Radiochemical Neutron Activation Analysis for Certification of Ion

Certification of Ion-Implanted Phosphorus in. Silicon. Rick L. Paul,*,† David S. Simons,† William F. Guthrie,‡ and John Lu‡. Chemical Science ...
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Anal. Chem. 2003, 75, 4028-4033

Radiochemical Neutron Activation Analysis for Certification of Ion-Implanted Phosphorus in Silicon Rick L. Paul,*,† David S. Simons,† William F. Guthrie,‡ and John Lu‡

Chemical Science and Technology Laboratory and Statistical Engineering Division, Information Technology Laboratory, National Institute of Standards and Technology, Gaithersburg, Maryland 20899

A radiochemical neutron activation analysis procedure has been developed, critically evaluated, and shown to have the necessary sensitivity, chemical specificity, matrix independence, and precision to certify phosphorus at ion implantation levels in silicon. 32P, produced by neutron capture of 31P, is chemically separated from the sample matrix and measured using a β proportional counter. The method is used here to certify the amount of phosphorus in SRM 2133 (Phosphorus Implant in Silicon Depth Profile Standard) as (9.58 ( 0.16) × 1014 atoms‚cm-2. A detailed evaluation of uncertainties is given. The U.S. semiconductor industry relies heavily on secondary ion mass spectrometry (SIMS) for characterization of the depth distribution of dopants in silicon. To achieve high accuracy in the concentration determination by SIMS, standards of known dopant concentration, conveniently provided by ion implants of known dose, are required. Standard reference materials (SRMs) of boron and arsenic implants in silicon have been developed previously by NIST as SIMS calibration standards.1,2 SEMATECH (a consortium of semiconductor manufacturers) recently listed an SRM implant of phosphorus in silicon as a high-priority need. The SIMS community in the United States undertook a round-robin study to calibrate the implanted dose of phosphorus in a silicon wafer by consensus. The wafer, implanted at a nominal dose of 8.5 × 1014 atoms‚cm-2, was sent to 16 different laboratories for SIMS analysis. Dose determinations among the laboratories varied by nearly a factor of 2 (Figure 1), reflecting primarily the errors of the respective in-house standards. These results demonstrate the need for a common phosphorus reference material to improve interlaboratory reproducibility. Certification of SRMs requires reliable methods of analysis. Such methods must have the necessary sensitivity, as well as chemical specificity and matrix independence. Radiochemical neutron activation analysis (RNAA) has been used previously at * Corresponding author. Phone: (301) 975-6287. Fax: (301) 208-9279. E-mail: [email protected]. † Chemical Science and Technology Laboratory. ‡ Statistical Engineering Division, Information Technology Laboratory. (1) SRM 2137 Boron Implant in Silicon Standard for Calibration of Concentration in a Depth Profile; Certificate of Analysis, NIST, Gaithersburg, MD 20899. Certificate issue date August 13, 1993. (2) Greenberg, R. R.; Lindstrom, R. M.; Simons, D. S. J. Radioanal. Nucl. Chem. 2000, 245, 57-63.

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Figure 1. Phosphorus measurements from the secondary ion mass spectrometry (SIMS) round-robin investigation. Each laboratory used in-house standards for measurements. The mean of the measurements from 16 laboratories was (8.40 ( 1.55) × 1014 cm-2 (1s uncertainty). Data provided by Charles Hitzman, Charles Evans and Associates.

NIST to value-assign phosphorus in SRM metals,3,4 and biological and geological SRMs. During irradiation, 31P undergoes neutron capture to form 32P, which must be separated and purified of other radionuclides before counting. The method is chemically specific and matrix-independent. Furthermore, because only 32P is measured, there is little chance of contamination during the postirradiation chemistry, hence the blank problems encountered with most analytical methods are avoided. An RNAA procedure for measuring phosphorus in silicon has recently been developed, critically evaluated, and shown to have the necessary sensitivity and precision to certify the phosphorus concentration at ion implantation levels.5,6 Analysis of 12 pieces of the silicon wafer used in the SIMS investigation yielded a phosphorus concentration of 8.30 × 1014 atoms‚cm-2 ( 0.14 × 1014 atoms‚cm-2 (expanded uncertainty), in agreement with the SIMS mean of 8.40 × 1014 (3) Paul, R. L. J. Radioanal. Nucl. Chem. 1998, 234 (1-2), 55-58. (4) Paul, R. L. J. Radioanal. Nucl. Chem. 2000, 245, 11-15. (5) Paul, R. L.; Simons, D. S. In Characterization and Metrology for ULSI Technology; Seiler, D. G., Diebold, A. C., Shaffner, T. J., McDonald, R., Bullis, W. M., Smith, P. J., Secula, E. M., Eds.; AIP Conference Proceedings 550; American Institute of Physics: Melville, NY, 2001; pp 677-681. (6) Paul, R. L.; Simons, D. S. In Silicon Front-End Junction Formation Technologies; Downey, D. F., Law, M. E., Claverie, A., Rendon, M. J., Eds.; MRS Symposium Proceedings 717; Materials Research Society: Warrendale, PA, 2002; pp 291-296. 10.1021/ac0342018 Not subject to U.S. Copyright. Publ. 2003 Am. Chem. Soc.

Published on Web 07/11/2003

( 1.55 × 1014 atoms‚cm-2 (1s uncertainty) (Figure 1). The procedure is used here to certify the phosphorus concentration in SRM 2133 (Phosphorus Implant in Silicon Depth Profile Standard). EXPERIMENTAL SECTION Wafer Implantation and Preparation of Samples and Blanks. A 200 mm diameter single crystal silicon wafer was implanted with 31P at a nominal dose of 8.5 × 1014 atoms‚cm-2. A nonimplanted wafer from the same lot was supplied as a blank. The implanted wafer was subjected to a nondestructive ThermaWave map to provide a measure of implantation uniformity and then diced into nominal 1 cm × 1 cm squares using a wafer dicing saw. (The identification of certain commercial equipment, instruments, or materials does not imply recommendation or endorsement by the National Institute of Standards and Technology. These identifications are made only in order to specify the experimental procedures in adequate detail.) Twelve specific wafer squares were chosen for analysis. The dimensions of these pieces were measured with a digital micrometer with a readout resolution of 0.0001 cm. The surface area of each piece was then calculated as length × width. The average area of the 12 pieces was 0.98761 cm2, with a standard deviation of 0.000 22 cm2. Six pieces of the nonimplanted wafer were prepared as blanks. Blanks are necessary for these measurements in order to correct the 32P produced from neutron capture by the implanted phosphorus (near-surface layer), from 32P produced by neutron capture by phosphorus impurity or silicon in the bulk of the sample. Neutron capture by the silicon produces 32P via the second order reaction, used in neutron transmutation doping: {30Si(n, γ) 31Si(β-, 2.6 h) 31P(n, γ) 32P}. Preparation of Targets for Irradiation. Silicon samples and blanks were cleaned of surface contamination by soaking in ultrapure water in an ultrasonic bath for 2 h, then rinsed with ethanol, and allowed to dry overnight in a clean hood. After cleaning, samples and blanks were sealed into polyethylene bags, which had previously been cleaned by soaking for 24 h in 1:1 ultrapure HNO3/ultrapure H2O, then for 24 h in ultrapure water, followed by drying in a clean hood for 2 d. Two different phosphorus solutions were used in the preparation of standards. The first solution, of concentration 0.5631 ( 0.0023 mg/mL (expanded uncertainty), was prepared by dilution of phosphorus SRM 3139a spectrometric solution, lot 890607, 9.99 ( 0.04 mg/g. The second, of concentration 0.5023 ( 0.0022 mg/mL (expanded uncertainty), was prepared by dilution of a phosphorus stock solution which was made by dissolving KH2PO4 (Alpha Aesar, Puratronic, 99.999%) in ultrapure water, and standardized by gravimetric analysis (as magnesium pyrophosphate) of replicate aliquots. All dilutions were performed using calibrated volumetric flasks and pipets. Standards for the current investigation were prepared by deposition of these solutions onto 25 mm × 25 mm aluminum foils (99.5% purity). Each standard was prepared by deposition of 1 drop of solution onto a foil, the solution being dispensed from a disposable polypropylene pipet, which was weighed to (0.000 01 g before and after dispensing. Each foil was allowed to dry overnight in a clean hood and was then folded and sealed into a clean polyethylene bag. The mass of phosphorus deposited on each foil was calculated from the mass of the dispensed solution, the solution concentration (mg/mL),

and the density of the solution (determined from replicate weighings of 2-mL aliquots). Six standards were used in the analysis, with three prepared from each of the two solutions. In order to correct the count rate of the standards for 32P produced from phosphorus impurity in the aluminum, three standard blanks were prepared by sealing folded 25 mm × 25 mm squares of aluminum foil into clean polyethylene bags. Thin disks of iron foil (6.35-mm diameter, mass ∼6.3 mg) were also sealed into polyethylene bags to be irradiated as flux monitors. Flux monitors are necessary to ensure that the neutron fluence is uniform throughout the irradiation vessel and to be able to compare data from targets irradiated in different vessels. After preparation, the targets were packaged into rabbits (polyethylene irradiation vessels). In order to minimize damage to the silicon pieces during irradiation, targets were first sealed into long strips of polyethylene. Each polyethylene strip was then folded accordion-style and placed into a rabbit, such that the samples were distributed along the vertical axis (length) of the rabbit. Styrofoam plugs were placed below and above the ribbon in order to hold it in place as well as to provide additional cushioning for the targets. A total of 3 rabbits were prepared, each rabbit containing 4 samples, 2 silicon blanks, 2 standards, a standard blank, and 3 flux monitors. Each rabbit was irradiated for 3 h in irradiation tube RT1 of the NIST 20 MW NBSR reactor at a neutron fluence rate of 1.05 × 1014 cm-2‚s-1. In order to ensure that the entire rabbit receive a uniform neutron fluence, rabbits were flipped (rotated 180°) at the midpoint of irradiation. The rabbits were allowed to sit for 2 d before processing of samples, in order to allow for decay of 31Si (t1/2 ) 2.62 h). Processing of Samples and Blanks. Silicon samples and blanks were processed using the method developed previously.5,6 (1) The silicon squares were removed from their irradiation bags, soaked in ultrapure water in an ultrasonic bath for 1 h, and then rinsed with ethanol to remove surface contamination. Each sample was then transferred to a Teflon beaker with 1.994 ( 0.003 mL of a phosphorus carrier solution containing 3.522 ( 0.008 mg/mL P, which was prepared by dilution of the stock solution of concentration 10.07 ( 0.02 mg/mL (1s uncertainties). (2) A mixture of 10 mL of HNO3 and 5 mL of HF was added to the beaker, and the beaker was covered with a Teflon watch glass and heated on medium heat for about 2 h in order to dissolve the silicon. (3) The cover was removed from the beaker and the solution evaporated to near dryness. The residue was dissolved in ∼5 mL of HClO4 and again evaporated to near dryness. This evaporation removes fluorides, which could interfere with the gravimetric determination of the carrier. (4) The residue was dissolved in 10 mL of HNO3, followed by addition of 30 mL of H2O, 5 g of NH4NO3, and 20 mL of ammonium molybdate solution (mass fraction 10%) to precipitate ammonium phosphomolybdate. The yellow precipitate was digested by heating and then allowed to settle and cool. (5) The ammonium phosphomolybdate precipitate was collected by centrifugation, washed with 15 mL H2O, and dissolved in 10 mL of NH4OH. To this solution were added 20 mL of H2O and 10 mL of NH4Cl solution (mass fraction ) 20%), followed by 15 mL of magnesia reagent (prepared by dissolving 30 g of MgCl2‚6H2O, 70 g of NH4Cl, and 65 mL of NH4OH in H2O and diluting to 500 mL). The magnesium ammonium phosphate precipitate was allowed to settle overnight. (6) The precipitate was Analytical Chemistry, Vol. 75, No. 16, August 15, 2003

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collected by suction filtration on medium porosity filter paper, washed with a 10% (vol/vol) solution of ammonium hydroxide and 100% ethanol, and air-dried in a clean hood for 3 d. The dried precipitate was weighed and transferred to a counting capsule for β counting of 32P. The yield (fraction of recovered phosphorus carrier) was determined gravimetrically, assuming a precipitate composition of MgNH4PO4‚6H2O. (7) After the counting cycle was completed, the precipitate was transferred quantitatively to a 5 mL porcelain crucible (previously heated for a few hours at 700 °C, cooled in a desiccator for 1 h, and then weighed on a microbalance). The crucible was heated in a muffle furnace to 650 °C for 2 h, cooled in a desiccator for 1 h, and again weighed using a microbalance. The yield was determined gravimetrically, this time assuming a precipitate composition of Mg2P2O7. Processing of Standards and Standard Blanks. Aluminum foils containing standards and standard blanks were transferred to Teflon beakers, mixed with 1.994 ( 0.003 mL of phosphorus carrier solution containing 10.07 ( 0.02 mg/mL (1s uncertainty), and each foil was dissolved in a mixture of 5 mL of HCl, 5 mL of HNO3, and 1 mL of HF. Each solution was transferred quantitatively to a 100 mL polypropylene volumetric flask and diluted to 100 mL with H2O. A 1.994 ( 0.003 mL aliquot of each 100 mL solution was transferred to a Teflon beaker and mixed with 1.994 ( 0.003 mL of phosphorus carrier solution containing 3.522 ( 0.008 mg/mL. HNO3 (10 mL) and 5 mL of HF were added, along with a nonirradiated piece of nonimplanted silicon (added to ensure that the chemical processing for the standard is identical to that of the samples). From here on, the sample procedure was followed, beginning with step 2. Counting of Flux Monitors. Iron flux monitors were counted using a high-purity germanium detector. The 59Fe count rate was determined by integration of the 1291 keV γ-ray peak. The count rate was corrected for decay time and pulse pileup and then divided by the mass of the iron foil to obtain counts‚s-1‚mg-1 Fe at the end of bombardment. The reason for irradiating and counting flux monitors was to check to make sure that each rabbit received a uniform neutron fluence (no top-to-bottom variation), as well as to be able to compare fluences received by different rabbits. The mean and standard deviation of the count rates measured for three flux monitors was determined for each rabbit: rabbit 1, 5.099 ( 0.043‚counts s-1‚mg-1; rabbit 2, 5.098 ( 0.010 counts‚s-1‚mg-1; and rabbit 3, 5.099 ( 0.006 counts‚s-1‚mg-1. In each case, the standard deviation was equal to or less than the uncertainty given by counting statistics. The overall mean and standard deviation of the count rates for the nine flux monitors was 5.099 ( 0.022 counts‚s-1‚mg-1. Hence, we can conclude that, within the statistical uncertainties, each rabbit not only received a uniform neutron fluence, but each rabbit received the same fluence as the others. Counting and Data Reduction. Counting of 32P (t1/2 ) 14.28 d) was performed using a β proportional counter. Each target was counted for 3 h. The efficiency of the counter was monitored by counting a calibrated 90Y-90Sr source at regular intervals. Purity of the 32P from other radionuclides was ascertained by γ-ray spectroscopy, and by monitoring β count rates over a period of 4 half-lives. Since most β-emitting nuclides also emit γ-rays during decay (notable exceptions being 32P, 35S, and 14C), the lack of γ radiation in the spectrum gives high confidence that other 4030

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Table 1. Results of Analysis of Phosphorus in SRM 2133 by RNAA sample

P (atoms‚cm-2)

1 2 3 4 5 6 7 8 9 10 11 12

9.50 × 1014 9.47 × 1014 9.62 × 1014 9.54 × 1014 9.42 × 1014 9.62 × 1014 9.73 × 1014 9.62 × 1014 9.63 × 1014 9.41 × 1014 9.61 × 1014 9.66 × 1014

average ( 1s relative standard deviation

(9.57 ( 0.10) × 1014 1.09%

radionuclides are not present at levels that will have any effect on the 32P measurement. Furthermore, since no other radionuclides have a half-life equal to that of 32P, a plot of ln(activity) versus time which yields a slope equal to the decay constant (0.002022 h-1) for 32P provides further evidence of the purity of the nuclide. For each target, the β count rate at time zero (end of neutron irradiation) was determined from the β decay curve. The count rates were then adjusted to account for differences in the amount of self-absorption of β particles in the different precipitates. Corrections were made using self-absorption curves (32P count rate vs milligrams of precipitate) derived from analyses of previous phosphorus standards.4 Count rates were also adjusted for blanks. The concentration (implantation dose per unit area) of phosphorus in each sample was calculated in atoms cm-2, using the following equation:

[P(atoms‚cm-2)] ) (NSTDI(ASA/AST)(YST/YSA))/ArSA (1) where NST is the number of phosphorus atoms in the irradiated standard; DI is the dilution factor for the standard; ASA is the 32P count rate of the sample, corrected for decay time, self-absorption, and blank; AST is the standard count rate, similarly corrected; YST is the yield (fraction of recovered carrier) for the standard; YSA is the yield for the sample; and ArSA is the surface area of the sample. RESULTS AND DISCUSSION Results. Table 1 gives phosphorus concentrations determined for SRM 2133. The relative standard deviation of 1.09% for the measurement method represents an observed improvement in reproducibility over the standard deviations of 2.35% and 1.35% obtained for measurements made during procedural development.5,6 The following discussion attributes some of this variability to spatial heterogeneity of the phosphorus concentration. No trends in the SRM results were observed due to sample position in the rabbit. Evaluation of Uncertainties. NIST currently certifies elemental concentrations in SRMs using one of three procedures: (1) a primary method at NIST with confirmation by other method(s), (2) two independent critically evaluated methods, and (3) one

method at NIST and different methods by outside laboratories.7 In order to establish RNAA as a primary method for certification of phosphorus concentration in silicon, it was necessary to evaluate all significant sources of uncertainty explicitly.2 In order to do this, we followed the method used by Greenberg et al. in the evaluation of uncertainties for arsenic by instrumental NAA.2 In the discussion that follows, all uncertainties for the determination of phosphorus in SRM 2133 are calculated as a relative percentage of the phosphorus concentration. Type A Uncertainties. Counting Uncertainties/Measurement Replication. Counting uncertainties arise from the fact that radioactive decay is a random process; hence, any measurement based on observing radiation emitted in nuclear decay is subject to some degree of statistical fluctuation. Uncertainties arising from counting statistics can be lowered by increasing the counting time. When counting uncertainties are larger than the spread of replicate measurements of the sample material, the counting uncertainty may be used as a measure of the uncertainty due to measurement replication. For the current measurements, however, the counting statistics were smaller than the spread of the data, so a better indicator of the uncertainty due to measurement replication can be obtained from an analysis of the 12 replicate samples. Because of the potential for variation in the phosphorus concentration across the wafer, a quadratic model in two dimensions was fit to the data using least squares regression. Although the quadratic terms in the model were not significantly different from zero, the linear and interaction terms did differ significantly from zero, indicating that the phosphorus concentration in each chip does depend on its spatial location in the wafer. Because of the spatial variation of the phosphorus concentrations, the best single value of phosphorus content to assign to this reference material would be the value at the center of the phosphorus concentrations predicted for the chips that will be distributed, rather than the mean of the measured values. Due to the nature of the model that was fit to the data, a good estimate of this value turns out to be the phosphorus content at the stationary point on the twodimensional surface fit to the data (9.58 × 1014 atoms‚cm-2). The uncertainty in the predicted value at this point due to measurement replication (0.0199 × 1014 atoms‚cm-2) was estimated using the residual standard deviation from the fit of the reduced model, without the quadratic terms. The uncertainty due to sample heterogeneity was estimated on the basis of the range of predicted values of phosphorus concentration across the area of the wafer from which SRM units will be taken (0.0573 × 1014 atoms‚cm-2). These two components were combined in quadrature to give a relative uncertainty of 0.63% for measurement replication and heterogeneity of samples. The measurement replication uncertainty for the standards was estimated using the standard deviation of the mean of the replicate samples and was computed to be s/xn ) 1.15/x6 or 0.47%. Measurement of Surface Area. Since the phosphorus concentration was normalized to surface area, the uncertainty in the determination of the surface area had to be evaluated. The silicon (7) May, W.; Parris, R.; Beck, C.; Fassett, J.; Greenberg, R.; Guenther, F.; Kramer, G.; Wise, S.; Gills, T.; Colbert, J.; Gettings, R.; MacDonald, B. Definitions of Terms and Modes Used at NIST for Value-Assignment of Reference Materials for Chemical Measurements; NIST Special Publication 260-136, National Institute of Standards and Technology, U. S. Government Printing Office: Washington, DC, 2000.

pieces were cut to nearly identical sizes. The length and width of each sample were measured using a micrometer with a resolution of 0.0001 cm. If we assume the pieces are precisely rectangular, the 0.0001 cm absolute accuracy is divided by 3 to obtain the 1s uncertainty, or 0.000058 cm. Therefore, the relative standard uncertainty in the accuracy basis for computing sample area (≈1 cm2) is equal to ((1.000058 cm/1 cm)2 - 1)/100 or 0.012%.2 Blank Correction. In order to calculate the implanted phosphorus dose, it is necessary to correct the sample count rate for 32P produced in the bulk of the sample. This was done by measuring phosphorus in the nonimplanted silicon blanks. The 32P measured in the blanks most likely arises from two sources: neutron capture by phosphorus impurity, and neutron capture by the silicon itself. ( 32P may also be produced via the fast neutron reactions 32S(n, p)32P and 35Cl(n, R)32P. Contributions from these reactions are negligible due to the low concentrations of S and Cl in the high purity silicon, and the low fast/thermal component of the neutron flux.) 32P is produced from silicon via a second order reaction:

Si(n, γ) 31Si(β-, 2.6 h) 31P(n, γ) 32P

30

The average blank correction of 11.1 ( 0.8 counts min-1, determined from measurement of 6 samples of nonimplanted silicon, is 0.65% of the mean 32P count rate (corrected for decay time and yield) of the samples, while the standard deviation of the blank measurements is equal to 0.06% of the mean sample count rate. The relative uncertainty in the blank correction is estimated as the standard deviation of the mean of the blank measurements, which is equal to 0.06/x6 ) 0.022% of the 32P count rate of the samples. The blank correction uncertainty for the standards was similarly computed, using data from the aluminum foil blanks. Quantity of Standard. Uncertainties in the quantity of irradiated phosphorus processed for each standard arise from 3 sources: the measured concentration of the original standard solutions, volumetric calibrations of glassware used for dilutions, and the mass of phosphorus solution dispensed on each aluminum foil. The uncertainties in the concentrations of the original (undiluted) phosphorus standard solutions were calculated as follows. Three of the standards were prepared from a diluted version of phosphorus spectrometric solution, SRM 3139a, which was certified at 9.99 ( 0.04 mg/g. The relative expanded uncertainty is 0.4%, which approximates a 95% confidence interval. Dividing this by the coverage factor (2) gives the 1s relative standard uncertainty of 0.2%. The other three standards were prepared from a diluted version of a phosphorus stock solution, with concentration 10.07 ( 0.02 mg/mL, or 0.2% relative, (1s, from measurement replication), which was standardized gravimetrically as MgP2O7 using 4 replicate aliquots of solution. The 1s uncertainty for this solution was taken to be s/xn ) 0.1%. Since an equal number of standards was prepared from each of these solutions, the 1s uncertainty due to calibration of standard solutions was calculated as the average of these two values, or 0.15%. The volumetric calibration uncertainty takes into account all uncertainties arising from dilutions and pipetting of solutions and was calculated by adding the individual uncertainties from all volumetric pipets and flasks used in the pre- and postirradiation Analytical Chemistry, Vol. 75, No. 16, August 15, 2003

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procedures. The total uncertainty due to volumetric calibrations was 0.12%. The uncertainty in the mass of phosphorus solution deposited on each foil was determined by dividing the weighing uncertainty of the balance by the mass of solution on the foil. The average of the six values determined in this manner was then determined as 0.037%. Radiochemical Impurity. Uncertainties arising from radiochemical impurity were assessed by comparing decay constants for 32P obtained from plots of ln[activity] versus time for samples and standards. For each target counted, the decay constant was determined as the slope of the curve. The average of the decay constants determined from the 12 sample plots was 0.002 025 1 ( 0.000 008 5 h-1 (1s uncertainty). The average of the decay constants determined from the 6 standard plots was 0.002 025 0 ( 0.000 002 3 h-1 (1s uncertainty). By comparison, the decay constant calculated from the best published value of the 32P halflife (14.262 ( 14 d8) is 0.002 025 0 ( 0.000 020 h-1; hence, the half-life difference between the samples and the best literature value is only 0.005%. Since samples and standards are being compared, any systematic errors cancel; thus, the best estimate of the uncertainty due to radiochemical impurity is equal to the difference in decay constants between samples and standards. The ratio of mean sample decay constant to mean standard decay constant was 1.000 074. The uncertainty in this ratio, taken as the uncertainty due to radiochemical impurity, is calculated as the sum of the relative standard deviation of the mean of the sample decay constants plus the relative standard deviation of the mean of the standard decay constants, which is equal to

[(100(0.0000085/0.0020251)/x12)2 + (100(0.0000023/0.0020250)/x6)2]1/2 ) 0.13%

Carrier Yield Determination. Uncertainties in the gravimetric determination of the carrier yield were evaluated by comparing the yield obtained from determination of phosphorus as MgNH4PO4‚6H2O (Y1) with that obtained by determination as Mg2P2O7 (Y2). Since sample yields are being compared with standard yields, what matters is not the absolute uncertainties in the yield determinations, but any differences in the yield determinations between samples and standards. We expect no bias in the yield determination between the two, since the chemistry that could lead to losses was the same for samples and standards. The uncertainty due to differences in yield determination between samples and standards was estimated by comparing the mean ratio of Y1/Y2 for the samples with that of the standards. For the samples, this ratio was equal to 0.9914 ( 0.0024 (1s), and for the standards it was 0.9900 ( 0.0017 (1s). Hence, the average difference between the two methods of yield determination was the same for both samples and standards. The ratio of the sample mean difference to standard mean difference is 0.9914/0.9900 ) 1.000 95 ( 0.000 98, where the uncertainty is calculated as the combined standard deviation of the mean of the two numbers. The relative uncertainty due to differences in yield determination is taken as 100 × 0.000 98/1.0095, which is approximately equal to 0.1%. (8) Lund/LBNL Nuclear Data Search, at http://nucleardata.nuclear.lu.se/ nucleardata/toi/.

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Type B Uncertainties. Neutron Self-Shielding. Neutron selfshielding arises from absorption of neutrons in a target during irradiation. If the target contains significant quantities of nuclei with large neutron absorption cross sections, then the inner portion of the target may experience a different effective neutron flux than the outer portion. Furthermore, targets that undergo different amounts of self-shielding may experience different effective neutron fluxes. The magnitude of the self-shielding effect depends on both the geometry and neutron absorbing power of the target. For our measurements, self-shielding effects were very small in both silicon targets and aluminum foil standards. Greenberg et al.2 estimated the self-shielding differences between silicon samples and arsenic standards deposited on filter paper (cellulose), which were stacked in an irradiation geometry similar to the one used here. The uncertainty arising from self-shielding differences was estimated at 0.024%, which we will use as an approximation for the uncertainty for our measurements. Our actual uncertainty is probably even lower than this, since our aluminum foil standards are closer in neutron absorbing power to silicon than are the cellulose standards used for arsenic. Irradiation Geometry. Irradiation geometry uncertainties arise from the fact that the neutron fluence rate is not the same at all locations in the rabbit, due to a flux gradient in the irradiation tube. Differences in total neutron fluence between targets were minimized by flipping the rabbit at the midpoint of the irradiation. We approximated the irradiation geometry uncertainty from previous measurements of the fluence rate gradient in the same irradiation facility.2 Previously, the smallest variation observed for samples irradiated in RT-1 was 0.2% (1s relative). While this may reflect some good-day effect, the observed standard deviation includes variation due to many factors other than irradiation geometry. For our measurements, the uncertainty in the mean value due to geometry differences for the 12 samples may be estimated as 0.2%/x12 ) 0.058%. Similarly, the uncertainty for the mean of the 6 standards is 0.2% /x6 ) 0.081%. The total irradiation geometry uncertainty is then calculated as x(0.0582 + 0.0812) ) 0.10%. β Self-Absorption. Corrections for self-absorption of β particles in the Mg(NH4)PO4‚6H2O precipitate are necessary to account for the fact that the thickness of the precipitate was not the same for all samples and standards. Since the ratio of sample to standard activities is calculated, it is not necessary that selfabsorption be zero, but only that the amount of self-absorption in standard and sample pellets be comparable, i.e., that the thickness of samples and standards be the same. Since the geometry of each sample or standard precipitate was a 12.7 mm diameter pellet, the thickness of the pellet was assumed to be proportional to the mass of the precipitate, which varied from about 41 to 51 mg in the samples and standards counted. Using β self-absorption curves, prepared from data obtained from phosphorus standards as described previously,4 the count rate for each sample or standard analyzed was adjusted to that expected for a 45 mg precipitate. Relative correction factors for β self-absorption ranged from 0.04% to about 1.1%. Uncertainties on these correction factors were calculated from curve fit uncertainties (given by the peak fit program). The overall uncertainty due to β self-absorption corrections was taken as the average of the 12 values, which was equal to 0.07%.

Table 2. Uncertainty Evaluation for Determination of Phosphorus in SRM 2133

uncertainty (1s) %

source of uncertainty

Type A Uncertainty measurement replication 0.63 and P heterogeneity measurement of area blank correction measurement replication (standards) (s/xn) volumetric calibrations (standards) concentrated std solution mass std solution on foil blank correction (standards) radiochemical purity carrier yield determination combined type A

0.19

0.012 0.022 0.47

0.012 0.022 0.47

0.12

0.12

0.15 0.037 0.019 0.13 0.10 0.83

0.15 0.037 0.019 0.13 0.10 0.57

Type B Uncertainty β self-absorption (samples) 0.07 β self-absorption (standards) 0.08 irradiation geometry 0.10 self-shielding correction 0.024 combined type B 0.14 combined uncertainty 0.84 coverage factor 2 expanded uncertainty 1.68 final value

uncertainty (1s) (%) assuming homogeneous P concentration

(9.58 ( 0.16) × 1014 atoms‚cm-2

0.07 0.08 0.10 0.024 0.14 0.59 2 1.17 (9.58 ( 0.11) × 1014 atoms‚cm-2

Expanded Uncertainty. Uncertainties associated with measurement of phosphorus in SRM 2133 are summarized in Table 2. An expanded uncertainty was evaluated using guidelines set by the International Organization for Standardization (ISO).9 All type A and B uncertainties were added in quadrature to obtain a combined uncertainty of 0.84% for the measurements. Using a coverage factor of 2, the relative expanded uncertainty was calculated as 1.68%. The final value for the phosphorus concentration in the silicon was determined to be 9.58 ( 0.16 atoms‚cm-2. (9) Guide to the Expression of Uncertainty in Measurement, 1st ed.; ISO: Switzerland, 1993.

The uncertainty associated with the measurement method alone, assuming that the phosphorus concentration was completely homogeneous across the wafer, is estimated in the second column of Table 2. These results use the residual standard deviation from the fit of the regression model (0.0635 × 1014 atoms‚cm-2), which estimates the amount of uncertainty as though there were no heterogeneity, divided by x12 to convert it to the standard uncertainty of the mean of 12 homogeneous samples. This estimate, with an expanded uncertainty of 1.17%, while not applicable to the SRM, provides an indication of the performance of the measurement method without including the spatial variation in the phosphorus concentration in the material. CONCLUSIONS We have successfully applied RNAA as a primary method for the certification of phosphorus concentration in SRM 2133. Certification of SRMs at NIST via a primary method requires evaluation of all sources of uncertainty as well as confirmation by another method. We have evaluated all potential sources of uncertainty. We have ascertained the radiochemical purity of the 32P by following the count rate of samples and standards for 4 half-lives. The method used to certify phosphorus concentration in SRM 2133 has been used to determine phosphorus in silicon measured in a round-robin SIMS investigation. A phosphorus concentration of (8.30 ( 0.14) × 1014 atoms‚cm-2 (expanded uncertainty) was obtained, in agreement with the SIMS mean of (8.40 ( 1.55) × 1014 atoms‚cm-2 (1s uncertainty). The observed relative expanded uncertainty of 1.68% obtained for SRM 2133 will allow the semiconductor industry to achieve low systematic error for dopant concentration measurements specified in the 2001 International Technology Roadmap for Semiconductors.10 ACKNOWLEDGMENT We gratefully acknowledge the staff of the NIST Center for Neutron Research for their assistance in performing the irradiations. We also thank Robert Greenberg for his advice and assistance with this work. Received for review February 27, 2003. Accepted May 27, 2003. AC0342018 (10) http://public.itrs.net/Files/2001ITRS/Home.htm, accessed Jan 2003.

Analytical Chemistry, Vol. 75, No. 16, August 15, 2003

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