NBSIR 74-496

Standard Measurements of the

Resistivity of Silicon by tlie

Four-Probe IMethod

W. Murray Bullis

Electronic Technology Division

Institute for Applied Technology

National Bureau of Standards

Washington, D. C. 20234

August 1974

Final

Orginally Prepared for Electronics Research Center

National Aeronautics and Space Administration

Cambridge, Massachusetts 02139

NBSIR 74-496

STANDARD MEASUREMENTS OF THE

RESISTIVITY OF SILICON BY THE

FOUR-PROBE METHOD

W. Murray Bullis

Electronic Technology Division

Institute for Applied Technology

National Bureau of Standards

Washington, D. C. 20234

August 1974

Final

Originally Prepared for Electronics Research Center

National Aeronautics and Space Administration

Cambridge, Massachusetts 02139

U. S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARD^. Richard W. Roberts. Director

PREFACE

This work represents the concluding phase of a long-range program

designed to establish the technical basis for a method of measuring

the resistivity of silicon slices with an interlaboratory precision

such that the relative standard deviation is 1 percent or less. This

phase was supported by the Electronics Research Center of the National

Aeronautics and Space Administration.. Mr. L. M. Pauplis of the

Qualifications and Standards Laboratory, Component Standards Branch,

Electronics Research Center, was project manager for NASA. Significant

contributions to the project were made by J. C. French, F. H. Brewer,

L. J. Swartzendruber , and W. M. Bullis (project leader). The participation and cooperation of many industry members of the Resistivity

Task Force, Subcommittee VI, ASTM Committee F-1 were instrumental in

the successful completion of the work, and their contributions of time,

materials, and suggestions are gratefully acknowledged.

iii
PREFACE TO SECOND PRINTING

' This report was originally printed on 29 December 1967 as NBS

Report 9666, NASA-CR-86032 and is available from the National Technical

Information Service by Accession Number N68^18067. The present printing

was prepared to provide the information in a slightly updated form. In

this printing, the original resistivity data from which were calculated

the temperature coefficients for silicon are added and the interim test

method (Revision 1 of ASTM Method of Test F-84-67T) and reference thereto are omitted. The present test method, which has been accepted as a

standard by the American Society for Testing and Materials, is designated F-84-73. It may be found in Part 8 of the 1973 edition of the

Annual Book of ASTM Standards. Beginning with the 1974 edition (available in November 1974), it may be' found in Part 43. With the exception

of the changes noted above, the text, tables, and figures in this printing are reproduced from the original masters.

iv
CONTENTS

Page

PREFACE iii

PREFACE TO SECOND PRINTING iv

LIST OF FIGURES .-. .

'. . . . . . . . . . . . vi

LIST OF TABLES vii

ABSTRACT

.

. 1

1. INTRODUCTION AND STATEMENT OF PROBLEM . 3

2. ANALYSIS OF THE ROUND-ROBIN EXPERIMENTS .... ... 7

2.1 Introduction 7

2.2 Experiment 1 8

2.2.1 Results 8

2.2.2 Error Analysis 15

2.3 Experiment 2 21

2.3.1 Results 21

2.4 Conclusions 27

2.5 Notes and References 28

3. EXPERIMENTAL STUDIES 29

3.1 Introduction 29

3.2 Thermal Equilibration Time 29

3.3 Effect of Non-Uniform Thickness 30

3.4 Effect of Probe Needle Wander 31

4. SUMMARY 33

APPENDIX A - Report on a Special Round Robin 35

APPENDIX B - Special Round Robin Procedure 45

APPENDIX C - Probe Separation Correction Factors for Thin

Slices 53

APPENDIX D - Errors Introduced Through Diameter and Thickness

Correction Factors , 57

APPENDIX E - Resistivity of Silicon over the Temperature Range

0 to 50°C 59

V
LIST OF FIGURES

FIGURE . , ,

. PAGE

1. Probe Methods of Measurement of Resistivity . ... . . .. ,4

2. Specimen Resistivity Profiles for the Slices Used in

Experim.ent 1 11

3. Specimen Resistivity Profiles for the Slices Used in

Experiment 2 22

4. History of the Interlaboratory Precision of Measurem.ent

of Silicon Resistivity in the Range 0.01 to 100 ^-cm

by the Four-Probe Method 32
LIST OF TABLES

TABLE PAGE

I - Average Resistivity (ft-cm) at 23°C (as Reported) 9

II - Average Resistivity (ft-cm) at 23°C (Recomputed) 9

III - Sample Standard Deviation in Per Cent (Recomputed) .... 9

IV - Per Cent Difference of One Reading from Overall Average

Resistivity 12

V - Per Cent Difference of Median of Three Readings from

Overall Average Resistivity . . . lU

VI - Probe Separation Measurement in Millimeters 14

VII - Average Resistance (Recomputed) 16

VIII - V/I (Q) Corrected to 23°C 16

IX - Specimen Diameter Measurement (Centimeters) 20

X - Specimen Thickness Measurement (Centimeters) 20

XI - Average Resistivity (fi-cm) at 23°C (as Reported) 23

XII - Average Resistivity (Q-cm) at 23°C (Recomputed) 23

XIII - Sample Standard Deviation in Per Cent (Recomputed) .... 23

XIV - Per Cent Difference of One Reading from Overall Average

Resistivity 24

XV - Per Cent Difference of Median of Three Readings from

Overall Average Resistivity 24

XVI - Probe Separation Measurement in Millimeters 24

XVII - Average Resistance (Recomputed) 25

XVIII - V/I (Q) Corrected to 23°C 25

XIX - Specimen Diameter Measurement (Centimeters) 26

XX - Specimen Thickness Measurement (Centimeters) ....... 26

XXI - Angle Lapped Wafer: Thickness and Resistivity ...... 30

vi i

STANDARD MEASUREMENTS OF THE RESISTIVITY

OF SILICON BY THE FOUR-PROBE METHOD

by

W. Murray Bullis

ABSTRACT

An improved standard procedure for measurement of circular silicon

slices with four in-line point probes has been developed in cooperation

with the Resistivity Task Force of ASTM Committee F-1. Detailed

analysis of a series of round-robin experiments showed that the procedure can attain a precision of t 2 per cent (three standard deviations)

for interlaboratory comparisons of slices with room temperature

resistivity between 0.005 and 120 ohm-cm. Resistivity non-uniformity

in the test slices was shown to be a significant factor in limiting

the precision which could be achieved. The importance of including

correction factors for temperature, finite thickness, finite diameter,

and unequal probe separations was demonstrated. The results of the

round-robin experiments also emphasized that the precision quoted can only be achieved if the measurements are carefully and correctly made

on a well maintained, accurately calibrated test system which meets the

requirements imposed by the test method. Determination of the precision to be expected from the method in non-referee applications such as routine production and quality control will require additional study

of such factors as surface conditions, probe force, current levels,

etc. Nevertheless, use of the various procedures of the method, in

particular the sections on probe and measuring circuit evaluations and

on thermal sinking of the wafer, would be expected to yield significantly improved precision in such applications. Use of these procedures on a regular and widespread basis should be encouraged.

1

1. INTRODUCTION AND STATEMENT OF PROBLEM

The resistivity of a semiconducting material is controlled principally by the density of free carriers which exists in the material. In

commercially useful semiconductors, such as silicon and , germanium, the

number of free carriers is tailored and controlled for particular applications by the addition of specific small quantities of impurity

dopants. The electrical characteristics of semiconductor, devices

depend in a critical way on the free' carrier density in the various

regions of the device structure. Because of this -basic importance of

resistivity, and of the relative" simplicity with which a resistance

measurement may be made by means of the four-probe method on a surface,

this parameter is the one most widely used in design and production

control of semiconductor devices.

For similar reasons^ resistivity is the parameter which is most

widely used in specifying semiconducting materials from which devices

are to be fabricated. Although the standard four-probe method is quite

simple in principle, the precision and reproducibility which were obtained in actual practice have been inadequate for some time. A

substantial expense to the industry, and ultimately to the users of

devices, arises from disagreements between vendors and users. In addit'ion "disagreements among different test sets within the same organization are frequently found when comparisons are made. Need for improved

precision in measuring resistivity of silicon for high quality, high "i reliability devices led the industry, through the Committee F-1 on Materials for Electron Devices and Microelectronics of the American

Society for Testing and Materials, to request the Electron Devices

Section of the National Bureau of Standards to assist in the development offjiev^., standards with the aim of achieving a precision of 1 per

cent or better (one standard deviation). The necessity for determining

variations in resistivity along a slice radius which developed in

connection with power devices and integrated circuits further emphasized the need for this precision.

When the project began, it was thought that sufficiently precise

measurements could only be made by using the two-probe method. In this

method, current of uniform density is passed through a rectangular bar

with metallic contacts completely covering the ends of the bar and the

potential drop is measured between two pointed or wedge shaped probes

applied to the side of the bar a known distance apart as shown in

Fig. 1. Careful comparisons of this method were made with the fourprobe method. In this latter method, the current is passed through

the outer two of four pointed probes in a linear array placed on a flat

semiconductor surface and the potential drop is measured between the

inner pair as shown in Fig. 1. Detailed investigations of the effects

of variations in specimen surface preparation, probe force, probe diameter, and probe material were carried out. The importance of allowing for the variation of resistivity with temperature was demonstrated

3
FOUR- PROBE METHOD

TWO -PROBE METHOD

r

w

r 1r i

^ I L

Where:

P' Resistivity

V = Potential

Difference

I = Current

S = Probe Spacing

W = Width

H = Height

L = Length

Figure 1. Probe Methods of Measurement of Resistivity.
and geometrical correction factors appropriate to both circular and

rectangular wafers were computed . .•>i«v#if>jj*.-i+,.>;. -.-^

These studies resulted. in the development of an improyed fourprobe method for measuring resistivity of semiconductor slices. A

round-robin experiment on several silicon slices in the 5 to 20 fi-cm

range indicated that a precision over an order of magnitude better than

that obtained using the earlier techniques could be achieved. Thus, it was demonstrated that the use of the expensive and destructive twoprobe method was not necessary to achieve the desired precision. The

correction factor tables which were published as a part of the program

have been widely used. These factors also enabled a significant imr

provement in off-center resistivity measurements to be made. This

improvement together with the increased precision of the method enabled

the accuracy of the determination of radial variations of resistivity

to be improved significantly.

The present project was undertaken in order to complete the

development, writing, and publication of a standard method for the

measurement of the resistivity of silicon wafers suitable for use throughout the electronics industry in cooperation with the ASTM and to

provide the additional effort which was necessary to extend and refine

the method for maximum usefulness. These added efforts involved:

(1) extension of the application of the four-probe method to the most

widely used resistivity ranges of silicon, (2) establishment of the

precision of the method in the various resistivity ranges, (3) more

precise establishment of the environmental control and geometrical

requirements of the method, and (4) participation with ASTM in the

writing of an industry standard for the measurement.

To accomplish these objectives the following tasks were performed:

1) Results of two round-robin experiments being carried out

by the Resistivity Task Force of Siibcommittee VI of ASTM

Committee F-1 were analyzed. These experiments were in

progress at the inception of this project and were completed in June 1967.

2) Experimental studies were carried out to establish the

environmental control and geometrical requirements of

.the method and the relative influence of these factors

on the precision of the method.

3) The procedure for making four-probe resistivity measurements on silicon slices was extended to include the

entire resistivity range between 0.0005 and 2000 fl-cm.

M-) A new draft of the resistivity standard based on the

results of the above study was written and submitted

to the Resistivity Task Force for review comment.

5
\ This draft appeared as an ASTM Tentative Method

in the 1968 Book of Standards.

Work on these various tasks is reported in detail in the following

sections.
2. ANALYSIS OF THE ROUND-ROBIN EXPERIMENTS

2.1 INTRODUCTION

Two round-robin experiments covering a wide spectrum of resistivity were carried out between November 1955 and June 1957 by the

Resistivity Task Force of Subcommittee VI, Committee F-1 on Materials

for Electron Devices and Microelectronics of the American Society for

Testing and Materials to determine the limits of precision for a Method

of Test for Resistivity of Silicon Slices Using Four-Point Probes.

Earlier, a preliminary round-robin experiment consisting only of three

specimens about 10 fi-cm had been carried out to establish the feasibility of the method. In the near future a fourth round-robin experiment is planned in order to extend the method to lower resistivity

('^' 0.001 f^-cm) and to recheck the precision in the 1000 ^2-cm range.

This section summarizes the results of the two wide-spectrum

experiments. For completeness, the report on the preliminary experiment which was originally presented to the Task Force at the Chicago

meeting June 1956 is included as Appendix A. Laboratories which had

participated in the preliminary experiment participated in one of the new experiments

:

Bell Telephone Labs., Allentown, Pa.,

Dow Corning Corp., Electronic Products Div., Hemlock, Mich.,

IBM Corp., Components Div., Hopewell Jen., N. Y.

Monsanto Co., Inorganic Chemicals Div., St. Louis, Mo., and

National Bureau of Standards, Washington, D. C. Except for NBS , laboratories in the second had not participated in

earlier experiments on resistivity:

Autonetics, Anaheim, Calif.,

Fairchild Semiconductor, Mountain View, Calif.,

General Electric Co., Syracuse, N. Y.,

NBS, Washington, D. C. Western Electric Co., Allentown, Pa., and

Westinghouse Electric Co., Youngwood, Pa.

The original plan was to have 5 n-type and 5 p-type slices in each

experiment. Each series was to include specimens with resistivity

about 0.01, 0.1, 10, 100, and 1000 Q-cm. Except for the fact that the

10 Q-cm n-type slices were improperly typed and turned out to be p-type,

this plan was carried out. Three slices were broken during the tests;

two were replaced so that data could be obtained. The 0.01 fl-cm p-type

slice in the second test was not replaced. Slices were prepared

according to the test method (see Method F-8M- - 119 Preparation of Test

Specimen) by the laboratory which supplied them. Analog circuits of

resistance 0.001, 0.01, 0.1, 10, 100, and 1000 were furnished for

both tests. The 10, 100, and 1000 Q analog circuits contained, as the

standard, commercial precision (tO.05 per cent) resistors and, as the

7
large series resistors, ordinary carbon composition (+10 per cent)

resistors. The other standard resistors were fabricated from various

wire of appropriate diameter and length.

2.2 EXPERIMENT 1.

In this experiment, the ten silicon wafers and six analog circuits

were furnished to each of the participants in turn together with suitable data sheets. Wafers from crystals 605503, 601333, 71983, and

71166 were n-type; the remainder were p-type. The procedure governing

the tests was "Proposed Method of Test for Resistivity of Silicon

Slices Using Four Point Probes", Third Draft, December 1, 1955.1

Since this procedure pertained only to specimens in the 10 to 20 Q,-cm

range a separate schedule of currents to be used in the test was also

supplied

:

Range (Q-cm or Q) 0.001 0.01 0.1 10 100 1000

Current (mA) 50 50 30 0.3 0.1 0.02

2.2.1 Results . The method specifies tests to evaluate both the

condition of the probe assembly and the accuracy of the electrical

measuring equipment in addition to the resistivity measurement itself.

All these facets of the method were studied as part of the round-robin

experiments. The results are summarized in Tables I through X. The

resistivity measurements themselves are considered first (Tables I through V), followed by the probe separation measurements (Table VI)

and the electrical analog circuit measurements (Table VII.) . Results

presented in Tables VIII through X are used in the error analysis

presented in paragraph 2.2.2.

2.2.1.1 Resistivity Tests . Table I lists the average resis- ' tivity of each wafer (based on ten measurements in both the forward

and reverse directions of current) reported by each laboratory. The

grand average (Avg.), the sample standard deviation (s) (of the averages), and the relative sample standard deviation (s(%)) (of the

averages) were calculated for each wafer. Results of the preliminary

round-robin had suggested that computation errors occur frequently.

Hence, the reported raw data were used to recompute the averages with

the use of an electronic desktop calculator programmed to yield average, sample standard deviation, and relative sample standard deviation.

Results are shown in Table II, from which it can be seen that the

relative sample standard deviation is less than 0.7 per cent in eight

of the eleven cases. This would suggest that a precision of t2 per

cent (R3S%) for these cases could be expected most of the time if the

experiment were repeated with the same care as exercised in this test.

Measurements on the 100 ^-cm and 1000 Q,-cm p-type wafers and one of

the 1000 ^2-cm n-type wafers had larger sample standard deviations.

Only the 1000 Q-cm p-type wafer significantly exceeded 1 per cent.

Without additional experiments it is not possible to include the 1000

fi-cm range in the ±2 per cent precision statement above.

8
TABLE I - Average Resistivity (Q-cm) at 23°C (as Reported)

Specimen Lab. 1 Lab. 2 Lab. 3 Lab1. 4 Lab. 5 Avg. s s (%)

605603-3 0. 008393 0.008378 0.008403 0. 00834 0. 0083343 0 .008370 0,.000031 0 .37 601333-2 0. 08540 0.08569 0. 085058 0. 08502 0. 085577 0 .08535 0,,00030 0 .35 71983-2 100. 32 101.19 95.8052 101. 725 100. 65 99 .94 2,,37 2 .37 71166-2a 836. 4 842.53 835. 0 838 .0 4,,0 0 .48 71166-2b 1136.27 1044. 13 932. 1037 .5 102,, 3 9 .86 600200-2 0. 007763 0.007824 0.007778 0. 00777 0. 0077425 0 .007775 0,,000030 0 .39 607075-2 0. 10927 0.10958 0.109189 0. 10883 0. 10881 0 .10914 0,,00032 0 . 30 70877-3 7. 916 7.937 8. 0403 8. 0410 7, 909 7 .969 0.,057 0 .83 H9445-2 11. 857 11.97 12.0045 11. 735 11. 877 11 .889 0.,106 0 . 89 66969-1 111. 91 114.03 113.150 112. 735 112. 59 112 .88 0,,78 0 .59 16603-2 940. 3 979.83 981.740 967. 31 941. 6 962 .2 20,,1 2 .09

TABLE II - Average Resistivity• (Q-cm) at 23°C (Recomputed)

Specimen Lab. 1 Lab. 2 Lab. 3 Lab>. 4 Lab. 5 Avg. s s (%)

605603-3 0. 008389 0,008452 0.008361 0. 00830 0. 0083343 0 .008357 0,,000058 0 .59 601333-2 0. 08540 0.08538 0.084723 0. 08462 0. 085577 0 .08514 0,,00044 0 .51 71983-2 100. 33 101.11 101.48 101. 9 100, 6 101 .08 0,,64 0 .63 71166-2a 836. 4 842.61 835. 0 838 .0 4 ,,0 0 .48 71166-2b 1122. 1040. 932, 1031 .3 95,,3 9 .24 500200-2 0. 007761 0.007818 0.007743 0. 007752 0. 0077426 0 .007763 0,,000031 0 .41 607075-2 0. 10928 0.10954 0.10873 0. 10901 0. 10881 0 .10909 0,,00037 0 . 34 70877-3 7. 915 7.930 7.991 8. 0267 7. 909 7 .954 0

,

,052 0 . 65 49445-2 11. 859 11.94 11.92 11. 747 11. 877 11 .869 0,,075 0 .63 55969-1 111. 90 114.09 112.5 liU. 20 112. 59 113 .05 1,,03 0 .91 16603-2 939.0 951.19 947.8 964. 9 944 . 8 949 , 5 9,,7 1 .02

TABLE III - Sample Standard Deviation in Per Cent (Recomputed)

Average

Resistivity

Specimen (Q -cm

)

Lab, 1 Labi. 2 Labi. 3 Lab. 4 Lab1. 5 Avg.

605603-3 0. 008367 0. 49 0, 91 1. 24 0.,97" 0. 71 0.59

601333-2 0. 08514 0..10 0. 19 0. 30 0.,29 0. 32 0.51

71983-2 101. 08 0,,13 0. 33 1. 41 2,,12 0. 38 0.53

71165-2a 838. 0 0.,27 0. 33 1. 53 0.48

71165-2b 1031. 3 8. 82 0,,30 1. 24 9.24

600200-2 0. 007763 0..15 0. 08 0. 18 0,,08 0. 20 0.41

607075-2 0. 10909 0,,16 0, 25 0. 16 0,,02 0. 16 0. 34 70877-3 7. 954 0,.11 0. 32 1. 30 0,,32 0. 34 0.55

49445-2 11. 869 0,.14 0, 12 1. 08 0,,21 0. 19 0.63

66969-1 113. 06 0,.10 0. 59 0. 43 0,,33 0. 25 0.91

15603-2 949, 5 0,,23 0. 87 0.. 82 0,,68 1. 15 1.02

9
2.2.1.2 Relative Single Laboratory Deviation . The relative

sample standard deviations obtained by each of the laboratories for

the sequence of ten measurements for each wafer are shown in Table III

together with the sample standard deviation of the grand average for

each wafer repeated from Table II. There appears to be little correlation between the deviation reported for a wafer by a particular laboratory and the per cent difference between that laboratory's average

resistivity value and the grand average. However, one of the limiting

factors in obtaining reproducible resistivity measurements is the uniformity of the wafer being measured. Although the method requires that

the measurements be made with the center of the probe array located

within 10.25 mm of the center of the wafer, there is no way to verify

from the data reported that this was actually done in every case. If

uniform wafers were used in the experiment, this source of variation

would not be present. That not all wafers used in the test were as uniform as would be desirable was shown by resistivity profiles of each

wafer which were made at the end of the round-robin series. These were made at NBS with the use of a four-point probe which had the probe

separation recommended in 115.3.4 of Method F-84 and which met the re- quirements of mo.l of the method. Measurements were taken at intervals of about 1 mm along two perpendicular diameters. Comparison of

the single laboratory deviations in Table III with these profiles which

are shown in Fig. 2 suggests that more uniform wafers show generally

smaller deviations. Differences in deviation between laboratories may

be due as much to differences in locating the center of the wafer as to other errors.

2.2.1.3 Single Readings. The procedure being tested by this

round robin calls for ten readings to be taken on each wafer measured.

Although this procedure is acceptable for referee and other comparative

measurements, single readings are much more practical in production

control and inspection applications. Hence several single readings

were analyzed to determine how the precision is affected in this case. The result of the analysis of the sixth, first, and tenth readings are shown in Table IV. In this table the per cent difference between

individual resistivity values and the overall grand average value for

that wafer are listed. The per cent difference between the value of

resistivity of a wafer as determined by averaging the individual values

reported by the various labs and the overall average value is listed

in the column headed "Avg" . The relative sample standard deviation

determined for each wafer is listed in the last column. If the 1000

fi-cm p-type wafer is excluded from the discussion it can be seen that over three-fourths of the values fall within 1 per cent of the appropriate grand average value. Less than 5 per cent of the values differ

by 2 per cent or more. Comparison of the sample standard deviations

with those in Table II shows that the reproducibility is only

moderately degraded.

2. 2.1. 4 Median of Three Readings . Sometimes it is possible

to improve the precision of a determination over that of a single

10
1

' -12 -8 0 4 8 12 -12 -8-4 0 4 8 12

Distance from Center (mm)

Figure 2. Specimen Resistivity Profiles for the Slices Used in

Experiment 1. (No profile was made on Slice 71166-2a.)

11
TABLE IV - Per Cent Difference of One Reading from Overall Average Resistivity

a) Sixth Reading

Specimen

Average

Resistivity

(n-cm) Lab. 1 Lab. 2 Lab. 3 Lab. 4 Lab. 5 Avp

.

s(%)

605603-3 0. 008367 -0.41 -0.55 + 0.42 -0. 80 + 0.49 -0.17 0.49

601333-2 0. 08514 + 0.45 + 0. 36 -0. 07 -0.80 + 0.41 + 0.07 0.53

71983-2 101. 08 -0.62 + 0.40 + 2.15 + 2.00 -0.42 + 0.70 1. 30 71166-2a 838. 0 -0.14 + 0.43 + 0.99 + 0.43 0. 56 71166-2b 1031.

3

+12.65 +0. 94 -10.60 + 1.00 11.51

600200-2 0.007763 -0. 21 + 0.63 -0.12 -0.13 -0.16 0.00 0. 35 607075-2 0. 10909 + 0. 11 + 0. 56 -0.11 -0.08 -0.24 + 0.05 0. 31 70877-3 7.954 -0.57 -0.11 + 2.29 + 0.83 -0.74 + 0.34 1.25

U941+5-2 11.869 + 0.08 + 0.68 +1.52 -0.99 + 0.38 + 0.33 0.91

66969-1 113.06 -1.11 + 1.67 -0.72 "

, +1.14 -0.53 +0.09 1.23

16503-2 949.5 -1.03 -0.70 -0.36 • +i.4Ch^,..l >0.41 -0i,22 0.95

Specimen

Average

Resistivity

(Q-cm) Lab. 1

b) First

Lab. 2

Rsading

Jjab .

" 3 Lab . 4' Lab . 5 Avs

.

s(%)

605603-3 0. 008367 +0. 80 + 1. 96 + 0;72 -2.12 -0.51 + 0.15 1.60

601333-2 0. 08514 + 0.15 -0.21 -0. 27 -0.94 + 0.26 -0.20 0.47

71983-2 101.08 -0. 90 + 0.33 -1 . 22 +1. 70 -0.87 -0.19 1.21

71166-2a 838. 0 -0. 56 + 1. 20 + 3.45 +1.36 1.98

7L166-2b 1031.

3

-3.46 +0.94 -8.95 -3. 82 5.15

600200-2 0.007763 -0. 08 + 0.71 -0.59 -0.03 -0.40 -0.08 0. 50 607075-2 0. 10909 -0.05 + 0.47 -0.49 * -0.06 -0. 33 -0.09 0.37

/Uo / / -

o

/ . yoH n 111 -U.Hi -U . D / J. 1 Q + 1 .00 n no J1 . nnuu 49445-2 11.869 -0.23 + 0.43 + 0.68 -1.60 ^ ^0.26

Ji +0.05

-0.20 0.75

66969-1 113.06 -1.21 + 0.58 + 0.22 + 1.07 + 0.14 0.85

16603-2 949.5 -0.98 ' +1.23 0.00 + 2.60 / -0.48 + 0.47 1.44

' . c) Tenth Reading

Average

Resistivity

Specimen (f2-cm) Lab. 1 Lab. 2 Lab. 3 Lab. 4 Lab. 5 Avg. s(%)

505603-3 0.008367 -0.13 + 1.54 + 0.78 -0.56 + 0.94 + 0.54 0.87

601333-2 0.08514 + 0.36 + 0.35 -0.41 -0.26 + 1.40 + 0.29 0.71

71983-2 101.08 -0.75 + 0.08 -0.19 -3.05 -0.10 -0.80 1.30

71166-2a 838.0 -0.45 + 0.58 -0.50 -0.12 0.61

71166-2b 1031.3 + 1.20 + 0.65 -7.40 -1.85 4.90

600200-2 0.007763 -0.12 + 0.61 -0.01 -0.15

' -0.25 + 0.02 0. 34 607075-2 0.10909 0.00 -0.47 -0.38 -0.07 -0.26 -0.24 0.30

70877-3 7.954 -0.58 -0.68 -0.38 + 0.92 -0.19 -0.18 0.65

49445-2 11.869 -0.20 + 0.43 + 0.35 -0.99 +0.25 -0.03 0.59

66969-1 113.06 -1.00 + 0.80 -1.20 + 1.03 -0.73 -0.22 1.05

16603-2 949.5 -1.16 + 0.78 + 0.26 + 1.66 -0.36 + 0.24 1.07

12
measurement by taking the median value of three measurements. This

involves no arithmetic and only a small amount of extra measurement

time while enabling isolated "wild" readings to be avoided. To test

the usefulness of this approach, the median value of the sixth, seventh,

and eighth readings was analyzed with the results shown in Table V. It can be seen by comparison with Table IV that the improvement is not

consistent enough over the single reading case to justify the extra

labor involved.

2.2.1.5 Probe Separation Measurement . In the preliminary

experiment the test for probe quality was shown to be adequate. In

that experiment each laboratory measured the separations on the same

probe. As a result of this experiment, it was concluded that a single

laboratory relative sample standard deviation greater than 0.25 per

cent in any measurement of probe separation would be considered grounds

for rejection of the probe. It was also concluded that the three

separations must be equal within 2 per cent for the probe to be

acceptable.

In the present experiment, each laboratory furnished its own

probe. The results of the probe separation measurements on the five

probes used are given in Table VI. The separation (S;j^), the sample

standard deviation (s^), and the relative sample standard deviation

(s^ (%)) are given for each of the three separations followed by the

average separation (S) and the probe separation correction factor (Fgp).

Two probes did not meet the requirements of the method. The probe used

by lab 2 had separations which differed by more than 2 per cent in

addition to slightly greater than acceptable deviation in two of the

three separations. The probe used by lab M- had one separation with

slightly greater than acceptable -deviation . Note that the probe used

by lab 2 had the probe separation correction factor nearest to unity

of all the probes used. No increase in the measurement spread could be

attributed definitely to either of these conditions. In neither case

were the requirements missed by a large amount

.

Details of the derivation of Fgp are given in Appendix C.

The 2 per cent requirement on probe separation difference is necessitated by the use of the approximate formula (C-7) for Fgp. Unless the

exact formula (C-5) is used to calculate Fgp, or (as in the case of the

probe used by lab 2), Fgp is within 0.1 per cent of unity, this requirement may not be relaxed. However, the results of this experiment

suggest that the allowed relative sample standard deviation for probe

separation can probably be increased to 0.30 per cent without producing

an observable increase in the overall sample standard deviation.

2.2.1.5 Electrical Equipment Tests . The electrical equipment test in the preliminary round robin was successful in identifying

one inadequate measuring system. The test circuit consisted of a precision resistor and four other resistors arranged as shown in Fig. 3

of the test method. The value of the other resistors, 300 times that

13
TABLE V - Per Cent Difference of Median of Three Readings from

Overall Average Resistivity

Average •

Specimen

Resistivity

(fi-cm) Lab. 1 Lab. 2 Lab. 3 Lab. 4 Lab. 5 Avg. s(%)

605603- 3 0. 008367 + 0. 33 + 0.96 + 0.42 -0.68 -0.83 + 0.04 0.77

601333- 2 0.08514 + 0. 40 + 0.35 —0.50 -0.52 + 0.41 + 0.03 0.49

71983- 2 101.08 -0.62 + 0.23 + 0.48 + 1.90 -0.40 + 0.32 0.99

71156- 2a 838.0 -O.IU + 0.43 -0.67 -0.13 0.55

71166- 2b 1031.3 +12.65 + 0.94 -9.82 + 1.26 11.10

600200- 2 0.007763 -0. 18 + 0.70 -0.12 -0.06 -0.36 -0.01 0.41

607075- 2 0.10909 + 0. 18 + 0.56 -0.37 -0.06 -0.33 0 0.39

70877- 3 7.95U -0.54 -0.11 + 2.29 + 0.78 -0.54 +0.38 1.19

494U5- 2 11. 869 + 0.08 + 0.68 + 1. 52 -0.99 + 0.15 + 0. 29 0.91

66969- 1 113.06 -1. 03 + 1.67 -0.66 + 1.14 -0.60 + 0.10 1.21

16603- 2 949.5 -1. 03 + 0.18 + 0.99 + 1.90 -0.41 + 0.32 1.14

TABLE VI'. - Probe Separation Measurement in Millimeters

Lab. 1 Lab. 2 Lab. 3 Lab. 4 Lab. 5

1. 5933 1.5657 1.5926 1.5903 1.5890

0.00053 0.00145 0.00262 0.0010 0.0010

; s^(%) 0. 03 0.09 0.16 0.06 0.06

^2 1. 5885 1.5850 1.5989 1.5938 1.5964

=

2 0. 00043 0.00409 0.00160 0.0033 0.0008

S2(%) 0. 03 0.26 0.10 0.21 0,05

^3 1. 5936 1.6096 1.5941 1.5870 1.5865

^3 0. 00069 n Anil ji 1 U • UUloU 0.0041 0.0008

S3(%) 0. 04 0.28 0.11 0.26 0.05

S 1. 5918 1.5867 1.5951 1.5903 1.5905

Fsp1. 0022 1.00107 0.99747 0.9976 0.99608

14
of the precision resistors, is based on the work of Logan^ who estimated that if the contact force is 0.25 N (25 gf) the spreading resistance at an osmium probe point is about 300 times the resistivity of

the specimen. Present conditions of tungsten carbide probe points and

a contact force of 1.75 N would be expected to reduce this ratio by

about a factor of two. The larger value was selected for the test in

order to allow for uncertainties in the estimate.

The results of measurements are shown in Table VII. There

were no problems encountered in measuring either the 10 or 100

resistors. Values reporred by two labs (3 and 5) for the 1000 Q. resistor fell outside the allowed band while three labs (1, 2, and 3)

reported relative sample standard deviations in excess of the allowed

0.3 per cent. Comparison of the measurements on the 1000 Q resistor

and measurements on specimens 71166-2 and 16603-2 shows the following

interesting but unexplained facts:

1) In measurements on the resistor lab 3 was low and lab 5 was high; in measurements on the wafers the reverse is

true. (A possible explanation of this inversion is that

the measurements were made by lab 5 closer to the center

of the wafers .

)

2) In measurements on the wafers, labs 1, 2, and 3 did not

show significantly larger relative sample standard

deviations than labs 4 and 5 except in one instance.

Problems of reproducibility were encountered in measuring the

three smaller resistors. Since the scatter in the resistor measurements much exceeded that in the wafer measurements it is suspected

that these analog test circuits were an inadequate test of the

electrical measuring equipment in the 0.01 and 0.1 9, ranges because of

unstable standard resistors. Further work will be required to eliminate

this problem which probably arises from either thermally generated

voltages or from temperature dependence of resistance or both. In

addition to the larger deviation of average values reported for the

0.001 Q resistor, all labs had relative sample standard deviations in excess of 0.3 per cent. No specimens in this resistivity range were included in this experiment. A separate round-robin to test this range

will be started as soon as an improved standard resistor in this range

can be assembled and tested.

2.2.2 Error Analysis . The following quantities are measured in

the experiment

:

Voltage (V)

Current ( I

)

Temperature (T)

Wafer Diameter (D)

Wafer Thickness (w)

Probe Spacing (S)

15
Analog

Circuit

No. No.

No.

No.

No.

No.

Lab.

0.000939

0. 010279

0.10061

10.015

100.04

1000.7

TABLE VII - Average Resistance (Recomputed)

a) Measured Values (fi)

Lab. 2 0.0009U3

0.01025H

0.10056

10.012

100.052

999.17

Lab. 3 0.000920

0.00998

0.0979

10.00

100.0

996.

Lab.

0.00096

0.01027

0.100U8

10.000

100.0

1000.

Lab.

0.0009280

0.010252

0.10051

10.013

100.01+ 1005.

Avg.

0.000938

0.010209

0.10001

10.008

100.03

1000.2

0.000015

0.000128

0.00118

0.007

0.02

3.2

s(%)

1.63

1.26

1.18

0.07

0.02

0.32

b) Sample Standard Deviation (Per Cent)

Analog

Circuit

No. 1 No.

No.

No.

No. No.

Lab. 1 1.02

0.02

0.02

O.OH 0.03

0.53

Lab. 2 0.60

0.14

0.02

0.01

0.12

2.70

Lab. 3 1.09

0.20

0.16

0.00

0.00

0.57

Lab. U 1.05

0.00

0.00

0.00

0.00

0.00

Lab. 5 0.36

0.01

0.01

0.01

0.02

0.11

Avg.

1.63

1.26

1.18

0.07

0.02

0.32

TABLE VIII - V/I (a) Corrected to 23°C

Specimen Lab. 1 Lab. 2 Lab. 3 Lab. 4 Lab. 5 Avg. s s(%)

605503-3 0 01665 0 01671 0 016654 0 01650 0 016650 0 .016633 0 .000785 0 .47 501333-2 0 16630 0 1658 0 165594 0 16515 0 16766 0 .16610 0 .000965 0 .58 71983-2 191 85 193.32 194 924 195 825 194 0 193 984 1 .5221 0 .78 71156-2a 1701 3 1711. 1718 8 1710 4 8 .77 0 .51 71156-2b 2323. 89 2123. 63 1915 2120 84 204 459 9 .64 500200-2 0. 011+83 0. 01490 0. 01485 0. 014819 0 014888 0 014857 0 000354 0 24 607075-2 0. 21202 0. 2115 0. 211751 0. 21207 0 21273 0. 21201 0 00046 0 22 70877-3 15. 452 15. 54 15. 6199 15. 7433 15. 556 15. 582 0 1082 0 69

49141+5-2 23, 538 23. 72 - 23. 7791 23. 734 23. 752 23. 705 0. 0957 0 40

65969-1 221. 31 225. 07 223. 169 226.628 224. 03 224. 04 1. 998 0. 89 15603-2 1809. 1831. 1837. 56 1862.62 1833. 9 1834. 82 19. 121 1. 04

16
with the use of the experience gained in the round-robin experiment

and the limits of error specified in the measurement method, it is

possible to estimate the contribution of error in each of these

quantities to the overall measurement error.

The resistivity of a thin, homogenous semiconductor slice at a reference temperature T^ is given by3

p =

I- w F_ F(w/S) F F^

o I z sp I

where the ratio of voltage V to current I is measured at a temperature

T, w is the slice thickness, Fj is a correction factor which accounts

for finite slice diameter and which decreased from Tr/ln2 as the ratio

S/D increases from 0, F(w/S) is a correction factor which accounts for

finite thickness and which decreases from 1.0 as the ratio w/S

increases, F^p is a correction factor which accounts for unequal probe

separations, and Frp is a temperature correction factor. The assumption

that the first three independent correction factors may be multiplied

together to obtain the total geometrical correction factor is valid

only when the deviations from the factors in their limiting cases

(D ^ <», w -> 0, and -> ^ S3 ^ S) are small. This is the reason

that certain geometrical restrictions (D ^ lOS, w < S and S-^, Sg

equal to 2 per cent) are employed in the method. Smits^ has considered

the factors F2 and F(w/S). More detailed tables of F2 have appeared

in the literature . 2

, 5 xhe factor F^p is discussed in Appendix C. The

temperature correction factor is discussed in Section 3 of this report.

Tables or formulas for the four factors are given in the method.

For small deviations from equilibrium values

:

dp dV dl dw ^^2 — dF(w/S) ^^sp '^^T

= -r, + + — + ~—- + —— + (1)

P V I ^ ^2 F(w/S) ^sp ^T

Since F-p = 1 - C (T - T^) ^ 1, one may write (dF^p/F^) = - C-pdT. Then,

with the use of (D-1) and (D-2) of Appendix D and (C-9) of Appendix C

(1) becomes:

dp dV dl ^ , , , dS ^^2 — ,^ dw dD pVIT = 1~ - C^^T -(a-b-c)^-c SS + (1 - ,wD b) — + a

where the coefficients a, b, and c are defined in Appendixes D and C.

If all these factors were independent, the analysis could precede in a straightforward manner. Uncertainty in the measured values of S, S2,

w, and D can be considered separately. However probe wander (resulting

in changes in S and S2) and uncertainty in temperature both affect the

uncertainty in voltage.

17
2.2.2.1 Errors in V, I, and T . These three quantities are lumped together since it is necessary to account for the temperature

variation of resistivity when voltage readings at different temperatures

are compared. Three factors contribute to the error in the V/I ratio

corrected to the reference temperature

:

1) direct measurement error of V/I ratio

2) effect of probe wander on V, and ^ 3) uncertainty of T.

The direct measurement error of the V/I ratio can be estimated

from the measurement of the resistor in the analog test circuit. The

relative standard deviation in this measurement is limited to 0.15 per

cent. Under good conditions, it is considerably smaller than this

limit as can be seen from Table VII.

Probe wander will affect the V/I ratio as discussed in

Appendix C. Since the probe is raised and lowered between each of the

ten independent readings of the ratio, the effect of probe wander on the uncertainty in the average value of the ratio is reduced to a negligible amount. Probe wander will be an important factor in the

single reading procedure (cf. 112.2.1.3) but, as will be seen below, it

is likely to be obscured by other effects.

Errors in temperature enter through uncertainties in the

appropriate correction factor. If the maximum linear temperature

coefficient is taken as 0.01 per deg uncertainties of temperature of

t0.2°C will be reflected as an tO.2 per cent error in tempera;ture

correction factor. In many cases the temperature coefficient is smaller

so this error will also be smaller. Uncertainties in linear temperature

coefficient (Crp) of tO.OOOl are reflected as errors of about tO.05 per

cent at the extremes of the allowed temperature interval [(23t5)°C].

This error is independent of and much smaller than the error due to

uncertainty in temperature so it can be neglected. With these assumptions and the assumption that the three sources of error are random

and independent, 6V/V becomes:

' ^ = /~0.15)^ + (0.2)^ = 0.25 per cent

The data in Table VIII demonstrates that this small a deviation is seldom obtained even in those cases where the resistors in the analog

test circuits were measured very accurately. The descrepancy probably

arises from the inhomogeneity of the wafers. Note that the two wafers

with the flattest resistivity profiles (Fig. 2) have the smallest

V/I standard deviations. The average observed value of 6V/V was

0.53 per cent if the 1000 Q-cm slices are excluded.

18
2.2.2.2 Errors in D . These errors enter into the calculation of resistivity only through the correction factor The diameter of the wafer is required to be constant to iD/5S per cent of D. The average diameter was determined with an average relative standard

deviation of less than 0.2 per cent as shown in Table IX. For the

usual diameter of 25 mm (16S for the recommended probe separation)

6F2/F is only 0.013 per cent. The maximum uncertainty in the proper

diameter correction factor to be used can be estimated by considering

inscribed and circumscribed circles. In this case AD/D = ±D/5S per

cent so the maximum AF/F2 becomes 10.21 per cent for D = 16S. Larger

deviations will occur in smaller diameter wafers and smaller deviations

in larger diameter wafers as discussed in Appendix D. The importance

of using the diameter correction factor on wafers with D ^ 25S is also

demonstrated in Appendix D.

2.2.2.3 Errors in w . These errors enter into the calculation of resistivity in two ways:

1) directly and

2) in the thickness correction factor [F(w/S)].

The second of these, 6F/F, is negative and has a value of -0.27 6w/w

when w = S (the maximum thickness allowed by the method) and decreases

in magnitude to zero as the thickness decreases. Some intermediate

values are listed in Appendix D. The permitted deviation on w in the

round-robin experiment was 0.16 per cent. From Table X it can be seen

that this was not achieved which suggests that instruments with the

required accuracy were not used. The value achieved was on the average about 0.3 per cent so that the total contribution to the error

arising from this source is between 0.3 per cent (for thin wafers) and

0.22 per cent (for the thickest wafer permitted). The wafers used in

the round-robin had a w/S ratio of about 0.75 so that the appropriate

value for the deviation due to thickness measurement errors is

(1 - 0.12) (6w/s) = 0.25 per cent.

2.2.2.4 Errors in S. There are two forms of this error.

First, there is an uncertainty in the measured values of the probe

separations which will depend on both probe wander on the polished test

wafer and the error in measuring the position of the impressions.

Second, there is the effect of probe wander on the measured V/I ratio.

The first of these will enter into the resistivity calculation through

the three correction factors F2, F(w/S), and Fgp as discussed in

Appendixes C and D. For typical slices (D = 16S, w = 0.75S) the con- tribution from F2 and F(w/S) can be neglected so the appropriate value

is 1.14(3/5) = 0.34 per cent. The effect of probe wander on the voltage measurement has been considered in 112.2.2.1 and Appendix C.

2.2.2.5 Summary . The total deviation may be found if it is

assumed that each type of error discussed above is random and independent. With this assumption, the total deviation is the square root

of the sum of the square of the individual deviations:

19'
TABLE IX - Specimen Diameter Measurement (Centimeters

)

Specimen Lab. 1 Lab. 2 Lab. 3 Lab. 4 Lab. 5 Avg. s s(%)

605603-3 2. 832 2. 832 2. 834 2. 832 2. 835 2 .833 0 .0014 0.05

601333-2 2. 908 2. 895 2. 902 2. 891 2. 888 2 .897 0.0082 0.28

71983-2 3. 343 3. 360 3. 3437 3. 350 3. 347 3 .349 0 .0069 0.21

71166-2a 2. 047 2. 051 2. 06 2 .053 0 .0067 0. 32 71155-2b 2. 0713 2. 070 2. 071 2 .071 0 .0007 0.03

600200-2 3. 099 3. 096 3. 104 3. 099 3. 103 3 .100 0 .0033 0.11

607075-2 2. 972 2. 969 2. 9696 2. 967 2. 971 2 .970 0 .0019 0.06

70877-3 3. 048 3. 045 3. 062 3. 058 3. 051 3 053 0 .0070 0.23

49445-2 2. 997 3. 000 3. 003 3. 000 3. 006 3 001 0 .0034 0.11

66969-1 3. 086 3. 0911 3. 10 3. 085 3. 094 3 091 0 ,0059 0.19

16603-2 2. 337 2. 344 2. 349 2. 347 2. 343 2 344 0.0046 0.20

TABLE X - Specimen Thickness Measurement (Centimeters

)

Specimen Lab. 1 Lab . 2 Lab . 3 Lab . 4 Lat . 5 Avg. s s(%)

605603-3 0. 1168 0.1173 0. 1168 0. 1171 0. 11662 0. 1169 0. 0003 0.23

601333-2 0. 1191 0. 1197 0. 1191 0. 1194 0. 11909 0 1193 0 0003 0.23

71983-2 0. 1207 0. 1209 0. 1207 0. 1206 0. 12043 0 1207 0 0002 0.14

71166-2a 0. 1166 0, 11696 0. 11581 0 1165 0. 0006 0.50

71156-2b 0. 1158 0. 1166 0. 11597 0. 1161 0. 0004 0.36

500200-2 0. 1214 0. 1218 0. 1214 0. 1219 0. 12121 0. 1215 0. 0003 0.24

607075-2 0. 1194 0. 1203 0. 1194 0. 1196 0, 11920 0. 1196 0.0004 0.36

70877-3 0. 1184 0. 1181 0. 1188 0. 1184 0. 11821 0. 1184 0. 0003 0.22

49445-2 0. 1163 0. 1163 0. 1163 0. 1145 0. 11610 0. 1159 0. 0008 0.68

66969-1 0. 1166 0. 11709 0. 1168 0. 1167 0. 11659 0. 1168 0. 0002 0.18

16603-2 0. 1224 0. 1227 0. 1222 0. 1227 0. 12222 0. 1224 0. 0002 0.20

20
^ = /(oTsS)^ + (0.013)^ + (0.26)^ + (0.34)^

P

'

.,

= /0.'+81 = 0.69 per cent.

This value is remarkably close to the value frequently found in the

round-robin experiment. It would appear to indicate that the errors

in the experiment can be accounted for by the various factors above.

Since the dominant error occurs in the V/I measurement, and since much

of this can be attributed to wafer non-uniformity, more uniform slices

must be available if increased precision is to be obtained.

2.3 EXPERIMENT 2.

This experiment was modeled after the preliminary experiment

described in Appendix A. The procedure given in Appendix B was used

with the addition of the table of currents described in connection with

Experiment 1. In addition to the ten silicon wafers and six analog

circuits the following equipment was furnished to each participant in

turn: (1) four-point probe and holder, (2) micrometer stage with

copper heat sink, mica insulator, and silicone heat-sink compound, (3)

calibrated thermometer, and (U) polished silicon blanks for the probe

separation measurement.

The analog circuits were similar to those used in Experiment 1.

Resistivity profiles made on the wafers at the end of the test are shown in Fig. 3. In most cases each is similar to the profiles of the

equivalent wafer used in Experiment 1.

2.3.1 Results . The results of the test are summarized in Table

XI through XX. These tables are arranged in the same order as Tables

I through X and present the data in a similar fashion. Much of the

discussion related to Experiment 1 can be carried over to the present

case. However, it is immediately obvious that the precision of the

measurement is considerably less (i.e., has a higher numerical value)

in Experiment 2. Examination of Tables XVII and XVIII shows that

significant difficulties with the electrical measuring apparatus were encountered in several of the labs. Unfortunately, these difficulties

render a quantitative, analysis of the experiment meaningless.

It can be noted that geometrical measurements on the wafers

(Tables XIX and XX) were made with nearly the precision attained in

Experiment 1. Since the same probe was supplied to all participants,

the data (Table XVI) yields an indication of the precision of the

measurement of probe separation. Labs 1, 2, and 3 appeared to have

problems in this area. The same probe was used in the preliminary

experiment. Comparison of Table XVI with Tables III and IV of Appendix

A shows that much of the spread in the present experiment is due to

measurement problems rather than probe problems.

21
Distance from Center (mm)

Figure 3. Specimen Resistivity Profiles for the Slices Used in

Experiment 2. (No profile was made on Slice 600200-3.)

22
TABLE XI - Average Resistivity (fi-cm) at 23°C (as Reported)

Specimen

505603-2

501333-1

71983-3

71166-3

500200-3

607075-3

BTL-4

49445-1

56969-2

15603-3

Lab. 1 Lab.

0.0079680 0.008

0.084034 0.086

96.693 135.8

1254.3 3373.

0.0077248 0.008

0.10800 0.103

10.187

11. 822 110.53

939.42

10.0

11.8

102.3

Lab. 3 0.0082

0.08436

100.75

Lab. 4 0.00808

0.08449

0.1074

10.220

11.7959

109.02

105.1

1057.9

0.1059

10.50

12.10

119.2

914.1

Lab. 5 0.00821

0.0851

86.64

704.6

0.107

9.58

11.83

108.3

735.07

Lab. 5 AVR . 0.0081977 0.008109

0.084731 0.08479

95.78 103.95

1152. 1512.

0.00786

0.10588

10.134

11.800

110.47

935.9

0.1054

10.12

11.85

110.0

881.4

s s(%)

0.000108 1.34

0.000694 0.82

17.31 16.55

1061. 70.16

0.00019 2.48

0.00179 1.68

0.271 2.68

0.119

5.4 98.2

1.01

4.95

11.14

TABLE XII - Average Resistivity (J2-cm) at 23°C (Recomputed)

Specimen

605603-2

501333-1

71983-3

71155-3

600200-3

607075-3

BTL-4

49445-1

55959-2

16603-3

Lab. Lab.

0.0079895 0.0080

0.084201 0.086

97.112 137. 1258.3 3350. 0.0077600 0.0078

0.10745 0.103

10.188 10.0

11.855

110.84

940.80

11.8

102.

Lab. 3 0.0082

0.0849

101.2

0.1079

10.23

11.85

110.8

Lab. Lab.

0.00823

0.08501

107.6

1075.

0.1067

10.55

12.26

119.0

923.4

0.00822

0.0850

86.5

704.

0.1070

9.70

11.83

107.9

734.

Lab . 6 Avg

.

0.0081977 0.008140

0.084731 0.08497

96.78 104.38

s(%)

1161.

0.10588

10.134

11.811

110.47

935.9

0.000113 1.39

0.000586 0.59

17.39 15.56

1514. 1054. 69.62

0.00778 0.000028 0.36

0.1065

10.13

11.90

110.2

883.8

0.00176 1.66

0.280 2.76

0.177 1.49

5.5 100.1

4.99

11.33

Specimen

TABLE Average

Resistivity

(fi-cm)

XIIILab. 1

Sample Standard

Lab. 2

Deviation

Lab. 3

in Per Cent

Lab. 4

(Recomputed)

Lab. 5 Lab1. 6 Avg.

605603-2 0. 008140 0.25 0.6 0.39 1.19 0. 73 0. 55 1.39

601333-1 0. 08497 0. 31 0.6 0.08 0.34 0. 42 0.13 0.59

71983-3 104. 38 0.38 2.5 0.89 1.22 4. 31 0. 24 16.55

71155-3 1514. 1.64 14.0 3.30 4. 44 1. 23 59.62

500200-3 0. 00778 0.15 0.0 .35 507075-3 0. 1055 0.15 0.0 1.17 0.50 0. 32 0. 08 1.55

BTL-4 10. 13 0.34 1.2 0.88 0.89 5. 05 0. 30 2.76

49445-1 11. 90 0.15 0.7 0.28 _ 0.24 0. 21 0. 10 1.49

66959-2 110. 2 0.14 2.5 . 0.25 1.74 0. 69 0. 18 4.99

15603-3 883. 8 0.31 1.66 0. 71 0. 14 11.33

23
TABLE XIV - Per Cent Difference of One Reading from Overall Average Resistivity

Average

Resistivity

Specimen !-cm

)

Lab. 1 Lab. 2 Lab. 3 Lab . 4 Lab. 5 Lab. 6 Ave

.

s(%)

605603-2 0. 008140 -1.65 -1. 72 + 0. 74 + 1. 35 + 0. 98 + 0. 18 0. 00 1.34

601333-1 0. 08497 -0.50 + 0. 04 + 0.04 + 0. 39 + 0. 04 -0. 11 -0.02 0. 29 71983-3 104. 38 -6. 96 + 8. 26 -3.81 + 5. 19 -25. 66 -7. 14 -5.02 12.60

71166-3 1514. -16.49 + 162. 88 -29. 66 -54. 76 -23. 18 + 7 . 73 81.59

600200-3 0, 00778 -0. 40 + 0. 26 -0.13 0.46

DU /U / D—

O

U . tU . y U -3. 29 T U • DO + 0. 66 + 0. 19 + 0. 38 n no

. BTL-4 10. 13 . . + 0. 69 + 0. 69 + 1.28 + 3. 95 -0. 99 + 0. 51 + 0.99 1.61

49HU5-1 11. 90 -0.35 -1. 68 -0.42 + 3. 28 -0. 75 -0. 83 -0.17 1.74

66969-2 110. 2 + 0.57 -4. 72 -8.53 + 7. 89 -2. 27 + 0. 19 -1.09 5.64

16603-3 883. 8 + 6.40 +4. 65 -16. 27 + 5. 83 + 0.16 10.96

TABLE XV - Per Cent Difference of Median of Three Readings from Overall Average Resistivity

Average

Resistivity

Specimen (fi-cm) Lat). 1 Lab. 2 Lab. 3 Lab. 4 Lab. 5 Lab. 6 Avj7 , s(%)

505603-2 0. 008140 -1. 84 -1.72 + 0.74 + 0. 98 + 1.84 + 0. 54 + 0,,12 1.52

601333-1 0. 08497 -0. 84 + 1.80 + 0.04 -0.20 +0.04 -0. 31 + 0,.08 0.90

71983-3 104. 38 -7. 05 +27.42 -3.81 + 3.56 -15.88 -7. 30 -0,.56 15.13

71165-3 1514. -15. 06 +132.50 -29.66 -54.76 -23. 65 + 1,.65 73.33

500200-3 0. 00778 -0. 29 + 0.26 0,,00 0.39

507075-3 0. 1055 + 0. 89 -3.29 + 0.56 +0.47 + 0.47 + 0. 39 -0,,09 1.58

BTL-4 10. 13 + 0. 35 -0.30 + 1.97 + 3.95 -0.99 + 0. 08 + 0.,89 1.80

49445-1 11. 90 -0. 35 -1.68 -0.42 + 3.03 -0.59 -0. 80 -0.,17 1.62

65959-2 110. 2 , . + 0. 67 -5.63 + 0. 54 + 7.71 -2.27 + 0. 33 + 0.,18 4.39

16603-3 883. 8 + 6. 80 +4.55 -15.72 + 5. 83 + 0.14 11.26

TABLE XVI - Probe Separation Measurement in Millimeters

s^(%)

S2(%)

S3(%)

sp

Lab. 1

1. 59156' 0.01387

0.87

1.58692

0.01651

1.04

1. 59720

0.01852

1.15

1.59189

1.0034

Lab. 2 1.5951

0.0020

0.13

1.5916

0.0046

0.29

1.5961

0.0055

0.35

1. 5944

1.0018

Lab. 3 1.5888

0.00729

0.45

1. 58534

0.00558

0.41

1.60528

0.00267

0.17

1.59314

1.0053

Lab. 4 1.58966

0.00122

0.08

1.58671

0. 001168

0,07

1.60099

0.00086

0.05

1.59245

1.0039

Lab. 5 1.5908

0.0015

0.10

1.5867

0.0008

0.05

1.6063

0.0023

0.14

1.5946

1.0053

Lab. 6 1.58775

0.001090

0.07

1.58796

0.001123

0.07

1.60210

0.001199

0.07

1.59261

1.0031

Avg.

1.5905

0.00259

0.16

1.5875

0.00213

0.13

1.6012

0.00411

0.26

1.59319

1.0038

24
Analog

Circuit

No. 1 No. 2 No. 3 No. 5 No. 6 No, 7

Lab. 0,0010032

0,010102

0,10025

10,015

99,853

990.75

TABLE XVII - Average Resistance (Recomputed)

a) Measured Values (Q)

Lab. 0.00107

0.0102

0.101

10.3

100.

880.

Lab. 3 0.00100

0.01001

0.1003

10.00

100.0

1001.

Lab. 4 0.001025

0.00996

0.1017

10.155

106.2

963.0

Lab. 5

0.01001

0.0995

9.96

93.8

Lab. 6 Avg.

0.0009799 0.001016

0.0101023 0.01006

0.10022 0.1005

10.017 10,07

99,94 99.97

1003.5 967.6

s s(%)

0,000034 3.38

0,00009 0.87

0,00076 0,75

0.129 1,28

3,92

51,6

3.92

5.33

b) Sample Standard Deviation (Per Cent)

Analog

Circuit

No.

No.

No.

No,

No.

No,

Lab. 1 0.11

0.02

0.00

0.01

0.05

0,11

Lab. 2 4.7

0.3

0.0

0.8

0.0

0.0

Lab. 3 0.80

0.28

0.2

0.0

0.0

0.33

Lab. 4 0.32

0.55

0.26

0.19

0,12

0.48

Lab. 5

0,22

0.13

0.28

0.04

Lab. 6 0.08

0.01

0.00

0,00

0,03

0.05

Avg,

3.38

0.87

0.75

1.28

3.92

5.33

TABLE XVIII - V/I (fi) Corrected to 23°C

Specimen

505503-2

501333-1

71983-3

71166-3

600200-3

607075-3

BTL-4

49445-1

55959-2

16603-3

Lab. Lab. Lab.

0.015798

0,15355

185,30

0.0159

0,1659

261.1

2583,0 6842.0

0,014799 0.015

0. 20849

21,990

23,432

217.72

1808.3

0.201

21.8

23.5

201.0

0,01629

0.1652

192.7

0.2095

22.078

23.563

217.0

Lab. 4 0.0164

0.166

206.6

2194.

0. 208

23.01

24.52

234.5

1782.

Lab. Lab, 6 Avg.

0.0163

0.166

165.

1438.

0.208

21.0

23.5

212.

1411.

0.016305 0.01617

0.16533 0,1655

185,2 199.3

s s(%)

0.000251 1,55

0.00110 0.65

33,13 16,62

2370, 3085, 2144. 69.48

0.014810 0,000142 0,95

0.20821 0.2072 0,00309 1.49

22,020 21.98 0,643 2.92

23.54 23.68 0.416 1.75

217.34 216.6 10.83 5,00

1805. 1702. 194.1 11,40

25
TABLE XIX - Specimen Diameter Measurement (Centimeters)

Specltnen Lab. 1 Lab. 2 Lab. 3 Lab. 4 Lab . 5 Lab. 5-.. AV2 . s s(%)

605503-2 2,.8313 2,, 81783 2.837 2.835 2. 835 2. 831 0,.0078 0.27

501333-1 2.,893 2,,858 2. 903 2.903 2. 901 2.892 0,.0192 0.67

71983-3 3,.340 3,,3338 3. 338 3.340 3. 331 3.337 0.. 0040 0.12

71166-3 2,.065 2,,064 2,068 2.064 2. 068 2.066 0..0020 0.10

600200-3 3,.0803 3,,09562 3.088 0,.0108 0. 35 607075-3 2,.9634 O ,93687 2.967 2.955 2. 965 2.959 0,.0127 0.43

BTL-4 oq .. Z z o

QO ."1 7 ^ 3.225 3.235 OO • £.00 3.221 0,.0260 0.81

49445-1 3,.0015 3,,0162 .. =

: 3.018 3.005 3. 0038 3.009 0,.0076 0.25

66909-2 3,.081 3.,01625 3.086 3.084 3. 087 3.071 0,.0306 1.00

16603-3. 9z • 2.355 2.347 . 2. 350 2.348 0.0063 0.27

TABLE XX -" Specimen Thickness Measurement (Centimeters ;

Specimen Lab. 1 . Lab. 2 Lab. 3 Lab. 4 Lab. 5 Lab. 6 Ave

.

s s(%)

605603-2 0..117057 0.,1166 0. 1158 0.1161 0. 11625 0.11636 0,.00048 0.42

601333-1 0,.119168 0.1189 0.11836 0.1186 0. 11865 0.11874 0,.00031 0.26

71983-3 0,,12082 0.,1209 '

' - ' 0.11989 0.1204 0. 12051 0.12050 0.,00040 0.33

71166-3 0.11618 0.,1165 0.11582 0.1156 0. 11589 0.11602 0..00039 0. 33 500200-3 0,.121376 0.,1212 0.12129 - 0..00012 0.10

c r\'~i r\'^ c o 60/0 /5-3 0.,119228 0,,1191

-

' ' 0.11835 0 . 1184 0. 11873 0 . 11876 0,,00040 (J . 00 BTL-4 0,,105537 0,.1054 0.10439 0.1049 0. 10480 0.10500 0. 00047 0.44

49445-1 0,,11568 0,.1156 - 0.11506 0.1156 0. 11561 0.11571 0. 00059 0.51

56909-2 • 0,,11734 0..1175 ' 0.11684 0.1171 0. 11716 0.11721 0. 00028 0.24

16603-3 0,,12257 0.12192 0.1222 0. 12219 0.12222 0. 00027 0.22

26
It should also be noted that some of the observed difficulties

arose because of inadequate resolution in the measuring equipment. As

a result, specific resolution requirements were added to the revision

of the method.

2.4 CONCLUSIONS

From the results of these experiments, it can be concluded that

resistivity measurements can be made according to the procedures of

the method under test with a precision of t2 per cent (3 standard

deviations). Relaxation of the requirement of averaging 10 pairs of

readings to permit a single pair to be used increases the 3 standard

deviation interval to ±4 per cent. It was shown that errors in the

six quantities measured during the test account for the overall deviation obtained if effects of wafer inhomogeneity are included. It was found that much of the error which entered in the determination of the

potential difference between the inner probes arose from this source.

This precision was not achieved over the entire resistivity range.

No very low ('v-O.OOl J^-cm) resistivity wafers were included in the

experiments. In addition gross inhomogeneity in the 1000 Q-cm p-type

wafers used in the tests prevented the acquisition of good data.

Accordingly, additional tests at both extremes are still needed. The

low resistivity test is scheduled to begin soon; in addition several

high resistivity wafers will be included in this test.

The poor precision achieved in Experiment 2 serves to emphasize

the need for adequate equipment and control procedures if precise

measurements are desired. Although resistivity is probably the most

widely measured semiconductor characteristic, the precise determination

of resistivity can only be done with facilities which are well maintained, accurately calibrated, and properly used.

The control procedures outlined in the test method appear to be

adequate to identify problems associated with the probe or electrical

measuring equipment. Additional study of the low resistance analog

circuits will be required before their usefulness can be fully documented. This will be done in connection with the forthcoming roundrobin experiment.

The results of the analysis indicate that aside from the single

measurement pair modification discussed above relaxation of the various

requirements of the method will reduce the precision of the measurement

to a value which is generally unacceptable. In particular if the permitted standard deviation were doubled the probe separation uncertainty

would become the dominant factor contributing to the variation in

measured resistivity. Furthermore, difficulties in thickness determination at the 1.1 mm level suggest that uncertainty in this deminsion

will also become a dominant factor if slices 0.25 to 0.5 mm are measured. Where +10 per cent measurements are sufficient, some relaxation in the geometrical requirements on the wafer would be feasible

27
even if only a single pair of readings is taken. It is possible to

incorporate the various geometrical correction factors into a direct

reading instrument. ^ It is also possible to include circuits which

incorporate the correction for temperature and unequal probe separations .

The method can be extended for use in sheet resistance measurements and for production control of slices. In order to determine the

limits of validity and the precision which might be anticipated in such

applications, additional studies of the effects of decreased probe

pressure and different surface conditions must be carried out.

2.5 NOTES AND REFERENCES

1. This draft version of the method differs only slightly

from the version published for information only in the

. back of Part 8 of the 1967 ASTM Book of Standards. In

subsequent revisions some paragraph numbers have been

f

? changed; references to paragraph numbers given in this

printing are appropriate to the current standard,

F- 84-7 3.

2. M. A. Logan, "An AC Bridge for Semiconductor Resistivity

Measurement Using a Four-Point Probe," Bell System Tech.

J. 40, 885-919 (1961). .

3. Note that this formula is appropriate to a thin slice

: while the formula shown in Fig. 1 is appropriate to a semi-infinite volume.

4. F. M. Smits, "Measurement of Sheet Resistivities with

the Four-Point Probe," Bell System Tech. J. 37, 711-718

(1958).

5. L. J. Swartzendruber, "Correction Factor Tables for FourPoint Probe Resistivity Measurements on Thin Circular

, v> Semiconductor Samples," NBS Technical Note 199, April

; .

. 15, 1964. , r

6.. L. J. Swartzendruber , F. H. Ulmer, and J. A. Coleman,

"Direct-Reading Instrument for Silicon and Germanium

Resistivity Measurement," Rev. Sci. Instrum. 39 , 1858-

,,., ..1863 (1968).- '., ,,,5^,,, ^j;^

28
3. EXPERIMENTAL STUDIES

3.1 INTRODUCTION

An experimental study of the temperature coefficient of resistivity of silicon and germanium was concluded during the project. A

report describing this work has been prepared for publication. The

data obtained are attached as Appendix E. In addition, initial experimental work was undertaken to establish thermal equilibration time, the

effect of non-vmiform thickness, and the effect of probe needle wobble.

-3.2 THERMAL EQUILIBRATION TIME

Two sets of experiments were run. In the first, a 0.1 Q,-cm n-type

germanium wafer cut into a "clover-leaf" shape for van der Pauw measurements was cooled below or heated above room temperature. After reaching

a suitable temperature it was placed on the copper heat sink of the

four-probe apparatus. The temperature of the wafer determined from its

resistivity was monitored as a function of time. It was found that the

wafer always approached a temperature somewhat greater than the heat

sink temperature but that it was within 0.5°C of the heat sink temperature in less than 3 minutes when initially at -50°C and in less than

7 minutes when initially at +35°C.

These results emphasize the importance of the use of proper current

levels when measuring resistivity. The current used (about 100 mA) was

large enough to cause sufficient joule heating in the wafer to raise

the temperature above the heat sink temperature. An auxiliary experiment, in which the wafer was not placed directly on the copper heat

sink but instead, inside a plastic box at the heat sink temperature,

showed that, in the absence of the heat sink, the wafer rises to a temperature nearly 9 deg above that of the heat sink in about 20

minutes. This current, which is larger than would normally be used on wafers of this resistivity, was selected in order to allow more rapid

measurements to be made.

Even with the larger current, it was not possible to follow the

initial stages of decay. Hence, a second series of measurements on a silicon wafer about 1.2 mm thick were made in which the temperature

difference between the top of the wafer and the copper block was measured with a differential copper-constantan thermocouple. One

junction of the thermocouple was attached to the wafer with galliumindium eutectic; mechanical support was provided by gluing the wires

just behind the junction to the wafer. The wafer was cooled or heated

to the desired initial temperature. After the reference junction of the

thermocouple was immersed in an oil-filled well in the copper heat sink

and the leads were connected to a recorder with a maximum sensitivity

of IpV/mm, the wafer was placed on a 12 \im thick mica sheet on the heat

sink and the probes were lowered. No current was passed through the

29
probes. In all cases the wafer temperature had reached within 0.2 deg

of the heat sink temperature in less than 30 s. Noise on the thermocouple leads prevented determination of smaller temperature differences.

In an auxiliary experiment, the wafer was placed near but not on the

heat sink; about 11 min elapsed before the wafer reached within 0.3

deg of the heat sink. , r. : . ^

3.3 EFFECT OF NON-UNIFORM THICKNESS "

'

These experiments were carried out oj\ an aluminum-doped silicon

wafer of about 0.245 ft-cm. The thickness was initially 1.022 mm and

the diameter, 26.85 mm. After measuring the resistivity with parallel

faces on the wafer, one side was angle lapped to 11 min., then 22 min.,

then 33 min., and finally parallel again. The average thickness in

each case was determined from five measurements , one at the center of

the wafer, and four on perpendicular radii about half way between the

center and the edge of the wafer. The resistivity was determined by

averaging the results of ten measurements at the center of the wafer,

five on each side. Between readings, the wafer was rotated about 15°.

Although a small increase in average resistivity was detected as the

taper angle was increased, the value in each case did not depart from

the average of the two parallel cases by more than 0.33%. The spread

on the averages of the two parallel cases was about 0.2%. Although

the dependence on taper angle may be statistically significant, it

would appear that it may be ignored as a practical matter at least

under conditions similar to those of this test. The results are summarized in Table XXI.

V Table XXI . , Angle lapped wafer: thickness and resistivity

thickness thickness

variation variation

thickness (edge-to-edge) (measured) resistivity

Condition (mm) ( mm) (%) (mm) (%) (n--cm)

parallel 1.0223 0.024 0.23 0. 24496 + 0. 00047

11 min. 0.9670 0. 0860 8.9 0.0356 3.7 0. 24534 + 0. 00066

22 min. 0. 8899 0. 1718 19. 3 0.1103 12.4 0. 24585 + 0. 00047

33 min. 0. 8449 0. 2577 30.4 0.1350 16.0 0. 24626 + 0. 00069

parallel 0.6083 0.032 0. 38 0. 24594 + 0. 00054

Nevertheless, uncertainty in thickness in very thin wafers causes equal uncertainty in resistivity. When the wafer has flat (though non- parallel) faces the uncertainty in thickness can be reduced considerably

below the variation in thickness over the wafer; however, this can not

be assumed always to be the case. As an example of an irregular shape,

a ring 2.674 mm wide and 0.130 mm deep was cut ultrasonically from the

30
outer edge of the wafer, leaving a "top-hat" structure. If the full

diameter and maximum thickness are used in the resistivity computation

a value about 1% larger than the average of the two parallel cases was obtained. A weighted value of average thickness yielded a resistivity

>about -H% lower i a 3% overcorrection..... No. conyejiient,mg^^n.s.^.pX,.Pt^fain

the effective diameter or thickness in this case has been found..^

3.4 EFFECT OF PROBE NEEDLE WANDER

Studies of the effect oF probe needle wander require the use of

probes with different amounts of needle wander. Several probe ,^ assemblies were tested during this reporting period but ^none which had

a sample standard deviation on probe spacing larger than that allowed

in the test method (cf. Method F-SU, HID. 1.3.1) was found.

Computations of expected effects of probe needle wander on thin J^. wafers were carried out as part of the error analysis of Experiment 1.

The results of these Computations have been summarized in Section 2.

'f

31
100

'INDUSTRIAL ROUND ROBIN PRIOR TO

AN A.S.TM STANDARD

UJO

UJ 10

>

UJ

o

o

on

<

o

CO

ROUND ROBIN TEST

OF A.S.T.M. STANDARD

ADOPTED 1964

VALUE DESIRED FOR MATERIALS

EXCHANGE - 1%

A.SIM. ROUND ROBIN \

'j

TEST OF N.B.S. METHOD -V

0.1 J L 1 J. 1

58

Figure U

,

60 62 64

YEAR

66

History of the Interlaboratory Precision of

Measurement of Silicon Resistivity in the

Range 0.01 to 100 fi-cm, by the Four-Probe

Method, Showing the Spread of Sample-Dependent Standard Deviations by the Vertical Bars

68

32
4. SUMMARY

During the period covered by this project, the Method of Test for

Resistivity of Silicon Slices Using Four Point Probes has been extended

to cover the useful resistivity range. The precision which can now be

obtained in the resistivity range between 0.005 and 120 ohm-cm is l2

per cent (3 standard deviations). This is significantly better than

was possible with earlier methods as shown by the plot of single

standard deviation against time in Fig. 4,

Detailed analysis of the round-robin experiments leads to the

following conclusions:

1) The desired precision can be obtained if correct

procedures are followed and if the equipment used

in the test meets the requirements of the test

method. The importance of a well maintained,

accurately calibrated, and properly used test

system can not be overemphasized.

2) The major contribution to measurement error

appears to be resistivity non-uniformity in the

specimen under study.

3) If a single pair of readings is taken rather than

the average of 10 pairs as required by the method,

the precision is degraded somewhat; the relative

standard deviation may double.

4) Relaxation of the geometrical requirements of the

method would be expected to reduce the precision

significantly. Some difficulty in maintaining

the required precision in the determination of the

thickness is expected if thin (0.25 to 0.5 mm)

slices are measured.

The importance of knowing the temperature of the slice being

measured and the effectiveness of the large copper heat sink in es- tablishing this temperature were also demonstrated. Slices initially

maintained at temperature well above or below the heat sink temperature

reached a temperature within 0.2°C of the heat sink temperature less

than 30 s after being placed on the mica insulator which electrically . isolates the slice being measured from the heat sink.

Slices with flat, but non-parallel, sides could be measured precisely at the center if the average thickness was used. However, an appropriate correction could not be found for a slice with a "top-hat"

configuration.

33
Difficulties with the low resistance analog circuits were attributed to instabilities in the circuits. New analog circuits of

improved design are being assembled but they have not yet been tested

to verify this conclusion. These tests are expected to be carried out

soon in connection with an additional round-robin experiment designed

to establish precision figures for very high and very low resistivity

slices.

Determination of the precision to be expected from the method in

non-referee applications such as routine production and quality control

will require additional study of such factors as surface conditions,

probe force, current levels, etc. Nevertheless, use of the various

procedures of the method, in particular the sections on probe and

measuring circuit evaluations and on thermal sinking of the wafer,

would be expected to yield significantly improved precision in such

applications. Use of these procedures on a regular and widespread basis

should be encouraged.

34
APPENDIX A

Report to the ASTM F~l

Subcommittee VI Semiconductor

Resistivity Task Force on

a Special Round Robin

Introduction

This is a report on a special round robin on 4-point probe resistivity measurements held March 1 to June 1, 1966. The report was originally made to the Task Force in a preliminary form at the Chicago

meeting in June 1966.

The idea for this round robin originated at a meeting of the Task

Force in Dallas in February. Final plans were formulated at a later

one-day meeting held at the National Bureau of Standards. The following

laboratories participated (listed in alphabetical order):

Bell Telephone Labs., Allentown, Pennsylvania

Dow Corning Corp., Electronic Products Div. , Hemlock, Michigan

International Business Machines Corp., East Fishkill Facility,

Hopewell Junction, New York

Monsanto Chemical Co., Inorganic Chemicals Div., St. Louis, Mo.

National Bureau of Standards, Washington, D. C.

A special vote of thanks go to K. Benson and C. Paulnak at BTL

for providing much of the material used and for making a special box to

facilitate shipping. Thanks also go to all the participants for their

expeditious handling of the measurements, allowing much useful data to

be obtained and analyzed between two Sub-VI meetings!

Purpose

Before F43-64T-'- (Tentative Methods of Test for Resistivity of Semiconductor Materials) can be properly revised, the need exists to

determine the contribution of each of the factors affecting the multilaboratory precision of resistivity measurements. The factors selected

for investigation in this round robin were the precision of the probe

spacing measurement and precision of electrical measuring equipment

when doing four-point probe measurements at the 10 Q-cm level. The

measuring process was to be that typical of a good industrial standards

laboratory. Future round robins including such factors as sample

temperature measurement and sample preparation will be necessary.

Method

The procedure that was used for the round robin is given in

Appendix B. As many of the variables as possible were controlled. The

35
mechanical equipment required for the measurement was supplied along

with silicon slices in the 10 Q,-cm range that had been prepared. Procedures for measuring the spacing of the probes in the furnished fourpoint probe and for determining the suitability of electrical equipment

(supplied by each laboratory) were specified. Independent measurements

of probe spacing, electrical equipment, slice diameter, and slice

thickness in each lab provided a means of evaluating "the reproducibility

obtainable in such measurements. Since the thermal sink and thermometer were both supplied and since the sample surfaces were prepared at

the beginning of the test, these conditions were not varied. .A round

robin "kit" encased in a sturdy wood box was shipped to each laboratory

in turn. This kit contained:

(a) a four-point probe, -

. . (b) a micrometer stage,

(c) a copper heat sink,

(d) a calibrated thermometer, •

. . (e) mica for use as an insulator on the copper block,

' (f) chemically polished silicon blanks for needle impressions,

(g) a four-point probe analog circuit,

(h) heat sink compound for making good thermal contact between

the heat sink and specimen, and

(i) three lapped silicon slices on which to measure resistivity.

Results . . ;, . .

First a word about notation. The abbreviation AVG will be used to

denote the sample mean and the symbol s to denote the square root of

the sample estimate of variance. (A capital S will be used to denote

probe spacing.) Also, the idea of a confidence interval will be used 2

Let us first look at the resistivity results as taken directly

from the data sheets which are shown in Table A-I . At first glance

this is discouraging; even disregarding lab. 5 the most probable multilaboratory precision is no better than ±2 per cent (R2S%).3 Lab.. 5

; is obviously in error but this was probably due to difficulty in

interpreting the instructions for use with direct reading equipment.

After finding out exactly how lab. 5 proceeded and recalculating

their results, and also correcting obvious errors on the rest of the

data sheets (e.g. misreading correction factors or errors like 49.9

mils = 0.1277 cm), the resistivities shown in Table A-II were obtained.

If we disregard lab. 5 we can assert ' that : 1) under the conditions of

this round robin, the most probable multi-lab precision for resistivity

measurement is +0.7 per cent (R2S%), and 2) the multi-laboratory

precision for resistivity measurement using the methods of this round

robin is better than ^2 per cent (R2S%) at a confidence level of 95

per cent. The latter statement means that if we were to repeat the

round robin a very large number of times, there is only a 5 per cent

36
Table A-I - Average Resistivity (J2-cm) at 23°C (as Reported)

Sample Sample Sample

Lab. No. BTL-2 BTL-U KN-H

1 10.24 10.36 lh.80

2 10.083 . 10.209 14.763

3 10.065 10.137 14.545

4 10.036 10.125 14.648

5 1.853 1.778 3.261

AVG 8.455 8.522 12.403

s 3.691 3.77 5.11

(44%) (44%) (41%)

AVG^^^ 10.106 10.208 14.689

s^^^ 0.091 0.108 0.116

(0.90%) (1.1%) (0.79%)

One-sided/^^ 1.22 0.26 0.27

95% confidence (2.2%) (2.5%) (1.9%)

upper bound on a

Table A-II - Average Resistivity (ft-cm) at 23^0 (Recomputed)

Sample Sample Sample

Lab. No. BTL-2 BTL-4 KN-4

1 10.042 10.148 14.513

2 10.081 10.209 14.538

3 10.055 10.148 14.557

4 10.036 10.125 14.523

5 9.899 10.036 14.338

AVG 10.023 10.133 14.494

s 0.071 0.063 0.089

(0.71%) (0.62%) (0.61%)

AVG^^^ 10.054 10.158 14.533

s^^^ 0.020 0.036 0.019

(0.2%) (0.36%) (0.13%)

One-sided, 0.058 0.105 0.045

95% confidence (0.6%) (1.0%) (0.3%)

upper bound on a

disregarding lab. 5 37
chance that the final precision obtained would be worse than 12 per

cent (R2S%).

Discussion ——^——^— t

\-

We wish to analyze and compare the relative magnitudes of the sources of error. The sources to be analyzed are:

(1) probe separation, ' (2) V/I measurement,

(3) temperature measurement, and

(4) slice thickness measurement.

(1) Probe separation measurement . Probe separation was measured

in each lab according to the specified procedure. The results, shown

in Table A-III, show that this is an excellent procedure for measuring

probe spacing, there being only a 5 per cent chance that the multi-lab

precision is worse than l0.5 per cent (R2S%). What part of this

precision is due to needle-wobble, what part to needle-tip condition,

and what part to the measuring apparatus is not certain.

Good probes can be selected by placing a maximum allowable singlelab s for the series of 10 determinations performed by each lab. The

values for s obtained by the individual labs in this round robin, using

a "good" probe, are given in Table A-IV. If we accept 0.16 per cent as

the "true" standard deviation for the probe used, and if we want only

a 5 per cent chance of rejecting a "good" probe (i.e. one at least as good as the one used in this round robin) we should require a singlelab s measurement of less than 0.26 per cent (see page 4-3 of Handbook

91). 2 . ..^^

In a measurement of the resistivity of a slice a probe spacing

error will show up in three places. The ratio*^ S2/r, where r is the

sample radius, is used to determine the correction factor for finite

diameter, F2. The ratio'^ W/S2, where w is the slice thickness, is used

to determine the correction factor, F(w/S). The individual values Sj^,

S2, and S3 are used to determine a correction for unequal probe spacing,

= 1 + 0.721(1 - S2/2S-L - S2/2S3).5

For the slice diameter and thickness used, an error of 0.1 per cent

in measuring S would cause

,

(1) an error of 0.01 per cent in F2,

(2) an error of 0.02 per cent in F(w/S), ^ (3) an error of 0.08 per cent in F„„.

P

Adding the effect of these errors directly (since they are not

independent) gives a total error of 0.11 per cent.

Note the following about Fgp, a correction factor which has not

been previously used. In this round robin the average value was 1.004.

This is a 0.4 per cent correction and thus should not be neglected.

38:v
Table A-III - Probe Separation Measurement in Millimeters

Lab 2

1 1.60160

2 1.60396

3 1.60134

4 1.6027

.5 " 1.6050

AVG 1.60294

s 0.00157

(0.10%)

One-sided, 95%

confidence upper 0.00373

bound on a (0.23%)

1.58572

1. 58590

1.58354

1.5857

1. 5880

1.58577

0.00157

(0.10%)

0.00373

(0.23%)

1.58572

1.58882

1.58496

1.5862

1.5875

1.58654

0.00152

(0.10%)

0.00361

(0.23%)

Table A-IV - Single Laboratory Relative Standard Deviation

Lab 1

1 0.10%

2 0.17

3 0.08

4 0.04

5 0.27

Overall AVG of

table above 0.12

Overall s of

table above 0.08

One sided, 95%

confidence upper

bound on AVG 0.15

_2

0.06%

0.10

0.07

0.04

0.24

__3

0.12%

0.21

0.06

0.04

0.19

39
(2) V/I measurement. The justification for disregarding the

lab. 5 results for V/I comes from measurement of the four-point probe

analog circuit. The resistor in the black box was a lOQ + 0.5%

resistor. The four other resistors were 3000^2 1 5% resistors. The

measured values are given in Table A-V. Note that lab. 5 measured a resistance about 3 per cent low and was also low on the slice resistivity measurement, (1.5 per cent, 1.2 per cent, and 1.2 per cent for

the 3 slices, respectively). This could have been caused by using a measuring system with an input impedance a little too low. Note that

a quantitative correction can not be determined from the analog circuit

measurement, but that a wrong measurement indicates that an incorrect

resistivity will probably be obtained.

The V/I values, in ohms corrected to 23°C, obtained by the first

four labs are shown in Table A-VI. Again this is good agreement, with

only a 5 per cent chance that the multi-lab precision is worse than

+ 0.9 per cent (R2S%), the most probable value being ± 0.3 per cent

(R2S%). One of the major aources of error here is probably specimen

nonuniformity although an attempt was made to reduce this as much as possible by selecting uniform slices and by recentering after each of

ten measurements in each lab, thus tending to average out the non- uniformity.

(3) Temperature measurement . This factor was largely eliminated

as a source of error by sending around the same thermometer and same

heat sink to every lab. Thus thermometer calibration and thermometer

specimen heat path were uniform. Correction for temperature was quite

important, however, corrections ranging as high as 3 per cent. A

future round robin should help determine the effect of using different

thermometers and heat sinks.

(4) Slice thickness measurement . The reported thicknesses are listed in Table A-VII. The thickness plays a double role in introducing error. For the sample thickness used an error of 0.11 per cent in w should cause,

(a) an error of 0.02 per cent due to the change of F(w/S), and

(b) an error of 0.11 per cent because w is a direct multiplier

in the formula for calculating resistivity.

These two errors are in opposite directions; this gives a resultant

error of about 0.09 per cent.

(5) Slice diameter measurement . The slices were not perfectly

round so everyone did not measure the same diameter. The reported

values, are listed in Table A-VIII. The last line shows the standard

deviation in the correction factor, F2, corresponding to the s in the

diameter measurement. It is smaller because, at the diameter used, the

correction factor is a slowly varying function of the diameter.

40
Table A-V - Average Resistance (^) (Recomputed)

Lab .

1

2

3

5

R

10.002

10.004

10.003

10.006

9.71

Table A- VI - V/I (Q) Corrected to 23°C

Lab. No,

1

2

3

4

AVG

s

Sample

BTL-2

21.743

21.752

21.796

21.718

21.752

0.032

Sample

BTL-4

22.045

22.083

22.035

21.970

22.033

0.047

Sample

KN-4

26.616

26.544

26.644

26.602

26.602

0.042

One sided, 95%

confidence upper

bound on o

(0.15%)

0.093

(0.44%)

(0.21%)

0.137

(0.51%)

(0.16%)

0.123

(0.47%)

41
Table A-VII - Slice Thickness (cm)

Lab. No . 1

2

3

4

5 AVG

s

One sided,

95% confidence

upper bound on o

Sample

BTL-2

0.1054

0.1054

0.10511

0.1052

0.1052

0.10522

0.00011

(0.10%)

0.00032

(0.29%)

Sample

BTL-4

0.1049

0.1052

0.10481

0.1049

0.1049

0.10494

0.00015

(0.14%)

0.00044

(0.41%)

Sample

KN-4

0.1267

0.1270

0.12672

0.1267

0.1267

0.12676

0.00013

(0.10%^

0.00038

(0.29%)

Table A-VIII - Slice Diameter (cm)

L ab. No . 1

2

3

.:. .4

•5 AVG

s

Resultant s in

correction factor

One sided, 95%

confidence upper

bound on a in

correction factor

Sample

BTL-2

3.215

3.228

3.223

3.236

3.221

3.224,8

0.0076

(0.23%)

(0.02%)

(0.05%)

Sample

BTL-4

3.236

3.236

3.2322

3.238

3.241

3.2367

0.0033

(0.10%)

(0.01%)

(0.02%)

Sample

KN-4

3.292

3.294

3.297

3.302

3.299

3.2969

0.0041

(0.12%)

(0.01%)

(0.02%)

42
(5) Total. If each source of error is assumed to be random and

independent, an estimate of the overall standard deviation may be found

as the square root of the sum of the measured sample variances

Probe spacing: s =- 0..11%

V/I measurement: s -- 0.,15%

Thickness measurement: s -= 0,,09%

Diameter measurement

:

s

- 0,,02%

Total s =- 0.,21%

Conclusions

The following conclusions can be drawn from the round robin

results

:

(1) We have a good method for the measurement of probe spacing.

(2) Using the black box analog circuit procedure is a good test

of the electrical equipment being used to measure resistivity,

at least for samples in the 10 fi-cm range finished with a 5 micron lapping compound.

(3) For an "industrial standards lab" procedure at the 10 ft-cm

level, the contribution of the electrical measurement

associated with a four-point probe method is on the

order of + 1/2 per cent (R2S%).

(4) The necessity for detailed, explicit instructions covering

every important detail of the measurement can not be over-

• emphasized. Neither can the need for detailed data sheets

that show all the data taken, all the porrection factors

used, and all the computations made in arriving at the final

values of the resistivity. When two laboratories are comparing

. resistivity measurements, these data sheets should be

exchanged along with the samples.

Notes and References

1. ASTM Book of Standards, Part 8.

2. For a full explanation of the statistical terras and the method of

computation used, see NBS Handbook 91, "Experimental Statistics",

by M. G. Natrella, Chapters 1 through 4.

3. (R2S%) is the two-sigma precision index expressed in relative

per cent.

43
There is an ambiguity in the method of Appendix B because it is

not specified whether to use S^, S3 or the average in

determining F2 and F(w/S). This is important since the difference

in the spacings is larger than the precision in their measurement.

This will affect the s in the final resistivity values, but not

the s for each factor.

Note that Fgp applies to slice measurement only . A more convenient

5orm of Fgp is: Fgp = 1 + 1.082(1 - S2/S), where S is the

average of S , S„,and S .

44
APPENDIX B

Special Round Robin Procedure

1. Measurement of Probe Spacing:

1.1 Measure the spacing of the probe provided (serial no. SI

62-lM-M-OC ) using the method below. If any deviations in

this technique are used, describe the deviations.

1.2 The technique used for measuring probe spacings is that of

observing and measuring the probes indentations in a polished

silicon surface.

1.3 Apparatus: The following apparatus is needed:

(1) A flat polished silicon surface. Some are provided but use your own if desired.

(2) A micrometer movement to move the probe or silicon in a direction perpendicular to a line through the probe points.

(3) A toolmaker's microscope for measuring

distances between the indentations.

The silicon surface can be that of a slice ov block which

can be conveniently placed under the probe. The surface

should be polished and reasonably flat. The micrometer

movement for moving the probe or silicon surface should be

capable of moving increments of 10 to 15 mils (0.25 to 0.375

mm) in a direction perpendicular to a line through the probe

points. The toolmaker's microscope should be capable of

measuring increments of 0.1% of the probe spacing S (0.06

mils (1.5ym) for a 62.5 mil (1.59 mm) probe spacing).

1.4 Procedure: With the four-point probe make a series of

indentations on a polished silicon surface. These indentations are made by applying the probe to the surface using

normal point pressures and measurement routine. The

probes are then lifted and the silicon surface or probe is

moved 10 to 15 mils (0.25 to 0.375 mm) in a direction

perpendicular to a line through the probe points. Again the

probe is applied to the silicon surface and the procedure

repeated until a series of 30 indentation sets is obtained.

The indentations obtained are often irregular in shape

and may show several areas of contact for each probe. Place

the silicon sample in the toolmaker's microscope. For 10

of the 30 indentation sets record the readings A through

45
H on the X axis of the toolmaker's microscope and the

readings Y. and Y„ on the Y axis for the locations shown

m the figure below:

rrTH G F E D C B A

The angle of placement of the silicon sample on the microscope should be such that the Y axis readings do not differ

more than 5 mils (0.125 mm). Record all readings on the

data sheet provided. Calculate S. sheet provided and the formulas

:

and using the data

S. = C + D A +

E + F _ C + D

2 2

G + H E + F

Measurements of the analog circuit.

2.1 In the apparatus to be used to measure sample resistivity

connect the four leads of the analog circuit. Leads A,

B, C, and D correspond to the four leads of a four-point

probe a, b, c, and d as shown below:

vVVt--'vVV-'^'\V^-

-IB

i

cX

C D

with A and D being the current leads and B and C being the

leads between which the voltage is measured.

2.2 At a current level of approximately one milliampere measure

the current and voltage first in one direction (the "forward"

direction) and then with current reversed (the "reverse"

direction). Record these values on the data sheet provided.

If the instrument being used measures V/I directly, record

this instead of currents and voltages. Repeat for ten

determinations. For each determination calculate V/I for

forward and reverse direction and the averages and record

on the data sheet provided.

46
Px^OBE SPACIfX; DATA

.1%

}5 J) V nU nn V

1

... . - _

2\

.

\' -J it

6 il

..•.! - i

-

10 1 PROBE spacii;g C.AJ.CUj;^".TIOjJ

%

C •'.-B G-fH

2 2 2 2 Si ^3

2

3h

5

ro

'l'.

e

10

!...

. Avei'a-2;e

:

LA"ROl^ATGHY:

DATE:

PROBE SERLAL No,

c _ C D ^1- "o

- A+_B

2

C -l-D

AI'!AT=OCr CIRCUIT MEASUREJ^iENT DATA

"3" 2

C/^iCULATJONS

?s

i

.."or ward

cm^rent

If

forvrarcl voltaic

LiV

reverse

current

I,,

iTia

reverse voltage

r.w

for ( readin-^

oh'."'.';

.irect

eouirj

.

v/i;

oni'is

3 J:

?

73

...2. i 10 !i

I -

d1i::;S

q 10

Ir

OiLMS

AvsraTG

:

avci-a

e£C;l run

hi
3. Sample Measurement

3.1 Measure each of the three samples by the method that follows

3.2 Use the heat sink and micrometer stage provided. Make sure each sample is electrically isolated from the heat sink

by measuring the resistance between sample and heat sink

with an ohmmeter. Electrical isolation is accomplished

with a mica layer (provided with heat sink). Measure the

sample temperature by placing one of the thermometers

provided in the hole in the heat sink. Two thermometers are provided. Please use thermometer NBS 63

(tagged with an H) unless it has been broken during the

round robin, in which case use thermometer NBS 64 (tagged

with an L). Note on the data sheet which thermometer

was used.

3.3 Before measuring each sample, clean ultrasonically in warm water and detergent, then rinse with flowing

deionized water. Then ultrasonically degrease in

acetone, rinse with alcohol and air dry.

3.4 Center the four point probe within 0.010" (0.25 mm) of the

center of the sample being measured.

3.5 Using the probe provided make ten determinations of

current, voltage, and temperature. Remove, replace, and

recenter the sample between each determination. Record

the following data in the data sheet provided:

(a) T, the temperature of the sample as measured by the

thermometer placed in the heat sink.

(b) I^, the current through the two outer probes.

(c) V^, the voltage across the two inner probes with

current in the direction of 1^.

(d) Ip, the current through the two outer probes when

the current direction is reversed.

(e) Vp, the voltage across the two inner probes with

the current in the direction of I . r

(f) For direct reading equipment only record V/I in

both forward and reverse directions instead of

I^, V^, I , and V . f f r r 3.6 Carry out the calculations on the data sheet. Obtain C-p from Figure B-1, F2 by linear interpolation of Table B-I

and F(w/S) (where w is the slice thickness) by linear

interpolation of Table B-II. Use the data on slice

thickness and diameter provided.

48
LABGRATORT

DATS:

SAJ-iPLF Ko.

Thickness = _

D:* PJ'.'ieter r.

Tber.r.or:\eter urcd:

DATA

1

T

T

rna.

V

X

IliV

T

,

' .-^ r]-] -

^1

^ ^ -t. V

. - . . .. -1

v,/i

: si

1

CALCULATIONS

Z 1

X

i T -

1 "'I

Avg olirn-ci.-i

1

2|

5

o|

7

3.05 Avera~e

:

14-0 .721(1-. 32/?Si-3p/2P^)

•(-A)r

F S ?o ^ F(v,"/r;) x F

- Avg X F,,-, X F

49
0.010

0.009

0.008

0.007

0.006

0.005

O.OOU

0.003

0.002

0.001,

10 10 10 10

p, fi-cm, calculated using = 1

Figure B-1,

50
Table B-I

s/r ^2 s/r ^2 s/

r

^2

0 , 532 0. 07 4

.

485 0.14 4,, 348

0 . 01 4

,

, 531 0 . 08 4. 470 0.15 4,. 322

0.02 4,, 528 0.09 • 4. 454 0.16 4,.294

0. 03 4,.524 0.10 4. 436 0.17 4,,265

0.04 4,, 517 0.11 4. 417 0.18 4,.235

0.05 4,, 508 0.12 4. 395 0.19 4..204

0.06 4,,497 0.13 4. 372 0.20 4.,171

Table B-II

w/s F(w/s)

0 .

5

0. 997

0 . 6 0. 992

0.7 0. 982

0.8 0. 966

0.9 0. 944

1.0 0. 921

51

APPENDIX C

Probe Separation Correction Factor for Thin Slices

The resistivity of large, thin sheets of homogeneous material when

measured with an in-line four-point probe is given by:

V 77 P ^ I 1^ w F , sp

(C-1)

where V is the potential difference between the inner probes, I is the

current through the outer probes, w is the slice thickness, and

2 In 2 sp

In

is^ t S^) (S^ -H S3)

^1^3

(C-2)

is the factor which corrects for unequal separations S-|_, S2> and S^

between adjacent probes. If the separations differ from their mean

value, S, by no more than a few per cent the expression can be

simplified to:-^

sp

1 + In 4

Ci - 2S, 2S,

-] = 1 + 0.721[1 - 2S, 2S (C-3)

where S2 is the separation between the inner probes. . This expression

was the one used in the Proposed Method. With the same assumptions it

is also possible to transform (C-2) into an equivalent form which

involves only S2 and S:

sp 1 ^ Ylh^^ - f>

1 + 1.082(1 - —).

S

(C-4)

If the slice thickness w exceeds S/2 or if the slice diameter is

less than 50S additional correction factors must be used with (C-1).

The effect of unequal probe separation on these factors has not been

investigated. If the probe separations do not differ from their means

by more than a few per cent (as required in the derivations above) it

is thought that significant error will not be introduced into the

result by multiplying the appropriate factors together for the w/S and

S/D ratios allowed by the method. Correction factors appropriate to

semi-infinite volumes have been discussed by Valdes^ and Hargreaves

and Millard.

3

The magnitude of Fgp will be affected by uncertainty in the

measured values of the probe separations. This effect can be

53
determined from the differential:

dF = ~ (SdS^ - S„dS), (C-5)

28^ in 4

2 2

If the measured positions of the four probes differ from their correct

values by small amounts dx^^ , dx2, dx3, dx^ (as a result either of

probe wander or error in measurement of an impression) then dS2 = dxg - dx2, dS = (dx^ - dx2_)/3 and (C-5) becomes:

dF = ^

(S_dx^ - 3Sdx^ + 3Sdx_ - S_dx ). (C-5)

. : , . ,^P.. , 25^ m 4 2 1 2 3 2 4

If it is assumed that the probe displacements are random and independent

each with standard deviation 6x and that S2 ^ S, then F^p 1 and

the relative standard deviation in F becomes

:

sp

ST

(c-7)

F In 4

, . •

. sp

One measures the standard deviation, s, of the probe separation rather

than the standard deviation of the probe displacement. These are related by s = /26x and (C-7) becomes:

sp

It should be noted that the standard- deviation of the mean probe

separation is 6S = s/3 because only the first and last probe displacements enter into the calculation. If the variation of S ahd S2 are considered to be independent, the following form of (C-5) is conve:pient

in the error analysis of 112.2.2:

dF dS- - sp 2 db / o r. \

• ' —=r^ - -c + c —, (C-9)

- - F Q s

'

- ;

. sp ^

^

where c = 3/2 In 4 = 1.082. In this form, the second term may be

combined directly with terms i n dS/S from Appendix D. (C-8) follows

directly since /l0/2 In 4 = 3/1 + (l/9)/2 In 4.

Probe wander will also influence the measured voltage. If all

other factors are held constant:

dV ^ _ ^

_ dS

^ ^ ^ . . ; V s s

54
where a and b are coefficients associated with the factors F2 and F(w/S)

discussed in Appendix D. For typical slices (D = 16S, w = 0.75S)

a = 0.07 and b = 0.12. Therefore (C-10) becomes:

4f = 1.13 ^+ 1.08

^

(C-11)

V s s

and the relative standard deviation in the voltage reading is

:

(1.08)2

V / 9 s s

It can be seen that the principal contribution to the deviation in V comes from F^p. Although (C-8) and (C-12) are similar in form, the

former refers to the uncertainty in measured probe separation which is

characterized by s while the latter refers to probe wander which is

characterized by 6S2. In general one would expect that 6S2 < s. If

data from a series of measurements are averaged to obtain the value of

voltage , the contribution to the uncertainty which arises from probe

wander is reduced significantly. For a single reading, it could be

as much as 0.34 per cent for the conditions of the method, but in

general would be expected to be significantly less than this.

REFERENCES

1. L. J. Swart zendruber , unpublished work.

2. L. B. Valdes , "Resistivity Measurements on Germanium for

Transistors," Proc. IRE 42, 420-427 (1954).

3. J. K. Hargreaves and D. Millard, "The Accuracy of Four-Probe

Resistivity Measurements on Silicon," Brit, J. Appl. Phys . 13,

231-234 (1962).

55

APPENDIX D

Errors Introduced Through Diameter and Thickness

Correction Factors

The correction factors F2 and F(w/S) are tabulated in Tables 1 and

2 of the Special Round Robin Procedure (see Appendix B). These factors

approach constant values as S/D and w/S, respectively, become small.

Hence the error introduced into the calculation for resistivity will be

a function of these ratios.

Consider first the factor Fj. The diameter, D, is permitted to

go from lOS to infinity. The effect of error in the determination

of D or S on is given by

^''2 dD

-a

S (D-1)

The coefficient, a, varies from 0.154 at D = lOS to 0.0 at D = «>. The following table lists several quantities of interest:

D a 6D AD &F^/F^a)

6S E(%)

lOS 4 171 0 .154 0.031 0. 31 -0.015 t8.7

16S 4 383 0 .066 0. 013 0. 21 -0.006 + 3.4

25S 4 470 0 .028 0. 006 0.14 -0.003 + 1.4

lOOS 4 528 0 .0015 0. 0003 0.03 -0. 0001 + 0.1

The first three columns list the diameter, the correction factor, and

the coefficient respectively. The fourth column lists the relative

deviation due to deviations in the measurement of average diameter

obtained in the round-robin experiment (0.2 per cent). The fifth

column lists the maximum uncertainty in the correction factor due to

permitted eccentricity of the wafer; the diameter is required to be

constant to within D/5S per cent. The sixth column lists the relative

sample standard deviation due to deviation in the measurement of average probe separation (0.1 per cent). The seventh column lists the error in computed resistivity if the diameter correction factor is not

used and clearly demonstrates the importance of using it if D ^ 25S.

It should be noted that this correction factor is appropriate only for

measurements taken at the center of the wafer. Additional factors are required if measurements are made elsewhere on the wafer. These have

been incorporated into an extended table of F„.-^

57
The factor F(w/S) is discussed in detail by Smits. The thickness,

w, is permitted in the present method to vary from S to as small a value as can be accurately controlled and measured. The effect of

error in the determination of w or S on F(w/S) is given by:

dF

F

-b

dw

b — w s (D-2)

The coefficient, b, varies from 0.27 for w = S to 0.018 for w

The following table lists several quantities of interest

:

= S/2,

w F(w/S) b 6F/F(%)

6w 6F/F(%)6S

s 0. 921 0 27 -0.081 0.027

0 75S 0.974 0 12 -0. 036 0.012

0 55S 0. 987 0 066 -0. 0020 0.0007

0 5S 0.997 0 018 -0.00054 0.0002

The first three columns list the thickness, the correction factor and

the coefficient. The fourth column lists the relative deviation due

to deviations in the measurement of thickness obtained in the roundrobin experiment (0.3 per cent). The fifth column lists the column

deviation due to deviations in the measurement of average probe

separation (0.1 per cent).

REFERENCES

1. L. J. Swart zendruber , "Correction Factor Tables for Four-Point

Probe Resistivity Measurements on Thin Circular Semiconductor

Samples," NBS Technical Note 199, April 15, 1964.

2. F. M. Smits, "Measurement of Sheet Resistivities with the FourPoint Probe," Bell System Tech. J. 37.' 711-718 (1958).

58
APPENDIX E

Resistivity of Silicon over the Temperature Range

0 to 50°C

The data in the following tables were taken on a selection of nand p-type silicon specimens in order to compute the temperature coefficient of resistivity of silicon at room temperature. Resistivity was measured over the temperature range 0 to 50°C on clover-leaf specimens

immersed in a bath of silicone oil by means of the van der Pauw method

as described in ASTM Method F-76. The calculations were made as described in the paper. Temperature Coefficient of Resistivity of Silicon

and Germanium near Room Temperature, the abstract of which is reproduced

following the data tables. Curves showing the temperature coefficient

of resistivity of silicon, valid between 18 and 28°C are also included

following the abstract.

Specimens were obtained from a wide variety of sources. All p-type

specimens were doped with boron except those identified with the symbol

(AL) which were doped with aluminum. There appears to be a significant

difference in temperature variation of resistivity between the boronand aluminum-doped specimens. No such difference in temperature variation was observed for the n-type specimens, most of which were doped

with phosphorus.

The data tables are arranged in the following fashion. The upper,

two-column table lists the resistivity calculated from the measured resistances. These data were fitted by the least squares method to the

relation

:

log p = log A + a log T

and the resistivity calculated at decade intervals from 0 to 50°C and at

23°C to yield the second table. The range in resistivity given in this

table is that associated with uncertainty in a only; the uncertainty in

the constant A was not computed since the study was concerned with temperature variation only.

59

SAMPLE NUMBER- f DATA COMPUTED ON 20 NOV 67

UAf/* TAKElj OiJ NOV 27» 1964

P-TYPE SI,. SAMPLE NUMBER- 2

DATA CO^IPUTEL) ON 20 NOV 67 DATA TAKE. J ON FED 9» 1965

P-TYPE SI

Pi is • cm

1 t, C p, SJ'Cm -.19 7.444-04 -.71 1.072-02

5.14 1.0 7^-02

9.05 7.5b3-04 9.95 1 .072-02

14.90 7.o4f!-04 1 4.57 1 .073-02 20.48 . 7.721-04 20.39 1.074-02 25.^6 7.706-04 25.14 1 .075-02

30.10 7.851-04 30.97 1.077-02

:)4.h3 7.918-04 34.95 1.079-02

40.12 7.994-04 39.92 1.081-02

45.49 8.070-04 44.92 1.084-02

50.04 8. 136-04 49.62 1.086-02

log p = log A + a log T

t + 273,16

t, °c p, ^I'Cm range Of P

. u 7.439-04 7.4UO-04 TO 7.470-04

10.0 7.580-04 7.537-04 TO 7.604-04

20.0 7.720-04 7.702-04 TO 7.738-04

23.

U

7.761-04 7.744-04 TO 7.779-04

30.0 7.857-04 7.838-04 TO 7.076-04

40.

U

7.992-04 7.9b7-04 TO 8.016-04

bO.O 8.125-04 8. 092-04 TO 8. 158-04

log p = 1 og A + a log T

t + 273.16

t. "C p, n-cm range of P

.0 1.069-02 1 . 066-02 ro 1.073- 02 10.0 1.072-02 1 .070-02 TO 1. 075- 0? 20.

U

1.075-02 1.073-02 TO 1

.

077- 02 23.0 1.076-02 1.074-02 TO 1. 078- 02 30.0 1.078-02 1 .076-02 TO 1. 080- 02 40.0 1.081-02 1. 070-02 TO 1. 084- 02 bO.O 1.084-02 1.080-02 TO 1

.

O07- 02

A = 3.915-05

ALPHA = .52485

W4 3.1962-02

ALPHA ± W4

A = 6.821-03

ALPHA = .08014

W4 = 2.5203-02

ALPHA ± W4

ranges given are 95% confidence limits ranges given are 95% confidence limits

SAMPLE NUMBER- -3 DmTA CUi^'PUTElJ ON 23 NOV 67 DATA TAKtij OIJ DEC 29. 1965

P-TYPE ST SAMPLE NUMBER- 'j 0/UA CO(';pUTt.lJ ON 2M NOV 67 UmFA TAKLM OlJ DEC. 21»196S

P-TYPE SI

t, °c P, il-cm .21 1 .454-02

5.03 1 .453-02

10.74 1 .452-02

15.04 1.453-02

20. db 1 .454-02 24.98 1.455-02

29.78 1 .457-02 34.95 1.460-02

39.65 1 .463-02

44.74 1 .466-02

49.44 1 .469-02

t, °c p , i"i • cm .61 1 . 4b9-U2

5.47 1 .4b(i-02 10.23 1 .457-02

15.29 1.458-02

20.19 1 .459-02 24.97 1 , 46U-02

29.78 1.4b2-02

34.78 1.464-02

39.90 1.467-02

45.90 1.471-02

50.02 1 .475-02

log p = log A + a log T T = t + 273,15

t, °c p , Q • cm range of p .0 1 .449-02 1 .445-02 TO 1 .454- 02 10.

u

1.453-02 1.450-1)2 T 0 1.456- 02 20.

d

1 .456-02 1 .454-1)2 TO 1.459- 0? 23.0 1 .457-02 1.455-02 TO 1 .460- 02 30.0 1.460-02 1.457-02 TO 1.462- 02 40. u 1 .463-02 1.459-02 TO 1.466- 02 50.0 1 .466-02 1.H61-02 TO 1.470- 02

log p = log A + a log T T = t + 273.16

t, °C

.0 10.

u

20.

u

23.0

30.0 40.0

50.0

p

, Q • cm I range of p 1.454-02

1 .458-02 1.461- 02 1.462- 02 1.464-02

1 .467-02

1 .470-02

1.44g-(j<d 1.454-02

1.458- 02 1.459- 112 1.462-02

1 .464-02

1 .466-02

TO 1.459-0?

TO 1.461-U2

TO 1.464-02

TO 1.465-02

TO 1.4t>7-0? TO 1.471-0?

TO 1.475-02

A ALPHA

9.970-03

.06671

W4 = z.saua-n?

ALPHA ± IJ4

A = 1.003-02

ALPHA = .06613

W4 = 2.5895-02

ALPHA ± W4

ranges given are 95% confidence limits ranges given are 95% confidence limits

61
SAMPLf-: NUMBEK- S

DA I A (.uMlnjlLU ON Zl\ lJUV h7

U/iTA TAKtiJ ON DtC ?1» 196^

P-TYPE SI SAMPLE NUMBER- &

DATA CO.-IPUTED ON 2H NOV 67 Data takeim on dec 29? ige-^

P-TYPE ST

t, °c p, fi'cm 1.11 2.511-02

5.87 2.515-02

10.51 2.520-02

15.45 2.527-02

20.25 2.535-02

25. 06 2.543-02

29.70 2.553-02

34.68 2.564-02

39.72 2.576-02

44.72 2.589-02

49.63 2.603-02

t, °C P, fi'Cm .37 2.849-02

5.11 2.854 —02

10. Bl 2.8b2-02

15.08 2.8b9-02

20.26 2.879-02

24.98 2.888-02

29.76 2. 899-02

34.91 2.912-02

39.58 2.925-02

44.67 2.941-02

49.36 2.956-02

log p = log A + a log T

T = t + 273,16

t, °c p, Si •cm range of p

. 0 2.500-02 2.490-02 TO 2.511-02

10.0 2.520-02 2.513-02 TO 2.528-02

20.0 2.540-02 2.534-02 TO 2.545-0?.

23. U 2.545-02 2.540-02 TO 2.551-02

30 .0 2.558-02 2.553-02 TO 2.564-02

40.0 2.577-02 2.569-02 TO 2.585-02

50.

U

2.595-02 2.585-02 TO 2.605-02

log p = log A + a log T T - t + 273.16

t, °c p, 17 •cm range of P

. u 2.838-02 2.827-02 TO 2.850- 0? iU.U 2.861-02 2.(<53-02 TO 2.870- 02 20.0 2.884-02 2.H77-02 To 2.890- 0? 23.0 2.890-02 2.884-02 TO 2.8Q6- 02 30.0 2.906-02 2.899-02 TO 2.912- 02 40,

u

2.927-02 2.918-02 TO 2.935- 0? 50.0 2.947-02 2.936-02 TO 2.959- 0?

A = 7.213-03

ALPHA

W4

,22158 3.1995-02

ALPHA ± W4

A = 8.064-03

ALPHA ,22432

W4 = 3.1516-02

a = ALPHA + W4

ranges given are 95% confidence limits ranges given are 95% confidence limits

SAMPLE NUMBER- 7 DATA CO.MpuTEU ON 28 NOV 67 DATA TAKE.I. org MAHCH 27» 1Q67

P-TYPE SI SAMPLE NUMBER- ©

DATA CO-IPUTED ON 28 NOV 67 DATA TAKtij ON OCT 6i 1966

P-TYPE 51

t, °c p, a- cm -.98 3.765-02

4.92 3.792-02

9.73 3.817-02

14.52 3.a43-0ci 19.23 3.870-02

24.30 3.900-02 29.66 3.934-02

34.67 3.968-02

39.84 4.004-02

44.80 4.040-02

49. 70 4.077-02

t. °c p, ii'cm .42 e. 044-02

5.67 9.012-02

10.23 9. 161-02

14.11 9.292-02

19.87 9.488-02

24.80 9.659-02

29.68 9.834-02

34.36 1 .000-01 39.24 1.019-01

44.17 1.037-01

49.31 1.057-01

log p = log A + o log T

t + 273.15

t, °c p, a- cm range of p

. u 3.757-02 3.739-02 TO 3.774- 0? 10.0 3.820-02 3.80;!-u2 TO 3.833- 02

<d0.0 3.883-02 3.873-02 TO 3.893- 0? 23 . 11 3.901-02 3.H92-02 TO 3.911- 02 30.0 3.944-02 3.934-02 TO 3.955- 0? 40.0 4.005-02 3.991-02 TO 4.019- 02 50.0 4. 064-02 4.046-02 TO 4. 083- 0?

log p = log A + a log T

t, °C

10.

u

20.0

<i3.o 30.

40. 50.

U

p, Q'Cm 8.806-02

9. 156-02 9.507-02 9.613-02

9.860-02

1.021-01 1.057-nl

T

= t + 273.16

range of p 8.779-02

9.136-02

9.492-02 9.596-02

9.H43-02

1 .019-01 1.054-01

TO 8.833-0?

TO 9.176-02

TO 9.523-0?

TO 9.628 0? TO 9.876-0?

TO 1.024-01

TO 1.060-01

A = 2.720-03

ALPHA = .^bSOi

W4 = 3.5649-02

ALPHA ± W4

f^ = 2.007-04

ALPHA = l»US'*bO

W4 = 2.3529-02

a = ALPHA ± W4

ranges given are 95% confidence limits ranges given are 95% confidence limits

62
SAMPLt NUMBER- 9 (AD

UATA COMPUTtU ON idH IjOV 67 DATA lAKblj Oil iguv 27. 1964

P-TYPE SI SAMPLE NUMUEH- /o UAIA CoWpUltU UfJ 2H UOV 67 UATA TAKtij ON Ftli 2> 1965

p-TYcr: ST

t, °c p » fi* cm t, °C p » • cm

.30 2.U70-U1 . t>5 2.236-01

4.15 2 . U 96-01 5.35 2 • 29b-U

1

9.23 2.139-01 10.14 2.35ti-Ul 15.02 2.1»jl-Ul 14. 'H 2.414-01

20.55 2.234-01 20 .5B 2.496-01 25.29 2.275-01 25.43 2.5l>2-01

30. Ul 2.317-01 30.09 2.628-01

34. b6 2.361-01 34.92 2.69b-01

39.96 2.412-01 39.59 2.764-01

45.29 2.462-01 44.76 2.840-01

49.78 2.510-01 49.26 2.907-01

log p = log A + a log T

t + 273,16

t, °C

.0 10.

u

20.0 23.0

30.0

40.0

50.0

2.057-01

2. 145-01 2.234-01

2.260-01 2.322-01

2.412-01

2.502-01

range of p 2.047-iJl

2. 137-01 2.227-01

2.254-01 2.316-01

2.403-01

TO 2.068-01

TO 2.153-01

TO 2.240-01

TO 2.266-01

TO 2.329-01

TO 2.421-01

2. 489-01 TO 2.514-01

log p = log A + a log T

T = t + 273.16

t, °c p, n-cm range of P .0 2.224-01 2.214-01 10 2,235- 01 10.0 2.356-nl 2.348-01 TO 2.364- 01 20.

u

2.491-nl 2.485-01 TO 2,498- 01 23.

u

2.532-01 2.526-01 TO 2.538- 01 30.0 2.629-01 2.622-01 TO 2.636- 01 40.

u

2.769-01 2.760-01 TO 2,779- 01 50.0 2.913-01 2.899-01 TO 2.926- 01

A

= 3.O22-0I*

ALPHA = 1 .16279

W4 = 4.0025-02

ALPHA ± W4

2.749-05 A ALPHA = 1.60397

W4 = 3.6735-02

a = ALPHA + W4

ranges given are 9S% confidence limits ranges given are 95% confidence limits

SAMPLE NUMBEfi- / ' (aU Data COf'.PuTEO ON 28 IJOV 67 OmTA TAKt,. ON OCT 31. 1966

P-TYPE SI SAMPLE NUMBER- //-/t"-/ (l^l) DaIA COMPUTED ON 28 IJOV 67 Data TAKEu oh AUG 16. 1967

P-TYPI^ ST

t, °c P. a- cm -.22 2 . oio-Ol

4.80 2.673-01

10.16 2.733-01

15. 06 2. 789-01 19.69 2. 843-01

25. 08 2.908-01

29.47 2.962-01

34.37 3.025-01

39.54 3. 093-01 44.33 3.157-01

48.99 3.222-01

t. °c

.60 5.36

10.53

15.22

19.92

24.t>8 29.50

34. b4 39.52

44.10 49.56

p, S2'cm 2.623- 01 2.b75- 01 2.732- 01 2.786- 01 2.842- 01 2.899- 01 2.960- 01 3.025- 01 3.089- 01 3.150- 01 3.226- 01

log p = log A + a log T T = t + 273.16 log p = log A + a log T

T

= t + 273.16

t. °c p , Ti • cm range of p t, °c p , si • cm range of P

, u 2,611-01 2.599-01 TO 2.624-01 .0 2.606-01 2.595-01 TO 2.618-01

10.

u

2,731-01 2.722-01 TO 2,741-01 10.0 2.727-01 2.718-01 TO 2.736-01

20. (J 2.853-01 2.845-01 TO 2.860-01 20.

U

2.849-01 2.842-01 TO 2.855-01

23.

u

2.889-01 2.882-01 TO 2.897-01 .23.0 2.885-01 2.879-01 TO 2.892-01

30.

u

2.975-01 2,967-01 TO 2.983-01 30.0 2.971-01 2.9b4-01 TO 2.979-01

40.

u

3.099-01 3. 088-01 TO 3.110-01 40.0 3.095-ni 3.085-01 TO 3.105-01

50.0 3.223-01 3.208-01 TO 3.238-01 50.0 3.220-01 3.206-01 TO 3.234-01

A = 2.313-04

ALPHA = 1.25294

W4 3.7154-02

ALPHA + W4

= 2.242-04

ALPHA = 1.25814

W4 = 3.4339-02

ALPHA ± W4

ranges given are 95% confidence limits ranges given are 95% confidence limits

63
SAMPLE NUMBER- "-V-.' ^^l] D/UA CO^IPUIEU ON 2fl m\J 67 UAIA TAKt.j OU AUli Ibt 1967

P-TYPE SI SAMPLE NUMgER- /Z CAL) (jHfA COMPUTED ON 2B NOV 67 UATA TAKEN ON NOV ?7t 1964

P-TYPF SI

t, °c p » i^' cm t °C n-cm

.6U 2.60H-01 .40 6.2bl-01

S • 3b 2 . 660-Ul 4 . 27 6.401-01

10. 2.717-ul 9.2U 6.602-01

15.22 2.771-01 15.04 6.038-Ul

19.92 2.8£:6-Ul 20.59 7.074-Ul

24.68 2.6b3-Ul 25.30 7.277-01

29.50 2.943-Ui 29.99 7.4b7-Ul

34.64 3.009-Ul 31.75 7.567-01

39.52 3.072-01 34.60 7.699-01

44. 10 3.133-01 39.80 7.945-01

49. b6 3.20(1-01 45.18 8.205-Ul

log p = log A + a log T

t + 273.16 log p = log A + a log T

T = t + 273.16

t. °c p, fi'cm range of p .0 2.592-ni 2.581-01 TO 2.603-01

10.0 2.712-nl 2 .7113-01 To 2.720-01

20.

U

2.833-01 2.826-01 TO 2.840-01

2.869-01 2. 863-01 TO 2.076-01

30. U 2.955-01 2.948-01 TO 2.962-01

40.0 3.078-ni 3.069-01 TO 3.088-01

bO.u 3.202-01 3.189-01 TO 3,216-01

t, °C p , a -cm range of P .0 6.221-01 6. 195-01 TO 6.247-01

10.0 6.634-01 6.615-01 TO 6.654-0]

20.

u

7.059-01 7.044-01 TO 7.074-01

23.

u

7.189-01 7.173-01 TO 7.2o4-01

30.0 7.496-01 7.478-01 TO 7.514-01

40.

u

7.944-01 7.918-01 TO 7.970-01

bO.O a. 403-01 8.366-01 TO 8.441-01

2.229-04

1.25817

W4 = 3.3588-02

A = ALPHA = a = ALPHA ± W4

A

= 2.721-05

ALPHA = 1-7B915

W4 = 3.Ufll9-n2

a = ALPHA ± W4

ranges given are 95% confidence limits ranges given are 95% confidence limits

SAMPLE NUMBER- /3 (n\J UaTA COMPuTEU ON 28 NOV 67 UATA TAKln Oij FEiJ 2tvi 1965

P-TYPE ST SAMPLE NUMBER- ' V

UmIA CONPUTEU ON 28 NOV 67 DATA TAKLiJ ON FEti 9> 1965

P-TYPE SI

t, °c p, ii'cm t, °c p, i2-cm

.78 8.396-01 -.62 8.593-01

5.42 8.665-01 5.2b a.983-Ul

10.31 6.9bb-01 10.02 9.308-01

14. Bb 9.246-01 14.62 9.626-01' 20.69 9.uiO-Ol 20.59 1 .005+00 25.29 9.906-01 25.16 1 .038+00 30.11 1 .021+00 29.96 1 ,074 + 00 35.13 1.054+00 34.89 1.111+00 39.88 1 . 086+00 39.86 1 , 150+00 44.82 1.119+00 44.85 1.186+00

49.56 1 . lbl+00 49.52 1 ,226+00

log log A + a log T

t, °C

. u lO.u 20.0

23.0, 30.0

40.0

50.

0

SI- cm 8.348-01

8.947-01 9.567-01

9.756-01

1,021+00

1.087+00

1.154+00

T = t + 273. 16

range of 8.319-01 8.925-01

9.549-01

9.739-01

1,019+00

1.084+00

1. 151+00

TO

TO

TO

TO

TO

TOTO

0.377-01 8.970-01 9.5H5-01 9.774-01

1 . 023+00

1 . 089+00

1 . 150+00

log p = log A + a log T

T = t + 273,16

t, °c p , s! • cm range of P

.0 8.629-01 8.596-01 TO 8.663-01

10.0 9.308-01 9.2fil-01 TO 9.334-01

20.0 1 .001+00 9.9^2-01 TO 1.003+00

23. 0 1. 023+00 1.021+00 TO 1 .025 + 00 30.0 1.075+00 1.072+00 TO 1 .077+00 40.0 1.151+00 1 . 147+00 TO 1 .154+00

bO.O 1.229+00 1.225+00 TO 1.234+00

A ALPHA

W4

1.669-05

1.92872 2.6704-02

a = ALPHA ± W4

^ ^ 6,425-06

ALPHA = 2.10'*76

W4 = 3.0142-02

ALPHA ± W4

ranges given are 95% confidence limits ranges given are 95% confidence limits

64
SAMPLE NUMBERUATA CO-ipUTEL) ON NOV 67 UAFA TAKEij Oil MAY 6i 1965

P-TYPE SI SAMPLt NUMBER- /fo DATA COkiRuTEU ON 2f) NOV 67 DATA TAKtu ON FE^i 26. 1965

P-TYPE 51

t, °c p, ri-cm t, °C Pi f2"cni -.19 8.599-Ul .62 1.332+OU

5.37 H.981-U1 5.31 1 . 383+00 10.37 9.324-Ul 10. 2U 1.438+00

14.87 9.635-01 14.81 1.490+00

19.82 9.990-01 2(J .67 1.557+00

25.32 1.039+00 25.26 1 .612 + 00 29.88 1 . 072+00 30.15 1,671+00 34.82 1 .109+00 35.20 1 .732+00 39.80 1.148+00 39.95 1,792+00

44.58 1.185+Ou 44.89 1,855+00

49.78 1.227+00 49.66 1.917+00

log p = log A + a log T = t + 273.16

t, °c J ,

.". • err. range of

. u 8.616-01 8.607-01 TO 0.625-01

10.0 9.295-01 9.2.11-01 TO 9.3n2-01

20.0 i.ono+00 9.995-01 TO i.oni+on

23.0 1.022+00 1. 021+00 TO 1.022+00

30,

U

1.073+00 1.073+00 TO 1.074+On

40.

U

1.150+00 1 . 149+00 TO 1 .150+00 50.0 1.228+00 1.227+00 TO 1.230+00

log p = log A + a log T

T = t + 273.16

t, °c p , n • cm range of P

. u 1 . 326+00 1.322+00 TO 1. 330+00

10.0 1.435+00 1 .43P+00 TO 1.433+00

20.0 1.550+00 1 . 547+00 TO 1.552+00

23.0 1.585+00 1.583+00 TO 1.5H7+00

30.

u

1.669+00 1.666+00 TO 1.671+00

40.0 1 .793+00 1.789+00 TO 1.796+00

50.0 1.922+00 1.916+00 TO 1. 927+00

A = 6.237-06

ALPHA = 2.10978

W4 = fl. 4114-03

ALPHA ± W4

A = 5.537-06

ALPHA = 2.20782

W4 = 2.141B-n2

ALPHA + W4

ranges given are 95;o confidence limits ranges given are 95% confidence limits

SAMPLE NUMBER- /S -« DATA CO^/RUTEl) ON 28 tiOV 67 DATA TAKE, J ON MAY 6. 1965

.P-TYPE 51 SAMPLE NUMBER- '

. DATA -COi'lPUTED ON 20 NOV 67 DATA TAKEN ON FED 9r 1965

..P-TYPE 51

t, °c p , -T • cm -.38 1.317+00

5.28 1.360+00

10.30 1 . 4jb+00

14.84 1 .487 + 00 19.80 1 . 545+UO 25.33 1.610+00

29.91 1 .665+00 34.87 1.725+00

39.68 1 .788+00

1+4.65 1 . 848+00

49.68 1.917+00

log p = log A + a log T

T = t + 273.16

t, °c p , Ti • cm range of c

. u 1.322+00 1.318+00 TO 1 .327 + 00 10,0 1.432+00 1.428+00 1 0 1.436+00

20,0 1.546+00 1.543+00 TO 1.549+00

23.0 1.582+00 1.579+00 TO 1.584+00

30.0 1.666+00 l.b62+00 TO 1 .669+00

40.

0

1.790+00 1 . 785+00 TO 1.794+00

50.

0

1.919+00 1.912+00 TO 1.925+00

A ALPHA

W4

5.344-06

2.21371

2.631fl-n2

ALPHA ± W4

log p

t, °c p, 12' cm -.80 3.978+00

5.04 4.182+00

9.68 4.357+00

14.53 4.529+00

20.59 4.759+00

25.12 4,934+00

29.99 5.127+00

35.00 5.329+00

40.01 5.536+00

45.01 5.744+00

49.72 5.946+00

A + a log T T

= t + 273.16

t. °c p , n • cm

. 0 4.007+00

10.0 4.362+00

20.0 4,735+00

23.0 4,850+00

30.0 5. 126+00

40.0 5.534+00

bO.O 5.961+00

A = 6.981-06

ALPHA = 2.363d5

W4 = 5.8606-03

range of p 4.004+00

4.360 + 00 . 4.733+00

4.848+00

5.123+00

5.531+00

5.957+00

010+00

364 + 00. 737+00 852+00

128+00

TO 5. 537+00

TO 5.966+00

TO

TOTO

TOTO

ALPHA ± W4

ranges given are 95% confidence limits ranges given are 95;i confidence limits

55
SAMPLE NUMBER- 17 P-TYPE ST

UrtIA COMPUIEU ON 20 IJOV 67 DATA TAKEN 0^4 FEU 2» 1965

SAMPLE NUMBER- ' S P-TYPE SI DATA CuMpuTEO ON 2M NOV 67 DATA TAKE, J ON SEPT 2Uf 1966 ^

t, °c Pi 12 * CfT]

• 10 fJ . 6 5 5 + 0

1

S. 11 9.096+01

10.01 9.510+01

It. J4 9.tJU9 + 01 20.54 1.048+02

25.32 1 . 091+02 30.09 1.137+02

35.02 1 . 1U5+02

39. UO . 1.233+02

45.03 1 .2b6+02 49.53 1.331+02

t, °c p, Q'Cm 6.36 9.40b+01

9.90 9.6o2+01

13.54 9.999+01

17. 14 1 . 030+02 20.53 1. 061+02

24. as 1.102+02

29.51 1.146+02

34.41 1.193+02

39.11 • 1 .239+02 44.39 1.292+02 49.66 1.347+02

log p = log A + a log T

T = t + 273.16

t, °c p, il'Cm range of P .0 a. 668+01 a. 635+01 TO 6.700+01

10.

u

9.513+ni 9.4tja+oi TO 9.533+01

20.

U

1.041+02 1.039+02 TO 1.043+02

23.0 1 . 069+02 1 . 0'i6 + 02 TO 1.071+02

30. u 1.135+02 1 .133+02 TO 1 . 137+02

40.0 1.235+02 1.231+02 TO 1.238+02

50.0 1.339+02 1.335+02 TO 1.344+02

A = t. 272-05

ALPHA = 2.56675

W4 = 2.B'+15-n2

.a = ALPHA ± W4

log p = log A + a log T

t + 273.16

t, °c p , ;7 • cm range of P .0 8.856+01 e. 809+01 TO 8.903+01

10.0 9.692+01 9.1)55 + 01 ro 9.720+01

20.

0

1 . 057+02 1 . 055+02 TO 1 . 060+02

23.

u

1. 005+02 1.082+02 TO 1.087+02

30.0 1.150+02 1.147+02 TO 1.153+02

40.0 1 .248+02 1.243+02 TO 1 .252+02

50.0 1.350+02 1.344+02 TO 1.356+02

A = 6.853-05

ALPHA = 2.50834

W4 = 4.0844-02

ALPHA ± W4

ranges given are 95:. confidence limits

SAMPLE NUMBER- /9 P-TYPE SI DATA COr-iPUTEQ ON 28 fjOV 67 Data TAKtij orj sept 2Sf 1966

ranges given are 95:; confidence limits

SAMPLE NUMOER- 2-0 P-TYPE SI Dm I A COMruTED ON .JH IJOV t>7 DATA TaKEI j ON SEPT 20 f 1966

t, °c p, fi'cm t, °c p, fi'cm 1.84 S. 958+02 6.10 9.862+02

6.35 9.459+02 9.77 1 .015+03

10.06 9.733+02 13.43 1.050+03

14.92 1.017+03 17.09 1.037+03

19.64 1 .050+03 20.50 1.133+03

24.80 1.096+03 24.82 1.179+03

29.73 1,143+03 29.55 1 .235+03 34.82 1 .193+03 34.47 1.293+03

38.94 1.234+03 39.22 1 .351+03

44. Ob 1 .285+03 44.54 1.415+03 49.37 1. 340+03 49.86 1.482+03

log p = 1 og A + a log T T = t + 273.16

t, °c p, n-cm range of P .0 8.674+02 8.810+02 TO 8.938+02

10.0 9.699+02 9,t)49 + 02 TO 9. 749+02

20.0 1.057+03 1.053+03 TO 1.061+03

23.0 1.064+03 1.080+03 TO 1.088+03

30.0 1.148+03 1 .144+03 TO 1 .153+03

40.0 1.244+03 1.238+03 TO 1 .250+03

50.

0

1.344+03 1.335+03 TO 1.354+03

A = ALPHA = W4 =

8.437-04

2.47164 5.5790-02

a = ALPHA ± W4

log p = log A + a log T

T = t + 273. 1 5

t, °c p, n-cm range of P .0 9.200+02 9. 126+02 TO 9.275+02

10.0 1.019+03 1 .014 + 03 TO 1 . 025 + 03 20.0 1.125+03 1.121+03 10 1.130+03

23.0 1.159+03 1.154+03 TO 1.163+03

30.0 1.238+03 1.234+03 TO 1.243+03

40.0 1.350+03 1.351+03 TO 1.365+03

50.

u

1.486+03 1.475+03 TO 1.497+03

A = 1.037-04

ALPHA = 2.051 78 W4 = 6. 1827-02

.a = ALPHA ± W4

ranges given are 95?i confidence limits ranges given are 95:^ confidence limits

66
SAMPLE NUMBEH- ^1

DATA COMPUTED ON 2H NOV 67 DATA TAKEU ON FED 2b< 1965

_P-TYPE SI SAMPLE NUMBER- ' DATA COi'lpUILU ON NOV t>7 DA I A TAKE. J ON DEC ?4 . 1904

N-TYPIT

t, °c p f

;"2 • cm .4 / 1. 217+03

5.17 1.283+03

in. 15 1 .349+03

14.76 1 .406+03

2n.b4 1 .483+03 25.23 1.544+03

30.18 1.613+03

35.27 1.685+03

40. U2. 1.752+03

44.98 1.823+03

49.77 1.894+03

log p = log A + a log T

t, °C.0 10.0

20.

u

23 . ij 30. 40.0

50.0

3 , • cm 221+03

342+03 471+03

511+03

608+03

752+03

904+03

T = t + 273.16

range of p 1.215+03

1.338+03

1 .468+03 1.508+03

1 .604+03

1 .746+03

1 .895+03

TO 1.226+03

TO 1.347+03

TO 1.475+03

TO 1.515+03

TO 1.612+03

TO 1.757+03

TO 1.912+03

t, °c p, -T-cm -.05 1.020-03

4.88 1.037-03

10.08 1.047-03

14.84 1.055-03 20.77 1 . 0b7-U3

25.31 1.075-03

30.04 1.083-03

35.37 1.093-03

40.10. 1.101-03

45.11 1.111-03

50.01 1.120-03

log p = 1 og A + a log T

T = t + 273.16

t, °c p , • cm range of P

. u 1 . 028-03 1 . 024-03 TO 1 .031-03

10.0 1.047-03 1.044-03 TO 1 .049-03

20.0 1.065-03 1 . 063-03 TO 1.067-03

23.0 1.071-03 1 . 069-03 TO 1.073-03

30.0 1 . 084-03 1.081-03 TO 1.0H6-03

40.0 1.102-03 1 . 099-03 TU 1. 104-03

50.0 1.119-03 1.116-03 TO 1 .123-03

A = '.427-04

ALPHA = 2.64343

W4 = 3.4265-02

ALPHA ± W4

A = 5.912-05

ALPHA = .50897

W4 = 2.7103-02

a = ALPHA ± W4

ranges given are 95:. confidence limits ranges given are 95^^ confidence limits

SAMPLE NUMBER- 2

DATA CO-lPUTED ON 2<^ igOV o7 DATA TAKLli ON OCT 6. 1966

N-TYPE St SAMPLE NUMBER- 3, Dai A Cul'iPUIb.lJ ON 28 NOV t.7 DATA fAKLij ON DLL 29r 196';

N-TYPE S

t, °c p > i2 • cm .18 5.045-02

5.52 5. 134-02

10.12 5.212-02

14.04 5.282-02

19. »3 5.387-02

24.80 5.479-02

29.70 5.575-02

34.42 5.670-02

39.34. 5.771-02

1 44.28 5.876-02

49.42 5.988-02

t, °c

.65 5.29 10.95

15.15

20.31 24.98

29.70 34.88

39.47 44.52

49.20

7.900-02

6. Ob6-02 8.271-02

R.42b-U2 8.623-02

8.807-02

a. 998-02

g. 209-02 9,406-02

9.625-02

9.834-02

log p = log A + a log T

T = t + 273.16

t, °C

.0 10.

u

20.

u

23.0

30. 40.0 50.0

iJ'Cm range of p 5.022-02 5.212-02

5.403-02 5.460-02

5.593-02 5.784-02

5.976-02

4.996-02

5. 193-02

5. 388-02 5.445-02

5.577-02

5.762-02

5.945-02

TO 5.048-0?

TO 5.232-02

5.418-0?

5.475-02

5.610-0?

5.8fl6-0?

TO

TO

TOTO

TO 6.006-02

= 1.517-04

ALPHA = 1.03421

W4 = 4.0205-02

ALPHA ± W4

log p = 1 og A + a log T

t, °c p, iT'Cm .0 7.050-02

10.0 8.237-02

20. U 6.630-02

23.0 8.748-02

30.0 9.026-02

40.0 9.428-02

bO.O 9.833-02

A

= 4.258-05

ALPHA = 1.34034

W4 = 3.6930-02

T

= t + 273.16

range of p 7.812-02

8.209-02 8.608-02

8.727-02

9.003-02 9.395-02

9.788-02

TO 7.888-0?

TO 8.266-02

TO e. 652-02

TO 6.770-02

TO 9.050-02

TO 9.460-02

TO 9.879-02

ALPHA ± W4

ranges given are 95"; confidence limits ranges given are 95% confidence limits

67
SAMPLE NUMBER- V- UAfA CUilPUTED ON 28 NOV 67 DATA TAKtiM ON DEC ?9r 1965

N-TYPE 51 SAMPLE NUMBER- S

DATA COilpUTED ON 28 NOV 67 DATA TAKEIJ OM DEC ?lt 196S

.._N-TYPE 'it

t, °c p, n-cra t, °c p, Si'cm .bi 7.934-02 .90 0.010-02

5.22 8. 103-02 5. 63 8. 183-02

10. aa 8.309-02 10.33 B. 356-02

15.12 8.468-02 15.34 8.547-02

20.29 8.fab7-02 20.22 0. 736-02 24.98 8.852-02 24.50 8.926-02

29.72 9. 04a-02 29.75 9. 122-02 34.87 9.2bl-02 34.71 9. 531-02 39.52

,

9.461-02 39.84. 9.552-02

44.59 9.681-02 44.90 9.777-02

49.27 9.693-02 49.08 1.000-01

log p = log A + o leg T T = t + 273.16

t, °c p, n-cm range of P .0 7.888-02 7.857-02 TO 7.919- 0? 10.0 8.279-02 B. 255-02 TO 0.302- 02 20.0 8.675-02 8.656-02 TO 8.693- 02 c3.u 8.794-02 8.776-02 TO 8.812- 02 30.

u

9.075-02 9. 056-02 TO 9.095- 02 40.0 9.481-02 9.454-02 TO 9.507- 02 bO.o 9.890-02 9.853-02 TO 9.928- 02

log p = log A + a log T T = t + 273.16

t, °c p

, a •cm range of P .0 7.951-02 7.913-02 TO 7.989-02

10.

u

8.347-02 8.318-02 TO 8.375- 02 20.

u

8.747-02 8.725-02 TO 0.770- 0? 23.0 8.868-02 B. 846-02 TO 8.890- 02 30.0 9.152-02 9. 129-02 TO 9.176- 0? 40.0 9.563-02 9.530-02 TO 9.595- 02 50.0 9.977-02 9.932-02 TO 1.002- 01

A

= 4.141-05

ALPHA = 1.34617

W4 = 3.0244-02

a = ALPHA ± W4

A = 4.075-05

ALPHA = 1.35046 M 3.6609-02

ALPHA ± W4

ranges given are 95" confidence limits ranges given are 95'. confidence limits

SAMPLE NUi1BE:R-. i& Data COMPUTED ON 20 NOV 67 DATA TAKtij OlJ DEC 21 1 19e'

N-TYPE 51 SAMPLE NUMBER- V

DATA COMPUTED ON 28 NOV 67 DATA TAKEIJ ON FEB 26. 1965

N-TYPE ST

t, °c P, il'cm t, °C p, a- cm .98 8.047-02 .91 8.405-02

5.75 fl. 219-02 5.50 0.582-02

10.44 0.393-0^ 10.39 8. 776-02

15.41 8.581-02 14.90 6.959-02

20.24 8.770-02 20.70 9. 202-02

25.03 8.962-02 25.30 9.400-02

29.73 9.158-02 30.07 9.611-02

34.70 9.360-02 35.07 9.037-02

39.77 9.508-02 39.62 1.006-01

44.81 9.812-02 44.76 1.029-01 49.75

1 1.004-01 49.47 1.052-01

log p = log A + a log T

t, °C

. 0 10.

u

20. 23.0

30.0 40.0

50.0

p , n • cm 7.980-02 8.377-02

8.780-02 8.901-02

9.187-02

9.598-02

1.001-01

T = t + 273.16

range of p 7.936-02

8.344-02

6.753-02 8.876-02

9. 159-02 9.560-02 9.961-02

TO 8.025-0?

TO 0.411-0?

TO 8.806-02

TO 6.927-0?

TO 9.214-0?

TO 9.636-02

TO 1.007-01

log p = 1 og A + a log T

T = t + 273.16

t, °c p , Q • cm range of P .0 ;>. 340-02 8. 294-02 TO 8.3fl5-0? 10.0 H. 763-02 8. 728-02 TO 0.797-02

20.0 9. 191-02 9.1fa5-02 TO 9.218-0?

23.

u

9.321-02 9.295-02 TO 9.347-0?

30.0 9.626-02 9.597-02 TO 9.654-02

40.0 1.007-01 1.003-01 TO 1.010-01

50.0 1.051-01 1.046-01 TO 1.057-01

A = 4.080-05

ALPHA = 1.350BB

W4 - 4.2721-02

ALPHA ± W4

3.694-05

1.37645 M = 4.1722-02

A ALPHA ALPHA ± W4

ranges given are 95;S confidence limits ranges given are 95% confidence limits

R8
SAMPLE NUMBER- 8

DATA COi'iPUT to Oti Z6 NOV 67 DATA TAKtU ON DEC 196'4

. N-TYPE 51 SAMPLE NUMBER- S -R

DATA CO-lpUTEU ON 20 NOV 67 DATA TAKEij ON MAY 6. 196b

.J\|-TYPE SI

t, °c P'

• lb 1 •6b7"'01 4.99 1.716-01 10.17 1.7o9-01

14.91 1 .til8-Ul 20.79 1.802-Ul

25.31 1 .932-01 30.01 1 .985-01 35.29 2. 047-01 40.29 2.103-01

44.99 2.164-01

49.04 2.225-01

t. °C p, •^•cm .07 1 . 6bb— U

1

5.43 1.720-01

10. bO 1.771-01

14.85 1. 814-01

19.86 1 .871-01 25.34 1.931-01

29. B2 1 .982-01 34.70 2.038-01

39.69 2.097-01 44.46 2.155-01

49.61 2.215-01

log p = log A + a log T

T = t + 273.16

t. °c p , n • cm range of p .0 1.661-01 1.654 -01 TO 1.668- 01 10.

u

1.767-ni 1.762 -01 TO 1.773- or 20.0 1.876-01 1.872 -01 " TO 1 .881- 01 23.0 1. 910-01 1.905 -01 TO 1.914- 01 30.

u

1 .988-01 1.984 -01 TO 1 .993- 01 40.0 2.103-ni 2.09o-01 TO 2.109- 01 bO.O 2.220-01 2.211 -01 TO 2.229- 01

log p = log A + a log T

T = t + 273.16

t, °c p, i7-cm range Of P .0 1.660-01 1 .652-01 TO 1.668- 01 10.0 1.766-01 1.700-01.. TO 1.772- 01 20.

u

1. 075-01 1 .870-01 TO 1.879- 01 23. u 1 .908-01 1.903-01 TO 1 .912- 01 30.0 1.986-01 1.981-01 TO 1.991- 01 40.0 2. 100-01 2.093-01 TO 2.ln7- 01 50.0 2.217-01 2.207-01 TO 2.227- 01

A = 1.044-05

ALPHA = 1.724D0 M = 3.1627-02

. ci = ALPHA ± W4

A

= 1.072-05

ALPHA = 1.71979

W4 = 3.5fle6-02

ALPHA ± W4

ranges given are 95°; confidence limits ranges given are 95?; confidence limits

SAMPLE NUMBER- 9

DmTA COMPUTED ON 28 NOV 67 DATA TAKEN ON DEC 24f 1964

N-TYPE SI sample number- 3 -/g. data coi-vutlu on 2h nov o7 Data taken qu may 6» 1965

, .N-TYPE SI

t, °c p, il'cm .38 4.040-01

5.11 4.179-01

10.25 4.333-01

14.98 4.478-01

20.82 4.663-01

25.31 4.808-01

29.98 4.963-01

35.21 5.1tl-ol

39.91 . 5.305-01

44.85 5.481-01

49.68 5.657-01

t, °c p, fi-cm -.05 4.017-01

5.46 4. 182-01 10.45 4.331-01

14.91 4.4b7-01

19.84 4.U23-01

25.32 4.798-01

29.85 4.948-01

34.75 5.114-01

39.73 5.287-01

44.52 5.456-01 49.69 5.646-01

log 1 og A + a log T T - t + 273.16

t, °c p, a •cm range of P .0 4.021-01 4. 008-01 TO 4.035- 01 10.

u

4. 326-01 4.316-01 TO 4.337- 01 20.0 4,642-01 4.634-01 TO 4.6.51- 01 23.

u

4.739-01 4.731-01 TO 4.747- 01 30.0 4.969-01 4.960-01 TO 4.979- 01 40.

u

5. 308-01 5.295-01 TO 5.321- 01 50.0 5.658-01 5.640-01 TO 5.676-01

log p = log A + a log T

t + 273.16

t, °c p, a-cm range of P .0 4.013-01 4.003-01 TO 4.023-01

10.0 4.317-01 4.309-01 TO 4.325-01

20.0 4.632-01 4.626-01 TO 4.638-01

23,0 4.729-01 4.723-01 TO 4.735-01

30.0 4.959-01 4.952-01 TO 4.965-01

40.0 5.297-01 5.287-01 TO 5.306-01

jO.O 5.646-01 5.632-01 TO b. 659-01

A = 4.521-06

ALPHA = 2.03132

W4 = 2.5ai0-n2

a = ALPHA ± W4

A = 4.528-06

ALPHA = 2.03067

^J4 1.9077-02

ALPHA ± W4

ranges given are 95;, confidence limits ranges given are 95^ confidence limits

69
SAMPLE NUMBER- lo

DATA CuMPUTEU ON 28 NOV 67 UaTA TAKEh on StPT 2a» 1966

N-TYPE sr SAMPLE NUMBER- //

U/»IA CUi'VUILL) ON 28 NOV 67

DA I A TAKtlJ ON FLU 2> 1965

N-TYPE SI

t, °c P. 12- cm t, °C p, n-cm 1.63 9.927-01 .52 3.225+00

6.14 1. 028+00 5.30 3.350+00

9.85 1.058+00 10.10 3.495+00

14.79 1.098+00 14.39 3.619+00

19. b9 1.130+00 20.57 3.004+00

24.81 1.102+00 25.40 3.952+00

29.80 1.225+00 30.09 I*. 100 + 00 34.89 1 .270+00 34.96 11.257 + 00 39.14 1.308+00 39.65 4.413+00

44.30 1.355+00 44.03 l+. 590 + 00 49.64 1. 405+00 49.35 4.74B+U0

log p = log A + a log T

T = t + 273,15

t, °c p , fJ • cm range of p .0 9.800-01 9,766-01 TO 9.836-01

la.o 1.059+nO 1 , 056+00 TO 1 . 0^2 + 00 20.

u

1.141+no 1 .139+00 TO 1 . 143+on

23.

u

1.167+00 1 . 164+00 TO 1.169+on

30.0 1.227+00 1.224+00 TO 1.229+On

40.0 1.316+00 1.312+00 TO 1.319+on

50.0 1.408+00 1.403+00 TO 1.413+00

log p = log A + a log T

T = t + 273. 15

t, °c p , Q • cm range of P .0 3.208+00 3.201+00 TO 3.2l5+0n

10.0 3.491+00 3.486+00 TO 3.497+On

20.

u

3.789+00 3. 784+00 TO 3. 793+00

23.0 3.880+00 3.876+00 TO 3. 885+00

30.0 4.100+00 4.095+00 TO 4.io5+on

40.0 4.425+00 4.418+00 TO 4.432+On

50,0 4.765+00 4.755+00 TO 4.775+On

A - ALPHA =

W4 =

5.491-06 . 2.15545

2.7607-02

ALPHA ± W4

^ , 5.912-06

ALPHA = 2.35366

W4 = 1.7186-02

ALPHA ± W4

ranges given are 95;^ confidence limits ranges given are 95" confidence limits

SAMPLE NUMBER- /2

D/UA CUI'IPUILU ON W NOV 67 DA I A IAKlu on Dt;C 24 f 1964

_N-TYPE SI SAMPLE NUMBER- z^-/?-/

U/UA tuMl-'Ull lJ ON 2M NOV 67 DATA TAK£ij ON AUo 16. 1967

M-TYPE SI

t, °c p, 12* cm t, °c p, fJ'cm

.71 7.412+00 .44 7.394+00

5.22 7.719+00 5.27 7.719+00

10.33 8.063+00 10.42 8.069+00

15.03 8.307+00 15.18 8.402+00

20.84 8. 001+00 19.92 8.736+00

25.31 9.127+00 24.69 9. 004+00 29.96 9.472+00 29.56 9.450+00

35. 15 9.871+00 34.72 9.846+00

39.81

.

1.024+01 39.61 , 1.023+01

44.71 1.0O3+01 44.17 1.059+01

49.50 1.103+01 49.69 1.105+01

log p = log A + a log T

T = t + 273,16

t, °c p , n • cm range of p .0 7.369+00 7.353+00 TO 7.3H5+0n

10.0 8.038+00 0.025+00 TO 8,051+00

20.0 8.741+00 8.731+00 TO 0.752+On

23.0 8.959+00 8.949+00 TO 8. 969+ on 30.0 9.479+00 9.468+00 TO 9.49l+0n

40.0 1.025+01 1.024+01 TO 1.027+01

50.0 1.106+OJ 1.104+01 TO l.in9+oi

log p = log A + a log T

T = t + 273,15

t, °c p, n-cm range Of P .0 7.368+00 7.346+00 TO 7. 39l+on

10.0 8.039+nO 6. 021+00 TO 8.056+00

20.0 8.744+00 8.730+00 TO 8.750+00

23.0 8.962+00 8.948+00 TO 0.976+00

30.0 9.484+00 9.4o8+00 TO 9. 500+00

40.0 1.026+01 1.024+01 TO 1.028+01

50.0 1.107+01 1.104+01 TO 1.110+01

A = 9.514-06

ALPHA = 2.41709

W4 = 1.6730-02

. a = ALPHA ± W4

A

= 9.214-06

ALPHA = 2.42279

W4 2.3417-02

ALPHA ± W4

ranges given are 95;i confidence limits ranges given are 95% confidence limits

70
SAMPLE NUMBEK- /2.-^-z

Da I A Coi'ipurLu ON rjov d? UATA TAKE., J ori AUb 16. 1967

N-TYPF ST iiAMPLE NUMQER- IS

UAIA CUl-lPUrLU ON 2M NOV b7 DAIA TAKtu ON Ftl) 2i 1966

N-TYPE SI

t, °c p . u • cm .44 7.377+00

5.27 7.701+00

10.42 0. 049+00

15.18 8.3U2+00

19.92 fj. 715+00 24.69 9. 062 + 00 29.56 9.420+00

34.72 9. 822+00

39.61 1.020+01

44.17 1.057+01

49.69 1.102+01 1

1- °r p i> cm

. O J 8 . 926+00

5.21 9.307+00

10.06 9.699+00

14.37 1 .006+01 20.56 1 . 058+01 25.37 1 .100+01 30.09 1.143+01 34.90 1 . 188+01

39.72. 1.233+01 44.92 1.264+01

j 49.43 1.328+01

log p = log A + a log T T = t + 273,16

t, °C

10.

u

20.

0

23 . u 30.0

40.

u

50.0

• err, 7.350+nO

a. 019+no 8.723+00 8.941+00

9.461+00

1.024+01

1.105+01

-7.325+00 8.000+00 8.707+00

8.y25+00

9.444+00

1.021+01 1.101+01

range or p TO 7. 376+00-

10 8.039+On

TO 8.739+00

TO 8.9S7+00

TO 9.479+00

TO 1.026+01

TO 1.108+01

1 og p = 1 og A + a 1 og T

T = t + 273. 1 6

t, °c p , • cm range Of P .0 0.891+00 0.865+00 TO 8.917+00

10.0 9.695+00 9.675+00 , 10 9.716+00

tiO.U 1.054+01 1.052+01 TO 1.056+01

23. u 1.000+01 1.079+01 TO 1.0H2+01

30.

u

1. 143 + 01 1.141+01 TO 1 . 145+01

40.

u

1.236+01 1.233+01 TO 1.238+01

bO.O 1.333+01 1.329+01 TO 1.337+01

A = 9.195-06

ALPHA = 2.42273

2.ft413-n2

ALPHA : IK

A = 1.201-05

ALPHA = 2.40098

w4 2.2603-02

ALPHA ± m

ranges given are 95, confidence limits

ranges given are 95,- confidence limits

SAMPLE NUMBERDATA CO"IPUTEO ON 20 IJoV 67 DATA TAKtN OlJ SEPT iJO, 1966

N-TYPE SI SAMPLE NUMBER- 15 DATA CO.^PuTED ON' 28 NOV 67 DATA TAKEN ON NOV 27. 1964

N-TYPE SI

log p = log A + c log T T = t + 273.16

t. °C p, .'i-cm range Of p .0 7.752+ni 7.736+01 TO 7.768+01

10.0 8.464+ni 8.452+01 TO 0, 477+0

1

2'3.0 9.214+01 9.204+01 TO 9.224+01

23.

u

9.447+01 9.437+01 TO 9.4S7+01

30.

u

1.000+02 9.991+01 TO 1.001+02

4U.0 1.083+02 1 . 081+02 TO 1.0a4+02

50.0 1.169+02 1.167+02 TO 1.172+02

t, °C 3, .I'cm t. °c 0, 9.-zm 1.73 7.871+01 .09 6.8^9+02

6.23 9.193+01 3.94 7. 0u4+02

9.95 8.4bl+ui 9.14 7.337+02

14.85 0.827+01 14.97 7.705+02

19. dO 9.1fj4 + ui 20.49 a. 151+02

24.00 9. 585+01 25.28 8.473+02

29.77 9.900+01 30.06 8.797+02

34.86 1.039+02 34.74 9.133+02

39.04 , 1.075+U2 40.05 . 9.509+02

44.18 1.110+02 45.39 9.891+02

49.51 1. 166+02 49.93

- 1.022+03

log p = log A + a log t + 273.16

t, °c p, iJ'cm range Of P .0 6.793+02 6.753+02 TO 6,832+02

10.

u

7.419+02 7. 380+02 TO 7.451+02

20.

U

6.079+02 8. 053+02 TO 8.105+02

23.

u

8. 284 + 02 8.25n+02 TO 0.310+0?

30.

U

0.773+02 8.744+02 TO 8.001+02

40.0 9.500+02 9.460+02 TO 9.541+02

50.

U

1.026+03 1.020+03 TO 1.032+03

fl. 508-05 2.44603

1.6106-02

i = ALPHA r l-X

^ 7.09n-04

ALPHA = 2.'»5480

i,;4 = 4.4718-02

ALPHA H4

rang-js given are 95.. confidence lin.its ranges given are 95, confidence limits

71

Solid-State Electronics Pergamon Press 1968. Vol. 11, pp. 639-646. Printed in Great Britain

TEMPERATURE COEFFICIENT OF RESISTIVITY OF

SILICON AND GERMANIUM NEAR ROOM TEMPERATURE*

W. M. BULLIS, F. H, BREWER, C. D. KOLSTADf and L. J. SWARTZENDRUBER

National Bureau of Standards, Washington, D.C. 20234, U.S.A.

(^Received 22 January 1968; in revised form 19 February 1968)

Abstract—Temperature coefficients for the resistivity of n- and ^-type germanium and silicon in the neighborhood of room temperature have been determined over a wide range of resistivity. Linear temperature coefficients have been found for the extrinsic exhaustion region (<5 fl-cm for germanium and < 5000 H-cm for siUcon). The results are presented as plots of temperature coeffi- cient against resistivity at 23°C. The plots may be used in connection with measurements of resistivity on extrinsic germanium and sihcon doped with the usual shallow impurities such as boron,

aluminum, gallium, phosphorus, arsenic, and antimony. Accurate linear coefficients cannot be

found for specimens doped with deep-lying impurities in sufficient amounts to affect the carrier density nor for specimens with resistivity in the transition region between extrinsic and intrinsic conduction. Resume—Les coefficients de temperature de la resistivite du germanium et du silicium des types n et p dans les environs de la temperature ambiante ont ete determines le long d'une gamme de resistivite etendue. Les coefficients de temperatures lineaires ont ete trouves dans la region extrinseque d' exhaustion (<5D-cm pour le germanium et <5000 O-cm pour le silicium). Les resultats sont presentes en forme de courbes de coefficients de temperature en fonction de la resistivite a 23°C. Les courbes peuvent etre employees en"coimexion aux mesures de la resistivite de germanium et silicium intrinseque

dopes d'impuretes peu profondes telles que le bore, I'aluminium, le gallium, le phosphore, I'arsenic et I'antimoine. Des coefficients lineaires exacts ne peuvent etre trouves, pour les specimens dop^s

d'impuretes profondes, en nombre suffisant pour affecter la densite de porteur ou pour des specimens

dont la resistivite se situe dans la region entre la conduction intrinseque et extrinseque.

Zusammenfassung—Die Temperaturkoeffizienten des spezifischen Widerstands von n- und pSilizium und Germanium nahe Raumtemperatur wurden iiber einen grossen Widerstandsbereich

bestimmt. Lineare Temperaturkoeffizienten ergeben sich fiir den Bereich der Storstellenerschopfung

(<2 H-cm fiir Germanium und <5000 O-cm fiir Silizium). Die Ergebnisse sind in Kurvenform

dargestellt. Aufgetragen ist der Temperaturkoeffi^ient gegen den spezifischen Widerstand bei 23°C. Die Kurven konnen in Verbindung mit Widerstandsmessungen an storstellenleitendem Silizium und

Germanium verwendet werden, wobei die iiblichen Dotierungsstoffe wie Bor, Aluminium, Gallium,

Phosphor, Arsen und Antimon herangezogen sind, welche niedrige Storstellenniveaus ergeben. Genau

lineare Koeffizienten finden sich weder fiir Proben, welche tiefliegende Storstellen in geniigenden

Anzahl enthalten um die Tragerdichten zu beeinflussen, noch fiir Proben mit Widerstanden im

tibergangsbereich zwischen Storstellenleitung und Eigenleitung.

* Supported in part by the NASA Electronics Research Center. Initial phases of the work were supported in part by- ARPA.

t Summer student, now at Bates College, Lewiston,

Maine 04240.

73
NBS-1 14A (REV. 7-7 f)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. ITHLU .A 1 ION OH \{\ I'OK I' NO. NBSIR 74-496

2. C mv 't Ac- If s s U)ll

4. I l l ANn suH rn i.iStandard Measurements of the Resistivity of Silicon by the

Four-Probe Method

3. K< ( ipu ill ' s Ai ( r,-.si(ili N,i.

5. I'iilili,-,iin.n l)at( August 1974

6. Pi- r f orm 111 Or^.ui i /.at k in ( udi

7. AU ril01<(Sl W. Murray Bullis

Per f or ni I n>; Ori;.in. Kt-pori Ni

9. pi-:ki-or.minc, orcanmza tion namk and AnnKi-:ss NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. IVojcct /Ta.sk/Work IJnil No.

11 . ( ornrai [ /Cirant Nc ER6576

12. .^ponsormp (Organization Name- and C'on-ipleti- AJJrc-ss (Street, City, State, ZIP)

NASA Electronics Research Center

(Now Defunct)

13. Type- of Report & Pi-riod Covered Supplementary

14. Spon.soring Agency ( ode

15. ,sl;ppi.i-;mi-:ntary noth.s

Supersedes NBS Report 9666 (NTIS Accession N68-18067)

.

report to the Electronics Research Center.

This report was the final

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

bibliography or literature survey, mention it here.) An improved standard procedure for measurement of circular silicon slices with

four in-line point probes has been developed in cooperation with the Resistivity Task

Force of ASTM Committee F-1. Detailed analysis of a series of round-robin experiments

showed that the procedure can attain a precision of ± 2 percent (three standard deviations) for interlaboratory comparisons of slices with room temperature resistivity between 0.005 and 120 ohm-cm. Resistivity non-uniformity in the test slices was shown to

be a significant factor in limiting the precision which could be achieved. The importance of including correction factors for temperature, finite thickness, finite diameter, and unequal probe separations was demonstrated. The results of the round-robin

experiments also emphasized that the precision quoted can only be achieved if the mea- surements are carefully and correctly made on a well maintained, accurately calibrated

test system which meets the requirements imposed by the test method. Determination of

the precision to be expected from the method in non-referee applications such as routine

production and quality control will require additional study of such factors as surface

conditions, probe force, current levels, etc. Nevertheless, use of the various procedures of the method, in particular the sections on probe and measuring circuit evaluations and on thermal sinking of the wafer, would be expected to yield significantly

improved precision in such applications. Use of these procedures on a regular and

widespread basis should be encouraged.

17. KEY WORDS (six to twelve entries; alphabetical order: capitalize only the first letter of the first key nord unless a proper name; separated by semic olons

)

ASTM Committee F-1; electronics; four-probe method; resistivity; semiconductors;

silicon.

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