Preparation and characterization of a reference aluminum mirror

Transcription

1 Preparation and characterization of a reference aluminum mirror Tzwetanka Babeva, Snejana Kitova, Borislav Mednikarov, and Ivan Konstantinov The preparation and characterization of a reference mirror of protected aluminum Al is reported. The mirror is made of nm-thick Al film, coated with several-nanometer-thick Al O 3 and 30-nm-thick film of AlN. The mirror characterization is based on reliable and precise reflectance measurements relative to a silicon- Si- wafer reference mirror. The simple phenomenological Drude Lorentz model is applied for modeling the diersion relations n and k of the Al film. The reflection of the protected Al mirror is determined in the nm ectral range with accuracy better than 0.01 for p- and s-polarized light at angles of incidence from 0 to 70. The accuracy has been confirmed with an evaporated thin silver film with known n, k, and d derived by photometric measurements at normal light incidence. 00 Optical Society of America OCIS codes: , , , , , The authors are with the Central Laboratories of Photoprocesses, Bulgarian Academy of Sciences, Acad. G. Bonchev Street, Building 109, Sofia 1113, Bulgaria. S. Kitova s address is skitova@clf.bas.bg. Received 3 July; revised manuscript received 3 January $ Optical Society of America 1. Introduction The major difficulty in precise ectrophotometric measurement at oblique light incidence with the most widely used accessories is the need for exact knowledge of the absolute reflection of the reference mirror. The accuracy of the sample reflection is determined mainly by the accuracy of the absolute reflection of the reference mirror. In addition, it is desirable that the mirror reflection be known for any polarization, wavelength, and angle of incidence. The preliminary tabulation of all these values is an extremely labor-consuming task, even if some standard is available. A more efficient approach is to calculate the absolute reflection of the reference mirror for any condition. For this purpose the optical constants and thicknesses of all layers of the mirror have to be known. In this study we describe a procedure for preparing and characterizing a mirror of protected aluminum Al and show how to obtain a reference mirror with an absolute ecular reflection known to have an accuracy of better than 0.01 for polarized and nonpolarized light at angles of incidence from 0 to 70 in the nm ectral range.. Preparation of the Protected Aluminum Mirror The Al film, nm thick, was deposited in a vacuum installation Z 700 P Leybold-Heraeus by dc magnetron uttering of Al in an argon Ar atmohere at a pressure of 10 1 Pa. Optical boronsilicate glass BK-7 plates that have been cleaned and highly polished were used as substrates. The protective AlN film, 30 nm thick, was deposited in the same vacuum cycle on the Al mirror by rf magnetron uttering in a gas medium of Ar and N with a pressure of 10 1 Pa and a ratio of partial pressures Ar:N 5:9. Auger electron ectroscopy analysis of the AlN film showed that it is stoichiometric. It is known that an oxide film several nanometers thick grows rapidly on the surface of freshly deposited Al films even at a vacuum higher than 10 6 Pa see, for example, Ref. 1. Therefore the structure of our mirror consists of a reflective Al film covered ontaneously by an Al O 3 on which an AlN film is uttered. 3. Characterization of the Protected Aluminum Mirror To calculate with sufficient accuracy the absolute reflection of the mirror as a function of and, we need to know the diersion relations of the optical constants of the Al and the protective films as well as their thicknesses APPLIED OPTICS Vol. 41, No July 00

2 Fig. 1. Diersion curves of n derived by ectrophotometric measurements at normal incidence points and fitted with the diersion formula of Selmeier curve for AlN film with d nm. Fig.. Reflections of eight Si wafers measured at normal incidence after the first and the second cleaning solid curve includes 16 experimental curves and calculated dashed curve with the optical constants taken from the literature. 6 The refractive index n and the thickness d of the AlN film were determined by the TR b R m, TR b, TR m, and T k 0 methods.,3 The transmission T, reflections R f and R b of the film deposited on a nonabsorbing substrate BK-7, and reflection R m of the film on an absorbing Si-wafer substrate were measured in the nm ectral range at normal light incidence with a Cary 5E ectrophotometer. Subscripts f and b denote reflection from the front and back sides of the film. It should be pointed out that the AlN films were obtained in the same vacuum cycle and conditions of mirror preparation. Figure 1 dilays the diersion curve of n derived by the method T k 0 at d nm. The solid curve illustrates the refractive index described by Sellmeier s diersion formula 4 Eq. 1, with the coefficients A and B : 1 A n 1. (1) 1 B For the refractive index of the Al O 3 we used data from the literature, 5 which were fitted by Sellmeier s diersion formula Eq. 1 with coefficients A and B As a result of deriving n and d of AlN and n of Al O 3, the problem of characterizing our threelayered mirror is reduced to determining n and k of Al and the thickness of the Al O 3. The unknown parameters can be obtained by the minimization of the objective function F including measured R meas and calculated R calc reflections: M N F R calc q, j R meas q, j, () q 1 j 1 N 81 is the number of points in the nm ectral range, and M is the number of suitably chosen ectral relationships of the reflection. We have found that for accurate determination of n and k of Al, several reliable ectral relationships of the reflection at oblique incidence should be known. However, for the measurements at any angle a reference mirror with reflection, known as accurately as possible at angles between 10 and 70,is required. We chose a Si wafer as a reference mirror, since its optical constants are known with sufficient accuracy and reproducibility in a wide ectral range if its surface is cleaned by the prescribed procedure. 6,7 A. Silicon Wafer as Reference Mirror Since we calculate the reflection R Si of the Si wafer at oblique incidence from its optical constants, it is necessary to make some assessments of the reliability and accuracy of R Si and of the influence of the R Si values and the correonding errors R Si on the measurements of the oblique reflection R Al of the Al mirror. Figure shows the measured reflections R Si at normal incidence of eight Si wafers after the first and the second cleaning; i.e., the solid curve includes 16 experimental curves. The dashed curve illustrates the reflection calculated with the optical constants taken from the literature. 6 The maximum difference R Si between the measured and the calculated values of R Si is at 600 nm. The question is whether this difference remains the same at oblique incidence of polarized light. To check this, we used the maximum absolute errors of R Si expressed by R Si R Si n R Si k. (3) n k Equation 3 does not take into account the error in angle, since the value of R Si is calculated for nominal values of with n and k, with known errors n and k. In Fig. 3 the contours of the error ratio R Si R Si calculated by Eq. 3 for a Si wafer are plotted in the plane. It is seen that the ratio is not higher than unity. This means that for obliquely incident s-polarized light the deviations R Si from the real values will not exceed provided that the deviations R Si between the measured and the calculated 1 July 00 Vol. 41, No. 19 APPLIED OPTICS 3841

3 Fig. 3. Contours of the error ratio R Si R Si with indicated values in the plane, calculated for a Si wafer at n 0.03 and k 0.0. R Si and R Si were calculated from Eq. 3 at normal and obliquely incident s-polarized light, reectively. values of R Si at normal incidence are the result of imprecise knowledge of the optical constants. If the deviations R Si result from the several-nanometerthick oxide film on the Si-wafer surface, similar assessments reveal that at oblique incidence possible oxide film with thicknesses as great as 5 6 nm would not change R Si values by more than Obviously, the problem now concerns which angles and polarization types would ensure the most accurate measurement of R Al, i.e., the lowest errors R Al. From two consecutive measurements R 1 and R of the standard reference mirror with known reflection R st and of the sample with R, reectively, the absolute reflection of the sample R is calculated by at R R R R st, (4) 1 R 1 ZR st, R ZR, (5) where Z is a coefficient that is proportional to the product of the reflections of the accessory mirrors and the transmission of the polarizer and depolarizer in the object beam and inversely proportional to the transmission of the attenuator in the reference beam. Since the quantities R 1 and R are independent measurements with errors R 1 and R, and R st is known with an error R st, the error R is calculated by the following expression: R R R R 1 1 R R st R st R R R. (6) Fig. 4. Error R as a function of the ratio R R st, calculated from Eq. 7 for the denoted values of Z. Using Eqs. 4 and 5 at R 1 R R, from Eq. 6 we obtain R R R R st st 1 R R Z R st. The error R in reflection measurements at oblique incidence is a result of the instrumentation error R instr and the experimental error in the angle ; i.e., (7) R R instr R. (8) For R instr 0.001, 0.5, and R 0.01 the change in sample reflection is 0.01 for 1, the error in R, calculated from Eq. 8, is R Figure 4 depicts the relationships R f R R st, calculated by Eq. 7 with the foregoing errors, R st R Si and R 0.007, and the indicated values of Z. It is seen that the error R of the measured sample in this particular case R Al will be less for higher reflection of R st, i.e., of R Si. Since the Si wafer has higher reflection for s-polarized light, we chose measurements at angles of incidence 30, 50, and 70 for characterizing the Al mirror. For the reflection value R Al of 0.8, expected for the Al mirror in the worst case at 30, we obtain R 30 R st 30 R Al 30 R Si 30. In this case, as seen in Fig. 4 at Z 0.6, the error in the measurement of R Al will be R Al In Figure 5 the solid curves are the values of R Al, measured with a Si wafer as a reference mirror for s-polarized light at the correonding angles of incidence and at normal light incidence. 384 APPLIED OPTICS Vol. 41, No July 00

4 Table 1. Initial and Fitted Values of the Drude- and Oscillator-Model Parameters and the Thicknesses of AlN and AlO Films Parameter Initial Value Calculated Value p ev ev p ev f f f ev ev ev ev ev ev d AlN nm d Al O 3 nm Here p is plasma frequency for intraband transitions and is the intraband relaxation time. The interband part Lor of the dielectric constant is a simple semiquantum model resembling the Lorentz result for insulators: K Lor j 1 f j p j i j, (11) where p is the plasma frequency and K is the number of interband transitions with frequency j, oscillator strength f j, and lifetime 1 j. Knowing the real r and the imaginary i parts of the dielectric constant, we determine the optical constants n Al and k Al of the Al film from the expressions Fig. 5. Reflection of protected Al mirror, calculated by the matrix method points and measured curves with a reference mirror of Si wafer at angles of incidence 30, 50, and 70 of s-polarized light as well as at normal light incidence. B. Deriving n and k of Aluminum and the Thicknesses of the Protective films As a result of the foregoing measurements and assessments for the reflections, we chose R q R, R s 30, R s 50, R s 70, i.e., M 4 in the objective function Eq.. For describing the diersion relations n and k of the Al film, we use the following model of the dielectric constant, which explicitly separates the intraband effects from the interband 8 : Dr Lor. (9) The intraband free-electron contribution Dr to the optical properties can be described by a Drude-model dielectric constant: Dr 1 p i. (10) n Al 0.5 r r i 1, k Al 0.5 r r i 1. (1) According to Rakic, 8 the interband transitions in Al, interpreted with oscillators in a semiquantum model, are located at approximately 0.4 ev 3100 nm, 1.56 ev 795 nm,.1 ev 590 nm, and 4.5 ev 75 nm. Our calculations confirmed that in the ectral range nm the contribution of the last oscillator to the dielectric constant is negligible less than 1% ; hence only the sum of three oscillators with approximate frequency values of ev, 1.56 ev, and 3.1 ev is sufficient for describing n Al and k Al in this ectral range. We have used a nonlinear subace trust region method combining the interior-reflective Newton method with a preconditioned conjugate-gradient method for the minimization of the goal function. 9 The initial guess values of the Drude- and oscillatormodel parameters and the thicknesses of the protective films, as well as the solutions obtained for the correonding parameters, are given in Table 1. Figure 6 compares the diersion curves of n Al and k Al, obtained with the initial and the fitted parame- 1 July 00 Vol. 41, No. 19 APPLIED OPTICS 3843

5 Fig. 6. Diersion curves of n Al and k Al obtained with the initial and the fitted parameters from Table 1, as well as data from the literature for Al deposited in ultrahigh vacuum. 10 ters from Table 1 and the literature data for Al deposited in ultrahigh vacuum. 10 The points in Fig. 5 represent the calculated values of Al mirror reflection for s-polarized light incident at angles of 0, 30, 50, and 70. The solid curves in the figure represent the correonding values of the measured reflections R meas used for minimizing the goal function F in Eq.. The figure illustrates the fairly good agreement between the measured and the calculated values of the mirror reflection. This gives us reason to assume that we could use the parameters thus determined for describing the mirror reflection of both types of light polarization at all angles of incidence with an accuracy not less than Figure 7 presents the reflectivity characteristics of the protected Al mirror calculated by the matrix method. 11 The contours of the reflection of s- and p-polarized light a and b, reectively with denoted values are depicted in the plane. It is seen that the reflection values vary between 0.7 and 0.9 depending on the light wavelength and angle of incidence. 5. Verification of the Protected Aluminum Mirror Thin Ag films were used for verifying the accuracy of R Al of the protected Al mirror. Ag films were deposited on cleaned glass substrates BK-7 and Si wafers by thermal evaporation under vacuum higher than 10 4 Pa. We should note that this is an unfavorable case in terms of the expected measurement error see Fig. 4, because at all angles of incidence the reflection of the Ag film is higher than that of the Al mirror; i.e., R R st 1. Since Ag films undergo changes on storage in air, all necessary photometric measurements were carried out immediately after taking the films out of the vacuum installation. The measurements of T, R f, R b, and R m at normal light incidence were used to determine n, k, and d of the Ag film. Figure 8 shows the diersion curves of n and k, obtained by the TR f R m, and TR f, and TR m methods.,3 R s and R p of the same film Fig. 7. Contours of the reflection of s- and p-polarized light a and b, reectively with denoted values in the plane for the protected Al mirror. at 30, 50, and 70 were measured with our reference mirror of protected Al. The values obtained are given with points in Fig. 9. In the figure the solid curves denote the correonding reflection values, calculated by the Fresnel equations 1 with n, k, and d determined by the photometric measurements at normal incidence see Fig. 8. It is seen that there is good agreement between the calculated and the measured reflections from the Ag film for both s-and p-polarized light and at all angles of incidence. The greatest difference, 0.011, between the calculated and the measured values is obtained at 800 nm for s-polarized light, incident at an angle of 70. Hence we can assume that the reflection of the pro- Fig. 8. Diersion curves of n and k of 36-nm-thick Ag film obtained by photometric measurements at normal light incidence APPLIED OPTICS Vol. 41, No July 00

6 Fig. 9. Reflections of Ag film. Points are R s and R p values measured at 30,50, and 70 with the reference Al mirror. Solid curves are the values calculated by the Fresnel equations with n, k, and d 36 nm, determined by the photometric measurements at normal light incidence. tected Al mirror is determined with an accuracy of up to 0.01 at all angles of incidence from 0 to 70 in the nm ectral range. 6. Conclusion In the present study we have described the procedure for preparing and characterizing a reference mirror of protected Al. As a result the absolute ecular reflection with an accuracy better than 0.01 for p- and s-polarized light at angles of incidence from 0 to 70 in the nm ectral range can be calculated by the matrix method. The mirror characterization is based on a reliable and precise measurement of the mirror reflection at angles of light incidence 0, 30, 50, and 70 in the nm ectral range with a Si wafer as a reference mirror. The reflections of the Al mirror, calculated by the matrix method, are fitted to the experimental data. The diersion relations employed reduce the variable parameters in the optimization procedure to a reasonable and feasible number and thus make the problem solvable. The simple phenomenological Drude Lorentz model was applied for modeling the dielectric constant and correonding n and k of the Al film. The model parameters as well as the thicknesses of the Al O 3 and AlN film were derived by minimization of the goal function. Once we have determined all needed parameters, the desired values of the reflection of the protected mirror can be easily calculated. The exact determination of the mirror reflections was confirmed by comparison of calculated and mea- 1 July 00 Vol. 41, No. 19 APPLIED OPTICS 3845

7 sured reflections of an evaporated Ag thin film with known n, k, and d, obtained by photometric measurements at normal light incidence. Finally, we would like to emphasize that the procedure described can be applied for the characterization of mirrors not only of Al but of other metals or with other protective films as well as for the preparation of mirrors with higher reflections or for other ectral ranges. References and Note 1. T. H. Allen, Study of Al with combined Auger electron ectrometer ellipsometer system, J. Vac. Sci. Technol. 13, V. Panayotov and I. Konstantinov, Algebraic determination of thin-film optical constants from photometric T, R, R f, R m and T, R b, R m measurements, in Optical Interference Coatings,F. Abelès, ed., Proc. SPIE 53, I. Konstantinov, Tz. Babeva, and S. Kitova, Analysis of errors in thin-film optical parameters derived from ectrophotometric measurements at normal light incidence, Appl. Opt. 37, H. Liddell, Computer-Aided Techniques for Design of Multilayer Filters Adam Hilger, Bristol, UK, 1981, p SOPRA measurements obtained from http: 6. D. E. Anes and A.A. Studna, Dielectric functions and optical parameters of Si, GaP, GaAs, GaSb, InP, InAs and InSb from 1.5 to 6.0 ev, Phys. Rev. B 7, T. Yasuda and D. E. Anes, Optical-standard surfaces of single-crystal silicon for calibrating ellipsometers and reflectometers, Appl. Opt. 33, A. Rakic, Algorithm for the determination of intrinsic optical constants of metal films: application to aluminum, Appl. Opt. 34, T. F. Coleman and Y. Li, An interior, trust region approach for nonlinear minimization subject to bounds, SIAM Soc. Ind. Appl. Math J. Optimization 6, D. Smith, E. Shiles, and M. Inokuti, Handbook of Optical Constants of Solids, D. Palik, ed. Academic, San Diego, Calif., 1985, pp Ref. 4, pp O. S. Heavens, Optical Properties of Thin Solid Films Butterworths Scientific, London, 1955, Chap. 4, pp , APPLIED OPTICS Vol. 41, No July 00