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A computer program for determination of thin films thickness and optical constants
Applied Surface Science 248 (2005) 440–445 www.elsevier.com/locate/apsusc
A computer program for determination of thin films thickness and optical constants A.P. Caricato, A. Fazzi, G. Leggieri * INFM, Dipartimento di Fisica, Universita` di Lecce, 73100 Lecce, Italy Available online 30 March 2005
Abstract A computer simulation program for processing transmission spectra of amorphous optical thin films deposited on weakly absorbing substrates and evaluation of the refractive index n, extinction coefficient k and thickness d was developed. The computer code is the implementation of an optical characterisation algorithm based on the determination of the upper and lower envelopes of the transmission spectrum interference fringes. Inhomogeneities in the thickness of the analysed films, which are responsible of a shrinking in the fringes amplitude, can be considered in the program. Relative errors in the calculated values of n, k and d have been determined using simulated transmission spectra in both cases of homogeneous and inhomogeneous films. The thickness and the refractive index of uniform films are calculated with an accuracy 0.5%, while the accuracy in the case including inhomogeneities is 2%. Simulation results for chalcogenide thin films deposited by pulsed laser deposition (PLD) on microscope slabs and glass slides are reported. # 2005 Elsevier B.V. All rights reserved. PACS: 07.05.Tp; 78.66.Bz; 78.20.c Keywords: Transmittance; Refractive index; Extinction coefficient; Pulsed laser deposition
1. Introduction The performance of planar optical devices depends on the wavelength dependence of the optical constants, i.e., the refractive index n(l) and the extinction coefficient k(l) of thin films, and their geometrical thickness, d. n(l), k(l) and d can be * Corresponding author. Tel.: +39 0832 297479; fax: +39 0832 297505. E-mail address: [email protected] (G. Leggieri).
calculated from the interference fringes present in the transmission, T(l), and reflection, R(l), spectra by using various techniques. Current methods for determining the optical constants of thin films are usually based on sophisticated computer iteration techniques [1–4]. A relatively simple method for the computation of the optical constants of dielectric films deposited on visible transparent substrates such as glass has been implemented by a computer program developed by us named ‘‘Refractor’’ and tested on optical films deposited by pulsed laser deposition
0169-4332/$ – see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2005.03.069
A.P. Caricato et al. / Applied Surface Science 248 (2005) 440–445
441
(PLD). The method is based on the determination of the upper and lower envelopes of the interference fringes in the measured transmission spectrum [5–7]. ‘‘Refractor’’ has several advantages: (1) it gives accurate results in a short time and reduces the number of arithmetic operations needed to compute n, k and d as compared to other computer techniques [1–4]; (2) it takes into account possible inhomogeneities in the film thickness; (3) it is not based on minimisation techniques as other commercial software; (4) using the obtained n, k and d values, the simulated spectrum as well as the experimental one can be graphically compared. The optical constants and thickness evaluated by ‘‘Refractor’’ for Pr3+-doped (2000 ppm) chalcogenide thin films, grown on microscope slabs and glass slides by PLD, are reported.
For k2 n2, the expression for the transmittance can be written [7]:
2. Preliminary theoretical considerations
N ¼ 2s
The considered optical system is a thin homogeneous film deposited on a weakly absorbing substrate (extinction coefficient ks 0) with a thickness several orders of magnitude greater than that of the film and with a refractive index s. The substrate surfaces are considered smooth but not perfectly parallel so that interference effects due to the substrate are negligible. When a monochromatic light beam impinges perpendicularly on the surface covered with a thin film, multiple reflections occur at the interfaces of the system. Assuming that these reflections are coherent in the film and incoherent in the substrate, both the reflected and transmitted beams are subject to interference phenomena. The interference fringes can be used to determine the optical constants and thickness of the film. The procedure is as follows. Taking into account all the multiple reflections at the three interfaces, the transmission T is a complex function of the variables l, n, k, s and d:
TM and Tm are the transmission maximum and the corresponding minimum at the wavelength l, one being measured from the spectrum and the other calculated. The Tm(l) and TM(l) envelopes are calculated using a ‘‘shape-preserving spline’’ interpolation algorithm [8]. The Tm(l) and TM(l) are considered continuous functions of the wavelength l. If n(l) is known, the constants in equation (2) are also known and x can be calculated in various ways [7]. If the film thickness is non-uniform, ‘‘Refractor’’ assumes that the thickness of the film varies linearly over the illuminated area: d ¼ d¯ Dd. In this case, from the work of Swanepoel [9], the equations describing the envelopes used in ‘‘Refractor’’ are:
T ¼ Tðl; n; k; s; dÞ:
(1)
If the substrate refractive index is known, it is convenient to write this equation in terms of n(l) and x(l) = exp(ad), the absorbance, where a = 4pk/l is the optical absorption coefficient of the thin film: T ¼ Tðn; xÞ:
TðlÞ ¼
Ax B Cx cos ’ þ Dx2
(2)
where A = 16n2s; B = (n + 1)3(n + s2); C = 2(n2 1) (n2 s2); D = (n 1)3(n s2); w = 4pnd/l. Since 1 cos w 1, T(l) values can vary Ax Ax between Tm ¼ BþCxþDx 2 and TM ¼ BCxþDx2 . A typical transmission spectrum at normal incidence has two spectral regions: the region of weak and medium absorption and the strong absorption region. In the weak and medium absorption region, a first approximation of the refractive index of the film can be calculated by the following expression [7]: n ¼ ½N þ ðN 2 s2 Þ1=2 1=2
(3)
where TM Tm s2 þ 1 : þ 2 TM Tm
l ax 2pn Dd ð1 b2 Þ1=2 x " # 1 þ bx 2pn Dd 1 tan tan l ð1 b2 Þ1=2
(4)
TM ðlÞ ¼
(5)
x
l ax 2pn Dd ð1 b2 Þ1=2 x " # 1 b 2pn Dd x 1 tan tan l ð1 b2 Þ1=2
Tm ðlÞ ¼
x
(6)
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where ax = Ax/(B + Dx2), bx = Cx/(B + Dx2). A–D and ¯ x are defined in equation (2) and x ¼ expðadÞ.
3. The algorithm The theory described above was implemented in the ‘‘Refractor’’ computer program written in C++ code. The flow chart in Fig. 1 shows the sequence of the elaboration steps starting from the experimental transmission data acquisition ending with the determination of the optical constants and the thickness of the film. To determine the upper and lower spectrum envelopes, a recursive algorithm was implemented. In each iteration, the algorithm determines, with increasing precision, the tangential points between the envelopes and the experimental values of the transmission spectrum. As starting point the interference maxima and minima are considered. Once the tangential conditions are matched, these points are interpolated with an appropriate function (a good fit is obtained choosing a monotonic piecewise cubic inter-
polation). The method produces envelopes that preserve a monotonic behaviour conforming to the experimental data. However, due to a non-unique choice of the monotonic conditions, the procedure occasionally produces visual artefacts. The problem was resolved, introducing an additional parameter to the expression of the interpolant function. In this way, much more control on the shape of envelopes is gained [10]. The accuracy of the algorithm depends on various factors. Critical is the determination of the envelope curves, TM(l) and Tm(l), and the accuracy of the substrate refractive index. To evaluate this intrinsic error, n, k and d were calculated in the case of a theoretical spectrum with the following parameters, which refer to a Si:H films deposited on SiO2 [7]: film thickness, d = 1000 nm; n(l) = (3 105)/l2 + 2.6; k(l) = l 10(1.5 106/l2)8/4p; and substrate refractive index s = 1.51. The values simulated by ‘‘Refractor’’ for this case were: d0 = 1001 4 nm; n0 (l) = (3.036 105)/ 2 (1.468 106/l2)7.904 0 l + 2.590; k (l) = l 10 /4p. These values are thus very close to the original ones, since the differences between Dn = jn(l j) n0 (l j)j and Dk = jk(l j) k0 (l j)j are less than 0.1%. The differences Dk increase with decreasing wavelength. This is due to the extrapolation of the refractive index values in the region of strong absorption. Thus, the choice of the appropriate function for the n(l) fit is important. The intrinsic error was evaluated also in the case of films non-uniform in thickness. The values of the refractive index and the extinction coefficient are the same as reported in [7], but using an average thickness of 1000 nm and a Dd = 30 nm. In this case, higher values in the differences were obtained (0.028 and 0.005 for Dn and Dk, respectively).
4. Experimental
Fig. 1. Flow chart of the algorithm used for optical constants and thickness evaluation in the case of uniform and non-uniform film.
Thin films of chalcogenide glasses were deposited by PLD. Pr3+-doped (2000 ppm) chalcogenide glass with nominal composition 70GeS2–15Ga2S3–15CsI (mol%) was ablated with a XeCl excimer laser at the fluence of 3 J/cm2 and repetition rate of 10 Hz. The films were deposited, at room temperature, on microscope slabs and glass slides placed at 60 mm in front of the ablating target. The 104 and 3.5 104
A.P. Caricato et al. / Applied Surface Science 248 (2005) 440–445
443
laser pulses were employed for the depositions. More details of the deposition parameters were reported elsewhere [11]. The transmission spectra were recorded with a Perkin-Elmer Lambda 900 spectrophotometer in the range 200–2500 nm. The refractive index, extinction coefficient and films thickness were determined by ‘‘Refractor’’ and compared with the values obtained by a commercial software. The film thickness was evaluated also by Rutherford backscattering spectrometry (RBS). The RBS spectra were recorded with 2.0 MeV He+ ions at a 1608 scattering angle. The acquired spectra were simulated with the Rutherford Universal Manipulation Program (RUMP) [12].
5. Results and discussion In Fig. 2, the transmittance spectra of two chalcogenide films are shown in the wavelength range of interest for optical applications (200–2500 nm). The spectrum in Fig. 2a (full line) refers to a film deposited with 104 laser pulses and the spectrum in Fig. 2b (full line) to a film deposited with 3.5 104 laser shots. For l 500 nm, the films exhibit a deep in the transmittance towards zero. The three regions selected for ‘‘Refractor’’ simulation are: (1) the strong absorption region from 500 to 600 nm; (2) the weakly absorption region from 600 to 1100 nm and from 600 to 750 nm for Fig. 2a and b, respectively; (3) the transparent region from 1100 to 2100 nm and from 750 to 2400 nm for Fig. 2a and b, respectively. The spectrum shown in Fig. 2a indicates that the film is quite uniform, since the minima in the transmittance decrease with decreasing wavelength. Hence, for the film with the uniform thickness ‘‘Refractor’’ gave the following results for d, n and k: d ¼ 0:436 0:005 mm; 0:117 þ 2:379; l2 b c d e f kðlÞ ¼ a þ 2 þ 4 þ 6 þ 8 þ 10 l l l l l nðlÞ ¼
(9)
with fitting parameters a = 0.0067, b = 0.0157, c = 0.0226, d = 0.0196, e = 0.0073 and f = 0.0008.
Fig. 2. Experimental (full line) and calculated (dashed line) transmission spectra of two chalcogenide films deposited with (a) 104 laser pulses and (b) 3.5 104 laser pulses.
In this case, a Cauchy dispersion function was used to fit the discrete values of the extinction coefficient. The calculated curves of n(l) and k(l) are shown in Fig. 3. The transmission spectrum calculated by ‘‘Refractor’’ using equation (2) is given in Fig. 2a by the dashed line. The relative percentage errors between the experimental points, Tiexp, and the calculated transmission Tcalc(l), ei ¼
Tiexp Tcalc ðlÞ 100, Tiexp
were determined by
‘‘Refractor’’ and are given in Fig. 4. A relative percentage error lower than 3% was obtained in the
444
A.P. Caricato et al. / Applied Surface Science 248 (2005) 440–445
Fig. 3. Refractive index (dashed line) and extinction coefficient (full line), determined by ‘‘Refractor’’, of the chalcogenide film deposited with 104 laser pulses.
region of high transparency, and about 7% in the region of strong absorption. These results obtained by ‘‘Refractor’’ were compared with those obtained by a commercial software, TFCalc (Scientific Software, Inc.), using a Sellmaier dispersion relation for n(l) and an exponential function for k(l) [11]: d ¼ 0:433 mm; nðlÞ ¼ 0:3244 þ
1=2 5:608l2 ; l2 0:07348 5:7309 kðlÞ ¼ 1:51 106 exp : l
(10)
There is a good agreement between the results obtained by the two softwares but the relative percentage errors ei are larger for TFCalc (5%) as compared to ‘‘Refractor’’ (3%). To test the film thickness obtained by Refractor (d = 0.436 0.005 mm), Rutherford backscattering spectrometry was performed giving the result of d = 0.50 0.03 mm. For the non-uniform film shown in Fig. 2b the optical parameters obtained were n(l) = 0.085/ l2 + 2.320; kðlÞ ¼ 3:14 108 expð7:689=lÞ with d¯ ¼ 1:961 0:006 mm and Dd =0.063 mm, where d¯ stands for the arithmetical average of the thickness and Dd for the maximum difference. The calculated transmission spectrum is given in Fig. 2b by the dashed line. As expected, a larger error was obtained in the region of strong absorption (8%), while in the weak and medium absorption region the errors were less than 2%. It must be noted that the error in these regions is lower than in the case of the uniform film because of the larger number of minima and maxima. The thickness values were also in good agreement with the RBS results which gave d = 2.0 0.1 mm [11].
6. Conclusions An optical simulation method for the accurate calculation of film thicknesses and optical constants was developed by a new software named ‘‘Refractor’’. The software tested on homogeneous and inhomogeneous chalcogenide films deposited by PLD gave good results in agreement with more complex commercially available computer programs. The thicknesses values were also close to those obtained by the RBS technique.
References
Fig. 4. Relative percentage error between the experimental points, Ti, and the calculated T(l) of the spectra in Fig. 2a.
[1] J. Szcyrbrowski, A. Czapla, Thin Solid Films 46 (1977) 127. [2] L. Vriens, W. Rippens, Appl. Opt. 22 (1983) 4105. [3] D.P. Arndt, R.M.A. Azzam, J.M. Bennett, J.P. Borgogno, C.K. Carniglia, W.E. Case, J.A. Dobrowolski, U.J. Gibson, T. Tuttle Hart, F.C. Ho, V.A. Hodgkin, W.P. Klapp, H.A. Macleod, E. Pelletier, M.K. Purvis, D.M. Quinn, D.H. Strome, R. Swenson, P.A. Temple, T.F. Thonn, Appl. Opt. 23 (1984) 3571. [4] D.A. Minkov, J. Mod. Opt. 37 (1990) 1977.
A.P. Caricato et al. / Applied Surface Science 248 (2005) 440–445 [5] D.A. Minkov, J. Phys. D: Appl. Phys. 22 (1989) 1157. [6] J. Ruı´z-Pe´ rez, E. Ma´ rquez, D. Minkov, J. Reyes, J.B. Ramı´rezMalo, P. Villares, R. Jime´ nez-Garay, Phys. Scr. 56 (1996) 76. [7] R. Swanepoel, J. Phys. E: Sci. Instrum. 16 (1983) 1214. [8] F.N. Fritsch, R.E. Carlson, SIAM J. Numer. Anal. 17 (1980) 238.
445
[9] R. Swanepoel, J. Phys. E: Sci. Instrum. 17 (1984) 896. [10] A. Fazzi, Comput. Programming 132 (2004) 75. [11] A.P. Caricato, M. De Sario, M. Ferna´ ndez, M. Ferrari, G. Leggieri, A. Luches, M. Martino, M. Montagna, F. Prudenzano, A. Jha, Appl. Surf. Sci. 208–209 (2003) 632. [12] L.R. Doolitle, Nucl. Instrum. Methods B 9 (1985) 344.
A computer program for determination of thin films thickness and optical constants A.P. Caricato, A. Fazzi, G. Leggieri * INFM, Dipartimento di Fisica, Universita` di Lecce, 73100 Lecce, Italy Available online 30 March 2005
Abstract A computer simulation program for processing transmission spectra of amorphous optical thin films deposited on weakly absorbing substrates and evaluation of the refractive index n, extinction coefficient k and thickness d was developed. The computer code is the implementation of an optical characterisation algorithm based on the determination of the upper and lower envelopes of the transmission spectrum interference fringes. Inhomogeneities in the thickness of the analysed films, which are responsible of a shrinking in the fringes amplitude, can be considered in the program. Relative errors in the calculated values of n, k and d have been determined using simulated transmission spectra in both cases of homogeneous and inhomogeneous films. The thickness and the refractive index of uniform films are calculated with an accuracy 0.5%, while the accuracy in the case including inhomogeneities is 2%. Simulation results for chalcogenide thin films deposited by pulsed laser deposition (PLD) on microscope slabs and glass slides are reported. # 2005 Elsevier B.V. All rights reserved. PACS: 07.05.Tp; 78.66.Bz; 78.20.c Keywords: Transmittance; Refractive index; Extinction coefficient; Pulsed laser deposition
1. Introduction The performance of planar optical devices depends on the wavelength dependence of the optical constants, i.e., the refractive index n(l) and the extinction coefficient k(l) of thin films, and their geometrical thickness, d. n(l), k(l) and d can be * Corresponding author. Tel.: +39 0832 297479; fax: +39 0832 297505. E-mail address: [email protected] (G. Leggieri).
calculated from the interference fringes present in the transmission, T(l), and reflection, R(l), spectra by using various techniques. Current methods for determining the optical constants of thin films are usually based on sophisticated computer iteration techniques [1–4]. A relatively simple method for the computation of the optical constants of dielectric films deposited on visible transparent substrates such as glass has been implemented by a computer program developed by us named ‘‘Refractor’’ and tested on optical films deposited by pulsed laser deposition
0169-4332/$ – see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2005.03.069
A.P. Caricato et al. / Applied Surface Science 248 (2005) 440–445
441
(PLD). The method is based on the determination of the upper and lower envelopes of the interference fringes in the measured transmission spectrum [5–7]. ‘‘Refractor’’ has several advantages: (1) it gives accurate results in a short time and reduces the number of arithmetic operations needed to compute n, k and d as compared to other computer techniques [1–4]; (2) it takes into account possible inhomogeneities in the film thickness; (3) it is not based on minimisation techniques as other commercial software; (4) using the obtained n, k and d values, the simulated spectrum as well as the experimental one can be graphically compared. The optical constants and thickness evaluated by ‘‘Refractor’’ for Pr3+-doped (2000 ppm) chalcogenide thin films, grown on microscope slabs and glass slides by PLD, are reported.
For k2 n2, the expression for the transmittance can be written [7]:
2. Preliminary theoretical considerations
N ¼ 2s
The considered optical system is a thin homogeneous film deposited on a weakly absorbing substrate (extinction coefficient ks 0) with a thickness several orders of magnitude greater than that of the film and with a refractive index s. The substrate surfaces are considered smooth but not perfectly parallel so that interference effects due to the substrate are negligible. When a monochromatic light beam impinges perpendicularly on the surface covered with a thin film, multiple reflections occur at the interfaces of the system. Assuming that these reflections are coherent in the film and incoherent in the substrate, both the reflected and transmitted beams are subject to interference phenomena. The interference fringes can be used to determine the optical constants and thickness of the film. The procedure is as follows. Taking into account all the multiple reflections at the three interfaces, the transmission T is a complex function of the variables l, n, k, s and d:
TM and Tm are the transmission maximum and the corresponding minimum at the wavelength l, one being measured from the spectrum and the other calculated. The Tm(l) and TM(l) envelopes are calculated using a ‘‘shape-preserving spline’’ interpolation algorithm [8]. The Tm(l) and TM(l) are considered continuous functions of the wavelength l. If n(l) is known, the constants in equation (2) are also known and x can be calculated in various ways [7]. If the film thickness is non-uniform, ‘‘Refractor’’ assumes that the thickness of the film varies linearly over the illuminated area: d ¼ d¯ Dd. In this case, from the work of Swanepoel [9], the equations describing the envelopes used in ‘‘Refractor’’ are:
T ¼ Tðl; n; k; s; dÞ:
(1)
If the substrate refractive index is known, it is convenient to write this equation in terms of n(l) and x(l) = exp(ad), the absorbance, where a = 4pk/l is the optical absorption coefficient of the thin film: T ¼ Tðn; xÞ:
TðlÞ ¼
Ax B Cx cos ’ þ Dx2
(2)
where A = 16n2s; B = (n + 1)3(n + s2); C = 2(n2 1) (n2 s2); D = (n 1)3(n s2); w = 4pnd/l. Since 1 cos w 1, T(l) values can vary Ax Ax between Tm ¼ BþCxþDx 2 and TM ¼ BCxþDx2 . A typical transmission spectrum at normal incidence has two spectral regions: the region of weak and medium absorption and the strong absorption region. In the weak and medium absorption region, a first approximation of the refractive index of the film can be calculated by the following expression [7]: n ¼ ½N þ ðN 2 s2 Þ1=2 1=2
(3)
where TM Tm s2 þ 1 : þ 2 TM Tm
l ax 2pn Dd ð1 b2 Þ1=2 x " # 1 þ bx 2pn Dd 1 tan tan l ð1 b2 Þ1=2
(4)
TM ðlÞ ¼
(5)
x
l ax 2pn Dd ð1 b2 Þ1=2 x " # 1 b 2pn Dd x 1 tan tan l ð1 b2 Þ1=2
Tm ðlÞ ¼
x
(6)
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A.P. Caricato et al. / Applied Surface Science 248 (2005) 440–445
where ax = Ax/(B + Dx2), bx = Cx/(B + Dx2). A–D and ¯ x are defined in equation (2) and x ¼ expðadÞ.
3. The algorithm The theory described above was implemented in the ‘‘Refractor’’ computer program written in C++ code. The flow chart in Fig. 1 shows the sequence of the elaboration steps starting from the experimental transmission data acquisition ending with the determination of the optical constants and the thickness of the film. To determine the upper and lower spectrum envelopes, a recursive algorithm was implemented. In each iteration, the algorithm determines, with increasing precision, the tangential points between the envelopes and the experimental values of the transmission spectrum. As starting point the interference maxima and minima are considered. Once the tangential conditions are matched, these points are interpolated with an appropriate function (a good fit is obtained choosing a monotonic piecewise cubic inter-
polation). The method produces envelopes that preserve a monotonic behaviour conforming to the experimental data. However, due to a non-unique choice of the monotonic conditions, the procedure occasionally produces visual artefacts. The problem was resolved, introducing an additional parameter to the expression of the interpolant function. In this way, much more control on the shape of envelopes is gained [10]. The accuracy of the algorithm depends on various factors. Critical is the determination of the envelope curves, TM(l) and Tm(l), and the accuracy of the substrate refractive index. To evaluate this intrinsic error, n, k and d were calculated in the case of a theoretical spectrum with the following parameters, which refer to a Si:H films deposited on SiO2 [7]: film thickness, d = 1000 nm; n(l) = (3 105)/l2 + 2.6; k(l) = l 10(1.5 106/l2)8/4p; and substrate refractive index s = 1.51. The values simulated by ‘‘Refractor’’ for this case were: d0 = 1001 4 nm; n0 (l) = (3.036 105)/ 2 (1.468 106/l2)7.904 0 l + 2.590; k (l) = l 10 /4p. These values are thus very close to the original ones, since the differences between Dn = jn(l j) n0 (l j)j and Dk = jk(l j) k0 (l j)j are less than 0.1%. The differences Dk increase with decreasing wavelength. This is due to the extrapolation of the refractive index values in the region of strong absorption. Thus, the choice of the appropriate function for the n(l) fit is important. The intrinsic error was evaluated also in the case of films non-uniform in thickness. The values of the refractive index and the extinction coefficient are the same as reported in [7], but using an average thickness of 1000 nm and a Dd = 30 nm. In this case, higher values in the differences were obtained (0.028 and 0.005 for Dn and Dk, respectively).
4. Experimental
Fig. 1. Flow chart of the algorithm used for optical constants and thickness evaluation in the case of uniform and non-uniform film.
Thin films of chalcogenide glasses were deposited by PLD. Pr3+-doped (2000 ppm) chalcogenide glass with nominal composition 70GeS2–15Ga2S3–15CsI (mol%) was ablated with a XeCl excimer laser at the fluence of 3 J/cm2 and repetition rate of 10 Hz. The films were deposited, at room temperature, on microscope slabs and glass slides placed at 60 mm in front of the ablating target. The 104 and 3.5 104
A.P. Caricato et al. / Applied Surface Science 248 (2005) 440–445
443
laser pulses were employed for the depositions. More details of the deposition parameters were reported elsewhere [11]. The transmission spectra were recorded with a Perkin-Elmer Lambda 900 spectrophotometer in the range 200–2500 nm. The refractive index, extinction coefficient and films thickness were determined by ‘‘Refractor’’ and compared with the values obtained by a commercial software. The film thickness was evaluated also by Rutherford backscattering spectrometry (RBS). The RBS spectra were recorded with 2.0 MeV He+ ions at a 1608 scattering angle. The acquired spectra were simulated with the Rutherford Universal Manipulation Program (RUMP) [12].
5. Results and discussion In Fig. 2, the transmittance spectra of two chalcogenide films are shown in the wavelength range of interest for optical applications (200–2500 nm). The spectrum in Fig. 2a (full line) refers to a film deposited with 104 laser pulses and the spectrum in Fig. 2b (full line) to a film deposited with 3.5 104 laser shots. For l 500 nm, the films exhibit a deep in the transmittance towards zero. The three regions selected for ‘‘Refractor’’ simulation are: (1) the strong absorption region from 500 to 600 nm; (2) the weakly absorption region from 600 to 1100 nm and from 600 to 750 nm for Fig. 2a and b, respectively; (3) the transparent region from 1100 to 2100 nm and from 750 to 2400 nm for Fig. 2a and b, respectively. The spectrum shown in Fig. 2a indicates that the film is quite uniform, since the minima in the transmittance decrease with decreasing wavelength. Hence, for the film with the uniform thickness ‘‘Refractor’’ gave the following results for d, n and k: d ¼ 0:436 0:005 mm; 0:117 þ 2:379; l2 b c d e f kðlÞ ¼ a þ 2 þ 4 þ 6 þ 8 þ 10 l l l l l nðlÞ ¼
(9)
with fitting parameters a = 0.0067, b = 0.0157, c = 0.0226, d = 0.0196, e = 0.0073 and f = 0.0008.
Fig. 2. Experimental (full line) and calculated (dashed line) transmission spectra of two chalcogenide films deposited with (a) 104 laser pulses and (b) 3.5 104 laser pulses.
In this case, a Cauchy dispersion function was used to fit the discrete values of the extinction coefficient. The calculated curves of n(l) and k(l) are shown in Fig. 3. The transmission spectrum calculated by ‘‘Refractor’’ using equation (2) is given in Fig. 2a by the dashed line. The relative percentage errors between the experimental points, Tiexp, and the calculated transmission Tcalc(l), ei ¼
Tiexp Tcalc ðlÞ 100, Tiexp
were determined by
‘‘Refractor’’ and are given in Fig. 4. A relative percentage error lower than 3% was obtained in the
444
A.P. Caricato et al. / Applied Surface Science 248 (2005) 440–445
Fig. 3. Refractive index (dashed line) and extinction coefficient (full line), determined by ‘‘Refractor’’, of the chalcogenide film deposited with 104 laser pulses.
region of high transparency, and about 7% in the region of strong absorption. These results obtained by ‘‘Refractor’’ were compared with those obtained by a commercial software, TFCalc (Scientific Software, Inc.), using a Sellmaier dispersion relation for n(l) and an exponential function for k(l) [11]: d ¼ 0:433 mm; nðlÞ ¼ 0:3244 þ
1=2 5:608l2 ; l2 0:07348 5:7309 kðlÞ ¼ 1:51 106 exp : l
(10)
There is a good agreement between the results obtained by the two softwares but the relative percentage errors ei are larger for TFCalc (5%) as compared to ‘‘Refractor’’ (3%). To test the film thickness obtained by Refractor (d = 0.436 0.005 mm), Rutherford backscattering spectrometry was performed giving the result of d = 0.50 0.03 mm. For the non-uniform film shown in Fig. 2b the optical parameters obtained were n(l) = 0.085/ l2 + 2.320; kðlÞ ¼ 3:14 108 expð7:689=lÞ with d¯ ¼ 1:961 0:006 mm and Dd =0.063 mm, where d¯ stands for the arithmetical average of the thickness and Dd for the maximum difference. The calculated transmission spectrum is given in Fig. 2b by the dashed line. As expected, a larger error was obtained in the region of strong absorption (8%), while in the weak and medium absorption region the errors were less than 2%. It must be noted that the error in these regions is lower than in the case of the uniform film because of the larger number of minima and maxima. The thickness values were also in good agreement with the RBS results which gave d = 2.0 0.1 mm [11].
6. Conclusions An optical simulation method for the accurate calculation of film thicknesses and optical constants was developed by a new software named ‘‘Refractor’’. The software tested on homogeneous and inhomogeneous chalcogenide films deposited by PLD gave good results in agreement with more complex commercially available computer programs. The thicknesses values were also close to those obtained by the RBS technique.
References
Fig. 4. Relative percentage error between the experimental points, Ti, and the calculated T(l) of the spectra in Fig. 2a.
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