Thin-film optical function acquisition from experimental measurements of the reflectance and transmittance spectra: a case study

The determination of the spectral dependence of thin-film optical functions from experimental measurements of the optical response, performed on thin-film/substrate structures, represents a problem of general interest to the thin-film community. In this paper, we present details related to thin-film optical function acquisition from experimental measurements of the reflectance and transmittance spectra, these being worked out through a case study performed on the optical functions associated with a series of thin-films of silicon that had been grown through ultra-high vacuum evaporation deposition on optical-quality-fused quartz substrates, this study being performed by Moghaddam et al. (J Mater Sci: Mater Electron 31:13186–13198, 2020). Following the presentation of our approach to deposition, details related to our optical response measurements, the uncertainty in our reflectance and transmittance results, the analytical framework within which the optical functions were acquired, how the ambient and the optical-quality-fused quartz substrate optical functions were modeled, and means through which unphysical solutions were identified and eliminated, are presented. We believe that the collection of these results will be of interest to those within the broader thin-film community.

1. A sequence of complementary UHV thin-film silicon growths, performed on c-Si and oxidized c-Si \((\hbox{SiO}_{2}\)/c-Si) substrates, were also performed, the growth rate for these growths being set to 0.2 nm/s for all cases. The optical responses of these thin silicon films could not be explored owing to the opacity of c-Si, i.e., joint determinations of the reflectance and transmittance spectra are required in order to determine the spectral dependence of the thin-film silicon optical functions. That is why the thin silicon films grown for the purposes of this optical function analysis were grown on optical-quality-fused quartz substrates, which are highly transparent.

2. SIMS was performed on the aforementioned UHV-prepared thin-films of silicon that were deposited on the c-Si and \(\hbox{SiO}_{2}\)/c-Si substrates, the deposition conditions being almost identical. SIMS could not be performed on the thin-films of silicon that had been grown on the optical-quality-fused quartz substrates, as sections of the thin-film would have been destroyed as a consequence. We expect that similar concentrations of hydrogen are present within our UHV-prepared forms of thin-film silicon grown on optical-quality-fused quartz substrates.

3. It is common to describe the experimental set-up used for these optical property measurements as configuring the spectrophotometer in the reflection mode or the transmission mode for the reflectance and transmittance measurements, respectively.

4. The PE-900 spectrophotometer could not accommodate the dual V–W accessory with the sample mounted in the horizontal plane, this particular geometry being required in order for the aforementioned 3-step NRC compensation method with a high reflectance standard to be performed. Hence, the need to use the PE-9 spectrophotometer for these measurements. Unfortunately, when configured in this particular geometry, the PE-9 spectrophotometer did not allow for a purge of the sample compartment.

5. While measurements of the reflectance and transmittance spectra are often used in order to determine the optical functions of thin-films deposited onto an underlying substrate, ellipsometry can also be used for such an application. A discussion on the strengths and weaknesses of these approaches is provided by Poelman and Smet [36].

6. There are a variety of alternate thin-film optical function formalisms that are available, including those of Manifacier et al. [37] and Swanepoel [38]. Some formalisms take into account effects that are not treated within the framework of Heavens [34], including light scattering. We used the formalism of Heavens [34], owing to its effectiveness, simplicity, and the clarity of insight that it offers.

7. The broad absorption feature that is observed in the transmittance spectrum of the optical-quality-fused quartz substrate is most probably due to absorption from hydroxide ion (\(\hbox{OH}^{-1}\)) impurities; UV-grade-fused silica has a dip in its transmission centered at about 1.4 microns (0.89 eV) whereas IR-grade-fused silica does not exhibit this dip. The sharp narrow features that are observed superimposed on this 0.89 eV feature in the unpurged reflectance spectrum of the bare (uncoated) optical-quality-fused quartz substrate is most likely due to atmospheric water vapor absorption (H-bonding) to the intrinsic hydroxide ion impurities. The residual features associated with these absorption features are also found in the other spectra, albeit to a lesser degree.

8. Based on the measurements of reflectance and transmittance spectra, we first calculated the following ratios: \(\frac{1+R}{T}\) and \(\frac{1-R}{T}\). Based on the values of refractive index for the ambient, i.e., \(n_{0} = 1\), and the substrate, i.e., \(n_{2} = 1.49\), and the extinction coefficient of 0 for the fused quartz, the Heaven’s equations for \(\frac{1+R}{T}\) and \(\frac{1-R}{T}\) have two unknowns. In order to solve this system of equations, the ‘solve’ function in Matlab is used for each photon energy considered in this analysis. The ‘solve’ function uses the Levenberg–Marquardt and trust-region methods, these being based on a number of non-linear least-squares solving algorithms. The trust-region algorithm is a sub-space trust-region method and is based on the interior-reflective Newton method. Each iteration involves the approximate solution of a large linear system using the method of preconditioned conjugate gradients. In addition, the Levenberg–Marquardt algorithm uses a search direction that stems from a solution of the linear set of equations.

9. For a uniform thin-film of thickness d, Cody et al. [47] assert that the maxima occur when

   $$\begin{aligned} \frac{4 \pi n \left( \lambda \right) d}{\lambda } = 2 m \pi , \end{aligned}$$

   where \(n \left( \lambda \right) \) is the spectral dependence of the refractive index, \(\lambda \) is the wavelength in vacuum, and m is an integer that can assume the values 0, 1, 2,.... Cody et al. [47] also assert that the minima occur when

   $$\begin{aligned} \frac{4 \pi n \left( \lambda \right) d}{\lambda } = 2 \left( m + \frac{1}{2} \right) \pi , \end{aligned}$$

   where, as with the condition for the maxima, m is an integer that can assume the values 0, 1, 2, .... The reflectance and transmittance spectral oscillations that are observed in Fig. 5a through d, inclusive, are mostly consistent with this expectation of Cody et al. [47]. Deviations from the principle of Cody et al. [47] arose from the inhomogeneities that are present within some of these thin-films of silicon.

10. There is evidence to suggest that there are inhomogeneities that occur in the optical properties as one goes deeper into a given thin-film, these inhomogeneities reflecting the structural inhomogeneities that are present. In effect, a given thin-film may be thought of as being composed of a stack of thin-film layers, each layer within such a stack having its own optical properties.

11. This Kramers–Kronig means of pruning out the unphysical solutions is still under development. For the purposes of this particular analysis, we found that it provided a rather effective means of performing this task.

Three of the authors (S. M., S. H. C., and S. K. O.) wish to thank the Natural Sciences and Engineering Research Council of Canada and MITACS for financial support. The authors would like to acknowledge the assistance of Dr. J. K. W. Ho of The Hong Kong Polytechnic University who assisted with the typesetting of the equations in the Appendix.

Authors and Affiliations

1. School of Engineering, The University of British Columbia, Kelowna, BC, V1V 1V7, Canada

   Saeed Moghaddam, Sin Hang Cheung & Stephen K. O’Leary

2. Metrology Research Centre, National Research Council of Canada, Ottawa, ON, K1A 0R6, Canada

   Mario Noël, Joanne C. Zwinkels & David J. Lockwood

3. Information and Communication Technologies, National Research Council of Canada, Ottawa, ON, K1A 0R6, Canada

   Jean-Marc Baribeau

Corresponding author

Correspondence to Stephen K. O’Leary.

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Appendix 1: Accounting for Eqs. (3) and (4) from the formalism of Heavens

An electromagnetic analysis of Heavens [34] demonstrated that

and

where the coefficients

and

where

and

where we employ the same nomenclature as Heavens [34]. Through the direct substitution of Eqs. (5) and (6), it is seen that

which, noting the common denominator in the expressions for R and T, i.e., Eqs. (5) and (6), respectively, may in turn be expressed as follows:

or ultimately

where all terms are as previously defined. Through the direct substitution of Eqs. (5) and (6), it is also seen that

which, noting the common denominator in the expressions for R and T, i.e., Eqs. (5) and (6), respectively, may in turn be expressed as follows:

or ultimately

where, as for Eq. (19), all terms are as previously defined.

In order to complete this analysis, we now develop the identities that are required the required for the simplification of Eqs. (19) and (22). In order to perform this analysis, we draw upon Eqs. (7), (8), (9), (10), (11), (12), (13), and (14). We start with expressions for \(1 - \left( g_{1}^{2} + h_{1}^{2} \right) \) and \(1 + \left( g_{1}^{2} + h_{1}^{2} \right) \). Through the use of Eqs. (11) and (12), it can be shown that

where similar terms are grouped using the symbols depicted in the above expressions, many such terms ultimately being eliminated; the symbols used are \(\boxed {\cdot }, \underline{\cdot }, \underbrace{\cdot }\), and \(\overbrace{\cdot }\). A straightforward analysis, stemming from Eq. (23), indicates that

and so it can be shown that

Other identities may be acquired directly from this result. In particular, noting that

and so, from Eqs. (11) and (25), it is seen that

In addition, from Eqs. (13) and (14), it can be seen that

Noting that

from Eqs. (13) and (28), we can see that

Finally, from Eqs. (7) and (9), it can be seen that

the final form of Eq. (31) drawing upon the use of Eqs. (11) and (13), a comparable analysis, from Eqs. (8) and (10), showing that

the final form of Eq. (32) drawing upon the use of Eqs. (11) and (14). Similarly, from Eqs. (7) and (9), it can be seen that

the final form of Eq. (33) drawing upon the use of Eqs. (12) and (14), a comparable analysis, from Eqs. (8) and (10), showing that

The final form of Eq. (34) drawing upon the use of Eqs. (12) and (13). So all of the identities required for the evaluation of Eqs. (19) and (22) have been developed.

Recalling, from Eq. (19) that

and so, from the identities in Eqs. (25), (27), (28), (30), (31), and (32), it can be shown that

Noting that \(2 \cosh \left( 2 \alpha _{1} \right) = \left[ \exp \left( 2 \alpha _{1} \right) + \exp \left( - 2 \alpha _{1} \right) \right] \) and that \(2 \sinh \left( 2 \alpha _{1} \right) = \left[ \exp \left( 2 \alpha _{1} \right) - \exp \left( - 2 \alpha _{1} \right) \right] \), from Eq. (36) we see that

which ultimately reduces to

which, through the use of Eqs. (15) and (16) for \(\alpha _{1}\) and \(\gamma _{1}\), respectively, may be shown to be identical to Eq. (3).

Recalling, from Eq. (22), that

and so, from the identities contained in Eqs. (23), (27), (28), (30), (33), and (34), it can be shown that

which ultimately reduces to

which, through the use of Eqs. (15) and (16) for \(\alpha _{1}\) and \(\gamma _{1}\), respectively, may be shown to be identical to Eq. (4).

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