ASTM-F723

Designation: F 723 – 99

Standard Practice forConversion Between Resistivity and Dopant Density forBoron-Doped, Phosphorus-Doped, and Arsenic-DopedSilicon 1

This standard is issued under the fixed designation F 723; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (e) indicates an editorial change since the last revision or reapproval.

INTRODUCTION

The ability to convert between resistivity and dopant density in a semiconductor is important for avariety of applications ranging from material inspection and acceptance to process and devicemodeling. Despite some experimental limitations, the conversion is more readily established from anempirical data base than from theoretical calculations. Resistivity may be unambiguously determinedthroughout the desired resistivity range regardless of the dopant impurity. However, it was necessaryto use a variety of techniques to establish the complete dopant density scale of interest; thesetechniques do not all respond to the same parameter of the semiconductor.

In the experimental work(1), (2), (3)2 supporting these conversions, capacitance-voltage measure-ments were used to determine the dopant density of both boron- and phosphorous-doped specimenswith dopant densities less than about 1018 cm−3. The specimens were assumed to be negligiblycompensated; hence, the data given by the capacitance-voltage measurements were taken to be a directmeasure of the dopant density in the specimen. Hall effect measurements were used to obtain dopantdensity values greater than 1018 cm −3. In addition, in this range neutron activation analysis andspectrophotometric analysis were used to determine phosphorus density, and the nuclear tracktechnique was used to determine boron density. Where there were discrepancies in the data from theanalytical techniques, more weight was given to the Hall effect results. Up to the highest densitiesmeasured, boron is expected to be fully electrically active. Therefore, the boron densities of thesespecimens were assumed equal to the carrier densities obtained from the Hall effect measurementswith the use of an estimate for the Hall proportionality factor based on the best available experimentaland theoretical information. In the case of specimens heavily doped with phosphorus, the Hallproportionality factor is unity, but there is considerable evidence that at densities above about5 3 10 19 cm−3 not all of the phosphorus is electrically active because of the formation of complexes.In the absence of data regarding the fraction of phosphorus atoms withdrawn from electrically activestates due to complexing, the values of carrier density taken from the Hall effect measurements wereassumed to be equal to the phosphorus density values. Consequently, the conversions based on thesedata may understate the total phosphorus density for stated values above about 53 1019 cm −3.

1. Scope

1.1 This practice3 describes a conversion between dopantdensity and resistivity for arsenic-, boron- and phosphorus-

doped single crystal silicon at 23°C. The conversions are basedprimarily on the data of Thurber et al(1), (2), (3)3 taken onbulk single crystal silicon having dopant density values in therange from 33 1013 cm−3 to 13 1020 cm−3 for phosphorus-doped silicon and in the range from 1014 cm−3 to 13 1020 cm−3 for boron-doped silicon. The phosphorus data base wassupplemented in the following manner: two bulk specimen datapoints of Esaki and Miyahara(4) and one diffused specimendata point of Fair and Tsai(5) were used to extend the data baseabove 1020 cm−3, and an imaginary point was added at1012 cm−3 to improve the quality of the conversion for lowdopant density values. A conversion for arsenic, distint fromthat of phosphorus, is presented for the range 1019 to 6 by 1020

cm−3.

1 This practice is under the jurisdiction of ASTM Committee F-1 on Electronicsand is the direct responsibility of Subcommittee F01.06 on Silicon Materials andProcess Control.

Current edition approved June 10, 1999. Published August 1999. Originallypublished as F 723 – 81. Last previous edition F 723 – 97.

2 Boldface numbers in parentheses refer to the list of references at the end of thispractice.

3 DIN 50434 is an equivalent method. It is the responsibility of DIN CommitteeNMP 221, with which Comittee F-1 maintains close technical liaison. DIN 50444,Testing of Materials for Semiconductor Technology: Conversion Between Resistiv-ity and Dopant Density of Silicon, is available from Beuth Verlag GmbHBurggrafenstrasse 4-10, D-1000 Berlin 30, Federal Republic of Germany.

1

AMERICAN SOCIETY FOR TESTING AND MATERIALS100 Barr Harbor Dr., West Conshohocken, PA 19428

Reprinted from the Annual Book of ASTM Standards. Copyright ASTM

COPYRIGHT American Society for Testing and MaterialsLicensed by Information Handling ServicesCOPYRIGHT American Society for Testing and MaterialsLicensed by Information Handling Services

1.2 The self consistency of the conversion (resistivity todopant density and dopant density to resistivity) (see AppendixX1) is within 3 % for boron from 0.0001 to 10 000V·cm, (1012 to 1021 cm−3) and within 4.5 % for phosphorus from 0.0002to 4000V·cm (1012 to 53 10 20 cm−3). This error increasesrapidly if the phosphorus conversions are used for densitiesabove 53 1020 cm−3 .

1.3 These conversions are based upon boron and phospho-rus data. They may be extended to other dopants in silicon thathave similar activation energies; although the accuracy ofconversions for other dopants has not been established, it isexpected that the phosphorus data would be satisfactory for usewith arsenic and antimony, except when approaching solidsolubility (see 6.3).

1.4 These conversions are between resistivity and dopantdensity and should not be confused with conversions betweenresistivity and carrier density or with mobility relations.

NOTE 1—The commonly used conversion between resistivity anddopant density compiled by Irvin(6) is compared with this conversion inAppendix X2. In this compilation, Irvin used the term “impurity concen-tration” instead of the term “dopant density.”

1.5 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to establish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use.

2. Referenced Documents

2.1 ASTM Standards:F 84 Test Method for Measuring Resistivity of Silicon

Wafers with an In-Line Four-Point Probe4

2.2 ASTM Adjuncts:Large Wall Chart5

3. Terminology

3.1 Definitions:3.1.1 carrier density, n(electrons);p (holes)—the number

of majority carriers per unit volume in an extrinsic semicon-ductor, usually given in number/cm3 although the SI unit isnumber/m3.

3.1.2 compensation—reduction in number of free carriersresulting from the presence of impurities other than themajority dopant density impurity. Compensation occurs whenboth donor and acceptor dopant impurities are present in asemiconductor; in this case the net dopant density (which isequal to the carrier density provided that all the dopantimpurities are ionized) is given by the absolute magnitude ofthe difference between the acceptor dopant density and thedonor dopant density. Compensation may also occur if deep-level impurities or defects are present in quantities comparablewith the dopant impurities; in this case the relationshipbetween the carrier density and the dopant density (under theassumption of full ionization of the dopant impurity) depends

on a variety of parameters(7). A semiconductor that exhibitscompensation is said to be “compensated.”

3.1.3 concentration—relative amount of a minority con-stituent of a mixture to the majority constituent (for example,parts per million, parts per billion, or percent) by either volumeor weight. In the semiconductor literature, often used inter-changeably with number density (for example, number per unitvolume).

3.1.4 deep-level impurity—a chemical element that whenintroduced into a semiconductor has an energy level (or levels)that lies in the mid-range of the forbidden energy gap, betweenthose of the dopant impurity species.

3.1.4.1 Discussion—Certain crystal defects and complexesmay also introduce electrically active deep levels in thesemiconductor.

3.1.5 dopant density—in an uncompensated extrinsic semi-conductor, the number of dopant impurity atoms per unitvolume, usually given in atoms/cm3 although the SI unit isatoms/m3. Symbols: ND for donor impurities and NA foracceptor impurities.

4. Summary of Practice

4.1 The conversions between resistivity and dopant densityare made using equations, tables, or graphs.

5. Significance and Use

5.1 Dopant density and resistivity of silicon are two impor-tant acceptance parameters used in the interchange of materialby consumers and producers in the semiconductor industry.Therefore, a particular method of converting from dopantdensity to resistivity and vice versa must be available sincesome test methods measure resistivity while others measuredopant density.

5.2 These conversions are useful in mathematical modelingof semiconductor processing and devices.

6. Interferences

6.1 Carrier Density— Attempts to derive carrier densityvalues from resistivity values by using these conversions aresubject to error. While dopant density and carrier densityvalues are expected to be the same at low densities (up to about1017 cm−3), the two quantities generally do not have the samevalue in a given specimen at moderate densities. At suchmoderate densities, (about 1017 cm−3 to 1019 cm −3), dopantdensities are larger than carrier densities due to incompleteionization. At densities above 1019 cm −3, the populationstatistics become degenerate, and carrier densities would nor-mally be equal to dopant densities. However, in this upperrange of densities, the possibility of formation of compoundsor complexes involving dopant atoms is more pronounced andwould prevent some of the dopant atoms from being electri-cally active. Such formation of compounds or complexes isparticularly likely in phosphorus- or arsenic-doped silicon.Precipitation occurs at dopant densities greater than solidsolubility.

6.2 Heavily Phosphorus-Doped Silicon—These conversionsare given as functions of resistivity and of dopant density. Forheavily phosphorus-doped specimens, primary emphasis wasplaced on Hall effect measurements for establishing the density

4 Annual Book of ASTM Standards, Vol 10.05.5 A large wall chart, “Conversion Between Resistivity and Dopant Density” is

available from ASTM, 100 Barr Harbor Drive, West Conshohocken, PA 19428.Request Adjunct PCN 12-607230-46.

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values. However, since the Hall effect measures carrier density,it was assumed for these heavily doped specimens that allatoms were electrically active; that is, the dopant density wasequal to the carrier density as measured by the Hall effect. Thepossible formation of phosphorus-vacancy pairs which areknown to reduce the electrically active phosphorus atoms athigh densities(5) was not tested or accounted for in the database or the resulting conversions. The existence of suchphosphorus-vacancy pairs would cause the Hall measurementsto understate the total dopant density for the heavilyphosphorus-doped specimens.

6.3 Other Dopant Species—The applicability of these con-versions to silicon doped with other than arsenic, boron orphosphorus has not yet been established. However, in thelightly doped range (<1017 cm−3) the conversions are expectedto be reasonably accurate for other dopants. Between 1017 and1019 cm −3, the difference in activation energy of differentdopant species will cause different resistivities to be measuredfor the same dopant density. In this range the differences inresistivities will be larger among thep-type dopants thanamong then-type dopants due to larger differences in theactivation energies among thep-type dopants. At high dopantdensities (>1019 cm−3), the formation of complexes involvingdopant atoms, lattice defects, and other impurities will lead toa modification in the number of charge carriers. The extent ofthis effect will depend upon the particular dopant species andis not well detailed in the literature for the various commondopants. Its onset is expected to be related to the density of thedopant compared to the solid solubility of that dopant insilicon. Therefore, in this upper dopant density range, theapplicability of these conversions to dopants other than boronand phosphorus is unclear.

6.4 Compensation— The specimens used to obtain the database for these conversions were assumed to be uncompensated.The measured net dopant density was taken to be the totaldensity of the intentional dopant in the specimen. For speci-mens in which significant compensation occurs, these conver-sions may not apply.

6.5 Temperature— The conversions in this practice hold fora temperature of 23°C. Resistivity varies with temperature, butdopant density does not.

NOTE 2—It is possible to obtain dopant density values from resistivityvalues that were not measured at 23°C by using Test Method F 84 tocorrect the resistivity values to 23°C. Also, the conversion from dopantdensity to resistivity may be made directly and the temperature correction

for resistivity then made following Test Method F 84 to obtain theresistivity at other than 23°C.

NOTE 3—References1, 2, and3 give values for the coefficients in theconversion equations at both 23°C and 300 K.

6.6 Other Electrically Active Centers—Numerous othermechanisms exist that may modify the number of free carriersor noticeably alter carrier mobility, either of which will changethe resistivity from the value given here for a given dopantdensity. Among these mechanisms are (1) lattice damage due toradiation (neutron transmutation doping or ion implantation), (2) formation of deep level centers due to chemical impurities(typically heavy metals, either unwanted or sometimes inten-tionally added for minority carrier lifetime control), and (3)unintentional doping due, for example, to electrically activatedoxygen. When any of these effects is known or expected to bepresent, the conversions given here may not apply.

6.7 Range of Arsenic-Doped Silicon Data—The conversiongiven for arsenic-doped silicon is from Fair and Tsai(8),covering the doping range of 1019 to 6 by 1020. This conversionwas generated using Hall effect measurements. The principalreference for neutron activation data is that of Newmanetal.(9). Neutron activation data give higher resistivity values fora given dopant density than do Hall data because of theassumption that the Hall coefficientRH 5 1/ne. A more com-plicated relationship betweenRH andn is given in Ref(9). Caremust be taken in using any conversion not to extrapolatebeyond the range of the data fitted, as the formulas will divergebeyond that range. Other studies support the use of theconversion given here Refs(10, 11, 12,and13 ), which meansthat the conversion to resistivity for phosphorus can be used forarsenic in this range. In the range 2 by 1019 to 1020, the Fair andTsai fit matches the Irvin formula. The range beyond 6 by 1020

is discussed only in Ref(13).

7. Procedure

7.1 Convert Resistivity Values to Dopant Density Values—Follow 7.1.1 (graphical method), 7.1.2 (tabular method), or7.1.3 (computation method).5

7.1.1 Graphical Method:7.1.1.1 For boron-doped silicon, use the curve labeled

“boron” in Fig. 1.7.1.1.2 For phosphorus-doped silicon, use the curve labeled

“phosphorus” in Fig. 1.7.1.2 Tabular Method:7.1.2.1 For boron-doped silicon use Table 1.

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7.1.2.2 For phosphorus-doped silicon use Table 2.7.1.3 Computational Method:7.1.3.1 For boron-doped silicon, calculate the dopant den-

sity value from the resistivity value as follows:

N 51.3303 10 16

r 11.0823 10 17

r @1 1 ~54.56r! 1.105#(1)

where:r 5 resistivity andN 5 dopant density.

7.1.3.2 For phosphorus-doped silicon, calculate the dopantdensity from the resistivity as follows:

N 56.2423 10 18

r 3 10 Z (2)

where:

Z 5A0 1 A1x 1 A2x

2 1 A3x3

1 1 B1x 1 B2x2 1 B3x

3 (3)

where:x 5 log10r,A0 5 −3.1083,A1 5 −3.2626,A2 5 −1.2196,A3 5 −0.13923,B 1 5 1.0265,B2 5 0.38755, andB3 5 0.041833.

7.2 Convert Dopant Density Values to Resistivity Values—Follow 7.2.1 (graphical method), 7.2.2 (tabular method), or7.2.3 (computational method).

7.2.1 Graphical Method:7.2.1.1 For boron-doped silicon, use the curve labeled

“boron” in Fig. 1.7.2.1.2 For phosphorus-doped silicon, use the curve labeled

“phosphorus” in Fig. 1.7.2.2 Tabular Method:7.2.2.1 For boron-doped silicon, use Table 3.

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7.2.2.2 For phosphorus-doped silicon, use Table 4.

7.2.3 Computational Method:

7.2.3.1 For boron-doped silicon, calculate the resistivityfrom the dopant density as follows:

r 51.3053 10 16

N 11.1333 10 17

N@1 1 ~2.583 10 219 N! 20.737#(4)

7.2.3.2 For phosphorus-doped silicon, calculate the resistiv-ity from the dopant density as follows:

r 56.2423 10 18

N 3 10 Z (5)

where:

Z 5A0 1 A1y 1 A2y

2 1 A3y3

1 1 B1y 1 B2y2 1 B3y

3 (6)

where:y 5 (log10N) − 16,A0 5 −3.0769,A 1 5 2.2108,A2 5 −0.62272,A3 5 0.057501,B1 5 −0.68157,B2 5 0.19833, andB3 5 −0.018376.

7.2.3.3 For arsenic-doped silicon, calculate the resistivityfrom the dopant density as follows:

log10r 5 26633.6671 AX1 BX2 1 CX3 1 DX4 1 EX5 1 FX6 1 GX7

1 HX8 1 IX9 1 JX10 (7)

where:X 5 log10N,A 5 768.2531,B 5 −25.77373,C 5 0.9658177,D 5 −0.05643443,E 5 −8.0085433 10 −4,F 5 9.0558383 10 −5,G 5 −1.7767013 10 −6,H 5 1.9532793 10 −7,I 5 −5.7545993 10 −9, andJ 5 −1.316573 10 −11.

8. Keywords

8.1 arsenic; boron; dopant density; phosphorus; resistivity;silicon

APPENDIXES

(Nonmandatory Information)

X1. DIFFERENCES IN CONVERSION SCALES ENCOUNTERED WHEN USING THE TABULAR OR COMPUTATIONALFORMS OF THIS PRACTICE

X1.1 The conversion equations and the resulting tables arederived by fitting the experimental data using either resistivityor dopant density as the independent variables. This leads tocomplementary equations, for example, 7.1.3.1 and 7.2.3.1.These complementary equations are not exactly equivalentmathematically and small discrepancies can be found whenusing the equations or the tables derived from them. A givenvalue of resistivity (for example, 1.00V· cm p-type) would

convert to a dopant density value (1.463 1016cm −3), but theuse of that dopant density in the complementary equation (ortable) gives a resistivity (1.02V·cm) which is different fromthe starting value. In this case, there is a 2 % (1.00V·cm versus1.02V·cm) self-consistency error.

X1.2 The self-consistency errors are plotted in Fig. X1.1.

X2. COMPARISON OF CONVERSIONS WITH THOSE DUE TO IRVIN

X2.1 Irvin (6) reported conversion relations between resis-tivity and dopant density forn- and p-type silicon at 300 K.Irvin chose to use the term impurity concentration instead ofthe term dopant density. Irvin’s analysis was based on acompilation of data by other authors based on several donorand acceptor dopants of similar energy levels in the siliconband gap and was supplemented by data taken by Irvin on

heavily arsenic- and boron-doped specimens. All specimenswere assumed to be uncompensated, but the possibility ofcompensation because of thermal activation of oxygen in theCzochralski crystals was recognized.

X2.2 A comparison of the conversions given by this methodwith those due to Irvin is shown in Fig. X1.2.

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REFERENCES

(1) Thurber, W. R., Mattis, R. L., Liu, Y. M., and Filliben, J. J.,“Resistivity-Dopant Density Relationship for Phosphorus-Doped Sili-con,” Journal of the Electrochemical Society, Vol 127, 1980, pp.1807–1812.

(2) Thurber, W. R., Mattis, R. L., Liu, Y. M., and Filliben, J. J.,“Resistivity-Dopant Density Relationship for Boron-Doped Silicon,”Journal of the Electrochemical Society, Vol 127, 1980, pp.2291–2294.

(3) Thurber, W. R., Mattis, R. L., Liu, Y. M., and Filliben, J. J.,Semiconductor Measurement Technology,“ Relationship Between Re-sistivity and Dopant Density for Phosphorus- and Boron-DopedSilicon,” NBS Special Publication 400-64 (April 1981).

(4) Esaki, L., and Miyahara, Y., “A New Device Using the TunnelingProcess in Narrowp-n Junction,”Solid-State Electronics, Vol 1, 1960,pp. 13–21.

(5) Fair, R. B., and Tsai, J. C. C., “A Quantitative Model for the Diffusingof Phosphorus in Silicon and the Emitter Dip Effect,”Journal of theElectrochemical Society,Vol 124, 1977, pp. 1107–1118.

(6) Irvin, J. C.,“ Resistivity of Bulk Silicon and of Diffused Layers inSilicon,” Bell System Technical Journal,Vol 41, 1962, pp. 387–410.

(7) Blakemore, J. S.,Semiconductor Statistics,Pergamon Press, Oxford,1962, pp. 153–161.

(8) Fair, R. B., and Tsai, J. C. C., “The Diffusion of Ion-Implanted Arsenicin Silicon,” Journal of the Electrochemical Society,Vol 122, No. 12,l975, p. 1689.

(9) Newman, P. F., Hirsch, M. J., and Holcomb, D. F., “A CalibrationCurve for Room-Temperature Resistivity versus Donor Atom Concen-tration in Si:As,”Journal of Applied Physics,Vol 58, No. 10, l985, p.3779.

(10) Furukawa, Y., “Impurity Effect Upon Mobility in Heavily DopedSilicon,” Journal of Physics Society,Japan, Vol 16, l961, p. 577.

(11) Murota, J., Arai, E., Kobayashi, K., and Kudo, K., “RelationshipBetween Total Arsenic and Electrically Active Arsenic Concentra-tions in Silicon Produced by the Diffusion Process.”Journal ofApplied Physics,Vol 50, No. 2, l979, p. 804.

(12) Matsumoto, S., Niimi, T., Murota, J., and Arai, E., “CarrierConcentration and Hall Mobility in Heavily Arsenic-Diffused Sili-con,” Journal of Electrochemical Society,Vol 127, No. 7, l980, p.1650.

(13) Masetti, G., Severi, M., and Solmi, S., “Modeling of CarrierMobility Against Carrier Concentration in Arsenic-, Phosphorus-, andBoron-Doped Silicon,”IEEE Trans. on Elec. Dev., ED-30,No. 7,l983, p. 764.

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TABLE 1 Dopant Density as a Function of Resistivity for Boron-Doped Silicon

NOTE 1—Entries in two significant digits are for regions of extrapolated data.

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TABLE 1 Continued

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TABLE 2 Dopant Density as a Function of Resistivity for Phosphorus-Doped Silicon


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TABLE 2 Continued

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TABLE 3 Resistivity as a Function of Dopant Density for Boron-Doped Silicon


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TABLE 3 Continued

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TABLE 4 Resistivity as a Function of Dopant Density for Phosphorus-Doped Silicon


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TABLE 4 Continued

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NOTE 1—The solid line shows the resistivity to dopant density conversion for the range of actual data. The chain dot line shows the dopant densityto resistivity conversion for the range of actual data. Dashed lines show regions of extrapolation from data.

NOTE 2—On the scale of the figure as reproduced in this book, the solid and chaindot lines cannot be distinguished visually. They are distinguishable,however, on the wall chart available as Adjunct PCN 12-607230-46, wherever the self-consistency error (see Appendix X1) becomes appreciable.

FIG. 1 Preferred Conversion Between Resistivity and Total Dopant Density Values for Boron- and Phosphorus-Doped Silicon

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FIG. X1.1 Differences in Conversion Scales Encountered When Using this Practice

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FIG. X1.2 Comparison of Conversions Between Resistivity and Dopant Density with Those of Irvin

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