2329
Anal. Chem. 1991, 63,2329-2330 (34) H e , R. chrcwMtogephy/~ovkKTensfom, Infrared S p e c t m w y and Its AppkaMDns; Marcel Dekker Inc: New York; 1990. (35) Wall, D. L.; Mantz, A. W. Appl. S p c m c . 1877, 31, 552-560. (38) K M a , L. J. mnd-m~ s s e y s Marcel ; Dekker Inc: MW york, 1985. (37) Salmaln, M. Ph.D. lhesls, UnlversR6 Parls V I , 1990. (38) phlbmkr, V. Ph.D. Thesis. UnhrersR6 Paris V I . 1991.
RECEIVED for review March 26,1991.Accepted June 14,1991.
This research was performed in part under the auspices of a France-Quebec exchange grant in biotechnology (A.V. and I.S.B.) and a France-QuBbec sabbatical fellowship than enabled I.S.B. to work in Paris. Research grants from the following research agencies are also gratefully acknowledged: ANVAR, CNRS, and Rggion Bourgogne (France); NSERC (Canada); FCAR (Quebec).
Achieving Transferable Multivariate Spectral Calibration Models: Demonstration with Infrared Spectra of Thin-Film Dielectrics on Silicon Indira 5. Adhihetty, Joseph A. McGuire, Boonsri Wangmaneerat, and Thomas M. Niemczyk* Department of Chemistry, University of New Mexico, Albuquerque, New Mexico 87131
David M. Haaland* Sandia National Laboratories, Albuquerque, New Mexico 87185
Muitlvarlate calkatlon methods are very useful in improving the p", accuracy, and roWabMty of quantitativeqwtrai analyses. I n thk study, the transfer of partial least-squares (PLS) cailkatbn models between spectrometerswas investigated for tho quantitatlve infrared analyrls of borophorphorliicate glass (BPSQ) thin films on silicon wafers. I n the cktermlnatbn of phosphorucr, boron, and thickness for BPSG films, sensitivlty studies showed that detector nonlinearity, frequency accuracy, incident angle, and variations in purge were important parameters to control to obtain transferable calibrations. The difficulty in achieving a transferable caiikatlon modd was found to increase as the compkxlty of the calibration model increased. A combination of controlling semltlve experimental parameters, selecting approprlato frequencies, and subtracting the spectra of purge gases improved the standard error of prediction for phosphorus determination from 0.75 to 0.18 wt % when predicting on qmctra collected on a spectrometer dmerent from that upon which the calibration model was based. This improvement in prediction ability resulted in a PLS calibration model that was transferable between two spectrometers. I t has been shown that sensitive outlier detection methods can be used to identify model transfer problems. The ability to achieve calibration modek that are robust to spectrometer variations implies that these calibrations can also be robust to spectrometer drift.
INTRODUCTION In almost any instrumental procedure designed to make a quantitative measurement, the first step involves performing a calibration. The calibration is performed by measuring the desired property or analyte concentration for a variety of "standard" samples using generally well-defined experimental parameters. The calibration standards are samples that should be representative of future unknown samples in terms of composition and are such that thedesired property or analyte concentration is known or can be determined by an inde-
pendent reference method. The calibration establishes an empirical model relating the instrumental measurement to the property or concentration of interest. This empirical calibration model is then used with the instrumental measurement of an unknown sample in a prediction phase to estimate the analyte concentration or sample property for which the calibration was developed. Spectrophotometry is an example of an instrumental technique commonly used to rapidly but indirectly measure analyte concentrations. The relationship between the sample absorbance, the instrumentally measured quantity, and the analyte concentration is often given by Beer's law
A = cbc where A is the absorbance, c is the molar absorptivity, b is the path length, and c is the component Concentration. Equation 1 is only valid for a system in which the component of interest is the only component to absorb at the selected wavelengths. In spectroscopy, a calibration is performed by measuring the absorption of a set of standards of known concentration using a fixed set of instrumental conditions, in particular the measurement frequency and bandwidth and a known sample path length. The measurement frequency is important because the molar absorption coefficient can be critically dependent on frequency. An unknown determination is performed by measurement of the unknown sample absorption using the same instrumental conditions and comparing the measured absorption to the calibration established with the standards. In theory, any spectrophotometer should be capable of measuring accurately the absorbance of a particular sample. According to Beer's law, the absorbance should be independent of measurement conditions if the frequency and bandwidths are fixed and the path length is known. If this were the case, the standardization step could be eliminated once c for the desired component was determined. c would be determined in the initial calibration and 'transferred" to other spectrophotometers. In practice, transfer of a calibration from spectrophotometer to spectrophotometer must be per-
000&2700/01/0363-2329502.5010 @ 1991 Amerlcan Chemlcal Soclety
2330 ANALYTICAL CHEMISTRY, VOL. 63,NO. 20, OCTOBER 15, 1991
formed with great caution. Even spectrophotometer drift with time can be considered comparable to changing spectrometers and must be carefully considered. Transfer of a calibration between spectrophotometers is a complex issue and has not received a great deal of attention. There are situations, however, where it is highly desirable, even necessary, to facilitate a calibration transfer between spectrometers. In any case, the elimination or reduction of the calibration procedure would minimize the expense of an often tedious and timeconsuming step. Of course, any measures developed for improving the transferability of the calibration model between spectrophotometers would simultaneously make the calibration more robust to spectrometer drift. This paper will give a detailed discussion of the transferability between spectrophotometers of a multivariate Calibration of Fourier transform infrared (FTIR) spectral data of borophosphosilicate glass (BPSG) thin films on silicon wafers. These thin dielectric films are of great importance in the manufacture of integrated circuits. Full-spectrum multivariate calibration methods used in this study can be quite sensitive to spectral detail. Because they can be sensitive to minor spectral variations, they may pick up the small instrument-dependent effects that are present in real spectrometers. Although the model used in the multivariate calibration is more complex than the simple univariate Beer's law model discussed above, the concepts involved in transferability o f t in eq 1 between spectrometers are analogous. The BPSG samples discussed here are relatively complex samples whose properties change dramatically with changes in the B and P concentrations. Because of the importance of these films to the microelectronics industry and the fact that the film properties can be tailored by controlling the B and P concentrations, a great deal of effort has been placed on performing determinations of the B and P content of both annealed and unannealed BPSG films (1-5). Analytical techniques employed have included inductively coupled plasma emission spectroscopy (6, 7), ion chromatography (4, 7,8), X-ray fluorescence spectroscopy (1,9), colorimetry (1, lo), and infrared spectroscopy (1, 4, 8, 11, 12). Fourier transform infrared spectral analysis is rapid and nondestructive; hence it offers a significant advantage when compared to the other analytical techniques. The infrared spectra of these films are, however, typical of thin-film glasses in that the bands are broad and overlapping. Quantitative measurements based on such spectra are difficult, especially if a traditional univariate measurement, e.g., peak intensity at a single frequency, is used to extract the desired information. We have shown, however, that it is possible to efficiently extract quantitative information from spectra if a multivariate calibration is employed (12-14). However, transfer of a calibration for a system as complex as BPSG is likely to be more difficult than transferring a simple univariate model. Previous discussions of transferability have dealt with near-infrared spectroscopy (15-17). Most of these methods focus on modifying the calibration model, e.g., software adjustments to allow for instrumental differences when going from one spectrometer to another. However, it is our opinion that experimental parameters that affect the transfer of the calibration should be identified and controlled before software adjustments are made. Therefore, this paper focuses on the experimental parameters that can be controlled or changed to make the calibration models more robust to transfer. In the discussion that follows, we have identified, in part through sensitivity studies, those instrumental variables that have a significant effect on the calibration transfer. Once the influential parameters have been identified, care can be taken to control them precisely. Finally, some data treatment steps that can be used to minimize some problems will be presented.
The combination of careful control of experimental parameters and judicious data treatment leads to a transferable calibration for the spectrometers used in these studies. EXPERIMENTAL SECTION A set of 38 samples derived from a 44-wafer BPSG sample set previously described (12)were used for our transferability studies. Six of the original samples were completely consumed during chemical analysis by inductively coupled plasma emission spectroscopy and were unavailable for further analysis. In this calibration set, the P concentration ranges from 2 to 6 wt %, and boron from 1 to 5 wt %. The thicknesses of the samples vary over a range of 430-1000 nm. Two different Nicolet FTIR instrumentswere used in the study. Primary studies involved two spectral data sets. One set was obtained by using the Nicolet 6000 FTIR spectrometer at the University of New Mexico (SPEC1)with a liquid nitrogen cooled Hg-Cd-Te (MCT) detector. The detectors are supplied by the manufacturer with an integral preamplifier. An MCT detector is inherently nonlinear (18,19).The response function of the detector considered here is the output of the detector/preamplifier combination versus input intensity. Some, especially older detector/ preamplifier combinations, were very nonlinear. Such a system is employed in some of the studies reported here and is referred to as "nonlinear". The manufacturers have developed preamplifiers that correct for the MCT nonlinearity to produce a more linear response function. The second detector/preampliiier used here and referred to as "linear" was such a detector. Ita response function is more linear, but there is evidence that some nonlinearity remains. The second spectrometer was a Nicolet 7199 FTIR at Sandia National Laboratories (SPEC2)equipped with the more linear MCT detector. Samples were mounted in either spectrometer on a rotatable sample holder that could be reproducibly adjusted to f O . l O . All the spectral data were collected at either 0 or 60Dincident angle by using a resolution of 4 cm-' and a 6-mm source aperture. A total of 32 scans were signalaveraged for each sample and background spectrum. To increase data collection efficiency, the sample compartment of the FTIR system was not purged at the time spectra were taken. However, the rest of the spectrometerwas purged with either dry air or dry nitrogen, While p-polarized light was used in ref 12, unpolarized spectra were collected in this study. SPEC1 was used for instrument-dependent sensitivity studies for those parameters considered to be potentially important for transfer of the multivariate calibration model between spectrometers. All data were analyzed on a DEC Microvax computer using multivariate calibration FORTRAN software written at Sandia National Laboratories (12-14). Partial least-squares (PLS) methods were used in the analyses with both concentration and spectral data mean-centered. RESULTS AND DISCUSSION Multivariate Calibration. Multivariate statistical methods can sometimes be used to obtain quantitative information from spectral data that would not produce good results if a univariate model was employed. Different multivariate statistical methods can be used for the determination of concentration or physical properties from IR spectral data. Among the multivariate methods are factor-analysis-based methods that include partial least squares (PLS) (13,14, 20-22)and principal component regression (PCR) (13,14,231 methods. PLS and PCR often have comparable prediction abilities (12,22,24). In our studies, PLS analysis was employed because it has greater computational efficiency and generally yields a simpler calibration model when compared to PCR. Uncorrected IR absorbance spectra of the 38 annealed BPSG films were analyzed from 450 to 1600 cm-' by using the PLS software. The precision of the results based on independent calibrations performed on data obtained from each spectrometer are presented in Table I for B concentration, P concentration, and film thickness. The results are expressed as the cross-validated standard error of prediction (SEP(CV))and standard error of calibration (SEC). Cross-
ANALYTICAL CHEMISTRY, VOL.
63,NO. 20, OCTOBER 15, 1991 2331
Table I. Calibration Results as a Function of Incidence Angle for Two FTIR Spectrometers Based on 38 BPSG Samplesa
calibration model SPECl O0
60'
amt of phosphorus,
amt of boron,
wt%
wt%
W 1.5
thickness, nm
SEP(CV) SEC SEP(CV) SEC SEP(CV) SEC 0.20 0.22
0.07 0.09
0.16 0.10
0.06 0.06
30 37
28 34
0.27 0.20
0.12 0.07
0.19 0.13
0.16 0.06
32 24
30 12
-
u
-
$ m
-
m
si-oh
-
Ya
-
a 1.0 -
SPEC2 O0
60'
SEP(CV) = cross-validated calibration standard error of prediction. SEC = standard error of calibration (for full calibration).
I
I
I
I
r
400
I
- I
I
,
so0
,
I
1
8
1200
FREQUENCY (cm-1)
validation (CV) is an internal validation method that yields independent estimates of the concentration of each of the samples in the calibration set (12, 13). Cross-validationwas accomplished in this ca8e by removing one sample, developing a calibration model based on the remaining samples, and then predicting the sample left out of the calibration. This process is repeated until all sample concentrations have been predicted. The SEC is determined on the basis of the concentrations of the samples estimated, while all calibration samples are retained in the model. SEP for the cross-validated model is defined by eq 2, where n n
Flguro 1. Infrared spectrum of film on a silicon wafer.
Table 11. Repeatability and Reproducibility of Concentration Predictions Based on Spectra from a Single Wafer Using SPECl with a Oo Incidence Angle std error of prediction
movement
position
fixed
same
inlout
same
inlout represents the number of samples used in the calibration model, ci represents the reference concentration of the ith sample, and c^i represents the estimated concentration of the ith sample based on the calibration model for the ith sample left out of the calibration. Notice that the results in Table I show that the full-calibration SEC is always smaller than the cross-validated SEP (SEP(CV)). This is to be expected since SEP(CV) involves predictions on independent samples and because the calibration model has to be extrapolated for some predictions during cross-validation. In addition, the full-calibration SEC represents an overfitted model since all the predicted samples are included in the calibration model. Therefore, SEP(CV) gives a better measure of the predictive ability of future unknown samples because the predicted sample is not included in the model. The cross-validation procedure permits a more sensitive detection of problem or outlier samples within the calibration set. Of course, predictions made on future unknown samples will be based on a calibration model developed from d l nonoutlier calibration samples. Table I includes the SEC, and SEP(CV) for spectra obtained a t both 60 and Oo incident angles. Notice that the analysis precision for spectra collected at 60° incident angle is generally better. At 60°, the effective path length through the film is greater when compared to the Oo data. In addition, the greater incident angle provides a different interaction between the film and the electromagnetic field of the radiation primarily through reflection changes resulting from the dispersion of the index of refraction that occurs a t each absorption band. Therefore, more molecular information can be extracted from the thin film at the 60° incident angle. The calibration precision in Table I is not as high as reported in ref 12 for thickness even though the same set of samples was used in both cases. One possible reason might be the nonuniformity of these samples. Each sample has a thickness variation of about 1 5 4 0 nm as well as small composition variations. In the previous study, both thickness measure-
a borophosphosilicate glass (BPSG)
diff
amtof phosphorus,
amtof boron,
wt %
wt%
nm
0.02 0.03 0.06
0.016 0.013 0.035
0.2 0.3 16.5
thickness,
ments by ellipsometry (the reference method) and FTIR were performed at the same position on the sample. But we were unable to measure the same exact spot in the sample that had been measured by ellipsometry; those portions of the wafers were destroyed to perform the concentration determinations by emission spectroscopy. Therefore, thickness results are not as precise as in the previous study. It can be commented that, in these thin-film samples on Si wafers, thickneas changes cause baseline variations and nonlinear interactions with the absorption bands that allow film concentrations and thicknesses to be independently determined (12). On the basis of the calibration results in Table I, we can conclude that the FTIR precision for the determination of boron is somewhat better than that of phosphorus. Poorer precision in the determination of phosphorus is probably a result of greater nonlinearities for the phosphorus absorptions combined with a low relative intensity for the P=O band and greater overlap with large B-O and S i 4 bands. This overlap and low relative intensity can be observed in the infrared spectrum of the BPSG film given in Figure 1. Phosphorus is also expected to have a greater tendency to change with time due to reactions with water vapor. This potential reaction might also affect the transferability of the calibration model for phosphorus. From the results in Table I, it is also clear that two different spectrometers yield different prediction abilities for the same sample set. This observation leads us to examine the precision of the FTIR spectral measurements within a spectrometer and the instrumental factors that affect the precision. Single-WaferPrecision. If two ideal spectrophotometers could be constructed, the absorption spectrum measured on each for the same sample would be identical within the noise limits of the measurements. Hence, it is important to establish what the effects of the noise limits are i.e., what is the best precision that can be achieved based on the presence of spectral noise. An assessment of the spectral noise limit and how this noise affects the prediction precision can be made
2332
ANALYTICAL CHEMISTRY, VOL. 63, NO. 20, OCTOBER 15, 1991
SOURCE
INTERFEROMETER
SAMPLE
D€rECTOR/AMPLIFIER
APERTURE
ALIGNMENT
INCIDENT
ALIGNMENT INTENSIN
RESOLUTIONS MIRROR VELOCIN
ANGLE PURGE
NPE LINEARIN GAIN ELECTRONIC FILTERS
FREQUENCY ACCURACY
SCAlTER
:
2.0 /I
tn
tn
tn 160 tn
n tn
I! I
q+ + o r
I-
IL
DIFFERENCES IN SPECTRAL MEASUREMENTS
CALIBRATION TRANSFER
Flgure 2. Spectrometer-relatedfactors affecting the transferability of the calibration model.
by performing repeat measurements/predictions on the same sample. The results of the repeat measurement/predictions of phosphorus, boron, and film thickness using a single wafer are given in Table 11. The first row of the table shows the precision of concentration predictions obtained by placing a sample in the spectrometer, measuring the spectrum and then performing the PLS prediction based upon the model derived from the calibration data. The process was repeated 10 times over a period of 20 min, without moving the sample between spectral acquisitions. These data yield the precision of concentration predictions based on FTIR measurements that is a result of spectral noise and short-term reproducibility of the spectrometer. As shown in Table 11, the inherent precision of the FTIR measurements is very high. This demonstrated precision is nearly 1 order of magnitude better than the reference methods used to measure the concentrations of the standards. Comparing the second row in Table I1 with the first row illustrates that moving the sample in and out of the spectrometer does not significantly decrease analysis precision when an attempt is made to reposition the sample such that the IR beam passes through the same location on the sample. The entries in the third row of the table are based on repeat measurements on 10 different locations on the sample, and these results coupled with those from the second row of Table I1 can yield a measure of the uniformity of the thin-film samples. The decrease in the precision indicated in the third row of the table illustrates problems associated with concentration and thickness variations over the wafer. The results presented were obtained by using one-fourth of a single 100-mm wafer that exhibited greater thickness uniformity than most of the calibration samples. Even though this is one of the more uniform samples, the film thickness variation is relatively large over the wafer. These thickness variations will affect the calibration model for film thickness. The precision results for 38 BPSG wafers given in Table I can be no better than the reproducibility meaaurements in the third row of Table I1 based on a single relatively uniform wafer since P and B concentrations and thickness vary over the samples. The analysis precision for the whole set is expected to be worse than that presented in the third row of Table I1 since the data in Table I1 are based on a relatively uniform sample. Sensitivity Studies. There are many instrumental variables that can cause significant or subtle differences in measured spectra. FTIR spectrometers are similar to all spectrometers in this sense. Figure 2 illustrates some of the spectrometer variables that might affect the measurement of a spectrum and hence the transferability of the calibration model. In Figure 2, the spectrometer variables are divided
-6
-4
+
da
+
-2
0
2
58
+
4
6
FREQUENCY SHIFT (cm' )
Figure 3. SEP dependence on wavenumber shift for a subset of the calibration samples. The y axes for the two variables have k n scaled so that they span the same variance range: (0) phosphorus; (+) thickness.
into four areas, each associated with a particular section of the instrument. Variables in each section were investigated as to the degree to which each affects the precision of the measurements. Fourier transform infrared spectrometers generally have very good frequency precision. The frequency accuracy is not as good as the frequency precision. The size of the source aperture and alignment of the source and interferometer optics can affect the frequency accuracy of the spectrometer. Any frequency shift between spectrometers can greatly influence the transfer of the calibration model, especially if sharp spectral features are included in the model. The effect of frequency shift on precision between calibration and prediction was studied by shifting a particular frequency from +3 to -3 data points (2 cm-'/data point) by adding/subtracting data points using software created in our laboratory. Figure 3 demonstrates the effect of spectral shifts on phosphorus and thickness precision for a subset of sample spectra. From this figure, it is clear that the wavelength accuracy can have a large impact on calibration transfer for phosphorus, but thickness precision is relatively insensitiveto frequency shifts. The two spectrometers used in this study differed in frequency by about 0.12 cm-l, as measured by the differences in the frequencies of the sharp water vapor bands in the spectra. If the linear relationship shown in Figure 3 for phosphorus is applicable for small frequency shifts, this 0.12-cm-' shift would cause a 0.1 w t % degradation in the SEP for phosphorus. In the sample section of the spectrometer, there are a number of parameters that could affect the calibration precision. The angle of incidence of the infrared beam on the sample and the quality of the sample compartment purge are factors that can be controlled. The samples studied here are BPSG films deposited on silicon wafers. Only the front sides of the wafers were polished, hence the wafers tend to scatter a significant amount of the infrared radiation. The degree of scattering is a sample-dependent phenomenon and cannot be controlled by any instrumental parameter. The critical factor is the amount of scattering that reaches the detector. If two different spectrometers collect a different fraction of the scattered radiation, the calibration will be affected. The likely result of a difference in the amount of scattered light collected would be a spectral baseline shift. The quality of purge can be controlled by fully purging the spectrometer and waiting a sufficient amount of time between sample changes and spectral acquisition. In order to accommodate high sample throughput, we left the sample compartment open to the atmosphere while spectra were collected. The main portion of the spectrometer was well purged, but the amount of C02and water vapor in the sample compartment could vary greatly during a day, or from day-to-day or site-to-site. As will be shown below, the changes in the atmospheric absorption peaks do affect the transfer of the
q
ANALYTICAL CHEMISTRY, VOL. 63, NO. 20, OCTOBER 15, 1991
Table 111. Summary of Predictions for Phosphorus, Boron, and Thickness"
t
5 2.0
2333
calibration prediction SEP(C), SEP(P), bias, SDP, wt % wt % wt % spectrometer spectrometer wt %
+
B
Phosphorus
z j
+
Cn
8 - 0 0
0
t
++
0 0 0
+ + h.h
@
Brh
0 0 0
I
SPECl SPEC2
SPECl SPECl
SPECl SPEC2
SPECl SPECl
SPECl SPEC2
SPECl SPECl
0.20 0.27
0.12 0.75
0.06 0.51
0.12 0.55
0.10 0.22
0.02 -0.13
0.09 0.18
Boron 0.16 0.19
Thickness
0
30 32
32 31
5.6 9.3
32 30
aSEP(C) = standard error of prediction for calibration step. SEP(P) = standard error of prediction for prediction step. SDP = standard deviation of prediction (corrected for intercept bias).
Systematic prediction errors or bias can be a major contributor to the SEP(P). The bias can be estimated by eq 3. n
bias =
(l/r~)z(C*~ - ci) i=l
(3)
If all errors are random, the bias should be approximately zero. The SEP after correcting for the bias is labeled standard deviation of prediction (SDP). SDP can be calculated for separate prediction samples by using the following equation: n
SDP = [(l/(n - l))C(Zi- ci - b i a ~ ) ~ ] ' / ~ (4) is1
Table I11 summarizes the results for a transferability experiment for spectra collected at Oo incident angle. The data in the rows of the table, identifying SPECl as the spectrometer used for both calibration and prediction, were obtained by performing the prediction step on spectra collected on the same samples 6 months after the original calibration spectra were obtained. Results from this experiment indicate a similar prediction ability for B, P, and thickness for spectra obtained six months after the calibration model was developed. Obviously, any spectrometer changes or drift over this 6-month period did not affect the calibration transfer even though the beam splitter alignment was changed, detectors were moved in and out, and the reference helium-neon laser was replaced during this time. In addition, these results would suggest that these annealed BPSG samples are stable over the 6-month time frame. When SPEC2 is used for calibration and SPECl is used for prediction, the results are not nearly as encouraging. These data were obtained with a linear detector/preamplifier in SPEC2 and a nonlinear detectorlpreamplifier in SPEC1. As shown in Table 111,the thickness calibration model appears to transfer between spectrometers, but the P calibration model transfers quite poorly. The boron model also does not transfer adequately since the SEP(P)is larger rather than smaller than the SEP(C), as expected for a transferable model. As we expected from our sensitivity studies, the results are significantly poorer for the transfer of calibration models for spectra collected at 60° incident angle on both spectrometers (Le., P prediction precision at 60° incident angle is 1.83 wt 9% for spectra obtained on the second spectrometer). It is clear that the P concentration model is most sensitive to differences in spectrometers. Therefore, we will focus on the transfer of the P concentration calibration model in the remainder of this paper. If a plot of the predicted property w reference property (e.g., phosphorus concentration versus the reference phosphorus determination) is made, the data should fall about a straight line with a slope of 1 and an intercept of 0. If the points fall
2334
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ANALYTICAL CHEMISTRY, VOL. 63, NO. 20, OCTOBER 15, 1991
I
8
- a 1.5 -
z
a
400
800
1200
1600
FREQUENCY (cm-1)
~~
400
1200 FREQUENCY (cm-1)
800
1600
Average spectra of 38 BPSG calibration samples obtained on SPEC2 (A) and SPECl (B). Also shown are dmerence spectra for SPECl minus SPEC2 on the same scale as the average spectra (C) and scale expanded (but not shlfted) by a factor of 10 (D). Figure 5.
consistently above or below this line, the predictions are said to be biased. The estimated bias (from eq 3) is included in Table I11 for each property determination. The biases in the predictions are small when both the calibration and prediction were performed by using SPEC1. Also, the estimated biases reported for thickness predictions baaed on SPEC2 calibration applied to SPECl spectral data in Table I11 are very small, again indicating that the model for thickness is transferable. Note, however, that SPECl predictions based on SPEC2 calibrations for both boron and phosphorus are significantly biased. One way to correct for potential calibration-transfer problems is to empirically correct for the bias. The biascorrected standard deviation of prediction (SDP) is included in Table 111. The SDP results for the thickness prediction do not improve since the bias makes only a small contribution to the SEP(P). Although the results for boron and phosphorus improve significantly, especially in the case of phosphorus, the model is not made transferable by this correction. It will be required to identify and eliminate the source of bias rather than attempt to correct for it. If the cause of the bias can be eliminated, the calibration model will certainly be more transferable. Insight into the problem of why the calibration model does not transfer can be obtained from several sources. One source of information is a comparison of an identical sample spectra, as obtained on the different spectrometers. Figure 5 contains such a comparison, but in this case, the average spectra for the 38 wafer sample set, as measured by using SPECl and SPEC2, are compared. The differences can be seen most dramatically in the difference spectrum scale expanded by a factor of 10. If the spectrometers were identical, the difference would show only spectrometer noise and perhaps any sample nonuniformities encountered. Using the average spectra minimizes the effect of sample nonuniformities. It is clear that the difference contains features that are much greater than the spectral noise. Two major bands in the spectra, the Si-Si phonon band of the Si wafer between 600 and 650 cm-' and the Si-0 stretching band of the BPSG film between lo00 and 1100 cm-' both show significant differences. Further, there is a baseline shift between the spectra obtained on the two spectrometers and evidence of differences in water vapor bands above 1300 cm-'. The presence of a difference in water vapor between the calibration and prediction steps is not surprising because no effort was made to eliminate water vapor from the sample compartment when the spectra were measured. Even if the amount of water vapor in the sample compartments was the same, the difference spectrum might still indicate problems.
400
800
1200
1600
FREQUENCY (cm-1) Flgufe 6. Effect of thickness on Si-0 band nonlinearttles: (A) infrared spectrum of a thick BPSG sample using SPEC1; (B) infrared Spectrum of the same thick BPSG sample using SPEC2; (C) infrared spectrum Infrared spectrum of the same of a thin BPSG sample using SPEC1; 0) thin WSG sample using SPEC2; (E) dlfference between A and B (X10); (F) difference between C and D (X10).
The sharp water vapor bands do not subtract perfectly due to the small frequency shift (0.12 cm-') between the spectrometers. The effect of this frequency shift is minimal for the broad peaks of the BPSG film and the baseline shift. The broad differences observed in Figure 5 are most likely related to differences in the detector/ preamplifiers and the detector optics employed in the two spectrometers. Peaks of high absorbance (e.g., the S i 4 peak) experience more spectrometer related nonlinearities than bands of low absorbance (14). The fact that SPECl employs a nonlinear detector/ preamplifier and SPEC2 a linear one probably accounts for much of the difference observed in this band. Figure 6 demonstrates that nonlinearities in the S i 4 region become more pronounced when the samples get thicker. The intensity of the Si-0 band in the difference spectrum E (difference between spectra recorded by SPECl and SPEC2 for a thick sample) is much more intense than in F (difference spectrum for a thin sample). Finally, the fact that these are scattering samples and the optics between the sample and the detector are not identical might contribute to the baseline shift. Another method to obtain insight into the nature of the calibration-transfer problem is to examine samples identified as outliers. Outliers in the calibration set are samples that are unlike the other samples or are unduly influential in the development of the calibration model. During prediction, outliers are samples that do not follow the model developed in the calibration. In the prediction step, outliers can be detected by a number of methods including spectral F ratios, Mahalanobis distances, and spectral residuals (12,13,23). The spectral F ratio indicates how well an unknown spectrum is fit by the calibration model. The spectral F ratio is the ratio of the sum of squared spectral residuals for the fit of the unknown sample spectrum to the average of the sum of squared residuals of the cross-validated calibration spectral estimates. In an earlier study, empirical evidence suggested that F ratios above 3 might be appropriate to flag outlier
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ANALYTICAL CHEMISTRY, VOL. 63, NO. 20, OCTOBER 15, 1991
+
h
E
v
w I-
1
++ /
6;
n
n
2335
4-
o -
s
400
E 2-
i
0
0
I
/
2
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4
S
REFERENCE P (W%)
Flguro 7. Predlcted versus reference concentration for phosphorus when the SPEC2 callbratlon model is applled to spectra of the same set of BPSG samples obtained on SPEC1. The llne represents the expected flt If the calibration model were fully transferable. 0 represents samples wlth an F ratio less than 2.5 and represents samples with an F ratlo greater than 2.5.
8
0.02
-
0.01
-
0.00
-
-0.01
-
-0.02
-
w
m
s3
+
400
samples for this original set of data (13). Using this criterion, 13 of the 38 samples were flagged as outliers when the phosphorus prediction was attempted. The presence of this large number of outliers is reflected in the poor (0.75wt %) SEP(P)for P in Table 111,further indicating a poor transfer of the calibration model between spectrometers for phosphorus. After removal of the prediction results from these 13 outliers, the calibration for P was recomputed, and the prediction precision improved to 0.37 w t % P. If we were to lower the empirical F ratio threshold to 2.5,then an additional 4 samples would be indicated as outliem. The 17 samples with larger F ratios also can be clearly identified (i.e., the samples that deviate from the partial least-squares model) in the P prediction model given in Figure 7. When these additional samples are removed from the analysis, the precision further improves to 0.23 wt %. This value is better than the calibration SEP obtained on the original spectrometer of 0.27wt %. Therefore, outlier detection not only identified a problem with calibration model transfer but, by removal of outlier samples with the spectral F ratio threshold set at 2.5 for this data set, the analysis of the remaining samples appears to be valid since the results are more precise than the calibration results on the parent spectrometer. Unfortunately, the appropriate spectral F ratio threahold for flagging outlier samples is data set dependent since it will be dependent on the correlation of the spectral errors. However, once selected for a given data set, the F ratio threshold should be valid for flagging future unknown samples as outliers. The problems the calibration model had in fitting these outlier prediction samples can be seen by examining the spectral residuals (i.e., the difference between the unknown spectrum and that predicted by the calibration model). Examples of some of the spectral residuals are shown ;a Figure 8. Parts b and c of Figure 8 demonstrate spectral residuals for outlier samples, while Figure 8a demonstrates a residual for a sample that is modeled well. Figure 8a shows fewer major systematic spectral features, and the sum of squared residuals is significantly smaller for the residuals in Figure 8a relative to those for Figure 8b or &. In all of the outlier residual spectra, the lack of spectral fit was largest either at low wavenumbers, the Si-Si and S i 4 bands, or the water vapor bands (or a combination). The spectral regression coefficients of the calibration model provide additional qualitative information that can be used to understand the calibration transferability issue. The dot product of the regression coefficient vector with the centered spectral vector can be used to predict the sample property
0.02
-C
0.00
.
-0.02
-
! 3
P
800 1200 FREQUENCY (cm-1)
1600
800 1200 FREQUENCY (cm-1)
1600
800 1200 FREQUENCY (cm-1)
1600
a ,
400
Spectral residuals: (a) nonoutlier sample; (b) and (c) outller samples based on the SPEC2 calibration model applied to BPSG spectra obtained on SPEC1. Flgwo 8.
(13). The regression coefficients determined during the calibration for phosphorus and thickness are shown in Figure 9. Note that the information used in predicting the thickness of the BPSG f i is primarily contained in the S i 4 stretching band. This is reasonable since the film is largely Si02,and the fraction of Si02 in the film does not vary as much as thickness (Si02ranges from 92 to 97 w t % while thickness varied from 430 to lo00 nm).Hence the intensity of thu S i 4 band correlates strongly with thickness. Note also that the regression coefficients for thickness is relatively noise free. This is primarily due to the fact that the thickness could be modeled with only four PLS factors. Thus, very little spectral noise is brought into the model for film thickness. In contrast, the regression coefficients for phosphorus show features located at the P-0 and the Si-0 stretching bands as well as in the carbon dioxide and water vapor regions. The phosphorus model required 10 PLS factors to achieve optimal predictions. When the complexity of the model increases, it incorporates more noise into the model. Compared to the regression coefficients for thickness, the coefficients for phosphorus are "noisy". This indicates that the model for P is more complex, and it contains more instrument-dependent variables. On the basis of the information in Figure 9,it is not surprising that the thickness model transfers between spectrometers while the phosphorus model does not (Table 111). Methods To Improve Transfer of the Model for BPSG Calibration. The preceding discussion suggests a number of steps that can be taken to improve calibration transfer. Perhaps the nmt obvious is to employ detectors/preamplifiem
2336
ANALYTICAL CHEMISTRY, VOL. 63, NO. 20, OCTOBER 15, 1991 0.10
1
D
400
800
1200
1600
FREQUENCY (cm-1)
A
t
-0.05 1
,
400
11,
! , I ,
I
, 800
I
,
I
in
,
, 1200
I
,
,
I .
,
I
1600
FREQUENCY (cm-1) Flgure 10. Differences of the average spectra for 38 BPSG samples. (A) difference between SPECl (nonlinear) and SPEC2 (linear); (6) difference between SPECl (linear) and SPEC2 (linear); (C) difference between SPECl (linear) and SPEC2 (linear)after removal of water; (D) difference between SPECl (linear) and SPEC2 (linear) after removal of water and clipping spectral regions.
400
800
1200
1600
FREQUENCY (cm-1) Flgus B. Regression coefficients for (a) phosphorus and (b) thickness based on the PLS calibration model for 38 BPSG samples obtained on SPECS.
Table IV. Phosphorus Results after Removing Water Vapor and Nonlinear Spectral Regions calibration
prediction
SPEC2
SPECl
MCT - linear MCT - linear MCT - linear -
waterO
MCT - linear water - clippedb MCT - MIX SPECl - SPEC2
SEP(CV), SEP(P), wt%
wt%
MCT - nonlinear MCT - linear MCT - linear water MCT - linear -
0.27 0.27 0.25
0.75 0.34 0.28
0.20
0.18
MCT - MIX SPECl - SPEC2
0.28
0.21
water - clipped
a Water: water spectrum was subtracted from sample spectrum. *Clipped: wavelength region 400-700 and 1050-1165 cm-* re-
moved.
in spectrometers that have similar response characteristics. To this end, the nonlinear detector/preamplifier in SPECl was replaced with a linear detector. The entire 38 wafer sample set was rerun with the new detector system, and the calibration model developed on SPEC2 was applied to the resulting spectra during prediction. The phosphorus results obtained in this experiment are summarized in the second row of Table IV. The first row contains the previous results using the nonlinear detector/preamplifier for comparison. The SEP for phosphorus prediction is greatly improved when the more linear detector is used in SPEC1, but the model is still not adequately transferable. The next step taken in an attempt to improve the transferability of the calibration was to perform a scaled subtraction of a water vapor spectrum from all spectra. The calibration and prediction steps were repeated after the water subtraction, and the slightly improved phosphorus calibration precision is presented in Table IV. The prediction SEP for spectra obtained on SPECl also improves following water vapor subtraction, but the model transfer is still far from ideal. This improvement is obtained because the 0.12-cm-' shift between spectrometers becomes important with the presence of sharp
water vapor bands but is not important for the broad BPSG bands in the absence of water bands. Nearly ideal calibration model transfer is achieved when the spectral regions that showed large residuals in the difference spectra are removed or clipped. The region below 700 cm-' was eliminated from the calibration. This region contains the Si-Si phonon band between 600 and 650 cm-' that is present in all difference spectra. The frequency region below the Si-Si band seems to contain primarily noise, water vapor, and an unstable baseline near the detector cutoff frequency. The other clipped region was between 1050 and 1165 cm-'. Again, this region always contains substantial signal in the difference spectra. The Si-0 band in this region is very intense and generally exhibits a nonlinear response that is concentration and thickness dependent. It should be noted that the absolute differences measured in this 1050-1165-cm-' region are very small, especially when one considers the intensity of this band. This Si-0 spectral region has little predictive power for phosphorus, so its elimination allows the model to be developed by using regions exhibiting a more linear response. The results for phosphorus following the clipping operation are also included in Table IV. Again, the calibration improves relative to the calibrations performed without clipping. Even more significant is the fact that the calibration model is now transferable between the two spectrometers. The improvement in precision presented in Table IV can also be seen in the spectral residuals in Figure 10. Figure 10 contains the average difference spectra obtained on the two spectrometers applying the methods used to obtain the results for each row in Table IV. Any spectral features above the noise observed in Figure 10 show where the spectra differ on the two spectrometers. The spectrum labeled A is the result of subtracting the average SPECl spectrum from the average SPEC2 spectrum. In this case, SPEC2 used a linear detector/preamplifier and SPECl used a nonlinear detector/ preamplifier. This is the same difference spectrum presented in Figure 5. Note that there is a significant mean difference between the average spectra; Le., the difference absorbance spectrum is negative at all frequencies. The difference spectrum labeled B results when the detector/preamplifier in SPECl is changed to one with more linear properties. Note that the difference spectrum is now nearly centered about zero absorbance. The difference spectrum labeled C shows the additional decrease in residuals between 1300 and 1500 cm-' due to subtraction of the water vapor features from the in-
ANALYTICAL CHEMISTRY, VOL. 63, NO. 20, OCTOBER 15, 1991
dividual spectrum used in obtaining the difference spectrum, B. Note that, for clarity, spectrum C has been offset from spectrum B by 0.05 absorbance units. Finally, spectrum D is the difference spectrum C after clipping spectral regions. Again, this spectrum has been offset from spectrum C for clarity. In the ideal case, the difference spectrum would show only random noise with no spectral features or baseline offset. Note that spectrum D represents primarily random noise, and this difference spectrum corresponds to the data in Table IV showing that the calibration model transfers following the detector change and data manipulations. The last row of Table IV contains the results of an experiment in which the sample set taken on both Spectrometers was divided in half. The PLS calibration was performed by using half of the spectra obtained on SPECl with a nonlinear detector and the other half on SPECP, without subtraction of water or removal of spectral regions. The predictions were carried out on sample spectra from both SPECl and SPEC2 that were not used in forming the calibration model. The results show that the model created in this fashion has good predictive properties for unknown spectra obtained on both spectrometers. Thus,by obtaining calibration spectra on both spectrometers, unique features of each spectrometer can be built into the model. This calibration can then be used to successfully predict concentrations and properties on the basis of sample spectra obtained on either spectrometer.
CONCLUSIONS Transfer of a calibration model between spectrometers has been shown to be possible. The experiments discussed here have demonstrated that the transfer of the calibration model can be greatly affected by spectrometer-dependent variables and the complexity of the model. Infrared spectra of thin solid films probably represent a rigorous case in which to test model transfer. The spectra contain overlapped bands and exhibit deviations from Beer's law. It can be difficult to perform precise quantitative determinations based on the spectra of these films. In order to make quantitative determinations under such conditions, the model developed is sometimes complex. The work presented here shows that a complex model, such as that developed for phosphorus, is more sensitive to a variety of experimental variables and hence is more difficult to transfer. A simple model, such as that for thickness, was found to be relatively insensitive to spectrometerdependent variables, and hence is more easily transferred between spectrometers. The spectrometers employed for these studies were shown, in the single-wafer studies, to be capable of producing highly reproducible spectra and predictions. These results demonstrate that the infrared-chemometric approach is capable of producing results more reproducible than the reference methods. It was also shown that many spectrometer variables can affect the transfer of the calibration model. When the more important variables for model transfer are identified, it is important that they be controlled carefully between spectrometers. It is also beneficial to operate under conditions where the model is not sensitive to small differences in spectrometer variables. It is possible that in order to achieve a transferable calibration model, precision of the analysis on the parent spectrometer may have to be sacrificed somewhat. For example, in order to avoid high precision sample alignment when working with thin solid film samples as used here, we chose to compromise prediction precision slightly by operating at Oo, where the calibration transfer is relatively insensitive to incidence angle. Detector nonlinearity was found to be an important factor limiting calibration model transfer. Even the newer more linear Hg-Cd-Te detector/preamplifiers were found to exhibit differences in linearity. In addition, frequency accuracy be-
2337
comes important if relatively sharp spectral features are incorporated into the calibration model. These and other problems in calibration model transfer could be addressed via several data treatment steps. What is needed, however, is some guide as to what the data treatment steps should entail because the steps necessary will likely vary depending on the data set. For example, a first step in identifying problem areas might involve simple subtractions of spectra obtained on the same sample with different spectrometers, a step which will quickly point out problems of detector nonlinearity or frequency accuracy. A more global method for identifying model transfer problems is the application of outlier detection methods such as spectral F ratios. However, even in the case where spectrometer differences were at their greatest, it was found that the analyses of those spectra collected on the second spectrometer yielded analysis precision comparable to the original calibration on the parent spectrometer if analyses were restricted to those spectra with spectral F ratios below 2.5. Examination of spectral residuals and spectral differences for spectra with high F ratios provides information that can effectively guide the analyst in determining what kind of data pretreatment steps should be employed to improve calibration model transfer. Finally, the spectral regression coefficients can also serve to guide the analyst in understanding the calibration model transfer problem. Modifications of the model with bias and slope corrections based upon measurement of a subset of calibration samples on the additional spectrometer can be useful for improving analysis precision for spectra collected on the new spectrometer. In addition, determining spectral transfer functions for a subset of calibration samples based on spectra obtained on the parent and secondary spectrometers will undoubtedly improve analysis precision on the additional spectrometers. However, the experimental variables that affect model transfer should be identified and controlled prior to any change in the model if complex models are to be reliably transferred between a variety of spectrometers. In fact, these other software correction methods may not even be necessary, as shown in this paper.
ACKNOWLEDGMENT We acknowledge M. T. Kay and G. R. Iben for providing the BPSG samples. A. R. Mahoney made the elipsometer measurements; M. Gonzales and W. B. Chambers performed the ICP analyses. A. Pimentel aided in the collection of spectra at Sandia National Laboratories. D. F. Taylor aided in a portion of the data analysis and facilitated the use of the multivariate software at the University of New Mexico. D. K. Melgaard contributed many software improvements throughout this work. Registry No. B, 7440-42-8; P, 7723-14-0. LITERATURE CITED (1) (2) (3) (4) (5) (6)
(7) (8) (9) (10) (11) (12)
(13) (14) (15)
Kern, W.; Schnabie, G. L. RCA Rev. 1982, 43, 423. Kern, W.; Smeker, R. K. Solid State Technol. 1985, 28 (6). 171. Avigal, I. SolM State Techno/. 1983, 26 (lo), 217. Tong, J. E.; Schertenieib, K.; Carplo, R. A. SolM State Techno/. 1984. 27(10), 161. Levy, R. A.; Nassau, K. SolM State rechnd. 1988. 29 (lo), 123. Welssman, S. H: Hallett, S. G. OuentHattve Anal)9k of Ffw@oMate a s s Films on Silicon Wafers; Sandia National Laboratories Report SAND82-0039; Sandia National Laboratories: Albuquerque, NM, 1989. Merrlll, R. M. LC-GC 1988. 6, 416. Becker, F. S.; Pawlik, D.; Schafer, H.; Staudlgil, G. J . Vac. Sci. Techno/. B 1986, 4 , 732. Tenney, A. S. J . Electrochem. Soc. 1973, 120(9), 1284. Grilleto. 0. Solid State Technd. 1977, 20 (2). 27. Huriey, K. H. Solid State Technol. 1987, 30 (3). 103. Haaland. D. M. Anal. Chem. 1988, 60, 1208. Haaiand, D. M.; Thomas, E. V. Anal. Chem. 1988. 60, 1193. Haaiand, D. M.; Thomas, E. V. Anal. Chem. 1988, 60. 1202. Mark, H.; Workman, J., Jr. Spectroscopy (€ugene,Orsg.) 1988, 3 ( l l ) , 26.
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(18) Oeborne, B. G.; Feam, T. J . Fwd Techno/. 1983, 18, 453. (17) She&, J. S.; Westehus, M. 0.;Templeton, W. C., Jr. Crop Scl. 1985, 25, 158. (18) chase, D. B. Appl. Spectrosc. 1984. 38(4),491, (18) Carter, R. O.,111: Lindsay, N. E.; Beduhn. D. Appl. Spectrosc. 1990, 44 (7),1147. (20) Fuller, M. P.; Rltter, 0. L.; Draper, C. S. Appl. Spectrosc. 1988, 42 (3).217 and 228. (21) Llndberg, W.; Persson. J. A.; Wold, S. Anal. Chem. 1983, 55, 643. (22) Cahn. F.; Compton. s. Appl. Spectrosc. 1988, 42. 865. (23) Frederlcks, P. M.; Lee, J. B.; Osborn, P. R.; Swinkels, D. A. Appl. Spectrosc. 1985, 39,303 and 311.
(24)Thomas, E. V.; kaland, D. M. Anal. Chem. 1990, 62 (lo),1091.
RECEIVED for review April 2, 1991. Accepted July 19, 1991. This work was performed in part at Sandia National Laboratories supported by the U. S. Dept. of Energy under Contract No. DE-AC04-76-DP00789 and by Semiconductor Research Corp*/SEMATECH through the University of New Mexico SEMATECH Center of Excellence.
Integrated Fluid Handling System for Biomolecular Interaction AnaIysis Stefan Sjolander and Csaba Urbaniczky*
Pharmacia Biosensor AB, Building F 61-1,S-751 82 Uppsala, Sweden
injection analysis might provide problems when conventional An Integrated fluid handling system used for muitlchannei bulk detectors are used (5). But with a surface-sensitive biomolecular interaction analysis is d e r c r W . Reactions detection method the sensitivity and detection limit are imbetween blologkai molecules are monltored In real time by proved by miniaturization, as will be shown in this paper. measuring changes In the angular porltlon where surface In the system described here, the biomolecular interaction plasmon resonance occurs at a biospectflc active surface. takes place on the biospecific active surface, which serves as The adsorptbn dfklency of the analyte onto the M o ~ c l f l c one of the walls in a thin-layer cell. This surface has such actlve surface Is up to =3%, due to the low channel height, properties that the adsorbed mass on it can be probed with 50 pm, In the flow cell. When a large part of the total blothe SPR technique when a part of it is illuminated from the rpedlic~emfaceformf~plermon~probln(l opposite side (2).For direct immunological sensing with SPR, (=0.15 m")b used, the sensttlvlty Ir hlgh. Sample dres In the thin-layer cell is used to transfer the analyte in the sample the order of 1-50 pL can be injected. The sample zone onto the sensing surface with a known mass-transfer rate and dkpembn b mhdmked by the low dead volume in the system efficiency. The mass-transfer characteristics for a thin-layer (=0.4 pL) a c w q H W d by udng lntwated sample bops and cell are well-known since that behavior is as for an electrothin conduits. An asset of thls integration is the low reagent chemical thin-layer cell when the active surface behaves as consumption. The sensor chlp wlth the blospeclflc actlve an infinite drain and the reaction proceeds with infinite resurface Is reusable and easily exchanged. Experimental reaction rate (6-8). An analytical solution for the mass-transfer suits obtalned with a theophyllh monoclonal antibody as the equation was given by Matauda. It is valid when the diffusion analyte are compared wtlh a theoretical model. The standard layer thickness next to the biospecific active surface is sufdeviation for the repeatablltty k =5 % typkaHy wlth 50 pL of ficiently thin relative to the channel height in the thin-layer 250 pM analyte, and the a m y t h e Is 10 mln. The detectlon cell (9). Some of the constrains in the boundary conditions were relaxed in a mathematical solution given by Weber and limit Is 4 0 pg of the analyte on the probed spot of the surPurdy (10)by using semiempirical constants. A mathematical face. P d b k hprovements of the sensitivity and detectlon solution consisting of power series was later given by Mollimit are dlscusmd.
INTRODUCTION With surface plasmon resonance (SPR) the kinetics for biomolecular interactions between, for instance, an antigen and an antibody can be followed directly without labeling (I). The SPR response is sensitive to changes in refractive index in the probed volume. The change in refractive index is proportional to changes in mass concentration. Since many biochemically active molecules have high molecular weight, samles with low concentrations of such analytes can be detected in situ with SPR (2). The penetration depth is so thin that the probed volume will be referred to as a probed area when it can not be misunderstood. The flow injection analysis technique (3) is suitable for reproducing qualitative and quantitative measurements with rapid response time. Sample and reagent volumes can be minimized by integrating and miniaturizing sample loops, valves, and conduits in a system. In addition, enhanced repeatability is achieved (4).Miniaturization as such with flow 0003-2700/91/0363-2338$02.50/0
doveanu and Anderson (1l) as well as by Roosendaal and Poppe (12).Some controversy about the validity of the formulas has been settled (6, 13). If low detection limit is the primary goal, the thin-layer cell should be designed to enable transfer of the analyte in the sample onto the sensing surface with (ultimately) 100% adsorption efficiency. Low detection limit is also easier achieved by using low flow rate. However, for kinetic studies a high mass-transfer rate is advantageous. If the mass-transfer rate is much higher than the heterogeneous reaction rate, then diffusion/convection terms in the mathematical model can be neglected. The reaction model is then simplified to include an ordinary differential equation instead of a partial differential equation. When the reaction rate of the antigen-antibody has a finite rate (in the same order as the mass-transfer constant) only numerical simulation methods are available (14, 15). Flow simulation is more common in electrochemistry (1617)and that approach is also valid for adsorption studies. Higher maas transfer of the analyte to the surface (by for instance higher flow rate) leads to a thinner diffusion layer, and thus, a lower relative amount of the analyte in the sample is adsorbed onto 0 1991 American Chemical Soclety
