SURFACE AND INTERFACE ANALYSIS Surf. Interface Anal. 28, 240–244 (1999) Complete Optical Characterization of the SiO2/Si System by Spectroscopic Ellipsometry, Spectroscopic Reflectometry and Atomic Force Microscopy I. Ohlı́dal,1 * D. Franta,2 E. Pinčı́k3 and M. Ohlı́dal4 1 2 3 4 Department of Solid State Physics, Faculty of Sciences, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic Department of Physical Electronics, Faculty of Sciences, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, 842 28 Bratislava, Slovak Republic Institute of Physical Engineering, Faculty of Engineering, Technical University, Technická 2, 616 69 Brno, Czech Republic In this paper results concerning optical analysis of the SiO2 =Si system performed by the combined ellipsometric and reflectometric method used in multiple-sample modification will be presented. This method is based on combining both the single-wavelength method and the dispersion method. Three models of the system mentioned, i.e. the model of the substrate and the layer with the smooth boundaries, the same model with a transition layer and the model of the substrate and the layer with rough boundaries, will be used to interpret the experimental data. The spectral dependences of the optical constants of silicon and SiO2 with the values of the other parameters will be determined. It will be shown that the simplest model with the smooth boundary is the most convenient with the experimental data. Copyright 1999 John Wiley & Sons, Ltd. KEYWORDS: optical constants of Si and SiO2 ; spectroscopic reflectometry; spectroscopic ellipsometry INTRODUCTION The optical properties of the SiO2 /Si system have been studied extensively because of its importance in practice. However, the results of the optical studies of this system presented in the literature are somewhat different. Considerable effort was therefore devoted to finding the correct and precise values of the optical parameters of the SiO2 /Si system in latter years. The basic problem of optical characterization of the SiO2 /Si system is to find correct and precise values for the silicon substrate. In practice, two basic groups of ellipsometric methods can be utilized for determining the optical parameters of the Si substrate. The methods of the first group are based on separate interpretation of the optical quantities of the system measured at individual wavelengths (single-wavelength methods). The methods of the latter group are based on interpreting the entire spectral dependences of the optical quantities of the system measured within a certain spectral region (dispersion methods). The advantages and disadvantages of the methods belonging to the two groups were discussed in our earlier paper.1 A good example of employing the single-wavelength method * Correspondence to: I. Ohlı́dal, Department of Solid State Physics, Faculty of Sciences, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic. E-mail: ohlidal@physics.muni.cz Contract/grant sponsor: Grant Agency of Czech Republic; Contract/grant number: 202/98/0988. Contract/grant sponsor: Grant Agency of Ministry of Education of Czech Republic; Contract/grant number: VS96084. CCC 0142–2421/99/130240–05 $17.50 Copyright 1999 John Wiley & Sons, Ltd. for determining the optical constants of a silicon singlecrystal substrate is presented by Aspnes and Theeten.2 The method used by Herzinger et al.3 is the typical dispersion method for characterizing this substrate but applied simultaneously to several samples of the system (multiplesample method). In this paper the results concerning optical characterization of samples of the SiO2 /Si system using a new combined optical method will be presented. PREPARATION OF SAMPLES AND EXPERIMENTAL ARRANGEMENTS Samples of the SiO2 /Si system studied were prepared as follows. The surfaces of Si wafers (n-type, (111)orientation) were treated in a standard way. The Si surfaces obtained in this way were treated by a special procedure based on a combination of thermal oxidation and etching to remove possible surface layers containing defects. The SiO2 layers were prepared by thermal oxidation of these Si surfaces at a temperature of 1040 ° C in an atmosphere of oxygen and HCl (25 : 1). Different values of the thicknesses of the SiO2 layers were achieved by different periods of oxidation of the Si surfaces. The samples of the Si surfaces covered with the SiO2 layers were finally heated in a nitrogen atmosphere at the same temperature (details of this procedure for creating the samples will be described in a forthcoming paper). The spectral dependences of the ellipsometric parameters (azimuth) and (phase change) of the samples were measured using a Jobin–Yvon ellipsometer Received 30 November 1998 Accepted 8 March 1999 OPTICAL CHARACTERIZATION OF SiO2 /Si (UVISEL) in the spectral range 240–830 nm at angles of incidence of 55–75° . The spectral dependences of the reflectance of these samples were measured by a spectrophotometer (Varian Cary 5E) in the same spectral region at an angle of incidence of 10° . Surface roughness of the SiO2 layers was measured by means of an atomic force microscope (Accurex, TopoMetrix Inc.). 241 E qj denotes the vectors whose components are where A formed by the complex amplitudes ACqj and Aqj of the electric fields belonging to the right-going and left-going waves inside the jth medium, respectively5 (the incident wave on the system is the right-going wave). It holds that j D 0, 1, 2, where the indices 0, 1 and 2 denote the ambient, layer and substrate, respectively. Then E qj D A DATA PROCESSING The ellipsometric parameters and and the reflectance R of isotropic thin-film systems are defined as tan ei D rOp , rOs RD jOrp j2 C jOrs j2 2 O ii D tan i eii , O ij D Ri Rj Ci Rj Ri E denotes the vector whose components are idenwhere X tical to the parameters sought. The index k represents the serial number of experimental value of the complex quanexp tity O k , wk are the weights of the experimental values and k and/or 0k is the wavelength and/or the angle of incidence of light falling onto the upper boundary of the system SiO2 /Si belonging to the corresponding experimental value (the indices i and j are functions of the index E k , 0k / is calculated using Eqns (1) k). The function O ij .X, and (2). The values of the parameters sought correspondE are considered ing to the minimum of the function S.X/ to be the true values of these parameters characterizing the samples studied. In the procedure of searching the minimum of S, the Marquardt–Levenberg algorithm was used.4 This LSM is employed for treating the experimental data in a combined way (see Results section). Mq21 Mq11 .6/ Substrate covered by a single film with smooth boundaries Q q of this system is expressed as The transfer matrix M Q q D BQ q1 TQ 1 BQ q2 M .7/ 1 1 BQ qj D Otqj rOqj rOqj 1 .8/ The symbol rOqj and/or Otqj represents the reflection and/or transmission Fresnel coefficient of the jth boundary for the wave incident on this boundary from the left side. The amplitudes of the waves inside the film corresponding to the first and second boundaries are connected by means of the elements of the phase matrix TQ 1 . i.e. TQ 1 D eik0 nO 1 d 0 0 e .9/ ik0 nO 1 d where the symbols d and/or n1 represent the thickness and/or refractive index of the film and k0 D 2/. Substrate covered by a transition layer and a single film For this system one can express the transfer matrix in the form Q q1 1 XQ q W Q q2 Q q D BQ q1 TQ 1 W .10/ M Q q1 1 XQ q W Q q2 represents the transition layer where the matrix W between the substrate and the single film. The matrices Q qk are expressed as W MODELS OF THE SiO2 /Si SYSTEM Let us consider an isotropic thin-film system placed on a semi-infinite substrate. The amplitudes of the waves propagating within both the ambient and the substrate then fulfil the following matrix equation5 Copyright 1999 John Wiley & Sons, Ltd. .5/ The matrix BQ qj (refraction matrix) is defined as k q D p, s rOq D .2/ where the indices i and j express the numbers of the samples .i, j D 1, 2 . . . 6/. For finding the values of the parameters characterizing the samples, the least-squares method (LSM) can be used. One can then construct the following merit function X E k , 0k / O kexp j2 wk E D jO ij .X, .3/ S.X/ E q0 D M E q2 , Q qA A Q q represents the transfer matrix of the sysThe symbol M tem for q-polarization. The reflection Fresnel coefficient of the whole system is then given as5 .1/ where rOp and / or rOs denotes the reflection Fresnel coefficient of the systems for p- and/or s-polarization. In our studies we treated the spectral dependences of the ellipsometric parameters and relative reflectances measured for the six samples and therefore we introduced the complex quantities O ij defined by ACqj Aqj .4/ Q sk D W 1 ik0 nO k cos O k and Q pk D W nO k ik0 cos O k 1 ik0 nO k cos O k nO k ik0 cos O k .11/ .12/ Surf. Interface Anal. 28, 240–244 (1999) 242 I. OHLÍDAL ET AL. If the thickness of the transition layer dT is much smaller than the wavelength, then matrices XQ q exhibit the forms 1 dT .13/ XQ s D dT k02 .n20 sin2 0 P/ 1 1 dT P .14/ XQ p D 2 2 2 dT k0 .Qn0 sin 0 1/ 1 where 1 PD dT Z dT 2 T nO .z/dz, 0 1 QD dT Z 0 dT 1 dz nO 2T .z/ .15/ by two quantities characterizing the roughness, i.e. the rms value of the heights of the irregularities and the autocorrelation length of these irregularities T. MODELS OF DISPERSION OF MEDIA OF THE SiO2 /Si SYSTEM The ambient is always formed by air and so its refractive index n0 is equal to unity. It is assumed that dispersion of the refractive indices of all of the thermal SiO2 layers studied is the same. Moreover, it is assumed that the formula expressing this dispersion is given as We assume that the refractive index of the transition layer nO T .z/ exhibits the profile expressed by the equation nO 2T .z/ D n21 C z 2 .nO dT 2 n21 / .16/ After integrating in Eqn (15) one obtains the following expressions for P and Q PD nO 22 C n21 , 2 QD ln nO 22 nO 22 ln n21 n21 .17/ Note that the matrix formalism presented is equivalent to the Drude approximate approach for inhomogeneous layers.5,6 Substrate covered by a single film with rough boundaries If the boundaries of the system are not ideally smooth then Fresnel coefficients of these boundaries differ from those describing the smooth boundaries. In the case when the root-mean-square (rms) values of the heights of the irregularities of the boundaries j are much smaller than the wavelength of the incident wave, the Rayleigh–Rice theory (RRT) can be employed for expressing the Fresnel coefficients.7 Moreover, if the autocorrelation length of roughness of the boundaries is larger than the thickness of the film, then the matrix formalism presented for the substrate covered by a single film with smooth boundaries cannot be used. One must then employ the formalism derived in our earlier paper.8 Within this formalism the Q q exhibits the general form transfer matrix M Qq D 1 M OtqR 1 rOqR OtqR OtqL rOqL rOqR rOqL .18/ In the foregoing equations the indices ‘R’ and ‘L’ mark the Fresnel coefficients of the whole system for the wave incident on this system from the left-side and right-side, respectively. From Eqn (6) it is evident that the reflection Fresnel coefficients of the system rOq that are important for our purposes are identical with the coefficients rOqR . Note that the expressions for the Fresnel coefficients considered above must be calculated using a numerical procedure.8 In our studies it is assumed that the film with the rough boundaries is represented by the model of the identical thin films whose boundaries are described by the Gaussian statistics.8 This model of the rough film is parameterized Surf. Interface Anal. 28, 240–244 (1999) n1 D A C B C C 4 2 .19/ where A, B and C are material parameters. As for the silicon substrate, no dispersion formulae are assumed for the refractive index n or the extinction coefficient k .nO 2 D n ik/. RESULTS First the model formed by the substrate covered by the single film with smooth boundaries was used to interpret the experimental data. In the first step we used the single-wavelength method for determining the values of all the optical parameters, i.e. the values of the refractive index n and the extinction coefficient k of the Si substrates and A, d1 , d2 , . . . , d6 of the SiO2 layers (B and C were fixed). In the second step we employed the dispersion method for determining the values of the parameters A, B, C, d1 , d2 , . . . , d6 . In the second step the values of both the refractive index and the extinction coefficient of the Si substrate were fixed for all the wavelengths in the values determined in the first step. These two steps were repeated in successive iterations, i.e. every iteration was formed by the two steps described above. In the first step of the second iteration the parameters d6 , B and C were fixed in the values obtained within the second step of the first iteration. In the first step of the third iteration the parameters d5 , d6 , B and C were fixed in the values found in the second step of the second iteration, etc. This means that from the eighth iteration the optical constants of the Si substrates n and k were only determined in the first step. After performing a sufficient number of iterations, this numerical procedure converges to the values sought for the thicknesses d1 , d2 , . . . , d6 and the spectral dependences of the optical constants of the system. In this way we obtained the following results: A D 1.45698 š 0.00008, B D .2689 š 21/ nm2 , C D .5.99 š 0.12/ ð 107 nm4 , d1 D .27.635 š 0.007/ nm, d2 D .44.654 š 0.005/ nm, d3 D .67.344 š 0.005/ nm, d4 D .86.057 š 0.005/ nm, d5 D .101.138 š 0.005/ nm and d6 D .13.853 š 0.006/ nm. The values of the spectral dependence of the refractive index of SiO2 calculated by means of the values of A, B and C presented above are greater than the values of bulk amorphous SiO2 9 (the difference between both sets of data is ¾0.007 over the entire spectral region of interest). Copyright 1999 John Wiley & Sons, Ltd. OPTICAL CHARACTERIZATION OF SiO2 /Si The spectral dependences of the optical constants of the silicon single-crystal substrate determined by the procedure described are not presented completely in this paper. (They will be published elsewhere.) Here we shall perform a comparison of our values of the silicon optical constants with those found by other researchers (Aspnes and Studna10 and Herzinger et al.3 ) at selected wavelengths (see Tables 1 and 2). Furthermore, in Fig. 1 a comparison of the values of the extinction coefficient determined by our procedure with those published by the other researchers is performed in the spectral region 450–830 nm. From Table 1 it can be seen that there are certain differences between the positions and magnitudes of the peaks of the imaginary part of the dielectric function ε2 .ε2 D 2nk/ corresponding to the critical points E1 and E2 . This same statement is also true for the values of the optical constants corresponding to the popular wavelengths in the visible region (see Table 2). Note that in the region of weak absorption of silicon our values of the extinction coefficient are greater than those presented by the researchers mentioned (see Table 2 and Fig. 1). However, our values of k agree with the values of this quantity published by Hulthèn11 determined using transmission measurements (see Fig. 1). If the model containing the transition layer is considered, the number of parameters sought is greater (the Table 1. The position and magnitudes of the "2 peaks of Si corresponding to critical points E1 and E2 Data Aspnes & Studna Herzinger et al.3 Our data 10 E1 (eV) ε2 .E1 / E2 (eV) ε2 .E2 / 3.429 3.418 3.415 36.44 36.62 37.70 4.249 4.239 4.233 46.81 47.64 46.91 Table 2. Values of the refractive index n and extinction coefficient k of Si for two selected walelengths D 546.1 nm n k Aspnes & Studna10 Herzinger et al.3 Our data 4.098 4.090 4.105 0.044 0.026 0.055 D 632.8 nm n k 3.882 3.874 3.879 0.019 0.015 0.030 243 thickness of the transition layer dT is added). When treating the experimental data by our numerical procedure we fixed the value of dT to 1 nm in the first nine iterations (the fixed value of dT was chosen on the basis of results achieved for the transition layer by other researchers2,3). From the tenth iteration the value of dT also was searched. By using this procedure we found that the thickness dT converges to zero and the other optical parameters converge to the values determined by means of the foregoing model. This fact thus implies that between the silicon substrates and thermal oxide layers there are relatively sharp boundaries. When treating the experimental data using the rough model, we applied our numerical procedure in a similar way to that employed in the case of the model with the transition layer. We found that the roughness parameters and T converge to the values corresponding to the smooth boundaries. One can thus state that roughness taking place in the boundaries cannot be observed within the optical methods. This conclusion is supported by our atomic force microscopy measurements performed for the samples of SiO2 /Si studied. From our atomic force microscopy measurements the following typical values of and T were determined: D 0.2 nm and T D 80 nm. From a numerical analysis it is clear that roughness characterized with such the values of and T cannot affect the ellipsometric and reflectometric data in a substantial way. CONCLUSION In this paper the results concerning optical analysis of the SiO2 /Si system by the combined ellipsometric and reflectometric method used in multiple-sample modification are presented. This method is based on combining both the single-wavelength method and the dispersion method. From the results one can imply that the best physical model for the experimental data is identical with the model formed by a silicon single-crystal substrate covered by as SiO2 layer with smooth and sharp boundaries. This conclusion is in contrast to the results of other researchers, who found that a transition layer existed between the Si substrate and the SiO2 layer on the basis of interpretation of ellipsometric data.2,3 This fact may be caused by the different technological conditions when preparing the samples of SiO2 /Si studied by the different researchers. Within our analysis the spectral dependences of the optical constants of Si and SiO2 are found, together with the values of the thicknesses of the SiO2 layers. Furthermore, it is shown that there are certain differences between our values of the optical constants mentioned and those presented by other researchers. The complete spectral dependences of the optical constants of silicon will be presented elsewhere. Acknowledgements Figure 1. Spectral dependence of the extinction coefficient k of Si determined by different researchers in the region of weak absorption of this material. Copyright 1999 John Wiley & Sons, Ltd. We wish to thank Dr C. M. Herzinger for providing us with his data of the optical constants of silicon and Dr K. Navrátil for measuring the reflectance data. Moreover, we wish to thank the researchers of SCB of the Masaryk University for their help with the numerical computations. This work was supported by the Grant Agency of the Czech Republic and the Grant Agency of the Ministry of Education of the Czech Republic under contract nos 202/98/0988 and VS96084, respectively. Surf. Interface Anal. 28, 240 244 (1999) 244 I. OHLÍDAL ET AL. REFERENCES 1. D. Franta, I. Ohlı́dal and D. Munzar, Acta Phys. Slov. 48, 451 (1998). 2. D. E. Aspnes and J. B. Theeten, J. Electrochem. Soc. 127, 1359 (1980). 3. C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam and W. Paulson, J. Appl. Phys. 83, 3323 (1998). 4. W. D. Marquardt, J. Soc. Ind. Appl. Math. 11, 431 (1963). 5. Z. Knittl, Optics of Thin Films. Wiley, Chichester (1976). Surf. Interface Anal. 28, 240–244 (1999) P. Drude, Wied. Ann. 43, 136 (1891). S. O. Rice, Commun. Pure Appl. Math. 4, 351 (1951). D. Franta and I. Ohlı́dal, J. Mod. Optics 45, 903 (1998). D. F. Edwards, in Handbook of Optical Constants of Solids, ed. by E. D. Palik, pp. 547 569. Academic Press, New York (1985). 10. D. E. Aspnes and A. A. Studna, Phys. Rev. B 27, 985 (1983). 11. R. Hulthèn, Phys. Scr. 12, 342 (1975). 6. 7. 8. 9. Copyright 1999 John Wiley & Sons, Ltd.

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