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Simple method for determining the optical constants of thin metallic films from transmittance measurements
Thin Solid Films 358 (2000) 166±171 www.elsevier.com/locate/tsf
Simple method for determining the optical constants of thin metallic ®lms from transmittance measurements E.E. Khawaja*, S.M.A. Durrani, A.M. Al-Shukri Center for Applied Physical Sciences, Research Institute, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia Received 17 December 1998; received in revised form 20 August 1999; accepted 20 August 1999
Abstract A method for the determination of the optical constants of thin metal ®lms on transparent substrates is proposed. It requires measurements at normal incidence of the transmittance from the specimen itself and when coated with a thin transparent layer. A procedure is given for determining the correct solutions for the indices of refraction and absorption and also for accurately ®xing the thickness of the ®lm. Advantage of the present method over existing methods is readily available measurement facilities. The method has been applied successfully to ®lms of gold. q 2000 Elsevier Science S.A. All rights reserved. Keywords: Gold; Optical coatings; Optical properties; Tungsten oxide
1. Introduction The optical properties of a homogeneous thin metal ®lm are fully described by its complex index of refraction N m nm l 2 ikm l where nm (refractive index) is the real part and km (absorption index) is the imaginary part while l is the wavelength. If a ®lm of known thickness dm and unknown Nm is deposited upon a transparent substrate of known ns we can, in principle, determine Nm from any pair of measurements with a spectral photometer. Such a pair of measurements could, for example, be either one re¯ection and one transmission measurement or two spectra recorded with different angles of incidence or different directions of polarization. Various methods, for the measurement of optical constants, have been described in Refs. [1,2]. In fact, a panel discussion on the measurement of optical constants of absorbing and non-absorbing thin ®lms, has been reported in Ref. [2]. Generally, in the case of metal ®lms, the methods include at least one measurement of re¯ection from the ®lms. However, in the present method we use measurements at normal incidence of the transmittance from the specimen (a gold ®lm on a glass substrate) itself and when coated with a thin transparent layer to determine the complex refractive index. The derivation is complicated (as in the existing methods [1,2]) by the existence of multiple solutions of * Corresponding author. Fax: 1966-3-860-4281. E-mail address: [email protected] (E.E. Khawaja)
the relevant equations. A procedure is given for determining the solutions for nm and km and also for accurately ®xing the thicknesses of the ®lms. Advantage of the present method over the other methods (involving re¯ection measurements) is readily available measurement facilities. In general, the commercially available spectrophotometers are applied for transmittance, T, measurement at normal incidence and only a few of them have re¯ectance, R, measuring facilities. The method has been applied successfully to ®lms of gold. 2. Principle of the method The transmittances at normal incidence from the specimen (a thin gold ®lm on a glass substrate), Tm, and when coated with a thin transparent layer, Tmt, were measured as functions of wavelength l . Equations relating these measurements to the optical constants and thicknesses of the metal (nm, km and dm) and transparent ®lms (nt and dt) and the refractive index (ns) of the substrate are given as single ®lm (for Tm) and double layer (for Tmt) formulae by Heavens [1]. These equations (reproduced in Appendix) are cumbersome; however, we can write these in functional form as ÿ 1 T m F1 n m ; k m ; d m ; n s ; l ÿ Tmt F2 nm ; km ; dm ; nt ; dt ; ns ; l
2
The equations given in Ref. [1] do not include the effects
0040-6090/00/$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S00 40-6090(99)0068 9-6
E.E. Khawaja et al. / Thin Solid Films 358 (2000) 166±171
of multiple re¯ection in the substrate. The use of a wedge substrate would ensure that re¯ection from the back of the substrate is de¯ected out of the optical path of the instrument so that the multiple re¯ections in the substrate do not affect the measurement. If the Tm and Tmt are measured as a function of wavelength and dm, nt, dt, and ns are known, then in principle, nm 2 ikm can be determined from the above relations. An expression giving an explicit value of nm or km (see Appendix) cannot be obtained. However, the equations for Tm and Tmt may be solved by a numerical method similar to that described by Denton et al. [3], as discussed below. A computer solution of equations for Tm and Tmt may be carried out by rearranging them in the forms f1 nm ; km 0 and f2 nm ; km 0, respectively. There is a single value of km for a given nm and Tm [3]. Therefore, one may readily solve f1 nm ; km 0, ®nding km for a given value of nm by successive approximation by using Newton's method, since 2f1/2km may be easily written down. It is necessary, as will appear from the example discussed below, to cover a wide range of value of nm. Each value of nm and the corresponding km is substituted in turn into f2 nm ; km 0. When a change in sign of f2(nm,km) occurs, a simple interpolation leads to the required solution of f2 nm ; km 0. Because of the existence of multiple roots [3] it is essential to continue the process for further values of nm and km, since the ®rst solution found may not be the true one. It is instructive to examine the numerical solution of these equations for nm and km for a hypothetical system. We have chosen a ®lm, approximating a gold ®lm, with the following characteristics nm 20:4 1 3:2 £ 105 =l2 ;
167
®lm thickness has a characteristic and easily distinguishable effect upon the computed solutions for nm, from which it is possible to infer, when using a set of experimental data, the estimated ®lm thickness is too high or too low, provided there is no systematic error in the measured transmittances. By adjusting the thickness, a solution can then be obtained yielding a continuous dispersion curve quite distinct from the unacceptable solutions. An important consideration for a selection of a transparent layer should be that it is ideally a perfectly plane parallel uniform layer. Our past experience suggests (see for example, Ref. [4]) that for crystalline ®lms the optical measurements (R and T) were not those appropriate to the perfectly plane parallel uniform thin ®lms such as those assumed for the derivation of the formulas used (such as those given in the Appendix). In contrast, amorphous ®lms can be treated as single uniform layers on substrates. In recent work [5] it was found that WO3 ®lms were uniform with smooth surfaces and transparent in the wavelength region (400± 800 nm) covered in the present method. Therefore in the present work WO3 was used for the transparent layer. Physical vapor deposited WO3 ®lms on unheated substrates are known to be amorphous in nature (see for example, Refs. [6,7]). Thus, no optical anisotropy in WO3 ®lms was anticipated. However, there is one drawback with the use of WO3
km 6:25 £ 1023 l 2 1;
and dm 20 nm where l is the wavelength in nm. Then, for the thin transparent layer, we have assumed nt 2:0 and independent of l and dt 80 nm. This gives a close representation to the WO3 ®lms actually used. For the substrate, it was assumed ns 1:5, independent of l . Using these data, transmittances Tm and Tmt were calculated from Eqs. (1) and (2). Then taking these values as given transmittance data, the two equations were solved for nm and km (assuming that nm and km were not known) by the above-described process of solution. The dispersion curve shown in Fig. 1(a) was obtained. In Fig. 1(a) one curve (corresponding to nm 20:4 1 3:2 £ 105 =l2 , used in the calculations) which is continuous over the entire spectral region is, of course, the correct solution, while another curve (intersecting the ®rst one) gives a wrong solution. Clearly, considering the existence of multiple solutions, measurements made at a single wavelength must be interpreted with care. Next the thickness dm of the metal ®lm was varied from its true value with the effect on the solutions for nm shown in Fig. 1(b) for an overestimation and Fig. 1(c) for an underestimation, both by 4%. Clearly, a small error in
Fig. 1. Calculated dispersion curves for a hypothetical set of ®lms, illustrating the effect of errors in dm, the thickness of a metal ®lm. Solutions calculated with (a) exact thickness, (b) increased thickness by 4%, and (c) decreased thickness by 4%.
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and that is that the measurements can not be extended far into the ultraviolet, because WO3 ®lms begin to absorb at a wavelength of about 400 nm, while most spectrophotometers go to 200 nm. In order to extend the measurements down to wavelength of 200 nm, it may be advisable to use SiO2 as a transparent layer (known to be amorphous [6]). Simulations, similar to the one shown in Fig. 1, were carried out where nt 2:0 (corresponding to WO3) was replaced by nt 1:46 and nt 1:38 (corresponding to SiO2 and MgF2, respectively). In all these simulations, the optical thickness of the transparent ®lm was kept constant. The results of these simulations were similar to those shown in Fig. 1, suggesting that the index of refraction of the transparent layer has an insigni®cant effect on the outcome of the results in the present method. In order to check the applicability of the method for very thin metal ®lms, similar simulations were carried out for thicknesses of metal ®lms of 10 and 5 nm. The dispersion curves thus obtained were similar to that shown in Fig. 1, suggesting that the method could also be used for very thin samples, provided these are uniform. A method given in Ref. [8] for determining both refractive index (as a function of wavelength) and the thickness of a transparent ®lm (WO3) on a transparent substrate (glass) from measurement of transmittance at normal incidence was used in the present work. In an earlier work [9], optical constants of a dielectric material were determined using two transmittance spectra of two ®lms of the dielectric with different thickness. A procedure was given [9] for determining the correct solutions for n and k, and also accurately ®xing the ®lm thickness by graphical methods, similar to the one followed in the present work (see Fig. 1). This method can not be applied to metallic ®lms, unless ®lm thickness is determined separately, as discussed below. These graphical methods of adjusting ®lm thickness can be only applied if there are multiple solutions (n and k) of the relevant equation and the optical thickness of the ®lm is greater than l /4. Multiple solutions arise due to multiple re¯ections in a dielectric ®lm, resulting in maxima and minima in the T vs. l curves. Such effects are absent in metallic ®lms. Therefore, we can not adjust the thickness of a metal ®lm by a graphical method similar to the one used here. The ®nal errors in the optical constants may be computed from the observed errors of measurement in the usual way since the necessary derivatives, though cumbersome, are readily obtainable from equations, given in Appendix. Explicit expressions for nm and km cannot be obtained, but we may write ÿ nm nm Tm ; Tmt ; dm ; dt ; nt ; ns ; l ÿ km km Tm ; Tmt ; dm ; dt ; nt ; ns ; l Then, to ®rst order the maximum error in nm is given by ÿ ÿ Dnm 2nm =2Tm DTm 1 2nm =2Tmt DTmt
ÿ 1 2nm =2dm Ddm ÿ ÿ ÿ 1 2nm =2dt Ddt 1 2nm =2nt Dnt 1 2nm =2ns Dns ÿ 1 2nm =2l Dl where DTm, etc. are the errors in the experimental quantities. A similar expression holds for Dkm. For a properly calibrated spectrophotometer Dl will be negligible. For a substrate such as BK7 glass the refractive indices are accurately known thus Dns may be small. The uncertainties in the measurement of nt and dt are both less than 1% [8]. The required partial derivatives can be expressed as functions of partial derivatives of Tm and Tmt; for example ÿ ÿ 2nm =2Tm 1=J 2Tm =2km 3 where ÿ ÿ ÿ ÿ J 2Tm =2nm 2Tmt =2km 2 2Tmt =2nm 2Tm =2km
4
3. Experimental Thin ®lms were prepared by evaporating gold (of 99.99% purity) from a tungsten boat in a vacuum of 10 26 mbar (using a Leybold model L560 box coater pumped by a turbomolecular pump), onto BK7 glass substrates. Pairs of such ®lms were deposited simultaneously and then exposed to air. One was coated with WO3 (of 99.99% purity) evaporated from a tungsten boat, and simultaneously a glass substrate was also coated with WO3. In this case, prior to evaporation the system was pumped to a base pressure of less than 10 26 mbar. Then oxygen was readmitted to raise the pressure to 10 24 mbar which was kept constant during ®lm deposition. In the simultaneous depositions it was ensured that the ®lms deposited on different substrates have the same thicknesses. All the ®lms were deposited onto substrates that were maintained at ambient temperature in the vacuum chamber. Substrates were rotated while deposition took place. The rates of deposition (0.1 nm/s for gold and 0.4 nm/s for WO3) were controlled by a quartz crystal thickness monitor and rate controller. The source-tosubstrate distance was 40 cm. Gold ®lms ranging in thickness from about 20 to 30 nm, were used. It is known that the optical constants of evaporated ®lms of gold depend on the thickness of the ®lm. However, these constants tend towards the same value as the thickness increases beyond 20 nm [1]. A range of thickness of WO3 ®lms was 60 to 150 nm. The measurements of transmittance on the samples at normal incidence were made over a 400±800 nm wavelength range using a double-beam Bausch and Lomb spectrophotometer, model Spectronic 2000. The samples were a thin gold ®lm on BK7 glass (a single layer system) and this system coated with a transparent ®lm of WO3 (a two layer system). It may be noted that transmittance Tm or Tmt given in Ref.
E.E. Khawaja et al. / Thin Solid Films 358 (2000) 166±171
Fig. 2. Measured transmittance from (a) a specimen (a gold ®lm of thickness of 26.6 nm on a glass substrate) and (b) the specimen coated with a WO3 ®lm of thickness 76.2 nm.
[1] is the transmittance into the substrate. The measured transmittance T T m ts , where ts is the transmittance across the back face of the substrate. The measured transmittance of the uncoated substrate (across its two faces) in terms of the refractive index of the substrate is given by nÿ ÿ o2 Ts ts2 4n0 ns = n0 1 ns 2 where n0 1 is the refractive index of air. The refractive index ns for various wavelengths was calculated from the measured Ts using the above equation. The present results for BK7 glass were found to be in agreement (within 0.2%) with the dispersion formula given for the BK7 glass in Ref. [10]. 4. Results for gold ®lms Results of the application of the above method to the experimental data of Fig. 2 for a gold ®lm on glass substrate and when coated with a thin transparent ®lm of WO3 are illustrated in Figs. 3 and 4. One can see the multiple solu-
Fig. 3. Dispersion curve for a gold ®lm showing multiple solutions and error bars, derived from the data of Fig. 2.
169
Fig. 4. Absorption curve for a gold ®lm, showing multiple solutions and error bars, derived from the data of Fig. 2.
tions in the ®gures, and that the right solutions are shown with a dispersion curve (Fig. 3) and absorption curve (Fig. 4). The thicknesses of the gold and WO3 ®lms (used for the generation of Figs. 3 and 4) were determined to be 26.6 and 76.2 nm, respectively. In Figs. 3 and 4, where the error bars are large they probably overestimate the error. As discussed by Denton et al. [3], in certain spectral regions the Jacobian J (Eq. (4)) which appears in the denominators of the expressions for the derivatives (such as in Eq. (3)), may become very small. Under these circumstances the derivatives become very large. It follows that the ®rst order calculation of error must be inadequate in these regions since it will grossly overestimate the overall error [3]. This is demonstrated by the smoothness of the computed curves in Figs. 3 and 4. The variances of indices of refraction and absorption, from ®lm to ®lm, in the 400±800 nm spectral region were less than 7 and 4%, respectively. In Figs. 5 and 6, the present results are compared with those reported in Ref. [11]. In Ref. [11], the results for evaporated Au and crystalline Au
Fig. 5. Comparison of the refractive index of the gold ®lms with the corresponding data taken from Ref. [11] on evaporated and crystalline gold.
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layer) of a single metal ®lm (of refractive index nm 2 ikm) on a non-absorbing substrate (ns) and Tmt, (double layer) of a transparent layer (nt) coated metal ®lm on the substrate, given by Heavens [1] are: ² The single layer: Tm at a wavelength l is ÿ Tm ns =n0 A=B
1
where n0 is the refractive index of air hÿ ihÿ i A 1 1 G1 2 1H12 1 1 G2 2 1H22 B e2am 1 G21 1 H12 G22 1 H22 e22am 1 Ccos2gm 1 Dsin2gm Fig. 6. Comparison of the absorption index of the gold ®lms with the corresponding data taken from Ref. [11] on evaporated and crystalline gold.
were taken from Refs. [12,13], respectively. In both the cases, the optical constants were deduced from re¯ection measurements only, and the results were found to be very sensitive to the quality of the surface of the sample. In an earlier stage of this work, we made an attempt to measure Ttm (i.e. transmittance of a metal ®lm on a transparent layer on a glass substrate) instead of present measurement of Tmt (i.e. a transparent layer on a metal ®lm on a glass substrate). However, in this case we were unable to obtain an acceptable dispersion curve. Lee et al. [14] have observed an anomalous layer, at the beginning of the deposition of Ag on deposited TiO2 ®lms, which was a composite of TiO2 and Ag. Such an anomalous layer was absent when TiO2 was deposited on Ag. Similar effects may be prevailing in the present work, however, these need to be con®rmed.
hÿ i 2 2 G1 n20 2 n2m 2 km = n0 1 nm 2 1km i ÿ hÿ H1 2n0 km = n0 1 nm 2 1km2 hÿ i 2 2 = nm 1 ns 2 1km G2 n2m 2 n2s 1 km i ÿ hÿ 2 H2 22ns km = nm 1 ns 2 1km ÿ C 2 G1 G2 2 H1 H2 ÿ D 2 G1 H2 1 G2 H1
am 2pkm dm =l gm 2pnm dm =l where dm is the thickness of the metal ®lm.
5. Conclusion
² The double layer: Tmt at a wavelength l is
A method to determine indices of refraction and absorption together with the ®lm thickness of metal ®lms based on simple measurements of transmittance at normal incidence has been proposed. The method has been successfully applied to ®lms of gold. However, this method could not be applied to thick metal ®lms because transmittance would be too small to be accurately measured.
Tmt
Acknowledgements This work is part of an internal project # CAPS1202, supported by the Research Institute of King Fahd University of Petroleum and Minerals. Appendix Formulas for normal incidence transmittance, Tm, (single
hÿ
i ns =n0 l213 1 m213 = p213 1 q213
where ÿ g1 n20 2 n2t = n0 1 nt 2 hÿ i 2 2 g2 n2t 2 n2m 2 km = nt 1 nm 2 1km hÿ i 2 2 = nm 1 ns 2 1km g3 n2m 2 n2s 1 km h1 0 hÿ i 2 h2 12nt km = nt 1 nm 2 1km hÿ i 2 h3 22ns km = nm 1 ns 2 1km
at 0
2
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gt 2pnt dt =l
p13 p12 p3 2 q12 q3 1 r12 t3 2 s12 u3
where dt is the thickness of the transparent layer.
q13 q12 p3 1 p12 q3 1 s12 t3 1 r12 u3
p3 eam cosgm q3 eam singm ÿ t3 e2am g3 cosgm 1 h3 singm ÿ u3 e2am h3 cosgm 2 g3 singm
171
ÿ ÿ ÿ ÿ l13 1 1 g1 1 1 g2 1 1 g3 2 h2 h3 1 1 g1 ÿ ÿ 2 h 3 h 1 1 1 g 2 2 h1 h 2 1 1 g 3 ÿ ÿ ÿ ÿ m13 h1 1 1 g2 1 1 g3 1 h2 1 1 g3 1 1 g1 ÿ ÿ 1 h 3 1 1 g 1 1 1 g2 2 h1 h 2 h 3
p2 cosgt q2 singt ÿ r2 g2 cosgt 2 h2 singt
References
ÿ s2 h2 cosgt 1 g2 singt
[1] O.S. Heavens, Optical Properties of Thin Solid Films, Dover Publications, New York, 1991. [2] D.P. Arndt, R.M.A. Azzam, J.M. Bennett, et al., Appl. Opt. 23 (1984) 3571. [3] R.E. Denton, R.D. Campbell, S.G. Tomlin, J. Phys. D: Appl. Phys. 5 (1972) 852. [4] E.E. Khawaja, F. Bouamrane, F. Al-Adel, A.B. Hallak, M.A. Daous, M.A. Salim, Thin Solid Films 240 (1994) 121. [5] E.E. Khawaja, S.M.A. Durrani, M.A. Daous, J. Phys, Condens. Matter 9 (1997) 9381. [6] F. Rauch, W. Wagner, K. Bange, Nucl. Instrum. Methods B 42 (1989) 264. [7] K. von Rottkay, M. Rubin, S.J. Wen, Thin Solid Films 306 (1997) 10. [8] E.E. Khawaja, J. Phys. D: Appl. Phys. 9 (1976) 1939. [9] E.E. Khawaja, F. Bouamrane, Appl. Opt. 32 (1993) 1168. [10] Optics Guide 4, Melles Griot, 1988, pp. 3±9. [11] American Institute of Physics Handbook, 3rd ed., McGraw-Hill, New York, 1972, pp. 6±138. [12] K. Weiss, Z. Naturforsch, 3a (1948) 143. [13] M. Otter, Z. Physik, 161 (1961) 163. [14] C.C. Lee, S.H. Chen, C.C. Jaing, Appl. Opt. 35 (1996) 5698.
ÿ t2 g2 cosgt 1 h2 singt ÿ u2 h2 cosgt 2 g2 singt v2 cosgt w2 2singt p12 p2 1 g1 t2 2 h1 u2 q12 q2 1 h1 t2 1 g1 u2 r12 r2 1 g1 v2 2 h1 w2 s12 s2 1 h1 v2 1 g1 w2
Simple method for determining the optical constants of thin metallic ®lms from transmittance measurements E.E. Khawaja*, S.M.A. Durrani, A.M. Al-Shukri Center for Applied Physical Sciences, Research Institute, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia Received 17 December 1998; received in revised form 20 August 1999; accepted 20 August 1999
Abstract A method for the determination of the optical constants of thin metal ®lms on transparent substrates is proposed. It requires measurements at normal incidence of the transmittance from the specimen itself and when coated with a thin transparent layer. A procedure is given for determining the correct solutions for the indices of refraction and absorption and also for accurately ®xing the thickness of the ®lm. Advantage of the present method over existing methods is readily available measurement facilities. The method has been applied successfully to ®lms of gold. q 2000 Elsevier Science S.A. All rights reserved. Keywords: Gold; Optical coatings; Optical properties; Tungsten oxide
1. Introduction The optical properties of a homogeneous thin metal ®lm are fully described by its complex index of refraction N m nm l 2 ikm l where nm (refractive index) is the real part and km (absorption index) is the imaginary part while l is the wavelength. If a ®lm of known thickness dm and unknown Nm is deposited upon a transparent substrate of known ns we can, in principle, determine Nm from any pair of measurements with a spectral photometer. Such a pair of measurements could, for example, be either one re¯ection and one transmission measurement or two spectra recorded with different angles of incidence or different directions of polarization. Various methods, for the measurement of optical constants, have been described in Refs. [1,2]. In fact, a panel discussion on the measurement of optical constants of absorbing and non-absorbing thin ®lms, has been reported in Ref. [2]. Generally, in the case of metal ®lms, the methods include at least one measurement of re¯ection from the ®lms. However, in the present method we use measurements at normal incidence of the transmittance from the specimen (a gold ®lm on a glass substrate) itself and when coated with a thin transparent layer to determine the complex refractive index. The derivation is complicated (as in the existing methods [1,2]) by the existence of multiple solutions of * Corresponding author. Fax: 1966-3-860-4281. E-mail address: [email protected] (E.E. Khawaja)
the relevant equations. A procedure is given for determining the solutions for nm and km and also for accurately ®xing the thicknesses of the ®lms. Advantage of the present method over the other methods (involving re¯ection measurements) is readily available measurement facilities. In general, the commercially available spectrophotometers are applied for transmittance, T, measurement at normal incidence and only a few of them have re¯ectance, R, measuring facilities. The method has been applied successfully to ®lms of gold. 2. Principle of the method The transmittances at normal incidence from the specimen (a thin gold ®lm on a glass substrate), Tm, and when coated with a thin transparent layer, Tmt, were measured as functions of wavelength l . Equations relating these measurements to the optical constants and thicknesses of the metal (nm, km and dm) and transparent ®lms (nt and dt) and the refractive index (ns) of the substrate are given as single ®lm (for Tm) and double layer (for Tmt) formulae by Heavens [1]. These equations (reproduced in Appendix) are cumbersome; however, we can write these in functional form as ÿ 1 T m F1 n m ; k m ; d m ; n s ; l ÿ Tmt F2 nm ; km ; dm ; nt ; dt ; ns ; l
2
The equations given in Ref. [1] do not include the effects
0040-6090/00/$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S00 40-6090(99)0068 9-6
E.E. Khawaja et al. / Thin Solid Films 358 (2000) 166±171
of multiple re¯ection in the substrate. The use of a wedge substrate would ensure that re¯ection from the back of the substrate is de¯ected out of the optical path of the instrument so that the multiple re¯ections in the substrate do not affect the measurement. If the Tm and Tmt are measured as a function of wavelength and dm, nt, dt, and ns are known, then in principle, nm 2 ikm can be determined from the above relations. An expression giving an explicit value of nm or km (see Appendix) cannot be obtained. However, the equations for Tm and Tmt may be solved by a numerical method similar to that described by Denton et al. [3], as discussed below. A computer solution of equations for Tm and Tmt may be carried out by rearranging them in the forms f1 nm ; km 0 and f2 nm ; km 0, respectively. There is a single value of km for a given nm and Tm [3]. Therefore, one may readily solve f1 nm ; km 0, ®nding km for a given value of nm by successive approximation by using Newton's method, since 2f1/2km may be easily written down. It is necessary, as will appear from the example discussed below, to cover a wide range of value of nm. Each value of nm and the corresponding km is substituted in turn into f2 nm ; km 0. When a change in sign of f2(nm,km) occurs, a simple interpolation leads to the required solution of f2 nm ; km 0. Because of the existence of multiple roots [3] it is essential to continue the process for further values of nm and km, since the ®rst solution found may not be the true one. It is instructive to examine the numerical solution of these equations for nm and km for a hypothetical system. We have chosen a ®lm, approximating a gold ®lm, with the following characteristics nm 20:4 1 3:2 £ 105 =l2 ;
167
®lm thickness has a characteristic and easily distinguishable effect upon the computed solutions for nm, from which it is possible to infer, when using a set of experimental data, the estimated ®lm thickness is too high or too low, provided there is no systematic error in the measured transmittances. By adjusting the thickness, a solution can then be obtained yielding a continuous dispersion curve quite distinct from the unacceptable solutions. An important consideration for a selection of a transparent layer should be that it is ideally a perfectly plane parallel uniform layer. Our past experience suggests (see for example, Ref. [4]) that for crystalline ®lms the optical measurements (R and T) were not those appropriate to the perfectly plane parallel uniform thin ®lms such as those assumed for the derivation of the formulas used (such as those given in the Appendix). In contrast, amorphous ®lms can be treated as single uniform layers on substrates. In recent work [5] it was found that WO3 ®lms were uniform with smooth surfaces and transparent in the wavelength region (400± 800 nm) covered in the present method. Therefore in the present work WO3 was used for the transparent layer. Physical vapor deposited WO3 ®lms on unheated substrates are known to be amorphous in nature (see for example, Refs. [6,7]). Thus, no optical anisotropy in WO3 ®lms was anticipated. However, there is one drawback with the use of WO3
km 6:25 £ 1023 l 2 1;
and dm 20 nm where l is the wavelength in nm. Then, for the thin transparent layer, we have assumed nt 2:0 and independent of l and dt 80 nm. This gives a close representation to the WO3 ®lms actually used. For the substrate, it was assumed ns 1:5, independent of l . Using these data, transmittances Tm and Tmt were calculated from Eqs. (1) and (2). Then taking these values as given transmittance data, the two equations were solved for nm and km (assuming that nm and km were not known) by the above-described process of solution. The dispersion curve shown in Fig. 1(a) was obtained. In Fig. 1(a) one curve (corresponding to nm 20:4 1 3:2 £ 105 =l2 , used in the calculations) which is continuous over the entire spectral region is, of course, the correct solution, while another curve (intersecting the ®rst one) gives a wrong solution. Clearly, considering the existence of multiple solutions, measurements made at a single wavelength must be interpreted with care. Next the thickness dm of the metal ®lm was varied from its true value with the effect on the solutions for nm shown in Fig. 1(b) for an overestimation and Fig. 1(c) for an underestimation, both by 4%. Clearly, a small error in
Fig. 1. Calculated dispersion curves for a hypothetical set of ®lms, illustrating the effect of errors in dm, the thickness of a metal ®lm. Solutions calculated with (a) exact thickness, (b) increased thickness by 4%, and (c) decreased thickness by 4%.
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and that is that the measurements can not be extended far into the ultraviolet, because WO3 ®lms begin to absorb at a wavelength of about 400 nm, while most spectrophotometers go to 200 nm. In order to extend the measurements down to wavelength of 200 nm, it may be advisable to use SiO2 as a transparent layer (known to be amorphous [6]). Simulations, similar to the one shown in Fig. 1, were carried out where nt 2:0 (corresponding to WO3) was replaced by nt 1:46 and nt 1:38 (corresponding to SiO2 and MgF2, respectively). In all these simulations, the optical thickness of the transparent ®lm was kept constant. The results of these simulations were similar to those shown in Fig. 1, suggesting that the index of refraction of the transparent layer has an insigni®cant effect on the outcome of the results in the present method. In order to check the applicability of the method for very thin metal ®lms, similar simulations were carried out for thicknesses of metal ®lms of 10 and 5 nm. The dispersion curves thus obtained were similar to that shown in Fig. 1, suggesting that the method could also be used for very thin samples, provided these are uniform. A method given in Ref. [8] for determining both refractive index (as a function of wavelength) and the thickness of a transparent ®lm (WO3) on a transparent substrate (glass) from measurement of transmittance at normal incidence was used in the present work. In an earlier work [9], optical constants of a dielectric material were determined using two transmittance spectra of two ®lms of the dielectric with different thickness. A procedure was given [9] for determining the correct solutions for n and k, and also accurately ®xing the ®lm thickness by graphical methods, similar to the one followed in the present work (see Fig. 1). This method can not be applied to metallic ®lms, unless ®lm thickness is determined separately, as discussed below. These graphical methods of adjusting ®lm thickness can be only applied if there are multiple solutions (n and k) of the relevant equation and the optical thickness of the ®lm is greater than l /4. Multiple solutions arise due to multiple re¯ections in a dielectric ®lm, resulting in maxima and minima in the T vs. l curves. Such effects are absent in metallic ®lms. Therefore, we can not adjust the thickness of a metal ®lm by a graphical method similar to the one used here. The ®nal errors in the optical constants may be computed from the observed errors of measurement in the usual way since the necessary derivatives, though cumbersome, are readily obtainable from equations, given in Appendix. Explicit expressions for nm and km cannot be obtained, but we may write ÿ nm nm Tm ; Tmt ; dm ; dt ; nt ; ns ; l ÿ km km Tm ; Tmt ; dm ; dt ; nt ; ns ; l Then, to ®rst order the maximum error in nm is given by ÿ ÿ Dnm 2nm =2Tm DTm 1 2nm =2Tmt DTmt
ÿ 1 2nm =2dm Ddm ÿ ÿ ÿ 1 2nm =2dt Ddt 1 2nm =2nt Dnt 1 2nm =2ns Dns ÿ 1 2nm =2l Dl where DTm, etc. are the errors in the experimental quantities. A similar expression holds for Dkm. For a properly calibrated spectrophotometer Dl will be negligible. For a substrate such as BK7 glass the refractive indices are accurately known thus Dns may be small. The uncertainties in the measurement of nt and dt are both less than 1% [8]. The required partial derivatives can be expressed as functions of partial derivatives of Tm and Tmt; for example ÿ ÿ 2nm =2Tm 1=J 2Tm =2km 3 where ÿ ÿ ÿ ÿ J 2Tm =2nm 2Tmt =2km 2 2Tmt =2nm 2Tm =2km
4
3. Experimental Thin ®lms were prepared by evaporating gold (of 99.99% purity) from a tungsten boat in a vacuum of 10 26 mbar (using a Leybold model L560 box coater pumped by a turbomolecular pump), onto BK7 glass substrates. Pairs of such ®lms were deposited simultaneously and then exposed to air. One was coated with WO3 (of 99.99% purity) evaporated from a tungsten boat, and simultaneously a glass substrate was also coated with WO3. In this case, prior to evaporation the system was pumped to a base pressure of less than 10 26 mbar. Then oxygen was readmitted to raise the pressure to 10 24 mbar which was kept constant during ®lm deposition. In the simultaneous depositions it was ensured that the ®lms deposited on different substrates have the same thicknesses. All the ®lms were deposited onto substrates that were maintained at ambient temperature in the vacuum chamber. Substrates were rotated while deposition took place. The rates of deposition (0.1 nm/s for gold and 0.4 nm/s for WO3) were controlled by a quartz crystal thickness monitor and rate controller. The source-tosubstrate distance was 40 cm. Gold ®lms ranging in thickness from about 20 to 30 nm, were used. It is known that the optical constants of evaporated ®lms of gold depend on the thickness of the ®lm. However, these constants tend towards the same value as the thickness increases beyond 20 nm [1]. A range of thickness of WO3 ®lms was 60 to 150 nm. The measurements of transmittance on the samples at normal incidence were made over a 400±800 nm wavelength range using a double-beam Bausch and Lomb spectrophotometer, model Spectronic 2000. The samples were a thin gold ®lm on BK7 glass (a single layer system) and this system coated with a transparent ®lm of WO3 (a two layer system). It may be noted that transmittance Tm or Tmt given in Ref.
E.E. Khawaja et al. / Thin Solid Films 358 (2000) 166±171
Fig. 2. Measured transmittance from (a) a specimen (a gold ®lm of thickness of 26.6 nm on a glass substrate) and (b) the specimen coated with a WO3 ®lm of thickness 76.2 nm.
[1] is the transmittance into the substrate. The measured transmittance T T m ts , where ts is the transmittance across the back face of the substrate. The measured transmittance of the uncoated substrate (across its two faces) in terms of the refractive index of the substrate is given by nÿ ÿ o2 Ts ts2 4n0 ns = n0 1 ns 2 where n0 1 is the refractive index of air. The refractive index ns for various wavelengths was calculated from the measured Ts using the above equation. The present results for BK7 glass were found to be in agreement (within 0.2%) with the dispersion formula given for the BK7 glass in Ref. [10]. 4. Results for gold ®lms Results of the application of the above method to the experimental data of Fig. 2 for a gold ®lm on glass substrate and when coated with a thin transparent ®lm of WO3 are illustrated in Figs. 3 and 4. One can see the multiple solu-
Fig. 3. Dispersion curve for a gold ®lm showing multiple solutions and error bars, derived from the data of Fig. 2.
169
Fig. 4. Absorption curve for a gold ®lm, showing multiple solutions and error bars, derived from the data of Fig. 2.
tions in the ®gures, and that the right solutions are shown with a dispersion curve (Fig. 3) and absorption curve (Fig. 4). The thicknesses of the gold and WO3 ®lms (used for the generation of Figs. 3 and 4) were determined to be 26.6 and 76.2 nm, respectively. In Figs. 3 and 4, where the error bars are large they probably overestimate the error. As discussed by Denton et al. [3], in certain spectral regions the Jacobian J (Eq. (4)) which appears in the denominators of the expressions for the derivatives (such as in Eq. (3)), may become very small. Under these circumstances the derivatives become very large. It follows that the ®rst order calculation of error must be inadequate in these regions since it will grossly overestimate the overall error [3]. This is demonstrated by the smoothness of the computed curves in Figs. 3 and 4. The variances of indices of refraction and absorption, from ®lm to ®lm, in the 400±800 nm spectral region were less than 7 and 4%, respectively. In Figs. 5 and 6, the present results are compared with those reported in Ref. [11]. In Ref. [11], the results for evaporated Au and crystalline Au
Fig. 5. Comparison of the refractive index of the gold ®lms with the corresponding data taken from Ref. [11] on evaporated and crystalline gold.
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layer) of a single metal ®lm (of refractive index nm 2 ikm) on a non-absorbing substrate (ns) and Tmt, (double layer) of a transparent layer (nt) coated metal ®lm on the substrate, given by Heavens [1] are: ² The single layer: Tm at a wavelength l is ÿ Tm ns =n0 A=B
1
where n0 is the refractive index of air hÿ ihÿ i A 1 1 G1 2 1H12 1 1 G2 2 1H22 B e2am 1 G21 1 H12 G22 1 H22 e22am 1 Ccos2gm 1 Dsin2gm Fig. 6. Comparison of the absorption index of the gold ®lms with the corresponding data taken from Ref. [11] on evaporated and crystalline gold.
were taken from Refs. [12,13], respectively. In both the cases, the optical constants were deduced from re¯ection measurements only, and the results were found to be very sensitive to the quality of the surface of the sample. In an earlier stage of this work, we made an attempt to measure Ttm (i.e. transmittance of a metal ®lm on a transparent layer on a glass substrate) instead of present measurement of Tmt (i.e. a transparent layer on a metal ®lm on a glass substrate). However, in this case we were unable to obtain an acceptable dispersion curve. Lee et al. [14] have observed an anomalous layer, at the beginning of the deposition of Ag on deposited TiO2 ®lms, which was a composite of TiO2 and Ag. Such an anomalous layer was absent when TiO2 was deposited on Ag. Similar effects may be prevailing in the present work, however, these need to be con®rmed.
hÿ i 2 2 G1 n20 2 n2m 2 km = n0 1 nm 2 1km i ÿ hÿ H1 2n0 km = n0 1 nm 2 1km2 hÿ i 2 2 = nm 1 ns 2 1km G2 n2m 2 n2s 1 km i ÿ hÿ 2 H2 22ns km = nm 1 ns 2 1km ÿ C 2 G1 G2 2 H1 H2 ÿ D 2 G1 H2 1 G2 H1
am 2pkm dm =l gm 2pnm dm =l where dm is the thickness of the metal ®lm.
5. Conclusion
² The double layer: Tmt at a wavelength l is
A method to determine indices of refraction and absorption together with the ®lm thickness of metal ®lms based on simple measurements of transmittance at normal incidence has been proposed. The method has been successfully applied to ®lms of gold. However, this method could not be applied to thick metal ®lms because transmittance would be too small to be accurately measured.
Tmt
Acknowledgements This work is part of an internal project # CAPS1202, supported by the Research Institute of King Fahd University of Petroleum and Minerals. Appendix Formulas for normal incidence transmittance, Tm, (single
hÿ
i ns =n0 l213 1 m213 = p213 1 q213
where ÿ g1 n20 2 n2t = n0 1 nt 2 hÿ i 2 2 g2 n2t 2 n2m 2 km = nt 1 nm 2 1km hÿ i 2 2 = nm 1 ns 2 1km g3 n2m 2 n2s 1 km h1 0 hÿ i 2 h2 12nt km = nt 1 nm 2 1km hÿ i 2 h3 22ns km = nm 1 ns 2 1km
at 0
2
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gt 2pnt dt =l
p13 p12 p3 2 q12 q3 1 r12 t3 2 s12 u3
where dt is the thickness of the transparent layer.
q13 q12 p3 1 p12 q3 1 s12 t3 1 r12 u3
p3 eam cosgm q3 eam singm ÿ t3 e2am g3 cosgm 1 h3 singm ÿ u3 e2am h3 cosgm 2 g3 singm
171
ÿ ÿ ÿ ÿ l13 1 1 g1 1 1 g2 1 1 g3 2 h2 h3 1 1 g1 ÿ ÿ 2 h 3 h 1 1 1 g 2 2 h1 h 2 1 1 g 3 ÿ ÿ ÿ ÿ m13 h1 1 1 g2 1 1 g3 1 h2 1 1 g3 1 1 g1 ÿ ÿ 1 h 3 1 1 g 1 1 1 g2 2 h1 h 2 h 3
p2 cosgt q2 singt ÿ r2 g2 cosgt 2 h2 singt
References
ÿ s2 h2 cosgt 1 g2 singt
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ÿ t2 g2 cosgt 1 h2 singt ÿ u2 h2 cosgt 2 g2 singt v2 cosgt w2 2singt p12 p2 1 g1 t2 2 h1 u2 q12 q2 1 h1 t2 1 g1 u2 r12 r2 1 g1 v2 2 h1 w2 s12 s2 1 h1 v2 1 g1 w2