Effects of detector nonlinearity on calibration and data reduction ofrotating-analyzer ellipsometers

William Ralph HunterIBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598

(Received 8 July 1975)

Small nonlinearity (-2%) of the light-flux detection system is shown to have appreciable effects on theaccuracy of calibration and data analysis in rotating-analyzer ellipsometers. Procedures for detecting andcorrecting these effects are presented.

The use of rotating-analyzer ellipsometers in automa-tic data-acquisition and analysis systems is increas-ing. l-5 Considerable attention has been given to theeffects of systematic errors of the calibration con-stants, including off-diagonal elements in the principal-frame Jones matrices, on measurement accuracy. 5-7However, almost no attention has been given to theeffects of departures from the ideal in the light-flux-detection system, in particular nonlinearity.

This paper presents data in which nonlinearity is notnegligible. The instrument used is identical to theellipsometric thickness analyzer (ETA) described else-where, 3 except that the polarizer and quarter-wave-plate compensator, if used, are precision componentsmounted in circles readable to 0. 010. The compensa-tor is an unsandwiched mica sheet that does not exhibitoptical activity. 6 Flux data are digitized for each of256 equally spaced angles per revolution of the analyzerand transmitted to an IBM 1800 computer, where theyare averaged at each angle for -10 revolutions, cor-rected for dc offset and nonlinearity, and Fourier ana-lyzed. The Fourier coefficients are used to computeeither t, A (from which optical constants and filmthicknesses are computed) in the polarizer-sample-analyzer (P-S-A) and polarizer-compensator-sample-analyzer (P-C-S-A) configurations, or T,, A, of thecompensator in the polarizer-compensator-analyzer(P-C-A) straight-through calibration configuration.

In Sec. I we discuss offset calibration and data ana-lysis. In Sec. I the results are discussed. In partic-ular, nonlinearity correction is introduced, which im-proves the accuracy of measurements for arbitrarysettings of the components.

I. OFFSET CALIBRATION AND ANALYSIS

We can write the component azimuthal angles as

P = Ps +oP,

A =AS+6A + 6P,

94 J. Opt. Soc. Am., Vol. 66, No. 2, February 1976

where P, C, andA are the angles that the principal frameof the polarizer, compensator, and analyzer make withplane of incidence in the Muller Nebraska convention. 8P5 Cs are the scale readings from the vernier cir-cles. As is the analyzer-scale angle; As= 0° corre-sponds to the first data point taken following the startpulse that is triggered by the optical encoder, which isrigidly connected to the analyzer. OP accounts for theoffset of the polarizer principal frame relative to theplane of incidence when Ps= 00. oC and 6A account forthe offsets of the compensator and analyzer principalframes relative to the polarizer principal frame. It isassumed for convenience that the offsets oP, oC, and6A are small.

The discrete flux data Ir at each of 256 angles perrevolution are corrected for background and non-linearity to give corrected data IC by solving for I, in

Io=Ao+I.+A 2 I, I (2)

where the coefficient of the linear term has been takento be 1 without loss of generality. Here, Ao0 tO if thereis a dc offset, e. g., in the ADC (analog-to-digital con-verter), and A2 . 0 is there is a nonlinearity some-where in the flux-detection system.

The discrete data Ic are then Fourier analyzed to findthe coefficients9 in

I,= ao +a2 cos2AS+ b2 sin2 A,. (3)

The offset oA can be determined by calculating theanalyzer-scale angle AsMINfor which minimum detectedflux occurs during a P-A straight-through measure-ment without the compensator. Under these conditions,AsmIN is given (in degrees) by

2ASMIN= 360° u(b,) - sgn(b2 ) cos- + 2 (4)

where u(x) = 0 for x < 0 and u(x) = 1 for x 0, sgn (x) = - 1(la) for x < 0 and sgn (x) = 1 for x 2 0, and the principal value(lb) for cos-1 (x), 0 x• 1800, is assumed. For these con-

ditions, the polarizer and analyzer are crossed and OA(lc) is given by

Copyright 0 1976 by the Optical Society of America 94

(5) diagonal elements. 6

In Eq. (5), and the subsequent discussion, it isunderstood that all angles are indeterminate to a mul-tiple of 180°.

In the P-S-A configuration, van der Meulen and Hien5describe a useful technique for determining OP basedon the use of their F function, Eq. (8) of Ref. 5. Asimilar F function, obtained by the replacementsP-(P-C), A-Ac, and 0-tan' (1/T,) in Eq. (8) of Ref.5, is useful in the P-C-A configuration for determining6C. Here, Tc and Ac are the relative amplitude atten-uation and relative phase retardation of the compensa-tor. These two F functions will be referred to asFPSA and FPCA.

Let (Ps) be the value of P8 that maximizes F. (Be-cause of symmetry properties, it can be the averageof two readings on either side of the maximum forwhich F is the same, optimally chosen so that I dF/dP, Iis maximized.) Then for the P-S-A configuration,

6P = - (Pd, (Ps) ° (6a)

orOP=90° -(P), Ps 90Q° ,

and for the P-C-A configuration,

6C=(Ps) -QC (Ps)C-1C 5

or

C= (P,) - (Cs + 900), (P,) 5 Cs +90°.

The offset 6C was determined in the P-C-A configura-tions by use of Eq. (7). With Ps set to (Ps), FPCA(max)should be 1. 0 but was also found to be about 1. 008. Fora given setting of Cs, data were taken for Ps varyingand were analyzed for Tc and c 2=Ac - 90° using Eq.(A3). The results are shown in Fig. 1 with (x). (Theuncorrected 6c are so close to the corrected 6d thatthey are not plotted. ) A systematic variation, particu-larly in Tc, can be seen. (The variation of 6c wasmuch larger prior to the polarizer-beam-deviationadjustment. )

Finally, in the P-S-A configuration, an oxidizedsilicon sample with 690 A S12 was used and 6P ob-.tained using Eq. (6). As before, with P. = (Ps), FPSA(max) was about 1. 008. Then data were taken for Psvarying; the variation of calculated index and thicknessof the SiOp layer is shown in Fig. 2 with (x). Again, asystematic variation with Ps is seen. FPSA becomesz0 (circularly polarized light incident on the analyzer)because And 900 for this particular sample.

F> 1 indicates that the dc level is smaller than the

(6b) amplitude of the second harmonic. Indeed, examinationof the uncorrected data invariably showed a maximumgreater than 2aO for the uncorrected data in those situa-

(7a) tions where F is expected to be 1. This suggests thatthe flux data be corrected with a value of A, >0 in Eq.(2). Accordingly, A2 was adjusted (with A,= ° lO) untilF was forced to be 1. 00000 in those situations for which

Cm) F should be 1, namely, P=0 or P=9O0 in P-S-A and

The usual ellipsometric parameters A, A for P-S-Aor P-C-S-A are calculated from the Fourier coefficientsassuming no off-diagonal elements in the principal-frame Jones matrix, e. g., no optical activity. 5-7 I amnot aware that the equations for TS, Ac in the P-C-Aconfiguration, although similar to those for 4 and A,have been given in the literature. They are given inthe Appendix.

II. DISCUSSION

All data discussed in this section have been takenwith all apertures between the polarizer (or compensa-tor, if used) removed. In particular, the control aper-ture of ETA3 was removed. Careful sample alignmentwas used in P-S-A.

Ideally, 6A obtained from Eq. (5) in the P-A config-uration should not be a function of Ps. Experimentally,however, OA showed a variation that was approximatelysinusoidal with 2Ps with an amplitude of 0. 020. Thereason for this variation is not clear, but it seems tobe related to polarizer-beam deviation. (Prior to amanufacturer adjustment of beam deviation, OA showeda much larger variation, ± 0. 100, also approximatelysinusoidal in 2Ps. ) In the P-A configuration, FPA, de-fined by Eq. (6) of Ref. 5, should be 1. 0, independentof the polarizer setting, but experimentally was foundto be 1. 008± 0. 001. This is the first indication of non-linearity, because even in the presence of off-diagonalelements in the component Jones matrix, FPA would beless than 1. 0 by quantities of second order in the off-

95 J. Opt. Soc. Am., Vol. 66, No. 2, February 1976

c UVb - +t 0.4 -

0.2 - + +I + + I I I I + + I

0.1 I I I I + I I IZD of -00 +<0 .Q + + - + ++11 -0.2 - + + + + + +

"0 -0.3 I I | + I I I

1.13

I -F I I

1.11 -

I-F

1.09 + + x + +xC +

+x

+ + + * 4x

+

1.07 -

1.051 | t x x- I I I I0 20 40 60 80 100 120 140 160 180

P

FIG. 1. FPcA and the compensator parameters 5. (in degrees)and Tc obtained in P-C-A are plotted against the polarizerazimuth P (in degrees) for no nonlinearity correction [AO = 0,A2= 0 in Eq. (2)], shown with Wc), and with nonlinearity correc-tion (AO = 0, A2 =- x 10-5) shown with (+). For the uncorrectedcase (x), 6cwere too close to corrected (+) to be plotted. Thecompensator was set at C = 90. 00'.

William Ralph Hunter 95

l l

x I

6A=P,+90-AsmN-

x

1.0+ TV

0.8 + I lIn + + + Ia- 0.6k

U- 04 - I +

Q2 + + + +l + l I I l I l + I

1.65

1.55

1.45

700

650

- 600

550

500

x x54 + + + + + x4 *x

+ + + + + + + + + * 4

0 20 40 60 80 100 120 140 160 18P

3

FIG. 2. Fps4 and the calculated SiO2 film index and thickness,nf and df (in ) obtained in P-S-A are plotted against polarizerazimuth P (in degrees) for no nonlinearity correction [AO = 0,A 2= 0 in Eq. (2)], shown with (x), and with nonlinearity correc-tion (AO = 0, A 2 = l, 28 x 10-5), shown with (-0.

P=C or P=C+90 0 in P-C-A. Optimum values of A2were about 1. Ox i0's + 20% resulting in 2% correctionfor (Ir)MAXr- 2000. Without a nonlinearity correction,the ratio of the fourth-harmonic amplitude to second-harmonic amplitude was S 0. 8%. With the nonlinearitycorrection, this ratio was usually reduced to < 0. 1%.

The data from which the uncorrected data of Figs. 1and 2 were calculated were reanalyzed by use of theoptimum value forA 2 andA 0 = 0. The new results are alsoplotted in Figs. l and 2 with (+). Numerical modelingindicates that the deviation of bc near P= 90° is probab-ly caused by an error in 6A of -0. 02°. Considerableimprovement of the consistency, and presumablyaccuracy, is apparent. Note that fairly good agreementbetween corrected and uncorrected results occurs whenF - 0, i. e., when the light incident on the analyzer iscircularly polarized, as mentioned by Hauge and Dill. 3

Several authors have pointed out other advantages of thecircularly polarized light condition in rotating-analyzerellipsometers. 6'

7" 11

Several comments are appropriate regarding the wayP-S-A data were taken. A2 was optimized at P= 90°with the amplified output of the photomultiplier at a2 V dc level. For P* 90°, the photomultiplier highvoltage was adjusted to maintain this dc level. Pre-vious experimentation showed that if this were not done,FPSA(P = 0) would be greater than 1. 0, i.e., that thevalue of A2 optimized at P = 90° would not be thesame as A2 optimized at P = 0. For the 690 A SiO2sample used, the signal would be down to - 0. 5 V dc atP= 0. Examination of the data here showed sin datatpoints at each of the two minima (out of 256 data pointsper revolution) for which the signal input to this par-

96 J. Opt. Soc. Am., Vol. 66, No. 2, February 1976

.-:- + + 7

Ic=caO+ aI cos2(A - C) +b•I sin2(A - C), (Al)

where

2'=a 2cos20 - b2sin2 0,

2 =a2sin2 0 + b2 cos20 ,

(A2a)

(A2b)

with 0 = (A - Cs - 5C) .

By use of Jones-matrix formalism, T6 and Aj can beobtained from solutions (if we assume that there are nooff-diagonal Jones-matrix elements) of the equations

William Ralph Hunter 96

A- +

c

ticular ADC was below its threshold. That is, the datanear zero flux were artificially reduced to zero by thedigital conversion. This could account for the decreaseof a0 implied by an increase of FPSA(P= 0). However,the disparity between uncorrected and corrected P-S-Adata in which the high voltage was kept constant wascomparable to the data presented in Fig. 2, except thatdf (corrected) was not as constant. To avoid this arti-fical increase of F, the high voltage was adjusted tokeep the dc level at 2 V. Even with this procedure,however, FPSA (P= 0 or P= 1800) was not quite thesame as FPSA(P= 900), which may have been caused bydrift. It has been found that the optimum value of A2,although constant over the short term, drifts over thelong term. This may be related to drift found by Haugeand Dill3 and attributed to a laser instability. For ex-ample, the value of A2 used to optimize to FPSA (P= 900)= 1. 00000 at the beginning of the run resulted in FPSA(P = 90°) = 0. 99680 at the end.

Empirically, AsMIN determined by Eq. (4) and hence6A determined by Eq. (5) are not affected by changesof A., to first order. Similarly, 6C and 6P can befound accurately by use of the procedures described,even when nonlinearity is present and uncorrected be-cause the distortion of F is the same for the two valuesof PS on either side of the F = 1 point. (However, 6Ccould be in error if Tc and Ac vary with Ps because ofbeam deviation. 12) This is true provided that it domi-nates the effects of optical activity or other off-diagonalJones-matrix elements. 6' 7 Geometric averaging of twoindependent circularly polarized light measurements 7

appears to be the best way of eliminating these twosources of inaccuracy.

In conclusion, nonlinearity can be detected and cor-rected (without eliminating it), resulting in im-proved accuracy for arbitrary settings of the compo-nents in rotating-analyzer ellipsometers. Ideally, itwould be desirable to eliminate the nonlinearity. How-ever, the light-flux-detection system of ETA is quitecomplex, consisting of a diffusing plate, fiber optics,a photomulitplier, amplifier, and ADC. No attemptwas made to find and correct the responsible compo-nent(s).

APPENDIX

In the P-C-A calibration configuration, it is conve-nient to use the Fourier coefficients in the expansion

T _ (a, - a') cos2 (C - P)CT- (a, + a') sin 2 (C - P)

b' cos(C - P)Tc(aO+a')sin(C -P)'

D. Cahan and R. F. Spanier, Surf.Greef, Rev, Sci. Instrum. 41, 532S. Hauge and F. H. Dill, IBM J. REE. Aspnes, Opt. Commun. 8, 222 (

T,

Copyright © 1976 by the Optical Society of America 97

'B.2R.3p

4 D.

-0 (A3a) by. J. van der Meulen and N. C. Hien, J. Opt. Soco Am. 64,804 (1974).

6 D. E. Aspnes, J. Opt. Soc. Am. 64, 812 (1974).7 R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 64,

Ac900 . (A3b) 1459 (1974).8 R. H. Muller, Surf, Sci. 16, 14 (1969).9Actually, the first, third, and fourth coefficients, which in-

dicate errors, are also calculated.10Previous measurements indicated thatA0 =0 was appropriate

Sci. 16, 166 (1969). for this particular instrument, However, small changes of(1970). AO (even with A 2 = 0) can appreciably affect F.es. Dev. 17, 472 (1973). 11D. E. Aspnes, J. Opt. Soc. Am. 64, 639 (1974),1973). l2Wo G. Oldham, J. Opt. Soc. Am. 57, 57 (1967).

97 J. Opt. Soc. Am., Vol. 66, No. 2, February 1976