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Application of the degree of polarization to film thickness gradients
Thin Solid Films 313]314 Ž1998. 102]107
Application of the degree of polarization to film thickness gradients Uwe Richter U SENTECH Instruments GmbH, Rudower Chaussee 6, D-12484 Berlin, Germany
Abstract The effect of thickness gradients is studied using unfocused and focused beams in spectroscopic ellipsometry. A measurement with an ellipsometer in the polarizer-compensator sample analyzer ŽPCSA. configuration is extended to a triplet of raw data Ž c , D, P . where P is the degree of polarization of the reflected beam. It is shown that an averaging scheme allows one to measure using a large beam diameter and that a correction using the measured degree of polarization works well for improving microspot measurements. Q 1998 Elsevier Science S.A. Keywords: Bonded silicon on insulator ŽSiO 2 . on silicon ŽBOSI.; Thickness gradient materials; PCSA spectroscopic ellipsometry; Degree of polarization; Microspot spectroscopic ellipsometry
1. Introduction The application of spectroscopic ellipsometry has moved from the well established analysis of ideal layers on well known substrates to more complicated systems with many non-ideal effects. The measurement of these non-ideal effects is important for correct results. It is necessary to detect these effects and to overcome the limitations introduced. A typical situation occurs in the case of a thickness gradient. On an ideal sample stack Ži.e. with zero thickness gradient. a layer causes a modulation in the spectra Ž c Ž l., DŽ l.. which is weakened for several reasons. One important reason and the topic of this article is the thickness gradient and other effects include focusing angles, stray light, backside reflections, etc. On most samples and under most measurement conditions only a small number of such non-ideal influences apply. The measurement of samples with non-ideal effects
can be divided into two parts: the detection of the non-ideality; and the removal or modelling of the effect. The detection of the effect is done in most cases after the measurement and during the fitting process. However, fitting non-ideal effects is very time-consuming due to more complicated modelling. Recently Jellison w1x reported the measurement of the degree of polarization using a two channel photoacoustic modulator ellipsometer. This article discusses how a polarizer-compensator sample analyzer Žin the following PCSA. ellipsometer can be used for measuring this additional parameter. The following results show how measurement with a spectroscopic ellipsometer in the PCSA arrangement can be used to detect thickness gradients and how the measurement can be improved for better fitting without modelling the non-ideality. The stack used for the experiments was bonded silicon on insulator ŽSiO 2 . on silicon ŽBOSI.. 2. Thickness gradient analysis with a PSA arrangement (no microspot)
U
Fax: q49 30 63925522; e-mail: sentech ] instr@compuserve. com 0040-6090r98r$19.00 Q 1998 Elsevier Science S.A. All rights reserved PII S0040-6090Ž97.01005-5
All measurements in this article are made with a
U. Richter r Thin Solid Films 313]314 (1998) 102]107
103
The basic idea of modelling a thickness distribution is that the detector is illuminated with partial beams reflected from areas with different thicknesses. Averaging these partial intensities will give the averaged Stokes vector analyzed by the rotating analyzer. For the PSA arrangement Žsuperscript p . the relations between the Fourier coefficients s1 and s2 and the ellipsometric angles are given by:
Fig. 1. Experimental data and fit Ždotted. to the measurement using a two layer model.
conventional PCSA ellipsometer ŽSENTECH SE 850 UVrVIS and Fourier transform NIR. with a polarizer prism at the position P relative to the plane of incidence. This section uses the PSA mode only, which is the typical setup in most ellipsometers. The advantage of this arrangement is the stability and precision in reading tan C and cos D. The disadvantage is that the readings of D near 08 or 1808 show poor precision and accuracies. Additionally the measurement of the degree of polarization is as difficult as measuring R p . A correction of ellipsometric measurement requires an accuracy of better than 10y3 which is not possible without a retarder. The extraction of sample imperfections can only be carried out indirectly by introducing a more complicated model. If the thickness gradient is a property of the sample, the analysis should give an average thickness and the thickness distribution as final results. Previous analysis of the BOSI system by El-Ghazzawi w2,3x have used a microspot to measure the thickness within a very small area and to obtain the thickness information at this single point on the sample. The measurement in Fig. 1 is typical and shows the influence of the gradient. The first analysis started with an ideal stack of c-Si on thermally grown SiO 2 on c-Si. Fig. 1 shows the measurement and fit using a large spot diameter of 4 mm. The main difference in the spectra is that the values of
s1p s
cos2 Py cos2 c 1 y cos2 P? cos2 c
Ž1.
s2p s
sin2 c ? sin2 P? cosD 1 y cos2 P? cos2 c
Ž2.
The measurement averages the Stokes vector Ži.e. on the intensity scale. for all thicknesses. The thickness distribution within the measurement spot varies with the angle of incidence f because the spot has an elliptical shape on the sample and its long diameter g l is related to its short diameter g s by: gl s
gs cos f
Ž3.
In the following, a linear thickness gradient w in the long axis g l is assumed without loss of generality: ws
dt dx
t Ž x . s tav q Ž x y x av . w 0
Ž4. Ž5.
where tav is the average thickness, x is the direction in the plane of incidence on the sample, x av is the position of the average thickness and w 0 is the gradient. The parameters to be fitted are tav and w 0 . The assumption of the index gradient must be proven by mapping with smaller spot sizes. The original idea of the linear gradient is derived from a reflectivity measurement with a FTP 500 film thickness probe which gave the top layer thickness. But this tool can only measure top layers and information on refractive indices is of low accuracy Žor not available. for some layer stacks. This makes the ellipsometer important for the general application, however, this study is limited to the thickness gradient as being the only unknown. The effective thickness range D t can be calculated from the above equations assuming the difference x max y x min sets the gradient along the long diameter of the spot g l ; see Eq. Ž3.. The elliptic shape is not taken into account because it must be weighted by the detection sensitivity within the beam profile and this is difficult to determine. The analysis in this section shows that such modelling is not required for analysing the gradient. First, D t is given by
U. Richter r Thin Solid Films 313]314 (1998) 102]107
104
D t Ž f , p0 , g s . s
g s p0 cos f
Ž6.
and t min Ž f , p 0 , g s . s tav y 12 D t Ž f , p 0 , g s .
Ž7.
t max Ž f , p 0 , g s . s tav q 12 D t Ž f , p 0 , g s .
Ž8.
These equations are very important for fitting multiple angle measurements because the long diameter of the spot and the averaging range changes with the angle of incidence. Using these equations, the averaging integral can be written as: t max t h Ž
² s jp : s
Ht
sj
t bosi , t oxide . R t h Ž t bosi , t oxide . dt bosi
min
t max
Ht
R t h Ž t bosi , t oxide . dt bosi
min
j s 1,2,3
Ž9.
where the following symbols are used: v
v v v v
² s jp :, effective s1 , s2 , s3 Žwhich should be the same as the measured values.; t bosi , thickness of the top c-Si layer; t oxide , thickness of the underlying silicon dioxide; s jt h , calculated s j of an ideal stack Žno gradient.; R t h , calculated reflectivity of an ideal stack.
The effective ² s jp : can be used to calculate the averaged ellipsometric angles: ²c p: s
cos2 Py ² s1p : 1 p y arccos 2 1 y ² s1p :cos2 P
² D p : s arccos
ž
ž
/
Ž cos2² c p : y cos2 P .² s2p : sin2 P? sin2² c p : ? ² s1p :
Ž 10 .
/
Ž 11 .
The beam diameter g s used for the theoretical calculations was 4 mm, which gives a good signal to noise ratio for measuring the NIR portion of the spectra. The fit is shown in Fig. 2. The resulting thickness gradient of 26 nmrmm requires accurate calculation of the weighting integral since the interference fringes are averaged over thickness periods. The thickness period at 633 nm wavelength and 708 angle of incidence is 85 nm for c-Si and the index gradient is D t s 164 nm within the spot. In the infrared Ž1600 nm. the period is 239 nm. These numbers explain the vanishing modulation in the VIS portion and the weak modulation in the NIR compared to the expected range 08 . . . 1808. However, the good agreement between theory and measurement shows that the gradient information is contained within the spectrum and can be fitted together with the thicknesses.
Fig. 2. Experimental data and fit Ždotted. for a measurement using a large spot Ž4 mm.. The fit assumes a thickness gradient for 3.5 m m thick c-Si on a 2-m m oxide layer on c-Si.
The next step is to check whether the fitted index gradient is a real effect and has the correct value. For these reasons two checks have been done. The first uses an independent reflectivity method for the thickness measurement. The second method will be discussed below in conjunction with the microspot measurement. The reflectivity measurement was performed with a spot of 10 m m at normal incidence within a spectral range of 500]900 nm using a SENTECH Instruments FTP 500 film thickness probe. This method allows a small spot size excluding any thickness gradients and converts the attenuated reflectivity interference pattern into thicknesses by a Fourier transformation method. The entire sample was mapped with high lateral resolution and the long diameter of the 708 angle of incidence spot was extracted from the map. The gradient within the measurement spot of the ellipsometer was determined to be 25 nmrmm which is in excellent agreement with the ellipsometer fit. 3. Measurement of the degree of polarization using a PCSA arrangement The above results gave valid information about the sample, but require a knowledge of the presence of the gradient and a very slow fitting procedure Ževen if an analytical formula can be derived for the averaging integral.. It would be better and much faster to detect such non-ideal effects during the measurement. A PCSA arrangement can be used to perform two measurements, i.e. with ŽPCSA mode. and without
U. Richter r Thin Solid Films 313]314 (1998) 102]107
the retarder ŽPSA mode.. The PSA mode allows only s1 and s2 to be measured, but the Stokes vector has four elements. Since the absolute intensity is not used in ellipsometry there are three parameters available as raw data. Many publications in ellipsometry measure the D value without sign information Ži.e. cos D ., since this information is not available from the PSA arrangement. Other systems use a retarder or modulator to obtain sign information for D. The third parameter left is the degree of polarization of the reflected beam. The PCSA mode uses a polarizer at P, and in addition, a retarder with fast axis at C and a phase difference T Žnear lr4.. The system is calculated using Mueller matrices ŽAzzam w4x.. In the following, the short syntax for trigonometric functions cos x s C x and sin xs S x are used. The resulting formulas for the PCSA setup are: s1c s
A y C2 c 1 y C2 c A
Ž 12 .
s2c s
S2 c SQqD 'D 2 q L2 1 y C2 c A
Ž 13 .
A s C2 P C22C q S22C CT
Ž 14.
D s S2 P S22C q C22C CT q C2 P S2 C C2 C Ž 1 y CT .
Ž 15 .
L s ST C2ŽCyP .
Ž 16 .
cosQ s
L
'D
2
qL
2
or
sinQ s
D
'D
2
q L2
Ž 17.
A superscript c indicates a measurement with the retarder and the superscript p in Eqs. Ž1., Ž2., Ž10. and Ž11. indicates a measurement without the retarder Žpolarizer only.. This allows one to calculate cos 2 c , cos D and sin D Ž Q f 908. from the four numbers s1p , s2p , s1c and s2c . It should be noted that the resulting values are Ž c p , D p . and Ž c c, Dc . with different accuracies. Since accuracy analysis is not the goal of this article, the method of calculating the final Ž c , D . is not discussed here. In an ideal measurement Ž c p , D p . and Ž c c, Dc . should agree within error limits. In the case of the thickness gradient studied here large differences were observed. The basic assumption of having totally polarized light reflected from the sample is wrong in this case. The light impinging on the detector consists of an unpolarized portion Iu and a polarized portion I p w5x: I s Iu q I p
Ip Pg r s I
where the following symbols are used: v v
Pg r , degree of polarization; I, total intensity of the beam;
and Pg r s s12 q s22 q s32
'
Ž 18.
Ž 19 .
The measurements in the PSA and PCSA modes give only two Fourier coefficients for each mode when a rotating analyzer is used. Together four values are measured. Since the two beams have different Strokes vectors we have six parameters characterizing the beams and two of those are unknowns s3p and s3c which cannot be measured directly. The method presented here combines the two measurements because the sample Ž c , D . is the same for both beams and the depolarization is linear with s1 , s2 and s3 Ži.e. the Mueller matrix for depolarization has diagonal elements Ž1, Pg r , Pg r , Pg r . and all other elements are zero w6x.. If depolarization occurs, the Fourier coefficients change with the degree of polarization Pg r : s j) s Pg r s j
q S2 P S2 C C 2 C Ž 1 y CT .
105
Ž 20 .
The starred values are measured and unstarred ones are the corrected Stokes parameters with partial polarization removed. Now sU2 is s2c for the PCSA mode and s2p for the PSA mode. The theoretical Fourier coefficients are calculated from the calibration constants for the system and from the ellipsometric angles of the sample. The expressions for s2 in the PCSA and PSA modes are taken from Eqs. Ž13. and Ž2.: 2 Ž s2c . 2 Ž 1 y C2 c A . 2 s S22c SQqD Ž D 2 q C 2 . Pg2r
Ž 21.
Ž s2p . 2 Ž 1 y C2 c C2 P . 2 s S22c S22 P CD2 Pg2r
Ž 22.
Both equations are the squares of the Fourier coefficient expressions given by Eqs. Ž2. and Ž13. for the two measurement steps and thus the equations can be added: Ž s2c . 2 Ž 1 y C2 c A . 2 q Ž s2p . 2 Ž 1 y C2 c C2 P . 2 2 Ž D2 q C 2 . s Pg2r S22c S22 P CD2 q SQqD
Ž 23.
The final formula for the degree of polarization is:
Pg r s
)
Ž s2c . 2 Ž 1 y C2 c A . 2 q Ž s2p . 2 Ž 1 y C2 c C2 P . 2 2 Ž D 2 q L2 . S22c S22 P CD2 q SQqD
Ž 24 .
U. Richter r Thin Solid Films 313]314 (1998) 102]107
106
If an ideal retarder is used Ž C s 08, T s 908, A s C2 P , D s 0, L s yS2 p , Q s 08., then this formula reduces to: Pg r s
'Ž s . q Ž s . c 2 2
p 2 2
?
1 y C2 c C2 P S2 c S2 P
Ž 25.
With the formulas above, it is possible to extend the typical ellipsometer readout from two raw data values Ž c , D 08 . . . 1808 . or Ž c , D 08 . . . 3608 . to a triplet Ž c , D 08 . . . 3608 , Pg t .. This third raw parameter is sensitive to effects such as thickness gradients and focusing as well as many other non-ideal effects. Now it is possible to define Ž c , D . for two cases. In the first case, the ellipsometric angles are calculated from Eqs. Ž1., Ž2.,Ž12. and Ž13. regardless of any depolarization with and without the retarder. In the second case, the measurement with and without the retarder are performed and a correction associated with the degree of polarization correction is applied. Ž s1p ,
s2p . ª Ž c
Ž s1p ,
s2p ,
s1c ,
, D 08 . . . 1808 .
s2c . ª Ž c
, D 08 . . . 3608 .
Ž s1p , s2p , s1c , s2c . ª Ž c corr , Dcorr . 08 . . . 3608
Ž 26 . Ž 27 . Ž 28 .
This has various consequences for the data fitting. In some cases the new Ž c corr, Dcorr . values exhibit an improved shape for better analysis using ideal models. If the fitting procedure uses any Stokes vector averaging, the degree of polarization must be calculated. Then the theoretical counterpart of the measured Ž c , D . must be calculated from either the averaged Fourier coefficients to produce effective Ž² c , ² D :. pairs or from the polarized portion of the averaged sUj : s j) ) s
s j)
'Ž s
2 2 ) .2 q Ž s2) . q Ž s3) . 1
Ž 29.
The theoretical model determines Ž c , D . values for a given sample. From Ž c , D . and the device calibration parameters Ž P, C, T ., the Fourier coefficients are calculated. If Stokes vector averaging is necessary Žexamples include thickness variations, transparent substrates, or beam focusing for microspots., then this produces the final Stokes vector which may contain depolarization. The polarized portion should be used Žsee Eqs. Ž20. and Ž29.. and the PCSA formulas are applied to obtain the final Ž c , D . values. This makes it clear, that, for ideal stacks and ideal ellipsometers, a Pg r value of unity should always be measured. Since Pg r is in the form of a raw measurement, it can be used as a detector for non-ideal effects; see, e.g., Fig. 3. Another advantage of this method is the removal of
Fig. 3. Measured degree of polarization with a 200-m m spot size, 58 focusing angle and 26 nmrmm thickness gradient for 3.5 m m c-Si on a 2-m m oxide layer on c-Si.
other minor effects such as stray light within the spectrometer used or focusing with microspots. 4. Microspot measurements The microspot used here had a spot size g s of 200 m m. The thickness variation visible by the spot on this sample is calculated to be 8.2 nm while the thickness period of c-Si is 85 nm at 633 nm and f s 708. Under such circumstances the measured spectra are expected to show drastically increased modulation in the UVrVIS range. This is verified by comparing Fig. 1 and Fig. 4. A comparison of the measurements with the microspot Ž Pg r correction applied. and without the spot show the expected increase in modulation. In fact the maximum value of D reaches the 1808 value, which is only possible in the absence of depolarizing effects as phase mixing. The unmapped data have the original range of 0]3608. This measurement is fitted using the same model as for the measurement without the microspot except that no thickness gradient is used. This ideal model applies for two reasons. First, the spot size is much smaller so that the effect of the gradient is much smaller and second, the polarization correction removes the residual phase mixing. The fit gives good agreement between model and measurement. The wavelength range 520]650 nm is highly sensitive to the reference absorption data of silicon near zero and differs for c-Si data reported by different authors. Here the data of Aspnes w7x are used. The correction method during the measurement gives the degree of polarization of the reflected beam. The Pg r Ž l. spectrum is shown in Fig. 3. This result shows Ps 1 over the spectral range where silicon is totally absorbing at 3.5 m m thickness. In this spectra range the sample behaves as a reflecting substrate and no thickness gradient is present. In the transparency region Pg r shows a modulation with wavelength as do Ž c , D . spectra. This results from the constructive or destructive superposition of the partial waves. If the superpo-
U. Richter r Thin Solid Films 313]314 (1998) 102]107
Fig. 4. Fit of microspot measurement Ž200 m m spot, 58 focussing angle. performed with correction for depolarization Ž3.5 m m c-Si on a 2-m m oxide layer on c-Si substrate measured at 708 angle of incidence.. The fit Ždotted. assumes an ideal sample structure without thickness gradient.
sition is destructive, only the first reflected partial beam gives a maximum in Pg r . At constructive superposition the non-planar interfaces lead to a reduction in the degree of polarization due to the incoherent partial beams. To compare these results with the measurements without a microspot lens, a line-scan has been performed along the long diameter g l of the large spot. All resulting spectra are fitted to obtain the thicknesses of the top and bottom layer. The resulting thickness profiles are given in Fig. 5. The profiles show the interesting result that the bottom oxide thickness is constant around 2 m m but the top c-Si shows a large thickness gradient. El-Ghazzawi et al. w2,3x reported a significant change in the spectra upon a slit. If this is compared with the shape of the spectra above, then it becomes clear that this difference must come from the degree of polarization correction. The slit reduces the focusing angle and the spot dimension, the latter effects creating depolarization and changing the spectra. Since the application of a slit largely reduces the intensity, it is found that the correction for the degree of polarization is a better approach because it gives a higher signal to noise ratio and at least the same accuracy in the Ž c , D . spectra. 5. Summary A thickness gradient sample of bonded silicon on silicon dioxide on silicon has been measured using a large spot diameter. The resulting attenuated modula-
107
Fig. 5. Thickness profiles measured with the microspot. The large spot used for the date in Fig. 1 has its long diameter from xs 1 mm to xs 8 mm.
tion in Ž c , D . has been successfully fitted to obtain both thicknesses and a thickness gradient. A new method for measuring the degree of polarization using a PCSA arrangement is presented. The advantage of this method is the application to spectroscopic ellipsometry where the retarder and polarizer are not available in the exact positions of T s Ž lr4. and Ps 458. The measurement of the degree of polarization gives a reliable sensor for non-ideal effects. The microspot measurement was fitted using an ideal two model with good agreement between theory and measurement. The thickness gradient fitted from the large spot data was reproduced with reflectivity and microspot spectroscopic ellipsometry measurements. A significant improvement of the microspot readings with the retarder is achieved due to the degree of polarization correction. References w1x G.E. Jellison, Jr., J.W. McCamy, Appl. Phys. Lett. 61 Ž1992. 512]514. w2x M.E. El-Ghazzawi, T. Saitoh, N. Hori, A. Sakai, T. Oka, Thin Solid Films 233 Ž1993. 218]222. w3x M.E.M. El-Ghazzawi, T. Saitoh, Opt. Eng. 34 Ž1995. 453]459. w4x Azzam, Bashara, Ellipsometry of Polarized Light, North-Holland, 1989. w5x A. Roseler, Infrared Spectroscopic Ellipsometry, Akademie¨ Verlag, Berlin, 1990. w6x J.T. Zettler, T. Trepk, L. Spanos, Y.-Z. Hu, W. Richter, Thin Solid Films 234 Ž1993. 402]407. w7x D.E. Aspnes, A.A. Studna, Phys. Rev. B 27 Ž1983. 985.
Application of the degree of polarization to film thickness gradients Uwe Richter U SENTECH Instruments GmbH, Rudower Chaussee 6, D-12484 Berlin, Germany
Abstract The effect of thickness gradients is studied using unfocused and focused beams in spectroscopic ellipsometry. A measurement with an ellipsometer in the polarizer-compensator sample analyzer ŽPCSA. configuration is extended to a triplet of raw data Ž c , D, P . where P is the degree of polarization of the reflected beam. It is shown that an averaging scheme allows one to measure using a large beam diameter and that a correction using the measured degree of polarization works well for improving microspot measurements. Q 1998 Elsevier Science S.A. Keywords: Bonded silicon on insulator ŽSiO 2 . on silicon ŽBOSI.; Thickness gradient materials; PCSA spectroscopic ellipsometry; Degree of polarization; Microspot spectroscopic ellipsometry
1. Introduction The application of spectroscopic ellipsometry has moved from the well established analysis of ideal layers on well known substrates to more complicated systems with many non-ideal effects. The measurement of these non-ideal effects is important for correct results. It is necessary to detect these effects and to overcome the limitations introduced. A typical situation occurs in the case of a thickness gradient. On an ideal sample stack Ži.e. with zero thickness gradient. a layer causes a modulation in the spectra Ž c Ž l., DŽ l.. which is weakened for several reasons. One important reason and the topic of this article is the thickness gradient and other effects include focusing angles, stray light, backside reflections, etc. On most samples and under most measurement conditions only a small number of such non-ideal influences apply. The measurement of samples with non-ideal effects
can be divided into two parts: the detection of the non-ideality; and the removal or modelling of the effect. The detection of the effect is done in most cases after the measurement and during the fitting process. However, fitting non-ideal effects is very time-consuming due to more complicated modelling. Recently Jellison w1x reported the measurement of the degree of polarization using a two channel photoacoustic modulator ellipsometer. This article discusses how a polarizer-compensator sample analyzer Žin the following PCSA. ellipsometer can be used for measuring this additional parameter. The following results show how measurement with a spectroscopic ellipsometer in the PCSA arrangement can be used to detect thickness gradients and how the measurement can be improved for better fitting without modelling the non-ideality. The stack used for the experiments was bonded silicon on insulator ŽSiO 2 . on silicon ŽBOSI.. 2. Thickness gradient analysis with a PSA arrangement (no microspot)
U
Fax: q49 30 63925522; e-mail: sentech ] instr@compuserve. com 0040-6090r98r$19.00 Q 1998 Elsevier Science S.A. All rights reserved PII S0040-6090Ž97.01005-5
All measurements in this article are made with a
U. Richter r Thin Solid Films 313]314 (1998) 102]107
103
The basic idea of modelling a thickness distribution is that the detector is illuminated with partial beams reflected from areas with different thicknesses. Averaging these partial intensities will give the averaged Stokes vector analyzed by the rotating analyzer. For the PSA arrangement Žsuperscript p . the relations between the Fourier coefficients s1 and s2 and the ellipsometric angles are given by:
Fig. 1. Experimental data and fit Ždotted. to the measurement using a two layer model.
conventional PCSA ellipsometer ŽSENTECH SE 850 UVrVIS and Fourier transform NIR. with a polarizer prism at the position P relative to the plane of incidence. This section uses the PSA mode only, which is the typical setup in most ellipsometers. The advantage of this arrangement is the stability and precision in reading tan C and cos D. The disadvantage is that the readings of D near 08 or 1808 show poor precision and accuracies. Additionally the measurement of the degree of polarization is as difficult as measuring R p . A correction of ellipsometric measurement requires an accuracy of better than 10y3 which is not possible without a retarder. The extraction of sample imperfections can only be carried out indirectly by introducing a more complicated model. If the thickness gradient is a property of the sample, the analysis should give an average thickness and the thickness distribution as final results. Previous analysis of the BOSI system by El-Ghazzawi w2,3x have used a microspot to measure the thickness within a very small area and to obtain the thickness information at this single point on the sample. The measurement in Fig. 1 is typical and shows the influence of the gradient. The first analysis started with an ideal stack of c-Si on thermally grown SiO 2 on c-Si. Fig. 1 shows the measurement and fit using a large spot diameter of 4 mm. The main difference in the spectra is that the values of
s1p s
cos2 Py cos2 c 1 y cos2 P? cos2 c
Ž1.
s2p s
sin2 c ? sin2 P? cosD 1 y cos2 P? cos2 c
Ž2.
The measurement averages the Stokes vector Ži.e. on the intensity scale. for all thicknesses. The thickness distribution within the measurement spot varies with the angle of incidence f because the spot has an elliptical shape on the sample and its long diameter g l is related to its short diameter g s by: gl s
gs cos f
Ž3.
In the following, a linear thickness gradient w in the long axis g l is assumed without loss of generality: ws
dt dx
t Ž x . s tav q Ž x y x av . w 0
Ž4. Ž5.
where tav is the average thickness, x is the direction in the plane of incidence on the sample, x av is the position of the average thickness and w 0 is the gradient. The parameters to be fitted are tav and w 0 . The assumption of the index gradient must be proven by mapping with smaller spot sizes. The original idea of the linear gradient is derived from a reflectivity measurement with a FTP 500 film thickness probe which gave the top layer thickness. But this tool can only measure top layers and information on refractive indices is of low accuracy Žor not available. for some layer stacks. This makes the ellipsometer important for the general application, however, this study is limited to the thickness gradient as being the only unknown. The effective thickness range D t can be calculated from the above equations assuming the difference x max y x min sets the gradient along the long diameter of the spot g l ; see Eq. Ž3.. The elliptic shape is not taken into account because it must be weighted by the detection sensitivity within the beam profile and this is difficult to determine. The analysis in this section shows that such modelling is not required for analysing the gradient. First, D t is given by
U. Richter r Thin Solid Films 313]314 (1998) 102]107
104
D t Ž f , p0 , g s . s
g s p0 cos f
Ž6.
and t min Ž f , p 0 , g s . s tav y 12 D t Ž f , p 0 , g s .
Ž7.
t max Ž f , p 0 , g s . s tav q 12 D t Ž f , p 0 , g s .
Ž8.
These equations are very important for fitting multiple angle measurements because the long diameter of the spot and the averaging range changes with the angle of incidence. Using these equations, the averaging integral can be written as: t max t h Ž
² s jp : s
Ht
sj
t bosi , t oxide . R t h Ž t bosi , t oxide . dt bosi
min
t max
Ht
R t h Ž t bosi , t oxide . dt bosi
min
j s 1,2,3
Ž9.
where the following symbols are used: v
v v v v
² s jp :, effective s1 , s2 , s3 Žwhich should be the same as the measured values.; t bosi , thickness of the top c-Si layer; t oxide , thickness of the underlying silicon dioxide; s jt h , calculated s j of an ideal stack Žno gradient.; R t h , calculated reflectivity of an ideal stack.
The effective ² s jp : can be used to calculate the averaged ellipsometric angles: ²c p: s
cos2 Py ² s1p : 1 p y arccos 2 1 y ² s1p :cos2 P
² D p : s arccos
ž
ž
/
Ž cos2² c p : y cos2 P .² s2p : sin2 P? sin2² c p : ? ² s1p :
Ž 10 .
/
Ž 11 .
The beam diameter g s used for the theoretical calculations was 4 mm, which gives a good signal to noise ratio for measuring the NIR portion of the spectra. The fit is shown in Fig. 2. The resulting thickness gradient of 26 nmrmm requires accurate calculation of the weighting integral since the interference fringes are averaged over thickness periods. The thickness period at 633 nm wavelength and 708 angle of incidence is 85 nm for c-Si and the index gradient is D t s 164 nm within the spot. In the infrared Ž1600 nm. the period is 239 nm. These numbers explain the vanishing modulation in the VIS portion and the weak modulation in the NIR compared to the expected range 08 . . . 1808. However, the good agreement between theory and measurement shows that the gradient information is contained within the spectrum and can be fitted together with the thicknesses.
Fig. 2. Experimental data and fit Ždotted. for a measurement using a large spot Ž4 mm.. The fit assumes a thickness gradient for 3.5 m m thick c-Si on a 2-m m oxide layer on c-Si.
The next step is to check whether the fitted index gradient is a real effect and has the correct value. For these reasons two checks have been done. The first uses an independent reflectivity method for the thickness measurement. The second method will be discussed below in conjunction with the microspot measurement. The reflectivity measurement was performed with a spot of 10 m m at normal incidence within a spectral range of 500]900 nm using a SENTECH Instruments FTP 500 film thickness probe. This method allows a small spot size excluding any thickness gradients and converts the attenuated reflectivity interference pattern into thicknesses by a Fourier transformation method. The entire sample was mapped with high lateral resolution and the long diameter of the 708 angle of incidence spot was extracted from the map. The gradient within the measurement spot of the ellipsometer was determined to be 25 nmrmm which is in excellent agreement with the ellipsometer fit. 3. Measurement of the degree of polarization using a PCSA arrangement The above results gave valid information about the sample, but require a knowledge of the presence of the gradient and a very slow fitting procedure Ževen if an analytical formula can be derived for the averaging integral.. It would be better and much faster to detect such non-ideal effects during the measurement. A PCSA arrangement can be used to perform two measurements, i.e. with ŽPCSA mode. and without
U. Richter r Thin Solid Films 313]314 (1998) 102]107
the retarder ŽPSA mode.. The PSA mode allows only s1 and s2 to be measured, but the Stokes vector has four elements. Since the absolute intensity is not used in ellipsometry there are three parameters available as raw data. Many publications in ellipsometry measure the D value without sign information Ži.e. cos D ., since this information is not available from the PSA arrangement. Other systems use a retarder or modulator to obtain sign information for D. The third parameter left is the degree of polarization of the reflected beam. The PCSA mode uses a polarizer at P, and in addition, a retarder with fast axis at C and a phase difference T Žnear lr4.. The system is calculated using Mueller matrices ŽAzzam w4x.. In the following, the short syntax for trigonometric functions cos x s C x and sin xs S x are used. The resulting formulas for the PCSA setup are: s1c s
A y C2 c 1 y C2 c A
Ž 12 .
s2c s
S2 c SQqD 'D 2 q L2 1 y C2 c A
Ž 13 .
A s C2 P C22C q S22C CT
Ž 14.
D s S2 P S22C q C22C CT q C2 P S2 C C2 C Ž 1 y CT .
Ž 15 .
L s ST C2ŽCyP .
Ž 16 .
cosQ s
L
'D
2
qL
2
or
sinQ s
D
'D
2
q L2
Ž 17.
A superscript c indicates a measurement with the retarder and the superscript p in Eqs. Ž1., Ž2., Ž10. and Ž11. indicates a measurement without the retarder Žpolarizer only.. This allows one to calculate cos 2 c , cos D and sin D Ž Q f 908. from the four numbers s1p , s2p , s1c and s2c . It should be noted that the resulting values are Ž c p , D p . and Ž c c, Dc . with different accuracies. Since accuracy analysis is not the goal of this article, the method of calculating the final Ž c , D . is not discussed here. In an ideal measurement Ž c p , D p . and Ž c c, Dc . should agree within error limits. In the case of the thickness gradient studied here large differences were observed. The basic assumption of having totally polarized light reflected from the sample is wrong in this case. The light impinging on the detector consists of an unpolarized portion Iu and a polarized portion I p w5x: I s Iu q I p
Ip Pg r s I
where the following symbols are used: v v
Pg r , degree of polarization; I, total intensity of the beam;
and Pg r s s12 q s22 q s32
'
Ž 18.
Ž 19 .
The measurements in the PSA and PCSA modes give only two Fourier coefficients for each mode when a rotating analyzer is used. Together four values are measured. Since the two beams have different Strokes vectors we have six parameters characterizing the beams and two of those are unknowns s3p and s3c which cannot be measured directly. The method presented here combines the two measurements because the sample Ž c , D . is the same for both beams and the depolarization is linear with s1 , s2 and s3 Ži.e. the Mueller matrix for depolarization has diagonal elements Ž1, Pg r , Pg r , Pg r . and all other elements are zero w6x.. If depolarization occurs, the Fourier coefficients change with the degree of polarization Pg r : s j) s Pg r s j
q S2 P S2 C C 2 C Ž 1 y CT .
105
Ž 20 .
The starred values are measured and unstarred ones are the corrected Stokes parameters with partial polarization removed. Now sU2 is s2c for the PCSA mode and s2p for the PSA mode. The theoretical Fourier coefficients are calculated from the calibration constants for the system and from the ellipsometric angles of the sample. The expressions for s2 in the PCSA and PSA modes are taken from Eqs. Ž13. and Ž2.: 2 Ž s2c . 2 Ž 1 y C2 c A . 2 s S22c SQqD Ž D 2 q C 2 . Pg2r
Ž 21.
Ž s2p . 2 Ž 1 y C2 c C2 P . 2 s S22c S22 P CD2 Pg2r
Ž 22.
Both equations are the squares of the Fourier coefficient expressions given by Eqs. Ž2. and Ž13. for the two measurement steps and thus the equations can be added: Ž s2c . 2 Ž 1 y C2 c A . 2 q Ž s2p . 2 Ž 1 y C2 c C2 P . 2 2 Ž D2 q C 2 . s Pg2r S22c S22 P CD2 q SQqD
Ž 23.
The final formula for the degree of polarization is:
Pg r s
)
Ž s2c . 2 Ž 1 y C2 c A . 2 q Ž s2p . 2 Ž 1 y C2 c C2 P . 2 2 Ž D 2 q L2 . S22c S22 P CD2 q SQqD
Ž 24 .
U. Richter r Thin Solid Films 313]314 (1998) 102]107
106
If an ideal retarder is used Ž C s 08, T s 908, A s C2 P , D s 0, L s yS2 p , Q s 08., then this formula reduces to: Pg r s
'Ž s . q Ž s . c 2 2
p 2 2
?
1 y C2 c C2 P S2 c S2 P
Ž 25.
With the formulas above, it is possible to extend the typical ellipsometer readout from two raw data values Ž c , D 08 . . . 1808 . or Ž c , D 08 . . . 3608 . to a triplet Ž c , D 08 . . . 3608 , Pg t .. This third raw parameter is sensitive to effects such as thickness gradients and focusing as well as many other non-ideal effects. Now it is possible to define Ž c , D . for two cases. In the first case, the ellipsometric angles are calculated from Eqs. Ž1., Ž2.,Ž12. and Ž13. regardless of any depolarization with and without the retarder. In the second case, the measurement with and without the retarder are performed and a correction associated with the degree of polarization correction is applied. Ž s1p ,
s2p . ª Ž c
Ž s1p ,
s2p ,
s1c ,
, D 08 . . . 1808 .
s2c . ª Ž c
, D 08 . . . 3608 .
Ž s1p , s2p , s1c , s2c . ª Ž c corr , Dcorr . 08 . . . 3608
Ž 26 . Ž 27 . Ž 28 .
This has various consequences for the data fitting. In some cases the new Ž c corr, Dcorr . values exhibit an improved shape for better analysis using ideal models. If the fitting procedure uses any Stokes vector averaging, the degree of polarization must be calculated. Then the theoretical counterpart of the measured Ž c , D . must be calculated from either the averaged Fourier coefficients to produce effective Ž² c , ² D :. pairs or from the polarized portion of the averaged sUj : s j) ) s
s j)
'Ž s
2 2 ) .2 q Ž s2) . q Ž s3) . 1
Ž 29.
The theoretical model determines Ž c , D . values for a given sample. From Ž c , D . and the device calibration parameters Ž P, C, T ., the Fourier coefficients are calculated. If Stokes vector averaging is necessary Žexamples include thickness variations, transparent substrates, or beam focusing for microspots., then this produces the final Stokes vector which may contain depolarization. The polarized portion should be used Žsee Eqs. Ž20. and Ž29.. and the PCSA formulas are applied to obtain the final Ž c , D . values. This makes it clear, that, for ideal stacks and ideal ellipsometers, a Pg r value of unity should always be measured. Since Pg r is in the form of a raw measurement, it can be used as a detector for non-ideal effects; see, e.g., Fig. 3. Another advantage of this method is the removal of
Fig. 3. Measured degree of polarization with a 200-m m spot size, 58 focusing angle and 26 nmrmm thickness gradient for 3.5 m m c-Si on a 2-m m oxide layer on c-Si.
other minor effects such as stray light within the spectrometer used or focusing with microspots. 4. Microspot measurements The microspot used here had a spot size g s of 200 m m. The thickness variation visible by the spot on this sample is calculated to be 8.2 nm while the thickness period of c-Si is 85 nm at 633 nm and f s 708. Under such circumstances the measured spectra are expected to show drastically increased modulation in the UVrVIS range. This is verified by comparing Fig. 1 and Fig. 4. A comparison of the measurements with the microspot Ž Pg r correction applied. and without the spot show the expected increase in modulation. In fact the maximum value of D reaches the 1808 value, which is only possible in the absence of depolarizing effects as phase mixing. The unmapped data have the original range of 0]3608. This measurement is fitted using the same model as for the measurement without the microspot except that no thickness gradient is used. This ideal model applies for two reasons. First, the spot size is much smaller so that the effect of the gradient is much smaller and second, the polarization correction removes the residual phase mixing. The fit gives good agreement between model and measurement. The wavelength range 520]650 nm is highly sensitive to the reference absorption data of silicon near zero and differs for c-Si data reported by different authors. Here the data of Aspnes w7x are used. The correction method during the measurement gives the degree of polarization of the reflected beam. The Pg r Ž l. spectrum is shown in Fig. 3. This result shows Ps 1 over the spectral range where silicon is totally absorbing at 3.5 m m thickness. In this spectra range the sample behaves as a reflecting substrate and no thickness gradient is present. In the transparency region Pg r shows a modulation with wavelength as do Ž c , D . spectra. This results from the constructive or destructive superposition of the partial waves. If the superpo-
U. Richter r Thin Solid Films 313]314 (1998) 102]107
Fig. 4. Fit of microspot measurement Ž200 m m spot, 58 focussing angle. performed with correction for depolarization Ž3.5 m m c-Si on a 2-m m oxide layer on c-Si substrate measured at 708 angle of incidence.. The fit Ždotted. assumes an ideal sample structure without thickness gradient.
sition is destructive, only the first reflected partial beam gives a maximum in Pg r . At constructive superposition the non-planar interfaces lead to a reduction in the degree of polarization due to the incoherent partial beams. To compare these results with the measurements without a microspot lens, a line-scan has been performed along the long diameter g l of the large spot. All resulting spectra are fitted to obtain the thicknesses of the top and bottom layer. The resulting thickness profiles are given in Fig. 5. The profiles show the interesting result that the bottom oxide thickness is constant around 2 m m but the top c-Si shows a large thickness gradient. El-Ghazzawi et al. w2,3x reported a significant change in the spectra upon a slit. If this is compared with the shape of the spectra above, then it becomes clear that this difference must come from the degree of polarization correction. The slit reduces the focusing angle and the spot dimension, the latter effects creating depolarization and changing the spectra. Since the application of a slit largely reduces the intensity, it is found that the correction for the degree of polarization is a better approach because it gives a higher signal to noise ratio and at least the same accuracy in the Ž c , D . spectra. 5. Summary A thickness gradient sample of bonded silicon on silicon dioxide on silicon has been measured using a large spot diameter. The resulting attenuated modula-
107
Fig. 5. Thickness profiles measured with the microspot. The large spot used for the date in Fig. 1 has its long diameter from xs 1 mm to xs 8 mm.
tion in Ž c , D . has been successfully fitted to obtain both thicknesses and a thickness gradient. A new method for measuring the degree of polarization using a PCSA arrangement is presented. The advantage of this method is the application to spectroscopic ellipsometry where the retarder and polarizer are not available in the exact positions of T s Ž lr4. and Ps 458. The measurement of the degree of polarization gives a reliable sensor for non-ideal effects. The microspot measurement was fitted using an ideal two model with good agreement between theory and measurement. The thickness gradient fitted from the large spot data was reproduced with reflectivity and microspot spectroscopic ellipsometry measurements. A significant improvement of the microspot readings with the retarder is achieved due to the degree of polarization correction. References w1x G.E. Jellison, Jr., J.W. McCamy, Appl. Phys. Lett. 61 Ž1992. 512]514. w2x M.E. El-Ghazzawi, T. Saitoh, N. Hori, A. Sakai, T. Oka, Thin Solid Films 233 Ž1993. 218]222. w3x M.E.M. El-Ghazzawi, T. Saitoh, Opt. Eng. 34 Ž1995. 453]459. w4x Azzam, Bashara, Ellipsometry of Polarized Light, North-Holland, 1989. w5x A. Roseler, Infrared Spectroscopic Ellipsometry, Akademie¨ Verlag, Berlin, 1990. w6x J.T. Zettler, T. Trepk, L. Spanos, Y.-Z. Hu, W. Richter, Thin Solid Films 234 Ž1993. 402]407. w7x D.E. Aspnes, A.A. Studna, Phys. Rev. B 27 Ž1983. 985.