Anal. Chem. 1988, 60, 1208-1217
1208
(10) Naes, T.; Martens, H. Commun. Statist.-Slmule. Computa. 1985, 14. 545.
(11) Haaland, D. M. Anal. Chem. 1988, 60. 1208. (12) wold, H. Muffhrkte Ana&&; Krlshnalah, P. R., Ed.; Academic: New York, 1986; p 391.
RECEIVEDfor review September 30,1987. Accepted February
8, 1988. This work was performed at Sandia National Laboratories and was supported by the U.S. Department of Energy under Contract DE-ACWDP00789. Portions of this work were presented at the Pittsburgh Conference & Exposition on Analytical Chemistry and Applied Spectroscopy in Atlantic City, NJ, March 10-14, 1986, Paper 1087.
Quantitative Infrared Analysis of Borophosphosilicate Films Using Multivariate Statistical Methods David M. Haaland Sandia National Laboratories, Albuquerque, New Mexico 87185
Infrared (IR) spectroscopy can serve as a rapld method for the quantnatlve anaiyW of borophosphoalllcate glass (BPSG) flhns on Si wafers for the microelectronics Industry. The advantages of uskrg statk#cdly d.dgned caHbratlon sels are emphasized. C l a ~ c a leastsquares l (CLS), partlal leastsquares (PLS), and prlnclpal component regresslon (PCR) methods are all found to provide Improved precislon over traditional peak-helght measurements. The quantltatlve results from spectral mecrsu.nwnk, taken in t r m mode at both 0' and 60' inckknt angles were also compared. PLS and PCR rmthods ylekled results that were comparable wtthln the sampUng error, and each exhbked a better analysls preclslon than that obtalned from the CLS analysls. Both PLS and PCR methods y W k d the W rosutb when applled to the orlglnal60' incldent angle data, whkh was not corrected for f h thkkness. PLS and PCR analyses each gave a standard error of predlcth (SEP) for boron of ~ 0 . wt 1 % and ~ 0 . 2 wt % forphosphon#rforasetof44callkatknsamplegwhkh spanned a range of concentratlons from 1 to 5 wt % B and 2 to 6 wt % P. The PLS and PCR methods applied to the I R spectra were also capable of monitoring ftlm thlckness with a SEP of 14 nm for Hlmr that varkd In thlckness from 430 to 1000 nm. The Importance of uslng these fullspectrum multlvarlate methods for outlier sample detectlon ls presented, and the aM#ty to extract quatltathre spectral Information from the CLS and PLS callbratlons Is demonstrated.
The continued decrease in dimensions of integrated circuits places greater demands on the oxide materials used as interlayer dielectrics. To achieve the smoothed topography required for metal interconnect continuity, these materials must be made to flow a t the appropriate processing temperatures. As devices are scaled smaller, back-end processing temperatures must be lowered in order to minimize lateral dopant diffusion in the transistors. Borophosphosilicateglass (BPSG) thin films (1-5) are currently being used as the interlayer dielectric material in higher density integrated circuits since these films can be processed at temperatures below 950 "C. However, their desired properties are inherently more dependent on composition than the previously used SiOzand two-component phosphosilicate glass (PSG) dielectric materials. For quality control of these films, a rapid and precise determination of the boron and phosphorus content is desired. 0003-2700/88/0360-1208$01.50/0
Inductively coupled plasma emission spectroscopy (ICP) (6), ion chromatography (IC) (4,6, 7),secondary ion mass spectroscopy (SIMS) (4,8), electron microprobe, and wet chemical colorimetric methods (1)have been used for film analysis, but these methods are relatively slow and/or destructive. X-ray fluorescence can monitor the P content of the film, but is relatively insensitive to B levels (1). Infrared (IR) spectroscopy has been used with varying degrees of success (1,4, 7,9-11) and is the best candidate for rapid BPSG thin-film analysis. The original quantitative infrared analyses of BPSG films involved peak-height measurements of the bands primarily responsible for B-0 and P=O stretching vibrations ( I , 4). Direct band-height measurements or band ratios were then used to determine boron and possibly phosphorus contents. More recently classical least-squares (CLS) methods (sometimes called K matrix methods) have been applied to the quantitative analysis of the IR spectra of BPSG films (9-11). New and more powerful partial least-squares (PLS) and principal component regression (PCR) methods (12-15) have recently been applied to infrared spectroscopy. These multivariate statistical methods have the advantage that they can model base-line variations and some types of nonlinearity in the Beer's law relationship. The numerous advantagesof these methods over previous IR analysis techniques make it profitable to apply them to the BPSG analysis. It will be shown that by coupling statistically designed calibration sets and novel sampling geometries with the new PLS or PCR statistical methods, large improvements in IR analysis of BPSG can be obtained. Analysis of the IR spectra of BPSG films using peak-height measurements and CLS methods will be compared with PLS and PCR methods. In addition, it will be shown that the standard inclusion of film-thickness measurements in the data can actually degrade the results in some cases. Thickness can be estimated directly from the infrared spectra, rather than measuring film thickness separately. Finally, methods for detecting outlier samples (Le., samples that are not representative of the calibration set and whose results must be considered suspect) will be presented, and the use of CLS or PLS calibrations to obtain useful qualitative information from the spectral data will be demonstrated.
EXPERIMENTAL SECTION The BPSG films were deposited on the polished side of the 10 cm diameter wafers of 1-10 Q cm Si using an atmospheric process. The wafers were held chemical vapor deposition (0) Ar, SiH.,, B,Hs, and at 420 "C in a flowing gas mixture of NO,02, PH3. The flows of B2Hsand PH3 were varied in different pro@ 1988 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 60, NO. 11, JUNE 1, 1988
duction runs to achieve variations in the boron and phosphorus contents. The B and P concentrations in the film were targeted for a two-factor, three-level calibration design (16). Two seta of calibration data were obtained by using the fadorial statistical design. Each calibration set involved 11processing runs with nine unique targeted design points. The center point was replicated 3 times. Two samples were taken from each processing run giving a total of 22 samples per calibration set. The processing run order was randomized. Each set had a different target value about which the B and P concentrationswere centered. However, experimental problems with flowmeters caused the second design not to be achieved as planned. The analyses presented here generally used all 44 samples (both calibration sets) to obtain a calibration with the widest range of B and P concentrations and thickness variations. These studies suggested that an optimal calibration design would be a three-factor design with film thickness being the third factor. All wafers were annealed in dry air at 850 "C for 30 min and thereafter stored in dry nitrogen. Annealing is necessary for the IR calibration since unannealed films have been shown to change structure with time (11). This change in structure, which greatly affects the IR band shapes and intensities, is primarily due to stress relaxation in the f i and reaction of the f i with moisture. Annealing in an oxidizing atmosphere was performed to achieve full conversion of the less stable PZO3 to P205 The more common practice of annealing in pyrogenic steam would also achieve this goal. Ion chromatography (IC) measurements confirmed that no Pz03was present in films annealed in air. Ellipsometer thickness measurements were made on the films at the center of the wafer. The infrared measurements were made at the same location and with the same spot size as used by the He-Ne laser beam (-6 mm) of the ellipsometer. The correct order used for the film thickness from the ellipsometer data was determined from the single interference fringe observed in the IR spectrum. The film thickness of the f i t sample set varied randomly from 676 to 1003 nm while that of the second set varied from 426 to 701 nm. The IR spectra of the BPSG samples were obtained by use of a Nicolet 7199 FT-IR spectrometer equipped with a liquid N2 cooled HgCd-Te detector. A total of 256 interferograms were signal averaged for both the sample and the separate background single-beam spectra. The absorbance spectra had a nominal resolution of 4 cm-'. The samples were mounted on a spring-loaded sample holder which was in turn mounted on a computer-controlled rotator and 10-cm linear translator. The rotator had a 0.01' step size, and the translator had a 1-pm resolution. Transmission IR spectra were collected separately at Oo and 60' incident angles with the IR beam impinging first on the BPSG film side of the Si water. A single background spectrum, with the sample translated out of the beam, was obtained between the two sample orientations. Computer control of the rotator and positioner minimized operator involvement and allowed a good N2purge of the spectrometerto be obtained before data collection was initiated. The linear translation of the sample could also be used to obtain IR spectra at other locations on the sample. The infrared beam was p-polarized (parallel polarization) by using a gold-wire-grid polarizer on an AgBr substrate. When the incident angle is 60°, p-polarized light minimizes the influence of the highly reflecting s-polarization in the spectrum ( 1 7). The spectral data were analyzed both as untreated absorbance data or pretreated spectral data. The pretreatment used consisted of subtracting the spectrum of the Si wafer (obtained at either 0' or 60' incident angle as appropriate) using the isolated 610 cm-' Si phonon band to obtain the proper subtraction factor. These spectra were then further pretreated by scaling to unit relative thickness. All data were transferred to an 8600 VAX and analyzed by CLS, PLS, and PCR Fortran software written at Sandia National Laboratories (12, 13, 18). After the IR data were collected, a 5 by 2.5 cm section of each wafer centered on the IR sampled area was cut for ICP and occasional IC analysis. The ICP and IC methods used have been described elsewhere (6). Unfortunately, the precisions of these methods are not accurately known since these methods are destructive and concentration variations are possible both between duplicate wafers and within samples on a single wafer. However, a m d e estimate of the relative precision of the ICP determinations of B and P concentrations can be made based on the fact that
1209
both IC and ICP measurements were made on each sample in the second set of 22 BPSG samples. The standard deviations of the differences between the IC and ICP data were 0.14 wt % for phosphorus and 0.06 wt % for boron. Since these analyses were each performed on the same solution prepared from the dissolution of the thin fiims, weighing errors, volumetric errors, and dilution errors are not included in this measure of precision. Thus, the ICP analysis precision could be lower than the above values if the weighing, volumetric, and dilution errors are relatively large, or the ICP precision could be higher if these errors are relatively small and more error is contained in the IC analyses. These numbers, therefore, only give a rough estimate of the precision of the reference method, but they suggest that phosphorus may have been determined at a lower precision than boron in the ICP analyses. The IR analysis, of course, can be no better, in terms of precision, than the reference analysis method used to calibrate the IR.
THEORY Statistical Calibration and Prediction Methods. A number of full-spectrum multivariate statistical methods for quantitative spectral analyses have recently been made available. These include CLS, PLS, and PCR methods which will be compared in this paper. Since these have been described in detail elsewhere (12, 13,18), only brief essentials will be given here. All three methods assume a linear relationship between absorbance and concentration and each involves a calibration step where the spectra of reference samples are related to the known concentrations obtained by a separate reference method. Each type of calibration can approximate nonlinearities as linear estimates of the nonlinearities over the narrow concentration range of the calibration set. Unknown samples can then be analyzed in the prediction step if they are indeed representative of the calibration samples. The classical least-squares method used here estimated B and P concentrations in a combined analysis which included a simultaneous least-squares fit of the base line assuming that the base line was linear over the entire spectral region analyzed. PLS and PCR methods used in this paper are similar to each other and are performed one component a t a time. They have the ability to model complex base lines and some types of nonlinearity in Beer's law at the expense of decreased precision. Of course, the use of linear models cannot account for true nonlinearities. However, over narrow concentration ranges, the nonlinearities often can be approximated with linear models. In addition, PLS and PCR can model interactions which are considered nonlinearities in Beer's law since the Beer's law model becomes inadequate in the presence of interactions. However, interactions arising from molecular interactions or reactions can still result in a linear additive model. PLS and PCR can adapt to interactions by increasing the number of factors used for prediction, whereas CLS methods either require explicit component concentrations for the products of the interaction or require a trial and error method of modeling the functional form of the interaction (19). PLS and PCR calibrations involve building a full-spectrum model of the calibration spectra. The model is increased one factor at a time until prediction on a separate validation set or cross validation (20) on the calibration samples shows that further factors do not improve predictive ability for concentration. We have previously used cross validation leaving one sample out of the calibration a t a time (12, 13). Thus, Calibrations are performed N times where N is the number of samples used in the calibration. The concentration of the sample left out in each calibration is predicted by the Calibration and can be compared with the known sample concentrations. The sum of the squared prediction errors (prediction error sum of squares or PRESS) is then used as a measure to select the optimal number of model factors. Rather than using the number of factors in the model associated with the minimum PRESS as optimal, we use ratios of PRESS
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ANALYTICAL CHEMISTRY, VOL. 60, NO. 11, JUNE 1, 1988
values (with the model with minimum PRESS as our reference) and the F statistic (12, 16) to determine the optimal number of factors for concentration prediction. We choose as the optimal model the one with the fewest number of fadors which has achieved an analysis precision that is not significantly different at the 0.50 significance level than that of the reference model. A significant difference a t the 0.50 level implies that the observed PRESS ratio would be larger less than 50% of the time, when there was, in fact, no difference in analysis precision between the reduced model and the reference model. This procedure has recently been presented elsewhere (12),and it optimizes the model while simultaneously reducing the potential for overfitting calibration data. Once the optimal number of factors in the model has been selected, all calibration samples are included in the final calibration to be applied to unknown samples. Because the samples used in this paper include a set of replicate samples which do not contain all the variability that would be associated with a true replicate sample and because cross validation will underestimate errors to be expected in a true unknown sample set if replicate design points are present, we applied cross-validation leaving out a design point (two samples) at a time. Therefore, the error in this crossvalidated prediction should be more representative of what one would obtain with an independent set of unknown samples that are truly representative of the calibration samples. This error should, in fact, be slightly greater than that obtained from a set of unknown glass samples whose average concentration is the same as that of the calibration set and whose concentrations are contained within the calibration concentration range. This is because some predictions are extrapolated rather than interpolated during cross-validated calibration, and there is generally a greater variability of concentrations in the designed calibration. Both of these factors tend to inflate the estimated errors during cross-validated calibration. Unfortunately, when either calibration or unknown samples are analyzed, there is always the possibility of outlier samples. In the calibration set, outlier samples may have large concentration errors or other factors such as unusual base lines or nonlinearities that are not modeled well. For unknown samples, any variation that is not properly represented in the calibration step (such as impurities, unusual base lines, concentrations outside the range of the calibration samples, etc.) can cause the samples to be outliers. Therefore, the results obtained on these samples must be considered suspect. It is important to identify these outlier samples in the calibration and unknown sample sets in order that they can be further studied to make sure that their analysis is correct. All three full-spectrum methods can be used to identify outliers by comparing the errors in either concentration or spectral space for each sample with that of the calibration set as a whole. Statistical F tests in concentration or spectral errors for the calibration samples or spectral errors for the unknown samples can be used as a simple method for identifying outliers. These F tests have been described in more detail elsewhere (12, 16). The results of the analyses will be presented as the standard error of prediction (SEP) which for cross-validation calibration is simply defined as N
SEP = (Ce,2/N)1/2 1=1
where each e, represents the difference between the reference concentration and the estimated concentration for the ith sample in the calibration set. SEP has concentration units and represents the equivalent of one standard deviation of the prediction error. Factorial Design of Calibration Sets. The use of factorial or other suitable statistically based calibration designs
is important in order to maximize the information that can be obtained from the smallest number of samples (16). This is especially important when the preparation and analysis of samples are time-consuming and expensive. A significant amount of information is available from statisticians for the efficient design of calibration sets, but this information is not often used by chemists. Since the proper design of the calibration set can be crucial to the success of any calibration, a short discussion of these designs is presented here. Generally, one wishes to maximize the concentration space spanned by the calibration samples. With the calibration samples primarily on the boundaries of the extremes in the concentrations and with those concentrations located in an orthogonal fashion, the information content is maximized and interaction effects can be determined. Use of large concentration range will enhance the chance that the original calibration will be useful when target concentrations change. In addition, large concentration variations are often important since a large concentration range of the components relative to the concentration error of the reference analysis method (ICP in this case) is needed for there to be sufficient information upon which to base the calibration. The design used depends on the relationship expected between concentrations and spectra. However, in this study the original experimental design was selected to evaluate the BPSG process conditions rather than to obtain the best IR calibration for B and P concentrations. Therefore, a two-factor, three-level factorial design was chosen and used for the analysis of the BPSG films. Each of the two concentrations has been measured at three levels in a structural fashion (see Results and Discussion). In addition, the run order of the analysis has been randomized to minimize the effect of experimental drift. Infrared Spectra. The infrared spectra of BPSG films on Si are usually taken in transmission mode with the IR beam normal to the film (0' incident angle) or in reflection mode. However, in transmission mode, more information becomes available as the incident angle becomes greater than Oo (22). New spectral features become apparent at higher incident angles due to the presence of the large dispersion in refractive index for strong absorption bands. Since these changes are dependent on polarization of the IR beam and since s-polarized radiation becomes highly reflecting at high incident angles, p-polarized radiation yields better signal-to-noiseratios when incident angles are increased. IR spectra of a few samples were collected at 15' incident angle intervals from 0 to 75". The 60' spectra were selected for further study since the spectra taken at this incident angle exhibit the most new spectral information relative to the 0" data without suffering the large increase in the noise present in the 7 5 O incident angle data. These thin films also cause interference fringes in the spectra which can result in absorbance changes that are not linear with path length even if the background fringe is removed. This is because the effect on the absorption spectrum of multiple reflections within the film is dependent on the intensity of the absorption band and the fraction of the IR beam reflected. Both of these factors are influenced by film thickness. Although this might at first seem detrimental to any quantitative IR analysis, PLS and PCR methods appear to be able to model these interactions in some cases. Thus, correcting the spectra (or concentrations) for path length variations with idealized Beer's law multiplicative factors may not always yield better results. This will be clearly shown to be true in the analysis of the 60' incident angle spectral data collected for BPSG films on Si.
RESULTS AND DISCUSSION The B and P concentrations determined for both sets of BPSG samples by ICP are displayed in Figure 1. The two-
ANALYTICAL CHEMISTRY, VOL. 60, NO. 11, JUNE 1, 1988
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SI-0
!
I
I
,
1
I
I
I
I
I
I
I
I
6.0-
3.0
m
t
t
0.01
0.0
1 . 0 2 . 0 3 . 0 4 . 0 5.0 6 . 0 7 . 0
w. x
P BY I C P
Flgue 1. ICP determinationof B and P in the BPSG fllms from sample sets 1 (triangles) and 2 (clrcles).
fador, three-level fadorial design is clearly evident in the f i t set of data. The target of the second set was not achieved as planned, but the set of samples still spans a large concentration range. An earlier set of 19 samples taken randomly from production runs a t an outside production facility had a variation in P and B concentrations of -0.8 wt % each. The small range of concentrations in these samples and their random distribution made calibrations based on this sample set unreliable for the P concentration. In addition, recent changes in target composition of the BPSG films rendered this calibration set inappropriate since the new target was outside the concentration range of the calibration samples. Therefore, it is clear that by spanning wide concentration ranges with an orthogonal set of concentrations in the calibration samples, a greater analysis precision and a more generally applicable calibration can be obtained. However, some trade-offs may be required since a large concentration range can cause problems if significant deviations from the assumed linear model are present. As discussed earlier, interactions can be often modeled with more PLS or PCR factors, but greater model complexity requires greater numbers of calibration samples to be used. Figure 2 presents representative spectra of the BPSG films on Si a t Oo incident angle with p-polarized IR radiation. Principal bands associated with B-0, P-0, Si-0, and Si-Si vibrations are noted in the figure (the high-energy band related to phosphorus is actually due to the P=O stretching vibration). As concentrations change in these samples, many of the bands shift in frequency indicating concentration-dependent interactions which will cause "nonlinearities" in the relation between absorbance and concentration. Note also that broad base-line variations are present which are principally due to the partial interference fringe present in the spectra of these thin films. The variations in the intensity of the S i 4 stretching vibration are approximately related to film thickness, since the Si concentration in the calibration set varies by less than 10% while the thickness varies by more than a factor of 2. Figure 3 demonstrates the additional information available when the spectra are obtained at an incident angle of 60°. The difference spectrum shows quite clearly the increase in the band at 1230 cm-' which corresponds to the minimum in refractive index of the film near the strong Si-0 stretching vibration. Its presence as a small shoulder in the Oo incident angle data is most likely the result of beam divergence in the focused beam causing a portion of the beam to strike the sample at an incident angle of greater than Oo. The high-
-
rboo
1400
1200
1boo
WRVENUMBER
boo
600
loo
Figure 2. I R spectra of representathre BPSG films at ' 0 incident angle: (A) 6.13 wt % P, 1.30 wt % B, 1003 nm; (B) 3.44 wt % P, 1.82 wt % B, 702 nm; (C) 4.72 wt % P, 4.58 wt % B, 618 nm; (D)2.40 wt % P, 4.44 wt % B, 426 nm.
L+ d
60';inus
1 00
00
1'100
*
1 00 1 00 WAVENUMBER
Figure 3. IR spectra of a BPSG film at 60' and '0 incident angle and their difference.
energy P=O and B-0 stretching bands and the low-energy P-0 bending vibration are all increased in relative intensity a t 60°. A portion of this increased sensitivity is due to the greater path length at 60° incident angle. This is not the complete source of the enhancement since the change in path length can be approximately calculated if it is assumed that the film has an index of refraction of 1.46 and the Si wafer has an index of 3.5. In this case, the increase in relative path length in the film is 24% but only 3% in the Si wafer (note that the angles in the film and wafer are less than 60' due to the bending of the beam toward the normal) (17). The complete set of reasons for the greater enhancement in these bands awaits model calculations of these samples using the Fresnel equations. Note, however, that the bands due to P and B do not exhibit the noticeable growth of new features as observed for the Si-0 stretching band since these minor
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ANALYTICAL CHEMISTRY, VOL. 60, NO. 11, JUNE 1, 1988
Table I. Classical Least-Squares Analysis of BPSG Thin Films" a
SEP, wt 9i
B
0 4 .
sample sets
Oo
60'
O0
60°
1
0.12 0.20 0.33
0.22
0.61 0.32 0.58
0.91 0.40 0.60
2
1and 2
A
P
0.11 0.14
d
" CLS applied to IR spectra that first had the spectrum of the silicon wafer removed followed by scaling for thickness. Spectra were analyzed from 425 to 1600 cm-I while simultaneously fitting a linear base line across this spectral region. components are present in a matrix dominated by the index of refraction of SiOz. The IR spectra at ' 0 and 60' incident angles were analyzed for all 44 samples displayed in Figure 1 by a variety of methods. Individual base-line-correctedpeak heights for the B-0 stretching band of the pretreated spectral data were plotted as a function of concentration and are presented in Figure 4 for the 60' data. It is clear that the scatter and slight nonlinearity in this plot limits the precision of the boron analysis that could be obtained from this calibratin set using this peak-height method. This scatter and nonlinearity are to be expected since Becker et al. (7)find that the absorptivity of the B-O stretching band decreases with phosphorus content and the phosphorus concentration in the samples plotted in Figure 4 range from 2 to 6 wt %. The relatively low intensity of the P=O stretching band and its overlap with the B-0 band prevented its intensity from being directly monitored with any reasonable precision. In fact, when the measured peak heights of the P=O stretching band were plotted as a function of the ICP determined P concentration, the plot was almost randomly distributed but with a very slight negative correlation. Therefore, P concentrations could not be determined directly from the infrared peak height of the P-0 stretching band. CLS methods were applied to the pretreated spectral data from 425 to 1600 cm-'. During prediction, a linear base line was simultaneously fit over the spectral range analyzed (18).
'/
0 0 O
0 0
1 0
2 0 WT
4 0
3 0
6 0
5 0
% B BY I C P
Figure 4. Base-line corrected peak height of the 6-0 stretching vibration vs the weight percent of B by ICP for the 44 BPSG thin-film spectra taken at 60' incident angle.
The nonlinearities and complex base line over the concentration-thickness range of the samples prevented highly accurate results in this case. The SEP from the CLS analyses for both B and P are presented in Table I for each calibration set analyzed individually and when all samples were analyzed as a single calibration set. For all CLS analyses, the Oo data yielded better results for P than the 6 0 O data while the results were variable for the B determination. By restriction of the analysis to each data set individually, improved precision might be expected since base-line variations and nonliinearities are reduced within a given calibration set. Although the second set yielded a more precise calibration than the combined sets, the first set was actually worse overall than the combined calibration. Therefore, in the remainder of the paper, we analyze only the combined data sets. PLS and PCR methods were applied to both the pretreated and uncorrected absorbance spectral data. In each case, the optimal number of factors were selected by using cross validation leaving out one design point (two replicate samples)
Table 11. P L S Analysis of 44 BPSG Thin-Film Samples SEP" P, w t 9i
B, w t %
O0 pretreated spectrab uncorrected spectra
60'
0.13 (10)' 0.19 (9)
0.12 (7) 0.11 (8)O
O0
60°
0.29 (10) 0.34 (9)
0.27 (13) 0.22 (9Y
thickness, nm O0 60' d
d
34 (3)
28 (4)e
a Standard error of prediction during cross validation of 44 calibration samples removing one design point at a time. PLS was applied to all spectra from 425 t o 1600 cm-'. Data were centered but not autoscaled. bPretreatment of spectral data included subtraction of silicon wafer spectrum followed by scaling for film thickness. 'Values in parentheses are the number of PLS factors used to model the calibration data. dThicknesswas not determined from pretreated spectra since the spectra were scaled for thickness. 'These data represent the lowest SEP values obtained during PLS analysis of the IR spectra.
Table 111. PCR Analysis of 44 BPSG Thin-Film Samples SEP" P, wt 70
B, wt 70 pretreated spectrab uncorrected spectra
thickness, nm O0 60°
00
60'
00
60°
0.13 (15)'
0.12 (10) 0.11 (12)e
0.28 (16) 0.37 (11)
0.27 (15)
d
d
0.21 (14Y
32 (4)
28 (4)e
0.17 (14)
Standard error of prediction during cross validation of 44 calibration samples removing one design point at a time. PCR was applied to all spectra from 425 to 1600 cm-'. Data were centered but not autoscaled. bPretreatment of spectral data included subtraction of silicon wafer spectrum followed by scaling for film thickness. Values in parentheses are the number of PCR factors used to model the calibration data. dThickness was not determined from pretreated spectra since the spectra were scaled for thickness. 'These data represent the lowest SEP values obtained during PCR analysis of the IR spectra.
ANALYTICAL CHEMISTRY, VOL. 60, NO. 11, JUNE 1, 1988
at a time. The results for PLS are presented in Table I1 and in Table I11 for PCR. Note that analysis of the pretreated Oo spectral data (corrected for the measured thickness and the presence of the Si wafer absorbance) yields slightly better precision than the uncorrected data at the same incident angle. F ratios based on the ratio of the squared SEP's indicate that the differences in analysis precision between these two data sets (uncorrected versus pretreated spectra for 0' incident angle) are significant at about the 0.05 level for the B determination by PLS or PCR and the P determination by PCR. In this case and in comparisons presented later, the concentration errors tend to be correlated rather than independent as required by the F statistic. Therefore, the given significance level is conservative (see ref 12, Appendix A). That is, the results are significant at an even higher level than quoted whenever errors are correlated. For the 60' data, the precisions in Tables I1 and I11 are slightly better when using the uncorrected spectra as opposed to the pretreated spectra, but these differences are not significant at the 0.25 level except for the PLS phosphorus determination. Although there is not often a significant difference between these prediction errors, the fact that analyses applied to the uncorrected spectra allow film thicknesses to be estimated makes the analysis of the uncorrected data preferable for the 60' spectra. An improvement in analysis precision of the uncorrected spectral data in going from 0 to 60° incident angles is significant to at least the 0.005 level for both B and P (PLS and PCR). The PLS and PCR analyses applied to the uncorrected data at 60' yield the lowest predicition error of any of the analyses. The differences between the PLS and PCR results presented in Tables I1 and I11 are not generally significant at the 0.25 level. However, note that the number of PLS factors required to model the data is always less than or equal to that required for PCR. In addition, the CPU time required for the PLS analysis was 5 vs 28 min for PCR on the 8600 VAX. The greater computational efficiency and simpler model obtained from PLS may make it preferable to the PCR method. The best PLS results are better than those from the CLS analysis of the combined sample sets (Table I) at the 0.1 significance level for B and the 0.001 level for P. The remainder of this section will, therefore, discuss primarily the results from the PLS analysis of the uncorrected spectra taken at the 60° incident angle. It should be noted, however, that any of these three calibration methods might be improved with an advantageous selection of spectral frequencies. Although several different frequency ranges were investigated without improvement in the quantitative results, an exhaustive or systematic selection of frequencieswas not attempted. The comparisons presented above are valid for the 425-1600 cm-' frequency range included in the analyses although a separate optimal selection of frequencies for each calibration method might result in different comparisons. It is interesting to note that eight and nine factors provide the optimal PLS model for B and P, respectively, for the BPSG films. This is in spite of the fact that this material presumably contains only three independent components (without thickness variations in the samples, there would be only two independent components since the sum of weight percents is constrained to be 100%). Therefore, additional factors are required to model chemical interactions and base-line variations over the concentrationand thickness range of these calibration samples. However, the optimal number of factors required for thickness prediction is only four for both PLS and PCR at 60' incident angle. This may be due to the fact that the largest source of spectral variation between these samples is a result of thickness differences. Many of the factors required to model B and P concentrations are not
6 0,
,
I
I
I
1213
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1 .-0
I
/d"!
1 //
0.0 0 0
1.0
2 0
3 0
4 0
5 0
6 0
W . Z E BY I C P
Flgure 5. Weight percent B determined during cross-validated PLS analyses of the uncorrected I R spectra of BPSG thin films at 60' inc+nt angle as a function of the weight percent B determined by ICP analyses. The solid line represents the ideal correlatlon with a slope of 1.0 and an intercept of 0.
useful for predicting thickness. A t first, it may seem surprising that film thickness can be estimated so precisely, and yet the results obtained from the 60' incident angle spectra are degraded when the spectra are normalized to constant thickness. However, this correction for f i thickness is made by assuming that thickness is purely a multiplicative Beer's law correction. As discussed in the Experimental Section, thickneas variations introduce base-line variations and interactions between the interference fringes and the absorption bands. Thus, the use of a simple Beer's law multiplicative correction can actually degrade the results since the necessary correction is not strictly multiplicative. Apparently, PLS and PCR can more adequately account for these interactions and base-line variations and thereby produce more precise boron and phosphorus concentrations from the uncorrected data. The presence of the base-line variations and interactions can also account for the high precision of the thickness estimates by PLS and PCR. This precision is higher than one would expect simply based on peak-height variations with thickness. This is demonstrated by the fact that even when the spectral data are scaled to unit thickness, a significant portion of the thickness variations can still be modeled by PLS and PCR (86% and 97% of the thickness variance is modeled by PLS using thickness scaled spectra for the Oo and 60' data, respectively). These results suggest that PLS and PCR may be the methods of choice when the quantitative IR analyses of samples of indeterminate path length are required. An illustration of the improved quality of the IR analysis of BPSG films using PLS is given by Figure 5 where the individual results for the PLS boron calibration of 44 samples are presented. The data are nicely clustered about the ideal line with zero intercept and slope of 1.0 indicating that predictions with a SEP of 0.11 wt % are possible for boron with this eight-factor PLS model. The precision of the PLS model is significantly better than the more commonly used B-0 peak-height method as can be seen by comparing the relative scatter of the data in Figure 4 and 5. The standard error calculated from a linear least-squares fit of the peak height vs B concentration data in Figure 4 is 0.24 wt % (or 0.23 wt % fitting a quadratic curve through the data). The results from these peak-height measurements should be compared with the SEP values of 0.14 wt % obtained with CLS and 0.11 wt % for both PLS and PCR. The high degree of scatter in Figure 4 and the resulting poorer precision are due to a variety
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ANALYTICAL CHEMISTRY, VOL. 60, NO. 11, JUNE 1, 1988
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of factors including base-line errors, variable error due to p=O band overlap with the B-0stretching band, and molecular interactions which vary with BPSG composition (e.g., see ref 7). Each data point in Figure 4 is also the result of an absorbance measurement at only one frequency,and they do not benefit from the signal averaging effects which are present in the full-spectrum methods. Note that Figure 5 presents the result of cross-validated PLS predictions. Each prediction was made on a sample not included in the calibration, and therefore, results with this precision or better are to be expected from a set of unknown samples that are representative of the calibration set. From the plot in Figure 5, it is not immediately clear whether any of the samples might be outliers. However, a plot of the boron concentration F ratios obtained during cross validation of the PLS analysis is presented in Figure 6. This figure serves as a dramatic demonstration of the ability of these methods to flag outlier samples. The sample with the highest B concentration is an outlier which has an analysis error that is significantly greater than that of the other samples as indicated by ita F ratio of 18. For the number of samples in the calibration set (i.e., the degrees of freedom),this F ratio is statistically significant at the 0.001 level (16). There are several possible reasons for this sample being an outlier. For one, it may have been analyzed incorrectly by ICP. Unfortunately, the sample was destroyed by the ICP analysis and could not be reanalyzed by either IR or ICP. It may also be an outlier simply because it was an extreme sample. This sample had the highest boron content and nearly the lowest phosphorus content while simultaneously having the greatest thickness of any of the samples near this concentration range. If it were important that this extreme region be adequately modeled, the proper course of action might be to simply add more samples which span this region of the calibration. Thus, this F-ratio test does not necessarily indicate that the sample was analyzed incorrectly. It only indicates that the infrared analysis of this sample had significantly greater error than other samples. Therefore, concentration F ratios should be used as a guide to flag samples in the calibration set that should be reanalyzed or investigated further. Spectral F ratios can also be used to flag samples in both the calibration and unknown sample sets. The spectral F ratios simply give a statistical comparison of how precisely a sample spectrum was fit relative to the other calibration samples. The spectral F ratios for B from each sample derived from cross-validated calibrations are presented in Figure 7. We have discussed elsewhere that spectral F ratios below 3
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Flgure 8. Weight percent P determined during cross-validated PLS analyses of the uncorrected IR spectra of BPSG thin films at 60' incident angle as a function of the weight percent P determined by ICP analyses. The solid line represents the ideal correlation with a slope of 1.0 and an intercept of 0.
do not appear to be significant for IR spectral data (12). The single outlier in this plot is a sample whose spectrum was fit significantly worse than the other samples. Although not flagged by the concentrationF ratio as a statistically significant outlier, this sample also had the second poorest concentration estimate in the set of 44 samples. An investigation of the spectrum of this sample and the full-spectrum residuals showed that it had an extreme base line that evidently was not modeled accurately during the PLS analysis. This particular sample was also flagged as the most significant outlier by the spectral F ratios obtained during the PLS analysis of the P content and the film thickness. Figure 8 illustrates the results of the P determination by PLS. The SEP for this PLS determination of P during cross validation was found to be 0.22 wt %. (The standard error obtained from the P=O peak-height measurement applied to the same data was 1.44 wt %, while it was 0.60 and 0.21 w t % for CLS and PCR, respectively.) The precision of this analysis is not as high as that found for B for a variety of reasons. As indicated in the Experimental Section, the ICP precision of the P determination may actually be a factor of 2 lower than than that obtained for B. Some of the poorer precision for the P determination during the PLS analysis of
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the spectra also may be related to the fact that the spectral features due to phosphorus are buried under the other spectral bands. In addition, it could be possible that the phosphorus-related bands exhibit poorer Beer's law linearity than the boron bands. These factors may contribute to the reduced precision of the IR analysis of phosphorus relative to that of boron. As in the case of B, the PLS analysis during the P determination yields one outlier in both the concentration and spectral F ratios. The outlier with a concentration F ratio of 11 was one of four samples that had a noticeable thickness variation at the location where the IR beam struck the sample. It may be an outlier for this reason or possibly due to an inaccurate ICP analysis. The only outlier indicated from the spectral F ratio (6.4) was the same sample which was flagged during the PLS determination of B because of its extreme base line. Figure 9 compares the PLS analysis of the film thickness with the reference ellipsometer measurement of thickness. The SEP for this PLS analysis of the IR spectra was 28 nm. The two samples in Figure 9 with ellipsometer thicknesses between 700 and 800 nm that appear above the line are shown to be significant outliers as determined by thickness F ratios (values of 32 and 10). These samples were two of four samples noted during ellipsometer measurements to have significant thickness variations at the location where the IR spectra and thickness measurements were made. Thus both the ellipsometer and the IR film thickness measurements might be expected to exhibit significant error in these samples. When these two outlier samples were removed from the calibration, the SEP obtained from the PLS determination of thickness from the infrared spectra was reduced to 14 nm. Thus the importance of outlier detection is demonstrated when samples which are flagged as outliers are known to be incorrectly analyzed. Unfortunately, the spectral F ratios of these two samples were
