Determination of optical constants of solgel-derived ... - OSA Publishing

Determination of optical constants of solgel-derived inhomogeneous TiO2 thin films by spectroscopic ellipsometry and transmission spectroscopy Md. Mosaddeq-ur-Rahman, Guolin Yu, Kalaga Murali Krishna, Tetsuo Soga, Junji Watanabe, Takashi Jimbo, and Masayoshi Umeno

Amorphous and nanocrystalline TiO2 thin films coated on a vitreous silica substrate by a solgel dip coating method are investigated for optical properties by spectroscopic ellipsometry ~SE! together with transmission spectroscopy. A method of analysis of SE data to determine the degree of inhomogeneity of TiO2 films has also been presented. Instead of the refractive index, the volume fraction of void has been assumed to vary along the thickness of the films and an excellent agreement between the experimental and calculated data of SE below the fundamental band gap has been obtained. The transmission spectrum of these samples is inverted to obtain the extinction coefficient k spectrum in the wavelength range of 300 –1600 nm by using the refractive indices and parameters of structure determined by SE. The nonzero extinction coefficient below the fundamental band-gap energy ~3.2 eV! has been obtained for the nanocrystalline TiO2 and shows the presence of optical scattering in the film. © 1998 Optical Society of America OCIS codes: 310.0310, 310.6860, 260.2130.

1. Introduction

Titanium dioxide is a large band-gap semiconductor of exceptional stability with diverse industrial applications. In recent years, solgel-derived nanocrystalline TiO2 thin films of anatase phase are becoming increasingly important because they are easy to prepare and have potential application in the areas of photovoltaics, sensing devices, photocatalysis, and microelectronics.1 Consequently, determination of the optical constants and evaluation of microstructural features such as surface roughness in these films are important. Wide variations in the optical and physical properAll the authors are with the Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466, Japan. M. Mosaddeq-urRahman and M. Umeno are with the Department of Electrical and Computer Engineering. G. Yu, M. Krishna, and J. Watanabe are with the Research Center for Micro-Structure Devices. T. Soga and T. Jimbo are with the Department of Environmental Technology and Urban Planning. Received 15 July 1997; revised manuscript received 26 September 1997. 0003-6935y98y040691-07$10.00y0 © 1998 Optical Society of America

ties of TiO2 thin films deposited by different techniques have been reported.2 The aim of this research is to study the optical constants and depth profile of solgelderived TiO2 thin films to better understand their optical properties. Spectroscopic ellipsometry ~SE!, known to be a very useful and nondestructive technique for investigating the optical properties of inhomogeneous films, was used together with transmission spectroscopy to study our films. The ellipsometric determination of the optical constants of inhomogeneous TiO2 films deposited by different evaporation techniques has been reported by various authors who assumed a linear variation in the refractive index along the thickness of the film.3,4 In this research, a fourphase model ~airyrough surface layeryinhomogeneous TiO2 layerysubstrate! has been used to fit the SE data, taken in the wavelength range of 260–830 nm. In the inhomogeneous layer, a void distribution has been assumed to vary instead of the variation in the refractive index n, and the unknown dielectric function of TiO2 is described by a single oscillator form. Excellent agreement between the calculated and the measured data of SE in the 335–830-nm wavelength range has been obtained. However, determination of the extinction coefficient k by fitting SE data with the di1 February 1998 y Vol. 37, No. 4 y APPLIED OPTICS

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electric function being described by a single oscillator is not very accurate near and above the fundamental band gap. To determine k accurately, Forouhi and Bloomer5 derived a formalism of k~l! above the fundamental band gap based on the quantum-mechanical theory of absorption.5 Kim used this formalism with SE and transmission spectroscopy for the simultaneous determination of n, k and void distribution of electron-beam ~EB!-evaporated amorphous TiO2 thin film.6 In this research we calculated the extinction coefficient for both the amorphous and the nanocrystalline TiO2 thin films from the transmission spectrum by using the refractive index, thickness, and void of each layer determined by SE. Our as-deposited and low-temperature-annealed ~400 °C! films were amorphous as observed by x-ray-diffraction analysis. At a higher annealing temperature ~600 °C!, films were found to be polycrystalline with a crystal structure matching the anatase modification of the TiO2 lattice. The scanning electron micrograph showed the 600 °C temperature-annealed sample to be nanocrystalline with a porous structure. Details of the experimental and theoretical procedures are described below, and the results are compared with those reported by others for TiO2 films deposited by different techniques. 2. Experimental

We deposited TiO2 thin films on the vitreous silica substrate by the solgel method, using titanium tetraisopropoxide, Ti~C3H7O!4 as the starting material for Ti. Details of the sol solution preparation are described elsewhere.7 Films were coated on the vitreous silica substrate by the dip-coating method with a pulling speed of 0.1 mmys. The coated films were dried at 80 °C for 15 min and heat treated at 400 °C for 1 h; the process was repeated 5 times. Two films thus obtained were annealed for 6 h at 400 and 600 °C and hereafter are referred to as Sample 1 and Sample 2, respectively. The measurements of spectroscopic ellipsometry were carried out at an angle of incidence of 75° in the wavelength range of 260–830 nm. The automatic ellipsometry used was of a rotating analyzer type, fitted with a 75-W xenon lamp as a light source. The back surface of the substrates was made rough and blackened before ellipsometric measurements were performed to eliminate back-surface light reflection. The transmission spectrum of the samples was taken before SE measurements were performed across a wavelength range of 300–1600 nm for the accurate determination of extinction coefficient k. The transmission spectrum of the bare vitreous silica substrate was also taken to determine its refractive index. All the measurements were performed at room temperature. 3. Theoretical Treatment

The ellipsometric parameters C and D are defined as usual from the ratio of reflected amplitudes for s and p polarization: r5 692

rp urpu 5 exp@i~dp 2 ds!# 5 tan C exp~iD!. rs ursu

APPLIED OPTICS y Vol. 37, No. 4 y 1 February 1998

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Fig. 1. Typical refractive-index profile of an inhomogeneous film of thickness d on a nonabsorbing substrate with refractive index n3 in a medium with refractive index n0. The refractive index of the film varies from n1 at the outer surface of the film to n2 at the inner surface.

In the case of an inhomogeneous film of thickness d on a nonabsorbing substrate with index of refraction n3, in a medium with refractive index n0, the refractive index of the film is not uniform but varies as a function of distance along the thickness of the film. The refractive-index profile for such a system is illustrated in Fig. 1. With the simplifying assumption that the reflection of light from the interior of the film can be ignored, the detailed expressions of C and D have been given by Carniglia.8 It has also been shown by Carniglia that ellipsometric data at halfwave ~HW! points, defined as wavelengths that are multiples of twice the optical thickness, in the tan C spectrum @hereafter referred to as tan C~HW!#, offers a sensitive measure of the degree of inhomogeneity. In the condition of extinction coefficient k 5 0, cos D 5 1, and tan C~HW! results in8 tan C~HW! 5

Y

~np0np2 2 np1np3! ~ns0ns2 2 ns1ns3! , ~np0np2 1 np1np3! ~ns0ns2 1 ns1ns3!

(2)

where nsi 5 ~ni 2 2 n02 sin2 u0!1y2, npi 5 ni 2y~ni 2 2 n02 sin2 u0!1y2. Here u0 is the angle of incidence, n1 and n2 are the refractive indices at the outer and the inner surfaces of the inhomogeneous layer, respectively. In particular, when the film is homogeneous, i.e., n1 5 n2, tan C~HW! is given by tan C~HW! 5

Y

~np0 2 np3! ~ns0 2 ns3! 5 tan Cs, ~np0 1 np3! ~ns0 1 ns3!

(3)

showing that for homogeneous film tan C~HW! is equal to tan Cs, the amplitude reflectance ratio of the uncoated substrate, and is independent of the index of the film. Thus, from the difference between tan Cs and tan C~HW!, one can get an idea about the degree of inhomogeneity of a film, which is defined by Dnyn# ; here n# is the average refractive index of the film and Dn 5 n1 2 n2. To demonstrate the effect of the degree of inhomogeneity on the tan C spectrum, a series of tan C spectra is shown in Fig. 2 for single-layer films ~solid curves!, together with the tan Cs spectrum ~dashed line!, with Dnyn# varying from 215% to 115% in steps of 7.5%, for an incidence angle of 75°. Here the av-

Fig. 2. Calculated tan C spectrum ~solid curves! for films together with the tan Cs spectrum ~dashed line! for the substrate with Dnyn# varying from 215% to 115% in steps of 7.5% for an angle of incidence of 75°. The average refractive index n# of the film is assumed to be 2.3 and that of the substrate to be 1.5.

Fig. 3. Calculated tan C spectrum ~solid curves! for films together with the tan Cs spectrum ~dashed line! for the substrate, with the average refractive index varying from 1.7 to 2.5. The degree of inhomogeneity Dnyn# for all the films is considered to be constant ~215%!.

erage refractive index of the film is assumed to be 2.3 and that of the substrate to be 1.5. It is clearly seen from Fig. 2 that the spectrum of tan C at the HW points shifts from values above that of the tan Cs spectrum to values below that of tan Cs as Dnyn# varies from 215% to 115%. Two conclusions can be drawn from Fig. 2: ~1! The larger the degree of inhomogeneity Dnyn# , the larger is the difference between tan Cs and tan C~HW!. ~2! The position of tan C~HW! points relative to the tan Cs spectrum depends on the sign of Dnyn# , i.e., for n1 . n2, tan C~HW! , tan Cs and vice versa. Note that there is no significant effect of the value of the average refractive index on the value of tan C~HW!, which has a strong dependence on the degree of inhomogeneity. This point is clearly depicted in Fig. 3 where the tan C spectra for single-layer films ~solid curves!, together with the tan Cs spectrum ~dashed line!, are drawn with an average refractive index varying from 1.7 to 2.5 and keeping the degree of inhomogeneity Dnyn# constant ~215%!. With an increase in the average refractive index, a small increase in the value of tan C~HW! is observed. However, compared with the large change in tan C~HW! ~;40%! for a relatively small change in Dnyn# ~15%!, this change in tan C~HW! ~;5%! for a relatively large change ~47%! in n# is quite small. Furthermore all tan C~HW! points remain on the same side of the tan Cs spectrum despite a large change in n# , and thus the sign of Dnyn# remains unaffected by the change in n# . Note that the above analysis is valid only for situations in which the substrate has a lower refractive index than that of the film, i.e., n3 , n# , and the incidence angle is 75°. This particular case is discussed because it matches our problem where we have used a vitreous silica substrate that has an average refractive index ~1.5! lower than that for TiO2 ~2.2–2.6!.2 For TiO2 films on substrates with higher refractive indices, such as Si, the relative positions of tan C and tan Cs spectra will change and the above discussion must be modified. A change in

the incidence angle will also affect the relative positions of tan C and tan Cs spectra. Using the same assumption invoked above in the calculations of tan C in SE, i.e., the reflection of light from the interior of the film can be ignored, we can write the amplitude transmission at normal incidence as t5

t10t23 exp~2id! , 1 1 r10r23 exp~2i2d!

(4)

where t10 and t23 are the Fresnel transmission coefficients at the medium–film and the film–substrate interfaces, respectively, and r10 and r23 are the corresponding Fresnel reflection coefficients and are given by t10 5

2n0 , n0 1 n1

t23 5

2n2 , n2 1 n3

r10 5

n0 2 n1 , n0 1 n1

r23 5

n2 2 n3 . n2 1 n3

Phase angle d is given by d5

2p l

*

d

n~z!dz.

0

The transmission T can be written as T5

n1n3 tt*. n0n2

(5)

The Bruggeman effective-medium theory ~EMT!, considered an important tool for investigating inhomogeneous film, is used to calculate the effective dielectric function of the film. The mathematical formula can be expressed as9 ~1 2 fn!

εm 2 ε 12ε 1 fn 5 0, εm 1 2ε 1 1 2ε

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Fig. 4. Schematic diagram of the film structure used in SE fitting.

where εm is the dielectric function of the main constituent material and fv is the volume fraction of the void. In this equation the dielectric function of the void is taken to be 1. In our SE data analysis, the unknown dielectric function of TiO2 is described by a single oscillator: ε 5 ε`9 1

~εs 2 ε`9!vt2 , vt2 2 v2 1 iG0v

(7)

where ε`9 represents the high-frequency dielectric constant, εs is the oscillator strength, vt is the frequency, and G0 is the damping factor of the oscillator. The unknown parameters can be numerically determined by minimizing the following mean-squares deviation with a regression program ~unbiased!: d2 5

1 2N 2 P

N

( @~tan C

i

exp

2 tan Ci cal!2

i

1 ~cos Di exp 2 cos Di cal!2#,

(8)

where N is the number of data points and P is the number of unknown model parameters. 4. Results and Discussion A.

Spectroscopic Ellipsometry

A four-phase structure ~airyrough surface layeryinhomogeneous TiO2ysubstrate! as shown in Fig. 4 has been used in the simultaneous fitting of measured parameters D and C of SE. The roughness layer on the surface was modeled as an effective mixture of 50% TiO2 and 50% void. Inhomogeneity of a film results from the nonuniform packing density of the film, which is usually expressed by the volume fraction of void fv. In our fitting analysis fv is varied from fvo 5 0 at the outer surface of the film to fvi at the inner surface. This assumption of varying the volume fraction of void along the thickness of the film simplifies the calculation rather than considering the variation in refractive index, because the refractive index is a function of both distance and wavelength, whereas fv is a function of distance only. However, note that our assumption of fv varying from zero at the outer surface to fvi at the inner surface is only for the sake of obtaining the distribution of the dielectric function along the depth of the film by fitting to el694


Fig. 5. Measured ~solid circles and plus sign! and fitted ~solid curves! cos D ~top! and tan C ~bottom! spectra of Samples 1 and 2 together with the calculated data of the bare substrate ~dashed lines!.

lipsometric data and shows only relative variation in the void fraction along the thickness of the film. The change in refractive index along the depth of the film will follow the change in fv. After obtaining the refractive index by fitting to SE data, we calculate the actual void distribution along the depth of the film, using the void-free refractive index of anatase TiO2 as Kim did.6 Measured ~solid circles and plus sign! and fitted ~solid curves! cos D ~top! and tan C ~bottom! spectra of Samples 1 and 2, together with the calculated data of the bare substrate ~dashed lines!, are shown in Fig. 5. An excellent agreement between the experimental and fitted cos D and tan C spectra for both the samples has been obtained. From this figure, HW fringes at 350 and 550 nm for Sample 1 and at 345 and 500 nm for Sample 2 are observed. At wavelengths of 550 and 500 nm, i.e., at the HW fringes for Samples 1 and 2, respectively, cos D ' 1 and tan C~HW! Þ tan Cs. cos D ' 1 illustrates that the extinction coefficient k of the films is close to zero in this region, whereas tan C~HW! Þ tan Cs shows the inhomogeneity of the films. For both films tan C~HW! , tan Cs, indicating n1 . n2 for both films according to our analysis discussed in Section 3, and the larger difference between tan C~HW! and tan Cs indicates a larger inhomogeneity for Sample 2, which has a nanocrystalline structure of anatase phase, than for Sample 1, which is amorphous, as observed by x-ray diffraction analysis. Table 1 shows the best-fit model parameters used in the simulation of cos D and tan C spectra. The high-frequency dielectric constants obtained are 2.83 and 3.7 for Samples 1 and 2, respectively. However, as pointed out by Gerfin and Gratzel for ITO films,10 here also ε`9 does not represent the true highfrequency dielectric constant ε`. The optimized dispersion formula can be regarded only as a

Table 1. Best-Fit Model Parameters of the Solgel-Derived TiO2 Thin Films on Vitreous Silica Substrate Determined by Spectroscopic Ellipsometrya

Sample

ε`9

εs

vt ~eV!

G0 ~eV!

fvi

d1 ~nm!

d2 ~nm!

d

1 2

2.83 6 0.03 3.70 6 0.06

4.07 6 0.05 4.52 6 0.1

4.39 6 0.03 4.06 6 0.07

0.20 6 0.01 0.25 6 0.02

0.097 6 0.002 0.210 6 0.003

4.74 6 0.03 2.19 6 0.03

148.74 6 0.001 133.41 6 0.001

0.005 0.008

The fvi is the volume fraction of void at the inner surface of layer 2. The 90% confidence limits are given with ~6!.

a

mathematical description of the optical properties below the band-edge region of the semiconductor. The values of ε` can be determined graphically, as shown in Fig. 6, and are found to be 4.16 and 4.39 for Samples 1 and 2, respectively, which are close to that determined by Kim for a void-free value ~4.8! of the EB-evaporated TiO2 thin film. The thickness of Sample 2 is smaller than that of Sample 1 ~from Table 1!, indicating that a higher packing density and a larger refractive index for Sample 2 than for Sample 1 may be due to the crystalline nature of Sample 2, and is consistent with the results reported by Vorotilov et al. for solgel-derived TiO2 thin films.11 In Fig. 7 we show the grading profiles of refractive indices obtained for Samples 1 and 2 at a 500-nm wavelength. An almost linear index gradient along the depth of the films has been obtained. Note that the larger index gradient and smaller refractive index at the inner surface of Sample 2 indicate that inhomogeneity of the nanocrystalline film is more severe, as was also found by Susan et al. for solgelderived Pb~Zr, Ti!O3 film.12 The higher refractive index at the outer surface of the films can be explained in terms of densification and crystallization of the as-deposited films that begin at the surface and gradually progress toward the depth of the film during the course of annealing.13 Actual void distribution along the thickness of the film has been calculated with the refractive index of anatase TiO2 ~Ref. 6! and is found to vary from 28% to 35% from the outer surface to the inner surface of the film for Sample 1 and from 25% to 38% for Sample 2 for the same. These values are much larger than that reported for an EB-evaporated TiO2 thin film ~16%!.6 The pres-

ence of a high percentage of void is a common feature for solgel-derived oxide thin films. Figure 8 shows the variation of the degree of inhomogeneity Dnyn# with photon energy for both samples, obtained by our fitting results. It shows a much larger degree of inhomogeneity Dnyn# for Sample 2 than for Sample 1 and is consistent with Fig. 2 where a larger difference between tan C~HW! and tan Cs has been observed for Sample 2 than for Sample 1, indicating a larger inhomogeneity for Sample 2. Furthermore, the degree of inhomogeneity of Sample 2 varying from 12% to 14% is much larger than those reported for TiO2 films deposited by EB evaporation ~3–5%! and rf sputtering ~5%! techniques,2 while that for Sample 1 varying from 5% to 6% is close to those reported values. This shows that high-temperature annealing severely degrades the homogeneity of the film. One reason for the higher degree of inhomogeneity for solgel-derived thin films may be that with this method films are coated layer by layer with each coating followed by drying of the films at a sufficiently high temperature ~.100 °C!, while in other techniques, such as EB evaporation and rf sputtering, films are deposited in a single run. However, the underlying mechanism that further deteriorates the homogeneity of the film on annealing at high temperatures is yet to be investigated.

Fig. 6. Graphic determination of the high-frequency dielectric constant of the two samples.

Fig. 7. Depth profile of the refractive indices at a 500-nm wavelength for the two samples.

B.

Transmission Spectroscopy

For the multilayer structure of our film, we use potential transmission c according to Berning et al., which is given by14 C5

T 5 12R

Ti

)12R , i

(9)

i


695

Fig. 8. Variation of the degree of inhomogeneity Dnyn# with photon energy for both samples.

where Ti , Ri are the transmission and the reflection of the ith layer, respectively. Figure 9 shows the measured ~solid circles and plus signs! and calculated ~solid curves! transmission spectra of both the samples together with the measured transmission spectrum of the bare substrate ~dashed line! in the wavelength range of 300 –1600 nm. The solid curves are obtained by using the extrapolated optical constants, thicknesses, and void distribution determined by SE. Good agreement between the experimental and calculated data in the long-wavelength region has been obtained whereas in the short-wavelength region the deviations in the calculation from the experimental data are due to the inaccurate extinction coefficient obtained by fitting SE data with a single oscillator, which does not accurately reflect the effect of interband transitions occurring at and above 3.2 eV. Note that the indirect band gap of anatase phase TiO2 is ;3.2 eV.

Fig. 10. Refractive index n and extinction coefficient k spectra of Samples 1 ~dotted curves! and 2 ~solid curves!.

We obtain the extinction coefficient k~l! for both samples across the wavelength range of 300 –1600 nm from the transmission spectra, using the extrapolated refractive indices and structure parameters of the films obtained by SE. In Fig. 10 we show the refractive index n and extinction coefficient k spectra for both the samples across the energy range of 0.77– 4.0 eV. Sample 2 shows a larger refractive index than Sample 1 and is attributed to the increase in packing density and crystallinity of the film at an elevated temperature, which are evident from the thickness measurement and X-ray-diffraction analysis. This result is in good agreement with that of Suhail et al., for magnetron sputtered TiO2 thin films deposited on vitreous silica substrate, where they reported a similar increase in the refractive index with an increasing annealing temperature.15 A sharper increase in k above the fundamental band gap for Sample 2 shows the crystalline nature of the film. Note that the nonzero extinction coefficient of Sample 2 at a much lower photon energy than the band-gap energy indicates the scattering effect in the nanocrystalline TiO2 thin film. 5. Conclusions

Fig. 9. Measured ~solid circles and plus sign! and calculated ~solid curves! transmission spectra of both samples in the wavelength range of 300 –1600 nm. The solid curves are obtained by using the extrapolated optical constants and thicknesses determined by SE. The dashed line shows the measured transmission spectrum of the bare substrate. 696


Solgel-derived TiO2 thin films, both amorphous and nanocrystalline, are investigated for optical properties by spectroscopic ellipsometry and transmission spectroscopy. A method for fitting simultaneously the measured parameters of SE, C and D, for inhomogeneous thin film has also been described. Instead of the refractive index, the volume fraction of the void has been assumed to vary along the thickness of the film in the fitting analysis, and an excellent fit has been obtained for both the amorphous and the nanocrystalline TiO2 films. A nearly linear refractive-index gradient has been obtained for both samples with Sample 2 having a larger index gradient than Sample 1, showing deterioration of homoge-

neity with an increase in annealing temperature. The degrees of inhomogeneity obtained are 5– 6% and 12–14% for Samples 1 and 2, respectively, and are higher than those reported for TiO2 films deposited by EB evaporation ~3–5%! and rf sputtering ~5%! techniques. Extinction coefficients k~l! of the samples are calculated from the transmission spectra. Nonzero k values for Sample 2 at photon energies much lower than the fundamental band-gap energy ~3.2 eV! show a scattering effect in the nanocrystalline TiO2 film. This research is partly supported by the Ministry of Education, Science and Culture, Government of Japan. References 1. M. Gratzel, “Nanocrystalline electronic junctions,” in Semiconductor Nanoclusters–Physical, Chemical and Catalytic Aspects, P. V. Kamat and D. Meisel, eds. ~Elsevier, The Netherlands, 1997!, pp. 353– 461. 2. J. M. Bennett, E. Pelletier, G. Albrand, J. P. Borgogno, B. Lazarides, C. K. Carniglia, R. A. Schmell, T. H. Allen, T. Tuttle-Hart, K. H. Guenther, and A. Saxer, “Comparison of the properties of titanium dioxide films prepared by various techniques,” Appl. Opt. 28, 3303–3316 ~1989!. 3. G. Parjadis de Lariviere, J. M. Frigerio, J. Rivory, and F. Abeles, “Estimate of the degree of inhomogeneity of the refractive index of dielectric films from spectroscopic ellipsometry,” Appl. Opt. 31, 6059 – 6061 ~1992!. 4. J. P. Borgogno, F. Flory, P. Roche, B. Schmitt, G. Albrand, E. Pelletier, and H. A. Macleod, “Refractive index and inhomogeneity of thin films,” Appl. Opt. 23, 3567–3570 ~1984!. 5. A. R. Forouhi and I. Bloomer, “Calculation of optical constants,

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n and k, in the interband region,” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. ~Academic, Toronto, 1991!, Chap. 7. S. Y. Kim, “Simultaneous determination of refractive index, extinction coefficient, and void distribution of titanium dioxide thin film by optical methods,” Appl. Opt. 35, 6703– 6707 ~1996!. M. M. Rahman, T. Miki, K. M. Krishna, T. Soga, K. Igarashi, S. Tanemura, and M. Umeno, “Structural and optical characterization of PbxTi12xO2 film prepared by solgel method,” Mater. Sci. Eng. B 41, 67–71 ~1996!. C. K. Carniglia, “Ellipsometric calculations for nonabsorbing thin films with linear refractive-index gradients,” J. Opt. Soc. Am. A 7, 848 – 856 ~1990!. D. E. Aspnes and J. B. Theeten, “Investigation of effectivemedium models of microscopic surface roughness by spectroscopic ellipsometry,” Phy. Rev. B 20, 3292–3302 ~1979!. T. Gerfin and M. Gratzel, “Optical properties of tin-doped indium oxide determined by spectroscopic ellipsometry,” J. Appl. Phys. 79, 1722–1729 ~1996!. K. A. Vorotilov, E. V. Orlova, and V. I. Petrovsky, “Solgel TiO2 films on silicon substrates,” Thin Solid Films 207, 180 –184 ~1992!. T. M. Susan, J. Chen, K. Vedam, and R. E. Newnham, “In situ annealing studies of solgel ferroelectric thin films by spectroscopic ellipsometry,” J. Am. Ceram. Soc. 78, 1907–1913 ~1995!. Y. Mishima, M. Takei, T. Uematsu, N. Matsumoto, T. Kakehi, U. Wakino, and M. Okabe, “Polycrystalline silicon formed by ultrahigh-vacuum sputtering system,” J. Appl. Phys. 78, 217– 223 ~1995!. H. A. Macleod, Thin-Film Optical Filters ~Adam Hilger, Bristol, UK, 1986!. M. H. Suhail, G. M. Rao, and S. Mohan, “dc reactive magnetron sputtering of titanium-structural and optical characterization of TiO2 films,” J. Appl. Phys. 71, 1421–1427 ~1992!.


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