 oC 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 oC temperature stability, the nominal working temperature is typically 23 o

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 co.mparable 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
~1 - Gauge Probe Terminates in Ball,

radius

Section of Wafer in Position for Measurement

(Texture Exaggerated)

Central Support Pedestal

on Thickness Gauge

Top Surface 2 mm x 2 nun

Baseplate

Figure 3. Conceptual drawing of wafer with two textured surfaces during thickness

measurement by electromechanical gauge.
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 11m

Measurement Accuracy :1:0. 1 11m

(At an Operating Temp of 19 oc to 21 O

Wafer Backside-Reference Pedestal 2 rom x 2 rom Probe Tip Radius 1 rom

Probe Working Force 1 N

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),

o(ti

p 

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

003 94 cm

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 11m (0.000 015 em) 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
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 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 11m 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 . Combining these two terms gives a variance of resistivity value directly due

to variance of thickness of 2.74 x 10-8 p

6 Evaluation of uncertainty due to thickness/probe separation scaling factor

A scaling factor F(IIS), 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),

2 0 (F(tls))

(tls)

where the denominator is effectively unity.

The scaling factor F(tlS), for dual-configuration measurements is similar to that F(wlS), for

single-configuration measurements in that they both asymptotically approach unity for values

of wlS just below 0.4. These scaling factors are virtually identical for values of the ratio wlS,

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 

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 11m (0.025 in) and a probe separation of 1587 11m (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 11m. 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 tiS 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
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 tiS, 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 ~m and with a typical

precision for 10 readings of about 0.06 % (1 ~m).

Wafer thickness and probe-spacing values for the 25 Q -em 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 tiS, ranges from 0.387 26

(F(tlS) = 0. 999 632), to 0.400 83 (F(tlS) = 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(tls) is 5. 07 x 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 -em SRMs. The second was

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

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

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 -em, 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 -em, fabricated from float-zone or neutrontransmutation-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 nun diameter SRM wafers were fabricated from boron-doped
Czochralski silicon from 0.01 Q-em to 10 Q-em 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 -em. 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 -em and 10 Q -em 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-em or

1 Q-em 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-em. 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 mill 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 -em boron-doped Cz crystal, these wafers were invalidated for use as SRMs. It was

possible to purchase a sufficient quantity of 10 Q -em wafers grown by the float-zone process

and phosphorus-doped by the NTD technique to be able to retain the 10 Q -em 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 -em 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-em

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 -em SRM level altogether and a judicious use of the

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

original 1 Q-em SRM wafers, i.e. , those wafers having the lowest amount of photosensitivity,
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 higherthan-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 ofthis 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, isa 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 ftom Sec;tion 5. It uses these

inputs to obtain the combined variance u;, the combined standard uncertainty c' 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.
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 need~d 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 and 5. 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 cmi, 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.

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 arid 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 is:

. . . .

2. = (S +s; +S: s; +S~ +S~g + a /3+ b2 /3+c /3

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

S(i

Selig

a/..J3

hl..J3

c/..J3

Short-term imprecision of certification probe

Run-to-run measurement variability

Longer-term measurement variability

Uncertainty of non-zero correction for bias of certification probe

Uncertainty of non-zero correction for probing induced drift (wafer-probe interaction)

Uncertainty of non-zero correction for probe-wiring configuration difference

Uncertainty of correction for probe wiring configuration (where the best correction = 0)

(Type A estimate, but based on limit of error)

Uncertainty of correction for bias of certification probe (where the best correction = 0)

(Type A estimate, but based on limit of error)

Uncertainty of correction for effect of illumination level (where the best correction = 0)

(Type A estimate, but based on limit of error).
40

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

in Appendices 2 through 8 and 10 through 12, mΩ.cm* # 

SRM Sε Sδ Sγ Sc S∆ Scfig a//3 b//3 c//3

2541 0.001 83 0.001 04 0.004 00 0. 0. 0.000 01 0. 0.000 47 0 

2541b2 0.002 32 0.001 04 0.001 17 0. 0. 0.000 19 0. 0.000 64 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 

2543b2 0.552 0.206 0. 0 0. 0.024 0 0.009 1.262 

2544 4.662 1.198 5.646 0. 0. 0.287 0. 0.204 0 

2544b2 4.666 1.095 0.0 0.26 0. 0. 0. 0. 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 whereas in the individual 

appendices the uncertainty components are expressed in ohm centimeters. 

 # Appendices 2-8 are in the sequence in which the SRMs were completed, not in the order of resistivity 

level 

Entries for 2541b2, 2543b2 and 2544b2, in this table and those following indicate “ second batch” 

certifications for these SRMs, with new control measurements and Type A uncertainty analyses for each of 

them (Type B uncertainty components remain unchanged from the first batch, “batch 1” of these SRMs). 

The revised certificates for the second batches are dated as follows: 

 2544b2: Sept 6, 2002; 2541b2: Jan 6, 2003; 2543b2: June 15, 2004. 

For the serial numbers to which these revised, batch 2, analyses and certificates apply, see Appendix 13. 

_______________________________________________________ 

 Table 8. Type A Standard Uncertainty Values, ui, Taken from Reports in Appendices 2 to 8 and 10 to 12 

 SHEET SHEET 

 RESISTIVITY RESISTIVITY RESISTANCE RESISTANCE 

SRM at center on circles at center on circles

 (mΩqcm) (mΩqcm) (mΩ) (mΩ) 

2541 "0.004 23 "0.0045 "0.067 3 "0.0716 

2541b2 "0.001 95 "0.0029 "0.0310 "0.0460 

2542 "0.045 "0.072 "0.725 "1.16 

2543 "1.72 "1.85 "27.5 "29.6 

2543b2 ±1.30 ±1.39 ±20.77 ±22.29 

2544 "6.09 "7.43 "96.9 "118.3 

2544b2 "2.03 "4.72 "32.3 "75.2 

2545 "7.8 "15.1 "125. "241. 

2546 "35.8 "74.8 "570. "1190. 

2547 "155. "199. "2470. "3180. 

Type A and Type B Variance Terms

In this section, Table 9 gives the variance terms constructed from the squares of the Type A standard 

uncertainty values listed in Table 8. Table 10 gives Type B variance values, obtained from the analyses in 

Section 5. Finally, Table 11 gives the combined variance, u

2

c , obtained by adding the Type A and Type B 

variances. 
41

 Table 9. Type A Variance Values, u

2

i, Obtained by Squaring the Entries in Table 8 

 SHEET SHEET 

 RESISTIVITY RESISTIVITY RESISTANCE RESISTANCE 

SRM at center on circles at center on circles 

 (Ω@cm)2

 (Ω@cm)2 (Ω2) (Ω2) 

2541 1.79 H 10-11 2.02 H 10-11 4.53 H 10-9 5.13 H 10-9 

2541b2 3.80 H 10-12 8.41 H 10-12 9.61 H 10-10 2.12 H 10-9

2542 2.02 H 10-9 5.18 H 10-9 5.26 H 10-7 1.35 H 10-6 

2543 2.96 H 10-6 3.42 H 10-6 7.57 H 10-4 8.76 H 10-4 

2543b2 1.68 H 10-6 1.93 H 10-6 4.31 H 10-4 4.97 H 10-4

2544 3.71 H 10-5 5.52 H 10-5 9.40 H 10-3 1.40 H 10-2 

2544b2 4.12 H 10-6 2.23 H 10-5 1.04 H 10-3 5.66 H 10-3 

2545 6.08 H 10-5 2.28 H 10-4 1.56 H 10-2 5.81 H 10-2 

2546 1.28 H 10-3 5.59 H 10-3 3.25 H 10–1 1.42 

2547 2.40 H 10-2 3.96 H 10-2 6.10 10.1 

----------------------------------------------------------------------------- 

 Table 10. Type B Variance Values, u

2

j, Calculated from Summation of Terms in Section 5.2 * 

 RESISTIVITY SHEET RESISTANCE 

 SRM at center & on circles at center & on circles 

 (Ω@cm) 2 (Ω2

) 

 2541 2.51 H 10-11 5.68 H 10-9 

2541b2 2.51 H 10-11 5.68 H 10-9

 2542 8.68 H 10-10 1.51 H 10-7

 2543 1.84 H 10-7 3.96 H 10-5 

 2543b2 1.84 H 10-7 3.96 H 10-5 

 2544 2.32 H 10-5 5.18 H 10-3 

2544b2 2.32 H 10-5 5.18 H 10-3 

 2545 1.47 H 10-4 3.30 H 10-2 

 2546 2.36 H 10-3 5.28 H 10-1 

 2547 9.43 H 10-3 2.11 

 * Type B variance components are the same for batches 1 and 2 of SRMs 2541, 2543 and 2544 

 

 -------------------------------------------------------------------------------- 

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 

 (Ω@cm) 2 (Ω@cm) 2 (Ω2

) (Ω2) 

2541 4.30 H 10-11 4.53 H 10-11 1.02 H 10-8 1.08 H 10-8 

2541b2 2.89 H 10-11 3.35 H 10-11 6.64 H 10-9 7.80 H 10-9 

2542 2.89 H 10-9 6.05 H 10-9 6.77 H 10-7 1.50 H 10-6

2543 3.14 H 10-6 3.61 H 10-6 7.97 H 10-4 9.16 H 10-4 
42

2543b2 1.86 H 10-6 2.11 H 10-6 4.71 H 10-4 5.37 H 10-4

2544 6.03 H 10-5 7.84 H 10-5 1.46 H 10-2 1.92 H 10-2 

2544b2 2.73 H 10-5 4.55 H 10-5 6.23 H 10-3 1.08 H 10-2 

2545 2.08 H 10-4 3.75 H 10-4 4.86 H 10-2 9.11 H 10-2 

2546 3.64 H 10-3 7.95 H 10-3 8.53 H 10-1 1.95 

2547 3.35 H 10-2 4.90 H 10-2 8.21 12.2 

 ----------------------------------------------------------------------------- 

Standard Uncertainty and Expanded Uncertainty

Table 12 gives values of the combined standard uncertainty, uc, and Table 13 gives 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 combined variance entries in Table 11. 

Table 12. Combined Standard Uncertainty Values, uc 

 SHEET SHEET 

 RESISTIVITY RESISTIVITY RESISTANCE RESISTANCE 

 SRM at center on circles at center on circles 

 (Ω@cm) (Ω@cm) (Ω) (Ω) 

2541 "0.000 006 56 "0.000 006 73 "0.000 101 "0.000 104 

2541b2 "0.000 005 38 "0.000 005 79 "0.000 081 "0.000 088 

2542 "0.000 0538 "0.000 077 8 "0.000 823 "0.001 22 

2543 "0.001 77 "0.001 90 "0.0282 "0.0303 

2543b2 "0.001 36 "0.001 45 "0.0217 "0.0232 

2544 "0.007 76 "0.008 85 "0.1208 "0.1385 

2544b2 "0.005 22 "0.006 74 "0.0789 "0.1039 

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) 

 SHEET SHEET 

 RESISTIVITY RESISTIVITY RESISTANCE RESISTANCE 

SRM at center on circles at center on circles 

 (Ω@cm) (Ω@cm) (Ω) (Ω) 

2541 0.000 013 1 0.000 013 5 0.000 202 0.000 208 

2541b2 0.000 010 8 0.000 011 6 0.000 163 0.000 177 

2542 0.000 108 0.000 156 0.001 65 0.002 45 

2543 0.003 54 0.003 80 0.0564 0.0605 

2543b2 0.002 73 0.002 90 0.0434 0.0464 

2544 0.015 53 0.017 70 0.241 0.277 

2544b2 0.0104 0.0135 0.1578 0.2078 

2545 0.0288 0.0387 0.441 0.604 

2546 0.121 0.178 1.85 2.78

2547* -0.498, +0.366 -0.575, +0.443 -7.83, +5.73 -9.08, +6.98 

* Asymmetry is due to a contribution of 0.132 Ω@cm, 2.10 Ω, from a wafer-probe interaction, i.e., a drift in 

value with repeated probing. 
43

7.2 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. 

Table 14. Probe Bias Corrections Applied to Measured Values 

SRM BIAS CORRECTION to Resistivity (Sheet resistance) SOURCE 

2541 Subtract 0.000 000 472 Ω@cm (0.000 007 52 Ω ) Wiring Configuration Bias 

 [This is a Negligible Amount] 

2541b2 Subtract 0.000 002 14 Ω@cm (0.000 034 08 Ω) Wiring Configuration Bias 

2542 Subtract 0.000 037 5 Ω@cm (0.000 597 Ω) Wiring Configuration Bias 

2543 Subtract 0.000 131 Ω@cm (0.002 10 Ω) Wiring Configuration Bias 

2543b2 Subtract 0.000 25 Ω@cm (0.004 00 Ω) Wiring Configuration Bias 

2544 Subtract 0.0011 Ω@cm (0.017 Ω) Wiring Configuration Bias 

2544b2 Subtract 0.000 816 Ω@cm (0.0130 Ω) Probe Bias 

2545 None ---------------------------- 

2546 Add 0.0393 Ω@cm (0.626 Ω) Probe Bias 

2547 Subtract 0.0490 Ω@cm (0.78 Ω) Probe Bias 

 [a wafer drift term of 0.132 Ω@cm, (2.10 Ω), is built into expanded uncertainty for SRM 2547] 

_______________________________________________________ 

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 

effect contributing 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 

 2541b2 24 109 

 2542 88 486 

 2543 16800 15415 

 2543b2 25820 21512 

 2544 6 14 

 2544b2 68 607 

 2545 52 528 

 2546 41 543 

 2547 26 66
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 MegaWorkshop, i. , 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. 11% to 0.25%, of the nominal resistivity for all SRMs except 2543 at 1 !lcm; for this

SRM the expanded uncertainty is 0.35%. The relative uncertainty ofindividual resistivity values

on the two circles increases slightly for several resistivity levels where probe imprecision was one

ofthe larger contributors to uncertainty, but it is still less than 0.2% for five ofthe SRM levels

and is 0.38% at 1 Q' cm and 0.28% at 200 Q' cm. 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 ITom a number of disparate-format pieces of source

material.

REFERENCES

(1) ASTMMethod F84-

, 

Standard Method for Measuring Resistivity of Silicon Wafers

with an In-Line Four-Point Probe, " Annual Book of ASTM Standards Vol. 10. , 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- , 1991

, p.

(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 92-67-10188- , 1st Ed.

ISO, Geneva, Switzerland, (1993).
(5) Taylor, B.N. and Kuyatt, C. E. , Guidelines for Evaluating and Expressing the

Uncertainty of NIST Measurement Results, NIST Technical Note 1297, U.

Government Printing Office, Washington DC (1994).

(6) ASTM Method F1529-

, 

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. , West Conshohocken, PA 19428..

(7)

(8)

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

Perloff, D. S., Gan, 1. N. and Wahl, F. E., Dose Accuracy and Doping Uniformity of

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

(9) 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).
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

2541

2542

2543

2544

2545

2546

2547

SRM level Crystal OrientiGrowthIDopant Supplier

(in Q-cm)

91905 (100) Cz Boron Recticon Corp.

0.1 91904 (100) Cz Boron Recticon Corp.

91907 (100) Cz Boron Recticon Corp.

29473 (111) FZ-NTD Phos. Wacker Siltronic

21565 (111) FZ-NTD Phos. Topsil Semi. AIS

100 51939 (111) FZ-NTD Phos. Topsil Semi. AIS

200 21566 (111) FZ-NTD Phos. Topsil Semi. AIS

Table 2. Electrical Measurement Conditions That Apply to Various Resistivity Levels

of the SRMs 2541 to 2547

SRM

2541

2542

2543

2544

2545

2546

2547

Probe

283

281

283

283

2062

2362

SRMl

Standard Resistor

Nominal Value/

Serial Number

1 Q / 1771494

1 Q / 1594503

10 Q / 1593079

100 Q / 1598893

100 Q / 1598893

1000 Q /1592167

1000 Q /1592167

Nominal

Measurement

Current

100 mA

28 mA

8 mA

260 IlA

11 0 IlA

29 IlA

14 IlA
Appendix 2. Analysis of Certification Data

and Control Experiments for SRM 2547

1. GENERAL COMMENTS AND SUMMARY OF TYPE A STANDARD

UNCERTAINTY COMPONENTS

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 of200 Q' , 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 rom, 5 rom, and 10 mm from the center of each wafer, with

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

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 SRMI 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 SRMI.

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

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

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 ofthe

terms that contribute to the Type A standard uncertainty. The details of the calculation of

the component terms are given in subsequent sections.

2 Certified Resistivities and Uncertainties

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

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

uncertainty associated with the certified value for the wafer center is

s/ s,./ 

+ "6 

s/ = 0. 155 o.' cm 

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

error of -8.. See Section 3.3 for details. Because the uncorrected systematic error is

always in one direction, the expanded uncertainty interval is nonsyrometric and is

expressed as

Certified value - (2 Ui l!), Certified value + 2 Ui 

where 2 Ui = 0.310 Q' , and l! = 0. 132 Q' cm .

Individual measurements on the 5 mm and 10 mm circles for each wafer, corrected for

the effect of probe SRMl , are reported as certified values on the certificates. The Type 

standard uncertainty associated with each of these individual certified values is

U. 

2 + 2 + 2 + 2 + S 2

1/2 = 0.20 0. cm 

The expanded uncertainty interval for individual measurements is then expressed 

Certified value - (2 Ui l!), Certified value + 2 Ui 

where 2 Ui for individual measurements = 0.40 g' cm.
Table 1. Components of Type A Standard Uncertainty for Crystal 21566, SRM 2547 with Probe SRM1 , Q'

RandomRandom

RandomRandomRandom

Source

Imprecision of probe SRMIRun-to-run measurement variability

Long-term measurement variability

Standard deviation of correction for probe SRMI biasStandard deviation of wafer drift with probing

Std dev Estimate

138

064

129

005010 S!:J.

.,J:..

SystematicSystematic

Systematic

Diff~ence between configurations bland b2

Difference between front and back faces of wafer

Neglected correction for wafer drift with probing

NegligibleNegligible

132

* A correction of - 049 Q'cm due to probe SRMI bias is applied to wafer center resistivity data for all wafers.
Table 2. Sources of Variation for Crystal 21566, SRM 2547 with Probe SRMl

, g

Source of error Experiment RMSEa DFb Relationship

SRM certifications 1374 685

Probe imprecIsiOn Pre-certification 1586 150

Post-certification 1134 150

Check standard 1367 105

Pooled 138 1090

Run-to-run Pre-certification 0857 1/2 ( So + tJ SE )

Post-certification 0926

Check standard 0772

Pooled 0854

Pre- and post-certification 1116 1 2 1 2 + :; S + jU S

Long-term #900 wafers 1477

Pooled 1350

aThe root-meaD-square error, RMSE, estimates within each source-of-error category are pooled in the table above. Standard deviations

associated with the individual effects, namely, imprecision, run-to-run variability, probe variability, and long-term variability, are computedusing the relationship shown in the last column, with the results summarized in Table 1 of this Appendix.bDegree of Freedom.

CThis column expresses the error components that comprise the pooled value in each" source of error" series of experiments; see reference at end

of this Appendix.
2. RANDOM COMPONENTS

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 == J.I. + Vi + oij + Eijk i=I 2; j=I

,...,

6; k=I

,...

(I)

where J.! is the average value, Vi is a component for long-term error; oij is a component of

run-to-run measurement error; and tijk 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,
Wafer

..

196. Iii

196.

196.

.&J

.r! 196.

::

.r! 196 .&J

.r! 195.

P:\ 195.

195.

Month/Date

Wafer 60

193.

..

Iii

193.

193.

.&J 193. .r!

::

.r! 193. .&J

.r!

193.4

P:\

193. 7 7. 8 8.

Month/Date

Wafer 100

192.

. 192.

192.

.&J

.r! 192.

.... ::

.&J 192.

'r!

: 192.

P:\

192. 7 7. 8 8.

Month/Date

Crystal 21566

Wafer 40

194

..

Iii

193.

.&J 193. .r!

::

.r!

.&J

193. 'r!

P:\

193. 7 7. 8 8.

Month/Date

Wafer 80

7 7. 8 8.

Month/Date

193

192.

t' 192.

'r!

::

:j 192.

.r!

V1 192.

P:\

192.

Figure 1, Resistivity (O'cm) on five control-wafers from crystal 21566 with probe

SRMI plotted versus the month/day of measurement showing change between

and within pre- and post-certification experiments,
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, Q'

Std. dev, Sf:, Std. dev, Sf:,

Probe Config hI Config b2

SRMI 1586 150 1907 150

281 0.2235 150 2468 150

Pre 283 2139 150 2389 150

2062 0.1645 150 0.2043 150

2362 1520 150 1635 150

SRMI 1134 150 1280 150

281 2102 150 2217 150

Post 283 0.1687 150 2115 150

2062 0.1568 150 1770 150

2362 0.1269 150 0.1374 150

Pooled

Value: SRMI 1379 300 0.1624 300

2 Run-to-runmeasurement 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 SRMI of 0.089 20 Q' cm with 40 degrees of

freedom incorporates both probe imprecision and day-to-day measurement error as shown

in the relationship column of Table 2,
Table 4, Run-to-Run Component of Error, Crystal 21566 

Averages and Standard Deviations for Last Five Runs

on Each Control-Wafer

, .

Pre-certification Post-certification

Wafer# Probe Average Std dev Average Std dev

SRMl 196. 2431 0875 196. 0078 1574

SRMl 193. 8663 0615 193. 7433 0605

SRMl 193. 5259 0795 193. 4869 0693

SRMl 192. 8096 1127 192. 6597 0777

100 SRMl 192. 5503 0835 192. 3768 0596

281 196. 2443 1380 196. 0423 2617

281 193. 8445 1105 193. 7325 0948

281 193. 5903 0935 193, 4285 1399

281 192. 7595 0 . 0826 192. 6768 1939

100 281 192. 5428 1189 192. 3991 0930

283 196. 1670 0937 195. 9598 1525

283 193. 7223 0 . 0499 193. 6426 0951

283 193. 4253 0536 193. 3253 0992

283 192. 7630 0396 192. 5120 0945

100 283 192, 4705 0545 192. 3259 0824

2062 196, 1481 1042 195. 9211 2248

2062 193. 8217 0 . 0957 193. 7494 0711

2062 193. 4647 0723 193. 4411 0355

2062 192. 7436 0727 192. 6205 1538

100 2062 192 . 4263 0412 192. 3818 0; 0644

2362 196. 1432 0884 195. 8630 2282

2362 193. 7696 0681 193. 7181 0667

2362 193. 4426 0581 193. 3722 0775

2362 192. 7206 0920 192. 5816 0810

100 2362 192. 4557 1279 192. 2694 1589

Pooled

Val ue: SRM1 085 092
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'

Wafer Pre-Certification Post -Certification Difference Std dev

100

196.2431

193. 8663

193.5259

192.8096

192,5503

196.0078 2353 1664

193,7433 1230 0870

193.4869 0390 0276

192,6597 1499 1060

192.3768 1735 1227

Pooled

Value: 0.1116

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-today 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,
Table 6. Long-Term Changes In Measurement Process with SRM1 , Q'

Wafer Face September October Difference Std dev

901 192,748 192.839 +0,091 0643

902 195,654 195,593 061 0431

903 200.866 201.096 +0.230 0.1626

904 190.982 191.381 +0.399 2821

901 192,704 192,974 +0.270 0.1909

902 195.478 195,533 +0.055 0389

903 200,955 201,098 +0.143 1011

904 191.198 191.367 +0.169 1195

Pooled

Value: 1477

2 Check-Standard Measurements

Twenty-three measurements (averages of six center measurements each) with probe

SRMI 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.
Crystal 21566

204 .

204.

,.

C 204.

2 0 4 . 5 

......

~ 204,

204 . 3 -

204 ,

Date In August

Figure 2. Resistivity measurements (O'cm) on check-wafer #150, with probe SRMI as a

function of time (date in August)
3. SYSTEMATIC COMPONENTS

1 Systematic Differences between Probe-Wiring Configurations bland b2

In the pre- and post-certification .experiments, six measurements at the center with the

probe in configuration b 1 were immediately followed by six measurements with the

probe in configuration b2. The differences between configurations b 1 andb2 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

t = -vro Average/Std dev ,

shows no evidence of a significant difference between configurations bland 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,

Table 7, Average Differences over All Control-Wafers

between Configurations bl and b2 for Probe SRMl, Q'

Probe

Pre-certification

Average Std dev

Post-certification

Average Std dev 

SRMI 002 44 044 49 020 56 031 38
Crystal 21566

O. 2

0, 15

I;j

0, 1

I;j

O. 05

...

Ci-i

..

- 0 ,

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 SRMl, Q.cm.

Legend: 0 = pre-certification; * = post-certific&tion
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 SRMI 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 SRMI (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 

049Q' cm. The standard deviation of the differences is divided by -ffU. . to

obtain the standard deviation of the correction Sc = 0.0050 Q'cm. The correction, C , 

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 SRMI for Each of the

Control-Wafers, Q.

Wafer

100

Pre-certification

054

061

036

050

061

Post-certification

049

026

076

050

026

Mean difference = - 049 g'

Standard deviation s= 0.0159 Q'

Correction to be applied = - 049 Q'

Standard deviation of correction s/VI0 = 0.005 Q'
Crystal 21566

O. 15

9., 0.

...

~ 0,

....

- 0.

- 0 ,

100 120

Control Wafer Number

Figure 4. Differences between individual probe responses and multiprobeaverage, for

pre-certification measurements on each of the control-wafers.

Plot symbol code: 1 == SRMl; 2 = 281; 3 = 283; 4 = 2062; 5 = 2362
Crystal 21566

0 ,

O. 05

....

Coi-i

....

- 0.

-0.

100 120

Control Wafer Number

Figure 5, Differences between individual probe responses and multiprobe average, for

post-certification measurements on each of the control-wafers,

Plot symbols: 1 = SRMl; 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 Q' cm 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 SRMl, the

resistivities always dropped after the first measurement for probe SRMI (see Fig. 1). The
average drop is A = 0, 132 Q' , and the standard deviation is = 0.0427 Q'cm, A

correction of - 1112 would assume an equal probability of initial damage between 0 and

8. Q' cm. 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 l! is added to the lower

limit of the expanded uncertainty. The standard deviation associated with 8. is s//20 

or 0, 010 Q' , 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 SRMl , Q'

Wafer

100

Pre-certification

086

0.122

174

177

0.113

Post-certification

185

072

122

093

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 SRMl , i, , ten times out often 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 often; 2362 - four out often; and 283 - six out often

possibilities. Thus, the effect is stronger for probe SRMI 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.
196.

196. 196. 196.

196.

196 . 196

195 . 9

Wafer 20

10 

Wafer 40

193 .

193 .

194

193.

193.

,....

a 193.

~ 193.

C 193. '-"193.

~ 193.

S: 

1 9 3 . 4

.,. 

1 93 . 3

..... 

6 0

rIl

.,.

rIl

192 .

192 .

40 

Wafer 60

70 

Wafer 80

193

100 105

Wafer 100

110 115 120

192.

192.

192 .

192 .

192 .

192.

192 .

192.

192 . 120 125 130 135 14 0 145 15 D

Cumulative Measurement Number

Figure 6. Resistivity (O'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 == +, SRMI 
192 .

192 . 192 .

192 . 192 .

192 . 192 .

192 191.9120 125 130 135 140 145 150

Cumulative Measurement Number

196 . 196 , 196. 196 .

196 . 196 195 , 195 . 195 , 195 .

194 .

194 . 194

193 .

193 .

193 .

193 . 6 193 . 193.

e 193. CJ 193.

At 193. '-".I 193.

~ 193.

....,.

1 93 . 2

s: 

1 93 . 1

ofllllll 

...

rIl

ofllllll

rIl

ell

193

192 .

192 .

192 .

192 .

192 .

192 .

192 .

Wafer 20

10 

Wafer 40

40 4.

Wafer 60

70 

Wafer 80

100 105

Wafer 100

110 115 120

Figure 7. Resistivity (O'cm) from post-certification measurements for five controlwafers from crystal 21566 vs. cumulative measurement run number,

Plot symbol code: 281 = 0; 2062 ::::: 1; 2362 = 2; 283 

SRMI =

Reference:

Graybill, F. A., An Introduction to Linear Statistical Models, Vol. 1 (McGraw-Hill, New

York, 1961), pp. 349-351.
Appendix 3. Analysis of Certification Data and Control Experiments

for SRM 2541

1. GENERAL COMMENTS AND SUMMARY OF TYPE A STANDARD

UNCERTAINTY COMPONENTS

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

01 Q' , 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. , and details of the analysis for a systematic bias for probe 283 in Section 2.

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.

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, 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

b2 

Ui = "3 + Sy + SF, + 6" = 0.000 004 2 Q'cm (0.0042 mQ.cm) .
Table 1. Components of Type A Standard Uncertainty for Crystal 91905, SRM 2541 with Probe 283

, g

RandomRandom

Source

Imprecision of probe 283Run-to-run measurement variability

Std dev Estimate Se 0.000001 83

Random Long-term measurement variability

00000104

000 004 00

0'1-....J

Systematic Uncertainty of zero correction for probe 283 bias b/43 00000047
Table 2. Sources of Variation for Crystal 91905, SRM 2541 with Probe 283, Q'cm

0'1

Source of error Experiment RMSE a DFb Relationship

SRM certifications 000001 70 650

Probe Imp~ecision Pre-certification 0:000 002 30 150

Post- certification 000 002 02 150

Check standard 000001 105

Pooled 000001 83 1055

Pre- certification 000001 59

Run-to-run Post-certification 000001 20

1/2

( So + 0 Se )

Check standard 000 000 87

Pooled 000001 28

Long-term Pre- and post-certification 000 004 03 1 2 1 2 1/2 ( Sy + 0" s + 30 SE )

aThe root-mean-square error, RMSE, estimates within each source-of-error category are pooled in the table above. Standard deviations

associated with the individual effects, namely, imprecision, run-to-run variability, probe variability, and long-term variability, are computed

using the relationship shown in the last column, with the results summarized in Table 1 of this Appendix.

bDegrees of Freedom.

cThis column expresses the error components that comprise the pooled value in each " source of error" series of experiments; see reference at the

end of Appendix 2.
The Type A standard uncertainty for individual resistivity values on the 5 mm and 10 rom

radius circles is .

ui ,13+s: +s; +s; = 0.0000045 Q'cm (0.0045 mO'cm).

2. SYSTEMATIC EFFECTS

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 - 000 000 68 g' cm with a standard deviation of the average of

000 000 21 g'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 ::J:b where

b = 0,00000082 Q' , and a contribution ofblf3 = 0.00000047 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

, g

Wafer Pre-certification Post-certification

000000 16 000 000 32

000 000 58 000 000 58

000 000 48 000001 10

000 001 44 000 000 86

144 000 000 94 000 000 26

Mean 000 000 53 000 000 82
Appendix 4, Analysis of Certification Data .and Control Experiments

for 8RM 2542

1, GENERAL COMMENTS AND SUMMARY OF TYPE A STANDARD

UNCERTAINTY COMPONENTS

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

10. 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. 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.

2 Certified Resistivities and Uncertainties

The averages of six measurements on the 0 mm circle, and individual measurements on the

5 mm and 10 rom 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

b2 1 2

Ui= 

3 + 

cftg + Sy + SI) + "6 S

& = 0,000 045 Q' cm .

The Type A standard uncertainty for the individual resistivity values on the 5 rom and

10 mm radius circles is

Ui 

= 3 + 

S~g s: si + s; == 0.000 072 Q' cm .
Table 1. Components of Type A Standard Uncertainty for Crystal 91904, SRM 2542 with Probe 281

, g

TypeRandomRandom

RanqomRandom

Source

Imprecision of probe 281Run-to-run measurement variability

Long-term measurement variability

Standard deviation of correction for wiring configurations

....) ..

Systematic Uncertainty ofa zero correction for probe 281 bias

Std dev Estimate

000 062000 032

000 004000011 cfig

b/13 000016

* A correction of - 0000375 g' cm due to wiring-configuration bias is applied to wafer center resistivity data for

all wafers.
Table 2. Sources of Variation for Crystal 91904, SRM 2542 with Probe 281 , Q.

.....)

Source of error Experiment RMSEa DFb Relationship

SRM certifications 000 046 645

Probe ImpreCIsion Pre-certification 000081 150

Post- certification 000 074 150

Check standard 000 086 120Pooled 000 062 1065

Pre- certification 000 028 9 1 2 1/2

Run-to-run Post-certification 000 042 5 + ?) SE )

Check standard 000 044 6Pooled 000 040 5

Long-term Pre- and post-certification 000017 1 2 1 2 ) 1/2 + ?) S + 30 S

Pooled 000017

aThe root-meaD-square error, RMSE, estimates within each source-or-error category are pooled in the table above. Standard deviations

associated with the individual effects, namely, imprecision, run-to-run variability, probe variability, and long-term variability, are computed

using the relationship shown in the last column, with the results summarized in Table 1 ofthis Appendix.bDegree of Freedom,

cThis column expresses the error components that comprise the pooled value in each " source of error" series of experiments; see reference at the

end of Appendix 2.
2. SYSTEMATIC EFFECTS

1 Differences between Wiring Configurations bland 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 bland b2 

t = ...p:o 

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 - 0000375 Q' , is applied to all certification measurements to obtain an

average over the two configurations, The standard deviation of the correction

- 1 

cjig 

2 +8 2 -

..fiO 

2 - 0.000 011 0. cm 

where SI 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

s4mdard uncertainty for the process.

Table 3. Average Differences and Standard Deviations

between Wiring Configurations bl and b2, Q'

Probe

Pre-certification Avg SD 

000 085 0.000 064 

Avg

Post-certification

SD 

281 000 065 000 072 19 4,
Appendix 5, Analysis of Certification Data and Control Experiments

for SRM 2545

1. GENERAL COMMENTS AND SUMMARY OF TYPE A STANDARD

UNCERTAINTY COMPONENTS

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' , 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. , 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,

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

1 2 112

Ui 1""+Sy +s = O,O08Q' cm.

The Type A standard uncertainty for the individual resistivity values on the 5mm and

10 mm radius circles is

1/2

Ui= ""3""+Sy +so +S = 0,015 Q'cm,
-...J

Table 1. Components of Type A Standard Uncertainty for Crystal 21565, SRM 2545 with Probe 2062

, g

Type

RandomRandom

Random

Systematic

Source

Imprecision of probe 2062

Run-to-run measurement variability

Long-term measurement variability

Std dev

Uncertainty of a zero correction for differences between wiring configurations a/13

Estimate

014 14

003 31

00301002 89
Table 2. Sources of Variation for Crystal 21565, SRM 2545 with Probe 2062, Q.cm

...)

0'1

Source of error Experiment RMSEa DFb Relationship

SRM certifications 01493 650

Probe imprecIsIon Pre-certification 011 150

Post-certification 01467 150

Check standard 01204 120

Pooled 014 14 1070

Pre-certification 006 968 1 2 1/2

Run-to-run Post-certification 004 588

( $

& + b SE )

Check standard 008 056

Pooled 006 655

Long-term Pre- and post-certification 004 057 1 2 112 ( Sy + b S + 30 S

Pooled 004 057

aThe root-meaD-square error, RMSE, estimates within each source-of-error category are pooled in the table above. Standard deviations

associated with the individual effects, namely, imprecision, run-to-run variability, probe variability, and long-term variability, are computed from

the relationships shown in the last column, with the results summarized in Table 1 of this Appendix.

bDegrees of Freedom.

cThis column expresses the error components that comprise the pooled value in each" source of error" series of experiments; see reference at the

end of Appendix 2.
2. SYSTEMATIC EFFECTS

1 Differences between Wiring Configurations b 1 .and b2

Differences are found between measurements in configurations bland 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 = 119 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 :l:a, where a = 0.005 Q'cm 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/ .J3 , or 0.002 89 Q'cm,

Table 3, Average Differences and Standard Deviations

between Wiring Configurationsb 1 and b2, O'

Pre-certification Post-certification

Probe Avg Avg

2062 00383 005 14 + 0,004 89 004 00
Appendix 6, Analysis of Certification Data and Control Experiments

for SRM 2546

L GENERAL COMMENTS AND SUMMARY OF TYPE A STANDARD

UNCERTAINTY COMPONENTS

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 , 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 ofSRM 2547, Section 2, 1 summarizes an

analysis of a probe bias correction that follows the procedures used in Appendix 2,

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

1 2 1/2

Ui = Sc y +S = O, 036g' cm.

The Type A standard uncertainty for individual resistivity values on the 5 mmand 10 

radius circles is

1/2

Ui 

( 

+ S o +S

& ) 

= O,075cm.
Table 1. Components of Type A Standard Uncertainty for Wafers from Crystal 51939, SRM 2546 with Probe 2362, Q'

RandomRandomRandom

Random

Source

Imprecision of probe 2362

Run-to-run measurement variabilityLong-term measurement variability

Standard deviation of the correction for probe 2362 bias

Std dev Estimate

0720

01340146

0051

......:J

*A correction of 0.0393 Q'cm due to probe 2362 bias is applied wafer-center resistivity data for all wafers.
Source of error Experiment RMSEa DFb Relationship

SRM certifications 074 6 650

Probe 1mprecisIOn Pre- and post-certification 071 0 300

Check standard 059 6 125

Pooled 072 0 1075

Run-to-run Pre- and post-certification 036 2

1 2 ilL.

( s& + 0" SE )

Check standard 026 8

Pooled 032 3

Long-term Pre- and post-certification 0197 1 2 1 2 112 + 0" + 30 S

Pooled 0197

Table 2. Sources of Variation for Wafers from Crystal 51939, SRM 2546 with Probe 2362, Q.

aThe root-meaD-square error, RMSE, estimates within each source-of-error category are pooled in the table above, Standard deviations

associated with the individual effects, namely, imprecision, run-to-run variability, probe variability, and long-term variability, are computed from

the relationships shown in the last column, with the results summarized in Table 1 of this Appendix.

bDegr~es of Freedom.

cThis column expresses the error components that comprise the pooled value in each " source of error" series of experiments; see refer~nce at the

end of Appendix 2.
2. SYSTEMATIC EFFECTS

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'

Wafer#

138

139

140

141

142

Pre-certification

0372

0094

0261

0252

0383

Post-certification

0507

0657

0398

0534

0469

Mean Bias, C 0.0393 Q'

Standard Deviation of Mean 0.0051 Q'
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' , 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

e. #11 , #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.
However, a correction for choice of probe wiring configuration was necessary and was

applied to all measurements on certified SRM wafers.

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 cent~ is:

b2 1 2 f"\

3" 3+Scfig +Sr +S

= 0. 001 721:a~' 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:

2 +S2 +S2 +S = 000185 Q'cm.

cfig r 
Table 1. Components of Type A Standard Uncertainty for Crystal 91907, SRM 2543, with Probe #283, Q' 

Random

RandomRandom

Random

Systematic

Systematic

Source

Imprecision of probe #283Run-to-run measurement variabilityLong-term measurement variability

Standard deviation of correction for wiring configurations

Stand. Dev. Estimate

000714000192

000154

000058 cfig

Uncertainty of a zero correction for probe #283 biasUncertainty of value due to ambient illumination conditions

bl.J3c/ ../3

000038

001682

* A correction of - 000 131 n. , due to probe wiring configuration differences, is applied to resistivity measurements

for all SRM wafers.

.,J:..
Table 2. Sources of Variation for Crystal 91907, SRM 2543, with Probe #283, Q'

Source of error Experiment RMSe Relationship

SRM certifications 000541 380

Pre-certification control wafer 000897 150

Probe imprecision Post-certification control wafer 000 866 150 Sf:

Check standard wafer 000724 125

Pooled 000714 805

Pre-certification control wafer 000312

Post-certification control wafer 000 336

Run-to-run Check standard wafer 000 395 (2 1 2 Y!2 6" 

Pooled 000349

Long-term Pre- and post-certification 000210 (2 1 2 1 2 

Y!2 + 6" 

a The individual root-mean-square error estimates (RMSE) within each source-of-error category are pooled in the table above.

Except for the experiments to estimate probe imprecision, the RMSE values are comprised of more than one component of

variation. Standard deviations associated with the individual effects, namely: imprecision, run-to-run variability, and long term

variability, listed in Table 1 , are computed from the pooled RMSE values using the relationships in the last column.

Degrees of Freedom.

This column expresses the components of variation that comprise the RMSE values for the experiments in each "source of

error" category (see the reference at the end of Appendix 2).
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), 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).

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 b 1 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.

Probe

Pre-certification

Config. b Config. b2

Post-certification

Config. b Config. h2

SRMI

281

283

2062

2362

000934

000767

000 897

000 908

001 129

001 619

000 878

000 981

001 793

000 788

000763

000 748

000 866

000 799

001 183

001 079

000 872

000 746

001 497

000816
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 exp~iments. The

pooled standard deviations, 0.000 3123 Q.cm and 0.000 3363Q. , 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 Run-to-Run Variability for Crystal 91907 Control Wafers Averages and Standard

Deviations for Six Runs

Pre-certification Post-certification Difference

Wafer Probe Resistivity Standard Dev. Resistivity Standard Dev. (pre.~ Post.

SRMI 073 76 0002436 1.073 94 000 355 0 - 0.000 18

SRMI 1.060 98 000 628 3 1.060 74 0002309 000 24

SRM1 1.046 49 000341 5 1.046 63 000 302 5 - 0. 000 14

131 SRMI 991 54 000274 6 99162 000 173 6 - 0.00008

208 SRM1 96249 0002122 962 34 000259 0 000 15

281 1.073 44 000405 5 07350 000265 1 - 0.000 06

281 060 73 000 266 8 1.060 63 000 174 1 000 10

281 1.04606 000457 3 1.04622 000 315 9 - 0. 000 16

131 281 991 46 000 312 3 991 34 0004978 000 12

208 281 962 07 0004146 962 36 000 1942 - 0.000 29

283 073 28 000 294 0 073 26 000 636 0 000 02

283 1.060 48 000 244 1 060 70 000 251 3 - 0.000 22

283 046 09 000 369 9 1.046 27 000 148 8 - 0.000 17

131 283 99112 000 328 3 991 60 000 140 4 - 0.000 48

208 283 961 96 000 314 1 96232 000 236 8 - 0.000 36

2062 1.072 90 000 477 0 1.073 28 000 326 2 - 0.000 38

2062 1.060 63 000 305 4 1.06049 000 509 5 000 14

2062 045 84 000 3992 1.045 85 000 166 3 - 0.000 02

131 2062 991 02 0004472 991 33 000405 8 - 0. 000 31

208 2062 961 65 000406 5 961 77 0007510 - 0. 000 12

2362 1.072 85 000 349 4 1.073 24 000482 3 - 0.000 39

2362 1.060 08 000 352 7 1.06051 0005185 - 0.000 43

2362 1.045 88 000 650 7 1.045 99 0002638 - 0. 00011

131 2362 99081 000621 5 991 32 000455 7 - 0.000 50

208 2362 961 75 000 180 8 961 87 000 493 8 - 0.000 11

Probe 283 (pooled across wafers) 000 312 3 000 336 3
1. 074

61, 0735

';: 1, 07T

......... ;;:

t; 1. 0725

Wafer 11

8 5

MonthlDate

Wafer 42

8 5

MontblDate

Wafer 208

062

1. 0616

'-' 1. 0612

;;:.

.;::: 1. 0608

1. 0604

Wafer 26

8 5

Month/Date

Wafer 131

8 5

MontblDate

Figure 1. Resistivity (Q'cm), 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 (0) plot symbols, control measurements.

1. 0 72'

1 046

,......

1 046

';: 1 046

..... ....

1 045

1 045

.....

0 961

O. 961

8 5

MonthlDate

1.06

992

,......

~ 0. 9915

. ... ,,.

0 . 991

.....

::i1 ~ 0. 9905
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.

Pre-cert. Post-cert.

Wafer Average Average Difference Stand. Dev.

1. 073 1.073 26 00002 000014

1.060 48 1.06070 00022 000 156

046 09 1.046 27 000 17 000 127

131 991 12 991 60 000 48 000339

208 961 96 96232 00036 000255

Standard deviation after pooling across control wafers 000210

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 Q' cm with 125 degrees of freedom. This pooled estimate is given

in Table 2 as theRMSE of the probe imprecision from check standard measurements. Also

given in Table 2 is the run-to-run standard deviation, 0.000 349 Q' , of the average values

from the 25 measurement runs.
s::

.....

I/.)

0011

0010

0009

0 . 0008

0007

0006

0005

0004

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.
3. SYSTEMATIC EFFECT COMPONENTS OF UNCERTAINTY

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 

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 SRMcertification

the functional bias could have fallen somewhere within the limits :I: b where

b = 0.000 0652Q.cm; .a standard uncertainty of b-fj = 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 cm.

Wafer Probe Pre-certification Post-certification

SRMI 0005140 0004960

281 0001940 000 056 0

283 000 034 0 000 184 1

2062 000 345 9 000 1640

2362 0003960 000 204 0

SRMI 000 399 9 000 1260

281 000 1500 0000160

283 000 100 0 000 086 1

2062 000 049 9 000 124 0

2362 000 500 0 000 1040

SRMI 000418 000438 

281 0000119 000 028 0

283 000 018 1 000 078 1

2062 0002319 000341 9

2362 - 0. 000 191 9 000201 9

131 SRMI 0003499 000 178 0

131 281 000 269 9 000 1020

131 283 - 0.000 070 1 000 158 0

131 2062 0001701 000 1120

131 2362 - 0.000 380 1 000 1220

208 SRMI 000 506 0 000 207 9

208 281 000085 9 0002279

208 283 - 0.000 024 0 000 187 9

208 2062 - 0. 000 334 0 000 362 1

208 2362 - 0. 000 234 1 000262 

Probe 283 (average) - 0. 000 028 4 +0.000065 2
001

i:Q

,...

0005

-0. 0005

-0. 001

0005

0004

0003

0 . 0002

0001

-0. 0001

0002

-0. 0003

0004

1 1

2 2

3 4 ~

3 ~

Pre-Certification Measurements

100 150 200 300

Control Wafer Number

Post-Certification Measurements

250

300

2 ~ 3

2 2

~ ~ 5

100 150 . 200

Control Wafer Number

Figure 3. Bias, in n' , of individual probes, relative to the multiprobe average value for

each of five control waf~s during the pre- and post-certification control measurements;

probe # 283 is symbol 
2 Difference Between Wiring Configurations b 1 & b2

Differences between measurements in wiring configurations b I & 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

b 1 & b2 is t = ..rw Avg/SD. The plots and t-statistics suggest a slight difference among

wiring configurations for this batch ofSRM 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' , is applied to all certification

measurements to obtain an average over the two configurations. The standard deviation of

the correction is:

cfig

:

b1 b2 = 0.000 058 Q' cm. 2 '" 30

Table 7. Differences Between Wiring Configurations bl & b2 for Six Days of

Control-Wafer Measurements with Probe #283, Q'

Wafer Pre-certification Post-certification

00028 00052

000 37 - 0. 000 10

000 13 - 0. 000 09

000 12 - 0. 000 17

000 38 000 35

001 55 000 16

000 57 00029

000 03 000 11

000 03 001 40

000 41 - 0. 000 11

000 60 000 35

000 35 00028

000 32 00048

000 65 000 71

00039 000 66

000 86 000 18

000 47 - 0.000 42

000 88 000 14
131 00049 00066

131 00003 00061

131 00002 001 26

131 00042 00003

131 000 23 00033

131 00092 000 06

208 00029 00030

208 000 30 000 56

208 00025 000 73

208 000 18 000 19

208 00026 00030

208 000 74 00036

Average 000 199 000326

Standard Dev. 000 498 000402

Table 8. Average Differences between Wiring Configurations bl & b2

for Probe #283, Q.

Pre-certification

Probe Average Stand. Dev.

Post-certification

Average Stand. Dev.

283 000 199 000498 0 .000 326 000 402 4.4
Pre-Certification Measurements

002

0 . 0015

Co)

001

.J5'

O. 0005

A A E E E

B B

.....

D D

-0. 0005

-0. 001

Measurement Run

Post-Certification Measurements

002

0 . 0015

001

D D

.IS O. 0005

B B

Co)

AA A

.....

-0, 0005

-0. 001

Measurement Run

Figure 4. Differences, a- , between wiring configurations bl and b2 for six

measurements on each of five control wafers, A to E.
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 tluoroescent 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 th~e 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' , and a component of c/.J3 = 0.001 682 . cm is included as a

systematic component of the uncertainty.

Table 9. Resistivity Values under Standard Illumination and in Dark, and their Difference

Wafer

Standard

Illumination Dark

102

104

106

107

108

123

124

9981

1.0095

1.0233

1.0047

9991

1.0057

9985

1.0013

1.0125

1.0261

1.0076

1.0018

1.0086

1.0014

Mean Difference

Standard Deviation

Difference

0032

0030

0028

0029

0027

0029

0029

00291

000 157
Table 10. Resistivity Values under Standard Illumination and in Very Bright Ambient

and their Difference, Q'

Wafer

Standard

Illumination

Very

Bright

102

104

106

107

108

123

124

9983

1.0094

1.0233

1.0047

9991

1.0057

9985

9976

1.0087

1.0225

1.0042

9984

0050

9978

Mean Difference

Standard Deviation

Difference

0007

0007

0008

0005

0007

0007

0007

000686

000090
Appendix 8. Analysis of Certification Data and Control Experiments

For SRM 2544

1. GENERAL COMMENTS AND SUMMARY OF TYPE A STANDARD UNCERTAINTY

COMPONENTS

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 ().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.

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 waf~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. , 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.
The Type A standard unc~ty for the average resistivity at the center of each SRM waf~ is:

b2 .-.2 . 2 1 2 f"\

3" + 

cjig 

+ So + = 0.00609b.l:' Cm.

The Type A standard uncertainty for individual resistivity values on the 5 mm and Omm radius

circles is:

b2 .-.2 f"\

cjig +Sr +s +.)~ = 0.00743 u.cm.
Table 1. Components of Type A Standard Uncertainty for Crystal 29473, SRM 2544, with Probe #283, n. 

TypeRandom

RandomRandom

Random

SystematicSystematic

Source

Imprecision of probe #283Run-to-run measurement variability

Long-term measurement variability

Standard deviation of correction for wiring configurations

Stand. dev Estimate

004 662

001 198

005 646

000 287 elig

Uncertainty of a zero correction for probe #283 biasUncertainty of value due to ambient illumination conditions

bl.J3 000204

Negligible

......

*A correction of - 001 085 Q. , due to probe wiring configuration differences, is applied to the resistivity

measurements for all SRM wafers.
Table 2. Sources of Variation for Crystal 29473, SRM 2544, with Probe #283, Q.

......

Source of error Exvenment RMSEa DFb Relationship C

SRM certifications 004910 720

Pre-certification control wafer 003 987 150

Probe imprecision Post-certification control wafer 004 025 150

Pooled 004 662 1020

Pre- certification 002 134

Run-to-run Post-certification 002 359 ( 2 1 2 

YI2 6"s

Pooled 002249

Long-term Pre- and post-certification 005 720 (2 1 2 1 2 

YI2 s & 6 8

a The individual root-mean-square error estimates (RMSE) within each source-of-error category are pooled in the table above.

Except for the experiments to estimate probe imprecision, the RMSE values are comprised of more than one component ofvariation. Standard deviations associated with the individual effects, namely, imprecision, run-to-run variability, and long term

variability, listed in Table 1 are computed from the pooled RMSE values using the relationships in the last column.

Degrees of Freedom.This column expresses the components of variation that comprise the RMSE values for the experiments in each "source of

error" category (see the reference at the end of Appendix 2).
2. SYSTEMMATIC EFFECTS

1 Bias of Probe 283

There is a small bias for the certification probe, #283, (relative to the average ov~ 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

bJ3 = 0.000 204Q. cm is included as a systematic component of the uncertainty.

Table 3. Bias of Probe #283 Relative to the Average for All Probes, Q.

Wafer Pre-certification Post-certification

108

120

000 6742

000 1497

0004139

0002584

000 3242

000 4902

001 7977

000 5331

000 2708

0004511

Mean Bias 000 3528 0002993

102
2 Difference Between Wiring Configurations b1 & bz

Differences are found between measurements in wiring configurations b1 & hz. Averages and

standard deviations are shown in Table 4. The t-statistic for testing for.a significant difference

between wiring configurations is t = .J3O AvglSD. 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 bland b2 

002 1605 g'cm. Certification measurements were taken using configuration b1 only.

A correction of minus one-half this difference, or - 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:

cjig 

r::::-n. b1 b2 = 0.000 287 Q' cm. 2 '" 30

Table 4. Average Differences and Standard Deviations Between Wiring Configurations

b1 & bz, Q.

Pre-certification Post-certification

Probe

283

Average Standard Dev. DF 

001 358 0.001 914 29 3.

Av~age Standard Dev. DF t

002963 0.002495 29 6.

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 oftime 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, 1::, 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, 0, 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- andpQst-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 SRMwafers to be issued, was applied.

2. Comments on the Results

For measurements on wafers at 0.01 Q-cm, 0. 1 Q-cm, 10 g-cm, 25 Q-cm, and 100 g-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 g-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 (20-) lower limit value for each wafer, which is calculated from the

base-line average value and the lower 20- uncertainty value given in Appendix 2. It can be

seen that the February 1997 values are clearly within the lower 20- limit for the 200 O-em

SRM level.

Table 1. Summary of the Six-Round Grand Averages and Standard Deviations from Preand 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

Feb 1997

Control Pre-certification Post-certification Feb. 1997 Minus

Wafer# Pavg Pavg Pavg PrelPost Avg

(Q'Cm) (1: (Q-em) (1: (0 -em) (cr, (Difference

, %)

002 011 286 021 011 275 015 011 281 046 +0,004

043 010 974 013 010 972 014 010 970 024 027

044 010 955 010 010 949 004 010 954 030 +0.018

053 010 923 015 010 926 006 010 923 023 014

144 010 350 014 010 352 012 010 352 015 +0, 010

CRYSTAL 91904 Probe 281 Elapsed Time 19 Months 

003 114 51 137 0.114 59 039 114 59 091 +0.035

066 113 88 123 113 82 063 113 80 110 044

097 112 61 143 112 52 016 112 56 062 004

161 10435 146 104 33 012 104 32 076 019

287 099 61 129 099 60 035 099 63 047 +0.025

CRYSTAL 91907 Probe 283 Elapsed Time 31 Months

011 0733 028 0733 059 1.0745 068 +0. 110

026 1.0605 023 1.0607 024 1.0613 022 +0,066

042 1.0461 035 1.0463 014 1.0470 078 +0,076

131 9911 033 9916 014 9916 077 +0,025

208 9619 033 9623 025 9630 125 +0.094

105
Table 1 (cont' d.

CRYSTAL 29473 . Probe 283 Elapsed Time 3 Months

Feb 1997

Control Pre-certifi cation Post-certification Feb 1997 Minus

Wafer# Pavg Pavg Pavg PrelPost Avg

(Q-cm) (1: (Q-cm) (1: (Q-cm) (0- (Difference, %)

016 10.085 014 10.080 033 10.085 079 +0.025

032 10. 105 025 10.096 021 10.109 049 +0.084

075 10.316 029 10.308 024 10.309 085 029

108 10. 186 017 10, 177 020 10. 181 041 005

120 10.082 017 10.073 014 10,077 056 005

CRYSTAL 21565 Probe 2062 Elapsed Time 29 Months

017 24.050 032 24,046 015 24.046 104 008

039 24.695 029 24.699 022 24.701 061 +0.016

063 24.509 016 24.517 011 24.495 058 073

103 24. 135 031 24.142 025 24. 124 044 060

125 24.052 032 24.056 019 24.054 068 +0.001

CRYSTAL 51939 Probe 2362 Elapsed Time 34 Months

138 95.093 038 95. 124 048 95. 131 125 +0.024

139 99.306 048 99.310 022 99,252 076 056

140 96.036 028 96.077 029 96. 103 072 +0.048

141 101.060 023 101.079 053 101.277 097 +0.205

142 94,215 029 94.244 039 94.309 080 +0.084

CRYSTAL 21566 Probe SRMI Elapsed Time 54 Months

Control Pre-certification

Wafer# Pavg

(Q-cm) (1:

Feb 1997

Post-certification Feb 1997 Minus Pavg Pavg PrelPost Avg

(Q-cm) (1:

%) 

(Q-cm) (0",%) (Difference

, %)

020

040

060

080

100

196.27 0,050

193.88 0.034

193.57 0,072

192.82 0,054

192.59 0.065

196,07 0.104

193.76 0.032

193.50 0.036

192.69 0,050

192.42 0.057

196.05 0.081

193.59 0.102

193.24 0.039

192.35 0.097

192.26 0.074

061

119

152

210

0.127

106

lcU.S. GOVERNMENT PRINI1NG OFfICE:1999-4S4-S31117108

20"

Lower

Limit

195.

193.39

193.

192.32

192.
107

 Appendix 10. Analysis of Certification Data and Control Experiments 

for SRM 2541, batch 2 

 

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 a second 

batch of 102 wafers from crystal 91905 for SRM 2541, batch 2, at 0.01 S@cm. It is taken 

from the NIST statistician’s Report of Analysis and follows the general procedures 

outlined in Appendix 3 of this document. That appendix documents general certification 

uncertainty analysis procedures and the results for the first batch of wafers from this crystal 

certified for this SRM. In comparison to the uncertainty analysis for that first batch, the 

various components of uncertainty from a Type A analysis of the check wafer, control 

wafer and certified SRM wafers for this certification batch will have different numerical 

values, due, principally, to accumulated changes of operating characteristics of the various 

probes available for the certification process. 

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 2541, batch 2. Tables 1 

and 2 give an executive summary of the terms that contribute to the Type A evaluation of 

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 was necessary. 

However a correction of - 0.000 002 14 Ω@cm, due to probe wiring configuration 

differences, is applied to resistivity measurements for all 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 expanded uncertainty associated with the certified value at the 

wafer center is: 
108

U =2 2 2 2 2

2

6

1

3 γ δ ε s s s s

b

ui = + cfig + + + = 0.000 003 9 S@cm . 

with a value of 24 for the effective number of degrees of freedom based on the WelchSatterthwaite formula. 

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 expanded uncertainty associated with each of 

these individual certified values is: 

 

 U=2 2 2 2 2

2

3 γ δ ε s s s s

b

ui = + cfig + + + = 0.000 005 8 S@cm. 

with a value of 109 for the effective number of degrees of freedom based on the WelchSatterthwaite formula. 
109

 Table 1. Components of Type A Standard Uncertainty for Crystal 91905, SRM 2541, batch 2, with Probe #283, 

 Type Source Stand. Dev. Estim

 Random Imprecision of probe #283 sε 0.000 

 Random Run-to-run measurement variability sδ 0.000  Random Long-term measurement variability sγ 0.000  Random Standard deviation of correction for wiring configurations* scfig 0.000  Systematic Uncertainty of a zero correction for probe #283 bias b 3 0.000 

 --------------- 

*A correction of -0.000 002 14 Ω@cm, due to probe wiring configuration differences, is applied to resis

 for all SRM wafers. 
110

Table 2. Sources of Variation for Certification of Crystal 91905, SRM 2541, batch 2, with Probe #283, S

Source of error Experiment RMSEa DFb RelationProbe imprecision 

SRM certifications Pre-certification control wafer Post-certification control wafer Check standard wafer Pooled 

0.000 001 50 510 0.000 003 26 150 0.000 003 05 150 0.000 002 64 135 0.000 002 32 945 

s

,

Run-to-run 

Pre-certification control wafer Post-certification control wafer Check standard wafer Pooled 

0.000 001 28 25 0.000 001 43 25 0.000 001 50 26 0.000 001 41 76 

⎝⎛ 

s2δ + 16 s

Long-term Pre- and post-certification 0.000 001 30 5 

⎝⎛ 

s2γ + 16 s2δ + 136 

a The individual root-mean-square error estimates (RMSE) within each source-of-error category are pooled in th

Except for the experiments to estimate probe imprecision, the RMSE values are comprised of more than one variation. Standard deviations associated with the individual effects, namely: imprecision, run-to-run variabiterm variability, listed in Table 1, are computed from the pooled RMSE values using the relationships in the last 

b Degrees of Freedom. c This column expresses the components of variation that comprise the RMSE values for the experiments in each 

error” category (see the reference at the end of Appendix 2). 
111

2. RANDOM COMPONENTS OF UNCERTAINTY 

Probe imprecision, represented by the standard deviation, s,, 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, 2) from a pooling, across control wafers and measurement replications, of the 

measurements with the certification probe, #283, taken during the pre- and postcertification control wafer measurements, and 3) from the pooled standard deviations of 

the check-standard measurements that were taken concurrently with SRM certification. 

Run-to-run measurement imprecision is estimated both from the pre- and post-certification 

control wafer measurements, and from the measurements on the check-standard wafer. 

Long-term imprecision is estimated from the control wafer measurements. These 

components are all summarized in Table 2. 
112

3. SYSTEMATIC EFFECT COMPONENTS OF UNCERTAINTY 

3.1 Effect of Using Probe #283 for Certification 

The five probes available for the SRM certification are assumed to be a random sample 

among similar four-point probes. Each is a commercial four-point probe of highest 

available quality of materials and design. Each has probe pins that have been used for 

previous measurements and that meet the requirements specified in ASTM Method F-84 

for measurements on bulk silicon wafers. However, only one of these probes, #283, was 

used for actual certification after selection using the criteria for analyzing initial control 

wafer measurements, as detailed in section 3.1 in the main body of this report. 

However, certification using only a single probe can have a small systematic effect on the 

measurements. From an analysis of the control wafer measurements, there is a small 

systematic bias for probe # 283 relative to the measurement averages over all probes. 

The average bias during those measurements was 0.000 001 13 S.

cm. No correction is 

made for this negligibly small effect. However, we assume that during certification the 

probe bias could fall uniformly within the limits ± b where b = 0.000 001 13 S.

cm. A 

standard uncertainty of b / /3= 0.000 000 64 S.

cm is included as a systematic component 

of the uncertainty. 

3.2 Difference Between Wiring Configurations b1 & b2 

In the control wafer experiments prior to, and following certification, measurements were 

taken with probe # 283 in each of the possible ways of connecting probe wires to achieve 

the second configuration of the dual configuration measurements (see notation on Fig. 2 of 

this report). The t-statistic for testing for a signficant difference between wiring 

configurations b1 & b2 is t = 30 Avg/SD. The t-statistics and plots of the data suggest a 

slight difference among wiring configurations for this batch of SRM wafers. The average 

difference between the pre- and post-configuration measurements in b1 and b2 is 

0.000 004 28 S@cm. A correction of minus one-half this difference, or - 0.000 002 14 

S@cm, is applied to all certification measurements to obtain an average over the two 

configurations. The standard deviation of the correction is: 

2

2

2

1

30

1

4

1

cfig b b s = s + s = 0.000 000 19 S@cm.
113

Appendix 11. Analysis of Certification Data and Control Experiments 

for SRM 2543, batch 2 

 

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 a second 

batch (of about 100) wafers from crystal 91907 for SRM 2543, batch 2, at 1 S@cm. It is 

taken from the NIST statistician’s Report of Analysis and follows the general procedures 

outlined in Appendix 7 of this document. That appendix documents general certification 

uncertainty analysis procedures and the results for the first batch of wafers from this crystal 

certified for this SRM. In comparison to the results for the first batch, the contibution of 

photosensitivity (see appendix 7 for a detailed discussion) to uncertainty was smaller for 

the wafers used in batch 2. Also, the various other components of uncertainty from a Type 

A analysis of check wafer, control wafer and certified SRM wafers will have different 

numerical values than for batch 1, due, principally, to accumulated changes of operating 

characteristics of the various probes available for the certification process. 

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, batch 2. Tables 1 

and 2 give an executive summary of the terms that contribute to the Type A standard 

uncertainty. Discussion 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 is 

required. However a correction of -0.000 25 Ω@cm, due to probe wiring configuration 

differences, is applied to resistivity measurements for all SRM wafers from this batch. 

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 
114

resistivity value. The Type A expanded uncertainty associated with the certified value at the 

wafer center is: 

U =2 2 2 2 2

2 2

6

1

3 3 γ δ ε s s s s

c b

ui = + + cfig + + + = 0.0026 S@cm . 

with a value of 25,820 for the effective number of degrees of freedom based on the 

Welch-Satterthwaite formula. 

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 expanded uncertainty associated with each of 

these individual certified values is: 

 

 U=2 2 2 2 2

2 2

3 3 γ δ ε s s s s

c b

ui = + + cfig + + + = 0.0028 S@cm. 

with a value of 21,512 for the effective number of degrees of freedom based on the 

Welch-Satterthwaite formula.
115

 Table 1. Components of Type A Standard Uncertainty for Crystal 91907, SRM 2543, batch 2, with Probe #283,  Type Source Stand. Dev. Estim

 Random Imprecision of probe #283 sε 0.000

 Random Run-to-run measurement variability sδ 0.000 Random Long-term measurement variability sγ 

 Random Standard deviation of correction for wiring configurations* scfig 0.000

 Systematic Uncertainty of a zero correction for probe #283 bias b 3 0.000

 Systematic Uncertainty of value due to ambient illumination conditions c / 3 0.001

 --------------- 

*A correction of -0.000 25 Ω@cm, due to probe wiring configuration differences, is applied to resistivit

 for all SRM wafers. 
116

Table 2. Sources of Variation for Crystal 91907, SRM 2543, batch 2, with Probe #283, S@

Source of error Experiment RMSEa DFb RelationProbe imprecision 

SRM certifications Pre-certification control wafer Post-certification control wafer Check standard wafer Pooled 

0.000 391 520 0.000 667 150 0.000 668 150 0.000 716 175 0.000 552 995 

s

,

Run-to-run 

Pre-certification control wafer Post-certification control wafer Check standard wafer Pooled 

0.000 300 25 0.000 337 25 0.000 285 34 0.000 306 84 

⎝⎛ 

s2δ + 16 s

Long-term Pre- and post-certification 0.000 117 5 

⎝⎛ 

s2γ + 16 s2δ + 136 

a The individual root-mean-square error estimates (RMSE) within each source-of-error category are pooled in th

Except for the experiments to estimate probe imprecision, the RMSE values are comprised of more than one variation. Standard deviations associated with the individual effects, namely: imprecision, run-to-run variabiterm variability, listed in Table 1, are computed from the pooled RMSE values using the relationships in the last 

b Degrees of Freedom. c This column expresses the functional relationship of the components of variation that comprise the RMSE value

experiments in each “source of error” category (see the reference at the end of Appendix 2). 
117

2. RANDOM COMPONENTS OF UNCERTAINTY 

Probe imprecision, represented by the standard deviation, s,, 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, 2) from a pooling, across control wafers and measurement replications, of the 

measurements with the certification probe, #283, taken during the pre- and postcertification control wafer measurements, and 3) from the pooled standard deviations of 

the check-standard measurements that were taken concurrently with SRM certification. 

Run-to-run measurement imprecision is obtained both from the pre- and post-certification 

control wafer measurements, and from the measurements on the check-standard wafer. 

Long-term imprecision is estimated from the control wafer measurements. These 

components are all summarized in Table 2. 
118

3. SYSTEMATIC EFFECT COMPONENTS OF UNCERTAINTY 

3.1 Effect of Using Probe #283 for Certification 

The five probes available for the SRM certification are assumed to be a random sample 

among similar four-point probes. Each is a commercial four-point probe of highest 

available quality of materials and design. Each has probe pins that have been used for 

previous measurements and that meet the requirements specified in ASTM Method F-84 

for measurements on bulk silicon wafers. However, only one of these probes, #283, was 

used for actual certification after selection using the criteria for analysing initial control 

wafer measurements, as detailed in section 3.1 of this report. 

However, certification using only a single probe can have a small systemmatic effect on the 

measurements. From an analysis of the control wafer measurements, there is a small 

systematic bias for probe # 283 relative to the measurement averages over all probes. 

The average bias during those measurements was 0.000015 S.cm. No correction for this 

negligibly small effect ia applied. However, we assume that during certification the probe 

bias could fall uniformly within the limits ± b where b = 0.000015 S.cm. A standard 

uncertainty of b / /3= 0.000009 S.

cm is included as a systemmatic component of the 

uncertainty. 

3.2 Difference Between Wiring Configurations b1 & b2 

In the control wafer experiments prior to, and following certification, measurements were 

taken with probe # 283 in each of the possible ways of connecting probe wires to achieve 

the second configuration of the dual configuration measurements (see notation on Fig. 2 of 

this report). The t-statistic for testing for a signficant difference between wiring 

configurations b1 & b2 is t = 30 Avg/SD. The t-statistics and plots of the data suggest a 

slight difference among wiring configurations for this batch of SRM wafers. The average 

difference between the pre- and post-configuration measurements in b1 and b2 is 

0.000 506 S@cm. A correction of minus one-half this difference, or - 0.000 253 S@cm, is 

applied to all certification measurements to obtain an average over the two configurations. 

The standard deviation of the correction is: 

2

2

2

1

30

1

4

1

cfig b b s = s + s = 0.000 024 S@cm.
119

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, i.e. they depends on the illumination level on the wafer at the 

time of, and shortly before, measurement. The full change in resistivity between lit and 

darkened room conditions takes several minutes to occur. The certification measurements 

for batch 2 were made in what is termed “standard conditions”, i.e. the same ambient 

fluoroescent room light that was used with the other resistivity level SRMs in this series. 

An experiment was performed on 4 wafers from the portion of crystal 91907 containing the 

wafers used for batch 2 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 very- bright illumination (bright enough for the 

photosensitivity effect to saturate in the direction of lowered resistivity). 

The resulting differences were asymmetric: there is a much larger change between standard 

and dark conditions than there is between standard and very-bright conditions. Because of 

the variety of lighting conditions that may exist in used laboratories, it is not meaningful to 

attempt to apply a correction to the NIST as-measured SRM resistivity values to correct 

those values to other illumination conditions. Instead, a component of uncertainty is 

evaluated to account for the range of 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 from the 4 wafer test, is used to evaluate the 

component of uncertainty. The photo-sensitivity was found to decrease with wafer serial 

number going from the seed end toward the tang end of the starting crystal, and we used 

data from test wafers in the low end of the serial number range for this batch to ensure a 

conservative estimate of the uncertainty due 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 about the NIST-measured value where 

c = 0.002 186 S@cm, and a component of c / 3 = 0.001 262 S.

cm is included as a 

systematic component of the uncertainty. 

 
120

Appendix 12. Analysis of Certification Data and Control Experiments 

for SRM 2544, batch 2 

 

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 a second 

batch (of 40) wafers from crystal 29473 for SRM 2544 at 10 S@cm. It is taken from the 

NIST statistician’s Report of Analysis and follows the general procedures outlined in 

Appendix 8 of this document. That appendix documents general certification uncertainty 

analysis procedures and the results for the first batch of wafers from this crystal certified 

for this SRM. In comparison to the uncertainty analysis for that first batch, the various 

components of uncertainty from a Type A analysis of the check wafer, control wafer and 

certified SRM wafers for this certification batch will have different numerical values, due, 

principally, to accumulated changes of operating characteristics of the various probes 

available for the certification process. 

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, batch 2. 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 due to probe wiring configuration probe used was 

necessary. However a correction of -0.000 816 Ω@cm, due to use of probe # 283, as 

opposed to one of the other available probes, is applied to resistivity measurements for all 

SRM wafers. 

1.2 Certified Resistivities and Uncertainties 

The average of six measurements at the center of each wafer, corrected for use of probe # 

283 for the certification measurements, is reported as a certified resistivity value. The Type 

A expanded uncertainty associated with the certified value at the wafer center is: 
121

U =2 2 2 2

6

1

δ ε u s s s i = c + + = 0.004 06 S@cm . 

with 68 effective degrees of freedom based on the Welch-Satterthwaite approximation. 

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 expanded uncertainty associated with each of 

these individual certified values is: 

 

 U=2 2 2 2

δ ε u s s s i = c + + = 0.009 44 S@cm 

with 607 effective degrees of freedom based on the Welch-Satterthwaite approximation. 
122

 Table 1. Components of Type A Standard Uncertainty for Crystal 29473, SRM 2544, batch 2 with Probe #283, S Type Source Stand. Dev. Estim Random Imprecision of probe #283 sε 0.004 

 Random Run-to-run measurement variability sδ 0.001  Random Long-term measurement variability sγ  Random Standard deviation of correction for wiring configurations ---- N Random Uncertainty of correction value applied for use of probe 283* sc 0.000 

 --------------- 

*A correction of -0.000 816 Ω@cm, due to use of probe 283 for certification, is applied to resistivity me for all SRM wafers. 
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Table 2. Sources of Variation for Certification of Crystal 29473, SRM 2544 , batch 2, with Probe #283, S@c

Source of error Experiment RMSEa DFb RelationProbe imprecision 

SRM certifications Pre-certification control wafer Post-certification control wafer Check standard wafer Pooled 

0.003 951 200 0.004 296 150 0.005 583 150 0.005 074 80 0.004 666 580 

s

,

Run-to-run 

Pre-certification control wafer Post-certification control wafer Check standard wafer Pooled 

0.002 370 25 0.002 355 25 0.001 518 15 0.002 197 65 

⎝⎛ 

s2δ + 16 s

Long-term Pre- and post-certification 0.000 180 5 ⎝⎛ 

s2γ + 16 s2δ + 136 

a The individual root-mean-square error estimates (RMSE) within each source-of-error category are pooled in th

Except for the experiments to estimate probe imprecision, the RMSE values are comprised of more than one variation. Standard deviations associated with the individual effects, namely: imprecision, run-to-run variabiterm variability, listed in Table 1, are computed from the pooled RMSE values using the relationships in the last 

b Degrees of Freedom. c This column expresses the components of variation that comprise the RMSE values for the experiments in each error” category (see the reference at the end of Appendix 2). 
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2. RANDOM COMPONENTS OF UNCERTAINTY 

Probe imprecision, represented by the standard deviation, s,, 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, 2) from a pooling, across control wafers and measurement replications, of the 

measurements with the certification probe, #283, taken during the pre- and postcertification control wafer measurements, and 3) from the pooled standard deviations of 

the check-standard measurements that were taken concurrently with SRM certification. 

Run-to-run measurement imprecision is estimated both from the pre- and post-certification 

control wafer measurements, and from the measurements on the check-standard wafer. 

Long-term imprecision is estimated from the control wafer measurements. These 

components are all summarized in Table 2. 
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3. SYSTEMATIC EFFECT COMPONENTS OF UNCERTAINTY 

3.1 Effect of Using Probe #283 for Certification 

The five probes available for the SRM certification are assumed to be a random sample 

among similar four-point probes. Each is a commercial four-point probe of highest 

available quality of materials and design. Each has probe pins that have been used for 

previous measurements and that meet the requirements specified in ASTM Method F-84 

for measurements on bulk silicon wafers. However, only one of these probes, #283, was 

used for actual certification after selection using the criteria for analyzing initial control 

wafer measurements, as detailed in section 3.1 of the main part of this report. 

However, certification using only a single probe can have a small systematic effect on the 

measurements. From an analysis of the control wafer measurements, there is a small 

systematic bias for probe # 283 relative to the measurement averages over all probes. 

The average bias during those measurements was - 0.000 816 S.cm. The uncertainty of 

this value is the standard deviation of the average bias, or 0.000 265 Ω•cm 

3.2 Difference Between Wiring Configurations b1 & b2 

In the control wafer experiments prior to, and following certification, measurements were 

taken with probe # 283 in both of the possible ways of connecting probe wires to achieve 

the second configuration of the dual configuration measurements (see notation on Fig. 2 of 

this report). The t-statistic for testing for a significant difference between wiring 

configurations b1 & b2 is t = 30 Avg/SD. Plots and numerical analysis of the data show 

no difference of statistical significance between these two wiring configurations. There is 

no correction to measured values needed, and no contribution to uncertainty from this 

source 
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Appendix 13. SRM Unit Serial Numbers included in the second certification batches of 

 SRMs 2541, 2543, 2544* #

SRM 2541, batch 2, SN’s: 

 66 68 71 80 88 91 106 142 148 156 

168 184 188 191 201 203 205 212 215 220 

229 238 253 256 263 266 267 268 274 277 

279 281 287 289 290 291 292 296 298 302 

303 304 305 308 309 310 311 312 314 315 

316 317 318 319 320 321 322 324 325 326 

327 332 333 334 338 340 342 343 344 345 

346 349 351 352 353 356 357 358 361 362 

365 369 370 371 372 374 376 381 383 384 

385 386 388 389 391 392 394 395 396 397 

SRM 2543, batch 2, SN’s: 

 62 229 232 234 240 241 245 248 250 254 

257 258 259 260 262 268 270 272 275 277 

279 285 287 288 291 292 293 294 295 296 

297 300 301 302 303 304 305 306 308 310 

311 314 317 318 319 321 322 328 336 337 

338 340 343 344 349 350 351 353 354 355 

357 363 364 366 367 368 369 371 372 373 

375 378 379 383 384 385 387 388 393 397 

398 399 400 401 405 406 407 408 409 410 

411 413 414 417 418 419 422 426 432 433 

SRM 2544, batch 2, SN’s 

211 212 213 214 215 216 218 219 220 221 

222 223 224 225 226 227 229 230 231 232 

234 235 236 237 238 239 240 241 242 243 

244 245 246 247 248 249 250 251 252 253 

* Components of Uncertainty for these SRM serial numbers are shown in Tables 7 to 15 (with the 

designation “b2”). Descriptive summaries of the analyses of the Type A components of uncertainty for 

the batch 2 certification exercises of these three SRMs are given in Appendices 10, 11, and 12. 

# All other wafer serial numbers for these three SRMs were certified as part of the original batch 

(batch 1) of these SRMs, and the uncertainty component entries for the first appearance of these SRMs in 

tables 7 to 15 apply to those other serial numbers.