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1

REFERENCE

NBS

PUBLICATIONS

NBS SPECIAL PUBLICATION 400-64

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

1

emiconductor Measurement Technology.

The Relationship Between Resistivity

and Dopant Density for Phosphorusand Boron-DoD«t Silicon

oc

100

U57

#400-64

1981

. laiiitMfliiaiiitiimriii^
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards' was established by an act ot Congress on March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology

and facilitate their effective application for public benefit. To this end, the Bureau conducts

research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and technological services for industry and government, (3) a technical basis tor equity in trade, and (4) technical services to promote public safety. The Bureau's technical work is per- formed by the National Measurement Laboratory, the National Engineering Laboratory, and

the Institute for Computer Sciences and Technology.

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Semiconductor Measurement Technology:

The Relationship Between Resistivity

and Dopant Density for Phosphorusand Boron-Doped Silicon

W. R. Thurber, R.L. Mattis, and Y.M. Liu

Electron Devices Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Bureau of Standards

Washington, DC 20234

and

J.J.Filliben

Statistical Engineering Division

Center for Applied Mathematics

National Engineering Laboratory

National Bureau of Standards

Washington, DC 20234

This activity was supported by:

Defense Advanced Research Projects Agency

1400 Wilson Boulevard

Arlington, VA 22209

and

National Bureau of Standards Washington, DC 20234

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

NATIONAL BUREAU OFSTANDARDS, Ernest Ambler, Director

Issued May 1981
Library of Congress Catalog Card Number: 81-600052

National Bureau of Standards Special Publication 400-64

Nat. Bur. Stand. (U.S.), Spec. Publ. 400-64, 53 pages (May 198 1

)

CODEN; XNBSAV

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1981

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

Price $3.25

(Add 25 percent for other than U.S. mailing)
TABLE OF CONTENTS

Page

Abstract 1

1. Introduction 2

2. Material Selection 2

3. Test Structure Measurements 3

4. Hall Effect Measurements 7

5. Results for Phosphorus-Doped Silicon 7

6. Results for Boron-Doped Silicon 17

7. Computer Curve Fits for Phosphorus-Doped Silicon 26

8. Computer Curve Fits for Boron-Doped Silicon 33

9. Conclusions 43

Acknowledgments 43

References 44

LIST OF FIGURES

Page

1. Schematic cross-sectional view of the four-probe resistor test

structure 4

2. Cross-sectional view of the base-collector diode used for the

junction C-V measurements of dopant density 6

3. Shape of van der Pauw specimens ultrasonically cut from bulk

silicon slices 6

4. Resistivity-dopant density product for phosphorus- doped silicon

at 300 K 15

5. Electron mobility as a function of electron density for phosphorusdoped silicon at 300 K 16

6. Electron mobility determined from Hall effect measurements for

electron densities greater than lO""^ cm-3 18

7. Resistivity-dopant density product as a function of resistivity

at 300 K for boron-doped silicon 25

iii
8. Hole mobility as a function of calculated hole density for borondoped silicon at 300 K 27

9. The qpN product as a function of dopant density for phosphorusdoped silicon at 300 K , . . . 29

10. The qpN product as a function of resistivity for phosphorus-doped

silicon at 300 K 30

11. Self-consistency of the product and mobility fits at 23°C for

phosphorus-doped silicon plotted versus dopant density for the

product fits and versus electron density for the mobility fits ... 31

12. Self-consistency of the product and mobility fits at 23°C for

phosphorus-doped silicon plotted versus resistivity 32

13. Electron mobility versus resistivity at 300 K for phosphorus- doped

silicon 34

14. The qPN product as a function of boron density for 300 K 37

15. Self-consistency of the product and mobility fits at 23°C for

boron-doped silicon plotted versus dopant density for the product fits and versus hole density for the mobility fits • 38

16. Self-consistency of the product and mobility fits at 23°C for

boron-doped silicon plotted versus resistivity 38

17. Hole mobility versus resistivity at 300 K for boron-doped silicon . 41

18. Dopant density Versus resistivity at 23°C for silicon doped with

phosphorus and with boron 42

LIST OF TABLES

Page

1. Data for Phosphorus-Doped Silicon Obtained from Test Patterns ... 8

2. Data for Phosphorus-Doped Silicon Obtained from van der Pauw

Specimens 13

3. Data for Silicon Slices with the Phosphorus Density Determined

by Neutron Activation Analysis 14

4. Data for Silicon Slices with the Phosphorus Density Determined

by the Photometric Technique 14

5. Data for Boron-Doped Silicon Obtained from Test Patterns ..... 19

6. Data for Silicon Slices with the Boron Density Determined by the

Nuclear Track Technique 22

iv
7. Data for Boron-Doped Silicon Obtained from van der Pauw Specimens . 24

8. Miscellaneous Data for the Computer Curve Fits of PhosphorusDoped Silicon 27

9. Coefficients and Residual Standard Deviation (R.S.D.) for the Fit

of the Product qpN for Phosphorus-Doped Silicon Using Eq ( 1 ) ... 29

10. Coefficients and Residual Standard Deviation (R.S.D. ) for the Fit

of Electron Mobility for Phosphorus-Doped Silicon Using Eq (2) . . 34

11. Parameters and Residual Standard Deviation (R.S.D.) for the Fit

of qpN versus p for Boron-Doped Silicon Using Eq (3) 36

12. Parameters and Residual Standard Deviation (R.S.D.) for the Fit

of qpN versus N for Boron-Doped Silicon Using Eq (4) 36

13. Parameters and Residual Standard Deviation (R.S.D.) for the Fit

of versus p Using Eq (5) 40

14. Parameters and Residual Standard Deviation (R.S.D.) for the Fit

of versus p Using Eq (6) 40

V

Semiconductor Measurement Technology

The Relationship Between Resistivity and Dopant Density for

Phosphorus- and Boron-Doped Silicon*

W. R. Thurber, R. L. Mattis, and Y. M. Liu"*" Electron Devices Division

National Bureau of Standards

Washington, DC 20234

and

J. J. Filliben

Statistical Engineering Division

National Bureau of Standards

Washington, DC 20234

New data have been obtained for the resistivity-dopant density relationship for silicon doped with phosphorus or boron for dopant densities in the range 10^^ to 10^^ cm"-^. For dopant densities

less than 10^^ cm~-^, results were calculated from resistivity

and junction capacitance-voltage measurements on processed wafers.

For more heavily doped material, data were obtained from Hall effect and resistivity measurements on specimens cut from bulk silicon slices. These primary methods were supplemented for

phosphorus-doped material by neutron activation analysis and a photometric technique and for boron-doped material by the nuclear

track technique. For phosphorus-doped silicon the results of this

work differ by 5 to 15 percent from the commonly used Irvin curve,

always in the direction of lower resistivity for a given dopant

density. For boron-doped silicon the results differ significantly

from the p-type Irvin curve for boron densities greater than

10^^ cm~^ with a maximum deviation of 45 percent at 5 x 10^^ cm"-^

in the direction of lower resistivity for a given dopant density.

Hole mobility values derived from the data are in reasonable

agreement with the Wagner expression with a maximum discrepancy of

10 percent in the 10^^ cm"-^ range. Analytical curves were fitted

to the resistivity-dopant density product as a function of resistivity and dopant density for temperatures of both 23°C and 300 K. Similar curves were obtained for the calculated carrier mobility as

a function of resistivity and carrier density.

Key Words: Boron; capacitance-voltage technique; dopant density;

electron mobility; Hall effect; hole mobility; Irvin curves;

phosphorus; resistivity; semiconductor; silicon.

* This work was conducted as part of the Semiconductor Technology Program at

NBS and was supported in part by the Defense Advanced Research Projects

Agency (Order No. 2 397).

t Now with Westinghouse Advanced Technology Laboratory, Baltimore, MD

21203.

1
1. INTRODUCTION I

i The conversion between resistivity and dopant density of silicon is widely

used in the semiconductor industry. In device design, the calculation of i various parameters such as breakdown voltage involves relating resistivity, j which can be readily measured, to dopant density, which is the desired quan- j tity but very difficult to measure directly. Applications using the conver- | sion over many decades of dopant density include the calculation of the sur- 1 face dopant density of a diffused layer from the sheet resistance- junction

j depth product and the determination of a dopant density profile from incremental sheet resistance measurements. Significant error in the results of ; these measurements occurs when incorrect expressions are used to relate re-

| sistivity and dopant density.

j

j During the past decade, the most frequently used conversion curves for both n- and p-type silicon are those formulated by Caughey and Thomas [1] based on

j the curves of Irvin [2]. Irvin's curve for n-type material includes measure- j ments on silicon doped with arsenic, antimony, and phosphorus. More recently

| Baccarani and Ostoja [3] published a conversion for phosphorus-doped silicon

| which agrees with that of Irvin at low phosphorus densities but departs at i high densities in the direction of lower resistivity for a given phosphorus

density with a difference of 30 percent at 10^*^ cm~^. Irvin's curve for

p-type material includes measurements on silicon doped with aluminum, gallium, and boron. More recently Wagner [4] published a conversion for boron- \ doped silicon based on ion implantation results which differs from that of i Irvin by up to 50 percent (in the direction of lower resistivity for a given ] boron density) for densities in the range 10^^ to 10^^ cm"-^. This report ! describes the results of a comprehensive redetermination of the resistivitydopant density relationships for both phosphorus-doped silicon and borondoped silicon and documents the results of the measurements upon which the ; redetermination was made. Papers have been published which summarize the re- i suits of this work on silicon doped with phosphorus [5] and with boron [5] , ] but these papers do not contain either a complete record of the measurements

| used for the redetermination or a comprehensive discussion of many of the ex- j perimental and analytical details, both of which are included in this report. j

2. MATERIAL SELECTION

The starting material, in both ingot and slice form, was obtained from several different suppliers and was selected for minimum resistivity gradients by

mechanical four-probe array measurements [7] prior to use. For phosphorus- | doped silicon, most of the results on high resistivity material were obtained i on neutron transmutation doped silicon which proved much more uniform than

material doped by more conventional techniques. The difference between cen-

' ter and half radius resistivity values ranged from -3 percent to +7 percent j with an average absolute difference of +2.5 percent for the selected ' phosphorus-doped material and even less for the selected boron-doped material. Spreading resistance measurements were made on a number of ingots and I slices to check on fine structure in the resistivity. All of the boron-doped ' material was very uniform, but a slight fine structure with a period of about

1 mm and a peak-to-valley variation of about 15 percent was seen in some of

the phosphorus- doped material.

2
3. TEST STRUCTURE MEASUREt4ENTS

For dopant densities of 10^^ cm~^ or less, resistivity and dopant density

data were obtained from microelectronic test structures fabricated on silicon

slices. The resistivity was measured directly on planar four-probe square

array structures described in detail elsewhere [8] . A cross-sectional view

of this structure is shown in figure 1. The dopant density was determined

from capacitance-voltage (C-V) measurements [9] on gated p-n junction diodes.

These test structures and other diagnostic structures were assembled on microelectronic test patterns NBS-3 [10] and NBS-4 [11] which were prepared es- pecially for this work.

The patterns were fabricated on the silicon slices by bipolar processing

[12]. Various measurements were made on the wafers to verify the fabrication

and to assure that leakage currents were low enough for proper operation of

the test structures. The resistivity data obtained from the square array

structures agreed to within a few percent with mechanical four-probe measurements [13] made prior to fabrication.

The dopant density was measured by the C-V method [9] on the gated basecollector diode structure shown schematically in figure 2. This method gives

the net dopant density, i.e., the number of dopant atoms in excess of any

compensating atoms on the more lightly doped side of the junction. For meaningful C-V measurements, it is important to bias the gate at the flat-band

voltage which was determined from Cj^^j^^^ - Cj^^^j, measurements on an MOS ca- pacitor structure of the test pattern [14]. The dopant density calculated

from the C-V method is particularly sensitive to the value used for the

diameter of the diode as the diameter enters as the fourth power. The diameter (nominally 432 ym) of several diodes was measured by the NBS linestandard scanning interferometer [15]. These results were then used as a calibration for determining from photomicrographs the diameters of other diodes from each processing run of wafers. The uncertainty in the measured diameter is estimated to be less than 1 ym. The base junction depth, used in

the analysis of the C-V data, was measured by the groove and stain technique

[16] with an estimated error of +0.1 ym.

Dopant density values were calculated from the C-V data using a computer program which includes corrections for peripheral effects and takes into account

the fraction of the total depletion depth which occurs in the diffused layer

[17]. A fitting procedure [18] was incorporated into the program in which a Gaussian shape is assumed for the base diffusion near the junction, and the

surface concentration of the diffusion and the background dopant density are adjusted to give a best fit to all the C-V data. The fitting procedure made

17 it possible to obtain results on wafers with dopant densities up to 7 x 10''

cm"-^; otherwise, wafers with dopant density greater than 10^^ cm"-^ could

not be measured because of compensation by the diffusion tail within the depletion depth obtainable in the substrate before voltage breakdown occurs. This fitting procedure was used for reducing the C-V data to dopant density

for all wafers except a few of the lightly doped ones. For these lightly

doped wafers, the program was used without the iterative fitting procedure as the diffused layer effects are negligible. Some lightly doped wafers were analyzed both ways, and the dopant density values were the same with the

3
I I COLLECTOR

mm EMITTER

BASE

r" 1 OXIDE

ALUMINUM

mnniniD gold

mmTimiimmiiiii iiimiiiiimmmmiiTTmTTTT

Figure 1. Schematic cross-sectional view of the fourprobe resistor test structure. The pipes are arranged

in a square configuration with a center-to-center spacing of 57.2 ym (2.25 mil).

4
fitting procedure as without. In the calculations, a value of 11.7 [19] was used for the relative dielectric constant of silicon.

The square array resistivity results were corrected to both 23°C and 300 K

from the actual measurement temperatures using published coefficients for the

change of resistivity with temperature [20] . The measurement temperature was determined by a copper-constantan thermocouple imbedded in the vacuum chuck

holding the wafer on a probe station. The C-V measurements were made at room temperature with no correction for temperature as the dopant is completely

ionized in the depletion layer.

The procedure used for correlating the resistivity and dopant density values was slightly different for the two test patterns. The one four-probe resistor structure contained on test pattern NBS-3 is not located in the same cell

as the diode used for the dopant density measurements. In order to minimize

the influence of resistivity variations over the wafer, the average of the

resistivities measured on the two four-probe resistors on either side of the

diode was taken as the resistivity corresponding to the dopant density at

that diode site. On test pattern NBS-4, there are four four-probe resistor

structures with different collector pipe sizes for measuring a wide range of

resistivities in the cell containing the base-collector diode. On this pattern, an average resistivity was obtained from measurements on two or more adjacent four-probe structures in the cell containing the diode. Differences

between minimum and maximum values of resistivity in a given cell rarely ex- ceeded 1 percent for boron-doped material but were in the range from 1 to 5 percent for phosphorus-doped material because of the localized resistivity

variations characteristic of most phosphorus- doped silicon. Typically, for

wafers fabricated with either test pattern, data were collected for six cells near the center of each wafer and the results averaged. Mobility values calculated for a few of the wafers were judged significantly below those from

other wafers with similar dopant densities. The wafers with low mobility

were omitted from the results so that the curves would represent good material without appreciable compensation or other problems.

The estimated error in the planar four-probe resistivity measurements is

about 1 percent. Factors which contribute to the error are finite wafer

thickness, pipe spacing uncertainty, instrument calibration, and temperature

measurement. Errors in the dopant density due to uncertainties in the sur- face dopant density of the diffusion, junction depth, and random measurement

error were examined in idealized data studies [18]. The errors caused by un- certainties in these parameters for the wafers in this work were shown to be

less than one percent. Additional error in the dopant density arises from

uncertainties in diode diameter, flat-band voltage, and instrument calibration, giving a total estimated error of 2 percent. Resistivity gradients in

the wafers are an additional source of error in that the resistivity at the

diode site may be different from that measured at the four-probe sites as discussed above. Consequently, for phosphorus-doped material, even after

averaging over six sites, the total error may be as large as 5 percent. For

the more uniform boron-doped material, the total error is in the range of 3 percent.

5
GATE

\^ 17 mil (432 Mm) -*-|

Figure 2. Cross-sectional view of the base-collector

diode used for the junction C-V measurements of dopant

density.

|-*^— 280 mil (7.1 mm) -—»-j

Figure 3. Shape of van der Pauw specimens

ultrasonically cut from bulk silicon slices.

6
4. HALL EFFECT MEASUREMENTS

For dopant densities greater than 10^^ cm~-^. Hall effect and resistivity

measurements were made on van der Pauw specimens [21] ultrasonically cut from

bulk silicon slices. The shape of the specimens is shown in figure 3. For

selected slices, a Greek cross design [22] was also cut. The measurements

were made at controlled temperatures of both 23° C and 300 K for a magnetic

flux density of 0.6 T (6 kG) following standard procedures [23]. The commercial Hall probe for measuring the flux density was calibrated using an NMR

gaussmeter. The integrating digital voltmeter used for all the voltage measurements was calibrated using a standard voltage source and a standard cell

calibrated at NBS. The standard resistors used for the determination of the

current through the specimens were calibrated against other standard resistors with known values. Resistivity values obtained from the van der Pauw

specimens were within a few percent of the four-probe values measured on the

respective wafers during the material selection process.

For phosphorus-doped silicon the electron density, n, was calculated from the

expression n = -r/qR-^, where q is the electronic charge, is the Hall

coefficient, and r is the Hall scattering factor [24] . For heavily doped

n-type material, r is unity.

For boron-doped silicon the hole density, p, was calculated from the expression p = r/qRj^. Due to the complex nature of the valence band of silicon,

the factor r for p-type material is less than unity, even for heavily doped

specimens. In this work r was taken equal to 0.8 as the majority of experimental results [25-28] and recent calculations [29] on moderately doped material suggest a value in this range. No measurements of r for dopant densities greater than 2 x 10^^ cm"-^ appear to be available for p-type silicon.

Results obtained on the van der Pauw specimens have an estimated error of

less than 3 percent for phosphorus-doped material. Most of this error is in

the Hall coefficient as it is more difficult to measure accurately than is

the resistivity. The measurement-related error is the same for boron-doped

material, but the uncertainty in r contributes an additional, unknown error. There is the probability that r differs from 0.8 by +0.1. This corresponds

to an error of up to 15 percent in p.

5. RESULTS FOR PHOSPHORUS-DOPED SILICON

Table 1 gives the data for phosphorus-doped silicon obtained from measurements on test patterns. Under wafer identification in column 1 are listed:

(1) the wafer number, (2) the initial resistivity at 23°C from four-probe ar- ray measurements at the center of the wafer before processing, (3) the test

pattern used for fabrication, and (4) the measured diameter, D, in micrometers of the diodes used for the C-V measurements of dopant density, N, in

column 3. The site listed in column 2 gives the x-y coordinates of the diode

on the wafer. The diode in the approximate center of the wafer is designated

0,0 and the others are referenced with respect to it. For example -1,1 specifies the diode located one unit (200 mils) in the -x direction and one unit

in the +y direction from the reference diode. Columns 4 and 5 give the re- sistivity values measured using the four-probe resistor structures and cor- rected to 23°C and 300 K, respectively. Column 6 gives the electron density

7
Table 1. Data for Phosphorus-Doped Silicon Obtained from Test Patterns.

Wafer

Identification Site 23»C

n*cm

300 K

fi'cm -3

'^23''C 2 cm /V "S

^300 K

2 cm /V"

G140Ph-1

= 137 NBS-4

D = 433.3

C100Ph-1

= 98 NBS-4

D = 433.3

G65Ph-1

p^ = 67.4 NBS-4

D = 433.3

C30Ph-1

P^ = 31.9 NBS-4

D = 433.3

Bl2Ph-1

p. = 11.8

1 NBS-3

D = 433.8

D9 . 5Ph

p^ = 9.51 NBS-3

D = 433.

0,0

-1,0

-1 /

1 0, 1

1 ,

1,0

0,0

-1,0

-1 ,

0, 1 1,1,0

2,0

Avg.

1,0

0,0

-1,0

-1,1

0,1

1,1 Avg.

1 ,0 0,0

0 , 1 -1,1

-1,0

-1,-1

Avg.

2.699

2.691 2.598

2.671 2.790 2.757

6.201

6. 147 5.892

6.141 6.358

6.127

6.007

6. 125 X

1 . 149 1.150 1.114

1. 126

1. 155

1 . 153

1 . 141

-1,0 3.557 X -1,1 3.248 X

0 , 1 3.623 X

1 ,

3. 174 X -1,2 3.253 X 0,2 3.363 X 1,2 3. 102 X -2 , 1 3.429 X -2,-1 3.206 X

1,-1 3.241 X Avg. 3.320 X

4.025 X 4.552 X 4.234 X 4.540 X 4.297 X 4.339 X 4.331 X

'I

.13

.13

14

I 14

I 14

I 14

I

. 14

14

14

I 14

I 14

I 14

I 14

I 14

14

14

14

I 14

14

14 'l4

1414

'l4

14

Avg. 2.701 X 158.6 163.6 2.701 X ioi3

1,0 3.893 X 110.3 113.8 3.893 X ioi3

;s"

ioi3

0,0 3.854 X 107.0 110.4 3.854 X -1,0 3.933 X 111.2 114.8 3.933 X

-1 ,

1 3.555 X 116. 1 119.8 3.555 X

0, 1 3.963 X 1 14.4 118.0 3.963 X

1,1 4.015 X 109. 7 113.2 4.015 X Avg. 3.869 X 111.4 115.0 3.869 X ioi3

,13 158.3 158.8

159.0

158.7

158.4

158. 1

163.3 163.9

164. 1 163.8

163.4

163. 2

,699

,691 ,598

,671 .790 2.757 X

68.07 70.25 6. 20 1 X 69.50 71.72 6. 147 X 69.70 71.93 5. 89 2 X 69.92 72. 15 6. 141 X 69.26 71.47 6. 358 X 69.70 71.93 6. 127 X 69.60 71 .82 6. 007 X 69.39 71.61 6. 125 X

37.42 38.62 1149 X 37.29 38.48 1. 150 X

37. 30 38.49 1114 X 37. 33 38.52 1 . 126 X 37.35 38.54 1

155 X 37.46 38.66 1. 153 X 37.36 38.55 1

141 X

13. 15 13.57 3. 557 X

12. 65 13.06 3. 248 X

12. 82 13.23 3. 623 X

13. 29 13.72 3. 174 X

13. 34 13.77 3. 253 X

13. 37 13.80 3. 363 X

12. 91 13.33 3. 102 X

12. 62 13.03 3. 429 X

13. 24 13.67 3. 206 X

12. 87 13.29 3. 241 X

13. 03 13.45 3. 320 X

10.01 10.34 4.025 X 10.03 10. 36 4.552 X 10.58 10.92 4.234 X

10. 23 10. 56 4. 540 X 9.94 10.26 4.297 X 10.04 10. 37 4. 339 X

10. 14 10.47 4.331 X

13

, 13

I 13

I 13

I 13

I 13

13

'l3

'l3

,

,,13

.13

13

,14

14

14

I 14

14

I 14

14

,14

14

, 1414

14

14

14

, 14

I 14

I 14

14

14

14

14

14

14

14

14

1461

1461

151 1 1472

1412 1432

1458 + 34

1454

1514

1427 1512

1377 1417

1450 + 55

1479

1461 1520 1454

1417

1462 1493

1469 + 32

1452 1455 1502

1485

1447 1445

1464 + 23

1334 1519

1344

1480 1438

1388

1559

1442

1470 1496

1447 + 74

1549

1367 1393 1344

1461 1433

1425 + 74

1413 1415 1464

1427

1369 1387

1413 + 33

1409

1467 1382

1466

1335 1373

1405 + 53

1433

1416

1473 1409

1374

1416

1447

1424 + 31

1407

1410 1456

1439 1402

1400

1419 + 23

1293

1471 1302

1433

1393

1345

1509

1397

1424

1449

1402 + 71

1500 1324

1350 1302

1416

1387

1380 + 72

8
Table 1 - Continued

Wafer

Identification -3 2300

•cm 300 K•cm -3 2300

cm /V 's

300 K

2 cm /V •s

F9.1Ph-1

P. = 9.1

1 NBS-4

D = 433.3

B5.9Ph-1

P. = 5.94

X NBS-4

D = 433.3

D2.4Ph-1

P. = 2.35

1 NBS-3

D = 433.

B2. 1Ph-2

= 2.14 NBS-4

D = 433.3

Bl.4Ph-3

p^ = 1.29 NBS-3

D = 433.

A1 .OPh-1

p^ = 1.01 NBS-3

D = 432.8

B0.80Ph-1

= 0.79 NBS-4

D = 433.3

0,0

-1,0

0,1

1 ,

1,0

-2,1

Avg.

0,0

-1,0

-1,1

0,1

1 , 1 1,0

Avg.

-1,1

0 , 1

1 ,

1 -1,0

0,0

1 ,0 -1,-1

0,-1

Avg.

0,0

1,0

1 , 1 0, 1 -1,1

-1,0

Avg.

0,-1

-1,0

-1,1

-2, 1 Avg.

1,0

0,0

1 ,

0,1

-1,1

-1,0

814

745

723 732

800

802 4.769 X

646

158 363 272

167 297

7. 150 X

946

968

058 966

842

1 .941

1 .878

1 .874 1.934 X

2. 185 X 2.321 X 2.368 X 2.281 X 2.438 X 2.389 X 2.330 X

-1,1 3. 749 X

0 , 1 3. 707 X

1 ,

3. 404 X -1,0 3. 460 X 0,0 3. 859 X 1,0 3. 278 X 0,-1 3. 553 X Avg. 3. 573 X

4.800 X 4.885 X

4.596 X 4.675 X 4.739 X

098 716

353 920 958

211

14

I 14

I 14

I 1414

I 14

14

14

'14

I 14

'14

I 14

I.14

14

I 15

I 15

I 15

I 15

I 15

I

, 15

15

15

I 15

'15

I 15

I 15

I 15

15

15

I 15

I 15

I 15

I 15

I 15

I 15

15

15

I 15

I 15

I 15

15

15

I 15 '15

'15

I 15

I.15

Avg. 6.210 X 10 15

9. 107 9.391 4.814 X

14

10^^

10^^

10^^

10^^

1424 1381 9.203 9.490 4.745 X 1429 1386 9.283 9.572 4.723 X 1424 1381 9.208 9.496 4.732 X 1432 1389 9.225 9.513 4.800 X 1410 1367 9.079 9.363 4.802 X 1432 1388

9. 184 9.471 4. 769 X 10^^ 1425 + 8 1382 + 8

5.699 5.875 7.646 X 10^^ 10^^

10^^

10^^

10^^

10^^

1432 1389 6.481 6.681 7. 158 X 1345 1305

6. 116 6.304 7.363 X 1386 1345 5.940 6. 123 7.272 X 1445 1402 6.327 6.522 7. 167 X 1376 1335 6.468 6.667 6.297 X 1532 1487

6. 172 6.362 7. 150 X

10^^ 1419 + 66 1377 + 65

2.353 2.426 1.946 X 1363 1322 2.275 2.346 1 .968 X 1394 1352 2.352 2.425 2.058 X 1289 1250 2.372 2.446 1 .966 X 1338 1298 2.429 2.504 1 .842 X 1395 1354 2.405 2.480 1 .941 X 1337 1297 2.432 2.508 1.878 X 1367 1325 2.455 2.531 1.874 X 1357 1316 2.384 2.458 1.934 X 1355 + 34 1314 + 33

2. 129 2. 192 2. 185 X 1342 1303 1.994 2.053 2.321 X 1349 1310

1 .976 2.035 2.368 X 1334 1295 2.003 2.062 2.281 X 1366 1327 1.950 2. 008 2.438 X 1313 1275 1.978 2. 037 2.389 X 1321 1283

2. 005 2.064 2. 330 X

15

10 1338 + 19 1299 + 19

1 . 358 1 . 398 3.749 X

,n 15 1226 1191 1.314 1 . 353 3.707 X 1281 1244

1 .301 1 . 340 3.404 X 1409 1368

1 .297 1.335 3.460 X 1391 1351

1 .253 1 .290 3.859 X 1291 1254

1 .296 1 . 334 3.278 X 1469 1427

1 .392 1 .433 3.553 X 1262 1226 1.316 1.355 3.573 X 1332 + 90 1295 + 87

1.019 1 .049 4.800 X -i

10^^

1276 1240

1 . 005 1 . 034 4.885 X 1271 1236

1 . 024 1 . 054 4. 596 X 1326 1288

1 . 040 1.071 4. 675 X 1284 1247

1 .022 1 .052 4. 739 X 10^^ 1289 + 25 1253 + 24

0. 7887 0.8106 6.098 X §

10^^

1298 1263 0.7643 0.7855 6.716 X 1216 1183 0.7834 0.8051 6.353 X 1254 1220 0.8122 0.8347 5.920 X 1298 1263 0.8067 0.8291 5.958 X 1299 1264 0.7807 0.8023 6.211 X 1287 1253 0.7893 0.8112 6.210 X 10^= 1275 + 34 1241 + 33

9
Table 1 - Continued

Wafer

Identification Site -3 2300

fi' cm

300K

S2 • cm

'^23''C 2 ^ cm /V's

0. 7912 0. 8138 6. 251

X

10^^ 10^^

0. 7771 0. 7993 6. 400

X

0. 7655 0. 7874 6. 630

X

0. 7647 0. 7865 6. 673

X

0. 7731 0. 7951 6. 257

X

0. 7990 0. 8218 5. 955

X 0 

7785 0 8007 6. 36 1 X 10 0 

4834 0 

4964 1 . 131

X

10^^

10^^ 10^^

10^^

10^^

10^^

0. 4856 0. 4987 1 . 174

X

0. 4778 0. 4907 1 . 121

X

0. 4912 0. 5044 1 071X

0. 5027 0. 5162 0. 973

X

0. 5133 0. 5271 1 026

X

0. 4735 0. 4862 1 . 1 14X

0. 4965 0. 5099 1. 031X 0 

4905 0. 5037 1 080X 10^^

0. 3586 0. 3677 1. 484

X

10^'

10^^

10^^

10^^ 0 

4442 0. 4555 1 . 144X

0. 4201 0.4308 1 302X

0. 4545 0. 4660 1 . 225X

0. 4542 0. 4657 1 . 185X

0. 3280 0. 3363 1 806X

0. 3605 0. 3697 1 602X

0. 4115 0. 4220 1 

197

X

0. 3532 0. 3622 1 623X 0 

3983 0. 4084 1 . 395X 10

0. 2845 0. 2915 2. 000X 10^"

10'^

0. 2869 0. 2940 1 941X

0. 2782 0. 2851 2. 106X

0. 2899 0. 2970 2. 021X

0, 2828 0. 2898 2. 056X

0. 2872 0. 2943 2. 042X

0. 2849 0. 2920 2. 029X

10^^

0. 2748 0. 2815 2. 165X 10^^

10^^

10^^ 10^^

0. 2708 0. 2774 2. 087

X

0. 2673 0. 2738 2. 088

X 0.2746 0. 2813 2. 084 X

0. 2719 0. 2785 2 

106X 10

0. 1900 0. 1943 3. 288X >>

0. 1927 0. 1970 3. 322X

0. 1924 0 . 1967 3. 247X

0. 1980 0. 2024 3. 158X

0. 1911 0. 1953 3. 336X

0. 1953 0. 1997 3. 215X

0. 1932 0. 1976 3 26 1 X 10

0. 1225 0. 1250 5. 557

X 10^^

10^^

10^^

10^^

10^^

0. 1221 0. 1245 5. 453X

0. 1249 0. 1274 6. 617

X

0. 1070 0. 1091 5. 604X

0. 1289 0. 1315 5. 776

X

0. 1235 0. 1260 4. 985X

0. 1152 0. 1175 5. 915X

0. 1296 0. 1322 5. 867X

0. 1217 0. 1242 5. 736

X 10^^

300K

cm /V'S

B0.77Ph-1 p . = 0.77 1 NBS-3 D = 433. 1

B0.47Ph-1 = 0.47 NBS-3 D = 432.8

B0.40Ph-1 P . = 0.40 1 NBS-4 D = 433.3

B0.30Ph-1

p^ = 0.285 NBS-4 D = 433.3

A0.27Ph-2

p^ = 0.269 NBS-3 D = 432.8

BO, 18Ph-2 p . = 0. 192 NBS-4 D = 433.3

BO. 12Ph-1 p . = 0. 124 1 NBS-4 D = 433.3 0 , 1 1 , 1 -1,0

0,0

1,0

0,-1

Avg.

0,0

0,1

-1,0

-1,1

-2,1

-2,0

-1,-1

1,0

Avg.

0,0

1,0 1 , 1 0 , 1 -1,1

-1,0

Avg.

0,0 0 , 1 -1,0

-1,1

Avg.

0,0

-1,0

-1,1

0,1 1 , 1 1,0

Avg.

6.251

X 6.400 X 5.630 X 6.673 X 5.257 X 5.955 X 6.361 X 1 . 132 1.175 1 . 122 1 . 072 0.974

1.027

1.115

1.032

1.081

X

-1,-1 1 .485

X 0,0 1 . 145 X -1,0 1 .303 X -1/1 1.225 X

0,1 1 , 185

X 1 , 

1 .807 X

1,0 1.603X 0,-1 1 . 198 X

1,-1 1 .624

X Avg. 1 .397 X

2.004

1.945 2.110

2.025

2.070

2.045

2.033

X

2.17 1 X 2.093 X 2.094 X 2.090 X 2.112 X

308

342 267

177 355

235 3.281

X

-1,0 5.730

X 0,0 5.514 X

1,0 6.691

X -1/1 5.667 X 0 / 1 5.840 X 1 , 

5.041

X -1,-1 5.981 X

1,-1 5.932

X Avg. 5.800 X

15

'15

15 '15

15 , 16

16

16 I 16 I 16

16 I 16 I 15

15 I 15 I.16

15

15 I 15

!l6

16

16 I 15

16 I 15

16

16 I

16

'15 I

.16

16

15

'I

15

1262 1255

1230

1223 1290

1312

1262 + 34 1 142 1095

1165

1186

1276

1185

1183 1219

1181 + 53

1173 1228 1 141 1121 1160 1054

1081 1257 1089

1145 + 70

1097

1121 1065 1065

1068 1054

1080 + 24

1049 1 104 1 118 1091

1090 + 30

999

975

999 998

979 994

991 + 11

899 937

755

1041 838

1014 916

821 903 + 97

1227

1220 1195 1 189 1255

1275

1227 + 33

1112

1065 1 135 1155

1243 1154 1 152 1 187 1150 + 52 1 144 1198

1270

1093 1 131 1028 1054

1235

1062

1101 + 75

1071 1094

1040

1040

1042 1039

1054 + 23

1024

1078 1092

1065

1065 + 29

977 954

977

975 958

972

969 + 10

881 919

740

1021 822

994

898

805

885 + 95

10
Table 1 - Continued

Wafer

Identification Site 23<>C 300 K

-3 23°C

2 ^ cm /V

"^300 K

2 „ cm /V .s

0. 09077 0. 09238 8. 831 X 10^^

10^^

10^^

0. 08933 0. 09091 9. 414 X

0. 08695 0. 08849 9. 589 X

0. 08959 0. 09118 8. 635 X

0. 09381 0. 09547 9. 412 X

0. 08953 0. 091 12 8. 942 X

0. 09000 0. 09159 9. 137 X

10^^

0. 07776 0. 07905 1

109 X

10^'

10^^

17 10

0. 07835 0. 07965 1

069 X

0. 07943 0. 08074 1

080 X

0. 08100 0. 08234 1

033 X

0. 07881 0. 08011 1 . 143 X

0. 08004 0. 08137 1. 065 X

0. 08058 0. 08191 1

019 X

0. 07891 0. 08022 1069 X

0. 07936 0. 08067 1 . 073 X 10^'

0. 06375 0. 06466 1421 X <

>

0. 06473 0. 06565 1. 504 X

0. 05920 0. 06004 1761 X

0. 06189 0. 06277 1

535 X

0. 06042 0. 06128 1

525 X

0. 05820 0. 05903 1. 863 X 0.06137 0. 06224 'i. 601 X

0. 05285 0. 05350 2. 160 X <10

0. 051 15 0. 05178 2. 074 X

0. 05439 0. 05506 2. 064 X

0. 05334 0. 05400 2. 044 X

0. 05453 0. 05520 2. 001 X

0. 05325 0, 05391 2. 069 X 10^^

0. 03410 0. 03438 4. 8 1 1 X s

10^' Z 17 °17

10

0. 03481 0. 03509 4. 483 X

0. 03527 0.03556 4. 03 1 X

0. 03422 0. 03450 4. 500 X

0. 03434 0. 03462 4. 781 X

0. 03508 0. 03536 5. 068 X

0. 03464 0. 03492 4. 612 X

17

10

0. 03152 0. 03172 4. 775 X

10^^

10^' 10^^

0. 03200 0. 03220 4. 758 X

0. 03149 0. 03169 4. 967 X

0. 03157 0. 03177 5. 105 X

0. 02991 0. 030 10 5. 390 X

0. 03297 0. 03318 4. 532 X

0. 03158 0. 03178 4. 921 X 10^^

B0.099Ph-3

p. = 0.092

i NBS-4

D = 433.3

B0.080Ph-1

= 0.080 NBS-4

D = 433.3

A0.060Ph-3

p^ = 0.062 NBS-4

D = 433.3

B0.050Ph-3

P. = 0.051 NBS-4

D = 433.3

B0.0 34Ph-1

p . = 0.034

1 NBS-4

D = 433.3

B0.031Ph-1

p^ = 0.032

NBS-4

D = 433. 1

-1,1

-1,0

0,0

1,0

1 , 1 0,1

Avg.

-1,0

-1,1

0,1

1 ,

1,0

0,0

Avg.

1,2

0,2

1,3

0,3

-1,3

Avg.

-1,0

-1,1

0, 1 1,1

1,0

0,0

Avg.

-1,1

0. 1

1 , 1 -1,0

0,0

1,0

Avg.

8.965

9.557

9.735

8.766

9.555

9.078

9.276 X

1 ,-1 1 . 131 X -1,-1 1 .090 X -1,1 1 . 101 X

0 ,

1 .053 X

1,1 1 . 165 X

1,0 1 .086 X 0,0 1 .039 X 0,-1 1 .090 X Avg. 1 .094 X

1.459 X

1 .544 X 1.808 X 1.576 X 1.566 X 1.913 X 1.644 X

2.234 X

2. 145 X 2.135 X

2. 114 X 2.070 X

2. 140 X

5.112 X 4.764 X 4.284 X 4.782 X 5.081 X 5.386 X 4.901 X

5.085 X 5.067 X 5.290 X 5.437 X 5.740 X 4.827 X 5.241 X 10

16

16

I

16

I.16

16

17

I 17

17

I 17

17

I 17

17

I 17

17

17

'l7

I 17

'l7 'l7

!l7

17

17

I 17

17

I 17

17

17

17

I 17

'17

I 17

I 17

I

. 17

17

17

I 17 'l7

'17

17

779 742

749 807

707 780

761 + 35

724

745

728

746

693 732

760 740 734 + 20

689

641 599 657

677 576

640 + 44

547 588

556

572 572

567 + 16

380

400 439

405

380

351 392 + 30

415

410 399

387

387 418

403 + 14

765 729

735

793 695

766

747 + 34

712

733 716

734

682

720 748

728

722 + 20

679 632

590 648

668

568 631 + 44

540

581 549

566

565

560 + 16

377 397 435

402

377 348

389 + 29

412

407 396

385

385 415

400 + 13

11
calculated from the dopant density using the results of Li and Thurber [30]

for the fraction of dopant which is ionized. Columns 7 and 8 give the electron mobility, y, calculated for temperatures of both 23°C and 300 K, respectively, using the relationship y = 1/qnp where q is the electronic charge, n is the electron density, and p is the resistivity.

Table 2 gives the Hall coefficient, electron density, resistivity, and mobility for the phosphorus-doped silicon van der Pauw specimens measured at both

23**C and 300 K. A comparison of the Hall coefficient results at the two temperatures gives a measure of the experimental reproducibility as the Hall co- efficient at 23°C should be essentially the same as that at 300 K. The re- sults are usually within 0.2 percent, and the largest discrepancy is 0.5 percent. In all cases the resistivity at 300 K is larger than at 23°C as ex- pected from the positive temperature coefficient of resistivity [20].

In addition to the C-V and Hall effect work discussed in this report, neutron

activation analysis and photometric measurements of total phosphorus density were made outside NBS on material used in this study. These techniques and a comparison of the results obtained from them are discussed in detail elsewhere [31]. A complete listing of data for slices analyzed by neutron activation analysis is given in table 3. Data for slices analyzed by the photometric technique are given in table 4.

Figure 4 shows the product of resistivity and dopant density as a function of

resistivity at 300 K. Since, to a first approximation, resistivity is inversely proportional to dopant density, a plot of the product allows deviations from this inverse relationship to be more easily seen, particularly for

low resistivities where the product is large. As seen in figure 4, there is

a systematic difference between the results from neutron activation analysis

and those from the photometric technique. The data points from the Hall effect, which measures only electrically active phosphorus, lie between the two

nonelectrical techniques. Neutron activation analysis and C-V agree within

experimental error for phosphorus densities in the range of 10^^ to 5 x 10^^

cm"-^. The data of Mousty et at. [32] , obtained from resistivity measurements and either neutron activation analysis or Hall effect for phosphorus

density, are also plotted in figure 4. Their results are in reasonable

agreement with this work except at the low resistivity end where differences

are as large as 15 percent.

Throughout the resistivity range, the data obtained in the present study are displaced from the Irvin curve [2], also shown in figure 4, in the direction

of lower dopant density for a given resistivity. The difference varies between 5 and 15 percent over the resistivity range with somewhat larger deviations below 1 ST^'cm than above.

Figure 5 is a graph of electron mobility at 300 K as a function of electron

density. The theoretical curve of Li and Thurber [30] is in good agreement

with the experimental results. The Caughey and Thomas [1] mobility expression for n-type silicon gives significantly lower mobility at low electron

density. The Caughey and Thomas expression is a fit to the Irvin curve with

the assumption that the electron density equals the dopant density. This is

true for lightly doped material (<10^^ cm"-^), but calculations [30] indicate

that the electron density is less than the dopant density for material in the

12
Table 2. Data for Phosphorus-Doped Silicon Obtained from van der Pauw Specimens.

I = 23 °c I = 300 K Specimen No. R

H

3 (cm /C) n( cm3

) p (fi-cm)

2

p {cm /V*s

)

3 R^(cm /C)

n

n( cm

3

) p{Q'cm) 2

p { cm /V • s

)

0. 0047Ph-A -0. 4815 1.296 X

19 10 0 .004766 101.0 -0.4812 1. 297 X

19

10 0.004781 100.6 0.0045Ph-39 -0. 4365 1 .430 X

19

10 0 .004335 100.7 -0.4367 1. 429 X

19

10 0.004353 100.3 0.0033Ph-L31 -0. 3167 1.971 X

19

10 0 .003304 95.8 -0.3172 1

.

968 X

19

10 0.003322 95.5 0.0026Ph-A -0. 2150 2.903 X

19

10 0 .002327 92.4 -0.2146 2. 908 X

19

10 0.002345 91.5 0.0020Ph-47 -0. 1802 3.464 X

19

10 0 .002000 90. 1 -0. 1792 3. 483 X

19

10 0.002015 88.9

0. 0020Ph-19 -0. 1783 3.501 X

19

10 0 .001984 89.9 -0. 1780 3. 506 X

19

10 0.001999 89. 1 0.0017Ph-A -0. 1431 4.362 X

19

10 0 .001597 89.6 -0. 1430 4. 365 X

19

10 0.001611 83.8 0.0015Ph-A -0. 1286 4.853 X

19

10 0 .00 1444 89. 1 -0. 1289 4. 842 X

19

10 0.00 1456 88.5

- 0

.

1 068 5.844 X 10^^ 0 .% 00\J \J 11 ^?PS^ .J 87 .

-0 . 1065 5

,

860 X 10^^ n nn 1 9

3

86 .

0. 0013Ph-46 -0. 1047 5.961 X 10^^ 0 .001200 87.3 -0. 1046 5. 967 X 10^^ 0.001210 86.4

0.001 IPh-A -0. 0879 7.10 1 X 10^^ 0 .001044 84.2 -0.0881 7. 084 X 10^^ 0.001054 83.6

0. 0009Ph-A -0. 0690 9.046 X 10^^ 0 .000857 80.5 -0.0693 9. 006 X 10^^ 0.000864 80.2

13
Table 3. Data for Silicon Slices with the Phosphorus Density Determined by Neutron Activation Analysis.

^1^=^ P230C

No.

46- 1 0 . 00 129 5.85 + 0.11 X 10 47-1 0.00201 3.66 + 0.10 X 10 39-1 0.00448 1.54 + 0.01 X 10 26-1 0.00697 9. 10 + 0. 09 X 10 52-1 0.0101 5.25 + 0. 10 X 10 52-3 0.0105 4.93 + 0. 08 X 10 25-1 0.0340 4.51 + 0.10 X 10 48-1 0.0520 2.28 + 0.08 X 10 48-2 0 . 053 1 2.12 + 0.05 X 10 34-1 0.0785 1.20 + 0.01 X 10 34-2 0.0759 1 . 19 + 0.04 X 10 51-1 0. 193 3.9 + 0.6 X 10 40-1 0.281 2.03 + 0.13 X 10 70-1 0.404 1 .50 + 0.05 X 10 70-2 0.387 1 .53 + 0.07 X 10 29-1 0.492 1 .16 + 0.08 X 10 38-1 0.805 7.35 + 0.35 X 10 49-2 1 .40 4.99 + 0.39 X 10

Table 4. Data for Silicon Slices

with the Phosphorus Density Determined by the Photometric Technique.

Slice

No.

p 23°C

'i' cm

Ncm

3

46-2 0.00130 5.17 X 10^^ 47-2 0.00219 3.02 X 10^^

L31-5 0.00378 1.65 X 10^^ 39-2 0. 00457 1.22 X 10^^ 26-2 0.00730 7.70 X 10^« 52-2 0.0101 4.44 X 10^«

14-A 0.0 144 2.40 X 10^«

25-L 0.0333 4.30 X 10^^ 48-L 0.0501 2,27 X 10^^

14
10x10'

S 4

10-

This Work

Activation Analysis * Photometric

° Hall Effect

° J. C V

Mousty et al. (32)

• Activation Analysis • Hall Effect

--0—o-o-o

10 10- 10- 10 10' 10^

RESISTIVITY (li cm)

- 0.0128

10^

Fiqure 4. Resistivity-dopant density product for phosphorus-doped

silicon at 300 K. Results of this work are compared with those of

Mousty et al. [32] and Irvin [2]. Values of the qpN product are on

the right ordinate.

15
1 1 1 r

0 I 1 I I I I I 10" 10'4 1015 1016 1017 1018 10'9 1020

ELECTRON DENSITY (cm-^)

Figure 5. Electron mobility as a function of electron density for phosphorusdoped silicon at 300 K. The junction C-V data have been corrected for deionization using the calculations of Li and Thurber [30]. The error bars are the standard deviation in the individual mobility values for that wafer.

When not shown, the error bars lie within the plotted symbol. Shown for com- parison with the computer fit to the data are the mobility expressions of

Caughey and Thomas [1], Baccarani and Ostoja [3], and tne theoretical calculation of Li and Thurber [30]

16
10'' cm"-^ to iiiid-10'° cm"-^ range. In the heavily doped range, the two densities are again equal except for the formation of phosphorus -vacancy pairs,

which give rise to electrically inactive atoms, at phosphorus densities

greater than about 5 x 10^^ cm"-^ [33].

Also shown in figure 5 is the mobility expression of Baccarani and Ostoja [3]

based on the data of Mousty et al. [32] , with the assumption that electron

and dopant densities are equal. For coitparison with the other curves the

Baccarani and Ostoja expression has been corrected to 300 K since the Mousty

data were taken at 23°C. At low dopant densities, the Baccarani and Ostoja

curve is about 7 percent below the data of this work. In the range 10^^ to

5 x 10^^ cm~3, their curve is parallel to, but about 5 percent above, the

results of this work. At high dopant densities, the Baccarani and Ostoja

curve gives values between 10 and 20 percent larger than those obtained in

this work as is seen more clearly in figure 6, which is a detailed expansion

of the high density portion of figure 5.

In figure 6, all of the results shown were obtained by Hall effect and resistivity measurements with the assumption that the Hall scattering factor was unity. In this dopant range, the Caughey and Thomas curve is lower and the

Baccarani and Ostoja curve is higher than the results of this work. The modification of the Caughey-Thomas expression shown in the figure consists of

multiplying the constant Pniin -'-'^ Caughey-Thomas expression by a term

which decreases y^iin high electron densities [34] . The computer fit to

the present data (open rectangular points) is also shown in figure 6.

With the exception of the data point of Fair and Tsai [35] , all of the re- sults in figure 6 were obtained on silicon doped during crystal growth. Recently, Masetti and Solmi [36] have made extensive measurements on silicon

heavily doped by thermal diffusion. Their mobility values obtained from incremental sheet resistance and Hall effect measurements are in good agreement

with those of this work. They concluded, from a comparison of their data

with those of Mousty et at. , that the mobility in diffused phosphorus-doped

silicon is always lower than that in uniformly doped silicon for n > 10 cm~-^. However, when the results of Masetti and Solmi are compared with

those of this work, the conclusion is that the mobility is the same in diffused and uniformly doped silicon.

6. RESULTS FOR BORON-DOPED SILICON

Table 5 gives the data for boron-doped silicon derived from measurements on test patterns. The items listed and the symbols used are the same as those

for table 1 except for the hole density, p, which was obtained from the dopant density using the ionization percentages calculated by Li [37] . Table 6

gives data on boron-doped silicon slices obtained by the nuclear track technique (NTT) [38] . Table 7 lists the results obtained from Hall effect mea- surements at both 23°C and 300 K. For calculating the hole density, the

scattering factor was taken equal to 0.8 as discussed in section 4. This

value was also used to obtain hole density from the Hall measurements of

Chapman et ol. [39] and Crowder [40]

.

figure 7 is a graph of the product of resistivity and dopant density as a function of resistivity at 300 K. The NTT results are seen to be in good

17
130

120

110

J^IOO

90

70

60

This Work

Fair & Tsai (35) • Mousty et al. (32) Omel'yanovskii et al.(44) • Yamanouchi et al. (45) A Logan et al. (46) • Esaki & Miyahara (43) ^Chapman et al. (39)

Baccarani & Ostoja

Caughey & Thomas

plus extra term

10 10

20

ELECTRON DENSITY (cm-3

Figure 6. Electron mobility determined from Hall effect measurements for

electron densities greater than 10-^^ cm~^. The computer fit to the data of

this work lies between the mobility expressions of Caughey and Thomas [1] and

Baccarani and Ostoja [3]. The extra term in the modified Caughey and Thomas

expression is given by Plunkett et al. [34].

18
Table 5. Data for Boron-Doped Silicon Obtained from Test Patterns.

Wafer

Identification Site -3 2300

'cm

'300 K

U'cm

'^23<'C 2 , cm /V "s

"^300 K

2 cm /V-s

B94B-3

= 94.0 NBS-4

D = 432.8

B49B-3

p. = 48.8

X NBS-4

D = 432.8

C2 lB-2

p. = 22.0

X NBS-4

D = 433.3

B9.0B-3

= 9.0 NBS-4

D = 433.3

B5.7B-4

P. = 5.7

1 NBS-4

D = 433.8

B3.0B-3

P^ = 3.0 NBS-4

D = 433.3

B1.8B-2

p^ = 1.79 NBS-4

D = 432.8

-1 ,

1 .284 X 0,1 1.362 X

1/

1.266 X -1,0 1.362 X 0,0 1.373 X

1,0 1.313 X

1,-1 1.328 X 0,-1 1.310 X Avg. 1.325 X

-1 ,

0, 1

1 , -1,0

0,0

1,0

Avg.

-1,1

0, 1

1 ,

1 -1,0

0,0

1,0

Avg.

-1,1

0 , 1

1 , 1 -1,0

0,0

1,0

Avg.

-1,1

0,1

-1,0

0,0

1,0

-1,-1

Avg.

547 537

640 625 648

608

2.601 X

258

267 454

248

203

219 6.275 X

0,0 1 .489 X

1,0 1 .453 X

1 ,

1 .488 X -1,0 1 .478 X -2,0 1.479 X -1,2 1 .489 X 0,2 1 .487 X 1,2 1.455 X Avg. 1 .477 X

.307

,294

,333

,312

,353

,342

2.324 X

,756

,731 ,643

,689

,673

,663

4.693 X

-1,1 8. 135 X 10 0, 1 8.035 X 10

1 ,

7.913 X 10 -1,0 8. 052 X 10 0,0 7.906 X 10 1,0 7.971 X 10 Avg. 8.002 X 10

14

I

.14

. 14

. 14

14

14

I 14

14

.14

14

'15

15

15

I 15

1 15

I 15

I 15

I 15

15

15

I 15

I 15

I 15

I 15

I 15

15

15

'15

I,15

1515

15

94.68

97.50 96.48

98.00 99.62

94.80

97.66 96.35

96.89

50.99 49.42

49.26

49.30

49.50 49.66

49.69

20.79

20.81 20.57

20.80 20.82 20.54

97.85

100.77 99.71

101.28

102.96 97.98

100.93 99.28

100. 10

52.68 51.06

50.89 50.93

51.14

51.30

51.33

21.48

21.50

21.25

21 .49 21.51 21.22

1.284

1.362 1.266

1.362

1.373

1.313 1.328

1.310

1.325 X

,547

,537

,640

,625 .648

,608 2.601 X

6.258

6.267 6.454

6.248

6.203

6.219

15

1.748 1.799 8.070 X 10

1 .772 1 .823 7.971 X 10 1.775 1 .827 7.850 X 10 1.764 1.8 14 7.988 X 10 1.778 1 .829 7.843 X 10 1.775 1.826 7.907 X 10

1 .769 1 .820 7.938 X 10

14

. 14

513, 470.

511, 467,

456,

501,

481. 494.

496.8

454.8

494.4

452.5

441.5

485.2

465.7 479.9

. 14

.14

486.9 + 21.3 471.4 + 20.7

480.6

497.8 479.9

482.3

476.2

481 .9

465.

481. 464.

466.

460. 466.

.14

14

I 14

.14 1^^

.14

483.1 + 7.5 467.6 + 7.3

479.7 478.6

470. 1 488. 1 483. 3 488.6

464.3

463.2

455. 1 464.8

467.8

473.0

20.72 21.41 6.275 X 10^^ 481.4 + 6. 9 464.7 + 5.9

8.971 9.254 1 .489 X i

10^^ >

467.2 453. 0 8.911 9.192 1 .453 X 482.0 467. 3 8.997 9.281 1.488 X 466.2 451 .9 8.948 9.230 1.478 X 471.9 457.5

8.848 9. 127 1.479 X 476.9 462.4

8.900 9.181 1 .489 X 471 .0 456.6 8.994 9.278 1 .487 X 466.7 452.4 8.966 9.249 1.455 X 478.4 463.8 8.942 9. 224 1 .477 X 10^^ 472.5 + 6. 0 458. 1 + 5.8

5.753 5.932 2.300 X 10^^

10^^ 10^^

471.7 457.5

5.795 5.976 2.287 X 470.9 456.7

5.767 5.947 2. 326 X 465.3 451.2

5.742 5.921 2.305 X 471.6 457. 3 5.648 5.824 2.346 X 471.0 456.8

5.731 5.910 2.335 X 466.4 452.3

5.739 5.918 2.317 X 10^^ 469.5 + 2. 9 455.3 + 2.8

2.946 3.037 4.733 X

Z 15

447.6 434. 2 2.949 3.040 4.708 X 449.5 436. 1 2.974 3.066 4.620 X 454.3 440.6

2.967 3.059 4.666 X 450.8 437.3

2.975 3.067 4.650 X 451.2 437.6

3.001 3.094 4.640 X 448.2 434.8

2.969 3.061 4.670 X 10'= 450.3 + 2. 4 436.8 + 2.3

15

'15

15

'15 '15

442

441 447 442

447 444

429.9

429.5

435.2

430.7

435. 1 432.3

15 444.6 + 2.6 432.1 + 2.5

19
Table 5 - Continued

l^l^' . Site «

, P230C ^300 K P, '^3°^

Identification -3 „ ^ -3 2 . 2 fi'cm fi-cm cm cm /V*s cm /V*s

B1. lB-3

P. = 1 . 14

1 NBS-3

D = 433. 1

B0.81B-1

P. = 0.81

1 NBS-4

D = 433.3

D0.58B-2

p . = 0.58

NBS- D = 3

433.6

E0.43B-3

p^ = 0.43

NBS-4

D = 433.8

B0.32B-2

p . = 0.32

1 NBS-4

D = 433.8

B0.23B-2

p . = 0. 22

1 NBS-4

D = 433.8

BO. 17B-1

= 0. 165 NBS-4

D = 433.8

-1,1 1.312 X

10^^

10 1^

1,1 1.262 X -2,0 1 .281 X 1,0 1 .279 X 2,0 1.281 X 0,-1 1.270 X Avg. 1 .28 1 X 10^^

-1,2 1 .908 X 10^^

10^^

10^^ 10^^

0,2 1 .908 X 1,2 1 .886 X -1,1 1.901 X

1 , 1 1 .855 X 0,0 1 .896 X

1,0 1 .909 X -1,-1 1.871 X Avg. 1 .892 X 10^^

-2,0 2.890 X

10^^

10^^

0,0 2.877 X

1,0 2.S97 X -2,-1 2.870 X -1,-1 2.906 X

1,-1 2.922 X Avg. 2.894 X 10^^

-1,1 4. 196 X 10^^ 10^^

10^^

10^^

0,1 4.311 X

1 , 1 4.215 X -1,0 4. 196 X 0,0 4. 139 X

1,0 4. 109 X Avg. 4. 194 X 10^^

-1,1 5.927 X 10^'

10^^

10^^

10^^

lo''^ 10^^

0,1 5.861 X

1 , 1 5.878 X -1,0 5.987 X 0,0 5.944 X

1 0 5.911

Avg. 5.918 X 10^^

-1,1 9. 020 X 10^^ 10^^

10^^

10^^

10^^ ,n16 10

0, 1 8.890 X

1 , 1 8. 79 1 X -1,0 8.775 X 0,0 8.783 X 1,0 9. 166 X Avg. 8.904 X 10^^

-1,1 1.369 X

10^^ 10"

0,1 1.366 X

1 , 1 1 .338 X -1,0 1.377 X 0,0 1 .395 X 1,0 1.397 X Avg. 1 .374 X 10^^

1. 141 1 . 173 1.297 X

1. 134 1 . 166 1.247 X

1 . 132 1 . 164 1 .266 X

1. 138 1.170 1 .264 X

1 . 146 1.178 1 .266 X

1 . 132 1. 164 1.255 X

1 . 137 1. 169 1 . 266 X

0.813 0.835 1 .874 X 0.805 0.827 1.874 X 0.802 0.824 1.852 X 0.800 0.822 1.867 X 0.810 0.832 1 .822 X 0.802 0.824 1.862 X 0.814 0.836 1.875 X 0.810 0.832 1.837 X 0.807 0.829 1 .858 X

0.5718 0. 5867 2.818 X 0.5723 0. 5872 2.805 X 0.5713 0. 5862 2.825 X 0.5777 0. 5927 2.799 X 0.5718 0. 5867 2.834 X 0.5677 0. 5825 2.849 X 0.5721 0. 5870 2.822 X

0.4068 0 .4165 4. 053 X 0.4104 0 .4202 4. 164 X 0.4072 0 .4169 4. 07 1 X 0.4078 0 .4175 4. 053 X 0.4179 0 .4279 3. 998 X 0.4071 0 .4168 3. 969 X 0.4095 0 .4193 4. 051 X

0.3085 0. 3154 5.661 X 0.3136 0. 3206 5.598 X

0 . 3 1 44 0. 3214 5.614 X 0.3105 0. 3174 5.718 X

0. 3096 0. 3165 5.677 X 0.3096 0. 3165 5.645 X 0.3110 0. 3180 5.646 X

0.2216 0.2261 8.461 X 0.2236 0. 2282 8.339 X 0.2213 0.2258 8.246 X 0.2234 0.2280 8.231 X 0.2241 0.2287 8. 239 X 0.2161 0. 2205 8.598 X 0.2217 0.2262 8.352 X

0. 1585 0. 1614 1 .254 X 10 0. 1604 0. 1634 1 .252 X 10 0. 1603 0. 1633 1 .226 X 10 0. 1588 0. 1617 1 .262 X 10 0. 1586 0. 1615 1 .278 X 10 0. 1595 0. 1624 1 .280 X 10

0. 1594 0. 1623 1 .259 X 10

16

I 16

'16

I 16

I 16

I 16

, 16

16

I 16

I 16

I 16

I 16

'16

I 16

I 16

, 15

16

I 16

I 16

I

16

I

. 16

16

16

I

, 15

' 16

I 16

I 16

, 16

16

I

16

I 16

I

16

16

16

I 16

I 16

I 16

I 16

I 16

16

17

I 17

17

1717

17

421 .8 441 .4 435.5

433.9

430.2

439.3

410.

429.

423.

422.

418.

427. 433.7 + 7.0 421.8 + 6.9

409

413

420 417

422 418.0 408.9

419.5

398.9

402.7

409.0

406.

411.

407.

398. 408. 416.4 + 5.1 405.3 + 4.9

387.3 388.8

386.7

386.0 385.2 385,9

377.

378. 376.

376.

375. 376. 386.7 + 1.3 376.8 + 1.2

378,

365. 376.

377.

373. 386.

369.7

356.7 367.7 368.9

364.8

377.3

376.3 + 6.9 367.5 + 6.7

357.4

355.5

353.6

351.5

355. 1 357. 1

349.6

347.8

345.9 343.9

347.4

349.3

355.0 + 2.2 347.3 + 2.

332.9

334,7

342.0 339.4

338.0 335.9

326. 3 328.0

335,2

332.6

331.2

329.2

337.2 + 3.3 330.4

314.0 310.8

317.6

311.4

307.9

305.7

308.4

305. 1 311.8

305.9

302.4 300.3

3.2

17 311.2 + 4.2 305.6 + 4. 1

20
Table 5 - Continued

Wafer

Identification Site -3 23">C 300 K

^23°C

2 cm /V'S

^^300 K

2 cm /V -s

0. 0968 0. 0982 2. 355 X 10^'

17

10^^

10^^

0. 0952 0. 0966 2. 509 X

0. 0942 0. 0955 2. 544 X

0. 0954 0. 0968 2. 469 X

0. 0945 0. 0958 2. 522 X

0. 0942 0. 0955 2. 481 X

0

.

095 1 0

.

0964 2, 480 X 10

0

.

0889 0

.

090 1 2

.

742 X ^°17

10

17

10 17

10

0. 0891 0. 0903 2. 788 X

0. 0893 0. 0905 2. 69 3 X

0. 0893 0. 0905 2. 760 X

0. 0891 0. 0903 2. 706 X

0. 0890 0. 0902 2. 752 X

0. 0890 0. 0902 2. 768 X

0

.

089 1 0

.

0903 2

.

744 X 10

0. 0548 0. 0553 5. 757 X

10 "

0. 0548 0. 0553 5. 696 X

0. 0553 0. 0558 5. 563 X

0. 0549 0. 0554 5. 677 X

0. 0546 0. 0551 5. 652 X

0. 0546 0. 0551 5. 637 X

0. 0546 0. 0551 5. 803 X

0. 0548 0. 0553 5. 684 X 10^^

B0.097B-2

= 0.096 NBS-4

D = 433.8

B0.090B-2

p . = 0.090

1 NBS-3

D = 433.6

B0.054B-3

p . = 0.055

1 NBS-4

D = 433.8

-1,1

0 , 1

1 , 1 -1,0

0,0

1,0

Avg.

-1,3

0,3

-2,2

0,2

1 ,2

0 , 1

1 , 1 Avg.

-1,1

0,1

1 , 1

1 ,0 -1,-1

0, -1

1, -1 Avg.

2.717

1.895 2.935

2.848

2.910

2.862

10 . 10

10 X 10 X 10 X 10

, 189

,242

, 132

,210

, 147

,200

,219

260

183 016

159

128 109 318

17

' 1717

1717

17

2.861 X 10 17

X 10 X 10 X 10 X 10 X 10 X 10 X 10

3. 191 X 10

17

17

17

1717

17

17

17

X 10 X 10 X 10 X 10 X 10 X 10

10

17

17

17

1717

1717

273.8

261.3 260.4

265.0 261.9

267. 1

269.9 257.5

256.9

261 . 1 258.3 263.4

264.9 + 5.0 261.2 + 4.9

256.0

251.3 259.5

253.2

258.9 254.8

253.4

252.6

247.9

256. 1 249.9

255.4

251 .4 250.0

255.3 + 3.0 251.9 + 3.0

197.8

200.0 202.9

200.3

202.2

202.8

197.0

196.0

198

201 198

200

200

195

7. 168 X 10

17 200.4 + 2.4 198.6 + 2.4

21
Table 5. Data for Silicon Slices with the Boron Density Determined by the Nuclear Track Technique.

H-n ^, P230C P3OOK P 23°C ^300 K Identification

cm

-3

i2*cm „

fi«cm cm-32,2 cm /V-s cm /V-s

A7 1 1.02 X <10^^ 10.9 11.2 1.02 X <10^^561 546

2 1.33 X 10.9 11.2 1.33 X 431 419

Avg •

1 1 Q la 1 0 X 1 u 1 u • y 11 0 1 1 Q 1 . 1 0 X 1 u 496 482

521-5 1 1.84 X

15 \>

10^^

10.8 11.1 1.84 X

15 314 306

2 1.76 X 10.8 11.1 1 . 76 X 328 320

3 1.71 X 10.8 11.1 1.71 X 338 329

Avg» X 10^^ in Q t U • 0

11 1 1

"7"7 X 327 318

521-22 1 5.01 X 10^^

f u 3,08 3. 17 4.99 X s\ u 406 395

2 fi • y U X J • U 1 J . i u 4.88 X

^ o c 413

Avg« X 10^^ 3.04 3.13 4.93 X 10^^ 415 404

521-21 1 6.89 X

10^^ 1.85 1.90 6.84 X 10^^ 493 480 A6 1 7.17 X

10^^ 1.84 1.89 7. 1

X

1 c; 10^^ 477 465 521-53 1 2.26 X 0.71 0.73 2.21 X s

398 387

2 2.26 X 0.71 0.73 2.21 X 398 387

3 2.27 X 0.71 0.73 2.22 X 396 385 Avg. 2.26 X 10^^ 0.71 0.73 2.21 X 10^^ 397 386

521-4 1 6.00 X s

0.28 0.29 5.72 X 390 376

2 6. 10 X 0.277 0.283 5.82 X 387 379

Avg. 6.05 X 10^^ 0.28 0.29 5.77 X 388 377

521-76 1 1 .29 X <10^^ 0.221 0. 225 1 . 19 X 10"237 233

2 1.12 X 0.221 0.225 1 .03 X 274 269 Avg• X

1 11 X in" TOR 0 R 1

521-57 1 4.55 X < 0.0959 0.0972 3.78 X 10" < 172.2 169.9

2 4.48 X 0.0959 0. 0972 3.72 X 175.0 172.6

Avg. X in" 1 u

c\ no RQ u • uy c\ n Q "7 0 X in" 1 U 1 / J • D 17 1 9

521-62 1 7.29 X

17

1 u

0.054 0.0545 5.78 X

17

in" 1 u

200.0 198.2

2 6 . 67 X 0 . 0 558 0.0 563 5.29 X 211.5 209 .

3 7.41 X 0 . 0558 0 . 0563 5 . 88 X 190.3 188.6 7.12 10" 0 . 0552 0 . 0557 5 . 65 10" 200.6 1 98 .

521-58 1 4.24 X 10^^ 10^^ 0.0204 0.0204 4. 24 X 10^^ 72.2 72.2

2 3.89 X 0.0204 0.0204 3.89 X 78.7 78.7

Avg. 4.06 X 10^^ 0.0204 0.0204 4. 06 X 10^« 75.5 75.5

A3 1^ 5.17 X 10^^ 0.0149 0.0149 5.17 X 10^^ 81.0 81.0 521-44 1 5.96 X 10^^ 10^«

0.0 130 0.0130 5.96 X 10^«

10^^ 80.6 80.6

2 5.80 X 0.0130 0.0130 5.80 X 82.8 82.8

Avg. 5.88 X 10^^ 0.0130 0.0 130 5.88 X 10^^ 81.7 81.7

'''Used for computer curve fit.

22
Table 6 - Continued

Sl^'^^ Run ^

, P230C P3OO K

P

, 230c ^^300 K Identification -3 -3 2 . , 2 „ cm Si-cm n-cm cm cm/V's cm/V"s

52 1-23 1 2.27 X lo''^

1 u

U • UU4

/

0 . 0047 2.27 X

1 u

58 .

5

58 .

5

2 2. 14 X 0,0047 0.0047 2. 14 X 62. 1 62. 1 0 in z • 1 u X u • u u ^ / u , u u** / 9 1 n X 63.2 63 . 2 Avg.

t 2.17 X .0" 0.0047 0.0047 2.17 X 61.3 61.3

A2 1 0 QQ X

1 u

n an Ar\ X

1 u

54 .

0

54 . 0 2 2*81 X U • U U4U n n n /I n U • U U 4U 2,81 X 55 .

5

55 .

5

3 2.58 X 0,0040 0.0040 2.58 X 60.5 60.5

X n n n /I U . U UfiU Z • oZ X 55.3 55.3 Avg • t 2 . 78 X

10^^ n n n /I n U . u u 4u z « /o X

10^^ 56 .

3

56 .

3

52 1 —45 1 2.80 X 1 u U . uuzy4 n n n o Q/i U . U Uzyfi 2 , 80 X 1 U 75.8 75 • 8

521-73 1 4.42 X 0.00176 0.00177 4.42 X 80.2 79.8

2 4.53 X 0.00176 0.00177 4.53 X 78.3 77.9

Avg. 4.47 X 0.00176 0.00177 4.47 X 10^^ 79.2 78.8

521-24 1 1.00 X

10^°

0.00105 0.00106 1.00 X

10^°

59.4 58.9

2 1 .23 X 0.00105 0.00106 1.23 X 48.3 47.9

3 1.23 X 0.00105 0.00106 1.23 X 48.3 47.9

Avg.

t 1.15 X

10^° 0.00105 0.00106 1. 15 X

10^° 52.0 51.6

Used for computer curve fit.

23
Table 7. Data for Boron-Doped Silicon Obtained from van der Pauw Specimens.

T = 23 "C T = 300 K Specimen No. R^(cm^/C)

n p(cm p (n-cm) y (cm^A's

R (cm^/C)

H

p( cm p (fi'cm)

2 p(cm /V's)

B0.010B-1 0.5839 8. 550 X 10^^ 0.009625 75.8 0.5823 8. 57 X 10^^ 0.009537 75.6 E0.0092B-1 0.5175 9. 650 X 10^^ 0.008865 73.0 0.5165 9. 67 X lo''^ 0.008873 72.7 A0.0082B-1 0.4455 1, 121 X io'^ 0.00798 69.8 0.4455 1. 1 18 X io'^ 0.00800 69.8 E0.0077B-2 0.4044 1, 235 X

19 10 0.00739 68.4 0.4041 1. 236 X

19

10 0.00741 68. 1 AO . 0072B-2 0 . 3788 1

318 X 1 0 0.00/ 054 67 .0 . 3784 1320 X 1 0 0 . 007080 56 .

E0.0064B-1 0.3163 1. 579 X

19 10^ 0.006083 65.0 0.3153 1. 584 X19 10^ 0.005100 64.6

A0.0061B-1 0.3037 1644 X 10^^ 0.005932 64. 0 0.3020 1

553 X 10^^ 0.005949 53. 5

AO .0059b- 1 0 . 2883 1

732 X 10. 0057 10 63.1 0 . 288 1 1733 X 1 0 0. 005727 62 .

AO0059b- 1C^ 0 . 2927 1

706 X 1 0 0 . 005830 52 .0 . 29 1

1 • 713 X 1 0 0 . 005845 62. 3 Avg. 0 . 2905 1

719 X 10 0. 005770 53 .0 . 2898 1723 X 10 0. 005786 52. 5

0.005B-R7370 0.2572 1. 941 X

19 10 0.00510 63. 1 0.2550 1. 950 X

19

10 0.005145 62.2

O. 0043B-23-1 0. 2030 X 1

0. 00424 59 .0 . 2047 2. X 10. 00425 60 . 1 0.0030B-45-1 0. 1305 3. 826 X

19 10^^ 0.002851 57.2 0. 1304 3. 829 X19 10^^ 0.002866 55.9

BO .001 9B— 2 0 . 0826. 080 X 10^^ 0.00 1849 55. 5 0. 0825 6. 045 X 10^^ 0.00 1862 55. 5

n n n 1 or— 7

0 • 0750 6660 X 10^^ 55 . 1 0 . 0743 5

720 X 10^^ 0u . 001719 U U 1 / 1 ^ 54 .

0.0018B-73-1C^ 0.0751 6. 650 X 10^^ 0.001712 54.8 0. 0752 5. 640 X 10^^ 0.001723 54.5

Avg. 0.0750 6. 655 X 10^^ 0.001707 55.0 0. 0747 6. 680 X 10^^ 0.001718 54.4

0.0010B-24-1 0.0449 ^• 1 12 X 10^° 0.001047 53.6 0. 0445 ^* 120 X 10^° 0.001054 52.9

0.0010B-74-1 0.0446 ^' 120 X 10^" 0.00 1036 53.8 0.0449 112 X

20

10 0.001043 53.8

( repeat

0.0447 117 X

10^° 0.001033 54. 1 0.0448 115 X

10^° 0.00 1040 53.8 0.0010B-74-1C^ 0.0446 120 X

10^° 0.001027 54.3 0.0444 125 X

10^° 0.001034 53.7

Avg. 0.0446 119 X 10^° 0.001032 54.

0.0447 117 X 10^° 0.001039 53.8

A0.00087B-1 0.0367 361 X 10^° 0.000851 53.9 0.0360 387 X

10^° 0.000857 52.5

•"C" denotes specimen in the shape of a Greek cross; all other specimens had shape shown in figure 3.

j

I

i

I

!

1

24 i
2 -

1 -

10

TRACK TECHNIQUE

A THURBER & CARPENTER (38)

HALL EFFECT

THIS WORK

CHAPMAN ET AL. (39)

A CROWDER (40)

J. C-V o THIS WORK

COMPUTER FIT

WAGNER

0.0208

0.0192

0.0176

0.0160

0.0144

0.0128

0.0112

0.0096

0.0080

0.0064

0.0048

0.0032

0.0016

0.0000

E

o

>-oo

o

O00

U3

C/) UJ

CD

10 10

-2 10-1 10 RESISTIVITY (n-cm)

10 10'

Figure 7. Resistivity-dopant density product as a function of resistivity at 300 K for

boron-doped silicon. The junction capacitance voltage and Hall effect measurements are compared with the published work of Thurber and Carpenter [38], Chapman et al . [39],

Crowder [40], Irvin [2], and Wagner [4]. The solid curve is an analytical fit to portions of these data as discussed in the text. Values of the product qpN are on the

right ordinate.

25
agreement with both the Hall effect and the C-V data. The curves shown are the Irvin relationship for p-type silicon, the Wagner relationship for

boron-implanted silicon, and an analytical curve fit to the data given in

tables 5, 5, and 7. As is evident from the figure, the results obtained

from measurements on boron-doped silicon depart considerably from the p-type

Irvin curve.

Figure 8 is a plot of hole mobility as a function of hole density at 300 K. The fairly recent data of Tsao and Sah [41] are significantly lower than the

data of this work. This may result from the fact that back-surface contact

resistance can strongly affect resistivity values obtained from the spreading

resistance structure used by Tsao and Sah. The Caughey and Thomas [1] curve

is a fit to the Irvin relationship with the assumption that the hole density

equals the dopant density. The same assumption was used to obtain the Wagner

curve; however, it was not a point-by-point determination but simply a change

in one parameter of the Caughey and Thomas expression as required to fit data

on boron-implanted silicon. Just as the Irvin curve is a poor fit to the

resistivity-dopant density data of this work, the Caughey and Thomas expression is a poor fit to the mobility data. The Wagner expression is considerably better, but there are still noticeable deviations with respect to the

experimental data. The theoretical calculation of Li [37] in figure 8, valid

for densities less than 3 x 10^^ cm"-^, is in excellent agreement with the C-V

data of this work.

7. COMPUTER CURVE FITS FOR PHOSPHORUS-DOPED SILICON

The curve fits of the data were done using the DATAPLOT language [42] for the

nonlinear least squares fitting. The simplest expression which gave a fit of

the same precision as the uncertainty in the experimental data was the log of

the resistivity-dopant density product equal to a third-degree polynominal

divided by a third-degree polynominal. Therefore, the equation for fitting

the resistivity-dopant density product was taken to have the form

2 3 Aq - A^X . A^X . A3X

^°^10 (^/^o^

=

2 T '

1 + B^X + B^X + B^X ( 1 )

with P = qpN where q is the electronic charge,* p is the resistivity, and N is

the electrically active dopant density. The normalization factor, Pq, was taken equal to 1 V«s/cm^. Fits were made for both X = log-|Q (p/pg) and X = log^Q (N/Nq) with Pq = 1 9,' cm and Nq =10^^ cm~^. With the use of the normalizing values, Pq and Nq, the magnitude of X is almost always less than 4.

This is desirable from a computational standpoint because the computation is

much less sensitive to the number of significant figures retained in the

coefficients.

The data used for the curve fitting consisted of 26 points obtained from the

C-V measurements, 12 points determined from the Hall effect measurements, 2 points from Esaki and Miyahara [43], and 1 point from Fair and Tsai [35].

The latter three points are listed in table 8 and were included to increase

* A value of 1.602 x 10"^^ C was used for q in all of the curve fits; consequently, this is the appropriate value to use for subsequent calculations.

26
21

HOLE DENSITY (cm-3)

Figure 8. Hole mobility as a function of calculated hole density for boron-doped

silicon at 300 K. All data points, except those in the heavily doped range, have

been corrected for incomplete ionization using the work of Li [37]. The CaugheyThomas [1] and Wagner [4] expressions were obtained without consideration of deionization and are plotted assuming equivalence of hole and dopant densities.

This gives lower mobility in the range 10^^ to 10^^ cm~3. The error bars on the

C-V data of this work show the standard deviation calculated from the individual

mobility values for that wafer. When not shown, the error bars lie within the

pflotted symbol.

Table 8. Miscellaneous Data for the Con5)uter Curve Fits of Phosphorus-Doped Silicon.

N P,, P,„„ n 23°C 300 K

23''C 300 K ^ ^ ^ -3 —3 2 2 cm n-cm fi'cm cm cm /V*s cm /V'

s

1 experimental extrapolation 1 X 10^^ 4276.0 4413.0 1 X 10^2 1460 1415

2 NAA/photometric 1.25 X 10^^ 0.020 0.020 1. 13 X 10^^ 276 276

3 NAA/photometric 2.57 X 10^« 0.014 0.014 2. 23 X 10^^ 200 200

4 NAA/photometric 4.70 X 10^« 0.010 0.010 4. 70 X 10^^ 133 133

5 NAA/photometric 7.75 X 10^^ 0.0073 0.0073 7. 75 X 10^^ 110 110

6 Esaki & Miyahara [43] 1.2 X 10^°^ 0.00067 0.00067 1. 2 X

10^° 78 78

7 Esaki & Miyahara [43] 1.6 X 10^°^ 0.00055 0.00055 1. 6 X 10^° 71 71

8 Fair & Tsai [35] 3 X 10^°^ 0.000298 0.000298 3 X 10^° * *

^Electrically active dopant density assumed equal to electron density. Not included in mobility fit.

27
the range of the fit at high densities beyond that justified by the data of

this work alone. In the region of the gap between the C-V and Hall effect

data (phosphorus densities between 10^^ and 10^^ cm~^), four resistivitydopant density pairs were generated for the data base to be used in the curve

fitting. These pairs, listed in table 8, were based on the NAA and photometric results, but actual values were chosen to give a smooth curve between the

C-V and Hall effect data. In addition, a point was added at N = 1 x 10^2

cm"-^ to improve the fit for calculating values in the low dopant density

range. This point was taken equal to the average of the four lowest dopant

density data points since the pN product is essentially constant in this

region. Also, the inclusion of this point helps insure that the fits as a function of log-|o (p/pg) and log^g (N/Ng) both give essentially the same calculated values in this region. The coefficients for the fit of the data to

eq ( 1 ) are given in table 9 for 23°C and 300 K. The table gives the approximate standard deviation for each coefficient and the residual standard deviation (R.S.D. ) in units of volts-second per square centimeter. The R.S.D. is

the square root of the quotient of the sum of the squared residuals,* divided

by the number of degrees of freedom.

The plot of the data used for the curve fit, with the exception of the point

at 10 '-^ cm ^ , and the least squares fit of the qpN product Versus N for 300 K

are shown in figure 9. This figure clearly shows that the "min-max" expressions of Caughey and Thomas [1] are not suitable for these data as there is no plateau or maximum at high dopant densities. The curve obtained for the

product as a function of resistivity is plotted in figure 10.

The two expressions, one a function of log-^Q (p/pg) and the other a function of log-|g (N/Ng), have a worst case self -consistency of 8 percent for all

values of N (and corresponding values of P) within the range 1 x 10 to

4 X 10^*^ cm~-^. That is, when the product qpN is calculated for a given N,

giving a p which is used in the qpN Versus p expression to again calculate

the product, the latter product will be within 8 percent of the former. The

maximum difference occurs only in the low 10^^ cm~^ range; elsewhere the fits are consistent within 4 percent. When the self-consistency is calculated

starting with a given p, the fits have a maximum difference of 4 percent for

resistivities from 2 x 10"^ to 3 x 10"^ fi*cm. The self-consistency is shown

(solid curves) as a function of dopant density in figure 11 and as a function

of resistivity in figure 12.

Curve fits were also obtained for electron mobility as a function of both

electron density, n, and resistivity. It was found that better and more

self-consistent fits were possible if the point at 3 x 10^*-* cm~-^ were

omitted. Otherwise the data used for the mobility fits corresponded to that

used for the product fits. As mentioned previously, the percent ionization

calculations of Li and Thurber [30] were used to obtain the electron density

from the measured dopant density prior to the computation of mobility. The

expression for calculating the electron mobility is of the form

2 3 A + AX + AX + AX

log,„ = . (2)

1 + B,X + B^X + B^X

1 2 3

* The difference between the measured point and the predicted value from the curve fit.

28
Table 9. Coefficients and Residual Standard Deviation (R.S.D. ) for the Fit of the Product qpN for Phosphorus-Doped Silicon Using Eq (1).

Ten^). 23'>C 23°C 300 K 300 K

X log^o (P/Pq) log^Q (N/Ng) log^g (p/p^) log^^ (N/N^)

A -3

1 083 0 .0038 -3, 0769 + 0 . 0027 -3

0951 + 0.0037 -3 . 0652

*i

-3. 2626 + 0.0952 2. 2108 + 0.0392 -3. 2303 + 0.0909 2. 1853 + 0.0415

*2 -1

2196 + 0.0341 -0. 62272 + 0.0159 -1. 2024 + 0.0325 -0.61080 + 0.0170

*3 -0. 13923 + 0.00468 0. 057501 + 0.00287 -0. 13679 + 0.00466 0.056189 + 0.00327

^1 1. 0265 + 0.0318 -0. 68157 + 0.0134 1. 0205 + 0.0305 -0.67642 + 0.0141

0. 38755 + 0.0109 0. 19833 + 0.00507 0. 38382 + 0.0105 0, 19542 + 0.00541 ^

0. 041833 + 0.00168 -0. 018376 + 0.000986 0. 041338 + 0.00 169 -0.018100 + 0.001 12

S.D. 2.78 X 10-^ 2.21 X 10-^ 2.67 X 10-^ 2. 09 X 10-^

0.015

0.010

^ 0.005

0.000

1 1 1 1 1

1

1 / • / o This Work, J.C-V A This Work, Chemical a This Work, Hall Effect • Esaki & Miyahara [43J Fair & Tsai (35)

cMD-o—o oa>o—crMXJO-oo-o--^^*''^^

1 1 1 1 1 1 1 1013 1014 1015 1016 1017 1018

DOPANT DENSITY (cm-3]

1019 1020 1021

Figure 9. The qpu product as a function of dopant density for phosphorus-doped silicon

at 300 K. The curve is the least squares fit to the data points shown using eq (1).

Coefficients are given in table 9- 29
0.000 I \ - L- \ \ I \ I

lO"^ 10"^ 10'^ 10"^ 10° 10^ 10^ 10^

RESISTIVITY (a - cm)

Figure 10. The qpN product as a function of resistivity for phosphorus-doped silicon

at 300 K. Equation (1) was used as the form of the curve fit to the data points

shown. Coefficients are given in table 9.

30
6.0

DOPANT/ELECTRON DENSITY (cm ^)

Figure 11. Self-consistency of the product and mobility fits at 23°C for phosphorusdoped silicon plotted versus dopant density for the product fits and versus electron

density for the mobility fits. The procedure for calculating the difference is discussed in the text.

31
32
where y is the mobility and ]Jq = 1 cm^/Vs. (In this equation, the A's

and B's represent coefficients to be determined; they are not constants from

previous fits.) As before, fits were made for both X = lo^io (p/Pq^ ^"'^ X = log^Q (n/ng) for temperatures of 23°C and 300 K. Values for the coefficients are given in table 10. The plot of the mobility data and fitted

curve as a function of resistivity for 300 K is given in figure 13. The fit

of mobility Vevsus electron density is shown in figures 5 and 6 for comparison with experimental data. An expression of the form suggested by Masetti

and Solmi [36] was also tried for the mobility fits, but the percent residuals in the low mobility range were significantly larger than with eq (2),

particularly for the fit as a function of resistivity.

The self-consistency of the mobility fits, plotted as dashed curves in

figures 11 and 12, shows a maximum difference of 5 percent for electron

densities in the range 10^^ to 10^^ cm~^. The expressions rapidly diverge

for densities greater than 2 x 10^*^ cm"-^. The fit as a function of resistivity is the better choice for approximating mobility values beyond the range

of the fitted data. When the self-consistency is calculated starting with a given resistivity, the fits have a maximum difference of 3 percent for

resistivities from 6 x 10"'* to 10*^ fl«cm. The self-consistency graphs shown

are for 23°C; calculations for 300 K closely parallel those for 23°C.

8. COMPUTER CURVE FITS FOR BORON-DOPED SILICON

The previous expressions for the fits to phosphorus- doped silicon are in the

form of quotients of two third-degree polynomials in order to fit the shape

of the product and mobility curves at high phosphorus densities. However,

for boron-doped silicon, the product is well behaved at very high densities,

such that "min-max" relationships similar to those proposed by Caughey and

Thomas [1] give satisfactory fits. Modifications are necessary for the fits

of mobility as will be discussed later. These relationships have the advantage that two of the parameters, (qpN)niin ^^"^ ^*3P^^max' have a readily identified physical significance in that they approximately equal the true limiting value of these quantities.

The expression for the product qpN as a function of p was taken as (qpN) -(qpN) . max mm , ^

qpN = (qpN) . + 7 -7^ , (3) min

' fc)

where q is the electronic charge, N is the boron density, p is the resistivity, (qpN)jj^in ^^P^^max asymptotic minimum and maximum values

of the product, P^ef ^ reference value corresponding to the resistivity

for which the derivative of the product is a maximum, and a is a measure of

the slope of the curve near the reference value. The data used for the fit,

given in tables 5, 6, and 7, were the junction C-V and Hall effect results of

this work, and selected NTT results for boron densities greater than 5 x 10^^ cm~^. The NTT data at lower densities were omitted because the data,

while in general agreement with the C-V results, showed significantly more

scatter. For high boron densities, the NTT results may be better than those

from the Hall effect because of the need to assume a value for the Hall

33
Table 10. Coefficients and Residual Standard Deviation (R.S.D.) for the Fit of Electron Mobility for Phosphorus -Doped Silicon Using Eq (2).

Temp. 23°C 23''C 300 K 300 K

X log^o (p/pg) log^g (n/n^) log^^ (p/p^) log^^^ (n/n^)

3. 1 122 + 0.0034 3.0746 + 0.0025 3.0985 + 0.0037 3.0629 + 0.0025

3. 3347 + 0.0951 -2.2679 + 0.0076 3.3257 + 0. 0923 -2.2522 + 0.0077

1. 2610 + 0.0510.62998 + 0.00245 1.2581 + 0.0481 0.62327 + 0.00249

0. 15701 + 0.0198 -0.061285 + 0.00087 0. 15679 + 0.0189 -0.060415 + 0.00087

1. 0463 + 0.0324 -0.70017 + 0.00290 1.0485 + 0.0317 -0.69851 + 0.00292

0. 39941 + 0.0167 0. 19839 + 0.001 13 0.40020 + 0.0156 0. 19716 + 0.00114

0. 049746 + 0.00547 -0.020150 + 0.00041 0.049883 + 0.00512 -0.019950 + 0.00041

R.S.D. 14.9 11 .6 14. 8 11 .2

1500

CME

u

1000

10-1 10°

RESISTIVITY Ifi-cm)

Figure 13. Electron mobility versus resistivity at 300 K for phosphorus-doped silicon. The curve is the least squares fit to the points using eq (2) with coefficients

given in table 10.

34
factor in the latter case. This same set of 300 K data, and its counterpart

at 23°C, was used for all of the curve fits. The resistivity values of the

39 fitted data points spanned the range 0.00085 to 100 Q-cm with corresponding dopant densities between 1.4 x 10^^ and 1.3 x 10^^ cm"^. The leastsquares curve at 300 K is shown in figure 8 for comparison with experimental

data and other curves. All four parameters as determined by the fit are giv- en in table 11 for both 23°C and 300 K. The approximate standard deviation

is given for each parameter in its own units, and the residual standard deviation (R.S.D.) for the fit is listed in units of qpN.

The expression for the product qpN as a function of dopant density has the same form as eq (3) except for a negative value of the exponent because of

the inverse relationship between p and N:

(qpN) - (qpN) . max min

qPN = (qPN) . + 7— r-^ . (4) min

(In this equation (qPN)j^in' ^^P^^max' ^ref' ^^'^ ^ represent the parameters

to be determined; they are "not constants from previous fits.) Values for

the four parameters are listed in table 12 for both 23° C and 300 K. The curve for 300 K and the experimental data used to obtain it are shown in

figure 14.

In comparing the product fits as a function of resistivity with those as a function of dopant density, it is evident that minimum and maximum values

which should be correspondingly equal are somewhat different. This is, of

course, a consequence of the independent fits where no parameters are fixed.

The (qpN)j^j.i^n values differ by 2 percent and the fits differ by this same percentage at N = 10 cm~-^. The (qpN)^^^^ values differ by 4 percent; however, these are asymptotic values, and at the upper limit of the experimental

data (NnjlO^'^ cm"-^) there is still considerable curvature. The difference

between the fits in this region is less than 1 percent. The product fits are

self -consistent within 3 percent for all dopant densities (and corresponding

resistivities) in the range 10 to 10^*^ cm~^; see the solid curves in fig- ures 15 and 16. That is, when the product qpN is calculated for a given dopant density with eq (4), a resistivity is derived which can be used in the

qpN versus p expression [eq (3)] to again calculate the product. The latter

product will be within 3 percent of the former.

Hole mobilities were calculated from resistivity and hole density values.

Except in the heavily doped range, where hole density is assumed to equal

dopant density, hole densities were obtained from dopant densities using the

percent ionization calculations of Li [37] . A suitable expression for the

hole mobility, y^, as a function of hole density, p, was found to be

P y = A exp(-p /p) + 7 r^— • (=>) h ^ ^c'

This form gave a significantly better fit than the simpler form of the type

used for fitting the product curves. The effect of the exponential factor in

35
Table 11 — parameters and Residual Standard Deviation (R.S.D.) for the

Fit of qpN versus p for Boron-Doped Silicon Using Eq (3).

Temperature 23° C 300 K

2

(qpN) ^ . (V's/cm ) min 0. 00213 + 0.00009 0.00220 + 0. 00009

2 (qpN) (V's/cm ) max

0. 01947 + 0.000 18 0.01973 + 0.00017

p ^(fi'cm) ref

0. 0 1833 + 0.00067 0. 0 1782 + 0.00060

a 1. 105 + 0.035 1.086 + 0.032

2 R.S.D. (V«s/cm ) 0. 000292 0.000266

Table 12 — parameters and Residual Standard Deviation (R.S.D.) for the Fit

of qpN versus N for Boron-Doped Silicon Using Eq (4).

Temperature 23°C 300 K

(qpN) . (V«s/cm^) 0.00209 + 0.00009 0.00215 + 0,00008 mm — —

(qpN) (V«s/cm^) 0.02024 + 0.00024 0.02053 + 0.00022 max — —

N ^ (cm ) 3.88 x 10 + 0.21 x 10 4.09 x 10 + 0.20 x 10

ref — —

a -0.737 + 0.024 -0.727 + 0.021

2 R.S.D. (V«s/cm ) 0.000268 0.000240

36
0.020

^ 0.015

>-

</5

OSo

X

>-

02

00

LUOSX

UJC5

OS

<xo

0.010

0.005 -

0.000

o THIS WORK, J.C-V THIS WORK, HALL EFFECT

A THURBER & CARPENTER, NTT (38)

10

14 10

15 1016 .17 18 10" 10

BORON DENSITY (cm-^

10

19 10

20 10

21

Figure 14. The qpU product as a function of boron density for 300 K. The curve is an analytical fit to the data points shown using eq (4). Parameter values are given in

table 12.

37
6.0

DOPANT /HOLE DENSITY (cm"^)

Figure 15. Self-consistency of the product and mobility fits at 23°C for boron-doped

silicon plotted versus dopant density for the product fits and versus hole density for

the mobility fits. The procedure for calculating the difference is discussed in the

text.

4.0 I

\ 1 1 \ \ r

-8.0 - I /

-10.0 - /

-12.0 I ^ I \ \ \ 1 \

10 * 10"^ 10"^ 10 ' 10° 10^ 10^ 10^

RESISTIVITY ( n • cm)

Figure 16. Self-consistency of the product and mobility fits at 23°C for boron-doped

silicon plotted versus resistivity. The procedure for calculating the difference is

discussed in the text.

38
the first term is to increase the mobility in the region of the curve where

the dopant and hole densities deviate significantly from each other. The

hole mobility approaches the constant value A for very large hole densities.

The five parameters, determined by the fit to the 39 data points, are listed

in table 13 for 23°C and 300 K. The curve for 300 K, calculated using the

parameters in table 13, is shown in figure 8 for comparison with experimental

data and other curves

.

During the course of fitting the hole mobility to eq (5), it was found that

convergence was obtained for two different values of p^. Further work con- firmed that there were two valleys, or local minima, for the variation of the

R.S.D. with p^, and the one located by the fitting procedure depended on the starting value for p^,. At 23°C one fit had p^ = 9.00 x 10 "l^, A = 45.2,

yjnax = 486. 1 , Pj-ef ^ 2. 11 x lo'''^, a = 0.718, and R.S.D. = 3.18; the other

had p^ = 1.09 X lo""^, A = 41.4, ii^^y. = 492.5, Pj-^f = 3.06 x lo'''^' a = 0.589,

and R.S.D. = 2.84. The former fit was selected in spite of its higher R.S.D.

because it gave better self -consistency with the fit as a function of resistivity.

The form for the hole mobility as a function of resistivity was taken as

It was also found that the simple "min-max" form did not fit the mobility

versus resistivity data well, particularly in the low resistivity range. The

inclusion of the exponential in the first term improved the fit considerably.

Its effect is similar to that of the exponential in eq (5). In this expression, the mobility is dominated by the first term at low resistivities and by

the second term at high resistivities. The parameters determined by the fit

are listed in table 14 for both 23°C and 300 K. The curve for 300 K and the

experimental data used for the fit are shown in figure 17. The curve fits

the shape of the experimental data very well at low resistivities so that ex- trapolation of eq (6), using the parameters in table 14, at the low resistivity end should give reasonable values. Thus, A can be considered to be approximately equal to the minimum value of the mobility. The mobility Versus

hole density fit is not as good at the low mobility end, as it is slightly

below the data at p = lO^'^ cm"-^ and goes to a lower asymptotic value than is

the case for eq (6). Consequently, an extrapolation of eq (5) at the low

mobility end is less likely to be correct.

The Uj^ax values for the mobility fits as a function of hole density differ

by about 0.7 percent from the y^iax values for the fits as a function of re- sistivity. The mobility fits are self-consistent within 2 percent for all

hole densities (and corresponding resistivities) in the range 10^^ to 5 x 10^

cm"-^; see the dashed curves in figures 15 and 16. In the 10^^ cm"-^ range,

there are differences of up to 8 percent. The self-consistency improves in

the low 10^^ cm"-^ range but increases again to 6 percent at 10^*^ cm"-^ as the fits depart for very heavy doping.

The results of this work are incorporated in the ASTM document F 723 entitled

"Practice for the Conversion Between Resistivity and Dopant Density for

y = A exp(-p/p ) + max

(6)

39
Table 13 — parameters and Residual Standard Deviation (R.S.D. ) for the

Fit of versus p Using Eq (5).

Temperature 23 °C 300 K

2 A (cm /V's) 45.2 + 1.3 44.9 + 1.2

p (cm~^) 9.00 X 10^^ + 2.3 x 10^^ 9.23 x 10^^ + 2.3 x 10^^

c — —

2

y (cm /V'S) 486.1 + 1.9 470.5 + 1.8 max — —

p ^ (cm""^) 2.11 X 10^^ + 0.09 x 10^^ 2.23 x

10^^ + 0.10 x

10^^

ref — —

a 0.718 + 0.013 0.719 + 0.013

2 R.S.D. (cm /V's) 3.18 3.07

Table 14 — parameters and Residual Standard Deviation (R.S.D.) for the

Fit of Vyi I^&^SUS P Using Eq (6).

Temperature 23 °C 300 K

2

A (cm /V*s) 52.4 + 3.8 51.6 + 3.7

p ( 9. 'cm) c 0.00409 + 0.00077 0.00406 + 0.00077

2 u (cm /V*s) max

482.8 + 2.4 467.3 + 2.4

p _ (fi'cm) ref

0.0825 + 0.0020 0.0794 + 0.0019

a -0.811 + 0.018 -0.808 + 0.019

2

R.S.D. (cm /V«s) 4.89 4.78

40
lO '* 10 '^ 10 ^ 10"' 10° 10 ' 10^

RESISTIVITY (n cm)

Figure 17. Hole mobility versus resistivity at 300 K for boron-doped silicon. The curve is an analytical fit to the data points shown using eq (6). Parameter values

are given in table 14.

41
CONVERSION BETWEEN

RESISTIVITY AND DOPANT DENSITY

Figure 18. Dopant density versus resistivity at 23°C for silicon doped with phosphorus and with boron. The solid line shows the dopant density calculated from the equation of the product as a function of resistivity for the region of actual data. The

chaindot line shows the resistivity calculated from the equation of the product as a function of dopant density for the region of actual data. Dashed lines show the ex- trapolation beyond the region covered by the data. On the scale of the figure shown

here, the solid and chaindot lines are indistinguishable. However, in regions where

the self-consistency error is appreciable, the lines are distinguishable on the wall

chart available from ASTM. (Graph is from ASTM document F 723 cited in the text.)

42
Boron- and Phosphorus-Doped Silicon" which has been approved by Committee F

on Electronics and will appear in the 1981 Annual Book of ASTM Standards,

Part 43. This document provides three procedures for converting between re- sistivity and dopant density at 23°C. They are: (1) graphical method - a graph, reproduced as figure 18 of this report, with curves for both boron and

phosphorus; (2) tabular method - a set of four tables covering the two dopants as functions of both resistivity and dopant density; and (3) computational method - the same equations and coefficients as given in this report

for use when exact calculations are most convenient or necessary. The document does not cover electron or hole mobility relationships.

9 . CONCLUSIONS

For phosphorus- doped silicon, the relationship between resistivity and dopant

density was determined for densities between 10^-^ and 10^'-' cm~-^. The results

differ from the n-type Irvin curve by 5 to 15 percent, always in the direction of lower resistivity for a given dopant density. The work is in agreement with that of Mousty et at. in the 10^^ to 10^^ cm"^ range, but differences exist for lower and for higher doping levels.

For boron-doped silicon resistivity and dopant density data were obtained for

boron densities between 10 and 10^*^ cm~-^. The results differ significantly

from the commonly used Irvin curve for densities greater than 10^^ cm~^ with

a maximum deviation of 45 percent at 5 x 10^^ cm"-^ in the direction of lower

resistivity for a given boron density. Hole mobility values, calculated from

this work with correction for percent ionization of the dopant atoms, are much larger than the experimental data of Tsao and Sah, but are in good

agreement with the theoretical calculations of Li. The hole mobility values

are in reasonable agreement with Wagner's work on boron-implanted silicon.

The maximum discrepancy occurs in the 10^^ cm~-^ range where the Wagner mobility is 10 percent lower because deionization effects were not included.

Analytical expressions, convenient for engineering calculations, were fitted

to the resistivity-dopant density data as a function of resistivity and dopant density for temperatures of both 23°C and 300 K. These fits are selfconsistent within a few percent so that conversion can go both directions

without iteration. Similar analytical expressions were fitted to the calculated hole mobility as a function of resistivity and hole density.

ACKNOWLEDGMENTS

The authors are indebted to D. R. Ricks for the mechanical four-probe

resistivity measurements, to F. H. Brewer for the Hall effect measurements,

and to J. Krawczyk and L. A. Robinson for specimen preparation. Special

thanks are due M. G. Buehler for designing the test structures used in this

work and R. D. Larrabee and W. M. Bullis for helpful comments on the

manuscript. The typing of the report with its many tables was expertly done

by Jane Walters and Jo Halapatz.

43
REFERENCES

1. Caughey, D. M. , and Thomas, R. E., Carrier Mobilities in Silicon Empiri

cally Related to Doping and Field, Pvoe. IEEE _55, 2192-2193 ( 1967),

2. Irvin, J. C. , Resistivity of Bulk Silicon and of Diffused Layers in

Silicon, Bell System Teeh. J. 4_1, 387-410 ( 1962).

3. Baccarani, G. , and Ostoja, P., Electron Mobility Empirically Related to

the Phosphorus Concentration in Silicon, Solid-State Electron. 18,

579-580 (1975).

4. Wagner, S., Diffusion of Boron from Shallow Ion Implants in Silicon, J»

Eleotroohem. Soo, 119, 1570-1576 (1972).

5. Thurber, W. R. , Mattis, R. L. , Liu, Y. M. , and Filliben, J. J.,

Resistivity-Dopant Density Relationship for Phosphorus-Doped Silicon,

J, Eleatroahem. Soo. 127, 1807-1812 (1980).

6. Thurber, W. R. , Mattis, R. L. , Liu, Y. M. and Filliben, J. J.,

Resistivity-Dopant Density Relationship for Boron-Doped Silicon, J.

Eleotroohem. Soo. 127, 2291-2294 (1980).

7. Standard Test Method for Measuring Radial Resistivity Variation on Sili con Slices, ASTM Method F 81, Annual Book of ASTM Standards, Part 43

(November 1980).

8. Buehler, M. G. , and Thurber, W. R. , A Planar Four-Probe Test Structure;

for Measuring Bulk Resistivity, IEEE Trans. Electron Devices ED-23 , 968-974 (1976).

9. Test Method for Net Carrier Density in Silicon Epitaxial Layers by

Voltage-Capacitance of Gated and Ungated Diodes, ASTM Method F 419,

Annual Book of ASTM Standards, Part 43 (November 1980).

10. Buehler, M. G. , Semiconductor Measurement Technology: Microelectronic

Test Pattern N3S-3 for Evaluating the Resistivity-Dopant Density Relationship of Silicon, NBS Spec. Publ. 400-22 (June 1976).

11. Thurber, W. R. , and Buehler, M. G. , Semiconductor Measurement Technology: Microelectronic Test Pattern NBS-4, NBS Spec. Publ. 400-32 (April

1978).

12. Leedy, T. F. , and Liu, Y. M. , Semiconductor Measurement Technology : Mi

croelectronic Processing Laboratory at NBS, NBS Spec. Publ. 400-53 (December 1978).

13. Standard Method for Measuring Resistivity of Silicon Slices with a Collinear Four-Probe Array, ASTM Method F 84, Annual Book of ASTM Standards, Part 43 (November 1980).

14. Grove, A. s.. Physics and Technology of Semiconductor Devices (John

Wiley and Sons, Inc., New York, 1967), pp. 278-285.

44
15. Gillilland, K. E. , Cook, H. D. , Mielenz, K. D. , and Stephens, R. B. , Use

of a Laser for Length Measurement by Fringe Counting, MetroZog-ia 2^, 95-98 (1966).

16. McDonald, B. , and Goetzberger, A., Measurement of the Depth of Diffused

Layers in Silicon by the Grooving Method, J. Eleetvoahem. Soo. 109 , 141-144 (1962).

17. Mattis, R. L. , and Buehler, M. G. , Semiaonduetov Measurement Teohnology

A BASIC Program for Calculating Dopant Density Profiles from

Capacitance-Voltage Data, NBS Spec. Publ. 400-11 (June 1975).

18. Mattis, R. L. , and Buehler, M. G. , A New Method for Calculating Background Dopant Density from p-n Junction Capacitance-Voltage Measurements, J. Eleatvoahem. Soo. 124, 1918-1923 (1977).

19. Dunlap, W. C. , Jr., and Watters, R. L. , Direct Measurement of the Dielectric Constants of Silicon and Germanium, Phys. ReV. 92., 1396-1397

( 1953)

20. Bullis, W. M. , Brewer, F. H. , Kolstad, C. D. , and Swartz en druber, L. J.,

Temperature Coefficient of Resistivity of Silicon and Germanium Near

Room Temperature, Solid-State Eleotvon. JM, 639-646 (1968).

21. van der Pauw, L. J., A Method of Measuring Specific Resistivity and Hall

Effect of Discs of Arbitrary Shape, Philips Res. Rep. jr3, 1-9 (1958).

22. David, J. M. , and Buehler, M. G. , A Numerical Analysis of Various Cross

Sheet Resistor Test Structures, Solid-State Eleotvon. 20, 539-543

( 1977)

23. Measuring Hall Mobility and Hall Coefficient in Extrinsic Semiconductor

Single Crystals, ASTM Method F 76, Annual Book of ASTM Standards, Part

43 (November 1980).

24. Beer, A. C. , Galvanomagnetic Effects in Semiconductors, Solid State

Phys.y F. Seitz and D. Turnbull, Eds., Suppl. 4, pp. 99-266 (Academic

Press, New York, 1963).

25. Morin, F. J., and Malta, J. P., Electrical Properties of Silicon Containing Arsenic and Boron, Phys. Rev. 96^, 28-35 (1954) .

26. Wolfstirn, K. B. , Hole and Electron Mobilities in Doped Silicon from

Radiochemical and Conductivity Measurements, J. Phys. Chem. Solids 16 , 279-284 (1960).

27. Schroder, D. K. , Braggins, T. T. , and Hobgood, H. M. , The Doping Concentrations of Indium-Doped Silicon Measured by Hall, C-V and JunctionBreakdown Techniques, J. Appl. Phys. 49, 5256-5259 (1978).

28. Linares, L. C. , The Mobility, Resistivity and Carrier Density in p-Type.

Silicon Doped with Boron, Gallium and Indium, Ph.D. Dissertation, University of Florida, 1979.

45
29. Nakagawa, N. , and Zukotynski, S. , Drift Mobility and Hall Coefficient

Factor of Holes in Germanium and Silicon, Can. J. Phys. 56, 364-372

(1978); and Lin, J. F. , Li, S. S. , Linares, L. C. , and Teng, K. W.

,

Theoretical Analysis of Hall Factor and Hall Mobility in p-Type Silicon

(to be published in Sotid-State Eleotvon., 1981).

30. Li, S. S., and Thurber, W. R. , The Dopant Density and Temperature Depen

dence of Electron Mobility and Resistivity in n-Type Silicon, Sol'idState Eleotvon. 20, 609-616 (1977).

3 1. Thurber, W. R. , Comparison of Measurement Techniques for Determining

Phosphorus Densities in Semiconductor Silicon, t/ . Eleotvon. Matev^als 9

551-560 (1980).

32. Mousty, F. , Ostoja, P., and Passari, L. , Relationship Between Resistivity and Phosphorus Concentration in Silicon, J. Appl. Phys. 45_, 4576-

4580 (1974).

33. Fair, R. B. , Analysis of Phosphorus-Diffused Layers in Silicon, J. Eleo

tVOOhem. SOO. 125, 323-327 (1978).

34. Plunkett, J. C. , Stone, J. L. , and Leu, A., A Computer Algorithm for Ac

curate and Repeatable Profile Analysis Using Anodization and Stripping

of Silicon, Solid-State Eleotvon. 20, 447-453 (1977).

35. Fair, R. B., and Tsai, J. C. C, A Quantitative Model for the Diffusion

of Phosphorus in Silicon and the Emitter Dip Effect, J. Eleotvoohem.

Soo. 124, 1107-1118 (1977).

36. Masetti, G. , and Solmi, S., Relationship between Carrier Mobility and

Electron Concentration in Silicon Heavily Doped with Phosphorus, Sol'tdState and Eleotvon. Devioes 3_f 65-68 (1979).

37. Li, S. S., The Dopant Density and Temperature Dependence of Hole Mobility and Resistivity in Boron Doped Silicon, Solid-State Eleotvon. 21 , 1109-1117 (1978). See also Li, S. S., Semioonduotov Measuvement Teohnology: The Theoretical and Experimental Study of the Temperature and

Dopant Density Dependence of Hole Mobility, Effective Mass, and Resistivity in Boron-Doped Silicon, NBS Spec. Publ. 400-47 (November 1979);

and Linares, L. C. , and Li, S. S., An Improved Model for Analyzing Hole

Mobility and Resistivity in p-Type Silicon Doped with Boron, Gallium,

and Indium, J. EleotVOohem. Soo. 128, 601-608 (1981).

33. Thurber, W. R. , and Carpenter, B. S., Boron Determination in Silicon by

the Nuclear Track Technique, J. Eleotvoohem. Soo. 125 , 654-657 (1978).

39. Chapman, P. W. , Tufte, O. N. , Zook, J. D. , and Long, D. , Electrical

Properties of Heavily Doped Silicon, J. Appl. Phys. 34, 3291-3295

( 1963)

40. Crowder, B. L. , The Influence of the Amorphous Phase on Ion Distributions and Annealing Behavior of Group III and Group V Ions Implanted

into Silicon, J. Eleotvoohem. Soo. 1 18, 943-952 (1971).

46
41. Tsao, K. Y. , and Sah, C. T. , Temperature Dependence of Resistivity and

Hole Conductivity Mobility in p-Type Silicon, Solid-State Electron. 19,

949-953 (1976). See also Woodley, T. J., and Sah, C. T., Deionization

Effect on the Evaluation of Hole Mobility in p-Si, Solid-State Eleetvon,

20, 385-388 (1977).

42. Filliben, J. J., DATAPLOT - Conunand Manual: ... Fit, NBS Statistical

Engineering Division Working Paper 80-1, 1980 (formal NBS publication to

follow)

43. Esaki, L. , and Miyahara, Y. , A New Device Using the Tunneling Process in

Narrow p-n Junctions, Solid-State Eleetvon. j_/ 13-21 ( 1960).

44. Omel'yanovskii, E. M. , Fistul', V. I., and Mil'vidskii, M. G. , Electron

Mobility in Heavily Doped Silicon, SOV. Phys. - Solid State 5, 676-680

( 1963)

45. Yamanouchi, C. , Mizuguchi, K. , and Sasaki, W. , Electric Conduction in

Phosphorus Doped Silicon at Low Temperatures, Jap, J. Phys, Soo. 22 , 859-854 (1967).

45. Logan, R. A., Gilbert, J. F. , and Trumbore, F. A., Electron Mobilities

and Tunneling Currents in Silicon, J. Appl. Phys, 32, 131-132 (1961).

47
NBS-114A (REV. 2-8C) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO. NBS SP 400-64

2. Performing Organ. Report No. 3. Publication Date May 1981

4. TITLE AND SUBTITLE

Semioonductov Measurement Technology : The Relationship Between Resistivity and

Dopant Density for Phosphorus- and Boron-Doped Silicon

5. AUTHOR(S)

W. R. Thurber, R. L. Mattis, Y. M. Liu, and J. J. Filliben

6. PERFORMING ORGANIZATION f/f joint or other than NBS. see instructions) 7. Contract/Grant No. ARPA Order No. 2 397

NATIONAL BUREAU OF STANDARDS Proaram Code 7Y10

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered WASHINGTON, D.C. 20234 Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City. State. Z/P)

Defense Advanced Research Projects Agency National Bureau of Standards

1400 Wilson Boulevard and Washington, DC 20234

Arlington, VA 22209

10. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number 81-600052,

Conducted as part of the NBS Semiconductor Technology Program.

Document describes a computer program; SF-185, FIPS Software Summary, is attached.

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

bi bl iography or literature survey, mention it here) New data have been obtained for the resistivity-dopant density relationship for sili- con doped with phosphorus or boron for dopant densities in the range lO"*-^ to 10^0 cm" 3. For dopant densities less than lO^S cm~3, results were calculated from resistivity

and junction capacitance- voltage measurements on processed wafers. For more heavily

doped material, data were obtained from Hall effect and resistivity measurements on specimens cut from bulk silicon slices. These primary methods were supplemented for

phosphorus- doped material by neutron activation analysis and a photometric technique

and for boron-doped material by the nuclear track technique. For phosphorus-doped

silicon the results of this work differ by 5 to 15 percent from the commonly used

Irvin curve, always in the direction of lower resistivity for a given dopant density.

For boron-doped silicon the results differ significantly from the p-type Irvin curve

for boron densities greater than 10 16 a 3 with a maximum deviation of 45 percent at

5 X 10-'-^ cm~3 in the direction of lower resistivity for a given dopant density. Hole

mobility values derived from the data are in reasonable agreement with the Wagner

expression with a maximum discrepancy of 10 percent in the 10-'-^ cm~-^ range. Analytical curves were fitted to the resistivity-dopant density product as a function of

resistivity and dopant density for temperatures of 23°C and 300 K. Similar curves were obtained for the calculated carrier mobility as a function of resistivity and

carrier density.

12. KEY WORDS (Six to twelve entries; alphabeti cal order; capitalize only proper names; and separate key words by semicolons)

Boron; capacitance- voltage technique; dopant density; electron mobility; Hall effect;

hole mobility; Irvin curves; phosphorus; resistivity; semiconductor; silicon.

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